# Dark matter in compact stars

Joseph Bramante<EMAIL_ADDRESS>Department of Physics, Engineering
Physics, and Astronomy, Queen’s University, Kingston, Ontario, K7N 3N6, Canada
The Arthur B. McDonald Canadian Astroparticle Physics Research Institute,
Kingston, Ontario, K7L 3N6, Canada  Perimeter Institute for Theoretical
Physics, Waterloo, Ontario, N2L 2Y5, Canada  Nirmal Raj<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of Science, C.V. Raman
Avenue, Bengaluru 560012, India

###### Abstract

White dwarfs and neutron stars are far-reaching and multi-faceted laboratories
in the hunt for dark matter. We review detection prospects of wave-like,
particulate, macroscopic and black hole dark matter that make use of several
exceptional properties of compact stars, such as ultra-high densities, deep
fermion degeneracies, low temperatures, nucleon superfluidity, strong magnetic
fields, high rotational regularity, and significant gravitational wave
emissivity. Foundational topics first made explicit in this document include
the effect of the “propellor phase” on neutron star baryonic accretion, and
the contribution of Auger and Cooper pair breaking effects to neutron star
heating by dark matter capture.

###### Contents

  1. 1 Introduction
  2. 2 The physics of compact objects
    1. 2.1 Fermi gas model and maximum masses
    2. 2.2 Structure equations and equation of state
    3. 2.3 Spin periods
    4. 2.4 Neutron star substructure
    5. 2.5 Thermonuclear explosions
    6. 2.6 Cooling
      1. 2.6.1 White dwarf cooling.
      2. 2.6.2 Neutron star cooling.
      3. 2.6.3 Comparison of white dwarf and neutron star late-stage cooling
    7. 2.7 Nucleon superfluidity
    8. 2.8 Neutron star magnetic field and spin-down
  3. 3 The white dwarf as a dark matter laboratory
    1. 3.1 Dark matter annihilation inside and heating white dwarfs
    2. 3.2 Non-annihilating dark matter converting white dwarfs into black holes
    3. 3.3 White dwarf explosions via dark matter
    4. 3.4 Dark matter’s influence on white dwarf equations of state
  4. 4 The neutron star as a dark matter laboratory
    1. 4.1 Dark matter kinetic and annihilation heating of neutron stars
      1. 4.1.1 Capture and kinetic heating
      2. 4.1.2 Dark matter self-annihilations, nucleon co-annihilations, and induced nucleon decay
      3. 4.1.3 Improvements and uncertainties
      4. 4.1.4 Dark matter models that heat neutron stars through scattering and annihilation
      5. 4.1.5 Neutron star reheating mechanisms not involving dark matter
    2. 4.2 Neutron stars and halo substructure
    3. 4.3 Dark matter inducing superbursts in neutron stars
    4. 4.4 Dark matter that implodes neutron stars into black holes
      1. 4.4.1 Dark matter thermalization in neutron stars
      2. 4.4.2 Collapse of dark matter and formation of small black hole
      3. 4.4.3 Growth or evaporation of dark matter-formed black hole in the neutron star
      4. 4.4.4 Signatures of dark matter that implodes neutron stars
    5. 4.5 Primordial black hole dark matter and neutron stars
    6. 4.6 Neutron stars admixed with dark sectors
      1. 4.6.1 Impact on nuclear equation of state
      2. 4.6.2 More admixed neutron stars
    7. 4.7 Exotic compact stars
    8. 4.8 Dark sectors leading to internal heating of neutron stars
    9. 4.9 Dark matter signals in gravitational waves from neutron star mergers
    10. 4.10 Dark matter signals in pulsar timing
      1. 4.10.1 Pulsar timing arrays
      2. 4.10.2 Binary pulsar timing
      3. 4.10.3 Pulsar spin-down
    11. 4.11 Axion-like and very light dark matter, and neutron stars
  5. 5 Conclusions and perspective
  6. Acknowledgments

##  1 Introduction

Dark matter is one of the foremost scientific mysteries of our times. Given
how little is known about its microphysical properties, its possible
identities seem limitless. This is famously encapsulated in the 90+ orders of
magnitude that dark matter (DM) masses could span, from $10^{-24}$ eV, set by
the maximum possible Compton wavelength containable within a dwarf galaxy, to
$10^{8}M_{\odot}\simeq 10^{74}$ eV, the mass of DM in a small galaxy. Over
this range of masses DM may be described as a wave/field, a particle, a
macroscopic object, or galactic substructure – including black holes and
topological defects. A promising strategy to confront such remarkable
diversity in possibility is to exploit physical systems with remarkable
diversity in characteristics.

Compact stars – white dwarfs (WDs) and neutron stars (NSs) typically formed as
relics of nuclear-powered stars – afford such an environment. Since their
quantum properties were first described in the 1920s$-$30s by Fowler [1],
Anderson [2], Stoner [3], Chandrasekhar [4], Zwicky and Baade [5] (and
possibly Landau [6]), our understanding of compact stars has been enriched at
the intersection of several branches of physics: astrophysics, general
relativity, particle physics, nuclear physics, statistical physics,
thermodynamics, and plasma physics. It is little wonder that they feature in
numerous tests of fundamental physics [7, 8], and it should come as no
surprise that they are also ideal laboratories to search for dark matter.
Indeed, DM hunters would do well to take advantage of their striking
properties: they have very high densities, with accompanying steep
gravitational potentials, sometimes deeply degenerate constituent fermions,
often very low temperatures, the presence of nucleon superfluidity, ultra-
strong magnetic fields, extreme regularity in rotation rivalling the precision
of atomic clocks, and powerful gravitational radiation emitted during binary
mergers, to name a few.

The use of stars to look for evidence of DM dates to proposals that weakly
interacting particle DM might alter nuclear reaction rates in the Sun [9, 10].
Shortly after, it was realized that NSs were useful for seeking out certain
models of DM that could form black holes in their interior [11]. One immediate
difference between a search for DM in compact stars and a terrestrial detector
is that, since DM is accelerated in the deep gravitational well of a compact
star, its interactions with stellar constituent particles occur at semi-
relativistic velocities: $\mathcal{O}(10^{-2}-10^{-1})~{}c$ for a WD and
$\mathcal{O}(0.5)~{}c$ for a NS. This high DM speed provides enhanced
sensitivity to theoretical models with velocity-suppressed rates for
scattering on Standard Model (SM) states, since in the Milky Way’s Galactic
halo (and by extension in terrestrial detectors) the velocity of DM particles
is only $\mathcal{O}(10^{-3})c$. In particular, the environs of a NS are
greatly suited to testing the origin of DM, since the kinetic energy of DM at
speeds $\sim 0.7c$ are similar to that during cosmological production,
particularly for “freeze-out” processes [12].

This review is organized as follows. In Section 2 we provide an overview of
the properties of NSs and WDs, emphasizing aspects that will be important for
dark matter searches. In Section 3, we describe WD searches for dark matter,
treating dark matter annihilation and heating of WDs, conversion of WDs to
black holes, ignition of Type Ia supernovae, and effects of dark matter on WD
equations of state. In Section 4, we describe NS searches for dark matter,
including dark matter heating NSs kinetically and via annihilations, models of
dark matter that convert NSs to black holes, exotic compact stars that
constitute dark matter, NSs admixed with dark matter, models of dark matter
that lead to internal heating of NSs, signals of dark matter in NS-related
gravitational waves and pulsar timing, and the utility of NSs in discovering
axion-like and primordial black hole dark matter. In Section 5, we briefly
discuss future research directions for dark matter in compact stars.

##  2 The physics of compact objects

A detailed account of the physical characteristics of WDs and NSs is beyond
the scope of this review, and for these we refer the reader to Refs. [13] and
[14]. Here we outline key properties of these stars that make them useful dark
matter detectors.

White dwarfs are stellar remnants formed from main sequence stars that undergo
a red giant phase not hot enough to fuse carbon. Depending on its mass, a WD
will be composed of some proportion of helium, carbon, oxygen, neon and
magnesium, which make up the bulk of the mass. A sea of electrons co-habiting
with nuclei provide, as we will see, the Fermi degeneracy pressure that
supports the WD against gravitational collapse.

Super-giant progenitors of mass around 10–25 $M_{\odot}$ that undergo core-
collapse leave behind short-lived “proto-NSs” through which neutrinos diffuse
out carrying away 99% of the star’s binding energy, following which NSs are
born. They are composed mainly of Fermi degenerate neutrons formed by
electron-proton capture, $e^{-}+p\rightarrow n+\nu_{e}$, at extreme densities
and temperatures. Due to beta chemical equilibrium, NSs are also thought to
contain populations of protons, electrons, and muons; it is in fact the filled
Fermi seas of these fermionic fields that keep neutrons from decaying to
protons, electrons, and muons inside NSs. The supernova collapse is
generically expected to be hydrodynamically asymmetric, resulting in a natal
“kick” to the NS at 450-1000 km/s speeds in a random direction [15, 16, 17,
18, 19]; a 1% fractional anisotropy in the momenta of escaping neutrinos could
be another source of the asymmetric kick [20, 21, 22].

###  2.1 Fermi gas model and maximum masses

Compact stars, especially WDs, are prevented from collapsing under their own
gravity by Fermi degeneracy pressure. In a low-temperature Fermi gas of Fermi
momentum $p_{F}$, the number of fermions (of spin degeneracy $g_{s}=2$)
filling up a real volume $V$ and Fermi sphere volume $V_{F}=4\pi p_{F}^{3}/3$,
is $N_{f}=g_{s}VV_{F}/(2\pi)^{3}$, from which we obtain:

$p_{F}=(3\pi^{2}n)^{1/3}~{},$ (1)

where $n$ is the fermion number density. The total energy of the Fermi gas
given the energy of a state $e(p)$ is

$E=4\pi g_{s}V\int_{0}^{p_{F}}dpp^{2}e(p)~{},$ (2)

and for an energy density $\varepsilon=E/V$ the pressure is obtained as

$P=-\bigg{(}\frac{\partial E}{\partial V}\bigg{)}_{N_{f}}=n^{2}\frac{{\rm
d}}{{\rm d}n}\bigg{(}\frac{\varepsilon}{n}\bigg{)}~{}.$ (3)

Setting $e(p)=m_{f}+p^{2}/(2m_{f})$ in the non-relativistic limit and $e(p)=p$
in the relativistic limit, and using Eqs. (1),(2) and (3), we get the Fermi
degeneracy pressure of a species as

$P=\begin{cases}[(3\pi^{2})^{2/3}/5m_{f}]\ n^{5/3}~{},~{}~{}p_{F}\ll
m_{f}~{},\\\ [(3\pi^{2})^{1/3}/4]\ n^{4/3}~{}~{}~{}~{}~{},~{}~{}p_{F}\gg
m_{f}~{}.\end{cases}$ (4)

The net pressure of the compact star is the sum of the contributions of
constituent species. In WDs, the electrons are unbound – the electron-nucleus
Coulomb energy $\simeq Ze^{2}(n_{e}/Z)^{1/3}$ is $\mathcal{O}(10^{-2})$ times
the typical kinetic energy, $(3\pi^{2}n_{e})^{1/3}$ – and thus form their own
Fermi gas system. It may be seen from Eq. (2) that due to their lightness and
abundance, it is the electrons that contribute the greatest to the pressure of
WDs. In contrast, in neutron stars the constituent neutrons contribute the
most to both the stellar mass and pressure.

The total energy of the star in the non-relativistic limit is given by

$\displaystyle E^{\rm non-rel}_{\rm tot}$ $\displaystyle\simeq$
$\displaystyle({\rm total\ kinetic})-({\rm gravitational\ binding})$ (5)
$\displaystyle=$ $\displaystyle\frac{3}{5}N_{\rm f}\frac{p_{F}^{2}}{2m_{\rm
f}}-\frac{3}{5}\frac{GM_{\star}^{2}}{R_{\star}}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{27\sqrt{3}\pi}{40\sqrt{10}}\bigg{)}^{2/3}\frac{1}{m_{f}R_{\star}^{2}}\bigg{(}\frac{ZM_{\star}}{Am_{N}}\bigg{)}^{5/3}-\frac{3}{5}\frac{GM_{\star}^{2}}{R_{\star}}~{},$

and in the relativistic limit,

$\displaystyle E^{\rm rel}_{\rm tot}$ $\displaystyle=$
$\displaystyle\frac{3}{4}N_{\rm
f}p_{F}-\frac{3}{5}\frac{GM_{\star}^{2}}{R_{\star}}$ (6) $\displaystyle=$
$\displaystyle\bigg{(}\frac{243\pi}{256}\bigg{)}^{1/3}\frac{1}{R_{\star}}\bigg{(}\frac{ZM_{\star}}{Am_{N}}\bigg{)}^{4/3}-\frac{3}{5}\frac{GM_{\star}^{2}}{R_{\star}}~{}.$

Eq. (5) shows that, in the ground state of the compact star where the virial
theorem (potential energy = -2 $\times$ kinetic energy) applies, we have
$R_{\star}\propto M^{-1/3}_{\star}$ as a mass-radius relation111This shows
that more compact stars are generally denser, which we will see is relevant to
determining the speed of DM passing through and the density of DM collected in
compact stars.. This implies that WDs, modeled accurately as a Fermi gas
system, become smaller with increasing mass. Hence the heaviest WDs are the
densest, and thus one expects electrons in them to be ultra-relativistic and
Eq. (6) to apply. In Eq. (6) both the potential and kinetic terms fall as
$R_{\star}^{-1}$, however the former grows faster with $M_{\star}$ than the
latter, implying a maximum WD mass above which the star will collapse. This
“Chandrasekhar limit” [23] (see also Sec. 2.5) is given by

$M_{\rm max-WD-
rel}=\sqrt{\frac{5\pi}{G^{3}}}\frac{15}{16}\bigg{(}\frac{Z}{Am_{N}}\bigg{)}^{2}\simeq
1.7\ \bigg{(}\frac{2Z}{A}\bigg{)}^{2}\ M_{\odot}~{}.$ (7)

A similar limit may be obtained for NS masses by setting $A\rightarrow 1$,
$Z\rightarrow 1$:

$M_{\rm max-NS-rel}=6.8\ M_{\odot}~{}.$ (8)

These estimates are not physically motivated: they assume relativistic
fermions constituting the entire volume of the star (true for neither WDs nor
NSs) and a non-interacting Fermi gas (not true for NSs). Nevertheless they
ballpark the true limit to within $\mathcal{O}(1)$ factors. A more precise
treatment must account for the stellar structure, which we will discuss below,
but first let us make two more estimates of the maximum mass of NSs.

(i) If we assume non-relativistic neutrons, the virial theorem using Eq. (6)
gives a mass-radius relationship:

$R_{\rm NS}\simeq 12\ {\rm km}\ \bigg{(}\frac{M_{\odot}}{M_{\rm
NS}}\bigg{)}^{1/3}~{}.$ (9)

In this picture the NS radius is a decreasing function of its mass; however it
cannot become smaller than the Schwarzschild radius corresponding to a maximum
mass, $R_{\rm Schw}=3\ {\rm km}\ (M/M_{\odot})$. This condition gives

$M_{\rm max-NS-nonrel}=2.8\ M_{\odot}~{}.$ (10)

(ii) Due to super-nuclear densities in the NS cores, strong interactions
cannot be neglected in considerations of NS structure. A maximum mass can be
obtained in the (unphysical) limit where such interactions solely support the
star against gravitational collapse [24]. Strong interactions become repulsive
at inter-nucleon distances roughly shorter than the reduced Compton wavelength
of the mediating pion, $m_{\pi}^{-1}$. This gives a maximum neutron density
$m_{N}m_{\pi}^{3}$, corresponding to a mass-radius relation of $M_{\rm
NS}=4\pi m_{N}m^{3}_{\pi}R^{3}_{\rm NS}/3$. For a surface escape speed $v_{\rm
esc}$, we have $R_{\rm NS}=R_{\rm Schw}v_{\rm esc}^{-1}=3~{}{\rm
km}~{}(M/M_{\odot})~{}v_{\rm esc}^{-1}$. Putting these together yields the
maximum NS mass as

$M_{\rm max-NS-strong}=\sqrt{\frac{3}{32\pi}}v^{3/2}_{\rm
esc}\bigg{(}\frac{M^{6}_{\rm Pl}}{m_{N}m_{\pi}^{3}}\bigg{)}^{1/2}\simeq
2~{}M_{\odot}\ \bigg{(}\frac{v_{\rm esc}}{0.5c}\bigg{)}^{3/2}~{}.$ (11)

As we will see below, this turns out to be an excellent estimate.

As argued in, e.g., Ref. [25], a more accurate reason for the existence of a
maximum mass for NSs is that the sound speed $c_{s}$ in NS material cannot be
arbitrarily large. In particular, $c^{2}_{s}/c^{2}=(\partial
P/\partial\varepsilon)_{\bar{s}}\leq 1$ must be satisfied everywhere in the
NS, where $\bar{s}$ is the specific entropy. Physically, increments in the
self-gravitating energy density result in increments in equilibrium-restoring
pressure, however this trend cannot extend forever due to the sound speed
limitation, putting a cap on NS masses. This is also an important criterion in
modelling the equation of state (EoS) of high-density NS matter.

Briefly, we review the argument for a _minimum_ NS mass. For a given EoS of
the NS core fluid, as the central density (hence mass) of the NS is decreased,
the gravitational binding energy will decrease, and at some minimum density,
the NS will be unstable to small radial perturbations. This EoS-dependent
minimum NS mass is typically $\sim 0.1M_{\odot}$ [26, 27]. Such an NS would be
primarily composed of an $\mathcal{O}(100)$ km crust zone, with a percent-
level fraction of mass in the central degenerate neutron fluid [27]. Be that
as it may, a realistic minimum mass of NSs is about $1M_{\odot}$, after
neutrino pressure and other thermal effects during the formation of a NS in a
core collapse supernova are considered222A compact object of mass
$0.77\begin{subarray}{c}+0.20\\\ -0.17\end{subarray}M_{\odot}$ has been
observed in the supernova remnant HESS J1731$-$347 [28], exciting speculations
as to its nature and the EoS of nuclear matter [29, 30]. [31].

###  2.2 Structure equations and equation of state

Detailed reviews of NS structure and the role of EoSs may be found in Refs.
[32, 33, 34], while we present here the essentials. Accurate estimates of
compact star macroscopic properties are best obtained by solving the
spherically symmetric stellar structure equations:

$\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{Gm\varepsilon}{c^{2}r^{2}}\bigg{(}1+\frac{P}{\varepsilon}\bigg{)}\bigg{(}1+\frac{4\pi
r^{3}P}{mc^{2}}\bigg{)}\bigg{(}1-\frac{2Gm}{c^{2}r}\bigg{)}^{-1}~{},$
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle
4\pi\frac{\varepsilon}{c^{2}}r^{2}~{},$ (12)

Here $m$ is the mass enclosed within a radial distance $r$, and all other
quantities are as defined above. The first equation, the Tolman-Oppenheimer-
Volkoff (TOV) equation, describes hydrostatic equilibrium, and the second
describes the conservation of mass in the star. Given an EoS $P(\varepsilon)$
and the boundary conditions $m(0)=0$, $\varepsilon(0)=\varepsilon_{c}$ (a
“central density”), the structure equations can be solved to obtain useful
quantities: a mass-radius relation (which would capture the maximum allowed
mass), radial profiles of pressure, energy or number density, chemical
potential, and so on.

Figure 1: Left. WD mass-radius relations derived from three different
equations of state for a 12C WD. The Chandrasekhar solution treats electrons
as a free gas. The Hamada and Salpeter equation of state incorporates Coulomb
corrections to the free gas approximation. The Relativistic FMT model
additionally accounts for WD relativistic corrections to the Wigner-Seitz cell
(Coulomb-corrected) treatment of the equation of state, [35]. Quantitatively
similar curves are obtained for 4He and 16O WDs. The point marked “$\beta$
instability” corresponds to where the central density is high enough that the
WD is unstable against electron capture of nuclei resulting in
$(Z,A)\to(Z-1,A)$ conversions. The point marked “GR instability” is where the
general relativistic corrections shift an otherwise infinite central density
to a finite value as the point at which the WD becomes gravitationally
unstable. Right. Same as the left panels, but for 56Fe WDs. These plots are
taken from Ref. [35]. Figure 2: NS mass-radius relations for various equations
of state for nuclear matter at high densities. The blue shaded region is
preferred by pulsar observations [33], the yellow region is a fit to the
observation of binary NS mergers using a hadronic EoS [36], the green region
is the 90% C.L. preferred region from an EoS-insensitive fit to GW170817 [37],
and the red regions are Bayesian fits at 90% C.L. from a combination of
gravitational wave and low-energy nuclear and astrophysical data [38]. The
horizontal thick-dashed lines depict the measured mass of the heaviest
observed pulsar MSP J0740+6620 [39]. The line-shaded bottom right region is
excluded by centrifugal mass loss, with the limit coming from observations of
the fastest-spinning (716 Hz) pulsar [40]. The line-shaded top left region is
excluded by the condition of causality: for any EoS the sound speed $c_{s}\leq
c$. The rectangular regions are simultaneous fits at 68% C.L. of NS mass and
radius by NICER (light brown by Refs. [41, 42] and dashed-green-enclosed by
Refs. [43, 44]). This plot is taken from Ref. [45].

A reliable estimate of WD properties may be gained by assuming a polytropic
EoS: $P(\varepsilon)=K\varepsilon^{\gamma}$. For WDs, one can set the second
and third terms to unity on the right-hand side of the first equation in Eq.
(12), as the $c$-dependent terms depict general relativistic corrections that
are only important for NSs. It is then straightforward to solve Eq. (12) for
polytropes [25, 46]. In particular, the cases of $\gamma=5/3$ and
$\gamma=4/3$, applicable respectively to the limit of non-relativistic and
relativistic electrons, result in the $M$-$R$ scaling relations we derived
from the virial theorem in Eqs. (5) and (6) with more refined numerical
factors. Notably, for the relativistic case we obtain the Chandrasekhar mass
as:

$M_{\rm Ch-WD}\simeq 1.4M_{\odot}~{}.$ (13)

Realistic EoSs are non-polytropes accounting for Coulomb corrections arising
from electron-ion interactions, e.g., the Feynman-Metropolis-Teller EoS [35].
Figure 1 shows representative $M$-$R$ relations for WDs of various nuclear
compositions, taken from Ref. [35]. A simple analytical fit to translate
between $\rho$ and WD masses $M_{\rm WD}\in[0.1,1.35]M_{\odot}$ is [47]

$\bigg{(}\frac{\rho_{\rm WD}}{1.95\times 10^{6}\ {\rm
g/cm}^{3}}\bigg{)}^{2/3}+1\approx\bigg{[}\sum_{i=0}^{6}c_{i}\bigg{(}\frac{M_{\rm
WD}}{M_{\odot}}\bigg{)}^{i}\bigg{]}^{-2}~{},$ (14)

with $\\{c_{i}\\}=\\{1.003,-0.309,-1.165,2.021,-2.060,1.169,-0.281\\}$.

In NSs the EoS of nuclear matter is non-trivial due to the non-perturbative
nature of QCD at the densities of the core. EoSs must account for nucleon-
nucleon interactions, far more uncertain than Coulomb interactions, and must
fit data on the per-nucleon binding energy in symmetric nuclear matter, the
so-called symmetry energy that accounts for the energy above the $N=Z$ ground
state, the nuclear compressibility, and much else. For these reasons a wide
range of EoSs has been proposed, resulting in multiple predictions for NS
configurations. Figure 2 displays $M$-$R$ curves obtained from a few popular
EoSs. The top left is a region where $c_{s}>c$ and hence causality is
violated; the NS mass for various EoSs is seen to reach a maximum close to
this region.

###  2.3 Spin periods

Celestial bodies have a maximum angular speed: the gravitational force on a
mass element on the equator must exceed the centrifugal force on it, giving a
minimum spin period

$P_{\rm min}\simeq\sqrt{\frac{3\pi}{G\rho}}=10^{4}~{}{\rm s}\sqrt{\frac{{\rm
g}/{\rm cm}^{3}}{\rho}}~{}.$ (15)

Thus for WDs with $\rho=\mathcal{O}(10^{6})$ g/cm3, $P_{\rm min}\simeq 10$ s,
and for NSs with $\rho=\mathcal{O}(10^{14})$ g/cm3, $P_{\rm min}\simeq
10^{-3}$ s. And indeed, the first pulsars historically espied were identified
as such by their gradually lengthening sub-second spin periods. Moreover, no
pulsars with spin periods smaller than $\mathcal{O}(\rm ms)$ have been
observed; those observed near this limit are called millisecond pulsars. The
bottom right region of Fig. 2 is excluded by the fastest spinning pulsar
observed with rotation frequency 716 Hz [40], a limit given by ($R$/10
km)${}^{3/2}\geq$ 1280 Hz $(M/1.5~{}M_{\odot})^{1/2}$ [32].

Figure 3: Top. Schematic of the internal structure of a NS, taken from Ref.
[48]. The layers of the crust are shown in the zoom. Bottom left. Density
profile of (various layers of) a NS crust. Right. Nucleon number as a function
of NS crust density. See Sec. 2.4 for further details.

###  2.4 Neutron star substructure

In the top panel of Figure 3 we show a schematic of the interior structure of
a NS. The physics of substructure is obtained by solving Eq. (12) with the
appropriate EoS for each stellar region. For illustration here we will make
use of the Brussels-Montreal “unified” equation of state (“BSk”) accounting
for all regions/densities in the NS, expressed in terms of analytic fits [49].
What follows is an overview of NS substructure; interested readers may gain
further details from Ref. [50] and the references listed in Ref. [48].

The crust, about $1\ \rm{km}$ thick, spans over 10 decades in density and
consists of several distinct layers corresponding to different phases of
nuclear matter. The bottom left panel of Figure 3 shows the density of
material as a function of the proper depth for the various crustal layers, and
the bottom right panel shows nucleon numbers of nuclei as a function of
densities spanning the entire crust; both plots were made using the EoS BsK21
[48]. These plots do not show the atmosphere (density $<10^{4}$ g/cm3,
thickness $\mathcal{O}(\mu$m), composed of hydrogen and lighter elements) and
ocean (density $<10^{10}$ g/cm3, thickness $\mathcal{O}(10)$ m, composed of
carbon and heavy metals); these layers affect the star’s thermal spectrum, and
are influenced by the star’s magnetic field.

The outer crust (density $10^{4}-10^{11}$ g/cm3) is composed of nuclei forming
a body-centered-cubic Coulomb crystal, interspersed with a degenerate and
nearly-free relativistic gas of electrons. À la WDs, electron degeneracy
contributes dominantly to the pressure, while nuclei contribute dominantly to
the mass. Deep in the crust, where the electron chemical potential is higher,
nuclei become increasingly neutron-rich due to inverse beta decay. The outer
crust terminates when the density and pressure become so high that free
neutron states begin to appear.

The transition to the inner crust is marked by the neutron drip line, marked
by density $\rho_{\rm drip}\simeq 4.2\times{10^{11}\ \rm{g/cm^{3}}}$ [51],
beyond which a fraction of neutrons becomes unbound from nuclei. Up to
densities about $0.1$ times the nuclear saturation density
$\rho_{0}\simeq{2\times{10^{14}}\ \rm{g/cm^{3}}}$, the inner crust comprises
of heavy, neutron-rich nuclei (also known as proton clusters) forming a
lattice, along with both an electron gas and a dripped-neutron gas. Such a
system is inaccessible to terrestrial experiments, hence the composition of
the inner crust is far more uncertain than the outer crust, and studies of
this region are limited to theoretical calculations, e.g., the Compressible
Liquid Drop Model, the Thomas-Fermi approximation, and many-body quantum
calculations. As the NS cools down, the dripped neutrons are expected to form
a superfluid phase.

Further down, the inner crust density approaches the nuclear saturation point,
and homogeneous nuclear matter appears [52, 53]. This has led to the
prediction of the so-called nuclear “pasta” phase at the bottom of the inner
crust [54, 55, 56, 57, 58, 59]. Intricate competition between nuclear
attraction and Coulomb repulsion forms these extended non-spherical phases of
nuclear matter; as the density increases, gnocchi, then spaghetti, and then
lasagna pasta phases become more prevalent. In the deepest layer of the inner
crust there are “inverted pasta phases” where nuclear density material
predominates over sparser, sub-nuclear density voids. This includes bucatini
(anti-spaghetti) and swiss cheese (anti-gnocchi) phases. Nuclear pasta is
confined to a thin layer, yet they constitute a significant fraction of the
crustal mass as they span densities of $0.1-1\ \rho_{0}$. They may also impact
several properties of the NS such as its thermal and electrical conductivity,
and the elasticity and neutrino opacity of the crust.

The inner crust terminates when the density reaches $\rho_{0}$, beyond which
nuclei “melt” into uniform nuclear matter, which form the core of the NS. The
core is further sub-divided into the outer core (densities 0.5$-$2
$\rho_{0}$), where the nuclear matter is expected to be comprised of neutrons,
protons, and electrons, and the inner core (densities 2$-$10 $\rho_{0}$),
where exotic states of matter may possibly be present. These could be meson
and hyperon condensates [60, 61, 62, 63]. These could also be deconfined quark
matter, which is possible when the bag constant is large in the QCD bag model
[63], either as $ud$ matter [64] or $uds$ matter [65, 66, 67, 68, 69, 70].
Deconfined quark matter is believed to be in a color superconducting (“CSC”)
phase [71, 72, 73], which could be crystalline [74]. If strange-quark pairing
is suppressed, a two-flavor superconducting (“2SC”) phase is formed involving
$u$ and $d$ quarks. If not, the $uds$ matter may exist in in a color-flavor-
locked (“CFL”) phase [75, 73] which may co-exist with confined states [76, 77,
78].

###  2.5 Thermonuclear explosions

Astrophysical situations may arise in which a WD exceeds its Chandrasekhar
mass (Eq. (7)). For carbon-oxygen WDs, this would lead to ignition of runaway
carbon fusion that unbinds the star. This is how Type Ia supernovae,
conventionally used as “standard candles” in cosmological distance
measurements, have been theorized to originate – via accreting material from a
binary companion and going super-Chandrasekhar. This picture, however, is
disputed by the lack of a specific “trigger” of the thermonuclear process
along with a number of other observational inconsistencies [79]. As will be
discussed later, other possible Type Ia progenitors include WD mergers and
pyconuclear reactions in sub-Chandrasekhar mass WDs.

Yet another setting in which thermonuclear chain reactions create an explosion
is in the ocean layer of NS crusts, and in particular the carbon component,
which could be ignited by mass accretion from a binary companion. For
accretion rates $>10\%$ of the Eddington limit, the result is “superbursts”,
x-ray bursts that spew $\mathcal{O}(10^{35})$ J of energy, lasting for hours,
and in some cases recurring about every year [80, 81, 82, 83]. This must be
distinguished from regular Type-I bursts in NSs, typically ignited by surface
accretion, emitting $10^{3}$ times less energy and lasting $10^{3}$ times
shorter.

Ref. [84] provides extended discussion on the physics of thermonuclear runaway
fusion, while we provide here a brief summary. Two generic conditions must be
satisfied: (1) a mimimum energy $Q_{\rm dep}$ must be deposited to raise the
temperature of a critical mass $M_{\rm crit}$ of density $\rho$ to a critical
temperature $T_{\rm crit}$ which can sustain fusion:

Condition 1 $\displaystyle Q_{\rm dep}\geq M_{\rm crit}(\rho,T_{\rm
crit})\bar{c}_{p}(\rho,T_{\rm crit})T_{\rm crit}~{}.$ (16)

The temperature prior to heating is here assumed $\ll T_{\rm crit}$, and
$\bar{c}_{p}\simeq c^{\rm e}_{p}/2+c^{\gamma}_{p}/4+c^{\rm ion}_{p}$ is the
average isobaric specific heat capacity, with

$c^{\ell}_{p}(\rho,T_{\rm
crit})=\frac{a_{\ell}b_{\ell}}{u}\bigg{(}\frac{T_{\rm crit}}{E_{\rm
F}}\bigg{)}^{\alpha_{\ell}}\bigg{[}1-\bigg{(}\frac{m_{e}}{E_{\rm
F}}\bigg{)}^{2}\bigg{]}^{\beta_{\ell}}~{}.$ (17)

Here $u$ is the atomic mass unit, $m_{e}$ the electron mass, and for the
{electronic, radiative, ionic} contributions,
$a_{\ell}=\\{\pi^{2},4\pi^{4}/5,5/2\\}$, $b_{\ell}=\\{\sum
X_{i}Z_{i}/A_{i},\sum X_{i}Z_{i}/A_{i},\sum X_{i}/A_{i},\\}$ (with $X_{i}$,
$Z_{i}$, $A_{i}$ the mass fraction, charge and atomic number of the ion
species $i$ respectively), $\alpha_{\ell}=\\{1,3,0\\}$, and
$\beta_{\ell}=\\{-1,-3/2,0\\}$ . The Fermi energy $E_{\rm
F}=[m^{2}_{e}+(3\pi^{2}n_{e})^{2/3}]^{1/2}$ with $n_{e}=\rho b_{\rm e}/u$ (Eq.
(1)). The trigger energy in Eq. (16) ranges $\mathcal{O}(10^{17})$ GeV
$\rightarrow$ $\mathcal{O}(10^{24})$ GeV for WD central densities
corresponding to WD masses ranging 1.4 $M_{\odot}\rightarrow$ 0.8 $M_{\odot}$.

Eq. (16) is necessary but not sufficient for runaway fusion. There is a second
condition, through which the critical mass $M_{\rm crit}=4\pi\rho\lambda_{\rm
trig}^{3}/3$ is also defined. To wit, the rate of energy gain via nuclear
fusion must exceed the rate of energy loss via diffusion over the volume set
by the “trigger length” $\lambda_{\rm trig}$:

Condition 2 $\displaystyle\dot{Q}_{\rm nuc}>\dot{Q}_{\rm diff}~{}.$ (18)

Here we have $\dot{Q}_{\rm nuc}=M_{\rm crit}\dot{S}_{\rm nuc}$ and
$\dot{Q}_{\rm diff}\simeq 4\pi k\lambda_{\rm trig}T_{\rm crit}$ for a nuclear
energy deposition rate per mass $\dot{S}_{\rm nuc}$ and thermal conductivity
$k$. Conductive diffusion from relativistic electrons provides the dominant
source of diffusion in WDs at the temperatures and densities relevant for
igniting thermonuclear fusion; see Ref. [85, 86] for analytic expressions for
$\dot{Q}_{\rm diff}$.

The estimation of $\dot{S}_{\rm nuc}$ involves numerical simulations of flame
propagation with a nuclear reaction network [84]. From this,

$\displaystyle\lambda_{\rm trig}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3kT_{\rm crit}}{\rho\dot{S}_{\rm nuc}(\rho,T_{\rm
crit})}}~{}$ (19) $\displaystyle=$
$\displaystyle\begin{cases}\lambda_{1}~{}(\frac{\rho}{\rho_{1}})^{-2}\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,\rho\leq\rho_{1}\\\
\lambda_{1}~{}(\frac{\rho}{\rho_{1}})^{\ln(\lambda_{2}/\lambda_{1})/\ln(\rho_{2}/\rho_{1})}~{},\rho_{1}<\rho\leq\rho_{2}\end{cases}$

where for WDs $\\{\lambda_{1}^{\rm WD},\lambda_{2}^{\rm WD}\\}=\\{1.3\times
10^{-4}~{}{\rm cm},2.5\times 10^{-5}~{}{\rm cm}\\}$ and
$\\{\rho_{1},\rho_{2}\\}=\\{2\times 10^{8}~{}{\rm g/cm}^{3},10^{10}~{}{\rm
g/cm}^{3}\\}$. This analytic form was obtained in Ref. [47] by fitting to
Figure 6 of Ref. [84] – that is restricted to $\rho_{1}\leq\rho\leq\rho_{2}$ –
and extrapolating to lower densities assuming plausible density-scalings of
$k$ and $\dot{S}_{\rm nuc}$. The fit is for $T_{\rm crit}$ = 0.5 MeV and
assumes equal carbon and oxygen masses in WDs. In the NS ocean, the mass
fraction of carbon is 10% [80], implying $\rho\to 0.1\rho$ in Eq. (19) if Eq.
(19) holds for pure carbon burning333It probably does, for the scalings of Eq.
(19) are seen to be similar to those in Table 3 of Ref. [84], for conductive
burning.. One could also fit a relation among the WD central density, critical
temperature and trigger mass [85]:

$T_{\rm crit}\gtrsim 10^{9.7}~{}{\rm K}\bigg{(}\frac{\rho}{10^{8}~{}{\rm
g/cm^{3}}}\bigg{)}^{3/140}\bigg{(}\frac{M_{\rm crit}}{{\rm
g}}\bigg{)}^{3/70}~{}.$ (20)

Figure 4: Cooling curves. Left. Luminosity versus time of WDs of various
masses, taken from Ref. [87]. The onset of crystallization at about $10^{8}$
yr takes cooling from the regime of thermal ions to the Debye regime. Right.
Surface temperature versus time of a benchmark WD and NS. Early cooling
dominated by emission of neutrinos is distinctly faster than that of photons.
See Sec. 2.6 for further details.

###  2.6 Cooling

As no nuclear fuel is burnt in compact stars, they cool down continually from
the moment of their birth unless energy is deposited into them by some means,
as discussed in Sections 3 and 4. Observations of compact star cooling are an
important handle on the physics governing their internal dynamics.

####  2.6.1 White dwarf cooling.

WDs initially cool by shedding the thermal energy of constituent ions. Given
the specific heat per ion $c_{v}=3/2$, the total WD energy in thermal ions is

$U=\frac{3T}{2}\bigg{(}\frac{M_{\rm WD}}{Am_{N}}\bigg{)}~{}.$ (21)

The WD luminosity $L=-dU/dt$, and the cooling curve can be obtained from an
independent expression for the luminosity in terms of the WD internal
temperature $T_{\rm int}$:

$L=0.2\ {\rm J/s}\ \bigg{(}\frac{M_{\rm
WD}}{M_{\odot}}\bigg{)}\bigg{(}\frac{T_{\rm int}}{{\rm K}}\bigg{)}^{7/2}~{},$
(22)

derived from photon diffusion in the WD surface layers assuming Kramer’s
opacity, and combining it with the EoS; see Ref. [26] for a detailed
treatment. The cooling timescale is then obtained as

$t_{\rm cool}\simeq{\rm
Gyr}~{}\bigg{(}\frac{M/M_{\odot}}{L/(10^{-3}L_{\odot})}\bigg{)}^{5/7}~{}.$
(23)

Thus the cooling times are long enough to keep WDs from becoming invisibly
faint today, yet short enough to make them fainter than main-sequence stars.
The above relation only holds for WDs with $T_{\rm int}>T_{\rm Debye}\simeq
10^{7}~{}$K, the typical Debye temperature below which the ions crystallize.
For smaller temperatures, corresponding to $L\lesssim 10^{-4}L_{\odot}$, the
specific heat comes from the vibration of the crystal lattice as opposed to
thermal motion of the ions. Obtaining WD cooling times accounting for this
effect involves a non-trivial treatment [26] that is beyond our scope. In Fig.
4 left panel we show a luminosity-vs-time cooling curve indicating the point
at which crystallization effects become important. In the right panel we show
a temperature-vs-time curve for a benchmark WD of mass 0.67 $M_{\odot}$
corresponding to a 7000 km radius.

####  2.6.2 Neutron star cooling.

NSs cool by emitting neutrinos (generated in weak processes) and photons; the
rate of neutrino cooling rate is initially larger and hence dominates up to a
point, before photon cooling takes over. In describing the cooling of NSs,
where GR effects are significant, it is necessary to distinguish between the
temperature in the frame of the NS, $T$, and in the frame of a distant
observer, $\widetilde{T}$, related by

$\displaystyle\widetilde{T}$ $\displaystyle\equiv$ $\displaystyle T/(1+z)~{},$
$\displaystyle 1+z$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{1-2GM_{\rm
NS}/R_{\rm NS}}}~{}.$ (24)

The temperature evolution during passive cooling is given by

$c_{\rm
v}(\widetilde{T})\frac{d\widetilde{T}}{dt}=-L_{\nu}^{\infty}(\widetilde{T})-L_{\gamma}^{\infty}(\widetilde{T})~{},$
(25)

where the neutrino luminosity of the NS as measured by a distant observer of
our benchmark NS is zenzizenzizenzic in NS temperature [88]:

$L_{\nu}^{\infty}(\widetilde{T})=1.33\times 10^{39}~{}{\rm
J/yr}~{}\bigg{(}\frac{\widetilde{T}}{10^{9}~{}{\rm K}}\bigg{)}^{8}~{},$ (26)

applicable for slow/modified Urca (“Murca”) processes such as $N+n\rightarrow
N+pe^{-}\bar{\nu}_{e}$ and $N+pe^{-}\rightarrow N+n\nu_{e}$ (with $N=n,p$),
the neutrino cooling mechanism as prescribed by the “minimal cooling” paradigm
[89]. In principle there could also be fast/direct Urca (“Durca”) processes
such as $n\rightarrow pe^{-}\bar{\nu}_{e}$ and $pe^{-}\rightarrow n\nu_{e}$
[90]. These processes dominate the NS cooling down to $\widetilde{T}=10^{8}$
K. It has also been suggested that cooling via $N\gamma\to N\nu\bar{\nu}$
induced by QCD anomaly-mediated interactions are comparable to Murca processes
in early-stage NS cooling [91, 92]. The luminosity of photon blackbody
emission from the NS surface is:

$L_{\gamma}^{\infty}(\widetilde{T}_{s})=4\pi(1+z)^{2}R_{\rm
NS}^{2}\widetilde{T}^{4}_{s}~{}.$ (27)

The NS heat capacity $c_{V}$ is given by [93]

$\displaystyle c_{V}(\widetilde{T})$ $\displaystyle=$ $\displaystyle 4.8\times
10^{26}~{}{\rm J/K}~{}\bigg{(}\frac{\widetilde{T}}{10^{4}~{}{\rm K}}\bigg{)}$
(28) $\displaystyle=$ $\displaystyle 2.7\times 10^{-21}~{}M_{\odot}/{\rm
K}~{}\bigg{(}\frac{\widetilde{T}}{10^{4}~{}{\rm K}}\bigg{)}~{}.$

Solving Eq. (25) requires a relation between the surface ($T_{s}$) and
internal ($T$) temperatures. Such a relation is highly sensitive to the
composition of the NS’ outermost envelope, which acts as an insulating layer
for temperatures $\gtrsim\mathcal{O}(10^{3})$ K, and becomes too thin for
insulation at smaller temperatures [90, 94]. For an iron envelope at high
temperatures [95, 96],

$T_{s}=10^{6}~{}{\rm K}\bigg{[}\bigg{(}\frac{M_{\rm
NS}}{1.5~{}M_{\odot}}\bigg{)}\cdot\bigg{(}\frac{10~{}{\rm km}}{R_{\rm
NS}}\bigg{)}^{2}\bigg{]}^{1/4}\bigg{[}\frac{T}{9.43\times 10^{7}~{}{\rm
K}}\bigg{]}^{0.55}~{}.$ (29)

One can then identify the thin-envelope regime by solving for $T_{s}=T$ in the
above equation, which gives $T_{\rm env}=3908$ K, below which one can simply
set $T_{s}=T$.

The solution of Eq. (25) can now be written down as the time for the NS to
cool to a temperature $\widetilde{T}_{\rm cool}$ ($\ll$ the initial
temperature) [97]:

$t_{\rm cool}(\widetilde{T}_{9})/{\rm yr}=\begin{cases}t_{\rm
env}=s_{1}^{-k}q^{-\gamma}\big{[}\big{(}1+(s_{1}/q)^{k}\widetilde{T}_{9}^{2-n}\big{)}^{-\gamma/k}-1\big{]},\
\widetilde{T}_{\rm cool}>\widetilde{T}_{\rm env}~{},\\\ t_{\rm
env}+(3s_{2})^{-1}(\widetilde{T}_{9}^{-2}-\widetilde{T}_{\rm env}^{-2}),\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \widetilde{T}_{\rm
cool}\leq\widetilde{T}_{\rm env}~{},\end{cases}$ (30)

where $\widetilde{T}_{9}=\widetilde{T}_{\rm cool}/(10^{9}~{}{\rm K})$,
$q=2.75\times 10^{-2}$, $s_{1}=8.88\times 10^{-6}$, $s_{2}=8.35\times 10^{4}$,
$k=(n-2)/(n-\alpha)$ and $\gamma=(2-\alpha)/(n-\alpha)$ with $\alpha=2.2$ and
$n=8$; $\widetilde{T}_{\rm env}\simeq 4000~{}$K corresponds to the time after
which the surface and internal temperatures equalize. The right-hand panel of
Figure 4 shows the NS cooling curve plotted using the above expression, with
the distinct regimes of neutrino and photon cooling labelled.

While in standard cooling scenarios NSs are expected to cool down to
$\mathcal{O}(10^{2})$ K over Gyr timescales (as in Fig. 4), temperatures as
high as $10^{4}$ K have been conjectured to persist if some additional
astrophysical source of NS heating is present: we discuss such speculative
reheating mechanisms in Sec. 4.1.5. We note that in the cases of magnetic
field decay and rotochemical heating, the expectation is that late-stage NSs
will still glow at below $10^{3}$ K.

####  2.6.3 Comparison of white dwarf and neutron star late-stage cooling

From the discussion above, and from the right-hand panel of Fig. 4, we see
that the temperature of an NS expected at late stages ($t_{\rm cool}\gtrsim$
Gyr), about $10^{2}$ K, is much smaller than the late-stage temperature of a
WD, about $10^{3.3-4}$ K. It is natural to wonder why, as one may expect both
to have a similar late-stage temperature since both are low-temperature
degenerate stars. The crucial difference is that the late-stage WD heat
capacity is determined by vibrational modes of the nuclear ionic Coulomb
lattice forming its interior, while the NS heat capacity is that of a
degenerate Fermi gas.

In the WD case, the heat capacity per ion is [26]

$c_{v}^{\rm WD}\simeq\int_{0}^{\kappa_{\rm
max}}\frac{\kappa^{2}d\kappa}{2\pi^{2}n}\sum_{\lambda=1}^{3}\frac{e^{\omega_{\lambda}(\kappa)/T}(\omega_{\lambda}(\kappa)/T)^{2}}{(e^{\omega_{\lambda}(\kappa)/T}-1)^{2}}~{},$
(31)

where $\omega_{\lambda}$ is the vibrational energy, $\kappa$ is the wavenumber
of the normal modes of the Coulomb lattice, and $\lambda$ labels transverse
and longitudinal modes. Using the linear approximation
$\omega_{\lambda}=\kappa c_{s}$, where $c_{s}$ is the sound speed in the
lattice, and changing variables in the integral to $\kappa\rightarrow
yT/c_{s}$, it can be immediately seen that $c_{v}^{\rm WD}\propto T^{3}$.

In the case of the NS, in the limit where the temperature $T$ drops below the
Fermi energy $E_{F}$, only a fraction $T/E_{F}$ of the fermions close to the
Fermi surface will be excited and raise the bulk temperature. Hence the energy
per fermion $\propto T(T/E_{F})$, and it follows that the heat capacity
$c_{v}^{\rm NS}\propto T$.

From Eq. (25), setting the photon luminosity $\propto T^{4}$, we obtain
$t_{\rm cool}\propto\log T$ for WDs and $t_{\rm cool}\propto 1/\sqrt{T}$ for
NSs. Thus, WDs indeed cool much slower than NSs at later stages.

###  2.7 Nucleon superfluidity

It has been recognized since 1959 that nucleons in a NS could be in a
superfluid state [98], a hypothesis supported by observational fits to cooling
curves [94]. Neutron superfluidity and proton superconductivity arise due to
their Cooper pairing with a 0.1 MeV energy gap, corresponding to a critical
temperature $T_{c}\simeq 10^{10}$ K [99, 63, 100]. Pairing occurs mainly close
to the Fermi surface, hence superfluidity does not influence the EoS of NS
matter (therefore bearing no consequence on NS mass-radius relations), but
does play a major role in setting the NS’ heat capacity and neutrino
emissivity. This is because these quantities are sensitive to particle-hole
excitations close to the Fermi surface: the energy gap exponentially
suppresses the nucleon contribution to the heat capacity for NS temperatures
$\ll T_{c}$.

Neutrons in the NS inner crust are expected to form Cooper pairs in the
singlet ${}^{1}S_{0}$ state and in the core in a triplet ${}^{3}P_{2}$ state:
at higher densities, singlet pairing becomes repulsive [100]. The less dense
protons are expected to pair in the singlet state in the NS core. A quark core
in the NS could give rise to “color superconductivity” with $ud$, $ds$, $su$
Cooper pairs carrying color charges [101]. Nucleon pairing models play a
central role in the possibility of rotochemical heating, as discussed in Sec.
4.1.5. The presence of superfluidity in NSs also gives rise to macroscopic
vortices and flux tubes, the former of which may play a role in late-stage
reheating of NSs (Sec.4.1.5).

Figure 5: Left. The “light cylinder” around a NS within which the co-rotating
magnetosphere is confined [102]. Acceleration of charges in this region are
thought to produce electromagnetic beams that are detected terrestrially as
regular pulses, making young NSs “pulsars”. Right. $P$-$\dot{P}$ diagram taken
from Ref. [103], illustrating the evolution of pulsars. For a description of
the types of pulsars displayed here, see Ref. [102]. See Sec. 2.8 for further
details.

###  2.8 Neutron star magnetic field and spin-down

When a progenitor star turns into an NS, its surface area shrinks by a factor
of about $10^{10}$. As a result, thanks to conservation of magnetic flux
(Gauss’ law for magnetism, $B\times R_{\rm NS}^{2}$ = constant) the stellar
magnetic fields increase by this factor, and thanks to conservation of angular
momentum the rotational speed also rises by this factor. Flux conservation
also implies that the total energy in the NS due to the magnetic field
decreases with the NS size:

$E^{\rm NS}_{B}=\frac{B^{2}}{8\pi}\cdot\frac{4\pi R_{\rm NS}^{3}}{3}=\frac{\rm
const.}{R_{\rm NS}}~{},$ (32)

hence the presence of $B$ fields tends to enlarge the NS. However $E^{\rm
NS}_{B}$ is bounded by the gravitational binding energy of the NS, giving the
condition

$B\leq\sqrt{\frac{18}{5}\frac{GM^{2}_{\rm NS}}{R^{4}_{\rm NS}}}\simeq
10^{18}~{}{\rm Gauss}~{}\bigg{(}\frac{M_{\rm
NS}}{M_{\odot}}\bigg{)}\bigg{(}\frac{10~{}{\rm km}}{R_{\rm
NS}}\bigg{)}^{2}~{}.$ (33)

A stricter upper limit can be obtained from considerations of hydromagnetic
stability [104]. Measurements from pulsar spin-down (discussed here) find that
millisecond pulsars typically have $B$ field strengths of about $10^{8}$
Gauss, classical pulsars about $10^{12}$ Gauss, and magnetars about $10^{15}$
Gauss. NSs have a “magnetosphere”, a region of plasma surrounding the NS and
co-rotating with it due to their coupling through the $B$ field. One can see
that this region is finite by simply locating the equatorial radius at which
its tangential speed $=c$ for a spin period $P$:

$R_{\perp}^{\rm LC}=\frac{cP}{2\pi}=48~{}{\rm km}\bigg{(}\frac{P}{\rm
ms}\bigg{)}~{}.$ (34)

This region defines the “light cylinder” shown in Figure 5, left panel. The
presence of strong moving magnetic fields in the light cylinder generates
electric fields that accelerate charged particles at the stellar surface,
leading to emission of electromagnetic beams from near the magnetic poles of
the NS. This beam, as we will soon see, is powered by the rotational energy of
the NS. The lighthouse-like sweep of the beam, detected as regular pulses on
Earth, serves to reveal NSs as pulsars444At the time of writing, two “white
dwarf pulsars” have been discovered [105, 106], but these refer to regular
pulsation corresponding to the beat frequency of orbital rotation with a
binary companion and the spin of the WD.. This is how NSs were historically
discovered by Bell and Hewish, and continues to be the primary method for
finding NSs in the sky [107].

The NS spin varies over the lifetime of the NS due to a number of factors,
chief among which is magnetic dipole radiation extracting rotational kinetic
energy, an effect known as pulsar spin-down. The radiation power of a rotating
magnetic dipole of moment $m$, with a component $m_{\perp}$ perpendicular to
the NS spin axis, and angular speed $\omega=2\pi/P$, is given by [102]

$\dot{E}_{\rm
rad,B}=\frac{2}{3c^{3}}m_{\perp}^{2}\omega^{4}=\frac{2}{3c^{3}}(B_{\perp}R_{\rm
NS})^{3}\bigg{(}\frac{2\pi}{P}\bigg{)}^{4}~{},$ (35)

where in the second equation we have used the expression for a sphere
uniformly magnetized with field strength $B$. The rotational power of an NS of
moment of inertia $I=2M_{\rm NS}R^{2}_{\rm NS}/5$ is given by

$\dot{E}_{\rm rot}=I\omega\dot{\omega}=-4\pi^{2}\frac{I\dot{P}}{P^{3}}~{}.$
(36)

For sub-kHz frequencies this radiation cannot penetrate the ISM nebula
surrounding the NS, and is hence deposited in it; the observed $P$, $\dot{P}$
and luminosities of supernova remnants such as the Crab Nebula ($P$ = 0.033
sec, $\dot{P}$ = 1 sec/80,000 yr, luminosity = $10^{5}L_{\odot}$, much higher
than that of the Crab Pulsar within) bears out the supposition that
$-\dot{E}_{\rm rot}\simeq\dot{E}_{\rm rad,B}$ [102].

NS spin-down provides a remarkably valuable handle on the age of an NS through
measurement of just its $P$ and $\dot{P}$, i.e. without requiring knowledge of
its radius, mass and $B$ field. Assuming the $B$ field remains constant, by
equating Eqs. (35) and (36) we see that $P\dot{P}$ is constant over time. For
an initial spin period $P_{0}$,

$\displaystyle\int_{0}^{\tau}dt(P^{\prime}\dot{P}^{\prime})$ $\displaystyle=$
$\displaystyle\int_{P_{0}}^{P}dP^{\prime}P^{\prime}$ $\displaystyle\Rightarrow
P\dot{P}\tau$ $\displaystyle=$ $\displaystyle\frac{P^{2}-P_{0}^{2}}{2}$
$\displaystyle\Rightarrow\ \ \ \tau$ $\displaystyle=$
$\displaystyle\frac{P}{2\dot{P}}~{},$ (37)

where the last equality assumed that the initial period $P_{0}\ll P$. This
characteristic age $\tau$ due to spin-down is often an excellent order-of-
magnitude estimate of an observed NS’s true age. It slightly overestimates the
latter for young NSs as an NS’ spin may initially decelerate via gravitational
radiation due to an oblate shape. For instance, for the Crab Pulsar, whose
supernova was observed in 1054 A.D., one finds $\tau$ = 1300 years. In the
case of older pulsars, Eq. (37) must again be used with special care,
specifically when being applied to NSs that are thought to have spun up at
some point in their life. These could be, e.g., millisecond pulsars that are
modelled as accreting mass and angular momentum from a binary companion; these
have been observed with a characteristic age older their actual age [108]. In
particular, there are millisecond pulsars with $\tau>$13.8 Gyr [107], the
measured age of the universe [109].

We note in passing that for NSs for which precise data on distances and proper
motions are available, their kinematic age may also be estimated by tracing
back their trajectories and locating a plausible birth site [110]. This
technique is possible thanks to the kick velocity imparted to the NS by the
asymmetric explosion of the progenitor, as mentioned in the beginning of Sec.
2.

The pulsar braking index $n$ is defined via $\dot{\omega}\propto\omega^{n}$.
With a little elementary calculus, it may be seen that

$n\equiv\frac{\omega\ddot{\omega}}{\dot{\omega}^{2}}=2-\frac{P\ddot{P}}{\dot{P}^{2}}~{}.$
(38)

For spin-down induced by magnetic dipole radiation, one finds by equating Eqs.
(35) and (36) that $n=3$, although pulsars with braking indices of 1.4$-$3
have been observed, suggesting other spin-down mechanisms [102].

It is useful to place observed pulsars on a $P$-$\dot{P}$ diagram such as the
one shown in Fig. 5 right panel. Pulsars typically begin life at the north-
west region of the diagram, and move south-east along contours of constant $B$
strengths while crossing contours of constant spin-down age. Eventually as
they age to about 10 Myr the rotational energy is insufficient to generate the
pulsar beam, and they cross the “pulsar death line”, sometimes referred to as
the “death valley”. However, the death line is not well-understood, for the
exact mechanism by which pulsar beams are created is still unknown and is an
active area of research. This is evident in the $P$-$\dot{P}$ diagram: quite a
few pulsars lie beyond various models of the death line [111, 112, 113, 114],
with PSR J2144-3933 lying well beyond all the canonical death lines. We will
re-encounter this oddball pulsar, which also happens to be the coldest NS
observed, in Sec. 4.1.

##  3 The white dwarf as a dark matter laboratory

WDs have been used as DM detectors via a number of mechanisms. There are four
main effects, which we will detail in the rest of the section: (1) DM can
collect and annihilate inside WDs, heating them to above the temperature that
would be expected from a standard WD cooling curve such as in Sec. 2.6. (2) So
much non-annihilating DM accumulates in a WD that the DM collapses and forms a
black hole deep in the WD interior. This small black hole can grow to accrete
the entire WD, thereby converting its host into a solar mass black hole. (3)
DM encounters with and collection in the WD can cause it to explode. (4) WDs’
internal structure could be altered if a substantial fraction of its mass were
comprised of DM.

In addition, resonant conversion of axion-like particle DM to photons in the
corona of a magnetic WD may be observed [115]; we relegate discussion of this
phenomenon in the context of NSs to Sec. 4.11.

Figure 6: Left. Upper bounds from Ref. [116] on DM density distributions in
the globular cluster M4, compared with an estimate of the DM densities
(labelled “1101.2737”) from Ref. [117] using a spherical collapse model. Also
shown are the range of DM densities required to match the observed
luminosities of WDs in M4 via DM annihilations within the WD as well as
kinetic heating by infalling DM; the horizontal range of the rectangles spans
the uncertainty in the positions of the WDs. Right. Bounds on dark matter
using an old WD in the Milky Way taken from [86]. See Secs. 3.1 and 3.2 for
further details.

.

###  3.1 Dark matter annihilation inside and heating white dwarfs

The possibility that dark matter can accumulate inside and change the internal
thermal properties of stars has long been appreciated [9, 10]. A number of
works has proposed that old WDs could have their late-time temperature altered
through accumulation and annihilation of DM in the interior [118, 117, 119,
120]. To a good approximation the amount of collisionless DM (for local DM
density $\rho_{\chi}$ and average DM-WD relative speed $v_{\rm rel}$) flowing
through a WD with mass $M_{\rm WD}=1.2~{}{\rm M_{\odot}}$, radius $R_{\rm
WD}=4000$ km, and surface escape velocity $v_{\rm esc}=\sqrt{2GM_{\rm
WD}/R_{\rm WD}}$ is

$\displaystyle\dot{M}$ $\displaystyle=\rho_{\chi}v_{\rm
rel}\times\pi\left(\frac{R_{\rm WD}v_{\rm esc}}{v_{\rm
rel}}\right)^{2}=10^{-7}~{}{\rm\frac{M_{\odot}}{\rm Gyr}}~{}\left(\frac{R_{\rm
WD}}{4000~{}{\rm km}}\right)^{2}\left(\frac{M_{\rm WD}}{1.2~{}{\rm
M_{\odot}}}\right)\left(\frac{\rho_{\chi}}{0.4~{}{\rm GeV/cm^{3}}}\right),$
(39)

where we have normalized to the mass accumulated over a gigayear to emphasize
that the DM mass accumulated inside the WD over the lifetime of the universe
is only a tiny fraction of the stellar mass. This expression assumes that all
DM incident on the WD is captured; for the DM-nucleon or DM-electron cross
section dependence of the capture rate, see Refs. [121, 86].

The late-time temperature of a benchmark WD described above, assuming it is
determined by the capture and annihilation of all DM transiting the WD, is
given by [116]

$T_{\rm WD}\approx 4000~{}{\rm K}\left(\frac{350~{}{\rm km/s}}{v_{\rm
rel}}\right)^{1/4}\bigg{(}\frac{\rho_{\chi}}{10^{3}~{}{\rm
GeV/cm^{3}}}\bigg{)}^{1/4},$ (40)

where here we have normalized this expression to a typical $v_{\rm rel}$, but
have chosen $\rho_{\chi}$ more than three orders of magnitude greater than the
inferred DM density near most WDs whose temperatures have been determined.
This is the DM density required for heating WDs above their expected late-time
temperature shown in Figure 4. In practice, this means that in order to find
or exclude DM this way, one would need to find an ancient WD in a region that
conclusively has a high DM density.

Reference [117] studied the heating effect that certain inelastic DM models
would have on the late-stage temperature of WDs, and found that for a
background DM density of $\rho_{\chi}\simeq 3\times 10^{4}~{}{\rm
GeV/cm^{3}}$, they would be sensitive to inelastic inter-state mass splittings
of about $10-10^{3}$ keV and per-nucleon scattering cross sections
$\sigma_{n\chi}\gtrsim 10^{-41}~{}{\rm cm^{2}}$. These authors proceeded to
investigate whether WDs observed in a very dense self-bound stellar system,
the globular cluster NGC 6121, a.k.a. Messier 4 (M4), might reside in a
background density of DM large enough to observe heating from DM. Assuming
that M4 was formed from a subhalo that was then tidally stripped by the Milky
Way parent halo, using a spherical collapse model first derived in Ref. [118],
adopting an NFW density profile, and accounting for the slight adiabatic
contraction of densities from the baryon potential, they estimated that the DM
density was approximately 800 GeV/cm3 at a cluster-centric distance $r=2.3$
pc, where the farthest WDs were observed in the Hubble Space Telescope.
Following this, a number of authors investigated the implications of DM in
globular clusters capturing in celestial bodies, under the assumption of a
large (103-104 GeV/cm3) DM density [122, 123, 124, 125, 126, 127, 128, 129].

A recent study [116] set empirical limits on the DM densities in M4 using
measurements of stellar line-of-sight velocities and performing a spherical
Jeans analysis; Figure 6 shows these limits on various DM density profiles
corresponding to upper bounds on NFW scale parameters. The density estimate of
Ref. [117], denoted by an asterisk, is safe from these limits. Nevertheless,
it was argued that the use of globular clusters as copious sources of DM
resulting in far-reaching conclusions about its microscopic properties is
problematic for several reasons.

  1. 1.

The origin story of globular clusters is unclear. While Ref. [117] echoed a
popular theory – corraborated in $N$-body simulations – that globular clusters
originate in DM subhalos that then strip tidally [130, 131, 132], alternative
simulations suggest they may form with no aid from dark matter via the
collapse of giant molecular clouds [133, 134, 135, 136].

  2. 2.

The V-band mass-to-light ratios of globular clusters in solar units is 1–5,
which is equivocal about the presence of DM in them, unlike, say, dwarf
galaxies (10–100), the Coma Cluster of galaxies (660) or the Milky Way (25)
which are known to harbor significant amounts of DM. In fact, a structure
defined as a stellar “cluster” is defined as a system whose dynamics need not
be explained by DM, unlike a “galaxy” [137]. Accordingly, studies of more than
20 globular clusters looking for DM in them have either failed to detect DM or
come to ambiguous conclusions [116].

  3. 3.

There is no guarantee that any invisible mass favored in globular cluster data
is in fact DM, as it may also be from faint stellar remnants [138].

  4. 4.

The interpretation of the presence or absence of DM in ambiguous datasets is
sensitive to priors and parametrizations. Ref. [139] found no evidence for DM
when analyzing NGC 2419 by fitting a Michie model for the stellar and a
generalized NFW profile for the DM distributions, but found strong evidence
for DM when fitting these quantities with no analytic form, floating instead
389 free parametes.

One could conclude that, due to these significant uncertainties and the
related infeasibility of determining DM density distributions in globulars
with current and imminent measurement sensitivities, globular clusters are
sytems that are far from robust for making statements about DM interactions.
On that note, there are proposals for finding WDs in dwarf galaxies like Segue
I and II [140].

###  3.2 Non-annihilating dark matter converting white dwarfs into black holes

If enough non-annihilating DM accumulates in WDs, the DM can collapse, and
subsequently form a small black hole that accretes surrounding WD material,
eventually consuming the entire WD [122, 86, 141]. Typically the DM is assumed
to be “asymmetric” since in such models DM typically does not self-annihilate
[142]. If in the process of accumulation and collapse DM self-annihilates
efficiently, too much of it may be lost to form a black hole in the WD core.

The routine by which DM could form a small black hole in the interior of a WD
is very similar to the more studied case of DM forming black holes in
NSs555See also Ref. [143, 144, 145], which study black hole formation in other
astrophysical bodies like the Earth, Sun, and Population III stars., which is
detailed in length in Section 4.4. To avoid repetition, here we will emphasize
aspects that are distinct from the case of the NS. The WD-to-BH conversion
process is as follows. First, DM accumulates in the WD over time, through
scattering on nuclei or electrons in its interior. Then, the captured DM
thermalizes with the WD interior, i.e., after repeated scattering it is
expected to localize within a small volume determined by the WD’s internal
temperature and gravitational potential.

One chief difference here between WDs and NSs is that during thermalization,
DM will scatter with a Coulomb lattice of ions in the core of the WD, which is
stabilized by relativistic electron degeneracy pressure. This effect
considerably suppresses DM-nucleus scattering rates at low momentum transfers,
the regime that determines the thermalization timescale $t_{\rm th}^{\rm WD}$.
For a carbon WD, this is given by [86]

$t_{\rm th}^{\rm WD}\simeq 20~{}{\rm yr}~{}\left(\frac{10^{-40}~{}{\rm
cm^{2}}}{\sigma_{n\chi}}\right)\left(\frac{m_{\chi}}{10^{6}~{}{\rm
GeV}}\right)^{2}\left(\frac{10^{7}~{}{\rm K}}{T_{\rm WD}}\right)^{5/2}~{}.$
(41)

Thus for $m_{\chi}>10^{10}$ GeV, it can take $>$ Gyr for DM to thermalize with
the WD interior. Another difference between DM collapsing to form black holes
in WDs and NSs is that, during the collapse inside a WD, DM may trigger a
star-destroying thermonuclear explosion. We now turn to this topic.

Figure 7: Illustration of mechanisms by which WDs may be prompted to explode
by dark matter. (a) DM acumulates to the point of collapse in the center of
the WD, then while collapsing (or after collapsing and forming a black hole)
heats the WD to a temperature inducing a thermonuclear chain reaction. (b) The
internal potential or mass energy of spatially extended DM is deposited as WD
nuclei enter its state, prompting local heating that initiates the
thermonuclear runaway. (c) Macroscopic DM transiting the WD transfers
explosive kinetic energy via scattering on WD constituents.

###  3.3 White dwarf explosions via dark matter

Dark matter accumulated inside WDs might trigger a Type Ia-like supernova
explosion through the deposition of enough energy to prompt runaway fusion
reactions in the carbon/oxygen/neon interior of the WD [85, 146]; see also
Ref. [147] for an early discussion of DM cores affecting Type Ia supernovae.

More generally, DM triggering WDs into supernovae can proceed in a number of
ways:

  * •

Attendant to DM converting WDs to black holes, DM can collect into a core
region of the WD, collapse, and as a result of the collapse, deposit enough
energy to ignite the WD. Ignition can occur either directly through nuclear
scattering during the collapse of the DM core [85, 146, 47] or through the
evaporation of a small black hole that forms out of the collapsed DM core
[148, 141, 47].

  * •

DM can have internal properties that result in energy being liberated as WD
particles enter the DM state. A simple example of this is captured (and
possibly thermalized) DM annihilating and depositing energy in the WD medium
with which it is admixed [149]. Other interesting possibilities are composite
DM with an internal potential for baryons [150], solitonic Q-ball DM that
absorbs baryonic charge and dissociates nuclei in the process [149], monopoles
that possibly induce nucleon decay in similar fashion (Sec. 4.1.2), and
accretion of WD carbon ions onto a black hole formed from collapse of
electrically charged DM states [47].

  * •

During an initial transit through the WD, DM can deposit kinetic energy gained
by falling into the WD’s gravitational potential. The DM could be in the form
of primordial black holes (PBHs), in which case energy is transferred via
dynamical friction [146, 151], or particles inducing nuclear scatter recoils
[149, 152, 153]. Tightly bound asteroid-like DM triggering WD explosions via
stellar shocks has also been suggested [154].

However the WD is heated, a number of requirements must be met for
thermonuclear reactions sparked to sustain themselves and cause the WD to
explode. These requirements are described in Sec. 3.3. We now discuss some
subtle aspects of this phenomenon as explored in the literature.

A detailed simulation of PBH energy deposition in a WD, including the effect
of turbulent flows in the wake of the passing PBH, found that heavier PBHs
were required to ignite WDs [151] compared to initial estimates [146]. This
study employed a 1D+1D hydrodynamic simulation of the shock front created by a
transiting PBH, and found that the development of hydrodynamic instabilities
dissipating heat deposited through dynamical friction appeared to occur more
rapidly than ignition processes, which were modeled using the same carbon
fusion reaction rates used in Ref. [84]. Instead of a burning reaction, the
prompt detonation of WD material by transiting PBHs was further studied in
Ref. [155]. Another study investigated ignition during DM core collapse using
a system of differential equations that track the evolution of per-particle
energies [156] \- this tracked carbon fusion inside the collapse region, and
found carbon depleted before the WD ignition temperature in Ref. [84] was
reached. Future work could build on this result in a number of directions,
$e.g.$ by studying C-O burning and employing the full nuclear reaction network
used in Ref. [84] to obtain WD ignition temperatures. In addition, future WD
ignition estimates should also consider convective flows of heated WD material
moving carbon through the collapse region replenishing carbon and oxygen, and
whether WD ignition occurs via thermal energy transported out of the collapse
region. This is especially important, since studies on carbon fusion occurring
inside DM bound states found that fusion can be induced in the region
surrounding the collapsing region that is the source of heat, either through
the evaporation of black holes of size much smaller than the ignition region,
or through effluence of thermal energy outside of the transiting DM composite
[86, 141, 150, 47]. Finally, the ignition of WD supernovae via oxygen burning
typically requires a temperature somewhat higher than that of carbon666We
thank Melissa Diamond for correspondence on this point. [84], and future
detailed treatments of WD ignition by collapsing DM should account for this
possibility.

Ref. [149] set limits on a wide range of DM-nucleus scattering cross sections
and DM masses assuming point-like elastic scattering of DM particles on carbon
in WDs. These constraints were placed using the condition for the minimum
stopping power,

$n_{\rm T}\sigma_{\rm T\chi}m_{\rm T}v^{2}_{\rm
esc}\gtrsim\rho\bar{c}_{p}T_{\rm crit}\lambda_{\rm trig}^{2}$ (42)

for a heating region of size $\leq\lambda_{\rm trig}$. However, this condition
does not account for the finite number of nuclei that the DM particle would
encounter during its transit through the heating region. Suitably modified,
the above condition should be

$\displaystyle N_{\rm hit}\frac{m_{\rm T}v^{2}_{\rm esc}}{\lambda_{\rm trig}}$
$\displaystyle\gtrsim$ $\displaystyle\rho\bar{c}_{p}T_{\rm crit}\lambda_{\rm
trig}^{2}~{},$ $\displaystyle N_{\rm hit}$ $\displaystyle=$
$\displaystyle\max[n_{\rm T}\sigma_{\rm T\chi}\lambda_{\rm trig},n^{1/3}_{\rm
T}\lambda_{\rm trig}]$ (43)

where $N_{\rm hit}$ is the number of point-like scatters on nuclei as the DM
particle traverses the length $\lambda_{\rm trig}$. One can see that Eq. (43)
reduces to Eq. (42) for $\sigma_{\rm T\chi}<n_{\rm T}^{-2/3}$. Ref. [149]
considers a 1.25 $M_{\odot}$ WD to set limits, for which $n_{\rm T}\simeq
10^{-31}$ cm-3, implying that $\sigma_{\rm T\chi}\lesssim 2\times 10^{-21}$
for Eq. (42) to be valid. However, $\sigma_{\rm T\chi}>10^{-12}$ cm2 is shown
to be excluded in Ref. [149]. One could also see the error in this result by
estimating the maximum energy transferred by DM elastic recoils by a linear
transit across a length $\lambda_{\rm trig}$. This is $(n^{1/3}_{\rm
T}\lambda_{\rm trig})(m_{\rm T}v^{2}_{\rm esc})\simeq=10^{4.5}$ GeV, which may
be compared with the trigger energies ranging across WD masses, $10^{17-24}$
GeV (Sec. 2.5). One could contrast this analysis against Refs. [152, 153],
which considered WD explosions triggered by the transit of macroscopic
composite DM. In these studies, the requisite number of WD nuclei within a
trigger volume may indeed be excited to ignite the region into runaway fusion.
In Fig. 11 bottom left panel we show the masses and radii of DM mini-clusters
constrained by the observed existence of WDs in our Galaxy, taken from Ref.
[153]. Overlaid here are contours of the minimum DM-nucleus elastic scattering
cross sections required to transfer sufficient kinetic energy to the WD
trigger volume to induce stellar explosion.

A number of phenomena have been linked to the DM-induced ignition of
thermonuclear explosions in WDs. It has been posited that DM core collapse in
WDs might account for a large fraction of observed Type Ia supernovae [85], as
a solution to the Type Ia progenitor problem [79] and consistent with the
apparent observation of sub-Chandrasekhar WDs as the origin of most Type Ia
supernovae [157]. Reference [85] also found that a trend in existing Type Ia
data [158], showing that more massive progenitors explode sooner, is
consistent with certain DM models that induce WD explosions through DM core
collapse, where this would occur sooner for heavier WDs. The accumulation in
certain sub-Chandrasekhar WDs of charged massive particles (CHAMPs) making up
DM, which might occur preferentially outside galaxies with magnetic fields
that serve to deflect CHAMPs, could be an explanation of the distribution of
calcium-rich gap transient WD supernovae [47] that do explode preferentially
on the outskirts of galaxies [159]. The distribution of Type Ia supernovae in
galaxies could be tied to local properties like velocity dispersion,
especially in the case of PBH-ignition [160]. Finally, a separate study has
investigated whether WD explosions from DM could explain the aforementioned
Ca-rich gap transient distribution, through the ignition of WDs in dwarf
spheroidal galaxies expected to be located at some distance from larger
galactic DM halos [161].

###  3.4 Dark matter’s influence on white dwarf equations of state

WDmass-radius relationships can also be observably impacted by DM. If a
substantially massive core of DM accumulated in the interior of a WD, its
stable configurations would be altered through revised TOV equations [147,
162, 163]. For a typical circumambient DM density, the amount of collisionless
DM required to induce these effects, $10^{-4}M_{\odot}-10^{-1}~{}M_{\odot}$,
well exceeds what could be collected in the WD over the lifetime of the
universe; see Eq. (39). However, future studies could investigate whether such
a large quantity of DM might be collected through collisional accretion,
analogous to the NS treatment in Ref. [164] (discussed in Sec. 4.2). Another
effect comes through the axion: its existence implies that its non-derivative
coupling to nucleons would displace it from its usual minimum in a finite-
density medium. This results in a reduction of the nucleon mass and alters the
TOV equations [165].

##  4 The neutron star as a dark matter laboratory

Figure 8: Top left. Cartoon showing the dark kinetic heating effect in NSs.
Scattering interactions of the infalling dark matter flux contribute to the
luminosity of a typical NS at the level of a $1500$ K blackbody temperature.
Top middle. The nucleon Auger effect that contributes to kinetic (and possibly
annihilation) heating by dark matter in NSs. The total energy deposited after
scattering turns out to be the dark matter energy transfer, although
physically it comes as the sum of two contributions: the energy spilled during
the rapid filling of the hole left behind by the struck target, and the energy
carried by the target in excess of the Fermi energy. Top right. The breaking
and re-pairing of Cooper pairs that contributes to kinetic (and possibly
annihilation) heating by dark matter in NSs. This phenomenon takes place for
dark matter with mass above about 35 MeV; for smaller masses, dark matter
capture proceeds through collective excitations in the nucleon superfluid
medium. Bottom left. Cartoon showing possible additional heating of NSs via
self-annihilations of dark matter possibly collected in a thermalized volume.
This highly model-dependent process could heat the NS to blackbody
temperatures around 2000 K. Bottom right. As a function of NS mass, NS
effective temperatures imparted by dark kinetic+annihilation heating that can
be measured at the James Webb Space Telescope at various signal-to-noise
ratios, taken from Ref. [45]. The band denotes variations over NS radii
predicted by numerous equations of state as well as NS-DM relative velocities
from estimates by various NS population models. See Sec. 4.1.1 for further
details.

###  4.1 Dark matter kinetic and annihilation heating of neutron stars

####  4.1.1 Capture and kinetic heating

Figure 9: Top. Capture cross section sensitivities for light dark matter scattering in a NS crust (left) (via excitation of superfluid phonons in the inner core) and in the NS core (via Pauli-blocked contact scattering on neutrons, although see Sec. 4.1.1 for a discussion on scattering in the superfluid core), and for heavier dark matter scattering in various layers of the crust and the core (right). These two plots are taken from Ref. [48]. See Sec. 4.1.1 for further details. Bottom. Sensitivities to the cutoff of effective CP-even scalar interactions of dark matter with relativistic, degenerate electrons in a NS, for DM that is spin-1/2 (left) and spin-0 (right). Also shown are the sensitivities for interactions with muons, protons and neutrons. The electron scattering limits are seen to widely complement terrestrial searches. These two plots are taken from Ref. [166]. See Sec. 4.1.1 for further details. effect | change in capture rate | applicability | reference
---|---|---|---
EoS of star effects |  $\mathcal{O}$(1): BSk20 $\to$ 21 | all $m_{\chi}$ |  [167]
none: QMC-2 $\to$ BSk24 | all $m_{\chi}$ |  [168]
mass-radius configuration |  $\mathcal{O}(100)$ as $1\to 2.2M_{\odot}$ | all $m_{\chi}$ |  [169]
nuclear self-energies | 30$-$100 |  $m_{\chi}>$ 100 MeV, any EoS | [170]
nucleon structure |  $\mathcal{O}(10^{3})$ for 2 $M_{\odot}$ NSs |  [168]
non-elastic scattering | subdominant | $-$ |  [168]
“collective” effects | $\mathcal{O}(1-10^{3})$ | 2 $M_{\odot}$ NS, |  [171]
|  |  $m_{\chi}<100$ MeV, |
|  |  $A^{\prime}$ mediator |
superfluidity: energy gap | maybe $\mathcal{O}$(1) |  $m_{\chi}\lesssim 35$ MeV, |  [48]
|  | single phonon excitation |  [164]
NS opacity/ extinction factor | $\mathcal{O}(1)$ |  $m_{\chi}>$ GeV |  [169]
relativistic kinematics | $\sim 4$ |  $m_{\chi}>$ GeV |  [169]
$\sim 10$ |  $m_{\chi}<$ GeV |  [169]
gravitational focusing | $<2$ | all $m_{\chi}$ |  [169]
light mediator kinematics | $\mathcal{O}(1)$ | $m_{\phi}/\mu_{\rm red}<10^{-1}$ | [172]
voided | $m_{\phi}/m_{\chi}<10^{-4}$
DM halo velocity distribution | $<2$ | all $m_{\chi}$ |  [173]
Table 1: Known effects that modify the rates of dark matter capture in NSs.
See Sec. 4.1.3 for further description.

NSs are excellent captors of particle dark matter by virtue of their extreme
densities and steep gravitational potentials, and also quite serviceable as
thermal detectors thanks to their typically small temperatures. While the
capture of DM in NSs and its subsequent thermal relaxation was first treated
in Ref. [11], it was only recently realized that this could be a minimal probe
of dark matter scattering on Standard Model (SM) states: the transfer of DM
kinetic energy to the NS’s constituent particles during the infall of DM at
semi-relativistic speeds overheats the NS [174]. It was also proposed that
upcoming infrared telescopes, e.g., the Thirty Meter Telescope (TMT) [175] and
the Extremely Large Telescope (ELT) [176] are sensitive to this “dark kinetic
heating” mechanism [174] for NSs out to about $100$ pc from Earth; a study has
also been dedicated to the sensitivity at the recently launched James Webb
Space Telescope (JWST) [177, 45], which has shown that finding an NS much
closer than 100 pc would likely be required. Thermal observations of nearer
pulsars could be made following the discovery of old, isolated NSs in radio
telescopes such as FAST [178], CHIME [179] and SKA [180]. Though their $B$
fields and rotational velocities are expected to be low, implying they
populate regions near the “pulsar death line” in $P$-$\dot{P}$ space beyond
which NSs are supposed to stop pulsing, NSs have been observed beyond the
death line [113, 111, 114, 112] calling into question models of NS pulsation
(as also discussed in Sec. 2.8). It is estimated that about $10^{5}$ NSs in
the Galaxy lie beyond the death line [113].

To illustrate the idea of dark kinetic heating let us consider the following
representative NS configuration:

$\displaystyle M_{\rm NS}$ $\displaystyle=$ $\displaystyle 1.5\
M_{\odot},~{}~{}R_{\rm NS}=12.85\ {\rm km}$ $\displaystyle\Rightarrow v_{\rm
esc}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2GM_{\rm NS}}{R_{\rm
NS}}}\simeq 0.59\ .$ (44)

where $v_{\rm esc}$ is the escape speed at the surface. This configuration is
obtained for a Quark Meson Coupling (QMC) EoS of matter [168].

For local DM density $\rho_{\chi}$ and average DM-NS relative speed $v_{\rm
rel}$ (which in the solar vicinity are $0.4$ GeV/cm3 and 350 km/s [181]), the
DM mass capture rate is given by [11]

$\displaystyle\dot{M}=m_{\chi}C_{n\chi}$ $\displaystyle=$
$\displaystyle\rho_{\chi}v_{\rm rel}\times\pi b_{\rm max}^{2}\times
p_{v}\times p_{\sigma}\ ,$ (45) $\displaystyle=$ $\displaystyle
p_{v}p_{\sigma}~{}\times~{}1.76\times 10^{25}~{}{\rm GeV/s}~{},$

where $b_{\rm max}=R_{\rm NS}(1+z)(v_{\rm esc}/v_{\rm rel})$ is the maximum
impact parameter of DM intersecting the NS, with $1+z=(1-v_{\rm
esc}^{2})^{-1/2}$ a blueshift factor magnifying the NS radius to a distant
observer, and $p_{v}$ is the probability that a scattered DM particle loses
sufficient energy to be captured. For instance, this probability $\simeq 1$
for scalar- or vector-mediated scatters, but may be suppressed for
pseudoscalar-mediated interactions that favor soft forward scatters [172]. Eq.
(45) is, of course, the DM capture rate for an isolated NS; an NS in a binary
system could capture DM at a rate greater by up to a factor of a few thanks to
gravitational assist [182].

The probability that incident DM is scattered is given by
$p_{\sigma}=1-e^{-\tau}\simeq\tau=\sigma_{n\chi}/\sigma_{\rm cap}$ where, for
optical depth $\tau$, the approximate equality in the first line holds in the
optically thin limit. The “capture cross section” above which $\tau>1$ in the
NS core is:

$\displaystyle\displaystyle\sigma_{\rm
cap}=\begin{cases}\sigma_{0}(\bar{m}_{n}/m_{\chi})\ &,\ \ m_{\rm
evap}<m_{\chi}<\bar{m}_{n}\ ,\\\ \sigma_{0}\ &,\ \ \bar{m}_{n}\leq
m_{\chi}\leq\text{PeV}\ ,\\\ \sigma_{0}(m_{\chi}/{\rm PeV})\ &,\ \
m_{\chi}>{\rm PeV}\ ,\end{cases}$ (46)

where the NS geometric cross section $\sigma_{0}=\pi(\bar{m}_{n}/M_{\rm
NS})R_{\rm NS}^{2}\simeq 2.2\times 10^{-45}\,{\rm cm}^{2}$. One understands
the dependence on $m_{\chi}$ in Eq. (46) by considering the typical neutron
recoil energy in the neutron rest frame:

$\displaystyle\Delta E_{\rm DM}$
$\displaystyle\simeq\frac{\bar{m}_{n}m_{\chi}^{2}(1+z)^{2}v^{2}_{\rm
esc}}{(\bar{m}_{n}^{2}+m_{\chi}^{2}+2(1+z)\bar{m}_{n}m_{\chi})}\ ,$ (47)

The above expression is a good approximation to describe DM-neutron scattering
in the stellar rest frame as well, since the neutrons are typically non-
relativistic: their Fermi momenta, varying over a few 100 MeV across the NS,
are smaller than their $\sim$GeV mass. For $m_{\chi}\\!<\\!\bar{m}_{n}$, only
a fraction $\simeq 3\Delta p/p_{F}$ of degenerate neutrons close enough to
their Fermi surface receive the typical momentum transfer $\Delta
p=\sqrt{2\bar{m}_{n}\Delta E_{\rm DM}}$ to scatter to a state above the Fermi
momentum $p_{F}\simeq 0.4~{}\text{GeV}$. This “Pauli-blocking” effect gives
$\sigma_{\rm cap}\propto\Delta E_{\rm DM}^{-1/2}\propto m_{\chi}^{-1}$. The
so-called evaporation mass,

$m_{\rm evap}\simeq 20~{}{\rm eV}\ \bigg{(}\frac{T_{\rm NS}}{10^{3}~{}{\rm
K}}\bigg{)}~{},$ (48)

is the DM mass below which the thermal energy of the NS would kinetically
eject the captured DM from the stellar potential well [167, 183]. For
$\bar{m}_{n}\\!\leq\\!m_{\chi}\\!\leq\\!10^{6}~{}\\!\text{GeV}$, a single
scatter suffices for capture: $\Delta E_{\rm DM}\simeq\bar{m}_{n}v_{\rm
esc}^{2}\gamma^{2}>$ KEhalo, the DM halo kinetic energy. For
$m_{\chi}>\text{PeV}$, multiple scatters are required for capture, so that
approximately $\sigma_{\rm cap}\propto{\rm KE}_{\rm halo}/\Delta E_{\rm
DM}\propto m_{\chi}$. The expression in Eq. (45) can be refined to account for
the velocity distribution of DM far from the NS [184].

The heating of the NS comes not only from the recoil of incident DM but from
two other secondary effects. As depicted in Fig. 8, a target neutron (or a
lepton) that is upscattered by DM leaves behind a hole in the Fermi sea. The
hole is filled up immediately by a nearby neutron from a higher energy level,
which in turn leaves a hole, and so on. This process spills over energy in the
form of radiation and kinetic energy, and is reminiscent of the Auger effect
observed in electron levels in superconductors; we will re-encounter this
effect as a means of NS internal heating in Sec. 4.8. The net energy deposited
in the NS by this effect, $E_{\rm Auger}$, is simply the difference in energy
between the Fermi surface and the position of the original hole. The energy
carried by the struck nucleon/lepton in excess of the Fermi energy, $E_{\rm
kin}$, is dissipated as kinetic energy above the Fermi surface. Thus the total
energy deposit $E_{\rm Auger}+E_{\rm kin}$ comes out to be simply the DM
recoil energy $\Delta E_{\rm DM}$. Yet another effect comes from the
superfluidity of nucleons (see Sec. 2.7). For $m_{\chi}\gtrsim 35$ MeV, DM
participates in elastic scattering by first breaking a nucleon Cooper pair,
which is bound with an energy given by the superfluidity energy gap $\sim$
MeV. The absorbed $\sim$ MeV energy is redeposited into the NS when the free
nucleon finds another and pairs up, liberating the gap energy. For
$m_{\chi}\lesssim 35$ MeV nucleons in the NS might not scatter elastically as
there isn’t enough energy transfer to break nucleon Cooper pairs, leaving DM
to capture via collective excitations instead [48, 164]. Light DM capture in
certain models through collective effects in NSs has been studied [171]. The
presence of DM self-interactions can enhance the capture rate by orders of
magnitude as initially captured DM particles can serve as captors of ambient
DM [185].

Once captured in the potential well, a DM particle repeatedly scatters on and
thermalizes with the NS until its orbit shrinks to within the radius of the
star, by which times most of its kinetic energy is transferred. Under
equilibrium, the kinetic power of the infalling dark matter, constituting the
NS heating rate, equals the rate at which photons are emitted from the NS
surface, constituting the NS cooling rate. The latter is dominated by such
photon emission for NSs older than $\sim$Myr, as we saw in Sec. 2.6. The NS
luminosity corresponding to a temperature $T$ (in the NS frame) is then
$L=z\dot{M}=4\pi R_{\rm NS}^{2}T^{4}$, which attains a maximum value $L_{\rm
max}$ for unit capture probabilities $p_{\sigma}$ and $p_{v}$. For our
representative NS configuration (Eq. (44)), $L_{\rm max}=7.6\times
10^{24}~{}{\rm GeV/s}$, corresponding to a NS temperature seen by a distant
observer $\widetilde{T}=T/(1+z)$ of $\widetilde{T}=1400~{}{\rm K}$.
Temperatures in this range are measurable within reasonable integration times
at current and imminent infrared telescope missions [174, 186], in particular
at the recently launched JWST [45], and the forthcoming ELT and TMT. For
instance, the NIRCam instrument at JWST could constrain the surface NS
temperature at 1750 K with a signal-to-noise ratio (SNR) of 2 in $27.8~{}{\rm
hr}(d/10\ {\rm pc})^{4}$, where $d$ is the distance to the NS [174]; the IRIS
instrument at TMT could do the same in $19.4~{}{\rm hr}(d/10\ {\rm pc})^{4}$.
In the bottom right panel of Fig. 8 are displayed the NS effective
temperatures constrainable at JWST at various SNRs for integration times of
5.5 hr and 24.3 hr, using the F150W2 filter on NIRCam. In this plot taken from
Ref. [45], the band spans the range of the NS radii (which determines the
range of DM capture rates) predicted by various EoSs, and integrates over the
NS-DM relative velocities predicted by various NS population models in Ref.
[187, 188]. These sensitivities are for the case of NSs being heated not only
by the kinetic energy of infalling DM but also by DM annihilations, which we
will discuss in Sec. 4.1.2. Searches for DM using NS thermal emissions are
best carried out with NSs whose “standard” temperatures are expected to be
below approx. $1000$ K. Thus one would need NSs older than 10 Myr (Fig. 4),
making the determination of their age via spin-down or kinematic
considerations (Sec. 2.8) crucial. One would also need them sufficiently
isolated to ensure no accretion of material from a binary companion.

The DM-nucleon scattering cross section may be so large that DM scatters
dominantly with the $\sim$km-thick low-density crust of the NS before reaching
the $\sim$20 km-diameter denser core. Moreover, the core may consist of exotic
phases of high-density matter such as meson condensates and deconfined $ud$ or
$uds$ quark matter, the latter of which may exist in any of the multiple
phases discussed in Sec. 2.4; in such cases, the dynamics governing DM
scattering cannot be unambiguously computed, whereas the better understood
crust can be treated robustly as a DM captor. DM scattering with the NS crust
leads to surface emission of photons under thermal equilibrium analogous to
capture in the NS core discussed above, hence the observational signatures of
NS heating are unchanged. In Figure 9 we show the DM capture cross section
$\sigma_{\rm cap}$ for every layer of the NS described in Sec. 2.4, derived in
Ref. [48] for a 1.8 $M_{\odot}$ mass, 12.5 km radius NS. For DM masses below
about 10 MeV (left panel), DM capture can occur by scattering on superfluid
neutrons in the inner crust, and exciting phonons. The single-phonon emission
mode is expected to dominate, which proceeds via a static structure function =
$\Delta p/(2m_{n}c_{s})$ that relates the per-nucleon cross section to the
phonon-excitation cross section. Here $c_{s}$ is the phonon speed. Due to the
proportionality to the transfer momentum, $\sigma_{\rm cap}\propto
m_{\chi}^{-1}$ similar to the Pauli-blocking regime of the NS core discussed
above. The latter sensitivity (applicable to when the core is populated mainly
by neutrons) is also shown for comparison in the plot. For DM masses above
about 100 MeV (right panel), DM capture can occur by scattering on individual
nucleons locked up in nuclei in the outer crust by transferring energies
greater than their $\sim$MeV binding energy. Scattering on nuclei is generally
suppressed: large $\Delta p$ leads to loss of nuclear coherence over multiple
nucleons, and small $\Delta p$ leads to loss of coherence over multiple
nuclei, described by a lattice structure function. Deeper down in the inner
crust, heavier-than-100-MeV DM capture proceeds by scattering on loosely bound
nucleons, and even further down, by scatterig on the pasta phase. Pasta
scattering may either be on individual nucleons at high DM masses or on
multiple nucleons at low DM masses as described by response functions
accounting for inter-nucleon correlations. A resonant peak in the response
function is seen to enhance the capture sensitivity near $m_{\chi}\simeq$ 100
MeV. For comparison is also shown the DM capture cross section for scattering
in an NS core dominated by neutrons.

Even in the absence of exotic phases, NS cores are expected to contain
$\sim$10% level populations of protons, electrons, and muons thanks to beta
chemical equilibrium. DM may be possibly be leptophilic, such that scattering
at tree level is solely on $e^{-}$ and/or $\mu^{-}$, or iso-spin violating,
such that scattering is dominantly on protons. NS capture and heating applies
to these scenarios, too [174]. While the Fermi momenta of protons and muons
are smaller than their mass, making them non-relativistic and amenable to the
above treatment, that of electrons are 1–2 orders of magnitude greater than
$m_{e}$, warranting relativistic kinematics to treat their DM capture in the
stellar rest frame [189, 167, 190, 191, 192, 193, 194, 166]. This also makes
the treatment of Pauli-blocking non-trivial [192, 166]. In particular, the
capture probability accounting for Pauli-blocking, relativistic scattering and
summing over multiple scatters is [166]

$df=\sum_{N_{\text{hit}}}\;d\sigma_{\rm CM}\,v_{\rm
Mol}\,dn_{\text{T}}\,\frac{\Delta t}{N_{\text{hit}}}\,\Theta\left(\Delta
E-\frac{E_{\text{halo}}}{N_{\text{hit}}}\right)\Theta\left(\frac{E_{\text{halo}}}{N_{\text{hit}}-1}-\Delta
E\right)\Theta\left(\Delta E+E_{p}-E_{\text{F}}\right)\ ,$ (49)

where $v_{\rm Mol}$ is the Möller velocity that relates the cross section in
any frame to that in the center of momentum frame ($d\sigma_{\rm CM}$),
$dn_{\text{T}}$ is the differential volume of the target momentum space
normalized to the Fermi volume, $E_{\rm halo}$ is the DM halo kinetic energy,
and $\Delta E$ is the energy transfer. We refer the reader to Ref. [166] for a
detailed formalism. In Figure 9’s bottom panels we show the NS capture
sensitivity to contact interaction cutoffs versus $m_{\chi}$ for scalar-type
operators involving spin-1/2 and spin-0 DM. For electron scattering the NS
capture reach is seen to be orders of magnitude greater than that of
terrestrial direct searches for $m_{\chi}>$ MeV, and indeed completely
complements the latter for sub-MeV DM masses.

NS capture-and-heating can also provide orders-of-magnitude improvement over
Earth-bound searches for DM with scattering that is

  1. 1.

spin-dependent, since scattering directly on fermions instead of nuclei does
not lead to the loss of nuclear coherence that limits spin-dependent searches
at direct detection [186, 169, 195],

  2. 2.

and/or velocity-dependent [186, 169, 195], since semi-relativistic DM speeds
at the point of capture overcome velocity-suppressed scattering rates,

  3. 3.

inelastic [174, 196, 197], since again the high DM speeds ensure that
$\mathcal{O}(100)$ MeV mass splittings between the DM and its excited state
can be probed, as opposed to $\mathcal{O}(100)$ keV at direct detection, and

  4. 4.

below the so-called neutrino floor at direct searches, coming from irreducible
neutrino backgrounds that are irrelevant for NS capture; see Fig. 9 top right
panel,

  5. 5.

with heavier-than-PeV DM, where DM capture proceeds mainly through multiple
scattering in transit [198, 199, 174].

####  4.1.2 Dark matter self-annihilations, nucleon co-annihilations, and
induced nucleon decay

While the discussion above focused NS heating from the transfer of captured DM
kinetic energy, applicable to any particulate dark matter model – in
particular to non-annihilating DM such as asymmetric DM – certain scenarios
may lead to DM annihilation inside the NS that further brightens it [184, 200]
and thereby facilitate observations [174, 186, 48, 45], in some cases reducing
telescope integration times by a factor of 10. For instance, JWST/NIRCam could
constrain a 2480 K NS, heated by local DM kinetic energy + annihilations, with
SNR 2 in 2.5 hr $(d/10{\rm pc})^{4}$, and TMR/IRIS could do so in 0.56 hr
$(d/10{\rm pc})^{4}$ [174]; compare these with kinetic heating-only exposure
times in Sec. 4.1.1. Fig. 8 shows JWST sensitivities in more detail, as
discussed in Sec. 4.1.1.

Self-annihilations of DM into most SM states would result in NS heating, the
exception being neutrinos with sub-100 MeV energies as their optical depth in
the NS material is too small to be trapped [201]. In any case, this phenomenon
relies intricately on whether or not the DM thermalizes with the NS within its
lifetime, since DM may possibly annihilate much more efficiently if it is
collected within a small volume in the NS core; this is a highly model-
dependent question [189, 194, 48] as discussed in Sec. 4.4.1. To understand
this, consider the evolution of the number of DM particles $N_{\chi}$ within a
volume $V$ of the NS self-annihilating with a thermally averaged cross section
$\langle\sigma_{\rm ann}v\rangle$, and its solution:

$\displaystyle\frac{dN_{\chi}}{dt}$ $\displaystyle=$ $\displaystyle
C_{\chi}-\frac{\langle\sigma_{\rm ann}v\rangle N_{\chi}^{2}}{V}~{},$
$\displaystyle N_{\chi}(t)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{C_{\chi}V}{\langle\sigma_{\rm
ann}v\rangle}}\tanh\bigg{(}\frac{t}{\tau_{\rm eq}}\bigg{)}~{},$ (50)
$\displaystyle\tau_{\rm eq}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{V}{C_{\chi}\langle\sigma_{\rm ann}v\rangle}}~{},$

where $C_{\chi}=C_{n\chi}+C_{\chi\chi}$ is the total DM capture rate via
scattering on nucleons (Eq. (45)) and, through self-interactions, on DM
already accumulated in the NS, and $\tau_{\rm eq}$ is the characteristic
timescale for equilibrium between capture and annihilation to establish, after
which $N_{\chi}(t)$ achieves a steady state ($dN_{\chi}/dt\rightarrow 0$).
Thus for $t>\tau_{\rm eq}$, the total annihilation rate equals the capture
rate. When $V$ is the thermal volume (Eq. (52)), one can then compute the
minimum annihilation cross section required for capture-annihilation
equilibrium to occur well within the age of an observed NS, $\tau_{\rm NS}$.
Using a partial-wave expansion $\langle\sigma_{\rm ann}v\rangle=a+bv^{2}$, the
condition may be written for $s$-wave and $p$-wave domination as [194]

$\displaystyle a>7.4\times 10^{-54}\ {\rm cm^{3}/s}\ \bigg{(}\frac{{\rm
Gyr}}{\tau_{\rm NS}}\bigg{)}^{2}\bigg{(}\frac{C_{\rm
max}}{C_{\chi}}\bigg{)}\bigg{(}\frac{\rm GeV}{m_{\chi}}\frac{T_{\rm
NS}}{10^{3}~{}{\rm K}}\bigg{)}^{3/2}~{},$ $\displaystyle b>2.9\times 10^{-44}\
{\rm cm^{3}/s}\ \bigg{(}\frac{{\rm Gyr}}{\tau_{\rm
NS}}\bigg{)}^{2}\bigg{(}\frac{C_{\rm max}}{C_{\chi}}\bigg{)}\bigg{(}\frac{\rm
GeV}{m_{\chi}}\frac{T_{\rm NS}}{10^{3}~{}{\rm K}}\bigg{)}^{1/2}~{},$ (51)

where $C_{\rm max}$ is the maximum capture rate achieved at the saturation
cross section.

Interestingly, a thermal Higgsino of 1.1 TeV mass, a largely unconstrained
true electroweak WIMP [140]), would thermalize with just the NS crust rapidly
enough to heat a reasonably old NS through annihilations in equilibrium with
the rate of capture [48]. We also remark that due to different scalings of the
NS luminosity from kinetic or annihilation heating on the NS mass and radius,
in principle it must be possible to distinguish between the two heating
mechanisms using an ensemble of NSs [174, 45].

An interesting way to probe DM self-annihilations in NSs is possible if the
primary annihilation products are feebly interacting DM-SM mediators that live
long enough to exit the star before decaying to SM states. One could search
for a flux of these states sourced by DM “focused” in celestial bodies via
capture. For gamma-ray final states, limits have been imposed with Fermi and
H.E.S.S. data on DM-nucleon scattering and DM self-annihilation cross sections
using brown dwarfs for sub-GeV DM and NSs for TeV mass DM [126]. For neutrino
final states, limits in the TeV-PeV range come from IceCube, KM3NeT and
ANTARES [202, 203].

Dark matter species that carry negative baryon number, arising for instance
from “hylogenesis” models, could annihilate with baryons in a NS post-capture
leading to possibly observable heating signals [204, 205, 206]. Such co-
annihilations with nucleons are also possible in models of “dark baryons” that
undergo quantum mixing with the neutron [207]. Yet another co-annihilation-
like scenario resulting in NS heating is when a component of the DM comes in
the form of magnetically charged black holes (MBHs) [208]. Subspecies that
come with electroweak-symmetric coronas are expected to be near-extremal in
mass, however upon encountering NSs they may become non-extremal: first they
may capture in the NS by stopping due to the Fermi-degenerate gas, then they
could absorb nucleons that are emitted back as (baryon number-violating)
Hawking radiation, overheating NSs. A smaller deposit of heat could come from
mergers of captured MBHs that enhance Hawking radiation, mimicking a self-
annihilation process. Energy depositions from DM annihilations may also
possibly nucleate quark matter bubbles in the NS interior, resulting in
emission of radiation, cosmic rays and gravitational waves [209].

The production of x-rays and other high-energy effluents emitted from NSs,
resulting from monopoles passing through and catalyzing nucleon decay, have
been studied [210, 211]. This provides a strong bound on the abundance of
monopole species that induce nucleon decay, which is a well-motivated class of
monopoles arising from symmetry-breaking in Grand Unified Theories.

####  4.1.3 Improvements and uncertainties

The above treatment has been improved by accounting for a number of physical
effects in the NS, which in some cases leads to observational uncertainties;
these effects are collected in Table 1. The largest uncertainty in the capture
rate, spanning two orders of magnitude, comes from the unknown mass of the NS
candidate that will be observed [169], unless some precise mass-radius
measurement is performed. Other effects that may modify the DM capture rate,
applicable to different DM and NS mass ranges, are variations in the EoS of NS
matter, self-energies from the nuclear potential, nucleon structure that
suppresses coherent scattering, nucleon superfluidity, extinction in the NS in
the optically thick regime, scattering on relativistic nucleons, gravitational
focusing in the NS interior layers, suppression of scattering via the
propagator of a mediator of mass smaller than the transfer momentum, and the
Galactic velocity distribution of DM. Table 1 lists the appropriate references
that treat these effects.

Figure 10: Sensitivities of self-consistent dark matter models to NS kinetic
heating; see Sec. 4.1.4. Top left. [174] Electroweakino singlet and doublet
mass parameters for various $\tan\beta\equiv$ ratio of Higgs VEVs, that may be
cornered through inelastic scattering of thermal Higgsino DM in the NS via
excitation to charged and neutral states (regions marked by “$\delta<$ GeV”).
Top right. [195] As a function of DM mass, gluonic coupling to an axion-like
particle that mediates velocity-dependent scattering interactions. The gray
region depicts limits from beam dumps, rare meson decays, and astrophysics. NS
capture can also proceed through mediation by a CP-even scalar in the theory,
which gives rise to limits from direct detection. Bottom left. [212] The
orange region can be probed for spin-0 DM scattering on muons in the NS by
exchanging a $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson. Also shown are constraints
from DM self-interactions. Bottom right. [207] NS temperatures achieved by
capture and heating of the anti-particle of DM carrying baryon number = 1, in
a scenario where DM self-interacts repulsively and annihilates to the mediator
$\phi$ that then decays to SM states that deposit heat.

####  4.1.4 Dark matter models that heat neutron stars through scattering and
annihilation

Making use of the general effects discussed above, specific UV-complete and
self-consistent DM models have been explored in the context of NS capture and
heating. These include the supersymmetric partner of the Higgs field, the
Higgsino, that captures through inelastic scattering to electrically neutral
and charged excited states [174, 48], a generalization of this to electroweak
multiplets [213], a model of DM with a vector force-carrier of a gauged
$L_{\mu}-L_{\tau}$ interaction [191], DM in the form of a GeV-scale “dark
baryon” that mixes with the neutron [207], simplified models of DM (specifying
a single state each for DM and the mediator) with various mediator species
[214, 215, 216], DM that arises as a pseudo-Goldstone boson [217], models of
dark sectors that can explain the muon $g-2$ anomaly [218], and consistent
models of DM interacting with nucleons through a pseudoscalar mediator: axion-
like particles and a CP-odd state that arises in a Two-Higgs doublet model
[195]. The sensitivities to parameters of some of these scenarios are shown in
Fig. 10.

####  4.1.5 Neutron star reheating mechanisms not involving dark matter

A search for DM reheating NSs in the late stages of their cooling must
encompass understanding other astrophysical mechanisms that could possibly do
the same. We discuss below those that feature prominently in the literature.

  1. 1.

DM capture in NSs would not encounter a “neutrino floor” due to very dilute
ambient neutrino densities that produce suppressed recoils/absorption on NS
constituents, owing to low cross sections and Pauli-blocking. However, it is
natural to ask if there is an “ISM floor” from accretion of interstellar
material. It turns out that old, isolated NSs that have spin periods $<$ 1000
seconds do not accrete ISM as they are in an ejector phase [219]: a “pulsar
wind” of ISM outflow powered by the NS’ magnetic field, being much denser than
the inflowing material attempting accretion, would pre-empt accretion via
kinetic pressure. Even if the pulsar wind happens to be weak enough for the
ISM to overcome it, there is a second barrier to accretion: the magnetosphere
co-rotating with the NS will impute centrifugal acceleration to the ISM,
spraying away the gas – the propellor phase. For NSs with unusually large spin
periods of $>1000$ seconds, these arguments do not apply, instead infalling
ISM would be deflected along the magnetic field lines of the NS and accretion
will be confined to a small polar region, which can be distinguished from all-
surface thermal emission. In any case, the ISM density in the local 100 pc is
10-3 GeV/cm3 [220] so that any ISM accretion will be outdone by present-day DM
capture near geometric cross sections.

  2. 2.

Rotochemical heating could result from an imbalance in chemical potentials as
the NS material is driven out of beta chemical equilibrium by deceleration in
the rotation of NSs. Reactions that seek to restore chemical equilibrium
deposit heat in the NS. This mechanism could occur for NSs with small (sub-7
ms) pulsar spin periods at birth for certain nucleon pairing models [221] – a
requirement in tension with studies that find that natal spin periods are
likely $\mathcal{O}(10-100)$ ms (see the references listed in Ref. [222]).

  3. 3.

Other astrophysical late-time NS reheating mechanisms include [223, 224]
magnetic field decay that dissipates energy into the NS material, crust
cracking which arises when the NS crust breaks as the NS relaxes from an
oblate to spherical shape, releasing accumulated strain energy, and vortex
creep which develops as superfluid vortex lines travel outward as the NS spins
down, and get pinned to the nuclear lattice in the inner crust thereby
introducing a velocity difference between the crust and the superfloid core,
which dissipates energy in the star.

We note that these mechanisms are speculative, and none have been
unequivocally observed. An exclusion set by non-observation of DM-induced
heating at imminent telescopes would also rule out these mechanisms. Another
notable point is that while the rotational power of NSs goes into dipole
radiation, which in turn illuminates the nebula surrounding the pulsar as we
saw in Sec. 2.8, it very likely does not contribute to the NS thermal
luminosity. This is already apparent in the Crab Nebula example discussed in
Sec. 2.8, but can also be inferred from x-ray emission bounds on observed
pulsars which have thermal luminosities 5-6 orders smaller than the rotational
power: see Table 5 of Ref. [225]. Further, the diffusion of the $B$ field in
the NS is also unlikely to heat the NS; as argued in Ref. [226], NSs older
than about Myr are cool enough for magnetic diffusion timescales to exceed the
NS age, effectively shutting off $B$ field dissipation regardless of the
initial strength of the field.

Figure 11:  Top. NS cooling timescale versus surface temperature (obtained
from Eq. (30)), superimposed on a plot of time between DM clump-NS encounters
versus the energy deposited by kinetic heating during the passage of a clump
for various NS radii (left). The ticks on either x-axis and either y-axis
correspond one-to-one to each other. This plot shows the region in which NSs
are expected to glow at a steady temperature, so that observing a single NS is
enough to set constraints, and the region where overheated NSs cool down
rapidly between clump encounters, so that astronomical surveys are required to
observe the fraction of overheated NSs in an ensemble. On the right are future
sensitivities of astronomical observations of NSs on DM clump radii and
masses, exploiting dark kinetic heating, seen to be complementary to limits
from other experiments. These limits are valid for DM-nucleon cross sections
greater than the values for which the effects in these searches are relevant.
These two plots are taken from Ref. [164]; see Sec.4.2 for further details.
Bottom left Dark clump masses and radii constrained by compact stellar
thermonuclear explosions, occurring for the minimum DM-nucleus cross sections
per DM mass overlaid; see Sec.4.2. Bottom right. For two different internal
density profiles, the mean flux density of transient radio signals at various
telescopes from encounters of axion miniclusters as a function of the transit
time (= signal burst duration), taken from Ref. [227]. See Ref. 4.11 for
further details.

###  4.2 Neutron stars and halo substructure

Numerous cosmologies predict enhanced small-scale power, for instance via an
early matter-dominated era or DM self-interactions assisting primordial
density perturbations, resulting in a substantial fraction of DM surviving in
substructure termed variously as clumps, subhalos, minihalos and miniclusters
[228, 229, 230, 231, 232, 233, 234, 235]. If DM has scattering interactions
with the SM, and if the interacting component resides in clumps, direct
searches may have observed no conclusive signal simply because the Earth has
yet to encounter a subhalo since their inception. In this senario, subhalo DM
may be observed by its heating of old, nearby NSs: the latter may travel
through DM clumps and capture constituent DM particles, giving rise to kinetic
and/or annihilation heating.

In the top left panel of Fig. 11, taken from Ref. [164], is shown the cooling
time of NSs as a function of the NS surface temperature in green, and in the
same plot is shown the energy deposited by clumps in NSs during encounters,
$E_{\rm meet}^{T}$, as a function of the time between NS-clump encounters for
various clump sizes. The $E_{\rm meet}^{T}$ in the top x-axis correspond to
the NS temperatures imparted in the bottom x-axis immediately following the
encounter. For encounter times shorter than cooling times, the NS will glow at
a steady-state luminosity, whereas for those longer than cooling times, NSs
would be expected to glow brightly for short durations following encounters
before dimming. In the latter case, sky surveys of large populations of NSs
may be able to pick out the fraction that is still above some temperature to
which the telescope is sensitive. In the top right panel, also taken from Ref.
[164], are shown clump mass vs radius regions that may discovered by observing
more than 100 NSs above 104 K in the local kiloparsec, e.g., by Roman/WFIRST
and Rubin/LSST, and excluded by observing a single NS with temperature $<$
1000 K, e.g. by JWST, ELT and TMT. Also shown is a region that is already
excluded by the observation of the coldest ($<$30,000 K) known NS PSR
J2144$-$3933 by the Hubble Space Telescope (HST) [236] for clumps made of
dissipative or strongly self-interacting DM, which would acrrete onto NSs
through the Bondi–Hoyle–Lyttleton mechanism [237, 238, 239].

In addition, in the presence of a long-range fifth force, NS heating by clumps
may be enhanced by greater focusing effects, greater DM kinetic energies upon
arrival at the NS surface, and seismic oscillations induced by an effective
tidal force. In the bottom right panel of Fig. 16, taken from Ref. [240], is
shown the limit from overheating PSR J2144$-$3933 on the effective NS-clump
coupling versus clump mass, for four values of the range of the fifth force
arising from a Yukawa potential [241]. (We do note that the DM need not be in
the form of a clump for these limits to apply, but could also be a tightly
bound composite.) The curve labelled “NS kinetic heating” corresponds to
having an additional short-range interaction enhance DM capture. These limits
are complementary to those coming from the Bullet Cluster on DM self-
interactions mediated by the light mediator, from weak equivalence principle
tests using Galacto-centric motions of celestial bodies on the inter-baryonic
force mediated by the same, and from the 15 year dataset of the NANOGrav
pulsar timing array (see also Sec. 4.10.1).

Yet another signature of clumps with nucleon scattering interactions is
thermonuclear explosions induced in compact stars, as discussed in Sec. 3.3.
These could be Type Ia-like supernovae in carbon-oxygen WDs or x-ray
superbursts in the carbon ocean layer in NS crusts (Sec. 2.5). Constraints
from the observed frequency of NS superbursts (Sec. 4.3) and from the
existence of WDs (Sec. 3.3) are shown in the left bottom panel of Fig. 11 in
the plane of clump size and mass; the contours overlaid are the minimum
reduced nuclear cross sections required to ignite a trigger mass of the
stellar material. This method of constraining clumps could be extended to
those with baryonic long-range forces discussed above. In that case, limits on
the effective coupling apply to far smaller values (all the way to unity) than
shown in Fig. 16 bottom right panel, and to much higher clump masses. See Ref.
[153].

Clumps encountering NSs can also be made of axions, leading to interesting
signatures depicted in the right bottom right panel of Fig. 11, which we
discuss in Sec. 4.11. We also note that the phenomenology of black hole
formation inside NSs (Sec. 4.4) would be applicable here if NS-clump
encounters are frequent enough.

###  4.3 Dark matter inducing superbursts in neutron stars

Superbursts in NS carbon oceans, described in Sec. 2.5, can be induced by
transiting DM if it is sufficiently heavy to deposit the requisite trigger
energy. Ref. [152] set limits on the cross sections and (super-Planckian)
masses of macroscopic DM opaque to nuclei by satisfying the runaway criteria
(Eqs. (16) and (18)) and requiring that the time between DM-NS encounters is
smaller than the inferred maximum recurrence time of the superburst 4U
1820+30. Ref. [153] set limits on the masses, radii and interaction strengths
of dark clumps (shown in Fig. 11 bottom left panel) and nuggets with long-
range baryonic forces, using inferred recurrence times of the six superbursts
(out of 16 detected in total) that have been observed to repeat [82, 83].

###  4.4 Dark matter that implodes neutron stars into black holes

Dark matter that is captured by an NS, after repeated re-scattering with the
NS medium, will settle into a small thermalized region at the center of the
NS. As more DM is collected, this spherical agglomeration can grow to a large
enough mass that it collapses and forms a small black hole, which may
(depending on its mass) subsequently accrete the entirety of the NS,
transforming it into a solar mass black hole [11]. The processes of DM capture
in NSs, thermalization, accumulation to the point of collapse, collapse,
formation of a black hole, and its possible evaporation via Hawking radiation
or growth consuming the NS, have been investigated in Refs. [242, 184, 118,
200, 243, 244, 245, 246, 247, 248, 249, 85, 250, 251, 252, 253, 254, 148, 141,
255, 256, 257, 258, 143, 150, 259, 260, 261]. In addition, possible
astrophysical signatures of DM converting NSs to black holes have been
identified in, $e.g.$, Refs. [249, 262, 251, 252].

The kind of DM that is by and large studied in this context is “asymmetric
dark matter”, DM primarily made of its particles as opposed to a symmetric
population of particles and anti-particles. This emulates the visible
universe, which is primarily matter (electrons, nucleons) and not anti-matter;
indeed, the asymmetry in DM may be linked to that of the visible sector [142,
263], but this is not necessary for the discussion that follows. The primary
feature that permits asymmetric DM to convert NSs into black holes is that it
is typically777For the exception, see Ref. [264]. non-annihilating, and so as
it collects inside the NS, it is not expected to annihilate to Standard Model
states. This may be compared with symmetric, annihilating DM discussed in Sec.
4.1.2. Investigation into what fraction of the DM may self-annihilate or co-
annihilate with nucleons, while still forming a black hole inside the NS, was
undertaken in Refs. [247, 248, 265].

Another kind of DM which could convert NSs into black holes is primordial
black holes [266, 267]. A PBH captured in an NS can settle inside, accrete NS
material, and convert the NS into another black hole [268, 269, 270, 271, 272,
251, 273, 274, 275, 145, 276, 277, 278]; this is detailed in Section 4.5. For
the remainder of this sub-section we will focus on particle DM.

We now turn to details of the processes leading asymmetric dark matter to
convert NSs into black holes. They proceed as follows: (1) DM is captured in
the NS and thermalizes with the NS interior, forming a small ball of DM at the
center, (2) the DM ball reaches a critical mass at which it collapses, and
through some cooling process continues to collapse until (3) a small black
hole forms which, provided accretion of NS material outstrips Hawking
radiation, will result in the conversion of the NS to a black hole. Figure 12
left panel shows a simple schematic of this process.

Figure 12: Left. Schematic of asymmetric dark matter converting a NS into a
black hole. Right. Dark matter per-nucleon scattering cross section versus
mass bounds on heavy fermionic asymmetric dark matter from the observation of
old Milky Way pulsars that have not been converted to black holes [251],
compared with terrestrial direct search limits and their neutrino floor. Also
shown are prospects for observating NS mergers with accompanying kilonovae,
localized to 1 kpc precision inside Milky Way-like spiral galaxies. A detailed
discussion of Milky Way pulsar ages, and in particular PSR J1738+0333, which
has a characteristic age confirmed by the age of its WD companion, can be
found in Ref. [279].

####  4.4.1 Dark matter thermalization in neutron stars

In step (1) above, the size of the thermalized region is determined by the
temperature of the NS, which sets the final temperature of the DM particles,
and by the central density of the NS, which sets the gravitational potential
binding energy. A simple application of the virial theorem yields an estimated
DM thermal radius of [247]

$r_{\rm th}\approx 20~{}{\rm cm}\left(\frac{{\rm
GeV}}{m_{\chi}}\right)^{1/2}\left(\frac{T_{NS}}{10^{3}~{}\rm
K}\right)^{1/2}\left(\frac{10^{15}~{}{\rm g/cm^{3}}}{\rho_{\rm
NSc}}\right)^{1/2},$ (52)

where $\rho_{\rm NSc}$ is the NS central density. The time it takes for DM to
sink to this region depends on a few timescales (see $e.g.$, Ref. [143],
Section 3 for a review), but usually the longest is the time it takes for DM
to scatter with its lowest velocities/temperatures on nucleons, after having
mostly settled inside the NS. A detailed calculation of this timescale
requires modeling the NS core, and so the result will depend on the density,
degeneracy, and possibly even new QCD phases in the NS interior. For neutrons
treated as a degenerate fluid, we have [189]

$t_{\rm th}\approx 3000~{}{\rm
yr}~{}\frac{\frac{m_{\chi}}{m_{n}}}{\left(1+\frac{m_{\chi}}{m_{n}}\right)^{2}}\left(\frac{2\times
10^{-45}~{}{\rm
cm^{2}}}{\sigma_{n\chi}}\right)\left(\frac{T_{NS}}{10^{5}~{}{\rm
K}}\right)^{2},$ (53)

where this expression assumes a momentum-independent cross section for
spin-1/2 DM scattering on nucleons via a heavy mediator. Extensions to spin-0
DM, Lorentz structures of DM-nucleon interactions leading to momentum-
dependent cross sections, and light mediators were investigated in Ref. [194].
In the above expression, the thermalization timescale counter-intuitively
decreases with increasing DM mass above $m_{n}$: one would naively expect that
heavier DM takes longer to thermalize. But the effect comes about because
$t_{\rm th}$ is set by the inverse of the energy loss rate (in turn depending
on the DM-nucleon scattering rate) in the NS degenerate medium with phase
space restrictions, and this rate goes as positive powers of the (continually
degrading) DM momentum $k_{\rm cold}$. For DM energies close to the NS
temperature, $k_{\rm cold}\simeq\sqrt{3m_{\chi}}T_{\rm NS}$, implying energy
is lost faster in the last few scatters for heavier DM, i.e., implying quicker
thermalization. In Figure 13 we show the per-nucleon cross section or
effective field theory coupling necessary for DM to thermalize inside an NS on
10 Gyr year timescales for certain models.

As discussed in Sec. 4.1.2, depending on the DM annihilation cross section,
thermalized DM collected within $r_{\rm th}$ can annihilate efficiently enough
to yield interesting signals.

Figure 13: Left. [189] Cross section for dark matter to thermalize in a
neutron star in 10 billion years, assuming a momentum-independent cross
section with neutrons. Right. [194] The case of scattering on neutrons through
the scalar current operator indicated in the figure with mediator mass
$m_{\phi}$. Thermalization in 10 Myr and 10 Gyr for different NS temperatures,
and a curve indicating parameters that lead to DM capture in the NS through
geometric cross sections, are shown.

####  4.4.2 Collapse of dark matter and formation of small black hole

In step (2), after enough DM has collected in the thermalized region in the
NS, it will reach a critical mass at which it collapses. The exact density of
DM required to initiate collapse will depend on its self-interactions, and by
extension its EoS and sound speed while contained in the NS. Assuming
negligible self-interactions, the critical mass required for collapse is

$M_{\rm crit}\approx 7\times 10^{46}~{}{\rm GeV}\left(\frac{10^{7}~{}{\rm
GeV}}{m_{\chi}}\right)^{3/2}\left(\frac{T_{\rm NS}}{10^{3}~{}\rm
K}\right)^{3/2}\left(\frac{10^{15}~{}{\rm g/cm^{3}}}{\rho_{\rm
NSc}}\right)^{1/2}.$ (54)

For a detailed review of the conditions for collapse see $e.g.$, Section 4 of
[143]. It is generally the case that if DM thermalizes rapidly through
scattering with neutrons in the NS interior, then when it reaches the point of
collapse it will also rapidly shed the gravitational energy required to form a
black hole. This is because the temperature is higher during collapse and
hence the time to shed gravitational energy is typically shorter. As the
shortness of this timeframe is common, this part of the collapse dynamics is
not always treated explicitly, but Refs. [85, 143, 148, 11] provide more
detailed treatment, both in compact stars and other astrophysical bodies. The
time for the DM sphere to collapse below its Schwarzschild radius will depend
on whether it cools via scattering with neutrons or through other radiative
process, $e.g.$, emission of a light particle in the dark sector [85].

An additional consideration is whether enough DM will have collected to exceed
the dark sector Chandrasekhar mass (analogous to Eq. (7)), parametrically of
order

$M_{\rm Chand,f}\approx\frac{M_{\rm Pl}^{3}}{m_{\chi}^{2}}\approx
M_{\odot}\left(\frac{{\rm GeV^{2}}}{m_{\chi}^{2}}\right)~{},$ (55)

while for bosons this is

$\displaystyle M_{\rm Chand,b}$ $\displaystyle\approx$
$\displaystyle\frac{2M_{\rm
Pl}^{2}}{m_{\chi}}\left(1+\frac{\lambda}{32\pi}\frac{m_{pl}^{2}}{m_{\chi}^{2}}\right)^{1/2}$
(56) $\displaystyle\rightarrow$ $\displaystyle\begin{cases}2\frac{M_{\rm
Pl}^{2}}{m_{\chi}}~{},\ \ \lambda\ll 1,\\\
\frac{\lambda}{2\sqrt{2}}\frac{M_{\rm
Pl}^{3}}{m_{\chi}^{2}}~{},\lambda>100m_{\chi}/M_{\rm Pl}~{},\end{cases}$

where $\lambda$ is the boson $\phi$’s repulsive self-interaction coupling
arising in the Langrangian $\mathcal{L}\supset-(\lambda/4!)\phi^{4}$.

Attractive DM self-interactions could alter the amount of asymmetric fermionic
DM necessary for collapse to a black hole [246, 265, 279]. The collapse of
light fermionic DM is in principle permitted by the attractive self-
interaction mediated by a light scalar, however, a detailed study of the final
stage collapse to a black hole has pointed out an important caveat [280]: for
a simple scalar field potential consisting only of a mass term and a coupling
to the fermions, the effective mass of the scalar could grow during DM fermion
collapse, preventing collapse to a black hole. Whether bosonic self-
interactions let DM form black holes in NSs is a non-trivial question. In
particular, a large value of $\lambda$ can shift bosonic asymmetric DM bounds
from old NSs to higher DM masses [245, 247, 248]. Bosonic asymmetric DM
forming black holes inside a NS do so by forming a Bose-Einstein condensate
(BEC) [245, 244, 281, 247, 248, 252], from which collapse will proceed for
GeV-mass DM. The dynamics of the BEC prior to and following collapse would
affect whether a black hole is produced and is an area of investigation [281,
247, 248, 252, 282].

####  4.4.3 Growth or evaporation of dark matter-formed black hole in the
neutron star

After a black hole is formed inside the NS, step (3) is to determine whether
it is so small that it will rapidly evaporate away via Hawking radiation, or
whether it is so large that through accumulating surrounding baryonic material
it will grow to consume the NS in a relatively short timeframe. Initial
studies of this process estimated whether Bondi accretion by the black hole
would proceed faster than Hawking radiation, which is entirely determined by
the initial mass of the black hole [245, 244]. Later studies incorporated the
accumulation of DM particles additionally collected into the NS and onto the
black hole, finding that this can substantially influence whether the black
hole would grow to consume the NS [247, 248, 265].

Altogether, the requirement that the black hole grows in the NS is given by

$\dot{M}^{(\rm NS~{}accretion)}+\dot{M}^{(\rm DM~{}accretion)}-\dot{M}^{(\rm
Hawking)}>0,$ (57)

where the first term is the NS accretion rate onto the black hole, the second
is the DM accretion rate onto the black hole, and the third is the Hawking
radiation rate. Each of these terms has been individually studied in the
context of asymmetric DM which causes NSs to implode:

  1. 1.

NS accretion: The simplest treatment of NS accretion onto the black hole
assumes Bondi accretion. In practice, angular momentum of the NS fluid around
the black hole, for a rapidly spinning NS, can diminish accretion relative to
naïve Bondi accretion, but the high viscosity of the NS fluid results in
infall rates consistent with spherical Bondi accretion despite angular
momentum effects [283]. Sufficiently small black holes will have a quantum
penalty to accumulation of neutrons due to the neutron de Broglie wavelength;
this effect can be pronounced for black hole masses near the edge of growth
vs. evaporation [260]. The accretion of the NS fluid onto the black hole
inside a NS has been studied in a detailed simulation that accounts for
hydrodynamic and general relativistic effects [255], finding that in the final
stages of accretion, the mass of NS fluid ejected from the accretion zone is
likely less than about $10^{-4}~{}M_{\odot}$.

  2. 2.

DM accretion: The accretion of DM onto the small black hole inside the NS can
substantially affect whether it grows or shrinks, especially when the DM
accretion rate onto the NS is maximized [247, 248, 265, 252].

  3. 3.

Hawking radiation: There is a correction to the evaporation rate of black
holes inside NSs, coming from the Pauli blocking of Hawking radiation, since
the region around the black hole will be inhabited by a degenerate sea of SM
fermions [284].

####  4.4.4 Signatures of dark matter that implodes neutron stars

Figure 14: Left. From a simulation of a NS accreting onto a black hole at is
center [255]. NS fluid density and velocity vectors are shown in the left half
of these figures, with angular momentum density shown on the right half, for a
different NS spin parameters $\Omega$. Right. Number of NSs converted to black
holes in a Milky Way-like galaxy, along with the current NS implosion rate for
dark matter models that would cause NSs near Earth to implode in 10 Gyr
(“ADM1”) and 50 Gyr (“ADM2”) [251].

A number of striking astrophysical signatures arise from DM that converts NSs
to black holes. Firstly, the oldest known pulsars can be used to set limits on
asymmetric DM, since for a given background DM density, the existence of these
pulsars limits the accumulation of DM [242, 243, 244, 245, 246, 247, 248, 249,
184, 200]. However, the “characteristic age” of pulsars comes with caveats: it
is not always a good indicator of the actual age of the pulsar as discussed in
Sec. 2.8. One particular pulsar, PSR J1738+0333, is in a binary system with an
old WD, and thus to go with its characteristic age, has an additional age-
marker in its WD companion, both of which point to a $\gtrsim 5$ Gyr-old NS.
Hence this pulsar has been used to set bounds on asymmetric DM [279, 251]. A
recent work [285] integrates over the density of DM that NSs traverse during
their orbits around the Milky Way, refining bounds that use characteristic
ages of old, nearby pulsars.

A number of prompt and delayed signatures may be sought if DM is converting
Gyr-old NSs to black holes in regions where DM is denser than in the outer
regions of the Milky Way. In particular, the absence of millisecond pulsars in
the Galactic Center [286, 287] has been linked to models of DM that would
convert old NSs to black holes [249, 200]. These studies predict a maximum