Combined analysis of Planck and SPTPol data favors the early dark energy models
IINR-TH-2020-019
Combined analysis of Planck and SPTPoldata favors the early dark energy models
Anton Chudaykin, a,b Dmitry Gorbunov, a,b Nikita Nedelko a,ca Institute for Nuclear Research of the Russian Academy of Sciences, b Moscow Institute of Physics and Technology,Institutsky lane 9, Dolgoprudny, Moscow region, 141700, Russia c Department of Particle Physics and Cosmology, Physics Faculty,M.V. Lomonosov Moscow State University,Vorobjevy Gory, 119991 Moscow, Russia
Abstract:
We study the implications of the Planck temperature power spectrum atlow multipoles, (cid:96) < , and SPTPol data. We show that this combination predictsconsistent lensing-induced smoothing of acoustic peaks within Λ CDM cosmology andyields the robust predictions of the cosmological parameters. Combining only thePlanck large-scale temperature data and the SPTPol polarization and lensing mea-surements within Λ CDM model we found substantially lower values of linear matterdensity perturbation σ which bring the late-time parameter S = σ (cid:112) Ω m / . . ± . into accordance with galaxy clustering and weak lensing measure-ments. It also raises up the Hubble constant H = 69 . ± .
00 km s − Mpc − thatreduces the Hubble tension to the . σ level. We examine the residual tension in theEarly Dark Energy (EDE) model which produces the brief energy injection prior torecombination. We implement both the background and perturbation evolutions ofthe scalar field which potential scales as V ( φ ) ∝ φ n . Including cosmic shear mea-surements (KiDS, VIKING-450, DES) and local distance-ladder data (SH0ES) to thecombined fit we found that EDE completely alleviates the Hubble tension while notdegradating the fit to large-scale structure data. The EDE scenario significantly im-proves the goodness-of-fit by . σ in comparison with the concordance Λ CDM model.The account for the intermediate-redshift data (the supernova dataset and baryonacoustic oscillation data) fits perfectly to our parameter predictions and indicatesthe preference of EDE over Λ CDM at σ . [email protected] [email protected] [email protected] a r X i v : . [ a s t r o - ph . C O ] M a y ontents Λ CDM constraints 9 A L test 12 n = 3 n to a free parameter 24 Measurements of the cosmic microwave background (CMB) have provided a profoundinsight into the nature of the Universe. Temperature and polarization anisotropiesof the CMB encode detailed information about the composition and evolution ofour Universe. Indeed, the primary CMB is one of the most powerful probes of theearly-Universe cosmology coming from the last scattering. It also provides cosmolog-ical (model-dependent) probe of the late-Universe composition through observationof the angular scale of the sound horizon at last scattering θ s . Besides, the late-time physics leaves distinctive imprints on the CMB anisotropy at small angularscales. In particular, the small scale distortions caused by gravitational lensing [1]and Sunyaev–Zeldovich effect [2] provide a unique probe of the Universe evolutionat late times. Our capability in tackling all these subtle effects strongly depends onthe quality and self-consistency of observed CMB maps.The most profound observation of CMB anisotropy over the full sky has beenaccomplished by the Planck satellite [3]. This mission provides with measurements ofcosmological parameters at the percent level accuracy thus exemplifying the poten-tial of CMB surveys as a high-precision probe of cosmology. However, the extensive– 1 –nalyses of the plethora of CMB data from this spacecraft have revealed internaltensions within the Planck dataset. The most prominent feature often called ’lensingtension’ refers to overly enhanced lensing smoothing of the CMB peaks compared tothe Λ CDM expectation [3–6]. A recently developed technique for probing the gravi-tational lensing potential from CMB data in a model-independent way reveals . σ tension between the full Planck dataset and Λ CDM expectation [7]. This anomalyraises the issue of whether the Planck dataset is internally consistent. The Planckcollaboration has thoroughly examined this problem and presented the cosmologicalparameter constraints from different multipole ranges [3, 6]. They found a moderatedisagreement ( (cid:38) σ ) in parameters extracted from multipoles (cid:96) < and (cid:96) > of CMB temperature anisotropy power spectrum (TT). The similar trend in drivingcosmological parameters has been revealed for the full Planck dataset including po-larization measurements albeit with a lower statistical significance ( (cid:46) σ ). Althoughthe Planck collaboration argued that the exclusion of power spectrum at high multi-poles does not change their baseline result, its impact might be crucial for extensionsof Λ CDM model. A considerably less tension originates from the overall deficit of thePlanck TT power spectra across the multipole range (cid:46) (cid:96) (cid:46) . Remarkably, thisfeature is strongly disfavoured by extra smoothing of acoustic CMB peaks observedby Planck on small angular scales which pull the amplitude of power spectra tolarger values. These internal tensions within the Planck dataset emphasize the needfor independent measurement of the CMB anisotropies, especially on small scales.An important consistency check of CMB anisotropy measurements at small scalescan be provided by ground-based telescopes. The small-scale CMB anisotropies canbe measured by the South Pole Telescope (SPT) [8–10] and the Atacama CosmologyTelescope (ACT) [11, 12]. These observations supplement the satellite-based mea-surements since the ground-based telescopes are sensitive to much smaller angularscales unattainable in full sky surveys. The most sensitive measurements to date ofsmall-angular scale temperature anisotropy have been performed by the SPT observa-tion of 2540 deg SPT-SZ survey [13]. The SPT measurements of CMB temperatureanisotropy at multipole range < (cid:96) < augmented with WMAP-7 observa-tions of large angular scales (cid:96) (cid:46) prefer a slightly smaller lensing amplitude whichis at the lower end, but within the σ prediction of Λ CDM model [13]. Thereby,SPT-SZ survey provides consistent temperature power spectrum down to arcminutescales which is compatible with theoretical predictions of the Λ CDM model.Measurements of CMB polarization anisotropies is a promising tool to obtainmore robust parameter inference. Since E-mode measurements are expected tobe fractionally less contaminated by foregrounds than temperature measurements[14, 15], the E-mode auto-power spectrum (EE) and the temperature-E-mode cor-relation (TE) have higher potential to study very small angular scales. In particu-lar, polarization measurements demonstrates high sensitivity to the photon diffusiondamping tail of the CMB power spectrum enabling tighter constraints on cosmolog-– 2 –cal parameters. Besides, TE and EE power spectra provide a powerful consistencycheck of lensing smoothing effect which can shed light on the internal tensions in thePlanck dataset.The most accurate measurements to date of TE and EE spectra have been pro-vided by SPTPol analysis of 500 deg survey [16]. These data are the most sensitivemeasurements of TE and EE spectra at (cid:96) > and (cid:96) > , respectively. Theseobservations reveal less smoothing of CMB acoustic peaks as allowed by Λ CDM whichlevel is . σ below the Λ CDM expectation and . σ lower than the value preferedby the Planck collaboration. It disproves stronger smoothing of the acoustic scalesobserved in the Planck maps. Altogether, the most sophisticated ground-based exper-iments provide consistent measurements of the temperature and polarization CMBanisotropies which do not find compelling evidence for the enhanced smoothing ofthe acoustic peaks. Conversely, they found a mild deficit of the lensing power in theCMB anisotropies on small scales which may be hints of new physics.Besides measurements of the CMB power spectrum, the lensing potential canbe directly extracted from quadratic estimators of T -, E - or B -fields. Independentmeasurement of lensing power spectrum C φφ(cid:96) represents an important cross-check ofthe CMB anisotropies on small scales and can verify the lensing tension observedin the Planck maps. Lensing potential power spectrum C φφ(cid:96) in the multipole range < (cid:96) < was recently constrained from the 500 deg SPTPol survey [17, 18].This measurement based on a minimum-variance estimator that combines both tem-perature and polarization CMB maps predicts a . σ lower value of lensing power ascompared to the Planck Λ CDM cosmology. Thereby, SPTPol measurements exhibitsimilar trends in both CMB power spectra and lensing potential power spectrumprobes.Alongside with the CMB measurements which bring only model-dependent pa-rameter constraints in the late Universe, the local measurements support direct ex-traction of cosmological parameters at low redshifts. These probes also provide an-other important consistency check of CMB results in a concrete cosmological model.Intriguingly, there is a modest level of discrepancy between the Planck results anddirect probes within the Λ CDM model. In particular, the amplitude of linear den-sity fluctuations σ deduced from galaxy cluster counts [19–23], cosmic shear and/orgalaxy clustering measurements [24–28] is distinctly lower than the value of σ sug-gested by Planck. Combining different cosmic shear surveys (KiDS, VIKING-450and DES) [29] leads to the substantially tighter constraint S = 0 . +0 . − . where S = σ (cid:112) Ω m / . is the principal-component parameter for weak gravitational lens-ing analyses. This measurement is in tension with the Planck value S = 0 . ± . [3] at the level of . σ . Recently, this joint analysis has been improved by including Actually, the DES redshifts in the joint analysis [29] was recalibrated using deep public spec-troscopic surveys. Adopting these revised redshifts, the results give a . σ reduction in the DES-inferred value of S making the DES results compatible with KiDS and VIKING [24, 25]. – 3 –dditional small-scale information yielding S = 0 . +0 . − . , that exacerbates thetension with the Planck to . σ [30]. In our work, we stick to the joint cosmic shearanalysis [29] which applies a more conservative approach of excluding the small-scaleinformation.More controversial tension between low-redshift and high-redshift data is at-tributed to the determination of the present-day expansion rate of the Universe.Planck measurements based on CMB temperature, polarization and lensing powerspectra accommodate the model-dependent estimate H = 67 . ± . [3]. Tra-ditional distance-ladder measurements made up Cepheids and Type Ia Supernovaefavour generally higher values of the Hubble constant [31–33]. Upon improved cal-ibration of the Cepheid distance-ladder in the Large Magellanic Cloud, the SH0EScollaboration presents the most severe constraint H = 74 . ± .
42 km s − Mpc − which tightens the tension with the Planck measurement to . σ [34]. Remarkably,the direct measurement based on Type Ia Supernovae is quite robust against thechoice of distance indicators. A number of other techniques which has been used tocalibrate the SN luminosity distances (distant megamasers [35], Miras or variable redgiant stars [36], strongly lensed quasars [37]) gives coherent results competitive withthe Cepheid-based measurement [34] albeit with lager H uncertainties, and onlythe Tip of the Red Giant Branch (TRGB) as a distance measure brings moderatelysmaller value of the Hubble constant H = 69 . ± . − Mpc − [38]. Intriguingly,all distance-ladder measurements are perfectly consistent with a completely differ-ent technique of the Hubble constant measurement based on strong gravitationallensing time delays which dictates H = 73 . +1 . − . km s − Mpc − [39]. A cosmologymodel-independent analysis for a flat Universe accumulating strong gravitation lens-ing and supernova data supports this estimate yielding H = 72 . +1 . − . km s − Mpc − [40].Besides direct measurements based on local distance anchors, there is a powerfulinverse-distance-ladder approach which makes use of the baryon acoustic oscillations(BAO) measurements and an independent determination of the sound horizon r drag .For the base- Λ CDM model, this approach can be used to constrain H withoutusing any CMB measurements. The combination of BAO data and primordial deu-terium abundance measurements augmented with supernova distance data [41] or thelate-time probe of the matter density [42] leads to the Hubble constraints, in per-fect agreement with the Planck measurements. The anisotropic BAO measurementscome from galaxy clustering and the Ly α forest along with a precise estimate ofthe primordial deuterium abundance yield nearly the same Hubble constant [43–45].It is worth nothing that the recent full-shape power spectrum analyses [46–48] alsobring the Hubble measurement into agreement with the Planck data. However, all The recent TRGB result [38] exhibit somewhat lower agreement with the analysis of gravita-tionally lensed quasars at the level of . σ . – 4 –hese measurements are obtained in the context of concordance Λ CDM cosmologicalmodel and hence they represent highly model-dependent estimates of H . Given thisreason, in our work we focus on the local Cepheid-based distance-ladder measure-ment [34] which does no rely on early universe physics and being the most widelyused local probe to date.In turn, the ground-based measurements of the CMB anisotropies dictate theestimates consistent with local measurements. Comparison between the Planck andSPT-SZ temperature power spectra reveals that adding the small-scale data fromSPT-SZ survey patch drives σ shifts in various cosmological parameters from thePlanck expectation values [49]. In particular, the temperature power spectrum withWMAP7-based Gaussian prior on τ yields H = 75 . ± . − Mpc − and σ =0 . ± . [13]. While the amplitude of density fluctuation σ agrees well withboth the Planck results and the direct probes at low redshifts, the H measurementexhibits more than σ discrepancy with the Planck value being fully consistent withthe distance-ladder estimate [34]. Measurements of CMB polarization power spectraexhibit a similar trend for the Λ CDM parameters to drift away from the Planckvalues as the SPTPol range is extended to higher multipoles. The SPTPol baselineresult reads H = 71 . ± .
12 km s − Mpc − and σ = 0 . ± . [16] that revealsa mild preference for lower σ and a . σ upward shift in H from the Planck value.Above findings demonstrate a perfect consistency of the CMB measurementsfrom the ground with the local probes at low redshifts . On the other hand, the pa-rameter fit based on the ground-based telescopes reveal a distinctive difference withthe Planck baseline cosmology. The bulk of this discrepancy is caused by an excessof the lensing-induce smoothing in the Planck maps at small scales. Since the SPTexhibits a higher sensitivity to small scales and it does not detect the excess of lensingpower at large multipoles, the idea of replacing the Planck spectrum to that observedby the SPT seems rather natural. Inspired by this simple idea, we present a com-bined data approach which utilises both Planck and SPT measurements providingconsistent parameter inference from both large and small angular scales. For that,we restrict the multipole range of the Planck TT spectrum to (cid:96) < and combineit with the SPTPol measurements of TE and EE power spectra at < (cid:96) ≤ .This allows us simultaneously to get rid of the lensing tension, which affects thePlanck spectrum at high multipoles, and take benefit from the ground-based experi-ments where their sensitivity surpass that of the Planck measurements. We proceedin similar fashion to Ref. [51] that combines the CMB damping tail from the par-tial SPT-SZ survey and seven-year WMAP temperature power spectrum to improveconstraints on cosmological parameters. We do not include the TT spectrum at the The similar concordance between Planck large-scale temperature anisotropies and direct mea-surements of linear matter density perturbation was obtained in Ref. [50]. They claim . σ tensionbetween the Planck CMB temperature power spectrum at high multipoles (cid:96) > and variousprobes of clustering statistics. – 5 –igh multipoles from SPT-SZ survey since this measurement has lower statisticalpower as compared to the Planck TT spectrum at (cid:96) < and does not influenceour parameter constraints.An alternative avenue in handling the tensions between the low and high redshiftmeasurements is a modification of the cosmological concordance model. Indeed,the parameter fit based on the CMB data is highly model dependent and thereforeinfluenced by possible extensions of Λ CDM model. Many attempts have been made torestore the concordance between different dataset by modifying either the early or thelocal Universe physics. The late-time solutions include modified [52–54], interacting[55–59], viscous [60–62] and phenomenologically emergent dark energy models [63,64], interaction between the dark sectors [65–68] and decaying dark matter [69–72]. Recent studies indicate that modification of the Universe expansion history inthe two decades of scale factor evolution prior to recombination is needed to solvethe H problem [73, 74]. The popular early-time solutions are Early Dark Energy(EDE) [75–79] and strong scattering interactions between the neutrinos or betweenother additional light relics [80–82]. A significant progress in clarifying differentdiscrepancies has been made with cosmology independent reconstruction techniqueof the Universe evolution [83–85]. For the present status of the Hubble tension onecan see [86].In this paper, we examine possible tensions between the Planck cosmology andastrophysical data with a suitable modification of the early-time cosmology. As areference, we consider the EDE model. The cornerstone of our research is the novelapproach of data analysis. Our framework allows one to combine the Planck andthe SPTPol data in a consistent way. This technique is free from the lensing tensioninherent in the Planck spectra on small scales and provides with robust determinationof the cosmological parameters.Previous EDE analyses [76, 77, 87] have shown this model can alleviate theHubble tension, but the late-time amplitude of density fluctuation σ increases ascompared to Λ CDM, increasing tension with large-scale structure data. Ref. [88]greatly updates previous analyses by considering large-scale structure data in detail.They found that additional weak gravitational lensing and galaxy clustering datasubstantially weaken the evidence for EDE, as result of the tension between thelarger values of S needed to fit the CMB and SH0ES data in the EDE scenario andthe lower values of this parameter measured by large-scale structure surveys. In thiswork, we show that the combined data approach restores concordance amongst thesemeasurements allowing for large H values without substantially degrading the fit tolarge-scale structure data. For the first time, we derive EDE constraints from thePlanck large-scale CMB anisotropy and the SPTPol data, which are consistent bothwith SH0ES data and weak lensing measurements along with other direct probes ofclustering statistics.The paper is organized as follows. In Section 2 we introduce all relevant datasets– 6 –nd describe our numerical procedure. In Section 3 we give a detailed description ofour analysis method and present parameter constraints in Λ CDM model. Section 4specifies the background and perturbed dynamics of EDE model. In Section 5 wepresent our final parameter constraints utilizing both the CMB and the low-redshiftsmeasurements within the EDE model. We discuss especially the importance of theintermediate redshift data composed of the supernova dataset and BAO measure-ments for cosmological inference. Finally, we conclude in Section 6.
In our analysis we exploit several different combinations of the following datasets. • The Planck temperature
Plik likelihood truncated at multipoles ≤ (cid:96) < and complemented with Commander power spectrum at low multipoles (cid:96) < [3]. We include all nuisance parameters and impose the same priors as in thePlanck analysis [3] for those. We refer to this likelihood as PlanckTT - low (cid:96) . • The Planck EE likelihood at low multipoles (cid:96) < [3]. Since the measurementof polarization anisotropies at large scales is essential to constrain the opticaldepth τ which breaks the harmful degeneracy between the amplitude of CMBspectra A s and τ we include this likelihood in all datasets. • CMB polarization measurements from the 500 deg
SPTPol survey which in-cludes TE and EE spectra in the multipole range < (cid:96) ≤ [16]. To obtaincosmological constraints we marginalise posteriors over six foreground parame-ters ( D PS EE , A T T , A EE , A T E , α T E , α EE ), the super-sample lensing variance κ ,two instrumental calibration terms ( T cal , P cal ) and two beam uncertainty shotnoises ( A , A ). We also impose flat priors on the first four of these andGaussian priors on the rest in full compliance with [16]. We apply appropriatewindow functions to transform theoretical spectra from unbinned to binnedbandpower space. We denote this likelihood as SPTPol . • Measurements of the lensing potential power spectrum C φφ(cid:96) in the multipolerange < (cid:96) < from the 500 deg SPTPol survey [17]. The lensingpotential is reconstructed from a minimum-variance quadratic estimator thatcombines both the temperature and polarization CMB maps. To incorporatethe effects of the survey geometry we convolve the theoretical prediction for C φφ(cid:96) with appropriate window functions at each point in the parameter space. Wealso perturbatively correct C φφ(cid:96) for changes due to the difference between therecovered lensing spectrum from simulation and the input spectrum followingRef. [18]. We refer to this measurement as Lens in our analysis.– 7 – Combined weak lensing measurements of S = 0 . ± . [29] provided bya homogeneous analysis of KiDS, VIKING-450 and DES cosmic shear surveys.Strictly speaking, the full large-scale structure likelihoods can be approximatedby a simple Gaussian prior only in the concordance Λ CDM model. However, theauthors in [88] have justified this procedure for EDE models showing that theinformation content in large-scale structure data is almost enterily containedin the S constraint. Guided by this observation, we impose the appropriateGaussian prior on S and refer to this measurement as S . • The Cepheid-based local measurement of the Hubble constant H = 74 . ± .
42 km s − Mpc − [34]. We implement the SH0ES measurement as the Gaus-sian prior on H and call it as H . • Baryonic Acoustic Oscillation measurements based on the consensus BOSSDR12 analysis [89] which combines galaxy samples at z = 0 . , . , and . .We name this likelihood BAO . • Luminosity distances of supernovae Type Ia coming from the Joint Light-curveAnalysis (JLA) using SNLS (Supernova Legacy Survey) and SDSS (Sloan Dig-ital Sky Survey) catalogues [90]. We denote this low-redshift probe as SN .As far as we implemented the SPTPol likelihoods within Montepython environ-ment and the publicly release was based on another MCMC sampler
CosmoMC itmakes sense to distribute our code among science community. Our likelihood codeis publicly available at https://github.com/ksardase/SPTPol-montepython and canbe used for various cosmological analyses.
All theoretical calculations are carried out in the publicly available Boltzmann
CLASS code [91]. We adopt spatially flat Universe and assume normal neutrino hierarchypattern with the total active mass (cid:80) m ν = 0 . eV. To investigate the EDE modelwe modify background and perturbation equations in CLASS in full compliance withSec. 4. To recover the posterior distributions in Λ CDM and EDE models we applythe Markov chain Monte Carlo (MCMC) approach. For that we use the publiclyavailable MCMC code
Montepython [92, 93]. Marginalized posterior densities, limitsand contours are produced with the latest version of the getdist package [94] .To calculate the effects of CMB lensing we use the Halofit suit [95, 96] whichallows for modelling the small-scale nonlinear matter power spectrum. These cor-rections are negligible for the lensed CMB power spectra but become relevant for https://pole.uchicago.edu/public/data/henning17 https://pole.uchicago.edu/public/data/lensing19 https://github.com/cmbant/getdist – 8 –arameter PlanckTT - low (cid:96) SPTPol PlanckTT - low (cid:96) +SPTPol Base100Ω b h . ± .
040 2 . ± .
048 2 . ± .
026 2 . ± . c h . ± .
003 0 . ± .
005 0 . ± .
002 0 . ± . H . ± .
72 70 . ± .
12 70 . ± .
15 69 . ± . τ . ± .
009 0 . ± .
010 0 . ± .
009 0 . ± . A s ) 3 . ± .
019 2 . ± .
031 3 . ± .
018 3 . ± . n s . ± .
012 0 . ± .
023 0 . ± .
008 0 . ± . r drag . ± .
64 146 . ± .
34 146 . ± .
52 145 . ± . m . ± .
020 0 . ± .
026 0 . ± .
013 0 . ± . σ . ± .
013 0 . ± .
021 0 . ± .
010 0 . ± . S . ± .
037 0 . ± .
052 0 . ± .
026 0 . ± . Table 1 . Parameter constraints in the standard Λ CDM model with σ errors. Polarizationmeasurements at low multipoles are included in all datasets, see Sec. 2.1. The Base datasetincludes PlanckTT - low (cid:96) +SPTPol+Lens . the calculation of the lensing potential power spectrum C φφ(cid:96) at high multipoles. Weinclude the Halofit module in all the data procedures. Λ CDM constraints
As a preliminary step, we examine a consistency between
PlanckTT - low (cid:96) and SPT-Pol datasets. For that we calculate posterior distributions for relevant cosmologicalparameters which are shown in Fig. 1. We reveal that the parameter constraintsinferred from PlanckTT - low (cid:96) (blue contours) and SPTPol (red contours) likelihoodsare perfectly consistent within σ . Thus, the two likelihoods are consistent witheach other, and we proceed to combine them. To quantify the difference between ourapproach and the standard data procedure we compare posterior distributions forthe combined likelihood PlanckTT - low (cid:96) + SPTPol (black contours) and the Planckbaseline analysis Planck 2018 (green contours). One may observe that the resultingposteriors deviate from each other at (cid:46) σ . In particular, our combined approachprovides substantially lower S and higher H as compared to the Planck baselineanalysis, thus alleviating the tensions with local measurements.Parameter constraints inferred from PlanckTT - low (cid:96) , SPTPol and its combina-tion are listed in Tab. 1. We found that combining PlanckTT - low (cid:96) and SPTPoldatasets one significantly improves constraints on all cosmological parameters ex-cept for τ which mean value and errorbar remain intact. It happens because thecorresponding constraint is driven by the CMB polarization measurements on largescales alone which, in turn, is included in all datasets. We stress that the value of S is now consistent with the homogeneous analysis of cosmic shear surveys [29] and– 9 – .
60 0 .
66 0 .
72 0 .
78 0 . S . . . Ω c h . . . . . . A s e − τ . . n s H . . . σ .
20 2 .
25 2 .
30 2 .
35 2 . Ω b h . . . . . S .
10 0 .
11 0 . Ω c h . . . . . . A s e − τ .
95 1 .
00 1 . n s
66 68 70 72 74 76 H .
70 0 .
75 0 . σ Planck 2018SPTPolPlanckTT - low ‘ PlanckTT - low ‘ +SPTPolPlanckTT - low ‘ +SPTPol+SPTLens (Base) Figure 1 . Marginalized parameter constraints for the Λ CDM model using differentdatasets. We explore independently
PlanckTT - low (cid:96) and SPTPol , combined likelihood
PlanckTT - low (cid:96) +SPTPol along with SPTPol lensing, PlanckTT - low (cid:96) +SPTPol+Lens . Forcomparison we include constraints from the baseline Planck analysis PlanckTTTEEE aswell. with other direct probes of clustering statistics [19–28]. The
PlanckTT - low (cid:96) +SPTPol dataset also mitigates the Hubble tension to the . σ level.To break the degeneracy between the matter power spectrum σ and the totalmatter density Ω m we include measurements of the lensing potential power spectrum.The Lens likelihood is primarily sensitive to the parameter combination σ Ω . m which– 10 –ataset Planck 2018 best-fit Base best-fit Planck TT , (cid:96) <
30 23 .
41 20 . Planck EE , (cid:96) <
30 396 .
19 396 . Planck TT , ≤ (cid:96) < .
93 404 . .
04 142 . .
79 5 . Total χ .
36 969 . Table 2 . χ values for the best-fit Λ CDM model to the baseline Planck 2018 cosmologyand Base dataset. breaks the degeneracy between σ and Ω m thus providing a more robust constrainton the late time parameter S . The posterior distribution for the PlanckTT - low (cid:96) +SPTPol+Lens dataset is depicted in Fig. 1 (dashed black lines) and the correspondingparameter constraints are tabulated in Tab. 1. We report that adding the Lenslikelihood does not induce any significant shifts in parameter constraints except for S which mean value increases by . σ with somewhat reduced error. It revealsthe remarkable agreement between PlanckTT - low (cid:96) +SPTPol and Lens datasets thathighlights a self-consistency of our approach. In what follows we refer to the combinedlikelihood
PlanckTT - low (cid:96) +SPTPol+Lens as the Base set. Final constraints from theBase dataset read S = 0 . ± . , H = 69 . ± .
00 km s − Mpc − We found that S value is completely consistent with the local probes, whereas theHubble tension persists at the . σ level. We address this residual tension with oneearly-time solution in Sec. 5.To justify statistical agreement amongst different likelihoods which constitute theBase dataset, we examine the goodness-of-fit to the CMB anisotropies as quantifiedby the χ -statistic. In Tab. 2 we list the χ values for each CMB likelihood for thebest-fit Λ CDM model to the baseline Planck 2018 cosmology (second column) andBase dataset (third column). We found that the Base dataset significantly improves χ -statistic for each likelihood except for the Planck EE (cid:96) < data with respectto the Planck 2018 baseline analysis. It is caused by internal tensions within Planckdata which results in different cosmological inference from low and high multipolesof Planck power spectra. The main culprit of this discrepancy is the overly enhancedlensing smoothing of the CMB peaks which pulls the late-time amplitude σ to ahigher value whilst the SPTPol measurements favour lower values of this parameter.Our data procedure is free from this tension and hence harvests a major improvementin the fit to both the Planck TT (cid:96) < and SPTPol likelihoods ( ∆ χ = − . ).Meanwhile, we found that the Base dataset notably worsens the χ -statistic for– 11 –PTPol likelihood ( ∆ χ = 3 . ) with respect to a pure SPTPol analysis whichgives χ = 138 . . A L test Here we present an important consistency check yielding the self-consistency of cos-mological constraints obtained using the combined data approach. Specifically, weverify that the Base dataset provides with a consistent lensing-induced smoothing ofacoustic peaks within the Λ CDM cosmology. For that, we introduce a free param-eter A L , which scales the C φφ(cid:96) at each point in the parameter space. Thus scaledlensing potential power spectrum is used to lens the CMB power spectra. Thereby, A L controls the theoretical prediction for the lensing-induced smoothing effect. Ifthe theory is correct and the data is not affected by systematic effects, one expects A L = 1 . In what follows we examine Λ CDM model by making use of the combineddata approach.Varying A L and 6 standard cosmological parameters of the Λ CDM concordancemodel we obtain almost identical posterior distributions with our baseline resultsoutlined in Fig. 1 (dashed black line) and listed in Tab. 1 (Base dataset). We find A L = 0 . ± . . (3.1)This result allows us to conclude that our combined data approach is free from thelensing tension and provides consistent measurements of cosmological parametersin Λ CDM model. In general, combining the Planck TT power spectrum in themultipole range < (cid:96) < with the SPTPol measurements of TE and EE spectraat < (cid:96) ≤ enables one to obtain an unbiased cosmological inference from bothlarge and small angular scales. To handle the residual tensions within Λ CDM model we resort to an early-time solu-tion. For the reference, we consider EDE that behaves like a cosmological constant atearly times and then dilutes away with the Universe expansion like radiation or fasterat later times. If the dilution starts near the matter-radiation equality it results in alarger Hubble constant and a smaller value of the baryon-photon sound horizon thusalleviating the tensions with the direct cosmological probes. In this Section, we statethe homogeneous and perturbed dynamics in the EDE sector pursuing generalityand simplicity aims for subsequent implementation in Sec. 5.First, we introduce one simple realization of EDE in the form of the scalar field.Then, we consistently describe the homogeneous dynamics of EDE field and evolutionof its linear perturbations within an effective-fluid approach [97]. We implement all EE polarization measurements at large scales are also considered in this analysis. – 12 –ecessary equations in
CLASS code and verify that they govern correct backgroundand perturbation dynamics. Finally, we obtain actual constraints on both Λ CDMand EDE parameters, examine various tensions and highlight the importance of
BAO+SN dataset for cosmological inference.It is worth noting that the effective fluid description is applicable framework totrack the scalar field dynamics as advocated in [87]. The previous work [77] claimedthe wrong conclusion about the validity of using the approximate fluid approach.Why that study could not fully recover the results of Ref. [76] is easily explained bythe different choice of the potential as shown in [87].
We consider EDE in the form of the cosmological scalar field φ with a power-lawpotential V n ( φ ) = V φ n n (4.1)where V denotes the potential amplitude and n states the power-law index. Atearly times, the Hubble friction dominates and the scalar field undergoes a "slow-roll" evolution. During this stage the scalar field is frozen at its initial value φ i actingas a pure dark energy with equation of state ω e (cid:39) − . Once the Hubble parameterdrops below a critical value m (which is determined by a form of potential through m (cid:39) ∂ V n /∂φ ), the field starts to oscillate around minima of its potential. Duringthe oscillating period the field amplitude decreases in time which dilutes the fieldenergy density. For n = 1 the field undergoes simple harmonic oscillations witha frequency which is independent of its amplitude and for n > the oscillationsare anharmonic and the frequency depends on the amplitude. Rapidly oscillatingsolutions can be modeled by an effective, averaged-over-cycle description with theeffective equation of state ω e (cid:39) ω n given by ω n = n − n + 1 . (4.2)To provide a smooth transition between the slow-roll and the oscillatory periodswe parameterise the energy density of the EDE field averaged over the oscillationperiod by ρ e ( a ) = 2 ρ e ( a c )1 + ( a/a c ) ω n +1) (4.3)where a c refers to a transition between the two regimes. An associated effectiveequation of state for the EDE fluid reads ω e ( a ) = 1 + ω n a c /a ) ω n +1) − (4.4)which asymptotically behaves as ω e ( a ) → − at a → and ω e ( a ) → ω n (4.2) for a (cid:29) a c . In Fig. 2 we show the evolution of EDE fraction in the total energy density– 13 – -7 -6 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 ρ φ / ρ t o t a n=2n=3n=4 -2.5-2.0-1.5-1.0-0.50.00.51.010 -5 -4 -3 -2 -1 a ω e c a2 c s2 k=0.001 Mpc -1 c s2 k=0.01 Mpc -1 c s2 k=0.1 Mpc -1 Figure 2 . Left panel: evolution of the energy density of the scalar field relative to totalenergy density of the universe for several values of n Right panel:
The scalar fieldequation of state, the adiabatic sound speed c a and the effective sound speed c s for various k as a functions of scale factor. Both panels: evolution of all quantities is obtained with f e = 0 . , z c = 3000 . The vertical dashed black line refers to z c = 3000 . of the Universe averaged over the scalar field oscillations (4.3) for several values of n (4.1) and the effective equation of state of EDE fluid (4.4). On can see that thescalar field contribution to the total energy budget of the Universe dilutes faster forlarger n .Ultimately, the background evolution of the EDE field is described by parameters ρ e ( a c ) and a c . In what follows it is convenient to use the maximum fraction of thetotal energy density in this field f e estimated over the all evolution history, f e , f e ≡ max (cid:26) ρ e ρ tot (cid:27) all a (4.5)We note that f e does not necessarily coincide with ρ e ( a c ) /ρ tot ( a c ) used in (4.3). The effective fluid approach is a powerful tool to describe not only a backgroundevolution but also a perturbed dynamics of the rapidly oscillating scalar field. Thisformalism provides with a set of approximate ’averaged-over-cycle’ equations in in-terms of fluid variables. In order to utilize the approximate perturbation equationsof motion one needs the adiabatic sound speed c a and the effective sound speed c s .The adiabatic sound speed can be straightforwardly calculated because it dependsonly on background quantities c a ≡ ˙ P e ˙ ρ e = ω e − ˙ ω e ω e ) H , (4.6)where H ≡ ˙ a/a and the dot refers to the derivative with respect to the conformaltime. In Fig. 2 we depict an evolution of the adiabatic sound speed (4.6) as a function– 14 –f the scale factor. During the slow-roll evolution of the scalar field using (4.4) theadiabatic sound speed equals c a = − n + 1 n + 1 . (4.7)Once the scalar field starts to oscillate, the adiabatic sound speed reaches ω e (4.4).In order to compose the approximate perturbation equations we also need thetime-averaged effective speed sound in the fluid’s rest frame. We adopt the resultfrom [97] c s ≡ (cid:68)(cid:68) δP e δρ e (cid:69)(cid:69) = 2 a ( n − (cid:36) ( a ) + k a ( n + 1) (cid:36) ( a ) + k , (4.8)where k denotes the conformal momentum and (cid:36) is the instantaneous oscillationfrequency of the field fluctuations, which for a pure power law potential is given by[97] (cid:36) ( a ) (cid:39) H ( a c ) (cid:114) πn (2 n −
1) Γ( n n )Γ( n n ) 1[1 + ( a/a c ) / ( n +1) ] n − . (4.9)We note that c s = 1 for a slowly rolling scalar field but it deviates from once thefield starts to oscillate. This feature of models with an early-time energy injection iscrucial since c s < over a large range of k is needed to resolve the Hubble tensionaccording to Ref. [78].In the right panel of Fig. 2 we plot the evolution of the effective sound speed(4.8) with the scale factor for three values of k . It is worth noting that the stan-dard effective fluid approach assumes an abrupt changing in the relevant quantities,specifically this implies c s = 1 and (4.7) at a < a c , and (4.8) with (4.6) at a > a c .We verify that using the approximate, average-over-cycle quantities, (4.8) and (4.6),rather than the exact ones during all Universe evolution does not lead to significanteffects on the predicted power spectra used to constrain the EDE model.Finally, the equations which govern the evolution of density and velocity pertur-bations in the synchronous gauge read ˙ δ e = − (cid:34) u e + (1 + ω e ) ˙ h (cid:35) − c s − ω e ) H δ e − c s − c a ) H u e k , ˙ u e = − (1 − c s ) H u e + 3 H ( ω e − c a ) u e + c s k δ e , (4.10)where we introduce the heat-flux u e ≡ (1 + ω e ) θ e since the linear perturbation equa-tions written in terms of the bulk velocity perturbation θ e exhibit rapid growth beinghard to track numerically. In principal, one should impose adiabatic initial conditionsfor δ e and u e on super-Hubble scales. However, the EDE component is always sub-dominant on superhorizon scales at early times and its perturbations fall inside thegravitational potential created by the radiation component immensely fast. Giventhis reason, we take δ e = u e = 0 initially since these quantities quickly approach ageneric solution at radiation dominated stage [98].– 15 – -14 -12 -10 -8 -6 -4 -2 -5 -4 -3 -2 -1 δ e a k=0.001 Mpc -1 k=0.01 Mpc -1 k=0.1 Mpc -1 -12 -10 -8 -6 -4 -2 -5 -4 -3 -2 -1 θ e ( + ω e ) a k=0.001 Mpc -1 k=0.01 Mpc -1 k=0.1 Mpc -1 Figure 3 . Left panel: the evolution of the density contrast of the scalar field for a set ofmomentum k . Right panel: the evolution of the heat-flux for a set of k . Both panels: theevolution of the quantities is obtained for f e = 0 . , z c = 3000 . The vertical dashed blackline refers to z c = 3000 . In Fig. 3 we depict the evolution of δ e and u e with the scalar factor for a set ofmomentum k . As long as c s = 1 , the pressure support leads to a strong decreasein the perturbation amplitude for both superhorizon and subhorizon modes. Once c s < , see (4.8), the field internal pressure support decreases, that yields nearlyconstant late-time density and velocity EDE perturbations at a > a c , seen in Fig. 3. EDE scenario can provide a larger Hubble constant and a smaller value of the baryon-photon sound horizon. While the former trend presents an opportunity to resolvethe Hubble tension, the latter feature allows one to reconcile the local measurementsof cosmic distance-ladder with the CMB observations. Indeed, H +BAO+SN bringsa model-independent late-time estimate of the baryon-photon sound horizon at thedrag epoch r drag (decoupling of protons from the cosmic plasma), which is signifi-cantly lower than the CMB-inferred value within the Λ CDM cosmology [99, 100].This mismatch indicates a need for modification of the early-time physics [73, 74].In particular, reducing the CMB-inferred sound horizon at the radiation drag epochby − would reconcile the CMB-inferred constraints with the local H and r drag determinations. It can be accomplished by an early energy ejection prior to recombi-nation [75, 76]. In our study we examine this possibility within the EDE frameworkusing the combined data analysis introduced in Sec. 3.In order to describe the background and perturbed dynamics of the EDE field wemodify the CLASS
Boltzmann code implementing (4.3), (4.4) and (4.10). We checkthat these equations produce sensible physical outputs such as power spectra of theEDE fluid. To derive the cosmological constraints within the MCMC approach oneneeds to specify the parameter space of the EDE model.– 16 –he EDE model with a pure power-law potential (4.1) is fully specified by threetheory parameters: the redshift z c when the scalar field starts to oscillate, the energydensity of the EDE field ρ e ( z c ) at z c and the power-law index n which parameterisesthe potential (4.1). Due to large scatter in z c and ρ e ( z c ) variables we consider loga-rithmic priors on these parameters . We note that absence of the initial scalar fieldvalue φ i in the definition of the effective sound speed parameter (4.8) is caused bythe power-law nature of our potential (4.1), see Ref. [87]. Eventually, the parameterspace of the EDE model is characterized entirely by Log ( z c ) , Log ( ρ e ( z c )) , n and6 standard parameters ω b , ω c , h , τ , ln(10 A s ) , n s . In what follows we consider n aseither fixed or free parameter in our fitting procedure.One comment is in order here. We emphasize that the fitting function calibrationimplemented in the Halofit module remains valid for EDE cosmology since the modelscapable of addressing the H tension require f e (cid:46) . which implies a small deviationfrom the Λ CDM expectation. More accurate justification for the validity of usingthe Halofit module is provided in [88]. Given this reason, we make use of the Halofitmodule in all EDE analyses. n = 3 To impose robust constraints on the cosmological parameters, in what follows we fixthe shape of energy injection which is determined by parameter n , see (4.1). The dataseem to favour lower values of the index n since they provide a larger peak energyinjection fraction for a fixed width of the transition from slow-roll to oscillatoryregime resulting in a larger Hubble constant and smaller value of the baryon-photonsound horizon [77, 87]. On the other hand, lower n models have slowly decayingwith the scale factor tail as shown in Fig. 2. Given this reason, it is difficult forthese models to inject sufficient energy in a relatively narrow time interval priorto recombination while not having significant residual energy towards low redshiftswhich, in turn, adversely affects the fit to the CMB data. The fit to cosmological datais thus driven by the two competing effects: injecting the largest possible amountof energy before recombination and dilution of this energy as quickly as possible inthe post-recombination era. As a compromise between low and high values of n wechoose n = 3 in our baseline analysis; this choice agrees with the recent EDE studies[87, 88]. It is worth mentioning that we do not consider n = 2 which corresponds to For the axion-like potential the uniform priors imposed on (physical) particle physics parameters(the axion decay constant and its mass) seriously downweight the preference for EDE models incomparison to uniform priors placed on the effective EDE parameters ( f e and Log ( z c ) ) [88]. Infact, this problem is attributed to the search of a proper theoretical solution that would matchthe prediction of our effective approach. Besides the axion-like potential which does not cover allphysical realisations of the EDE scenario, there is a great variety of other EDE setups [79, 101, 102]which have different physical priors. In our analysis, we hold a phenomenological point of viewand vary the effective parameters, f e , Log ( z c ) , which parameterise the EDE dynamics in a modelindependent way. – 17 –arameter ΛCDM EDE n = 3Base+ S +H Base+S +H Base+S +H +BAO+SN100Ω b h . ± .
022 2 . ± .
037 2 . ± . c h . ± .
001 0 . ± .
005 0 . ± . H . ± .
72 73 . ± .
26 72 . ± . τ . ± .
008 0 . ± .
009 0 . ± . A s ) 3 . ± .
017 3 . ± .
017 3 . ± . n s . ± .
006 0 . ± .
008 0 . ± . ( z c ) − . ± .
20 3 . ± . ( ρ e ( z c )) − . ± .
70 2 . ± . f e − . ± .
025 0 . ± . r drag . ± .
35 140 . ± .
19 139 . ± . m . ± .
008 0 . ± .
008 0 . ± . σ . ± .
007 0 . ± .
012 0 . ± . S . ± .
015 0 . ± .
018 0 . ± . Table 3 . Parameter constraints in the standard Λ CDM and EDE models with σ errors.Polarization measurements at low multipoles are included to all datasets, see Sec. 2.1. TheBase dataset includes PlanckTT - low (cid:96) +SPTPol+Lens . the EDE diluting away like radiation: it induces the phenomenon of self-resonanceresulting in exponential growth of EDE perturbations [87]. Since we solve onlylinear perturbation equations for the EDE fluid, our approximate framework can notaddress the resonance phenomenon properly.The posterior distributions of the relevant cosmological parameters based on Base + S + H likelihood are shown in Fig. 4 (red contours). The correspondingparameter constraints are listed in Tab. 3. We found that the Hubble tension is fullyresolved in the EDE scenario. Moreover, the fit does not degradate the S constraintremaining within σ interval with its local prediction [29]. Eventually, the dataset Base+S +H imposes S = 0 . ± . and H = 73 . ± .
26 km s − Mpc − . Ouroutcomes are qualitatively similar to results of the previous EDE analyses [77, 87, 88]but, in contrast to them, we claim a perfect consistency with the local measurementsowing to the combined data approach introduced in Sec. 3.A better fit to the local measurements is provided by the early-energy injec-tion just before recombination. Indeed, the marginalized constraint on the redshifttransition reads Log ( z c ) = 3 . ± . which indicates the relatively brief energyinjection before recombination. In turn, the maximal injected EDE fraction equals f e = 0 . ± . . We emphasize that a quite brief period of the energy injectionaround recombination minimizes the impact on other successful Λ CDM predictionshence being essential for the good fit to CMB. The most peculiar background feature– 18 – .
03 0 .
06 0 .
09 0 .
12 0 . f e . . . H . . . . . σ . . . . . . S r d r ag . . . . L og ( z c ) .
108 0 .
120 0 . Ω c h . . . . . f e . . . . . H .
76 0 .
78 0 .
80 0 .
82 0 . σ . . . . . . S
132 138 144 r drag .
25 3 .
50 3 .
75 4 .
00 4 . Log ( z c ) ΛCDM BaseΛCDM Base+S +H EDE Base+S +H EDE Base+S +H +BAO+SN Figure 4 . Marginalized parameter constraints for the Λ CDM and EDE models usingvarious datasets. We explore the parameter space using the
Base + S + H dataset withand without intermediate redshift probe BAO+SN . For comparison we include constraintsfrom the Base data set only in Λ CDM model. The Base dataset includes
PlanckTT - low (cid:96) +SPTPol+Lens . in models with early-time energy injection is a positive correlation between ω c and f e as plotted in Fig. 4. This behaviour is attributed to the early integrated Sachs–Wolfeffect which fixes the height of the first CMB temperature acoustic peak. Indeed,the extra energy injected by the oscillating scalar field can be compensated by largerdark matter abundance thereby keeping the Sachs–Wolf amplitude intact. At the– 19 –ataset Λ CDM EDE
Planck TT , (cid:96) <
30 20 .
22 20 . Planck EE , (cid:96) <
30 396 .
24 395 . Planck TT , ≤ (cid:96) < .
00 404 . .
68 142 . .
67 4 . .
13 0 . .
58 0 . Total χ .
52 968 . Table 4 . χ values for the best-fit Λ CDM and EDE models to the
Base+S +H dataset. level of perturbations, the bigger dust component in the early Universe shifts thematter-radiation equality to an earlier epoch. It elongates the matter dominationstage and affects the evolution of matter perturbations. In particular, it increases alate-time amplitude of matter fluctuations probed by σ . Thus, obtaining a biggerHubble parameter and a smaller sound horizon at the drag epoch within the modelswith energy injection near recombination is accompanied by an increase of σ that isshown in Fig. 4. Furthermore, the late-time parameter S grows as well, that wouldhelp to distinguish EDE models from other competing solutions.To justify the inclusion of Gaussian priors on S and H parameters we resortto statistical analysis. The χ -statistic for each likelihood in the Λ CDM and EDEfits to the
Base+S +H dataset is given in Tab. 4. First, it is instructive to comparethe Λ CDM fits to CMB measurements for the Base and
Base+S +H datasets, the χ statistic for the former set is provided by Tab. 2 (third column). We found thatthe goodness-of-fit to CMB data is moderately degraded upon imposing Gaussianpriors on S and H ( ∆ χ = 3 . ), predominantly driven by the worsened fitto the Planck TT ≤ (cid:96) < data. This change indicates that the distance-ladder Hubble measurement and CMB data are in tension within Λ CDM cosmology.Second, we confront the CMB fits to the
Base+S +H likelihood in Λ CDM and EDEmodels. We found a significantly improved CMB fit in the EDE cosmology comparedto Λ CDM ( ∆ χ = − . ), driven primarily by the restored concordance of thePlanck TT ≤ (cid:96) < likelihood. Furthermore, the EDE model notably improvesthe χ -statistic for CMB data using Base+S +H dataset with respect to the Λ CDMfit to the Base dataset ( ∆ χ = − . ). It implies that EDE restores concordanceamongst CMB and SH0ES measurements providing even better fit to CMB data incomparison with the Λ CDM model without any priors on S and H . We emphasizethat the EDE model allows for larger H values without substantially degrading thefit to the cosmic shear measurements. This result justifies the inclusion of additionallarge-scale structure data within EDE cosmology. It is instructive to compare ourresults with Ref. [88] that hints at the potential for additional large-scale structure– 20 –ikelihoods to substantially constrain the EDE models. The tight EDE constraintsfound there arise from the use of the full Planck likelihoods which pull the late-timeamplitude σ to higher values [3, 5, 6]. Thus, in order to simultaneously fit theCMB and SH0ES data in the EDE model, one needs even higher values of the late-time amplitude, thereby conflicting with weak lensing and other large-scale structureprobes. In contrast, when combining the Planck temperature power spectrum at lowmultipoles (cid:96) < and SPTPol data, we found substantially lower values of σ allowing for the EDE solution of the Hubble tension being consistent with large-scale structure data.An important cross-check of our parameter constraints can be provided by in-termediate redshift astrophysical data. The BAO+SN dataset represents a late-timeprobe of the parameter combination r drag h . An appealing characteristic of this mea-surement consists in its direct nature since the supernova data allow for translatingthe cosmic distance scale from the BAO observations to a given redshift in a model-independent way. Then, one can calibrate the r drag h measurement with the helpof the local Hubble probe [34] to obtain a model-independent determination of thebaryon-photon sound horizon at the drag epoch r drag . On the other hand, the stan-dard ruler r drag can be obtained through the CMB measurements but this inferencestrongly depends on the cosmological model assumption. If the late-time probe of r drag given by BAO+SN+H matches its early-time estimate within the EDE scenarioprovided by the CMB, it will be strong evidence in favour of the EDE model. Weexamine this possibility by including the BAO+SN dataset in what follows.Resulting constraints from the
Base+S +H +BAO+SN dataset are shown in Fig. 4(blue contours). We observe that the Hubble constant measurement is completelyconsistent with that from the Base+S +H dataset, H = 72 . ± .
26 km s − Mpc − .On the contrary, the value of S underwent substantial raising by . σ upwards to S = 0 . ± . . The further increase of S is disfavoured by the direct probes ofclustering statistics. It means that the upcoming galaxy and weak lensing surveyswill elucidate the potential of the EDE fluid to completely reconcile the cosmologicaltensions.The main advantage of the intermediate redshift astrophysical data is that ityields the absolute scale for the distance measurements (anchor) at the oppositeend of the observable Universe. In particular, H +BAO+SN provides with a late-time model-independent probe of r drag which value can be contrasted with the CMBinference under the assumption of the EDE model. We find that inclusion of BAO+SN drives the sound horizon at the drag epoch downwards by . σ with the nearlyidentical error, r drag = 139 . ± . . This value demonstrates a perfect consistencywith the EDE prediction. Reducing of r drag with respect to the concordance Λ CDMmodel reflects the main property of early-time energy injection models. For instance,the sound horizon at the drag epoch in the EDE scenario is diminished in comparisonwith the Λ CDM consideration by for both Base and
Base+S +H datasets that– 21 –ataset Λ CDM EDE
Planck TT , (cid:96) <
30 21 .
35 20 . Planck EE , (cid:96) <
30 395 .
66 397 . Planck TT , ≤ (cid:96) < .
94 405 . .
90 141 . .
37 4 . .
01 1 . .
16 0 . .
06 4 . .
38 683 . Total χ .
83 1658 . Table 5 . χ values for the best-fit Λ CDM and EDE models to the
Base+S +H +BAO+SN data. agrees with the previous investigations [73, 100] . Regarding the EDE sector, weobtain f e = 0 . ± . which indicates a . σ evidence for nonzero EDE fraction.To assess the concordance of BAO + SN data with the rest measurements, weprovide the χ -statistic for each likelihood in the Λ CDM and EDE fits to the
Base+S +H +BAO+SN dataset in Tab. 5. The Λ CDM fit exhibits a slight tension withthe BAO measurements at . σ level . This mild tension is completely alleviatedin the EDE scenario. The fit to supernova data is not worsened either in the EDEmodel compared to Λ CDM. Regarding the EDE fit to
Base+S +H dataset given inTab. 4 (third column), we observe nearly the same contributions of all likelihoods tothe χ -statistic except for the cosmic shear measurements which notably degrade theEDE fit ( ∆ χ = 1 . ) highlighting the specific role of future large scale structuredata. These results emphasize the remarkable agreement between BAO + SN and
Base+S +H likelihoods which justifies the combination of these measurements inone dataset.Finally, it is instructive to examine the parameter constraints in the plane ( S , H ) with and without BAO + SN dataset. We show the corresponding pos-terior distributions for EDE and Λ CDM models in Fig. 5. The negative correlationbetween S and H in Λ CDM model can be readily understood. The parameterconstraints in this case is mostly driven by the CMB measurements which impose a It worth noting that our constraints on the sound horizon r drag in the Λ CDM cosmology given inTab. 1 dictate substantially lower values of this parameter with respect to the Planck measurements[3]. It is caused by higher values of H dictated by our combined data approach which, in turn,results in lower values of r drag . Given this reason, the reduction in sound horizon at drag epoch by will suffice to reconcile the local Hubble measurements with CMB. The value χ = 7 . is distributed as χ N with effective degrees of freedom N = 4 givenby number of data points ( D A and H at three different redshifts) minus sum of fitting parameterswhich parametrize the theory prediction ( ω c and h ). – 22 – . . . . . . . H . . . . . . . . . . S EDE Base+S +H ΛCDM Base+S +H . . . . . . . f e
68 69 70 71 72 73 74 75 76 77 H . . . . . . . . . S EDE Base+S +H +BAO+SNΛCDM Base+S +H +BAO+SN . . . . . . . f e Figure 5 . The marginalized posterior distribution in the plane ( S , H ) for two data setcombinations Base+S +H ( left panel ) and Base+S +H +BAO+SN ( right panel ) in the Λ CDM and EDE models. The scattered points represent values of f e . The gray bandsrepresent the σ and σ constraints on S and H coming from [29] and [34]. The Basedataset includes PlanckTT - low (cid:96) +SPTPol+Lens . tight constraint on the observed angular size of the sound horizon at last scattering θ ∗ . For the base Λ CDM model, the main parameter dependence of θ ∗ is approxi-mately described by ω c · h which should be constant. Given this reason, the higherHubble constant implies lower ω c . Since ω c is positively correlated with σ , the CMBanisotropy in the Λ CDM model yield smaller values of Ω m and σ thus providing alower value of S for higher H , see Fig. 5 (dashed black line). In the EDE case, thedegeneracy direction in the plane ( S , H ) is altered. The reason of that consists indifferent perturbation dynamics in the EDE model witch results in amplified late-time fluctuations of matter density probed by σ . It ensures a negative correlationbetween S and H parameters shown in Fig. 5 (black line). Once we include theintermediate redshift data BAO + SN , the positive correlation between S and H parameters within the EDE scenario becomes more pronounced.To understand quantitatively which model ( Λ CDM or EDE) is preferable, wecorroborate our analysis based on posterior distributions with χ -analysis. For that,we compare the differences in logarithmic likelihoods log L calculated for these twomodels in their respective best-fit points for the same datasets. Each difference ∆ log L is distributed as χ with effective degrees of freedom equal to the differencein the number of free parameters in Λ CDM and EDE models. Herein this numberequals (recall in this Section we fix n = 3 ) corresponding to two extra parametersin the EDE model, Log ( ρ e ( z c )) and Log ( z c ) . Resulting improvements are shownin Tab. 6.Our statistical analysis reveals that the EDE scenario strongly improves thegoodness-of-fit by . σ compared to Λ CDM. The preference does not change if one– 23 –ata set ∆ χ p-value Improvement Base+ S +H . . . σ Base+ S +H +BAO+SN 12 .
04 0 . σ Base+ S +H +BAO+SN ( n free) .
04 0 . . σ Table 6 . A statistical improvement of EDE over Λ CDM in fitting the several data sets. Inthe top panel we set the power-law index to n = 3 (2 extra degrees of freedom) wheres inthe bottom line n is treated as a free parameter (3 extra degrees of freedom). includes the astrophysical data at intermediate redshifts BAO+SN that indicates theremarkable agreement between the early and the late-time cosmological inferenceswithin the EDE model. We argue that the early-time energy injection model pro-vides a much better description of the CMB and the low-redshift measurements ofcosmological parameters. n to a free parameter So far, we have considered n = 3 , based on the cosmological considerations. However,from the phenomenological point of view the shape of power-law potential (4.1) is notknown a priory . Hence, it is instructive to explore a broader range of energy injectionshapes which are controlled by n . To this end we allow the power-law index n tofloat freely in the fit. To choose the appropriate range for n we emphasize that fora significantly large n there is a specific class of no oscillatory (power law) solutionswith an asymptotically constant equation of state [103, 104]. Since the resolutionof the Hubble tension requires oscillatory solutions which ensure c s < over a largerange of k [78], we expect that the region of large n is strongly disfavoured by thedata [87]. In our fitting procedure we impose the following flat prior on n ∈ [1 , .Our resulting posterior distributions are shown in Fig. 6. As discussed above, thedata tends to favor lower values of the index n , since they allow for a larger peak en-ergy injection fraction. On the other hand, models with lower n have slowly decayingtail which adversely affect the fit to CMB data. Given this reason, the models with n < dilute the injected energy too slow and therefore are disfavoured, see the leftpanel of Fig. 6. The right panel of Fig. 6 clearly illustrates that larger values of theHubble constant are favoured for < n < . Our outcome is qualitatively consistentwith the previous EDE analyse [77] but extends this study to smaller values of n . Ithas been established owing to effective fluid description which allows for averagingthe rapid field oscillations no matter how high the rate is. An explicit numericalsolution of the field equations is very time consuming exercise that can make the os-cillating field dynamics intractable. For instance, computational complexity restrictsthe previous analysis performed in Ref. [77] to the region n > . It explains why ourparameter constraints are somewhat different at n ∼ as compared to those in [77].– 24 – . . . . . . . . . . n . . . . . . . . f e EDE Base+S +H +BAO+SN . . . . . . . . . . n H EDE Base+S +H +BAO+SN .
00 0 .
04 0 .
08 0 .
12 0 . f e . . . . . Log ( z c ) n Figure 6 . A marginalized posterior distribution in the planes ( f e , n ) and ( H , n ) us-ing the Base + S + H + BAO + SN dataset in the EDE model. The gray bands repre-sent the σ and σ constraints on H coming from [34]. The bottom panel represents 1dmarginalized distributions for f e , Log ( z c ) and n parameters. The Base dataset includes PlanckTT - low (cid:96) +SPTPol+Lens . It is worth noting that the existence of the resonance at n = 2 does not effect ourresults since the resonance width is too narrow to be captured in our analysis [87].We verified that the EDE perturbations remain linear and never become comparableto the homogeneous amplitude during the Universe evolution.We carry out the χ statistical analysis for the case of varying power-low index n . Our result is represented in Tab. 6. We do not find any improvement withvarying index n compared to the case n = 3 . Somewhat lowering of the overallimprovement for the free n analysis is caused by the penalty of adding one additionalEDE parameter, i.e. n . We confirm that the final result is almost insensitive to thevalue of n > supporting the claim of Ref. [76]. We formulated a new method to analyse the CMB data, that combines the Planckand the SPTPol data in a consistent way. This approach benefits from both full-sky observations and ground-based experiments, and yields an unbiased parameter– 25 –nference. Using this approach we examine various cosmological tensions in the Λ CDM and the early-time energy injection model. We list our conclusions below. • . σ tension between the CMB constraints and the cosmic shear measurementspreviously declared in Ref. [29] is completely alleviated in the Λ CDM model.It also concerns other local probes of the late-time amplitude of matter densityperturbation [19–28]. The upward shift of σ inferred from the Planck analysisis solely driven by an excess of the lensing-induce smoothing of acoustic peaksin the Planck spectra which is absent in our approach. Our resulting constrainton the late-time parameter read S = σ (cid:112) Ω m / . . ± . . • Accounting for only the CMB measurements substantially diminishes the Hub-ble tension with local distance-ladder [34] from . σ to . σ in the Λ CDMcosmology. The combined fit to the Planck and the SPTPol data drives the Λ CDM fit to the remarkably higher value of the Hubble constant, H =69 . ± .
00 km s − Mpc − . • The residual tension with the local distance-ladder measurement of the Hubbleconstant [34] is completely alleviated in the EDE scenario. Using
Base+S +H dataset we find H = 73 . ± .
26 km s − Mpc − . At the same time, it does notdegrade the fit to the direct probe of the late-time amplitude [29] leading to S = σ (cid:112) Ω m / . . ± . . It relieves the conflict between the SH0ES-resolving EDE cosmologies and large-scale structure data recently claimed inRef. [88]. • The intermediate redshifts data
BAO+SN present important consistency checkof the EDE model since it provides an independent probe of r drag h at latetimes. We found that the parameter constraints driven by H +BAO+SN are inexcellent agreement with the EDE prediction. In particular, the Base+S +H +BAO+SN dataset yields somewhat lower value of the Hubble constant H =72 . ± .
26 km s − Mpc − . At the same time, it results in a substantially higheramplitude of late-time matter perturbation characterized by S = 0 . ± . ,but remaining within σ interval with its local measurement [29]. • The cosmological data disfavour the early-time injection models with the power-law indexes n < . Furthermore, setting free the power-law index n we do notfind any improvement in comparison with the case of n = 3 .We emphasize that ongoing and future galaxy probes such as DESI, Euclid andLSST will clarify the potential of early-time energy injection model to completelyalleviate various cosmological tensions.– 26 – cknowledgments We thank Mikhail Ivanov for helpful discussions. We also thank Vivian Poulin forvaluable comments. The work was supported by the RSF grant 17-12-01547. All nu-merical calculations were performed on the Computational Cluster of the Theoreticaldivision of INR RAS and the MVS-10P supercomputer of the Joint SupercomputerCenter of the Russian Academy of Sciences (JSCC RAS).
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