Constraining gravity with synergies between radio and optical cosmological surveys

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Santiago Casasa,b, Isabella P. Caruccic,d, Valeria Pettorinoa, Stefano Camerac,d,e, Matteo Martinellif,g

aUniversit´e Paris-Saclay, Universit´e Paris Cit´e, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France

bInstitute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany

cDipartimento di Fisica, Universit`a degli Studi di Torino, via P. Giuria 1, 10125 Torino, Italy

dINFN – Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy

eINAF – Istituto Nazionale di Astroﬁsica, Osservatorio Astroﬁsico di Torino, strada Osservatorio 20, 10025 Pino Torinese, Italy

fINAF – Istituto Nazionale di Astroﬁsica, Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monteporzio Catone, Italy

gINFN – Istituto Nazionale di Fisica Nucleare, Sezione di Roma, piazzale Aldo Moro 2, 00185 Roma, Italy

Abstract

In this work we present updated forecasts on parameterised modiﬁcations of gravity that can capture deviations

of the behaviour of cosmological density perturbations beyond ΛCDM. For these forecasts we adopt the

SKA Observatory (SKAO) as a benchmark for future cosmological surveys at radio frequencies, combining a

continuum survey for weak lensing and angular galaxy clustering with an Higalaxy survey for spectroscopic

galaxy clustering that can detect baryon acoustic oscillations and redshift space distortions. Moreover, we

also add 21cm Hiintensity mapping, which provides invaluable information at higher redshifts, and can

complement tomographic resolution, thus allowing us to probe redshift-dependent deviations of modiﬁed

gravity models. For some of these cases, we combine the probes with other optical surveys, such as the Dark

Energy Spectroscopic Instrument (DESI) and the Vera C. Rubin Observatory (VRO). We show that such

synergies are powerful tools to remove systematic eﬀects and degeneracies in the non-linear and small-scale

modelling of the observables. Overall, we ﬁnd that the combination of all SKAO radio probes will have the

ability to constrain the present value of the functions parameterising deviations from ΛCDM (µand Σ) with

a precision of 2.7% and 1.8% respectively, competitive with the constraints expected from optical surveys and

with constraints we have on gravitational interactions in the standard model. Exploring the radio-optical

synergies, we ﬁnd that the combination of VRO with SKAO can yield extremely tight constraints on µand

Σ(0.9% and 0.7% respectively), which are further improved when the cross-correlation between intensity

mapping and DESI galaxies is included.

Keywords: dark energy, modiﬁed gravity, galaxy clustering, weak lensing, radio surveys, optical surveys

1. Introduction

The current concordance cosmological model,

ΛCDM, has been able to pass a variety of tests along

the years, and as of today it still is a very good ﬁt

to present available data (see [1–7] among many oth-

ers). Despite the success of ΛCDM, the nature of

dark energy and dark matter remains unknown: from

a theory point of view, there is no convincing predic-

tion of the value of the cosmological constant Λ; it

requires a high level of ﬁne tuning in the initial con-

ditions, and it marks our epoch as a very special time

in the evolution of the Universe. Recent observations

have highlighted tensions between low redshift mea-

surements of cosmological parameters and their value

inferred from high redshift observations through the

assumption of a ΛCDM expansion history (see e.g.

[8] for a recent review). While such tensions may

have an origin in unknown systematic eﬀects, sev-

eral works have been in parallel investigating whether

scenarios alternative to ΛCDM are able to overcome

these shortcomings, also extending General Relativ-

ity (GR) beyond Einstein’s theory, at cosmological

scales (see [9] and references therein for a recent re-

view.)

In this paper we rely on phenomenological param-

eterisations of departures from GR, and forecast our

ability to test them with cosmological surveys with

Preprint submitted to Elsevier 13 October 2022

arXiv:2210.05705v1 [astro-ph.CO] 11 Oct 2022

the upcoming SKA Observatory1(SKAO), alone and

in synergy with other surveys at optical/near-infrared

wavelengths. We use parameterisations of the evolu-

tion of cosmological perturbations that modify the

standard equations for the gauge-invariant gravita-

tional potentials, Φand Ψ. Perturbations are fully

deﬁned with two free functions of redshift and scale

that modify the Poisson equation and the behaviour

of the two gravitational potentials [10,11]. While

there is no unique choice for such free functions, we

follow here the approach of [12], where the parame-

terised functions are µ, modifying the Poisson equa-

tion for the Newtonian potential Ψ, and η, which de-

termines the ratio between Φand Ψ.

Even within this theoretical framework, there is no

unique choice for such functions, and diﬀerent ap-

proaches can be taken to parameterise them. For

instance, one can assume they are scale-independent

[1], or include extra parameters controlling how these

functions change with scale [12,13], or use as free

parameters the values of these functions in redshift

bins [14–16]. Extensions of this binned approach for

parameterised modiﬁed gravity have been worked out

in [17] and implemented into N-body simulations in

[18]. Moreover, a purely phenomenological investiga-

tion is not the only possible choice, and several results

were obtained within the framework of the so-called

eﬀective ﬁeld theory of dark energy [19], which allows

us to study departures from GR in the context of the

Horndeski class of theories [20–23].

The common line of all these approaches is to study

how departures from GR modify the evolution of cos-

mological perturbations. These studies will there-

fore particularly beneﬁt of the increased sensitivity

of galaxy surveys planned for the current decade (see

e.g. [24] and references therein). Galaxy clustering

(GC) and weak lensing cosmic shear (WL) data are

particularly sensitive to modiﬁcations of the theory of

gravity. The former probes the growth of cosmolog-

ical structures and is sensitive to the evolution of Ψ,

while the latter can probe the distribution of matter

through its gravitational eﬀects on the path of pho-

tons, and it is therefore sensitive to the combination

Φ+Ψ, which sources the lensing potential.

Furthermore, a new technique to probe cosmo-

logical structures has been advocated over the last

decade: line intensity mapping (IM) [25–29]. Do-

ing IM of a particular galactic emission line means

1https://www.skao.int

measuring the integrated radiation from unresolved

sources in large patches of the sky. This way, we

map the underlying dark matter ﬁeld with excellent

redshift resolution, making IM a sensitive probe of

Ψ, and therefore a useful tool to constrain cosmo-

logical parameters and deviations from GR [30–32].

For instance, we can focus on the 21-cm line emit-

ted by atomic neutral hydrogen (Hi), the most abun-

dant baryonic element in the Universe and an optimal

tracer of its structure. For cosmology, we are inter-

ested in the largest scales we can probe. Hence, we

can perform HiIM surveys with radio telescopes in

the so-called single-dish mode. Each antenna/dish

operates as a single telescope, not in interferometry,

and maps are co-added. As a result, the angular res-

olution is low, but the area coverage unprecedented

[33–35].

In this work, we focus on the extensive radio sur-

veys that the SKAO’s Mid Telescope, located in the

Karoo desert in South Africa, will be able to carry

out. Thanks to these, we can exploit all the probes

described above, through galactic radio continuum

emission and 21-cm line emission from resolved galax-

ies and in IM [36].

This paper is organized as follows: in section 2

we review the main equations used to describe phe-

nomenologically deviations from GR and specify our

choice of the parameterisation. In section 3 we

present the Fisher matrix method used to obtain our

forecasts, we describe the observational probes con-

sidered and highlight the experimental setup. Our

forecast results are shown in section 4 and we draw

our conclusions in section 5.

2. Parameterising Modiﬁed Gravity

We choose to work in the conformal Newtonian

gauge and in a ﬂat Universe, with the line element

given by

ds2=−(1 + 2 Ψ) dt2+a2(1 −2Φ) dxidxi,(1)

where ais the scale factor, related to the redshift z

via 1 + z= 1/a. In this gauge, the two scalar metric

perturbations Φand Ψ, functions of time and scale,

coincide with the gauge-invariant Bardeen potentials

[37].

In theories with extra degrees of freedom (dark

energy, DE) or modiﬁcations of General Relativity

(modiﬁed gravity, MG), the normal linear perturba-

tion equations are altered with respect to the stan-

dard case, thus leading to diﬀerent values of Φand Ψ

for a given matter source. Such departures from the

standard behaviour of the two potentials can gener-

ally be encoded in two functions of time and scale.

Several choices are possible and have been adopted

in the literature for these functions, see e.g. [12] for a

limited overview. The choice we do in this work is to

introduce the two functions through a modiﬁcation of

the Poisson equation for Ψand a gravitational slip.

While the former changes the evolution in time and

scale of the Ψpotential, the latter introduces a diﬀer-

ence between Ψand Φ(the equivalent of anisotropic

stress) already at the linear level and for pure cold

dark matter: ΛCDM is retrieved when Ψ=Φ.

The expressions that deﬁne µ(a, k) and η(a, k) as

the functions encoding the modiﬁed behaviour of the

potentials are

−k2Ψ(a, k) = 4 π G a2µ(a, k)ρ(a)∆(a, k) ; (2)

η(a, k) = Φ(a, k)

Ψ(a, k).(3)

Here ρ(a) is the average dark matter density and

∆(a, k)≡δ(a, k) + 3 a H(a)∇·v(a, k) is the comov-

ing density contrast with δthe fractional overdensity,

Hthe Hubble rate, and vthe peculiar velocity ﬁeld.

We neglect here relativistic particles and radiation as

we are only interested in modeling the perturbation

behaviour at late times. Under these assumptions,

η, which is eﬀectively a model independent observ-

able [38], is closely related to modiﬁcations of GR

via the gravitational potentials [39,40], while µen-

codes deviations in gravitational clustering, especially

in redshift-space distortions, as non-relativistic parti-

cles are accelerated by the gradient of Ψ.

In this work, we will also consider weak lensing ob-

servations, which are instead sensitive to deviations

in the lensing or Weyl potential Υ= (Φ+Ψ)/2, since

it is this combination that aﬀects null-geodesics (rela-

tivistic particles). To this end we introduce a function

Σ(a, k) so that

−k2Υ(a, k)=4π G a2Σ(a, k)ρ(a)∆(a, k).(4)

Note that, as such, Σplays the role of µin a Poisson-

like equation for the Weyl potential (cf. Equation 2).

As metric perturbations are fully speciﬁed by two

functions of time and scale, this latter function Σ

is not independent from µand η, and one can relate

the three functions through

Σ(a, k) = µ(a, k)

2[1 + η(a, k)] .(5)

Throughout this work, we will denote the standard

ΛCDM model, deﬁned through the Einstein-Hilbert

action with a cosmological constant, simply as GR.

For this case we have that µ=η=Σ= 1. All other

cases in which these functions are not unity will be

considered as MG models.

The advantage of using phenomenological functions

such as µand ηis that they allow to model any de-

viations of the perturbation behaviour from ΛCDM

expectations, they are relatively close to observations,

and they can also be related to other commonly used

parameterisations [41]. On the other hand, they are

not easy to map to an action (as opposed to ap-

proaches like eﬀective ﬁeld theories that are based

on an explicit action) and in addition they contain so

much freedom that we normally restrict their param-

eterisation to a subset of possible functions.

In this work we assume a simple parameterisation,

based on the one used in the Planck analysis [12]:

µ(a, k) = 1 + E11 ΩDE(a),(6)

η(a, k) = 1 + E22 ΩDE(a).(7)

This parameterisation is usually referred to as ‘late-

time parameterisation’, as it depends on the DE en-

ergy density ΩDE(a), and therefore allows a departure

from GR mainly at low redshift where DE dominates.

We neglect here any scale dependence; the amplitude

of the deviations from the GR limit is modulated by

the parameters E11 and E22, while the time evolu-

tion of the MG functions is related to the DE density

fraction.

For the forecasts presented below, we will show the

constraints on µand Σdeﬁned as the values that the

functions deﬁned in Equation 5 and Equation 6 take

at z= 0, which in our parametrization is directly

related to ΩDE,0≡ΩDE(a= 1).

3. Fisher forecasts

In this work we aim at forecasting the constraints

that SKAO will be able to obtain on modiﬁcations

of gravity. To achieve this goal we rely on a Fisher

matrix analysis, and in this section we review its fun-

damentals, as well as how it can be applied to the

observables of interest for the SKAO.

3.1. Fisher formalism

Given a theoretical model describing a target ob-

servable and a set of experimental speciﬁcations for

its measurement, the Fisher formalism provides us

with a simple recipe to forecast marginal errors on the

estimation of the model parameters. Starting from a

likelihood function L(Θ)≡P(d|Θ), representing the

probability of the data, d={da}, given the model

parameters Θ={Θα}, the Fisher matrix [42,43] can

be deﬁned as

Fαβ =−∂2ln L(Θ)

∂Θα∂Θβﬁd

,(8)

where ‘ﬁd’ means that the derivatives are computed

at the ﬁducial values of the model parameters, Θﬁd.

Now, let us assume that L(Θ) is a multivariate

Gaussian distribution, namely

−2 ln L(Θ)=[d−t(Θ)]TC−1[d−t(Θ)]

+ ln det (2 πC),(9)

where t(Θ) is the theoretical prediction, depending

upon the model parameters, and C={Cab}is the

data covariance matrix, which we assume does not

depend on Θ. Under these assumptions, Equation 8

applied to Equation 9 gives

Fαβ =∂tT

∂Θα

C−1∂t

∂Θβ

.(10)

In other words, the Fisher matrix is the inverse of the

covariance matrix of the parameters. For this reason,

it provides us with the expected errors around their

ﬁducial values—in turn, an estimate of the ability of

an experiment (or a combination of experiments) to

constrain the parameters of the model.

In this work, we obtain our Fisher matrices using

the CosmicFish code [44,45]. We use an upgraded

python implementation of this code that is not pub-

licly available yet, but that will be released in the

near future2. The cosmological functions used within

CosmicFish to compute the observables are instead

obtained from MGCAMB [46–48], which is able to obtain

such functions in the MG model we consider in this

work3.

3.2. Spectroscopic galaxy clustering

GC probes the correlation among the three-

dimensional positions of galaxies, which represent bi-

ased tracers of the distribution of matter in the Uni-

verse. The correlator of the Fourier transform of the

2The public version of CosmicFish is available at https:

//cosmicfish.github.io/.

3In this work we use our own public fork of the MGCAMB

repository, available at https://github.com/santiagocasas/

MGCAMB.

matter density contrast at a given redshift z,δm(z, k),

with itself is the matter power spectrum Pδδ(z, k).

What we can measure through galaxy surveys, how-

ever, is the power spectrum of galaxies, rather than

directly the one of matter. On large enough scales

and in conﬁguration space, the galaxy (number) den-

sity contrast δgis related to that of matter through

δg=bgδm, where bg(z) is the so-called linear galaxy

bias, and is assumed to be scale-independent in that

regime.

The cosmological information in GC is mostly con-

tained in the shape of the baryon acoustic oscillations

(BAO), which appear as wiggles in the power spec-

trum, and in the redshift space distortions (RSD),

which induce anisotropies in galaxy number density

ﬂuctuations as a function of the angle with respect

to the line of sight. While BAO are very sensitive to

the baryonic content and the geometry of the Uni-

verse, RSD are very sensitive to the growth of den-

sity perturbations and the peculiar velocity ﬁeld of

matter and galaxies. In redshift space, we then write

δg=bgδm+(1+z)/H(z)ˆ

n·∇(ˆ

n·v), with ˆ

nthe line-

of-sight direction and vthe peculiar velocity ﬁeld,

whose radial component contributes to the measured

redshift.

The observed power spectrum of galaxies is then

given in terms of the matter power spectrum as [49–

51]

Pgg(z, k, µθ) = AP(z)×Pδδ,zs(z, k, µθ)

×exp −k2µ2

θσ2

z(z)c2/H2(z)

+Pshot(z),(11)

where µθ≡ˆ

n·k/k, i.e. it is the cosine of the angle

θbetween the wave vector kand ˆ

n. The ﬁrst term

in Equation 11 corresponds to the Alcock-Paczynksi

eﬀect [52], viz.

AP(z)≡[dA,ref (z)]2H(z)

A(z)Href (z),(12)

where dA(z) is the angular diameter distance, and the

subscript ‘ref’ means that the corresponding quan-

tity is calculated at the reference ﬁducial cosmology.

The exponential term in Equation 11 is a line-of-sight

damping due to redshift uncertainty, modelled by its

error σz(z). Then, the additive term Pshot(z) is the

extra contribution to account for incorrect subtrac-

tion of shot noise, which is usually set to zero. Lastly,

Pδδ,zs is the redshift-space power spectrum,

Pδδ,zs(z, k, µθ) = FoG(z, k, µθ)×K2

rsd(bg;z, k, µθ)

×Pdw(z, k, µθ)

σ2

8(z),(13)

where the ﬁrst term is due to non-linear RSD. It is

called ‘Finger-of-God’ (FoG) eﬀect and models the

damping of power on small scales due to the incoher-

ent peculiar motions of galaxies,

FoG(z, k, µθ)≡1

1 + k2µ2

θσ2

p(z).(14)

In the above equation, the strength of the FoG ef-

fect is modulated by the pairwise velocity dispersion,

which we model as

σ2

p(z) = 1

6π2Zdk Pδδ(k, z)f2(k, z),(15)

where f≡d ln δ/d ln ais the growth rate of mat-

ter perturbations and we have taken into account the

possibility of a scale-dependent growth induced in a

general modiﬁed gravity parametrization. We com-

pute this term at each redshift bin and evaluate it at

the ﬁducial cosmology, keeping it ﬁxed in our anal-

ysis, which corresponds to the optimistic settings in

[43]. The Krsd(bg, z, k, µθ) term represents the Kaiser

term, which accounts for linear redshift space distor-

tions and is given by

Krsd(bg;z, k, µθ)≡bg(z)σ8(z) + f(z, k)σ8(z)µ2

θ.

(16)

Both the linear and non-linear RSD terms arise due

to the transformation between redshift space and real

space, when observing galaxies using redshift surveys.

Here, σ8(z) the amplitude of matter ﬂuctuations as a

function of redshift. Finally, Pdw(z, k, µθ) stands for

the ‘de-wiggled’ power spectrum, modelling the eﬀect

of BAO damping on the matter power spectrum. We

refer the reader to [43] for a more detailed description

of this term. In Figure 1 we plot the term Pgg(z=

0.6, k, µθ) as a function of kfor two diﬀerent values

of µθ, namely µθ= 0 and 1 (solid and dashed blue

lines, respectively). This is a theoretical model of

the galaxy power spectrum; dependence on a speciﬁc

survey will enter in the spectroscopic redshift error

σz(z), which will be speciﬁed in subsection 3.5.

A GC survey in a redshift bin of width ∆z, centred

on redshift ¯zcovers a volume Vsurvey, and observes

galaxies with comoving (volumetric) number density

N(z), depending on the survey speciﬁcations. The

survey provides information for Fourier modes only

in a range [kmin, kmax], which also depends on the

survey’s speciﬁcations or on the scale until which one

can accurately model non-linear scales.

Considering a GC survey carried out for a redshift

range discretized into Nbredshift bins, we evaluate

Pgg(z, k, µ;Θ) and its derivatives at the centre ¯zmof

each redshift bin m, and at the ﬁducial value for each

of the Nθcosmological parameters. While we may

expect the Fisher matrix to be an Nθ×Nθmatrix, it

is in practice more complicated due to the presence of

bg(z) and Pshot(z), which are in general unknown. In

order to address this problem, we discretize bg(z) and

Pshot(z) into Nbredshift bins, assuming them to be

mutually independent and considering them at each

redshift bin as additional independent model param-

eters with some ﬁducial values. Thus our full Fisher

matrix is of dimension (Nθ+ 2 Nb)×(Nθ+ 2 Nb),

and we can in the end marginalize it over the 2 Nb

nuisance parameters.

Given the full (cosmological + nuisance) parame-

ter set Θ={θα, bg,m, Pshot,m}, where θαare the cos-

mological parameters, bg,m ≡bg(¯zm) and Pshot,m ≡

Pshot(¯zm), the total Fisher matrix for a GC survey

over all redshift bins can be written as [43]

FAB

αβ =

m,n=1 X

a,b,c,d,n

∂PAB (¯zm, ka, µb)

∂Θα

×∂PAB (¯zn, kc, µd)

∂ΘβCAB (¯zm,¯zn)−1

abcd ,(17)

where A, B label the probe under scrutiny, i.e. A=

B= g for galaxy clustering. Above, kaand µbrepre-

sent the discretised values of kand µθthe signal has

been binned into, and Cis the covariance matrix be-

tween a set of measurements of PAB(¯zm, ka, µb) and

one of PAB(¯zn, kc, µd). Again, in full generality, it

reads

CAB

abcd(¯zm) = 4π2δK

ac δK

bd δK

a∆ka∆µbVsurvey

×h˜

PAA(¯zm, ka, µb)˜

PBB(¯zm, ka, µb)

+˜

PAB(¯zm, ka, µb)˜

PAB(¯zm, ka, µb)i,(18)

where ˜

PAB =PAB +PAB,noise δK

AB.

The power spectrum and its derivatives appearing

in Equation 17 are evaluated at the ﬁducial values

of the parameters, and the ﬁnal Fisher matrix is the

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ConstraininggravitywithsynergiesbetweenradioandopticalcosmologicalsurveysSantiagoCasasa,b,IsabellaP.Caruccic,d,ValeriaPettorinoa,StefanoCamerac,d,e,MatteoMartinellif,gaUniversiteParis-Saclay,UniversiteParisCite,CEA,CNRS,AIM,91191,Gif-sur-Yvette,FrancebInstituteforTheoreticalParticlePhysicsandCosm...

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