Fitting a Self-Interacting Dark Matter Model to Data Ranging From Satellite Galaxies to Galaxy Clusters Sudhakantha Girmohanta12and Robert Shrock1

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Fitting a Self-Interacting Dark Matter Model to Data Ranging From Satellite

Galaxies to Galaxy Clusters

Sudhakantha Girmohanta1,2and Robert Shrock1

1 C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy,

Stony Brook University, Stony Brook, New York 11794, USA and

2 Tsung-Dao Lee Institute and School of Physics and Astronomy,

Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China

We present a ﬁt to observational data in an asymmetric self-interacting dark matter model using

our recently calculated cross sections that incorporate both t-channel and u-channel exchanges in

the scattering of identical particles. We ﬁnd good ﬁts to the data ranging from dwarf galaxies to

galaxy clusters, and equivalent relative velocities from ∼20 km/sec to >

∼103km/s. We compare

our results with previous ﬁts that used only t-channel exchange contributions to the scattering.

I. INTRODUCTION

There is strong evidence for dark matter (DM), com-

prising about 85 % of the matter in the universe. Cold

dark matter (CDM) can account for structures on length

scales larger than ∼10 Mpc [1–6] (reviews include [7–

13].) However, problems have been noted with ﬁts to

observational data on shorter length scales of ∼1−100

kpc using early CDM simulations without baryon feed-

back [14–16]. These problems included the prediction of

greater density in the central region of galaxies than was

observed (the core-cusp problem), a greater number of

dwarf satellite galaxies than were seen (the missing satel-

lite problem), and the so-called “too big to fail” problem

pertaining to star formation in dwarf satellite galaxies.

Models with self-interacting dark matter (SIDM) have

been shown to avoid these problems (some reviews in-

clude [17–19]). The extension of cold dark matter N-

body simulations to include baryon feedback can amelio-

rate these problems with pure CDM simulations [20–33].

Nevertheless, cosmological models with self-interacting

dark matter (SIDM) are of considerable interest in their

own right and have been the subject of intensive study

[17–19, 34–89].

In the framework of a particle theory of dark mat-

ter, the rate of DM-DM scatterings is given by Γ =

(σ/mDM)vrelρDM , where σ,mDM,vrel, and ρDM are the

DM-DM scattering cross section, DM particle mass, rela-

tive velocity of two colliding DM particles, and DM mass

density, respectively. Fits to observational data on the

scale of ∼1−10 kpc, with velocities vrel ∼20−200 km/s,

yield values σ/mDM ∼1 cm2/g, while ﬁts to observations

of galaxy clusters on distance scales of several Mpc and

vrel ∼O(103) km/s yield smaller values of σ/mDM ∼0.1

cm2/g. This implies that viable SIDM models should

have cross sections that decrease as a function of vrel.

This property can be achieved in models in which DM

particles, denoted χhere, interact via exchange of a light

(Lorentz scalar or vector) mediator ﬁeld, generically de-

noted ξ.

In models with asymmetric dark matter (ADM), after

the number asymmetry is established in the early uni-

verse, the DM self-interaction occurs via the reaction

χ+χ→χ+χ . (1.1)

Because of the identical particles in the ﬁnal state, a

proper treatment necessarily includes both the t-channel

and the u-channel contributions to the scattering ampli-

tude. In [89], we presented diﬀerential and integrated

cross sections for the reaction (1.1) with both the t-

channel and u-channel terms included and discussed the

diﬀerences with respect to previous calculations that in-

cluded only the t-channel term. Identical-particle eﬀects

have also been noted in [58, 88] in a ﬁeld-theoretic con-

text and in [48, 79] in the context of solutions of the

Schr¨odinger equation for potential scattering. An inter-

esting question raised by our work in [89] is the following:

how do the ﬁts to observational data change when one

uses the cross section with both t-channel and u-channel

contributions to the scattering, as contrasted with pre-

vious ﬁts that used only the t-channel contributions? In

the present paper we address this question using the same

observational data set that was analyzed in [52].

II. CROSS SECTIONS

First, we review the basic properties of the SIDM

model with asymmetric dark matter that we used in [89].

In this model, the dark matter particle χis a spin-1/2

Dirac fermion, and the mediator, ξ, is a real scalar, ξ=φ,

or a vector, ξ=V. Both χand ξare singlets under the

Standard Model (SM). For the version of the model with

a real scalar mediator, we take the χ-φinteraction to be of

Yukawa form, as described by the interaction Lagrangian

LYuk =yχ[ ¯χχ]φ. In the version with a vector media-

tor, the DM fermion χis assumed to be charged under

a U(1)Vgauge symmetry with gauge ﬁeld Vand gauge

coupling g. Since only the product of the U(1)Vcharge

of χtimes goccurs in the covariant derivative in this the-

ory, we may, without loss of generality, take this charge

to be unity and denote the product as gχ. The corre-

sponding interaction Lagrangian is L¯χχV =gχ[¯χγµχ]Vµ.

A Higgs-type mechanism is assumed to break the U(1)V

symmetry and give a mass mVto the gauge ﬁeld V. For

arXiv:2210.01132v2 [hep-ph] 14 Feb 2023

compact notation, we use the same symbol, αχ, to denote

χ/(4π) for the case of a scalar mediator and g2

χ/(4π) for

the case of a vector mediator. We assume that the ki-

netic mixing of Vwith the SM hypercharge gauge boson

is negligibly small. (For an example of how this mixing

can be suppressed in a DM model with speciﬁed ultravi-

olet physics, see, e.g., [90].) In [89] the illustrative set of

values mχ= 5 GeV, mξ= 5 MeV, and αχ= 3 ×10−4

was used. Below we will show that this choice is consis-

tent with the ﬁt to astronomical data that we perform

here.

We restrict to the case where αχis small enough so

that lowest-order perturbation theory provides a reliable

description of the physics. As was shown in [89], the pa-

rameter choice used there satisﬁes this restriction while

simultaneously yielding suﬃcient depletion of the ¯χnum-

ber density in the early universe to produce the assumed

number asymmetry in our ADM model. For further de-

tails on our model, we refer the reader to [89].

The amplitude for the reaction (1.1) is M=M(t)−

M(u), where M(t)and M(u)are the t-channel and u-

channel contributions and the minus sign embodies the

eﬀect of interchange of identical fermions in the ﬁnal

state. Let us deﬁne a prefactor σ0and dimensionless

ratio ras

σ0=α2

χm2

, r =βrelmχ

mξ2

.(2.1)

where βrel =vrel/c. For all the relevant data, the val-

ues of vrel are nonrelativistic (NR). In [89] we calculated

the diﬀerential cross section in the center-of-mass (CM),

dσCM/dΩ, for the reaction (1.1) with both scalar and

vector mediators in the regime where the Born approxi-

mation is valid. In the NR limit relevant to ﬁtting data,

the results for the scalar and vector mediators are equal

and are [89]

dσ

dΩCM,NR

=σ01

(1 + rsin2(θ/2))2+1

(1 + rcos2(θ/2))2−1

(1 + rsin2(θ/2))(1 + rcos2(θ/2)) .

(2.2)

The terms on the right-hand side of Eq. (2.2) are from

|M(t)|2,|M(u)|2, and [M(t)∗M(u)+M(u)∗M(t)], respec-

tively. The angular integrals of these terms are corre-

spondingly denoted as σ(t),σ(u), and σ(tu). Because of

the identical particles in the ﬁnal state, a scattering event

in which a scattered χparticle emerges at angle θis in-

distingishable from one in which a scattered χemerges at

angle π−θ. The total cross section for the reaction (1.1)

thus involves a symmetry factor of 1/2 to compensate for

the double-counting involved in the integration over the

range θ∈[0, π]:

σ=1

2ZdΩdσ

dΩCM

.(2.3)

Owing to the symmetry dσ

dΩCM (θ) = dσ

dΩCM (π−θ),

this is equivalent to a polar angle integration from 0 to

π/2.

To describe the thermalization eﬀects of DM-DM scat-

tering, cross sections that give greater weight to large-

angle scattering have also been used in ﬁts to data.

These include the transfer (T) cross section dσT/dΩ =

(1 −cos θ)(dσ/dΩ)CM and the viscosity (V) cross sec-

tion, dσV/dΩ = (1 −cos2θ)(dσ/dΩ)CM. These have

the respective weighting factors wT(θ)=1−cos θand

wV(θ)=1−cos2θ, as indicated. Ref. [46] suggested the

use of the viscosity cross section σVfor studies of SIDM

thermalization eﬀects, and recently, Ref. [88] ﬁnds that

σVprovides a very good description of thermalization

eﬀect of SIDM scattering. However, since σThas been

used in a number of past ﬁts to observational data, we

include results for it here for completeness. We obtained

the integrated cross sections (given as Eqs. (4.30) and

(4.38) in [89])

σ=σT= 4πσ01

1 + r−ln(1 + r)

r(2 + r)(2.4)

and

σV=8πσ0

r2−5 + 2(5 + 5r+r2) ln(1 + r)

(2 + r)r.

(2.5)

For a given weighting factor, the integrals of the terms

(dσ/dΩ)CM were denoted σ(t),σ(u), and σ(tu)and ana-

lytic expressions for these were given for σ=σTand σV

in our previous work [89]. We note that (when one in-

cludes both t-channel and u-channel contributions) since

the weighting factor wT(θ) = (1 −cos θ) has no net ef-

fect in suppressing contributions from scattering events

that do not produce thermalization, it follows that σT

could overestimate the thermalization eﬀect from SIDM

self-scattering.

In the literature, in the same NR Born regime for the

reaction (1.1) a formula was used for the diﬀerential cross

section of reaction (1.1) that implicitly assumed that the

colliding particles were distinguishable (e.g., Eq. (5) in

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FittingaSelf-InteractingDarkMatterModeltoDataRangingFromSatelliteGalaxiestoGalaxyClustersSudhakanthaGirmohanta1;2andRobertShrock11C.N.YangInstituteforTheoreticalPhysicsandDepartmentofPhysicsandAstronomy,StonyBrookUniversity,StonyBrook,NewYork11794,USAand2Tsung-DaoLeeInstituteandSchoolofPhysicsandAst...

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Fitting a Self-Interacting Dark Matter Model to Data Ranging From Satellite Galaxies to Galaxy Clusters Sudhakantha Girmohanta12and Robert Shrock1

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