# Calibrating baryonic feedback with weak lensing and fast radio bursts

Robert Reischke  Dennis Neumann  Klara Antonia Bertmann  Steffen Hagstotz  and
Hendrik Hildebrandt

###### Abstract

One of the key limitations of large-scale structure surveys of the current and
future generation, such as Euclid, LSST-Rubin or Roman, is the influence of
feedback processes on the distribution of matter in the Universe. This effect,
called baryonic feedback, modifies the matter power spectrum on non-linear
scales much stronger than any cosmological parameter of interest. Constraining
these modifications is therefore key to unlock the full potential of the
upcoming surveys, and we propose to do so with the help of Fast Radio Bursts
(FRBs). FRBs are short, astrophysical radio transients of extragalactic
origin. Their burst signal is dispersed by the free electrons in the large-
scale-structure, leading to delayed arrival times at different frequencies
characterised by the dispersion measure (DM). Since the dispersion measure is
sensitive to the integrated line-of-sight electron density, it is a direct
probe of the baryonic content of the Universe. We investigate how FRBs can
break the degeneracies between cosmological and feedback parameters by
correlating the observed Dispersion Measure with the weak gravitational
lensing signal of a Euclid-like survey. In particular we use a simple one-
parameter model controlling baryonic feedback, but we expect similar findings
for more complex models. Within this model we find that $\sim 10^{4}$ FRBs are
sufficient to constrain the baryonic feedback 10 times better than cosmic
shear alone. Breaking this degeneracy will tighten the constraints
considerably, for example we expect a factor of two improvement on the sum of
neutrino masses.

##  1 Introduction

Cosmic shear, the weak gravitational lensing effect imprinted on distant
galaxies by the large-scale structure (LSS), is one of the primary science
goals for the the upcoming stage-4 surveys Euclid111https://www.euclid-
ec.org/, Rubin-LSST222https://www.lsst.org/ and the Roman
telescope333https://roman.gsfc.nasa.gov/. The foundations for these missions
has been laid down by their predecessors: the Kilo-Degree Survey [1, 2,
KiDS]444https://kids.strw.leidenuniv.nl/, the Dark Energy Survey [3, 4,
DES]555https://www.darkenergysurvey.org/ and the Subaru Hyper Suprime-Cam [5,
6, HSC]666https://hsc.mtk.nao.ac.jp/ssp/.

The stage-4 surveys in particular target the highly non-linear regime of the
matter distribution. Even though there is a wealth of information to be
gathered at $k>1\,\mathrm{Mpc}^{-1}h$, baryonic feedback (BF) processes
imprint large theoretical uncertainties on the total matter power spectrum.
While the distribution of collisionless cold dark matter is well understood,
at least on the level of numerical simulations, baryons remain the cause of
large uncertainties. The displacement of baryons is due to nonlinear effects
sourced by star formation, Supernovae, Active Galactic Nuclei (AGN) or cooling
[see e.g. 7, 8, for reviews]. In turn, the displaced baryons affect the total
matter power spectrum causing a reduction of power on intermediate scales due
to outflows and energy injection by e.g. AGN, and an increase in power on
small scales due to cooling. As these effects span a multitude of scales and
are highly non-linear, they are almost impossible to model analytically and
are thus extracted from hydrodynamical simulations which include a model with
a fixed set of parameters to control the feedback strength. The main challenge
arises from the fact that feedback processes originate from physics which is
far below the resolution of the numerical simulation. Each simulation
therefore has sub-grid models implemented. However, different models can lead
to significantly different predictions for the matter clustering statistics at
small scales [9, 10]. For example, in the CAMELS simulation suite [11] two
very distinct sub-grid models are implemented. However, the differences are
largely due to higher gas temperature in the inter-galactic medium [11].
Another suite of simulations are the BAHAMAS simulations [12] whose power
spectra have been calibrated to a hydrodynamical version of the halo model
[13], specifically designed for weak lensing cross correlations studies with
the thermal Sunyae-Zel’dovich (tSZ) effect [14].

Any independent measurement of the baryon distribution can help to
independently constrain feedback models. It was shown in [14] that e.g. the
tSZ effect has the potential to partially calibrate the feedback strength and
breaking degeneracies (see also a discussion [15]). At the moment, the tSZ
effect is the only way to measure the electron distribution in the Universe.
It is therefore important to add additional probes for $(i)$ more constraining
power and $(ii)$ different systematic effects in the measurements.

An alternative probe of the electron distribution are fast radio bursts
(FRBs). FRBs are short transients, lasting typically a few milliseconds, with
frequencies ranging from $\sim 100$ MHz to several GHz. Since the radio signal
is travelling through the ionized intergalactic medium (IGM), each frequency
of the pulse experiences a delay characterised by the dispersion measure (DM)
proportional to the integrated free electron density along the line-of-sight
[e.g. 16, 17, 18, 19, 20]. The mechanism responsible for the radio emission is
currently unknown, but the isotropic occurrence of the events detected so far
shows no alignment with the Milky Way disk and the measured DM for most events
is very large, requiring an extragalactic origin. By now, a subset of events
has been localised to galaxies up to redshifts $z\sim 1$. In general, the
total DM associated with an extragalactic FRB event consists of contributions
from the host galaxy, the Milky Way and the diffuse electrons in the large-
scale structure (LSS). Several authors therefore proposed to use the DM
inferred from FRBs as a cosmological probe using either the averaged signal
[e.g. 21, 22, 23, 24, 25, 26, 27, 28] or the statistics of DM fluctuations
[e.g. 29, 30, 31, 32, 33, 34, 35].

Currently roughly $\sim 800$ FRBs777See https://www.herta-
experiment.org/frbstats/ for an exhaustive list. are detected, but estimated
rates of $\sim 1000$ events/night/sky allow surveys such as CHIME, UTMOST,
HIRAX, ASKAP or SKA [36, 37, 38, 39] to provide at least $\sim 10^{4}$ FRBs
per decade. Although FRBs do not show spectral features that allow for
accurate redshift estimation, the accumulated DM can be translated into a
noisy distance estimate [29]. Since FRBs are observed with $\mathrm{DM}\geq
1000\;\mathrm{pc}\;\mathrm{cm}^{-3}$ and models of the Milky way suggest
$\mathrm{DM}\leq 50\;\mathrm{pc}\;\mathrm{cm}^{-3}$ due to gas in our Galaxy
for most directions on the sky [40], the majority of the DM signal originates
from the diffuse electrons in the IGM, which follows the total matter
distribution on large scales. Although the host galaxy contribution is still
under debate, its magnitude is expected to be comparable to that of the Milky
Way and hence usually smaller than the contribution from the LSS. While the
host dispersion can reach several hundreds $\mathrm{pc}\;\mathrm{cm}^{-3}$ in
rare cases, particularly when the FRB is originates in the trailing edge of
the host galaxy (as seen by the observer), one typically finds that
$\mathrm{DM}_{\mathrm{LSS}}>\mathrm{DM}_{\mathrm{host}}$ for FRBs at redshifts
$z\sim 1$. The dispersion of FRBs tests, similar to cosmic shear, an
integrated quantity along the line-of-sight. However, since host galaxies
typically contribute less to the DM than the LSS, FRBs are way less affected
by shot noise compared to cosmic shear. Combining FRBs and cosmic shear will
accordingly probe similar structures, but testing the baryon distribution and
the total matter distribution respectively, enabling strong constraints on the
different clustering behaviour of both components. It should be noted that the
cosmic shear signal has a vanishing expectation value, unlike the DM of FRBs.
However, since we are interested in the statistical properties of the DM this
is not that important for our study here.

In this paper we will hence investigate the potential gains from combining a
stage-4 cosmic shear measurement with the statistical properties of the DM
measured from an FRB sample which should be available on the same timescale as
cosmic shear data from Euclid and Rubin-LSST. To this end, we simulate mock
data for these kind of surveys and carry out a Fisher and MCMC analysis to
investigate the impact of adding FRBs in a cosmological analysis with a
particular focus on the baryonic feedback and on the cosmological parameters
that are most affected by feedback uncertainties, such as the sum of neutrino
masses. The paper is structured as follows: In Section 2 we summarise the LSS
structure probes and describe the methodology for our analysis. Section 3 is
devoted to the discussion of the results and a comparison with other probes of
the electron distribution. Lastly, we conclude in Section 4 and discuss
potential ways forward.

##  2 Large-Scale-Structure Probes

In this section we summarise the two probes used to test the matter
distribution: cosmic shear and FRB dispersion measure. We introduce cosmic
shear in Section 2.1 and DM correlations in Section 2.2, which are sensitive
to the full matter and electron distribution respectively and can hence be
used to break degeneracies between baryonic feedback and cosmological
parameters.

###  2.1 Cosmic Shear

Cosmic shear is the gravitational lensing effect of the LSS on an ensemble of
background sources (galaxies) and is thus sensitive to the projected matter
distribution along the line-of-sight. The cosmological information of
gravitational lensing by the LSS is contained in the traceless part of the
cosmic shear tensor:

$\boldsymbol{\gamma}_{i}=2\int\mathrm{d}\chi
W_{\gamma_{i}}(\chi)\eth\eth\Phi(\hat{\boldsymbol{x}},\chi)\;,$ (2.1)

in absence of anisotropic stress. Here $\gamma_{i}$ denotes the shear in the
$i$-th tomographic bin, $W_{\gamma_{i}}$ is a geometrical weighting function
defined below, $\Phi$ are the scalar metric perturbations, $\chi$ the co-
moving distance and $\eth$ is the eth-derivative for a spin-2 field. Using
Poisson’s equation, one can relate the metric perturbations to the density
contrast, yielding the following expression for the angular power spectrum:

$\displaystyle
C^{\gamma_{i}\gamma_{j}}_{\ell}=\frac{(\ell+2)!}{(\ell-2)!}\frac{2}{\uppi}\int\mathrm{d}\chi_{1}\int\mathrm{d}\chi_{2}\int
k^{2}\mathrm{d}k\;K_{i}(\chi_{1})K_{j}(\chi_{2})\frac{j_{\ell}(\chi_{1}k)}{(k\chi_{1})^{2}}\frac{j_{\ell}(\chi_{2}k)}{(k\chi_{2})^{2}}\sqrt{P_{\delta}(k,\chi_{1})P_{\delta}(k,\chi_{2})}\;.$
(2.2)

Approximating the Bessel function by a Dirac distribution [41, 42], the
integrations over $\chi_{1,2}$ can be carried out to find the commonly used
expression for for the angular power spectrum:

$C^{\gamma_{i}\gamma_{j}}_{\ell}=\int_{0}^{\chi_{\mathrm{H}}}\frac{\mathrm{d}\chi}{\chi^{2}}W^{(i)}_{\gamma}(\chi)W^{(j)}_{\gamma}(\chi)P_{\delta}\left(\frac{\ell+1/2}{\chi},\chi\right)\;,$
(2.3)

which will be accurate enough for cosmic shear even for the next generation of
surveys [43] due to the broad lensing kernel. Here $P_{\delta}$ is the matter
power spectrum, for which we use the emulated spectrum from HMcode[44].
$W^{(i)}_{\gamma}(\chi)$ is the lensing weight of the $i$-th tomographic bin
as given by:

$W^{(i)}_{\gamma}(\chi)=\frac{3\Omega_{\mathrm{m0}}}{2\chi_{\mathrm{H}}^{2}}\frac{\chi}{a(\chi)}\int_{\chi}^{\chi_{\mathrm{H}}}\mathrm{d}{\chi^{\prime}}n^{(i)}_{\mathrm{s}}({\chi^{\prime}})\frac{{\chi^{\prime}}-\chi}{{\chi^{\prime}}}\;.$
(2.4)

Here $a$ is the scale factor, $\Omega_{\mathrm{m}0}$ the matter density
parameter today, $\chi_{\mathrm{H}}$ the Hubble radius and
$n^{(i)}_{\mathrm{s}}$ is the distribution of sources.

Lastly, due to the finite number of background galaxies, observed spectra
obtain a shape-noise contribution:

$C^{\gamma_{i}\gamma_{j}}_{\ell}\;\to\;C^{\gamma_{i}\gamma_{j}}_{\ell}+\frac{\sigma_{\epsilon,i}^{2}}{2\bar{n}_{i}}\;,$
(2.5)

with the total ellipticity dispersion $\sigma_{\epsilon,i}^{2}$ and the
average number of sources $\bar{n}_{i}$ in the $i$-th tomographic bin.

Survey |  area $f_{\mathrm{sky}}$ | number of sources |  intrinsic noise $\sigma$ | tomographic bins
---|---|---|---|---
FRB | 0.7 | $10^{4}$ | $50\;\mathrm{pc}\;\mathrm{cm}^{-3}$ | 1
Euclid | 0.3 | $30\;\mathrm{arcmin}^{-2}$ | 0.3 | 10
KiDS | 0.025 | $7.55\;\mathrm{arcmin}^{-2}$ | 0.28 | 5
Table 1: Survey settings assumed in the analysis. The number of sources for
the cosmic shear surveys is an average density over all tomographic bins. The
actual number in each tomographic bin can be slightly different. The same
holds for the ellipticity dispersion, for more details see [45] and [46].

###  2.2 FRB statistics

FRB pulses undergo dispersion while travelling through the ionized IGM,
causing a frequency-dependent (proportional to $\nu^{-2}$) offset of arrival
times. The corresponding time delay measured $\delta
t(\hat{\boldsymbol{x}},z)$ for an FRB at redshift $z$ in direction
$\hat{\boldsymbol{x}}$ proportional to the observed dispersion measure:
$\delta
t(\hat{\boldsymbol{x}},z)=\mathrm{DM}_{\mathrm{tot}}(\hat{\boldsymbol{x}},z)\nu^{-2}$.
The observed DM can be broken up into different components:

$\mathrm{DM}_{\mathrm{tot}}(\hat{\boldsymbol{x}},z)=\mathrm{DM}_{\mathrm{LSS}}(\hat{\boldsymbol{x}},z)+\mathrm{DM}_{\mathrm{MW}}(\hat{\boldsymbol{x}})+\mathrm{DM}_{\mathrm{host}}(z)\;,$
(2.6)

where $\mathrm{DM}_{\mathrm{LSS}}(\hat{\boldsymbol{x}},z)$ is the DM caused by
the electron distribution in the LSS, while
$\mathrm{DM}_{\mathrm{MW}}(\hat{\boldsymbol{x}})$ and
$\mathrm{DM}_{\mathrm{host}}(z)$ describe the contributions from the Milky Way
and the host galaxy, respectively. Here we made the dependence on redshift and
direction explicit. Note that the rest-frame DM of the host,
$\mathrm{DM}_{\mathrm{host,rf}}$, is observed as
$\mathrm{DM}_{\mathrm{host}}(z)=(1+z)^{-1}\mathrm{DM}_{\mathrm{host,rf}}$.

Current models of the galactic electron distribution predict
${\mathrm{DM}_{\mathrm{MW}}\sim 60\,\mathrm{pc}\,\mathrm{cm}^{-3}}$ over most
of the sky [40, 47]. Since the galactic electron distribution also depends on
the direction $\hat{\boldsymbol{x}}$, it is in principle a large contaminant
for DM correlation measurement. Here, however, we assume that it can be
modelled accurately enough to be subtracted from the observed DM without
leaving a significant imprint on the measured DM correlations. Naively one can
expect $\mathrm{DM}_{\mathrm{host}}(z)$ to be very similar to
$\mathrm{DM}_{\mathrm{MW}}$. However, in practice it depends strongly on the
host galaxy’s properties and is thus treated as a free parameter. Since it
does not correlate with the LSS, it will not leave an imprint on the DM
correlation but rather act as a noise term as we will discuss later. Finally,
the LSS contribution to the DM is given by [34]

$\mathrm{DM}_{\mathrm{LSS}}(\hat{\boldsymbol{x}},z)=\chi_{\mathrm{H}}\int_{0}^{z}\mathrm{d}z^{\prime}\,n_{\mathrm{e}}({\boldsymbol{x}},z^{\prime})\,\frac{1+z^{\prime}}{E(z^{\prime})}\,\;,$
(2.7)

where $E(z)$ is the expansion function and $n_{e}$ the electron density which
depends on the local matter over-density,
$\delta_{\mathrm{m}}(\boldsymbol{x},z)$ and the electron bias
$b_{\mathrm{e}}({\boldsymbol{x}},z)$:

$n_{\mathrm{e}}({\boldsymbol{x}},z)=\frac{\bar{\rho}_{\mathrm{b}}(z)}{m_{\mathrm{p}}}F(z)\left[1+b_{\mathrm{e}}({\boldsymbol{x}},z)\,\delta_{\mathrm{m}}({\boldsymbol{x}},z)\right]\;.$
(2.8)

Here we used the mean baryon density $\bar{\rho}_{\mathrm{b}}(z)$, the proton
mass $m_{\mathrm{p}}$ and the mass fraction $F(z)$ of electrons in the IGM.
The latter can be expressed as:

$F(z)=f_{\mathrm{IGM}}(z)[Y_{\mathrm{H}}X_{\mathrm{e},\mathrm{H}}(z)+Y_{\mathrm{He}}X_{\mathrm{e},\mathrm{He}}(z)]\;,$
(2.9)

with the mass fraction of hydrogen and helium $Y_{\mathrm{H}}=0.75$ and
$Y_{\mathrm{He}}=0.25$ respectively, as well as their ionization fractions
$X_{\mathrm{e},\mathrm{H}}(z)$ and $X_{\mathrm{e},\mathrm{He}}(z)$. Since both
hydrogen and helium are both fully ionised for $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3$ [48, 49] we set
$X_{\mathrm{e},\mathrm{H}}=X_{\mathrm{e},\mathrm{He}}=1$. It should be noted
that for the survey settings considered here the majority of the signal arises
from $z<3$, so the last assumption is indeed well justified. The fraction of
baryons in the IGM, $f_{\mathrm{IGM}}(z)$, has a slight redshift dependence
[50], with $10\%\;(20\%)$ locked up in galaxies at $z\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}1.5\;(\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.4)$. Rewriting eq. 2.7 leaves us
with the following expression:

$\\!\\!\mathrm{DM}_{\mathrm{LSS}}(\hat{\boldsymbol{x}},z)=\mathcal{A}\\!\int_{0}^{z}\\!\\!\\!\mathrm{d}z^{\prime}\frac{1+z^{\prime}}{E(z^{\prime})}F(z^{\prime})[1\\!+\\!b_{\mathrm{e}}({\boldsymbol{x}},z^{\prime})\delta_{\mathrm{m}}({\boldsymbol{x}},z^{\prime})]\;,$
(2.10)

with the amplitude

$\mathcal{A}\equiv\frac{3H_{0}^{2}\Omega_{\mathrm{b}0}\chi_{H}}{8\uppi
Gm_{\mathrm{p}}}\;.$ (2.11)

A very intriguing aspect of FRBs is that the cosmological contribution is
typically larger than the contribution from the host or the Milky Way. This is
in large contrast to cosmic shear, where the ellipticity imprinted by
gravitational lensing is a percent effect of the intrinsic ellipticity.
Therefore, far fewer FRBs are required for a significant detection of DM
correlations compared the number of galaxies in weak lensing studies.
Following [34], we rewrite the LSS contribution to the DM as a background
contribution and a perturbation:

$\mathrm{DM}_{\mathrm{LSS}}(\hat{\boldsymbol{x}},z)=\langle\mathrm{DM}_{\mathrm{LSS}}\rangle(z)(1+\mathcal{D}(\hat{\boldsymbol{x}},z))\;,$
(2.12)

where $\mathcal{D}(\hat{\boldsymbol{x}},z)$ is the effective DM induced by the
fluctuations in the LSS. In full analogy to cosmic shear, we assume a source-
redshift distribution $n_{\mathrm{FRB}}(z)$. To be accurate, one is using a
source-DM distribution, which in turn is converted to
$n_{\mathrm{FRB}}(z(\mathrm{DM}))$. This procedure was discussed in [34] and
[51]. Note that we do not make use of the fraction of FRBs that will most
likely have an identified host with redshift estimates, which additionally
improves the expected constraints. By rearranging integration limits one finds

$\mathcal{D}(\hat{\boldsymbol{x}})=\int_{0}^{\chi_{H}}\\!\\!\mathrm{d}\chi\;W_{\mathcal{D}}(\chi)\delta_{\mathrm{e}}\big{(}\hat{\boldsymbol{x}},z(\chi)\big{)}\;,$
(2.13)

with the averaged weighting function

$W_{\mathcal{D}}(\chi)=W(\chi)\int_{\chi}^{\chi_{H}}\mathrm{d}\chi^{\prime}n_{\mathrm{FRB}}(\chi^{\prime})\;,$
(2.14)

and $W(\chi)$ being defined via eq. 2.10:

$W(\chi)=\mathcal{A}\frac{F\big{(}z(\chi)\big{)}\big{(}1+z(\chi)\big{)}}{E\big{(}z(\chi)\big{)}}\left|\frac{\mathrm{d}z}{\mathrm{d}\chi}\right|\;.$
(2.15)

In complete analogy to cosmic shear, the angular power spectrum for the DM
fluctuations $\mathcal{D}$ is given by:

$C^{\mathcal{D}\mathcal{D}}(\ell)=\;\frac{2}{\uppi}\int\mathrm{d}\chi_{1}\int\mathrm{d}\chi_{2}\int
k^{2}\mathrm{d}k\;W_{\mathcal{D}}(\chi_{1})W_{\mathcal{D}}(\chi_{2})\sqrt{P_{\mathrm{ee}}(k,\chi_{1})P_{\mathrm{ee}}(k,\chi_{2})}j_{\ell}(k\chi_{1})j_{\ell}(k\chi_{2})\;.$
(2.16)

As discussed in [34], the Limber approximation is suitable for DM correlations
as well. Hence, we again use eq. 2.3. When cross-correlating cosmic shear and
the DM, we replace one of the weight functions in Equation 2.3 with
$W_{\mathcal{D}}$ and the matter power spectrum by the electron-matter cross
power spectrum $P_{\delta\mathrm{e}}$.

Lastly, the host galaxy contribution acts as a shot noise contribution with
variance $\sigma^{2}_{\mathrm{host}}$ per event, thus adding a white noise
contribution to the observed spectra:

$C^{\mathcal{D}\mathcal{D}}(\ell)\;\to\;C^{\mathcal{D}\mathcal{D}}(\ell)+\frac{\sigma^{2}_{\mathrm{host}}}{\bar{n}}\;.$
(2.17)

###  2.3 Power Spectra

The angular power spectra in Equation 2.2 and Equation 2.16 are integrals over
the matter power spectrum and the electron power spectrum. Cross-correlating
the probes also requires the matter-electron power spectrum. We use pyhmcode
[44, 13, 14] which was fitted to the BAHAMAS [12] and COSMO-OWLS [52]
simulation suites to reproduce the power spectra of matter, gas, stars and
pressure assuming a halo model. The baryonic feedback in pyhmcode is
controlled by a single parameter, the strength of AGN encapsulated in
$T_{\mathrm{AGN}}$. This AGN temperature should just be seen as a nuisance
parameters without physical correspondence in the real Universe. Larger AGN
temperatures lead to different suppression on large $k$ (and rise again on
even smaller scales). For current cosmological surveys, this description is
usually flexible enough, since more extended models are likely to be prior
dominated and therefore offer too much flexibility. For future surveys,
however, more flexible emulators might be needed [53, 54, 55], allowing for
more freedom at non-linear scales due to baryonic feedback.

Figure 1: Weighting function for FRBs (dotted black line) and for cosmic shear
with the KiDS redshift distribution (left plot) or Euclid’s redshift
distribution (right plot). The colour bar shows the tomographic bin index. All
weighting functions have been normalised to the peak value. In the case of
cosmic shear the peak of the last tomographic bin was used.

###  2.4 Statistics

Full sky surveys measure $2\ell+1$ independent modes per multipole $\ell$. For
any projected fields $f_{i}(\hat{\boldsymbol{x}})$, one would derive modes
$f_{i,\ell m}$ from a spherical harmonic decomposition. Assuming that the
likelihood for the set of modes $\\{f_{i,\ell m}\\}$ is a multivariate
Gaussian with zero mean and covariance (or 2-point correlation)
$\boldsymbol{C}$ with components

$(\boldsymbol{C}_{\ell})_{ij}=\langle f_{i,\ell
m}f^{*}_{j,\ell^{\prime}m^{\prime}}\rangle\equiv\delta^{\mathrm{K}}_{mm^{\prime}}\delta^{\mathrm{K}}_{\ell\ell^{\prime}}C^{f_{i}f_{j}}_{\ell}\;,$
(2.18)

the Fisher matrix is given by [56]:

$F_{\alpha\beta}=f_{\mathrm{sky}}\sum_{\ell}\frac{2\ell+1}{2}\mathrm{tr}\left(\boldsymbol{C}^{-1}_{\ell}\partial_{\alpha}\boldsymbol{C}_{\ell}\boldsymbol{C}^{-1}_{\ell}\partial_{\beta}\boldsymbol{C}_{\ell}\right)\;.$
(2.19)

Here greek indices label parameters, and the observed sky fraction
$f_{\mathrm{sky}}$ accounts for incomplete sky coverage. The inverse of the
Fisher matrix $\boldsymbol{F}^{-1}$ yields the covariance matrix of the
parameters and serves as a lower limit for obtainable errors given the survey
settings by means of the Cramér-Rao bound.

Figure 2: Angular power spectra for the KiDS redshift distributions and an FRB
sample with $\alpha=3.5$ as defined in Equation 3.1. The solid black line
shows the auto-correlation spectrum of the DM, while the solid lines from blue
to yellow depict the cosmic shear power spectrum for the five tomographic bins
of KiDS. Dashed lines show the cross-correlation between the DM and cosmic
shear. The shaded areas indicate the noise contribution, ignoring sample
variance, for both measurements. Note that the noise can vary slightly across
the tomographic bins for KiDS.

Since the derivative in eq. 2.19 is calculated numerically, small difference
can propagate into the matrix inversion to obtain possible constraints.
Therefore, we check the stability of the derivative in appendix A. Lastly, we
are also calculating the signal-to-noise ratio (SNR) of the measurement, i.e.
we assume that one wants to measure the amplitude, $A$, of the cosmological
term in eqs. 2.5 and 2.17. Using eq. 2.19 for
$\boldsymbol{C}=A\boldsymbol{S}+\boldsymbol{N}$, one finds

$\Sigma^{2}(\leq\ell)=f_{\mathrm{sky}}\sum_{\ell^{\prime}}^{\ell}\frac{2\ell^{\prime}+1}{2}\mathrm{tr}\left(\boldsymbol{C}^{-1}_{\ell^{\prime}}\boldsymbol{S}_{\ell^{\prime}}\boldsymbol{C}^{-1}_{\ell^{\prime}}\boldsymbol{S}_{\ell^{\prime}}\right)\equiv\sum_{\ell^{\prime}}^{\ell}\mathcal{S}^{2}(\ell^{\prime})\;.$
(2.20)

In the case here, the covariance takes the form:

$\boldsymbol{C}_{\ell}=\begin{pmatrix}\boldsymbol{C}^{\mathcal{D}\mathcal{D}}_{\ell}&\boldsymbol{C}^{\mathcal{D}\gamma}_{\ell}\\\
\boldsymbol{C}^{\gamma\mathcal{D}}_{\ell}&\boldsymbol{C}^{\gamma\gamma}_{\ell}\end{pmatrix}\;.$
(2.21)

Note that noise between the DM and the shear are uncorrelated, hence
$\boldsymbol{C}^{\gamma\mathcal{D}}_{\ell}$ does not have a flat noise
contribution. Lastly, Equation 2.19, assumes that all probes are measured over
the same footprint $f_{\mathrm{sky}}$. In order to account for different
footprints we modify the Fisher matrix as follows:

$\boldsymbol{F}^{\mathrm{FRB}+\mathrm{lensing}}\to
f_{\mathrm{sky,\mathrm{min}}}\boldsymbol{F}^{\mathrm{FRB}+\mathrm{lensing}}+(f_{\mathrm{sky,\mathrm{max}}}-f_{\mathrm{sky,\mathrm{min}}})\boldsymbol{F}^{\mathrm{FRB}/\mathrm{lensing}}_{\mathrm{max}}\;$
(2.22)

where $f_{\mathrm{sky,min}}$ and $f_{\mathrm{sky,max}}$ identify the small and
larger footprint of the two surveys respectively and
$\boldsymbol{F}^{\mathrm{FRB}/\mathrm{lensing}}_{\mathrm{max}}$ is the
corresponding Fisher matrix of the larger footprint. This replacement assumes
that the footprints of from which the angular power spectra are estimated
overlap completely, making it equivalent to studying the power spectra
estimators with a purely Gaussian covariance.

##  3 Results

###  3.1 Survey Specifications

The survey specifications are given in Table 1 and the the following redshift
distribution for the FRBs is assumed

$n_{\mathrm{FRB}}(z)\propto z^{2}\mathrm{e}^{-z\alpha}\;,$ (3.1)

where $\alpha=3.5$ controls the depth of the survey. For the weak lensing
surveys, the redshift distributions are discussed in [45] and [57]. We show
the weighting functions $W(\chi)$ for the individual surveys in Figure 1. The
number of FRBs used to estimate the correlation is $10^{4}$, which should be
well in reach of SKA. The colour bar in fig. 1 indicates the tomographic bin
index for the weak lensing surveys. All weights are normalised to the maximum
of the respective highest redshift bin. There is substantial overlap with
Euclid (apart from the last tomographic bin) and complete overlap with a
survey such as KiDS.

Figure 2 shows the angular power spectra. The DM auto-correlation is shown in
black, while the cosmic shear spectra (also auto-correlation and not between
different tomographic bins) is shown again with the same colour code as in
Figure 1. For better visibility, only the spectra for the KiDS redshift
distribution are shown. The dashed lines correspond to the DM-shear cross-
correlation spectrum for the five tomographic bins in KiDS. Finally, the
shaded regions indicate the pure noise contribution (excluding cosmic
variance) for the DM and shear spectra. Due to the noise properties, the
majority of the signal resides well below $\ell\sim 1000$. Furthermore, one
can see that the DM correlation flattens out at high $\ell$ due to the
suppression by baryonic feedback.

Figure 3: Signal-to-Noise ratio (SNR) for a joint analysis of a cosmic shear
survey with DM statistics. dashed lines depict the squared SNR at each
multipole, while solid lines show the cumulative SNR below the indicated
multipole, $\ell$ (compare to Equation 2.20). Left: for KiDS, Right: for
Euclid. Note that the same FRB sample was considered in both cases. Figure 4:
Cumulative SNR as a function of the Noise specification in Equation 2.17,
where $\bar{n}=N_{\mathrm{FRB}}/(4\uppi)$. White lines indicate selected
curves of constant SNR.

To obtain a better understanding of the significance of each measurement, the
SNR curves are shown in Figure 3 as a function of multipole $\ell$. Dashed
lines illustrate the squared differential SNR at each multipole, while solid
lines show the integrated SNR. In the case of a stage-3 survey such as KiDS,
the SNR from the DM correlations is roughly three times as large and picks up
signal at larger scales. This is mainly due to the large $f_{\mathrm{sky}}$ in
case of the FRBs. Comparing this with Figure 2 we see that, while the noise
level is similar for both observables, the smaller area probed by KiDS reduces
the overall SNR relative to the FRBs. The green line shows that mainly the
small scales add additional signal due to the large correlation between FRBs
and lensing. For the state-4 survey the signal is now dominated by cosmic
shear. Especially at high multipoles only cosmic shear adds any signal.

Since the DM scatter of the host galaxy, $\sigma_{\mathrm{host}}$, is quite
uncertain, the total SNR is shown for different noise levels in Figure 4 where
both $\sigma_{\mathrm{host}}$ and $N_{\mathrm{FRB}}$ are varied while the rest
of the survey specifications is kept fixed (i.e. the distribution of FRBs
along the line-of-sight). It should also be noted that the host contribution
follows rather a log-normal than a Gaussian distribution [58, 59] so
$\sigma_{\mathrm{host}}$ should be seen as the corresponding variance to that
distribution which is an accurate description as long as one is considering
2-point correlations only. The fiducial value of $\sigma_{\mathrm{DM}}=50$ is
on the lower end of the plot and it might be considerably larger. On the other
hand $N_{\mathrm{FRB}}=10^{4}$ is a conservative estimate for the number of
FRBs available at later data releases by Euclid or LSST-Rubin at the end of
the decade. We would therefore argue that the SNR assumed here is in
reasonable reach in that time span.

Probe | $h$ | $\Omega_{\mathrm{cdm}}$ | $\sigma_{8}$ | $\Omega_{\mathrm{b}}$ | $w_{0}$ | $w_{1}$ |  a$\;\mathrm{log}_{10}T_{\mathrm{AGN}}$ | $\sum\\!m_{\nu}[\mathrm{eV}]$ | $n_{\mathrm{s}}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | Fisher |  |  |  |
Euclid | 0.1 | 0.019 | 0.039 | 0.015 | 0.215 | 0.908 | 0.208 | 0.334 | 0.092
KiDS | 0.678 | 0.026 | 0.08 | 0.117 | - | - | 1.145 | - | -
FRB | 1.483 | 1.471 | 0.958 | 0.128 | 9.955 | 53.246 | 1.219 | 2.559 | 2.018
Euclid+FRB | 0.064 | 0.014 | 0.022 | 0.006 | 0.17 | 0.677 | 0.014 | 0.191 | 0.037
|  ($85\;\%)$ |  ($45\;\%)$ |  ($95\;\%)$ |  ($679\;\%)$ |  ($47\;\%)$ |  ($57\;\%)$ |  ($2153\;\%)$ |  ($122\;\%)$ |  ($340\;\%)$
KiDS+FRB | 0.041 | 0.026 | 0.05 | 0.006 | - | - | 1.145 | - | -
|  |  |  | MCMC |  |  |  |
Prior | $[0.64,0.82]$ | $[0.051,0.255]h^{-2}$ | $[0.7,1.2]$ | $[0.019,0.026]h^{-2}$ | $[-1,0]$ | $[-1-w_{0},-w_{0}]$ | none | $[0,\infty]$ | none
Euclid | 0.113 | 0.01 | 0.016 | 0.013 | 0.101 | 0.237 | 0.196 | 0.137 | 0.117
Euclid + FRB | 0.065 | 0.009 | 0.012 | 0.005 | 0.092 | 0.21 | 0.017 | 0.089 | 0.051
| $(73\;\%)$ | $(5\;\%)$ | $(26\;\%)$ | $(172\;\%)$ | $(10\;\%)$ | $(12\;\%)$ | $(1020\;\%)$ | $(54\;\%)$ | $(129\;\%)$
Table 2: One dimensional marginal constraints for 68$\;\%$ confidence on the
cosmological and feedback parameters using different survey settings. The
first part of the table uses a Fisher analysis with no prior. In the lower
part we fit a noiseless data vector with using MCMC and flat priors on the
some of the parameters. For the case of Euclid+FRB we also quote the relative
reduction of error in percent. Note that the FRB case is only given for the
full parameter set.  Figure 5: Fisher matrix forecast for Euclid and an FRB
survey with $N_{\mathrm{FRB}}=10^{4}$ and $\alpha=3.5$ (compare right panel of
Figure 3). Blue contours show the constraints obtainable with, while green
contours show the improvement when adding DM correlations. The footprints over
which the obsvervables are estimated are assumed to overlap completely.
Contours from the FRBs are not shown in this Figure, as they are outperformed
for most parameters by cosmic shear in this case.

###  3.2 Cosmological Constraints

We now analyse a set of 9 (5) cosmological and feedback parameters for a
stage-4 and a stage-3 survey respectively, using the setup from the previous
section and Table 1. The parameters and the corresponding one dimensional
marginal constraints are shown in Table 2. An analogous corner plot is shown
in Figure 5 for the case of a stage-4 survey. In general we distinguish two
cases: $(i)$ FRB and cosmic shear auto-correlations separately and $(ii)$ FRB
plus cosmic shear, also including the cross-correlation measurements. From
Figure 3 we already know that a stage-4 cosmic shear survey outperforms the
considered FRB sample by almost a factor of 10 in SNR. It is thus expected
that constraints on parameters modifying the amplitude of both spectra
simultaneously will be dominated by the cosmic shear measurement. This can be
seen clearly in Figure 5, where the amplitude of the power spectrum,
$\sigma_{8}$, is very well constrained by Euclid. Similarly, one finds that
the dark energy equation of state parameters $w_{0}$ and $w_{1}$ assume all
the constraining power from the lensing signal as well. This is due to the
fact that there is a longer redshift baseline for the cosmic shear survey and
more tomographic information. The FRB sample on the other hand requires a lot
of its signal to be put into constraining the total amplitude eq. 2.11,
leaving almost no constraining power for the other parameters. Note that this
drastically changes by using FRBs with host identification [22, 23]. However,
we would like to stress that also unlocalised FRBs still add information to
the cosmic shear measurements, even to those parameters typically well
constrained by lensing alone. As explained below in more detail, this is
caused by the tighter constraints on feedback from DM correlations.

Typical parameters weakly constrained by lensing are the Hubble constant, $h$
and the baryon density $\Omega_{\mathrm{b}}$. Since both enter in the
amplitude of the DM, it is clear that they can be very well measured with the
help of FRBs. The following feature, however, is striking: the constraints on
the feedback $\log_{10}T_{\mathrm{AGN}}$ are an order of magnitude better
including FRBs compared to cosmic shear alone. The main driver for this effect
is that the baryon distribution itself is much more affected by feedback than
the total matter distribution which consists mostly of dark matter, making it
more inert to feedback. Similar performance can be expected for more
complicated feedback models with more model parameters [53, 55]. In fact, for
more free parameters of the feedback external constraints will become more and
more important. Due to the degeneracy between the sum of neutrino masses,
$\sum m_{\nu}$, and the feedback parameter, $\log_{10}T_{\mathrm{AGN}}$, the
former will benefit from the increased precision of the latter. Another
interesting degeneracy is the primordial slope of the power spectrum,
$n_{\mathrm{s}}$. Fixing this slope is important to connect the clustering
strength on small scales with the one measured on larger scales. Cosmic shear
alone can only achieve this if the feedback is constrained, showing that FRBs
help to break this degeneracy is well. Measuring $n_{s}$ precisely is very
important for the measurement of scale dependent growth on large scales, as is
predicted in many theories of modified gravity.

To mimic a more realistic situation, we create a noiseless data vector for the
three spectra and estimate the posterior distribution using Markov Chain Monte
Carlo (MCMC) sampled by emcee [60]. We impose some restrictions to the
parameter space by imposing priors on the cosmological parameters. This
particularly ensures that the nonlinear power spectrum and electron bias are
not extrapolated beyond the range of simulations used to calibrate the model
(see also table [14]). We also summarise the choices for the flat priors in
the lower half of Table 2 together with the marginal $68\%$ confidence
intervals obtained by cosmic shear alone and cosmic shear combined with FRBs.
The trends are very similar, however, we get in general less improvement. The
reason for this is that the prior range injects additional information. It can
be seen for example that the posteriors for $h$ or $\Omega_{\mathrm{b}}$ are
prior dominated in the cosmic shear case. Also the restriction to non-phantom
dark energies and no crossing of $w>0$ reduces the parameter space volume.
These effects reduce the gain from adding FRB measurements since degeneracies
are already slightly broken by the prior. Nonetheless, we still find an order
of magnitude improvement for the constraints on $\log_{10}T_{\mathrm{AGN}}$
and $50\%$ for the neutrino mass. Finally it is worth pointing out that the
addition of FRBs also allows cosmic shears to be a more independent probe from
the Cosmic Microwave Background (similar to using tSZ) as external priors on
the baryon density are not required and the parameter can be fitted
simultaneously.

Figure 6: Forecast using a noiseless data vector for Euclid and an FRB survey
with $N_{\mathrm{FRB}}=10^{4}$ and $\alpha=3.5$ (compare right panel of Figure
3). Line colours and styles are the same as in Figure 5 with the caveat that
the priors from Table 2 are used. The two contours correspond to the standard
$68$ and $95\;\%$ confidence intervals.

##  4 Conclusion

The ability of future weak lensing surveys to address their key scientific
questions, including the search for the mass scale of neutrinos, is largely
limited by our understanding of the effect of baryonic physics on the
clustering of matter on non-linear scales. While these effects can be measured
from simulations, they are vastly different depending on the sub-grid model
used. In this work, we investigated an alternative probe of the electron power
spectrum, which has been accessible only via the tSZ effect so far, the DM
angular correlations measured from FRBs. DM correlations are an integrated
quantity very similar to cosmic shear, they therefore provides access to very
similar scales and can efficiently break degeneracies between cosmological and
feedback parameters. Compared to the tSZ effect, DM correlations give
different weights to different redshifts and scales. We used a simple feedback
prescription with one parameter to model the matter, electron and matter-
electron power spectra and investigated the improvements when cross-
correlating a stage-4 cosmic shear survey with the DM from $10^{4}$ FRBs, a
number easily achievable within the next years. Our main findings can be
summarised as follows:

  1. i)

The DM of FRB observations can improve the constraints on baryonic feedback by
an order of magnitude, depending on the SNR of the detection of the DM
correlations. Simultaneously, FRBs also help to constrain the Hubble constant
and the baryon density. Combining these effects, the constraints on the
neutrino mass can be improved by 50 to 100 percent. The SNR used in this work
is well in reach within the next decade considering upcoming observations and
dedicated searches for FRBs.

  2. ii)

DM correlations have a different radial weighting function than tSZ
measurements, they are therefore sensitive to different redshifts. Furthermore
the observations do not rely directly on the CMB and are hence an entirely
different probe at low redshifts with different systematics.

There are a few limiting factors in our analysis. These include a very simple
feedback model, Gaussian statistics and the absence of systematic effects. We
did not make use of any redshift distribution of FRBs provided by host
identification. The scope of this paper, however, is to show that FRBs can be
a powerful cosmological probe in particular in conjunction with tracers of the
full matter distribution, leveraging the different clustering properties of
baryons and dark matter. In the future we intend to tackle the mentioned
limitations by considering a more realistic setup for the inference process.

## Acknowledgments

RR is supported by the European Research Council (Grant No. 770935). SH was
supported by the Excellence Cluster ORIGINS which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2094 - 390783311.

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##  Appendix A Stability of the Fisher matrix

To ensure the stability of the Fisher matrix, we vary the step size with which
we evaluate the derivative. This exercise is shown in Figure 7 for two
different choices for $\epsilon$. In particular we take the central form of
the finite difference and vary the parameters by

$\delta\theta_{i\pm}=\left\\{\begin{array}[]{ll}\theta_{i,0}(1\pm\epsilon)&\;\;\text{if
}\;\;\theta_{i,0}\neq 0\\\ \pm\epsilon&\;\;\text{else }\end{array}\right.\;,$
(A.1)

where $\theta_{i,0}$ is the parameter’s fiducial value.

Figure 7: Same as Figure 5 but with varying step size for the numerical
derivative. In particular we choose $\epsilon_{3}=10^{-3}$ and
$\epsilon_{1}=10^{-2}$ for the green and dark violet lines respectively. The
fiducial case in Figure 5 assumes $\epsilon_{1}=5\times 10^{-3}$.