Early Dark Energy at High Redshifts: Status and Perspectives
aa r X i v : . [ a s t r o - ph . C O ] A p r Early Dark Energy at High Redshifts: Status and Perspectives
Jun-Qing Xia ∗ and Matteo Viel , † Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste, Italy INAF-Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy and INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy (Dated: October 29, 2018)Early dark energy models, for which the contribution to the dark energy density at high redshiftsis not negligible, influence the growth of cosmic structures and could leave observable signaturesthat are different from the standard cosmological constant cold dark matter (ΛCDM) 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),Ω
EDE ( z lss ) < . × − (95% C.L.). When we include higher redshift observational probes, such asmeasurements of the linear growth factors, Gamma-Ray Bursts (GRBs) and Lyman- α forest (Ly α ),this limit improves significantly and becomes Ω EDE ( z lss ) < . × − (95% C.L.). Furthermore, wefind that future measurements, based on the Alcock-Paczy´nski test using the 21cm neutral hydrogenline, on GRBs and on the Ly α forest, could constrain the behavior of the dark energy componentand distinguish at a high confidence level between early dark energy models and pure ΛCDM. 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 γ ,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) at z < DE ( z lss ) ≃
0, The differences between early dark en-ergy models and pure ΛCDM 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] † Electronic 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], highredshift SNIa [23], the Ly α 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 ΛCDMmodel, 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 < α 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 α constraintsand a measurement that could be potentially very im-portant of the so-called Alcock-Paczy´nski (AP, [29]) testusing 21cm maps [30, 31]. As we will see, the partic-ular parameterization of EDE chosen will be such thatthe higher the redshift the tighter the constraints will beon the parameters describing the given model. Thereby,futuristic intergalactic medium Ly α 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 inthis 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 DATAA. 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 ΛCDM 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: w EDE ( a ) = − (cid:20) − w w a C (cid:21) − , (1)where a = 1 / (1 + z ) is the scale factor, w is the presentequation-of-state of dark energy and C characterizes the“running” of the equation of state. Consequently, theevolution of dark energy density can easily be obtainedvia energy conservation as: ρ EDE ( a ) ρ EDE (1) = (cid:2) (1 + w ) a − C − w (cid:3) /C . (2) w E D E ( z ) E D E ( z ) log (1+z) EDE1 EDE2 CDM G ( a ) = / a EDE1 EDE2 CDMa sFigure 1: Left panel: the evolution of dark energy densityand equation of state for three models, w = − .
972 and C =1 .
858 (EDE1), w = − .
95 and C = 2 . G ( a ) = δ ( a ) /a normalized at z = ∞ for the three models. The shaded arearepresents the redshifts that we will be mainly probing in thepresent work ( z ∼ − 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 ΛCDM (dotted blue line), EDE1 (solidblack line) which has ( w , C ) = ( − . , . w , C ) = ( − . , . 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 low z 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 ( a eq is thematter-radiation equality scale factor):Ω EDE , sf = − (ln a eq ) − Z a eq Ω EDE ( a ) d ln a , (3)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 α 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 α 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 FieldGalaxy Redshift Survey (2dFGRS) [19, 20, 39, 40]. TheBAO can directly measure not only the angular diameterdistance, D A ( 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 of D A ( z ) and H ( z ) separately[42], one can only determine an effective distance [19]: D v ( z ) ≡ (cid:20) (1 + z ) D A ( z ) czH ( z ) (cid:21) / . (4)In this paper we use the Gaussian priors on the distanceratios r s ( z d ) /D v ( z ): r s ( z d ) /D v ( z = 0 .
20) = 0 . ± . ,r s ( z d ) /D v ( z = 0 .
35) = 0 . ± . , (5)with a correlation coefficient of 0 .
39, extracted from theSDSS and 2dFGRS surveys [40], where r s is the comovingsound horizon size and z d 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 . .
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 results .Furthermore, we make use of the Hubble Space Tele-scope (HST) measurement of the Hubble parameter H ≡ h km s − Mpc − by a Gaussian likelihoodfunction centered around h = 0 .
72 and with a standarddeviation σ = 0 .
08 [46].In order to constrain the early dark energy model athigh redshifts, we also include the GRBs, linear growthfactors and Ly α 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 . − .
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 σ inferred from theSDSS Ly α power spectrum [49, 50]; ii) the linear growthrate f ≡ Ω γ m from galaxy power spectrum at low redshift[51] and Ly α growth factor measurement obtained from 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. These data at different redshifts are not totally independent. Thederived σ 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. TABLE I. σ values and growth rates with 1 σ error barsfrom which we derived the linear growth factors used in ouranalysis. z σ σ σ Ref.2.125 0.95 0.17 [49]2.72 0.92 0.172.2 0.92 0.16 [50]2.4 0.89 0.112.6 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 0.17 z f σ f Ref.0.15 0.51 0.11 [51]3.00 1.46 0.29 [25] the SDSS Ly α power spectrum at z = 3 by Ref.[25].We have used two Ly α 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 α for-est sample presented in Ref.[54]. The SDSS Ly α forestdata set consists of 3035 QSO spectra with low resolu-tion ( R ∼ <
10 per pixel) span-ning a wide range of redshifts ( z = 2 . − . R ∼ >
50 per pixel)QSO spectra with median redshifts of z = 2 .
125 and z = 2 .
72, respectively. The flux power spectrum of theLy α 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 α ” together with the SDSS Ly α power spec-trum “SDSSLy α ”, since they are in agreement and thislatter has a stronger constraining power. More precisely,the VHSLy α 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 at z = 2 .
72, in the range 0 . < k (s/km) < .
03, while theSDSSLy α consists of a single measurement at z = 3 and k = 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 µ K-arcmin. f sky l max (GHz) θ fwhm ∆ T ∆ P two data sets, derived using numerical simulations of theobserved flux power, and we will label this as GFLy α .
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 . < z < .
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 takesthe magnitude dispersion to be 0 .
15 and the systematicerror σ sys = 0 . × z/ .
7. The whole error for each datais given by σ mag ( z i ) = q σ ( z i ) + 0 . /n i , where n i isthe number of supernovae of the i ′ th 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 < . σ µ = 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 α 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 α power spectrum, consistingof three data points at z = 2 , H ( z ) D A ( z )] − 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 ∼ − H ( z ) D A ( z )] − , will be measured at z = 6 . , ,
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 for z >
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 densityΩ c h ∈ [0 . , . b h ∈ [0 . , . n s ∈ [0 . , . A s ] ∈ [2 . , .
0] and theangular diameter of the sound horizon at last scattering θ ∈ [0 . , k s = 0 .
05 Mpc − .When CMB data are included, we also vary the opticaldepth to reionization τ ∈ [0 . , . −0.95 −0.9 −0.8500.20.40.60.81 w C σ −0.8 −0.6 −0.4 −0.2 000.20.40.60.81 w EDE (z lss ) −3 Ω EDE (z lss ) Ω EDE,sf
Figure 2: 1D current (marginalized) constraints onthe dark energy parameters w , C , as well as σ , w EDE ( z lss ), Ω EDE ( z lss ) and Ω EDE , sf , from different currentdata combinations: WMAP5+BAO+SN (red solid lines),WMAP5+BAO+SN+GRB+Ly α (blue dash-dot lines) andWMAP5+BAO+SN+GRB+GFLy α (black dashed lines). parameter H , the present matter fraction Ω m0 , σ , andΩ EDE , 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 . , . γ ,defined as: d ln δ ( a ) d ln a = Ω m ( a ) γ . (6)This parameter, introduced by [66], provides a good fitto the growth rate for many different cosmological mod-els and depends on the growth of perturbations, H ( z )and Ω 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 ΛCDM model ( γ ∼ .
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 w < − .
906 and
C < . TABLE III. Constraints on the early dark energy model from the current observations. Here we show the mean and the bestfit values. For some parameters that are only weakly constrained we quote the 95% upper limit.Parameter WMAP5+BAO+SN +GRB+Ly α +GRB+GFLy α Mean BestFit Mean BestFit Mean BestFit w < − . − . < − . − . < − . − . C < .
711 1 . < .
613 1 . < .
245 0 . m . ± .
014 0 .
258 0 . ± .
014 0 .
279 0 . ± .
014 0 . σ . ± .
049 0 .
734 0 . ± .
022 0 .
863 0 . ± .
024 0 . EDE ( z lss ) < . . < . . × − < . . × − Ω EDE , sf . ± . . . ± . . . ± . . γ − − . ± .
139 0 . − − on the present equation of state of dark energy compo-nent is still consistent with other recent results in theliterature [1, 45]. However, current observational data,could not determine the “running” of equation of statevery well. The reason is that the data we use are onlythe low redshift BAO and SNIa ( z < z ∼ < z < C , is rather poor.Consequently, current observations still allow verylarge amount of early dark energy at high redshift z ∼ EDE ( z lss ) < . w EDE ( z lss ). The first peak around w EDE ( z lss ) = 0is in fact caused by the poor constraint on C . When C islarger than one, the equation of state w EDE will approachzero at high redshift. The larger C is, the faster w EDE approaches w = 0 (we will come back to this point inSect. IV). Considering the weak constraint on C , thereis significant probability that allows w EDE ( z lss ) = 0. Thesecond peak around w EDE ( z lss ) = − w is close to − C isclose to 0, which is consistent with the data, w EDE ( z lss )reaches the cosmological constant value. Therefore, inorder to remove the first peak, we need a much morestringent constraint on C from future measurements.As we showed in Sect. II, early dark energy modelsalso affect structure formation [9]. From Fig. 1 we cansee that the linear growth factor of early dark energymodel is suppressed significantly by the large value ofparameter C . The non-negligible dark energy densityat high redshifts slows down the linear growth functionof the matter perturbation and leads to a low value of σ today. Using CMB+BAO+SNIa data combination,illustrated in Fig. 2, we obtain the limit on σ todayof σ = 0 . ± .
049 (68% C.L.), which is obviouslylower than one obtained in the pure ΛCDM framework: w σ −0.95 −0.9 −0.850.550.60.650.70.750.80.850.90.95 Ω EDE (z lss ) Ω E D E , s f −3 C σ Figure 3: 2D (marginalized) contours of someparameters from different current data combi-nations: WMAP5+BAO+SN (red solid lines),WMAP5+BAO+SN+GRB+Ly α (blue dash-dot lines)and WMAP5+BAO+SN+GRB+GFLy α (black dashedlines). σ = 0 . ± .
026 [1], while the error bar is enlargedby a factor of two. Therefore, the σ today and C areanti-correlated, as shown in Fig. 3. On the other hand,the σ and equation of state w at present also have ananti-correlation. When w is increased, the fraction ofdark energy density becomes large, which also leads thelower σ today.Finally, we discuss the average value of the fractionof dark energy in the structure formation era Ω EDE , sf .In Table III we can find that the current constraint isΩ EDE , sf = 0 . ± . σ confidence level. FromEq.(3) and Fig. 3 it is easy to see that a large Ω EDE ( z lss )leads to the high Ω EDE , sf . In Fig. 4 we also plot thetwo dimensional plots between Ω EDE , sf and other relatedparameters. As we discussed before, when w and C become large, the fraction of dark energy density at highredshifts will also be large. On the other hand, whenΩ EDE , sf increases, the linear growth function of matterperturbation will be slowed down and the value of σ today will decrease consequently: Ω EDE , sf and the σ w Ω E D E , s f −0.95 −0.9 −0.850.040.050.060.070.080.090.1 C Ω E D E , s f σ Ω E D E , s f w EDE (z lss ) Ω E D E , s f −0.8 −0.4 00.040.050.060.070.080.090.1 Figure 4: 2D (marginalized) contours of Ω
EDE , sf ver-sus w , C , σ and w EDE ( z lss ) from different currentdata combinations: WMAP5+BAO+SN (red solid lines),WMAP5+BAO+SN+GRB+Ly α (blue dash-dot lines) andWMAP5+BAO+SN+GRB+GFLy α (black dashed lines). today are then obviously anti-correlated.To sum up, using CMB, BAO and SNIa, we constrainthe 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 ( ∼ 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 α andGFLy α data, to improve our constraints. Because mostof GFLy α data are obtained from the Ly α forest powerspectrum, in order to avoid overestimating the weightof Ly α data, in our analysis we do not use GFLy α andLy α 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 α data.When adding the high-redshift probes, the constraintson w and C improve significantly: w < − .
958 and
C < .
245 at 95% confidence level, while their best fitvalues are very close to the pure ΛCDM model. Natu-rally, the 95% upper limit of Ω
EDE ( z lss ) has also beentightened by a factor of ten, Ω EDE ( z lss ) < . × − . 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 α improves the constraints on Ω EDE ( z lss ) 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 inthe distribution of w EDE ( z lss ) is still present, which im-plies that the current data are not accurate enough tomeasure w EDE ( z lss ), even when high-redshift probes areadded.Another effect of adding high-redshift probes is thelarger value of σ today. As we know, the Ly α forestdata favor a larger σ ∼ . . When we add the GFLy α or Ly α forest data, weobtain a higher σ today than before: σ = 0 . ± . σ ). However, this is still smaller than the value favoredby Ly α forest data alone in the pure ΛCDM model andconsistent with WMAP5+BAO+SN measurement of [1].We also note that if we use only VHSLy α data sets andadding them to WMAP5+BAO+SN and GRBs the re-sults do not change significantly as compared to havingSDSSLy α and VHSLy α together. This is mainly due tothe constraining power of the growth factors measure-ment obtained with VHSLy α .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 . C. Growth Index γ Finally, we extend our discussion by briefly addressingthe perspective of constraining the linear growth index γ .Measurements of the growth history of cosmic structurescombine information on both cosmic expansion and theunderlying theory of gravity, the parameter γ 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 γ = 0 . ± .
139 (1 σ ) error bar for our EDE model,which is in agreement with the ΛCDM and with thevalues obtained by Refs.[67, 68] (with slightly smallererror bars). The present error on γ is also similar tothat inferred from future weak lensing and SNIa data byRefs.[69]. This fact does not imply a contradiction or a failure of the ΛCDMmodel 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- α forest and weak-lensing measurements of the COSMOS-z survey [24] point to avalue of σ = 0 . ± . γ L w γ −0.98 −0.96 −0.940.30.40.50.60.70.80.91 C γ γ w EDE (z lss ) γ Ω EDE (z lss ) −3 γ Ω EDE,sf
Figure 5: The posterior distribution of γ and 2D (marginal-ized) contours of γ and w , C , w EDE ( z lss ), Ω EDE ( z lss ) andΩ EDE , sf from WMAP5+BAO+SN+GRB+Ly α (blue dottedlines).TABLE IV. Constraints on the early dark energy modelfrom future measurements. We also list the standarddeviation of these parameters based on the future mock datasets. For parameters that are only weakly constrained wequote the 95% upper limit instead.Parameter Fiducial ΛCDM Fiducial EDE w < − .
975 [ − . , − . C < .
635 [0 . , . m . . σ . . EDE ( z lss ) < . × − < . EDE , sf . . γ .
026 0 . In Figure 5 we show the 2-dimensional contours of γ vs. σ , C, w and Ω EDE ( z lss ) to address possible degeneraciesof this parameter. For the current data sets, we note thatthe degeneracies are weak and only the degeneracy in the γ − Ω EDE , sf plane is significant. This is easily understoodsince a larger γ 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 ΛCDM 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 −0.98 −0.94 −0.900.20.40.60.81 w C σ −0.8 −0.6 −0.4 −0.2 000.20.40.60.81 w EDE (z lss ) Ω EDE (z lss ) Ω EDE,sf
Figure 6: 1D current (marginalized) constraints on the darkenergy parameters w , C , as well as σ , w EDE ( z lss ), Ω EDE ( z lss )and Ω EDE , sf from future measurements with the fiducial mod-els: ΛCDM (red dashed lines), EDE1 model (black solidlines). w ( z ) =-0.95 w =-0.98 w =-0.99 w =-0.999 w =-0.9999 Figure 7: Evolution of different equation of state for theMocker model when fixing C = 2. All these models fit lowredshift ( z <
2) and CMB constraints but have very differentevolution at intermediate redshifts. in Fig.1.
A. Fiducial Λ CDM
Firstly, we choose the pure ΛCDM as the fiducialmodel. In Table IV we list the forecasts for some pa-rameters using all the future measurements described inSection II B 2: GRBs, SNIa, Ly α (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 w improves significantly, namely the95% upper limit is now w < − . w C −0.99 −0.97 −0.95 −0.93 −0.91 −0.890.511.522.53 Figure 8: 2D marginalized contours C vs w from future mockmeasurements of the fiducial EDE1 model. early dark energy density has been limited very stringent,Ω EDE ( z lss ) < . × − (95% C.L.), which is reducedby another factor of ten, compared with the present con-straint. The error on the γ parameter that we forecastis 0 . . × − . 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 C has not been im-proved, C < . w and keeping C = 2. We note that the smaller the red-shift at which w approaches the cosmological constant,the smaller the redshift at which w ( z ) flattens to zero inthe high redshift universe. For example, if w = − . w ( z = 10) ∼ − . C = 2 has been ruledout obviously. On the other hand, if w = − . w ( z = 10) ∼ −
1. Consequently, C = 2 is stillfavored by the data. Therefore, even if the constraint on w 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 of w EDE ( z lss ) 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, w = − .
972 and C = 1 . w and C . The 95% confidence level is − . 920 and 0 . < C < . σ confidence level. We also showthe contour in the ( w , C ) plane in Fig. 8. We note thateven with these futuristic data sets simulated in a con-servative way the value C = 0 is still allowed at a 2 σ confidence level. When w increases, C must decrease inorder to match the current observations. Thus, w and C are anti-correlated. Consequently, the allowed value ofΩ EDE ( z lss ) could be very large: Ω EDE ( z lss ) < . × − (95% C.L.). Furthermore, with this fiducial model, thedouble peak in the distribution of w EDE ( z lss ) disappears:because the pure ΛCDM is disfavored at more than 5 σ confidence level.In Fig. 6 we clearly appreciate the very distinctive pre-dictions of Ω EDE , sf and σ for the ΛCDM and EDE1 mod-els. The values obtained for Ω EDE , sf and σ are differentby 20% and 10%, respectively, and can be measured atvery high precision in the two cases.We are also interested in understanding which of thedifferent cosmological probes give the most stringent con-straints. In order to investigate this we perform threedifferent runs: we find that the best constraints are ob-tained when future Ly α and GFLy α are added to Planckand SNAP data, while adding AP only to Planck andSNAP results in the the worst constraints. In the thirdrun, we consider adding only GRBs mock data to Planckand SNAP and in this case the results are in betweenthose inferred from Ly α and AP test. The degeneraciesbetween different parameters are similar in three cases.We also checked that degrading the accuracy of the APtest to 10% as opposed to the chosen 5% has not a sig-nificant impact on the recovered parameters.We also consider the results in terms of γ has done inSection III C. We find that we could constrain γ to highsignificance in the two cases: the standard deviations are0 . 026 and 0 . γ are more precise and degeneracieswith Ω EDE ( z lss ) and Ω EDE , sf are stronger. V. CONCLUSIONS AND DISCUSSION With the availability of new data from SNIa, and theadvent of relatively new cosmological probes, such asLy α and GRBs in the high redshift universe, there hasbeen an increasing interest in the study of dynamical darkenergy models with a significant energy contribution inthe structure formation era. In this paper we present con-straints on a particular early dark energy model using thelatest observations. We use the Mocker parameterizationproposed by Ref.[32], but our results are general since theamount of dark energy at the last scattering surface andin the structure formation era have also been quoted inorder to better compare with other possible dark energyparameterizations [72].0 γ L w γ −0.98 −0.94 −0.90.450.50.550.60.650.7 C γ γ w EDE (z lss ) γ Ω EDE (z lss ) γ Ω EDE,sf Figure 9: The posterior distribution of γ and 2D (marginal-ized) contours of γ and w , C , w EDE ( z lss ), Ω EDE ( z lss ) andΩ EDE , sf from future measurements with the fiducial models:ΛCDM (red dashed lines), EDE1 model (black solid lines). We find that current constraints based on CMB, BAOand SNIa data limit the amount of dark energy density atthe last scattering surface to be 2%, while in the structureformation period is of the order of 6%. If we add highredshift GRBs and the Ly α forest data, in the form of amatter power spectrum or in the form of a growth factormeasurement, the constraints improve by a factor fiveand fifteen, respectively. The ΛCDM model is still a verygood fit to the data currently available and only upperlimits on the amount of dark energy at the last scatteringcould be quoted.We also forecast what could be done with future datasets in particular: Planck for the CMB; GRBs and SNIaluminosity distances as could be measured by SNAP; abetter measurement of Ly α matter power spectrum andgrowth factors as could be obtained from SDSS-III andthe AP test performed on HI 21 cm maps from LOFAR.With these futuristic mock data sets we simulate twodifferent fiducial models that are currently in agreementwith observations a ΛCDM model and an EDE modelwith about 6% contribution to the energy density in thestructure formation epoch. We find that the two mod-els will be clearly distinguishable at more than 5 σ con-fidence level. Thereby, the role of these high redshiftprobes is crucial in discriminating between pure ΛCDMand quintessence and/or modified gravity models.Furthermore, motivated by the fact that the growthfactors at high redshift seem to have the highest con-straining power, we also address the perspectives of mea-suring the parameter γ that parameterizes the growthrate of structures and is usually a very precise predic- tion for any quintessence and/or modified gravity model.Even in this case, we find that in the two models we couldconstrain γ to high significance: the standard deviationsare reduced by a factor of five and three, when comparedwith the current constraint, respectively.We conclude this Section with some words of cau-tion: while high redshift probes are important in con-straining early dark energy models, the obtained valuesfrom present and future constraints rely mainly on theLy α forest and its derived growth factors. These mea-surements, summarized partly in Table I, have been ob-tained in the framework of pure ΛCDM model and arevalid only when small deviations from this model are con-sidered. We believe that we are in such a situation here,where the amount of dark energy has little impact onobserved Ly α quantities and this effect is possibly de-generate with other parameters that have been alreadymarginalized over to obtain the final estimate. However,further work using both numerical hydrodynamical sim-ulations as long with an accurate control of systematicsshould be done in order to better check this (see for ex-ample Ref.[73]). Along the same lines, while the roleof SNIa as cosmological probes is somewhat reasonablywell established, those of GRBs is still controversial. Wetried to circumvent this problem by marginalizing overthe nuisance parameters that enter the calibration of theGRBs distance moduli but, even in this case, more obser-vational and theoretical work is needed in order to trustquantitatively the results. Even regarding the forecast-ing using the AP test, we tried to implement these con-straints in a conservative way, in a range of redshift thatcould be observed by LOFAR, without assuming that afull matter power spectrum could be measured from theobserved brightness temperature (this of course will havea fundamental impact especially at even higher redshiftsthan those considered here).However, even considering these cautious remarks, ourwork clearly emphasizes the capabilities of present andfuture data in an intermediate redshift range which ispoorly probed by the observations, to quantitatively con-strain the dark energy component and its evolution. 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