# Constraining dark energy with the integrated Sachs-Wolfe effect

Emeric Seraille  Institute of Cosmology & Gravitation, University of
Portsmouth, Portsmouth, PO1 3FX, U.K.  École Normale Supérieure, 45 Rue d’Ulm,
F-75230 Paris, France  Johannes Noller  Department of Physics & Astronomy,
University College London, London, WC1E 6BT, U.K.  Institute of Cosmology &
Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, U.K.  DAMTP,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.  Blake D.
Sherwin  DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA,
U.K.

###### Abstract

We use the integrated Sachs-Wolfe (ISW) effect, by now detectable at $\sim
5\sigma$ within the context of $\Lambda{}$CDM cosmologies, to place strong
constraints on dynamical dark energy theories. Working within an effective
field theory framework for dark energy we find that including ISW constraints
from galaxy-CMB cross-correlations significantly strengthens existing large-
scale structure constraints, yielding bounds consistent with $\Lambda{}$CDM
and approximately reducing the viable parameter space by $\sim 70\%$. This is
a direct consequence of ${{\cal O}(1)}$ changes induced in these cross-
correlations by otherwise viable dark energy models, which we discuss in
detail. We compute constraints by adapting the $\Lambda{}$CDM ISW likelihood
from Stolzner:2017ged  for dynamical dark energy models using galaxy data from
2MASS, WISE $\times$ SuperCOSMOS, SDSS-DR12, QSOs and NVSS, CMB data from
Planck 18, and BAO and RSD large scale structure measurements from BOSS and
6dF. We show constraints both in terms of EFT-inspired $\alpha_{i}$ and
phenomenological $\mu/\Sigma$ parametrisations. Furthermore we discuss the
approximations involved and related aspects of bias modelling in detail and
highlight what these constraints imply for the underlying dark energy
theories.

###### Contents

  1. I Introduction
  2. II EFTs of dark energy and Horndeski gravity
    1. II.1 Horndeski gravity
    2. II.2 Linear cosmology
    3. II.3 The Quasi-Static Approximation
  3. III Integrated Sachs-Wolfe effect
  4. IV Data sets and likelihood implementation
    1. IV.1 Source catalogues
    2. IV.2 Complementary data sets
    3. IV.3 Bias modelling
  5. V Cosmological parameter constraints
    1. V.1 Baseline CMB constraints
    2. V.2 Qualitative ISW constraints
    3. V.3 Observational ISW constraints I: EFT parametrisation
    4. V.4 Observational ISW constraints II: $\mu/\Sigma$ parametrisation
  6. VI Conclusions
  7. A Cosmological constraints and bias models
  8. B ISW priors from observational constraints
  9. C Different $\alpha_{i}$ parametrisations
  10. D Cosmological parameter constraints beyond EFT parameters

##  I Introduction

While dark energy contributes approximately 70% of the Universe’s effective
energy budget Planck:2018vyg , its precise nature still eludes us. However,
there has recently been significant progress in continuously zeroing in more
and more on constraining and therefore understanding dark energy. Doing so
inevitably takes many forms, ranging from the investigation of specific
‘lamppost’ models to more general parametrised approaches that aspire to be
model-independent. One particularly promising avenue in this context have been
effective field theory approaches to parametrising dark energy (henceforth
EFTDE) that e.g. construct general dark energy interactions assuming dark
energy can be modelled as a scalar field and using fundamental theoretical
principles to restrict the nature of these interactions Gubitosi:2012hu ;
Bloomfield:2012ff ; Gleyzes:2013ooa ; Gleyzes:2014rba ; Bellini:2014fua ;
Lagos:2016wyv ; Lagos:2017hdr .111Restricting the new dark energy physics to a
single scalar field is motivated by the fact that, if dark energy is indeed
dynamical in nature, it generically comes in the form of new particles/fields.
So this is a minimal choice in that sense. At the linear level this results in
a small number of parameters (really: time-dependent functions) controlling
the dynamics of dark energy perturbations imprinted on the large scale
structure of the Universe. Notably, when working to second order in
derivatives, this neatly matches the linearised version of Horndeski scalar-
tensor theories Bellini:2014fua ; Kobayashi:2011nu , so working with
linearised Horndeski or a (linearised) EFTDE is equivalent in this specific
context. The small number of parameters controlling linear physics in these
theories then is sufficiently rich to capture a wide range of interesting
phenomenology, yet suitably restricted so that meaningful constraints on these
parameters can be derived from observations. With these EFT/Horndeski-inspired
parametrisations having a firm theoretical footing by construction, placing
data-driven constraints on them then indeed allows us to zero in on a
theoretically well-motivated parameter space of dark energy theories.

Given its cosmological nature, large scale structure data sets are the most
obvious place to turn to for deriving observational constraints on dark
energy. Within the context of EFTDE approaches, some of the tightest
observational constraints have been obtained e.g. using CMB data, baryon
acoustic oscillations, weak-lensing, and redshift space distortions – see
Noller:2018wyv ; Bellini:2015xja ; Hu:2013twa ; Raveri:2014cka ;
Gleyzes:2015rua ; Kreisch:2017uet ; Zumalacarregui:2016pph ; Alonso:2016suf ;
Arai:2017hxj ; Frusciante:2018jzw ; Reischke:2018ooh ; Mancini:2018qtb ;
Brando:2019xbv ; Arjona:2019rfn ; Raveri:2019mxg ; Perenon:2019dpc ;
Frusciante:2019xia ; SpurioMancini:2019rxy ; Bonilla:2019mbm ; Baker:2020apq ;
Joudaki:2020shz ; Noller:2020lav ; Noller:2020afd ; Gsponer:2021obj  for
related constraints. For specific scalar-tensor theories within the Horndeski
class it has also been shown that galaxy-CMB temperature cross-correlations,
i.e. the integrated Sachs-Wolfe effect (ISW), can significantly improve
observational constraints on such models, e.g. for Kinetic Gravity Braiding
models Kimura:2011td , for the covariant Galileon Renk:2017rzu , and for
Galileon Ghost Condensates, Generalized Cubic Covariant Galileons and
K-mouflage models Kable:2021yws . Here we therefore investigate how
incorporating galaxy-CMB temperature cross-correlations into the analysis
affects constraints at the level of general EFTDE/Horndeski parametrisations.
We do so by adapting the $\Lambda{}$CDM ISW likelihood from Stolzner:2017ged
for dynamical dark energy models (incorporating measurements of galaxy-CMB
cross-correlations from 2MASS, WISE $\times$ SuperCOSMOS, SDSS-DR12, QSOs and
NVSS)) and combining it with the large scale structure likelihoods used in
Noller:2018wyv  (temperature and polarisation data from Planck 18, baryon
acoustic oscillations from BOSS, and redshift space distortions from 6dF and
BOSS). Notably (and differently to some of the aforementioned ISW constraints
in the dynamical dark energy context), we perform a full Markov chain Monte
Carlo (MCMC) exploration of the parameter space and derive the resulting
constraints.

This paper is structured as follows: In section II we briefly review the
relevant aspects of Horndeski gravity and EFTDE approaches. In section III we
then recall the way in which the ISW effect manifests itself in galaxy-CMB
cross-correlations, how bias can be modelled and calibrated and what
approximations can be used in evaluating the resulting expressions. We pay
particular attention to what changes in the evaluation when going from a
$\Lambda{}$CDM cosmology to the dynamical dark energy scenarios encoded by
EFTDE approaches. We give details on the data sets used and the likelihood
implementation in section IV. The main results, i.e. cosmological parameter
constraints, are presented in section V. Here we also present a number of
cross-checks and qualitative constraints that provide helpful intuition for
how precisely the ISW constraints arise. Finally, we conclude in section VI
and collect additional details in the appendices.

##  II EFTs of dark energy and Horndeski gravity

###  II.1 Horndeski gravity

As discussed in the introduction, modelling dark energy as a scalar field is a
minimal choice when considering dynamical dark energy models, in the sense
that such models generically introduce novel degrees of freedom/fields.
Assuming that a single such degree of freedom (encoded in the scalar)
dominates the evolution then is a minimal choice in this context. Horndeski
gravity Horndeski:1974wa ; Deffayet:2011gz ,222For the equivalence between the
formulations of Horndeski:1974wa  and Deffayet:2011gz , see Kobayashi:2011nu .
is a very general framework when investigating the resulting scalar-tensor
theories, describing the most general Lorentz-invariant ST action that gives
rise to second order equations of motion. Here we will for simplicity focus on
the subset of such scalar-tensor theories where gravitational waves propagate
with the speed of light.333This trivially satisfies existing observational
constraints TheLIGOScientific:2017qsa ; 2041-8205-848-2-L14 ;
2041-8205-848-2-L15 ; 2041-8205-848-2-L13 ; 2041-8205-848-2-L12 ;
Jimenez:2015bwa  – c.f. Creminelli:2017sry ; Sakstein:2017xjx ;
Ezquiaga:2017ekz ; Baker:2017hug  and references therein – but we also point
the interested reader to deRham:2018red ; LISACosmologyWorkingGroup:2022wjo ;
Baker:2022eiz ; Harry:2022zey  for more detailed discussions of the (subtle)
connection between this assumption and experimental constraints, as well as of
models beyond (1) consistent with existing constraints. The resulting
Lagrangian ${\cal L}_{i}$ for a scalar $\phi$ and a massless tensor
$g_{\mu\nu}$ is described by

$\displaystyle{\cal
L}=\Lambda_{2}^{4}G_{2}(\phi,X)+\frac{\Lambda_{2}^{4}}{\Lambda_{3}^{3}}G_{3}(\phi,X)\Box\phi+M_{\rm
Pl}^{2}G_{4}(\phi)R,$ (1)

where $X=-\tfrac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi/\Lambda_{2}^{4}$ is the
scalar kinetic term, and the $G_{i}$ are free dimensionless functions of
$\phi/M_{\rm Pl}$ and $X$. Note that we effectively have three mass scales:
$M_{\rm Pl},\Lambda_{2}$ and $\Lambda_{3}$. In cosmology they are
conventionally taken to satisfy $\Lambda_{2}=M_{\rm Pl}H_{0}$ and
$\Lambda_{3}=M_{\rm Pl}H_{0}^{2}$, which ensures that all interactions can
give ${{\cal O}(1)}$ contributions to the background evolution.

###  II.2 Linear cosmology

In this paper we will focus on investigating and constraining dark energy
phenomenology on the largest scales, well described by cosmological
perturbation theory. Treating (1) as a candidate description of dark energy on
those scales, we are therefore particularly interested in its linearised
perturbations around a cosmological FLRW background. Here we will assume this
to be a $\Lambda{}$CDM background (motivated by the observed proximity to such
a solution) and focus on constraining the behaviour of perturbations around
this. The freedom in the dynamics of such perturbations is primarily
controlled by two functions of time, $\alpha_{M}$ and $\alpha_{B}$. More
specifically, they are given by Bellini:2014fua

$\displaystyle\frac{M^{2}}{M_{\rm Pl}^{2}}\alpha_{M}$
$\displaystyle=2\frac{\dot{\phi}}{H}G_{4,\phi},$
$\displaystyle\frac{M^{2}}{M_{\rm Pl}^{2}}\alpha_{B}$
$\displaystyle=-2\frac{\dot{\phi}}{H}\left(XG_{3,X}+G_{4,\phi}\right).$ (2)

Here $G_{i,\phi}$ and $G_{i,X}$ denote the partial derivatives of the $G_{i}$
(with respect to these dimensionless arguments) and $M^{2}\equiv 2M_{\rm
Pl}^{2}G_{4}$ is the effective Planck mass seen by tensor
perturbations.444Note that there is a (conventional) sign difference for
$G_{3}$ in the expressions for the $\alpha_{i}$ here compared to
Bellini:2014fua . Differences in factors of $H_{0}$ and/or $M_{\rm Pl}$ are
due to our dimensionless definition of the $G_{i}$ (as opposed to the
dimensionful $G_{i}$ in Bellini:2014fua ). $\alpha_{M}$ is also known as the
“running” of the effective Planck mass $M$ and $\alpha_{B}$ as the “braiding”
that quantifies kinetic mixing between the metric and scalar perturbations in
Jordan frame (the frame the $\alpha_{i}$ are implicitly defined in). Note that
there are three other relevant functions of time: 1) the Hubble scale $H$,
which encapsulates the background evolution, 2) $\alpha_{T}$, the difference
between the speed of gravitational waves and the speed of light, which is zero
here by definition (i.e. by virtue of working with (1)), and 3) the
‘kineticity’ $\alpha_{K}$, which does not affect constraints on other
parameters at leading order at the level of linear perturbations
Bellini:2015xja ; Alonso:2016suf .

Note that, while here we have summarised the route from an underlying
covariant theory to the parameters governing linear perturbations around a
cosmological background for this theory, the resulting perturbation theory
need not arise out of Horndeski theory and can be seen as an effective
description of linear perturbations on a cosmological background in its own
right. As discussed above, the same such description arises in different
effective field theory approaches Gubitosi:2012hu ; Bloomfield:2012ff ;
Gleyzes:2013ooa ; Gleyzes:2014rba ; Bellini:2014fua ; Lagos:2016wyv ;
Lagos:2017hdr , which construct general interactions of a (perturbative)
scalar degree of freedom (in practice: the Goldstone boson associated with the
breaking of time translation invariance in cosmology) on an FLRW
background.555This applies when working up to second order in derivatives.
When working to higher orders in (derivatives) of perturbations, the results
naturally recover perturbative regimes arising out of higher derivative parent
covariant theories. While we will work in the $\alpha_{i}$ basis outlined
above here, our results can therefore straightforwardly be mapped onto
different EFTDE formulations – for notational differences between the two
approaches we refer to the summary tables detailing the mapping between
different notational choices in Bellini:2014fua .

Fully specifying the dynamics of linear cosmological perturbations in the
context outlined above requires us to be more explicit about the precise
parametric form of the relevant free functions. For a full fiducial underlying
Horndeski theory this would require specifying the functional form of the
$G_{i}$, but in the EFT spirit we are adopting here we will instead directly
parameterize the $\alpha_{i}$ that control the dynamics of linear
perturbations. While numerous parametrisations exist, here we will focus on
arguably the most commonly used parametrisation

$\displaystyle\alpha_{i}=c_{i}\Omega_{\rm DE},$ (3)

where all $\alpha_{i}$ are proportional to the fractional dark energy density
$\Omega_{\rm DE}$ which determines their time dependence, while all the
additional functional freedom is condensed into a single proportionality
constant $c_{i}$ for each given $\alpha_{i}$. We refer to Bellini:2014fua ;
Bellini:2015xja ; Linder:2015rcz ; Linder:2016wqw ; Denissenya:2018mqs ;
Lombriser:2018olq ; Gleyzes:2017kpi ; Alonso:2016suf ; Noller:2018wyv ;
Noller:2020afd ; Traykova:2021hbr  for a more detailed discussion of the
relative merits of different such parametrisations.

###  II.3 The Quasi-Static Approximation

The vast majority of scales that we have observational access to are sub-
horizon. As a result we can employ the quasi-static approximation (QSA) to
accurately recover the full evolution of gravitational perturbations
Sawicki:2015zya .666 Note that the accuracy of the QSA can be related to the
proximity of the background evolution to that of $\Lambda{}$CDM delaCruz-
Dombriz:2008ium ; Noller:2013wca . The QSA then amounts to assuming
$|\dot{X}|\sim{H}|X|\ll|\partial_{i}X|$ for any gravitational perturbation
field $X$, and hence to a sub-horizon regime where time derivatives of
gravitational perturbations can be neglected in comparison to their spatial
derivatives.

Using the QSA we can significantly simplify the Einstein equations at the
level of linear perturbations. We work in Newtonian gauge and focus on scalar
perturbations, so we have

$ds^{2}=-\left(1+2\Phi\right)dt^{2}+a^{2}\left(1-2\Psi\right)dx_{i}dx^{i},$
(4)

where $\Phi$ and $\Psi$ are the standard Bardeen potentials. Manipulating the
perturbed Einstein equations derived from (1) and applying the QSA, one can
now obtain a simple effective Poisson equation as well as an analogous
equation for the evolution of the ‘lensing potential’
$\Phi_{L}\equiv\Phi+\Psi$

$\displaystyle\frac{k^{2}}{a^{2}}\Phi$ $\displaystyle=-4\pi
G\mu(a)\rho\Delta,$ $\displaystyle\frac{k^{2}}{a^{2}}\Phi_{L}$
$\displaystyle=-8\pi G\,\Sigma(a)\rho\Delta,$ (5)

where $\Delta$ is the comoving gauge invariant density perturbation, $G$ is
Newton’s constant satisfying $G=1/(8\pi M_{P}^{2})$, and we have ignored any
anisotropic stress source in the matter sector. Here $\mu$ quantifies
modifications to the effective strength of gravity (i.e. an effective
different Newton’s constant) and $\Sigma$ parametrises modifications to the
effective lensing potential (also sometimes called the Weyl potential) probed
by gravitational lensing. GR corresponds to the case when $\mu,\Sigma$ are
both unity, so deviations away from unity for either of these parameters are
probing the presence of new gravitational physics.777Sometimes a third
parameter is defined in this context: $\gamma$, which measures the presence of
an effective gravitational anisotropic stress (often called gravitational
“slip”). This is simply related to the above parameters by
$\Sigma=\frac{1}{2}(1+\gamma)\mu$.

The $\mu$ and $\Sigma$ functions can explicitly be linked to the $\alpha_{i}$
that control the evolution of linear perturbations (2). Doing so, at leading
order one finds888Note that, even if we had kept $\alpha_{K}$ in our analysis,
it drops out of these expressions at leading order Bellini:2015xja ;
Alonso:2016suf , justifying its omission a posteriori.

$\displaystyle M^{2}\mu$
$\displaystyle=\frac{2\beta_{3}}{2\beta_{1}+\beta_{2}(2-\alpha_{B})},$
$\displaystyle M^{2}\Sigma$
$\displaystyle=\frac{\beta_{1}+\beta_{2}+\beta_{3}}{2\beta_{1}+\beta_{2}\left(2-\alpha_{B}\right)}.\;$
(6)

where we have followed Alonso:2016suf ; Lagos:2017hdr  and introduced the
$\beta_{i}$ shorthand functions

$\displaystyle\beta_{1}$ $\displaystyle\equiv-\frac{3(\rho_{\rm tot}+p_{\rm
tot})}{H^{2}M^{2}}-2\frac{\dot{H}}{H^{2}}+\frac{\tfrac{d}{dt}{(\alpha_{B}H)}}{H^{2}},$
$\displaystyle\beta_{2}$
$\displaystyle\equiv\alpha_{B}+2\alpha_{M},\quad\quad\quad\beta_{3}\equiv\beta_{1}+(1+\alpha_{M})\beta_{2},$
(7)

where $\rho_{\rm tot}$ and $p_{\rm tot}$ are the total energy density and
pressure in the Universe, and we note we have set $8\pi G=1$ (and re-scaled
all densities and pressures by a factor of 3, using CLASS conventions
Blas:2011rf ). In the following section we will find that working with the
intermediate $\mu,\Sigma$ functions greatly simplifies the evaluation of
galaxy-CMB cross-correlations, so the quasi-static expressions summarised here
will be particularly useful for us in this context.

##  III Integrated Sachs-Wolfe effect

Figure 1: Evolution of the lensing potential $\Phi_{L}=\Phi+\Psi$ as a
function of redshift for various choices of the dark energy parameter $c_{b}$,
where $\alpha_{i}=c_{i}\Omega_{DE}$ – see equation (2). All $\Lambda{}$CDM
parameters are fixed to the P18 best-fit cosmology Planck:2018vyg  and in this
plot we show results for a fiducial choice of the second dark energy parameter
$c_{m}=0$. We can clearly see the standard $\Lambda{}$CDM prediction of a
decaying lensing potential $\Phi_{L}$, but also that $\Phi_{L}$ can both decay
or grow in the presence of non-trivial dark energy dynamics. Note that, for
such cosmologies, $\Phi_{L}$ can in principle also decay faster than in
$\Lambda{}$CDM, but this requires $c_{m}\neq 0$ so cannot be seen in this
plot, but see figure 9 for comparison.  Figure 2: Here we show the
$C_{\ell}^{Tg}$ quantifying cross correlations between galaxies and CMB
temperature measurements. As in figure 1 above, we vary $c_{b}$ while keeping
$c_{m}=0$ fixed. We can neatly see that a decaying lensing potential leads to
positive cross-correlations between galaxies and CMB temperature and vice
versa, as expected and discussed more explicitly in section III. The strong
${{\cal O}(1)}$ change in the $C_{\ell}^{Tg}$ signal produced by different
models (as shown here) ultimately gives rise to the powerful ISW constraints
we find below. the Note that, in order to compute the cross-correlations shown
in this figure, we need to choose a fiducial galaxy density distribution –
here we have chosen the distribution from the SDSS-DR12 photo-z sample
Beck_2016 , but the qualitative behaviour of $C_{\ell}^{Tg}$ is robust under
changes of this choice.

The Integrated Sachs-Wolfe (ISW) effect arises due to the time variation of
the gravitational potentials along the lines of sight of CMB photons. This
gives rise to (secondary) CMB temperature anisotropies via

$\displaystyle\frac{\delta T_{ISW}(\hat{n})}{\langle
T\rangle}=\int_{0}^{z^{*}}dz\;e^{-\tau(z)}\frac{\partial{\Phi_{L}}}{\partial{z}}(z,\hat{n}\chi(z))$
(8)

Where $\hat{n}$ is a unit vector along the line of sight of an observer,
$\tau$ is the optical depth, $z$ is redshift, $z^{*}$ is its value at
recombination, and $\chi(z)$ is the comoving distance to redshift $z$. During
matter-domination $\partial\Phi_{L}/\partial z=0$, so this effect is a natural
probe of the dark energy-induced evolution of $\Phi_{L}$ at late times and
hence of dark energy itself. The underlying physics is straightforward: On
their way towards an observer, CMB photons will, e.g., fall into the potential
well associated with a galaxy and gain energy. Having traversed the potential
well the photons will climb back out of the well. If $\Phi_{L}$ has not
evolved in the meantime (as during matter domination), there will be no net
gain or loss of energy. However, if $\Phi_{L}$ has decayed as it does in
$\Lambda{}$CDM, then the photon will gain energy leading to a positive
correlation of CMB temperature anisotropies with the positions of galaxies.
Naturally the opposite, i.e. an anti-correlation, will be the consequence of a
growing $\Phi_{L}$.

Since dark energy only becomes a main driver of cosmic evolution for
$z\lesssim 1.5$, the ISW effect primarily affects large scales and so e.g.
affects the CMB temperature power spectrum for large angular scales. While
this effect is only ${\cal O}(10\%)$ for $\Lambda{}$CDM and is hence swamped
by cosmic variance in that case Huffenberger:2004tv , it nevertheless already
places powerful constraints on models of dynamical dark energy or modified
gravity, where it can be enhanced significantly – see figure 3 in
Noller:2018wyv  for an explicit illustration of this point. However, as
suggested above, a more powerful probe of the ISW effect comes from cross-
correlations of CMB temperature with galaxies or quasars (or indeed any other
tracer of the underlying gravitational potentials) - see a variety of past ISW
analyses employing various tracers of the underlying matter distribution
Afshordi:2003xu ; Boughn:2003yz ; Fosalba:2003ge ; Vielva:2004zg ;
Cabre:2007rv ; Cabre:2006qm ; Giannantonio:2006du ; Pietrobon:2006gh ;
McEwen:2006my ; Raccanelli:2008bc ; Rassat:2006kq ; Ho:2008bz ; Xia:2009dr ;
Dupe:2010zs ; Hernandez-Monteagudo:2013vwa ; Planck:2013owu ; Ferraro:2014msa
; Planck:2015fcm ; Shajib:2016bes ; Giannantonio:2008zi ; Giannantonio:2012aa
; Xavier:2018owe ; DES:2018nlb ; Calore:2020bpd ; Krolewski:2021znk ;
Kovacs:2021mnf ; Hang:2021kfx . This is what we will exploit here in the
context of EFTDE approaches as outlined in the previous section. A few points
are worth stressing before proceeding: 1) We will assume that $\tau$ is
sufficiently small for the relevant redshift range to approximate
$e^{-\tau(z)}\sim 1$. This approximation is known to introduce a $\sim 10\%$
error Stolzner:2017ged , i.e. below the accuracy with which the ISW effect is
typically determined. 2) While $\Phi_{L}\equiv\Phi+\Psi=2\Phi$ in
$\Lambda{}$CDM, for the dark energy models considered here this need not be
the case and indeed generically $\Phi\neq\Psi$. Unsurprisingly, this can
strongly affect the ISW signal and, as we will see later on, indeed leads to
strong constraints on the $\alpha_{i}$ dark energy parameters. 3) Note that
$\Phi_{L}$ can grow or decay in an expanding Universe with dynamical dark
energy, so the background evolution does not mandate either a growing or
decaying $\Phi_{L}$. 4) When selecting source catalogues for the computation
of cross-correlations, focusing on the range $1.5\gtrsim z\gtrsim 0$ is well
motivated given that $\partial\Phi_{L}/\partial z\sim 0$ during matter
domination. In practice, most of the presently available source catalogues
come from within this redshift range and so naturally the majority of the data
sets we will employ here do as well, but we will use some data points going
all the way to $z=5$ – see section IV for details on the precise data sets we
will use.

In the same way we present CMB temperature anisotropies due to the ISW effect
in equation (8), we can also describe the fractional overdensity of large-
scale structure tracers as a function of direction on the sky as follows:

$\displaystyle\frac{\delta N_{g}(\hat{n})}{\langle
N_{g}\rangle}=\int_{0}^{z^{*}}dz\delta_{g}(z,\hat{n}\chi(z))\frac{dN_{g}}{dz}(z)$
(9)

where $\frac{dN_{g}}{dz}$ represents the normalized selection function for a
given kind of compact object tracer (e.g. galaxies, quasars, radio sources)
and $\delta_{g}$ is the corresponding density contrast. So while the label $g$
suggests we only apply this to galaxies, this has a wider context in mind.
Assuming simple linear biasing at each redshift, we can then link this density
contrast to the matter density contrast through

$\displaystyle\delta_{g}(z,\hat{n}\chi(z))=b_{g}(z)\delta_{m}(z,\hat{n}\chi(z))$
(10)

where $\delta_{m}$ is the total matter density contrast and $b_{g}$ is the
bias, which can in general be a function of redshift and depends on the type
of compact matter tracer under investigation. We can now decompose equations
(8) and (9) on the base of spherical harmonics $Y_{l}^{m}(\theta,\phi)$ as
usual, where $\phi$ and $\theta$ are the two angles defining the vector
$\hat{n}$, and write

$\displaystyle\frac{\delta T_{ISW}(\hat{n})}{\langle T\rangle}$
$\displaystyle=\sum_{l,m}a_{lm}^{ISW}Y_{lm}(\hat{n}),$
$\displaystyle\frac{\delta N_{g}(\hat{n})}{\langle N_{g}\rangle}$
$\displaystyle=\sum_{l,m}a_{lm}^{g}Y_{lm}(\hat{n}).$ (11)

Fourier transforming, applying the Rayleigh plane wave identity and isolating
the multipole coefficient multiplying each spherical harmonic, we then obtain

$\displaystyle a_{lm}^{ISW}$ $\displaystyle=-\frac{i^{l}}{2\pi^{2}}\int
d^{3}k\int_{0}^{z^{*}}dze^{-\tau(z)}\frac{\partial{\Phi_{L}}}{\partial{z}}(z,\vec{k})j_{l}[k\chi(z)]Y_{lm}^{*}(\hat{k}),$
$\displaystyle a_{lm}^{g}$ $\displaystyle=\frac{i^{l}}{2\pi^{2}}\int
d^{3}k\int_{0}^{z^{*}}dzb_{g}(z)\frac{dN_{g}}{dz}(z)\delta_{m}(k,z)j_{l}[k\chi(z)]Y_{lm}^{*}(\hat{k}).$
(12)

where $k$ is the wavenumber and $j_{l}$ is the spherical Bessel function of
order l. Equipped with these expressions, we can write down expressions for
the cross-correlation angular power spectrum between CMB temperature and the
galaxy field (or field of other compact object tracers) $g$, finding

$\displaystyle
C_{l}^{Tg}=\frac{3{H_{0}}^{2}\Omega_{m,0}}{\left(D_{0}(l+\frac{1}{2})\right)^{2}}\int_{0}^{z^{*}}dzH(z)\frac{dN}{dz}(z)b_{g}(z){D(z)}^{2}A(k_{l},z)P(k_{l}),$
(13)

where we have defined
$A(k,z)\equiv\Sigma(k,z)\bigl{(}1+(1+z)\frac{d}{dz}ln(D(z)\Sigma(k,z))\bigr{)}$,
the scale $k_{l}$ satisfies $k_{l}=(l+\tfrac{1}{2})/\chi(z)$, and
$D(z)=(P(k,z)/P(k))^{1/2}$ is the linear growth factor of density
fluctuations. Note that, in deriving this expression, we have used (5) to
solve for the lensing potential and also used the Limber approximation
Limber:1954zz ; Kaiser:1991qi ; Kaiser:1996tp ; LoVerde:2008re ; Ho:2008bz

$\displaystyle\int
dkk^{2}h(k)j_{l}[k\chi]j_{l}[k\chi^{\prime}]\simeq\frac{\pi}{2}\frac{\delta(\chi-\chi^{\prime})}{\chi^{2}}h\left(\frac{l+\tfrac{1}{2}}{\chi}\right)$
(14)

Finally, proceeding along the same lines as above, we can also obtain an
expression for the auto-spectrum of the large-scale structure tracer

$\displaystyle{C_{l}}^{gg}=\int_{0}^{z^{*}}dz\frac{H(z)}{{(\chi(z)(1+z))}^{2}}\left(\frac{dN}{dz}(z)b(z)D(z)\right)^{2}P(k_{l}).$
(15)

##  IV Data sets and likelihood implementation

Catalogue | redshift bins
---|---
2MPZ | $[0.01,0.105],[0.105,0.195],[0.195,0.3]$
WISE $\times$ SuperCOSMOS | $[0.09,0.21],[0.21,0.30],[0.30,0.60]$
SDSS-DR12 photo-z | $[0.1,0.3],[0.3,0.4],[0.4,0.5],[0.5,0.7],[0.7,1.0]$
QSO | $[0.5,1.0],[1.0,2.0],[2.0,3.0]$
NVSS | $[0,5.0]$
Table 1: Summary of the source catalogues and corresponding redshift bins from
Stolzner:2017ged . This is used in our analysis as detailed in section IV.1.

Having established the formalism for computing ISW correlations in the
previous section, we know briefly summarise the data sets we will use in order
to constrain the underlying parameter space. Given that we are focusing on
CMB-galaxy cross-correlations, our data sets naturally include several galaxy
catalogues as well as CMB measurements and we complement this with additional
BAO and RSD measurements.

###  IV.1 Source catalogues

In order to compute galaxy-CMB cross-correlations we adapt the ‘ISW’
likelihood developed for $\Lambda{}$CDM cosmologies by Stolzner:2017ged ,
extending it to be able to handle dark energy and modified gravity theories as
outlined in the previous section. The galaxy data sets used are therefore
identical to those employed by Stolzner:2017ged . They are the 2MASS
photometric redshift catalogue (2MPZ) Bilicki:2013sza  which samples local
large scale structures in the range $z\in[0.00,0.30]$, the WISE $\times$
SuperCOSMOS photo-z catalogue Bilicki:2016irk , an all-sky extension of 2MPZ
also going to higher redshift $z\in[0.00,0.60]$, the Sloan Digital Sky Survey
Data Release 12 (SDSS-DR12) photo-z sample Beck_2016  with $z\in[0.00,1.00]$ ,
the catalogue of photometric quasars (QSOs) Richards:2008eq  tracing high
redshift large scale structures in the range $z\in[0.00,3.00]$, and finally
the NRAO VLA Sky Survey (NVSS) Condon:1998iy  sampling mostly extragalactic
radio sources in the very broad range $z\in[0.00,5.00]$. All the data sets are
binned according to redshift, where we summarise all the relevant redshift
bins in table 1. Note that the likelihood in particular uses the
${C_{l}}^{Tg}$ and ${C_{l}}^{gg}$ observables discussed in the previous
section and the correlation data used are precisely as described in
Stolzner:2017ged , where they were computed using Planck 2015 CMB maps and
masks as well as additional specific masks for each catalogue used – for
further details on the binning as well as on masks applied to these data sets
see Stolzner:2017ged . Note that the binning has important implications for
the way in which we will model bias, which we discuss in more detail below.

###  IV.2 Complementary data sets

We complement the catalogues of galaxies and other compact sources discussed
above by CMB temperature, lensing and low-$\ell$ polarisation data from Planck
2018 Planck:2015mym ; Planck:2015bpv ; Planck:2015fie  Note that working with
a $\Lambda{}$CDM background here enables us to work with the Plik_lite
likelihoods, where most of the internal nuisance parameters are pre-
marginalised – see Noller:2018wyv  for an explicit justification for this
choice. As alluded to above, note that it is the late-time ISW effect on the
TT power spectrum that is driving CMB constraints on the $\alpha_{i}$
parameters. In addition we use baryon acoustic oscillation (BAO) measurements
from SDSS/BOSS Anderson:2014 ; Ross:2015  as well as redshift space distortion
(RSD) measurements from BOSS and 6dF Beutler:2012 ; Samushia:2014 . Note that
there are subtleties related to the observational modelling required in order
to extract measurements of $f\sigma_{8}$ from galaxy surveys, which are
particularly relevant in the dynamical dark energy context we are considering
here. We will discuss those in more detail below.

###  IV.3 Bias modelling

In the previous section we introduced the redshift-dependent bias $b(z)$,
relating overdensities in galaxies (or other sources we are using as matter
tracers) to the underlying total matter distribution. In practice the galaxy-
galaxy auto-correlations will tightly constrain the bias in a manner that is
very robust to small changes in other cosmological parameters. Computing these
constraints then necessitates choosing a bias model, with modelling
uncertainties arising analogously to our above discussion for the $\alpha_{i}$
parametrisation. In order to quantify the dependence of our results on those
uncertainties in bias modelling and evolution, we will therefore employ two
bias models in what follows.

  * •

Bias model I: Here we adopt the bias modelling of Stolzner:2017ged , which
amounts to assigning an independent constant bias factor to each individual
bin in all the data sets considered – see table 1. This amounts to 17 bias
parameters.

  * •

Bias model II: Here we adapt the model discussed in Ferraro:2014msa  and fix
the redshift-dependence according to $b_{g}(z)=b_{\rm survey}(1+z)$ for each
survey. Given the five source catalogues we use, this results in 5 bias
parameters.

As we show in appendix A, the second approach produces nearly identical
constraints on dark energy and other cosmological parameters, while naturally
speeding up the computation of constraints (given the much smaller parameter
space dimension). So while (by design) it does not precisely recover the exact
redshift evolution preferred by the data within the context of the more
elaborate bias model I, model II nevertheless is an excellent approximation to
use here.

##  V Cosmological parameter constraints

We are now in a position to place observational constraints on the underlying
dark energy parameters. We will do so via a Markov chain Monte Carlo (MCMC)
analysis, computing constraints on the modified gravity/dark energy parameters
$c_{M}$ and $c_{B}$ introduced in (3) while marginalising over the standard
$\Lambda{\rm CDM}$ parameters $\Omega_{\rm cdm},\Omega_{\rm
b},\theta_{s},A_{s},n_{s}$ and $\tau_{\rm reio}$. Note that we impose a prior
$\tau_{\rm reio}\geq 0.04$, corresponding to $z_{\rm reio}\gtrsim 6$,
motivated by observations of the Gunn-Peterson trough SDSS:2001tew . Following
Planck:2015fie , our fiducial model includes two massless and one massive
neutrino eigenstate with the minimal sum of masses allowed by oscillation
experiments, $\sum_{\nu}m_{\nu}=0.06\;{\rm eV}$. For related cosmological
parameter constraints on deviations from GR using general parameterised
approaches and a variety of (current and forecasted) experimental data, see
Noller:2018wyv ; Bellini:2015xja ; Hu:2013twa ; Raveri:2014cka ;
Gleyzes:2015rua ; Kreisch:2017uet ; Zumalacarregui:2016pph ; Alonso:2016suf ;
Arai:2017hxj ; Frusciante:2018jzw ; Reischke:2018ooh ; Mancini:2018qtb ;
Noller:2018eht ; Perenon:2019dpc ; Frusciante:2019xia ; Arai:2019zul ;
Scott:2022fev ; Andrade:2023pws ; Wen:2023bcj ; Castello:2023zjr ;
Sugiyama:2023tes . Finally we will work with a fiducial $c_{k}=0.1$ choice
throughout. As discussed above, the ‘kineticity’ $\alpha_{K}$ does not affect
cosmological constraints at leading order, and indeed it has been explicitly
verified that fixing a fiducial $c_{k}$ as we do here does not bias the
constraints on the other $c_{i}$ Bellini:2015xja . For reference throughout
the remainder of this section, we adopt the following shorthands in referring
to the datasets detailed in the previous section:

  * •

P18B: This prior includes Planck 2018 CMB temperature, CMB lensing and
low-$\ell$ polarisation data Planck:2015mym ; Planck:2015bpv ; Planck:2015fie
as well as baryon acoustic oscillation (BAO) measurements from SDSS/BOSS
Anderson:2014 ; Ross:2015

  * •

RSD: Here we include redshift space distortion (RSD) constraints from BOSS and
6dF Beutler:2012 ; Samushia:2014 . Note that this is a minimal set of RSD
constraints chosen to be conservative and eliminate the risk of introducing
unwanted correlations between (RSD) data sets – see Noller:2018wyv  for
details.

  * •

ISW: This label denotes the constraints coming from the auto- and CMB cross-
correlations computed using the 2MPZ, WISE $\times$ SuperCOSMOS, SDSS-DR12
photo-z, QSO, and NVSS data sets detailed in IV.1.

###  V.1 Baseline CMB constraints

Figure 3: Here we show marginalised constraints for the dark energy
$\\{c_{b},c_{m}\\}$ parameters using Planck 18 (P18) and Planck 18 + BAO data
(P18B) – for details on the data sets used see section IV. Contours mark
$68\%$ and $95\%$ confidence intervals and the origin corresponds to the
$\Lambda{}$CDM limit (here consistent at $\sim 2\sigma$). For the
$\Lambda{}$CDM background we are considering here, incorporating BAO data
visibly only very mildly affects overall constraints on $\\{c_{b},c_{m}\\}$
and CMB-only constraints are approximately of the (as we will see: relatively
weak) order $c_{i}\lesssim{\cal O}(10)$.

As a first step, we briefly recap CMB constraints on the modified gravity/dark
energy parameters $c_{M}$ and $c_{B}$ – for details on the derivation of these
constraints in the present context of a $\alpha_{i}=c_{i}\Omega_{\rm DE}$
parametrisation see e.g. Bellini:2015xja ; Noller:2018wyv . We summarise these
constraints in figure 3. As we can see there, CMB-only constraints (more
specifically, temperature, lensing and low-$\ell$ polarisation data from
Planck 18) already provide significant constraints on the marginalised
$\\{c_{m},c_{b}\\}$ parameter space, to a first approximation enforcing
$c_{i}\lesssim{\cal O}(10)$ without any strong degeneracies in this parameter
space. In this figure we also show that the inclusion of BAO constraints only
has a minimal effect, consistent with the finding of Noller:2018wyv  and
largely due to the fact that we assume a $\Lambda{}$CDM background and the BAO
measurements considered here do not add significant constraining power on top
of the CMB data sets detailed above in this context. As such, we will only
refer to the combined CMB+BAO constraints (P18B) in the following sections.

###  V.2 Qualitative ISW constraints

Figure 4: Qualitative ISW constraint on the (marginalised) $\\{c_{b},c_{m}\\}$
plane. We again plot the CMB+BAO constraints from figure 3, but here also show
how what effectively amounts to a naive ISW prior impacts the parameter space.
Black areas are those plagued by gradient instabilities, while grey hashed
regions are those where the lensing potential $\Phi_{L}$ grows (between
redshifts $5$ and $0$). Given current constraints within $\Lambda{}$CDM detect
the ISW effect, in other words a decaying lensing potential, at $\sim 5\sigma$
Stolzner:2017ged , one would expect that the grey hashed regions of parameter
space are strongly disfavoured and incorporating full ISW constraints allows
placing yet tighter bounds. We will indeed find this to be the case in a more
complete analysis below, but the qualitative constraints outlined here already
provide useful intuition for the kind of constraints one can expect.

Before computing the observational constraints from the LSS data sets we
detail above (both ISW and RSD), it is worth to consider where and how we
expect ISW constraints to act. In other words, to identify which parts of
parameter space one should expect to be ruled out/restricted by such
constraints prior to any detailed likelihood calculation. To this end it is
important to recall that current measurements of the ISW effect establish that
there is a positive cross-correlations between CMB temperature anisotropies
and (the position of) galaxies at $\sim 5\sigma$ Stolzner:2017ged . As
discussed in section III, this is ultimately telling us that the lensing
potential has a positive derivative in redshift, i.e. the lensing potential is
decaying as a function of time in the late Universe. While this decay is
indeed a prediction in standard $\Lambda{}$CDM cosmologies, this is not
generically true for the EFTDE theories we are considering here. We show this
explicitly in figure 1 where we inspect a slice of theory space with $c_{m}=0$
and varying $c_{b}$ values broadly consistent with the P18B constraints
discussed and shown above in figure 3. The ${{\cal O}(1)}$ changes to the
predicted cross-correlations shown immediately suggest that measurements of
the ISW effect will yield powerful constraints on dark energy physics. To
calibrate expectations it makes sense to first demarcate regions of parameter
space where the lensing potential is indeed decaying vs. regions where it is
growing. At the very least we would expect observational constraints to rule
out the latter region. We perform this qualitative check in figure 4.
Comparing with the P18B constraints discussed above and reproduced in the
figure, it is clear that significant regions of the marginalised
$\\{c_{m},c_{b}\\}$ parameter space that are consistent with CMB-only
constraints are inconsistent with a decaying lensing potential in the late
Universe and so we would indeed expect incorporating ISW constraints into the
analysis to yield powerful constraints on the landscape of dark energy
theories investigated here.

Figure 5: Left panel: Here we show marginalised constraints for the dark
energy $\\{c_{b},c_{m}\\}$ parameters for different combinations of data sets.
Contours mark $68\%$ and $95\%$ confidence intervals, and are computed using
Planck 18 and BAO data (P18B), as well as measurements of galaxy-galaxy auto-
and galaxy-CMB temperature cross correlations (ISW), and redshift space
distortion measurements (RSD) – for details on the data sets used see section
IV. Note that ISW constraints effectively enforce $c_{b}\lesssim 0.5$ and
incorporating them drastically improves CMB+BAO-only bounds. RSD measurements
in addition impose a somewhat tighter bound on $c_{m}$, namely $c_{m}\lesssim
2$, so that a combination of bounds from all data sets used here allows
another (more mild improvement) of constraints. Right panel: Analogous plot to
the left panel, but for the more phenomenological $\mu/\Sigma$
parametrisation, where $\mu_{0}/\Sigma_{0}$ are derived from the
$c_{i}$/$\alpha_{i}$ via (6). The $0$ subscript denotes the value of
$\mu/\Sigma$ today. Projected into the $\mu_{0}/\Sigma_{0}$ parameter space,
the constraining power of adding ISW measurements is highlighted even more
than for the $c_{i}$ parametrisation in figure 5, with a stark improvement in
going from P18B to P18B+ISW constraints, while there is only a marginal
difference between P18B+ISW and P18B+ISW+RSD bounds. We emphasise that the
strong $\mu_{0}/\Sigma_{0}$ constraints found here incorporate a ‘theoretical
prior’ of working with an underlying EFT $c_{i}$ parameter basis – see main
text for a more detailed discussion.

###  V.3 Observational ISW constraints I: EFT parametrisation

Our main results are shown in figure 5. Here we compare constraints in the
$\\{c_{m},c_{b}\\}$ plane obtained via different combination of the data sets
detailed above. We see that ISW data in particular rule out $c_{b}\gtrsim 1$
when combined with P18B constraints. As expected, this is in excellent
agreement with the expectation from the qualitative ISW constraints shown in
figure 4 and discussed in detail above. As we stressed there, current
measurements establish that there a decaying lensing potential (i.e. a
positive cross-correlations between CMB temperature anisotropies and the
position of galaxies) at $\sim 5\sigma$ Stolzner:2017ged . It is therefore
unsurprising that the ISW constraints shown in figure 5 are significantly
stronger than the conservative qualitative estimate above – we show this
comparison explicitly in figure 6. As we discuss in more detail in appendix B,
the data constraints rule out additional regions of parameter space where the
lensing potential does not decay sufficiently quickly (including cases where
it does decay at early and late times, but plateaus at intermediate
redshifts). It is also worth noting that the addition of the ISW-related data
sets discussed here, while significantly tightening constraints in the
$\\{c_{m},c_{b}\\}$ plane, only has a very small effect on the other varied
(standard $\Lambda{}$CDM) parameters – see appendix D for more details on this
point. This is because changing these parameters within existing CMB
constraints does not qualitatively alter the evolution of the lensing
potential and hence these parameter changes only mildly affect the observables
probed by the ISW effect. The ISW measurements used here are therefore a
particularly clean and powerful way to constrain the dark energy/modified
gravity effects we are focusing on in this paper.

We also compare and contrast constraints derived from P18B+ISW with those
derived using another set of LSS constraints, namely RSD constraints as
detailed in Noller:2018wyv . Note that these RSD constraints are more subtle
in that extracting $f\sigma_{8}$ measurements from the data sets in question
is somewhat cosmology-dependent. More specifically, note that observational
modeling is required to extract measurements of $f\sigma_{8}$ from galaxy
surveys, which are typically constructed based on GR and validated using GR
mock catalogues. Using such standard measurements has been shown to give
accurate results e.g. in the context of DGP models Barreira:2016ovx , but not
in at least some models displaying scale-dependent linear growth
Taruya:2013quf . It is therefore worth emphasising that the ISW constraints
discussed here are not affected by an analogous modelling proviso and are
therefore more robust in this specific sense. In terms of the marginalised
$\\{c_{m},c_{b}\\}$ parameter space, RSD constraints in particular constrain
$c_{m}$ to be bounded by $c_{m}\lesssim 2$. This is intuitive since clustering
as measured by RSD constraints is highly sensitive to changes in the effective
Newtonian constant $G_{\rm eff}$ and hence to changes in the effective Planck
mass as encoded by $c_{m}$. Because of the different and orthogonal ways in
which they constrain the parameter space, ISW and RSD constraints are highly
complementary and so combining them can significantly improve the overall
constraining power. For the minimal RSD constraints shown here, adding RSD to
P18B+ISW constraints only mildly improves bounds, but this will change when
considering more comprehensive RSD constraints in the future redshift .999Note
that going beyond the minimal and conservative RSD data set used here while
avoiding using correlated data sets (that can bias constraints) requires
careful data selection and modelling. It is worth pointing out that the
inclusion of ISW data also results in constraints that are ‘more’ consistent
with $\Lambda{}$CDM, removing the $\sim 2\sigma$ preference for
non-$\Lambda{}$CDM parameter values displayed by P18B and P18B + RSD
constraints. Overall we find that the viable region/volume in $c_{i}$
parameter-space is significantly restricted by including ISW constraints, with
that region shrinking by $\sim 70\%$ when comparing P18B + ISW + RSD
constraints with P18B + RSD constraints.101010By the “volume” in $c_{i}$
parameter space we mean the very simple measure $\Delta c_{b}\Delta c_{m}$ ,
where $\Delta c_{i}$ denotes the $95\%$ confidence interval for $c_{i}$ (note
that this measure is not unique and many alternative measures exist). For
example, for the P18B + ISW + RSD constraints shown in table 2 we have $\Delta
c_{b}=0.28+0.29=0.57$.

Data sets | $c_{b}$ | $c_{m}$ | $\mu_{0}-1$ | $\Sigma_{0}-1$
---|---|---|---|---
P18B | $1.17^{+1.27}_{-0.85}$ | $1.87^{+4.09}_{-2.37}$ | $0.38^{+0.45}_{-0.28}$ | $0.07^{+0.42}_{-0.30}$
P18B + ISW | $0.16^{+0.33}_{-0.33}$ | $0.86^{+1.15}_{-0.84}$ | $0.08^{+0.09}_{-0.08}$ | $-0.08^{+0.12}_{-0.13}$
P18B + RSD | $0.84^{+0.74}_{-0.66}$ | $0.13^{+1.26}_{-0.86}$ | $0.27^{+0.30}_{-0.20}$ | $0.21^{+0.36}_{-0.30}$
P18B + ISW + RSD | $0.12^{+0.28}_{-0.29}$ | $0.54^{+0.90}_{-0.60}$ | $0.07^{+0.09}_{-0.07}$ | $-0.04^{+0.11}_{-0.12}$
Table 2: Here we show $2\sigma$ bounds (marginalised 1D posteriors) on the
dark energy/modified gravity $c_{i}$ parameters for different combinations of
data sets, as discussed in section V (and especially subsections V.3 and V.4).
We also show the corresponding bounds on $\mu_{0}$ and $\Sigma_{0}$, where the
$0$ subscript denotes the value of these parameters today and the parameters
themselves are inferred from the $c_{i}$ (and ultimately $\alpha_{i}$) using
the expressions discussed around equation (6).

###  V.4 Observational ISW constraints II: $\mu/\Sigma$ parametrisation

Above we discussed observational constraints in the $\alpha_{i}$ parameter
space when using different data sets. Here we would like to map these
constraints into the $\\{\mu,\Sigma\\}$ parameter space introduced in section
II.3. This parameter space is often seen as a more phenomenological
alternative to the more theory-informed $\alpha_{i}$ parametrisation. Indeed,
constraints on $\\{\mu,\Sigma\\}$ are frequently obtained in this way, by
parametrising these parameters directly – see e.g. recent survey results from
Planck 2018 and the Dark Energy Survey Y3 in Planck:2018vyg ; DES:2022ccp .
Instead here we always ultimately work with the theory-informed $\alpha_{i}$
basis, parametrising this directly and hence constraining the corresponding
$c_{i}$ parameter space, but we use the fact that a well-defined mapping
exists from this parameter space to $\\{\mu,\Sigma\\}$ – as detailed in
section II.3. We stress that the choice of implicit (as in our case via the
underlying $\alpha_{i}$ basis) or explicit (as in Planck:2018vyg ; DES:2022ccp
) parametrisation of $\\{\mu,\Sigma\\}$ has a significant effect on the
precise constraints obtained and one should therefore not expect identical
constraints when using different parametrisations. We illustrate this point in
appendix C, where we show analogous constraints to those shown in the main
text, but for a $\alpha_{i}=c_{i}a$ parametrisation instead of the
$\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation (3) employed in the main
text.

In the right panel of figure 5 we show the constraints on $\\{\mu,\Sigma\\}$
implied by our underlying theory-informed $c_{i}$ constrains as shown in the
left panel of the same figure. It is instructive to think of this mapping as
constituting a theoretical prior impacting the resulting constraints on
$\\{\mu,\Sigma\\}$.111111We therefore re-emphasise that care ought to be taken
when comparing these constraints with constraints that are obtained for
different, direct parametrisations of $\\{\mu,\Sigma\\}$ encountered in more
phenomenological approaches, since differences in the resulting constraints
may be driven by the parametrisations employed rather than e.g. the data sets
being used. Figure 5 then clearly shows how powerful the addition of ISW
constraints is, even more so than when constraints are expressed in the
$c_{i}$ basis. The inclusion of ISW constraints significantly tightens
constraints on $\Sigma$, which is intuitive since the (time derivative of the)
lensing potential is precisely what the ISW effect probes. c.f. equation (13).
What is less intuitive is that $\mu$ is likewise strongly constrained, which
is (intuitively) more linked to clustering and is therefore indeed
significantly constrained when e.g. including RSD measurements, as can also be
seen in figure 5. That the inclusion of ISW constraints also places further
strong constraints on $\mu$ should therefore be interpreted as a consequence
of the ‘theoretical prior’ imposed by working with an underlying $c_{i}$
parameter basis. Both $\alpha_{M}$ and $\alpha_{B}$ affect $\mu$ and $\Sigma$
and are constrained by all the data sets being used here, so strong additional
constraints on $\mu$ are a consequence of using this underlying theory-
informed parametrisation here. The resulting overall improvement in
constraints is remarkable. We find that the viable region/volume in
$\\{\mu_{0},\Sigma_{0}\\}$ parameter-space is significantly shrunk by
including ISW constraints, namely by close to $\sim 90\%$ when comparing P18B
+ ISW + RSD constraints with P18B + RSD constraints.121212In analogy to our
previous discussion for the $c_{i}$ parameter space, by the “volume” in
$\mu/\Sigma$ parameter space we mean the very simple measure
$\Delta\mu_{0}\Delta\Sigma_{0}$, where $\Delta\mu_{0}$ denotes the $95\%$
confidence interval for $\mu$ etc.

Figure 6: Comparison between the P18B and ISW observational constraints shown
in figure 5 and the minimal ‘qualitative’ constraints shown in figure 4. Black
regions are excluded by the presence of (gradient) instabilities, while the
hashed out grey area corresponds to a ‘naive’ ISW exclusion prior ruling out
regions of parameter space where the lensing potential does not decay (between
redshift 5 and 0, although the precise choice of this redshift range has a
minimal effect). Since the ISW effect is currently detectable at $\sim
5\sigma$, the data driven ISW constraints are (as expected) significantly
stronger than just applying this naive prior to P18B constraints would yield.

##  VI Conclusions

In this paper we have investigated to what extent the ISW effect can be used
to tighten the observational constraints we can place on dynamical dark energy
theories. Working within an effective field theory formalism that captures the
behaviour of scalar perturbations in a wide range of dynamical dark
energy/modified gravity theories around a cosmological background space-time,
we indeed found that ISW constraints from galaxy-CMB cross-correlations
significantly improve constraints. More specifically, our key findings
include:

  * •

Including ISW constraints reduces the viable parameter space from CMB+BAO+RSD
constraints by $\sim 70\%$ for theoretically motivated ‘EFT’ parametrisations
of the underlying dark energy/modified gravity physics. This corresponds to a
$\sim 90\%$ improvement when mapped to the frequently-used phenomenological
$\mu/\Sigma$ parameters. The constraints we find are consistent with
$\Lambda{}$CDM and remove $\sim 2\sigma$ preferences for a departure from
$\Lambda{}$CDM found when only employing CMB+BAO+RSD constraints.

  * •

When working with the $\alpha_{i}$ EFT functions, we find that ISW
measurements in particular constrain the ‘braiding’ function $\alpha_{B}$. In
terms of the $\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation, this
quantitatively means $c_{b}\lesssim 0.5$ at the $2\sigma$ confidence level.

  * •

We show that these constraints are robust when employing different bias models
(see appendix A for details), but quantitatively depend on which precise
parametrisations are employed for the free functions controlling the dynamics
of (linear) perturbations (see appendix C for details).

Going forward, obvious next steps include the incorporation of larger data
sets, both in terms of the survey catalogues used to compute ISW cross-
correlations as well as for complementary constraints from other large scale
structure probes – in particular see redshift  for an upcoming discussion of
improved constraints from redshift space distortions. Finally, note that we
have only considered (linear) large scale structure constraints here,
implicitly constraining a dark energy EFT for the associated large scales
only. When going beyond this, additional subtleties in connecting constraints
from a large range of scales arise, additional constraints can e.g. be found
by 1) considering gravitational-wave induced dark energy instabilities
Creminelli:2019kjy , significantly further tightening cosmological parameter
constraints on dark energy Noller:2020afd , 2) deriving complementary
constraints tightly linking local solar system constraints to cosmology
Babichev:2011iz ; Burrage:2020jkj ; Noller:2020lav , 3) demanding that such
theories have a sensible UV completion – see Melville:2019wyy ;
Kennedy:2020ehn ; deRham:2021fpu ; Melville:2022ykg  and references therein
for a discussion of how the resulting bounds can affect the cosmological
parameter constraints discussed here, 4) going beyond linear dynamics and
using non-linear information to further constrain the dark energy/modified
gravity theories considered here, e.g. along the lines discussed in
Cusin:2017mzw ; Cusin:2017wjg ; Thomas:2020duj ; Srinivasan:2021gib ;
Fiorini:2022srj ; Brando:2022gvg ; Wright:2022krq ; Bose:2022vwi , 5)
incorporating additional theoretical priors into the analysis, e.g. related to
radiative stability Noller:2018eht ; Heisenberg:2020cyi  or to improved
theoretical modelling of the EFT functions Traykova:2021hbr . All these
different probes constrain dark energy-related physics in a variety of
complementary and powerful ways and it will be interesting to better
understand their interplay in the future. Returning our focus to the large
scale structure regime we have investigated in this paper, with a plethora of
new large scale structure surveys and the associated data coming up, it will
be fascinating to see how constraints on the nature of dark energy evolve in
the years to come. What we hope to have shown here is that, in the context of
theoretically well-motivated models, CMB/galaxy cross-correlations can play a
very significant role in this enterprise.

## Acknowledgments

JN is supported by an STFC Ernest Rutherford Fellowship (ST/S004572/1). In
deriving the results of this paper, we have used: CLASS Blas:2011rf , corner
corner , hi_class Zumalacarregui:2016pph ; Bellini:2019syt , MontePyton
Audren:2012wb ; Brinckmann:2018cvx  and xAct xAct .

Data availability Supporting research data are available on reasonable request
from the authors.

##  Appendix A Cosmological constraints and bias models

In section IV.3 we introduced two different bias models, one identical to the
setup employed by Stolzner:2017ged  and a second, simplified model with a
specific ansatz for the redshift dependence of the bias function and hence
fewer bias parameters. In figure 7 we show that the differences between the
two bias models do not have a significant effect on cosmological parameter
constraints in the $\\{c_{b},c_{m}\\}$ space (marginalised over all standard
$\Lambda{}$CDM parameters). We have also checked that this remains true for
the standard $\Lambda{}$CDM parameters themselves.

Figure 7: Comparison between constraints for different bias models in the
$\\{c_{m},c_{b}\\}$ plane. Here we use CMB + BAO + ISW data, but employ bias
model I and II as detailed in section IV.3. We can clearly see that the
simplified bias model II recovers near-identical constraints on the
$\\{c_{m},c_{b}\\}$ parameters and have checked that this remains true for
other cosmological parameters as well.  Figure 8: Analogous plot to figure 6,
but showing a different (more restrictive) ISW exclusion prior. Black regions
are excluded by the presence of (gradient) instabilities, while the hashed out
grey area corresponds to an ISW exclusion prior requiring that the lensing
potential decays by at least $9\%$ between redshifts $5$ and $0$. In terms of
the $\Delta\Phi$ notation introduced in equation (16), this means we require
$\Delta\Phi_{L}>0.09$, i.e. rule out regions of parameter space where
$\Delta\Phi_{L}\leq 0.09$. Here we also show the effect of including RSD
constraints explicitly. Noticeably, most of the P18B + RSD favoured parameter
space is inconsistent with the ISW exclusion prior shown.

##  Appendix B ISW priors from observational constraints

In the main text we investigated both qualitative ISW constraints derived from
the simple requirement that the lensing potential $\Phi_{L}$ is decaying as
well as observational constraints derived by computing the precise (ISW)
cross-correlations between galaxies and CMB temperature, cf. figures 5 and 6.
Here we now wish to investigate whether we can identify a more restrictive
criterion/theory prior to demarcate the region of parameter space consistent
with ISW constraints than the conservative bound discussed in the main text
that simply mandated a decaying lensing potential $\Phi_{L}$. Ideally, such a
criterion can recover the observational constraints more closely even before
computing any galaxy-CMB temperature cross-correlations and hence serve as an
effective theoretical prior. Such a prior would provide a powerful shortcut in
the analysis of dark energy models in the future, so has the potential to be
very helpful in practice.

In figure 6 we showed P18B and P18B+ISW data constraints in the marginalised
$\\{c_{b},c_{m}\\}$ plane, combined with the qualitative ‘exclusion prior’
from demanding a decaying lensing potential. It is clear that the P18B+ISW
constraints are highly consistent with this prior, also providing a helpful
consistency check for the ISW likelihood implementation. Let us formalise an
exclusion prior like the one discussed above by defining $\Delta\Phi_{L}$ as
follows

$\displaystyle\Delta\Phi_{L}\equiv\frac{\Phi_{L}{}|_{z=5}-\Phi_{L}{}|_{z=0}}{\Phi_{L}{}|_{z=5}},$
(16)

where $X|_{z=z_{0}}$ denotes the evaluation of the function $X$ at redshift
$z_{0}$. $\Delta\Phi_{L}$ in effect measures whether the lensing potential
overall decays/grows between redshifts $5$ and $0$ and, if so, by how much.
The qualitative ISW constraints discussed in section V.2 therefore amount to
placing a $\Delta\Phi_{L}>0$ prior (since a decay in time implies growth with
increasing redshift). Note that the precise choice of the upper redshift
chosen will only very mildly affect the analysis that follows. Conceptually
speaking, what is important is to choose a redshift before dark energy plays a
significant role in determining the Universe’s expansion – we choose redshift
$5$ here as an illustrative example.

Figure 9:  Left plot: Evolution of the lensing potential $\Phi_{L}=\Phi+\Psi$
as a function of redshift for various choices of the dark energy parameter
$c_{b}$, where $\alpha_{i}=c_{i}\Omega_{DE}$ – see equation (2). All
$\Lambda{}$CDM parameters are fixed to the P18 best-fit cosmology
Planck:2018vyg  and in this plot we show results for a fiducial choice of the
second dark energy parameter $c_{m}=1$ (compare with figure 1 where this was
fixed to $c_{m}=0$). Here we also explicitly see that $\Phi_{L}$ can decay
faster than in $\Lambda{}$CDM. Right plot: Here we show the $C_{\ell}^{Tg}$
quantifying cross correlations between galaxies and CMB temperature
measurements for the same cosmologies as on the left – c.f. figure 2 in the
main text.

Figure 6 in the main text clearly showed that a $\Delta\Phi>0$ prior
unsurprisingly and significantly underestimates the strength of current ISW
constraints. In figure 8 we here show the analogous exclusion region for a
$\Delta\Phi>0.09$ prior, i.e. the effect of a prior that demands a decay of
the lensing potential by at least $9\%$ between redshifts $5$ and $0$. Two
observations then jump out: 1) This prior indeed successfully rules out more
of the parameter space ultimately excluded by ISW constraints and we can see
that cosmologies with less than a $9\%$ decay in the lensing potential are
indeed ruled out at $2\sigma$ here, 2) ISW data constraints clearly do more
than just place a constraint on $\Delta\Phi$. In particular one can see that
for $c_{b},c_{m}\gtrsim 1$ regions consistent with P18B constraints at
$1\sigma$ are consistent with $\Delta\Phi>0.09$, yet are ruled out by ISW
data. We find that such cases are e.g. associated with cosmologies, where
$\Phi_{L}$ does indeed decay at early and late times, but plateaus for
intermediate redshifts. Such evolutions are qualitatively different to e.g.
those shown in figure 9 and produce a significantly worse fit to the observed
$C_{\ell}$’s.

Overall we therefore see how, while imperfect, the above $\Delta\Phi_{L}$ can
help to understand which regions of parameter space are consistent with ISW
constraints. We caution that the more restrictive prior discussed here has
been derived in the context of the Horndeski scalar-tensor theories we have
focused on here, with implicit assumptions on the form of the underlying
functional freedom encoded in the parametrisations used. We leave a more
global analysis for different theories and parametrisations to future work,
but caution that a specific $\Delta\Phi_{L}>X$ prior for some value $X$ should
not be blindly applied to other analyses without ideally first verifying its
accuracy in this new context in a similar fashion to what has been shown here.
Finally, note that we have clearly tuned the $\Delta\Phi_{L}>0.09$ prior
discussed above a posteriori given the data constraints shown e.g. in figures
6 and 8, so this is directly impacted by the constraining power of current
data and an analogous prior should be adjusted as future data become
available.

##  Appendix C Different $\alpha_{i}$ parametrisations

As discussed in the main text, numerous parametrisations exist for the
$\alpha_{i}$ EFT functions encoding the dynamics of linear perturbations and
we have presented constraints focused on the commonly used
$\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation. To complement this, here we
show constraints analogous to what is displayed and discussed in the main text
in section V for another parametrisation that is commonly used, namely for
$\alpha_{i}=c_{i}\cdot a$. As in the main text we show constraints in the
$\\{c_{m},c_{b}\\}$ plane as well as when those constraints are mapped to the
$\\{\mu,\Sigma\\}$ parameter space.

Our main results for the $\alpha_{i}=c_{i}\cdot a$ parametrisation are shown
in figure 10 and table 3. Differences to constraints obtained using the
$\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation in figure 5 and table 2 are
immediately apparent and neatly illustrate a point stressed in the main text:
The choice of underlying parametrisation has a significant effect on the
precise constraints obtained and one should therefore not expect identical
constraints when using different parametrisations.131313A more detailed
investigation of this parametrisation dependence is beyond the scope of this
paper, but we here simply report the corresponding bounds using the
$\alpha_{i}=c_{i}\cdot a$ for comparison and to illustrate this point. For the
$\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation we saw that the novel ISW
constraints we focus on here (when combined with P18B bounds) were
significantly stronger than the RSD constraints we compared them with.
Nevertheless those two sets of constraints were complementary and P18B + RSD +
ISW constraints were noticeably improved over P18B + ISW. For the
$\alpha_{i}=c_{i}\cdot a$ parametrisation this is not the case, with near
identical bounds for both scenarios. for this parametrisation, the P18B + ISW
constraints almost fully preempt any further improvements from adding RSD
constraints in our setup. This is primarily due to the much tighter
$\\{c_{m},c_{b}\\}$ contours for this parametrisation (a consequence of the
different time-dependence for the $\alpha_{i}$) and is mirrored when
inspecting constraints in the $\\{\mu,\Sigma\\}$ plane, as shown in figure 10.

Finally, a quick note about prior boundaries. Already in the
$\alpha_{i}=c_{i}\Omega_{\rm DE}$ parametrisation discussed in the main text,
we implicitly encountered some hard prior boundaries, where there is an
excellent fit to the data on the boundary. For example in figure 6 gradient
stability constraints place hard cuts in some directions in parameter space.
Similarly, for the $\alpha_{i}=c_{i}\cdot a$ parametrisation considered in
this appendix, $\alpha_{M}$ and hence $c_{m}$ cannot become negative due to
gradient stability constraints and so while the data in fact prefer a near-
zero value for $c_{m}$, stable deviations are only possible in the positive
$c_{m}$ direction. In addition, we here encounter a second hard prior
boundary: We have restricted $\alpha_{B}$ to remain smaller than 2 at all
times. Crossing the $\alpha_{B}=2$ point (which would happen at some point in
the evolution for all models with $c_{B}\geq 2$) is associated with a number
of discontinuities and so evolutions crossing this point have frequently been
conservatively excluded in the literature Noller:2018wyv ;
SpurioMancini:2019rxy ; Noller:2020afd . Since this issue only affects the
pure P18B constraints discussed in this appendix and becomes irrelevant once
RSD or ISW constraints are added, we follow the same conservative methodology
here and refer to Bellini:2014fua ; Lagos:2017hdr ; Ijjas:2017pei ;
Noller:2018wyv ; SpurioMancini:2019rxy ; Noller:2020afd  for more in-depth
discussions of this point.

Figure 10: Left panel: Here we show constraints in the $\\{c_{m},c_{b}\\}$ plane for the $\alpha_{i}=c_{i}a$ parametrisation. So this is the analogue to the left panel of figure 5, where this was shown for $\alpha_{i}=c_{i}\Omega_{\rm DE}$. In the parametrisation shown here, ISW constraints noticeably fully preempt any additional constraints placed by RSD measurements. Notice that we impose $\alpha_{B}\leq 2$ here – see appendix C for a related discussion – but this only affects the P18B-only bounds. Right panel: Analogous plot to the left panel, but for the more phenomenological $\mu/\Sigma$ parametrisation, where $\mu_{0}/\Sigma_{0}$ are related to the $c_{i}$/$\alpha_{i}$ via (6). Note that the $0$ subscript denotes the value of $\mu/\Sigma$ today.  Data sets | $c_{b}$ | $c_{m}$ | $\mu_{0}-1$ | $\Sigma_{0}-1$
---|---|---|---|---
P18B | $0.29<c_{b}<2^{*}$ | $0^{*}<c_{m}<1.23$ | $0.57^{+0.53}_{-0.51}$ | $0.35^{+0.33}_{-0.30}$
P18B + ISW | $0.24^{+0.42}_{-0.21}$ | $0*<c_{m}<0.24$ | $0.07^{+0.16}_{-0.06}$ | $0.05^{+0.10}_{-0.04}$
P18B + RSD | $0.51^{+0.83}_{-0.42}$ | $0*<c_{m}<0.68$ | $0.19^{+0.41}_{-0.16}$ | $0.12^{+0.24}_{-0.10}$
P18B + ISW + RSD | $0.21^{+0.40}_{-0.19}$ | $0*<c_{m}<0.22$ | $0.06^{+0.15}_{-0.05}$ | $0.04^{+0.10}_{-0.04}$
Table 3: Here we show $2\sigma$ bounds (marginalised 1D posteriors) on the
dark energy/modified gravity $c_{i}$ parameters in the $\alpha_{i}=c_{i}\cdot
a$ parametrisation for different combinations of data sets, as discussed in
appendix C. We also show the corresponding bounds on $\mu_{0}$ and
$\Sigma_{0}$, where the $0$ subscript denotes the value of these parameters
today and the parameters themselves are inferred from the $c_{i}$ (and
ultimately $\alpha_{i}$) using the expressions discussed around equation (6).
When the $c_{i}$ distribution is strongly non-Gaussian and there is an
excellent fit to the data on a prior-induced boundary, we do not give a mean
value and denote limit values with an asterisk – see appendix C for details.

##  Appendix D Cosmological parameter constraints beyond EFT parameters

Figure 11:  Here we show parameter constraints for the full dark energy
$\\{c_{b},c_{m}\\}$ \+ standard $\Lambda{}$CDM parameter space for the same
combinations of data sets as in the main text. Contours mark $68\%$ and $95\%$
confidence intervals, and are computed using Planck 18 and BAO data (P18B), as
well as measurements of galaxy-galaxy auto- and galaxy-CMB temperature cross
correlations (ISW), and redshift space distortion measurements (RSD) – for
details on the data sets used see section IV. See appendix D for a discussion
of the constraints shown.

Throughout the above we have focused on constraints for the $c_{i}$ parameters
controlling the $\alpha_{i}$ functions and the corresponding constraints when
mapped to the $\\{\mu,\Sigma\\}$ plane, always marginalised over the standard
cosmological $\Lambda{}$CDM parameters as well as any ‘nuisance’ parameters
associated with the bias modelling. In figure 11 we now for completeness also
show constraints for the standard cosmological $\Lambda{}$CDM parameters for
one of the parametrisations used, namely $\alpha_{i}=c_{i}\Omega_{DE}$. As
pointed out in the main text, we see that the addition of ISW constraints only
has a mild effect on the standard $\Lambda{}$CDM parameters, most noticeably a
small shift in $\omega_{\rm cdm}$ as a result of larger $c_{b}$ values mildly
preferring smaller $\omega_{\rm cdm}$. However, no strong degeneracies between
the $c_{i}$ and standard $\Lambda{}$CDM parameters are present. For more
details on how these constraints compare to pure $\Lambda{}$CDM constraints in
the absence of the deviations encoded in the $c_{i}$, we refer to the analysis
of Noller:2018wyv . Finally we note that, motivated by observations of the
Gunn-Peterson trough (see e.g. SDSS:2001tew ), in our analysis we have imposed
a prior $\tau_{\rm reion}\geq 0.04$, corresponding to $z_{\rm reion}\gtrsim
6$.

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