Galaxy clusters in high denition a dark matter search Geo Beckand Michael Sarkis School of Physics and Centre for Astrophysics University of the Witwatersrand Johannesburg Wits 2050 South Africa.

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Galaxy clusters in high deﬁnition: a dark matter search

Geoﬀ Beck∗and Michael Sarkis

School of Physics and Centre for Astrophysics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa.

Recent radio-frequency probes, with the ATCA and ASKAP telescopes, have proven themselves

to be at the forefront of placing indirect limits on the properties of dark matter. The latter being

able to substantially exceed the constraining power of Fermi-LAT data. However, these observations

were based only on dwarf galaxies, where magnetic ﬁeld uncertainties are large. Here we re-examine

the case for galaxy clusters, often ignored due to substantial diﬀuse radio backgrounds, by consid-

ering the extrapolation of known cluster surface brightness proﬁles down to scales observable with

MeerKAT. Despite large baryonic backgrounds, we ﬁnd that clusters can be competitive with dwarf

galaxies. Extrapolated Coma data being able to rule out WIMPs of mass <700 GeV annihilating

via b-quarks. This is while having lesser uncertainties surrounding the magnetic ﬁeld and diﬀusive

environment. Such compelling results are possible due to a clash between the inner shape of the

dark matter halo and the ﬂat inner proﬁle of radio halos which is most pronounced for NFW-like

Einasto proﬁles, the presence of which having some supporting evidence in the literature.

I. INTRODUCTION

The nature of Dark Matter (DM) remains a major

open question in modern cosmology and particle physics.

So-called ‘indirect’ probes of DM, via the consequences

of annihilation or decay in cosmic structures, have made

major strides in ruling out annihilation models domi-

nated by b-quarks and τ-leptons for Weakly Interact-

ing Massive Particles (WIMPs) with masses .100 GeV

through gamma-ray telescopes like Fermi-LAT [1, 2].

Recently, radio-frequency probes have begun to realise

the potential [3–10] that had previously been argued

for [5, 11–14]. However, the majority of the radio ef-

forts have focused on dwarf galaxies, as galaxy clusters,

despite being heavily DM dominated, tend to host rela-

tively large baryonic background emissions. In Chan et

al 2020 [7], the authors look to produce tight constraints

on DM in a high redshift galaxy cluster via characterising

the cosmic-ray synchrotron contribution. In this work we

aim to explore how making use of high angular resolu-

tion radio observations of galaxy clusters can contribute

to placing powerful limits on DM despite the presence of

baryonic backgrounds. This has a signiﬁcant advantage

over the previous method [7]: it is far less uncertain.

The signiﬁcance of this work is that new, high-

resolution, radio observatories like MeerKAT are now

online (with the SKA to follow shortly). Thus, explor-

ing their potential in multiple DM dominated environ-

ments is a matter of urgency. Currently, the observed

trend in the diﬀuse radio halos of galaxy clusters is that

their spatial proﬁle is exponential [15]. This presents

an opportunity to constrain the properties of DM if the

halo density proﬁle follows a non-cored proﬁle, as the

clash between the shape of predicted DM emissions and

the observed proﬁle at small radii could be highly limit-

ing to potential annihilation cross-sections. It is there-

fore necessary to consider what the halo shape of clus-

∗Email: geoﬀrey.beck@wits.ac.za

ters tends to be. In Newman et al 2013 and Collet et

al 2017 [16, 17] the authors’ observational results sug-

gest either shallow inner slopes for DM halos or a cored

Navarro-Frenk-White (NFW) halo, in contradiction to

cold DM simulations that indicate cuspy halos. How-

ever, the authors in Mamon et al 2019 [18], ﬁnd some-

what diﬀerent results: weak evidence from the WINGS

cluster sample ruling out cored NFW proﬁles in low red-

shift clusters and no evidence for deviation from plain

NFW and NFW-like Einasto proﬁles. Notably, these re-

sults hold down to 0.03r200 which, since clusters typi-

cally have a concentration parameter ∼5, means that no

evidence for deviation from NFW proﬁles emerges even

below the characteristic scale of the halos. In He at al

2020 [19], the authors also ﬁnd steeper inner halo slopes

from simulated halos than the aforementioned observed

values [16, 17]. This is reconciled by noting the diﬀer-

ence between asymptotic and mass-weighted mean proﬁle

slopes, with the steeper asymptotic values from simula-

tion [19] having means consistent with shallower values

found in observations [16, 17] in the observed regions of

the clusters. It seems clear then that there is some evi-

dence in favour of proﬁles like NFW in galaxy clusters.

Notably, the clusters we focus on, being Coma and Ophi-

uchus, do not display any statistically signiﬁcant prefer-

ence between cored and cuspy NFW halos [20, 21], sug-

gesting the data is not suﬃcient to probe substantially

below the scale radius of the DM halos. This all means

we can have some conﬁdence in using NFW and NFW-

like Einasto proﬁles, while also considering a shallowly

cusped proﬁle for good measure.

In this work we ﬁnd that, with NFW proﬁles, nearly

an order of magnitude improvement in annihilation cross-

section upper limits is possible from cluster halo data in

the Coma and Ophiuchus clusters when the smallest ob-

servable scale is set 10 arcseconds. Since MeerKAT is

capable of imaging substantially smaller scales, down to

around 5 arcseconds [22], we ﬁnd that high-resolution re-

observation of halo-hosting clusters can provide highly

competitive limits on WIMP DM, even exceeding a re-

cent ASKAP study of the Large Magellanic Cloud [10]

arXiv:2210.00796v2 [astro-ph.CO] 17 Nov 2022

which, under reasonable assumptions about the diﬀusive

environment, rules out WIMPs with masses below 500

GeV annihilating via quarks.

This paper is structured to ﬁrst review the formal-

ism of radio emissions from WIMP annihilation in sec-

tion II. The galaxy cluster data will then be detailed in

section III. Results are presented in section IV and dis-

cussed in V.

II. RADIO EMISSIONS FROM DARK MATTER

Radio emissions are produced by DM annihilation

when relativistic electrons/positrons (electrons from here

on) are products of this process. These electrons are in-

jected continuously into the DM halo over a long period

of time. In addition to this, halo environments are com-

monly magnetised. Thus, we need to consider long-term

evolution of the injected electrons before we determine

the resulting synchrotron emissions.

A. Diﬀusion of electrons - Green’s functions

To determine our required electron distributions ψ, we

must solve an equilibrium form of the diﬀusion-loss equa-

tion:

∇·D(E, ~x)~

∇ψ+∂

∂E [b(E, ~x)ψ] + Qe(E, ~x) = 0 ,(1)

where Qeis the source function from DM injection,

D(E, ~x) is the diﬀusion function, and b(E, ~x) is the en-

ergy loss function. Note that Eq. (1) does not prevent

super-luminal diﬀusive motion [23]. However, this ef-

fect only becomes substantial for ultra-high energy cos-

mic rays [23]. The source function is given by

Qe=1

2ρχ(~x)

Mχ2

hσV iψ|inj ,(2)

where ρχis the DM density, Mχis the DM mass, hσV i

is the velocity-averaged annihilation cross-section, and

ψ|inj is the spectrum of electrons/positrons injected per

DM annihilation. To facilitate solution of the equation

we will use a Green’s function method, which requires the

diﬀusion and loss functions have no spatial dependencies.

Therefore, we deﬁne the diﬀusion function, under the

assumption of Kolmogorov turbulence, via [24]:

D(E) = 3 ×1028 d2

0,kpcEGeV

BµG!1/3

cm2s−1,(3)

where d0∼15 kpc is the coherence length of the magnetic

ﬁeld [25], d0,kpc =d0

1kpc,Bis the average magnetic

ﬁeld, BµG=B

1µG, and EGeV =E

1 GeV . We will also

explore a Bohmian diﬀusion case where [24]

DBohm(E)=3.3×1022 EGeV

BµG

cm2s−1.(4)

The energy-loss function is given by

b(E) = bICE2

GeV +bsyncE2

GeVB2

µG

+bCoulncm3 1 + 1

75 log γ

ncm3 

+bbremncm3EGeV ,

(5)

where γ=E

mec2with mebeing the electron mass, n

is the average gas density, and ncm3 =n

1cm−3. The

coeﬃcients bIC,bsync,bCoul,bbrem are the energy-loss

rates from ICS, synchrotron emission, Coulomb scatter-

ing, and bremsstrahlung. These coeﬃcients are given

by 0.25 ×10−16(1 + z)4(for CMB target photons),

0.0254 ×10−16, 6.13 ×10−16, 4.7×10−16 in units of

GeV s−1. The average quantities ¯nand ¯

Bare computed

within r≤rs(rsbeing the characteristic scale of the DM

halo), ensuring they accurately reﬂect the environment of

the majority of annihilations.

The equilibrium solutions to Eq. (1) are given by [11,

26, 27]

ψ(r, E) = 1

b(E)ZMχ

dE0G(r, ∆v)Qe(r, E0),(6)

where Gis the Green’s function, given by:

G(r, ∆v) = 1

√4π∆v

∞

n=−∞

(−1)nZrmax

dr0r0

fG,n ,(7)

fG,n =e−(r0−rn)2

4∆v−e−(r0+rn)2

4∆vQe(r0)

Qe(r),(8)

here

∆v=v(u(E)) −v(u(E0)) ,(9)

with

v(u(E)) = Zu(E)

umin

dx D(x),

u(E) = ZEmax

b(x).

(10)

B. Diﬀusion of electrons - ADI method

As an alternative solution approach we implement an

Alternating Direction Implicit (ADI) method, in which

the diﬀusion and loss functions keep their full spatial de-

pendence. In this method we set up a multi-dimensional

grid over space and energy, and make use of an operator-

splitting technique to solve for the equilibrium distribu-

tion iteratively.

It is worth noting, at this point, that although this

method is referred to in the literature as ADI, this label is

slightly misleading. We will use the ADI term, in keeping

with existing literature, but note that this method would

be more appropriately referred to as an operator splitting

method.

1. Diﬀusion and loss functions

For this method we replace Band n, in the deﬁnition

of the loss function b, with B(r) and n(r) respectively.

The diﬀusion function we deﬁne as

D(E) = 1029 B(r)

B(0)−1

GeV cm2s−1,(11)

following Regis et al 2017 [4] in their method of includ-

ing the spatial dependence of the magnetic ﬁeld. The

constant coeﬃcient was chosen to closely match that of

Green’s function method.

2. Crank-Nicolson scheme

The solution for each dimension in our grid uses a gen-

eralised Crank-Nicolson scheme [28], which is a method

of ﬁnite-diﬀerencing that uses an average of both explicit

and implicit diﬀerencing terms. This allows it to gain

the stability of an implicit method, while maintaining

second-order accuracy. In fact, this method turns out

to be unconditionally stable for any time-step size ∆t,

which is useful since the diﬀusion-loss equation consid-

ered here contains processes that operate on vastly dif-

ferent time-scales. For arbitrary diﬀusion and loss func-

tions, we write the general scheme, as in Regis et al.

[29] and Strong and Moskalenko [30],

ψn+1

i−ψn

∆t=α1ψn+1

i−1−α2ψn+1

i+α3ψn+1

i+1

2∆t

+α1ψn

i−1−α2ψn

i+α3ψn

i+1

2∆t+Qe,i ,(12)

where temporal indices are given by nand dimensional

(space, energy) indices by i. Each of the αcoeﬃcients

contain the diﬀusion and loss functions, and are found

by matching this equation to the ﬁnite-diﬀerenced form

of the diﬀusion-loss equation. By isolating the implicit

and explicit terms, we obtain the following:

−α1

2ψn+1

i−1+1 + α2

2ψn+1

i−α3

2ψn+1

i+1

=Qe,i∆t+α1

2ψn

i−1+1−α2

2ψn

i+α3

2ψn

i+1 .(13)

This now represents a system of linear equations in the

overall updating equation Aψn+1 =Bψn+Qe, where

Aand Bare tri-diagonal matrices containing the α-

coeﬃcients.

3. Operator Splitting

Since the diﬀusion-loss equation considered here is 2-

dimensional, the Crank-Nicolson scheme as presented

above would need to be generalised further, and would

thus lose the relative simplicity of the tridiagonal matrix

equation. Instead, we make use of an operator splitting

technique in which the two dimensions are treated inde-

pendently, and then solved in alternating steps using the

1-dimensional Crank-Nicolson scheme. This method has

been successfully used before, in the public code package

galprop [30] and in Regis et al. [29].

This method is implemented as follows. Firstly, we

make the simplifying assumption of spherical symmetry,

so that ~x →r, where ris the radius from the centre of

the halo. We then transform the variables Eand rto

use a logarithmic scale, to better account for the large

physical scales involved, i.e. ˜

E= log10(E/E0) and ˜r=

log10(r/r0), where E0and r0are chosen scale parameters.

With these modiﬁcations to the diﬀusion-loss equation,

we then ﬁnd a ﬁnite-diﬀerence scheme for each of the

diﬀusion and energy loss operators as follows:

∂

∂r r2D∂ ψ

∂r →

(r0log(10)10˜ri)−2ψi+1 −ψi−1

2∆˜rlog(10)D+∂D

∂˜ri

+ψi+1 −2ψi+ψi−1

∆˜r2D|i,(14)

for radius and

∂

∂E (bψ)→(E0log(10)10 ˜

Ej)−1bjψj+1 −bjψj

∆˜

E,(15)

for energy, where ∆˜rand ∆ ˜

Erepresent the radial and

energy grid spacings, respectively. Note that in the case

of energy losses, we only consider upstream diﬀerencing.

This corresponds to each grid point’s energy loss only

depending on those points with equal or higher energies

(or, to only jand j+ 1 terms entering into the updating

equation).

If these schemes are represented by Ψ, we can sum-

marise the solution method with the following steps:

ψn+1/2= Ψ ˜

E(ψn),(16)

ψn+1 = Ψ˜r(ψn+1/2).(17)

These steps are then computed on each iteration of the

algorithm in turn, updating the value of ψuntil con-

vergence is reached and the equilibrium distribution is

found.

The forms of Ψ ˜

Eand Ψ˜rare also used to ﬁnd the

values of the α-coeﬃcients present in the matrices A

and B. By equating coeﬃcients with the general 1-

dimensional Crank-Nicolson scheme given above, these

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Galaxyclustersinhighdenition:adarkmattersearchGeoBeckandMichaelSarkisSchoolofPhysicsandCentreforAstrophysics,UniversityoftheWitwatersrand,Johannesburg,Wits2050,SouthAfrica.Recentradio-frequencyprobes,withtheATCAandASKAPtelescopes,haveproventhemselvestobeattheforefrontofplacingindirectlimitsonthep...

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Galaxy clusters in high denition a dark matter search Geo Beckand Michael Sarkis School of Physics and Centre for Astrophysics University of the Witwatersrand Johannesburg Wits 2050 South Africa.

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