Jinglan Zheng, Gong-Bo Zhao , Yuting Wang, Xiaoyong Mu, Ruiyang Zhao , Weibing Zhang, Shuo Yuan,David Bacon, and Kazuya Koyama

1 National Astronomy Observatories, Chinese Academy of Sciences, Beijing, 100101, China; jinglan.zheng@myport.ac.uk, gbzhao@nao.cas.cn

2 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China

3 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK

Received 2022 April 12; revised 2022 April 19; accepted 2022 April 22; published 2022 May 26

Abstract We propose a new consistency test for the ΛCDM cosmology using baryonic acoustic oscillations (BAO) and redshift space distortion (RSD) measurements from galaxy redshift surveys. Specifically, we determine the peak position of fσ8(z) in redshift z offered by an RSD measurement, and compare it to the one predicted by the BAO observables assuming a flat ΛCDM cosmology. We demonstrate this new test using the simulated data for the DESI galaxy survey, and argue that this test complements those using the background observables alone, and is less subject to systematics in the RSD analysis, compared to traditional methods using values of fσ8(z) directly.

Key words: Cosmology – (cosmology:) dark energy – galaxies: distances and redshifts

1. Introduction

The ΛCDM model, in which the cold dark matter (CDM)and a cosmological constant, Λ, contribute to roughly 1/3 and 2/3 energy budget of the current universe respectively, has become the standard cosmological paradigm(Riess et al.1998;Perlmutter et al. 1999). Although this “vanilla” model is favored by most observations available so far in terms of model selection, it is being challenged, especially by the “Hubble crisis” (see Di Valentino et al. 2021 for a recent review).

Performing consistency tests for the ΛCDM model is one of the most efficient ways to discover new physics,if any,beyond the standard cosmological scenario,and efforts have been made along these lines.For example,the quantity Om(z)(Sahni et al.2008), derived from H(z) using cosmic chronometers measuring the age of passive galaxies at various redshifts (Moresco et al. 2012), complemented with the local H0measurement(Riess et al.2021)can be used for a consistency test.Om(z)is a constant and coincides with ΩM,0only if the underlying cosmology is ΛCDM,while it evolves with redshift otherwise.This quick test relies on measuring both H(z)and H0precisely,which is challenging. Further, this test only accounts for the expansion history of the universe.

In this paper, we propose a new consistency test for the Λ CDM model only using observables delivered by spectroscopic galaxy surveys; namely, we use baryonic acoustic oscillations (BAO) (Eisenstein et al. 2005) and redshift space distortion (RSD) (Kaiser 1987) measurements at multiple redshifts to hunt for deviations from the ΛCDM model at both the background and perturbation levels, and demonstrate our method using simulated Dark Energy Spectroscopic Instrument(DESI) measurements (DESI Collaboration et al. 2016).

The new method is presented in Section 2, including the relevant formalism and procedure; we then show the main result in Section 3, before concluding in Section 4.

2. Methodology

2.1. The Idea and Procedure

We start from the well-known relations for the evolution of the matter density parameter and growth of structure(Dodelson 2003),

where H denotes the expansion rate of the universe, ΩMis the fractional matter density, δ is the overdensity of matter and σ8is the root-mean-square (rms) matter fluctuation on a scale of 8 h-1Mpc. Symbols with a subscript 0 mean quantities at redshift 0, and Equation (2) is a reasonable approximation relating the expansion history with structure growth, through the growth index γ (Linder 2005).

Combining Equations (1)–(3), we obtain

Figure 1. The orange and blue curves show quantities γ(1-2q) and ΩM γ as a function of a,respectively.The collections of curves represent models with five values of w,ΩM,0 and γ each,evenly distributed in ranges of w ∈[-1.2,-0.8],ΩM,0 ∈[0.28, 0.35] and γ ∈[0.45, 0.65], so that there are 125 curves in total for both γ(1-2q) and ΩM γ .

According to Equation(4),fσ8(a)has a peak in a(and in z)at a specific redshift, namely, a=ap, if

where the last equation holds in the ΛCDM scenario. To investigate whether the peak of fσ8exists for a range of cosmologies,we allow the growth index γ,the equation of state of dark energy w(assumed to be a constant),and the fractional matter density at current epoch ΩM,0, to vary within a wide range. Figure 1 shows two groups of curves ofΩMγand γ(1-2q) as functions of the scale factor a, calculated using Equations (1) and (2) with H0=67.4 km s-1Mpc-1,ΩM=0.31, γ=6/11, which are the values favored by the Planck 2018 observations in the ΛCDM model (Planck Collaboration et al. 2020).

The two groups of curves intersect in all cases, which means that the peak exists for all these cosmologies. This is confirmed by Figure 2, where fσ8for various cosmologies is shown.

Given ΩM,0, w and γ, the position of the peak in fσ8(a) can be found by solving Equation (5) explicitly. In ΛCDM, where w=-1 and γ=6/11 (Linder 2005), the peak position of fσ8(a) in a is determined once ΩM,0is known. On the other hand, ΩM,0in a flat ΛCDM universe can be found from DA(z)H(z),the product of the radial and transverse distances at redshift z,which is provided by a BAO measurement.Note that in a flat ΛCDM universe, DA(z)H(z) only depends on ΩM,0.

This motivates a new consistency test for ΛCDM model using BAO and RSD measurements, which are provided by galaxy surveys. The procedure for this new test is as follows:

2.2. Determine ,and

Given a cosmological model, which in this work is considered to be wγ CDM (a CDM model with a constant w for dark energy and a growth index γ for matter in a flat universe), we first perform a Fisher matrix forecast for DAH and fσ8(with all relevant correlation coefficients) at 18 effective redshifts (uniform in z from z=0.05 to 1.75) jointly covered by the LRGs and ELGs to be observed by a 14,000 deg2DESI survey, with dN/dz specified in Table 2.3 in the official DESI forecast paper (DESI Collaboration et al. 2016).

Figure 2.The quantity fσ8 as a function of a for various cosmologies.In all panels,the fiducial ΛCDM model is shown in the middle for reference.“Variations”in the bottom right panel means fσ8 for models (125 in total) with all parameters varied.

3. Results

In this section, we demonstrate the proposed new consistency test using the simulated BAO and RSD data assuming a DESI sensitivity.We perform the new consistency test on four different models:

Figure 3.The derived , and for the ΛCDM model.Top:The 1D distribution of at 18 redshifts(blue)and (black).The green vertical line denotes the fiducial ΩM,0,which is consistent with the Planck 2018 cosmology used for the forecast.Bottom:The forecast fσ8(blue data points with error bars),(thin green band) and (thick orange).

1. Model I: w=-1, γ=γGR=6/11 (ΛCDM)

2. Model II:w=-1, γ=γGR-0.095=0.45

3. Model III: w=-1, γ=γGR+0.105=0.65

Figure 4. Same as Figure 3 but for model II (w=-1, γ=0.45).

Figure 5. Same as Figure 3 but for model III (w=-1, γ=0.65).

Table 1 The Significance ofDerived from Different Redshifts not Being a Constant (Middle Column), and of ≠ (Right Column) for Various Models as Listed

Table 1 The Significance ofDerived from Different Redshifts not Being a Constant (Middle Column), and of ≠ (Right Column) for Various Models as Listed

Models(S/N)B(S/N)P w=-1, γ=γGR00.41 w=-1, γ=γGR-0.09502.43 w=-1, γ=γGR+0.10504.14 w=-1.2, γ=γGR8.473.86 w=-1.2, γ=γGR-0.0958.474.71 w=-1.2, γ=γGR+0.1058.470.23 w=-0.8, γ=γGR7.361.47 w=-0.8, γ=γGR-0.0957.361.96 w=-0.8, γ=γGR+0.1057.366.30

4. Model IV: w=-0.8, γ=γGR+0.105=0.65 We carry out all of the procedures described in Section 2 for these models; the results are summarized in Figures 3–6 and Table 1.(S/N)Bwith a small (S/N)Pmay suggest a deviation from the ΛCDM model at both the background and perturbation level.The cosmological implication given (S/N)Band (S/N)Pis summarized in Table 2.

Table 2 The Cosmological Implication given (S/N)B and (S/N)P

4. Discussion and Conclusions

In this work we propose a new consistency test for the standard ΛCDM paradigm using the BAO and RSD measurements directly accessible from galaxy redshift surveys.

This new test contains two essential ingredients:a test for the expansion history and a test for the structure growth.Consistency tests for the expansion history have been proposed in the literature, e.g., the Om statistic, which also checks the constancy of ΩM,0derived from observables at different redshifts. However, the key difference between Om and our test is that Om relies on measurements of H(z) and H0, while ours only requires the BAO measurement. Direct H(z)measurements are performed using the age of passive galaxies,and may be subject to large statistical and systematical uncertainties. The local H0measurement, on the other hand,is in serious tension with the indirect inference from the CMB,which may suggest new physics beyond ΛCDM, or unknown systematics. This makes our new test for the background more robust—the BAO measurements are known to be less contaminated by systematics (Ross et al. 2012), and are easier to access from existing galaxy surveys, including 2dFGRS(Lahav et al. 2002), SDSS-III BOSS (Dawson et al. 2013),SDSS-IV eBOSS (Dawson et al. 2016), WiggleZ (Blake et al.2011) and ongoing galaxy surveys such as DESI (DESI Collaboration et al. 2016), PFS (Takada et al. 2014) and the Euclid mission (Amendola et al. 2018).

Acknowledgments

J. Z. is supported by the Chinese Scholarship Council and STFC.J.Z.,X.M.,R.Z.,W.Z.and G.B.Z.are supported by NSFC grants 11925303,11720101004 and 11890691.Y.W.is supported by NSFC Grant 11890691,by the Youth Innovation Promotion Association CAS, and by the Nebula Talents Program of NAOC. G. B. Z. is also supported by the National Key Basic Research and Development Program of China,and a grant of CAS Interdisciplinary Innovation Team. We also acknowledge the science research grants from the China Manned Space Project with No. CMS-CSST-2021-B01.

Appendix Derivation of Equation (4)

Let us start from

which is exactly Equation (4).

ORCID iDs

Gong-Bo Zhao https://orcid.org/0000-0003-4726-6714

Ruiyang Zhao https://orcid.org/0000-0002-7284-7265