Test the effects of H0 on fσ8 tension with Gaussian Process method

En-Kun Li,∗ Minghui Du,† Zhi-Huan Zhou,‡ Hongchao Zhang,§ and Lixin Xu¶

Institute of Theoretical Physics, School of Physics,Dalian University of Technology, Dalian, 116024, China

Using the fσ8(z) redshift space distortion (RSD) data, the σ08 −Ω0

m tension is studied utilizing aparameterization of growth rate f(z) = Ωm(z)γ . Here, f(z) is derived from the expansion historyH(z) which is reconstructed from the observational Hubble data applying the Gaussian Processmethod. It is found that different priors of H0 have a great influence on the evolution curve ofH(z) and the constraint of σ0

8 − Ω0m. When using a larger H0 prior, the low redshifts H(z) deviate

significantly from that of the ΛCDM model, which indicates that a dark energy model differentfrom the cosmological constant can help to relax the H0 tension problem. The tension betweenour best-fit values of σ0

8 − Ω0m and that of the Planck 2018 ΛCDM (PLA) will disappear (less than

1σ) when taking a prior for H0 obtained from PLA. Moreover, the tension exceeds 2σ level whenapplying the prior H0 = 73.52 ± 1.62 resulted from the Hubble Space Telescope photometry.

PACS numbers: 95.36.+x, 98.80.Es, 95.35.+d

I. INTRODUCTION

The past and present analyses of various cosmologi-cal observations converge to the fact that our universe isundergoing an accelerated expansion phase [1–7]. To ex-plain this phenomenon, two kinds of interpretations havebeen raised. One is proposing an unknown componentwith negative pressure called dark energy in the contextof General Relativity (GR), and the other is modifyingthe laws of gravity (MG). Based on these two branches,numerous models have been presented. Among thesemodels, Lambda Cold Dark Matter (ΛCDM) model is themost simple one and can excellently fit with almost allobservational data, such as the cosmic microwave back-ground (CMB) radiations [4, 6, 7], the baryon acousticoscillations (BAO) [8–10], and the type Ia Supernovae(SNIa) [11–14], etc.

Nonetheless, it is becoming exceedingly apparent thatthere are some discrepancies between the Planck ΛCDMresults and some independent observations in intermedi-ate cosmological scale [15]. These discrepancies includethe estimates of the Hubble constantH0 [16–20], the mat-ter density parameter Ω0

m and the amplitude of the powerspectrum on the scale of 8h−1Mpc (σ0

8) [21–25], etc. Inorder to solve these discrepancies, different methods andcosmology models have been reported, including viscousbulk cosmology [26], assuming a variable Newton con-stant [27, 28], considering the interaction between neu-trinos and dark matter [29], introducing interacting darkenergy [30, 31], model-independent method [32–34], andso on.

Precise large-scale structure measurements are helpfulto distinguish different models because these models may

∗ ekli 091@mail.dlut.edu.cn† angelbeats@mail.dlut.edu.cn‡ scott@mail.dlut.edu.cn§ zhanghc@mail.dlut.edu.cn¶ Corresponding author: lxxu@dlut.edu.cn

have different growth histories of structure. As a startingpoint, in the subhorizon (k aH), the equation thatdescribes the evolution of the linear matter growth factorδ = δρm/ρm in the context of GR and most MG modelshas the form [28]

δ′′ +

(H ′

H− 1

1 + z

)δ′ ' 3

2(1 + z)

H20

H2

Geff(z, k)

GNΩ0mδ,

(1.1)where ρm is the background matter density, H = a/ais the Hubble expansion rate at scale factor a = 1/(1 +z), GN is the Newton’s constant, Geff is the effectiveNewton’s constant which in general may depend on boththe redshift z and the cosmological scale k, and “ ′ ”denotes a derivative with respect to z. In GR we haveGeff = GN while in MG Geff/GN may vary with bothcosmological redshift and scale.

Although it is difficult to give the analytical solutionof Eq. (1.1), a good parameterization of the growth ratef(a) ≡ d ln δ/d ln a is given by [35–40]

f(a) ' Ωm(a)γ , (1.2)

where Ωm(a) = Ω0ma−3/(H(a)/H0)2 is the fractional

matter density, and γ is the growth index. The growth in-dex differs between different cosmological models [39, 41].In the ΛCDM model, γ = 6/11 is a solution to Eq. (1.1)where the terms O(1−Ωm(a))2 are neglected [35], whileγ ' 0.55 is that of dark energy models with slowly vary-ing equation of state [36]. For MG models, different val-ues are predicted, e.g., γ ' 0.68 for Dvali-Gabadadze-Porrati (DGP) braneworld model [42, 43]. Applyingsome model-independent methods, authors in Refs. [44–46] found that the value of γ is consistent with that ofthe flat-ΛCDM model, and Yin & Wei [47, 48] also in-vestigated the time varying γ(z). Since most of the in-formation on linear clustering is expected to come fromthe epoch of equality of matter and dark energy, it isreasonable to use this parameterization to approximatef(z) [49].

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In particular, most of the growth rate measurementscan be obtained from redshift space distortion (RSD)measurements via the peculiar velocities of galaxies [50].However, f(z) is sensitive to the bias parameter b,which makes the observation of f(z) data unreliable [27].Therefore, most growth rate measurements are reportedas the combination f(z)σ8(z) = fσ8(z) instead of f(z),where σ8(z) = σ0

8δ(z)/δ0 is the matter power spectrumnormalization on scales of 8h−1Mpc. In addition, thejoint measurement of expansion history and growth his-tory provides an important test of GR and can help tobreak the degeneracies between MG theories and darkenergy models in GR [36, 51]. In this paper, using theRSD data and the observational Hubble data (OHD), wewill investigate the σ0

8−Ω0m tension utilizing the Gaussian

Process method. We reconstruct the expansion historyH(z) firstly using the OHD data with priors for Hub-ble constant H0, and then derive the theoretical value offσ8(z) applying the parameterization f(z) = Ωm(z)γ .Finally, by adopting the Markov Chain Monte Carlo(MCMC) method, the constraints on the free parame-ters in fσ8(z) are given using the RSD data corrected bythe fiducial model corrections.

The layout of this work is as follows. In section II,we introduce the basic methodology adopted to derivefσ8(z) and the observational data combinations used toconstrain free parameters. And then, in section III, weshow the reconstruction of Hubble parameter H(z) andfσ8(z) under different combinations of OHD. Our resultsand discussions are displayed in section IV. At last, wesummarize our conclusions in section V.

II. METHODOLOGY AND OBSERVATIONALDATA

A. Methodology

As we known that most growth rate measurements arereported as the combination fσ8(z). Using the defini-tions of f(z), σ8(z) and Eq. (1.2), one can obtain

fσ8(z) = σ08

(Ω0mH

20

)γ ( (1 + z)3

H(z)2

)γ· exp

[−(Ω0mH

20

)γ ∫ z

0

(1 + z′)3γ−1

H(z′)2γdz′]. (2.1)

Thus, given an expansion history function H(z) orH(z)/H0, we can reconstruct the observable quantityfσ8(z), assuming σ0

8 , Ω0m and γ are known.

The Gaussian Process method [52–54] can provide asmooth reconstructed H(z) using the combination ofOHD without assuming a parametrisation of the func-tion. So we can get a full model-independent recon-structed fσ8(z) with three free parameters σ0

8 ,Ω0m, γ

using Eq. (2.1).Now, we can use a χ2 minimization to constrain the

three free parameters,

χ2 = ∆VTCov−1∆V, (2.2)

∆Vi = fσ8,obs(zi)− fσ8,rec(zi) (2.3)

Cov = Covobs + Covrec, (2.4)

where Covobs is the covariance matrix of fσ8,obs andCovrec is the covariance matrix of the reconstructedfσ8,rec(z) which is defined in Eq. (2.1). The likeli-hood of the free parameters can be obtained from L ∝exp[−χ2/2]. The constraints on the free parametersare performed using the Markov Chain Monte Carlo(MCMC) sampling method. It’s easy to do this by usingthe publicly available code Cobaya 1, which calls theMCMC sampler developed for CosmoMC [55, 56].

Furthermore, in order to quantify the tension betweendifferent estimate of parameter ξ, we need to introducea quantization function of the tension level. Assumingthe 68% confidence level ranges of parameter ξ is ξ ∈[ξ1−σ1,low, ξ1 +σ1,up] from observation O1, and ξ ∈ [ξ2−σ2,low, ξ2+σ2,up] from observation O2. Then, the simplestand most intuitive way to measure the degree of tensioncan be written as [57]

s ≡ ξ1 − ξ2√σ2

1,low + σ22,up

, (2.5)

for the case ξ1 > ξ2. This means that the tension of ξbetween O1 and O2 is at sσ level.

B. fσ8(z) measurements

Table II shows a sample consisting of 63 observationalfσ8,obs(z) RSD data points collected by Kazantzidis, etal. [28]. It comprises the data published by various sur-veys from 2006 to the present and the parameters of thecorresponding fiducial cosmology model are also shownin this table. For more details please refer to Ref. [28]and references therein.

The covariance matrix of the 63 fσ8 data points areassumed to be diagonal except for the WiggleZ subset ofthe data (three data points). The covariance matrix ofthe three points of WiggleZ has been published as

CWiggleZi,j =

0.00640 0.002570 0.0000000.00257 0.003969 0.0025400.00000 0.002540 0.005184

. (2.6)

One should note that all the fσ8,obs(z) data listed inTable II are obtained assuming a fiducial ΛCDM cos-mology [28]. Thus, the Alcock-Paczynski (AP) effect[58] should be considered. In the present paper, we will

1 https://github.com/CobayaSampler/cobaya

3

use the following rough approxmation of the AP effect[28, 59]

fσ8,ap(z) ' H(z)DA(z)

Hfid(z,Ωm)DfidA (z,Ωm)

fσ8,obs(z), (2.7)

where DA(z) is the angular diameter distance, and it canbe written as

DA(z) =c

1 + z

∫ z

0

dz′

H(z′), (2.8)

in the spatially flat universe.

C. Observational Hubble data

The Hubble parameter H(z) is usually evaluated as afunction of the redshift z

H(z) = − 1

1 + z

dz

dt. (2.9)

It can be seen that H(z) depends on the derivative ofredshift with respect to cosmic time. The H(z) measure-ments can be obtained via two approaches. One is cal-culating the differential ages of passively evolving galax-ies [60] providing H(z) measurements that are model-independent. This method is usually called the cosmicchronometers (CC). The other method is based on theclustering of galaxies or quasars, which is firstly proposedby [61], where the BAO peak position is used as a stan-dard ruler in the radial direction.

Here, we use the compilation of OHD data points col-lected by Magana, et al. [62] and Geng, et al. [63],including almost all H(z) data reported in various sur-veys so far. The 31 CC H(z) data points are listed inTable III and the 23 H(z) data points obtained fromclustering measurements are listed in Table IV. One mayfind that some of the H(z) data points from clusteringmeasurements are correlated since they either belong tothe same analysis or there is overlap between the galaxysamples. Here in this paper, we mainly take the centralvalue and standard deviation of the OHD data into con-sideration. Thus, just as in Ref. [63], we assume thatthey are independent measurements.

In addition, there is no observation for H0 in theseOHD data points mentioned above, so we also con-sider two different priors of H0. One is H0 = 67.27 ±0.60 km/s/Mpc [7] provided by Planck 2018 power spec-tra (TT,TE,EE+lowE) measurements by assuming baseΛCDM model (hereafter P18). The other isH0 = 73.52±1.62 km/s/Mpc presented by the Hubble Space Tele-scope photometry of long-period Milky Way Cepheidsand GAIA parallaxes [20] (hereafter R18).

III. MODEL-INDEPENDENTRECONSTRUCTION

GP method provides a technique to reconstruct a func-tion using the observational data without assuming a spe-

cific parameterization. It is easy to reconstruct the Hub-ble parameters directly from the OHD data applying afreely available GaPP package 2. The numerical programwritten by ourselves is used in this paper, and there isno difference between our code and that of GaPP, whichalso indicates our program is credible.

GP are characterized by mean and covariance func-tions, which are defined by a small number of hyper-parameters. Throughout this work, we assume a priorimean function equal to zero, and use the squared expo-nential covariance function:

k(x, y) = σ2f exp

(− (x− y)2

2l2

), (3.1)

where σf and l are two hyperparameters which can bedetermined by the observational data. Supposing an ob-servational data-set xi, yi, σi, where xi is the locationof data point i, yi = f(xi)+εi is the corresponding actualobserved value which is assumed to be scattered aroundthe underlying function f(xi) and Gaussian noise withvariance σi is assumed. Using the GP method, the re-constructed mean value and covariance of the underlyingfunction f(x) can be written as [53]

f(x) =

N∑i,j=1

k(x, xi)(M−1)ijyi, (3.2)

Cov(fx, fy) = k(x, y)−N∑

i,j=1

k(x, xi)(M−1)ijk(xj , y),

(3.3)

where Mij = k(xi, xj) + Cij and Cij is the covariancematrix of the observational data.

A. Reconstruction of Hubble parameter

By using Eqs. (3.2) and (3.3), one can easily get the re-constructed Hubble parameters H(z) and its covariancematrix between different redshifts. The propagated co-variance [41, 64–66] of the reconstructed E(z) can be cal-culated with the reconstructed H(z), and its covariancematrix is

Cov[Ei, Ej ] = EiEi

(Cov[Hi, Hj ]

HiHj− Cov[Hi, H0]

HiH0

−Cov[H0, Hj ]

H0Hj+

Cov[H0, H0]

H0H0

)(3.4)

where Ei = E(zi) = H(zi)/H0 is the dimensionless Hub-ble parameter, and Hi = H(zi) is the reconstructed Hub-ble parameter at zi.

2 http://www.acgc.uct.ac.za/~seikel/GAPP/index.html

4

Next, we will examine the differences between the re-constructed H(z) under various OHD data combinations.In Figs. 1 and 2, we have ploted the reconstructed H(z)and E(z) within 2σ region using different data combi-nations, and the reconstructed H0 is also shown in theFig. 5. Here the indexes CC, BAO and ALL represent theOHD data obtained from CC, clustering measurements,and CC+clustering measurements, respectively. We alsoconsider three different priors of H0, i.e., no prior (indexNone), H0 of R18 and H0 of P18. The three panels ofthe first column in Figs. 1 and 2 are the reconstructedresults from the three data combinations with no prioron H0. From the legend labels in Fig. 1 and the error barplots in Fig. 5 one may find that OHD from BAO prefersa much smaller H0 than P18 or R18, and the mean valueof the derived H0 from the CC data is much close to thatof P18. However, we should note that the tension level ofthe three reconstructed H0 with P18 or R18 are all lessthan 2σ as a result of the big error bars.

Comparing the three panels in the same rows of Figs. 1and 2, one can find that adding a prior on H0 can signif-icantly reduce the error bars of the reconstructed H(z)at low redshifts, because the H0 measurements of R18and P18 have much smaller variances than the rest OHDdata points. It also can be found that the slope of thereconstructed H(z) varies when choosing different priorsof H0, especially at low redshifts z < 0.5. For the casesof using the same OHD data points but with different H0

priors, we find that the evolution curves of H(z) underthe P18 prior are similar to that without H0 prior, thisis due to the fact that they have similar mean values ofH0. Meanwhile, from Figs. 1 and 2, we can see that CCdata gives a much looser reconstruction of H(z) at higherredshifts than BAO data, which is because the BAO datapoints have much smaller variances at high redshifts.

B. fσ8(z) data after fiducial model correction

In this work, we will use the reconstructed model-independent H(z) and DA(z) for fiducial model correc-tion. Thus, the central value of fσ8,ap(z) and its co-variance matrix can be calculated according to Eq. (2.7).The covariance matrix will be

Covapij = qiqjCov∗ij + pipjCovHDij , (3.5)

where Cov∗ and CovHD are the covariance of obser-vational RSD data and the reconstructed H(z)DA(z),

respectively, qi = HiDA,i/(Hfidi Dfid

A,i ), and pi =

fσ8,i,obs/(Hfidi Dfid

A,i ).

The original fσ8,obs(z) data and the fiducial model cor-rection data fσ8,ap(z) are ploted in Fig. 3. As shown inFig. 3 this correction has little effect on the mean val-ues of fσ8(z). After some calculations, we find that thelargest corrections on the mean values and the variancesare less than 11% and 18%, respectively. The correla-tions between different data points also need to be taken

into account when constraining on the free parameters.

C. Reconstruction of fσ8(z)

Using the reconstructed H(z) or E(z), the recon-structed fσ8(z) can be obtained through Eq. (2.1). Themean value of fσ8(z) can be calculated using Eq. (2.1)with the reconstructed mean value of H(z). The meanvalue and the propagated covariance [41, 64–66] of thereconstructed fσ8(z) can be written as

fσ8(zi) = σ08(Ω0

m)γ(1 + z)3γ

E2γ(z)exp

[−(Ω0

m)γIi], (3.6)

Cov[fσ8,i, fσ8,j ] = 4γ2fσ8,i, fσ8,j

Cov[Ei, Ej ]

EiEj

− (Ω0m)γMij − (Ω0

m)γMji + (Ω0m)2γNij

,

(3.7)

where fσ8,i = fσ8(zi),

Ii =

∫ zi

0

(1 + z)3γ−1

E2γ(z)dz, (3.8)

Mij =

∫ zj

0

(1 + z)3γ−1

H2γ(z)

Cov[E(z), Ei]

E(z)Eidz, (3.9)

Nij =

∫ zi

0

∫ zj

0

[(1 + x)(1 + y)]3γ−1

[E(x)E(y)]2γCov[E(x), E(y)]

E(x)E(y)dxdy.

(3.10)

In Fig. 4, we plot the variation of fσ8(z) reconstructedunder different data combinations with respect to red-shift z. As shown in this figure, the uncertainties of thereconstructed fσ8(z) from BAO or CC+BAO data com-bination are much smaller than the observational uncer-tainties, which is due to the smaller variances of H(z)from BAO data can significantly reduce the reconstructederrors of fσ8(z) at high redshifts. Meanwhile, one canalso find that the reconstructed fσ8(z) with P18 prioror without prior on H0 are consistent with that of theΛCDM model within 2σ region. However, when adoptingR18 prior, the trends of the reconstructed fσ8(z) change.This change is consistent with the reconstruction of H(z)described in section III A.

IV. RESULTS AND DISCUSSION

In this section, we describe the main results acquiredfrom the model-independent method considered in thiswork. As a comparison, we also give constrain on theΛCDM model under the OHD and the RSD data points.The flat-linear priors for the free parameters we chooseare σ0

8 ∈ [0.5, 1.5],Ω0m ∈ [0.0, 1.0], and γ ∈ [0.0, 1.5]. For

the ΛCDM model, we choose H0 ∈ [55, 90] and the priorsof the other three parameters are the same as that of theGP method.

5

50

100

150

200

250

H(z)

CC+NoneH0 = 67.36±4.765

CDM

CC+R18H0 = 73.24±1.553

CDM

CC+P18H0 = 67.26±0.595

CDM

50

100

150

200

250

H(z)

BAO+NoneH0 = 63.22±3.420

CDM

BAO+R18H0 = 73.15±1.560

CDM

BAO+P18H0 = 67.22±0.593

CDM

0.0 0.5 1.0 1.5 2.0 2.5z

50

100

150

200

250

H(z)

ALL+NoneH0 = 66.68±3.098

CDM

0.0 0.5 1.0 1.5 2.0 2.5z

ALL+R18H0 = 72.62±1.478

CDM

0.0 0.5 1.0 1.5 2.0 2.5z

ALL+P18H0 = 67.25±0.589

CDM

FIG. 1. The reconstructed H(z) as function of redshift z within 2σ region. The parameter for the ΛCDM model is Ωm = 0.3and H0 is the reconstructed mean value from the corresponding data combinations. The red error bar plot is OHD data.

The summary of the observational constraints on thefree parameters using different data combinations is dis-played in Table I. The results show that in the ΛCDMmodel, the different OHD data combinations and differ-ent priors of H0 have little influence on σ0

8 and γ, buthave significant impacts on Ω0

m and H0. Unlike that inthe ΛCDM model, the various data combinations and thepriors of H0 have great influence on the constraints of allthe free parameters of the GP method.

In Fig. 5, one can find that the ΛCDM model givesa much tighter constraint on H0 than the GP method,which is due to H0 in the GP method is reconstructedusing only the OHD data. By comparing our results ofthe ΛCDM model with the PLA results, where σ0

8 =0.8120 ± 0.0073 and Ω0

m = 0.3166 ± 0.0084 [7], we findthat σ0

8 is consistent with the PLA results in 1σ regionunder different data combinations with or without prioron H0. Meanwhile, all the constraints of γ in ΛCDMmodel are consistent with the GR prediction γ = 0.55 in1σ region.

From Table I, we should also note that in the ΛCDMmodel and the GP reconstruction method, the increasingof H0 results in a decreasing of Ω0

m, which increases thetension level between our constraint Ω0

m and that of thePLA results. To better understand the relationship and

tension level between the free parameters, in Figs. 6, 7and 8 we display the 2-D contour plots and the 1-D pos-terior distributions of the corresponding free parametersunder different cases. The results of the ΛCDM modelunder the same data combinations with P18 priors on H0

are also shown in Figs. 6, 7, and 8 as a comparison.

The contour plots of the GP method in the three fig-ures suggest that σ0

8 is anti-correlated with Ω0m and γ,

but Ω0m is positively correlated with γ, which are quite

different from that of ΛCDM model. As can be seen fromFigs. 6 - 8 and the results in Table I, the OHD data fromBAO prefers larger σ0

8 , Ω0m and γ than the CC data if we

do not take any prior on H0. However, using all the OHDdata points without prior onH0, the best-fit of our resultsare closer to that of PLA results than using only OHDdata from CC or BAO. Comparing the results under thesame data combination, it can be found that the threefree parameters vary greatly under different priors of H0,especially the matter density parameter Ω0

m, which sug-gests that H0 has a great influence on constraining thethree parameters.

From the Ω0m − σ0

8 contour plots in Figs. 6, 7 and 8,one can find that the tension level between our constraintΩ0m − σ0

8 and the PLA values are within 1σ level whentaking P18 H0 prior and out of 2σ level when taking

6

1.0

1.5

2.0

2.5

3.0

3.5

4.0

E(z)

CC+NoneCDM

CC+R18CDM

CC+P18CDM

1.0

1.5

2.0

2.5

3.0

3.5

4.0

E(z)

BAO+NoneCDM

BAO+R18CDM

BAO+P18CDM

0.0 0.5 1.0 1.5 2.0 2.5z

1.0

1.5

2.0

2.5

3.0

3.5

4.0

E(z)

ALL+NoneCDM

0.0 0.5 1.0 1.5 2.0 2.5z

ALL+R18CDM

0.0 0.5 1.0 1.5 2.0 2.5z

ALL+P18CDM

FIG. 2. The reconstructed E(z) as function of redshift z within 2σ region. The parameter for the ΛCDM model is Ωm = 0.3.

R18 prior. The dashed lines in the Ω0m − σ0

8 contourplots represent the best-fit values of PLA results. Be-sides, though our constrainton the growth index γ variesgreatly in different cases, the tension level between ourconstraint result and the GR prediction are almost lessthan 2σ.

V. SUMMARY

In this work, we consider the constraints on the matterfluctuation amplitude σ0

8 , the matter density parameterΩ0m, and the growth index γ by using the latest OHD

and RSD data combinations. To be model-independent,we use GP method to reconstruct the Hubble parameterH(z) from different OHD data combinations, and thenobtain a theoretical fσ8(z) function. We then use thereconstructed fσ8(z) and the RSD data to sample thefree parameters by means of the MCMC method. Here,to reduce the impact of the AP effect, we also correct theRSD data using the H(z) and DA(z) reconstructed byGP method.

From the curves of the reconstructed H(z) and E(z)shown in Figs. 1 and 2 respectively, one can find that theexpansion history varies greatly from the ΛCDM modelwhen taking R18 prior on H0, especially at the low red-

shifts. This indicates that to solve or relax the H0 tensionproblem, it is necessary to have a model with a differentexpansion history from the ΛCDM model or a dynamicaldark energy model which is different from the cosmolog-ical constant at low redshifts.

As Fig. 4 shows, the reconstructed fσ8(z) fits well withthe ΛCDM model when using the same cosmological pa-rameters. Meanwhile, we also find that both the expan-sion history H(z) and the prior of H0 have great influ-ences on the reconstruction of the fσ8(z), which is alsosupported by the MCMC sampling results obtained inthis paper.

Our results show that the tension level between ourbest-fit of Ω0

m − σ08 and that of the PLA results are no

more than 2σ in the cases of no H0 prior. And the ten-sion will even disappear (because of less than 1σ) whenadopting P18 H0 prior. However, we should note that ourconstraints on Ω0

m and σ08 are much looser compared with

the PLA results, which means that the small tension levelmay be caused by the larger uncertainties. Nonetheless,using all the OHD data points and the RSD data with-out H0 prior, our constraint result of Ω0

m and σ08 are very

close to the PLA values, and the tension level betweenthe growth index γ and the GR prediction γ = 0.55 is at1σ level.

7

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

CC+None ObsAP

CC+R18 ObsAP

CC+P18 ObsAP

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

BAO+None ObsAP

BAO+R18 ObsAP

BAO+P18 ObsAP

0 10 20 30 40 50 60No. of data

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

ALL+None ObsAP

0 10 20 30 40 50 60No. of data

ALL+R18 ObsAP

0 10 20 30 40 50 60No. of data

ALL+P18 ObsAP

FIG. 3. Here the index “Obs” is for the original data and “AP” is the data after fiducial model correction. The fσap8 (z) datain different graphs are obtained with different reconstructions of H(z).

ACKNOWLEDGMENTS

This work is supported in part by National NaturalScience Foundation of China under Grant No. 11675032(People’s Republic of China). E.-K. Li thanks J. Gengfor suggestions of English language revising.

8

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

CC+NoneCDM

Rec

CC+R18CDM

Rec

CC+P18CDM

Rec

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

BAO+NoneCDM

Rec

BAO+R18CDM

Rec

BAO+P18CDM

Rec

0.0 0.5 1.0 1.5 2.0No. of data

0.2

0.3

0.4

0.5

0.6

0.7

f8(

z)

ALL+NoneCDM

Rec

0.0 0.5 1.0 1.5 2.0No. of data

ALL+R18CDM

Rec

0.0 0.5 1.0 1.5 2.0No. of data

ALL+P18CDM

Rec

FIG. 4. The blue region is the reconstruction of fσ8(z) using the reconstructed H(z) from OHD data through Eq. (3.6) within2σ region, the solid black line is fσ8(z) derived form Eq. (2.1) under ΛCDM model. The red error bar plot are the measurementsof RSD data after fiducial model correction. The parameters used here are Ωm = 0.3, σ8 = 0.8, and γ = 0.545.

9

TABLE I. 68% and 95% confidence level constraints on the free parameters for various observational data combinations. Notethat H0 of the GP method is directly reconstructed from the OHD data.

Method/Model Data combination σ08 Ω0

m γ H0

GP

RSD+CC+None 0.703+0.057−0.16 < 0.969 0.46+0.16

−0.10+0.23−0.27 0.91+0.19

−0.19+0.38−0.36 67.36 ± 4.8 ± 9.6

RSD+CC+R18 0.92+0.13−0.24

+0.42−0.33 0.169+0.057

−0.15 < 0.380 0.48+0.10−0.13

+0.23−0.21 73.24 ± 1.5 ± 3.0

RSD+CC+P18 0.754+0.068−0.19 < 1.07 0.41+0.19

−0.13+0.27−0.32 0.74+0.17

−0.17+0.34−0.32 67.26 ± 0.60 ± 1.20

RSD+BAO+None 0.719+0.061−0.16

+0.27−0.22 0.507+0.16

−0.095+0.23−0.28 1.01+0.20

−0.20+0.38−0.38 63.22 ± 3.4 ± 6.8

RSD+BAO+R18 1.14+0.19−0.19 > 0.828 0.095+0.023

−0.077+0.13−0.089 0.463+0.079

−0.099+0.18−0.17 73.15 ± 1.6 ± 3.2

RSD+BAO+P18 0.92+0.12−0.25

+0.43−0.32 0.26+0.11

−0.18+0.25−0.23 0.62+0.13

−0.15+0.27−0.26 67.22 ± 0.59 ± 1.18

RSD+ALL+None 0.86+0.10−0.23

+0.40−0.31 0.32+0.14

−0.14+0.25−0.26 0.73+0.15

−0.18+0.32−0.31 66.68 ± 3.1 ± 6.2

RSD+ALL+R18 1.04+0.18−0.26

+0.41−0.34 0.123+0.035

−0.11 < 0.300 0.460+0.083−0.11

+0.21−0.18 72.62 ± 1.5 ± 3.0

RSD+ALL+P18 0.91+0.12−0.25

+0.42−0.33 0.26+0.11

−0.18+0.25−0.25 0.61+0.13

−0.15+0.30−0.26 67.25 ± 0.59 ± 1.18

ΛCDM

RSD+CC+None 0.80+0.13−0.15

+0.25−0.25 0.332+0.051

−0.070+0.13−0.11 0.68+0.28

−0.32+0.56−0.55 67.7+3.1

−3.1+5.9−6.0

RSD+CC+R18 0.78+0.12−0.15

+0.24−0.24 0.261+0.032

−0.032+0.066−0.062 0.65+0.28

−0.33+0.56−0.55 72.2 ± 1.4 ± 2.8

RSD+CC+P18 0.80+0.12−0.15

+0.25−0.24 0.335+0.032

−0.032+0.064−0.062 0.67+0.27

−0.32+0.56−0.55 67.31+0.59

−0.59+1.1−1.2

RSD+BAO+None 0.78+0.12−0.15

+0.24−0.24 0.261+0.017

−0.020+0.040−0.035 0.65+0.28

−0.32+0.55−0.55 70.2+1.2

−1.2+2.3−2.4

RSD+BAO+R18 0.77+0.12−0.15

+0.24−0.24 0.245+0.015

−0.015+0.030−0.028 0.64+0.27

−0.33+0.56−0.55 71.31+0.93

−0.93+1.8−1.8

RSD+BAO+P18 0.79+0.12−0.15

+0.24−0.24 0.297+0.012

−0.012+0.025−0.023 0.66+0.28

−0.28+0.54−0.54 67.83+0.55

−0.55+1.1−1.1

RSD+ALL+None 0.78+0.12−0.15

+0.24−0.24 0.267+0.017

−0.019+0.035−0.035 0.65+0.28

−0.325+0.55−0.56 69.9+1.1

−1.1+2.1−2.2

RSD+ALL+R18 0.77+0.11−0.15

+0.24−0.23 0.251+0.014

−0.014+0.029−0.028 0.64+0.27

−0.33+0.55−0.54 71.01+0.90

−0.90+1.7−1.8

RSD+ALL+P18 0.79+0.12−0.15

+0.25−0.24 0.299+0.012

−0.012+0.023−0.022 0.67+0.28

−0.33+0.56−0.55 67.86 ± 0.53 ± 1.0

10

TABLE II. A compilation of RSD data that reported in different surveys.Index Dataset z fσ8(z) Refs. Year Fiducial Cosmology

1 SDSS-LRG 0.35 0.440± 0.050 [67] 30 October 2006 (Ω0m,ΩK , σ8 [68])= (0.25, 0, 0.756)2 VVDS 0.77 0.490± 0.18 [67] 6 October 2009 (Ω0m,ΩK , σ8) = (0.25, 0, 0.78)3 2dFGRS 0.17 0.510± 0.060 [67] 6 October 2009 (Ω0m,ΩK) = (0.3, 0, 0.9)4 2MRS 0.02 0.314± 0.048 [69], [70] 13 Novemver 2010 (Ω0m,ΩK , σ8) = (0.266, 0, 0.65)5 SnIa+IRAS 0.02 0.398± 0.065 [71], [70] 20 October 2011 (Ω0m,ΩK , σ8) = (0.3, 0, 0.814)6 SDSS-LRG-200 0.25 0.3512± 0.0583 [72] 9 December 2011 (Ω0m,ΩK , σ8) = (0.276, 0, 0.8)7 SDSS-LRG-200 0.37 0.4602± 0.0378 [72] 9 December 20118 SDSS-LRG-60 0.25 0.3665± 0.0601 [72] 9 December 2011 (Ω0m,ΩK , σ8) = (0.276, 0, 0.8)9 SDSS-LRG-60 0.37 0.4031± 0.0586 [72] 9 December 201110 WiggleZ 0.44 0.413± 0.080 [73] 12 June 2012 (Ω0m, h, σ8) = (0.27, 0.71, 0.8)11 WiggleZ 0.60 0.390± 0.063 [73] 12 June 2012 Cij = Eq. (2.6)12 WiggleZ 0.73 0.437± 0.072 [73] 12 June 201213 6dFGS 0.067 0.423± 0.055 [74] 4 July 2012 (Ω0m,ΩK , σ8) = (0.27, 0, 0.76)14 SDSS-BOSS 0.30 0.407± 0.055 [75] 11 August 2012 (Ω0m,ΩK , σ8) = (0.25, 0, 0.804)15 SDSS-BOSS 0.40 0.419± 0.041 [75] 11 August 201216 SDSS-BOSS 0.50 0.427± 0.043 [75] 11 August 201217 SDSS-BOSS 0.60 0.433± 0.067 [75] 11 August 201218 Vipers 0.80 0.470± 0.080 [76] 9 July 2013 (Ω0m,ΩK , σ8) = (0.25, 0, 0.82)19 SDSS-DR7-LRG 0.35 0.429± 0.089 [77] 8 August 2013 (Ω0m,ΩK , σ8 [2])= (0.25, 0, 0.809)20 GAMA 0.18 0.360± 0.090 [78] 22 September 2013 (Ω0m,ΩK , σ8) = (0.27, 0, 0.8)21 GAMA 0.38 0.440± 0.060 [78] 22 September 201322 BOSS-LOWZ 0.32 0.384± 0.095 [79] 17 December 2013 (Ω0m,ΩK , σ8) = (0.274, 0, 0.8)23 SDSS DR10 and DR11 0.32 0.48± 0.10 [79] 17 December 2013 (Ω0m,ΩK , σ8 [80])= (0.274, 0, 0.8)24 SDSS DR10 and DR11 0.57 0.417± 0.045 [79] 17 December 201325 SDSS-MGS 0.15 0.490± 0.145 [81] 30 January 2015 (Ω0m, h, σ8) = (0.31, 0.67, 0.83)26 SDSS-veloc 0.10 0.370± 0.130 [82] 16 June 2015 (Ω0m,ΩK , σ8 [83])= (0.3, 0, 0.89)27 FastSound 1.40 0.482± 0.116 [84] 25 November 2015 (Ω0m,ΩK , σ8 [4])= (0.27, 0, 0.82)28 SDSS-CMASS 0.59 0.488± 0.060 [85] 8 July 2016 (Ω0m, h, σ8) = (0.307115, 0.6777, 0.8288)29 BOSS DR12 0.38 0.497± 0.045 [10] 11 July 2016 (Ω0m,ΩK , σ8) = (0.31, 0, 0.8)30 BOSS DR12 0.51 0.458± 0.038 [10] 11 July 201631 BOSS DR12 0.61 0.436± 0.034 [10] 11 July 201632 BOSS DR12 0.38 0.477± 0.051 [86] 11 July 2016 (Ω0m, h, σ8) = (0.31, 0.676, 0.8)33 BOSS DR12 0.51 0.453± 0.050 [86] 11 July 201634 BOSS DR12 0.61 0.410± 0.044 [86] 11 July 201635 Vipers v7 0.76 0.440± 0.040 [87] 26 October 2016 (Ω0m, σ8) = (0.308, 0.8149)36 Vipers v7 1.05 0.280± 0.080 [87] 26 October 201637 BOSS LOWZ 0.32 0.427± 0.056 [88] 26 October 2016 (Ω0m,ΩK , σ8) = (0.31, 0, 0.8475)38 BOSS CMASS 0.57 0.426± 0.029 [88] 26 October 201639 Vipers 0.727 0.296± 0.0765 [89] 21 November 2016 (Ω0m,ΩK , σ8) = (0.31, 0, 0.7)40 6dFGS+SnIa 0.02 0.428± 0.0465 [90] 29 November 2016 (Ω0m, h, σ8) = (0.3, 0.683, 0.8)41 Vipers 0.6 0.48± 0.12 [91] 16 December 2016 (Ω0m,Ωb, ns, σ8 [6])= (0.3, 0.045, 0.96, 0.831)42 Vipers 0.86 0.48± 0.10 [91] 16 December 201643 Vipers PDR-2 0.60 0.550± 0.120 [92] 16 December 2016 (Ω0m,Ωb, σ8) = (0.3, 0.045, 0.823)44 Vipers PDR-2 0.86 0.400± 0.110 [92] 16 December 201645 SDSS DR13 0.1 0.48± 0.16 [93] 22 December 2016 (Ω0m, σ8 [83])= (0.25, 0.89)46 2MTF 0.001 0.505± 0.085 [94] 16 June 2017 (Ω0m, σ8) = (0.3121, 0.815)47 Vipers PDR-2 0.85 0.45± 0.11 [95] 31 July 2017 (Ωb,Ω0m, h) = (0.045, 0.30, 0.8)48 BOSS DR12 0.31 0.469± 0.098 [96] 15 September 2017 (Ω0m, h, σ8) = (0.307, 0.6777, 0.8288)49 BOSS DR12 0.36 0.474± 0.097 [96] 15 September 201750 BOSS DR12 0.40 0.473± 0.086 [96] 15 September 201751 BOSS DR12 0.44 0.481± 0.076 [96] 15 September 201752 BOSS DR12 0.48 0.482± 0.067 [96] 15 September 201753 BOSS DR12 0.52 0.488± 0.065 [96] 15 September 201754 BOSS DR12 0.56 0.482± 0.067 [96] 15 September 201755 BOSS DR12 0.59 0.481± 0.066 [96] 15 September 201756 BOSS DR12 0.64 0.486± 0.070 [96] 15 September 201757 SDSS DR7 0.1 0.376± 0.038 [97] 12 December 2017 (Ω0m,Ωb, σ8) = (0.282, 0.046, 0.817)58 SDSS-IV 1.52 0.420± 0.076 [98] 8 January 2018 (Ω0m,Ωbh

2, σ8) = (0.26479, 0.02258, 0.8)59 SDSS-IV 1.52 0.396± 0.079 [99] 8 January 2018 (Ω0m,Ωbh

2, σ8) = (0.31, 0.022, 0.8225)60 SDSS-IV 0.978 0.379± 0.176 [100] 9 January 2018 (Ω0m, σ8) = (0.31, 0.8)61 SDSS-IV 1.23 0.385± 0.099 [100] 9 January 201862 SDSS-IV 1.526 0.342± 0.070 [100] 9 January 201863 SDSS-IV 1.944 0.364± 0.106 [100] 9 January 2018

11

60 62 64 66 68 70 72 74 76H0 [km/s/Mpc]

GP: CC+NoneCDM: CC+None

GP: CC+R18CDM: CC+R18

GP: CC+P18CDM: CC+P18

GP: BAO+NoneCDM: BAO+None

GP: BAO+R18CDM: BAO+R18

GP: BAO+P18CDM: BAO+P18GP: ALL+None

CDM: ALL+NoneGP: ALL+R18

CDM: ALL+R18GP: ALL+P18

CDM: ALL+P18

FIG. 5. Whisker plot with the 68% confidence level of theHubble constant H0 obtained from the GP method and theMCMC sampling results with ΛCDM model using differentdata combinations. The two grey vertical bands correspond tothe P18 (left) and R18 (right) values for the Hubble constant.

0.3 0.6 0.9 1.2

0.2

0.4

0.6

0.8

0 m

0.8 1.0 1.208

0.3

0.6

0.9

1.2

0.2 0.4 0.6 0.80m

RSD+CC+P18( CDM)RSD+CC+NoneRSD+CC+P18RSD+CC+R18

FIG. 6. The 68% and 95% confidence level contour plots be-tween various combinations of the free parameters using thereconstructed H(z) from CC data points, and the correspond-ing one dimensional marginalized posterior distributions. Thevertical and horizontal dashed lines represent σ0

8 = 0.8120,Ω0m = 0.3166 and γ = 0.545.

0.3 0.6 0.9 1.2

0.2

0.4

0.6

0.8

0 m

0.8 1.0 1.208

0.3

0.6

0.9

1.2

0.2 0.4 0.6 0.80m

RSD+BAO+P18( CDM)RSD+BAO+NoneRSD+BAO+P18RSD+BAO+R18

FIG. 7. The 68% and 95% confidence level contour plotsbetween various combinations of the free parameters usingthe reconstructed H(z) from BAO data points, and the cor-responding one dimensional marginalized posterior distribu-tions. The vertical and horizontal dashed lines representσ08 = 0.8120, Ω0

m = 0.3166 and γ = 0.545.

0.3 0.6 0.9 1.2

0.2

0.4

0.6

0 m

0.8 1.0 1.208

0.3

0.6

0.9

1.2

0.2 0.4 0.60m

RSD+ALL+P18( CDM)RSD+ALL+NoneRSD+ALL+P18RSD+ALL+R18

FIG. 8. The 68% and 95% confidence level contour plots be-tween various combinations of the free parameters using thereconstructed H(z) from CC+BAO data combination, andthe corresponding one dimensional marginalized posterior dis-tributions. The vertical and horizontal dashed lines representσ08 = 0.8120, Ω0

m = 0.3166 and γ = 0.545.

12

TABLE III. The latest Hubble parameter measurements H(z)(in units of km s−1 Mpc−1) and their errors σH at redshift zobtained from the differential age method (DA).

Index z H(z) σH Reference1 0.07 69.0 19.6

[101]2 0.12 68.6 26.23 0.2 72.9 29.64 0.28 88.8 36.65 0.1 69 12

[102]

6 0.17 83 87 0.27 77 148 0.4 95 179 0.48 97 6010 0.88 90 4011 0.9 117 2312 1.3 168 1713 1.43 177 1814 1.53 140 1415 1.75 202 4016 0.1797 75 4

[103]

17 0.1993 75 518 0.3519 83 1419 0.5929 104 1320 0.6797 92 821 0.7812 105 1222 0.8754 125 1723 1.037 154 2024 0.3802 83 13.5

[104]25 0.4004 77 10.226 0.4247 87.1 11.227 0.4497 92.8 12.928 0.4783 80.9 929 1.363 160 33.6

[105]30 1.965 186.5 50.431 0.47 89 34 [106]

TABLE IV. The latest Hubble parameter measurements H(z)(in units of km s−1 Mpc−1) and their errors σH at redshift zobtained from the radial BAO method (clustering).

Index z H(z) σH Reference1 0.24 79.69 2.65

[61]2 0.43 86.45 3.683 0.3 81.7 6.22 [107]4 0.31 78.17 4.74

[108]

5 0.36 79.93 3.396 0.40 82.04 2.037 0.44 84.81 1.838 0.48 87.79 2.039 0.52 94.35 2.6510 0.56 93.33 2.3211 0.59 98.48 3.1912 0.64 98.82 2.9913 0.35 82.7 8.4 [77]14 0.38 81.5 1.9

[10]15 0.51 90.4 1.916 0.61 97.3 2.117 0.44 82.6 7.8

[73]18 0.6 87.9 6.119 0.73 97.3 720 0.57 96.8 3.4 [80]21 2.33 224 8 [109]22 2.34 222 7 [9]23 2.36 226 8 [110]

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