The implications of an extended dark energy cosmology with massive neutrinos for cosmological tensions
Vivian Poulin, Kimberly K. Boddy, Simeon Bird, Marc Kamionkowski
TThe implications of an extended dark energy cosmology with massive neutrinos forcosmological tensions
Vivian Poulin, Kimberly K. Boddy, Simeon Bird, and Marc Kamionkowski
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA andDepartment of Physics and Astronomy, UC Riverside, Riverside, CA 92512, USA
We perform a comprehensive analysis of the most common early- and late-Universe solutions tothe H , Ly- α , and S discrepancies. When considered on their own, massive neutrinos provide anatural solution to the S discrepancy at the expense of increasing the H tension. If all extensionsare considered simultaneously, the best-fit solution has a neutrino mass sum of ∼ . H tension, while weakened, remains unresolved. Motivated bythis result, we perform a non-parametric reconstruction of the evolution of the dark energy fluiddensity (allowing for negative energy densities), together with massive neutrinos. When all datasetsare included, there exists a residual ∼ . σ tension with H . If this residual tension remains inthe future, it will indicate that it is not possible to solve the H tension solely with a modificationof the late-Universe dynamics within standard general relativity. However, we do find that it ispossible to resolve the tension if either galaxy BAO or JLA supernovae data are omitted. We findthat negative dark energy densities are favored near redshift z ∼ .
35 when including the Ly- α BAO measurement (at ∼ σ ). This behavior may point to a negative curvature, but it is mostlikely indicative of systematics or at least an underestimated covariance matrix. Quite remarkably,we find that in the extended cosmologies considered in this work, the neutrino mass sum is alwaysclose to 0 . H tension is solved orsignificantly decreased. I. INTRODUCTION
The concordance ΛCDM model of cosmology is verysuccessful in explaining the large-scale structure (LSS)of the Universe; it passes a number of precision tests anddescribes well observations of the cosmic microwave back-ground (CMB) from the
Planck satellite [1]. However,with the increasing precision and sensitivity of variousinstruments, interesting tensions have emerged. A recentdirect measurement of the local value of the present dayHubble rate H [2] shows a > σ tension with the inferredvalue from CMB observations [1]. Furthermore, there is along-standing discrepancy between LSS surveys and theCMB determination of the quantity S = σ (Ω M / Ω ref M ) α ,where σ is the amplitude of matter density fluctuationsin spheres with radius of 8 h − Mpc, Ω M is the relic den-sity of matter in the Universe today, and Ω ref M is a normal-ization. Measurements of S from galaxy clustering andweak lensing surveys (such as CFHTLenS [3], KiDS [4, 5],DES [6], and Planck SZ cluster counts [7]) are all smaller(between 2 σ and 4 σ ) than the CMB prediction. Finally,the BOSS DR11 baryon acoustic oscillation (BAO) mea-surements from the Ly- α auto-correlation analysis andcross-correlation with quasars have a reported ∼ . σ tension with the flat ΛCDM Planck prediction [8]. Thesignificance of this discrepancy is reduced by recent in-creases in the size of the dataset, perhaps suggestinga statistical fluctuation combined with a mildly non- The values of Ω ref M and α vary between experiments, but they areoften set to 0 . .
5, respectively.
Gaussian covariance matrix [9], but a 2.3 σ tension re-mains with the latest DR12 data [10]. There have beenvarious efforts to resolve these tensions with different cos-mological models, usually classified as either early- orlate-Universe solutions [11–22]. These attempts often fo-cus on solving one of the tensions, using specific datasetsto fit simple extensions of ΛCDM. However, these exten-sions are inconsistent when additional datasets constrain-ing late-Universe expansion quantities, such as the BAOscale or the luminosity distance from type Ia supernovae(SNe Ia), are incorporated [2, 16, 19].In this paper, we consider a wide range of datasetsmeasuring both the early- and late-Universe propertiesto see if a coherent model emerges. We focus on massiveneutrino solutions to the S problem, because they arethe less “theoretically costly”: oscillation experiments in-dicate that neutrinos must have non-zero masses. More-over, massive neutrinos reduce the growth of perturba-tions below their free-streaming length [23], and dedi-cated studies point to a neutrino mass sum (cid:80) m ν ∼ . H measurements: the valueof (cid:80) m ν results in a lower Hubble rate inferred fromthe CMB, ultimately exacerbating the H tension. Weapproach the H , Ly- α , and S tensions in two ways.We first attempt to solve all tensions simultaneously bycombining the most common early- and late-Universe ex-tensions of ΛCDM. We incorporate massive neutrinos,and we allow for an additional ultra-relativistic species Another class of potential solutions involves interacting [14, 20,21] or decaying dark matter [11–13] in an isolated dark sector. a r X i v : . [ a s t r o - ph . C O ] S e p with ∆ N eff and an arbitrary effective sound speed c and viscosity speed c . We model the dark energy (DE)sector as a fluid whose equation of state is given by theCPL parameterization w ( a ) = w + (1 − a ) w a [29]. Us-ing Planck
CMB data [30],
Planck
SZ data [7], and therecent H measurement [2], we find that resolving the H and S tensions simultaneously require phantom-likeDE [31] and (cid:80) m ν ∼ . H ( z ) [32] and the DE equation of state w ( z ) [33], ouranalysis differs in several ways. In the former analy-sis [32], only data measuring the late-Universe expansionare considered. This requires a prior on the sound hori-zon at baryon drag r drag s and diminishes the constrainingpower on the matter and baryon energy densities, ω m and ω b . In the latter analysis [33], the behavior of w ( z )strongly deviates from the nominal case of a cosmologicalconstant with w = − ExDE ( z ) to take on both positive andnegative values. Although we assign this energy densityto the DE sector, it can also be thought of as a proxy forany number of new species that could collectively giverise to the arbitrarily complicated dynamics favored bythe CMB and low-redshift data. Hence, it can indicatethat the energy density in another sector must decrease(as is the case, for instance, if part of the dark matter isdecaying or if the Universe has an open geometry). Nat-urally, this can also indicate a strong inconsistency in thedata.With our formalism, we are able to solve the H , Ly- α , and S tensions and achieve compatibility with theCMB, LSS, and either galaxy measurements of the BAOscale or measurements of SNe Ia. There is a ∼ . σ ten-sion with H that persists when all datasets are includedin our analysis, a finding consistent with previous stud-ies [32, 33]. This is because the BAO and SNe Ia dataprefer slightly different expansion histories at late times,ultimately forcing the behavior of the ExDE to be veryclose to that of a cosmological constant below z < .
6. Ifthis residual tension remains in the future, it would indi-cate that it is not possible to solve the H tension solelywith a modification of the late-Universe dynamics withinstandard general relativity. We have additionally allowedfor an extra ultra-relativistic fluid, but it neither affectsthe reconstruction nor helps reduce the tension. More-over, we find that the Ly- α BAO measurements favor negative values of Ω
ExDE ( z ) at z ∼ .
5. We discuss pos- sible explanations of such behavior, but stress that thismay point to systematics in the data. Last but not least,we find that the neutrino mass sum is close to 0 . H tension is solved or significantly decreased. Wehave verified that this finding remains true when includ-ing A lens as a free parameter [34].This paper is organized as follows. Section II is devotedto a preliminary discussion on the H and S tensions andparticular solutions. We perform an in-depth analysis ofa combination of the most common extensions to ΛCDMadvocated to solve these tensions in Section III, followedby an agnostic approach in Section IV. From this recon-struction, we discuss in Section V models that explainthis behavior and therefore provide a solution to the S , H , and Ly − α tensions without spoiling the successfuldescription of other probes. II. PRELIMINARY CONSIDERATIONS
In this section, we discuss how the present-day Hub-ble rate H and the quantity S are measured or inferredfrom observations, and we comment on the discrepanciesseen between experiments. We then discuss the standardextentions of ΛCDM that are most often invoked in theattempt to reconcile these discrepancies. Although cer-tain cosmological models may lessen tensions with spe-cific data sets, no solutions are robust to the inclusion ofadditional datasets such as the BAO or SNe Ia. A. Datasets and analysis procedure
We summarize the various datasets considered in theremainder of this work. • CMB: In Section III, we use the
Planck (cid:96)
TT, TE, and EE power spectra [30] with a gaussianprior on τ reio = 0 . ± . Planck lens-ing likelihood [36]. In Section IV, we instead usethe lite version of this dataset to decrease the con-vergence time of our likelihood analysis. We haveverified that doing so has no impact on our conclu-sions, apart from slightly increasing the error barson the fitted cosmological parameters. • LSS: We use the measurement of the halo powerspectrum from the Luminous Red Galaxies SDSS-DR7 [37] and the full correlation functions fromthe CFHTLenS weak lensing survey [3]. We alsouse the S measurement from the Planck
SZ clus-ter counts [38], since it is at the heart of the claimed S discrepancy. Although not included in our like-lihood analysis, we later assess whether our bestfit model can accommodate the S measurementsfrom KiDS [5] and DES1 [39]. • SH0ES: We use the SH0ES measurement of thepresent-day Hubble rate H = 73 . ± .
174 [2]. • BAO: We use measurements of the volume distancefrom 6dFGS at z = 0 .
106 [40] and the MGS galaxysample of SDSS at z = 0 .
15 [41], as well as therecent DES1 BAO measurement at z = 0 .
81 [42].We include the anisotropic measurements from theCMASS and LOWZ galaxy samples from the BOSSDR12 at z = 0 .
38, 0 .
51, and 0 .
61 [43]. The BOSSDR12 measurements also include measurements ofthe growth function f , defined by f σ ≡ (cid:104) σ ( vd )8 ( z ) (cid:105) σ ( dd )8 ( z ) , (1)where σ ( vd )8 measures the smoothed density-velocity correlation, analogous to σ ≡ σ ( dd )8 thatmeasures the smoothed density-density correlation. • Ly- α : The latest lyman- α BAO (auto and cross-correlation with quasars) at z = 1 . z =2 .
33 [9] and z = 2 . • JLA: We use the SDSS-II/SNLS3 Joint Light-Curve Analysis (JLA) data compilation of > . < ∼ z < ∼ . Monte Python [47], we runMonte Carlo Markov chain analyses with the Metropolis-Hastings algorithm and assume flat priors on all param-eters. Our ΛCDM parameters are { ω cdm , ω b , θ s , A s , n s , τ reio } . There are many nuisance parameters for the
Planck [30]and JLA [46] likelihoods that we analyze together withthese cosmological parameters. We use a Cholesky de-composition to handle the large number of nuisance pa-rameters [48]. Using the Gelman-Rubin criterion [49], weapply the condition R − < .
05 to indicate our chainshave converged. For the nuisance parameters, we use the default priors that areprovided by
MontePython . B. The discrepancy between local distancemeasurements of H and the CMB Observations of the CMB provide a firm measurementof the distance scale at decoupling: d s ( z dec ) = 11 + z dec (cid:90) ∞ z dec c s H ( z ) dz . (2)This represents an early-time anchor of the cosmic dis-tance ladder. The CMB also provides an estimate ofa late-time anchor of the distance ladder: H , the ex-pansion rate today (see, e.g. , Chapter 5.1 in Ref. [50] formore details). However, this measurement is indirect anddepends on the assumed cosmological model. Thus, the direct determination of H at low-redshift is essential tofirmly calibrate the distance ladder in a model indepen-dent fashion.The SH0ES survey measured the value of the present-day Hubble rate to a precision of 2 . z < .
15. Their final result is H =73 . ± .
74 km / s / Mpc [2]. This direct measurementof H is discrepant at the ∼ . σ level with the inferredvalue of H = 66 . ± .
62 km / s / Mpc from
Planck [35](from the TT+TE+EE+SIMlow measurements at the68% confidence level).
1. Early-time solutions
To resolve the tension between the
Planck and SH0ESdetermination of H within ΛCDM by modifying thedistance ladder at early times, the CMB-inferred valueof d s ( z dec ) must be reduced by a factor of ∼
6% to10 Mpc [32]. As a result, either the sound speed in thephoton–baryon plasma must decrease or the redshift ofrecombination must increase [see Eq. (2)]. To achievethese effects, a higher primordial helium fraction Y p or anextra ultra-relativistic species are often invoked. How-ever, both these possibilities are ruled out. The CMBand big-bang nucleosynthesis (BBN) constrain Y p to beclose to 0 .
25 [32]. Extra relativistic degrees of freedomsufficient to recover the low-redshift value of H are ruledout within ΛCDM by Planck polarization data and BAOmeasurements [17, 32].
2. Late-time solutions
Late-time solutions for this discrepancy rely on alter-ing the expansion history, such that the expansion ratematches the CMB at decoupling and the local rate today. In principle, any species affecting the background expansion atearly times could be used. See, e.g. , Ref. [51] for an alternativeattempts at solving the H discrepancy via an early DE compo-nent. Within ΛCDM it is not possible to accomodate both H and BAO data, which fix the expansion history be-tween z = 2 . z = 0 .
15 ; the only extra low-redshiftdegree of freedom is the ratio between Ω Λ and Ω m , whichis insufficient to allow the expansion history to changesignificantly between z = 0 .
15 and z = 0.Alternative standard extensions attempting to solvethe H discrepancy include a phantom-like dark en-ergy (DE) component with an equation of state w < − H measurements if BAO and SNeIa data in agreement with Planck are included in theanalysis.
C. The discrepancy between the power spectrumamplitude from the CMB and LSS
There is a moderate tension within ΛCDM betweenthe value of S measured by LSS surveyGalaxy cluster-ing and weak lensing surveys (such as CFHTLenS [3],KiDS [4, 5], DES [6], and Planck
SZ cluster counts [7])measure a value of S between 2 σ and 4 σ smaller thanthat inferred from the CMB. Note that, through lensing,the CMB measures the power spectrum amplitude notonly at z = 1100, but also over a redshift range centredat z ≈
2. These two
Planck measurements are internallyinconsistent, and the nuisance parameter A lens is used toallow them to vary freely. Marginalising over A lens re-duces the significance of the S tension but does not re-move it, because the lensing 4-point correlation estimator C φφl itself does not favor high value of A lens . Indeed, theamount of lensing measured from the smoothing of highmultipole peaks in the TT spectrum is higher than thatmeasured from C φφl , the latter being compatible with theΛCDM expectation [7, 36]. Weak lensing measurementsprobe a lower redshift range, z ≈ . − .
0, comparedto CMB lensing. Furthermore, weak lensing surveys andgalaxy clusters measure S on smaller scales than the Planck
CMB, k ∼ . ∼ z < S problem [20, 21, 54],but are in tension with Ly- α data [20, 55]. Here, we focuson another possibility; massive neutrinos, which reducepower on small scales by reducing the growth rate.
1. Solutions due to massive neutrinos
There is some weak evidence from cosmology support-ing a non-zero neutrino mass sum. For example, Ref. [26]found a 2 . σ preference for a non-zero neutrino massfrom SDSS, and S constraints from galaxy cluster countsgive similar results [24, 25]. Recently, Ref. [28] com-bined Planck
CMB measurements with thermal Sunyaev-Zeldovich (tSZ), BAO, and lensing data. They used asuite of hydrodynamic simulations calibrated to producerealistic cluster gas profiles [27]. Central to their analysiswas removing the internal tension between
Planck
CMBand
Planck lensing by marginalising over A lens . Theirconclusions are in striking agreement with those of thiswork, finding that a neutrino mass sum (cid:80) m ν ∼ . α forest flux powerspectrum can be combined with Planck to constrain theneutrino mass sum to be (cid:80) m ν < .
12 eV [56]. Notethat the forest alone constrains only (cid:80) m ν < α forest is sensitive to the matter power spectrumon non-linear scales of k = 0 . h/ Mpc, this constraintrequires simulations for calibration and assumes a ΛCDMcosmology. Given that our models include substantialdeviations from ΛCDM even at z >
2, along with thelack of a public likelihood function code, we chose not touse this Ly- α forest dataset.However, we note that the Ly- α forest measures aspectral index n s = 0 . ± .
01, 2–3 σ lower than the n s = 0 . ± . Planck [1, 56]. Thus, theLy- α forest, in agreement with the rest of our analysis,does prefer reduced power on small scales compared tothe CMB. A Ly- α forest analysis allowing for a moregeneral dark energy model would be an interesting checkon our conclusions, and we may address this in futurework. We also note that constraints on (cid:80) m ν usuallydepends on the assumed DE equation of state; they canbe very strong when w ≥ w (asfavored by the combination of CMB and SHOES data)are allowed [58]. III. COMBINING THE MOST COMMONEXTENSIONS TO Λ CDM
We have argued that the most common extensions toΛCDM invoked in order to solve the H and S problems,when considered separately, are not able to accommodateall datasets currently available. In this section, we con-sider a combination of these extensions to see if they canachieve in concert what they could not alone. We retainthe basic framework of ΛCDM throughout this section,considering only well-motivated extensions. A. Models
We denote the standard ΛCDM cosmology with mass-less neutrinos as ν ΛCDM, and we consider the followingmodifications: • Massive neutrinos: We consider a degenerate masshierarchy for the neutrinos, as we find the speci-fication of the mass hierarchy to be irrelevant forcurrent datasets. The exception is if one of the neu-trinos is massless, in which case the matter powerspectrum is significantly altered [23]. • DE as a scalar field: We use the CPL parameter-ization w ( a ) = w + (1 − a ) w a [29], with a pa-rameterized post-Friedmann treatment to allow thecrossing of the phantom divide [59]. We set thesound speed in the rest frame of the scalar fieldto unity and use the priors w ∈ [ − , .
3] and w a ∈ [ − ,
2] [16]. • Additional ultra-relativistic species: There aremany models that introduce additional relativisticdegrees of freedom ∆ N eff . For example, extra ac-tive or sterile neutrinos, light scalar fields, or darkradiation in a dark sector. For a given ∆ N eff , all ofthese models have the same background effects onthe CMB, but there are a number of perturbation effects that are model dependent (for instance, afree-streaming species is known to induce a shift ofshifts CMB peaks towards larger scales, or smallerangles—an effect known as ”neutrino drag”).To keep the discussion as general and model-independent as possible, there is a postulated lin-ear and time-independent relation between theisotropic pressure perturbations and density per-turbations δp/δρ = c (defined in the rest frameof the ultra-relativistic species); similarly, there is aviscosity coefficient c that enters the source termof the anisotropic pressure [7, 32, 60, 61]. We addan ultra-relativistic species, which does not sharethe same mass as the active neutrinos, by modifying N eff , the effective sound speed c , and the viscositysound speed c . We use the priors ∆ N eff ∈ [ − , c , c ∈ [0 , ν M w CDM+ N fluid . B. Results
1. Restricted Datasets
First, we perform an analysis that includes only theCMB, the SH0ES, and
Planck
SZ datasets. With these datasets alone, an extended model can solve the ten-sion between the CMB and SH0ES and the tension be-tween the CMB and
Planck SZ simultaneously . We find H = 72 . ± .
8, in agreement with local measurements,while ( σ , Ω M ) = (0 . +0 . − . , . +0 . − . ), in agree-ment with the low- z measurements. This is possible be-cause the extra freedom allowed by our extended cos-mological model is absorbed by the CMB. What waspreviously a tension thus appears as extended parame-ters which deviate strongly from ΛCDM. We have a neu-trino mass sum (cid:80) m ν = 0 . +0 . − . eV and DE param-eters ( w , w a ) = ( − . +0 . − . , − . +0 . − . ). The good-ness of fit is ∆ χ = χ ( ν ΛCDM) − χ ( ν M w CDM+ N fluid ) = − .
08, showing that the χ does improve bymore than the additional number of free parameters.These parameters deviate strongly from their ΛCDMvalues and are statistically compatible with the resultsfrom previous literature, introduced in Section II. Wenote that in this restricted analysis, the neutrino masssum is higher than the 0 . Planck
SZ cluster measurement. We also find that∆ N eff is consistent with zero, and ( c , c ) are uncon-strained, indicating that these datasets are not sensitiveto this model extension.
2. Full Datasets
We turn to a full analysis that includes all datasetsoutlined in Section II A. We compare the posterior dis-tribution of { H , σ , Ω m , (cid:80) m ν , w , w a , ∆ N fluid } to thatobtained in ΛCDM in Figure 1. In Tables I and II, wereport constraints on cosmological parameters, as well asthe χ contribution from each dataset. These addi-tional datasets restrict the ability of our ExDE modelto resolve the tensions. The BAO and JLA data, asshown in Table I, constrain the DE parameters to bevery close to ΛCDM. Additional ultra-relativistic speciesare still disfavored by the data: (∆ N eff , c , c ) =( − . +0 . − . , . +0 . − . , . +0 . − . ).As a result, the central value of the H measurementdoes not significantly change between the extended cos-mology and ΛCDM. The tension between the CMB andthe SH0ES measurement is reduced to the 2 . σ level onlybecause of the increase in error bars. This is reflected in amodest change in ∆ χ = − .
19 with respect to ΛCDMat the expense of 5 new parameters. The improvementto the fit is primarily due to a reduced S tension be-tween the CMB and the Planck
SZ data: ∆ χ = − . (cid:80) m ν = 0 . +0 . − . . Note that the χ of the power spectrum measurements from SDSS andCFHTLenS is almost unchanged, indicating that theyare consistent with this value of the neutrino mass.In conclusion, it is possible to solve the S tension withmassive neutrinos even when the H measurement is in-cluded in the analysis. However, it is not possible to fullysolve the H tension within the ν M w CDM+ N fluid model.The values of ( w , w a ) required to make the SH0ES valueof H compatible with the CMB prediction are ruled outby BAO and supernovae, even when considering a com-bination of early- and late-Universe modifications. Model ν ΛCDM ν M w CDM + N fluid ω b . +0 . − . . +0 . − . ω cdm . +0 . − . . +0 . − . θ s . +0 . − . . +0 . − . ln 10 A s . +0 . − . . +0 . − . n s . +0 . − . . +0 . − . τ reio . +0 . − . . +0 . − . (cid:80) m ν . +0 . − . w -1 − . +0 . − . w a − . +0 . − . ∆ N eff − . +0 . − . c . +0 . − . c . +0 . − . σ . +0 . − . . +0 . − . Ω m . +0 . − . . +0 . − . H . +0 . − . . +0 . − . TABLE I. Constraints at 68% C.L. on cosmological param-eters in various models including (cid:80) m ν , N eff and ( w , w a )using all datasets considered in this work.Model ν ΛCDM ν M w CDM + N fluid Planck high- (cid:96) τ SIMlow 0.24 0.17
Planck lensing 11.25 11.32SDSS DR7 45.77 46.11CFHTLenS 97.92 98.60BAO (DES1) z ∼ . z ∼ . − .
15 2.82 2.82BAO z ∼ . − . α +QSOs 8.71 9.40JLA 683.95 683.94SH0ES 5.29 6.63 Planck
SZ 9.14 4.89Total χ χ χ per experiment for the standard ν ΛCDM model and the ν M w CDM + N fluid . IV. MINIMALLY PARAMETRICRECONSTRUCTION OF THE DARK ENERGYDYNAMICS
In Section III, we restricted possible DE dynamics tothose allowed by the simple ( w , w a ) parameterization ofthe DE equation of state. We found that this parame-terization did not allow enough freedom in the expansionrate to reconcile BAO and local H measurements. Inthis section, therefore, we consider what expansion ratewould be required. We use a fully general, minimallyparametric model for the ExDE density as a functionof redshift. This allows the expansion rate to changeessentially arbitrarily as a function of redshift. In par-ticular the expansion rate can match that expected for H = 69 at z > .
15, and thus match BAO, and thenmatch H = 72 at z = 0. We emphasise that the best fitparameters may not necessarily be realizable in a physicalmodel. In this section we are interested in determiningwhat the data requires, partly to allow an assessment ofthe relative plausibility of explanations based on experi-mental systematics.We write the Hubble expansion rate as H ( z ) = H (cid:112) Ω m (1 + z ) + Ω r (1 + z ) + Ω ExDE ( z ) , (3)where Ω ExDE ( z ) corresponds to an unknown exotic DEspecies with an arbitrary density and equation of state.Note that we do not restrict Ω ExDE ( z ) to be positive.This allows us to include complicated dynamics resultingfrom, for example, a reduction in matter density fromdecaying dark matter or curvature. This ExDE sectoris implemented by modifying the expansion rate modulein the Boltzmann code CLASS [62]. We neglect perturba-tions in the exotic fluid and change only the backgroundexpansion rate. Ω
ExDE ( z ) is given by a cubic spline inter-polated between a series of values at different redshifts,called z knots . We place a weak prior on the energy densityof the exotic fluid at the knots to be | Ω ExDE ( z knot ) | < F ExDE F ExDE = (cid:90) z max z min (Ω ExDE ( z )) (cid:48)(cid:48) dz . (4)In practice, we minimize the following quantity M = − ln L + λF ExDE , (5)where λ is chosen according to the CV procedure. Weremove part of the data and perform a parameter fit forseveral values of λ on the remaining datasets. The best-fit parameters obtained from this limited dataset are then σ Ω m P m ν -1.23-0.954-0.676 w -2-0.7580.483 w a -0.341 -0.06 0.221 ∆ N fluid H -0.341-0.060.221 ∆ N fl u i d σ Ω m P m ν -1.23 -0.954 -0.676 w -2 -0.758 0.483 w a Λ CDM ν M w CDM+ N fluid FIG. 1. The posterior distribution of { H , σ , Ω m , (cid:80) m ν , w , w a , ∆ N fluid } when fitting to all datasets considered in this work,compared to the ΛCDM fit of the same dataset. used to compute the χ associated with the removed part.The value of λ that minimizes the χ calculated on the setof data not included in the runs is λ ∼ .
1. We investigatewhether or not it is possible to solve the H and S dis-crepancies, accommodating all datasets in Section II A,and we investigate how changes in the background evo-lution influence the measurement of the neutrino masssum. All analyses include the CMB, LSS, SH0ES, andLy- α BAO datasets. We show results of fits including only a single z <
A. Reconstruction from all datasets
Since we use CMB data, we include a knot at z =1100 and a knot at the initial redshift considered in CLASS , namely z = 10 , whose only purpose is to en- . . . . . . . . . H E x D E / H Λ C D M P m ν = 0 . eV P m ν = 0 . eV . . . . . . z − − − Ω E x D E Ω ExDE full prior Ω ExDE positive prior . . . . . . . . . H E x D E / H Λ C D M P m ν free P m ν free . . . . . . z − − − Ω E x D E Ω ExDE full prior Ω ExDE positive prior
FIG. 2. Reconstructed ExDE energy density and Hubble expansion rate (compared to the ΛCDM prediction from PlanckTT,TE,EE+SIMlow, black line) with (cid:80) m ν = 0 .
06 eV (left panel) or (cid:80) m ν left as a free parameter (right panel), whenincluding all datasets considered in this work and for different choice of prior on Ω ExDE (see text). The thick solid lines showthe best fit spline in each case, while the thin lines show samples from the 68% confidence region. The vertical arrows show thepositions of the knots. The orange band indicates the uncertainty on the Hubble parameter as measured by SH0ES (strictlyspeaking it is only valid a z = 0). sure a smooth interpolation. We also include a knotat z = 0 for the H data and at z = 2 . α BAO. The remaining knots are spaced linearly atlow redshift and logarithmically at high redshift: z =(0 . , . , . , . , . , . H ExDE (normalized to ΛCDM, using
Planck
TT,TE,EE+SIMlow [35]) and reconstructed en-ergy density Ω
ExDE as a function of z , along with 500curves chosen at random from the 68% confidence region.The left panel shows the result with the neutrino masssum set to (cid:80) m ν = 0 .
06 eV, while the right panel showsthe result with (cid:80) m ν as a free parameter. We show ex-pansion histories in which the neutrino mass sum is set to (cid:80) m ν = 0 .
06 and those in which it is a free parameter.We also show reconstructions which enforce a positivevalue for Ω
ExDE ( z ) and those which allow Ω ExDE ( z ) tobe negative.Ω ExDE ( z ) is roughly constant when Ω ExDE ( z ) > ExDE ( z ) is allowed to be neg-ative, the Ly- α BAO data make the best-fit Ω
ExDE ( z )negative for 2 < ∼ z < ∼ .
5. The significance of this isgreater than 68%, but does not quite reach 95%. Thisis unaffected by whether the neutrino mass is fixed, al-though fixing the neutrino mass causes an increase in energy density at z = 1 .
5. While it is possible that thiscould result from a modified gravity model, or potentiallya decay in the dark matter density [11], the most likelyestimate is systematics in the Ly- α BAO data. Notethat by z = 1100 Ω ExDE ( z ) is again positive, which ar-gues against a cosmological explanation. If we removethe Ly- α BAO, there is no data at z = 2 . ExDE ( z )is consistent with zero and ΛCDM at this redshift. Notethat because the DR12 BAO likelihood is not yet public,we are using a Gaussianized version, which may underes-timate the errors. The best explanation for this discrep-ancy thus appears to be statistical.If we weaken the effect of the Ly- α BAO data by, forexample, enforcing Ω
ExDE ( z ) >
0, we see that the expan-sion history is consistent with ΛCDM within the errorbars. Thus, even when arbitrary DE dynamics are al-lowed, the tension between H measured by SH0ES andthat measured by BAO and the CMB remains. Notehowever that the increased freedom in the model meansthat the tension is significantly weakened to less than ∼ . σ . One reason for this is that, given the valueof H , the JLA and galaxy BAO measurements are inslight (1 − σ ) tension. This is illustrated in Figure 3:at z < ∼ . ExDE ( z ) in a slightlydifferent direction, forcing an overall compromise valueclose to that of a cosmological constant. The JLA datagenerally agree with the local H data, while the BAOmeasurements agree with that from the CMB. We em-phasize that there is not necessarily any tension beyondstatistical variation between these datasets. Their agree-ment is well within the 2 σ level. The different behavioris mostly driven by the fact that fits to JLA data are Model ΛCDM ExDE + (cid:80) m ν = 0 .
06 ExDE + (cid:80) m ν freePrior on Ω ExDE − Full Positive Full Positive
Planck lite 217.35 214.20 215.98 209.20 212.66 τ SIMlow 0.24 0.06 0.06 0.11 0.01
Planck lensing 11.25 10.03 10.06 8.86 10.71SH0ES 4.75 5.4 3.32 4.28 5.10
Planck
SZ 9.14 5.88 8.64 0.12 2.58SDSS DR7 45.78 44.97 45.05 46.67 45.55CFHTLenS 97.92 97.06 97.22 97.90 97.52DES1 BAO 0.01 0.05 0.05 0.01 0.09BAO Ly- α +QSOs 8.71 3.88 5.86 6.08 7.17BAO iso DR11 2.81 3.03 2.33 2.05 2.39BAO + fσ DR12 7.14 4.08 4.11 4.68 5.37JLA 683.95 686.4 687.27 683.58 684.85 χ χ TABLE III. The best χ per experiment for the reconstructedDE dynamics with and without the neutrino mass sum as anextra free parameter when all datasets are included. insensitive to the value of H [46]. Moreover, when com-bined together, their respective χ stays very good (seeTable III).Interestingly, even with all datasets included, the neu-trino mass sum is (cid:80) m ν = 0 . +0 . − . eV, driven by animprovement in the χ with the Planck
SZ data, asin Section III. We have checked explicitly that the pre-ferred neutrino mass changes by less than 1 σ when omit-ting galaxy BAO or JLA from the datasets, even if theExDE dynamics is very different from that of a cosmo-logical constant. This is illustrated in Figure 4, wherewe show the posterior distribution of { Ω m , σ , H , (cid:80) m ν } obtained when allowing for a free neutrino mass (rightpanel) or fixing it to the minimal value indicated by os-cillation experiments (left panel).Table III shows the χ for each dataset, fitting toΛCDM, ExDE with the neutrino mass sum fixed to (cid:80) m ν = 0 .
06, and ExDE with the neutrino mass sumleft as a free parameter. For each ExDE case, the priorΩ
ExDE ( z ) is either restricted to be positive or is allowedto take on its full range of positive and negative values.The Ly- α data near z ∼ .
35 are better fit with the “fullprior”, pulling Ω
ExDE to negative values: the χ in the“full prior” case is improved compared to the “positiveprior” by ∆ χ = − .
88 when (cid:80) m ν = 0 .
06 eV andby ∆ χ = − .
08 when the neutrino mass sum is leftfree. Finally, we perform an analysis of all datasets, in-cluding an extra ultra-relativistic fluid (∆ N eff , c , c )and letting the neutrino mass sum vary. We find thatthis additional ultra-relativistic species does not reducethe tension further, nor does it affect the reconstructionat low- z or the determination of the neutrino mass sum. B. Robustness of the result
We have performed a number of additional tests to as-sess the robustness of our conclusions. First, we havechecked explicitly that our results are robust to the addi-tion of an extra high redshift knot at z ∼
4. As expected,we find that adding knots at this redshift and higher hasno impact. Indeed, there are no datasets sensitive to suchredshifts (except for the CMB in a very mild way throughthe integrated Sachs-Wolfe effect). Moreover, our prioron Ω
ExDE ensures that the Universe is largely matterdominated at these times. We have also made several al-terations to the position of the low-redshift knots [e.g. weset them at z = (0 . , . , . , . , , . , . Planck lite like-lihood for the full likelihood. Although we did not imple-ment the full KiDS and DES likelihoods for this analysis,we checked that when the data from these experimentsare reduced to a Gaussian prior on S our best-fits arefully compatible with these measurements. On the otherhand, when removing the Planck
SZ likelihood, we findthat (cid:80) m ν < .
48 eV (at the 95% confidence level) with abest-fit around 0 . (cid:80) m ν ∼ . Planck data and usingBBN data instead. As expected, doing so has no strongimpact on the late-Universe reconstruction; it simply in-creases the uncertainty on the densities of the variouscomponents in our Universe and reduces the H tensionto ∼ . σ .Finally, we have tested our results by introducing theextra free parameter, A lens , which rescales the global am-plitude of the lensing potential [34]. Ref. [28] found thatthis can affect the constraining power of the lensing like-lihood on (cid:80) m ν . We still find (cid:80) m ν = 0 . +0 . − . , in verygood agreement with our previous fit within error bars.We additionally find A lens = 1 . +0 . − . , in agreementwith the value found by the Planck analysis [64]. Thisvalue is discrepant at 2 σ with the expected ΛCDM valueof 1 and thus represents an internal tension in the Planck data due to an extra smoothing of the CMB high multi-poles, as argued previously.
V. CONCLUSIONS
In this paper we have examined two well-known ten-sions in the ΛCDM cosmology: the tension between localmeasurements of H and the CMB-inferred value, andthe tension between CMB measurements of the powerspectrum amplitude σ and that measured by galaxyclusters in Planck
SZ. Many papers have focused on pos-0 σ Ω m P m ν H P m ν σ Ω m Λ CDMCPL-parametrization with P m ν freereconstruction with P m ν = 0 . eVreconstruction with P m ν free . . . . . . . . . H E x D E / H Λ C D M P m ν free P m ν free . . . . . . z − − − Ω E x D E BAOJLA
FIG. 3.
Left panel:
A comparison between the 1D and 2D posterior distributions of ( σ , Ω m , H , (cid:80) m ν ) obtained in variousmodels when using all datasets considered in this work. The grey band shows the R16 measurement, the purple band is thePlanck SZ determination of S . Right panel:
Reconstructed DE energy density and Hubble expansion rate (compared to theΛCDM prediction from Planck TT,TE,EE+SIMlow, black line) with (cid:80) m ν left as a free parameter. We include either theBAO (red) or JLA data (blue). The thick solid lines show the best fit spline in each case, while the thin lines show drawsfrom the 68% most likely fits. The red arrows pointing upwards show the locations of the BAO knots, while the blue arrowspointing downwards show the positions of the JLA knots. The orange band indicates the uncertainty on the Hubble parameteras measured by SH0ES (strictly speaking it is only valid a z = 0). σ Ω m H Ω m σ BAOJLAAll Data H σ Ω m P m ν Ω m H σ BAOJLAAll Data
FIG. 4. 1D and 2D posterior distributions of ( σ , Ω m , H , (cid:80) m ν ) with a fixed neutrino mass sum (left panel) and a free neutrinomass sum (right panel) when using SDSS DR7 CFHTLens, SH0ES, CMB, Ly- α BAO DR11, and either galaxy BAO DR12(red curves) or JLA (blue curves). The grey band shows the SH0ES measurement, while the purple band is the
Planck
SZdetermination of S . { Ω m , σ , H , (cid:80) m ν } in the left panel of Figure 3 forthe various cosmological models considered in this workwhen including all datasets.We first examined whether these tensions could be re-solved by the simultaneous adoption of standard exten-sions to ΛCDM. These extensions include massive neu-trinos, extra relativistic degrees of freedom, and a fluidmodel of dark energy parameterized by a power law equa-tion of state. Several authors have previously used theseextensions individually to resolve these tensions, but weconsider enabling them at once. We find that none of theextensions significantly reduce the tensions, with the ex-ception of massive neutrinos. We find that the addition ofextra relativistic degrees of freedom does not reduce thetensions. Since the galaxy BAO and JLA data measurethe expansion history at relatively low redshift, there isinsufficient freedom in the power law equation of state toreduce the tension with local H measurements.We found that a neutrino mass sum of 0 . S tension, and this resolution persists for thedatasets we considered, as long as a model with enoughfreedom to reduce the significance of the H tension wasused. The extra model freedom is important, because aside-effect of a non-zero neutrino mass sum is that it in-creases the tension between local H measurements andthe CMB by decreasing the inferred value of H fromthe CMB. However, a non-zero neutrino mass sum iswell-motivated theoretically. Whenever the H tensionis solved or greatly decreased, the S value from Planck
SZ cluster count drives the neutrino mass sum to be closeto 0 . H tension, which indicates thatit is relatively robust.Since explaining the total sum of cosmological datasetsrequires additional freedom in the expansion history, weincluded an exotic dark energy sector, which we allowedto have an energy density varying arbitrarily with red-shift. We emphasize that although we have assigned thissector to dark energy, it can be viewed as a proxy forother more physically motivated models, such as decay-ing dark matter or curvature. We have not attemptedto identify these models, treating the exotic dark energysector as a purely phenomenological parameterization ofthe expansion rate. We use cross-validation to avoid over-fitting the data. We found that the best-fit model whenall datasets was included was an expansion history rel-atively close to ΛCDM. Thus the H tension was notfully solved, although the extra model freedom did re-duce the significance of the tension to less than 2 σ . Inorder to fully solve this tension, it was necessary to alsoomit either the JLA data or the galaxy BAO data. Ei-ther dataset allowed for a non-ΛCDM expansion historysolution, but these solutions were inconsistent with eachother. We found that the Ly- α BAO dataset preferred a neg-ative density of exotic dark energy at z ∼ .
3, a be-haviour that cannot be recovered with an equation ofstate. This result is not so cosmologically bizarre as itat first seems: for example, it could potentially be ex-plained by an open Universe with a negative curvaturecomponent. Although curvature is highly constrained bythe CMB, these constraints are dependent on assumingΛCDM and weaken significantly with more general mod-els. The presence of a negative curvature, as is the caseif the Universe presents an open geometry, can naturallylead to apparent negative energy density for the darksector.Another possibility is that the exotic dark energy sec-tor could include a decaying dark matter component. Ifthe decay products dilute faster than matter, the expan-sion rate can be reduced around z ∼ .
3. However, thesimplest such model, a dark matter component decayinginto dark radiation with constant lifetime [11, 65], is inconflict with observations of the late integrated Sachs-Wolfe effect and lensing power spectrum [12, 13]. More-over, we find Ω
ExDE becomes positive again at z < . α BAO, is by far the most likely explanation. Toaccommodate the data, Ω DE would then need to followa dynamics very close to that obtained when restrictingthe analysis to positive priors on Ω ExDE . Such behaviorcan be obtained from a scalar field with a peculiar phan-tom behavior. Of course, it would be theoretically moreappealling to find a solution for which this behavior isnot due to decoupled sectors, but arise from the commondynamics of several species related to each other. Mea-surements of the expansion history at redshifts higherthan those currently probed (for instance via future in-tensity mapping or 21cm BAO experiments) can allow usto understand whether the preference for exotic dark en-ergy is real. If this behavior persists at higher redshifts,it can give important insights on the dark sector. How-ever, if it does not continue, it can cast serious doubtsregarding the validity of this interpretation of the Ly- α measurement.While even our most general ExDE model was unableto solve the H tension, there are classes of solutions notconsidered here. For example, a modification of grav-ity such as Horndeski’s theory [66], gravity theories withhigher derivatives (e.g. f ( R ) gravity [67], tele-parallel” f ( T ) gravity [68] or Galileon gravity [69, 70]) or nonlocalgravity ([71]). The recently discussed “redshift remap-ping” is another potential solution that is not coveredby our reconstruction [72]. Our reconstruction can serveas a guide to build a model, successfully explaining alldatasets, and we may examine this in a future study. Fi-nally, we note that it is interesting that, whenever the H tension was solved or weakened, the best fit neutrino2mass sum was around 0 . ACKNOWLEDGEMENTS
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