# Gravitational scattering of the sterile neutrino halo dark matter

Man Ho Chan
Department of Science and Environmental Studies, The Education University of
Hong Kong, Tai Po, Hong Kong<EMAIL_ADDRESS>
(Accepted XXXX, Received XXXX)

###### Abstract

A recent study shows that gravitational scattering of dark matter, in the form
of massive objects with mass $m\sim 10^{3}-10^{4}M_{\odot}$, could provide a
possible solution to alleviate the small-scale structure problems of cold dark
matter. The scattering cross section is velocity-dependent so that this
scenario can explain why self-interaction of dark matter is significant in
dwarf galaxies, but not in massive galaxies and galaxy clusters. In this
Letter, we show that this kind of dark massive objects could be made of
sterile neutrinos with a possible rest mass range $m_{\nu}\sim 7.6$ keV $-$ 71
MeV. This mass range generally satisfies most of the current observational
constraints. The entire structure of the sterile neutrino halos can be simply
predicted from standard physics.

###### keywords:

(cosmology:) dark matter; cosmology: theory

††pagerange: Gravitational scattering of the sterile neutrino halo dark
matter–References††pubyear: XXXX

##  1 Introduction

The idea of self-interacting dark matter (SIDM) has been proposed for more
than two decades. The main reason for introducing SIDM is to solve the small-
scale structure problems of cold dark matter (CDM). The CDM model predicts
that the central cold dark matter density exhibits a cusp-like profile
(Navarro, Frenk & White, 1997). This is generally true for massive galaxies
(Iocco, Pato & Bertone, 2015; Sofue, 2015) and galaxy clusters (Viola et al.,
2015) while observations indicate that many dwarf galaxies exhibit core-like
density profiles (Zackrisson et al., 2006; de Blok, 2010). The discrepancy
between the CDM prediction and the core-like structures observed in dwarf
galaxies is now known as the core-cusp problem (de Blok, 2010). In view of
this problem, Spergel & Steinhardt (2000); Firmani et al. (2000) show that
dark matter with self-interaction could modify the central density cusp to
core. This effect has been verified by many recent numerical simulations
(Sameie et al., 2020; Silverman et al., 2022). Moreover, the SIDM model can
also help solve the too-big-to-fail problem (Vogelsberger, Zavala & Loeb,
2012; Kaplinghat, Valli & Yu, 2019). Therefore, the SIDM model has become a
popular scenario to solve the small-scale structure problems in the CDM model.

To solve the small-scale structure problems, the self-interacting cross
section per unit mass should be $\sigma/m\sim 1-10$ cm2/g (Kaplinghat, Valli &
Yu, 2019; Silverman et al., 2022). However, recent observations have placed
stringent constraints on the upper limit of the self-interacting cross section
per unit mass. For example, The study in Randall et al. (2008) analyze the
data of the Bullet Cluster and obtain a strong upper limit $\sigma/m\leq 1.25$
cm2/g (68% confidence). Recent analyses have improved the limit to
$\sigma/m\leq 0.35$ cm2/g (Peter et al., 2013; Sagunski et al., 2021).
Besides, there is no apparent discrepancy with the CDM model for massive
galaxies and galaxy clusters. Therefore, Loeb & Weiner (2011) propose that the
self-interacting cross section might be velocity-dependent so that self-
interaction of dark matter is important in dwarf galaxies only (with low
velocity). In particular, if the self-interaction is mediated by a Yukawa
potential, the self-interacting cross section would be inversely proportional
to the fourth power of dark matter velocity $v$ (Loeb & Weiner, 2011). This
idea can perfectly address the small-scale structure problems and maintain the
success of the CDM model in large-scale structures simultaneously (Loeb &
Weiner, 2011; Chan, 2013).

Recently, a study has suggested a new scenario of velocity-dependent SIDM
(Loeb, 2022). By assuming that dark matter exists in the form of massive
halos, the gravitational scattering among the massive halos can provide the
required self-interaction to solve the small-scale structure problems. The
term ‘self-interaction’ here refers to the gravitational scattering of the
massive halos. The cross section per unit mass has a velocity dependence of
$v^{-4}$ and the possible mass range of the halos is $m\sim
10^{3}-10^{4}M_{\odot}$ (Loeb, 2022). In this Letter, by following standard
quantum and gravitational physics, we show that sterile neutrinos can form
this kind of dark massive halos. The possible range of sterile neutrino mass
is $\sim 7.6$ keV $-$ 71 MeV, which is consistent with the proposals of the
cosmological warm dark matter (Dodelson & Widrow, 1994; Shi & Fuller, 1999).

##  2 Sterile neutrino halos

Observations indicate that active neutrinos have rest mass (Fukuda et al.,
1998), which possibly suggests that right-handed neutrinos might also exist.
Many particle physics models propose that there exist the 4th-type neutrinos
called sterile neutrinos with mass larger than keV (Adhikari et al., 2017).
They can be produced via resonant mechanism (active-sterile neutrino
conversion) (Shi & Fuller, 1999) or non-resonant mechanism (oscillation
between active and sterile neutrinos) (Dodelson & Widrow, 1994) in the early
universe. They are commonly proposed as a candidate of warm dark matter. Many
recent studies have focused on the decaying properties of sterile neutrinos
(Chan & Chu, 2008, 2011; Bulbul et al., 2014; Boyarsky et al., 2014).

Sterile neutrinos are fermions. They have non-zero rest mass and they would
collapse gravitationally. However, this gravitational collapse would not
necessarily form a black hole because sterile neutrinos would exert quantum
degeneracy pressure $P(r)$ when their density $\rho_{\nu}(r)$ is high:

$P(r)=\frac{h^{2}}{5m_{\nu}^{8/3}}\left(\frac{3}{4\pi
g_{s}}\right)^{2/3}[\rho_{\nu}(r)]^{5/3}=K[\rho_{\nu}(r)]^{5/3},$ (1)

where $g_{s}$ is the particle spin degeneracy of sterile neutrinos. In the
followings, we assume $g_{s}=2$ for simplicity (Gómez, 2019). This quantum
degeneracy pressure can balance the gravitational attraction of the sterile
neutrinos to achieve a hydrostatic equilibrium:

$\frac{dP(r)}{dr}=-\frac{Gm(r)\rho_{\nu}(r)}{r^{2}},$ (2)

where

$m(r)=4\pi\int_{0}^{r}r^{\prime 2}\rho_{\nu}(r^{\prime})dr^{\prime}.$ (3)

We can combine the above three equations to form the Lane-Emden equation with
the polytropic index $n=3/2$ (Domcke & Urbano, 2015; Gómez, 2019):

$\frac{1}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}\frac{d\theta}{d\xi}\right)=-[\theta(\xi)]^{3/2},$
(4)

where $\rho_{\nu}(\xi)=\rho_{c}[\theta(\xi)]^{3/2}$,
$r=\xi(5K\rho_{c}^{-1/3}/8\pi G)^{1/2}$, and $\rho_{c}$ is the central
density.

Solving the Lane-Emden equation, the radius $R$ and the total mass $m$ of the
sterile neutrino halo are (Domcke & Urbano, 2015; Gómez, 2019)

$R=3.654\left(\frac{5K}{8\pi G}\right)^{1/2}\rho_{c}^{-1/6},$ (5)

and

$m=34.11\left(\frac{5K}{8\pi G}\right)^{3/2}\rho_{c}^{1/2}$ (6)

respectively. By combining Eqs. (5) and (6), we can get the relation between
$R$ and $m$ (Domcke & Urbano, 2015):

$R=11.85\left(\frac{5K}{8\pi G}\right)m^{-1/3}=192~{}{\rm
pc}\left(\frac{m_{\nu}}{1~{}\rm
keV}\right)^{-8/3}\left(\frac{m}{10^{4}M_{\odot}}\right)^{-1/3}.$ (7)

Generally speaking, the time required for the gravitational collapse of
sterile neutrinos can be estimated by the free-falling time (Phillips, 1994):

$t_{ff}=\sqrt{\frac{3\pi}{32G\langle\rho(z)\rangle}},$ (8)

where $\langle\rho(z)\rangle$ is the average cosmological dark matter density
at redshift $z$. As discussed in Loeb (2022), the minimum formation redshift
of the dark matter halo could be $z>700$. If the sterile neutrinos begin to
form the halos at $z=700$, we have $\langle\rho(z)\rangle\sim 1.4\times
10^{10}M_{\odot}$ kpc-3 and $t_{ff}\sim 2$ Myr ($\sim 7\times 10^{13}$ s).
Based on this estimated time $t_{ff}\sim 2$ Myr, the formation of the halos
would be finished before $z\sim 200$, which satisfies the criterion of the
first virialized mini-halo formation at $z\sim 70$ (Loeb, 2022). In fact,
similar studies have been done previously in examining the behaviors of the
Fermion ball dark matter (Munyaneza & Viollier, 2002; Domcke & Urbano, 2015;
Gómez, 2019) or neutrino halo dark matter (Viollier, Trautmann & Tupper, 1993;
Chan & Chu, 2008). However, the scopes of these studies are entirely different
from that in this study.

##  3 Constraints of the sterile neutrino mass

According to the gravitational scattering model proposed by Loeb (2022), the
physical size of the dark massive halo must be smaller than the minimum impact
parameter $b_{\rm min}=2Gm/v^{2}$ for the gravitational scattering. Therefore,
we have $R<2Gm/v^{2}$. The first constraint is thus given by:

$223\left(\frac{m_{\nu}}{1~{}{\rm keV}}\right)^{-8/3}\left(\frac{v}{10~{}{\rm
km/s}}\right)^{2}<\left(\frac{m}{10^{4}M_{\odot}}\right)^{4/3}.$ (9)

Generally speaking, a small value of $m_{\nu}$ would give a large $R$.
Therefore, the constraint in Eq. (9) sets the lower bound of the sterile
neutrino mass $m_{\nu}$.

On the other hand, if the sterile neutrino density is very high, the
degenerate sterile neutrinos are relativistic and the equation of state would
change to $P(r)\propto[\rho_{\nu}(r)]^{4/3}$. In this case, the situation
would be similar to the case of a white dwarf (Phillips, 1994). There exists
an upper limit of the mass (the Chandrasekhar mass) for the sterile neutrino
halos. Therefore, we can write the ‘Chandrasekhar limit’ of the sterile
neutrino halos as

$M_{\rm Ch}\approx 5.0\times 10^{12}M_{\odot}\left(\frac{m_{\nu}}{1~{}\rm
keV}\right)^{-2}.$ (10)

Since the value of $m$ must be smaller than the Chandrasekhar limit (i.e.
$m<M_{\rm Ch}$), we can get the second constraint:

$m<5.0\times 10^{12}M_{\odot}\left(\frac{m_{\nu}}{1~{}\rm keV}\right)^{-2}.$
(11)

Here, as a large value of $m_{\nu}$ would give a small $M_{\rm Ch}$, the
constraint in Eq. (11) sets the upper bound of the sterile neutrino mass
$m_{\nu}$. Note that the Chandrasekhar limit here is the mass limit of the
case with the largest possible mass density of sterile neutrinos (infinitely
large), which is equivalent to binding the relativistic sterile neutrinos
together. For binding non-relativistic sterile neutrinos, the maximum possible
mass of the halo must be smaller than the Chandrasekhar limit. Nevertheless,
it is very difficult to determine the robust upper mass limit for non-
relativistic sterile neutrinos as the actual maximum central mass density is
uncertain. Therefore, the Chandrasekhar limit used here is a very conservative
limit.

As pointed out in Loeb (2022), the dark matter self-interacting (gravitational
scattering) cross section per unit mass is given by

$\frac{\sigma}{m}=10~{}{\rm
cm^{2}/g}\left[\frac{(m/10^{4}M_{\odot})}{(v/10~{}{\rm km/s})^{4}}\right].$
(12)

To solve the core-cusp problem in dwarf galaxies, the expected cross section
per unit mass is $\sigma/m\sim 1-10$ cm2/g (Kaplinghat, Valli & Yu, 2019;
Silverman et al., 2022). Assuming the fiducial value of the velocity in dwarf
galaxies $v\sim 10$ km/s (Loeb, 2022), the possible range of $m$ should be
$10^{3}-10^{4}M_{\odot}$. Note that the value of $\sigma/m$ is proportional to
$mv^{-4}$. A smaller value of $v$ would give a smaller possible upper limit of
$m$. Nevertheless, some values of $v$ obtained by observational data are
smaller than $v=10$ km/s (e.g., Carina dwarf galaxy) (Walker et al., 2007).
Therefore, the upper limit of $m\sim 10^{4}M_{\odot}$ considered in this study
based on the fiducial value $v=10$ km/s is somewhat conservative. By
considering the range $m=10^{3}-10^{4}M_{\odot}$ and the two constraints in
Eqs. (9) and (11), the possible range of $m_{\nu}$ is $\sim 7.6$ keV $-$ 71
MeV, assuming $v=10$ km/s (see Fig. 1).

Moreover, we can also estimate the possible range of $m_{\nu}$ by using the
Jeans mass analysis. The sterile neutrinos would undergo gravitational
collapse when the mass of halo $m$ is larger than the Jeans mass $M_{J}$. The
Jeans mass is given by:

$M_{J}\approx\frac{\pi c_{s}^{3}}{6G^{3/2}\langle\rho_{\nu}\rangle^{1/2}},$
(13)

where $\langle\rho_{\nu}\rangle$ is the average sterile neutrino density at
the formation time and
$c_{s}=\sqrt{dP/d\rho_{\nu}}=\sqrt{(5/3)K\langle\rho_{\nu}\rangle^{2/3}}$ is
the sound speed. If we take the typical value of the central density of a
dwarf galaxy as the average sterile neutrino density
$\langle\rho_{\nu}\rangle=3\times 10^{7}M_{\odot}$ kpc-3 for an estimation
(Loeb, 2022), we get

$M_{J}=1.4\times 10^{4}M_{\odot}\left(\frac{m_{\nu}}{1~{}\rm
keV}\right)^{-4}.$ (14)

Therefore, for $M_{J}=10^{3}M_{\odot}-10^{4}M_{\odot}$, we get $m_{\nu}\sim
1-2$ keV. In other words, for $m_{\nu}<1$ keV, the halo mass with
$m<10^{4}M_{\odot}$ would be difficult to form. Therefore, our proposed range
of $m_{\nu}$ can satisfy the Jeans criterion for the gravitational collapse of
sterile neutrino halos. Generally speaking, there is no upper limit of
$m_{\nu}$ based on the Jeans mass analysis.

##  4 Discussion

In this Letter, we have investigated the possibility of the degenerate sterile
neutrino halos being the self-interacting dark massive objects suggested in
Loeb (2022). Note that sterile neutrinos do not have self-interaction except
gravity. The term ‘self-interacting’ here refers to the gravitational
scattering of the massive objects formed by the sterile neutrinos. The sterile
neutrinos formed in the early universe would collapse into the dark massive
objects due to self-gravitational attractive force. The required formation
time is much shorter than the current cosmological age. By using the known
properties of neutrinos (can exert quantum degeneracy pressure) and following
standard gravitational physics, we can get the intrinsic relation of $m$ and
$R$ and constrain the allowed range of $m_{\nu}$. The sterile neutrino halos
can be viewed as a large dark matter particle (with size $<1$ pc) and they
behave like cold dark matter. Moreover, the gravitational scattering of these
dark massive objects can help explain the small-scale problem in dwarf
galaxies because the cross section is strongly velocity-dependent. Here, we
have shown that a wide range of sterile neutrino mass ($\sim 7.6$ keV $-$ 71
MeV) can form the suggested dark massive objects. This range of $m_{\nu}$ can
also satisfy the Jeans criterion for the formation of $\sim
10^{3}-10^{4}M_{\odot}$ sterile neutrino halos at the centre of a dwarf
galaxy.

The theoretical model-independent lower limit of sterile neutrino mass can be
constrained by the Tremaine-Gunn bound (Tremaine & Gunn, 1979). Using current
data, the Tremaine-Gunn lower bound for fermionic dark matter is about $\sim
100$ eV (Davoudiasl, Denton & McGady, 2021). For the specific non-resonant
production mechanism for sterile neutrinos, the most stringent Tremaine-Gunn
lower bound is $m_{\nu}>2.79$ keV (Boyarsky, Ruchayskiy & Iakubovskyi, 2009).
Therefore, our constrained range is far above these lower bounds. Moreover, if
sterile neutrino can decay, the sterile neutrino mass $m_{\nu}$ can be
constrained by the X-ray/gamma-ray flux limit observations. However, since the
decay rate depends on the model-dependent mixing angle $\theta$ (Barger,
Phillips & Sarkar, 1995), the X-ray/gamma-ray flux limits can only constrain a
certain parameter space of $\theta$ and $m_{\nu}$ (Boyarsky, Ruchayskiy &
Iakubovskyi, 2009; Roach et al., 2020), unless there is a sharp excess line
observed. For some specific production mechanisms, the mixing angle determines
both the sterile neutrino dark matter abundance and decay rate so that the
parameter space of $\theta$ and $m_{\nu}$ can be further constrained
(Boyarsky, Ruchayskiy & Iakubovskyi, 2009; Ng et al., 2019; Roach et al.,
2020). In these cases, X-ray/gamma-ray flux limits can provide constraints for
$m_{\nu}$. Nevertheless, the most stringent constraints based on the X-ray
data of the M31 galaxy and our Galaxy still allow $m_{\nu}\geq 7$ keV (Ng et
al., 2019; Roach et al., 2020). Generally speaking, our proposed range of
$m_{\nu}$ can satisfy most of the current bounds (Gelmini, Lu & Takhistov,
2019).

If our model is correct, the dark matter halo mass $m$ is not constant because
it depends on the central mass density $\rho_{c}$. Therefore, there might
exist a distribution of $m$ inside a galaxy or a galaxy cluster. Moreover, it
is also possible that some sterile neutrinos are completely free (without
forming any massive halos). If the amount of the free sterile neutrinos is
small, then it would be very difficult to get any positive signals from the
direct-detection experiments of sterile neutrinos (Campos & Rodejohann, 2016;
Shoemaker & Wyenberg, 2019).

Some previous studies have claimed the discoveries of the decaying sterile
neutrino signals with $m_{\nu}\approx 7$ keV (Bulbul et al., 2014; Boyarsky et
al., 2014) or $m_{\nu}\approx 17$ keV (Prokhorov & Silk, 2010; Chan & Chu,
2011). However, these discoveries are quite controversial now (Bhargava et
al., 2020; Dekker et al., 2021; Silich et al., 2021). In particular, the
claimed value of $m_{\nu}\approx 7$ keV is just marginally consistent with our
possible range obtained. This value has also been challenged by the recent
studies of the Lyman-$\alpha$ forest (Garzilli et al., 2019; Enzi et al.,
2021), strong lensing (Vegetti et al., 2018; Enzi et al., 2021), satellite
galaxy count (Cherry & Horiuchi, 2017; Nadler et al., 2021), and the 21-cm
signal (Vipp, Hektor & Hütsi, 2021). In particular, the 21-cm data have placed
a lower limit $m_{\nu}\geq 63^{+19}_{-35}$ keV for the non-resonant production
mechanism (Vipp, Hektor & Hütsi, 2021). Our results combining with the other
recent bounds generally favor $m_{\nu}\geq 10$ keV. In view of this, the decay
signals of sterile neutrinos might provide a direct evidence of the existence
of sterile neutrinos. These potential signals could be observed by future
X-ray or MeV gamma-ray telescopes.

Figure 1: The unshaded region represents the allowed parameter space for
$m_{\nu}$ and $m$. The shaded regions are the forbidden regions constrained by
the conditions $R<b_{\rm min}$, $m<M_{\rm Ch}$ and
$10^{3}M_{\odot}<m<10^{4}M_{\odot}$. Here, we have assumed $v=10$ km/s.

##  5 Acknowledgements

We thank the anonymous referee for useful constructive feedback and comments.
The work described in this paper was partially supported by the Seed Funding
Grant (RG 68/2020-2021R) and the Dean’s Research Fund of the Faculty of
Liberal Arts and Social Sciences, The Education University of Hong Kong, Hong
Kong Special Administrative Region, China (Project No.: FLASS/DRF 04628).

##  6 Data availability statement

The data underlying this article will be shared on reasonable request to the
corresponding author.

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