Jun-Qing Xia and Matteo Viel- Early Dark Energy at High Redshifts: Status and Perspectives

8/3/2019 Jun-Qing Xia and Matteo Viel- Early Dark Energy at High Redshifts: Status and Perspectives

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Early Dark Energy at High Redshifts: Status and Perspectives

Jun-Qing Xia1 and Matteo Viel2,31Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste, Italy

2INAF-Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy and3INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy

(Dated: April 2, 2009)

Early dark energy models, for which the contribution to the dark energy density at high redshifts

is not negligible, influence the growth of cosmic structures and could leave observable signaturesthat are different from the standard cosmological constant cold dark matter (CDM) model. Inthis paper, we present updated constraints on early dark energy using geometrical and dynamicalprobes. From WMAP five-year data, baryon acoustic oscillations and type Ia supernovae luminositydistances, we obtain an upper limit of the dark energy density at the last scattering surface (lss),EDE(zlss) < 2.3 10

2 (95% C.L.). When we include higher redshift observational probes, such asmeasurements of the linear growth factors, Gamma-Ray Bursts (GRBs) and Lyman- forest (Ly),this limit improves significantly and becomes EDE(zlss) < 1.4 10

3 (95% C.L.). Furthermore, wefind that future measurements, based on the Alcock-Paczynski test using the 21cm neutral hydrogenline, on GRBs and on the Ly forest, could constrain the behavior of the dark energy componentand distinguish at a high confidence level between early dark energy models and pure CDM. In thiscase, the constraints on the amount of early dark energy at the last scattering surface improve by afactor ten, when compared to present constraints. We also discuss the impact on the parameter ,the growth rate index, which describes the growth of structures in standard and in modified gravitymodels.

I. INTRODUCTION

Current cosmological observations, such as the cosmicmicrowave background (CMB) measurements of temper-ature anisotropies and polarization [1] and the redshift-distance measurements of Type Ia Supernovae (SNIa) atz < 2 [2], have demonstrated that the Universe is nowundergoing an accelerated phase of expansion and thatits total energy budget is dominated by the dark energycomponent. The nature of dark energy is one of thebiggest unsolved problems in modern physics and has

been extensively investigated in recent years, both un-der the theoretical and the observational point of view.The simplest candidate of dark energy is the cosmologi-cal constant (CDM): however, this model suffers of thefine-tuning and coincidence problems [3]. To lift thesetensions, many alternative dynamical dark energy mod-els, such as quintessence [4], phantom [5], K-essence [6]and quintom [7], have been proposed (for a review seeRef.[8]).

Among all the dynamical dark energy models, we willfocus here on early dark energy (EDE) ones, in whicha small fraction of dark energy is present up to thelast scattering surface (lss), unlike CDM for which

DE(zlss) 0, The differences between early dark en-ergy models and pure CDM are particularly evident athigh redshifts, over a large fraction of the cosmic time,when the first structures form. EDE has been shown toinfluence the growth of cosmic structures (both in thelinear and in the non-linear regime), to change the age

Electronic address: [email protected] address: [email protected]

of the universe, to have an influence on CMB physics,to impact on the reionization history of the universe, tomodify the statistics of giant arcs in strong cluster lensingstatistics [9, 10, 11, 12, 13, 14, 15, 16, 17].

The reason for addressing this particular model of darkenergy is also driven by the increasing availability of highredshift observations that somewhat bridge the gap be-tween the CMB and very local cosmological probes. Darkenergy studies rely mainly on: the clustering propertiesof luminous red galaxies [18], baryonic acoustic oscilla-tions (BAO) [19, 20], weak lensing data [21, 22], high

redshift SNIa [23], the Ly forest [24, 25], GRBs [26, 27]as standard candles, and also the number of high redshiftgalaxies and clusters of galaxies [28] that could be studiedwith deep field observations. Of course, since the presentdata sets seem to be in good agreement with pure CDMmodel, we hope that these observations could open up anew high redshift window on the properties of dark en-ergy and possible allow to confirm or disproof its redshiftevolution. Such a difficult measurement would have pro-found implications for physical cosmology and particlephysics.

In this paper we extend our studies of dark energybehavior into the redshift range 2 < z < 1100, whichcovers the so-called dark ages, and present constraintson EDE from current and future cosmological observa-tions. We will use CMB, BAO, SNIa, Ly forest ob-servations of growth factors and matter power spectrumand the GRBs from current available data sets, while forforecasts we will add to Planck-like observations, somehigher redshift GRBs, some improved Ly constraintsand a measurement that could be potentially very im-portant of the so-called Alcock-Paczynski (AP, [29]) testusing 21cm maps [30, 31]. As we will see, the partic-ular parameterization of EDE chosen will be such that
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the higher the redshift the tighter the constraints will beon the parameters describing the given model. Thereby,futuristic intergalactic medium Ly forest data, GRBsdistance moduli and the 21cm maps, before or aroundreionization, are expected to be very promising probes ofdynamical dark energy models (see also Ref.[13] for theimpact of these models on the reionization history of theuniverse). We note that the behavior of dark energy in

this redshift range has been exploited by Ref.[32], whoconstrained the EDE density of to be less than 2% of thetotal energy density. Here, we will improve this limit byadding several different observations: as mentioned be-fore this is not only important to get stronger constraintson some cosmological parameters but also to understandthe consistency of different probes and how systematiceffects impact on the final derived measurements (e.g.Refs.[33, 34]). Of course, we stress that while our ap-proach has the great advantage of using dynamical andgeometrical cosmological probes in the linear or quasi-linear regime, other approaches, based for example onhaloes concentrations, could also be envisaged and mustrely on an accurate comparison with numerical simula-tions [16].

Our paper is organized as follows: in Sec. II we de-scribe the theoretical framework of the early dark energymodel and the datasets we used. Sec. III contains ourmain global fitting results from the current observations.In Sec. IV we present the forecasts for the future mea-surements while Sec. V is dedicated to the conclusionsand discussion.

II. METHOD AND DATA

A. Parameterization of early dark energy

We decided to use the mocker model introduced inRef.[32] motivated by the following two observationalfacts: i) the best-fit model is pure CDM model andthe amount of dark energy at the last scattering sur-face is constrained to be close to zero from CMB and bigbang nucleosynthesis (BBN) observations; ii) an equationof state of the dark energy component which is rapidlyevolving (i.e. dw/dz 1) seems to be ruled out by high-redshift SNIa [23].

For these EDE models the parameterization reads:

wEDE(a) = 1 +

1

w0

1 + w0 aC

1

, (1)

where a = 1/(1 + z) is the scale factor, w0 is the presentequation-of-state of dark energy and C characterizes therunning of the equation of state. Consequently, theevolution of dark energy density can easily be obtainedvia energy conservation as:

EDE(a)

EDE(1)=

(1 + w0)aC

w03/C

. (2)

0 1 2 3

- 1 . 0

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

w

E

D

E

(

z

)

E

D

E

(

z

)

( 1 + z )

E D E 1

E D E 2

C D M

G

(

a

)

=

/

a

E D E 1

E D E 2

C D M

Figure 1: Left panel: the evolution of dark energy densityand equation of state for three models, w0 = 0.972 and C =1.858 (EDE1), w0 = 0.95 and C = 2.5 (EDE2) and pure

CDM. Right panel: the linear growth factor G(a) = (a)/anormalized at z = for the three models. The shaded arearepresents the redshifts that we will b e mainly probing in thepresent work (z 2 20).

In Fig. 1 we plot the dark energy density (upper partof the left panel) and equation of state (bottom part ofthe left panel) as a function of redshift for three differ-ent models: pure CDM (dotted blue line), EDE1 (solidblack line) which has (w0, C) = (0.972, 1.858) andEDE2 (dashed red line) which has (w0, C) = (0.95, 2.5).All these models fit the CMB and the lower redshift SNIa

constraints very well. In the right panel, we also show thelinear growth rate (growth factors divided by the scalefactor) normalized at z = and compare the differentevolutions with redshifts in the three cases: differences ofthe order of 20% are visible at z 3. In the two panelswe also show as a shaded vertical band the region in theredshift range z = 2 20, a period which is roughly 25%of the age of the universe. This is the redshift range wewill be (mainly) focussing on in the rest of the paper inorder to discriminate between the different dark energymodels.

We stress that this is just one of the possible parame-terizations for early dark energy models, another one hasbeen suggested by Ref.[35] and recently used in Ref.[17].

However, we prefer to use the mocker model in order tocompare with the findings of Ref.[32] and because thisparameterization has a smooth redshift derivative at lowz for w(z). We note that one of the most important pa-rameters is the amount of dark energy during the struc-ture formation period and this is given by (aeq is thematter-radiation equality scale factor):

EDE,sf = (ln aeq)1

0lnaeq

EDE (a) d l n a , (3)


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and we will also quote this value in the rest of the paperin order to compare with other works and constraints aswell (e.g. Refs.[9, 15]).

We will also use some growth factors measurementsobtained mainly through Ly forest observations andone lower redshift from Sloan Digital Sky Survey (SDSS)galaxies. For this purpose we will compute the growthfactors for the EDE models with a subroutine imple-

mented in the publicly available Monte Carlo MarkovChains (MCMC) package CosmoMC [36] that solves thesecond order differential equations for the growth factoras in Ref.[10]. Furthermore, we also modify CosmoMCto include the perturbations of dynamical dark energymodels generally as done in Refs.[37].

B. Current and Future Datasets

1. Current Observations

We will rely here on the following cosmological probes:

i) CMB anisotropies and polarization; ii) baryonic acous-tic oscillations in the galaxy power spectra; iii) SNIa dis-tance moduli; iv) GRBs distance moduli; v) the Ly for-est growth factors and matter power spectrum measure-ments.

In the computation of CMB power spectra we haveincluded the WMAP five-year (WMAP5) temperatureand polarization power spectra with the routines forcomputing the likelihood supplied by the WMAP team[38]. Besides the WMAP5 information, we also use somedistance-scale indicators.

BAOs have been detected in the current galaxy redshiftsurvey data from the SDSS and the Two-degree Field

Galaxy Redshift Survey (2dFGRS) [19, 20, 39, 40]. TheBAO can directly measure not only the angular diameterdistance, DA(z), but also the expansion rate of the uni-verse, H(z), which is powerful for studying dark energy[41]. Since current BAO data are not accurate enough forextracting the information ofDA(z) and H(z) separately[42], one can only determine an effective distance [19]:

Dv(z)

(1 + z)2D2A(z)

cz

H(z)

1/3. (4)

In this paper we use the Gaussian priors on the distanceratios rs(zd)/Dv(z):

rs(zd)/Dv(z = 0.20) = 0.1980 0.0058 ,rs(zd)/Dv(z = 0.35) = 0.1094 0.0033 , (5)

with a correlation coefficient of 0.39, extracted from theSDSS and 2dFGRS surveys [40], where rs is the comovingsound horizon size and zd is drag epoch at which baryonswere released from photons given by Ref.[43].

The SNIa data provide the luminosity distance as afunction of redshift which is also a very powerful mea-surement of dark energy evolution. The supernovae data

we use in this paper are the recently released Union com-pilation (307 samples) from the Supernova Cosmologyproject [2], which include the recent samples of SNIa fromthe (Supernovae Legacy Survey) SNLS and ESSENCEsurvey, as well as some older data sets, and span the red-shift range 0 z 1.55. In the calculation of the likeli-hood from SNIa we have marginalized over the nuisanceparameter as done in [44] and ignored the systematic er-

rors to improve our results1

.Furthermore, we make use of the Hubble Space Tele-

scope (HST) measurement of the Hubble parameterH0 100 h km s

1 Mpc1 by a Gaussian likelihoodfunction centered around h = 0.72 and with a standarddeviation = 0.08 [46].

In order to constrain the early dark energy model athigh redshifts, we also include the GRBs, linear growthfactors and Ly forest data.

GRBs can potentially be used to measure the luminos-ity distance out to higher redshift than SNIa. Recently,several empirical correlations between GRB observableswere reported, and these findings have triggered inten-

sive studies on the possibility of using GRBs as cosmo-logical standard candles. However, due to the lack oflow-redshift long GRBs data to calibrate these relations,in a cosmology-independent way, the parameters of thereported correlations are given assuming an input cos-mology and obviously depend on the same cosmologicalparameters that we would like to constrain. Thus, apply-ing such relations to constrain cosmological parametersleads to biased results. In Ref.[47] the circular problemis naturally eliminated by marginalizing over the free pa-rameters involved in the correlations; in addition, someresults show that these correlations do not change signif-icantly for a wide range of cosmological parameters [48].

Therefore, in this paper we use the 69 GRBs sample overa redshift range from z = 0.176.60 published in Ref.[26]but we keep in mind the issues related to the circularproblem that are more extensively discussed in Ref.[47].

In Table I we list two types of linear growth factorsdata we use: i) the normalization 8 inferred from theSDSS Ly power spectrum2 [49, 50]; ii) the linear growthrate f m from galaxy power spectrum at low redshift[51] and Ly growth factor measurement obtained from

1 In this paper, we find that the low redshifts probes could notconstrain on the early dark energy models very well. Therefore,we focus on the constraints on the EDE model from high red-shifts probes. Furthermore, in Ref.[45] we discussed the effect ofconstraints on dark energy models with or without systematicerror. When including the systematic error, the constraints areslightly weaker, but the main results are not changed.

2 These data at different redshifts are not totally independent. Thederived 8 values are obtained by using the covariance matrixas measured from the data of the SDSS flux power spectrumand there are (very weak) correlations in the flux power betweendifferent redshift bins. So, although not totally independent, weregard the derived values of the power spectrum amplitude fromthe SDSS flux power as nearly independent.


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TABLE I. 8 values and growth rates with 1 error barsfrom which we derived the linear growth factors used in our

analysis.

z 8 8 Ref.2.125 0.95 0.17 [49]2.72 0.92 0.172.2 0.92 0.16 [50]2.4 0.89 0.11

2.6 0.98 0.132.8 1.02 0.093.0 0.94 0.083.2 0.88 0.093.4 0.87 0.123.6 0.95 0.163.8 0.90 0.17z f f Ref.

0.15 0.51 0.11 [51]3.00 1.46 0.29 [25]

the SDSS Ly power spectrum at z = 3 by Ref.[25].

We have used two Ly forest data sets: i) the highresolution QSO absorption spectra presented in Ref.[49]consisting of the LUQAS sample [52] and the reanalyzeddata in Ref.[53] (C02, this data set as a whole will belabelled as VHS in the following); ii) the SDSS Ly for-est sample presented in Ref.[54]. The SDSS Ly forestdata set consists of 3035 QSO spectra with low resolu-tion (R 2000) and low S/N (< 10 per pixel) span-ning a wide range of redshifts (z = 2.2 4.2), whilethe LUQAS and the C02 samples contain 57 high reso-lution (R 45000), high signal-to-noise (> 50 per pixel)QSO spectra with median redshifts of z = 2.125 andz = 2.72, respectively. The flux power spectrum of the

Ly forest is the quantity which is observed and needsto be modeled at the percent or sub-percent level us-ing accurate numerical simulations that incorporate therelevant cosmological and astrophysical processes, in or-der to extract the underlying (linear) dark matter powerspectrum. In this paper, we will use the derived linearpower spectrum measured by the data set of VHS, basedon VHSLy together with the SDSS Ly power spec-trum SDSSLy, since they are in agreement and thislatter has a stronger constraining power. More precisely,the VHSLy power spectrum consists of estimates of thelinear dark matter power spectrum at nine values in thewavenumber space k at z = 2.125 and nine values atz = 2.72, in the range 0.003 < k (s/km)< 0.03, while theSDSSLy consists of a single measurement at z = 3 andk = 0.009 s/km of amplitude, slope and curvature. Theestimate of the uncertainty of the overall amplitude ofthe matter power spectrum is 29% for the first datasetand 14% for the second. This estimate takes into ac-count possible systematic and statistical errors (see therelevant tables of VHS for a detailed discussion). Thecode assigns a Gaussian prior to the corresponding nui-sance parameter and marginalize over it. Finally, we willalso rely on the growth factor measurements from the

TABLE II. Assumed experimental specifications for themock Planck-like measurements. The noise parameters T

and P are given in units of K-arcmin.

fsky lmax (GHz) fwhm T P0.65 2500 100 9.5 6.8 10.9

143 7.1 6.0 11.4217 5.0 13.1 26.7

two data sets, derived using numerical simulations of theobserved flux power, and we will label this as GFLy.

2. Future Measurements

In order to forecast future measurements we will usethe same observables as before without BAO but exploit-ing the role of 21cm maps at high redshifts.

For the simulation with PLANCK [55], we follow themethod given in Ref.[56] and mock the CMB temper-

ature (TT) and polarization (EE) power spectra andtemperature-polarization cross correlation (TE) by as-suming a given fiducial cosmological model. In Table II,we list the assumed experimental specifications of the fu-ture (mock) Planck measurement.

The proposed satellite SNAP (Supernova / Acceler-ation Probe) will be a space based telescope with aone square degree field of view that will survey thewhole sky [57]. It aims at increasing the discovery rateof SNIa to about 2000 per year in the redshift range0.2 < z < 1.7. In this paper we simulate about 2000SNIa according to the forecast distribution of the SNAP[58]. For the error, we follow the Ref.[58] which takes

the magnitude dispersion to be 0.15 and the systematicerror sys = 0.02 z/1.7. The whole error for each data

is given by mag(zi) =

2sys(zi) + 0.152/ni , where ni is

the number of supernovae of the ith redshift bin. Fur-thermore, we add as an external data set a mock set of400 GRBs, that mimic SWIFT [60] observations, in theredshift range 0 < z < 6.4 with an intrinsic dispersion inthe distance modulus of = 0.16 and with a redshiftdistribution very similar to that of Figure 1 of Ref. [59].

For the linear growth factors data, we simulate themock data from the fiducial model of Table I with theerror bars reduced by a factor of two. This is probablyreasonable given the larger amounts of Ly forest data(e.g. SDSS-III or the X-shooter spectrograph [61]) thatwill become available soon as long with a better controlof several systematics errors and more importantly onthe thermal history of the intergalactic medium.

We also simulate the Ly power spectrum, consistingof three data points at z = 2, 3 and 3.5 and at wavenum-bers of 0.002 s/km, 0.009 s/km and 0.02 s/km, with 1%fractional errors in a similar way as done by Ref.[62], inwhich a forecasting on the perspective of constraining theneutrino masses was presented.


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The availability of 21cm (1420 MHz) maps with fu-ture experiments like LOFAR [63] will allow to mea-sure in a given frequency interval and for a given po-sition in the sky the differential brightness temperatureof a patch of gas against the CMB. Thus a completelynew window on the neutral hydrogen content of the uni-verse, regarded as a tracer of cosmic structures and ofastrophysical processes, will be opened soon. The AP

test [29] assumes that the for an isotropically distributedset of astronomical objects the separations along andacross the line-of-sights scale differently and thus gen-erate anisotropies. The geometrical distortion dependson the quantity [H(z) DA(z)]

1 that thereby can bemeasured and provide constraints on some parameters.The extended LOFAR will operate between 30 and 240MHz, with spatial resolution of the order of arcseconds,probing the universe in the redshift range z 6 45.For our purposes here we will assume that the quantity[H(z) DA(z)]

1, will be measured at z = 6.5, 8, 12 with5% precision. Although other measurements could beperformed using 21cm datasets, such as (quite optimisti-cally) a full measurement of the matter power spectrumat very high redshifts, we will concentrate here only onthe AP test, since this appears to be easier to be per-formed and it is less sensitive to the astrophysical uncer-tainties that relate the differential brightness tempera-ture to the matter density (see however Ref.[64] for otherpossible and very promising constraints). It is neverthe-less important to note that LOFAR sensitivity drops forz > 12 and useful HI maps at these redshifts could onlybe obtained with future ground based telescopes like SKA[65] and with a better understanding of the foregroundsthat contaminate the cosmological signal (see for exam-ple the discussion in Ref.[30]).

Here, we will use a somewhat more conservative ap-

proach and simulate a measurement of the quantity de-scribed above in a redshift range accessible to the LOFARexperiment. Although challenging, we believe that theAP test could be performed and give constraints similarto those obtained here.

III. RESULTS FROM PRESENT DATASETS

In our analysis, we perform a global fitting using theCosmoMC package. We assume purely adiabatic ini-tial conditions and a flat universe, with no tensor con-tribution. We vary the following cosmological parame-ters with top-hat priors: the dark matter energy densitych2 [0.01, 0.99], the baryon energy density bh2 [0.005, 0.1], the primordial spectral index ns [0.5, 1.5],the primordial amplitude log[1010As] [2.7, 4.0] and theangular diameter of the sound horizon at last scattering [0.5, 10]. For the pivot scale we set ks0 = 0.05 Mpc

1.When CMB data are included, we also vary the opticaldepth to reionization [0.01, 0.8]. We do not considerany massive neutrino contribution. From the parame-ters above the MCMC code derives the reduced Hubble

0.95 0.9 0.850

0.2

0.4

0.6

0.8

1

w0

1 2 3 40

0.2

0.4

0.6

0.8

1

C0.6 0.7 0.8 0.9

0

0.2

0.4

0.6

0.8

1

8

0.8 0.6 0.4 0.2 00

0.2

0.4

0.6

0.8

1

wEDE

(zlss

)0 5 10

x 103

0

0.2

0.4

0.6

0.8

1

EDE

(zlss

)0 .06 0. 08 0. 1

0

0.2

0.4

0.6

0.8

1

EDE,sf

Figure 2: 1D current (marginalized) constraints onthe dark energy parameters w0, C, as well as 8,wEDE(zlss), EDE(zlss) and EDE,sf, from different currentdata combinations: WMAP5+BAO+SN (red solid lines),WMAP5+BAO+SN+GRB+Ly (blue dash-dot lines) andWMAP5+BAO+SN+GRB+GFLy (black dashed lines).

parameter H0, the present matter fraction m0, 8, andEDE,sf: so, these parameters have non-flat priors andthe corresponding bounds must be interpreted with somecare. In addition, CosmoMC imposes a weak prior on theHubble parameter: h [0.4, 1.0]. For part of the anal-ysis, we also present constraints on the growth index ,defined as:

d ln (a)

d ln a= m(a)

. (6)

This parameter, introduced by [66], provides a good fit

to the growth rate for many different cosmological mod-els and depends on the growth of perturbations, H(z)and m0. It has been recently measured by Refs.[67, 68]using mainly the observed growth factors at high andlow redshifts and values different from that expected forthe pure CDM model ( 0.55) could be importantfor constraining both quintessence models and modifiedgravity scenarios (see Ref.[69, 70] for forecasting and amore extensive discussion on linear growth rate in mod-ified gravity models).

A. CMB+BAO+SNIa

In Table III we show our main global constraints onthe early dark energy model from different current datacombinations.

Firstly, we present the constraints from the data com-bination: CMB+ BAO+SNIa. In Fig. 2 we show the onedimensional distributions of some cosmological parame-ters from this data combination (solid red lines). For theparameters describing the equation of state of early darkenergy, the constraints are still weak, namely the 95%upper limits are w0 < 0.906 and C < 2.711. The limit


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w0

EDE,s

f

0.95 0.9 0.850.04

0.05

0.06

0.07

0.08

0.09

0.1

C

EDE,s

f

0.5 1 1.5 2 2.5 3 3.50.04

0.05

0.06

0.07

0.08

0.09

0.1

8

EDE,s

f

0.6 0.7 0.8 0.90.04

0.05

0.06

0.07

0.08

0.09

0.1

wEDE

(zlss

)

EDE,s

f

0.8 0.4 00.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 4: 2D (marginalized) contours of EDE,sf ver-sus w0, C, 8 and wEDE(zlss) from different currentdata combinations: WMAP5+BAO+SN (red solid lines),WMAP5+BAO+SN+GRB+Ly (blue dash-dot lines) andWMAP5+BAO+SN+GRB+GFLy (black dashed lines).

today are then obviously anti-correlated.To sum up, using CMB, BAO and SNIa, we constrain

the amount of dark energy at the last scattering surfaceto be less than 2% and in the structure formation erato be of the order of 6%. This last number is in roughagreement with the results of [72], who used WMAP 3year data and some external data sets, although theirvalues are somewhat smaller ( 4%) and this is due tothe slightly different definition of EDE,sf.

B. Adding High-Redshift Probes

From the results above, we can see that without ob-servations in or around the so called dark ages, the con-straints on the early dark energy model are interestingbut not very tight. In this subsection we add somehigh-redshift observational data, such as GRBs, Ly andGFLy data, to improve our constraints. Because mostof GFLy data are obtained from the Ly forest powerspectrum, in order to avoid overestimating the weightof Ly data, in our analysis we do not use GFLy andLy data together. And the results, as we expect, fromthese two data combinations are consistent between eachother. Therefore, in the following we mainly present theglobal fitting results obtained from those that includeGRB and GFLy data.

When adding the high-redshift probes, the constraintson w0 and C improve significantly: w0 < 0.958 andC < 2.245 at 95% confidence level, while their best fitvalues are very close to the pure CDM model. Natu-rally, the 95% upper limit of EDE(zlss) has also beentightened by a factor of ten, EDE(zlss) < 1.4 10

3. Alarge number of early dark energy models have been ruled

out by the high-redshift probes. The high-redshift obser-vations are indeed effective in constraining further theearly dark energy models investigated here. Note thatadding only the power spectrum information as providedby Ly improves the constraints on EDE(zlss) by a fac-tor five, when compared to CMB+BAO+SNIa, so thegrowth factor information is really fundamental in con-straining the EDE model. However, the double peak in

the distribution of wEDE(zlss) is still present, which im-plies that the current data are not accurate enough tomeasure wEDE(zlss), even when high-redshift probes areadded.

Another effect of adding high-redshift probes is thelarger value of 8 today. As we know, the Ly forestdata favor a larger 8 0.9 than the CMB measure-ment3. When we add the GFLy or Ly forest data, weobtain a higher 8 today than before: 8 = 0.836 0.024(1). However, this is still smaller than the value favoredby Ly forest data alone in the pure CDM model andconsistent with WMAP5+BAO+SN measurement of [1].We also note that if we use only VHSLy data sets and

adding them to WMAP5+BAO+SN and GRBs the re-sults do not change significantly as compared to havingSDSSLy and VHSLy together. This is mainly due tothe constraining power of the growth factors measure-ment obtained with VHSLy.

The conclusion of this section is that adding higher red-shift probes constrains the energy density of dark energyat the last scattering surface to be around 0.1%.

C. Growth Index

Finally, we extend our discussion by briefly addressing

the perspective of constraining the linear growth index .Measurements of the growth history of cosmic structurescombine information on both cosmic expansion and theunderlying theory of gravity, the parameter is thus aunique prediction of any modified gravity model and asimultaneous fitting of this parameter and other dynam-ical probes could provide constraints on modified gravityscenarios and dynamical dark energy models.

By using all the current data available we obtain = 0.622 0.139 (1) error bar for our EDE model,which is in agreement with the CDM and with thevalues obtained by Refs.[67, 68] (with slightly smallererror bars). The present error on is also similar tothat inferred from future weak lensing and SNIa data byRefs.[69].

3 This fact does not imply a contradiction or a failure of the CDMmodel but there is indeed a tension between small scale con-straints and the large scale ones. However, more recent stud-ies that combine WMAP year 3, the Lyman- forest and weak-lensing measurements of the COSMOS-z survey [24] point to avalue of8 = 0.8000.023, which is in good agreement with thefindings of WMAP year 5 plus BAO and SNe [1].


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Figure 5: The posterior distribution of and 2D (marginal-ized) contours of and w0, C, wEDE(zlss), EDE(zlss) andEDE,sf from WMAP5+BAO+SN+GRB+Ly (blue dottedlines).

TABLE IV. Constraints on the early dark energy modelfrom future measurements. We also list the standard

deviation of these parameters based on the future mock datasets. For parameters that are only weakly constrained we

quote the 95% upper limit instead.

Parameter Fiducial CDM Fiducial EDEw0 < 0.975 [0.985,0.920]C < 2.635 [0.256, 2.300]

m 0.0029 0.00328 0.0078 0.0145

EDE(zlss) < 1.02 104 < 0.011

EDE,sf 0.0011 0.0037 0.026 0.048

In Figure 5 we show the 2-dimensional contours ofvs.8, C , w0 and EDE(zlss) to address possible degeneraciesof this parameter. For the current data sets, we note thatthe degeneracies are weak and only the degeneracy in the EDE,sf plane is significant. This is easily understoodsince a larger implies a faster growth of structures andthereby a smaller amount of dark energy.

IV. RESULTS FROM FUTURE DATA SETS

Since the present data do not give very stringent con-straints on the parameters of early dark energy model,it is worthwhile discussing whether future data could de-termine these parameters conclusively. For this purposewe have performed a further analysis and we have cho-sen two fiducial models in perfect agreement with currentdata: a pure CDM model and an EDE model with pa-rameters taken to be the best-fit values of Table III fromthe current constraints of CMB+BAO+SNIa. This lattermodel will be labelled EDE1, and its evolution is shown

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in Fig.1.

A. Fiducial CDM

Firstly, we choose the pure CDM as the fiducialmodel. In Table IV we list the forecasts for some pa-rameters using all the future measurements described inSection IIB2: GRBs, SNIa, Ly (matter power spec-trum and growth factors) and the AP test.

Due to the smaller error bars of the mock data sets,the constraint on w0 improves significantly, namely the95% upper limit is now w0 < 0.975. The fraction of


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early dark energy density has been limited very stringent,EDE(zlss) < 1.02 104 (95% C.L.), which is reduced

by another factor of ten, compared with the present con-straint. The error on the parameter that we forecastis 0.026, while the error on the amount of EDE in thestructure formation era is about 1.1 103. Therefore,we can foresee a very precise determination of the amountof dark energy in the dark ages (or in the structure for-mation era) with this future data set.

By contrast, the 95% upper limit on Chas not been im-proved, C < 2.635. This is due to the chosen parameteri-zation of early dark energy. In Fig. 7 we illustrate the dif-ferent evolution of several EDE equations of state varyingw0 and keeping C = 2. We note that the smaller the red-shift at which w0 approaches the cosmological constant,

the smaller the redshift at which w(z) flattens to zero inthe high redshift universe. For example, if w0 = 0.95,we have w(z = 10) 0.1 and C = 2 has been ruledout obviously. On the other hand, if w0 = 0.9999, westill have w(z = 10) 1. Consequently, C = 2 is stillfavored by the data. Therefore, even if the constraint onw0 becomes stringent, a large value of C is still allowed,and then the constraint on C will be somewhat poorer.Moreover, due to the allowed large value of C, the firstpeak ofwEDE(zlss) around w = 0 is still present, as shownin Fig. 6, where the 1D marginalized constraints at thelast scattering surface are shown.

B. Fiducial EDE

For our EDE fiducial model we choose the best fit val-ues of current constraint as the fiducial model to simulatethe future measurements, w0 = 0.972 and C = 1.858(EDE1).

In Fig. 6, we plot the one dimensional distributionsof w0 and C. The 95% confidence level is 0.985

Jun-Qing Xia and Matteo Viel- Early Dark Energy at High Redshifts: Status and Perspectives

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Transcript of Jun-Qing Xia and Matteo Viel- Early Dark Energy at High Redshifts: Status and Perspectives