aainstitutetext: Astrophysics Research Center of the Open University, The Open
University of Israel, Ra’anana, Israel

# Sunyaev-Zeldovich signatures from non-thermal, relativistic electrons using
CMB maps

Sandeep Kumar Acharya<EMAIL_ADDRESS>
###### Abstract

Relativistically energetic, non-thermal population of electrons can give rise
to unique CMB spectral distortion signatures which can be significantly
different from thermal Sunyaev-Zeldovich signal or $y$-distortion. These
signatures depend upon the spectrum of non-thermal electrons, therefore, a
detection can inform us about the existence and abundance of non-thermal
electrons in our universe. Using public CMB maps and data, we derive upper
limits on non-thermal $y$-parameter for a relativistic, power-law electron
distribution. With future CMB experiments, we may be in a position to detect
or put significantly tighter constraints on these signals which can affect our
understanding of non-thermal electron distributions in our universe.

##  1 Introduction

The Sunyaev-Zeldovich (SZ) effect ZS1969  has become an indispensable tool for
Cosmology as well as astrophysics in recent times due to the precise
observation of Cosmic Microwave Background (CMB) radiation wmap2011 ;
PlanckSZ_2016 ; SPT2022 ; ACT2023 . The SZ effect can be of two types, (i)
thermal SZ effect ZS1969 , which is produced due to the inverse Compton
scattering of CMB photons by hot, energetic electrons, (ii) kinetic SZ effect
SZ1980 , due to Doppler boosting by bulk flow of electrons. The thermal SZ
effect is predominantly produced at $z\lesssim 2$, once massive structures
such as galaxy clusters forms KS2002 . Due to the bulk motion of these galaxy
clusters, they also produce the kinetic SZ effect, however, it is subdominant
compared to the thermal SZ effect by an order of magnitude SBDH2010 . During
reionization epoch at $6\lesssim z\lesssim 10$, formation of ionized bubbles
also contribute to the total kinetic SZ effect GH1998 . The readers are
referred to B1999 ; M2019  for a detailed review on the SZ effect.

Gravitational collapse of matter leads to conversion of potential energy to
thermal energy which heats up the electrons to $\sim$ keV temperatures inside
massive objects such as galaxy clusters. CMB photons while passing through
these structures get upscattered with probability given by the optical depth
or $n_{e}\sigma_{\rm T}R$ where $n_{e}$ is the electron density, $\sigma_{\rm
T}$ is the Thomson cross section and $R$ is the line of sight dimension of the
object. On each collision, the thermal electrons impart a fraction of their
energy ($\sim\frac{k_{\rm B}T_{\rm e}}{m_{\rm e}{\rm c^{2}}}$) to the CMB
photons, where $T_{\rm e}$ is the electron temperature. This effect distorts
the CMB blackbody Fixsen1996  with a distinct spectral shape which is
classically termed as $y$-distortion ZS1969 . This characteristic shape help
us in distinguishing the thermal SZ signature from kinetic SZ as well as CMB
and other astrophysical foregrounds.

The spectral distortion shape is sensitive to the kinematics of Compton
scattering between the electron and the CMB photons. For $T_{\rm e}\lesssim$ 1
keV, the thermal SZ signature is independent of the temperature as Compton
scattering is non-relativistic. At higher temperatures, relativistic
corrections become important due to Klein-Nishina terms IKN1998 ;
Challinor1998 ; SS1998 . These computations can be done extremely fast using
the public code SZpack CNSN2012 . For a non-thermal relativistic electron
distribution, perturbative corrections terms are not sufficient and one needs
to resort to a more general formulation EK2000 ; C2008 . The non-thermal SZ
effect carries distinct information of the electron spectrum which produced it
in the first place. Therefore, this is an ideal tool to study the abundance of
relativistic electrons in our universe using CMB maps. Although, in this
paper, we only consider scattering of electrons with the CMB photons, in
general, these electrons will scatter all photons in the way which need not be
just CMB photons. Recent works have discussed scattering of Cosmic Infrared
Background SHB2022 ; AC2023  and radio background HC2021  photons with the
electrons residing inside galaxy clusters which can give rise to interesting
signatures.

Possible sources of non-thermal electrons include cosmic ray electrons and
relativistic electrons in radio galaxy lobes FR1969 ; S1974 ; BC1989 ; N1995 ;
ZPP2014 . The electron distribution is typically a power law CMBM2013  which
is inferred from radio and X-ray observations ITGHNHSM2009 . For a
relativistic electron with Lorentz factor $\gamma$, a photon with energy
$E_{\gamma}$ is boosted to $\gamma^{2}E_{\gamma}$ via inverse Compton
scattering BG1970 . For electrons with $\gamma\lesssim 10$, the boosted CMB
photons still show up in CMB band ($\sim 100-500$ GHz). Electrons in same
population but with $\gamma\sim 10^{4}-10^{5}$ can boost the CMB photons to
X-ray bands. These eletrons in extreme high energy tail also produce radio
signature via synchrotron emission. Therefore, a detection of non-thermal SZ
signature will nicely complement the radio and X-ray probes of relativistic
electrons. Even a non-detection can put interesting upper limits, rule out
some parameter space and can constrain our understanding of particle
acceleration processes. In a recent work CCAT2023 , the authors claim to
obtain competitive constraints on magnetic field inside clusters hosting radio
halos by constraining the abundance of non-thermal electrons.

In this paper, we aim to constrain non-thermal SZ signal for a few
representative electron spectra using Planck full sky CMB data Pl2016 . Using
a stacking approach, we obtain upper limits on the amplitude of the signal.
While the current limits may be uninteresting, with future CMB experiments, we
may be in a position to detect or set interesting upper limits on these
signals. This paper is organized as follows. In Sec. 2, we describe non-
thermal SZ signature in detail. We discuss the datasets used and matched
filtering method in Sec. 3 and 4 respectively. We detail our modelling of data
in Sec. 5 and explain our results in Sec. 6. We conclude with some potential
future directions in Sec. 7.

##  2 Sunyaev-Zeldovich (SZ) effect from non-thermal electron distribution

Figure 1: Spectral distortion shape $g_{NT}(x)$ with $x=\frac{{\rm
h}\nu}{k_{\rm B}T_{\rm cmb}}$ and for different $p_{\rm min}$. The spectral
index $\alpha_{p}$ is assumed to be 3. The dotted and solid lines correspond
to negative and positive part of $g_{NT}(x)$ respectively. We show thermal
$y$-distortion for reference.

The CMB photons while passing through a distribution of energetic electrons
can get upscattered by the electrons which imprints a characteristic
distortion signal on top of CMB. For a distribution of thermal electrons, the
CMB distortion signal is given by the well-known $y$-distortion (ZS1969, ).
The distortion signal is sensitive to electron energy distribution as it
depends upon the kinematics of scattering process involved which in this case
is inverse Compton scattering. The number of CMB photons which get upscattered
from energy $x^{\prime}$ to $x$ is given by,

$N(x^{\prime}->x)=P(t,p)\times
2\frac{(k_{\rm{B}}T_{\rm{cmb}})^{2}}{({hc})^{2}}\frac{x^{\prime
2}dx^{\prime}}{e^{x^{\prime}}-1},$ (1)

where $P(t,p)$ is the kernel of the inverse Compton scattering which captures
the kinematics of photon scattering with the electrons with electron momentum
$p$, $x$ is the dimensionless frequency of photons which is given by
$x=\frac{E_{\gamma}}{k_{\rm{B}}T_{\rm{cmb}}}$, $T_{\rm cmb}$ being the CMB
temperature and $t=\frac{x}{x^{\prime}}$. The kernel satisfies the relation,
$\int{\rm d}tP(t,p)=1$, since Compton scattering is a photon number conserving
process. The expression for $P(t,p)$ is given by (EK2000, ),

$P(t;p)=\frac{-3\left|(1-t)\right|}{32p^{6}t}[1+(10+8p^{2}+4p^{4})t+t^{2}]\\\
+\frac{3(1+t)}{8p^{5}}\left[\frac{3+3p^{2}+p^{4}}{\sqrt{1+p^{2}}}-\frac{3+2p^{2}}{2p}(2\operatorname{arcsinh}{p}-\left|\ln{t}\right|)\right],$
(2)

Exact expressions for Compton scattering kernels across all energy ranges can
be found in SCL2019 . By taking into account the number of photons which
escape or enter into a particular energy bin and summing over all $x$ bins, we
obtain the spectral shape of distorted CMB. The intensity of this distortion
is given by,

$I_{NT}=y_{NT}g_{NT}(x),$ (3)

where $I_{NT}$ is the change in intensity of distorted CMB with respect to the
undistorted Planckian spectrum. The amplitude of distortion signal is given by
$y_{NT}$ (where $NT$ stands for non-thermal) whose expression is given by,

$y_{NT}=\int\frac{p_{c}}{m_{\rm e}{\rm c}^{2}}\sigma_{\rm T}{\rm d}R,$ (4)

where $p_{c}$ is the pressure and ${\rm d}R$ is a length element along the
line of sight. The spectral shape $g_{NT}(x)$ is the spectral distortion shape
from a normalized electron distribution and as explained above is sensitive to
the non-thermal electron distribution. In this work, we assume the non-thermal
electrons to have a power-law shape with electron number density
$f_{e}(p)\propto p^{-\alpha_{p}}$ where $p$ is the dimensionless momentum with
$p=\sqrt{\gamma^{2}-1}$, $\gamma$ being the Lorentz factor. We assume the
spectrum to be extended from $p_{\rm min}$ to $p_{\rm max}\sim 10^{6}$ though
reasonable change to $p_{\rm max}$ will not affect results significantly. We
fix $\alpha_{p}$ to be 3 which is the average spectral index inferred from a
population of radio galaxies CMBM2013 .

In Fig. 1, we plot the dimensionless spectral distortion shape for few
different $p_{\rm min}$ and assuming $y_{NT}=1$. With increase in electron
energy, an electron can boost the CMB photons to higher energy, therefore,
moving the null points to the right. Since $y_{NT}$ captures the fraction of
energy injected to the scatterd CMB photons with respect to unscattered
photons, keeping $y_{NT}$ fixed and increasing $p_{\rm min}$, reduces the
amplitude of $g_{NT}$. This is beacuse a higher energy electron can impart a
larger fraction of energy to the CMB photons in one scattering. Therefore, as
the photons move to higher energy, their overall amplitude in the CMB band
reduces. It is interesting to note that below $\nu\sim 200$ GHz, the
distortion shapes looks like scaled version of each other. This is expected
since the target photon distribution, which is the CMB, is same for all the
electrons. Therefore, the negative part of distortion or removal of photons
mimics the spectrum of CMB. This also illustrates a non-trivial result that
one can not distinguish different non-thermal electron distribution using
multifrequency CMB experiments with frequency channels below $\sim 200$ GHz
and higher frequency channels are critical for such science case.

We should also point out that the intensity distortion of CMB ($I_{NT}$) is
the physical observable. Therefore, a detection or non-detection of non-
thermal spectral distortion will result in constraints which are degenerate
combination of $y_{NT}$ and $g_{NT}$. This is opposed to the case of thermal
distortion for which one assumes the distortion shape to be so-called
$y$-distortion (ZS1969, ) which is universal and has no parameter dependence
in it. Therefore, one can directly translate the constraints on the intensity
to thermal $y$-parameter, $y_{T}$, or just $y$ which is the notation used in
literature. However, things will not be that simple once we take into account
relativistic corrections which has a dependence of gas temperature
(Wright1979, ; Fabbri1981, ; Rephaeli1995, ; Challinor1998, ; Sazonov1998, ;
Itoh98, ) and is relevant for the environment inside the biggest galaxy
clusters in the universe (RBRC2019, ). Then, one needs the temperature
information to get constraint on thermal $y$-parameter.

In this work, we choose a few representative $p_{\rm min}=1$, 10 and 100. As
can be noted from Fig. 1, chances of detecting non-thermal SZ signature goes
down with increasing $p_{\rm min}$. However, such cases will generate signals
in radio, X-ray or may be even gamma ray bands which can provide complementary
constraints on radio galaxy parameters. Therfore, a non-detection in CMB bands
while a correlated detection in higher energy bands can constrain our
understanding of particle acceleration processes.

###  2.1 Sources of non-thermal electrons inside galaxy clusters

Compression of matter due to gravity during collapse of matter or structure
formation, releases energy which heat gas. Since structure formation is a
complex process, it is not impossible to believe that a significant portion of
electrons do not follow thermal or Maxwell-Boltzmann distribution and remain
relativistic with a power-law like spectrum. Besides, cosmic ray electrons and
emission of relativistic electrons from accreting supermassive black holes can
produce a non-negligible population of non-thermal electrons EK2000 ; C2008 ;
CMBM2013 . Non-thermal electrons in radio galaxy lobes produce radio emission
through synchrotron emission as well as X-rays via inverse Compton process on
the CMB photons CHHBBW2005 ; ITGHNHSM2009 . Therefore, a distortion in CMB
bands is expected from the same population of electrons. Studying the
correlated signatures of these non-thermal electrons across CMB, radio and
X-ray bands can potentially improve our knowledge of these energetic processes
significantly. In this paper, we assume that all non-thermal electrons are
hosted within the galaxy clusters and use the available CMB data to constrain
their abundance.

##  3 Datasets

###  3.1 Planck full-sky maps

In this work, we have used 2nd data release of Planck full sky map Pl2016 . We
have used the maps for 5 frequency channels (70,100,143,217 and 353 GHz) and
ignored the lower and higher frequency channels. The lower frequency channels
(30 and 44 GHz) have much lower angular resolution, 32 and 27 arcmin
respectively, and are expected to be contaminated by radio point sources.
Similarly, 545 and 857 GHz are severely contaminated by dust and infrared
point sources. The resolution of Planck frequency maps are tabulated in Table
1. The maps are provided in HEALPIX (healpix2005, ) format with $N_{\rm
side}=2048$. We have reduced the maps to $N_{\rm side}=1024$ for computational
simplicity. The maps between 30-353 GHz are provided in the units of ${\rm
K_{\rm cmb}}$ and we have converted to the units of ${\rm MJy/sr}$ using the
transformation,

$I_{\nu}=\frac{2{\rm h}\nu^{3}}{\rm
c^{2}}\frac{xe^{x}}{(e^{x}-1)^{2}}\frac{\Delta T}{T_{\rm cmb}},$ (5)

where $x=\frac{{\rm h}\nu}{k_{B}T_{\rm cmb}}$ and the maps are provided in the
format of $\frac{\Delta T}{T_{\rm cmb}}$ K. We have also used the ${\tt COM-
Mask-CMB-IQU-common-field}$ mask, in order to reduce the galactic
contamination.

###  3.2 Galaxy cluster catalogue

Our goal of this work is to constrain non-thermal SZ contribution which we
assume to be sourced inside the massive structures in the universe such as
galaxy clusters. Therefore, we use the Planck second Sunyaev-Zeldovich source
Union catalog PlanckSZ_2016  which provides a sample of $\sim 1650$ candidate
galaxy cluster objects. The Union catalogue combines the result of three
independent galaxy cluster extraction algorithms. Several of properties such
as angular coordinates, redshift, mass are listed in the catalogue, however,
in this work, we use only the angular coordinates of these galaxy clusters.
Our use of galactic mask reduces the number of galaxy clusters unaffected by
the mask which turns out to be $\sim 1400$. We use all of these objects in the
mask-free region for our stacking computation.

Frequency (GHz) | FWHM (arcmin)
---|---
70 | 13.31
100 | 9.66
143 | 7.27
217 | 5.01
353 | 4.86
Table 1: Characteristics of Planck maps used in this work.

##  4 Matched filtering method

In this work, we have used the matched filtering approach to extract the
temperature fluctuation along the direction of galaxy clusters. The matched
filtering approach was developed in the works of HT1996 ; HSHBDML2002 ;
MBD2006  and was subsequently applied to CMB data to extract galaxy cluster
catalogues, as an example see PlanckSZ_2016 . For more recent and interesting
applications, the readers are referred to SGEPS2017 ; EBCB2018 . In this
approach, we use a template for the signal we are looking for in order to
extract this signal from a map containing signal along with foregrounds and
noise. In general, one can use both a spatial and frequency template to make
the filtering process more effective. Indeed, it is typically assumed that the
hot gas inside the galaxy clusters give rise to thermal SZ signal
predominantly and its spectral independence from all other foregrounds along
with CMB as well as the spatial profile of clusters is used to identify the
galaxy cluster candidates. Here, we use only the spatial information in the
filtering step for individual galaxy cluster candidate and use frequency
template to isolate non-thermal SZ signal only after stacking all the objects
in our galaxy catalogue.

The signal at an angular location $x$ in sky can be modelled as,

$M(x)=s_{0}T_{\theta_{c}}(x-x_{0})+N(x),$ (6)

where $T_{\theta_{c}}$ is the spatial template, parameterized by $\theta_{c}$
and centered at $x_{0}$, $s_{0}$ is the magnitude of the signal component,
that we are after, at the center of the object and $N(x)$ is the noise which
refer to every other signal that we are not interested in including
instrumental noise. We want to construct a filter which when applied to this
data will reliably give us back the signal we are after or concretely,

$s_{0}=\int d^{2}x\Psi_{\theta_{c}}(x-x_{0})M(x)$ (7)

In the flat sky approximation, the filter in the fourier space is given by
(MBD2006, ),

$\Psi_{\theta_{c}}(k)=\sigma_{\theta_{c}}^{2}P^{-1}(k)T_{\theta_{c}}(k),$ (8)

where $\sigma_{\theta_{c}}=\left[\int
d^{2}k\frac{T_{\theta_{c}}(k)^{2}}{P(k)}\right]^{-1/2}$. In the flat-sky
approximation, there is a one-to-one correspondence between the angular
multipole $l$ and fourier mode $k$ as $l=2\pi k$. Also the noise power
spectrum can be identified with $C_{l}$ which can be be directly extracted
from the map. Technically, the noise power spectrum extracted from the data
includes the contribution from the signal that we are looking after. However,
in case of galaxy clusters, this contribution is negligible compared to the
CMB as well as the foregrounds. A recent work looks into a more accurate
computation using an iterative approach ZRCB2023  which can be important for
future CMB experiments.

The filter needs to convolved with the beam since one can not resolve sources
with size less than the width of the beam. Including the effect of beamwidth,
the expression for filter becomes,

$\Psi_{\theta_{c}}(l)=\frac{C_{l}^{-1}T_{\theta_{c}}(l)B_{l}}{\left[\frac{1}{2\pi}\int
ldl\frac{T_{\theta_{c}}(l)^{2}}{C_{l}}\right]}$ (9)

The beam is assumed to be gaussian with FWHM given in Table 1.

Figure 2: Matched filter in position or pixel space at $\nu=143$ GHz.

We assume that the non-thermal SZ signature is correlated with the shape of
galaxy clusters which is typically a spherical beta model. A signal correlated
with the shape of the cluster can be expressed as,

$s(\theta)=s_{0}\left[1+\left(\frac{\theta}{\theta_{c}}\right)^{2}\right]^{(1-3\beta)/2},$
(10)

where $\theta_{c}$ is the core radius and $\beta$ is assumed to be 1. The
analytic spherical transform of the profile with $s_{0}=1$ is given by
SGEPS2017 ,

$T_{\theta_{c}}(l)=2\pi\theta_{c}^{2}K_{0}(l\theta_{c}),$ (11)

where $K_{0}$ is the modified Bessel function of the second kind.

The galaxy clusters resides inside dark matter halos whose size are denoted by
$\theta_{500}$ which can be related to its mass and redshift as,

$\theta_{500}=6.997\left[\frac{h}{0.7}\right]^{-2/3}\left[\frac{(1-b)M_{500}}{3\times
10^{14}M_{\odot}}\right]^{1/3}E(z)^{-2/3}\left[\frac{D_{A}(z)}{\rm
500Mpc}\right]\hskip 14.22636pt(\rm arcmin),$ (12)

where $M_{500}$ is the mass of the halo, $E(z)=H(z)/H_{0}$ and $D_{A}(z)$ is
the angular diameter distance and $b$ is the hydrostatic bias
Planck2014_powerspectrum . For the redshift and mass distribution of galaxy
cluster catalogue, $\theta_{500}$ varies between $\sim 1$ to 10 arcmin. The
core size is related to halo size by the expression,
$\theta_{c}=c_{500}\theta_{500}$, where $c_{500}\sim 0.2$ using the
concentration-mass relation HK2003 ; DSKD2008 . In this work, we fix
$\theta_{c}=1$ arcmin to be the representative size of the galaxy clusters and
do not bin the objects by their sizes. We do not expect this choice to
significantly affect our results as we do a stacking computation which should
average out these effects.

In Fig. 2, we show the matched filter in pixel space which we use for the 143
GHz map. The filter is polar symmetric which is expected due to our choice of
cluster profile. The filter shows ringing pattern with positive and negative
values which helps in removing large scale foregrounds, uncorrelated to the
galaxy clusters. Essentially, our matched filter acts as a high pass filter.
We apply the filter to appropriate frequency maps and then stack all the
unmasked galaxy clusters. In principle, one can bin clusters in mass or
redshift. However, we expect the signal to be weak and binning will make it
even more weaker. Therefore, in this work, we have stacked all the clusters
without any binning. This is useful in providing upper limits and once we have
a detection, we can look for variation of signal as a function of mass or
redshift of galaxy clusters.

In Fig. 3, we show the spectral intensity of signal at the location of
clusters by stacking them. One can clearly see the stacked signal with high
significance at the central pixel. One can also see that the spectral
intensity is negative at $\nu<217$ GHz while positive at higher frequency. We
now move to fitting frequency template to the matched filtered data and
discuss our modelling in the next section.

(a) 70 GHz

(b) 100 GHz

(c) 143 GHz

(d) 217 GHz

(e) 353 GHz

Figure 3: Matched filtered, stacked intensity maps of galaxy clusters for the
different frequency channels. The numbers are shown in ${\rm MJy/sr}$ unit.

##  5 Data modelling

After filtering, the only surviving contribution from galaxy clusters are
thermal SZ and infrared emission correlated to the clusters, in addition to
non-thermal SZ, if any. Therefore, we fit a three component model to our data.
The thermal $y$-distortion is given by the analytic formula ZS1969 ,

$I_{T}=y_{T}\frac{2{\rm h}\nu^{3}}{\rm c^{2}}\frac{x{\rm e}^{x}}{({\rm
e}^{x}-1)^{2}}\left[x\frac{{\rm e}^{x}+1)}{{\rm e}^{x}-1}-4.0\right],$ (13)

where $x=\frac{{\rm h}\nu}{k_{B}T_{\rm cmb}}$ and $y_{T}$ is the amplitude
which is a free parameter. Similarly, the template for infrared component is
given by a modified blackbody function,

$I_{{\rm IR}}=A_{IR}\times 10^{-6}\frac{2{\rm h}\nu^{3}}{\rm
c^{2}}\left(\frac{\nu}{\nu_{0}}\right)^{\beta}\frac{1}{({\rm e}^{{\rm
h}\nu/k_{B}T_{d}}-1)}.$ (14)

We have fixed the value of $\nu_{0}=353$ GHz, $\beta=1.5$ and $T_{d}=18$ K.
This simple choice of template and parameters fit the data well
Planck_infrared  but we do not claim that it captures all the physics
correctly. In SHKO2012  and MM2021 , the authors describe a halo model
approach which motivate a similar spectral template within the context of
current data. The non-thermal distortion is parameterized as in eq. 3,

$I_{NT}=y_{NT}g_{NT}(x)$ (15)

In our modelling, we allow only the amplitude of the three components to vary.
To estimate the noise covariance matrix, we stack uniformly distributed random
positions, with same number as many as galaxy clusters in our sample, in the
map. We repeat the exercise for $10^{4}$ times to obtain a distribution which
sets the variance for individual frequency channel. We ignore correlation
across different frequency channel, therefore, our covariance matrix is
diagonal. Assuming gaussian likelihood, we use Markov Chain Monte Carlo (MCMC)
sampling to obtain posterier distribution on the free parameters. In Fig. 4,
we show our matched filtered, stacked data and best fit combination to data
with just thermal $y$ and IR. In addition, we show the marginalized posterior
distribution on $y_{T}$ and $A_{IR}$ in Fig. 5. The best fit $y_{T}$ and
$A_{IR}$ turns out to be $\sim 10^{-4}$ and 2.2 respectively. In the next
section, we include non-thermal SZ and derive marginalized constraints on
$y_{NT}$.

Figure 4: Fit to matched filtered, stacked spectral intensity data. As a
starting point, we fit thermal $y$-ditortion, IR spectrum to these data points
and show the best fit combination.

Figure 5: Marginalized constraints on $y_{T}$ and $A_{\rm IR}$ by fitting only
thermal $y$ and IR component to the data.

##  6 Results

In this section, we provide our main results in the form of upper limits on
$y_{NT}$ for a few different non-thermal electron spectrum parameterized by
$p_{\rm min}$ as in Sec. 2. We use flat prior on $y_{T}$, $y_{NT}$ and $A_{\rm
IR}$ for MCMC sampling.

###  6.1 Constraining $y_{NT}$ with varying $y_{T}$ and $A_{IR}$

In Fig. 6, we show the marginalized posterior on $y_{NT}$ by simultaneously
varying $y_{T}$, $y_{NT}$ and $A_{IR}$. For $p_{\rm min}=1$, we obtain upper
limit on $y_{NT}$ which is of the order of $10^{-4}$. This is expected as the
spectral shape for $p_{\rm min}=1$ is similar to thermal $y$-distortion (at
least for $\nu\lesssim 200$ GHz), therefore, our upper limit is of the same
order as the best fit $y_{T}$ as obtained in the previous section. The
constraints weaken significantly for higher $p_{\rm min}$ which again follows
from behaviour as seen in Fig. 1.

We further consider a case, where we fix $y_{T}=10^{-4}$ and $A_{IR}=2.2$.
This corresponds to the case where we have a strong prior on both of these
parameters. This might be possible with future CMB experiments such as SO2019
; CMB_HD2019 ; CCAT2023  where we will detect thermal SZ signature with very
high significance and can model IR component much better with more high
frequency channels. As expected, we obtain much stronger constraints with
$y_{NT}\lesssim 10^{-5}$ for $p_{\rm min}=1$.

Figure 6: (a) Marginalized posterior on $y_{NT}$ with varying $y_{T}$ and
$A_{IR}$ (left), (b) by fixing $y_{T}=10^{-4}$ and $A_{IR}=2.2$ obtained in
previous section (right). We show 68 percent confidence interval in dot-dashed
vertical lines.

###  6.2 Constraining $y_{NT}$ with varying $y_{T}$ only

As described in the previous section, we have used a simple description for IR
component where we have fixed some of the parameters (dust temperature as an
example). Depending on the complexity of the modelling, constraints on
$y_{NT}$ obtained above might change. Therefore, in this section, we do not
fit for IR component to the data. In Fig. 7, we show the marginalized
posterior on $y_{NT}$ for this case. We obtain upper limits which are
quantitatively similar compared to the previous subsection.

In this work, we have used a sample of $\sim 10^{3}$ galaxy clusters while, in
future, we might have a sample of $10^{5}$ clusters CMB_HD2019 . Assuming the
noise in Fig. 4 goes down by $\frac{1}{\sqrt{N}}$, where $N$ is the number of
galaxy clusters, the constraints can further tighten up or may lead to a
potential detection. We simulate such a situation where we have simply scaled
down the noise in Fig. 4 by a factor of 10 and repeating the analysis. The
results are shown in the right hand panel of Fig. 7. We find that keeping up
with the noise level, the constraints also tighten up by an order of magnitude
and reach a level of $y_{NT}\lesssim 10^{-5}$ for $p_{\rm min}=1$. An upper
limit of $y_{NT}\lesssim 10^{-5}$ would constraint non-thermal electron
pressure to less than about 10 percent of thermal electrons. Even though
higher $p_{\rm min}$ results in weaker constraints, a non-detection will
result in ruling out some parameter space in conjunction with radio or X-ray
observations. This will have interesting implications for particle
acceleration mechanisms in cosmic objects like galaxy clusters.

Figure 7:  (a) Marginalized posterior on $y_{NT}$ with varying $y_{T}$ (left),
(b) same with rescaling the noise down by a factor of 10 (right).

##  7 Conclusion

In this paper, we have obtained constraints on non-thermal SZ signatures with
a power-law electron population, assuming that these non-thermal electrons are
hosted inside the galaxy clusters. We have used Planck public data and galaxy
cluster catalogue to obtain the upper limits on non-thermal SZ for a few
representative electron populations. Unsurprisingly, these upper limits depend
upon the electron spectrum and our strongest upper limit is of the order of
$y_{NT}\sim 10^{-4}$. This would imply that the maximum possible non-thermal
electron pressure is about the same order of magnitude compared to thermal
electrons. This may be physically non-interesting as thermal electrons are
expected to dominate non-thermal electrons inside the galaxy clusters.
However, with increase in number of galaxy cluster detections with future CMB
experiments such as CMB-HD, we could tighten the upper limits by an order of
magnitude. Even with eROSITA galaxy cluster catalogue erosita2012  with about
$10^{5}$ detections, we may be able to reach that goal in near future. While a
detection will be exciting, a non-detecton of non-thermal SZ signature in
combination with radio, X-ray surveys can put strong constraints on parameter
space of potential non-thermal electron spectra and expand our knowledge of
particle acceleration mechanism inside galaxy clusters.

##  8 Acknowledgements

This work is supported by ARCO fellowship. We acknowledge discussions with
Vyoma Muralidhara in the early part of the project. We acknowledge Jens Chluba
for comments on the draft and Rishi Khatri for discussions on some aspects of
calculations.

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