11institutetext: SISSA-International School for Advanced Studies, Via Bonomea
265, 34136 Trieste, Italy 22institutetext: INFN, Sezione di Trieste, Via
Valerio 2, I-34127 Trieste, Italy 33institutetext: IFPU, Institute for
Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
44institutetext: Dipartimento di Fisica, Sapienza Universitá di Roma, P.le
Aldo Moro 2, 00185, Roma, Italy 55institutetext: INAF - Osservatorio
Astronomico di Roma, Via Frascati 33, 00040, Monteporzio Catone, Roma, Italy
66institutetext: INFN- Sezione di Roma 1, P.le Aldo Moro 2, 00185, Roma, Italy

# Generalised scalar-tensor theories of gravity and pressure profiles of
galaxy clusters

Balakrishna S. Haridasu 112233<EMAIL_ADDRESS>Purnendu Karmakar 44
Marco De Petris 445566 Vincenzo F. Cardone 5566 Roberto Maoli 4455

(August 27, 2024)

###### Abstract

In the current proceedings, we summarise the results presented during the mm
Universe@NIKA2 conference, taken from our main results in Haridasu:2021hzq .
We test the Degenerate higher-order scalar-tensor(DHOST) theory as a
generalised platform for scalar-tensor theory at galaxy cluster scales to
predict in such static systems small scale modification to the gravitational
potential. DHOST theory is not only a good alternative to $\Lambda$CDM for the
background evolution but also predicts small-scale modification to the
gravitational potential in static systems such as galaxy clusters. With a
sample of 12 clusters with accurate X-ray Intra Cluster Medium (ICM) data
(X-COP project) and Sunyaev-Zel’dovich (SZ) ICM pressure (Planck satellite),
we place preliminary constraints on the DHOST parameters defining the
deviation from GR. Moreover, we also collect a few supplementary analyses we
have performed during the course: i) Gaussian process reconstruction without
parametric assumptions, ii) $P_{\rm SZ}$-only data analysis not aided by the
X-ray data. Finally, we present possible extensions to the current work which
may benefit from future high sensitivity and spatial resolution observations
such as the ongoing NIKA2 camera.

##  1 Introduction

Einstein developed the standard general relativity (GR) by considering gravity
as warps and curves in the fabric of geometric space-time, and that geometry
of space-time is described only by a metric. GR can explain the majority of
the cosmological observations, but challenged to explain the acceleration in
the cosmological distance. Therefore, if we plan to keep the basic metric
structure because of the large success of GR, and add one scalar degrees of
freedom to explain the acceleration, then the degenerate higher-order scalar-
tensor (DHOST) theory is the best possible platform to test the largest
classes of scalar-tensor theories Crisostomi:2016czh ; Achour:2016rkg .

The additional degree of freedom must be suppressed in within the small scales
to explain the observations, is the so-called screening mechanism
Babichev:2013usa . The modified Newtonian potential for the DHOST theory in
the galaxy cluster, which produces the gravitational waves of the velocity
($c_{g}$) equal to the velocity of light ($c$), is

$\frac{d\Phi(r)}{dr}=\frac{G_{\rm N}^{\rm eff}M_{\rm
HSE}(r)}{r^{2}}+\Xi_{1}G_{\rm N}^{\rm eff}\frac{d^{2}M_{\rm
HSE}(r)}{dr^{2}}\,,$ (1)

while assuming the hydrostatic equilibrium (HSE), where $M_{\rm HSE}(r)$ is
the total mass within the radial distance $r$. Both, $\Xi_{1}$ and $G_{\rm
N}^{\rm eff}$ parameters track the departure of DHOST theory from GR, which
modify the ICM thermal pressure profile as

$P^{\rm th}(r)=P^{\rm th}(0)-\int_{0}^{r}\rho_{\rm
gas}(\tilde{r})\frac{d\Phi(\tilde{r})}{dr}{\rm d}\tilde{r}\,,$ (2)

where $\rho_{\rm gas}(r)$ is the gas density. Our formalism is a straight-up
implementation of the so called forward-method, where the pressure profile is
computed while assuming empirical profiles for the mass, $M_{\rm HSE}(r)$, and
the electron density, $n_{e}(r)$, radial profiles which in our case are the
standard NFW Navarro:1995iw  and simplified Vikhlinin parametric model
Vikhlinin:2005mp , respectively. In this context, several previous works have
implemented similar approach either using stacked clusters and/or having
complementary weak lensing data Sakstein:2016ggl ; Wilcox:2015kna ;
Terukina:2013eqa .

Throughout the current discourse we assume $H_{0}=70$ km/s Mpc-1and
$\Omega_{\rm m}=0.3$. $R_{500}$ and $M_{500}$ carry usual definitions, i.e,
total mass $M_{\Delta}$ within the radius $R_{\Delta}$, with a mean mass
density $\Delta$ times the critical density ($\rho_{\rm c}(z)=3H^{2}(z)/8\pi
G_{\rm N}$).

##  2 Main Results

Firstly, we figure that only 8 clusters out of the 12 available clusters in
the dataset prefer the NFW mass profile in a comparison through information
criteria (see Fig 2, of Ettori:2018tus ). We refer to the other 4 clusters
(A644, A1644, A2319 and A2255) as non-NFW clusters for brevity. As a
consequence of which, we find that the non-NFW clusters fit the data within
the DHOST scenario with statistical preferences reaching
$\Delta\log(\mathcal{B})>>25$ (further details in Haridasu:2021hzq  and
Ettori:2018tus ), which imply a very high preference yet providing very low
values of the parameter $\Xi_{1}$, in complete disagreement with the
theoretical limits Saito:2015fza ; Sakstein:2015aac ; Sakstein:2015zoa . In
Figure 1, we show the fits of the GR and DHOST scenarios for A644 (non-NFW)
and A1795 cluster and the $1\sigma$ dispersion to provide a comparison. We
show the posteriors for $\Xi_{1}$ obtained using the individual cluster and
the joint constraint in left panel of Figure 2 and a comparison with earlier
constraints in the right panel.

Figure 1: Mass (upper) and pressure (lower) radial profiles for the cluster
A644 (Left) and A1795 (Right) in the cases of GR (blue) and DHOST (red),
normalised to GR. The former cluster is an example case of the 4 non-NFW
clusters, while the latter is one of the 4 clusters we retain in estimating
our final conservative result. The vertical dashed (GR) and dot-dashed (DHOST)
lines show $R_{500}$. The 3 inner $P_{\rm SZ}$ data points (black) are
excluded in the main analysis.

As the main result of our analysis we updated the limits on the modified
gravity parameter $\Xi_{1}=-0.030\pm 0.043$, which is a much more stringent
constrain in comparison to the earlier result of
$\Xi_{1}\sim-0.028^{+0.23}_{-0.17}$, reported in Sakstein:2016ggl  using a
similar formalism with stacked X-ray cluster profile. In a more conservative
case where we utilise only 4 clusters (A1795, A3158, RXC1825 and ZW1215), we
find a less stringent $\Xi_{1}=-0.061\pm 0.074$, yet twice as tighter
constraint than the earlier analysis. These 4 conservative clusters are
restricted within a redshift range of $0.0597<z<0.0766$. In the right panel of
Figure 2, we show a comparison of the earlier constraints, our results and the
theoretical limits. As can be seen, our results are in excellent agreement
with the latter and an improvement for the constraints on $\Xi_{1}$. The lower
theoretical limit is marked by the requirement of stable static solutions of
non-relativistic stars Saito:2015fza ; Saltas:2018mxc  and the upper limits
are obtained on the based on the minimum mass required for hydrogen burning in
low mass red dwarfs Sakstein:2015aac ; Sakstein:2015zoa .

An added advantage of our current analysis is that having individual clusters
spread over a redshift range of $0.04<z<0.1$, we are able to present for the
first time the necessary assessment of time-evolution of the parameter
$\Xi_{1}(a\equiv 1/(1+z))$. We utilise a simple first order Taylor expansion
of $\Xi_{1}(a)$ as the scale factor, $a\to 1$ and perform a post-MCMC analysis
on the inferred posteriors of the $\Xi_{1}$ obtained for individual clusters.
Using 8 clusters we find a mild deviation from a constant behaviour for
$\Xi_{1}$, at present is however driven only by two clusters (A85 and
A2142)and the advantage of extending the analysis over a larger dataset that
covers a wider redshift range is clear.

Figure 2: Left: Posterior distribution of the parameter $\Xi_{1}$, over-
plotted by smooth Gaussian kernel density profiles. The vertical dashed line
marks the GR ($\Xi_{1}=0$) case. The black curve represents our joint
constraint using 8 clusters. Right: Constraints on the parameter $\Xi_{1}$,
from an earlier analysis of 58 stacked clusters (blue) (Sakstein:2016ggl, )
and our results with 8 clusters (green) and conservative 4 clusters (orange).
The grey shaded region shows the allowed region for the same parameter
obtained from theoretical limits Sakstein:2015aac ; Sakstein:2015zoa ;
Saito:2015fza .

##  3 Supplementary Results

In this section we present a few supplementary analyses we have performed,
also constituting possible future extension of the current work.

Figure 3:  Mass radial profiles with $1\sigma$ dispersion for clusters A644
(left) and A1795 (right) for the GR (blue), DHOST (red) and GP (green). A644
and A1795 represent one of the non-NFW cluster and one of the 4 conservative
clusters, respectively. The vertical dashed, dot-dashed and dotted lines show
the $R_{500}$ in GR, DHOST and GP, respectively. The $R_{500}$ from the GP
case coincides very well with that in GR case. The mass profiles obtained
utilising only the $P_{\rm SZ}$ data for the GR case are shown in orange. Here
the GR case corresponds to $\tilde{\gamma}_{\rm N}=1$.

###  3.1 Gaussian Process

In addition to the parametric analysis, we also perform a non-parametric
model-independent analysis based on the Gaussian Process (GP) formalism,
please see Haridasu:2018gqm  and references therein for details on the method.
In essence, once could reconstruct the underlying functional form [$f(x_{i})$]
for a collection of Gaussian data points at
$\mathbf{x}\equiv\\{x_{1},x_{2},..x_{N}\\}$, assumed belonging to the same
process, which now represents a GP. This is implemented by modelling a
covariance $K(x_{i},x_{j})$, amongst the data points, instead of functional
form for the process itself. For this purpose we independently reconstruct the
electron density and the $P_{\rm SZ}$ pressure profiles assuming the squared-
exponential (SE) 111The functional form for the SE kernel is
$K(x_{i},x_{j})=\sigma_{f}^{2}\exp{\left[{(x_{i}-x_{j})^{2}}/{2l^{2}}\right]}$,
where $\\{\sigma_{f},\,l\\}$ are now free parameters inferred by optimising a
log-marginal likelihood.  kernel. Note that here we cannot extrapolate the
profile beyond the range of the available data, as GP tends to retrieve broad
prior regions, with no constrain on the posterior. We intend the analysis with
GP as an alternate verification for the fits based on the profile themselves
as the model-independent method is agnostic with no assumptions made for the
electron density profile and the mass profile. Therefore, a parametric method
based on assumed empirical profiles is expected to serve better if it is in a
good agreement with the GP method.

In Figure 3, we show the mass reconstructions obtained using the NFW profile
in both the GR (blue) and DHOST (red) scenarios and the GP (green)
reconstructed result. For the GP method we firstly find a lowering of the mass
profile in the inner regions, which is an immediate consequence of not
excluding the inner most 3 data-points of $P_{\rm SZ}$ data (see
Ettori:2018tus ; Ghirardini:2018byi ). However, being a non-parametric
estimation and that it only follows the data points, we find the GP to be in
better agreement towards the outskirts of the mass profile. In the case of the
A644, a non-NFW cluster, we find that the model-independent reconstruction is
in better agreement with the GR case than with the DHOST scenario, which is a
clear indication of our results for the latter being biased when utilising the
NFW mass profile. The $R_{500}=1.15\pm 0.06\,[{\rm Mpc}]$ and
$M_{500}=4.57^{+0.73}_{-0.65}\,[10^{14}\,M_{\odot}]$ estimated using GP here
are in excellent agreement with our constraints in the GR case. In the case of
A1795 however, the deviation between the GR and the DHOST cases is minimal and
is also reflected in the GP reconstruction. Once again the GP based
$R_{500}=1.12\pm 0.04\,[{\rm Mpc}]$ and
$M_{500}=4.24^{+0.045}_{-0.040}\,[10^{14}\,M_{\odot}]$ are in excellent
agreement with the parametric method assuming the NFW profile.

###  3.2 $P_{\rm SZ}$ only analysis

ndeed, one of the major advantages in our analysis is that we perform the
joint-fit to the $T_{\rm X}$( or $P_{\rm X}$) and the $P_{\rm SZ}$ data, which
provides a better estimation of the mass profile to the outskirts ($R\sim 1.2$
Mpc) of the clusters. In the screening mechanism, the GR is recovered in the
innermost region of the galaxy cluster, while the gravity gradually deviates
from GR in the outskirt of the cluster. Given that, the inclusion of the
$P_{\rm SZ}$ data actually provides a very suitable opportunity to asses the
mass profile in a larger radial range and hence the modifications to gravity.

To contrast our constraints against the combined analysis of $P_{\rm
SZ}+P_{\rm X}$, we also perform an MCMC analysis using only the $P_{\rm SZ}$
data (without excluding the 3 inner most data points). We also show the
inferred mass profiles (orange) of the same in Figure 3. One can immediately
notice that for the cluster A644, all the 4 contrasted scenarios are almost
discordant at the face-value. This in turn indicates that a more careful
assessment has to be made when combining the data sets and/or trying to assess
the modifications to physics using this cluster. On the other hand, the A1795
cluster shows that all the approaches are are in very good agreement,
especially in the outskirts of the cluster. However, comparing the $P_{\rm
SZ}$ parametric fit and the GP based reconstruction we find that the former
tends to show higher masses, also in agreement with the $P_{\rm X}+P_{\rm SZ}$
analysis. The results for the posterior distributions on $\Xi_{1}$ are
summarised in Appendix C. of Haridasu:2021hzq .

Within the joint analysis however, one needs to check internally for a
consistency between the X-ray and SZ data, see Appendix A of
Ghirardini:2018byi  for more details. Therefore in a similar approach we
introduce an intrinsic dispersion ($\sigma_{\rm P,int}$) as a free parameter
which is scaled by the modelled pressure ($\sigma_{P,\rm int}\sim P^{\rm
model}\sigma_{\rm int}$) (Appendix D of Ghirardini:2017apw ), added in
quadrature to the error and inferred through the MCMC analysis. Alongside the
dispersion, we also include a rescaling parameter $\eta$, such that $P^{\rm
data}_{\rm X}\to\eta\times P^{\rm data}_{\rm X}$. We show the posterior
distributions of these two consistency parameters in Figure 4 for the A644 and
A1795 clusters. Clearly, both clusters show a good agreement between $P_{\rm
SZ}$ and $P_{\rm X}$ data, while A644 shows a larger dispersion within the
$P_{\rm X}$ data points. This in turn provides additional correlations with
$M_{500}$ and increases the error-bars, however note also that we fix $\eta=1$
in the main analysis. Please refer to original analysis in Ettori:2018tus ;
Ghirardini:2017apw ; Ghirardini:2018byi , where these arguments are
elaborated.

Figure 4: We compare here the posteriors for the intrinsic dispersion in the
X-ray data and the scaling parameter. The dashed grey line marks the case of
$\eta=1$, which implies good agreement between the $P_{\rm X}$ and $P_{\rm
SZ}$ data.

In summary, as we have shown here, having the galaxy cluster physics well-
mapped in a larger radial range and having several of these objects also
distributed along the redshift will be crucial to assess the constraints on
the $\Xi_{1}$ in individual clusters and its time evolution. We intend to
explore the same with soon to be available future data. For this goal, NIKA2
SZ observations could be helpful to extend this analysis.

## Acknowledgements

Authors are grateful to the ‘mm Universe $@$NIKA2’ conference for providing an
oppurtunity to present the work. Authors acknowledge receiving useful comments
and feedback on the use of data from Stefano Ettori and Dominique Eckert. BSH
is supported by the INFN INDARK grant. PK, MDP, VC,RM acknowledge support from
INFN/Euclid Sezione di Roma. PK, MDP and RM also acknowledge support from
Sapienza Universitá di Roma thanks to Progetti di Ricerca Medi 2018, prot.
RM118164365E40D9 and 2019, prot. RM11916B7540DD8D.

## References

  * (1) B.S. Haridasu, P. Karmakar, M. De Petris, V.F. Cardone, R. Maoli (2021), 2111.01101
  * (2) M. Crisostomi, K. Koyama, G. Tasinato, JCAP 1604, 044 (2016), 1602.03119
  * (3) J. Ben Achour, D. Langlois, K. Noui, Phys. Rev. D93, 124005 (2016), 1602.08398
  * (4) E. Babichev, C. Deffayet, Class. Quant. Grav. 30, 184001 (2013), 1304.7240
  * (5) J.F. Navarro, C.S. Frenk, S.D.M. White, Astrophys. J. 462, 563 (1996), astro-ph/9508025
  * (6) A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M. Markevitch, S.S. Murray, L. Van Speybroeck, Astrophys. J. 640, 691 (2006), astro-ph/0507092
  * (7) J. Sakstein, H. Wilcox, D. Bacon, K. Koyama, R.C. Nichol, JCAP 1607, 019 (2016), 1603.06368
  * (8) H. Wilcox et al., Mon. Not. Roy. Astron. Soc. 452, 1171 (2015), 1504.03937
  * (9) A. Terukina, L. Lombriser, K. Yamamoto, D. Bacon, K. Koyama, R.C. Nichol, JCAP 04, 013 (2014), 1312.5083
  * (10) S. Ettori, V. Ghirardini, D. Eckert, E. Pointecouteau, F. Gastaldello, M. Sereno, M. Gaspari, S. Ghizzardi, M. Roncarelli, M. Rossetti, Astron. Astrophys. 621, A39 (2019), 1805.00035
  * (11) R. Saito, D. Yamauchi, S. Mizuno, J. Gleyzes, D. Langlois, JCAP 1506, 008 (2015), 1503.01448
  * (12) J. Sakstein, Phys. Rev. D92, 124045 (2015), 1511.01685
  * (13) J. Sakstein, Phys. Rev. Lett. 115, 201101 (2015), 1510.05964
  * (14) I.D. Saltas, I. Sawicki, I. Lopes, JCAP 05, 028 (2018), 1803.00541
  * (15) B.S. Haridasu, V.V. Luković, M. Moresco, N. Vittorio, JCAP 10, 015 (2018), 1805.03595
  * (16) V. Ghirardini et al., Astron. Astrophys. 621, A41 (2019), 1805.00042
  * (17) V. Ghirardini, S. Ettori, D. Eckert, S. Molendi, F. Gastaldello, E. Pointecouteau, G. Hurier, H. Bourdin, Astron. Astrophys. 614, A7 (2018), 1708.02954