New Early Dark Energy

Florian Niedermann1, ∗ and Martin S. Sloth1, †

1CP3-Origins, Center for Cosmology and Particle Physics PhenomenologyUniversity of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark

New measurements of the expansion rate of the Universe have plunged the standard model ofcosmology into a severe crisis. In this letter, we propose a simple resolution to the problem thatrelies on a first order phase transition in a dark sector in the early Universe, before recombination.This will lead to a short phase of a New Early Dark Energy (NEDE) component and can explainthe observations. We model the false vacuum decay of the NEDE scalar field as a sudden transitionfrom a cosmological constant source to a decaying fluid with constant equation of state. Thecorresponding fluid perturbations are covariantly matched to the adiabatic fluctuations of a sub-dominant scalar field that triggers the phase transition. Fitting our model to measurements of thecosmic microwave background (CMB), baryonic acoustic oscillations (BAO, and supernovae (SNe)yields a significant improvement of the best-fit compared with the standard cosmological modelwithout NEDE. We find the mean value of the present Hubble parameter in the NEDE model to beH0 = 71.4 ± 1.0 km s−1 Mpc−1 (68 % C.L.).

PACS numbers: 98.80.Cq,98.80.-k

INTRODUCTION

Recent measurements of the expansion of the Universehave led to an apparent crisis for the standard model ofcosmology, the ΛCDM model. Within the ΛCDM model,we can calculate the evolution of the Universe from theearliest times until today, and until recently all our mea-surements were consistent with the model. In particular,we can use the measurements of the CMB radiation toinfer the present value of the Hubble parameter H0. Ifthe ΛCDM model is correct, this value will have to agreewith the value obtained by directly measuring the expan-sion rate today using supernovae redshift measurements.Now, the problem is that the measurements, direct andindirect, do not agree, and this puts the ΛCDM modelin a crisis.

The most precise measurements we have of the temper-ature fluctuations, polarization and lensing in the CMBradiation are from the Planck satellite, which, assum-ing the ΛCDM model, infers the value of the expansionrate today to be H0 = 67.36 ± 0.54 km s−1 Mpc−1 [1].Comparing that with the expansion rate measured fromCepheids-calibrated supernovae by the SH0ES team [2],H0 = 74.03±1.42 km s−1 Mpc−1, there is a 4.4 σ discrep-ancy. Other measurements of the current Hubble rate,such as H0LiCoW [3], are also significantly discrepantwith the Planck measurement [4].

The Planck measurement of the CMB is a very cleanexperiment with the systematics well under control, andit is therefore unlikely that there is non-understood sys-tematics in the CMB measurements that can explainthe discrepancy. The local supernova observations, onthe other hand, involves astronomical distance measure-ments, which are notoriously difficult, and have beenplagued by non-understood systematic errors in the past.Various possible sources of systematics have been consid-

ered extensively in the literature already [5–8]. So far,astronomers have no commonly accepted idea of pos-sible systematic effects to explain the discrepancy, andan often echoed conclusion is that new physics beyondthe ΛCDM model is required to resolve the tension (seee.g., [9, 10]). While it is important to continue to look forpossible systematic effects, in the present paper, we willrather consider a simple solution in terms of new physics.

We will study the possibility that a first order phasetransition in a dark sector at zero temperature happenedshortly before recombination in the early Universe. Sucha phase transition will have the effect of lowering an ini-tially high value of the cosmological constant in the earlyUniverse down to the value today, inferred from the mea-surement of H0. Effectively this means that there hasbeen an extra component of dark energy in the earlyUniverse, providing a short burst of additional repulsion.Currently, an extra component of Early Dark Energy(EDE) seems to be a promising way to resolve the tensionbetween the early and late measurements of H0 [10–17].So far, people have typically considered a dynamical EDEcomponent that disappears due to a second order phasetransition of a slowly rolling scalar field.1 Such scenar-ios have complications if monomial potentials are usedboth at background and perturbative level [13, 20], asone needs the potential to be steep and anharmonic atthe bottom to end up with a sufficiently stiff fluid butalso flat initially to achieve a sound speed c2s < 0.9 fora large enough range of sub-horizon modes. While thisproblem can be overcome by using specific terms fromthe non-perturbative form of the axion potential [11–13],it represents a non-generic choice [14].2

1 The general idea of having an early dark energy component isolder and dates back to [18, 19].

2 For other proposals to address the Hubble tension operational at

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On the other hand, we believe that a first order phasetransition holds in it the potential to fully resolve thediscrepancy between the early and late measurements ofH0 much more naturally. In addition, a first order phasetransition will lead to different experimental signaturesin the details of the CMB and large-scale structure aswell as gravitational waves.

Below we explore the simplest NEDE model. For moredetails and generalizations of the model, as well as a de-tailed comparison with other models, we refer the readerto our longer subsequent paper [46].

THE MODEL

In order to have a change in the vacuum energy due to afield that undergoes a first order phase transition, we willconsider a scalar field with two non-degenerate minima atzero temperature. However, if the tunneling probabilityfrom the false to the true vacuum is initially high, thefield will tunnel immediately and NEDE never makes asizable contribution. On the other hand, once tunnelingcommences, we need a large rate in order to produceenough bubbles of true vacuum that will quickly collide.If the rate is too small, then part of the Universe will be inthe true and part of it in the false vacuum, which will leadto large inhomogeneities ruled out by observations. Wetherefore require an additional sub-dominant trigger fieldthat, at the right moment, makes the tunneling rate veryhigh. Analogous to previously considered mechanismsfor ending inflation in [47–50], we will therefore considermodels with a general potential of the form,

V (ψ, φ) =λ

4ψ4 +

1

2βM2ψ2 (1)

−1

3αMψ3 +

1

2m2φ2 +

1

2λφ2ψ2 ,

where ψ is the tunneling field and φ is the trigger field.The sub-dominant trigger field will be frozen as long as itsmass is smaller than the Hubble rate, but as soon as theHubble rate drops below its mass, it will start decayingand this will trigger the tunneling of the ψ field. For asecond minimum to develop after the point of inflection,we need to impose α2 > 4βλ, β > 0. In Fig. 1, we showa 3D visualization of the evolution of the potential asthe trigger field, φ, starts evolving along the orange pathopening up the new vacuum for ψ, to which it tunnelswith high probability.

The decay rate per unit volume is Γ = K exp (−SE),where K is a determinant factor which is generically setby the energy scale of the phase transition [51, 52] andSE is the Euclidian action corresponding to a so-called

late and/or early times see [21–45].

FIG. 1. Schematic plot of the two-field potential in (1). ForH <∼ m, the field rolls along the orange line correspondingto ψ = 0. At the inflection point (blue dot) the potential(in ψ direction) develops a second minimum which becomesdegenerate shortly after (orange dot). The nucleation prob-ability increases towards φ = 0 (red dot). The true vacuumcorresponds to the white dot.

bounce solution [53]. While it is possible to find an an-alytic expression in the thin wall limit for a single field,the general case requires a numerical approach. In [46]we argue that a good approximation of the Euclidianaction (describing the potential as being effectively one-dimensional) can be written as

SE ≈4π2

3λ(2− δeff)

−3 (α1δeff + α2δ

2eff + α3δ

3eff

), (2)

with numerically determined coefficients [54] α1 =13.832, α2 = −10.819, α3 = 2.0765 and

δeff(t) = 9λ

α2

(β + λ

φ2(t)

M2

). (3)

We see that SE becomes large as δeff → 2 and vanishesas δeff → 0. As a result, the tunneling rate is suppressedwhen φ is frozen at a sufficiently large initial field value(corresponding to δeff > 9/4 ∼ 2) and becomes maximalas φ→ 0 once the Hubble drag is released (correspondingto δeff → 9λβ/α2 < 9/4).

At early times, we require the transition rate to behighly suppressed, which fixes the initial value of the trig-ger field, φini, and can be satisfied consistently with thecondition that φini/Mpl � 1, which is sufficient to ensurethat the contribution of φ to the total energy density issub-dominant.

Now, we also have to ensure that NEDE, given by thepotential energy in the ψ field, gives a sizable contri-bution to the energy budget at the time t∗ where bub-ble percolation of the ψ vacuum becomes efficient. Wecan quantify it in terms of the ratio fNEDE = ∆V/ρ(t∗),where ∆V is the liberated vacuum energy and ρ the totalenergy density. If the transition occurs at a redshift oforder z ∼ 5000, λ ∼ 0.1, α ∼ β ∼ O(1) and fNEDE ∼ 0.1,we have M ∼ eV and an ultra-light mass scale of order

3

m ∼ 10−27eV. A microphysical model explaining themass hierarchy between the M and the m scale wouldbe a model of axion monodromy with two axion fields(see [55] for a field theory version). Here, the masses areprotected by softly broken shift symmetries.

We also have to make sure that the nucleation itselfhappens sufficiently quickly. To that end, we definethe percolation parameter p = Γ/H4 ∼ M4/m4 e−SE ,where we approximated K ∼ M4. Provided p � 1, alarge number of bubbles is nucleated within one Hub-ble patch and one Hubble time. In fact, for the abovechoice of parameters, the huge hierarchy between thescalar masses, M4/m4 ∼ 10108, implies that p � 1 onlyrequires SE < 250, which according to (2) and (3) canbe easily satisfied as φ → 0. This means that percola-tion is extremely efficient and will cover the entire spacewith bubbles of true vacuum in a tiny fraction of a Hub-ble time. Therefore, we can treat it as an instantaneousprocess on cosmological time scales, which takes place attime t∗.

As the space is being filled with bubbles of true vac-uum, they expand and start to collide when they are ofphysical size today � Mpc. Thus, they do not induceanisotropies on scales large enough to be probed usingCMB measurements. This phase is governed by compli-cated dynamics, which can be studied analytically only insimplified two-bubble scenarios as in [56]. As part of thecollision process, the complicated ψ condensate starts todecay. Microscopically, the released free energy gets con-verted into anisotropic stress on small scales, which weexpect, after partially being converted to gravitationalradiation, to decay as 1/a6, similar to a stiff fluid com-ponent. We leave it for future work to substantiate thispicture, which assumes a decoupling of small and largescales, through explicit numerical studies.

MATCHING CONDITIONS

We use a simple background model describing the in-stantaneous transition from a background fluid with anequation of state (e.o.s.) parameter that changes from −1to w∗NEDE,

wNEDE(t) =

{−1 for t < t∗ ,

w∗NEDE for t > t∗ ,(4)

where the transition happens at time t∗. In terms of ourfield theory model in (1), this corresponds to a situationwhere all of the liberated vacuum energy is transferred toa fluid with e.o.s. parameter w∗NEDE, and where accordingto the considerations above, we expect 1/3 ≤ w∗NEDE ≤ 1.Describing the bubble wall condensate in terms of afluid with a constant w∗NEDE is a simplifying assump-tion, which should ultimately be tested with lattice field

theory techniques resolving the scalar field ψ and its per-turbations explicitly.3

Background Matching

The above condition fixes the evolution of the back-ground energy density uniquely,

ρNEDE(t) = ρ∗NEDE

(a(t∗)

a(t)

)3[1+wNEDE(t)]

, (5)

where ρ∗NEDE = fNEDE ρ∗ = const. The energy densityof NEDE is normalized with respect to the true vacuumand continuous across the transition. The discontinuityof a time dependent function f(t) across the transitionsurface at time t∗ is denoted as

[f ]± = limε→0

[f(t∗ − ε)− f(t∗ + ε)] ≡ f (−) − f (+) . (6)

Applying this operation to the Friedmann equations, wethen find

[H]± = 0 , (7a)[H]±

= 4πG(1 + w∗NEDE)ρ∗NEDE , (7b)

where we used the continuity of the background energydensity, [ρ]± = 0, which holds due to (5) and the instan-taneous character of the transition. The derivation of(7b) also assumes that the e.o.s. of all other fluid com-ponents (except for NEDE) is preserved during the tran-sition. Besides the NEDE component, we also track theevolution of the sub-dominant field φ to turn on the phasetransition.

Perturbation Matching

Before the decay we can set the perturbations of theNEDE fluid to zero as it behaves as a (non-fluctuating)cosmological constant. This raises the issue of how toinitialize perturbations at time t∗. Moreover, since thetransition is allowed to happen at a relatively late stagein the evolution of the primordial plasma (in the extremecase right before recombination), we cannot assume thatall relevant modes are outside the horizon. In the specific

3 The importance of tracking field perturbations was highlightedrecently in the case of the second-order, single-field EDE modelin [20], although after the NEDE transition, the frequency ofthe fluctuations is much higher, and the course grained fluiddescription is expected to be a better approximation.

4

case of our two-field model, we use the trigger field todefine the transition surface Σ, explicitly,

φ(t∗,x)|Σ = const . (8)

This is motivated by the φ dependence of the param-eter δeff in (3) which controls the exponential in thetunneling rate through (2). As a consequence, fluc-tuations in φ lead to spatial variations of the timet∗ at which the decay takes place. These variations,δφ(t∗,x)= φ(t∗,x)− φ(t∗), then provide the initial con-ditions for the fluctuations in the NEDE fluid after thephase transition.

In order to match the conventions used in the Boltz-mann code community, we work in synchronous gauge,

ds2 = −dt2 + a(t)2 (δij + hij) dxidxj , (9)

where in momentum space

hij =kikjk2

h+

(kikjk2− 1

3δij

)6η , (10)

and h = δijhij . In the following we will make use of theequations for the metric perturbations that are first orderin time derivatives [57],

1

2Hh− k2

a2η = 4πGδρ , (11a)

k2

a2η = 4πG (ρ+ p)

θ

a, (11b)

where (ρ+ p) θ =∑i (ρi + pi) θi and δρ =

∑i δρi are

the total divergence of the fluid velocity and the total en-ergy density perturbation, respectively. The dynamicalequations have to be supplemented with Israel’s match-ing conditions [58, 59]. They relate the time derivativesof η and h before and after the transition,

[h]±

= −6 [η]± = 6[H]±

δφ(t∗,k)˙φ(t∗)

, (12)

where[H]±

is specified in (7b), and we used the resid-

ual gauge freedom in the synchronous gauge to bring thematching conditions on this simple form. We furtherfind that all perturbations without a derivative, includ-ing the fluid sector, are continuous, i.e. [h]± = [η]± =[δi]± = [θi]± = 0, where δi = δρi/ρi. This does notapply to NEDE perturbations because the derivation as-sumed that the e.o.s. of a particular matter componenti is not changing during the transition, in contrast with(4). As argued before, we can consistently set

δNEDE = θNEDE = 0 for t < t∗ . (13)

We further introduce the notation δ(+)NEDE ≡ δ∗NEDE and

θ(+)NEDE ≡ θ∗NEDE to denote the fluctuations right after the

transition. We can now evaluate the discontinuity of Ein-stein’s equations (11) in order to fix δ∗NEDE and θ∗NEDE,providing the initial conditions for the NEDE perturba-tions after the transition. Using (12) and (7b), we have

δ∗NEDE = −3 (1 + w∗NEDE)H(t∗)δφ(t∗,k)

˙φ(t∗), (14a)

θ∗NEDE =k2

a(t∗)

δφ(t∗,k)˙φ(t∗)

. (14b)

These two equations together with the junction condi-tions (12) will allow us to consistently implement ourmodel in a Boltzmann code. In order to close the dif-ferential system of the perturbed fluid equations, we setthe rest-frame sound speed [60] in the NEDE fluid toc2s = w∗NEDE.

DATA ANALYSIS AND RESULTS

In order to fit the NEDE model to the CMB data,we have incorporated it into the Boltzmann code4 CLASS

[61, 62]. To that end, we made the simplifying assump-tion that all liberated vacuum energy is ultimately con-verted to small scale anisotropic stress and gravitationalradiation described as a fluid with 1/3 ≤ w∗NEDE ≤ 1.As a specific choice for our data fit, we take the midpointw∗NEDE = 2/3(= c2s), which we relax in our subsequentpaper [46]. In accordance with our microscopic model thedecay is triggered shortly before φ = 0, where for defi-niteness we takeH/m = 0.2 (which avoids a tuning and isstill compatible with a quick decay). The sub-dominanttrigger field and its perturbations are evolved explicitlyand matched to the fluid perturbations through (14).This scenario also assumes that there are no sizeableoscillations in φ around the true vacuum, which couldgive rise to an additional sub-dominant dust component.A more detailed discussion of the corresponding micro-scopic constraints is provided in Sec. IID in [46].

The cosmological parameters are then extracted withthe Monte Carlo Markov chain code MontePython [63,64], employing a Metropolis-Hastings algorithm. Weperform a model comparison by computing the differ-ence in Bayesian evidence ∆ logB = logB(NEDE) −logB(ΛCDM) using the MultiNest algorithm (evidencetolerance 0.1 and 1000 live-points) [65–67]. Compared toΛCDM, we introduce two new parameters: the fractionof NEDE before the decay, fNEDE = ρ∗NEDE/ρ(t∗), andthe logarithm of the mass of the trigger field log10(m ×Mpc), which defines the redshift at decay time, z∗,

4 The adapted CLASS code is publicly available on GitHub: https:

//github.com/flo1984/TriggerCLASS

5

via H(z∗) = 0.2m. In total, we vary eight parame-ters {ωb, ωcdm, h, ln 1010As, ns, τreio , fNEDE, log10(m×Mpc)}, on which we impose flat priors. The neutrino sec-tor is modeled in terms of two massless and one massivespecies with Mν = 0.06 eV. We impose the initial valueφini/Mpl = 10−4 to make sure that the trigger field is al-ways sub-dominant and the tunneling rate at early timessufficiently suppressed.

We will use the following data sets: the most re-cent SH0ES measurement, which is H0 = 74.03 ±1.42 km s−1Mpc−1 [2]; the Pantheon data set [68] com-prised of 1048 SNe Ia in a range 0.01 < z < 2.3; thelarge-z BOSS DR 12 anisotropic BAO and growth func-tion measurements at redshift z = 0.38, 0.51 and 0.61based on the CMASS and LOWZ galaxy samples [69], aswell as small-z, isotropic BAO measurements of the 6dFGalaxy Survey [70] and the SDSS DR7main Galaxy sam-ple [71] at z = 0.106 and z = 0.15, respectively (collec-tively referred to as BAO); the Planck 2018 TT, TE, EEand lensing likelihood [72] with all nuisance parameters;constraints on the primordial helium abundance from [73](referred to as BBN). We perform one likelihood analysiswith all data sets combined (see red contours in Fig. 2),one where we only exclude the SH0ES value (turquoisecontours) and one with Planck (TT, TE, EE and lens-ing) alone (orange contours). For the latter two, we fixlog10(m×Mpc) = 2.58 in order to avoid sampling volumeartifacts in the fNEDE → 0 limit. While we provide anexhaustive discussion of this issue and also results with-out fixing log10(m) in our subsequent paper [46], here,we highlight the main findings [74].

For the analysis with all data sets, the best-fit im-proves by ∆χ2 = −15.6 compared to ΛCDM. This im-provement is shared between SH0ES [∆χ(SH0ES) = -13.8] and the other data sets [∆χ(w/o SH0ES) = -1.8].This observation is crucial at it shows that NEDE doesnot lead to new tensions. Instead, it also improves theoverall fit to the other data sets.5 Moreover, we findH0 = 71.4 ± 1.0 km s−1 Mpc−1. The decay takes placeat z∗ = 4920+620

−730, and there is a non-vanishing NEDE

fraction fNEDE = 12.6+3.2−2.9 %, excluding fNEDE = 0

with a 4.3σ significance. This is also supported by theBayesian evidence measure, which amounts to ∆B = 5.5,corresponding to a “very strong” evidence on Jeffreys’scale [76].

This picture is further solidified by our runs with-out SH0ES, which lead to a (very similar) fit improve-ment of ∆χ2 = −3.1 (Planck) and ∆χ2 = −2.9(Planck+BAO+Pantheon+BBN). In both cases, we finda 1.9σ evidence for a non-vanishing value of fNEDE anda “weak” (but positive) Bayesian evidence of 0.6 <

5 There is a slight degradation of the large-z BAO data set of∆χ2(large-z BAO + LSS) = 0.9, which we will explore in ourfuture work about large-scale structure within NEDE [75].

66 70 74

H0 [km s−1Mpc−1]

0.12

0.13

0.14

ωcdm

4000

6000

8000

z ∗

0.05

0.10

0.15

0.20

f NEDE

0.1 0.2

fNEDE

4000 6000

z∗

0.12 0.13 0.14

ωcdm

NEDE base [Planck+BAO+Pantheon+BBN+SH0ES]

NEDE base; fixed m [Planck]

NEDE base; fixed m [Planck+BAO+Pantheon+BBN]

ΛCDM [Planck]

FIG. 2. Covariances and posteriors of H0, fNEDE, z∗ and ωcdm

for our combined analyses. The 68% and 95% C.L. correspondrespectively to the light and dark shaded regions. The SH0ESvalue is represented as the vertical gray bands.

∆B < 0.7 [when fixing log10(m)]. The mean val-ues are H0 = 69.5+1.1

−1.5 km s−1 Mpc−1 and H0 =

69.6+1.0−1.3 km s−1 Mpc−1, respectively, which brings the

Hubble tension down to 2.5σ, in turn justifying our jointanalysis. In short, NEDE introduces an approximate de-generacy in the fNEDE vs H0 plane (see Fig. 2). TheSH0ES measurement is then needed to select NEDE asthe favored model.

CONCLUSIONS

We have studied a NEDE model where the decay ofNEDE happens through a first order phase transition.This makes our model unique compared to older EDEmodels (which all rely on a second order phase transition)both from a theoretical and phenomenological perspec-tive. The NEDE model holds in it the potential to fullyresolve the discrepancy in H0 as inferred from early CMBand BAO measurements and late time distance laddermeasurements. Our first most simplified implementationof the model (fixing as many free parameters as possibleby making simple assumptions) already yields a signif-icant improvement in the fit over the ΛCDM model of∆χ2 = −15.6 when including the SH0ES measurementof H0. Crucially, this does not compromise the fit to theother data sets. Correspondingly, without including theSH0ES prior on H0, the Hubble tension is reduced to the2.5σ level. We expect that the model will fit the dataeven better when the simplifying assumptions made in

6

the present short paper are dropped in future work.

We thank Edmund Copeland, Jose Espinosa, NemanjaKaloper, Antonio Padilla and Kari Rummukainen fordiscussions and/or useful comments on the manuscript.This work is supported by Villum Fonden grant 13384.CP3-Origins is partially funded by the Danish NationalResearch Foundation, grant number DNRF90.

∗ niedermann@cp3.sdu.dk† sloth@cp3.sdu.dk

[1] N. Aghanim et al. [Planck Collaboration],arXiv:1807.06209 [astro-ph.CO].

[2] A. G. Riess, S. Casertano, W. Yuan, L. M. Macriand D. Scolnic, Astrophys. J. 876 (2019) no.1, 85doi:10.3847/1538-4357/ab1422 [arXiv:1903.07603 [astro-ph.CO]].

[3] G. C.-F. Chen et al., doi:10.1093/mnras/stz2547arXiv:1907.02533 [astro-ph.CO].

[4] L. Verde, T. Treu and A. G. Riess, Nature Astronomy2019 doi:10.1038/s41550-019-0902-0 [arXiv:1907.10625[astro-ph.CO]].

[5] T. M. Davis, S. R. Hinton, C. Howlett and J. Cal-cino, Mon. Not. Roy. Astron. Soc. 490 (2019) no.2,2948-2957 doi:10.1093/mnras/stz2652 [arXiv:1907.12639[astro-ph.CO]].

[6] R. Wojtak, A. Knebe, W. A. Watson, I. T. Iliev,S. Heß, D. Rapetti, G. Yepes and S. Gottlober,Mon. Not. Roy. Astron. Soc. 438 (2014) no.2,1805 doi:10.1093/mnras/stt2321 [arXiv:1312.0276 [astro-ph.CO]].

[7] I. Odderskov, S. Hannestad and J. Brandbyge, JCAP1703 (2017) 022 doi:10.1088/1475-7516/2017/03/022[arXiv:1701.05391 [astro-ph.CO]].

[8] H. Y. Wu and D. Huterer, Mon. Not. Roy. Astron.Soc. 471 (2017) no.4, 4946 doi:10.1093/mnras/stx1967[arXiv:1706.09723 [astro-ph.CO]].

[9] J. L. Bernal, L. Verde and A. G. Riess, JCAP1610 (2016) 019 doi:10.1088/1475-7516/2016/10/019[arXiv:1607.05617 [astro-ph.CO]].

[10] L. Knox and M. Millea, Phys. Rev. D 101 (2020)no.4, 043533 doi:10.1103/PhysRevD.101.043533[arXiv:1908.03663 [astro-ph.CO]].

[11] V. Poulin, T. L. Smith, D. Grin, T. Karwal andM. Kamionkowski, Phys. Rev. D 98 (2018) no.8, 083525doi:10.1103/PhysRevD.98.083525 [arXiv:1806.10608[astro-ph.CO]].

[12] V. Poulin, T. L. Smith, T. Karwal and M. Kamionkowski,Phys. Rev. Lett. 122 (2019) no.22, 221301doi:10.1103/PhysRevLett.122.221301 [arXiv:1811.04083[astro-ph.CO]].

[13] T. L. Smith, V. Poulin and M. A. Amin,Phys. Rev. D 101 (2020) no.6, 063523doi:10.1103/PhysRevD.101.063523 [arXiv:1908.06995[astro-ph.CO]].

[14] N. Kaloper, Int. J. Mod. Phys. D 28 (2019)no.14, 1944017 doi:10.1142/S0218271819440176[arXiv:1903.11676 [hep-th]].

[15] S. Alexander and E. McDonough, Phys. Lett. B797 (2019) 134830 doi:10.1016/j.physletb.2019.134830

[arXiv:1904.08912 [astro-ph.CO]].[16] M. X. Lin, G. Benevento, W. Hu and

M. Raveri, Phys. Rev. D 100 (2019) no.6, 063542doi:10.1103/PhysRevD.100.063542 [arXiv:1905.12618[astro-ph.CO]].

[17] E. Hardy and S. Parameswaran, Phys. Rev. D 101(2020) no.2, 023503 doi:10.1103/PhysRevD.101.023503[arXiv:1907.10141 [hep-th]].

[18] C. Wetterich, Phys. Lett. B 594 (2004), 17-22doi:10.1016/j.physletb.2004.05.008 [arXiv:astro-ph/0403289 [astro-ph]].

[19] M. Doran and G. Robbers, JCAP 06 (2006),026 doi:10.1088/1475-7516/2006/06/026 [arXiv:astro-ph/0601544 [astro-ph]].

[20] P. Agrawal, F. Y. Cyr-Racine, D. Pinner and L. Randall,[arXiv:1904.01016 [astro-ph.CO]].

[21] E. Di Valentino, A. Melchiorri and J. Silk, Phys. Lett. B761 (2016), 242-246 doi:10.1016/j.physletb.2016.08.043[arXiv:1606.00634 [astro-ph.CO]].

[22] S. Kumar and R. C. Nunes, Phys. Rev. D 94(2016) no.12, 123511 doi:10.1103/PhysRevD.94.123511[arXiv:1608.02454 [astro-ph.CO]].

[23] S. Kumar and R. C. Nunes, Phys. Rev. D 96(2017) no.10, 103511 doi:10.1103/PhysRevD.96.103511[arXiv:1702.02143 [astro-ph.CO]].

[24] E. Di Valentino, A. Melchiorri and O. Mena,Phys. Rev. D 96 (2017) no.4, 043503doi:10.1103/PhysRevD.96.043503 [arXiv:1704.08342[astro-ph.CO]].

[25] E. Di Valentino, A. Melchiorri, E. V. Linder andJ. Silk, Phys. Rev. D 96 (2017) no.2, 023523doi:10.1103/PhysRevD.96.023523 [arXiv:1704.00762[astro-ph.CO]].

[26] G. E. Addison, D. J. Watts, C. L. Bennett,M. Halpern, G. Hinshaw and J. L. Weiland, Astrophys.J. 853 (2018) no.2, 119 doi:10.3847/1538-4357/aaa1ed[arXiv:1707.06547 [astro-ph.CO]].

[27] M. A. Buen-Abad, M. Schmaltz, J. Lesgourgues andT. Brinckmann, JCAP 01 (2018), 008 doi:10.1088/1475-7516/2018/01/008 [arXiv:1708.09406 [astro-ph.CO]].

[28] E. Mortsell and S. Dhawan, JCAP 09 (2018), 025doi:10.1088/1475-7516/2018/09/025 [arXiv:1801.07260[astro-ph.CO]].

[29] S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goo-bar and O. Mena, Phys. Rev. D 98 (2018) no.8, 083501doi:10.1103/PhysRevD.98.083501 [arXiv:1801.08553[astro-ph.CO]].

[30] V. Poulin, K. K. Boddy, S. Bird and M. Kamionkowski,Phys. Rev. D 97 (2018) no.12, 123504doi:10.1103/PhysRevD.97.123504 [arXiv:1803.02474[astro-ph.CO]].

[31] W. Yang, S. Pan, E. Di Valentino, R. C. Nunes,S. Vagnozzi and D. F. Mota, JCAP 09 (2018), 019doi:10.1088/1475-7516/2018/09/019 [arXiv:1805.08252[astro-ph.CO]].

[32] F. D’Eramo, R. Z. Ferreira, A. Notari andJ. L. Bernal, JCAP 11 (2018), 014 doi:10.1088/1475-7516/2018/11/014 [arXiv:1808.07430 [hep-ph]].

[33] R. Y. Guo, J. F. Zhang and X. Zhang, JCAP02 (2019), 054 doi:10.1088/1475-7516/2019/02/054[arXiv:1809.02340 [astro-ph.CO]].

[34] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathanand W. L. K. Wu, Astrophys. J. 874 (2019) no.1, 4doi:10.3847/1538-4357/ab0898 [arXiv:1811.00537 [astro-

7

ph.CO]].[35] C. D. Kreisch, F. Y. Cyr-Racine and O. Dore,

Phys. Rev. D 101 (2020) no.12, 123505doi:10.1103/PhysRevD.101.123505 [arXiv:1902.00534[astro-ph.CO]].

[36] M. Raveri, Phys. Rev. D 101 (2020) no.8, 083524doi:10.1103/PhysRevD.101.083524 [arXiv:1902.01366[astro-ph.CO]].

[37] R. E. Keeley, S. Joudaki, M. Kaplinghat andD. Kirkby, JCAP 12 (2019), 035 doi:10.1088/1475-7516/2019/12/035 [arXiv:1905.10198 [astro-ph.CO]].

[38] E. Di Valentino, R. Z. Ferreira, L. Visinelli andU. Danielsson, Phys. Dark Univ. 26 (2019), 100385doi:10.1016/j.dark.2019.100385 [arXiv:1906.11255 [astro-ph.CO]].

[39] G. B. Gelmini, A. Kusenko and V. Takhistov,[arXiv:1906.10136 [astro-ph.CO]].

[40] S. Pan, W. Yang, E. Di Valentino, E. N. Sari-dakis and S. Chakraborty, Phys. Rev. D 100(2019) no.10, 103520 doi:10.1103/PhysRevD.100.103520[arXiv:1907.07540 [astro-ph.CO]].

[41] S. Vagnozzi, Phys. Rev. D 102 (2020) no.2, 023518doi:10.1103/PhysRevD.102.023518 [arXiv:1907.07569[astro-ph.CO]].

[42] L. Visinelli, S. Vagnozzi and U. Danielsson, Sym-metry 11 (2019) no.8, 1035 doi:10.3390/sym11081035[arXiv:1907.07953 [astro-ph.CO]].

[43] S. Pan, W. Yang, E. Di Valentino, A. Shafielooand S. Chakraborty, JCAP 06 (2020) no.06, 062doi:10.1088/1475-7516/2020/06/062 [arXiv:1907.12551[astro-ph.CO]].

[44] E. Di Valentino, A. Melchiorri, O. Mena andS. Vagnozzi, Phys. Dark Univ. 30 (2020), 100666doi:10.1016/j.dark.2020.100666 [arXiv:1908.04281 [astro-ph.CO]].

[45] N. Arendse, R. J. Wojtak, A. Agnello, G. C. F. Chen,C. D. Fassnacht, D. Sluse, S. Hilbert, M. Millon,V. Bonvin and K. C. Wong, et al. Astron. Astro-phys. 639 (2020), A57 doi:10.1051/0004-6361/201936720[arXiv:1909.07986 [astro-ph.CO]].

[46] F. Niedermann and M. S. Sloth, Phys. Rev. D 102(2020) no.6, 063527 doi:10.1103/PhysRevD.102.063527[arXiv:2006.06686 [astro-ph.CO]].

[47] A. D. Linde, Phys. Lett. B 249 (1990), 18-26doi:10.1016/0370-2693(90)90521-7

[48] F. C. Adams and K. Freese, Phys. Rev. D 43 (1991),353-361 doi:10.1103/PhysRevD.43.353 [arXiv:hep-ph/0504135 [hep-ph]].

[49] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stew-art and D. Wands, Phys. Rev. D 49 (1994) 6410doi:10.1103/PhysRevD.49.6410 [astro-ph/9401011].

[50] M. Cortes and A. R. Liddle, Phys. Rev. D 80(2009) 083524 doi:10.1103/PhysRevD.80.083524[arXiv:0905.0289 [astro-ph.CO]].

[51] C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16(1977) 1762. doi:10.1103/PhysRevD.16.1762

[52] A. D. Linde, Nucl. Phys. B 216 (1983) 421 Erra-tum: [Nucl. Phys. B 223 (1983) 544]. doi:10.1016/0550-3213(83)90293-6, 10.1016/0550-3213(83)90072-X

[53] S. R. Coleman, Phys. Rev. D 15 (1977) 2929Erratum: [Phys. Rev. D 16 (1977) 1248].doi:10.1103/PhysRevD.15.2929, 10.1103/Phys-RevD.16.1248

[54] F. C. Adams, Phys. Rev. D 48 (1993) 2800doi:10.1103/PhysRevD.48.2800 [hep-ph/9302321].

[55] N. Kaloper and L. Sorbo, Phys. Rev. Lett. 102(2009) 121301 doi:10.1103/PhysRevLett.102.121301[arXiv:0811.1989 [hep-th]].

[56] S. W. Hawking, I. G. Moss and J. M. Stewart, Phys. Rev.D 26 (1982) 2681. doi:10.1103/PhysRevD.26.2681

[57] C. P. Ma and E. Bertschinger, Astrophys. J. 455 (1995)7 doi:10.1086/176550 [astro-ph/9506072].

[58] N. Deruelle and V. F. Mukhanov, Phys. Rev. D 52 (1995)5549 doi:10.1103/PhysRevD.52.5549

[59] W. Israel, Nuovo Cim. B 44S10 (1966) 1 [Nuovo Cim.B 44 (1966) 1] Erratum: [Nuovo Cim. B 48 (1967) 463].doi:10.1007/BF02710419, 10.1007/BF02712210

[60] W. Hu, Astrophys. J. 506 (1998), 485-494doi:10.1086/306274 [arXiv:astro-ph/9801234 [astro-ph]].

[61] J. Lesgourgues, arXiv:1104.2932 [astro-ph.IM].[62] D. Blas, J. Lesgourgues and T. Tram, JCAP

1107 (2011) 034 doi:10.1088/1475-7516/2011/07/034[arXiv:1104.2933 [astro-ph.CO]].

[63] B. Audren, J. Lesgourgues, K. Benabed andS. Prunet, JCAP 1302 (2013) 001 doi:10.1088/1475-7516/2013/02/001 [arXiv:1210.7183 [astro-ph.CO]].

[64] T. Brinckmann and J. Lesgourgues, arXiv:1804.07261[astro-ph.CO].

[65] F. Feroz, M. P. Hobson and M. Bridges, Mon. Not. Roy.Astron. Soc. 398 (2009), 1601-1614 doi:10.1111/j.1365-2966.2009.14548.x [arXiv:0809.3437 [astro-ph]].

[66] F. Feroz, M. P. Hobson, E. Cameron and A. N. Pet-titt, Open J. Astrophys. 2 (2019) no.1, 10doi:10.21105/astro.1306.2144 [arXiv:1306.2144 [astro-ph.IM]].

[67] J. Buchner, A. Georgakakis, K. Nandra, L. Hsu,C. Rangel, M. Brightman, A. Merloni, M. Sal-vato, J. Donley and D. Kocevski, Astron. Astrophys.564 (2014), A125 doi:10.1051/0004-6361/201322971[arXiv:1402.0004 [astro-ph.HE]].

[68] D. M. Scolnic et al., Astrophys. J. 859 (2018) no.2, 101doi:10.3847/1538-4357/aab9bb [arXiv:1710.00845 [astro-ph.CO]].

[69] S. Alam et al. [BOSS Collaboration], Mon.Not. Roy. Astron. Soc. 470 (2017) no.3, 2617doi:10.1093/mnras/stx721 [arXiv:1607.03155 [astro-ph.CO]].

[70] F. Beutler et al., Mon. Not. Roy. Astron. Soc.416 (2011) 3017 doi:10.1111/j.1365-2966.2011.19250.x[arXiv:1106.3366 [astro-ph.CO]].

[71] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden and M. Manera, Mon. Not. Roy. Astron.Soc. 449 (2015) no.1, 835 doi:10.1093/mnras/stv154[arXiv:1409.3242 [astro-ph.CO]].

[72] N. Aghanim et al. [Planck], [arXiv:1907.12875 [astro-ph.CO]].

[73] E. Aver, K. A. Olive and E. D. Skillman, JCAP07 (2015), 011 doi:10.1088/1475-7516/2015/07/011[arXiv:1503.08146 [astro-ph.CO]].

[74] See Supplemental Material at [URL] for a comprehensivelist of all mean, best-fit and χ2 values.

[75] F. Niedermann and M. S. Sloth, [arXiv:2009.00006 [astro-ph.CO]].

[76] B. Efron and A. Gous, Lecture Notes - Monogr. Ser.. 38.(2001), 208-246 doi:10.1214/lnms/1215540972.