Cosmological constraints from the power spectrum of eBOSS quasars Chudaykin Anton1 2and Mikhail M. Ivanov3 4y

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Cosmological constraints from the power spectrum of eBOSS quasars

Chudaykin Anton1, 2, ∗and Mikhail M. Ivanov3, 4, †

1Department of Physics & Astronomy, McMaster University,

1280 Main Street West, Hamilton, ON L8S 4M1, Canada

2Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

3School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

4NASA Hubble Fellowship Program Einstein Postdoctoral Fellow

We present the eﬀective-ﬁeld theory (EFT)-based cosmological full-shape analysis of the

anisotropic power spectrum of eBOSS quasars at the eﬀective redshift zeﬀ = 1.48. We perform

extensive tests of our pipeline on simulations, paying a particular attention to the modeling of ob-

servational systematics, such as redshift smearing, ﬁber collisions, and the radial integral constraint.

Assuming the minimal ΛCDM model, and ﬁxing the primordial power spectrum tilt and the phys-

ical baryon density, we ﬁnd the Hubble constant H0= (66.7±3.2) km s−1Mpc−1, the matter

density fraction Ωm= 0.32 ±0.03, and the late-time mass ﬂuctuation amplitude σ8= 0.95 ±0.08.

These measurements are fully consistent with the Planck cosmic microwave background results. Our

eBOSS quasar S8posterior, 0.98 ±0.11, does not exhibit the so-called S8tension. Our work paves

the way for systematic full-shape analyses of quasar samples from future surveys like DESI.

1. INTRODUCTION

The distribution of luminous objects on large cosmo-

logical scales (large-scale structure) is one of the key ob-

servables that allows us to understand the expansion his-

tory and constituents of our Universe. Large-scale struc-

ture traces Universe’s evolution at low redshifts that are

especially important for dark energy studies. As such, it

provides important information complementary to that

of the cosmic microwave background radiation (CMB)

measurements (e.g. Planck [1], ACT [2], SPT-3G [3]),

whose primary ﬂuctuations are most sensitive to physics

at a much earlier epoch, during recombination.

The role of various large-scale structure probes in mea-

suring cosmological parameters has become even more

prominent in the recent years due to the appearance

of tensions between various cosmological datasets. The

most critical ones are the Hubble and growth tensions,

which manifest themselves in the diﬀerence between di-

rect and indirect probes of the Hubble constant H0

and the structure formation parameter S8(or equiva-

lently, the mass ﬂuctuation amplitude σ8), see [4] for

a recent review. Large scale structure surveys such as

BOSS [5], eBOSS [6], DES [7,8], KIDS [9], HSC [10],

are already contributing very signiﬁcantly in eﬀorts

∗Chudayka@mcmaster.ca

†ivanov@ias.edu

to understand these tensions. In particular, the ex-

tended Baryon acoustic Oscillation Spectroscopic Sur-

vey (eBOSS) has provided crucial probes of the expan-

sion rate through baryon acoustic oscillations (BAO)

and the matter growth through redshift-space distortions

(RSD) [6].

An important development in spectroscopic survey

data analysis has appeared recently thanks to theoret-

ical eﬀorts in the eﬀective ﬁeld theory (EFT) of large-

scale structure [11,12], see [13] for a review. This theory

gives an accurate and mathematically consistent theo-

retical model for clustering of matter and various lumi-

nous tracers in the mildly-nonlinear (quasilinear) regime.

This regime is relevant for shot-noise limited spectro-

scopic surveys, which contain the bulk of cosmological

information precisely on the quasilinear scales character-

ized by wavenumber k.0.5hMpc−1. It should be em-

phasized that the EFT is systematic and consistent, i.e.

it is a program of successive approximations that allows

us to compute the nonlinear clustering of galaxies to any

desired accuracy. This can be contrasted with popular

phenomenological models, whose regime of validity, and

hence accuracy, is ultimately limited even on large scales.

Importantly, the EFT model allows to describe the en-

tire shape of the galaxy power spectrum, and hence ex-

tract the information which is not accessible with the

conventional BAO/RSD techniques [14,15]. The EFT-

based full-shape analysis has been successfully applied to

arXiv:2210.17044v2 [astro-ph.CO] 6 Feb 2023

the power spectra and bispectra of BOSS luminous red

galaxies (LRGs) [14–18] and also the power spectrum of

eBOSS emission line galaxies (ELGs) [19]. The main

goal of our paper is to extend the EFT-based full-shape

analysis to the eBOSS quasars (QSO) [20–23].

Quasars are interesting for multiple reasons. Thanks

to their high luminosity, they can be used to trace

the matter distribution at high redshifts. In particu-

lar, the eBOSS quasar sample covers the redshift range

0.8< z < 2.2. This gives us a new window onto the

physics of our Universe just before the onset of dark en-

ergy. In addition, the large comoving volume covered

by quasars makes them ideal to constrain local primor-

dial non-Gaussianity [24,25], which, if present, should

manifest itself as a scale-dependent bias on very large

scales [26]. These are the reasons that made eBOSS

QSO a subject of intense research over last years [20–

25,27–29]. Application of the EFT framework to the

QSO promises to multiply the scientiﬁc gain from this

sample, which can be estimated by a successful execu-

tion of the EFT program on the eBOSS LRG and ELG

catalogs [17,19]. In this work we make a step towards

systematic studies of the eBOSS QSO sample with the

EFT of LSS.

Our paper is structured as follows. We present a sum-

mary of main results in Section 2. We describe the data

and our method in details in Section 3. Extensive tests

on simulations and mock catalogs are presented in Sec-

tion 4. Our ﬁnal results are presented in Section 5and

discussed in Section 6. Several appendices contain im-

portant additional material, such as the full parameter

and constraint tables, further tests of systematics, and

prior volume eﬀects.

2. SUMMARY OF MAIN RESULTS

We start with a short summary of our results. We have

analysed the publicly available redshift-space power spec-

trum monopole (`= 0), quadrupole (`= 2) and hexade-

copole (`= 4) moments of the eBOSS quasar sample [23].

This sample covers the redshift range 0.8< z < 2.2 (ef-

fective redshift zeﬀ = 1.48) and is spreads across two dif-

ferent patches of the sky: North Galactic Cap (NGC) and

South Galactic Cap (SGC). In our baseline QSO analysis

we ﬁx the current physical baryon density ωb= 0.02268

using the BBN prediction [30,31], and adopt the primor-

dial power spectrum tilt ns= 0.9649 from the Planck

CMB measurements [1]1. We vary three other cosmo-

logical parameters in our Markov Chain Monte Carlo

(MCMC) chains: the physical density of cold dark mat-

ter (ωcdm), the reduced Hubble constant (h) and the log-

arithm of the primordial scalar power spectrum ampli-

tude, ln(1010As). We approximate the neutrino sector

with a single state of mass mν= 0.06 eV and two mass-

less states. Alongside cosmological parameters, we vary

a set of 26 nuisance parameters (13 per each cap) cap-

turing non-linear clustering of matter, galaxy bias, and

redshift-space distortions.

The posterior distribution of the QSO eBOSS data in

Ωm−H0−σ8space is shown in the left panel of Fig. 1

along with the BOSS full-shape analysis of the red lu-

minous galaxy sample and the Planck 2018 results. The

right panel of Fig. 1shows the QSO P`(k) measure-

ments along with the best-ﬁt theoretical model. The 1d

marginalized constraints are listed in Tab. I.

3. DATA AND METHODOLOGY

3.1. Data

QSO. Our main analysis is based on the power spectra

from the QSO SDSS DR16 catalogue [23]. The QSO sam-

ple was selected from the SDSS-I-II-III optical imaging

data in the ugriz photometric pass bands [32] and from

the Wide-ﬁeld Infrared Survey Explorer [33]. For details

of the selection algorithm see Ref. [34]. The eBOSS QSO

power spectra were estimated by drawing the redshifts of

the random catalogues from the data catalogue, which

suppresses the radial modes on large scales. We refer the

reader to Refs. [23,35] for details on the QSO catalogue

and its systematics.

The power spectrum was built from the galaxies col-

lected from two diﬀerent patches of the sky. The NGC

and SGC patches have the following eﬀective redshifts

zeﬀ , eﬀective shot noise term Pshot

0of the Yamamoto es-

1We checked that imposing tight Gaussian priors on ωband ns

from BBN and Planck, respectively, leads to identical parameter

constraints.

0.3 0.4

Ωm

0.8

1.0

1.2

σ8

60 65 70 75

0.8 1.0 1.2

σ8

eBOSS QSO

BOSS FS+BAO+BBN

Planck 2018

●ℓ=0

●ℓ=2

●ℓ=4

0.05 0.10 0.15 0.20 0.25 0.30

-500

500

1000

k, h Mpc-1

kPℓ(k),(Mpc/h)2

QSO NGC sample, zeff=1.48

●ℓ=0

●ℓ=2

●ℓ=4

0.05 0.10 0.15 0.20 0.25 0.30

-500

500

1000

k, h Mpc-1

kPℓ(k),(Mpc/h)2

QSO SGC sample, zeff=1.48

FIG. 1. Left panel: marginalized constraints on the matter fraction Ωm, Hubble constant H0and mass ﬂuctuation amplitude

σ8in the ΛCDM model from the power spectra of eBOSS quasars (QSO). For comparison we also show the Planck CMB 2018

[1] and BOSS DR12 full-shape analysis of the galaxy power spectra and the bispectrum monopole [17]. Right panel: QSO

power spectrum monopole (`= 0), quadrupole (`= 2) and hexadecapole (`= 4) moments from North and South Galactic

Caps of the eBOSS data. The best-ﬁt ΛCDM model from the combined analysis of both caps is shown by solid lines. The error

bars correspond to the square root of diagonal elements of the covariance matrix.

Param.

Dataset eBOSS QSO BOSS FS+BAO+BBN Planck 2018

ωcdm 0.1184+0.009

−0.011 0.1262+0.0053

−0.0059 0.1200+0.0012

−0.0012

102ωb− − 2.237+0.015

−0.015

h0.6669+0.031

−0.034 0.6832+0.0083

−0.0086 0.6736+0.0055

−0.0053

ln(1010As) 3.375+0.18

−0.18 2.742+0.096

−0.099 3.044+0.014

−0.015

ns− − 0.9649+0.0042

−0.0042

Ωm0.3205+0.03

−0.038 0.3196+0.01

−0.01 0.3153+0.0071

−0.0077

σ80.9445+0.081

−0.083 0.7221+0.032

−0.037 0.8112+0.006

−0.0062

S80.976+0.1

−0.12 0.745+0.037

−0.041 0.832+0.013

−0.013

TABLE I. Mean values and 68% CL minimum credible intervals for the parameters of the ΛCDM model. The BBN prior on

ωband the Planck prior on nsare assumed in the two LSS analyses, and the corresponding posteriors are not displayed. The

top group are the parameters directly sampled in the MCMC chains, the bottom ones are derived parameters.

timator [23,36]2and comoving geometric volumes V,

NGC : zeﬀ = 1.48 , P shot

0= 6.3·104[Mpc/h]3,

V= 13.62 h−3Gpc3,(1)

SGC : zeﬀ = 1.48 , P shot

0= 7.1·104[Mpc/h]3,

V= 8.76 h−3Gpc3.(2)

In our analysis we use the publicly available pre-

reconstructed eBOSS QSO power spectrum miltipoles

computed from the FKP-weighted density ﬁeld [37]

using the Yamamoto estimator [36] implemented in

the nbodykit package [38].3We use the monopole,

quadrupole and hexadecope moments with the scale cuts

kmin = 0.02 and kmax = 0.3hMpc−1following the eBOSS

analysis [23] and our own tests presented later. The

power spectrum multipoles are binned in momentum

space spheres of width ∆k= 0.01 hMpc−1. The redshifts

and angular scales are converted into comoving coordi-

nates using a ﬂat ΛCDM cosmology with Ωm= 0.31.

BOSS FS+BAO+BBN. We also present the results

of the full-shape BOSS DR12 LRG analysis based on

the power spectrum and the bispectrum monopole data

from [5,39] combined with the BBN prior on ωband

the Planck prior on ns. We use the same likelihood

as in Ref. [17] based on the window-free estimators. 4

The power spectrum analysis includes the monopole,

quadrupole and hexadecope moments with kmin = 0.01

and kmax = 0.2hMpc−1along with the BAO measure-

ments performed with the post-reconstructed power spec-

tra. We compute the cross-covariance between the BAO

and full-shape measurements from the Patchy mocks ac-

cording to the procedure presented in Ref. [16]. As far

as the bispectrum is concerned we use kmin = 0.01 and

kmax = 0.08 hMpc−1with step ∆k= 0.01 which corre-

sponds to 62 unique triangular conﬁgurations following

the recent analyses [17,40].

2Note that the shot noise term includes various corrections for

survey geometry and systematic eﬀects. Thus, it does not iden-

tically correspond to a physical number density of the sample.

3Publicly available at https://svn.sdss.org/public/data/

eboss/DR16cosmo/tags/v1_0_1/dataveccov/lrg_elg_qso/QSO_

Pk/

4Publicly available at https://github.com/oliverphilcox/

BOSS-Without-Windows

3.2. Theory model

We ﬁt the QSO power spectrum multipoles’ data us-

ing the one-loop perturbation theory (eﬀective ﬁeld the-

ory). The theoretical predictions are calculated using the

CLASS-PT code [41]. This code is based on the path inte-

gral formulation of the EFT of LSS known as time-sliced

perturbation theory [42–46]. In this work we vary the

following set of power spectrum nuisance parameters in

our MCMC chains,

{b1, b2, bG2, bΓ3, c0, c2, c4,˜c, a2, Pshot, Pshot,2},(3)

where in addition to the standard one-loop parameters

[14,47] we introduce a constant contribution to the

quadrupole moment, Pshot,2, which serves to model sys-

tematic eﬀects present in the QSO sample. We will val-

idate our model with N-body and approximate simula-

tions. The priors on these nuisance parameters will be

discussed shortly. Detailed information about the stan-

dard EFT of LSS theoretical model and nuisance param-

eters can be found in Refs. [14,41].

3.3. Window function

Survey geometry. To take survey geometry eﬀects

into account one needs to fold the true power spectrum

with a survey-speciﬁc selection function. We use the pub-

licly available survey window functions from [23]. The

eBOSS QSO window function multipoles for the diﬀer-

ent galactic caps are shown in Fig. 2.

ℓ=0 NGC ℓ=2 NGC ℓ=4 NGC

ℓ=0 SGC ℓ=2 SGC ℓ=4 SGC

1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s, h-1Mpc

Wℓ2

Window function multipoles

FIG. 2. Window function multipoles of the baseline eBOSS

QSO sample.

ℓ=2

ℓ=4

noW (ℓ=0)

NGC W

SGC W

0.05 0.10 0.15 0.20 0.25 0.30

200

400

600

800

1000

k, h Mpc-1

kPℓ(k),(Mpc/h)2

EZ mocks window function effect

FIG. 3. Eﬀect of the window functions on typical power spec-

trum multipoles of the EZ mocks for NGC and SGC foot-

prints.

We implement window function by employing the stan-

dard approach based on the plane parallel approxima-

tion, see Ref. [48,49]. This method amounts to a simple

multiplication of the true correlation functions ξtrue

`in

position space with the window function multipoles W2

In our analysis we use the quasar multipole moments up

to the hexadecopole, so we have

ξwin

0=ξtrue

0W2

0+1

5ξtrue

2W2

2+1

9ξtrue

4W2

ξwin

2=ξtrue

0W2

2+ξtrue

2W2

0+2

7W2

2+2

7W2

4

+ξtrue

42

7W2

2+100

693W2

4,

ξwin

4=ξtrue

0W2

4+ξtrue

218

35W2

2+20

77W2

4

+ξtrue

4W2

0+20

77W2

2+162

1001W2

4.

(4)

We have checked that the eﬀect of higher order window

functions is negligible. Given this reason, we do not in-

clude multipole window functions with `= 6 and higher.

The eﬀect of the window functions on the power spec-

trum multipoles is shown in Fig. 3.

Radial integral constraint. Another important ef-

fect is due to the scheme used to assign redshifts to ran-

doms from the mock data redshift distribution which sup-

presses radial modes on large scales. To take this eﬀect

into account, we implement the radial integral constraint

(RIC) using the formalism introduced in Ref. [35]. The

full RIC correction is to be subtracted from the windowed

correlation function in conﬁguration space,

ξicc

`(s) = ξwin

`(s)−PshotWsn

`(s)

−X

4π

2`0+ 1 Zs02ds0W``0(s, s0)ξtrue

`0(s0),(5)

where W``0are Legendre multipole moments of the 3-

point correlator of the selection function, and Wsn

`de-

notes the shot noise contribution to the integral con-

straint, for details we refer the reader to Ref. [35]. Fol-

lowing this work, we treat separately the deterministic

piece of the power spectrum and the term produced by

the constant shot noise contribution. This procedure in-

creases numerical accuracy. Thus, the correlation func-

tions ξtrue

`in Eq. (5) does not contain the shot noise

contribution, i.e. one needs to perform Fourier trans-

forms of power spectrum without the shot noise correc-

tion Pshot

0. Note that according to Ref. [23], the impact

of the RIC constraint on cosmological inference is below

one percent for the scale cut kmin = 0.02 hMpc−1. Nev-

ertheless, keeping in mind future analyses of primordial

non-Gaussianity sensitive to large scales, we prefer to ac-

count for the RIC in the theory model by implementing

Eq. (5). The eﬀect of the RIC for the NGC footprint is

shown in Fig. 4.

W(ℓ=0)

W+RIC

ℓ=2

ℓ=4

0.001 0.005 0.010 0.050 0.100

-10 000

10 000

20 000

30 000

40 000

50 000

k, h Mpc-1

Pℓ(k),(Mpc/h)3

NGC EZ mocks

FIG. 4. The impact of the integral constraint on typical power

spectrum multipoles of the EZ mocks for the NGC footprint.

Fiber collisions. Finally, we implement the correc-

tion due to ﬁber collisions. We follow the eﬀective win-

dow approach proposed in [50]. In this method, two col-

liding targets are modeled by a top-hat window function

that depends on the physical scale of ﬁber collisions, Dfc.

In Fourier space the eﬀect of ﬁber collisions on the true

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CosmologicalconstraintsfromthepowerspectrumofeBOSSquasarsChudaykinAnton1,2,andMikhailM.Ivanov3,4,y1DepartmentofPhysics&Astronomy,McMasterUniversity,1280MainStreetWest,Hamilton,ONL8S4M1,Canada2PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,N2L2Y5,Canada3SchoolofNaturalSciences,Institutefor...

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Cosmological constraints from the power spectrum of eBOSS quasars Chudaykin Anton1 2and Mikhail M. Ivanov3 4y

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