MNRAS 000, 1–6 (2020) Preprint 7 August 2020 Compiled using MNRAS LATEX style file v3.

Black Dwarf Supernova in the Far Future

M. E. Caplan,1⋆
1 Department of Physics, Illinois State University, Normal, IL 61761 USA
arXiv:2008.02296v1 [astro-ph.HE] 5 Aug 2020

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

In the far future long after star formation has ceased the universe will be populated
by sparse degenerate remnants, mostly white dwarfs, though their ultimate fate is an open
question. These white dwarfs will cool and freeze solid into black dwarfs while pycnonuclear
fusion will slowly process their composition to iron-56. However, due to the declining electron
fraction the Chandrasekhar limit of these stars will be decreasing and will eventually be below
that of the most massive black dwarfs. As such, isolated dwarf stars with masses greater than
∼ 1.2M ⊙ will collapse in the far future due to the slow accumulation of iron-56 in their cores.
If proton decay does not occur then this is the ultimate fate of about 1021 stars, approximately
one percent of all stars in the observable universe. We present calculations of the internal
structure of black dwarfs with iron cores as a model for progenitors. From pycnonuclear fusion
rates we estimate their lifetime and thus delay time to be 101100 years. We speculate that high
mass black dwarf supernovae resemble accretion induced collapse of O/Ne/Mg white dwarfs
while later low mass transients will be similar to stripped-envelope core-collapse supernova,
and may be the last interesting astrophysical transients to occur prior to heat death.
Key words: dense matter – supernovae: general – white dwarfs – cosmology: miscellaneous

1 INTRODUCTION that a large amount of baryonic matter will be found in degenerate

remnants.
While it is now expected that all stars will evolve toward degenerate
Low mass stars, comprising the bulk of all stars today, will be
remnants (such as neutron stars (NS) or white dwarfs (WD)) or black
abundant among these remnants having evolved to WDs. van Horn
holes, the fate of these objects in the far future is an open question.
(1968) argues that as WDs cool their cores freeze solid and re-
Observations of the accelerating expansion and ΛCDM now sug-
lease latent heat which has recently been observationally confirmed
gest that our universe will expand forever, becoming increasingly
with Gaia (Tremblay et al. 2019). On cosmological timescales WDs
dark energy dominated while the temperature and matter density
without a heat source will fully crystallize and cool to equilibrium
asymptotically approach zero. In such a future it is expected that
with the cosmic background radiation, at sub-Kelvin temperatures
the universe will exhaust all gas for star formation and that almost
in the far future, though there may be a period of heating due to halo
all stars will become degenerate WDs when they exhaust their fuel
dark matter annihilation of 1025 up to 1047 years depending on the
over the next 1014 years (Dyson 1979; Adams & Laughlin 1997).
exact dark matter candidate considered (Adams & Laughlin 1997).
The ultimate fate of these ∼ 1023 WDs is an open question.1
Any future evolution of these near absolute zero ‘black dwarfs’ then
Dyson (1979) and Adams & Laughlin (1997) both describe a far fu-
depends on the stability of the proton.
ture in which all galaxies evaporate through gravitational scattering
If the proton decays then Adams & Laughlin (1997) expect
which ejects most objects, while occasional collisions or encounters
these black dwarfs to decay on timescales of 1032 to 1049 years. In
with black holes destroy a minority of remnants. Those in binaries
the alternative scenario, where the proton is stable, we can expect
merge due to gravitational wave radiation if close, but are more
black dwarfs to become iron rich in the far future. Pycnonuclear
likely disrupted by encounters during the period of scattering that
fusion reactions, driven by quantum tunneling of adjacent nuclei
evaporates galaxies. Therefore, virtually all surviving degenerate
on the crystal lattice within black dwarfs, will tend to process the
remnants are eventually isolated on sufficiently long timescales (ap-
matter toward iron-56 (which is possibly the ground state of baryonic
proximately 1020 years). In the far future it therefore seems likely
matter (Page 1992)). Dyson (1979) estimates a tunneling timescale
of

⋆ E-mail: mecapl1@ilstu.edu
1 Assuming order 1012 galaxies and 1011 stars per galaxy in the observable
universe. T = eS T0 (1)

© 2020 The Authors

2 M. E. Caplan et al.
where T0 is a characteristic nuclear timescale (~/m p c2 ≈ 10−25 s) 2 THE MAXIMUM MASS OF ELECTRON
and S is an action integral approximated for fusion by DEGENERATE STARS
The Chandrasekhar mass can be calculated easily from the ultra-
S ≈ 30A1/2 Z 5/6 (2) relativistic equation of state, P ∝ ρ4/3 . Integrating the structure
profile of a WD (i.e. its density radially outward from the core) yields
with A and Z the mass and charge of the fusion product. Dyson its mass (Chandrasekhar 1931, 1935; Maoz 2016; Koonin 2018).
(1979) obtains these expressions by approximating the action For this equation of state one finds that there is a fixed mass for
S ≈ (8MUd 2 /~2 )1/2 using mean barrier height U and width d all WDs, regardless of the core density (i.e. the boundary condition
for a tunneling particle of mass M. This barrier is then taken to be of integration). This equation of state will yield arbitrarily small
the Coulomb barrier screened over a distance d = Z −1/3 (~/me2 ) radii for ever increasing core densities, and the convergence to zero
with a reduced mass M = Am p /4 for two A/2 and Z/2 nuclei. radius at infinite central density is clearly shown by Chandrasekhar
This treatment excludes the sensitive density dependence for py- (1935). This of course is not physical and other physics must set an
cnonuclear fusion, which occurs most quickly at higher densities upper limit on the central density.
Schramm & Koonin (1990); Afanasjev et al. (2012); Meisel et al. As a practical matter there is an upper limit on possible core
(2018). For silicon-28 nuclei fusing to iron group elements, Dyson’s pressures and densities which is set by the condition for electron
scheme gives an approximate 101500 year timescale for tunneling to capture onto iron-56 nuclei (Cardall 2008; Warren et al. 2019). This
process black dwarfs to iron-56. capture produces manganese-56 (Z = 25) and a neutrino and due
WDs (and black dwarfs) are electron degenerate which sup- to odd-even staggering this capture is almost immediately followed
ports matter up to a finite mass limit, the Chandrasekhar mass, given by another electron capture to chromium-56 (Z = 24) whenever it
by (Chandrasekhar 1931, 1935; Maoz 2016): occurs. So while the ultrarelativistic equation of state gives an upper
limit on the mass of a degenerate electron gas of fixed Ye , reactions
exist in the cores of degenerate remnants which, if the threshold is
MCh ≈ 1.44(2Ye )2 M ⊙ . (3) met, further reduce Ye thereby triggering collapse. These reactions
give us a maximum realizable mass of a cold and isolated degener-
For a canonical WD (Ye = 0.5) we obtain the known Chandrasekhar ate remnant which is slightly below the Chandrasekhar mass. This
mass limit. If a WD exceeds the Chandrasekhar mass it will collapse condition is one of several possible triggers for a supernova pro-
(Howell 2011; Maoz 2016). genitor today, though given the low temperatures considered here it
As a white dwarf evolves toward a black dwarf, and eventually is our only relevant mechanism (Bethe et al. 1979; Nomoto 1984b;
an iron black dwarf, its equation of state is always that of a rela- Langanke & Martínez-Pinedo 2014; Warren et al. 2019).
tivistic electron gas. However, in the far future when light nuclides The threshold electron Fermi energy needed to spontaneously
are converted to iron-56 the electron fraction of the core will have drive electron capture can be determined by following Bahcall
decreased to Ye = 26/56 = 0.464. Thus, by Eq. 3 the maximum (1964):
mass will be smaller by a factor (0.464/0.5)2 ≈ 0.86. Therefore, the
effective upper mass limit of these degenerates remnants is decreas-
m(Z, A)c2 + ǫF = m(Z − 1, A)c2 + me c2
(e)
ing with time down toward about 1.2M ⊙ as they become increas- (4)
ingly iron rich. Indeed, it is long known in the supernova literature
(e)
that iron cores above 1.12 M ⊙ will collapse Baron & Cooperstein where m(Z, A) is the mass of the nuclide with (Z, A), ǫF is the
(1990). As black dwarfs above this mass limit accumulate iron in electron Fermi energy, and me is the electron mass. This is a typi-
their cores they will necessarily exceed their Chandrasekhar mass cal criteria for finding capture layers in accreted neutron star crusts
precisely because the Chandrasekhar mass is decreasing with Ye . As (Haensel & Zdunik 1990a,b; Fantina et al. 2018). For iron-56 and
(e)
a result, one may expect catastrophic collapse of these black dwarfs manganese-56 we find ǫF = 4.207 MeV. This Fermi energy corre-
in the far future, powering transients similar to supernova today. sponds to pressures of P = 5.57 × 1026 dyn/cm2 , or equivalently a
In this work we consider this fate for the most massive isolated central mass density of 1.19 × 109 g/cm3 . Thus, iron black dwarfs
white dwarfs, between approximately 1.2 and 1.4 solar masses. with core densities exceeding this will collapse.
These white dwarfs are the evolutionary endpoint of stars just be-
low the zero-age main sequence mass (ZAMS) for core collapse,
roughly 7 to 10 solar masses, and are now thought to be composed
3 EVOLUTION OF BLACK DWARFS
of O/Ne/Mg Nomoto (1984a,b); Woosley et al. (2002); Heger et al.
(2003). There are two phases of evolution to be considered. First, We now consider the evolution of a WD toward an iron black dwarf
there is a cosmologically long phase of pycnonuclear burning which and the circumstances that result in collapse. Going beyond the
occurs at near zero temperature which is the primary focus of simple order of magnitude estimates of Dyson (1979), we know
this work. Once the Chandrasekhar mass is reached, gravitational pycnonuclear fusion rates are strongly dependent on density so they
contraction may proceed on hydrodynamical and nuclear burning are greatest in the core of the black dwarf and slowest at the surface.
timescales, and the transient begins which we only briefly speculate Therefore, the internal structure of a black dwarf evolving toward
on here. collapse can be thought of as an astronomically slowly moving
In Secs. 2, 3, and 4 we consider the internal structure and ‘burning’ front growing outward from the core toward the surface.
determine the mass-radius relation for progenitors at the time of This burning front grows outward much more slowly than any hy-
collapse. We calculate the abundance of progenitors in Sec. 5 and drodynamical or nuclear timescale, and the star remains at approx-
their lifetimes in Sec. 6, and speculate on some general properties of imately zero temperature for this phase. Furthermore, in contrast
the astrophysical transients associated with collapse in 7. In Sec. 8 to traditional thermonuclear stellar burning, the later reactions with
we also consider how the rate and cosmological state of the universe higher Z parents take significantly longer due to the larger tunneling
at the time. We conclude in Sec. 9 barriers for fusion.

MNRAS 000, 1–6 (2020)

 Black Dwarf Supernova in the Far Future 3
The dwarf will undergo progressive stages of burning likely 1.5
converting the star from O/Ne/Mg to heavier symmetric nuclei (i.e.
Z = N), likely including silicon-28. Complex fusion pathways may
exist depending on the relative timescales for other reactions, which
future authors may seek to explore with stellar evolution codes such
as MESA using a pycnonuclear reaction network. For example, α- 1
particle exchange (or similar transfer reactions) between adjacent
nuclides may be one mechanism for achieving a more uniform 1.3
lower energy composition when evolving a mixture of light nuclides
(Chugunov 2018). The lattice structure of the mixture may also 0.5
impact pycnonuclear fusion rates (Caplan 2020). Whatever the exact 1.2
fusion pathway, we do not anticipate any disruption or contraction
(or any increase in core pressure) of the white dwarf during this 0.35 0.37
process as Ye is unchanged. 0
Finally, once the core has largely burnt to silicon-28 (or a simi- 0 0.1 0.2 0.3 0.4
lar nuclide), we expect it will fuse to produce nickel-56 (or a similar
iron group element) which can then decay through positron emis-
sions to iron-56, with Ye = 0.464. The annihilation of two electrons Figure 1. (Color online) Profiles of black dwarf supernova progenitors. All
from these reactions slowly deleptonizes the star and reduces the profiles have the same central density, and thus the Ye = 0.464 part of the
core Ye , resulting in contraction which increases core pressure. profile is identical, following the bottom curve for the pure iron black dwarf
Unlike typical supernova progenitors, this contraction will not (red). The profile is uniquely determined up to some transition pressure at
the interface between Ye = 0.464 and Ye = 0.5 matter (squares). A density
heat the black dwarf considerably due to the astronomically long
discontinuity is found there; we report the density ρ6 on the Ye = 0.5 side of
fusion timescales. Furthermore, the reduction in radius is of order the interface for our most massive progenitors. At radii above the transition
ten percent (YeFe /YeSi = 0.928) over timescales we estimated above we see the profiles rise off that of a pure iron black dwarf as the equation of
to be of order 101500 years. Any cooling mechanism, such as black- state stiffens at higher Ye . In the inset we zoom in on the ends of our curves
body emission, will proceed much more quickly. To determine when to show the progenitors mass-radius relationship.
exactly these objects collapse we must calculate how long it takes
to evolve to the critical core pressure for electron capture, which
requires us to first know their internal structure. Therefore, we de- dwarfs. Each line shows the profile of a progenitor of a different
termine the stellar profiles of the progenitors just prior to collapse mass just prior to collapse. As the profile of the star within Ptr ans
before proceeding. is identical, we plot points (squares) at Ptr ans for each profile to
show where it begins to rise off of the profile of a fully iron-56
progenitor (red). For reference, we include the mass density on the
silicon side of the interface for the three greatest transition pressures
4 INTERNAL STRUCTURE OF BLACK DWARFS
shown, with ρ6 the density in units of 106 g/cm3 . As expected, the
We calculate the profiles of the progenitors from the equation of most massive progenitors are approximately 1.35M ⊙ and Ptr ans
state for relativistic electrons. This smoothly connects the low den- is very near the core while the least massive progenitors are nearly
sity nonrelativistic equation of state P ∝ ρ5/3 to the ultrarelavistic entirely iron-56 and have masses of 1.16M ⊙ .
limit where P ∝ ρ4/3 . We follow the procedure of Koonin (2018). In the inset we show the progenitor mass-radius curve (dotted).
Our equation of state is defined piecewise with Ptr ans being a Note that the mass of the progenitor is most sensitive to small
sharp interfacial transition with a density discontinuity. For pres- changes in Ptr ans when it is found at high pressure, which can
sures greater than Ptr ans we use Ye = 0.464 (for iron-56) and at be seen from profiles with labeled core densities (blue-green). For
pressures below we use Ye = 0.50. We are not actually sensitive to the purely iron black dwarf (red) we find M = 1.167M ⊙ and R =
whether the outer layers are a pure composition or mixture, or even 0.347R ⊕ while the progenitor with an infinitesimal iron core has
if there are onion layers as in core collapse supernova (CCSN) pro- M = 1.355M ⊙ and R = 0.374R ⊕ (purple). These correspond to the
genitors, as an electron gas for Z = N nuclei results in an identical minimum and maximum progenitor masses and radii respectively.
stellar profile; as argued above, stable Z = N nuclei are found up to As expected both the masses and radii of progenitors only vary by
silicon-28, which fuses to iron group nuclides. approximately ten percent, of order the difference in Ye between
Integrating the classical equations of stellar structure require iron-56 and silicon-28 (YeF e /YeSi = 0.464/0.5 = 0.928). It can be
the core density as a boundary condition (Chandrasekhar 1935; shown for a relativistic equation of state that R ∝ Ye and M ∝ Ye2 ,
Maoz 2016; Koonin 2018); it suffices to observe that the core density which is what we find here (Koonin 2018).
we are interested in is always the critical density for electron capture Our estimate for a fully Ye = 0.5 profile is consistent with
onto iron-56. We therefore have only one free parameter, which is observations. The most massive WD presently known is REJ 0317-
Ptr ans . Interior to this interface the structure of the star is identical 853 whose mass and radius are reported to be 1.35M ⊙ and 0.38R ⊕
just prior to collapse for all progenitors, having an iron-56 core by Barstow et al. (1995); Burleigh & Jordan (1998); Külebi et al.
with a central density of 1.19 × 109 g/cm3 . Furthermore, this means (2010) gives a mass range of 1.32 − 1.38M ⊙ due to uncertainty in
there is a one-to-one mapping from the mass of the progenitor to the the core composition, which is consistent with what we find here.
density of the silicon-28 at the interface where we find our burning As a black dwarf is processed toward iron its mass will decrease
to front, which will be useful for calculating lifetimes below. As the by releasing energy and neutrinos, though this effect is small and
equation of state stiffens above the transition interface, those with does not affect our treatment above. The burning of alpha cluster
transitions nearest to the core will be the most massive. nuclei such as O, Ne, and Mg (binding energies ∼ 8 MeV/nucleon)
In Fig. 1 we show the mass-radius profiles of progenitor black to iron group elements (binding energies ∼ 9 MeV/nucleon) during

MNRAS 000, 1–6 (2020)

4 M. E. Caplan et al.
the pycnonuclear phase only releases ∼ 1 MeV/nucleon. Mass loss for collapse ultimately determines the lifetime of the black dwarf.
during the pycnonuclear burning phase will therefore only reduce Because of this, the most massive black dwarfs will collapse first.
the mass of the star by, at most, 1 MeV/939 MeV ∼ 10−3 which is Only a small amount of matter already at high density has to fuse to
small. Thus, the mass of the WD is basically equivalent to the mass iron-56 before the threshold Fermi energy is reached. Less massive
of the black dwarf supernova progenitor. black dwarfs take longer to reach this threshold, requiring ever larger
iron cores, while black dwarfs at the lower mass limit only collapse
once the outermost layers of the star have fused to iron.
To obtain a coarse estimate of the lifetime of these progenitors
5 NUMBER OF PROGENITORS IN THE FAR FUTURE
we calculate the lifetime of a silicon-28 nucleus against pyconu-
The initial-final mass relation (IFMR) for WDs and a Salpeter ini- clear fusion at the burning front. Following Schramm & Koonin
tial mass function (IMF) are sufficient to estimate the number of (1990)2 , the pycnonuclear fusion reaction rate in the Wigner-Seitz
presently observable stars which will evolve to progenitors that may approximation for a bcc crystal is given by
collapse and explode in the far future.
While the initial-final mass relation (IFMR) is now well con- −1/2 )
strained in the range of 2M ⊙ < Minitial < 4M ⊙ , the behavior for R = (1.06 × 1045 )S ρAZ 4 λ 7/4 e(6.754−2.516λ cm−3 s−1 (6)
Minitial > 4M ⊙ is less well understood. Extrapolating the IFMR where S is the astrophysical S-factor in MeV barns, ρ is the mass
from the 2M ⊙ < Minitial < 4M ⊙ range would tend to underestimate density of the silicon (A = 28, Z = 14) at the interface, and λ is
the minimum Minitial as the onset of dredge-up above 4M ⊙ tends to the ratio of the nuclear Bohr radius and lattice spacing a (for a bcc
reduce the core mass (Andrews et al. 2015). For this work, we use lattice γ = 2) such that
the IFMR reported by Cummings et al. (2016) fit to Minitial > 4M ⊙ :

1/3
λ = 0.0245A−4/3 Z −2 γ −1/3 ρ6 (7)
Mfinal ≈ 0.1Minitial + 0.5M ⊙ . (5)
The behavior of this IFMR is typical for the Minitial > 4M ⊙ range which for a bcc silicon-28 lattice evaluates to
and, when extrapolated to higher masses, is the intermediate case λ = 1.167 × 10−7 ρ6 .
1/3
(8)
when compared to other IFMRs in the literature (Cummings et al.
2016; Salaris et al. 2009; Catalan et al. 2008). This IFMR sug- We obtain the astrophysical S-factor following the analytical proce-
gests that stars with a ZAMS mass of 6.5M ⊙ or greater will dure in Afanasjev et al. (2012) assuming E = 0, finding S = 3×1063
have WD masses greater than the progenitor minimum of 1.16M ⊙ MeV barns. The precise value is unimportant as the rate uncertainty
found above. This is also consistent with measurements of the is overwhelmingly dominated by the exponential dependence on
IFMR with Gaia from El-Badry et al. (2018). Meanwhile, the max- density.
imum Minitial ∼ 10M ⊙ is determined by the minimum ZAMS The lifetime τ of a silicon nucleus at the core-interface can be
mass for core collapse supernova progenitors (Heger et al. 2003). calculated using reaction rate in a volume which is the Wigner-Sietz
Díaz-Rodríguez et al. (2018) use observed supernova remnants in cell, Vs = 2m/ρ = 4.7 × 10−29 ρ−1 6
cm3 . As our only free parameter
M31 and M33 to determine a minimum ZAMS mass for single-star is ρ6 , we express the lifetime as
CCSN progenitors of 7.3M ⊙ , which we will use as a conservative
upper limit.
−7/12 7365ρ6−1/6
The stellar mass density is only expected to grow by about 5 τ ≈ (Vs R)−1 ≈ 1083 ρ6 e s. (9)
percent above current values (Sobral et al. 2013), so the total number
For the most massive black dwarfs we find that the core be-
of stars in the universe today represent nearly the total number that
gins fusing to iron in approximately 101100 years (M = 1.35M ⊙ ,
will ever exist. Integrating the Salpeter IMF (α = 2.35) for 6.5M ⊙
ρ6 = 1.19 × 103 ). For an intermediate mass black dwarf approx-
to 7.7M ⊙ (using 1023 stars in the observable universe) gives us
imately half the mass is iron at the time of collapse which occurs
order 1021 progenitors. Extending the upper bound to 10M ⊙ only
in approximately 101600 years (M = 1.24M ⊙ , ρ6 = 102 ), compa-
increases this by a factor of 2. Order of magnitude uncertainties
rable to the rudimentary estimate from Dyson (1979). For the least
in the number of stars in the universe prevent us from improving
massive black dwarfs the lifetime will be set by the pycnonuclear
upon this constraint. As above, stars with a ZAMS in this range are
fusion rates at the surface. Here we expect that the electron gas is
expected to evolve toward O/Ne/Mg WDs (Heger et al. 2003)
not degenerate so atomic silicon may be expected with terrestrial
densities for which we find lifetimes of 1032000 (M = 1.16M ⊙ ,
ρ6 ∼ 10−6 ), though at such low densities this treatment may not
6 LIFETIME OF BLACK DWARFS AGAINST COLLAPSE be appropriate. The exponent of these lifetimes has been rounded
to the nearest hundred so our choice of units (years) aren’t strictly
We can estimate the lifetime of black dwarfs, and thus the delay time
accurate, however the difference between seconds and the current
for transients associated with collapse, from the timescale for the
age of the universe is ultimately small for our purposes.
pycnonuclear burning front to grow radially outward. The slowest
For comparison, a 1011 M ⊙ black hole (of order the upper limit
fusion reaction in the evolution will be the one occurring at the
for supermassive black holes from King 2015) has a lifetime due to
lowest density and with the highest Z parent nuclides. Due the
evaporation from Hawking radiation of 10100 years. If black dwarf
strongly exponential density and Z dependence for pycnonuclear
collapse produces transients in the far future, they are likely to be
fusion, most of the lifetime of the black dwarf will be spent in the
among the last to occur prior to the heat death of the universe.
stages just prior to this collapse.
Conveniently, this condition is satisfied by the burning front
for the progenitors shown above. Therefore, the fusion rate at the
silicon burning front when the core reaches the critical pressure 2 The factor of A2 in their eq. 33 is corrected to A in the erratum.

MNRAS 000, 1–6 (2020)

 Black Dwarf Supernova in the Far Future 5
7 BLACK DWARF SUPERNOVA TRANSIENTS black dwarfs to become isolated with exponentially growing separa-
tions relatively soon. Therefore, when black dwarf supernova begin
Without detailed stellar evolution calculations to determine com-
the proper-distance radius of the observable universe will have very
positions at the time of collapse it is difficult to make precise 1100
predictions about the nature of transients. Nevertheless, order of roughly grown by ∼ e10 (H ∼ 10−11 years−1 which is in the error
magnitude arguments and analogs with various presently occur- of the timescale, discussed above). With such large separations, it
may not be intuitive or meaningful to attempt to report a volumetric
ring transients may be informative. For example, we expect that the
transients associated with black dwarf supernova will evolve with rate.
cosmic time, as the delay time is related to the mass. Instead, to get a sense of the cosmological conditions where
black dwarf supernova are found, consider that the cosmic event
The most massive progenitors will be the first to explode,
horizon for each comoving observer in a de Sitter universe is found
and thus their composition will be largely unprocessed with only
at reh = c/H ≈ 1010 lightyears (Melia 2007; Neat 2019), so after
a small iron-56 core, possibly similar to accretion induced col-
becoming gravitationally unbound we may expect all objects to
lapse of O/Ne/Mg stars. It is an open question whether such ex-
plosions produce NSs or low mass WDs with iron cores, as these recede beyond their mutual cosmic event horizons on a timescale
similar to the current age of the universe (Adams & Laughlin 1997).
explosions depend sensitively on the hydrodynamic burning prop-
We therefore expect that every degenerate remnant in the universe
agation and explosive ignition (Isern et al. 1991; Canal et al. 1992;
Panei et al. 2000; Schwab et al. 2015). Recent 3D simulations of will be causally disconnected from every other degenerate remnant,
which will all see each other red-shifted to infinity long before
electron-capture supernova by Jones et al. (2016) suggest that an
any transients considered in this work may occur Maoz (2016);
iron core white dwarf might be more likely than a neutron star with
analogous progenitors, as in Isern et al. (1991). Davis & Lineweaver (2004).
However, the analog with accretion induced collapse studied in
the literature is imperfect. For example, given the sensitive depen-
dence of pycnonuclear fusion rates on density and Z it is possible 9 CONCLUSIONS
that much of the core has been processed to silicon while a frac-
tion of the lightest nuclei in the outer layers have been processed to Black dwarf supernova progenitors with masses between about 1.16
heavier nuclei. Therefore, there is less energy available for nuclear and 1.35 M ⊙ are at near zero temperature which allows us to de-
burning, especially in the core, and so the transition to a simmering termine their internal structure with fair accuracy using only a rel-
phase, deflagration, and eventual supernova may be much different ativistic Fermi gas. Furthermore we can calculate their lifetimes
than currently modeled accretion induced collapse. with simple analytical formulas for the pycnonuclear reaction rates
at the surfaces of their iron cores. If proton decay does not occur
The low mass progenitors will meanwhile have compact iron
cores but thin envelopes which may result in dynamics comparable then in the far future we expect approximately one percent of all
stars today, about 1021 stars, to collapse and explode in supernova
to stripped-envelope core-collapse supernovae (CCSNe) today, and
beginning in approximately 101100 years and lasting no more than
with little accretion after core bounce the shock may expand almost
unimpeded. Exact luminosities are difficult to determine without about 1032000 years. At such advanced time it is difficult to imagine
any other astrophysical processes occurring, which may make black
simulations, but the low masses may similarly reduce electromag-
dwarf supernova the last transients to occur in our universe prior to
netic yields. Compared to CCSNe today we might expect a similar
neutrino luminosity as the iron core collapse should be compara- heat death.
Ultimately black dwarfs will explode because pycnonuclear
ble. Gravitational wave emission of the proto-NS may be similar, but
without excitation from convection in the infalling matter one might fusion reactions slowly process their interiors to iron, decreasing Ye
expect less driving of the oscillation. Furthermore, the progenitor and their Chandrasekhar mass. This is amusingly like the inverse
of accretion induced supernova today; whereas accretion induced
is likely highly spherical which will weaken the gravitational wave
signal. For all masses considered it seems unlikely that a black hole collapse involves a WD increasing its mass up to near the Chan-
should form. drasekhar limit, a black dwarf supernova involves a dwarf decreas-
ing its Chandrasekhar limit down to its mass. Unfortunately, due to
the timescales involved there are few observable consequences for
this work, but this result may nevertheless be of popular interest.
However, given the simplicity of the progenitors described in this
8 TRANSIENT RATE AND FAR FUTURE COSMOLOGY
work it may be interesting and straightforward to simulate the evo-
In principle, a volumetric supernova rate can be obtained knowing lution of the structure and composition of progenitors with nuclear
the number of progenitors as well as their lifetimes from Secs. 5 reaction networks accounting for pycnonuclear fusion, such as with
and 6. However, given the enormous delay times the consequences MESA. Furthermore, with detailed calculations of the evolved com-
of the accelerating expansion of the universe makes it difficult to positions from MESA it may be possible to explore these transients
report this value meaningfully. in more detail with supernova codes.
If black dwarf supernova will not begin for another 101100
years then it is relevant to consider how much the observable uni-
verse will grow in that time from ΛCDM. As the universe is now
ACKNOWLEDGEMENTS
entering a dark energy-dominated expansion phase, distances be-
tween gravitationally unbound objects will approximately grow like M.C. thanks Neil Christensen, Zach Meisel, MacKenzie Warren,
R(t) ∝ e H t as in a de Sitter universe (with constant Hubble
√ param- and the students in the Spring 2020 PHY 308 Astrophysics course
eter proportional to some cosmological constant H ∝ Λ, Maoz at Illinois State University for conversation and inspiration. This
2016). Now, recall that Adams & Laughlin (1997); Dyson (1979) work benefited from support by the National Science Foundation
show that all orbits become unbound from galaxies (or decay by under Grant No. PHY-1430152 (JINA Center for the Evolution of
gravitational wave emission) in only 1020 years, so we expect all the Elements).

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6 M. E. Caplan et al.
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