# Electromagnetic Signatures of Mirror Stars

Isabella Armstrong* Department of Astronomy and Astrophysics, University of
Toronto, Toronto, Ontario M5S 1A7, Canada  Berkin Gurbuz* Department of
Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada  David Curtin
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
Christopher Matzner Department of Astronomy and Astrophysics, University of
Toronto, Toronto, Ontario M5S 1A7, Canada

###### Abstract

Mirror Stars are a generic prediction of dissipative dark matter models,
including minimal atomic dark matter and twin baryons in the Mirror Twin
Higgs. Mirror Stars can capture regular matter from the interstellar medium
through extremely suppressed kinetic mixing interactions between the regular
and the dark photon. This accumulated “nugget” will draw heat from the mirror
star core and emit highly characteristic X-ray and optical signals. In this
work, we devise a general parameterization of mirror star nugget properties
that is independent of the unknown details of mirror star stellar physics, and
use the Cloudy spectral synthesis code to obtain realistic and comprehensive
predictions for the thermal emissions from optically thin mirror star nuggets.
We find that mirror star nuggets populate an extremely well-defined and narrow
region of the HR diagram that only partially overlaps with the white dwarf
population. Our detailed spectral predictions, which we make publicly
available, allow us to demonstrate that optically thin nuggets can be clearly
distinguished from white dwarf stars by their continuum spectrum shape, and
from planetary nebulae and other optically thin standard sources by their
highly exotic emission line ratios. Our work will enable realistic mirror star
telescope searches, which may reveal the detailed nature of dark matter.

##  1 Introduction

†† _First authors are indicated with *._

While the existence of dark matter (DM) is essentially undisputed, very little
is known about its detailed nature or interactions. The prevailing paradigm is
cold dark matter (CDM). Assumed to be collisionless on astrophysical scales,
and typically imagined as a single type of new subatomic particle like the
WIMP (Weakly Interacting Massive Particle) (Battaglieri et al., 2017), CDM
provides an excellent description of the large-scale structure of our universe
(Aghanim et al., 2020), and accounts for the invisible mass component inferred
through its gravitational influence in galaxies and galaxy clusters. While CDM
has been a successful benchmark, and seems on the surface to be a pleasingly
minimal model, there is ample motivation to go beyond this simple framework.

Empirically, discrepancies between CDM simulations and observational data at
dwarf galaxy scales (Bullock & Boylan-Kolchin, 2017; Governato et al., 2010;
Garrison-Kimmel et al., 2019; Relatores et al., 2019) suggest the possibility
of DM self-interactions. Theoretically, it is highly plausible that DM is part
of a more complex dark sector that includes multiple particle species and
forces, which could account for the small-scale discrepancies (Tulin et al.,
2013). From a bottom-up point of view, this is motivated by analogy to the
highly non-minimal structure of the Standard Model (SM) of particle physics.
This possibility is studied by using simplified benchmark models like Atomic
Dark Matter (aDM) (Kaplan et al., 2010), which postulate the existence of a
dark analogue of electromagnetism mediated by a massless (and to us invisible)
dark photon, under which a heavier and a lighter dark matter state (dubbed
dark proton and dark electron) are oppositely charged.

Importantly, many theories of beyond-SM (BSM) physics that address various
long-standing mysteries in fundamental physics predict the existence of
exactly such a dark sector. The most relevant examples are models of Neutral
Naturalness such as the Mirror Twin Higgs (MTH: Chacko et al., 2006, 2017;
Craig et al., 2017), which solve the electroweak hierarchy problem in a
different manner from canonical frameworks like supersymmetry (Martin, 1998),
making them compatible with null results from the Large Hadron Collider: see
reviews by Batell et al. (2022) and Craig (2023). The MTH includes a complex
dark sector related to the SM by a discrete symmetry, thereby predicting the
existence of twin protons, twin neutrons, and twin electrons interacting via
twin versions of the SM forces, essentially implementing aDM with nuclear
interactions. This discrete symmetry also makes it highly plausible that aDM,
or something like it, makes up a _fraction_ of the total DM abundance, since
whatever unknown mechanism of baryogenesis produces SM baryons in the early
universe, which make up about 1/6 of the total matter budget, would be
expected to have a dark-baryon equivalent as well (see Farina (2015); Earl et
al. (2020); Alonso-Álvarez et al. (2023) for concrete theoretical proposals).
Such aDM sub-components are compatible with overall DM self-interaction
constraints (Randall et al., 2008), even if they behave very differently from
CDM.

aDM is not only self-interacting: it can form dark atomic bound states and is
_dissipative_ , meaning it can cool by emitting dark photons and collapse to
form structure. While aDM would therefore leave its imprint in early-universe
cosmology (Cyr-Racine et al., 2014; Bansal et al., 2023, 2022; Zu et al.,
2023) and galactic dynamics (Fan et al., 2013a; Ghalsasi & McQuinn, 2018; Roy
et al., 2023; Gemmell et al., 2023),111See also Fan et al. (2013b); McCullough
& Randall (2013); Randall & Scholtz (2015); Schutz et al. (2018); Buch et al.
(2019); Ryan et al. (2022a); Gurian et al. (2022a); Ryan et al. (2022b); Foot
(2014a); Foot & Vagnozzi (2015); Foot (2016); Chashchina et al. (2017); Foot
(2018a, b); Foot & Vagnozzi (2016); Foot (2013). one of its most spectacular
predictions is the existence of _Mirror Stars_.

Mirror Stars are, in perfect analogy to regular stars, collapsed clumps of aDM
that radiate their internal energy away as invisible dark photons. These
mirror stars can capture regular matter from the interstellar medium due a
tiny mixing between dark and SM photons, which quickly settles into the
center, in analogy to the hypothetical capture of DM by the Sun (Press &
Spergel, 1985; Gould, 1987). It was recently established (Curtin & Setford,
2020a, b) that the captured SM matter is heated by the core and gives off
highly characteristic optical and X-ray emissions. While faint, emissions by
these captured _nuggets_ are detectable in our stellar neighborhood,
representing a unique discovery opportunity for atomic dark matter using
telescope observations.

Previous work demonstrated how mirror stars could be searched for in public
$Gaia$ data (Howe et al., 2022), since the nuggets’ high temperature $(T\sim
10^{4}K)$ but extreme faintness distinguishes mirror stars from standard
astrophysical point sources. However, this analysis used highly simplified
bremsstrahlung-only calculations for the emissions of the captured nuggets.
While this gives a passable estimate of the overall magnitude, it does not
represent realistic emission spectra. Improved estimates are needed to obtain
quantitatively correct predictions for the colour of mirror stars in a
Hertzsprung-Russell diagram and conduct realistic $Gaia$ searches. Detailed
emission calculations will also enable searches for mirror stars that rely
purely on their unique spectral characteristics, without requiring parallax
measurements.

In this paper, we present realistic predictions for the electromagnetic
emission spectra of mirror stars (via their captured SM nuggets).
Astrophysically, optically thin nuggets behave much like compressed planetary
nebula, and just like planetary nebulae, their emissions are dominated by
atomic processes, which determine both their temperatures and their spectral
properties. We therefore use the nebular synthesis code Cloudy (Ferland et
al., 1998), a sophisticated platform for computing the thermal, ionization,
and dynamical equilibria of interstellar clouds, along with their emitted
spectra. Cloudy can be adapted to solve for the equilibrium configuration of
SM gas that sits in the gravitational well and is heated by the mirror star
core, making it ideal for the task at hand. The optical mirror star signal can
be parameterized in a very general way in terms of the heating rate from the
core to the nugget; the total nugget mass; and the core density of the mirror
star. This allows us to capture the full range of mirror star emission
signals, without having to specify any details of mirror star stellar physics,
lifetime, capture processes, etc.

We make our Cloudy input cards and full results publicly
available.222https://github.com/davidrcurtin/mirror_star_emissions This will
enable a wide range of new optical searches for mirror stars.

This paper is structured as follows. We briefly review the physics of mirror
stars, how they capture SM matter, and the resulting electromagnetic emissions
in Section 2. We also discuss how to generally parameterize the mirror star
signal without specifying all the details of the mirror star or its
constituent dark sector or aDM particles. In Section 3, we discuss how to use
the Cloudy code to model optically thin nuggets captured by mirror stars, and
compute their emissions. The physical properties of our nugget solutions are
also discussed. Our predictions for mirror star optical signatures across the
possible parameter space are presented in Section 4. Section 4.1 presents
mirror star emissions in a Hertzsprung-Russell diagram to establish a robust
signal region for mirror star with optically thin SM nuggets, and Section 4.2
presents the resulting emissions spectra and how they can be distingushed from
standard astrophysical sources like planetary nebulae or white dwarfs. We
conclude by discussing the implications of our work, including direct
application to a variety of possible mirror star telescope searches, in
Section 5.

##  2 Mirror Star Review

Since atomic dark matter is dissipative and self-interacting, it will cool and
collapse and form compact objects, in direct analogy to star formation for SM
baryons. If the aDM additionally has dark nuclear interactions, as it does in
models like the Twin Higgs, then these mirror stars may undergo dark fusion
processes in their cores and generically shine dark photons for
astrophysically long periods of time. If the aDM is minimal and does not
feature dark nuclear reactions, or those reactions exist but do not ignite, a
mirror star will still evolve for Kelvin-Helmholtz timescales, before either
forming a cooling, degenerate object or (if its mass exceeds the dark
Chandrasekhar limit) collapsing to a dark neutron star or black hole. In any
case, low-mass mirror stars, which probably dominate the population as they do
for normal stars, are likely to collect and heat SM material for $\sim
10^{8-10}$ years or more, for a broad range of aDM parameters.

The range of possible properties of these mirror stars will be determined by
the microphysics parameters of the aDM: its constituent masses, and its dark-
electromagnetic coupling. The actual distribution of mirror stars is
additionally determined by the detailed cosmological and astrophysical
evolution of the atomic dark matter density perturbations from the early
universe through to galaxy formation and local star formation today.
Predicting the former from the microphysics would be challenging but possible;
predicting the latter with a degree of reliability is as difficult as
predicting the shape of the Milky Way and the stellar mass function from the
SM Lagrangian. Fortunately, we can understand the general properties of mirror
stars and their possible discovery signals without relying on such detailed
model-specific predictions.

We focus on electromagnetic signals of mirror stars, but microlensing surveys
will also be highly sensitive in the near future, see Winch et al. (2022).
Furthermore, mirror stellar relics are expected to give rise to distinctive
gravitational wave signals from merger events, including mirror white dwarfs
(Ryan & Radice, 2022), mirror neutron stars (Hippert et al., 2022, 2023) and
black holes with unusual masses sourced by aDM collapse (Pollack et al., 2015;
Shandera et al., 2018; Singh et al., 2021; Gurian et al., 2022b; Fernandez et
al., 2022).

###  2.1 Capture and Heating of SM Matter

We now briefly summarize the most important points of the analyses by Curtin &
Setford (2020a, b), which demonstrated the existence of observable optical and
X-ray signatures of mirror stars.333See Foot (1999); Foot et al. (2001); Foot
(2004, 2014b) for earlier work discussing the possibility of mirror stars.

If aDM or something like it exists, then it is highly likely that the massless
dark photon _kinetically mixes_ with the SM photon. This is expressed as the
Lagrangian operator $\mathcal{L}\supset\frac{1}{2}\epsilon
F_{\mu\nu}F_{D}^{\mu\nu}$, with $F,F_{D}$ being the field strength of the SM
and dark photon respectively. While the kinetic mixing $\epsilon$ can formally
be of any size, it is generally expected that this parameter is _dynamically
generated_ by quantum mechanical loop effects connecting the SM and dark
sector at some order. Estimates of quantum-gravity contributions motivate
$\epsilon\sim 10^{-14}-10^{-13}$ (Gherghetta et al., 2019), while the
accidental symmetries of the MTH with a minimal UV completion suggest
$\epsilon\sim 10^{-13}-10^{-10}$ (Koren & McGehee, 2020). Such tiny mixings
have negligible cosmological and astrophysical effects, except for the fact
that they enable capture of SM matter in mirror stars (and vice versa, see
e.g., Curtin & Setford, 2021), and allow for direct detection of ambient aDM
(Chacko et al., 2021; Barak et al., 2020; Albakry et al., 2022).

The kinetic mixing interaction results in the capture of interstellar SM
matter as the mirror star traverses our Milky Way. For mirror stars with
similar properties to regular stars, this is already highly efficient for
$\epsilon\gtrsim 10^{-12}$, resulting in asteroid-mass-scale amounts of
interstellar gas accumulating in the mirror star and forming a small _nugget_
in the center. The SM gas is then heated through $\epsilon$-suppressed
electromagnetic interactions with the hot mirror star core. The heating rate
per SM atom is (in natural particle physics units)

$\Gamma_{n}^{coll}\approx
n^{\prime}\frac{2\pi\epsilon^{2}\alpha\alpha^{\prime}Z^{2}{Z^{\prime}}^{2}}{m}\Bigg{\langle}\frac{1}{v_{rel}}\left(\log\frac{8\mu^{2}v_{rel}^{2}}{(1/a_{0})^{2}}-1\right)\Bigg{\rangle}\
.$ (1)

This expression applies for collisions involving a single SM (aDM) gas species
with nuclear mass $m$ ($m^{\prime}$), atomic number $Z$ ($Z^{\prime}$) and
atomic number density $n$ $(n^{\prime})$, where the SM gas is assumed to be
mostly non-ionized (resulting in an approximate screening length444A more
precise treatment would replace $a_{0}$ by the Debye length of the SM plasma
if it is highly ionized, but as the exact screening length has only a very
minor effect on the signal, and since the overwhelming majority of the optical
nugget signal is generated at temperatures with relatively low ionization
fractions, Eqn. (1) is an acceptable approximation. of the Bohr radius
$a_{0}$), and much colder than the mirror star core temperature $T_{{\rm
core}}$. For nuggets and mirror star cores with multiple (dark) atomic
species, the above expression is generalized to sums of SM-dark collision
pairs.

These interactions heat up the captured nugget to temperatures of order
$T_{{\rm nugget}}\sim 10^{4}$ K, at which point atomic and other cooling
processes become very efficient and easily radiate away the energy siphoned
from the mirror star core through $\epsilon^{2}$-suppressed collisions. Since
the heating rate is independent of the nugget temperature, the total
luminosity can be simply found by multiplying Eqn. (1) by the number of
constituent SM atoms for each species. For mirror stars that are similar to
regular stars, this can be as bright as a faint white dwarf, or orders of
magnitudes dimmer, depending on $\epsilon$ and the mirror star lifetime (which
proportionally determines the accumulated mass).

While the nugget luminosity is fairly straightforward to determine, and could
clearly be detectable in our stellar neighborhood with various telescope
surveys, more careful calculations are required to predict the detailed
emissions spectrum. The nuggets sit in an external gravitational potential,
which can be approximated by a constant mass density given by the central
mirror star core density $\rho_{{\rm core}}$. This assumes that the nugget is
much smaller than the mirror star core, but the resulting scale height

$h_{{\rm nugget}}\sim\sqrt{\frac{kT_{{\rm nugget}}}{G\rho_{{\rm
core}}\bar{m}}}$ (2)

is $\mathcal{O}(10^{3}\mathrm{km})$ for $\rho_{{\rm core}}=\rho_{{\rm
core},\odot}=160\,$g/cm3, our sun’s core density. This assumption is therefore
very likely to be satisfied. There is only modest dependence on the nugget
composition through the average nuclear constituent mass $\bar{m}$. With
nugget size and temperature only weakly depending on the total nugget mass, it
is therefore clear that nuggets below some mass threshold will be diffuse
enough to be optically thin, and cool via volume emission from atomic and
free-free processes, while heavier nuggets are optically thick, and cool via
surface black-body emission, analogous to tiny regular stars. In this work, we
will focus on optically thin nuggets.

One of the potentially most exciting signals of mirror stars are the X-ray
emissions of the captured nugget. The presence of regular matter and aDM in
the same space allows for _X-ray conversion_ , where thermal dark photons from
the mirror star core elastically scatter off SM electrons and nuclei with
$\epsilon^{2}$ suppressed rates, to be re-emitted as a SM photon of the same
energy. In other words, the captured nugget shines with a faint, approximately
black body X-ray spectrum, given by the mirror star core temperature $T_{{\rm
core}}$, in addition to its own thermal emissions set by $T_{{\rm nugget}}$.
The conjunction of these two unusual signals is one of the most robust smoking
guns of mirror stars (Curtin & Setford, 2020a, b).

There are subtleties in computing the X-ray signal, since the nugget is
generally optically thick at such high frequencies, so only X-rays emitted in
a shell near the nugget surface escape with some modulation of their spectrum.
However, we will focus on the regime where the amount of energy transferred to
the nugget via X-ray conversion relative to the collisional heating of Eqn.
(1) is small. This ratio is given by

$\frac{\Gamma_{n}^{conv}}{\Gamma_{n}^{coll}}\sim 10^{-5}\left(\frac{\rho_{{\rm
core},\odot}}{\rho_{{\rm
core}}}\right)\left(\frac{m_{H^{\prime}}}{m_{H}}\right)\left(\frac{T_{{\rm
core}}}{10^{7}K}\right)^{9/2}\ .$ (3)

For core densities and temperatures somewhat similar to regular stars, as
normalized above, it is indeed valid to ignore the contribution of X-ray
conversion to the nugget heating for the purpose of computing its optical
signature. However, the steep $T_{{\rm core}}$ dependence means that X-ray
heating may dominate for aDM scenarios with much hotter mirror star interiors.
We leave this interesting possibility for future investigation, and focus in
this paper exclusively on carefully determining the thermal emissions of
optically thin nuggets due to the collisional heating of Eqn. (1).

###  2.2 Parameterization of Mirror Star Signal

The properties and emissions of the captured nuggets depend on the properties
of the mirror star, the aDM microphysics (masses and couplings), and the size
of the photon kinetic mixing portal. Fortunately, a very compact subset of
parameters is sufficient to uniquely determine the characteristics of the
nugget, and we can vary them without specifying the underlying dark sector
physics:

  1. 1.

_Nugget mass_ $M_{{\rm nugget}}$, with more massive nuggets extracting more
energy from the mirror star core and being therefore more luminous.

  2. 2.

_Mirror star central core density_ $\rho_{{\rm core}}$, which determines the
gravitational potential in which the nugget settles into hydrostatic
equilibrium.

  3. 3.

_Heating rate_ $\xi$ from the mirror star to the nugget, which determines the
total luminosity.

The last parameter requires some explanation. First, we choose a convention
where rather than summing over each constituent species (H, He, C, …) in the
captured gas of the nugget, we express the _total_ heating rate into the
nugget from all types of collisions in terms of the local hydrogen number
density:

$\Gamma_{H}\ \ \approx\ \left(\frac{\rho_{{\rm core}}}{\rho_{{\rm
core},\odot}}\right)\ \xi\ (\mathrm{J/s})\ .$ (4)

The total heating rate of the nugget per unit volume is therefore
$\Gamma_{H}n_{H}$, which is compatible with how Cloudy handles heating rates
(all per-atom heating rates and non-hydrogen number densities are given in
terms of $n_{H}$). We define the dimensionless heating rate parameter $\xi$,
which is taken as an input parameter to determine nugget properties, but it
can be related to microphysical parameters, as well as the detailed nugget and
mirror star core composition, using Eqn. (1):

$\displaystyle\xi$ $\displaystyle\approx$
$\displaystyle(0.45)\epsilon^{2}\left(\frac{\alpha_{D}}{\alpha}\right)\left(\frac{1.5\times
10^{7}K}{T_{{\rm
core}}}\right)^{\frac{1}{2}}\left(\frac{m_{H}}{m_{H^{\prime}}}\right)^{\frac{1}{2}}\left(\frac{L_{HH^{\prime}}}{14}\right)$
(5)
$\displaystyle\times\left[\sum_{i,i^{\prime}}\frac{q_{i}\eta^{\prime}_{i^{\prime}}Z_{i}^{2}{Z^{\prime}_{i^{\prime}}}^{2}}{A_{i}\sqrt{A^{\prime}_{i^{\prime}}}}\frac{L_{ii^{\prime}}}{L_{HH^{\prime}}}\right]\
.$

where we adopt the convention of labeling the lightest dark nuclear state as
“dark hydrogen” $H^{\prime}$ (without necessarily assuming that its dark
electric charge is 1). The dimensionless term in square brackets accounts for
the composition of the nugget and the mirror star core beyond pure (dark)
hydrogen, and is generally expected to be $\mathcal{O}(1)$.555The exact
physical interpretation of a given heating rate parameter $\xi$ is therefore
slightly dependent on the nugget composition, but since $\xi$ varies over many
orders of magnitude due to the unknown tiny kinetic mixing factor
$\epsilon^{2}$, the convenience of this choice justifies a slight convolution
of SM and BSM parameters. We parameterize the local nugget composition
relative to the hydrogen number density $n_{i}=q_{i}n_{H}$;
$\eta^{\prime}_{i^{\prime}}$ is the mass fraction of dark nuclear component
$i^{\prime}$ in the core; we define generalized atomic numbers
$A_{i}=m_{i}/m_{H},A^{\prime}_{i^{\prime}}=m_{i^{\prime}}/m_{H^{\prime}}$; and
the log factor is

$L_{ii^{\prime}}\equiv\log\left[8a_{0}^{2}\mu_{ii^{\prime}}^{2}\frac{3kT_{{\rm
core}}}{m_{i^{\prime}}}\right]-1\ .$ (6)

where $\mu_{ii^{\prime}}$ is the reduced mass of $i-i^{\prime}$ collisions.
This log factor only has very slight variation with aDM and mirror star
parameters: $L_{HH^{\prime}}\sim 14$ for core temperatures and dark hydrogen
masses close to their SM counterparts, and the ratios
$L_{ii^{\prime}}/L_{HH^{\prime}}$ in square brackets above will generally be
very close to 1. The usefulness of this parameterization is now clear: it is
completely general with respect to nugget and mirror star composition, but for
even vaguely SM-like aDM parameters and mirror star properties, we can roughly
interpret $\xi\sim\epsilon^{2}$.

This parameterization covers both optically thin and optically thick nuggets,
even as we focus on the former in this work. Though Eqn. (4) is obviously
optimized to match the dependencies of collisional heating, our results can
likely be applied, with some re-interpretation of $\xi$ to separate out the
$\rho_{{\rm core}}$ dependence of Eqn. (3), to determine the optical signal of
nuggets dominantly heated by X-ray conversion as well (assuming most of the
X-rays are absorbed by the nugget). To additionally describe the X-ray signal
itself we would only have to minimally extend our parameterization to include
separately the mirror star core temperature $T_{{\rm core}}$. We will actually
make use of this flexibility to investigate the corona of the nugget in the
next sections.

While our parameterization is sufficient to uniquely determine the signal of
captured nuggets, it is useful to relate the nugget mass $M_{{\rm nugget}}$ to
a rough _mirror star lifetime_ $\tau_{MS}$ required to actually collect this
amount of SM interstellar material. For the kinds of small kinetic mixings we
are most interested in, $\epsilon\sim 10^{-14}-10^{-10}$, the fastest possible
accumulation mechanism is capture of ISM material by collisions with the
nugget itself.666Indeed, this capture process quickly becomes dominant early
in the lifetime of the SM-like mirror stars studied in Curtin & Setford
(2020a). Assuming an approximate nugget size given by the scale height in Eqn.
(2), geometric capture is valid for $M_{{\rm
nugget}}\gtrsim(10^{10}g)(\rho_{{\rm core},\odot}/\rho_{{\rm core}})$, which
is smaller than any nugget size we consider in our analysis. Assuming an
interstellar medium density of one hydrogen atom per $cm^{3}$ and a one-
dimensional velocity dispersion of $\sim$30 km/s characteristic of our local
stellar environment (Binney & Tremaine, 2011, Table 1.2), the nugget mass
accumulated from geometric capture during a mirror star lifetime of
$\tau_{MS}$ is

$M_{{\rm nugget}}\sim\left(10^{16}g\right)\left(\frac{\rho_{{\rm
core},\odot}}{\rho_{{\rm core}}}\right)\left(\frac{\tau_{MS}}{10^{9}\
\mathrm{years}}\right)\ .$ (7)

Our analysis will enable us to understand the full range of $M_{{\rm
nugget}}$, $\rho_{{\rm core}}$ and $\xi$ that give rise to observable
optically thin nuggets in mirror stars. This will correspond to core densities
within two orders of magnitude of our sun, and nugget masses in the range
$M_{{\rm nugget}}\sim 10^{14}-10^{18}g$. Our results therefore correspond to
potentially observable nuggets that would require mirror star ages in the
million to billion year range, very reasonable possibilities for mirror star
lifetimes both with and without dark nuclear fusion.

##  3 Modeling the captured SM Nugget

The baryonic mirror star nuggets were modeled using Cloudy release C22
(Ferland et al., 2023), a physical and spectral synthesis code that solves for
the ionization, thermal state, radiative transfer, and (optionally)
hydrostatic equilibrium of diffuse nebulae, subject to the requirement of low
to moderate optical depth. While mirror star nuggets are orders of magnitude
more dense than planetary nebulae, they are often sufficiently diffuse and
optically thin for Cloudy to be able to find physical solutions to the nugget
properties and emissions. The calculation is exactly analogous to that of
planetary nebulae, with the external gravitational potential of stars in the
nebulae replaced by the external gravitational potential of the mirror star
core, and the heating from stars replaced by the constant heating rate of Eqn.
(4).

###  3.1 Numerical Integration with Cloudy

The full details of how the mirror star nuggets are modeled are documented in
our public results repository [link]. For our purposes, we work in a
spherically symmetric geometry and impose the external gravitational potential
generated by the constant background density $\rho_{{\rm core}}$. Cloudy
starts with an initial hydrogen density at the center (which determines the
nugget mass $M_{{\rm nugget}}$) and integrates outwards to self consistently
compute the structure of the cloud as well as predicting its observed
spectrum. Cloudy divides the cloud into a series of zones $k$ at radii $r_{k}$
and thickness $dr_{k}$, with each zone having constant temperature $T_{k}$,
pressure $P_{k}$, density $\rho_{k}$, and ionization state $\chi_{k}$. Working
outwards, the code self-consistently solves for hydrostatic equilibrium,
continuity, radiative and heating balance, applying the heating rate of Eqn.
(4) and from the CMB separately within each zone according to its density. We
adopt Cloudy’s default ISM abundances, which are an average for the warm and
cold phases of the ISM (Ferland et al., 2023). The composition is
predominantly H and He, with additional contributions from heavier elements
(Savage & Sembach, 1996; Meyer et al., 1998; Snow et al., 2007; Mullman et
al., 1998). Dust grains are ignored since, if they are captured, they will
settle within the nugget.777They may subsequently be vaporized by heating from
the mirror star, which raises questions beyond the scope of this paper.

This yields self-consistent solutions for the nugget’s structure (enclosed
mass, temperature, pressure, and other physical properties as functions of
radial distance $r$ from the mirror star center) as well as the emission
spectrum differential in frequency.

The integration starts at the center and proceeds outwards until the stopping
condition is met. This stopping condition deserves a careful discussion.

###  3.2 Stopping condition and X-ray corona

Figure 1:  Cooling rate of baryonic gas at densities of $10^{8}$, $10^{13}$
and $10^{18}$ H atoms$/$cm3 as a function of temperature, calculated with
Cloudy.

The total cooling rate must equal the heating rate in the nugget at
equilibrium. For optically thin nuggets, this equilibrium condition must be
satisfied locally, i.e. in each zone separately. The heating from the mirror
star core is unusual, in that it is $\epsilon^{2}$-suppressed and can hence be
very small, but since it originates from a thermal bath at some presumably
very high core temperature $T_{{\rm core}}\gg 10^{4}$ K, this heating rate is
essentially constant and independent of nugget temperature until $T_{{\rm
nugget}}\sim T_{{\rm core}}$ (see Curtin & Setford 2021 for the detailed
dependence on $T_{{\rm nugget}}/T_{{\rm core}}$).

Figure 1 shows the cooling rate per hydrogen atom for ISM gas obtained from a
one-zone Cloudy calculation at different densities. We can imagine finding the
equilibrium temperature of each zone by finding the lowest temperature at
which the cooling rate $\Lambda$ equals the total heating rate $\Gamma_{H}$.
Since the nugget density decreases monotonically with increasing $r$, leading
to less efficient cooling in the outer zones, $T_{{\rm nugget}}(r)$ will
increase with $r$. However, the sharp rise in the cooling rate at $\sim
10^{4}$ K implies that most of the nugget will be at about this temperature
for a wide range of heating rates. Note that atomic cooling processes create a
a local maximum in the cooling rate at $T_{{\rm max}}\approx 8\times 10^{4}$
K. If the heating rate exceeds this maximum, which it will for zones of
sufficiently low density at some $r_{{\rm max}}$, then the equilibrium
temperature (assuming only standard-model cooling) automatically jumps to
$\sim 10^{9}$ K. If this is higher than the mirror star core temperature, then
all zones with $r>r_{{\rm max}}$ will effectively thermalize with the mirror
star core. However, since our simplified parameterization of the heating rate
does not include the inverse of the heating process nor its
$T_{\mathrm{core}}$ dependence, we cannot use Cloudy to solve for zones that
equilibrate with the mirror star.

We therefore terminate the Cloudy integration as soon as one of the zones
exceeds $T_{{\rm max}}$. Physically, we can expect that for $r>r_{{\rm max}}$,
the nugget gas would be extremely hot $(T\sim T_{{\rm core}})$ but also
extremely diffuse, dominantly cooling by X-ray bremsstrahlung. In other words,
it forms a _corona_.

Realistically, this corona would roughly track the temperature profile and
radial extent of the mirror star core. This is physically very interesting,
and could reveal detailed information about mirror stellar physics if it could
be observed. To estimate the maximum possible impact of this corona on the
electromagnetic nugget signal, we approximate it by an isothermal solution of
hydrostatic equilibrium with $T=T_{{\rm core}}\gg 10^{4}$ K for $r>r_{{\rm
max}}$ in the gravitational potential of constant $\rho_{{\rm core}}$ for
arbitrarily high $r$. We evaluate this for a variety of mirror star core
temperatures that are high enough for the corona to be completely ionized, and
we match the corona pressure at $r=r_{{\rm max}}$ to the outermost zone solved
by Cloudy. This analytical corona solution adds to the nugget mass and
emission spectrum, with the latter contribution given by the one-zone Cloudy
simulations we used to obtain the cooling curves in Figure 1. However,
ultimately this corona contribution to both nugget mass and luminosity is
completely negligible by many orders of magnitude.

We can hence conclude that the thermal signal and total mass $M_{{\rm
nugget}}$ of the nugget for a given $\rho_{{\rm core}},\xi$ are well
determined by the $T_{{\rm core}}$-independent Cloudy solution terminating at
$r=r_{{\rm max}}$, where $T=T_{{\rm max}}=8\times 10^{4}$ K. We therefore
ignore the corona of the optically thin nuggets we study from here on.888If we
wanted to also determine the observable X-ray signal, direct compton
conversion, see Eqn. (3), would completely dominate the contribution of the
corona as well.

---
Figure 2:  Example of radial density, temperature, and ionization profiles for
a nugget with mass $M_{{\rm nugget}}=4\cdot 10^{12}$g with mirror star core
density $\rho_{{\rm core}}=1\rho_{{\rm core,\odot}}$ and heating rate
parameter $\xi=10^{-16}$, see Eqn. (4). The total nugget luminosity is $6\cdot
10^{-7}L_{\odot}$. At radii beyond the shown range, the nugget forms an
unobservably faint and diffuse X-ray corona that is thermalized with the
mirror star core, see discussion in Section 3.2.

###  3.3 Properties of Mirror Star Nuggets

$\rho_{{\rm core}}$ |  $T_{{\rm nugget}}(r=0)$ (K) | $L/L_{\odot}$ |  $r_{{\rm max}}$ (km) | $\log_{10}H_{\alpha}/NII$
---|---|---|---|---
|  |  |  |
aaa $0.01\rho_{{\rm core},\odot}$ |  |  |  |
aaaa $0.1\rho_{{\rm core},\odot}$ |  |  |  |
aaaaaa $\rho_{{\rm core},\odot}$ |  |  |  |
aaaa $10\rho_{{\rm core},\odot}$ |  |  |  |
aaa $100\rho_{{\rm core},\odot}$ |  |  |  |
Figure 3:  Parameter space of optically thin mirror star nuggets in the plane
of heating rate parameter $\xi$, see Eqn. (4), and nugget mass $M_{{\rm
nugget}}$, for different $\rho_{{\rm
core}}=(10^{-2},10^{-1},10^{0},10^{1},10^{2})\times\rho_{{\rm core},\odot}$,
with $\rho_{{\rm core},\odot}\approx 160g/cm^{3}$, in different rows from top
to bottom. Color indicates central nugget temperature (1st column), total
nugget luminosity (2nd column), effective nugget radius $r_{{\rm max}}$ (3rd
column) and the emission line ratio H$\alpha$/N II (4th column). Circles are
Cloudy solutions, squares are obtained by interpolating the shown physical
nugget property across the gap where Cloudy cannot complete the integration.

We show a representative nugget solution produced by Cloudy in Figure 2. The
scale height of the density column is $\sim 10^{3}$ km as expected from Eqn.
(2). Temperature is lowest near the center, where the transparent nugget can
cool most efficiently due to the higher central density, but increases only
very modestly until close to $r_{{\rm max}}$, where it jumps sharply towards
the integration limit $T_{{\rm max}}=8\times 10^{4}$ K. The unobservably faint
and diffuse X-ray corona exists beyond $r_{{\rm max}}$.

We run Cloudy for mirror star core densities $\rho_{{\rm
core}}=(10^{-2},10^{-1},10^{0},10^{1},10^{2})\times\rho_{{\rm core},\odot}$.
For each $\rho_{{\rm core}}$, nugget solutions are found in a grid of central
hydrogen density $n_{H}$ (determining $M_{{\rm nugget}}$) and heating rate
parameter $\xi$, and our range of parameters covers the entire observable
range where Cloudy is able to find stable equilibrium nugget solutions. This
parameter space is shown in Figure 3.

Away from our scan points in Figure 3, nuggets generated by Cloudy fail to
pass various consistency checks, either Cloudy’s or those we impose in our
analysis. Specifically, the white region in the upper left corresponds to
optically thick nuggets where Cloudy cannot exactly solve for full radiative
balance and hydrostatic equilibrium. In the upper right, Cloudy generates
seemingly stable solutions, but we find that they do not obey global energy
balance, i.e., the total luminosity significantly deviates from the total
heating rate. This is likely due to the initial central density being too high
for Cloudy to self consistently solve. Although there is no strict upper
density limit imposed by Cloudy, its prescriptions for self consistency break
down in the optically thick limit. This explains why Cloudy can generate a
full solution while failing our additional consistency checks. We impose the
requirement that total luminosity and total heating rate agree within a factor
of 2. For nuggets where this ratio is different from 1 but within a factor of
2, we shift the nugget’s assumed $\xi$-value in the parameter space to agree
with its luminosity. This results in some deviations from a regular grid near
the top right corners of the plots in Figure 3.

In the lower right, the density is so low and the heating rate so high that
nugget temperatures violate the non-relativistic assumptions of Cloudy. On the
other hand, nuggets in the lower left are unobservably faint, which informs
the limits of our scan range. For mirror star core densities of
$0.1\rho_{\mathrm{core,\odot}}$ or higher, Cloudy cannot find stable solutions
for the lowest heating rates and nugget masses we explore, likely due to the
steep dependence of the cooling rate at very low temperatures in Figure 1.
Finally, we also remove any Cloudy solutions that only have three or fewer
solved zones. These nuggets lie along the diagonal boundary of the lower right
corner of our parameter space, and reach Cloudy’s temperature limit of
$10^{9}$ K within a few zones. This is due to the balance between the nugget
heating $\xi$, and the ability of the nugget to efficiently cool. These
regions correspond to extremely low-mass nuggets that are never observable in
practice, and would represent only an extremely transient early stage in
nugget evolution as the mirror star accumulates more SM matter.

For the remaining solutions, we show how central nugget temperature, total
nugget luminosity, nugget radius before transitioning to the corona at
$r_{max}$, and the H$\alpha$/N II emission line ratio, vary with $(\xi,M_{{\rm
nugget}})$. These quantities vary essentially as expected across this
parameter space. Nugget temperature will increase as $\xi$ increases, but will
decrease as mass increases at a fixed $\xi$ due to the increased cooling
efficiency with increased nugget density. Nugget luminosity increases as
$(\xi,M_{nugget})$ increase, and the trend is uniform across different values
of $\rho_{core}$. Larger values of $\rho_{core}$ means the nugget is situated
in a stronger gravitational potential generate, and therefore have a smaller
radii. The H$\alpha$/N II emission line ratio becomes larger with increasing
nugget luminosity, as H$\alpha$ emission is the dominant cooling channel.

Cloudy raises an error if the Compton optical depth $\int_{0}^{r_{\rm
max}}\sigma_{\rm es}n_{e}dr$ becomes large. We additionally compare the
nugget’s luminosity to that of an optically thick blackbody of similar size
and temperature. Valid solutions should satisfy $L_{\rm nugget}/(4\pi R_{\rm
nugget}^{2}\sigma T_{\rm nugget}^{4})\ll 1$, and we have confirmed that this
is indeed the case for nuggets that pass the various consistency checks
described above.

Here we must point out that there is another region where Cloudy does not
produce stable nugget solutions, the “gap” at moderate heating rates and
masses, identified with square markers in Figure 3. This region is surrounded
by regions where Cloudy returns stable solutions, and after thorough
investigation we were unable to identify any physical reason why the nuggets
in this gap would not be optically thin and solvable by Cloudy. We suspect
that an unknown numerical issue causes the code to fail in these regions, and
we deal with this issue by interpolating the various physical nugget
properties (including $M_{nugget}$, which is derived from the central density
initial condition) of interest across the gap using Delaunay triangulation.
All physical characteristics vary very smoothly across the nugget parameter
space, seemingly with powers of linear combinations of $M_{{\rm nugget}}$ and
$\xi$, and the interpolation output does not significantly depend on the exact
method of interpolation employed.999Some of the emmission line ratio
predictions for heavy nuggets have slightly non-trivial variation across the
parameter space, and therefore interpolation could introduce some quantitative
uncertainty, but the differences between the line ratios of nuggets and
planetary nebulae are so drastic that this will not affect any of our
conclusions. While first-principles Cloudy solutions would obviously be
preferable, we find this to be a satisfactory solution, and clearly mark those
parameter points obtained by interpolation in subsequent plots.

Generally, higher mirror star core densities yield fewer observable optically
thin nuggets: with total nugget luminosity determined by its mass, higher core
densities means nuggets of the same mass are denser and therefore more likely
to be optically thick, placing them beyond the scope of our current analysis.
For $\rho_{{\rm core}}=100\rho_{{\rm core},\odot}$ we find very few optically
thin nuggets with magnitude $<30$, meaning considering higher core densities
would not yield additional observable optically thin nuggets. Of course, we
could consider core densities below our minimum of $10^{-2}\rho_{{\rm
core},\odot}$, but this would merely continue the trends in nugget observables
that will be clearly identified in the next section.

---
aaaa $\rho_{\mathrm{core}}=0.01\rho_{{\rm core},\odot}$
aaaaa $\rho_{\mathrm{core}}=0.1\rho_{{\rm core},\odot}$
aaaaaa $\rho_{\mathrm{core}}=1\rho_{{\rm core},\odot}$
aaaaa $\rho_{\mathrm{core}}=10\rho_{{\rm core},\odot}$
aaaa $\rho_{\mathrm{core}}=100\rho_{{\rm core},\odot}$
Figure 4:  Hertzsprung-Russell diagram of sources from the $Gaia$ DR3
catalogue and optically thin mirror star nuggets for different mirror star
core densities. For the nuggets, marker color (size) indicates heating rate
$\xi$ ($M_{{\rm nugget}}$). Circles are Cloudy solutions to nuggets, whereas
squares are interpolated.

##  4 Electromagnetic Mirror Star Signatures

We now describe the electromagnetic signatures of optically thin nuggets in
Mirror Stars from near-ultraviolet to near-infrared wavelengths. This will
establish a clear mirror star signal region in the HR diagram, but importantly
also establish purely spectral methods of distinguishing mirror stars from
other astrophysical objects like white dwarfs and planetary nebulae.

###  4.1 Distribution of Mirror Stars in HR Diagram

A Hertzsprung–Russell (HR) diagram comprised of stars with parallex of 1 to 80
mas and error less than 5% selected from the $Gaia$ DR3 is shown in Figure 4.
We show nuggets brighter than $G=30$, with dimmer nuggets deemed unobservable.
The $Gaia$ passband functions (Jordi et al., 2010) were used to obtain color
and absolute magnitude values for our optically thin mirror star nuggets and
place them in the HR diagram, with marker color (size) indicating heating rate
$\xi$ ($M_{{\rm nugget}}$). As expected, optically thin nuggets have lower
maximum luminosity for higher mirror star core densities, since they
transition to being optically thick at lower masses/luminosities. These
optically thick nuggets will be studied in a future investigation.

Remarkably, there is an extremely well-defined “mirror star signal region”.
Regardless of $\rho_{{\rm core}}$, the optically thin nuggets populate a
narrow band in colour space,

$G_{BP}-G_{RP}\in(0.4,1.4)\ .$ (8)

In absolute magnitude, the nuggets can range from arbitrarily low luminosity
up to the luminosities of white dwarfs. Note that these luminous objects must
be at least moderately optically thick. This is notably different from the
signal region assumed in the $Gaia$ search demonstration Howe et al. (2022),
but this is expected due to the toy-emission model employed in that early
analysis. The results of our work would now enable a realistic mirror star
search in $Gaia$ data to be conducted.

The position of an optically thin nugget in the HR diagram does not directly
reveal its physical properties, though nugget brightness scales with total
heating rate and nugget mass and lower heating rates tend to be clustered
towards the redder end of the signal region.

###  4.2 Detailed Nugget Emission Spectra

There are two important reasons to study the detailed spectra of optically
thin mirror star nuggets. First, the mirror star region of the HR diagram
overlaps that of white dwarfs, making distinguishing the two an observational
priority. Second, it would be desirable to identify purely spectral methods of
identifying mirror stars and distinguishing them from standard astrophysical
point sources, without requiring a parallax measurement. This would enable
mirror star searches using spectroscopic surveys and other instruments that
may have much greater sensitivity than Gaia.

---
Figure 5:  Comparison of several representative white dwarf spectra from the SDSS (BOSS) public data set with emission spectra from observable optically thin nuggets captured by mirror stars. The difference between the black-body-like white dwarf spectrum shape and the continuum of the nuggets is apparent, as are the prominent Balmer and Paschen jumps at 3645 Å and 8206 Å, respectively.  |
---|---
a $\rho_{\mathrm{core}}=0.01\rho_{{\rm core},\odot}$ |
aa $\rho_{\mathrm{core}}=0.1\rho_{{\rm core},\odot}$ |
aaa $\rho_{\mathrm{core}}=1\rho_{{\rm core},\odot}$ |
aa $\rho_{\mathrm{core}}=10\rho_{{\rm core},\odot}$ |
a $\rho_{\mathrm{core}}=100\rho_{{\rm core},\odot}$ |
Figure 6:  Correlation between the H$\alpha$/N II and H$\alpha$/S II ratios on
the left, and between H$\alpha$/N II and H$\alpha$ and $[S6717]/[S6731]$ on
the right, for mirror star nuggets (colored markers) and planetary nebulae
(data from Riesgo & López (2006)), for different $\rho_{{\rm core}}$ (rows).

In Figure 5, we show a few representative white dwarf spectra from the public
SDSS (BOSS) data set (Kepler et al., 2021) with spectral classes DA and DB.
White dwarfs showing only hydrogen absorption lines are classified as spectral
type DA, and are the most common spectral class. The next most common white
dwarf spectra class is DB, which are characterized by He I lines (e.g., Kepler
et al., 2021). The WD spectra are compared to nugget spectra for several
$(\rho_{{\rm core}},M_{{\rm nugget}},\xi)$ values. The two types of objects
are easily distinguished by their spectra, as WDs have hot blackbody continua
with pressure-broadened absorption lines, whereas nugget spectra show
significant emission lines on a bright nebular continuum. Any spectral
measurement of a mirror star’s optically thin nugget would conclusively
demonstrate that this is no standard star.

On the other hand, the emission of optically thin nuggets is similar in
character to that of planetary nebulae, with collisionally-excited and
recombination lines superimposed on a nebular continuum. Planetary nebulae are
routinely characterized by the relative strengths of specific emission lines
(Baldwin et al., 1981; Riesgo & López, 2006), which allow the physical
conditions within the nebula (such as density, electron temperature, and
abundances), to be determined. Our nuggets are distinguished from planetary
nebulae by their high densities, heating and ionization by collisions rather
than by photo-ionization or the photoelectric effect, and the absence of any
blended or scattered stellar continuum. These properties influence the Balmer
and Paschen jumps, as well as lines of critical density below that of the
nebula (Zhang et al., 2004; Guo et al., 2022). We expect our mirror star
nuggets to exhibit different line ratios from planetary nebulae, and
especially to exhibit relatively bright H lines because these are less
affected by collisional de-excitation (i.e., their critical densities are very
high) as well as a lack of fluorescence lines and lines from ions with super-
thermal ionization potentials. These expectations is borne out in Figure 6,
where we show the correlation between the H$\alpha$/N II and H$\alpha$/S II
ratios on the left, and between H$\alpha$/N II and H$\alpha$ and [S 6717]/[S
6731] on the right, for mirror star nuggets (colored markers) and planetary
nebulae (data from Riesgo & López 2006). Again, nugget marker color and size
shows heating rate and nugget mass, but unobservable nuggets ($M_{G}>30$) are
separately shown as purple markers to demonstrate the continuous nature of the
nugget distribution in this parameter space. As expected, the H$\alpha$/metal
line ratios are much higher for observable nuggets than for planetary nebulae
owing to the former’s much higher density.

##  5 Conclusions

Mirror stars are a spectacular signature of dissipative dark matter models
like atomic dark matter and twin baryons in the Mirror Twin Higgs. Their
discovery would provide unambiguous evidence of physics beyond the Standard
Model, and reveal detailed information about the nature of the dark sector.
Mirror Stars can emit faint electromagnetic signals through the capture,
heating and excitation of SM matter from the interstellar medium. This would
allow telescopes to discover mirror stars by direct observation, but this
requires a realistic and detailed prediction of their emission spectrum.

In this work, we supply comprehensive predictions for the emissions of
optically thin mirror star nuggets. This not only defines a remarkably narrow
mirror star signal region on the HR diagram (Figure 4), it also informs purely
spectral methods of searching for mirror stars and distinguishing them from
standard astrophysical point sources. Optically thin nuggets have a continuum
spectrum that is very different from the black-body like emissions of white
dwarfs and other stars; while they can be distinguished from planetary nebulae
and other optically thin sources by their emission line ratios and continuum
features, indicative of their much higher density than other optically thin
gas accumulations at $T\sim 10^{4}$ K.

Future work will analyze the properties and emissions of optically thick
nuggets, with accumulated masses and mirror star core densities higher than
the ranges studied here. This will ultimately lead to a comprehensive
understanding of mirror star optical and X-ray signals.

Our results enable a variety of new searches for mirror stars, in astrometry
data, spectral surveys, and other observational catalogues. Dark matter could
therefore be directly observed in telescopes, opening up exciting new
discovery opportunities.

##  6 Acknowledgments

We thank Peter van Hoof for conversations regarding Cloudy.

The work of IA, BG and DC was in part supported by Discovery Grants from the
Natural Sciences and Engineering Research Council of Canada, the Canada
Research Chair program, the Ontario Early Researcher Award, and the University
of Toronto McLean Award. BG was additionally supported by the University of
Toronto Excellence Award. The work of CM was supported by an NSERC Discovery
Grant. This research was enabled by computing resources and support provided
the Digital Research Alliance of Canada (alliancecan.ca).

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