PACS numbers: 95.30.Sf, 95.35.+d, 95.36.+x, 98.62.Gq, 98.80.-k

arX

iv:2

109.

0596

3v1

[gr

-qc]

13

Sep

2021

K-essence Lagrangians of polytropic and logotropic unified dark matter

and dark energy models

Pierre-Henri Chavanis1, ∗

1Laboratoire de Physique Theorique, Universite de Toulouse, CNRS, UPS, France

We determine the k-essence Lagrangian of a relativistic barotropic fluid. The equation of stateof the fluid can be specified in different manners depending on whether the pressure is expressedin terms of the energy density (model I), the rest-mass density (model II), or the pseudo rest-mass density for a complex scalar field in the Thomas-Fermi approximation (model III). In thenonrelativistic limit, these three formulations coincide. In the relativistic regime, they lead todifferent models that we study exhaustively. We provide general results valid for an arbitraryequation of state and show how the different models are connected to each other. For illustration,we specifically consider polytropic and logotropic dark fluids that have been proposed as unified darkmatter and dark energy models. We recover the Born-Infeld action of the Chaplygin gas in models Iand III and obtain the explicit expression of the reduced action of the logotropic dark fluid in modelsII and III. We also derive the two-fluid representation of the Chaplygin and logotropic models. Ourgeneral formalism can be applied to many other situations such as Bose-Einstein condensates witha |ϕ|4 (or more general) self-interaction, dark matter superfluids, and mixed models.

PACS numbers: 95.30.Sf, 95.35.+d, 95.36.+x, 98.62.Gq, 98.80.-k

I. INTRODUCTION

Baryonic matter constitutes only 5% of the content ofthe universe today. The rest of the universe is made of ap-proximately 25% dark matter (DM) and 70% dark energy(DE) [1, 2]. DM can explain the flat rotation curves ofthe spiral galaxies. It is also necessary to form the large-scale structures of the universe. DE does not cluster butis responsible for the late time acceleration of the uni-verse revealed by the observations of type Ia supernovae,the cosmic microwave background (CMB) anisotropies,and galaxy clustering. Although there have been manytheoretical attempts to explain DM and DE, we still donot have a robust model for these dark components thatcan pass all the theoretical and observational tests.

The most natural and simplest model is the ΛCDMmodel which treats DM as a nonrelativistic cold pres-sureless gas and DE as a cosmological constant Λ possi-bly representing vacuum energy [3, 4]. The effect of thecosmological constant is equivalent to that of a fluid witha constant energy density ǫΛ = Λc2/8πG and a negativepressure PΛ = −ǫΛ. Therefore, the ΛCDM model is atwo-fluid model comprising DM with an equation of statePdm = 0 and DE with an equation of state Pde = −ǫde.When combined with the energy conservation equation,the equation of state Pdm = 0 implies that the DM den-sity decreases with the scale factor as ǫdm ∝ a−3 and theequation of state Pde = −ǫde implies that the DE densityis constant: ǫde = ǫΛ. Therefore, the total energy densityof the universe (DM + DE) evolves as

ǫ =ǫdm,0

a3+ ǫΛ, (1)

∗Electronic address: [email protected]

where ǫdm,0 is the present density of DM.1 DM domi-nates at early times when the density is high and DEdominates at late times when the density is low. Thescale factor increases algebraically as a ∝ t2/3 during theDM era (Einstein-de Sitter regime) and exponentially as

a ∝ e√

Λ/3t during the DE era (de Sitter regime). At thepresent epoch, both components are important in the en-ergy budget of the universe.

Although the ΛCDM model is perfectly consistentwith current cosmological observations, it faces two mainproblems. The first problem is to explain the tiny value ofthe cosmological constant Λ = 1.00× 10−35 s−2. Indeed,if DE can be attributed to vacuum fluctuations, quan-tum field theory predicts that Λ should correspond to thePlanck scale which lies 123 orders of magnitude above theobserved value. This is called the cosmological constantproblem [6, 7]. The second problem is to explain why DMand DE are of similar magnitudes today although theyscale differently with the universe’s expansion. This isthe cosmic coincidence problem [8, 9], frequently trigger-ing anthropic explanations. The CDM model also facesimportant problems at the scale of DM halos such as thecore-cusp problem [10], the missing satellite problem [11–13], and the “too big to fail” problem [14]. This leads tothe so-called small-scale crisis of CDM [15].

For these reasons, other types of matter with nega-tive pressure that can behave like a cosmological con-stant at late time have been considered as candidatesof DE: fluids of topological defects (domain walls, cos-mic strings) [16–20], x-fluids with a linear equation of

1 For simplicity of presentation we ignore the contribution of bary-onic matter in the present discussion. We also assume that theuniverse is spatially flat in agreement with the inflation paradigm[5] and the measurements of the CMB anisotropies [1, 2].

http://arxiv.org/abs/2109.05963v1

mailto:[email protected]

2

state [21–23], quintessence – an evolving self-interactingscalar field (SF) minimally coupled to gravity – [24] (seeearlier works in [25–33]),2 k-essence fields – SFs with anoncanonical kinetic term [37–39] that were initially in-troduced to describe inflation (k-inflation) [40, 41] – andeven phantom or ghost fields [42, 43] which predict thatthe energy density of the universe may ultimately in-crease with time. Quintessence can be viewed as a dy-namical vacuum energy following the old idea that thecosmological term could evolve [44–47]. However, thesemodels still face the cosmic coincidence problem3 becausethey treat DM and DE as distinct entities. Accountingfor similar magnitude of DM and DE today requires veryparticular (fine-tuned) initial conditions. For some kindof potential terms, which have their justification in super-gravity [48], this problem can be solved by the so-calledtracking solution [9, 49]. The self-interacting SF evolvesin such a way that it approaches a cosmological constantbehaviour exactly today [48]. However, this is achievedat the expense of fine-tuning the potential parameters.This unsatisfactory state of affairs motivated a searchfor further alternatives.In the standard ΛCDM model and in quintessence

CDM models, DM and DE are two distinct entities intro-duced to explain the clustering of matter and the cosmicacceleration, respectively. However, DM and DE couldbe two different manifestations of a common structure, adark fluid. In this respect, Kamenshchik et al. [50] haveproposed a simple unification of DM and DE in the formof a perfect fluid with an exotic equation of state knownas the Chaplygin gas, for which

P = − A

ǫ/c2, (2)

where A is a positive constant. This gas exhibits a neg-ative pressure, as required to explain the acceleration ofthe universe today, but the squared speed of sound is pos-itive (c2s = P ′(ǫ)c2 = Ac4/ǫ2 > 0). This is a very impor-tant property because many fluids with negative pressureobeying a barotropic equation of state suffer from insta-bilities at small scales due to an imaginary speed of sound[18, 19].4 This is not the case for the Chaplygin gas.

2 SFs have been used in a variety of inflationary models [34] to de-scribe the transition from the exponential (de Sitter) expansionof the early universe to a decelerated expansion. It was thereforenatural to try to understand the present acceleration of the uni-verse, which has an exponential behaviour too, in terms of SFs[35, 36]. However, one has to deal now with the opposite situa-tion, i.e., describing the transition from a decelerated expansionto an exponential (de Sitter) expansion.

3 A class of k-essence models [38, 39] has been claimed to solve thecoincidence problem by linking the onset of DE domination tothe epoch of DM domination.

4 To explain the accelerated expansion taking place today, the uni-verse must be dominated by a fluid of negative pressure violat-ing the strong energy condition, i.e., P < −ǫ/3. For a fluidwith a linear equation of state P = ωǫ, where ω is a constant,

Integrating the energy conservation equation with theChaplygin equation of state (2) leads to

ǫ/c2 =

√

A

c2+B

a6, (3)

where B is an integration constant. Therefore, the Chap-lygin gas smoothly interpolates between pressureless DM(P ≃ 0, ǫ ∼ a−3, cs ≃ 0) at high redshift and a cosmo-

logical constant (P = −ǫ, ǫ→√Ac2, cs → c) as a tends

to infinity. There is also an intermediate phase whichcan be described by a cosmological constant mixed witha stiff matter fluid (P ∼ ǫ, ǫ ∼ a−6, cs ∼ c) [50]. Inthe Chaplygin gas model, DM and DE are different man-ifestations of a single underlying substance (dark fluid)that is called “quartessence” [52]. These models wherethe fluid behaves as DM at early times and as DE atlate times are also called unified models for DM and DE(UDM) [52]. This dual behavior avoids fine-tuning prob-lems since the Chaplygin gas model can be interpreted asan entangled mixture of DM and DE. In this cosmolog-ical context, Kamenshchik et al. [50] introduced a realSF representation of the Chaplygin gas and determinedits potential V (ϕ) explicitly.The Chaplygin gas model has an interesting history

that we briefly sketch below. Chaplygin [53] introducedhis equation of state P = −A/ρ in 1904 as a convenientsoluble model to study the lifting force on a plane wingin aerodynamics. The same model was rediscovered laterby Tsien [54] and von Karman [55]. It was also real-ized that certain deformable solids can be described bythe Chaplygin equation of state [56]. The integrabilityof the corresponding Euler equations resides in the factthat they have a large symmetry group (see [57–60] fora modern description). Indeed, the Chaplygin gas modelpossesses further space-time symmetries beyond those ofthe Galileo group [57]. In addition, the Chaplygin gasis the only fluid which admits a supersymmetric general-ization [61–65]. The Chaplygin equation of state involvesa negative pressure which is required to account for theaccelerated expansion of the universe.5 It is possible todevelop a Lagrangian description of the nonrelativisticChaplygin gas [57–59, 66–68] leading to an action of theform

LChap = −(2A)1/2√

θ +1

2(∇θ)2, (4)

we need ω < −1/3 to have an acceleration. But, in that case,c2s = ωc2 < 0, yielding instabilities at small scales [18]. Thisis the usual problem for a fluid description of domain walls(ω = −2ǫ/3) and cosmic strings (ω = −ǫ/3). Quintessence mod-els with a standard kinetic term do not have this problem becausethe speed of sound is equal to the speed of light [51]. K-essencemodels [37–39] with a nonstandard kinetic term are different inthis respect [41], but they still have a positive squared speed ofsound (note that cs can exceed the speed of light).

5 Negative pressures arise in different domains of physics such asexchange forces in atoms, stripe states in the quantum Hall effect,Bose-Einstein condensates with an attractive self-interaction etc.

3

where θ is the potential of the flow. The relativistic gen-eralization of the Chaplygin gas model leads to a Born-Infeld-type [69] theory for a real SF [58, 59, 68, 70, 71].The Born-Infeld action

LBI = −(Ac2)1/2√

1− 1

c2∂µθ∂µθ (5)

possesses additional symmetries beyond the Lorentz andPoincare invariance and has an interesting connectionwith string/M theory [72]. The Chaplygin gas modelcan be motivated by a brane-world interpretation (see[73] for a review on brane world models). Indeed, the“hidden” symmetries and the associated transformationlaws for the Chaplygin and Born-Infeld models may begiven a coherent setting [59] by considering the Nambu-Goto action [74] for a d-brane in (d + 1) spatial dimen-sions moving in a (d+ 1, 1)-dimensional spacetime. TheGalileo-invariant (nonrelativistic) Chaplygin gas action(4) is obtained in the light-cone parametrization and thePoincare-invariant (relativistic) Born-Infeld action (5) isobtained in the Cartesian parametrization [58, 59].6 Afluid with a Chaplygin equation of state is also necessaryto stabilize the branes [90] in black hole bulks [91, 92].This is how Kamenshchik et al. [93] came across this fluidand had the idea to consider its cosmological implications[50]. Bilic et al. [71] generalized the Chaplygin gas modelin the inhomogeneous case and showed that the real SFthat occurs in the Born-Infeld action can be interpretedas the phase of a complex self-interacting SF describedby the Klein-Gordon (KG) equation. This SF may begiven a hydrodynamic representation in terms of an ir-rotational barotropic flow with the Chaplygin equationof state. This explains the connection of the Born-Infeld

6 A d-brane is a d-dimensional extended object. For example, a(d = 2)-brane is a membrane. d-branes arise in string theory forthe following reason. Just as the action of a relativistic pointparticle is proportional to the world line it follows, the action ofa relativistic string is proportional to the area of the sheet that ittraces by traveling through spacetime. The close connection be-tween a relativistic membrane [(d = 2)-brane] in three spatial di-mensions and planar fluid mechanics was known to J. Goldstone(unpublished) and developed by Hoppe and Bordemann [66, 70](the Chaplygin equation of state P = −A/ρ appears explicitly in[66] and the Born-Infeld Lagrangian associated with the actionof a membrane appears explicitly in [70]). These results weregeneralized to arbitrary d-branes by Jackiw and Polychronakos[58]. The same Lagrangian appears as the leading term in Sun-drum’s [75] effective field theory approach to large extra dimen-sions. The Born-Infeld Lagrangian can be viewed as a k-essenceLagrangian involving a nonstandard kinetic term. The Chaply-gin equation of state is obtained from the stress-energy tensorTµν derived from this Lagrangian. Therefore, the Chaplygin gasis the hydrodynamical description of a SF with the Born-InfeldLagrangian. A more general k-essence model is the string theoryinspired tachyon Lagrangian with a potential V (θ) [76–87]. Itcan be shown that every tachyon condensate model can be in-terpreted as a 3 + 1 brane moving in a 4 + 1 bulk [88, 89]. TheBorn-Infeld Lagrangian is recovered when V (θ) is replaced by aconstant [81].

action with fluid mechanics in the Thomas-Fermi (TF)approximation. Bilic et al. [71] determined the potentialV (|ϕ|2) of the complex SF associated with the Chaplygingas. This potential is different from the potential V (ϕ) ofthe real SF introduced by Kamenshchik et al [50] whichis valid for an homogeneous SF in a cosmological context.A generalized Chaplygin gas model (GCG) has been

introduced. It has an equation of state7

P = − A

(ǫ/c2)α(7)

with A > 0 and a generic α constant in the range 0 ≤α ≤ 1 in order to ensure the condition of stability c2s ≥ 0and the condition of causality cs ≤ c (the quantity c2s/c

2

goes from 0 to α when a goes from 0 to +∞). Combinedwith the energy conservation equation, we obtain

ǫ/c2 =

[

A

c2+

B

a3(α+1)

]1

α+1

. (8)

This model interpolates between a universe dominatedby dust and de Sitter eras via an intermediate phase de-scribed by a linear equation of state P ∼ αǫ [50, 95].The original Chaplygin gas model is recovered for α = 1.Bento et al. [95] argued that the GCG model correspondsto a generalized Nambu-Goto action which can be in-terpreted as a perturbed d-brane in a (d + 1, 1) space-time. Bilic et al. [100] mentioned that the generalizedNambu-Goto action lacks any geometrical interpretation,but that the generalized Chaplygin equation of state canbe obtained from a moving brane in Schwarzschild-anti-de-Sitter bulk [101].For α = 0, the generalized Chaplygin equation of state

(7) reduces to a constant negative pressure

P = −A. (9)

In that case, the speed of sound vanishes identically(c2s = 0). It can be shown [102, 103] that this modelis equivalent to the ΛCDM model not only to 0th orderin perturbation theory (background) but to all orders,even in the nonlinear clustering regime (contrary to the

7 This generalization was mentioned by Kamenshchik et al. [50],Bilic et al. [71] and Gorini et al. [94], and was specifically workedout by Bento et al. [95]. Equation (7) can be viewed as a poly-tropic equation of state P = K(ǫ/c2)γ with a polytropic indexγ = −α and a polytropic constant K = −A. A further general-ization of the GCG model has been proposed. It has an equationof state

P = ωǫ−A

(ǫ/c2)α, (6)

where ω is a constant. This is called the Modified ChaplyginGas (MCG) model [96]. It can be viewed as the sum of a linearequation of state and a polytropic equation of state. This gener-alized polytropic equation of state has been studied in detail in acosmological framework in Refs. [97–99]. The potential V (ϕ) ofits real SF representation generalizing the result of Kamenshchiket al. [50] has been determined explicitly in these papers.

4

initial claim made in Ref. [104]). Therefore, the ΛCDMmodel can either be considered as a two-fluid model in-volving a DM fluid with Pdm = 0 and a DE fluid withPde = −ǫde, or as a single dark fluid with a constantnegative pressure P = −ǫΛ [98]. In this sense, it may beregarded as the simplest UDM model one can possiblyconceive in which DM and DE appear as different man-ifestations of a single dark fluid. As a result, the GCGmodel includes the original Chaplygin gas model (α = 1)and the ΛCDM model (α = 0) as particular cases.

The GCG model has been successfully confronted withvarious phenomenological tests such as high precisionCosmic Microwave Background Radiation data [105–109], type Ia supernova (SNIa) data [52, 94, 110–113],age estimates of high-z objects [112] and gravitationallensing [114]. Although the GCG model is consistentwith observations related to the background cosmology(the Hubble law is almost insensitive to α) [52, 110, 113],Sandvik et al. [103] showed that it produces unphysicaloscillations or even an exponential blow-up which are notseen in the observed matter power spectrum calculatedat the present time. This is caused by the behaviourof the sound speed through the GCG fluid. At earlytimes, the GCG behaves as DM and its sound speed van-ishes. In that case, the GCG clusters like pressurelessdust. At late times, when the GCG behaves as DE, itssound speed becomes relatively large yielding unphysicalfeatures in the matter power spectrum. To avoid such un-physical features, the value of α must be extremely closeto zero (|α| < 10−5), so that the GCG model becomesindistinguishable from the ΛCDM model. Similar con-clusions were reached by Bean and Dore [115], Carturanand Finelli [108] and Amendola et al. [109] who studiedthe effect of the GCG on density perturbations and oncosmic microwave background (CMB) anisotropies andfound that the GCG strongly increases the amount ofintegrated Sachs-Wolfe effect. Therefore, CMB data aremore selective than SN Ia data to constrain α. The GCGis essentially ruled out except for a tiny region of param-eter space very close to the ΛCDM limit. This conclusionis not restricted to the GCG model but is actually validfor all UDM models.8 Some solutions to this problemhave been suggested (see a short review in Sec. XVI of[116]) but there is no definite consensus at the presenttime.

These results show that a viable UDM model should beas close as possible to the ΛCDM model, but sufficientlydifferent from it in order to solve its problems. This is thebasic idea that led us to introduce the logotropic modelin Ref. [117] (see also [118–121]). The logotropic dark

8 If a solution to these problems cannot be provided, this wouldappear as an evidence for an independent origin of DM and DE(i.e., they are two distinct substances) [103].

fluid has an equation of state

P = A ln

(

ρmρ∗

)

, (10)

where ρm is the rest-mass density. This equation of statecan be obtained from the polytropic (GCG) equation ofstate P = Kργm by considering the limit γ → 0 andK → ∞ with A = Kγ constant (see Sec. 3 of [117] andAppendix A of [116]). This yields

P = Keγ ln ρm ≃ K(1 + γ ln ρm + ...) ≃ K +A ln ρ, (11)

which is equivalent to Eq. (10) up to a constant term.9

Since the ΛCDM model (viewed as a UDM model) isequivalent to a polytropic gas of index γ = 0 (constantpressure), one can say that the logotropic model whichhas γ → 0 is the simplest extension of the ΛCDM model.It is argued in Ref. [117] that ρ∗ = ρP = c5/(~G2) =5.16 × 1099 g/m3 is the Planck density. It is also ar-gued that A/c2 = BρΛ = 2.10 × 10−26 gm−3 (whereB = 3.53 × 10−3 and ρΛ = 5.96 × 10−24 gm−3) is afundamental constant of physics that supersedes the cos-mological constant Λ. The logotropic model is able toaccount for the transition between a DM era and a DEera and is indistinguishable from the ΛCDM model, forwhat concerns the evolution of the cosmological back-ground, up to 25 billion years in the future when itbecomes phantom [117–121]. Very interestingly, the lo-gotropic model implies that DM halos should have a con-stant surface density and it predicts its universal valueΣth

0 = 0.01955c√Λ/G = 133M⊙/pc2 [117–121] with-

out adjustable parameter. This theoretical value is ingood agreement with the value Σobs

0 = 141+83−52M⊙/pc2

obtained from the observations [122]. The logotropicmodel also predicts the value of the constant Ωdm,0 thatis usually interpreted as the present proportion of DMin the Universe: it gives Ωth

dm,0 ≃ 1/(1 + e) = 0.269

[120, 121] in good agreement with the measured valueΩobs

dm,0 = 0.260.10 Unfortunately, the logotropic modelsuffers from the same problems as the GCG model re-garding the presence of unphysical oscillations in thematter power spectrum [121, 123]. It is not clear howthese problems can be circumvented (see the discussionin [121]). Anyway, the logotropic dark fluid (LDF) re-mains an interesting UDM model, especially because ofits connection with the polytropic (GCG) model.

9 This procedure is not well-defined mathematically because ityields an infinite additional constant K → +∞. This constantdisappears if we take the gradient of the pressure as in [117].However, in general, an infinite constant term remains. There-fore, the above procedure simply suggests a connection betweenthe polytropic and logotropic equations of state, but this con-nection is rather subtle.

10 There is no DM and no DE in the logotropic model, just a singledark fluid. In that case, Ωdm,0 represents the constant thatappears in the asymptotic expression ǫ/ǫ0 ∼ Ωdm,0/a

3 of theenergy density versus scale factor relation for a ≪ 1.

5

The aim of the present paper is to develop the La-grangian formulation of the polytropic (GCG) and lo-gotropic models. We point out that the equation of statecan be specified in different manners, yielding three sortsof models. In model I, the pressure is a function P (ǫ) ofthe energy density; in model II, the pressure is a functionP (ρm) of the rest-mass density; in model III, the pres-sure is a function P (ρ) of the pseudo rest-mass densityassociated with a complex SF (in the sense given below).In the nonrelativistic regime, these three formulations co-incide. However, in the relativistic regime, they lead todifferent models. In this paper, we describe these mod-els in detail and show their interrelations. For example,given P (ǫ), we show how one can obtain P (ρm) and P (ρ),and reciprocally. We also explain how one can obtain thek-essence Lagrangian (action) for each model. We firstprovide general results that can be applied to an arbi-trary barotropic equation of state. Then, for illustration,we obtain explicit analytical results for a polytropic and alogotropic equation of state. We recover the Born-Infeldaction of the Chaplygin gas and determine the expres-sion of the action of the GCG of type I, II and III. Wealso explicitly obtain the logotropic action in models IIand III. We show that it can be recovered from the poly-tropic (GCG) action in the limit γ → 0 andK → ∞ withA = Kγ constant.

The paper is organized as follows. In Sec. II, we con-sider a nonrelativistic complex self-interacting SF whichmay represent the wavefunction of a Bose-Einstein con-densate (BEC) described by the Gross-Pitaevskii (GP)equation. We determine its Madelung hydrodynamic rep-resentation and show that it is equivalent to an irrota-tional quantum fluid with a quantum potential and abarotropic equation of state P (ρ) determined by the self-interaction potential. In the TF limit, it reduces to anirrotational classical barotropic fluid. We determine itsLagrangian and “reduced” Lagrangian for an arbitraryequation of state. The reduced Lagrangian has the formof a k-essence Lagrangian L(x) with x = θ+(1/2)(∇θ)2,where θ is the potential of the velocity field. In Sec.III, we consider a relativistic complex self-interacting SFdescribed by the KG equation which may represent thewavefunction of a relativistic BEC. We determine its deBroglie hydrodynamic representation and show that it isequivalent to an irrotational quantum fluid with a covari-ant quantum potential and a barotropic equation of stateP (ρ) determined by the self-interaction potential. In theTF limit, it reduces to an irrotational classical barotropicfluid. We determine its Lagrangian and reduced La-grangian for an arbitrary equation of state. Its reducedLagrangian has the form of a k-essence Lagrangian L(X)with X = (1/2)∂µθ∂

µθ, where the real SF θ is played bythe phase of the complex SF. In Sec. IV, we introducethree types of equations of state (models I, II and III)and explain their physical meanings. We provide gen-eral equations allowing us to connect one model to theother and to determine the reduced Lagrangian L(X) foran arbitrary equation of state. In Secs. V and VI, we

illustrate our general results by applying them to a poly-tropic (GCG) equation of state and a logotropic equationof state. We recover the Born-Infeld action of the Chap-lygin gas and determine the reduced action of the GCGand of the logotropic gas. In Appendix A, we establishgeneral identities valid for a nonrelativistic cold gas. InAppendix B, we consider a general k-essence Lagrangianand specifically discuss the case of a canonical and tachy-onic SF. In Appendix C, we define the equation of stateof model I and detail how one can obtain the potentialV (ϕ) of a homogeneous real SF in an expanding universe.In Appendix D, we define the equation of state of modelII and detail how one can obtain the corresponding in-ternal energy. In Appendix E, we define the equation ofstate of model III (see also Sec. III) and detail how thebasic equations of the problem can be simplified in thecase of a homogeneous SF in a cosmological context. InAppendix F, we discuss the analogies and the differencesbetween the internal energy and the potential of a com-plex SF in the TF limit. In Appendix H, we list somestudies devoted to polytropic and logotropic equationsof state of type I, II and III. In Appendix I, we detailthe Lagrangian structure and the conservation laws of anonrelativistic and relativistic SF. Applications and gen-eralizations of the results of this paper will be presentedin future works [124].

II. NONRELATIVISTIC THEORY

A. The Gross-Pitaevskii equation

We consider a complex SF ψ(r, t) whose evolution isgoverned by the GP equation

i~∂ψ

∂t= − ~

2

2m∆ψ +mh(|ψ|2)ψ. (12)

This equation describes, for example, the wave functionof a BEC at T = 0 [125]. For the sake of generality,we have introduced an arbitrary nonlinearity determinedby the effective potential h(|ψ|2) instead of the quadraticpotential h(|ψ|2) = g|ψ|2 = (4πas~

2/m3)|ψ|2, where asis the s-scattering length of the bosons, arising from paircontact interactions in the usual GP equation [126–129](see, e.g., the discussion in Ref. [125]). In this manner,we can describe a larger class of systems.11 The GPequation (12) can also be derived from the Klein-Gordon(KG) equation

ϕ+m2c2

~2ϕ+ 2

dV

d|ϕ|2ϕ = 0, (13)

11 We could also consider the case of self-gravitating BECs. In thatcase, one has to introduce a mean field gravitational potentialΦ(r, t) in the GP equation which is produced by the particlesthemselves through a Poisson equation (see Ref. [125] for moredetails).

6

which governs the evolution of a complex SF ϕ(r, t) witha self-interaction potential V (|ϕ|2). The GP equation(12) is obtained from the KG equation (13) in the non-relativistic limit c → +∞ after making the Klein trans-formation

ϕ(r, t) =~

me−imc2t/~ψ(r, t). (14)

In that case, the effective potential h(|ψ|2) that appearsin the GP equation is related to the self-interaction po-tential V (|ϕ|2) present in the KG equation by (see Refs.[130, 131] and Appendix C of Ref. [117])

h(|ψ|2) = dV

d|ψ|2 with |ψ|2 =m2

~2|ϕ|2. (15)

As a result, the GP equation (12) can be rewritten as

i~∂ψ

∂t= − ~

2

2m∆ψ +m

dV

d|ψ|2ψ. (16)

Remark: The GP equation (16) can also be derivedfrom the KG equation governing the evolution of a realSF but, in that case, the potential V (|ψ|2) that appearsin the GP equation does not coincide with the potentialV (ϕ) present in the KG equation. Indeed, V (|ψ|2) is aneffective potential obtained after averaging V (ϕ) over thefast oscillations of the SF (see Secs. II and III of [132]and Appendix A of [133] for details).

B. The Madelung transformation

We can write the GP equation (12) under the formof hydrodynamic equations by using the Madelung [134]transformation. To that purpose, we write the wave func-tion as

ψ(r, t) =√

ρ(r, t)eiS(r,t)/~, (17)

where ρ(r, t) is the density and S(r, t) is the action. Theyare given by

ρ = |ψ|2 and S = −i~2ln

(

ψ

ψ∗

)

. (18)

Following Madelung, we introduce the velocity field

u =∇Sm

. (19)

Since the velocity derives from a potential, the flow isirrotational: ∇× u = 0. Substituting Eq. (17) into Eq.(12) and separating the real and the imaginary parts, weobtain

∂ρ

∂t+∇ · (ρu) = 0, (20)

∂S

∂t+

1

2m(∇S)2 +mh(ρ) +Q = 0, (21)

where

Q = − ~2

2m

∆√ρ

√ρ

= − ~2

4m

[

∆ρ

ρ− 1

2

(∇ρ)2ρ2

]

(22)

is the quantum potential. It takes into account theHeisenberg uncertainty principle. Eq. (20) is similarto the equation of continuity in hydrodynamics. It ac-counts for the local conservation of mass M =

∫

ρ dr.Eq. (21) has a form similar to the classical Hamilton-Jacobi equation with an additional quantum potential.It can also be interpreted as a quantum Bernoulli equa-tion for a potential flow. Taking the gradient of Eq.(21), and using the well-known identity of vector anal-ysis (u · ∇)u = ∇(u2/2)− u× (∇× u) which reduces to(u · ∇)u = ∇(u2/2) for an irrotational flow, we obtain

∂u

∂t+ (u · ∇)u = −∇h− 1

m∇Q. (23)

Since h = h(ρ) we can introduce a function P = P (ρ) sat-isfying ∇h = (1/ρ)∇P . Eq. (23) can then be rewrittenas

∂u

∂t+ (u · ∇)u = −1

ρ∇P − 1

m∇Q. (24)

This equation is similar to the Euler equation with apressure force −(1/ρ)∇P and a quantum force − 1

m∇Q.Since P (r, t) = P [ρ(r, t)] is a function of the density,the flow is barotropic. The equation of state P (ρ) isdetermined by the potential h(ρ) through the relation

h′(ρ) =P ′(ρ)

ρ. (25)

Equation (25) can be integrated into

P (ρ) = ρh(ρ)− V (ρ) = ρV ′(ρ)− V (ρ) = ρ2[

V (ρ)

ρ

]′,

(26)where V is a primitive of h. This notation is consistentwith Eq. (15) which can be rewritten as

h(ρ) = V ′(ρ), (27)

where V (ρ) is the potential in the KG equation (13) or inthe GP equation (16). Eq. (26) determines the pressureP (ρ) as a function of the potential V (ρ). Inversely, thepotential is determined as a function of the pressure by12

V (ρ) = ρ

∫

P (ρ)

ρ2dρ. (28)

12 We note that the potential is defined from the pressure up to aterm of the form Aρ, where A is a constant. If we add a termAρ in the potential V , we do not change the pressure. On theother hand, if we add a constant term C in the potential V , thisadds a term −C in the pressure. However, for nonrelativisticsystems, this constant term has no observable effect since onlythe gradient of the pressure matters.

7

The speed of sound is c2s = P ′(ρ) = ρV ′′(ρ). The GPequation (12) is equivalent to the hydrodynamic equa-tions (20), (21) and (24). We shall call them the quan-tum Euler equations. Since there is no viscosity, theydescribe a superfluid. In the TF approximation ~ → 0,they reduce to the classical Euler equations.Remark: We show in Appendix A that the effective

potential h appearing in the GP equation can be inter-preted, in the hydrodynamic equations, as an enthalpy(or as a chemical potential by unit of mass h = µ/m)and that its primitive V (ρ), equal to the potential in theKG equation, can be interpreted as the internal energydensity u. Thus, we have

u(ρ) = V (ρ), h(ρ) =P (ρ) + V (ρ)

ρ. (29)

On the other hand, if we define the energy by

E(r, t) = −∂S∂t, (30)

the Hamilton-Jacobi equation (21) can be rewritten as

E(r, t) =1

2mu

2 +mh(ρ) +Q. (31)

C. Lagrangian of a quantum barotropic gas

The action of the complex SF associated with the GPequation (16) is given by

S =

∫

Ldt (32)

with the Lagrangian

L =

∫

i~

2m

(

ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

)

− ~2

2m2|∇ψ|2−V (|ψ|2)

dr.

(33)We can view the Lagrangian (33) as a functional of ψ,

ψ and ∇ψ. The least action principle δS = 0, which isequivalent to the Euler-Lagrange equation

∂

∂t

(

δL

δψ

)

+∇ ·(

δL

δ∇ψ

)

− δL

δψ= 0, (34)

returns the GP equation (16). The Hamiltonian (energy)is obtained from the transformation

H =

∫

i~

2m

(

ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

)

dr− L (35)

yielding

H =~2

2m2

∫

|∇ψ|2 dr+∫

V (|ψ|2) dr. (36)

The first term is the kinetic energy Θ = − ~2

2m2

∫

ψ∗∆ψ drand the second term is the self-interaction energy U .

Since the Lagrangian does not explicitly depend on time,the Hamiltonian (energy) is conserved.13 The GP equa-tion (16) can be written as

i~∂ψ

∂t= m

δH

δψ∗ , i~∂ψ∗

∂t= −mδH

δψ, (37)

which can be interpreted as Hamilton equations (see Ap-pendix I 3).Using the Madelung transformation, we can rewrite

the Lagrangian in terms of hydrodynamic variables. Ac-cording to Eqs. (17) and (18) we have

∂S

∂t= −i~

2

1

|ψ|2(

ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

)

(38)

and

|∇ψ|2 =1

4ρ(∇ρ)2 + ρ

~2(∇S)2. (39)

Substituting these identities into Eq. (33), we get

L = −∫

ρ

m

∂S

∂t+

ρ

2m2(∇S)2 + ~

2

8m2

(∇ρ)2ρ

+V (ρ)

dr.

(40)

We can view the Lagrangian (40) as a functional of S, S,∇S, ρ, ρ, and ∇ρ. The Euler-Lagrange equation for theaction

∂

∂t

(

δL

δS

)

+∇ ·(

δL

δ∇S

)

− δL

δS= 0 (41)

returns the equation of continuity (20). The Euler-Lagrange equation for the density

∂

∂t

(

δL

δρ

)

+∇ ·(

δL

δ∇ρ

)

− δL

δρ= 0 (42)

returns the quantum Hamilton-Jacobi (or Bernoulli)equation (21) leading to the quantum Euler equation(24). The Hamiltonian (energy) is obtained from thetransformation

H = −∫

ρ

m

∂S

∂tdr− L (43)

yielding

H =

∫

1

2ρu2 dr+

∫

~2

8m2

(∇ρ)2ρ

dr+

∫

V (ρ) dr. (44)

This expression is equivalent to Eq. (36) as can be seenby a direct calculation using the Madelung transforma-tion [see Eq. (39)]. The first term is the classical kinetic

13 Similarly, the invariance of the Lagrangian with respect to spatialtranslation implies the conservation of linear momentum. In rela-tivity theory, these apparently separate conservation laws are as-pects of a single conservation law, that of the energy-momentumtensor (see Sec. III).

8

energy Θc, the second term is the quantum kinetic energyΘQ (we have Θ = Θc + ΘQ), and the third term is theself-interaction energy U . The quantum kinetic energycan also be written as ΘQ =

∫

ρQm dr [135]. The continu-

ity equation (20) and the Hamilton-Jacobi (or Bernoulli)equation (21) can be written as

∂ρ

∂t= m

δH

δS,

∂S

∂t= −mδH

δρ, (45)

which can be interpreted as Hamilton equations (see Ap-pendix I 4).If we substitute the quantum Hamilton-Jacobi (or

Bernoulli) equation (21) into the Lagrangian (40) anduse Eqs. (22) and (26) we find that

L =

∫

P dr. (46)

This shows that the Lagrangian density is equal to thepressure: L = P . Actually, the Lagrangian density is

equal to L = P (ρ) − ~2

4m2∆ρ. There is an additionalterm which disappears by integration. The same resultis obtained by substituting the GP equation (16) into theLagrangian (33) and using Eq. (26).

D. Lagrangian of a classical barotropic gas

Introducing the notation θ = S/m, so that ψ =√ρeimθ/~,14 and taking the limit ~ → 0 in Eq. (40),

we obtain the classical Lagrangian15

L = −∫ [

ρθ +1

2ρ(∇θ)2 + V (ρ)

]

dr. (47)

We can view the Lagrangian (47) as a functional of θ, θ,∇θ, ρ, ρ, and ∇ρ. The Euler-Lagrange equation for theaction leads to the equation of continuity

∂ρ

∂t+∇ · (ρu) = 0, (48)

with the velocity field

u = ∇θ. (49)

The Euler-Lagrange equation for the density leads to theBernoulli (or Hamilton-Jacobi) equation

θ +1

2(∇θ)2 + V ′(ρ) = 0. (50)

14 The variable θ is related to the phase (angle) Θ = S/~ of the SFby θ = ~Θ/m.

15 In our framework, the limit ~ → 0 corresponds to the TF ap-proximation where the quantum potential can be neglected.

Taking the gradient of Eq. (50) and using Eq. (26), weobtain the Euler equation

∂u

∂t+ (u · ∇)u = −1

ρ∇P. (51)

The Hamiltonian (energy) is obtained from the trans-formation

H = −∫

ρθ dr− L (52)

yielding

H =

∫

1

2ρu2 dr+

∫

V (ρ) dr. (53)

The first term is the classical kinetic energy Θc and thesecond term is the self-interaction energy U .Remark: In our presentation, we started from a quan-

tum fluid (or from the hydrodynamic representation ofthe GP equation) and finally considered the classical limit~ → 0. Alternatively, we can obtain the equations of thissection directly from the classical Euler equations by as-suming that the fluid is barotropic (so that P = P (ρ))and that the flow is irrotational (so that the velocity de-rives from a potential: u = ∇θ) [59]. The Lagrangians(40) and (47) were first obtained by Eckart [136] for aclassical fluid and from the hydrodynamic representationof the Schrodinger equation (see also [57, 137]). The La-grangian (47) with the potential V = A/(2ρ) correspond-ing to the Chaplygin equation of state P = −A/ρ (seebelow) also appeared in the theory of membranes (d = 2)[57, 66]. It was later generalized to a d-brane moving ina (d+ 1, 1) space-time [58, 59].

E. Reduced Lagrangian L(x)

We introduce the Lagrangian density

L = −[

ρθ +1

2ρ(∇θ)2 + V (ρ)

]

, (54)

so that L =∫

L dr and S =∫

L drdt. Using the Bernoulliequation (50) and the identity (26), we can eliminate θfrom the Lagrangian and obtain

L = ρV ′(ρ)− V (ρ) = P (ρ). (55)

Therefore, the Lagrangian density is equal to the pres-sure:

L = P. (56)

We now eliminate ρ from the Lagrangian. Introducingthe notation

x = θ +1

2(∇θ)2, (57)

9

the Bernoulli equation (50) can be written as

x = −V ′(ρ). (58)

Assuming V ′′ > 0, this equation can be reversed to give

ρ = F (x) (59)

with F (x) = (V ′)−1(−x).16 As a result, the equation ofcontinuity (48) can be written in terms of θ alone as

∂

∂t

F

[

θ +1

2(∇θ)2

]

+∇ ·

F

[

θ +1

2(∇θ)2

]

∇θ

= 0.

(60)On the other hand, according to Eqs. (26) and (59), wehave P = P (x). Therefore, recalling Eq. (56), we get

L = P (x). (61)

In this manner, we have eliminated the density ρ fromthe Lagrangian (54) and we have obtained a reduced La-grangian of the form L(x) that depends only on x. Thiskind of Lagrangian, called k-essence Lagrangian, is dis-cussed in Appendix B. We show below that

ρ = F (x) = −L′(x) = −P ′(x). (62)

Using Eq. (62), we can write Eq. (48) in terms of L′(x)like in Eq. (B47) of Appendix B.Proof of Eq. (62): From Eqs. (26) and (58) we have

P ′(ρ) = ρV ′′(ρ) (63)

and

dx

dρ= −V ′′(ρ). (64)

Starting from Eq. (61) and using Eqs. (63) and (64) weobtain

L′(x) = P ′(x) = P ′(ρ)dρ

dx= ρV ′′(ρ)

dρ

dx= −ρdx

dρ

dρ

dx= −ρ,(65)

which establishes Eq. (62).The preceding results are general. In the following

sections, we consider particular equations of state.

F. Polytropic gas

We first consider the polytropic equation of state [138]

P = Kργ . (66)

16 The condition V ′′ > 0 is necessary for local stablity since c2s =P ′(ρ) = ρV ′′(ρ). When the system is subjected to a potential Φ,the function F determines the relation between the density andthe potential at equilibrium (see Sec. 3.4 of [135]).

It can be obtained from the potential [135]

V (ρ) =K

γ − 1ργ . (67)

As discussed in [135] this potential is similar to the Tsal-lis free energy density −Ksγ, where the polytropic con-stant K plays the role of a generalized temperature andsγ = − 1

γ−1(ργ − ρ) is the Tsallis entropy density. The

Lagrangian and the Hamiltonian are given by Eqs. (47)and (53) with Eq. (67). The Bernoulli equation (50)takes the form

θ +1

2(∇θ)2 + Kγ

γ − 1ργ−1 = 0. (68)

From this equation, we obtain

ρ =

[

−γ − 1

Kγ

(

θ +1

2(∇θ)2

)]1

γ−1

, (69)

which is similar to the Tsallis distribution. The reducedLagrangian L(x) corresponding to the polytropic gas is

L = P = K

[

−γ − 1

Kγ

(

θ +1

2(∇θ)2

)]γ

γ−1

. (70)

The equation of motion is

∂

∂t

(

∣

∣

∣

∣

θ +1

2(∇θ)2

∣

∣

∣

∣

1γ−1

)

+∇·(

∣

∣

∣

∣

θ +1

2(∇θ)2

∣

∣

∣

∣

1γ−1

∇θ)

= 0.

(71)We note that the polytropic constant K does not appearin this equation.

G. Isothermal gas

The case γ = 1, corresponding to the isothermal equa-tion of state [138]

P = Kρ, (72)

must be treated specifically (here K plays the role ofthe temperature kBT/m). It can be obtained from thepotential [135]

V (ρ) = Kρ [ln(ρ/ρ∗)− 1] . (73)

As discussed in [135] this potential is similar to the Boltz-mann free energy density −KsB, where K plays the roleof the temperature and sB = −ρ[ln(ρ/ρ∗) − 1] is theBoltzmann entropy density. The Lagrangian and theHamiltonian are given by Eqs. (47) and (53) with Eq.(73). The Bernoulli equation (50) takes the form

θ +1

2(∇θ)2 +K ln(ρ/ρ∗) = 0. (74)

From this equation, we obtain

ρ = ρ∗e− 1

K [θ+ 12 (∇θ)2], (75)

10

which is similar to the Boltzmann distribution. The re-duced Lagrangian L(x) corresponding to the isothermalgas is

L = P = Kρ∗e− 1

K [θ+ 12 (∇θ)2]. (76)


∂

∂t

(

e−1K [θ+ 1

2 (∇θ)2])

+∇·(

e−1K [θ+ 1

2 (∇θ)2]∇θ)

= 0. (77)

We note that the constant K (temperature) cannot beeliminated from this equation contrary to the polytropiccase.

H. Chaplygin gas

The Chaplygin equation of state writes [53]

P =K

ρ. (78)

The ordinary Chaplygin gas corresponds to K < 0. Thecase K > 0 is called the anti-Chaplygin gas. Eq. (78) isa particular polytropic equation of state (66) correspond-ing to γ = −1.17 It can be obtained from the potential

V (ρ) = −K

2ρ. (79)

The Lagrangian and the Hamiltonian are given by Eqs.(47) and (53) with Eq. (79). The Bernoulli equation (50)takes the form

θ +1

2(∇θ)2 + K

2ρ2= 0, (80)

yielding

ρ =

√

√

√

√

−K2[

θ + 12 (∇θ)2

] . (81)

The reduced Lagrangian L(x) corresponding to theChaplygin gas is

L = P = K

√

2

−K

[

θ +1

2(∇θ)2

]

. (82)


∂

∂t

1√

|θ + 12 (∇θ)2|

+∇ ·

∇θ√

|θ + 12 (∇θ)2|

= 0.(83)

17 For that reason, the polytropic equation of state (66) is alsocalled the GCG [95].

The Chaplygin constant K does not appear in this equa-tion. If we consider time-independent solutions, thisequation reduces to

∇ ·(

∇θ√

(∇θ)2

)

= 0. (84)

The same equation is obtained by taking the masslesslimit (recalling that θ = S/m). In the theory of d-branes,this equation means that the surface θ(x1, x2, ..., xd) =const has zero extrinsic mean curvature [68]. This solu-tion exists only when K < 0.Remark: The Lagrangian (82) was obtained by [58, 59]

in two different manners: (i) starting from the La-grangian (47) with Eq. (79) and using the Bernoulliequation (80) to eliminate ρ as we have done here; (ii)for a d-brane moving in a (d + 1, 1) space-time. In thatsecond case, it can be obtained from the Nambu-Gotoaction in the light-cone parametrization. This explainsthe connection between d-branes and the hydrodynamicsof the Chaplygin gas.

I. Standard BEC

The potential of a standard BEC described by the or-dinary GP equation is

V (|ψ|2) = 2πas~2

m3|ψ|4, (85)

where as is the scattering length of the bosons (the in-teraction is repulsive when as > 0 and attractive whenas < 0). This quartic potential accounts for two-bodyinteractions in a weakly interacting microscopic theoryof the superfluid. The corresponding equation of state is

P =2πas~

2

m3ρ2. (86)

This is the equation of state of a polytrope of indexγ = 2 and polytropic constant K = 2πas~

2/m3. TheLagrangian and the Hamiltonian are given by Eqs. (47)and (53) with Eq. (85). The Bernoulli equation (50)takes the form

θ +1

2(∇θ)2 + 2Kρ = 0, (87)

yielding

ρ = − 1

2K

[

θ +1

2(∇θ)2

]

. (88)

The reduced Lagrangian L(x) corresponding to the stan-dard BEC is

L = P =1

4K

[

θ +1

2(∇θ)2

]2

. (89)


∂

∂t

[

θ +1

2(∇θ)2

]

+∇ ·([

θ +1

2(∇θ)2

]

∇θ)

= 0. (90)

The BEC constant K does not appear in this equation.

11

J. DM superfluid

The potential of a superfluid (BEC) with a sextic self-interaction is

V (|ψ|2) = 1

2K|ψ|6. (91)

This potential accounts for three-body interactions in aweakly interacting microscopic theory of the superfluid[132]. The potential (91) may also describe a more exoticDM superfluid [139]. In that case, it has a completelydifferent interpretation. The corresponding equation ofstate is

P = Kρ3. (92)

This is the equation of state of a polytrope of index γ = 3.The Lagrangian and the Hamiltonian are given by Eqs.(47) and (53) with Eq. (91). The Bernoulli equation (50)takes the form

θ +1

2(∇θ)2 + 3K

2ρ2 = 0, (93)

yielding

ρ =

√

− 2

3K

[

θ +1

2(∇θ)2

]

. (94)

The reduced Lagrangian L(x) corresponding to the su-perfluid is

L = P = K

− 2

3K

[

θ +1

2(∇θ)2

]3/2

. (95)


∂

∂t

[

√

|θ + 1

2(∇θ)2|

]

+∇ ·(

√

|θ + 1

2(∇θ)2|∇θ

)

= 0.

(96)The superfluid constant K does not appear in this equa-tion. If we consider time-independent solutions, thisequation reduces to

∇ · (|∇θ|∇θ) = 0. (97)

This solution exists only when K < 0. Interestingly,there is a connection between a superfluid described byEq. (97) and the modified Newtonian dynamics (MOND)theory (see, e.g., [139, 140] for more details).

K. Logotropic gas

Finally, we consider the logotropic equation of state[117]

P = A ln

(

ρ

ρ∗

)

, (98)

which can be obtained from the potential [135]

V (ρ) = −A ln

(

ρ

ρ∗

)

−A. (99)

The Lagrangian and the Hamiltonian are given by Eqs.(47) and (53) with Eq. (99). The Bernoulli equation (50)takes the form

θ +1

2(∇θ)2 − A

ρ= 0, (100)

from which we get

ρ =A

θ + 12 (∇θ)2

. (101)

The reduced Lagrangian L(x) corresponding to the lo-gotropic gas is

L = P = −A ln

[

ρ∗A

(

θ +1

2(∇θ)2

)]

. (102)


∂

∂t

(

1

|θ + 12 (∇θ)2|

)

+∇ ·(

∇θ|θ + 1

2 (∇θ)2|

)

= 0. (103)

The logotropic constant A does not appear in this equa-tion.Remark: We can recover these results from the poly-

tropic equation of state of Sec. II F by considering thelimit γ → 0, K → ∞ with A = Kγ constant [117, 141].Starting from Eq. (70), we get

L = K

[

−γ − 1

Kγx

]γ

γ−1

≃ Ke−γ ln( xKγ )

≃ K

[

1− γ ln

(

x

Kγ

)

+ ...

]

≃ K −A ln( x

A

)

, (104)

which is equivalent to Eq. (102) up to a constant term(see footnote 12).

L. Summary

For a polytropic equation of state P = Kργ with γ 6= 1,the reduced Lagrangian is

L(x) = K

(

−γ − 1

Kγx

)γ

γ−1

. (105)

It is a pure power-law L ∝ xγ

γ−1 . In particular, for theChaplygin gas (γ = −1), for the standard BEC (γ = 2)and for the DM superfluid (γ = 3) we have L ∝ x1/2,L ∝ x2 and L ∝ x3/2 respectively. For the unitary Fermigas (γ = 5/3) we have L ∝ x5/2. For an isothermalequation of state P = Kρ, the reduced Lagrangian is

L(x) = Kρ∗e−x/K . (106)

For a logotropic equation of state P = A ln(ρ/ρ∗), thereduced Lagrangian is

L(x) = −A ln(ρ∗Ax)

. (107)

12

III. RELATIVISTIC THEORY

A. Klein-Gordon equation

We consider a relativistic complex SF ϕ(xµ) =ϕ(x, y, z, t) which is a continuous function of space andtime. It can represent the wavefunction of a relativisticBEC. The action of the SF can be written as

S =

∫

L√−g d4x, (108)

where L = L(ϕ, ϕ∗, ∂µϕ, ∂µϕ∗) is the Lagrangian densityand g = det(gµν) is the determinant of the metric tensor.We consider a canonical Lagrangian density of the form

L =1

2gµν∂µϕ

∗∂νϕ− Vtot(|ϕ|2), (109)

where the first term is the kinetic energy and the secondterm is minus the potential energy. The potential energycan be decomposed into a rest-mass energy term and aself-interaction energy term:

Vtot(|ϕ|2) =1

2

m2c2

~2|ϕ|2 + V (|ϕ|2). (110)

The least action principle δS = 0 with respect to vari-ations δϕ (or δϕ∗), which is equivalent to the Euler-Lagrange equation

Dµ

[

∂L∂(∂µϕ)∗

]

− ∂L∂ϕ∗ = 0, (111)

yields the KG equation

ϕ+ 2dVtotd|ϕ|2ϕ = 0, (112)

where = Dµ∂µ = 1√−g

∂µ(√−g gµν∂ν) is the

d’Alembertian operator in a curved spacetime. It canbe written explicitly as

ϕ = Dµ∂µϕ = gµνDµ∂νϕ = gµν(∂µ∂νϕ− Γσ

µν∂σϕ)

=1√−g∂µ(

√−g∂µϕ). (113)

For a free massless SF (Vtot = 0), the KG equation re-duces to ϕ = 0.The energy-momentum (stress) tensor is given by

Tµν =2√−g

δS

δgµν=

2√−g∂(√−gL)∂gµν

= 2∂L∂gµν

− gµνL. (114)

For a complex SF, we have

T νµ =

∂L∂(∂νϕ)

∂µϕ+∂L

∂(∂νϕ∗)∂µϕ

∗ − gνµL. (115)

For the Lagrangian (109), we obtain

Tµν =1

2(∂µϕ

∗∂νϕ+ ∂νϕ∗∂µϕ)

− gµν

[

1

2gρσ∂ρϕ

∗∂σϕ− Vtot(|ϕ|2)]

. (116)

The conservation of the energy-momentum tensor, whichresults from the invariance of the Lagrangian density un-der continuous translations in space and time (Noethertheorem [142]), writes

DνTµν = 0. (117)

The energy-momentum four vector is Pµ =∫

T µ0√−g d3x. Its time component P 0 is the en-ergy. Each component of Pµ is conserved in time, i.e., itis a constant of motion. Indeed, we have

Pµ =d

dt

∫

T µ0√−g d3x = c

∫

∂0(Tµ0√−g) d3x

= −c∫

∂i(Tµi√−g) d3x = 0, (118)

where we have used Eq. (117) with DµVµ =

1√−g∂µ(

√−gV µ) to get the third equality.

The current of charge of a complex SF is given by

Jµ =m

i~

[

ϕ∂L

∂(∂µϕ)− ϕ∗ ∂L

∂(∂µϕ∗)

]

. (119)


Jµ = − m

2i~(ϕ∗∂µϕ− ϕ∂µϕ

∗). (120)

Using the KG equation (112), one can show that

DµJµ = 0. (121)

This equation expresses the local conservation of thecharge. The total charge of the SF is

Q =e

mc

∫

J0√−g d3x, (122)

and we easily find from Eq. (121) that Q = 0. The chargeQ is proportional to the number N of bosons providedthat antibosons are counted negatively [143]. Therefore,Eq. (121) also expresses the local conservation of theboson number (Q = Ne). This conservation law resultsvia the Noether theorem from the global U(1) symme-try of the Lagrangian, i.e., from the invariance of theLagrangian density under a global phase transformationϕ → ϕe−iθ (rotation) of the complex SF. Note that Jµvanishes for a real SF so the charge and the particle num-ber are not conserved in that case.The Einstein-Hilbert action of general relativity is

Sg =c4

16πG

∫

R√−g d4x, (123)

13

where R the Ricci scalar curvature. The least actionprinciple δSg = 0 with respect to variations δgµν yieldsthe Einstein field equations

Gµν ≡ Rµν − 1

2gµνR =

8πG

c4Tµν . (124)

The contracted Bianchi identity DνGµν = 0 implies the

conservation of the energy momentum tensor (DνTµν =

0).

B. The de Broglie transformation

We can write the KG equation (112) under the formof hydrodynamic equations by using the de Broglie [144–146] transformation. To that purpose, we write the SFas

ϕ =~

m

√ρeiStot/~, (125)

where ρ is the pseudo rest-mass density18 and Stot is theaction. They are given by

ρ =m2

~2|ϕ|2 and Stot =

~

2iln

(

ϕ

ϕ∗

)

. (126)

For convenience, we define θ = Stot/m (see footnote 14)so that Eq. (125) can be rewritten as

ϕ =~

m

√ρeimθ/~. (127)

Substituting this expression into the Lagrangian density(109), we obtain

L =1

2gµνρ∂µθ∂νθ +

~2

8m2ρgµν∂µρ∂νρ− Vtot(ρ) (128)

with

Vtot(ρ) =1

2ρc2 + V (ρ). (129)

The Euler-Lagrange equations for θ and ρ, expressing theleast action principle, are

Dµ

[

∂L∂(∂µθ)

]

− ∂L∂θ

= 0, (130)

Dµ

[

∂L∂(∂µρ)

]

− ∂L∂ρ

= 0. (131)

They yield

Dµ (ρ∂µθ) = 0, (132)

18 We stress that ρ is not the rest-mass density ρm = nm (seebelow). It is only in the nonrelativistic regime c → +∞ that ρcoincides with the rest-mass density ρm.

1

2∂µθ∂

µθ − ~2

2m2

√ρ

√ρ

− V ′tot(ρ) = 0. (133)

The same equations are obtained by substituting the deBroglie transformation from Eq. (127) into the KG equa-tion (112), and by separating the real and the imaginaryparts. Equation (132) can be interpreted as a continuityequation and Eq. (133) can be interpreted as a quantumrelativistic Hamilton-Jacobi (or Bernoulli) equation witha relativistic covariant quantum potential

Q =~2

2m

√ρ

√ρ. (134)

Introducing the pseudo quadrivelocity19

vµ = −∂µStot

m= −∂µθ, (135)

we can rewrite Eqs. (132) and (133) as

Dµ (ρvµ) = 0, (136)

1

2mvµv

µ −Q−mV ′tot(ρ) = 0. (137)

Taking the gradient of the quantum Hamilton-Jacobiequation (137) we obtain [131]

dvνdt

≡ vµDµvν =1

m∂νQ+ ∂νV

′(ρ), (138)

which can be interpreted as a relativistic quantum Eu-ler equation (with the limitations of footnote 19). Thefirst term on the right hand side can be interpreted as aquantum force and the second term as a pressure force(1/ρ)∂νP given by (1/ρ)P ′(ρ) = h′(ρ) = V ′′(ρ), whereh is the pseudo enthalpy. We note that the pressure isdetermined by Eqs. (25)-(28) as in the nonrelativisticcase.The energy-momentum tensor is given by Eq. (114)

or, in the hydrodynamic representation, by

T νµ =

∂L∂(∂νθ)

∂µθ +∂L

∂(∂νρ)∂µρ− gνµL. (139)

For the Lagrangian (128) we obtain

Tµν = ρ∂µθ∂νθ +~2

4m2ρ∂µρ∂νρ− gµνL. (140)

The current of charge of a complex SF is given by

Jµ = − ∂L∂(∂µθ)

(141)

19 The pseudo quadrivelocity vµ does not satisfy vµvµ = c2 so itis not guaranteed to be always timelike. Nevertheless, vµ can beintroduced as a convenient notation.

14


Jµ = − ρ

m∂µStot = −ρ∂µθ = ρvµ. (142)

This result can also be obtained from Eq. (120) by usingEq. (125) coming from the de Broglie transformation.We then see that the continuity equation (132) or (136)is equivalent to Eq. (121). It expresses the conservationof the charge Q of the SF (or the conservation of theboson number N)

Q = Ne = − e

mc

∫

ρ∂0θ√−g d3x. (143)

Assuming ∂µθ∂µθ > 0, we can introduce the fluid

quadrivelocity20

uµ = − ∂µθ√

∂µθ∂µθc, (144)

which satisfies the identity

uµuµ = c2. (145)

Using Eqs. (142) and (144), we can write the current as

Jµ =ρ

c

√

∂µθ∂µθ uµ (146)

and the continuity equation as

Dµ

[

ρ√

∂µθ∂µθ uµ]

= 0. (147)

The rest-mass density ρm = nm (which is proportionalto the charge density ρe) is defined by

Jµ = ρmuµ. (148)

Using Eq. (145), we see that ρmc2 = uµJ

µ. The conti-nuity equation (121) can be written as

Dµ(ρmuµ) = 0. (149)

Comparing Eq. (146) with Eq. (148), we find that therest-mass density ρm = nm of the SF is given by

ρm =ρ

c

√

∂µθ∂µθ. (150)

Using the Bernoulli equation (133), we get

ρm =ρ

c

√

~2

m2

√ρ

√ρ

+ 2V ′tot(ρ). (151)

Remark: More generally, we can define the quadrive-locity by

uµ =Jµ

√

JµJµc, (152)

20 It differs from the pseudo quadrivelocity vµ introduced in Eq.(135).

which satisfies the identity (145). Using Eq. (148) wefind that the rest-mass (or charge) density is given by

ρm =1

c

√

JµJµ. (153)

We note that J0 is not equal to the rest-mass density ingeneral (ρm 6= J0/c) except if the SF is static in whichcase uµ = c δµ0 and J0 = ρmc.

C. TF approximation

In the classical or TF limit (~ → 0), the Lagrangianfrom Eq. (128) reduces to

L =1

2gµνρ∂µθ∂νθ − Vtot(ρ). (154)

The Euler-Lagrange equations (130) and (131) yield theequations of motion

Dµ (ρ∂µθ) = 0, (155)

1

2∂µθ∂

µθ − V ′tot(ρ) = 0. (156)

The same equations are obtained by making the TF ap-proximation in Eq. (133), i.e., by neglecting the quantumpotential. Equation (155) can be interpreted as a con-tinuity equation and Eq. (156) can be interpreted as aclassical relativistic Hamilton-Jacobi (or Bernoulli) equa-tion. In order to determine the rest mass density, we canproceed as before. Assuming V ′

tot > 0, and using Eq.(156), we introduce the fluid quadrivelocity

uµ = − ∂µθ√

2V ′tot(ρ)

c, (157)

which satisfies the identity (145). Using Eqs. (142) and(157), we can write the current as

Jµ =ρ

c

√

2V ′tot(ρ)uµ (158)

and the continuity equation (155) as

Dµ

[

ρ√

2V ′tot(ρ)u

µ]

= 0. (159)

Comparing Eq. (158) with Eq. (148), we find that therest-mass density ρm = nm is given, in the TF approxi-mation, by

ρm =ρ

c

√

2V ′tot(ρ). (160)

In general, ρm 6= ρ except when V is constant, corre-sponding to the ΛCDM model (see below), and in thenonrelativistic limit c→ +∞.

15

The energy-momentum tensor is given by Eq. (114)or, in the hydrodynamic representation, by Eq. (139).For the Lagrangian (154) we obtain

Tµν = ρ∂µθ∂νθ − gµνL (161)

or, using Eq. (157),

Tµν = 2ρV ′tot(ρ)

uµuνc2

− gµνL. (162)

The energy-momentum tensor (162) can be written underthe perfect fluid form21

Tµν = (ǫ+ P )uµuνc2

− Pgµν , (164)

where ǫ is the energy density and P is the pressure, pro-vided that we make the identifications

P = L, ǫ+ P = 2ρV ′tot(ρ). (165)

Therefore, the Lagrangian plays the role of the pressureof the fluid. Combining Eq. (154) with the Bernoulliequation (156), we get

L = ρV ′tot(ρ)− Vtot(ρ). (166)

Therefore, according to Eqs. (165) and (166), the energydensity and the pressure derived from the Lagrangian(154) are given by

ǫ = ρV ′tot(ρ) + Vtot(ρ) = ρc2 + ρV ′(ρ) + V (ρ), (167)

P = ρV ′tot(ρ)− Vtot(ρ) = ρV ′(ρ)− V (ρ), (168)

where we have used Eq. (129) to get the second equal-ities. Eliminating ρ between these equations, we obtainthe equation of state P (ǫ). Equation (168) for the pres-sure is exactly the same as Eq. (26) obtained in thenonrelativistic limit. Therefore, knowing P (ρ), we canobtain the SF potential V (ρ) by the formula [see Eq.(28)] 22

V (ρ) = ρ

∫

P (ρ)

ρ2dρ. (169)

The squared speed of sound is

c2s = P ′(ǫ)c2 =ρV ′′(ρ)c2

c2 + ρV ′′(ρ) + 2V ′(ρ). (170)

21 We note that

ǫ =uµuν

c2Tµν . (163)

22 We note that the potential is defined from the pressure up to aterm of the form Aρ, where A is a constant. If we add a termAρ in the potential V , we do not change the pressure P but weintroduce a term 2Aρ in the energy density. On the other hand,if we add a constant term C in the potential V (cosmologicalconstant), this adds a term −C in the pressure and a term +Cin the energy density.

Remark: In [147] we have considered a spatially homo-geneous complex SF in an expanding universe describedby the Klein-Gordon-Friedmann (KGF) equations. Inthe fast oscillation regime ω ≫ H , where ω is the pulsa-tion of the SF andH the Hubble constant, we can averagethe KG equation over the oscillations of the SF (see Ap-pendix A of [147] and references therein) and obtain avirial relation leading to Eqs. (167) and (168). Theseequations can also be obtained by transforming the KGequation into hydrodynamic equations, taking the limit~ → 0, and using the Bernoulli equation (see Sec. IIof [147]).23 This is similar to the derivation given here.However, the present derivation is more general since itapplies to a possibly inhomogeneous SF [71]. An interestof the results of [147] is to show that the fast oscilla-tion approximation in cosmology is equivalent to the TFapproximation.

D. Reduced Lagrangian L(X)

In the previous section, we have used the Bernoulliequation (156) to eliminate θ from the Lagrangian, lead-ing to Eq. (166). Here, we eliminate ρ from the La-grangian. Introducing the notation

X =1

2∂µθ∂

µθ, (171)

the Bernoulli equation (156) can be written as

X = V ′tot(ρ). (172)

Assuming V ′′tot > 0, this equation can be reversed to give

ρ = G(X) (173)

with G(X) = (V ′tot)

−1(X). As a result, the equation ofcontinuity (155) can be written as

Dµ[G(X)∂µθ] = 0. (174)

According to Eqs. (167), (168) and (173), we have ǫ =ǫ(X) and P = P (X). Therefore,

L = P (X). (175)

In this manner, we have eliminated the pseudo rest-massdensity ρ from the Lagrangian (154) and we have ob-tained a reduced Lagrangian of the form L(X) that de-pends only on X . This kind of Lagrangian, called k-essence Lagrangian, is discussed in Appendix B. We showbelow that

ρ = G(X) = P ′(X) = L′(X). (176)

23 We cannot directly take the limit ~ → 0 in the KG equation. Thisis why we have to average over the oscillations. Alternatively, wecan directly take the limit ~ → 0 in the hydrodynamic equationsassociated with the KG equation. This is equivalent to the WKBmethod.

16

Using Eq. (176), we can rewrite Eq. (174) in terms ofL′(X) as in Eq. (B8). We also show below that

ǫ = 2XP ′(X)− P. (177)

If we know ǫ = ǫ(P ) we can solve this differential equationto obtain P (X), hence L(X).Proof of Eq. (177): According to Eqs. (168) and (172),

we have

P ′(ρ) = ρV ′′tot(ρ) (178)

and

dX

dρ= V ′′

tot(ρ). (179)

Starting from Eq. (175) and using Eqs. (178) and (179)we obtain

L′(X) = P ′(X) = P ′(ρ)dρ

dX= ρV ′′

tot(ρ)dρ

dX

= ρdX

dρ

dρ

dX= ρ, (180)

which establishes Eq. (176). On the other hand, accord-ing to Eqs. (167) and (168), we have

ǫ+ P = 2ρV ′tot(ρ). (181)

Using Eqs. (172) and (180), we obtain

ǫ+ P = 2ρX = 2XP ′(X), (182)

which establishes Eq. (177).Remark: We can obtain the preceding results in a more

direct and more general manner from a k-essence La-grangian L(X) by using the results of Appendix B. Thepresent calculations show how a k-essential Lagrangianarises from the canonical Lagrangian of a complex SF ϕin the TF limit. In that case, the real SF θ representsthe phase of the complex SF ϕ.

E. Nonrelativistic limit

To obtain the nonrelativistic limit of the foregoingequations, we first have to make the Klein transforma-tion (14) then take the limit c → +∞. In this manner,the KG equation (112) reduces to the GP equation (12)and the relativistic hydrodynamic equations (136)-(138)reduce to the nonrelativistic equations (20)-(24). Thesetransformations are discussed in detail in [130, 131, 148]for self-gravitating BECs. Here, we consider the nongrav-itational case and we focus on the nonrelativistic limit ofthe Lagrangien L(X) from Sec. III D leading to the La-grangian L(x) from Sec. II E.Since L = P in the two cases, we just have to find the

relation between X and x when c → +∞. Making the

Klein transformation24

θ = θNR − c2t (183)

in Eq. (171), we obtain

X =1

2∂µθ∂

µθ

≃ 1

2c2

(

∂θ

∂t

)2

− 1

2(∇θ)2

≃ 1

2c2

(

∂θNR

∂t

)2

− ∂θNR

∂t+c2

2− 1

2(∇θNR)

2.

(184)

Taking the limit c→ +∞, we find that

X ∼ c2

2. (185)

The nonrelativistic limit is then given by

c2

2−X → θNR +

1

2(∇θNR)

2. (186)

Therefore, when c→ +∞, we can write

X ≃ c2

2− x, (187)

where x is defined by Eq. (57).Using Eq. (187) we can easily check that the equations

of Sec. III D return the equations of Sec. II E in thenonrelativistic limit. For example, using Eq. (129), therelation X = V ′

tot(ρ) reduces to x = −V ′(ρ). On theother hand, using ǫ ∼ ρc2, Eq. (177) reduces to

ρ ∼ P ′(x)dx

dX∼ −P ′(x) ∼ −L′(x), (188)

which, together with Eq. (176), returns Eq. (62).

F. Enthalpy

Using Eqs. (D5), (167) and (168) we find that theenthalpy is given by

h = 2ρ

ρmV ′tot(ρ). (189)

Using Eq. (160), we obtain

h =√

2V ′tot(ρ) c. (190)

According to Eq. (156), the enthalpy can be written as

h = c√

∂µθ∂µθ = c√2X. (191)

Substituting Eq. (129) into Eq. (190), subtracting c2,and taking the nonrelativistic limit c→ +∞, we recoverEq. (27).

24 Substituting Eqs. (17) and (125) into Eq. (14), we obtain Stot =S −mc2t which is equivalent to Eq. (183).

17

IV. GENERAL EQUATION OF STATE

In this section, we provide general results valid for anarbitrary equation of state. We consider three differentmanners to specify the equation of state depending onwhether the pressure P is expressed as a function of (i)the energy density ǫ; (ii) the rest-mass density ρm; (iii)the pseudo rest-mass density ρ. In each case, we deter-mine the pressure P , the energy density ǫ, the rest-massdensity ρm, the internal energy u, the pseudo rest-massdensity ρ, the SF potential Vtot(ρ) and the k-essence La-grangian L(X).

A. Equation of state of type I

We first consider an equation of state of type I (seeAppendix C) where the pressure is given as a function ofthe energy density: P = P (ǫ).

1. Determination of ρm, P (ρm) and u(ρm)

Using the results of Appendix D, we can obtain therest-mass density ρm = nm and the internal energy u asfollows. According to Eq. (D4), we have

ln ρm =

∫

dǫ

P (ǫ) + ǫ, (192)

which determines ρm(ǫ). Eliminating ǫ between P (ǫ) andρm(ǫ) we obtain P (ρm). On the other hand, accordingto Eq. (D2), we have

u = ǫ − ρm(ǫ)c2. (193)

Eliminating ǫ between Eqs. (192) and (193), we obtainu(ρm). We can also obtain u(ρm) from P (ρm), or theconverse, by using Eqs. (D7) and (D8).

2. Determination of ρ, P (ρ) and Vtot(ρ)

Using the results of Sec. III C, we can obtain thepseudo rest-mass density ρ and the SF potential Vtot asfollows. According to Eqs. (167) and (168), we have

ǫ− P = 2Vtot(ρ), (194)

ǫ+ P = 2ρV ′tot(ρ). (195)

Differentiating Eq. (194) and using Eq. (195), we get

d(ǫ − P ) = 2V ′tot(ρ)dρ =

ǫ+ P

ρdρ. (196)

This yields

ln ρ =

∫

1− P ′(ǫ)

ǫ+ P (ǫ)dǫ, (197)

which determines ρ(ǫ). Eliminating ǫ between P (ǫ) andρ(ǫ) we obtain P (ρ). On the other hand, according toEq. (194), we have

Vtot =1

2[ǫ− P (ǫ)]. (198)

Eliminating ǫ between Eqs. (197) and (198), we obtainVtot(ρ). We can also obtain V (ρ) from P (ρ), or the con-verse, by using Eqs. (168) and (169).Remark: If the relation ǫ(P ) is more explicit than P (ǫ),

we can use

ln ρ =

∫

ǫ′(P )− 1

ǫ(P ) + PdP (199)

and

Vtot =1

2[ǫ(P )− P ], (200)

instead of Eqs. (197) and (198). The first equation givesρ(P ). Eliminating P between Eqs. (199) and (200), weobtain Vtot(ρ).

3. Lagrangian L(X)

If we know ǫ = ǫ(P ) then, according to Eq. (177), wehave

lnX = 2

∫

dP

ǫ(P ) + P, (201)

which determines X(P ). If this function can be invertedwe get P (X) hence L(X). If we know P = P (ǫ), we canrewrite Eq. (201) as

lnX = 2

∫

P ′(ǫ)

ǫ + P (ǫ)dǫ, (202)

which determines X(ǫ). If this function can be invertedwe get ǫ(X), then P (X) = P [ǫ(X)], hence L(X).

B. Equation of state of type II

We now consider an equation of state of type II (seeAppendix D) where the pressure is given as a function ofthe rest-mass density: P = P (ρm). We can then deter-mine the internal energy u(ρm) from Eq. (D7). Inversely,we can specify the internal energy u(ρm) as a function ofthe rest-mass density and obtain the equation of stateP (ρm) from Eq. (D8).

1. Determination of ǫ and P (ǫ)

According to Eqs. (D2) and (D8), the energy densityand the pressure are given by

ǫ = ρmc2 + u(ρm), (203)

18

P = ρmu′(ρm)− u(ρm). (204)

Eliminating ρm between Eqs. (203) and (204), we obtainP (ǫ).


According to Eq. (197), we have

ln ρ =

∫

ǫ′(ρm)− P ′(ρm)

ǫ(ρm) + P (ρm)dρm. (205)

Then, using Eqs. (203) and (204), we obtain

ln ρ =

∫

c2 + u′(ρm)− ρmu′′(ρm)

ρmc2 + ρmu′(ρm)dρm, (206)

which determines ρ(ρm). This equation can be integratedinto

ρ =ρm

1 + 1c2u

′(ρm), (207)

where the constant of integration has been determinedin order to obtain ρ = ρm in the nonrelativistic limit.Identifying ǫ+ P = 2ρV ′

tot(ρ) from Eqs. (167) and (168)with ǫ+P = ρm(c2 + u′(ρm)) from Eqs. (203) and (204)we see that Eq. (207) is equivalent to Eq. (160). Elim-inating ρm between P (ρm) and ρ(ρm), we obtain P (ρ).On the other hand, according to Eqs. (198), (203) and(204), we get

Vtot =1

2

[

ρmc2 + 2u(ρm)− ρmu

′(ρm)]

. (208)

Eliminating ρm between Eqs. (206) and (208) we ob-tain Vtot(ρ). We can also obtain V (ρ) from P (ρ), or theconverse, by using Eqs. (168) and (169).

3. Lagrangian L(X)

According to Eq. (201), we have

lnX = 2

∫

P ′(ρm)

ǫ(ρm) + P (ρm)dρm. (209)


lnX = 2

∫

u′′(ρm)

c2 + u′(ρm)dρm, (210)

which can be integrated into

X =1

2c2[

c2 + u′(ρm)]2. (211)

We have determined the constant of integration so that,in the nonrelativistic limit, X ∼ c2/2 (see Sec. III E).From Eq. (211) we obtain X(ρm). If this function can beinverted we get ρm(X), then P (X) = P [ρm(X)], henceL(X).

C. Equation of state of type III

Finally, we consider an equation of state of type III (seeSec. III and Appendix E) where the pressure is given asa function of the pseudo rest-mass density: P = P (ρ).We can then determine the SF potential V (ρ) from Eq.(169).25 Inversely, we can specify the SF potential V (ρ)and obtain the equation of state P (ρ) from Eq. (168).


According to Eqs. (167) and (168), the energy densityand the pressure are given by

ǫ = ρV ′tot(ρ) + Vtot(ρ), (212)

P = ρV ′tot(ρ)− Vtot(ρ). (213)

Eliminating ρ between Eqs. (212) and (213), we obtainP (ǫ).


According to Eq. (D4), we have

ln ρm =

∫

ǫ′(ρ)

ǫ+ P (ρ)dρ. (214)


ln ρm =

∫

ρV ′′tot(ρ) + 2V ′

tot(ρ)

2ρV ′tot(ρ)

dρ, (215)

which determines ρm(ρ). This equation can be integratedinto

ρm =ρ

c

√

2V ′tot(ρ), (216)

where the constant of integration has been determinedin order to obtain ρm = ρ in the nonrelativistic limit.This relation is equivalent to Eq. (160). Eliminating ρbetween P (ρ) and ρm(ρ) we obtain P (ρm). On the otherhand, according to Eq. (D2), we have

u = ǫ − ρmc2. (217)


u = ρV ′tot(ρ) + Vtot(ρ)− ρm(ρ)c2

= ρV ′tot(ρ) + Vtot(ρ)− ρc

√

2V ′tot(ρ). (218)

Eliminating ρ between Eqs. (215) and (218) we obtainu(ρm). We can also obtain u(ρm) from P (ρm), or theconverse, by using Eqs. (D7) and (D8).

25 We note that the expression of V (ρ) for an equation of stateP (ρ) of type III coincides with the expression of u(ρm) for anequation of state P (ρm) of type II of the same functional formprovided that we make the replacements u → V and ρm → ρ(see Appendix F).

19

3. Lagrangian L(X)

According to Eq. (172) we have

X = V ′tot(ρ), (219)

which determines X(ρ). If this function can be invertedwe get ρ(X), then P (X) = P [ρ(X)], hence L(X).

V. POLYTROPES

In this section, we apply the general results of Sec. IVto the case of a polytropic equation of state.

A. Polytropic equation of state of type I

The polytropic equation of state of type I writes [149]

P = K( ǫ

c2

)γ

, (220)

where K is the polytropic constant and γ = 1 + 1/n isthe polytropic index. This is the equation of state of theGCG [95]. In the nonrelativistic regime, using ǫ ∼ ρc2,we recover Eq. (66).(i) For γ = −1, we obtain

P =Kc2

ǫ. (221)

This is the equation of state of the Chaplygin (K < 0)or anti-Chaplygin (K > 0) gas [50, 71, 86, 98].(ii) For γ = 2, we obtain

P = K( ǫ

c2

)2

. (222)

This is the equation of state of the standard BEC with re-pulsive (K > 0) or attractive (K < 0) self-interaction.26

In that case, K = 2πas~2/m3 (see Sec. II I).

(iii) For γ = 0, we obtain

P = K. (223)

This is the equation of state of the ΛCDM (K < 0) oranti-ΛCDM (K > 0) model [86, 98, 102, 103]. In thatcase K = −ρΛc2, where ρΛ = Λ/(8πG) is the cosmologi-cal density.

26 The true equation of state of a relativistic BEC is given by Eq.(329) or (344) corresponding to a polytrope of type III with in-dex γ = 2 [147, 150–154]. However, in order to have a unifiedterminology throughout the paper, we shall always associate thepolytropic index γ = 2 to a BEC even if this association is notquite correct for models of type I and II in the relativistic regime(see Appendix H). As explained in Appendix F, all the modelscoincide in the nonrelativistic limit.

(iv) For γ = 3, we obtain

P = K( ǫ

c2

)3

. (224)

This is the equation of state of a superfluid with repulsive(K > 0) or attractive (K < 0) self-interaction (see Sec.II J).(v) The case γ = 1 must be treated specifically. In

that case, we have a linear equation of state [155–159]

P = αǫ, (225)

where we have defined

α =K

c2. (226)

This linear equation of state describes pressureless matter(α = 0), radiation (α = 1/3) and stiff matter (α = 1).The nonrelativistic limit corresponds to α → 0. Usingǫ ∼ ρc2, we recover the isothermal equation of state (72).


The rest-mass density is determined by Eq. (192) withthe equation of state (220). We have

ln ρm =

∫

dǫ

K(

ǫc2

)γ+ ǫ

. (227)

The integral can be calculated analytically yielding

ρmc2 =

ǫ[

1 + Kc2

(

ǫc2

)γ−1]1/(γ−1)

. (228)

We have determined the constant of integration so thatǫ ∼ ρmc

2 in the nonrelativistic limit. Eq. (228) can beinverted to give

ǫ =ρmc

2

(

1− Kργ−1m

c2

)1

γ−1

. (229)

Substituting this result into Eq. (220), we obtain

P =Kργm

(

1− Kργ−1m

c2

)γ

γ−1

. (230)

The internal energy is given by Eqs. (193) and (229)giving

u =ρmc

2

(

1− Kργ−1m

c2

)1

γ−1

− ρmc2. (231)

These results are consistent with those obtained in Ap-pendix B.3 of [152]. In the nonrelativistic limit, using

20

ǫ ∼ ρmc2[1+ K

(γ−1)c2 ργ−1m ], we recover Eqs. (66) and (67)

[recalling Eq. (29)].(i) For γ = −1 (Chaplygin gas), we obtain

ǫ =√

(ρmc2)2 −Kc2, (232)

P =Kc2

√

(ρmc2)2 −Kc2, (233)

u =√

(ρmc2)2 −Kc2 − ρmc2. (234)

(ii) For γ = 2 (BEC), we obtain

ǫ =ρmc

2

1− Kρm

c2

, (235)

P =Kρ2m

(

1− Kρm

c2

)2 , (236)

u =Kρ2m

1− Kρm

c2

. (237)

For K > 0 there is a maximum density (ρm)max = c2/K.The equation of state (236) was first obtained in [152].(iii) For γ = 0 (ΛCDM model), we obtain

ǫ = ρmc2 −K, (238)

P = K, (239)

u = −K. (240)

(iv) For γ = 3 (superfluid), we obtain

ǫ =ρmc

2

(

1− Kρ2m

c2

)1/2, (241)

P =Kρ3m

(

1− Kρ2m

c2

)3/2, (242)

u =ρmc

2

(

1− Kρ2m

c2

)1/2− ρmc

2. (243)

ForK > 0 there is a maximum density (ρm)max = c/√K.

(v) For γ = 1, Eq. (192) can be integrated into

ρm =

[

αǫ

K(α)

]1

1+α

, (244)

where K(α) is a constant that depends on α. In the non-relativistic limit α → 0, the condition ǫ ∼ ρmc

2 impliesK(α) → αc2 = K. Combining Eq. (244) with Eq. (225),we obtain

P = K(α)ρ1+αm . (245)

This is the equation of state of a polytrope of type II(see Sec. VB) with a polytropic index Γ = 1 + α (i.e.n = 1/α) and a polytropic constant K(α).27 In the non-relativistic limit α → 0, we obtain an isothermal equationof state P = Kρm with a “temperature”K. The internalenergy (193) is given by

u = ρmc2

[K(α)

αc2ραm − 1

]

. (246)

It is similar to a Tsallis free energy density −Ksq (wheresq = − 1

q−1ρqm) of index q = 1 + α with a “polytropic”

temperature K(α). In the nonrelativistic limit α → 0 (i.e.q → 1), Eq. (246) reduces to u = Kρm ln ρm (up to anadditive constant) and we recover Eq. (73) [recalling Eq.(29)]. It is similar to the Boltzmann free energy density−KsB (where sB = −ρm ln ρm) with the temperatureK.In the present context, the Tsallis entropy arises fromrelativistic effects (α 6= 0 ⇒ q 6= 1).


The pseudo rest-mass density and the SF potential aredetermined by Eqs. (197) and (198) with the equation ofstate (220). We have

ln ρ =

∫

1− Kγc2

(

ǫc2

)γ−1

ǫ+K(

ǫc2

)γ dǫ. (247)

The integral can be calculated analytically yielding

ρc2 = ǫ

[

1 +K

c2

( ǫ

c2

)γ−1](1+γ)/(1−γ)

. (248)

We have determined the constant of integration so thatǫ ∼ ρc2 in the nonrelativistic limit. The SF potential isgiven by

Vtot =1

2

[

ǫ−K( ǫ

c2

)γ]

. (249)

Eqs. (220), (248) and (249) define P (ρ) and Vtot(ρ) inparametric form with parameter ǫ. In the nonrelativistic

27 The linear equation of state (225) with α = Γ−1 corresponds tothe ultrarelativistic limit of the equation of state (284) associatedwith a polytrope of type II with index Γ. Indeed, for a polytropeP = KρΓm, Eq. (284) yields P ∼ (Γ− 1)ǫ in the ultrarelativisticlimit. The index Γ = 4/3 corresponds to α = 1/3 (radiation)and the index Γ = 2 corresponds to α = 1 (stiff matter).

21

limit, using ǫ ∼ ρc2[1 − K(1+γ)(1−γ)c2 ρ

γ−1] and Eq. (129), we

recover Eqs. (66) and (67).(i) For γ = −1 (Chaplygin gas), we obtain

ǫ = ρc2, (250)

P =K

ρ, (251)

Vtot(ρ) =1

2

(

ρc2 − K

ρ

)

. (252)

Expression (252) of the SF potential was first given in[71]. We note that the energy density ǫ coincides withthe pseudo rest-mass energy density ρc2 [see Eq. (250)].As a result, P (ρ) is a Chaplygin equation of state oftype III (see Sec. VC). Therefore, the models I and IIIcoincide in that case.(ii) For γ = 2 (BEC), the energy density is determined

by a cubic equation

ρc2 =ǫ

(

1 + Kǫc4

)3 . (253)

The solution ǫ(ρ) can be obtained by standard means.The total potential Vtot(ρ) is then given by

Vtot =1

2

[

ǫ −K( ǫ

c2

)2]

(254)

with ǫ replaced by ǫ(ρ). Eqs. (253) and (254) also deter-mine Vtot(ρ) in parametric form.(iii) For γ = 0 (ΛCDM model), we obtain

ǫ = ρc2 −K, (255)

P = K, (256)

Vtot(ρ) =1

2ρc2 −K. (257)

Comparing these results with Eqs. (238)-(240), we notethat ρ = ρm and V = u = −K. The potential V isconstant.(iv) For γ = 3 (superfluid), the total potential Vtot(ρ)

is given in parametric form by

ρc2 =ǫ

[

1 + Kc2

(

ǫc2

)2]2 , (258)

Vtot =1

2

[

ǫ−K( ǫ

c2

)3]

. (259)

It is not possible to obtain more explicit expressions.

(v) For γ = 1, Eq. (197) can be integrated into

ρ =

[

αǫ

K(α)

](1−α)/(1+α)

, (260)

where K(α) is a constant that depends on α. In thenonrelativistic limit α→ 0, the condition ǫ ∼ ρc2 impliesK(α) → αc2 = K. Combining Eq. (260) with Eq. (225),we obtain

P = K(α)ρ(1+α)/(1−α). (261)

This is the equation of state of a polytrope of type III (seeSec. VC) with a polytropic index Γ = (1+α)/(1−α) (i.e.n = (1− α)/(2α)) and a polytropic constant K(α).28 Inthe nonrelativistic limit α → 0, we obtain an isothermalequation of state P = Kρ with a “temperature” K. TheSF potential (198) is given by

Vtot =1− α

2αK(α)ρ(1+α)/(1−α). (262)

This is a power-law potential.29 In the nonrelativisticlimit α → 0, Eq. (262) reduces to Vtot = Kρ ln ρ (upto an additive constant) and we recover Eq. (73). Forα = 1 (stiff matter) we obtain Vtot = 0, corresponding toa free massless SF satisfying ϕ = 0.

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eq. (201) orEq. (202) with the equation of state (220). We have

lnX = 2

∫

dP(

PK

)1/γc2 + P

(263)

or

lnX = 2

∫ Kγc2

(

ǫc2

)γ−1

ǫ +K(

ǫc2

)γ dǫ. (264)

The integrals can be calculated analytically yielding

L(X) = P = K

c2

−K

[

1−(

2X

c2

)γ−12γ

]

γγ−1

. (265)

28 The linear equation of state (225) with α = (Γ−1)/(Γ+1) corre-sponds to the ultrarelativistic limit of the equation of state (338)associated with a polytrope of type III with index Γ. Indeed, fora polytrope P = KρΓ, Eq. (338) yields P ∼ [(Γ − 1)/(Γ + 1)]ǫin the ultrarelativistic limit. The index Γ = 2 corresponds toα = 1/3 (radiation) and the index Γ = ∞ corresponds to α = 1(stiff matter).

29 Equation (262) is similar to a Tsallis free energy density of indexq = (1 + α)/(1 − α). Comments similar to those following Eq.(246) appy to the present situation.

22

We have determined the constant of integration so that,in the nonrelativistic limit, X ∼ c2/2 (see Sec. III E). Inthe nonrelativistic limit, using Eq. (187), we recover Eq.(70). The Lagrangian (265) was first obtained in [95] inrelation to the GCG by using the procedure of [71] thatwe have followed.(i) For γ = −1 (Chaplygin gas), we obtain

L(X) = P = K

√

c2

−K

(

1− 2X

c2

)

. (266)

The Lagrangian of the Chaplygin gas (K < 0) is of theBorn-Infeld type. Indeed, setting K = −A so that P =−Ac2/ǫ we get the Born-Infeld Lagrangian

LBI = −(Ac2)1/2√

1− 1

c2∂µθ∂µθ. (267)

In the nonrelativistic limit, using Eq. (187), it reducesto

LNR = −(2A)1/2√

θ +1

2(∇θ)2, (268)

corresponding to Eq. (82). Using Eq. (174) or Eq. (B8),we obtain the equation of motion

Dµ

∂µθ√

|1− 1c2 ∂νθ∂

νθ|

= 0. (269)

In the nonrelativistic limit, it reduces to Eq. (83). TheBorn-Infeld Lagrangian (267) was obtained by [58, 59]in two different manners: (i) starting from a heuristicrelativistic Lagrangian

L = −∫

[

ρθ + ρc2

√

1 +A

ρ2c2

√

1 +(∇θ)2c2

]

dr (270)

which generalizes the nonrelativistic Lagrangian fromEqs. (47) and (79), writing the equations of motion, andeliminating ρ with the aid of the Bernoulli equation; (ii)for a d-brane moving in a (d + 1, 1) space-time. In thesecond case, it can be obtained from the Nambu-Goto ac-tion in the Cartesian parametrization. The Born-InfeldLagrangian (267) was also obtained in [71] for a complexSF in the TF regime by developing the procedure that wehave followed. It can also be directly obtained from thek-essence formalism applied to the Chaplygin equationof state (see Appendix B).(ii) For γ = 2 (BEC), we obtain

L(X) = P = K

c2

−K

[

1−(

2X

c2

)1/4]2

. (271)

(iii) For γ = 0 (ΛCDM model) the k-essence La-grangian is ill-defined.


L(X) = P = K

c2

−K

[

1−(

2X

c2

)1/3]3/2

. (272)

(v) For γ = 1, Eq. (201) or Eq. (202) can be easilyintegrated yielding

L(X) = P = A(α)

(

2X

c2

)α+12α

, (273)

where A(α) is a constant that depends on α. The La-grangian is a pure power-law. It was first given in [41].Using Eq. (174) or Eq. (B8), we obtain the equation ofmotion

Dµ

[

(

1

c2∂νθ∂

νθ

)(1−α)/2α

∂µθ

]

= 0. (274)

In the nonrelativistic limit corresponding to α → 0, usingEq. (187), we recover Eqs. (76) and (77) with A(α) →Kρ∗. In the case α = 1, we obtain L ∝ X and θ = 0(free massless SF).

B. Polytropic equation of state of type II

The polytropic equation of state of type II writes [164]

P = Kργm. (275)

Using Eq. (D7), the internal energy is

u =K

γ − 1ργm. (276)

It is similar to the Tsallis free energy. In the nonrela-tivistic limit, using ρm ∼ ρ, we recover Eqs. (66) and(67) [recalling Eq. (29)]. Actually, Eqs. (275) and (276)coincide with Eqs. (66) and (67) with ρm in place of ρ.We note that P = (γ − 1)u.(i) For γ = −1 (Chaplygin gas), we obtain

P =K

ρm, u = − K

2ρm. (277)


P = Kρ2m, u = Kρ2m. (278)

(iii) For γ = 0 (ΛCDM model), we obtain

P = K, u = −K. (279)


P = Kρ3m, u =1

2Kρ3m. (280)

23

(v) The case γ = 1, corresponding to an isothermalequation of state

P = Kρm, (281)

must be treated specifically. Using Eq. (D7), the internalenergy is

u = Kρm

[

ln

(

ρmρ∗

)

− 1

]

. (282)

It is similar to the Boltzmann free energy. In the nonrel-ativistic limit, using ρm ∼ ρ, we recover Eqs. (72) and(73) [recalling Eq. (29)]. Actually, Eqs. (281) and (282)coincide with Eqs. (72) and (73) with ρm in place of ρ.


The energy density is determined by Eq. (203) withEq. (276). We obtain

ǫ = ρmc2 +

K

γ − 1ργm. (283)

The pressure is determined by Eq. (204) with Eq. (276).This returns Eq. (275). Eliminating ρm between Eq.(275) and Eq. (283) we obtain P (ǫ) under the inverseform ǫ(P ) as

ǫ =

(

P

K

)1/γ

c2 +P

γ − 1. (284)

In the nonrelativistic limit, using ǫ ∼ ρc2, we recover Eq.(66).(i) For γ = −1 (Chaplygin gas), we obtain

ǫ = ρmc2 − K

2ρm, (285)

ǫ =Kc2

P− P

2, (286)

ρmc2 =

ǫ±√ǫ2 + 2Kc2

2, (287)

P = −ǫ±√

ǫ2 + 2Kc2. (288)

We note that the Chaplygin gas of type II is differentfrom the Chaplygin gas of type I.(ii) For γ = 2 (BEC), we obtain

ǫ = ρmc2 +Kρ2m, (289)

ǫ =

√

P

Kc2 + P, (290)

ρm =−c2 ±

√c4 + 4Kǫ

2K, (291)

P =1

4K

[

−c2 ±√

c4 + 4Kǫ]2

. (292)

The equation of state (292) was first obtained in [152,165].(iii) For γ = 0 (ΛCDM model), we obtain

ǫ = ρmc2 −K, (293)

P = K. (294)


ǫ = ρmc2 +

1

2Kρ3m, (295)

ǫ =

(

P

K

)1/3

c2 +1

2P. (296)

This is a third degree equation which can be solved bystandard means to obtain P (ǫ).(v) For γ = 1, the energy density is determined by Eq.

(203) with Eq. (282). We obtain

ǫ = ρmc2 +Kρm

[

ln

(

ρmρ∗

)

− 1

]

. (297)

The pressure is determined by Eq. (204) with Eq. (282).This returns Eq. (281). Eliminating ρm between Eq.(281) and Eqs. (297) we obtain P (ǫ) under the inverseform ǫ(P ) as

ǫ =P

Kc2 + P

[

ln

(

P

Kρ∗

)

− 1

]

. (298)

Remark: For the index γ = 1/2, we can inverse Eq.(284) to obtain

P =K2

c2±K

√

K2

c4+

ǫ

c2. (299)

For the index γ = 3/2, Eq. (283) becomes

ǫ = ρmc2 + 2Kρ3/2m . (300)

This is a third degree equation for√ρm which can be

solved by standard means. One can then obtain P (ǫ)explicitly.


The pseudo rest-mass density and the SF potential aredetermined by Eqs. (207) and (208) with Eq. (276). Weget

ρ =ρm

1 + γγ−1

Kc2 ρ

γ−1m

, (301)

24

Vtot =1

2

[

ρmc2 −K

γ − 2

γ − 1ργm

]

. (302)

Equations (275), (301) and (302) determine P (ρ) andVtot(ρ) in parametric form with parameter ρm. In thenonrelativistic limit, using ρm ∼ ρ[1 + γ

γ−1Kc2 ρ

γ−1] and

Eq. (129), we recover Eqs. (66) and (67).(i) For γ = −1 (Chaplygin gas), ρm is determined by

a cubic equation

2ρ3m − 2ρρ2m − Kρ

c2= 0, (303)

which can be solved by standard means. The SF poten-tial is given by

Vtot =1

2

(

ρmc2 +

3K

2ρm

)

. (304)

One can then obtain P (ρ) and Vtot(ρ) explicitly.(ii) For γ = 2 (BEC), we obtain

ρ =ρm

1 + 2Kc2 ρm

, (305)

ρm =ρ

1− 2Kc2 ρ

, (306)

P =Kρ2

(

1− 2Kc2 ρ)2 , (307)

Vtot =1

2ρc2 +

Kρ2

1− 2Kρc2

. (308)

(iii) For γ = 0 (ΛCDM model), we obtain

ρ = ρm, P = K, (309)

Vtot =1

2ρc2 −K. (310)


ρ =ρm

1 + 3K2c2 ρ

2m

, (311)

ρm =c2

3Kρ

(

1±√

1− 6K

c2ρ2

)

, (312)

P = K

(

c2

3Kρ

)3(

1±√

1− 6K

c2ρ2

)3

, (313)

Vtot =c4

6Kρ

(

1±√

1− 6K

c2ρ2

)

×[

4

3− c2

9Kρ2

(

1±√

1− 6K

c2ρ2

)]

. (314)

(v) For γ = 1, the pseudo rest-mass density and the SFpotential are determined by Eqs. (207) and (208) withEq. (282). We get

ρ =ρm

1 + Kc2 ln

(

ρm

ρ∗

) , (315)

Vtot =1

2

[

ρmc2 +Kρm ln

(

ρmρ∗

)

− 2Kρm

]

. (316)

Equations (281), (315) and (316) determine P (ρ) andVtot(ρ) in parametric form with parameter ρm. In thenonrelativistic limit, using ρm ≃ ρ[1 + (K/c2) ln(ρ/ρ∗)]and Eq. (129), we recover Eq. (73).Remark: For the index γ = 1/2, Eq. (301) can be

written as

ρ3/2m − ρρ1/2m +Kρ

c2= 0. (317)


solved by standard means. One can then obtain P (ρ)and Vtot(ρ) explicitly. For the index γ = 3/2 we findthat

P = K

(

3K

2c2ρ±

√

9K2

4c4ρ2 + ρ

)3

, (318)

Vtot =1

2

(

3K

2c2ρ±

√

9K2

4c4ρ2 + ρ

)2

c2

×[

1 +K

2c2ρ

(

3K

c2±√

9K2

c4+

4

ρ

)]

. (319)

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eqs. (211),(275) and (276). We get

X =1

2c2

[

c2 +Kγ

γ − 1ργ−1m

]2

. (320)

This equation can be inverted to give

ργ−1m = −γ − 1

γ

c2

K

[

1−(

2X

c2

)1/2]

. (321)

We then obtain

L(X) = P = K

−γ − 1

γ

c2

K

[

1−(

2X

c2

)1/2]

γγ−1

.

(322)In the nonrelativistic limit, using Eq. (187), we recoverEq. (70).

25

(i) For γ = −1 (Chaplygin gas), we obtain

L(X) = P = K

√

√

√

√

2c2

−K

[

1−(

2X

c2

)1/2]

. (323)


L(X) = P = K

c2

−2K

[

1−(

2X

c2

)1/2]2

. (324)

(iii) For γ = 0 (ΛCDM model), the k-essence La-grangian is ill-defined.(iv) For γ = 3 (superfluid), we obtain

L(X) = P = K

2c2

−3K

[

1−(

2X

c2

)1/2]3/2

. (325)

(v) For γ = 1, the Lagrangian L(X) is determined byEqs. (211), (281) and (282). This yields

X =1

2c2

[

c2 +K ln

(

ρmρ∗

)]2

. (326)

This equation can be inverted to give

ρm = ρ∗e− c2

K

[

1−( 2Xc2)1/2

]

. (327)

We then obtain

L(X) = P = Kρ∗e− c2

K

[

1−( 2Xc2)1/2

]

. (328)

In the nonrelativistic limit, using Eq. (187), we recoverEq. (76).

C. Polytropic equation of state of type III

The polytropic equation of state of type III writes [147,154]

P = Kργ . (329)

Using Eq. (169), the SF potential is

Vtot(ρ) =1

2ρc2 +

K

γ − 1ργ . (330)

It is similar to the Tsallis free energy. In the nonrela-tivistic limit, we recover Eqs. (66) and (67) [recallingEq. (129)]. Actually, Eqs. (329) and (330) coincide withEqs. (66) and (67). The SF potential V (ρ) correspondsto a pure power-law. We note that P = (γ − 1)V .(i) For γ = −1 (Chaplygin gas), we obtain

P =K

ρ, Vtot =

1

2ρc2 − K

2ρ, (331)

as found in [147, 154]. We recover the potential from Eq.(252) for the reason explained in Sec. VC1.(ii) For γ = 2 (BEC), we obtain

P = Kρ2, Vtot =1

2ρc2 +Kρ2. (332)

The SF potential V (ρ) = Kρ2 from Eq. (332) with K =2πas~

2/m3 corresponds to the standard |ϕ|4 potential ofa BEC [147, 150, 154].(iii) For γ = 0 (ΛCDM model), we obtain

P = K, Vtot =1

2ρc2 −K. (333)

The SF potential V (ρ) = −K from Eq. (333) is constant.Using K = −ρΛc2, we see that V (ρ) = ρΛc

2 is equal tothe cosmological density [154].(iv) For γ = 3 (superfluid), we obtain

P = Kρ3, Vtot =1

2ρc2 +

1

2Kρ3. (334)

(v) The case γ = 1, corresponding to a linear equationof state

P = Kρ, (335)

must be treated specifically. Using Eq. (169), the SFpotential is

Vtot(ρ) =1

2ρc2 +Kρ

[

ln

(

ρ

ρ∗

)

− 1

]

. (336)

It is similar to the Boltzmann free energy. In the nonrela-tivistic limit, we recover Eqs. (72) and (73) [recalling Eq.(129)]. Actually, Eqs. (335) and (336) coincide with Eqs.(72) and (73). The potential (336) was first obtained in[133, 154].


The energy density is determined by Eq. (212) withEq. (330). This yields

ǫ = ρc2 +γ + 1

γ − 1Kργ . (337)

The pressure is determined by Eq. (213) with Eq. (330).This returns Eq. (329). Eliminating ρ between Eqs.(329) and (337), we obtain P (ǫ) under the inverse formǫ(P ) as

ǫ =

(

P

K

)1/γ

c2 +γ + 1

γ − 1P. (338)

In the nonrelativistic limit, using ǫ ∼ ρc2, we recover Eq.(66).(i) For γ = −1 (Chaplygin gas), we obtain

ǫ = ρc2, (339)

26

P =Kc2

ǫ. (340)

This returns the Chaplygin gas of type I (see Sec. VA).Therefore, the Chaplygin gas models of type I and IIIcoincide.(ii) For γ = 2 (BEC), we obtain

ǫ = ρc2 + 3Kρ2, (341)

ǫ =

√

P

Kc2 + 3P, (342)

ρ =−c2 ±

√c4 + 12Kǫ

6K, (343)

P =1

36K

[

−c2 ±√

c4 + 12Kǫ]2

. (344)

This equation of state was first obtained in [150] (see also[147, 148, 154]).(iii) For γ = 0 (ΛCDM model), we obtain

P = K, ǫ = ρc2 −K. (345)


ǫ = ρc2 + 2Kρ3, (346)

ǫ =

(

P

K

)1/3

c2 + 2P. (347)

This is a third degree equation which can be solved bystandard means to obtain P (ǫ).(v) For γ = 1, the energy density is determined by Eq.

(212) with Eq. (336). This yields

ǫ = ρc2 + 2Kρ ln

(

ρ

ρ∗

)

−Kρ. (348)


ǫ =P

Kc2 + 2P ln

(

P

Kρ∗

)

− P. (349)

In the nonrelativistic limit, using ǫ ∼ ρc2, we recover Eq.(72).Remark: For the index γ = 1/2, we can inverse Eq.

(338) to obtain

P =3K2

2c2±K

√

9K2

4c4+

ǫ

c2. (350)

For the index γ = 3/2, Eq. (337) becomes

ǫ = ρc2 + 5Kρ3/2. (351)


solved by standard means. One can then obtain P (ǫ)explicitly.


The rest-mass density and the internal energy are de-termined by Eqs. (216) and (218) with Eq. (330). Weget

ρm = ρ

√

1 +2γ

γ − 1

K

c2ργ−1, (352)

u = ρc2 +γ + 1

γ − 1Kργ − ρm(ρ)c2. (353)

Eqs. (329), (352) and (353) define P (ρm) and u(ρm) inparametric form with parameter ρ. In the nonrelativisticlimit, using ρm ∼ ρ[1+ γ

γ−1Kc2 ρ

γ−1], we recover Eqs. (66)

and (67) [recalling Eq. (29)].(i) For γ = −1 (Chaplygin gas), we obtain

ρmc2 =

√

(ρc2)2 +Kc2, (354)

ρc2 =√

(ρmc2)2 −Kc2, (355)

P =Kc2

√

(ρmc2)2 −Kc2, (356)

u =√

(ρmc2)2 −Kc2 − ρmc2. (357)

(ii) For γ = 2 (BEC), ρ is determined by a cubic equa-tion

4K

c2ρ3 + ρ2 − ρ2m = 0, (358)

which can be solved by standard means. The internalenergy is given by

u = ρc2 + 3Kρ2 − ρm(ρ)c2. (359)

One can then obtain P (ρm) and u(ρm) explicitly.(iii) For γ = 0 (ΛCDM model), we obtain

P = K, ρm = ρ, u = −K. (360)

We note that the rest-mass density coincides with thepseudo rest-mass density (ρm = ρ).(iv) For γ = 3 (superfluid), we obtain

ρm = ρ

√

1 +3K

c2ρ2, (361)

ρ =

(

− c2

6K± c2

6K

√

1 +12K

c2ρ2m

)1/2

, (362)

P = K

(

− c2

6K± c2

6K

√

1 +12K

c2ρ2m

)3/2

, (363)

27

u = c2

(

− c2

6K± c2

6K

√

1 +12K

c2ρ2m

)1/2

×(

2

3± 1

3

√

1 +12K

c2ρ2m

)

− ρmc2. (364)

(v) For γ = 1 the rest-mass density and the internalenergy are determined by Eqs. (216) and (218) with Eq.(336). This gives

ρm = ρ

√

1 +2K

c2ln

(

ρ

ρ∗

)

, (365)

u = ρc2 + 2Kρ ln

(

ρ

ρ∗

)

−Kρ− ρm(ρ)c2. (366)

Equations (335), (365) and (366) determine P (ρm) andu(ρm) in parametric form with parameter ρ. In the non-relativistic limit, using ρm ∼ ρ[1 + K

c2 ln (ρ/ρ∗)], we re-cover Eqs. (72) and (73) [recalling Eq. (29)].

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eq. (219) withEq. (330). We get

X =1

2c2 +

Kγ

γ − 1ργ−1. (367)

This relation can be reversed to give

ργ−1 = −γ − 1

2γ

c2

K

(

1− 2X

c2

)

. (368)

We then obtain

L(X) = P = K

[

−γ − 1

2γ

c2

K

(

1− 2X

c2

)]

γγ−1

. (369)

In the nonrelativistic limit, using Eq. (187), we recoverEq. (70). Actually, Eq. (369) coincides with Eq. (70)with c2/2−X in place of x. Interestingly, the Lagrangian(369) corresponds to the Lagrangian introduced heuristi-cally in [166] in relation to the GCG [see their Eq. (33)].In [166] it was obtained from a heuristic relativistic La-grangian

L = −∫

[

ρθ + ρc2

√

1 +2K

γ − 1

1

ρ1−γc2

√

1 +(∇θ)2c2

]

dr

(370)which generalizes the Lagrangian from Eq. (270). Ourapproach provides therefore a justification of the La-grangian (369) from a more rigorous relativistic theory.We note that this Lagrangian differs from the Lagrangian(265) introduced in [95] except for the particular indexγ = −1 corresponding to the Chaplygin gas (see below).

It is also different from the Lagrangian (322) even forγ = −1. This is an effect of the inequivalence betweenthe equations of state of types I, II and III.(i) For γ = −1 (Chaplygin gas), we obtain

L(X) = P = K

√

c2

−K

(

1− 2X

c2

)

, (371)

like in Eq. (266).(ii) For γ = 2 (BEC), we obtain

L(X) = P = K

[

c2

−4K

(

1− 2X

c2

)]2

. (372)

(iii) For γ = 0 (ΛCDM model), the k-essence La-grangian is ill-defined.(iv) For γ = 3 (superfluid), we obtain

L(X) = P = K

[

c2

−3K

(

1− 2X

c2

)]3/2

. (373)

(v) For γ = 1, the Lagrangian L(X) is determined byEq. (219) with Eq. (336). We get

X =1

2c2 +K ln

(

ρ

ρ∗

)

. (374)

This relation can be reversed to give

ρ = ρ∗e− c2

2K (1− 2Xc2). (375)

We then obtain

L(X) = P = Kρ∗e− c2

2K (1− 2Xc2). (376)

In the nonrelativistic limit, using Eq. (187), we recoverEq. (76). Actually, Eq. (376) coincides with Eq. (76)with c2/2−X in place of x.

VI. LOGOTROPES

In this section, we apply the general results of Sec. IVto the case of a logotropic equation of state.

A. Logotropic equation of state of type I

The logotropic equation of state of type I writes [117]

P = A ln

(

ǫ

ǫ∗

)

. (377)


The rest-mass density and the internal energy are de-termined by Eqs. (192) and (193) with the equation ofstate (377) yielding

ln ρm =

∫

dǫ

A ln(

ǫǫ∗

)

+ ǫ, (378)

28

u = ǫ − ρm(ǫ)c2. (379)

Equations (377)-(379) determine P (ρm) and u(ρm) inparametric form with parameter ǫ. Unfortunately, theintegral in Eq. (378) cannot be calculated analytically.


The pseudo rest-mass density and the SF potential aredetermined by Eqs. (197) and (198) with the equation ofstate (377) yielding

ln ρ =

∫

1− Aǫ

ǫ+A ln(

ǫǫ∗

) dǫ, (380)

Vtot =1

2

[

ǫ− A ln

(

ǫ

ǫ∗

)]

. (381)

They can also be determined by Eqs. (199) and (200)with Eq. (377) yielding

ln ρ =

∫ ǫ∗A e

P/A − 1

ǫ∗eP/A + PdP, (382)

Vtot =1

2

(

ǫ∗eP/A − P

)

. (383)

Equations (377) and (380)-(383) determine P (ρ), orρ(P ), and Vtot(ρ) in parametric form with parameter ǫ orP . Unfortunately, the integrals in Eqs. (380) and (382)cannot be calculated analytically.

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eq. (201) orEq. (202) with Eq. (377) yielding

lnX = 2

∫

dP

ǫ∗eP/A + P(384)

or

lnX = 2

∫ Aǫ

ǫ+A ln(

ǫǫ∗

) dǫ. (385)

These equations determine P (X), thus L(X). Unfortu-nately, the integrals in Eqs. (384) and (385) cannot becalculated analytically.

B. Logotropic equation of state of type II

The logotropic equation of state of type II writes [117]

P = A ln

(

ρmρ∗

)

. (386)

Using Eq. (D7), the internal energy is

u = −A ln

(

ρmρ∗

)

−A. (387)

In the nonrelativistic limit, using ρm ∼ ρ, we recover Eqs.(98) and (99) [recalling Eq. (29)]. Actually, Eqs. (386)and (387) coincide with Eqs. (98) and (99) with ρm inplace of ρ.


The energy density is determined by Eq. (203) withEq. (387). We obtain

ǫ = ρmc2 −A ln

(

ρmρ∗

)

−A. (388)

The pressure is determined by Eq. (204) with Eq. (387).This returns Eq. (386). Eliminating ρm between Eqs.(386) and (388) we obtain P (ǫ) under the inverse formǫ(P ) as

ǫ = ρ∗c2eP/A − P −A. (389)

In the nonrelativistic limit, using ǫ ∼ ρc2, we recover Eq.(98).


The pseudo rest-mass density and the SF potential aredetermined by Eqs. (207) and (208) with Eq. (387). Weget

ρ =ρ2m

ρm − Ac2

, (390)

Vtot =1

2

[

ρmc2 − 2A ln

(

ρmρ∗

)

−A

]

. (391)

Equation (390) can be inverted to give

ρm =ρ±

√

ρ2 − 4Aρc2

2. (392)

Combined with Eqs. (386) and (391) we explicitly obtainP (ρ) and Vtot(ρ) under the form

P = A ln

ρ±√

ρ2 − 4Aρc2

2ρ∗

, (393)

Vtot =ρ±

√

ρ2 − 4Aρc2

4c2−A ln

ρ±√

ρ2 − 4Aρc2

2ρ∗

− A

2.

(394)In the nonrelativistic limit, using ρm ∼ ρ(1−A/ρc2) andEq. (129), we recover Eqs. (98) and (99).

29

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eqs. (211),(386) and (387). We get

X =c2

2

(

1− A

ρmc2

)2

. (395)

This relation can be inverted to give

ρm =A/c2

1−(

2Xc2

)1/2. (396)

We then obtain

L(X) = P = −A ln

[

ρ∗c2

A

(

1−√

2X

c2

)]

. (397)

In the nonrelativistic limit, using Eq. (187), we recoverEq. (102).Remark: Starting from Eq. (322), taking the limit

γ → 0, K → +∞ with Kγ = A constant, and proceedingas in Eq. (104), we obtain Eq. (397) up to an additionalconstant. More generally, we can recover in the samemanner the other equations of this section.

C. Logotropic equation of state of type III

The logotropic equation of state of type III writes [121]

P = A ln

(

ρ

ρ∗

)

. (398)

Using Eq. (169), the SF potential is

Vtot(ρ) =1

2ρc2 −A ln

(

ρ

ρ∗

)

−A. (399)

In the nonrelativistic limit, we recover Eqs. (98) and (99)[recalling Eq. (129)]. Actually, Eqs. (398) and (399)coincide with Eqs. (98) and (99). The SF potential V (ρ)is logarithmic.


The energy density is determined by Eqs. (212) withEq. (399). This yields

ǫ = ρc2 −A ln

(

ρ

ρ∗

)

− 2A. (400)


ǫ = ρ∗c2eP/A − P − 2A. (401)

In the nonrelativistic limit, using ǫ ∼ ρc2, we recover Eq.(98).


The rest-mass density and the internal energy are de-termined by Eqs. (216) and (218) with Eq. (399). Weget

ρm =

√

ρ

(

ρ− 2A

c2

)

, (402)

u = ρc2 −A ln

(

ρ

ρ∗

)

− 2A− ρm(ρ)c2. (403)

Equation (402) can be inverted to give

ρ =A

c2+

√

A2

c4+ ρ2m. (404)

Combined with Eqs. (398) and (403) we explicitly obtainP (ρm) and u(ρm) under the form

P = A ln

(

A

ρ∗c2+

√

A2

ρ2∗c4+ρ2mρ2∗

)

, (405)

u =√

A2 + ρ2mc4−A ln

(

A

ρ∗c2+

√

A2

ρ2∗c4+ρ2mρ2∗

)

−A−ρmc2.

(406)In the nonrelativistic limit, using ρm ≃ ρ − A/c2, werecover Eqs. (98) and (99) [recalling Eq. (29)].

3. Lagrangian L(X)

The Lagrangian L(X) is determined by Eq. (219) withEq. (399). We get

X =1

2c2 − A

ρ. (407)

This equation can be reversed to give

ρ =A

12c

2 −X. (408)

We then obtain

L(X) = P = −A ln

[

ρ∗c2

2A

(

1− 2X

c2

)]

. (409)

In the nonrelativistic limit, using Eq. (187), we recoverEq. (102). Actually, Eq. (409) coincides with Eq. (102)with c2/2−X in place of x. Using Eq. (174) or Eq. (B8),we obtain the equation of motion

Dµ

[

∂µθ

1− 1c2 ∂νθ∂

νθ

]

= 0. (410)

Remark: Starting from Eq. (369), taking the limitγ → 0, K → +∞ with Kγ = A constant, and proceedingas in Eq. (104), we obtain Eq. (409) up to an additionalconstant. More generally, we can recover in the samemanner the other equations of this section.

30

VII. CONCLUSION

In this paper, we have showed that the equation ofstate of a relativistic barotropic fluid could be specified indifferent manners depending on whether the pressure P isexpressed in terms of the energy density ǫ (model I), therest-mass density ρm (model II), or the pseudo rest-massdensity ρ (model III). In model II, specifying the equationof state P (ρm) is equivalent to specifying the internalenergy u(ρm). In model III, specifying the equation ofstate P (ρ) is equivalent to specifying the potential V (ρ)of the complex SF to which the fluid is associated inthe TF limit. In the nonrelativistic limit, these threeformulations coincide.

We have shown how these different models are con-nected to each other. We have established general equa-tions allowing us to determine [ǫ, P (ǫ)], [ρm, P (ρm),u(ρm)] and [ρ, P (ρ) V (ρ)] once an equation of state isspecified under the form I, II or III.

In model III, we have determined the hydrodynamicrepresentation of a complex SF with a potential V (|ϕ|2)and the form of its Lagrangian. In the TF approximation,we can use the Bernoulli equation to obtain a reduced La-grangian of the form L(X) with X = 1

2∂µθ∂µθ, where θ

is the phase of the SF. This is a k-essence Lagrangianwhose expression is determined by the potential of thecomplex SF. We have established general equations al-lowing us to obtain L(X) once an equation of state isspecified under the form I, II or III.

For illustration, we have applied our formalism to poly-tropic, isothermal and logotropic equations of state oftype I, II and III that have been proposed as UDM mod-els. We have recovered previously obtained results, andwe have derived new results. For example, we have estab-lished the general analytical expression of the k-essenceLagrangian of polytropic and isothermal equations ofstate of type I, II and III. For γ = −1 (Chaplygin gas),the models of type I and III are equivalent and returnthe Born-Infeld action, while the model of type II leadsto a different action. We have also established the generalanalytical expression of the k-essence Lagrangian associ-ated with a logotrope of type II and III (the k-essenceLagrangian associated with a logotrope of type I cannotbe obtained analytically).

In a future contribution [167], we will apply our generalformalism to more complicated equations of state whichcan be viewed as a superposition of polytropic, isother-mal (linear) and logotropic equations of state.

The mixed equation of state of type I generically writes

P = K( ǫ

c2

)γ

+ αǫ − ǫΛ +A ln

(

ǫ

ǫP

)

, (411)

where we can add several polytropic terms with differentindices γ. More specifically, we can consider generalized

polytropic models of type I of the form

P = −(α+ 1)ǫ

(

ǫ

ǫP

)1/|ne|+ αǫ − (α+ 1)ǫ

(ǫΛǫ

)1/|nl|,

(412)or

P = −(α+ 1)ǫ2

ǫP+ αǫ− (α + 1)ǫΛ. (413)

Polytropic and isothermal (linear) equations of state oftype I have been studied in the context of relativistic stars[149, 152, 153, 155–159] and cosmology [50, 86, 95, 97–99]. Mixed models of type I of the form of Eqs. (411)-(413) have been introduced and studied in cosmology inRefs. [97–99, 168–171]. The equation of state (413) de-scribes the early inflation, the intermediate deceleratingexpansion, and the late accelerating expansion of the uni-verse in a unified manner.The mixed equation of state of type II generically

writes

P = Kργm + αρmc2 − ρΛc

2 +A ln

(

ρmρP

)

, (414)

where we can add several polytropic terms with differentindices γ. It is associated with an internal energy of theform

u =K

γ − 1ργm + αρmc

2

[

ln

(

ρmρ∗

)

− 1

]

+ρΛc2 −A ln

(

ρmρP

)

−A. (415)

Polytropic, isothermal (linear) and logotropic equationsof state of type II have been studied in the context ofrelativistic stars [152, 164] and cosmology [117, 165]. Itis often assumed that DM is pressureless (P = 0) sothat α = 0. However, a nonvanishing value of α canaccount for thermal effects as in [172–174]. In that caseαc2 = kBT/m. Mixed models of type II of the form ofEqs. (414) and (415) have been introduced and studiedin Refs. [117, 165].The mixed equation of state of type III generically

writes

P = Kργ + αρc2 − ρΛc2 +A ln

(

ρ

ρP

)

, (416)

where we can add several polytropic terms with differentindices γ. It is associated with a complex SF potentialof the form

Vtot =1

2ρc2 +

K

γ − 1ργ + αρc2

[

ln

(

ρ

ρ∗

)

− 1

]

+ρΛc2 −A ln

(

ρ

ρP

)

−A, (417)

where we recall that ρ = (m/~)2|ϕ|2. Polytropic, isother-mal (linear) and logotropic equations of state of typeIII have been studied in the context of relativistic stars[150, 152, 153] and cosmology [116, 147, 151, 154]. Mixedmodels of type III of the form of Eqs. (416) and (417)have been introduced and studied in Refs. [116, 154].

31

Appendix A: General identities for a nonrelativistic

cold gas

The first principle of thermodynamics for a nonrela-tivistic gas can be written as

d

(

u

ρ

)

= −Pd(

1

ρ

)

+ Td

(

s

ρ

)

, (A1)

where u is the density of internal energy, s the densityof entropy, ρ = nm the mass density, P the pressure,and T the temperature. For a cold (T = 0) or isentropic(s/ρm = cst) gas, it reduces to

d

(

u

ρ

)

= −Pd(

1

ρ

)

=P

ρ2dρ. (A2)

Introducing the enthalpy per particle

h =P + u

ρ, (A3)

we get

du = hdρ and dh =dP

ρ. (A4)

For a barotropic gas for which P = P (ρ), the foregoingequations can be written as

P (ρ) = −d(u/ρ)d(1/ρ)

= ρ2[

u(ρ)

ρ

]′= ρu′(ρ)− u(ρ), (A5)

c2s = P ′(ρ) = ρu′′(ρ), h(ρ) =P (ρ) + u(ρ)

ρ, (A6)

h(ρ) = u′(ρ), h′(ρ) =P ′(ρ)

ρ, (A7)

where c2s = P ′(ρ) is the squared speed of sound. Eq.(A5) determines the equation of state P (ρ) as a functionof the internal energy u(ρ). Inversely, the internal energyis determined by the equation of state according to therelation

u(ρ) = ρ

∫

P (ρ)

ρ2dρ, (A8)

which is the solution of the differential equation

ρdu

dρ− u(ρ) = P (ρ). (A9)

Comparing Eq. (A8) with Eq. (28), we see that thepotential V (ρ) represents the density of internal energy

u(ρ) = V (ρ). (A10)

We then have

P (ρ) = ρ2[

V (ρ)

ρ

]′= ρV ′(ρ)− V (ρ), (A11)

c2s = P ′(ρ) = ρV ′′(ρ), h(ρ) =P (ρ) + V (ρ)

ρ, (A12)

h(ρ) = V ′(ρ), h′(ρ) =P ′(ρ)

ρ, (A13)

V (ρ) = ρ

∫

P (ρ)

ρ2dρ. (A14)

Remark: The first principle of thermodynamics can bewritten as

du = Tds+ µdn. (A15)

This can be viewed as the variational principle (δs/kB −βδu + αδn = 0 with β = 1/kBT and α = µ/kBT ) as-sociated with the maximization of the entropy densitys at fixed energy density u and particle density n [175].Combined with the Gibbs-Duhem relation [175]

s =u+ P − µn

T, (A16)

we obtain Eq. (A1) and

sdT − dP + ndµ = 0. (A17)

If T = cst, then dP = ndµ. For T = 0, the foregoingequations reduce to

du = µdn, µ =u+ P

n, dP = ndµ, (A18)

which are equivalent to Eqs. (A3) and (A4) with µ = mh.Therefore, the enthalpy h(r) is equal to the local chemicalpotential µ(r) by unit of mass: h(r) = µ(r)/m.

Appendix B: K-essence Lagrangian of a real SF

1. General results

We consider a relativistic real SF ϕ(xµ) = ϕ(x, y, z, t)characterized by the action

S =

∫

L√−g d4x, (B1)

where L = L(ϕ, ∂µϕ) is the Lagrangian density and g =det(gµν) is the determinant of the metric tensor. TheLagrangian of a relativistic real SF ϕ is usually writtenunder the canonical form as

L = X − V (ϕ), (B2)

where

X =1

2∂µϕ∂

µϕ (B3)

32

is the kinetic energy and V is the potential energy.30 Inthat case, all the physics of the problem is contained inthe potential term. However, some authors have pro-posed to take V = 0 and modify the kinetic term. Thisleads to a Lagrangian of the form

L = L(X) (B4)

that is called a k-essence Lagrangian [38]. In that case,the physics of the problem is encapsulated in the non-canonical kinetic term L(X).31 Eq. (B4) is a pure k-essence Lagrangian. More general Lagrangians

L = L(X,ϕ) (B5)

can depend both on X and ϕ [40, 41]. The particularforms L = V (ϕ)F (X) and L = F (X)− V (ϕ) have beenspecifically introduced in Refs. [40] and [176–178] respec-tively.The least action principle δS = 0, which is equivalent

to the Euler-Lagrange equation

Dµ

[

∂L∂(∂µϕ)

]

− ∂L∂ϕ

= 0, (B6)

yields the equation of motion

Dµ

(

∂L∂X

∂µϕ

)

− ∂L∂ϕ

= 0. (B7)

For the Lagrangian (B4), it reduces to

Dµ [L′(X)∂µϕ] = 0. (B8)

For the Lagrangian (B4) the current is given by

Jµ = − ∂L∂(∂µϕ)

, (B9)

yielding

Jµ = −L′(X)∂µϕ. (B10)

Equation (B8) can then be written as DµJµ = 0. It

can therefore be viewed as a continuity equation ex-pressing the local conservation of the charge (or the lo-cal conservation of the boson number) given by Q =emc

∫

J0√−g d3x, i.e.,

Q = − e

mc

∫

L′(X)∂0ϕ√−g d3x. (B11)

30 Note that V represents here the total potential including the restmass term. For brevity, we write V instead of Vtot.

31 K-essence Lagrangians were initially introduced to describe in-flation (k-inflation) [40, 41]. They were later used to describeddark energy [37–39]. K-essence Lagrangians can also be obtainedfrom a canonical complex SF in the TF limit ~ → 0 [71, 89]. Inthat case, the real SF ϕ corresponds to the action (phase) θ ofthe complex SF (see Sec. III D).

In the present context, the conservation of the charge(or boson number) is related to the invariance of the La-grangian density under the constant shift ϕ→ ϕ+cst ofthe SF (Noether theorem).32 Introducing the quadrive-locity

uµ = − ∂µϕ√2X

c, (B12)

which satisfies by construction the identity uµuµ = c2,

we get

Jµ = L′(X)√2X

uµc. (B13)

We can therefore rewrite the continuity equation as

Dµ

[

L′(X)√2Xuµ

]

= 0. (B14)

Comparing Eqs. (B13) and (B14) with Jµ = ρmuµ andDµ(ρmu

µ) = 0, we find that the rest-mass density isgiven by

ρm = L′(X)√2X

1

c. (B15)

The energy-momentum tensor is given by Eq. (114).It satisfies the conservation law DµT

µν = 0. For a realSF we have

T νµ =

∂L∂(∂νϕ)

∂µϕ− gνµL. (B16)

The energy-momentum tensor associated with the La-grangian (B5) is

Tµν =∂L∂X

∂µϕ∂νϕ− gµνL. (B17)

Introducing the quadrivelocity from Eq. (B12) we get

Tµν = 2X∂L∂X

uµuνc2

− gµνL. (B18)

The energy-momentum tensor (B17) can be written un-der the perfect fluid form

Tµν = (ǫ+ P )uµuνc2

− Pgµν , (B19)

where ǫ is the energy density and P is the pressure, pro-vided that we make the identifications

P = L and ǫ+ P = 2X∂L∂X

. (B20)

As a result, the pressure and the energy density associ-ated with the Lagrangian (B5) are given by

P = L(X,ϕ), (B21)

32 Note that the more general Lagrangian (B5) does not conservethe charge (or the boson number).

33

ǫ = 2X∂P

∂X− P. (B22)

The Lagrangian plays the role of an effective pressure.If the Lagrangian satisfies the condition X∂P/∂X ≪ Pfor some range of X and ϕ, then the equation of stateis P ≃ −ǫ (vacuum energy) and we have an inflationarysolution [179]. On the other hand, for the LagrangianL = V (ϕ)X corresponding to P ∝ X i.e. X∂P/∂X = Pwe obtain the stiff equation of state P = ǫ. In that case,the equation of motion (B7) becomes Dµ(V (ϕ)∂µϕ) −12V

′(ϕ)∂µϕ∂µϕ = 0. It reduces to ϕ = 0 when L =AX .The equation of state parameter and the squared speed

of sound are given by [41]

w =P

ǫ=

P

2X ∂P∂X − P

(B23)

and

c2s =∂P∂X∂ǫ∂X

c2 =∂P∂X

∂P∂X + 2X ∂2P

∂X2

c2. (B24)

We note that cs ≃ c if 2X∂2P/∂X2 ≪ ∂P/∂X . Thisis the case in particular for the Lagrangian L = V (ϕ)Xdiscussed above for which cs = c exactly.In the general case, we have P = P (X,ϕ) and ǫ =

ǫ(X,ϕ) so that the fluid is not necessarily barotropic.However, for a k-essence SF described by a Lagrangianof the form of Eq. (B4), we have P = P (X) and ǫ = ǫ(X)implying P = P (ǫ). In that case, the fluid is barotropicand c2s = P ′(ǫ)c2. On the other hand, using Eqs. (D5),(B15), (B21) and (B22), we find that the enthalpy isgiven by

h = c√2X. (B25)

Remark: For the Lagrangian L = V (ϕ)X(α+1)/2α, weobtain the linear equation of state P = αǫ. This includesstiff matter (α = 1; L = V (ϕ)X), radiation (α = 1/3;L = V (ϕ)X2) and a cosmological constant (α = −1;L = V (ϕ)). The equation of motion (B7) becomes

α+ 1

2αDµ

[

V (ϕ)X1−α2α ∂µϕ

]

− V ′(ϕ)Xα+12α = 0. (B26)

It reduces to

Dµ

(

X1−α2α ∂µϕ

)

= 0 (B27)

when L = AX(α+1)/2α. For the Lagrangian L =V (ϕ) lnX (which can be viewed as a limit of the La-grangian V (ϕ)X(α+1)/2α for α → −1 and V → +∞ withV (α+1) finite), we obtain P = 2V (ϕ)− ǫ which reducesto the affine equation of state P = 2A − ǫ [98, 99] whenV (ϕ) = A. In that case, the equation of motion (B7)becomes

Dµ

[

V (ϕ)

X∂µϕ

]

− V ′(ϕ) lnX = 0. (B28)

It reduces to

Dµ

(

1

X∂µϕ

)

= 0 (B29)

when L = A lnX .

2. Canonical SF

The Lagrangian of a canonical SF is

L =1

2gµν∂µϕ∂νϕ− V (ϕ), (B30)

where the first term is the kinetic energy and the secondterm is minus the potential energy. It is of the formL = X − V (ϕ). The least action principle δS = 0, whichis equivalent to the Euler-Lagrange equation (B6), leadsto the KG equation

ϕ+dV

dϕ= 0, (B31)

where = Dµ∂µ is the d’Alembertian. A canonical real

SF does not conserve the charge.The energy-momentum tensor (B16) associated with

the canonical Lagrangian (B30) is

Tµν = ∂µϕ∂νϕ− gµνL. (B32)

Repeating the procedure of Appendix B 1 we find thatthe energy density and the pressure are given by

ǫ =1

2∂µϕ∂

µϕ+ V (ϕ), (B33)

P =1

2∂µϕ∂

µϕ− V (ϕ). (B34)

Since ǫ = X + V (ϕ) and P = X − V (ϕ), we find thatP = ǫ− 2V (ϕ), w = [X − V (ϕ)]/[X + V (ϕ)] and cs = c.For a canonical SF, the speed of sound is equal to thespeed of light.When X ≫ V , we obtain ǫ = X and P = X leading to

the equation of state P = ǫ corresponding to stiff matter.This is the so-called kination regime [180]. This regimeis achieved in particular when V = 0. In that case, theLagrangian L = X describes a noninteracting masslessSF and the KG equation reduces to ϕ = 0. WhenX ≪ V , we obtain ǫ = V and P = −V leading to theequation of state P = −ǫ corresponding to the vacuumenergy. This regime is achieved in particular when ϕ =ϕ0 is constant (X = 0) and lies at the bottom of thepotential (V ′(ϕ0) = 0). In cosmology, this equation ofstate leads to a de Sitter era where ǫ = V (ϕ0) is constantand the scale factor increases exponentially rapidly withtime as a ∝ exp[(8πGǫ/3c2)1/2t]. The condition X ≪ Vcorresponds to the slow-roll regime [179].

34

When V (ϕ) = V0 is constant, the Lagrangian L =X − V0 describes a massless SF in the presence of a cos-mological constant (ǫΛ = V0). In that case, ǫ = X + V0and P = X − V0 leading to the affine equation of stateP = ǫ − 2V0 = ǫ− 2ǫΛ [98]. In cosmology, when V0 > 0,this equation of state generically leads to a stiff matterera followed by a de Sitter era (or a de Sitter era alonewhen X = 0, i.e., ϕ = cst). When V0 = 0 we recoverthe Lagrangian L = X a free massless SF. In that case,ǫ = X and P = X leading to the stiff equation of stateP = ǫ. In cosmology, we have a pure stiff matter era.Remark: The Lagrangian of a particle of mass m and

position q(t) in Newtonian mechanics is L = (1/2)mq2−V (q). Its impulse is p = ∂L/∂q = mq and its energyis E = pq − L = q∂L/∂q − L = (1/2)mq2 + V (q), i.e.,E = p2/(2m) + V (q). Its equation of motion is given bythe Euler-Lagrange equation (d/dt)(∂L/∂q)−∂L/∂q = 0yielding mq = −V ′(q). The Lagrangian equations of acanonical SF are similar to the Lagrangian equations ofa nonrelativistic particle in which the SF ϕ(xµ) plays therole of q(t) and the SF potential V (ϕ) the role of V (q).

3. Tachyonic SF

The Lagrangian of a tachyonic SF is

L = −V (ϕ)

√

1− 1

c2gµν∂µϕ∂νϕ. (B35)

This corresponds to the Born-Infeld Lagrangian (5)multiplied by V (ϕ). It is of the form L =

−V (ϕ)√

1− 2X/c2. This Lagrangian was introducedby Sen [77–79] in the context of string theory and d-branes and further discussed in [80–87]. The relationto k-essence fields was made in [81, 83, 87]. The leastaction principle δS = 0, which is equivalent to the Euler-Lagrange equation (B6), leads to the equation of motion

Dµ

V (ϕ)/c2√

1− 1c2 ∂µϕ∂

µϕ∂µϕ

+V ′(ϕ)

√

1− 1

c2∂µϕ∂µϕ = 0

(B36)or, equivalently,

Dµ∂µϕ+

Dµ∂νϕ

1− ∂µϕ∂µϕ∂µϕ∂νϕ+ (ln V )′c2 = 0. (B37)

A real tachyonic SF does not conserve the charge.The energy-momentum tensor (B16) associated with

the tachyonic Lagrangian (B35) is

Tµν =V (ϕ)/c2

√

1− 1c2 ∂µϕ∂

µϕ∂µϕ∂νϕ− gµνL. (B38)

Repeating the procedure of Appendix B 1 we find thatthe energy density and the pressure are given by

ǫ =V (ϕ)

√

1− 1c2 ∂µϕ∂

µϕ, (B39)

P = −V (ϕ)

√

1− 1

c2∂µϕ∂µϕ. (B40)

Since ǫ = V (ϕ)/√

1− 2X/c2 and P =

−V (ϕ)√

1− 2X/c2, we find that P = −V (ϕ)2/ǫ,w = −(1− 2X/c2) and c2s = (1− 2X/c2)c2 = −wc2.When V (ϕ) = V0 is constant, the Lagrangian (B35)

reduces to the Born-Infeld Lagrangian (5) and we obtain

ǫ = V0/√

1− 2X/c2 and P = −V0√

1− 2X/c2 leading tothe Chaplygin equation of state P = −V 2

0 /ǫ (inversely,the Chaplygin equation of state P = −Ac2/ǫ leads to theBorn-Infeld Lagrangian (5) corresponding to Eq. (B35)with V (ϕ) = V0 constant). Therefore, the Chaplygingas can be considered as the simplest tachyon modelwhere the tachyon field is associated with a purely ki-netic Lagrangian. The relation between the tachyonicLagrangian with a constant potential (reducing to theBorn-Infeld Lagrangian) and the Chaplygin gas [50] wasfirst made by [81]. The fact that the Chaplygin gasis associated with the Born-Infeld Lagrangian was un-derstood by [58, 59, 70, 71]. The relation between theChaplygin gas, the Born-Infeld Lagrangian, k-essence La-grangians and tachyon fields were further discussed in[86, 88, 89, 94, 181].Remark: The Lagrangian of a particle of mass

m and position q(t) in special relativity is L =

−mc2√

1− q2/c2. Its impulse is p = mq/√

1− q2/c2

and its energy is E = mc2/√

1− q2/c2, i.e., E2 =p2c2+m2c4. Its equation of motion is given by the Euler-Lagrange equation (d/dt)(∂L/∂q) − ∂L/∂q = 0 yielding

(d/dt)(mq/√

1− q2/c2) +m′(q)c2√

1− q2/c2 = 0. TheLagrangian equations of a tachyonic SF are similar to theLagrangian equations of a relativistic particle in whichthe SF ϕ(xµ) plays the role of q(t) and the SF potentialV (ϕ) the role of the mass m that would depend on q inthe general case.

4. Nonrelativistic limit

The action of a nonrelativistic real SF is

S =

∫

L d3xdt, (B41)

where L = L(ϕ, ϕ,∇ϕ) is the Lagrangian density. Weconsider a pure k-essence Lagrangian of the form

L = L(x), (B42)

where

x = ϕ+1

2(∇ϕ)2. (B43)

More general k-essence Lagrangians

L = L(x, ϕ) (B44)

35

can depend both on x and ϕ. The least action princi-ple δS = 0, which is equivalent to the Euler-Lagrangeequation

∂

∂t

(

∂L∂ϕ

)

+∇ ·(

∂L∂∇ϕ

)

− ∂L∂ϕ

= 0, (B45)

yields the equation of motion

∂

∂t

(

∂L∂x

)

+∇ ·(

∂L∂x

∇ϕ)

− ∂L∂ϕ

= 0. (B46)

For the Lagrangian (B42), it reduces to

∂

∂t[L′(x)] +∇ · [L′(x)∇ϕ] = 0. (B47)

For the Lagrangian (B42) the current is given by

Jµ = − ∂L∂(∂µϕ)

. (B48)

It determines the mass density

ρ ≡ −∂L∂ϕ

= −L′(x) (B49)

and the mass flux

J ≡ − ∂L∂(∇ϕ) = −L′(x)∇ϕ. (B50)

Equation (B47) can be written as

∂ρ

∂t+∇ · J = 0. (B51)

It expresses the local conservation of mass. This conser-vation law is associated with the invariance of the La-grangian density under the transformation ϕ → ϕ + cst(Noether theorem). Introducing the velocity

u = ∇ϕ, (B52)

we get

J = ρu, (B53)

and the continuity equation

∂ρ

∂t+∇ · (ρu) = 0. (B54)

The energy-momentum tensor is given by

T νµ = ∂µϕ

∂L∂(∂νϕ)

− Lδνµ. (B55)

The local conservation of energy and impulse can be writ-ten as

∂T00∂t

− ∂iT0i = 0, (B56)

−∂Ti0∂t

+ ∂jTij = 0. (B57)

For the Lagrangian (B44) we obtain the energy density

T00 ≡ ϕ∂L∂ϕ

− L =∂L∂x

ϕ− L(x), (B58)

the momentum density

−Ti0 ≡ −∂iϕ∂L∂ϕ

= −∂L∂x

∂iϕ, (B59)

the energy flux

−T0i ≡ ϕ∂L

∂(∂iϕ)=∂L∂x

ϕ∂iϕ, (B60)

and the momentum fluxes (stress tensor)

Tij ≡ −∂iϕ∂L

∂(∂jϕ)+Lδij = −∂L

∂x∂iϕ∂jϕ+Lδij . (B61)

Introducing the velocity from Eq. (B52), we get

Tij = −∂L∂x

uiuj + Lδij . (B62)

The energy-momentum tensor Tij can be written underthe perfect fluid form

Tij = ρuiuj + Pδij (B63)

provided that we make the identifications

P = L(x, ϕ) (B64)

and

ρ = −∂L∂x

= −∂P∂x

. (B65)

The equation of state parameter and the squared speedof sound are given by

w =P

ρc2= − P

∂P∂x c

2, (B66)

c2s =∂P∂x∂ρ∂x

= −∂P∂x∂2P∂x2

. (B67)

Using Eqs. (B64) and (B65), we can rewrite Eq. (B46)and Eqs. (B58)-(B60) as

∂ρ

∂t+∇ · (ρu) + ∂L

∂ϕ= 0, (B68)

T00 = −ρϕ− L(x), (B69)

−Ti0 = ρui, (B70)

36

−T0i = −ρϕui. (B71)

For the Lagrangian (B42), Eq. (B68) reduces to the con-tinuity equation (B54). In that case, the momentum den-sity is equal to the mass flux: −Ti0 = Ji. On the otherhand, using Eq. (B43), the energy density can be writtenas

T00 =1

2ρu2 + V (ρ), (B72)

where we have defined the potential V (ρ) by the Legendretransform V (ρ) = −ρx − L(x). Using Eq. (B65), weget V ′(ρ) = −x. Then, using Eq. (B64), we obtainP (ρ) = ρV ′(ρ) − V (ρ) returning Eqs. (26) and (58).Therefore, V (ρ) coincides with the potential introducedin Sec. II. Similarly, the energy flux can be written as

−T0i = ρ

[

1

2u2 + V ′(ρ)

]

ui. (B73)

These results are consistent with the results obtained inAppendix I when ~ = 0.Remark: Eqs. (B64) and (B65) are the counterparts

of Eqs. (B21) and (B22) in the relativistic case. Indeed,using Eqs. (187) and ǫ ∼ ρc2 valid in the nonrelativisticlimit, Eqs. (B21) and (B22) imply P = L(x, ϕ) and

ρ ∼ ǫ

c2∼ −∂P

∂x∼ −∂L

∂x, (B74)

returning Eqs. (B64) and (B65).

5. Cosmological evolution

We now consider a spatially homogeneous SF describedby a k-essence Lagrangian in an expanding universe. Inthat case33

X =1

2ϕ2. (B75)

On the other hand, the energy-momentum tensor is di-agonal T ν

µ = diag(ǫ,−P,−P,−P ).34 The energy densityand the pressure of the SF are given by

ǫ = T 00 =

∂L∂ϕ

ϕ− L, P = −T ii = L, (B76)

returning Eqs. (B22) and (B21).For a Lagrangian density of the form L = L(X,ϕ), the

equation of motion (B7) of the SF becomes

(

∂L∂X

+ 2X∂2L∂X2

)

ϕ+ 3H∂L∂X

ϕ+ 2X∂2L∂X∂ϕ

− ∂L∂ϕ

= 0.

(B77)

33 In this Appendix and in Appendices C2 and C 3, t stands for ct.34 In the homogeneous case, using Eq. (B19) with u0 = c and

ui = 0 we get T 00= ǫ and T i

i= −P .

This equation is equivalent to the energy conservationequation (C4). Indeed, taking the time derivative of ǫfrom Eq. (B22) and substituting the result into Eq. (C4)we get

(

∂L∂X

+ 2X∂2L∂X2

)

X +

(

2X∂2L∂X∂ϕ

− ∂L∂ϕ

)

ϕ

+ 6HX∂L∂X

= 0. (B78)

Recalling Eq. (B75), we obtain Eq. (B77). We cancheck that this equation returns Eqs. (C12) and (C36)for a canonical and a tachyonic SF respectively.For a Lagrangian density of the form L = V (ϕ)F (X),

Eq. (B77) reduces to

(FX+2XFXX)ϕ+3HFXϕ+(2XFX−F )V′

V= 0. (B79)

In the particular case L = V (ϕ)X , corresponding to astiff equation of state [see the comment after Eq. (B21)],we get

ϕ+ 3Hϕ+XV ′

V= 0. (B80)

For a Lagrangian density of the form L = F (X)−V (ϕ),Eq. (B77) reduces to

(FX + 2XFXX)ϕ+ 3HFX ϕ+ V ′(ϕ) = 0. (B81)

For a pure k-essence Lagrangian L = L(X), Eq. (B77)reduces to

(L′ + 2XL′′)ϕ+ 3HL′ϕ = 0. (B82)

We also have [see Eq. (B78)]

(L′ + 2XL′′)dX

da+

6

aXL′ = 0. (B83)

This equation integrates to give

√XL′(X) =

k

a3. (B84)

Using Eq. (B15), we see that this equation is equivalentto the conservation of the rest-mass: ρm ∝ a−3. Eq.(B84) was first obtained by Chimento [182] and Scherrer[183] but they did not realize the relation with the rest-mass density. Our approach provides therefore a physicalinterpretation of their result.

Appendix C: Equation of state of type I

In this Appendix, we consider a barotropic fluid de-scribed by an equation of state of type I where the pres-sure P = P (ǫ) is specified as a function of the energydensity. We show that, in a cosmological context, it ispossible to associate to this fluid a real SF with a po-tential V (ϕ) which is fully determined by the equationof state. As an illustration, we determine the real SFpotential associated with a polytropic equation of stateof type I.

37

1. Friedmann equations

If we consider an expanding homogeneous backgroundand adopt the Friedmann-Lemaıtre-Robertson-Walker(FLRW) metric, the Einstein field equations reduce tothe Friedmann equations

H2 =8πG

3c2ǫ, (C1)

2H + 3H2 = −8πG

c2P, (C2)

where H = a/a is the Hubble parameter and a(t) is thescale factor. To obtain Eq. (C1), we have assumed thatthe universe is flat (k = 0) in agreement with the infla-tion paradigm [5] and the observations of the cosmic mi-crowave background (CMB) [1, 2]. On the other hand, wehave set the cosmological constant to zero (Λ = 0) sincedark energy can be taken into account in the equationof state P (ǫ) or in the SF potential V (ϕ) (quintessence).Eq. (C2) can also be written as

a

a= −4πG

3c2(3P + ǫ), (C3)

showing that the expansion of the universe is deceleratingwhen P > −ǫ/3 and accelerating when P < −ǫ/3.Using Eqs. (C1) and (C2), we obtain the energy con-

servation equation

dǫ

dt+ 3H (ǫ+ P ) = 0. (C4)

This equation can be directly obtained from the conser-vation of the energy-momentum tensorDµT

µν = 0 whichresults from the Bianchi identities. The energy densityincreases with the scale factor when P > −ǫ and de-creases with the scale factor when P < −ǫ. The lattercase corresponds to a phantom universe.For a given equation of state P (ǫ) we can solve Eq.

(C4) to get

ln a = −1

3

∫

dǫ

ǫ+ P (ǫ). (C5)

This equation determines ǫ(a). We can then solve theFriedmann equation (C1) to obtain the temporal evolu-tion of the scale factor a(t).A polytropic equation of state of type I is defined by

P = K( ǫ

c2

)γ

with γ = 1 + 1/n. (C6)

Assuming 1 + (K/c2)(ǫ/c2)1/n ≥ 0, i.e., P ≥ −ǫ corre-sponding to a nonphantom universe,35 the energy con-servation equation (C4) can be integrated into [97, 98]

ǫ =ρ∗c2

[

(a/a∗)3/n ∓ 1]n , (C7)

35 The case of a phantom universe is treated in [99].

where ρ∗ = (c2/|K|)n and a∗ is a constant of integration.The upper sign corresponds to K > 0 and the lower signto K < 0.The case γ = 1, corresponding to a linear equation of

state

P = αǫ (C8)

with α = K/c2, must be treated specifically. In that case,the solution of Eq. (C4) can be written as

ǫ =ρ∗c2

(a/a∗)3(1+α)

, (C9)

where ρ∗a3(1+α)∗ is a constant of integration.

Remark: Unfortunately, for the logotropic equation ofstate of type I [see Eq. (377)], the energy conservationequation

ln a = −1

3

∫

dǫ

ǫ+A ln(ǫ/ǫ∗)(C10)

cannot be integrated explicitly.

2. Canonical SF

We consider a spatially homogeneous canonical SF inan expanding universe with a Lagrangian

L =1

2ϕ2 − V (ϕ). (C11)

It evolves according to the KG equation [see Eq. (B77)]

ϕ+ 3Hϕ+dV

dϕ= 0 (C12)

coupled to the Friedmann equation (C1). The SF tendsto run down the potential towards lower energies whileexperiencing a Hubble friction. The energy-momentumtensor is diagonal T ν

µ = diag(ǫ,−P,−P,−P ). The energydensity and the pressure of the SF are given by [see Eq.(B76)]

ǫ =1

2ϕ2 + V (ϕ), (C13)

P =1

2ϕ2 − V (ϕ). (C14)

We note that, here, V represents the total SF potentialincluding the rest-mass term. When the kinetic termdominates we obtain the stiff equation of state P = ǫ.When the potential term dominates, we obtain the equa-tion of state P = −ǫ corresponding to the vacuum en-ergy. We can easily check that the KG equation (C12)with Eqs. (C13) and (C14) implies the energy conserva-tion equation (C4) (see Appendix G). Inversely, the theenergy conservation equation (C4) with Eqs. (C13) and

38

(C14) implies the KG equation (C12). The equation ofstate parameter w = P/ǫ is given by

w =12 ϕ

2 − V (ϕ)12 ϕ

2 + V (ϕ). (C15)

It satisfies −1 ≤ w ≤ 1. The speed of sound is equal tothe speed of light (cs = c).Using standard techniques [83, 184–186], we can obtain

the SF potential as follows [98]. From Eqs. (C13) and(C14), we get

ϕ2 = (w + 1)ǫ. (C16)

Then, using ϕ = (dϕ/da)Ha and the Friedmann equation(C1), we find that the relation between the SF and thescale factor is given by

dϕ

da=

(

3c4

8πG

)1/2 √1 + w

a. (C17)

We note that ϕ is a monotonic function of a. We haveselected the solution where ϕ increases with a. On theother hand, according to Eqs. (C13) and (C14), we have

V =1

2(1− w)ǫ. (C18)

Therefore, the potential V (ϕ) of the canonical SF is de-termined in parametric form by the equations

ϕ(a) =

(

3c4

8πG

)1/2 ∫√

1 + w(a)da

a, (C19)

V (a) =1

2[1− w(a)] ǫ(a). (C20)

We note that ϕ is defined up to an additive constant.The canonical SF potential corresponding to a poly-

tropic equation of state of type I [see Eq. (C1)] has beendetermined in Sec. 8.1. of [98]. It is given by

V =1

2ρ∗c

2 cosh2 ψ + 1

cosh2γ

γ−1 ψ(K < 0), (C21)

V =1

2ρ∗c

2 sinh2 ψ − 1

sinh2γ

γ−1 ψ(K > 0), (C22)

where

ψ =

(

8πG

3c4

)1/23

2(γ − 1)ϕ. (C23)

The relation between the scale factor and the SF is

a

a∗= sinh

23(γ−1) ψ (K < 0), (C24)

a

a∗= cosh

23(γ−1) ψ (K > 0), (C25)

These expressions are valid for ψ ≥ 0.(i) For γ = −1 (Chaplygin gas), we get

V =1

2ρ∗c

2

(

coshψ +1

coshψ

)

(K < 0), (C26)

V =1

2ρ∗c

2

(

sinhψ − 1

sinhψ

)

(K > 0), (C27)

with ρ∗ =√

|K|/c2. This SF potential was first obtainedin [50].(ii) For γ = 2 (BEC), we get

V =1

2ρ∗c

2 cosh2 ψ + 1

cosh4 ψ(K < 0), (C28)

V =1

2ρ∗c

2 sinh2 ψ − 1

sinh4 ψ(K > 0), (C29)

with ρ∗ = c2/|K|. This SF potential was first obtainedin [98].(iii) For γ = 0 (ΛCDM model), we get

V =1

2ρ∗c

2(cosh2 ψ + 1) (K < 0), (C30)

V =1

2ρ∗c

2(sinh2 ψ − 1) (K > 0), (C31)

with ρ∗ = |K|/c2. This SF potential was first obtainedin [86] and rediscovered independently in [98].(iv) For γ = 3 (superfluid), we get

V =1

2ρ∗c

2 cosh2 ψ + 1

cosh3 ψ(K < 0), (C32)

V =1

2ρ∗c

2 sinh2 ψ − 1

sinh3 ψ(K > 0), (C33)

with ρ∗ =√

c2/|K|.(v) For γ = 1, we get

V (ϕ) =1

2ρ∗c

2(1− α)e−3√α+1( 8πG

3c4)1/2

ϕ. (C34)

This exponential potential was obtained in [83] but itappeared in earlier works on inflation and quintessence[26, 27, 33, 187, 188]. In that case, the relation betweenthe scale factor and the SF is

ϕ(a) =

(

3c4

8πG

)1/2 √1 + α ln

(

a

a∗

)

. (C35)

For α = 1 (stiff matter), we find that V (ϕ) = 0. Onthe other hand, for α = −1 (vacuum energy), we findthat ϕ = 0 so that ϕ = ϕ0 is constant. This is possibleaccording to the equation of motion (C12) provided thatV ′(ϕ0) = 0. Therefore, ϕ0 must be at the minimum ofthe potential V (ϕ). In that case, ǫ = V (ϕ0) = V0 andP = −V (ϕ0) = −V0, yielding P = −ǫ. Note that V (ϕ)is not necessarily constant but it must have a minimumV0 > 0.

39

3. Tachyonic SF

We consider a spatially homogeneous tachyonic SF [80–87] in an expanding universe with a Lagrangian

L = −V (ϕ)√

1− ϕ2. (C36)

It evolves according to the equation [see Eq. (B77)]

ϕ

1− ϕ2+ 3Hϕ+

1

V

dV

dϕ= 0 (C37)

coupled to the Friedmann equation (C1). The SF tendsto run down the potential towards lower energies whileexperiencing a Hubble friction. The energy-momentumtensor is diagonal T ν

µ = diag(ǫ,−P,−P,−P ). The den-sity and the pressure of the SF are given by [see Eq.(B76)]

ǫ =V (ϕ)

√

1− ϕ2, (C38)

P = −V (ϕ)√

1− ϕ2. (C39)

We can easily check that the equation of motion (C37)with Eqs. (C38) and (C39) implies the energy conser-vation equation (C4) (see Appendix G). Inversely, theenergy conservation equation (C4) with Eqs. (C38) and(C39) implies the equation of motion (C37). The equa-tion of state parameter w = P/ǫ is given by

w = ϕ2 − 1. (C40)

It satisfies −1 ≤ w ≤ 0. The squared speed of sound isgiven by c2s/c

2 = 1− ϕ2 = −w. It satisfies 0 ≤ w ≤ 1.Using standard techniques [83, 184–186], we can obtain

the SF potential as follows [98]. From Eqs. (C38) and(C39), we obtain

ϕ2 = 1 + w. (C41)

Using ϕ = (dϕ/da)Ha, and the Friedmann equation(C1), we get

dϕ

da=

(

3c4

8πG

)1/2 √1 + w√ǫa

. (C42)

We note that ϕ is a monotonic function of a. We haveselected the solution where ϕ increases with a. On theother hand, from Eqs. (C38) and (C39), we have

V 2 = −wǫ2. (C43)

Therefore, the potential V (ϕ) of the tachyonic SF is de-termined in parametric form by the equations

ϕ(a) =

(

3c4

8πG

)1/2 ∫ √

1 + w(a)√

ǫ(a)

da

a, (C44)

V (a) =√

−w(a)ǫ(a). (C45)

The tachyonic SF potential corresponding to a poly-tropic equation of state of type I [see Eq. (C1)] has beendetermined in Sec. 8.2. of [98]. It is defined only forK < 0. It is given in parametric form by

V =ρ∗c2

(x2 + 1)1+γ

2(γ−1)

, (C46)

ψ =

∫

(x2 + 1)2−γ

2(γ−1) dx, (C47)

where we have introduced the variable

ψ =√

ρ∗c2(

8πG

3c4

)1/23

2(γ − 1)ϕ. (C48)

The relation between the scale factor and the SF is givenby Eq. (C47) with

x =

(

a

a∗

)32 (γ−1)

. (C49)

The integral in Eq. (C47) can be expressed in terms ofhypergeometric functions. Simple analytical expressionscan be obtained in special cases.(i) For γ = −1 and K < 0 (Chaplygin gas), we find

that V (ϕ) = ρ∗c2 with ρ∗ =√

|K|/c2. In that case, thepotential is constant [86]. This leads to the Born-InfeldLagrangian (see Appendix B3).(ii) For γ = 2 and K < 0 (BEC), we get

V (ψ) =ρ∗c2

(ψ2 + 1)3/2(C50)

with ρ∗ = c2/|K|. We have a/a∗ = ψ2/3 with ψ ≥ 0.This potential was first obtained in [98].(iii) For γ = 0 and K < 0 (ΛCDM model), we get

V (ψ) =ρ∗c2

cosψ(C51)

with ρ∗ = |K|/c2. We have a/a∗ = 1/ tan(ψ)2/3 with0 ≤ ψ ≤ π/2. This potential was first obtained in [86]and rediscovered independently in [98].(iv) For γ = 3 andK < 0 (superfluid), it is not possible

to obtain explicit expressions.(v) For γ = 1 and −1 < α < 0, we get

V (ϕ) =

√−α

1 + α

c4

6πG

1

ϕ2. (C52)

This inverse square law potential was first obtained in[83, 87]. In that case, the relation between the scalefactor and the SF is

ϕ =2

3

1√

ρ∗c2

(

3c4

8πG

)1/21√

1 + α

(

a

a∗

)3(1+α)/2

.

(C53)

40

For α = −1 (vacuum energy), we find that ϕ = 0 sothat ϕ = ϕ0 is constant. This is possible according tothe equation of motion (C37) provided that V ′(ϕ0) = 0.Therefore, ϕ0 must be at the minimum of the potentialV (ϕ). In that case, ǫ = V (ϕ0) = V0 and P = −V (ϕ0) =−V0, yielding P = −ǫ. Note that V (ϕ) is not necessarilyconstant but it must have a minimum V0 > 0.Remark: For γ = 1/2 and K < 0, we get

V (ψ) =ρ∗c2

(1− ψ2)3/2(C54)

with ρ∗ = (|K|/c2)2. We have (a/a∗)3/4 =√

1− ψ2/ψwith 0 ≤ ψ ≤ 1. This potential was first obtained in [98].

Appendix D: Equation of state of type II

In this Appendix, we consider a barotropic fluid de-scribed by an equation of state of type II where the pres-sure P = P (ρm) is specified as a function of the rest-mass density. After recalling general results, we applythis equation of state to a cosmological context.

1. General results

The first principle of thermodynamics for a relativisticgas can be written as

d

(

ǫ

ρm

)

= −Pd(

1

ρm

)

+ Td

(

s

ρm

)

, (D1)

where

ǫ = ρmc2 + u(ρm) (D2)

is the energy density including the rest-mass energy den-sity ρmc

2 (where ρm = nm is the rest-mass density) andthe internal energy density u(ρm), s is the density of en-tropy, P is the pressure, and T is the temperature. Weassume that Td(s/ρm) = 0. This corresponds to cold(T = 0) or isentropic (s/ρm = cst) gases. In that case,Eq. (D1) reduces to

d

(

ǫ

ρm

)

= −Pd(

1

ρm

)

=P

ρ2mdρm. (D3)

This equation can be rewritten as

dǫ

dρm=P + ǫ

ρm(D4)

where the term in the right hand side is the enthalpy h.We have

h =P + ǫ

ρm, h =

dǫ

dρm, dh =

dP

ρm. (D5)

Eq. (D3) can be integrated into

ǫ = ρmc2 + ρm

∫

P (ρm)

ρ2mdρm (D6)

establishing that

u(ρm) = ρm

∫

P (ρm)

ρ2mdρm. (D7)

This equation determines the internal energy as a func-tion of the equation of state P (ρm). Inversely, the equa-tion of state is determined by the internal energy u(ρm)from the relation

P (ρm) = −d(u/ρm)

d(1/ρm)= ρ2m

[

u(ρm)

ρm

]′= ρmu

′(ρm)−u(ρm).

(D8)We note that

P ′(ρm) = ρmu′′(ρm). (D9)

The squared speed of sound is

c2s = P ′(ǫ)c2 =ρmǫ

′′(ρm)

ǫ′(ρm)c2 =

ρmu′′(ρm)c2

c2 + u′(ρm). (D10)

Remark: The first principle of thermodynamics can bewritten as [175]

dǫ = Tds+ µdn. (D11)

Combined with the Gibbs-Duhem relation [175]

s =ǫ+ P − µn

T, (D12)

we obtain Eq. (D1) and

sdT − dP + ndµ = 0. (D13)

If T = cst, then dP = ndµ. For T = 0, the foregoingequations reduce to

dǫ = µdn, µ =ǫ+ P

n, dP = ndµ, (D14)

which are equivalent to Eq. (D5) with µ = mh.

2. Cosmology

Let us apply these equations in a cosmological con-text, namely for a homogeneous fluid in an expandingbackground. Combining the energy conservation equa-tion (C4) with Eq. (D4), we obtain

dρmdt

+ 3Hρm = 0. (D15)

This equation expresses the conservation of the parti-cle number (or rest-mass). It can be integrated intoρm ∝ a−3. Inserting this relation into Eq. (D2), we seethat ρm represents DM while u represents DE. Therefore,DM represents the rest-mass energy density and DE rep-resents the internal energy density. This decompositionprovides therefore a simple (and nice) interpretation of

41

DM and DE in terms of a single DF [117, 118]. Owingto this interpretation, we can write

ρmc2 =

Ωm,0ǫ0a3

(D16)

and

ǫ =Ωm,0ǫ0a3

+ u

(

Ωm,0ǫ0c2a3

)

, (D17)

where ǫ0 is the present energy density of the universe andΩm,0 is the present proportion of DM. For given P (ρm)or u(ρm) we can get ǫ(a) from Eq. (D17). We can thensolve the Friedmann equation (C1) to obtain the tempo-ral evolution of the scale factor a(t).Remark: Eqs. (D2) and (D8) determine the equation

of state P = P (ǫ). As a result, we can obtain Eq. (D16)directly from Eqs. (D2), (D8) and the energy conserva-tion equation (C4) . Indeed, combining these equationswe obtain Eq. (D15) which integrates to give Eq. (D16).

3. Two-fluid model

In the model of type II, we have a single dark fluid withan equation of state P = P (ρm). Still, the energy density(D2) is the sum of two terms, a rest-mass density termρm which mimics DM and an internal energy term u(ρm)which mimics DE. It is interesting to consider a two-fluidmodel which leads to the same results as the single darkfluid model, at least for what concerns the evolution ofthe homogeneous background. In this two-fluid model,one fluid corresponds to pressureless DM with an equa-tion of state Pm = 0 and a density ρmc

2 = Ωm,0ǫ0/a3

determined by the energy conservation equation for DM,and the other fluid corresponds to DE with an equationof state Pde(ǫde) and an energy density ǫde(a) determinedby the energy conservation equation for DE. We can ob-tain the equation of state of DE yielding the same resultsas the one-fluid model by taking

Pde = P (ρm), ǫde = u(ρm). (D18)

In other words, the equation of state Pde(ǫde) of DE in thetwo-fluid model corresponds to the relation P (u) in thesingle fluid model. Explicit examples of the correspon-dance between the one and two-fluid models are givenbelow. We note that although the one and two-fluid mod-els are equivalent for the evolution of the homogeneousbackground, they may differ for what concerns the for-mation of the large-scale structures of the Universe andfor inhomogeneous systems in general.In the two-fluid model associated with the Chaplygin

gas of type I (or III), the DE has an equation of state

Pde =2Kc2ǫdeǫ2de −Kc2

, (D19)

which is obtained by eliminating ρm between Eqs. (233)and (234), and by identifying P (u) with Pde(ǫde).

In the two-fluid model associated with the BEC of typeI, the DE has an equation of state

Pde =4Kǫ2de

[

−Kǫdec2 ±

√

(

Kǫdec2

)2+ 4Kǫde

]2 , (D20)

which is obtained by eliminating ρm between Eqs. (236)and (237), and by identifying P (u) with Pde(ǫde).In the two-fluid model associated with the ΛCDM

model, the DE has an equation of state

Pde = −ǫde. (D21)

In the two-fluid model associated with a polytrope oftype II, the DE has an equation of state

Pde = (γ − 1)ǫde, (D22)

which is obtained by eliminating ρm between Eqs. (275)and (276), and by identifying P (u) with Pde(ǫde).In the two-fluid model associated with a logotrope of

type II, the DE has an equation of state [119]

Pde = −ǫde −A, (D23)

which is obtained by eliminating ρm between Eqs. (386)and (387), and by identifying P (u) with Pde(ǫde).

Appendix E: Equation of state of type III

In this Appendix, we consider a barotropic fluid de-scribed by an equation of state of type III where thepressure P = P (ρ) is specified as a function of the pseudorest-mass density. As explained in Sec. III C this hydro-dynamic description arises naturally when considering acomplex SF with a potential V (|ϕ|2) in the TF approxi-mation. Here, we consider the case of a spatially homo-geneous complex SF in an expanding background.

1. General results

Let us first establish general results that are valid be-yond the TF approximation.We consider a spatially homogeneous complex SF in

an expanding universe with a Lagrangian

L =1

2c2|ϕ|2 − Vtot(ϕ). (E1)

Its cosmological evolution is governed by the KGF equa-tions

1

c2d2ϕ

dt2+

3H

c2dϕ

dt+ 2

dVtotd|ϕ|2ϕ = 0, (E2)

H2 =8πG

3c2ǫ. (E3)

42

The energy density ǫ(t) and the pressure P (t) of the SFare given by

ǫ ≡ T 00 =

∂L

∂ϕϕ+

∂L

∂ϕ∗ ϕ∗ − L, P ≡ −T i

i = L, (E4)

yielding

ǫ =1

2c2

∣

∣

∣

∣

dϕ

dt

∣

∣

∣

∣

2

+ Vtot(|ϕ|2), (E5)

P =1

2c2

∣

∣

∣

∣

dϕ

dt

∣

∣

∣

∣

2

− Vtot(|ϕ|2). (E6)

When the kinetic term dominates we obtain the stiffequation of state P = ǫ. When the potential term dom-inates, we obtain the equation of state P = −ǫ corre-sponding to the vacuum energy.In the following, we use the hydrodynamic representa-

tion of the SF (see Sec. III C and [147]). The Lagrangianis

L =1

2c2ρθ2 +

~2

8m2ρc2ρ2 − Vtot(ρ). (E7)

The energy density ǫ(t) and the pressure P (t) of the SFare given by

ǫ ≡ T 00 =

∂L

∂θθ +

∂L

∂ρρ− L, P ≡ −T i

i = L, (E8)

yielding

ǫ =1

2c2ρθ2 +

~2

8m2ρc2ρ2 + Vtot(ρ), (E9)

P =1

2c2ρθ2 +

~2

8m2ρc2ρ2 − Vtot(ρ). (E10)

The equation DνTµν = 0 leads to the energy conser-

vation equation

dǫ

dt+ 3H(ǫ+ P ) = 0. (E11)

This equation can also be obtained from the KG equa-tion (E2) with Eqs. (E11) and (E6) (see Appendix G).Inversely, the energy conservation equation (E11) withEqs. (E11) and (E6) implies the KG equation (E2).The equation DµJ

µ = 0, which is equivalent to thecontinuity equation (132), can be written as

d

dt

(

Etotρa3)

= 0, (E12)

where

Etot = ~ω = −mθ = −Stot (E13)

is the energy of the SF (ω = −Θ with Θ = mθ/~ is itspulsation). Eq. (E12) expresses the conservation of the

charge of the complex SF (or equivalently the conserva-tion of the boson number). It can be written as

ρEtot =Qm2c2

a3, (E14)

where Q = Ne is a constant of integration representingthe charge of the SF which is proportional to the bosonnumber N [147, 148, 151, 189–192]. Indeed, according toEq. (122), the charge of the SF is defined by36

Q =1

mc

∫

J0√−g d3x, (E15)

where J0 is the time component of the quadricurrentJµ = −ρ∂µθ (see Sec. III B). For a spatially homoge-neous SF in an expanding background, we have

J0 = −1

cρθ =

~

mcρω =

1

mcρEtot (E16)

and Q = J0a3/mc = ρEtota3/m2c2 yielding Eq. (E14).

The quantum Hamilton-Jacobi (or Bernoulli) equation(133) takes the form

E2tot = ~

2 1√ρ

d2√ρ

dt2+ 3H~

2 1√ρ

d√ρ

dt+ 2m2c2V ′

tot(ρ).

(E17)Finally, we have established in the general case (see

Sec. III B) that the rest-mass density is given by

ρm =ρ

c

√

∂µθ∂µθ. (E18)

For a spatially homogeneous SF in an expanding back-ground, we get

ρm = −ρc∂0θ = − 1

c2ρθ =

~

mc2ρω =

1

mc2ρEtot. (E19)

Using Eq. (E19), Eqs. (E12) and (E14) can be rewrittenas

dρmdt

+ 3Hρm = 0, (E20)

and

ρm =Qm

a3. (E21)

Equations (E20) and (E21) can also be obtained from thefirst law of thermodynamics for a cold fluid (T = 0) ina homogeneous background (see Appendix D). They ex-press the conservation of the particle number. Inversely,Eq. (E19) can be directly obtained from Eq. (E14) using

36 We have taken e = 1 so that the charge of the SF coincides withthe boson number.

43

Eq. (E21). Comparing Eqs. (E16) and (E19), we notethat

ρm =J0

c. (E22)

This relation is not generally valid (see Sec. III B). Inthe present case, it arises from the general identity Jµ =ρmu

µ and the fact that uµ = cδµ0 since the fluid (SF) isstatic in the expanding background.

2. TF approximation

In the TF approximation (~ → 0), the Lagrangian (E7)reduces to

L =1

2c2ρθ2 − Vtot(ρ), (E23)

the energy density ǫ(t) and the pressure P (t) of the SFreduce to

ǫ =1

2c2ρθ2 + Vtot(ρ), (E24)

P =1

2c2ρθ2 − Vtot(ρ), (E25)

and the quantum Hamilton-Jacobi (or Bernoulli) equa-tion (E17) reduces to37

E2tot = 2m2c2V ′

tot(ρ). (E26)

Combining Eqs. (E14) and (E26), we obtain

Qmc

a3= ρ√

2V ′tot(ρ). (E27)

This equation determines the relation between thepseudo rest-mass density ρ and the scale factor a. Onthe other hand, according to Eqs. (E19) and (E26), therest-mass densitity is given by

ρm =ρ

c

√

2V ′tot(ρ). (E28)

According to Eqs. (E13), (E24) and (E25) the energydensity and the pressure of the SF in the TF approxima-tion are given by

ǫ = ρV ′tot(ρ) + Vtot(ρ), (E29)

P = ρV ′tot(ρ)− Vtot(ρ). (E30)

37 For a spatially homogeneous SF, it is shown in Ref. [147] thatthe TF approximation is equivalent to the fast oscillation ap-proximation ω ≫ H.

Inversely, the SF potential is determined by the equationof state P (ρ) according to

Vtot(ρ) = ρ

∫

P (ρ)

ρ2dρ. (E31)

Equations (E28)-(E31) are always true in the TF ap-proximation even for inhomogeneous systems (see Sec.III C).For given P (ρ) or Vtot(ρ), we can obtain ρ(a) fromEq. (E27) and ǫ(a) from Eq. (E29). We can then solvethe Friedmann Eq. (E3) to obtain the temporal evolu-tion of the scale factor a(t). Actually, since it is notalways possible to invert Eq. (E27), we can proceed dif-ferently (see [147]). Taking the logarithmic derivative ofEq. (E27), we get

a

a= −1

3

ρ

ρ

[

1 +ρV ′′

tot(ρ)

2V ′tot(ρ)

]

. (E32)

Then, using Eqs. (E3) and (E29), we obtain

c2

24πG

(

ρ

ρ

)2

=ρV ′

tot(ρ) + Vtot(ρ)[

1 +ρV ′′

tot(ρ)2V ′

tot(ρ)

]2 . (E33)

For given Vtot(ρ) this is just a first order differential equa-tion that can be solved by integration.Remark: Eqs. (E29) and (E30) determine the equation

of state P = P (ǫ). As a result, we can obtain Eq. (E27)directly from Eqs. (E29), (E30) and the energy conserva-tion equation (E11). Indeed, combining these equationswe obtain

[2V ′tot(ρ) + ρV ′′

tot(ρ)]dρ

dt= −6HρV ′

tot(ρ). (E34)

leading to

∫

2V ′tot(ρ) + ρV ′′

tot(ρ)

ρV ′tot(ρ)

= −6 lna. (E35)

Eq. (E35) integrates to give Eq. (E27).

Appendix F: Analogies and differences between u

and V

For a relativistic fluid of type II, we have establishedthe identities (see Appendix D)

ǫ = ρmc2 + u(ρm), (F1)

P = ρmu′(ρm)− u(ρm), (F2)

u(ρm) = ρm

∫

P (ρm)

ρ2mdρm, (F3)

where ρm is the rest-mass density and u is the internalenergy.

44

For a relativistic fluids of type III, we have establishedthe identities (see Sec. III C)

ǫ = ρc2 + ρV ′(ρ) + V (ρ), (F4)

P = ρV ′(ρ)− V (ρ), (F5)

V (ρ) = ρ

∫

P (ρ)

ρ2dρ, (F6)

where ρ is the pseudo rest-mass density and V is thepotential of the complex SF.We note that Eqs. (F5) and (F6) are identical to Eqs.

(F2) and (F3) with ρ instead of ρm and V instead ofu. In general, the variables ρ and V are different fromthe variables ρm and u. However, they coincide in thenonrelativistic limit. For a nonrelativistic complex SF(BEC), Eqs. (F1)-(F6) reduce to

ǫ ∼ ρc2, (F7)

P = ρV ′(ρ)− V (ρ), (F8)

V (ρ) = ρ

∫

P (ρ)

ρ2dρ, (F9)

where ρ = ρm is the mass density and V = u is the poten-tial of the SF or the internal energy of the correspondingbarotropic fluid (see Appendix A).

Appendix G: Energy conservation equation for a SF

1. Complex SF

We consider a spatially homogeneous complex SF inan expanding background (see Appendix E). Taking thetime derivative of the energy density given by Eq. (E5),we get

dǫ

dt=

1

2c2d2ϕ

dt2dϕ∗

dt+ V ′

tot(|ϕ|2)dϕ

dtϕ∗ + c.c. (G1)

Using the KG equation (E2), we obtain after simplifica-tion

dǫ

dt= −3H

c2

∣

∣

∣

∣

dϕ

dt

∣

∣

∣

∣

2

. (G2)

From Eqs. (E5) and (E6) we have

ǫ+ P =1

c2

∣

∣

∣

∣

dϕ

dt

∣

∣

∣

∣

2

. (G3)

Combining Eqs. (G2) and (G3), we obtain the energyconservation equation (E11). Inversely, from Eqs. (E5),(E6) and (E11), we can directly derive the KG equation(E2).

2. Real canonical SF

We consider a spatially homogeneous real canonical SFin an expanding background (see Appendix C2). Takingthe time derivative of the energy density given by Eq.(C13), we get

dǫ

dt=d2ϕ

dt2dϕ

dt+ V ′(ϕ)

dϕ

dt. (G4)

Using the KG equation (C12), we obtain after simplifi-cation

dǫ

dt= −3Hϕ2. (G5)

From Eqs. (C13) and (C14) we have

ǫ+ P = ϕ2. (G6)

Combining Eqs. (G5) and (G6), we obtain the energyconservation equation (C4). Inversely, from Eqs. (C4),(C13) and (C14) we can directly derive the KG equation(C12).

3. Real tachyonic SF

We consider a spatially homogeneous real tachyonic SFin an expanding background (see Appendix C3). Takingthe time derivative of the energy density given by Eq.(C38), we get

dǫ

dt=

V ′(ϕ)√

1− ϕ2ϕ+

V (ϕ)

(1− ϕ2)3/2ϕϕ. (G7)

Using the field equation (C37), we obtain after simplifi-cation

dǫ

dt= −3H

V (ϕ)√

1− ϕ2ϕ2. (G8)

From Eqs. (C38) and (C39) we have

ǫ+ P =V (ϕ)

√

1− ϕ2ϕ2. (G9)

Combining Eqs. (G8) and (G9), we obtain the energyconservation equation (C4). Inversely, from Eqs. (C4),(C38) and (C39), we can directly derive the field equation(C37).

Appendix H: Some studies devoted to polytropic

and logotropic equations of state of type I, II and III

In this Appendix, we briefly mention studies devoted topolytropic and logotropic equations of state of type I, IIand III in the context of stars, DM halos and cosmology.

45

The study of nonrelativistic stars described by a poly-tropic equation of state dates back to the paper of Lane[193]. Isothermal stars were first considered by Zollner[194]. A very complete study of polytropic and isother-mal stars is presented in the books of Emden [195] andChandrasekhar [138]. Nonrelativistic logotropic starswere studied by McLaughlin and Pudritz [196]. The lo-gotropic equation of state was applied to DM halos byChavanis [116, 117, 141].

General relativistic stars described by a polytropicequation of state of type I were first considered by Tooper[149]. Polytropes of type I with index γ = 2 were specif-ically studied by Chavanis and Harko [152, 153] in re-lation to general relativistic BEC stars (however, this isnot the correct equation of state for these systems – seebelow). General relativistic stars described by a linearequation of state, extending the models of Newtonianisothermal stars, were studied by Chandrasekhar [155](see also [156–159] and references therein). Cosmologicalmodels based on a polytropic equation of state of type Iwith an arbitrary index γ were studied in [97–99]. Thespecific index γ = −1 corresponds to the Chaplygin gas[50, 86] and the indices −1 ≤ γ ≤ 0 correspond to theGCG [95].

General relativistic stars described by a polytropicequation of state of type II were first considered byTooper [164]. Polytropes of type II with index γ = 2were specifically studied by Chavanis [152] and Latifahet al. [197] in relation to general relativistic BEC stars(however, this is not the correct equation of state forthese systems – see below). Cosmological models basedon a polytropic equation of state of type II with an ar-bitrary index γ were studied in [165] (the index γ = 2of a BEC is specifically treated in the main text of [165]and the case of a general index is treated in Appendix Dof [165]). A cosmological model based on the logotropicequation of state of type II was studied in [117].

General relativistic stars described by a polytropicequation of state of type III were studied by Colpi et al.[150] and Chavanis and Harko [152, 153] for the particularindex γ = 2 corresponding to BECs. This is the hydrody-namic representation, valid in the TF regime, of a com-plex SF with a repulsive |ϕ|4 self-interaction described bythe KGE equations [150]. Therefore, a polytropic equa-tion of state of type III with index γ = 2, leading to theequation of state (344), is the correct equation of state ofa relativistic BEC with a quartic self-interaction in theTF regime. Cosmological models based on a polytropicequation of state of type III with an arbitrary index γwere studied in [147, 154] (the index γ = 2 of a BECis specifically treated in the main text of [147, 154] andthe case of an arbitrary index is treated in Appendix I of[147] and in [154]). This is the hydrodynamic represen-tation, valid in the TF regime or in the fast oscillationregime, of a complex SF with a |ϕ|4 potential describedby the KGE equations [151]. A cosmological model basedon the logotropic equation of state of type III has beenstudied recently in [116].

Appendix I: Conservation laws for a nonrelativistic

SF

In this Appendix, we establish the local conservationlaws of mass, impulse and energy for a nonrelativistic SF(see Sec. II).

1. Conservation laws in terms of hydrodynamic

variables

The equation of continuity (20) can be written as

∂ρ

∂t+∇ · J = 0, (I1)

where ρ is the mass density and

J = ρu (I2)

is the density current. This equation expresses the localconservation of mass M =

∫

ρ dr.Using the continuity equation (20), the quantum Euler

equation (24) can be rewritten as

∂

∂t(ρu) +∇(ρu⊗ u) +∇P +

ρ

m∇Q = 0. (I3)

On the other hand, the quantum force can be writtenunder the form (see Sec. 2.5 of [135])

− ρ

m∂iQ = −∂jPQ

ij , (I4)

where the anisotropic quantum pressure tensor PQij is

given by

PQij = − ~

2

4m2ρ∂i∂j ln ρ =

~2

4m2

(

1

ρ∂iρ∂jρ− ∂i∂jρ

)

(I5)

or, alternatively, by

PQij =

~2

4m2

(

1

ρ∂iρ∂jρ− δij∆ρ

)

. (I6)

Substituting Eq. (I4) into Eq. (I3), we obtain

∂

∂t(ρu) +∇(ρu⊗ u) +∇P + ∂jP

Qij = 0. (I7)

Introducing the momentum density

−Ti0 = ρu, (I8)

we can rewrite Eq. (I7) as

−∂Ti0∂t

+ ∂jTij = 0, (I9)

where

Tij = ρuiuj + Pδij + PQij (I10)

46

is the stress tensor. Using Eqs. (I5) and (I6), we get

Tij = ρuiuj + Pδij −~2

4m2ρ∂i∂j ln ρ

= ρuiuj + Pδij +~2

4m2

(

1

ρ∂iρ∂jρ− ∂i∂jρ

)

(I11)

or, alternatively,

Tij = ρuiuj +

[

P (ρ)− ~2

4m2∆ρ

]

δij +~2

4m2

1

ρ∂iρ∂jρ.

(I12)Eq. (I9) expresses the local conservation of the momen-tum P =

∫

ρu dr.Introducing the energy density

T00 = ρe = ρu2

2+ ρ

Q

m+ V (ρ) (I13)

and combining the equation of continuity (20) and thequantum Euler equation (24), we obtain (see AppendixE of [135])

∂

∂t(ρe) +∇ · (ρeu) +∇ · (Pu) +∇ · JQ = 0, (I14)

where

JQ =~2

4m2ρ∂∇ ln ρ

∂t(I15)

is the quantum current. Introducing the energy current

−T0i = ρeu+ Pu+ JQ

= ρ

[

u2

2+Q

m+V (ρ) + P

ρ

]

u+ JQ

= ρ

[

u2

2+Q

m+ V ′(ρ)

]

u+ JQ, (I16)

where we have used Eq. (26) to obtain the last equality,we can rewrite Eq. (I14) as

∂T00∂t

− ∂iT0i = 0. (I17)

This equation expresses the local conservation of energyE =

∫

ρe dr. We also recall that h(ρ) = V ′(ρ) is theenthalpy.For classical systems (~ = 0), or for BECs in the TF

limit, the foregoing equations reduce to

T00 = ρe = ρu2

2+ V (ρ), (I18)

−T0i = ρ

[

u2

2+ V ′(ρ)

]

u, (I19)

−Ti0 = ρu, (I20)

Tij = ρuiuj + Pδij . (I21)

Remark: We note that T0i 6= Ti0 because the theoryis not Lorentz invariant. By contrast, Tij = Tji becausethe theory is invariant against spatial rotations. We alsonote that the momentum density is equal to the massflux:

−Ti0 = ρu = J. (I22)

2. Conservation laws in terms of the wave function

Using Eqs. (17)-(19), the density current (I2) can beexpressed in terms of the wave function as

J =~

2im(ψ∗∇ψ − ψ∇ψ∗) . (I23)

As a result, the equation of continuity (I1) takes the form

∂|ψ|2∂t

+~

2im∇ · (ψ∗∇ψ − ψ∇ψ∗) = 0. (I24)

Similarly, the momentum density (I8) and the stresstensor (I12) can be written in terms of the wave functionas (see Appendix A of [135])

−Ti0 = J =~

2im(ψ∗∇ψ − ψ∇ψ∗) (I25)

and

Tij =~2

m2Re

(

∂ψ

∂xi

∂ψ∗

∂xj

)

+

[

P (|ψ|2)− ~2

4m2∆|ψ|2

]

δij

(I26)or, alternatively,

Tij =~2

2m2Re

(

∂ψ

∂xi

∂ψ∗

∂xj− ψ

∂2ψ∗

∂xi∂xj

)

+ Pδij . (I27)

Finally, the energy density (I13) and the energy current(I16) can be written in terms of the wave function as

T00 =~2

2m2|∇ψ|2 + V (|ψ|2) (I28)

and

−T0i =[

~2

2m2

|∇ψ|2|ψ|2 + V ′(|ψ|2)

]

J+ JQ (I29)

with

JQ =~2

4m2|ψ|2 ∂∇ ln |ψ|2

∂t. (I30)

3. Conservation laws from the Lagrangian

expressed in terms of the wave function

The current of a complex SF is given by

Jµ =m

i~

[

ψ∂L

∂(∂µψ)− ψ∗ ∂L

∂(∂µψ∗)

]

. (I31)

47

For the Lagrangian (33) we obtain the mass density

J0 = |ψ|2 = ρ (I32)

and the mass flux

J = − i~

2m(ψ∗∇ψ − ψ∇ψ∗). (I33)

The energy-momentum tensor of a complex SF is givenby

T νµ = ∂µψ

∂L∂(∂νψ)

+ ∂µψ∗ ∂L∂(∂νψ∗)

− Lδνµ. (I34)

For the Lagrangian (33) we obtain the energy density

T00 =~2

2m2|∇ψ|2 + V (|ψ|2), (I35)


−Ti0 = − i~

2m(ψ∗∇ψ − ψ∇ψ∗) = J, (I36)

the energy flux

−T0i = − ~2

2m2(ψ∇ψ∗ + ψ∗∇ψ), (I37)

and the momentum fluxes (stress tensor)

Tij =~2

m2Re(∂iψ∂jψ

∗) + Lδij . (I38)

These are their general expressions. If we use the GPequation (16), which is obtained after extremizing theaction, we can rewrite Eq. (I37) as Eq. (I29). Similarly,if we use the expression (46) of the Lagrangian which alsorelies on the GP equation, we can rewrite Eq. (I38) asEq. (I26). In this manner, we recover the equations ofAppendix I 2 (up to terms that vanish by integration).Remark: The energy density is

T00 = ψ∂L∂ψ

+ ψ∗ ∂L∂ψ∗

− L

= πψ + π∗ψ∗ − L

=i~

2m

(

ψ∗ψ − ψψ∗)

− L, (I39)

where π = ∂L/∂ψ = i~2mψ

∗ is the conjugate momentumto ψ. This leads to Eqs. (35) and (36). On the otherhand, Eq. (37) can be rewritten in the form of Hamiltonequations

∂ψ

∂t=

1

2

δH

δπ,

∂π

∂t= −1

2

δH

δψ. (I40)

They are equivalent to the GP equation (16) and its com-plex conjugate.

4. Conservation laws from the Lagrangian

expressed in terms of hydrodynamic variables

The current of a complex SF in its hydrodynamic rep-resentation is given by

Jµ = −m ∂L∂(∂µS)

. (I41)

For the Lagrangian (40) we obtain the mass density

J0 = ρ (I42)

and the mass flux

J = ρ∇Sm

= ρu. (I43)

The energy-momentum tensor of a complex SF in itshydrodynamic representation is given by

T νµ = ∂µρ

∂L∂(∂νρ)

+ ∂µS∂L

∂(∂νS)− Lδνµ. (I44)

For the Lagrangian (40) we obtain the energy density

T00 =ρ

2m2(∇S)2 + ~

2

8m2

(∇ρ)2ρ

+ V (ρ)

=1

2ρu2 + ρ

Q

m+ V (ρ), (I45)


−Ti0 =ρ

m∇S = ρu = J, (I46)

the energy flux

−T0i = −∂ρ∂t

~2

4m2

∇ρρ

− ∂S

∂t

ρ

m2∇S, (I47)

and the momentum fluxes

Tij =~2

4m2

1

ρ∂iρ∂jρ+

ρ

m2∂iS∂jS + Lδij

= ρuiuj +~2

4m2

1

ρ∂iρ∂jρ+ Lδij . (I48)

These are their general expressions. If we use the quan-tum Hamilton-Jacobi (or Bernoulli) equation (21), whichis obtained after extremizing the action, we can rewriteEq. (I47) as Eq. (I16). Similarly, if we use the expression(46) of the Lagrangian which also relies on the quantumHamilton-Jacobi (or Bernoulli) equation, we can rewriteEq. (I48) as Eq. (I12). In this manner, we recover theequations of Appendix I 1 (up to terms that vanish byintegration).Remark: The energy density is

T00 = S∂L∂S

− L = πS − L = − ρ

mS − L, (I49)

48

where π = ∂L/∂S = −ρ/m is the conjugate momentumto S. This leads to Eqs. (43) and (44). On the otherhand, Eq. (45) can be rewritten in the form of Hamiltonequations

∂S

∂t=δH

δπ,

∂π

∂t= −δH

δS. (I50)

They are equivalent to the continuity equation (20) andto the quantum Hamilton-Jacobi (or Bernoulli) equation(21).

Appendix J: Conservation laws for a relativistic

complex SF

In this Appendix, we establish the local conservationlaws of boson number (charge), impulse and energy for arelativistic complex SF (see Sec. III). For simplicity, weconsider flat metric and a static background.

1. Relativistic Lagrangian in terms of the wave

function

The density Lagrangian of a complex SF is [see Eq.(109)]

L =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

− 1

2|∇ϕ|2 − Vtot(|ϕ|2). (J1)

The components of the current (119) or (120) are

J0 = − m

2i~c

(

ϕ∗ ∂ϕ

∂t− ϕ

∂ϕ∗

∂t

)

, (J2)

J i =m

2i~(ϕ∗∂iϕ− ϕ∂iϕ

∗) . (J3)

The local conservation of the boson number (charge) canbe written as

1

c

∂J0

∂t+ ∂iJ

i = 0. (J4)

The components of the energy-momentum tensor (115)are

T 00 =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

+1

2|∇ϕ|2 + Vtot(|ϕ|2), (J5)

T 0i = − 1

2c

∂ϕ∗

∂t∂iϕ− 1

2c

∂ϕ

∂t∂iϕ

∗, (J6)

T ij =1

2∂iϕ

∗∂jϕ+1

2∂iϕ∂jϕ

∗ + δijL. (J7)

We note that T ii = |∇ϕ|2 + 3L. The conservation of theenergy-momentum tensor [see Eq. (117)] can be writtenas

1

c

∂T 00

∂t+ ∂iT

0i = 0, (J8)

1

c

∂T i0

∂t+ ∂jT

ij = 0. (J9)

The Euler-Lagrange equation (111) yields the KGequation

1

c2∂2ϕ

∂t2−∆ϕ+ 2

dVtotd|ϕ|2ϕ = 0 (J10)

which displays the d’Alembertian operator = 1c2

∂2

∂t2 −∆.In the homogeneous case, the foregoing equations re-

duce

L =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

− Vtot(|ϕ|2), (J11)

T 00 =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

+ Vtot(|ϕ|2), (J12)

T 0i = 0, (J13)

T ij = δijL, (J14)

1

c2∂2ϕ

∂t2+ 2

dVtotd|ϕ|2ϕ = 0. (J15)

The energy-momentum tensor is diagonal T µν =diag(ǫ, P, P, P ). The energy density and the pressure aregiven by

ǫ =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

+ Vtot(|ϕ|2), (J16)

P =1

2c2

∣

∣

∣

∣

∂ϕ

∂t

∣

∣

∣

∣

2

− Vtot(|ϕ|2). (J17)

To obtain the nonrelativistic limit, we first make theKlein transformation (14) and use Eq. (110). We get

L =~2

2m2c2

∣

∣

∣

∣

∂ψ

∂t

∣

∣

∣

∣

2

+i~

2m

(

ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

)

− ~2

2m2|∇ψ|2 − V (|ψ|2), (J18)

J0 = − ~

2icm

(

ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t− 2imc2

~|ψ|2

)

, (J19)

49

J i =~

2im(ψ∗∂iψ − ψ∂iψ

∗) , (J20)

T 00 =~2

2m2c2

∣

∣

∣

∣

∂ψ

∂t

∣

∣

∣

∣

2

+~2

2m2|∇ψ|2 + c2|ψ|2 + V (|ψ|2)

+i~

2m

(

∂ψ

∂tψ∗ − ψ

∂ψ∗

∂t

)

, (J21)

T 0i = − ~2

2m2c

(

∂ψ∗

∂t∂iψ +

∂ψ

∂t∂iψ

∗)

− i~c2m

(ψ∗∂iψ − ψ∂iψ∗) ,

(J22)

T ij =~2

m2Re (∂iψ∂jψ

∗) + δijL. (J23)

If we take the limit c → +∞ in Eq. (J18) we recoverEq. (33). If we divide Eq. (J19) by c and take the limitc → +∞, we obtain J0/c = |ψ|2 leading to Eq. (I32).Equation (J20) is equivalent to Eq. (I33). To leadingorder, Eq. (J21) gives T 00 ∼ ρc2. If we subtract therest mass term cJ0 (see Ref. [143]) and take the limitc → +∞ in Eq. (J21), we recover Eq. (I35). If wemultiply or divide Eq. (J22) by c and consider the termsthat are independent of c (see Ref. [143]) we get

T 0i

c= − i~

2m(ψ∗∂iψ − ψ∂iψ

∗) , (J24)

T 0ic = − ~2

2m2

(

∂ψ∗

∂t∂iψ +

∂ψ

∂t∂iψ

∗)

. (J25)

This returns Eqs. (I36) and (I37). Equation (J23) re-turns Eq. (I38). Finally, the KG equation (J10) becomes

i~∂ψ

∂t− ~

2

2mc2∂2ψ

∂t2+

~2

2m∆ψ −m

dV

d|ψ|2ψ = 0. (J26)

In the nonrelativistic limit c → +∞, we recover the GPequation (16).Remark: We note that the energy-momentum tensor

is symmetric in relativity theory (T µν = T νµ) while it isnot symmetric in Newtonian theory (T0i 6= Ti0). This isbecause space and time are not treated on equal footingin Newtonian theory. This is also why we have to considerthe two terms T 0i/c and T 0ic in the nonrelativistic limit[see Eqs. (J24) and (J25)].

2. Relativistic Lagrangian in terms of

hydrodynamic variables

The density Lagrangian of a complex SF in its hydro-dynamic representation is [see Eq. (128)]

L =~2

8ρm2c2

(

∂ρ

∂t

)2

+1

2c2ρ

(

∂θ

∂t

)2

− ~2

8ρm2(∇ρ)2 − 1

2ρ(∇θ)2 − Vtot(ρ). (J27)

The components of the current (142) are

J0 = −ρc

∂θ

∂t, J i = ρ∂iθ. (J28)

The components of the energy-momentum tensor (139)are

T 00 =~2

8ρm2c2

(

∂ρ

∂t

)2

+1

2c2ρ

(

∂θ

∂t

)2

+~2

8ρm2(∇ρ)2 + 1

2ρ(∇θ)2 + Vtot(ρ), (J29)

T 0i = −1

cρ∂θ

∂t∂iθ −

~2

4ρm2c

∂ρ

∂t∂iρ, (J30)

T ij =~2

4ρm2∂iρ∂jρ+ ρ∂iθ∂jθ + δijL. (J31)

The Euler-Lagrange equations (130) and (131) yieldthe continuity equation

1

c2∂

∂t

(

ρ∂θ

∂t

)

−∇ · (ρ∇θ) = 0 (J32)

and the quantum Hamilton-Jacobi (or Bernoulli) equa-tion

1

2c2

(

∂θ

∂t

)2

− 1

2(∇θ)2 − Q

m− V ′

tot(ρ) = 0, (J33)

with the quantum potential

Q ≡ ~2

2m

√ρ

√ρ

=~2

4ρmc2∂2ρ

∂t2− ~

2

8ρ2mc2

(

∂ρ

∂t

)2

− ~2

4ρm∆ρ+

~2

8ρ2m(∇ρ)2. (J34)

According to Eq. (135), the pseudo energy and thepseudo velocity are

v0 = v0 = −1

c

∂θ

∂t, vi = −vi = ∂iθ. (J35)

Recalling that θ = Stot/m, we get

Etot = −∂Stot

∂t, v =

∇Stot

m. (J36)

We also have

J0 =ρEtot

mc, J = ρv. (J37)

The continuity equation (J32) and the quantumHamilton-Jacobi (or Bernoulli) equation (J33) can berewritten as

∂

∂t

(

ρEtot

mc2

)

+∇ · (ρv) = 0 (J38)

50

and

E2tot

2m2c2− v

2

2− Q

m− V ′

tot(ρ) = 0. (J39)

Taking the gradient of Eq. (J39), we obtain the Eulerequation

Etot

mc2∂v

∂t+ (v · ∇)v = − 1

m∇Q− 1

ρ∇P. (J40)

In the homogeneous case, the foregoing equations re-duce to

L =~2

8ρm2c2

(

dρ

dt

)2

+1

2c2ρ

(

dθ

dt

)2

− Vtot(ρ), (J41)

T 00 =~2

8ρm2c2

(

dρ

dt

)2

+1

2c2ρ

(

dθ

dt

)2

+ Vtot(ρ), (J42)

T 0i = 0, (J43)

T ij = δijL, (J44)

d

dt

(

ρdθ

dt

)

= 0, (J45)

1

2c2

(

dθ

dt

)2

− Q

m− V ′

tot(ρ) = 0, (J46)

with the quantum potential

Q ≡ ~2

2mc2√ρ

d2√ρ

dt2=

~2

4ρmc2d2ρ

dt2− ~

2

8ρ2mc2

(

dρ

dt

)2

.

(J47)The energy-momentum tensor is diagonal T µν =diag(ǫ, P, P, P ). The energy and the pressure are givenby

ǫ =~2

8ρm2c2

(

dρ

dt

)2

+1

2c2ρ

(

dθ

dt

)2

+ Vtot(ρ), (J48)

P =~2

8ρm2c2

(

dρ

dt

)2

+1

2c2ρ

(

dθ

dt

)2

− Vtot(ρ). (J49)

To obtain the nonrelativistic limit, we first make theKlein transformation (see Sec. III E)

mθ = S −mc2t, (J50)

and use Eq. (129). We get

L =~2

8ρm2c2

(

∂ρ

∂t

)2

+ρ

2m2c2

(

∂S

∂t

)2

− ρ

m

∂S

∂t

− ~2

8ρm2(∇ρ)2 − ρ

2m2(∇S)2 − V (ρ), (J51)

J0 = − ρ

mc

∂S

∂t+ ρc, (J52)

J i = ρ∂iS

m= ρui, (J53)

T 00 =~2

8ρm2c2

(

∂ρ

∂t

)2

+ρ

2m2c2

(

∂S

∂t

)2

+ ρc2 − ρ

m

∂S

∂t

+~2

8ρm2(∇ρ)2 + ρ

2m2(∇S)2 + V (ρ), (J54)

T 0i = ρ∂iS

mc− ρ

m2c

∂S

∂t∂iS − ~

2

4ρm2c

∂ρ

∂t∂iρ, (J55)

T ij =~2

4ρm2∂iρ∂jρ+

ρ

m2∂iS∂jS + δijL. (J56)

If take the limit c → +∞ in Eq. (J51) we recover Eq.(40). If we divide Eq. (J52) by c and take the limitc → +∞, we obtain J0/c = ρ leading to Eq. (I42).Equation (J53) is equivalent to Eq. (I43). To leadingorder, Eq. (J54) gives T 00 ∼ ρc2. If we subtract therest mass term cJ0 (see Ref. [143]) and take the limitc → +∞ in Eq. (J54), we recover Eq. (I45). If wemultiply or divide Eq. (J55) by c and consider the termsthat are independent of c (see Ref. [143]) we get

T 0i

c= ρ

∂iS

m, (J57)

T 0ic = − ρ

m2

∂S

∂t∂iS − ~

2

4ρm2

∂ρ

∂t∂iρ. (J58)

This returns Eqs. (I46) and (I47). Equation (J56) re-turns Eq. (I48). Finally, the continuity equation (J32)and the quantum Hamilton-Jacobi (or Bernoulli) equa-tion (J33) become

− 1

mc2∂

∂t

(

ρ∂S

∂t

)

+∂ρ

∂t+∇ ·

(

ρ∇Sm

)

= 0 (J59)

and

− 1

2mc2

(

∂S

∂t

)2

+∂S

∂t+

1

2m(∇S)2 +Q+mV ′(ρ) = 0.

(J60)Using the identities

Stot = S −mc2t, Etot = E +mc2, (J61)

E = −∂S∂t, u =

∇Sm

, (J62)

J0 =ρE

mc+ ρc, J = ρu, (J63)

51

they can be rewritten as

1

mc2∂

∂t(ρE) +

∂ρ

∂t+∇ · (ρu) = 0 (J64)

and

− E2

2mc2− E +

1

2mu

2 +Q+mV ′(ρ) = 0. (J65)

Taking the gradient of Eq. (J65) we obtain the Eulerequation

∂u

∂t+(u ·∇)u = −1

ρ∇P − 1

m∇Q+

1

2m2c2∇(E2). (J66)

In the limit c → +∞, we recover Eqs. (20), (21) and(24).

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