Cosmological inference from the EFTofLSS the eBOSS QSO full-shape analysis Théo Simon1 Pierre Zhang2345 Vivian Poulin1

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Cosmological inference from the EFTofLSS:

the eBOSS QSO full-shape analysis

Théo Simon1, Pierre Zhang2,3,4,5, Vivian Poulin1

1Laboratoire Univers & Particules de Montpellier (LUPM), CNRS & Université de

Montpellier (UMR-5299), Place Eugène Bataillon, F-34095 Montpellier Cedex 05, France

2Department of Astronomy, School of Physical Sciences,

University of Science and Technology of China, Hefei, Anhui 230026, China

3CAS Key Laboratory for Research in Galaxies and Cosmology,

University of Science and Technology of China, Hefei, Anhui 230026, China

4School of Astronomy and Space Science,

University of Science and Technology of China, Hefei, Anhui 230026, China

5Institut fur Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

Abstract

We present cosmological results inferred from the eﬀective-ﬁeld theory (EFT) analysis of the full-

shape of eBOSS quasars (QSO) power spectrum. We validate our analysis pipeline against simu-

lations, and ﬁnd overall good agreement between the analyses in Fourier and conﬁguration space.

Keeping the baryon abundance and the spectral tilt ﬁxed, we reconstruct at 68% CL the fractional

matter abundance Ωm, the reduced Hubble constant h, and the clustering amplitude σ8, to respec-

tively Ωm= 0.327 ±0.035,h= 0.655 ±0.034, and σ8= 0.880 ±0.083 from eBOSS QSO alone.

These constraints are consistent at ≲1.8σwith the ones from Planck and from the EFT analysis

of BOSS full-shape. Interestingly S8reconstructed from eBOSS QSO is slightly higher than that

deduced from Planck and BOSS, although statistically consistent. In combination with the EFT

likelihood of BOSS, supernovae from Pantheon, and BAO from lyman-αand 6dF/MGS, constraints

improve to Ωm= 0.2985±0.0069 and h= 0.6803±0.0075, in agreement with Planck and with similar

precision. We also explore one-parameter extensions to ΛCDM and ﬁnd that results are consistent

with ﬂat ΛCDM at ≲1.3σ. We obtain competitive constraints on the curvature density fraction

Ωk=−0.039 ±0.029, the dark energy equation of state w0=−1.038 ±0.041, the eﬀective number

of relativistic species Neﬀ = 3.44+0.44

−0.91 at 68% CL, and the sum of neutrino masses Pmν<0.274eV

at 95% CL, without Planck data. Including Planck data, contraints signiﬁcantly improve thanks

to the large lever arm in redshift between LSS and CMB measurements. In particular, we obtain

the stringent constraint Pmν<0.093eV, competitive with recent lyman-αforest power spectrum

bound.

arXiv:2210.14931v2 [astro-ph.CO] 17 Jul 2023

Contents

1 Introduction 2

2 Analysis pipeline 5

2.1 Two-point function at the one loop ........................ 5

2.2 Cosmological inference setup ............................ 8

2.3 Scale cut from governing scales .......................... 11

2.4 Assessing systematics beyond the EFT reach ................... 14

3 Constraints on ﬂat ΛCDM 16

3.1 Flat ΛCDM from the EFT analysis of eBOSS .................. 17

3.2 Comparison with several LSS probes ....................... 20

3.3 Comparison with Planck .............................. 22

4 Extensions to the ﬂat ΛCDM model 24

4.1 ΩkΛCDM ...................................... 24

4.2 w0CDM ....................................... 28

4.3 νΛCDM ....................................... 29

4.4 Neﬀ ΛCDM ...................................... 31

5 Conclusion 32

A What happens if we vary nsand ωbin the LSS analyses? 35

B Redshift resolution uncertainties 37

1 Introduction

The distribution of matter at large scales contains a wealth of cosmological information, from

the initial conditions of the Universe to the gravitational collapse of late-time objects. The

program of cosmic microwave background (CMB) experiments has now matured to a state

where ΛCDM parameters have been measured to percent level with the Planck satellite [1],

and with similar precision by subsequent experiments, e.g., ACT [2] and SPT [3,4]. At

the same time, the data volume gathered by large-scale structure (LSS) surveys has been

continuously growing. As those surveys probe vastly diﬀerent epochs in the history of the

Universe, they allow for a crucial consistency test of the ΛCDM model and have delivered

independent cosmological determinations at precision comparable to CMB measurements,

see e.g., the recent results from the photometric surveys DES [5] and KIDS [6], or from the

spectroscopic surveys BOSS [7]. In addition, LSS data have become paramount to break

degeneracies of the ΛCDM model and extensions when combined with CMB.

Recently, as the accuracy of observations has improved, various cosmological probes have

delivered cosmological parameter measurements with a growing level of inconsistency. The

most statistically signiﬁcant cosmological discrepancy is the “Hubble tension” [8], correspond-

ing to a diﬀerence of ∼5σbetween the determination of the Hubble constant from Planck

data analyzed under ΛCDM [1] and its local determination from the cosmic distance lad-

der based on cepheid-calibrated SNIa by the SH0ES team [9]. Another intriguing cosmo-

logical puzzle, the “S8tension” (∼2−3σ), has emerged between weak lensing measure-

ments [6,5,10,11] and CMB [1,2] determinations of the local matter ﬂuctuations, param-

eterized as S8=σ8pΩm/0.3, where σ8is the root mean square of matter ﬂuctuations on

a8Mpc h−1scale and Ωmthe fractional matter density today. Spectroscopic surveys probe

the distribution of matter at similar redshifts as the photometric ones, but rely essentially on

scales that in average are larger than the one probed in weak lensing. Thus, spectroscopic

surveys have the potential to play a key role in shedding light on these tensions. In par-

ticular, an agreement between clustering and CMB data would have, under the assumption

that there is no systematic error, signiﬁcant impact on the interpretation of these tensions.

Regarding the S8tension, this would hint that the origin lies in the scales beyond the (large)

scales included in clustering or CMB analyses (see e.g., Ref. [11]). As for the H0tension, a

resolution would then require modiﬁcations to the concordance model that can lift both the

values measured in the CMB and in the LSS.

The large amount of LSS data available provides us with new opportunities to extract

additional cosmological information, by making use of the full-shape of summary statistics

built from clustering data. Among the spectrocopic surveys, the Extended Baryon Oscil-

lation Spectroscopic Survey (eBOSS), combined with previous phases of the Sloan Digital

Sky Survey (SDSS), has mapped more than 11 billion years of cosmic history, providing an

unprecedented map of the matter clustering in the Universe [12] through diﬀerent tracers

of the underlying matter density distribution, e.g., galaxies, quasars or the lyman-αforest.

To extract cosmological information from these surveys, the (e)BOSS collaboration follows

the convention of compressing information from these surveys into simple parameters that

can be easily compared with cosmological models. These are usually expressed in the form

of the Alcock-Paczynski (AP) parameters measured from the BAO angles [13] and the fσ8

parameter, where fis the growth factor, measured from redshift space distortions (RSD)

[14]. However, the large amount of LSS data available provides us with new opportunities

to extract additional cosmological information, by making use of the full-shape of summary

statistics built from clustering data. Given the increasing data volume and the variety of

tracers probed, new methods to make reliable predictions for the full-shape are necessary to

extract the cosmological parameters in a robust and systematic ways.

Thankfully, the underlying density and velocity ﬁelds of any tracer respect a set of sym-

metries in the long-wavelength limit known as Galilean invariance [15,16,17]. Moreover, we

are interested in objects that are non-relativistic, allowing us to deﬁne a nonlinear scale as

the average distance travelled by the objects during the age of the Universe, under which

the underlying ﬁelds and their dynamics can be smoothed out [18]. Building on those con-

siderations, the Eﬀective Field Theory of Large-scale Structure (EFTofLSS) has emerged as

a systematic way to organize the expansions in ﬂuctuations and derivatives of the density

and velocity ﬁelds of the observed tracers at long wavelengths [18,19,20,21,22]. 1The

prediction at the one-loop order for the power spectrum of biased tracers in redshift space

from the EFTofLSS [24] (see also Ref. [25]) has been used to analyze the full-shape of BOSS

clustering data in Refs. [26,27]. These works have shown that: i) higher wavenumbers beyond

the linear regime in good theoretical control can be accessed, bringing additional cosmolog-

ical information (see also Ref. [28]), and ii) with reliable predictions, as the cosmological

parameters (together with the nuisance parameters) are scanned the template can be varied

instead of being held ﬁxed, exploiting the full information of the full-shape beyond the one

from geometrical distortions (see Refs. [29,30] for earlier works where the full-shape pre-

dictions, yet not from the EFTofLSS, were varied at each point in parameter space). EFT

analyses of BOSS data have provided precise and robust determination of ΛCDM parame-

ters [26,27,31,32,33,34,35,36], and pushed down limits on extensions, such as neutrino

masses and eﬀective number of relativistic species [31,37,38,39,40], dark energy [41,42,43],

curvature [44,45], early dark energy [46,47,48,49], non-cold dark matter [50,51], inter-

acting dark energy [52], and more [53,54,55,56]. Some EFT analyses of BOSS data have

also included the bispectrum at tree-level [26,57] and at one-loop [23] (see also Ref. [58]),

pushing down uncertainties on ΛCDM parameters and setting new bounds from the LSS on

non-Gaussianities [59,60,61]. See also, e.g., Refs. [62,63,64,65,66,67] for results from

BOSS and/or eBOSS full-shape analyses using methods diﬀerent from the EFTofLSS. In ad-

dition, the EFTofLSS has made possible the development of a new consistency test of the

ΛCDM and alternative models based on a sound horizon-free analysis [68,69,70], providing

a new way to probe beyond ΛCDM models [71].

In this paper, we analyze the eBOSS quasars (QSO) full-shape using the prediction from

the EFTofLSS. There are two main motivations behind this work. First, the EFTofLSS has

only been used to analyze BOSS luminous red galaxies (LRG) and more recently eBOSS

emission line galaxies (ELG) [72]. As QSO are diﬀerent tracers than LRG, and selected by

SDSS at an overall higher redshift than LRG, the eBOSS QSO full-shape analysis complements

previous BOSS full-shape analysis, providing yet another important consistency test of ΛCDM

at a diﬀerent epoch and for another tracer (while also allowing us to test the assumptions

behind the EFTofLSS such as the aforementioned Galilean invariance symmetries).

Second, the eBOSS QSO full-shape once combined with other cosmological probes can

shed light on extensions to ΛCDM model. Here, we explore four one-parameter extensions to

the ﬂat ΛCDM model, namely the curvature density fraction Ωk, the dark energy equation

of state w0, neutrino masses Pmν, and the eﬀective number of relativistic species Neﬀ . We

1See also the introduction footnote in, e.g., Ref. [23] for relevant related works on the EFTofLSS.

compare the obtained limits with the ones from Planck and with the ones from the standard

BAO/fσ8technique, in order to assess both the consistency of the results and the potential

improvements brought by the EFT analysis.

Our paper is organized as follow. In section 2, we describe the EFT analysis pipeline

for eBOSS QSO that we built. In particular, we review the theoretical prediction of the

EFTofLSS in 2.1, and present the dataset, likelihood, and prior chosen for our analysis in 2.2.

In 2.3, we assess the highest wavenumbers kmax that can be included in the analysis of eBOSS

QSO full-shape data by making use of a general method that consists in evaluating the size of

the theoretical error through the insertion of the dominant next-to-next leading order terms

in the EFTofLSS prediction at one-loop. In 2.4, we address known observational system-

atic eﬀects and provide tests against simulations. In section 3, we present and discuss the

constraints on ﬂat ΛCDM from the EFT analysis of the eBOSS QSO full-shape, both in

Fourier and conﬁguration space, and in combination with other cosmological probes. Re-

sults on extensions to ΛCDM are presented and discussed in section 4. A summary of our

results and concluding remarks are given in section 5. Additional material can be found in

the appendices. Appendix Ais dedicated to exploring the impact of ﬁxing the spectral tilt

nsand the baryons abundance ωbin the base-ΛCDM analysis of the eBOSS QSO full-shape.

In appendix B, we discuss uncertainties in the redshift determination of quasars, and argue

that the main correction happens to be degenerate with some EFT counterterms, justifying

that our analysis is free from those potential systematics.

2 Analysis pipeline

2.1 Two-point function at the one loop

Power spectrum At linear order, the power spectrum of galaxies in redshift space is given

by the famous Kaiser formula [14]:

Pg(z, k, µ) = b1(z) + fµ22P11(z, k),(1)

where P11(z, k)corresponds to the linear matter power spectrum that can be calculated with

a Boltzmann code such as CLASS [73] or CAMB [74], fis the growth factor, b1(z)is the linear

galaxy bias parameter, and µ= ˆz·ˆ

kis the cosine of the angle between the line-of-sight ⃗z and

the wavevector of the Fourier mode ⃗

k. At one-loop order, the formula is improved to [24]:

Pg(k, µ) = Z1(µ)2P11(k)+2Z1(µ)P11(k)cct

+cr,1µ2k2

+cr,2µ4k2

r(2)

+ 2 Zd3q

(2π)3Z2(q,k−q, µ)2P11(|k−q|)P11(q)+6Z1(µ)P11(k)Zd3q

(2π)3Z3(q,−q,k, µ)P11(q)

¯ngcϵ,0+cmono

+ 3cquad

ϵµ2−1

3k2

m,

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CosmologicalinferencefromtheEFTofLSS:theeBOSSQSOfull-shapeanalysisThéoSimon1,PierreZhang2,3,4,5,VivianPoulin11LaboratoireUnivers&ParticulesdeMontpellier(LUPM),CNRS&UniversitédeMontpellier(UMR-5299),PlaceEugèneBataillon,F-34095MontpellierCedex05,France2DepartmentofAstronomy,SchoolofPhysicalSciences,U...

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Cosmological inference from the EFTofLSS the eBOSS QSO full-shape analysis Théo Simon1 Pierre Zhang2345 Vivian Poulin1

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