# Constraints on the variation of fine structure constant from joint SPT-SZ
and XMM-Newton observations

Kamal Bora  E-mail<EMAIL_ADDRESS>Shantanu Desai  E-mail:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology,
Hyderabad, Kandi, Telangana-502285, India

###### Abstract

We search for a variation of the electromagnetic fine structure constant
($\alpha\equiv e^{2}/\hbar c$) using a sample of 58 SZ selected clusters in
the redshift range ($0.2<z<1.5$) detected by the South Pole Telescope, along
with X-ray measurements using the XMM-Newton observatory. We use the ratio of
the integrated SZ Compto-ionization to its X-ray counterpart as our observable
for this search. We first obtain a model-independent constraint on $\alpha$ of
about 0.7%, using the fact that the aforementioned ratio is constant as
function of redshift. We then look for logarithmic dependence of $\alpha$ as a
function of redshift: $\Delta\alpha/\alpha=-\gamma\ln(1+z)$, as this is
predicted by runaway dilaton models. We find that $\gamma$ = $-0.046\pm 0.1$,
which indicates that blackthere is no logarithmic variation of $\alpha$ as a
function of redshift. We also search for a dipole variation of the fine
structure constant using the same cluster sample. We do not find any evidence
for such a spatial variation.

##  I Introduction

Ever since Dirac’s ingenuous argument that Newton’s Gravitational Constant
could vary with time Dirac (1937), a number of theories beyond General
Relativity and the Standard Model of Particle Physics have predicted the
variation of fundamental constants, including the fine-structure constant
($\alpha\equiv e^{2}/\hbar c$) Uzan (2011); Martins (2017). Therefore, a
plethora of searches have been undertaken using both laboratory and
astrophysical observations to search for variations of $\alpha$. For theories
which predict a variation of $\alpha$, Einstein’s equivalence principle is
violated in the electromagnetic sector. These theories usually involve
coupling between the electromagnetic part of matter fields and scalar fields
Bekenstein (1982); Damour and Polyakov (1994); Sandvik et al. (2002); Barrow
and Lip (2012).

Another strong impetus in searching for a varying $\alpha$ using different
methods, comes from a claimed variation of $\alpha$ using quasar absorption
systems Webb et al. (2001) with the Keck/HIRES and VLT/UVES telescopes.
Subsequently, the same group also argued for a spatial variation in $\alpha$
at 4.2$\sigma$ significance, which is consistent with a dipole Webb et al.
(2011); King et al. (2012). The best-fit position of this dipole is at
RA=$(17.5\pm 0.9)$ hr and Declination=$(-58\pm 9)^{\circ}$ Webb et al. (2011);
King et al. (2012). However, other groups have failed to confirm this result
(eg. Srianand et al. (2004)). blackRecent studies however indicate that this
dipole variation could be due to the distortion of the wavelength scale over
long wavelength ranges ($\geq 1000$ Å) Whitmore and Murphy (2015). These
distortions of the wavelength calibration were demonstrated using asteroid and
iodine cell star exposures from both Keck-HIRES and VLT-UVES Whitmore and
Murphy (2015), and the aforementioned work also showed that simulated quasar
spectra can mimic some of the VLT-UVES features seen in King et al. (2012),
although they cannot explain the Keck-HIRES results. Recently, measurements of
zinc and chromium absorption lines in quasars, (which are not sensitive to the
aforementioned wavelength distortions) were not consistent with no variations
in $\alpha$ within $1\sigma$ Murphy et al. (2016, 2017).

Another motivation in recent years to search for variation in $\alpha$ comes
from the Hubble tension conundrum related to the discrepancy in Hubble
constant values between the low-redshift and high redshift probes Verde et al.
(2019); Bethapudi and Desai (2017). Although current studies indicate that a
variation of $\alpha$ can only play a minor role in resolving the Hubble
tension Knox and Millea (2020); Hart and Chluba (2020), a variation in
$\alpha$ could have implications for this problem.

Therefore, a large number of searches for variations of $\alpha$ have been
carried out using a plethora of astrophysical/cosmological probes such as CMB
Menegoni et al. (2012); Ade et al. (2015); Hart and Chluba (2018); Smith et
al. (2019), Big-Bang Nucleosynthesis Clara and Martins (2020), supermassive
black hole at the galactic center Hees et al. (2020), white dwarf spectra
Berengut et al. (2013), strong gravitational lensing Colaço et al. (2020), and
also Earth-based measurements using the Oklo natural reactor Damour and Dyson
(1996), blackatomic clocks Godun et al. (2014) etc. A recent review
summarizing the latest results from all these searches can be found in Martins
(2017) and references therein. All these other searches have come up with null
results, and failed to corroborate any claims for a variation in $\alpha$.
However, given the positive claim from one group, it is important to continue
searching for a variation of $\alpha$ using multiple independent sources, as
we continue to gather new data.

Here, we use galaxy clusters in testing for variations in $\alpha$. Galaxy
clusters are the most massive, gravitationally collapsed objects in the
universe and have proved to be wonderful laboratories for studying cosmology,
structure formation, galaxy evolution, neutrino mass, graviton mass, various
modified gravity theories etc Allen et al. (2011); Vikhlinin et al. (2014);
Desai and Gupta (2020). In the past two decades, a large number of galaxy
clusters have been discovered upto very high redshifts thanks to multi-
wavelength surveys in optical, X-ray, and Sunyaev-Zeldovich (SZ, hereafter) at
mm wavelengths which have mapped out large-area contiguous regions of the
universe. The first test for the variation of $\alpha$ using cluster SZ
observations was implemented by Galli Galli (2013), who showed that the ratio
of integrated Compto-ionization in SZ ($Y_{SZ})$ to its X-ray counterpart
($Y_{X}$) scales as $\alpha^{3.5}$. A limit on the variation of $\alpha$ was
obtained from the fact this ratio for the 2011 Early Planck Release Planck
Collaboration et al. (2011)) catalog is constant. Holanda et al. (2016a) then
showed that the ratio of gas fraction from SZ and X-ray measurements scales as
$\alpha^{3}$, after assuming that a variation of $\alpha$ leads to a violation
of cosmic distance-duality relation (CDDR) Etherington (1933). They compared
the gas fraction measurements for 29 clusters in the redshift range
$0.14<z<0.89$ LaRoque et al. (2006) to constrain $\alpha$ variation at these
redshifts. In a follow-up work, Holanda et al Holanda et al. (2016b) combined
the angular diameter distance of galaxy clusters along with luminosity
distance measurements from type Ia supernovae in the redshift range
$0.023<z<0.784$ to constrain variations in $\alpha$. A similar idea was
thereafter applied to the combination of gas fraction measurements of Atacama
Cosmology Telescope selected clusters in the redshift range $0.12<z<1.36$ and
Type Ia supernovae Holanda et al. (2017). Martino et al de Martino et al.
(2016) looked for spatial evolution of $\alpha$ from Planck SZ data, by
looking for variations in the CMB temperature as a function of redshift at the
location of the clusters. Colaco et al Colaço et al. (2019) (C19, hereafter)
carried out a similar analysis as in Galli (2013) by looking for variations in
the ratio of $Y_{SZ}$ to $Y_{X}$ as a function of redshift. The main
difference with respect to Galli (2013), is that they assumed (similar to
Holanda et al. (2016a, b); Holanda et al. (2017)) that a variation in $\alpha$
also leads to a violation of the CDDR. Motivated by runaway dilaton models
Damour et al. (2002a, b); Martins et al. (2015), C19 thereafter modeled the
variation in $\alpha$ as a logarithmic function of the redshift, and used the
Planck Early SZ catalog to constrain this variation. All these searches have
failed to find any evidence for the variation of $\alpha$.

In this work, we first implement the same procedure as Galli (2013), to
constrain the model-independent variation of $\alpha$. Then, similar to C19,
we look for logarithmic variations of $\alpha$ as a function of redshift, by
considering South Pole Telescope-selected clusters (with joint X-ray and SZ
observations) in the redshift range 0.2 $\leq$ z $\leq$ 1.5. This spans a
wider redshift range than in C19, which looked for clusters with $z<0.5$. With
the same dataset, we also look for a spatial variation in $\alpha$.

This paper is organized as follows. The basic theory behind the X-ray and SZ
observables used for the analysis is presented in Sect. II. Our model for the
variation of $\alpha$ is described in Sect. III. The dataset used for our
analysis is discussed in Sect. IV. A model-independent constraint on variation
of $\alpha$ can be found in Sect. V. Our results for time-varying (using
runaway dilaton model) and spatial searches for $\alpha$ can be found in Sect.
VI and Sect. VII respectively. We conclude in Sect. VIII.

##  II Method

The basic observable which we use for this work is the dimensionless ratio of
$Y_{SZ}$ to $Y_{X}$ (after suitable scalings). More precise definitions will
be given in forthcoming subsections. Both $Y_{SZ}$ and $Y_{X}$ are different
proxies for the thermal energy of the cluster and are proportional to cluster
mass. blackThey also provide insights on the amount of inhomogeneity and
clumping of the intra-cluster medium Planelles et al. (2017). Therefore,
studying the relation between the two is important for both astrophysics and
cosmology. Consequently, a large number of works Bonamente et al. (2008);
Andersson et al. (2011); Bonamente et al. (2012); Rozo et al. (2012, 2014a,
2014b); Liu et al. (2015); Chiu et al. (2018); De Martino and Atrio-Barandela
(2016); Biffi et al. (2014); Planelles et al. (2017); Bender et al. (2016);
Zhu et al. (2019); Pratt and Bregman (2020); Henden et al. (2018) have studied
the scaling relations between $Y_{X}$ and $Y_{SZ}$ using both data and
simulations to characterize the systematics in the mass determination as well
as any departures from self-similarity. Here, we use this ratio to test for a
variation in $\alpha$.

###  II.1 SZ effect

The thermal SZ effect refers to the distortion in the CMB spectrum due to the
inverse Compton scattering between the hot gas present in the intracluster
medium and the CMB photons Sunyaev and Zeldovich (1972); Birkinshaw (1999);
Carlstrom et al. (2002); Mroczkowski et al. (2019). Since the SZ effect is a
spectral distortion, which is independent of redshift, it has become a
powerful probe to detect galaxy clusters upto very high redshifts. In the past
decade, there have been three primary experiments: South Pole Telescope (SPT)
Carlstrom et al. (2011), Atacama Cosmology Telescope Swetz et al. (2011), and
the Planck satellite Planck Collaboration et al. (2016), which have carried
out a blind SZ survey to detect galaxy clusters upto very high redshifts. We
now discuss how the SZ signal depends on $\alpha$, for which we follow the
same outline as in C19.

The distortion measured in SZ experiments is proportional to a parameter,
called the Compton parameter $y$, which is given by Birkinshaw (1999);
Carlstrom et al. (2002),

$y=\frac{{\sigma_{T}}k_{B}}{m_{e}c^{2}}\int n_{e}Tdl$ (1)

Here, $k_{B}$ is the Boltzmann constant, $c$ is the speed of light, $m_{e}$ is
mass of the electron, $n_{e}$ is the number density of electrons, $T$ is the
electron temperature, and $\sigma_{T}$ is the Thompson scattering cross
section which can be written in terms of $\alpha$ as,

$\sigma_{T}=\frac{8\pi}{3}\left(\frac{\epsilon^{2}}{m_{e}c^{2}}\right)^{2}=\frac{8\pi}{3}\left(\frac{\hbar^{2}\alpha^{2}}{m_{e}^{2}c^{2}}\right)$
(2)

Now, the integrated Compton parameter $Y_{SZ}$ (over the solid angle of a
galaxy cluster) can be written as,

$Y_{SZ}\equiv\int_{\Omega}y{d\Omega},$ (3)

where ${d\Omega}=dA/D_{A}^{2}$, blackand $D_{A}$ is the angular diameter
distance. Assuming an ideal gas equation of state given by $P=n_{e}k_{B}T$,
where $P$ is the pressure of the intracluster medium, one can combine Eq. 1
and Eq. 3 to obtain,

$Y_{SZ}D_{A}^{2}\equiv\frac{\sigma_{T}}{m_{e}c^{2}}\int PdV$ (4)

Since $\sigma_{T}\propto\alpha^{2}$ (cf. Eq. 2), we get

$Y_{SZ}D_{A}^{2}\propto\alpha^{2}.$ (5)

If we model the variation in $\alpha$ as $\alpha(z)\equiv\alpha_{0}\phi(z)$,
where $\alpha_{0}$ is the present value of $\alpha$, the fractional variation
in $\alpha$ is given by,

$\frac{\Delta\alpha}{\alpha_{0}}=\phi(z)-1$ (6)

Eq. 5 can then be re-written as

$Y_{SZ}D_{A}^{2}\propto\phi(z)^{2}$ (7)

###  II.2 X-rays

At high temperatures, gas present in the intracluster medium emits in X-rays
mainly through the thermal bremsstrahlung process Allen et al. (2011). The
thermal energy of the gas can be parameterized by the $Y_{X}$ parameter
Kravtsov et al. (2006), which can obtained from X-ray surface brightness
observations and is given by,

$Y_{X}=M_{g}(R)T_{X}$ (8)

where $T_{X}$ is the X-ray temperature of the gas and $M_{g}(R)$ is mass of
the gas within the radius $R$. Kravtsov et al Kravtsov et al. (2006) have
shown that $Y_{X}$ is strongly correlated with the cluster mass with an
intrinsic scatter of about 5-8%, and hence is a very robust proxy for the
cluster mass.

The gas mass $M_{g}(R)$ is scales with $\alpha$ as Gonçalves et al. (2012);
Galli (2013); Colaço et al. (2019),

$M_{g}(<\theta)\propto\phi(z)^{-3/2}D_{L}D_{A}^{3/2},$ (9)

where $D_{L}$ is the luminosity distance. From Eq. 9, we can see that
$M_{g}(R)$ depends upon both $D_{L}$ and $D_{A}$. Both of them are linked by
CDDR, $D_{L}=(1+z)^{2}D_{A}$ Etherington (1933). As shown in Hees et al.
(2014); Gonçalves et al. (2020), any variation in $\alpha$ is intertwined with
the violation of CDDR, which must be included in any searches for variation of
$\alpha$ Colaço et al. (2019). Similar to C19, if we parameterize a violation
of CDDR using $D_{L}$ = $\eta(z)$ $(1+z)^{2}$ $D_{A}$, then the variation of
$Y_{X}$ scales according to,

$Y_{X}\propto\phi(z)^{-3/2}\eta(z)$ (10)

As argued in C19 (and references therein) for modified theories of gravity
where the scalar field couples to the electromagnetic sector and breaks the
equivalence principle, $\alpha(z)$ is related to $\eta(z)$ according to
$\alpha(z)=\eta(z)^{2}$ Colaço et al. (2020); Hees et al. (2014). Therefore,
Eq. 10 can be recast as

$Y_{X}\propto\phi(z)^{-1}$ (11)

##  III Model for $\alpha$ variation

The dimensionless ratio of $Y_{SZ}D_{A}^{2}$ to $Y_{X}$ can be combined from
Eq. 4 and Eq. 10 Galli (2013)

$\frac{Y_{SZ}D_{A}^{2}}{Y_{X}}=C_{XSZ}\frac{\int n_{e}(r)T(r)dV}{T(R)\int
n_{e}(r)dV},$ (12)

where

$C_{XSZ}=\left(\frac{\sigma_{T}}{m_{e}m_{p}\mu_{e}c^{2}}\right)\\\ \approx
1.416\times 10^{-19}\left(\frac{Mpc^{2}}{M_{\odot}keV}\right)$ (13)

The numerator and denominator in Eq. 12 are different proxies for the thermal
energy of the cluster. Simulations show that the ratio in Eq. 12 is expected
to be a constant with redshift with a scatter of approximately 5-15% Stanek et
al. (2010); Fabjan et al. (2011); Kay et al. (2012); Biffi et al. (2014);
Planelles et al. (2017). For clusters with isothermal or universal temperature
profile Loken et al. (2002), this ratio should be constant with redshift and
equal to unity Galli (2013); Colaço et al. (2019). Therefore, assuming no new
Physics, there should not be any variation for this ratio as a function of
redshift.

Following C19, we combine Eq. 7 and Eq. 11 to rewrite the ratio in Eq. 12 in
terms of the variation in $\alpha$ as,

$\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}}=C\phi^{3}$ (14)

where $C$ is an unknown constant, which encapsulates all the cluster
astrophysics in the ratio in Eq. 12. A value of $C$ close to one further
indicates that the cluster gas has an isothermal profile. Similar to C19 (see
also Holanda et al. (2016a, b); Holanda et al. (2017)), we use runaway dilaton
models Damour et al. (2002a, b); Martins et al. (2015) to parameterize the
variation in $\phi$ as

$\phi(z)=1-\gamma\ln(1+z)$ (15)

In the dilaton model proposed by Damour et al Damour et al. (2002a, b), Eq. 15
was derived by assuming that the velocity of the dilaton field is constant in
both matter and dark-energy eras (with different constants for both). With
these assumptions, Eq. 15 is valid only in the dark energy dominated era and
needs to be augmented by an additional constant term in the matter dominated
era. However, De Martinis et al Martins et al. (2015); Martins (2017) have
shown that Eq. 15 can also be obtained by directly integrating the Friedman
and scalar field equations for the runaway dilaton model, followed by
linearizing the field evolution. As pointed out in Martins et al. (2015), this
approximation is valid for low redshifts, but breaks down at large redshifts
(close to $z=5$). At these redshifts, the variation with respect to $z$ is
model-dependent, depending on the couplings of dilaton field to baryons, dark
matter, and dark energy. These variations have been calculated for the
different couplings, and can be found in Martinelli et al. (2017) and can be
parametrized by polynomial functions of $z$.

Since our sample extends only to redshift of 1.5, we use Eq. 15 to constrain
any putative variation in $\alpha$, as this function has also been used in
previous works. However, it is straightforward to extend this analysis to any
other parametric model for the variation of $\alpha$. From Eq. 15, the
fractional variation in $\alpha$ can be written as

$\frac{\Delta\alpha}{\alpha}=-\gamma\ln(1+z)$ (16)

Therefore, the relation in Eq. 14 now becomes

$\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}}=C(1-\gamma\ln(1+z))^{3}$ (17)

We now fit for this relation using the SPT data. blackWe note that we have
used $D_{A}$ (needed for evaluating Eq 17) using the Planck 2018 $\Lambda CDM$
cosmological parameters Planck Collaboration et al. (2020) and evaluated using
the astropy Astropy Collaboration et al. (2018) module. However, that the
Friedman equations themselves get corrections due to the
$H^{2}{\phi^{\prime}}^{2}$ term in the presence of a dilaton field Damour et
al. (2002a, b); Martins (2017), where the prime indicates the derivative with
respect to the logarithm of the scale factor. As shown in Martins (2017),
$\frac{1}{H}\frac{\dot{\alpha}}{\alpha}\sim\frac{\alpha_{had}}{40}\phi^{\prime}$,
where $\alpha_{had}$ is the coupling of the dilaton field to hadronic matter.
Based on reported limits of $\frac{\dot{\alpha}}{\alpha}<10^{-17}$/yr from
atomic clocks Godun et al. (2014), one expects negligible corrections to
Friedmann equations from the dilaton couplings, over the redshift range used
for the SPT sample. Therefore, it is reasonable to neglect the contribution of
the dilaton field to Friedmann equation, and to use the standard Friedmann
equation to compute angular diameter distance.

Figure 1: The observed values of $Y_{SZ}$ $D_{A}^{2}$ / $C_{XYZ}$ $Y_{X}$ for
the SPT galaxy cluster sample together with X-ray measurements from XMM-Newton
observations Bulbul et al. (2019).

##  IV SPT Cluster Sample

For this work, we use joint X-ray and SZ data for 58 SPT galaxy clusters
Bulbul et al. (2019). The SZ data have been obtained by SPT, which is a 10 m
telescope at the South Pole, that has imaged the sky at three different
frequencies, viz. 95 GHz, 150 GHz and 220 GHz Carlstrom et al. (2011). SPT
carried out a 2500 square degree survey between 2007 and 2011 to detect galaxy
clusters using the SZ effect. This SPT-SZ survey detected 516 galaxy clusters
with mass threshold of $3\times 10^{14}M_{\odot}$ upto redshift of 1.8 Bleem
et al. (2015). Detailed properties of the SPT clusters (confirmed until 2015)
are discussed in Bleem et al. (2015). Their redshifts have been obtained using
a dedicated optical follow-up campaign, consisting of pointed spectroscopic
and photometric observations Song et al. (2012); Ruel et al. (2014); Bayliss
et al. (2017), and also using data from surveys mapping out contiguous regions
of the sky such as BCS Desai et al. (2012) and DES Saro et al. (2015). For
every SPT cluster, $Y_{SZ}$ has been estimated by averaging in a cylindrical
volume within an aperture radius of 0.75′ Bleem et al. (2015). These
cylindrically averaged $Y_{SZ}$ values for the confirmed SPT clusters are
reported in Bleem et al. (2015) and recently updated in Bocquet et al. (2019).
In order to search for a variation of $\alpha$, one needs to compare the
$Y_{X}$ measured within a radius $R$, with the $Y_{SZ}$ obtained by averaging
over a spherical volume within the same radius $R$ Galli (2013); Holanda et
al. (2016a). Since the SPT $Y_{SZ}$ are obtained from cylindrical averaging
over an angular aperture of 0.75′, we first need to convert these values to
spherically averaged $Y_{SZ}$ Arnaud et al. (2010) within the same radius at
which $Y_{X}$ was measured. To do this conversion we follow the prescription
in Arnaud et al. (2010) (see also Melin et al. (2011)). The cylindrical and
spherically averaged $Y$ can be related using:

$\displaystyle Y_{cyl}(R_{1})$ $\displaystyle=$ $\displaystyle
Y_{sph}(R_{b})-\frac{\sigma_{T}}{m_{e}c^{2}}\int_{R_{1}}^{R_{b}}4\pi
P(r)\sqrt{r^{2}-R_{1}^{2}}rdr$ (18) $\displaystyle Y_{sph}(R_{2})$
$\displaystyle=$
$\displaystyle\frac{\sigma_{T}}{m_{e}c^{2}}\int_{0}^{R_{2}}4\pi P(r)r^{2}dr$
(19)

where $Y_{cyl}$ is the SZ signal within a cylindrical aperture of radius
$R_{1}$; $R_{b}$ is the radial extent of the cluster; and $P(r)$ is the gas
pressure in the intra-cluster medium. Similarly, $Y_{sph}$ is the
corresponding integrated SZ flux within a sphere of radius $R_{2}$. To do the
conversion, we use the Universal Pressure Profile to model $P(r)$ Arnaud et
al. (2010). We choose $R_{b}=10R_{500}$. Since the SPT $Y_{X}$ are reported at
$R_{500}$ and $Y_{SZ}$ values calculated for an aperture of 0.75′, we assume
$R_{1}=0.75^{\prime}D_{A}$ in Eq. 18 and $R_{2}=R_{500}$ in Eq. 19. Therefore,
from Eq. 18 and Eq. 19, we can estimate the ratio of $Y_{sph}(R_{500})$ to
$Y_{cyl}(0.75^{\prime}D_{A})$, which is then used to estimate
$Y_{sph}(R_{500})$. As pointed out by C19, $Y_{SZ}$ values could also be
affected by a modification of the adiabatic evolution of the CMB temperature
as a function of redshift. The SPT collaboration has looked for such a
violation, and their results are consistent with the standard model of CMB
temperature variation with redshift Saro et al. (2014).

SPT has also been undergoing a massive X-ray followup campaign using both the
Chandra and XMM-Newton telescopes. Some details of the Chandra-based followups
can be found in McDonald et al. (2013, 2014). Previous studies for the scaling
relation between SPT $Y_{SZ}$ and $Y_{X}$ (with Chandra measurements) as well
as other Physics results have been reported in Andersson et al. (2011); Semler
et al. (2012); McDonald et al. (2013, 2014, 2017); Chiu et al. (2018);
McDonald et al. (2019).

Here, we shall use XMM-Newton followup observations of 58 SPT clusters
described in Bulbul et al Bulbul et al. (2019). These XMM-Newton observations
have been obtained using a combination of targeted X-ray followup programs,
led by SPT collaboration members and also other non-SPT based small programs.
The redshift range of this sample is given by 0.2 $<$ $z$ $<$ 1.5. The
exposure time for each cluster is $\mathcal{O}(10)$ ks. More details of the
observations and XMM-Newton data reduction can be found in Bulbul et al.
(2019). $Y_{X}$ values for all the 58 clusters have been provided at $R_{500}$
in Bulbul et al. (2019). blackThe temperatures used for SPT $Y_{X}$
evaluations are emission-weighted temperatures Bulbul et al. (2019). Note that
these Bulbul et al. (2019) provide both the core-excised as well as the core-
included $Y_{X}$ data for all the 58 clusters. For our analysis, we combined
these blackcore-included $Y_{X}$ estimates along with the spherically averaged
$Y_{SZ}$ (converted from those in Bleem et al. (2015)), to construct the ratio
$(Y_{SZ}$ $D_{A}^{2}$/ $C_{XYZ}$ $Y_{X})$ as a function of redshift. This
variation is shown in Fig. 1. By eye, we see no trends with redshift, thus
indicate that the ratio is constant as a function of redshift. We quantify
this in the next sections.

##  V Model-independent constraint

We first report a model-independent constraint on the variation of $\alpha$,
along the same lines as Galli Galli (2013), by using the fact that the ratio
of SZ to X-Ray Compto-ionization ratio is constant as a function of redshift.

Similar to Galli (2013), we use the modified weighted least-squares method
(MWLS) to calculate the new weighted mean for the ratio and its intrinsic
scatter. From the ordinary weighted least-squares method we find from the
measurements in Fig. 1 that $\ln(\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}})=0.05\pm
0.02$, with $\chi^{2}/dof=237/57$. Therefore, using the MWLS, we keep both the
weighted mean and the intrinsic scatter as free parameters. For each value of
the new weighted mean, the intrinsic scatter is chosen so that the reduced
$\chi^{2}$ is equal to one. Among these values, the weighted mean which gives
the minimum value for the intrinsic scatter was chosen as the new weighted
mean. Using this procedure, we get
$\ln(\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}})=0.046\pm 0.02$, with an intrinsic
scatter of 29%. This corresponds to
$(\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}})=1.05\pm 0.02$

The uncertainty in the variation of $\alpha$ is then given by Galli (2013)

$\frac{\sigma(\alpha)}{\alpha_{0}}=\frac{1}{X}\frac{\sigma(Y_{SZ}D_{A}^{2}/Y_{X}C_{XSZ})}{(Y_{SZ}D_{A}^{2}/Y_{X}C_{XSZ})}$
(20)

where $X$ is equal to 3 or 3.5, depending on whether the theoretical model for
$\alpha$ variation violates CDDR or not. From this we get
$\frac{\sigma(\alpha)}{\alpha_{0}}$ $\sim$ 0.7% (violation of CDDR) or 0.6%
(no violation of CDDR). This is about the same level of precision as that
obtained using the Planck ESZ sample Galli (2013). Similar to Galli (2013),
this limit also assumes no astrophysical evolution of the X-ray to SZ Compto-
ionization ratio. However, here we have included the error in cosmological
parameters while calculating the observed Compto-ionization ratio, unlike
Galli (2013).

##  VI Constraints on runaway dilaton models

To look for variations in $\alpha$, we fit the observed ratio (Eq. 14) for the
SPT cluster sample to the model in Eq. 17, and obtain the best-fit values of
$C$ and $\gamma$ by maximization of the log-likelihood. The log-likelihood
function ($\mathcal{L}$) used to test for the variation in $\alpha$ can be
written as,

$-2\ln\mathcal{L}=\sum_{i}\ln
2\pi{\sigma_{i}^{2}}+\sum_{i=1}^{N}\frac{(\phi_{obs,i}-C(1-\gamma\ln(1+z))^{3})^{2}}{\sigma_{i}^{2}}$
(21)

where $\phi_{obs,i}$ denote the observed values of ratio in Eq. 12 for SPT
clusters, $N$ is the total number of clusters, and $\sigma_{i}$ denotes the
total error calculated as

$\sigma_{i}^{2}=\sigma_{\phi}^{2}+\sigma_{int}^{2}$ (22)

Here, $\sigma_{\phi}$, which is the error in $\phi_{obs,i}$ is obtained by
propagating the error in $Y_{X}$, $Y_{SZ}$, and $D_{A}$. We also added an
intrinsic scatter term ($\sigma_{int}$) in quadrature, which we keep as a free
parameter, while maximizing the log-likelihood. blackWe note that the
intrinsic scatter is often added as a free parameter in linear regression
problems, since the relation between $y$ and $x$ may not always be 100%
deterministic even for noise-free observations, for a variety of reasons Kelly
(2007); Hogg et al. (2010). The use of intrinsic scatter is ubiquitous in
galaxy cluster astrophysics to quantify how well a given scaling relation is
obeyed (eg. Pratt and Bregman (2020); Liu et al. (2015); Chiu et al. (2020)).
A small value for this quantity indicates negligible deviations from the
scaling relation and vice-versa. Adding the intrinsic scatter term as a free
parameter to the observational errors is similar to the method adopted in Tian
et al. (2020), who used a similar expression to test the tightness of the
radial acceleration relation for galaxy clusters. We note however that our
treatment of systematic error differs from C19, who used a fixed intrinsic
scatter of 17%. For calculating $D_{A}$ and its error, as mentioned earlier,
we used the Planck 2018 cosmological parameters Planck Collaboration et al.
(2020) ($H_{0}$ = $67.4\pm 0.5$ km/sec/Mpc and $\Omega_{m}$ = $0.315\pm
0.007$).

Figure 2: Constraints on $\gamma$ , $C$ (defined in Eq. 14) and $\sigma_{int}$
(defined in Eq. (22)), which are obtained by maximizing the likelihood in Eq.
21 using the emcee module. The plots along the diagonal are the 1-D
marginalized likelihood distributions. The red line represents the mean value
of the distribution quoted right above the histogram. The off-diagonal plots
are the two-dimensional marginalized constraints showing the 68%, 95%, and 99%
credible regions, obtained using the Corner module. Our results are consistent
with no variation of $\alpha$ and an isothermal profile for clusters.

We maximize the likelihood using the emcee MCMC sampler Foreman-Mackey et al.
(2013). The 68%, 95%, and 99% confidence level plots along with the
marginalized one-dimensional likelihoods for each of the three parameters are
displayed in Fig. 2. The best fits which we get are $\gamma$ =
$-0.046_{-0.1}^{+0.097}$ and $C$ = $0.95_{-0.11}^{+0.12}$, with an intrinsic
scatter of about 26%. Therefore, our results indicate that there is no
evolution of $\gamma$ with redshift, implying no variation in $\alpha$.
Furthermore, since the best-fit value of $C$ is consistent with 1.0 within
1$\sigma$ (similar to C19), we conclude that our cluster sample can be
adequately described by the isothermal temperature profile.

blackWe note that simulations predict an intrinsic scatter of 5-15% Stanek et
al. (2010); Fabjan et al. (2011); Kay et al. (2012); Biffi et al. (2014);
Planelles et al. (2017). Our estimated intrinsic scatter of 26% is larger than
these estimates. Other observational results for the intrinsic scatter in
$Y_{X}-Y_{SZ}$ relation range from negligible scatter of 5% Rozo et al.
(2012), to scatters between 15-20% Planck Collaboration et al. (2011); Zhu et
al. (2019) which are comparable to our estimate, and also extremely large
scatter of 60-90% De Martino and Atrio-Barandela (2016); Pratt and Bregman
(2020). These estimates also depend upon the sample used (including
significant differences between cool-core and non cool core Zhu et al.
(2019)), as well as the regression method employed. We discuss some possible
reasons for our large scatter compared to the expectations from simulations.

black Biffi et al. (2014) showed using the MUSIC dataset of simulated clusters
that the emission-weighted temperatures show deviations of about 20%, compared
to spectroscopic-like or mass-weighted temperatures, which show deviations of
only about 10% and 5% respectively. The aforementioned work also showed that
the $Y_{X}-Y_{SZ}$ scaling relation shows a scatter of 5% for $Y_{X}$ computed
using mass-weighted temperatures, which is comparable to the deviations with
respect to the true temperatures. Therefore, one would expect a scatter of
about 20%, if $Y_{X}$ was estimated using emission-weighted temperatures.
Since the $Y_{X}$ estimates were obtained using the emission-weighted
temperatures, that could be one possible reason for our large scatter. Other
possibilities could be because of the use of Universal Pressure Profile Arnaud
et al. (2010), which is known to overestimate the thermal SZ signal by upto
15-20% depending upon the angular aperture De Martino and Atrio-Barandela
(2016). However, a detailed investigation of these and other causes for the
large intrinsic will be deferred to future works.

Table. 1 shows the constraints on $\gamma$ from different cosmological
observations along with our results. We note that the most stringent bound on
$\gamma$ are however obtained from tests of violations of weak equivalence
principle and are $\mathcal{O}(10^{-6})$ Martins and Vacher (2019).

Data Set | $\gamma$ | Reference
---|---|---
Only Gas Mass Fractions | $+0.065\pm 0.095$ |  Holanda et al. (2016a)
Angular Diameter Distance + SNe Ia | $-0.037\pm 0.157$ |  Holanda et al. (2016b)
Gas Mass Fractions + SNe Ia | $+0.010\pm 0.030$ |  Holanda et al. (2017)
${Y_{SZ}D_{A}^{2}}/{C_{XYZ}Y_{X}}$ | $-0.15\pm 0.10$ |  Colaço et al. (2019)
Strong Gravitational Lensing + SNe Ia | $-0.013_{-0.09}^{+0.08}$ |  Colaço et al. (2020)
$\mathbf{{Y_{SZ}D_{A}^{2}}/{C_{XYZ}Y_{X}}}$ | $\mathbf{-0.046_{-0.104}^{+0.097}}$ | This work
Table 1:  A summary of the current astrophysical/cosmological constraints from
different works (including the result from this work) on a possible time
evolution of $\alpha$ for a class of dilaton runaway models ($\phi$ =
1-$\gamma$ $\ln$(1 + $z$)).

##  VII Search for Dipole variation of $\alpha$

To confirm the dipole variation of $\alpha$ found using quasar data Webb et
al. (2011), Galli Galli (2013) also searched for a spatial variation of
$\alpha$. Due to their different redshift and spatial distributions compared
to quasars, clusters offer a complementary probe to test the claim in Webb et
al. (2011). The model for the dipole variation posited in Galli (2013) is
given by

$\frac{\Delta\alpha}{\alpha}=Arcos(\theta)$ (23)

where $A$ represents the dipole amplitude, $\theta$ is the angular separation
between clusters and best-fit dipole position found in Webb et al. (2011), $r$
is the lookback time which is given by,

$r=\int_{0}^{z}\frac{cdz^{{}^{\prime}}}{H(z^{{}^{\prime}})}$ (24)

For a flat $\Lambda$CDM universe, $H(z^{{}^{\prime}})$ is given by:

$H(z^{{}^{\prime}})=H_{0}\sqrt{\Omega_{m}(1+z^{{}^{\prime}})^{3}+\Omega_{\Lambda}}$
(25)

Therefore, Eq. 12 becomes

$\left(\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}}\right)^{1/3}=a+brcos(\theta),$
(26)

where $a$ = $Y_{0}^{1/3}$ and $b$ = $A$ $Y_{0}^{1/3}$; $Y_{0}$ refers to a
fixed reference value for $\left(\frac{Y_{SZ}D_{A}^{2}}{Y_{X}C_{XSZ}}\right)$;
and $\cos(\theta)$ is the angular position between every SPT cluster and the
best-fit dipole position as reported in Webb et al. (2011). Note that we have
$1/3$ in the exponent of Eq. 26, unlike $1/3.5$ in Galli (2013), as we have
assumed (similar to the analysis in Sect. VI) that a variation in $\alpha$
also leads to a violation of CDDR. Fig. 3 shows the ratio
$\frac{Y_{SZ}D_{A}^{2}}{C_{XYZ}Y_{X}}$ as a function of $rcos(\theta)$ for the
SPT cluster sample.

To get the best-fit value of $Y_{0}$ and $A$, we maximize a log-likelihood
similar to that in Eq. 21, after including an intrinsic scatter. Again, we
have used the emcee MCMC sampler for the optimization. From this analysis, we
obtained $Y_{0}$ = $1.0_{-0.045}^{+0.047}$ and $A$ =
$-0.003_{-0.003}^{+0.003}$ $(GLy^{-1})$ with an intrinsic scatter of about
26%. The likelihood distributions of our parameters (after assuming violation
of CDDR) along with 2-D marginalized contours are displayed in Fig. 4. To
compare our results with Galli (2013), We also repeated our analysis for
dipole variation of $\alpha$, assuming no violation in CDDR. So we redid the
analysis by replacing the exponent of 3 in Eq. 26 with 3.5. With this
assumption, we find $Y_{0}$ = $1.0_{-0.046}^{+0.048}$ and $A$ =
$-0.002_{-0.003}^{+0.002}$ $(GLy^{-1})$ with an intrinsic scatter of about
26%. Therefore, our results are also consistent with no dipole variation. A
comparison of parameter $A$ with the previous studies in literature is
summarized in Table. 2.

Data Set | $A(GLy^{-1})$ | Reference
---|---|---
Quasar | ($1.1\pm 0.25$) $\times$ $10^{-6}$ |  Webb et al. (2011)
Planck ESZ Clusters | ($-5.5\pm 7.9$ )$\times$ $10^{-3}$ |  Galli (2013)
SPT Clusters | ($\mathbf{-3.0\pm 3.0)\times 10^{-3}}$ | This work
SPT Clusters (without CDDR violation) | ($\mathbf{-2.0_{-3.0}^{+2.0})\times 10^{-3}}$ | This work
Table 2: . A comparison of different studies showing the dipole variation of
$\alpha$. We note that the study in Galli (2013) using Planck ESZ clusters
does not assume a violation of CDDR.  Figure 3: $Y_{SZ}$ $D_{A}^{2}$ /
$C_{XYZ}$ $Y_{X}$ as a function of $rcos(\theta)$ for the SPT galaxy cluster
sample using XMM-Newton observations Bulbul et al. (2019). Figure 4:
Constraints on $Y_{0}$, $A$ and $\sigma_{int}$ (defined in Eq. 26) used to
model a spatial variation in $\alpha$. Eq. 26 assumes that CDDR is also
violated when $\alpha$ varies. The plots along the diagonal are the one-
dimensional marginalized likelihood distributions. The off-diagonal plots are
the two-dimensional marginalized constraints showing the 68%, 95%, and 99%
credible regions.

##  VIII Conclusions

We search for a variation in $\alpha$ as a function of redshift, using the
dimensionless ratio of the integrated Compto-ionization to its X-ray
counterpart ${Y_{SZ}D_{A}^{2}}/{C_{XYZ}Y_{X}}$. For this study, we use data
from 58 SPT-SZ selected galaxy clusters Bleem et al. (2015) in the redshift
range 0.2 $\leq$ $z$ $\leq$ 1.5, along with X-ray measurements from XMM-Newton
Bulbul et al. (2019). We first do a model-independent test for the variation
of $\alpha$ (similar to Galli (2013)), by using the fact that this ratio is
constant as a function of redshift. The variation of $\alpha$ is constrained
to within 0.6-0.7% in the redshift range ($0.2<z<1.5$).

Then, similar to C19, we assumed that a variation of $\alpha$ also leads to a
violation of cosmic distance duality relation, and parameterized the variation
in $\alpha$ as a logarithmic function of redshift and encoded in the parameter
$\gamma$ (defined in Eq. 16). This logarithmic variation is characteristic of
runaway dilaton models Damour et al. (2002a, b); Martins et al. (2015).

When we fit the $Y_{SZ}$ to $Y_{X}$ ratio for the SPT-SZ data to this model,
our best-fit values are given by $\gamma=-0.046_{-0.1}^{+0.097}$ and
$C=0.95_{-0.11}^{+0.12}$. Therefore, our results show no significant variation
of the $\alpha$ with redshift, in accord with previous results using clusters.
Similar to C19, our inferred value of $C$ also indicates that our samples are
well approximated by the isothermal temperature profile. A comparison of our
results with previous results in literature are summarized in Table 1.

Finally, similar to Galli (2013), we also search for a dipole variation of
$\alpha$ with the best-fit direction equal to the value found using quasar-
based searches Webb et al. (2011). The model we use to test for dipole
variation is given in Eq. 26. Our results show no such spatial variation of
$\alpha$, in agreement with previous studies (cf. Table 2).

The first generation SZ surveys from the SPT and ACT telescopes have been
superseded by SPTpol, SPT-3G Benson et al. (2014); Bleem et al. (2020) and
ACTpol Thornton et al. (2016) respectively, and should detect about an order
of magnitude more clusters upto redshift of 2.0. On the X-ray side, the
recently launched eROSITA satellite should discover upto 100,000 clusters
Hofmann et al. (2017). Therefore, further robust tests of both the temporal
and spatial variations of $\alpha$ should soon be possible.

## ACKNOWLEDGEMENT

KB acknowledges Department of Science and Technology, Government of India for
providing the financial support under DST-INSPIRE Fellowship program. blackWe
are grateful to the anonymous referee for useful and constructive feedback on
the manuscript.

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