11institutetext: Jodrell Bank Centre for Astrophysics, Department of Physics
and Astronomy, The University of Manchester, Manchester M13 9PL, UK

# Relativistic SZ maps and electron gas temperature spectroscopy

Mathieu Remazeilles 11<EMAIL_ADDRESS>
###### Abstract

While third-generation CMB experiments have allowed to release the first maps
of Compton-$y$ distortion due to thermal Sunyaev-Zeldovich (SZ) effect, next-
generation CMB experiments should allow us to map also the electron gas
temperature, $T_{\rm e}$, across the sky through the detection of relativistic
corrections to the thermal SZ effect. We discuss about experimental
requirements to break the $y$-$T_{\rm e}$ degeneracy of the observed SZ
intensity, and propose a new component separation approach based on moment
expansion to disentangle the $y$ and $T_{\rm e}$ observables of the
relativistic SZ effect while mitigating foregrounds. We show how our approach
offers a new spectroscopic view of the clusters not only across frequencies
but now also across temperatures. We also show how the relativistic electron
temperature power spectrum provides a new cosmological observable which may
complement the Compton-$y$ map power spectrum to break some of the parameter
degeneracies in future cosmological SZ analyses.

##  1 Introduction

By crossing the hot gas of electrons in galaxy clusters and filaments, cosmic
microwave background (CMB) photons get scattered while receiving a boost of
energy as the result of inverse Compton scattering. This is known as the
thermal Sunyaev-Zeldovich (SZ) effect sz , which causes $y$-type spectral
distortions to the primary CMB blackbody spectrum prominently in the direction
$\hat{n}$ of galaxy clusters:

$I_{\nu}^{\rm SZ}(\hat{n})\equiv\frac{\Delta I_{\nu}^{\rm CMB}}{I_{\nu}^{\rm
CMB}}(\hat{n})=g(\nu)\,y(\hat{n})\,.$ (1)

The amplitude of the distortion given by the Compton parameter $y$ depends on
the integrated pressure profile $P_{\rm e}$ of the electron gas along the
line-of-sight as

$y=\frac{\sigma_{T}}{m_{\rm e}c^{2}}\int P_{\rm e}(l)dl\,,$ (2)

where $m_{\rm e}$ is the electron mass, $c$ is the speed of light, and
$\sigma_{T}$ is the Thomson scattering cross-section. The spectral shape
$g(\nu)$ of the distortion across the frequencies $\nu$ is given in the _non-
relativistic_ limit, i.e. $kT_{\rm e}/m_{\rm e}c^{2}\ll 1$, by sz

$g(\nu)\equiv f(\nu,T_{\rm e}\simeq
0)=\frac{x^{4}e^{x}}{\left(e^{x}-1\right)^{2}}\left[x\coth\left(\frac{x}{2}\right)-4\right]\,,$
(3)

where $T_{\rm e}$ is the electron gas temperature, $h$ is the Planck constant,
$k$ is the Boltzmann constant, $T_{\rm CMB}=2.7255$ K is the CMB blackbody
temperature, and $x\equiv h\nu/kT_{\rm CMB}$.

The peculiar spectral signature Eq. (3) of the thermal SZ effect, as
illustrated in Fig. 1 by the black solid line, with a decrement at low
frequencies, a null around $218$ GHz and an increment at high frequencies,
enables to detect galaxy clusters by spectroscopy across frequencies
Planck_sz_2011 . Hence, the thermal SZ effect has now become a routine tool of
third-generation CMB experiments to detect thousands of galaxy clusters (e.g.
Planck_sz_2015 ) and trace the hot gas in the Universe through the mapping of
Compton-$y$ parameter anisotropies across the sky Planck_ymap_2015 .

Figure 1: _Left_ : Variations of the thermal SZ spectrum with the electron
temperature $T_{\rm e}$ at fixed $y=10^{-4}$, using SZpack Chluba2013 .
_Right_ : A varying Compton-$y$ parameter is used to rescale the amplitudes of
the spectra and make them matching at low frequencies, which allows us to
highlight that the distortion of the SZ spectral shape due to relativistic
electron temperature corrections actually arises at high frequencies.

However, galaxy clusters are massive, therefore they are hot objects with
typical electron gas temperature of $kT_{\rm e}\simeq 5$ keV for typical
cluster mass of $M_{500}=3\times 10^{14}\,h^{-1}M_{\odot}$ Arnaud2005 . Hence,
the thermal velocity of electrons in the gas approach the speed of light,
${v_{\rm e}^{\rm th}=\sqrt{2kT_{\rm e}/m_{e}}\gtrsim 0.1c}$. Therefore,
relativistic electron temperature corrections to the thermal SZ effect become
relevant:

$I_{\nu}^{\rm SZ}(\hat{n})=f(\nu,T_{\rm e}(\hat{n}))\,y(\hat{n})\,.$ (4)

The spectral signature $f(\nu,T_{\rm e}(\hat{n}))$ of the _relativistic_ SZ
(rSZ) effect thus varies depending on the local electron gas temperature in
the sky (Fig. 1). While at Planck sensitivity relativistic temperature
corrections are not perceptible on individual clusters, they become
statistically significant for a large number of clusters Hurier2016 ;
Erler2018 ; Remazeilles2019 . With larger sensitivity and larger sets of
frequencies, next-generation CMB experiments Hazumi2020 ; Hanany2019  should
allow us to not only map the thermal SZ Compton parameter $y(\hat{n})$ but
also the relativistic electron temperature field $T_{\rm e}(\hat{n})$ across
the sky. In this proceeding paper we describe a new component separation
approach allowing us to disentangle the $y$ and $T_{\rm e}$ observables of the
rSZ effect Remazeilles2020 , and show how the extracted $T_{\rm e}$ field can
complement the Compton-$y$ observable to probe cosmology and astrophysics with
galaxy clusters.

##  2 Disentangling the $y$ and $T_{\rm e}$ observables of the relativistic SZ
effect

As shown in the left panel of Fig. 1, relativistic corrections reduce the
overall intensity of the thermal SZ signal. Therefore, for a measured
intensity of the signal across the frequencies, the assumption of the non-
relativistic limit (black line) of the spectrum to detect the SZ effect from
hot galaxy clusters will result in underestimating the Compton-$y$ parameter.
As shown in Remazeilles2019 , neglecting relativistic SZ corrections in the
_Planck_ data analysis may have led to an underestimation of the overall
amplitude of the _Planck_ $y$-map power spectrum, and thereby the inferred
$\sigma_{8}$ value. Future cosmological analyses will thus have to account for
rSZ effects.

###  2.1 Experimental requirements

In the right panel of Fig. 1, we rescaled the various rSZ spectra of different
temperatures $T_{\rm e}$ by varying the parameter $y$ so that they match at
low frequency. This rescaling allows us to highlight that relativistic
electron temperatures actually distort the spectral shape of the thermal SZ
signal mostly in the high-frequency range $\gtrsim 300$ GHz. At low
frequencies $<200$ GHz, the rSZ effect does not cause any spectral distortion
to the signal (right panel of Fig. 1) but only a change in overall amplitude
(left panel of Fig. 1). Therefore, there is a $y$-$T_{\rm e}$ degeneracy of
the rSZ signal at low frequencies, where a lower amplitude of the measured
intensity can either be attributed to lower $y$ or larger $T_{\rm e}$ without
any possible distinction. In order to break this spectral degeneracy and
disentangle the $y$ and $T_{\rm e}$ observables of the rSZ effect, it is thus
recommended to observe the sky at high frequencies $>200$ GHz. Fourth-
generation CMB satellite concepts, e.g. _LiteBIRD_ Hazumi2020  or _PICO_
Hanany2019 , which are designed to observe the sky from space at frequencies
above $300$ GHz with high sensitivity, are thus best suited for our purpose.

###  2.2 Sky map simulations

We generated full-sky map simulations in 21 frequency bands ranging from $21$
to $800$ GHz using the instrumental specifications of _PICO_ Hanany2019  in
order to test the ability of our component separation method to disentangle
the $y$ and $T_{\rm e}$ observables of the rSZ effect. Our sky simulations
include the relativistic SZ signal in each frequency channel, which we
generated from all-sky template maps of $y$ and $T_{\rm e}$ (see figure 3 in
Remazeilles2020 ) using the frequency scaling Eq. (4) computed with SZpack
Chluba2013 . Our simulations also include several foreground emissions:
kinetic SZ effect, CMB anisotropies, cosmic infrared background anisotropies,
synchrotron, anomalous microwave emission, free-free and thermal dust (see
Remazeilles2020  for details).

###  2.3 Component separation method

The observed sky data, $d_{\nu}(\hat{n})$, in the line-of-sight $\hat{n}$ and
frequency channel $\nu$ can be written as

$d_{\nu}(\hat{n})=I_{\nu}^{\rm SZ}(\hat{n})+n_{\nu}(\hat{n})\,,$ (5)

where $I_{\nu}^{\rm SZ}(\hat{n})$ is the rSZ signal intensity (Eq. 4) and
$n_{\nu}(\hat{n})$ is the overall nuisance term including foregrounds and
instrumental noise. Following Chluba2013 , we can perform the Taylor expansion
of the rSZ spectral energy distribution (SED) around some pivot temperature
$\overline{T}_{\rm e}$ so that

$d_{\nu}(\hat{n})=f(\nu,\overline{T}_{\rm e})\,y(\hat{n})+\frac{\partial
f(\nu,\overline{T}_{\rm e})}{\partial\overline{T}_{\rm e}}\,\left(T_{\rm
e}(\hat{n})-\overline{T}_{\rm e}\right)y(\hat{n})+\tilde{n}_{\nu}(\hat{n})\,,$
(6)

where the nuisance term $\tilde{n}_{\nu}$ now includes higher-order terms in
$T_{\rm e}(\hat{n})$ in addition to foregrounds and noise. The expansion Eq.
(6) thus highlights two distinct components of emission, the Compton parameter
component $y(\hat{n})$ and a _temperature-modulated_ component ${\left(T_{\rm
e}(\hat{n})-\overline{T}_{\rm e}\right)y(\hat{n})}$, with different spectral
signatures $f(\nu,\overline{T}_{\rm e})$ and $\partial f(\nu,\overline{T}_{\rm
e})/\partial\overline{T}_{\rm e}$, respectively. Therefore, it is possible in
principle to disentangle the $y$ and ${y(T_{\rm e}-\overline{T}_{\rm e})}$
fields through multi-frequency observations and tailored component separation
methods.

Given that both signals $y$ and ${y(T_{\rm e}-\overline{T}_{\rm e})}$ are
correlated with each other, it is essential to fully deproject one signal from
the estimated map of the other signal, i.e. nulling out $y$ in the
reconstructed map of the ${y(T_{\rm e}-\overline{T}_{\rm e})}$ field and vice-
versa, by using a _constrained_ ILC approach Remazeilles2011 ; Remazeilles2020
; Remazeilles2021 . Denoting the temperature-modulated Compton-$y$ signal as
${z(\hat{n})\equiv y(\hat{n})\left(T_{\rm e}(\hat{n})-\overline{T}_{\rm
e}\right)}$, we thus form our constrained-ILC estimate $\hat{z}(\hat{n})$
through the following weighted linear combination of the data across
frequencies:

$\hat{z}(\hat{n})=\sum_{\nu}w_{\nu}\,d_{\nu}(\hat{n})\quad\textrm{such
that}\quad\begin{cases}\textrm{the variance
}\langle\,\hat{z}^{\,2}\rangle\textrm{ is minimum}\,,\\\\[4.2679pt]
\sum_{\nu}w_{\nu}\,\frac{\partial f(\nu,\overline{T}_{\rm
e})}{\partial\overline{T}_{\rm e}}=1\,,\\\\[4.2679pt]
\sum_{\nu}w_{\nu}\,f(\nu,\overline{T}_{\rm e})=0\,.\end{cases}$ (7)

The unit constraint in Eq. (7) ensures the conservation of the signal of
interest, ${z\equiv y(T_{\rm e}-\overline{T}_{\rm e})}$, during variance
minimization, while the null constraint guarantees full cancellation of $y$
residuals in the recovered ${y(T_{\rm e}-\overline{T}_{\rm e})}$-map. Finally,
the minimum-variance condition in Eq. (7) ensures the mitigation of
foregrounds and noise. Note that the $y$-map can be obtained in a similar way,
simply by interchanging the unit and null constraints in Eq. (7). In addition
to above constraints in Eq. (7) we add two extra nulling constraints to
deproject the kinetic SZ effect (hence the CMB) and bulk of the thermal dust
emission in the recovered rSZ maps (see Remazeilles2020  for details). The
expression of the constrained-ILC weights $\boldsymbol{w}\equiv\\{w_{\nu}\\}$
for our specific component separation problem Eq. (7) can be derived using
Lagrange multipliers (see Remazeilles2020 ):

$\boldsymbol{w}^{\rm T}=\boldsymbol{e}^{\rm T}\left({\rm A}^{\rm T}{\rm
C}^{-1}{\rm A}\right)^{-1}{\rm A}^{\rm T}{\rm C}^{-1}\,,$ (8)

where the first two columns of matrix ${\rm
A}=\left(\partial\boldsymbol{f}/\partial\overline{T}_{\rm
e}\,\,\boldsymbol{f}\,\,\boldsymbol{g}_{\rm kSZ}\,\,\boldsymbol{g}_{\rm
dust}\right)$ collect the effective SED vectors of the rSZ signals (see Eq. 6)
and the last two columns the SED vector $\boldsymbol{g}_{\rm kSZ}$ of the
kinetic SZ effect and the average SED vector $\boldsymbol{g}_{\rm dust}$ of
the thermal dust, while $\boldsymbol{e}^{\rm T}=\left(1\,0\,0\,0\right)$, and
$C=\langle\boldsymbol{d}\boldsymbol{d}^{\rm T}\rangle$ is the channel-to-
channel covariance matrix of the data.

##  3 Overview of the results

###  3.1 Electron-temperature spectroscopy of galaxy clusters

Figure 2: Temperature spectroscopy of galaxy clusters: Gnomonic view of the
reconstructed $y(T_{\rm e}-\overline{T}_{\rm e})$ all-sky map around Coma for
different pivot temperatures $\overline{T}_{\rm e}$.

The new rSZ observable ${z(\hat{n})\equiv y(\hat{n})(T_{\rm
e}(\hat{n})-\overline{T}_{\rm e})}$ is particularly interesting to find out
the actual temperature $T_{\rm e}(\hat{n})$ of a cluster. Indeed, using
different pivot temperatures $\overline{T}_{\rm e}$ for the Constrained-ILC
filter Eq. (8) allows us to perform the temperature spectroscopy of the
clusters: the recovered signal ${z(\hat{n})\equiv y(\hat{n})(T_{\rm
e}(\hat{n})-\overline{T}_{\rm e})}$ will show a _decrement_ at the position of
a cluster in the map if the actual temperature of that cluster is smaller than
the pivot temperature, i.e. $T_{\rm e}(\hat{n})<\overline{T}_{\rm e}$, while
it will show an _increment_ if $T_{\rm e}(\hat{n})>\overline{T}_{\rm e}$ and a
_null_ if $T_{\rm e}(\hat{n})\simeq\overline{T}_{\rm e}$. In the latter case
the pivot temperature must be the actual temperature of the cluster.

Such temperature spectroscopy is illustrated in Fig. 2 where we show the
reconstructed map of ${y(\hat{n})\left(T_{\rm e}(\hat{n})-\overline{T}_{\rm
e}\right)}$ around the Coma cluster as obtained for different pivots
$k\overline{T}_{\rm e}=5$, $7$, and $10$ keV after component separation on the
_PICO_ map simulations. The recovered rSZ signal informs us that the
temperature of Coma is hotter than $5$ keV but colder than $10$ keV, and
closer to $7$ keV (null), which is the fiducial temperature of Coma in the
current simulation.

###  3.2 Electron gas temperature observables from the rSZ maps

From the reconstructed $y$ and $y(T_{\rm e}-\overline{T}_{\rm e})$ maps after
component separation, we can easily create the $yT_{\rm e}$-map simply by
mutiplying the $y$-map by the pivot temperature $\overline{T}_{\rm e}$ and
adding it to the $y(T_{\rm e}-\overline{T}_{\rm e})$-map: ${yT_{\rm
e}(\hat{n})=\overline{T}_{\rm e}\,y(\hat{n})+z(\hat{n})=\overline{T}_{\rm
e}y(\hat{n})+y(\hat{n})\left(T_{\rm e}(\hat{n})-\overline{T}_{\rm e}\right)}$.
Using the $y$-map and the $yT_{\rm e}$-map we can then derive several
observables which we outline hereafter.

First, by computing the ratio of the flux in the $yT_{\rm e}$-map and the flux
in the $y$-map within $R_{500}$ around a cluster, we obtain the $y$-weighted
average temperature of that cluster over $R_{500}$:

$T_{\rm e}^{y}\left[R_{500}\right]=\frac{\langle yT_{\rm
e}(\hat{n})\rangle_{|\hat{n}-\hat{n}_{c}|\leq R_{500}}}{\langle
y(\hat{n})\rangle_{|\hat{n}-\hat{n}_{c}|\leq R_{500}}}\,,$ (9)

where $\hat{n}_{c}$ denotes the line-of-sight corresponding to the centre of
the cluster. Figure 3 (left panel) shows our successful recovery of the
electron temperatures of more than 800 galaxy clusters across the entire sky
for the _PICO_ sky simulation after foreground cleaning and component
separation Remazeilles2020 . The recovered rSZ temperatures could be used in
principle as a new proxy for determining cluster masses without relying on
X-ray measurements.

Second, we can also derive the electron temperature profile of a cluster
simply from the ratio of the cluster profile in the $yT_{\rm e}$-map and the
cluster profile in the $y$-map:

$T_{\rm e}^{y}(r)=\frac{yT_{\rm e}(r)}{y(r)}\,,$ (10)

where $r$ is the radius from the centre of the cluster. Figure 3 (middle
panel) shows as an example our recovery of the electron temperature profile of
the Coma cluster up to large radii, with a $10\sigma$ measurement of the
average temperature of Coma over $R_{500}$ for _PICO_ Remazeilles2020 .

Finally, the cross-power spectrum between the reconstructed $y$\- and $yT_{\rm
e}$-maps relative to the auto-power spectrum of the $y$-map measures the
$y^{2}$-weighted average electron temperature over the full sky across
multipoles:

$T_{\rm e}^{yy}(\ell)=\frac{\langle\left(yT_{\rm e}\right)_{\ell
m}\,y^{*}_{\ell m}\rangle}{\langle y_{\ell m}\,y^{*}_{\ell
m}\rangle}=\frac{C_{\ell}^{y,yT_{\rm e}}}{C_{\ell}^{yy}}\,,$ (11)

thus providing an effective power spectrum of the electron temperature field
Remazeilles2019 . Figure 3 (right panel) shows accurate recovery of the
electron temperature power spectrum with _PICO_ over a large range of
multipoles after foreground cleaning and component separation Remazeilles2020
.

Concluding remarks. The electron temperature power spectrum $T_{\rm
e}^{yy}(\ell)$ provides a new map-based observable, complementing the $y$-map
power spectrum $C_{\ell}^{yy}$, to constrain cosmological parameters, since
the exact shapes of the power spectra $T_{\rm e}^{yy}$ and $C_{\ell}^{yy}$
have different scaling and dependence on cosmological parameters. To gain
intuition on this, we may look into theoretical expressions:

$\displaystyle C_{\ell}^{yy}$ $\displaystyle=\int_{\,0}^{\,z_{\rm
max}}dz{dV\over dz}\int_{\,M_{\rm min}}^{\,M_{\rm max}}dM\,{dn(M,z)\over
dM}\,|y_{\ell}(M,z)|^{2}\,,$ (12) $\displaystyle C_{\ell}^{y,yT_{\rm e}}$
$\displaystyle=\int_{\,0}^{\,z_{\rm max}}dz{dV\over dz}\int_{\,M_{\rm
min}}^{\,M_{\rm max}}dM\,{dn(M,z)\over dM}\,T_{\rm
e}(M,z)\,|y_{\ell}(M,z)|^{2}\,.$ (13)

While $C_{\ell}^{yy}$ and $C_{\ell}^{y,yT_{\rm e}}$ have same scaling with the
mass bias $b$ through exact same dependence on the profile
$|y_{\ell}(M,z)|^{2}$, they have different scaling with $\sigma_{8}$ due to
the modulation of the halo mass function in $C_{\ell}^{y,yT_{\rm e}}$ by the
mass-dependent temperature $T_{\rm e}$. Therefore, while the $y$-map power
spectrum $C_{\ell}^{yy}$ depends on $\sigma_{8}$ and $b$ in a degenerate form,
the electron temperature power spectrum $T_{\rm e}^{yy}(\ell)$, as the ratio
of $C_{\ell}^{y,yT_{\rm e}}$ and $C_{\ell}^{yy}$ (Eq. 11), must depend only on
$\sigma_{8}$ and be quite insensitive to the mass bias Remazeilles2020 . Such
new rSZ observable thus provides a possible avenue to break some of the
parameter degeneracies with future cosmological SZ analyses.

_Acknowledgements_ : This work is supported by the ERC Consolidator Grant
CMBSPEC (No. 725456) as part of the European Union’s Horizon 2020 research and
innovation program.

Figure 3: _Left_ : Recovered electron temperatures $T_{\rm e}[R_{500}]$ across
the sky of a sample of 811 clusters after foreground cleaning. _Middle_ :
Reconstructed temperature profile $T_{\rm e}(r)$ of Coma. _Right_ :
Reconstructed diffuse electron gas temperature power spectrum $T_{\rm
e}^{yy}(\ell)$ after foreground cleaning.

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