# Binary interactions enhance the Diffuse Supernova Neutrino Background

Shunsaku Horiuchi<EMAIL_ADDRESS>Center for Neutrino Physics, Department of
Physics, Virginia Tech, Blacksburg, VA 24061, USA  Tomoya Kinugawa  Institute
for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba, Japan
Tomoya Takiwaki  National Astronomical Observatory of Japan, Mitaka, Tokyo
181-8588, Japan  Koh Takahashi  Max Planck Institute for Gravitational
Physics, D-14476 Potsdam, Germany  Kei Kotake  Department of Applied Physics &
Research Institute of Stellar Explosive Phenomena, Fukuoka University, Fukuoka
814-0180, Japan

###### Abstract

Binary interactions, especially mass transfer and mergers, can strongly
influence the evolution of massive stars and change their final properties and
the occurrence of supernovae. Here, we investigate how binary interactions
affect predictions of the diffuse flux of neutrinos. By performing stellar
population syntheses including prescriptions for binary interactions, we show
that the resulting detection rates of the diffuse supernova neutrino
background is enhanced by 15–20% compared to estimates without binary
considerations, but this may be as large as 75%. The large range of possible
enhancement is driven by the presently sparse knowledge of the evolution of
rapidly rotating carbon-oxygen cores, especially those created as a result of
mergers near the white dwarf to core collapse boundary.

##  I Introduction

A star with a sufficiently massive core ends its evolution with a core
collapse, where its core, unable to sustain itself against gravity, collapses
on dynamical time scales to a compact object. During this violent collapse a
shock can be formed, which can lead to a core-collapse supernova explosion
(CCSN) if the shock reaches the photosphere of the star (see, e.g., reviews
Langanke and Martínez-Pinedo (2003); Mezzacappa (2005); Kotake  _et al._
(2006); Woosley and Janka (2005); Burrows (2013); Janka (2012); Foglizzo  _et
al._ (2015); Janka (2017); Burrows and Vartanyan (2020)). The collapsed core
emits a copious number of neutrinos of tens of MeV in energy, which carry away
the majority of the gravitational binding energy liberated in the collapse.
These neutrinos are therefore critical for understanding the energetics and
evolution of the core from collapse to potential explosion, as well as provide
an unique opportunity to study the system otherwise hidden from view by the
stellar envelope. Neutrino detectors around the globe are poised to detect
tens of thousands of neutrinos from the next core collapse occurring in the
Milky Way galaxy, offering many opportunities for studying the properties of
massive stars, core collapse, as well as neutrinos (see, e.g., reviews
Scholberg (2012); Mirizzi  _et al._ (2016); Horiuchi and Kneller (2018)).
However, the occurrence rate of core collapse in the Milky Way is not very
high, being a few per century Diehl  _et al._ (2006); Rozwadowska  _et al._
(2021).

A complementary strategy is to study extragalactic core collapses. Future
Mton-class neutrino detectors will be sensitive to neutrinos from core
collapses occurring in nearby galaxies Ando  _et al._ (2005); Kistler  _et
al._ (2011); Horiuchi  _et al._ (2013); Nakamura  _et al._ (2016). Another
strategy is to search for the diffuse flux of neutrinos caused by past core
collapses, called the diffuse supernova neutrino background (DSNB, see reviews
Beacom (2010); Lunardini (2016)). The predicted DSNB is isotropic, constant in
time, and its flux is roughly within a factor of two of the latest upper
limits placed by Super-Kamiokande (Super-K) Bays  _et al._ (2012).
Importantly, Super-K has recently completed a major upgrade—enhancing its
detector volume with gadolinium Watanabe  _et al._ (2009) as proposed in Ref.
Beacom and Vagins (2004)—which allows a vastly improved differentiation
between the DSNB and background signals. With this upgrade, the search for the
DSNB with Super-K has transitioned from a background-limited one Bays  _et
al._ (2012) to a signal-limited one.

Predictions of the DSNB have improved over the years Krauss  _et al._ (1984);
Dar (1985); Totani and Sato (1995); Totani  _et al._ (1996); Malaney (1997);
Hartmann and Woosley (1997); Kaplinghat  _et al._ (2000); Ando and Sato
(2003); Fukugita and Kawasaki (2003); Strigari  _et al._ (2004); Iocco  _et
al._ (2005); Strigari  _et al._ (2005); Lunardini (2006); Daigne  _et al._
(2005); Yuksel  _et al._ (2006); Horiuchi  _et al._ (2009); Lunardini (2009);
Lien  _et al._ (2010); Keehn and Lunardini (2012); Vissani and Pagliaroli
(2011); Lunardini and Tamborra (2012); Nakazato (2013); Mathews  _et al._
(2014); Yuksel and Kistler (2015); Nakazato  _et al._ (2015); Hidaka  _et al._
(2016); Priya and Lunardini (2017); Horiuchi  _et al._ (2018); Møller  _et
al._ (2018); Riya and Rentala (2020); Kresse  _et al._ (2020). However, the
impacts of stellar binaries have mostly been neglected. Recent observations of
stars Sana  _et al._ (2012); Zapartas  _et al._ (2020) show that the majority
of massive stars have stellar companions and experience binary interactions.
This includes massive stars which are the progenitors for CCSNe, and hence the
majority of core collapses should be affected by binary interactions. Binary
interaction, especially mass transfer, can significantly change the masses of
core-collapse progenitors and strongly influence the type of CCSN
Podsiadlowski  _et al._ (1992); Woosley  _et al._ (2002). Furthermore, if
stars in a binary merge together, a rapidly rotating star remains. This effect
opens a novel channel for stars that are initially too low mass to undergo
core collapse to become a viable core-collapse progenitor. Both this and mass
transfer would affect the landscape of CCSNe and impact the DSNB.

Recently, Ref. Kresse  _et al._ (2020) made a first assessment of the impacts
of binary interactions on the DSNB. The authors simulated the core collapse of
progenitors stripped of its envelope due to mass loss Woosley (2019) as a
proxy for close-binary interactions. Due to mass stripping, these stars have
less massive cores than their isolated counterparts, leading to lower neutrino
emissions and larger ZAMS mass threshold for core collapse. Both of these
result in lower DSNB predictions. However, the fraction of such hydrogen-
stripped stars was determined by a fraction parameter, which was varied
between 33% to 100% of the core-collapse progenitor population. Also, the
impacts of binary mergers and other types of binary interactions were not
included.

In this paper, we quantify the impacts of binary interactions on the DSNB flux
using binary population synthesis calculations which model the effects of both
mass transfer and mergers. Based on these synthetic populations, we compute
the DSNB flux and discuss consequences for current and future neutrino
detectors. The paper is organized as follows. In Section II, we summarize the
important binary interactions and our binary population synthesis simulations.
In Section III, we make predictions for the DSNB and discuss detection rates
at neutrino detectors. We finish with discussion and conclusions in Section
IV. Throughout, we adopt a $\Lambda$ cold dark matter cosmology with
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70$ km s-1 Mpc-1.

##  II Binary Treatment

Stars in binaries experience various binary interactions which can change the
mass of the star during its evolution (e.g., Ref. Eggleton (2006)). In
particular, stable mass transfer and the common envelope (CE) phase are
important to understand the changing mass of CCSN progenitors (e.g., Refs.
Podsiadlowski  _et al._ (1992); Podsiadlowski (2010)). In this section, we
focus on binary effects which affect the evolution of CCSN progenitors.

###  II.1 Binary effects / treatment

If one of stars in a binary system fulfils its Roche lobe, some of the stellar
material is transferred to the companion star. This process is called the
Roche lobe overflow (RLOF). The radius of the Roche lobe around the donor star
is approximately expressed as (Eggleton, 1983),

$R_{\rm L,1}\simeq
a\frac{0.49q_{1}^{2/3}}{0.6q_{1}^{2/3}+\ln(1+q_{1}^{1/3})},$ (1)

where $a$ and $q_{1}=M_{1}/M_{2}$ are the binary separation and the mass
ratio, where $M_{1}$ is the donor and $M_{2}$ is the companion masses,
respectively. The behavior of the mass transfer is determined by the response
of the Roche lobe radius and the stellar radius of the donor star when it
loses its material (Paczynski, 1976). Assuming no mass is lost from the binary
system during the mass transfer, i.e., the conservative mass transfer, the
response of the Roche lobe radius is characterized by (Tout  _et al._ , 1997),

$\zeta_{\rm L}\equiv\frac{d\ln R_{\rm L,1}}{d\ln M_{1}}\simeq 2.13q_{1}-1.67$
(2)

For $\zeta_{\rm L}<\zeta_{\ast}(\equiv d\ln R_{1,ad}/d\ln M_{1})$, where
$R_{1,ad}$ is the adiabatic radius of donor star, the radius of the donor star
shrinks and becomes smaller than the Roche lobe radius after the mass of the
donor star is transferred. In this case, stable mass transfer occurs. On the
other hand, for $\zeta_{\rm L}>\zeta_{\ast}$, the mass transfer would be
unstable and the two stars would enter the CE phase. The value of
$\zeta_{\ast}$ depends on the stellar envelope of the donor star. When the
donor star is in the red giant phase with a convective envelope,
$\zeta_{\ast}$ is given by,

$\zeta_{\ast}=-1+\frac{2}{3}\frac{M_{1}}{M_{\rm env,1}},$ (3)

where $M_{\rm env,1}$ is the envelope mass of the donor giant. When the donor
star is in the main sequence (MS), giant phase with a radiative envelope
(Hjellming, 1989), naked-He MS, or naked-He giant star (Ivanova  _et al._ ,
2002; Belczynski  _et al._ , 2008), we use $\zeta_{\ast}$ = 2.59, 6.85, 1.95
and 5.79, respectively.

In the case of stable mass transfer ($\zeta_{\ast}>\zeta_{\rm L}$), we use the
transfer rate which is expressed as Hurley  _et al._ (2002),

$\dot{M_{1}}=F(M_{1})\left[\ln\left(\frac{R_{1}}{R_{\rm
L,1}}\right)\right]^{3}~{}M_{\odot}~{}\rm yr^{-1},$ (4)

where $R_{1}$ is the radius of the donor star, and

$F(M_{1})=3\times 10^{-6}\left\\{{\rm
min}\left[\left(\frac{M_{1}}{1M_{\odot}}\right),5.0\right]\right\\}^{2}.$ (5)

Under our assumed conservative mass transfer, the accretion rate onto the
companion star is the same as the mass loss rate of the donor star. This would
not be valid if the accretion rate exceeds the Eddington limit of the
companion star, but in reality this condition is not reached in any of our
binary systems which are not compact objects.

In the case of the CE phase ($\zeta_{\ast}<\zeta_{\rm L}$), we use the
$\alpha\lambda$ formalism (Webbink, 1984),

$\alpha\left(\frac{GM_{\rm{c,1}}M_{2}}{2a_{\rm{f}}}-\frac{GM_{1}M_{2}}{2a_{\rm{i}}}\right)=\frac{GM_{\rm{1}}M_{\rm{env,1}}}{\lambda
R_{1}},$ (6)

for a binary of a mass-losing giant star and a companion star, where $a_{\rm
i}$, and $M_{\rm c,1}$ are the initial binary separation just before the CE
phase and the core mass of the mass-losing giant, respectively. In order to
calculate the separation just after the CE phase, $a_{\rm f}$, we use Eq. (6).
Here, $\alpha$ is the efficiency parameter parameterizing how much of the
orbital energy is transferred to the unbound envelope of the giant. The
parameter $\lambda$ is for the envelope’s binding energy of the giant. If the
companion star is also a giant Nelemans  _et al._ (2001), Eq. (6) is replaced
by,

$\displaystyle\alpha\left(\frac{GM_{\rm{c,1}}M_{c,2}}{2a_{\rm{f}}}-\frac{GM_{1}M_{2}}{2a_{\rm{i}}}\right)=$
$\displaystyle\frac{GM_{\rm{1}}M_{\rm{env,1}}}{\lambda R_{1}}$ (7)
$\displaystyle+\frac{GM_{\rm{2}}M_{\rm{env,2}}}{\lambda R_{2}},$

where $R_{2}$, $M_{\rm c,2}$, and $M_{\rm env,2}=M_{2}-M_{\rm c,2}$ are the
radius, the core mass, and the envelope mass of the companion giant star,
respectively. The CE parameters $\alpha$ and $\lambda$ are not understood very
well (Ivanova  _et al._ , 2013). We adopt the CE parameter values adopted in
previous binary population studies, $\alpha\lambda=1$ and $0.1$ (e.g., Refs.
Belczynski  _et al._ (2007); Kinugawa  _et al._ (2014)). Physically, the
product $\alpha\lambda$ is related to how hard it is to unbind the envelope
due to the CE. If $\alpha\lambda$ is small, a large orbital energy is need to
unbind the envelope. Thus, it is easier to shrink the binary orbit and the
binary more easily merges during the CE phase.

When the CE phase is over, we calculate the separation just after the CE phase
$a_{\rm f}$, and check whether the binary has merged within the CE phase or
not. If $a_{\rm f}$ is smaller than the sum of the remnant stellar radii, we
consider the binary has merged. Additionally, when the post-MS star does not
reach the Hayashi track or ignite helium burning, such a star, so-called a
Hertzsprung gap star, may not have a clear core-envelope structure. In this
case, we assume the binary merges Taam and Sandquist (2000); Ivanova and Taam
(2004); Belczynski  _et al._ (2008). If a binary merges before CCSNe, we treat
the merged product as a rapidly rotating star.

Fast rotations can affect stellar evolution in various ways. Namely, the
centrifugal force changes the pressure balance, the wind mass-loss rate can be
enhanced, and the rotation induced instability can enhance the material mixing
inside the star. Theoretical estimates of the integrated effect for solar-
metallicity single stars are shown in Figure 1, where the percentage increase
of the carbon-oxygen (CO) core mass with respect to the non-rotating case is
shown. For massive stars with $M_{\rm ZAMS}$ (the total mass at zero-age main
sequence) $>13M_{\odot}$, we consider the results of fast rotators with an
initial rotation velocity of $v_{\rm rot,ini}=300$ km s-1 studied in Limongi
et al. Limongi (2017). For $M_{\rm ZAMS}<13M_{\odot}$, models with $v_{\rm
rot,ini}/v_{\rm Kep}$ = 0.6, where $v_{\rm Kep}=\sqrt{GM/R}$ is the Kepler
rotation velocity, is newly calculated using the code described in Takahashi
et al. Takahashi  _et al._ (2014). The Takahashi models have initial rotation
velocities of $v_{\rm rot,ini}\sim 500$ km s-1. The chief effect of stellar
rotation in stars with ZAMS masses of $\lesssim 20M_{\odot}$ is the
enhancement in material mixing. This leads to an increase of the core mass,
but the efficiency is smaller for less massive models since the thermal
(Kelvin-Helmholtz) timescale is relatively longer compared with the stellar
lifetime. On the other hand, the enhancement of wind mass-loss due to the
$\Omega$–$\Gamma$ effect becomes more significant for more massive stars with
$\gtrsim 20M_{\odot}$, which reduces the total mass, counterbalancing the
core-mass increase due to rotation-induced mixing.

We assume that stellar mergers result in the formation of fast rotators;
otherwise the stars are non-rotating. The evolution of the fast rotating
merger remnant is approximately treated by enhancing the CO core mass with
respect to the non-rotating counterpart having the same total mass. From the
results shown in Figure 1, we derive simple fitting formulae as a function of
the ZAMS mass,

$f_{L}=53.4M_{\rm ZAMS}^{-3/2}+0.847$ (8)

from Limongi’s models and

$f_{T}=0.123M_{\rm ZAMS}+0.392$ (9)

from Takahashi’s models, which are also shown by the green and red solid
lines, respectively. Considering the large uncertainty in the theory of
rotating stellar evolution, we consider following three schemes:

  * •

Fiducial: we apply $f_{T}$ for merger remnants with $M_{\rm ZAMS}<13M_{\odot}$
and $f_{L}$ for $M_{\rm ZAMS}>13M_{\odot}$.

  * •

Extrapolated: we only apply $f_{L}$ for all merger remnants, i.e., for $M_{\rm
ZAMS}<13M_{\odot}$ we perform an extrapolation of $f_{L}$.

  * •

No rotation: we do not increase the CO core mass of a merger remnant, hence,
we ignore the effect of rotation.

Note that in all cases, if the CO core mass estimated by the above schemes
exceed the total mass, we limit the CO core mass to the total stellar mass.

Figure 1: The core mass increase due to rotational effects. The horizontal
axis is the ZAMS mass. The vertical axis is the percentage of increase. Blue
points are from the Limongi models Limongi (2017), while the orange points are
for our fiducial model Takahashi  _et al._ (2014). The green line is the
fitting to the Limongi models, while the red line is the fitting to our
fiducial models. We adopt the combination of the two fits for our fiducial
calculation.

###  II.2 Population synthesis

In order to calculate the binary effect for core-collapse progenitors, we use
the binary population synthesis method developed by Ref. Hurley  _et al._
(2002). Here, we give only a brief outline, and refer the reader to Refs.
Hurley  _et al._ (2002); Kinugawa  _et al._ (2014) for details.

Figure 2: Distributions of CO mass for our binary population synthesis
compared to the case with no binary interactions (red solid). Three types of
binary treatment with different post-merger evolution scenarios are shown:
extending the study of Limongi et al (black dot-dashed-dashed) Limongi (2017),
our fiducial model (green dot-dashed) Takahashi  _et al._ (2014), and a low
version neglecting post-merger rotation (dashed blue). CE parameter is
$\alpha\lambda=0.1$.

First, we set initial binary parameters such as the primary mass $M_{1}$, mass
ratio $M_{2}/M_{1}$, separation $a$, and eccentricity $e$, using the Monte
Carlo method based on the initial distribution functions which are obtained by
observations. We use a Salpeter IMF from $3M_{\odot}$ to $140M_{\odot}$ for
the primary IMF Salpeter (1955), the flat distribution from
$0.1M_{\odot}/M_{1}$ to 1 for the mass ratio Kobulnicky and Fryer (2007), the
logflat distribution from $a_{\rm min}$ to $10^{6}R_{\odot}$ for the
separation Abt (1983), and the thermal distribution ($\propto e$) from 0 to 1
for the eccentricity Heggie (1975), where $a_{\rm min}$ is the minimum
separation where the binary cannot occur a mass transfer. Note that while the
primary stars follow the Salpeter IMF by construction, after combining with
the secondary stars the effective IMF is flatter. The power-law index of the
combined stellar population is actually approximately ${-2.2}$.

Next, we calculate each stellar evolution and judge whether stars experience
binary interactions such as tidal friction, Roche lobe overflow, and the CE
phase, and change the parameters $M_{1},M_{2},a$, and $e$ in each time step
accordingly Hurley  _et al._ (2000, 2002); Kinugawa  _et al._ (2014). We
calculate $10^{6}$ binaries for each run. We consider two CE parameters cases
$\alpha\lambda=0.1$ and $\alpha\lambda=1$, and three schemes for rotating
mergers stars (fiducial, extrapolated, and no rotation), for a total of 6
synthetic stellar populations. In addition, we perform a comparison set with
binary separation large so binary effects do not operate.

Figure 3: Same as Figure 2, but showing the breakdown into different binary
evolution channels. In the top panel we show non-merge populations: single
(light-blue dot-dot-dashed) and double systems (blue dot-dashed). In the
bottom panel we show the merger channel, separately for the extending the
study of Limongi et al (black dot-dashed-dashed), our fiducial model (green
dot-dashed), and a low version neglecting post-merger rotation (dashed blue).
Total numbers in each channel are summarized in Table 1.

###  II.3 Effect on CCSN progenitors

| Merger | Non-merger | Ratio wrt
---|---|---|---
|  | (rotation) | Double | Single | no binary, $f_{b}$
No binary evolution | 0 | 0 | 122,600 | 171,002 | 1
Binary $\alpha\lambda=0.1$ Extrapolated | 155,235 | 315,722 | 75,723 | 109,276 | 1.76
Binary $\alpha\lambda=0.1$ Fiducial | 155,235 | 50,102 | 75,723 | 109,276 | 1.24
Binary $\alpha\lambda=0.1$ No rotation | 155,235 | 0 | 75,723 | 109,276 | 1.00
Binary $\alpha\lambda=1$ Extrapolated | 140,467 | 196,983 | 83,070 | 131,679 | 1.53
Binary $\alpha\lambda=1$ Fiducial | 140,467 | 39,869 | 83,070 | 131,679 | 1.24
Binary $\alpha\lambda=1$ No rotation | 140,467 | 0 | 83,070 | 131,679 | 1.05
Table 1: Counts of core-collapse progenitor systems predicted by binary
population syntheses, separated into the merger and non-merger populations.
For the merger population, we separately show numbers due to rapid rotation
(column 3). For non-merger population, we separately shown numbers of double
and single systems. Note that these are system numbers, i.e., the number of
progenitors in double is twice the quoted numbers. The final column shows the
ratio of the total number of core-collapse progenitors with respect to the no
binary evolution.

Figure 2 shows the binary effect for the CO core mass distributions compared
with the no binary case. We see that the maximum CO core mass is significantly
increased by the binary effect. This feature is seen in all rotation schemes
of the merger remnants. This is because the highest mass increase is caused by
mass accretion due to the Roche lobe overflow. Figure 3 shows the breakdown of
the binary effect for different evolution channels. Here, we split the stellar
population depending on their state prior to core collapse:

  * •

Non-merger population: binary systems where the stars do not merge before core
collapse. We further split this into binary systems where both stars are core-
collapse progenitors (double) and binary systems where only one of the stars
are core-collapse progenitors (single).

  * •

Merger population: binary systems where the stars merge before core collapse.
This includes progenitors which become core-collapse progenitors as a result
of rotational effects. We use three different treatments for this population,
as described in Section II.1.

The upper panel of Fig. 3 shows the CO core mass distributions of non-merger
binaries, while the bottom panel shows the CO core mass distribution of the
merger population, separately for each rotation scheme. We can observe from
these panels that the highest CO core mass population arises from mass
accretion due to the Roche lobe overflow in the non-merger double and single
systems. On the other hand, the distribution of lower CO core mass stars
depend on the rotation scheme (lower panel of Fig. 3). In particular, the rate
of increase of the CO core mass is very high for the extrapolated scheme for
progenitors with $M_{\rm initial}<13M_{\odot}$ (see Fig. 1). The effect is to
almost double the CO core mass of $M_{\rm ZAMS}\sim 13M_{\odot}$ stars from
its original $M_{\rm CO}\sim 3M_{\odot}$, which explains the peak seen in the
final CO core mass distribution at $\sim 5M_{\odot}$ (bottom panel of Fig. 3).

---
Figure 4: Cumulative bar charts of SN types, showing whether the progenitor
merged or not for each. The horizontal axis shows the initial mass of
progenitors. For merged progenitors, the horizontal axis shows the total
initial mass of the binary.

In Table 1, we show the breakdown of the total number of binary systems for
our 6 synthetic binary populations as well as the non-binary population for
comparison. For the merger population we show in a separate column the
contribution from rotating stars, and for the non-merger population we show
the numbers separately for double and single systems. We see that in the no
rotation scheme, the number of core-collapse progenitors is almost unchanged
from the no binary case. Mergers or mass accretion by binary interactions can
make low-mass stars become core-collapse progenitors and increase the tally.
On the other hand, the mergers of massive stars will reduce the number of
core-collapse progenitors. Even though the IMF is steeply falling with mass,
the total number of stars capable of becoming core-collapse progenitors
through mergers (i.e., say, with initial mass between 4–$8M_{\odot}$) is not
that much larger than the total number of stars above $8M_{\odot}$. For our
binary synthesis setup, with an effective IMF slope of $-2.2$ and upper mass
$140M_{\odot}$, this ratio is $\approx 4:3$. In the case of no rotation
scheme, we therefore see that these effects are roughly balanced.

However, the merged remnant should gain spin angular momentum from the orbital
angular momentum. Figure 1 shows the CO core mass gain due to rotation for our
fiducial scheme. This gain provides additional core-collapse progenitors.
Thus, in our fiducial and extrapolated schemes, the ratio with respect to no
binary is greater than one. Especially in the extrapolated scheme where the
mass gain is large, it is easier to form core-collapse progenitors than our
fiducial scheme. As for the CE parameter $\alpha\lambda$, a small
$\alpha\lambda$ makes it easier to merge than a large $\alpha\lambda$. Thus,
our $\alpha\lambda=0.1$ populations show more merger progenitors that our
$\alpha\lambda=1$ populations.

Figure 4 shows the cumulative bar charts for the CCSN type and whether the
progenitor is a merger or non-merger, separately for each rotation scheme. In
no binary evolution (i.e., single stellar evolution), whether the SN type is
II or Ibc switches at $M_{\rm ZAMS}\sim 24M_{\odot}$. However, Fig. 4 shows
for binary populations there are some Type Ibc SNe whose progenitor ZAMS
masses are less than $24M_{\odot}$; these are due to the envelope loss by a
mass transfer or a CE phase. Especially, the number of Type Ibc SNe arising
from merged remnants is higher in the $\alpha\lambda=1$ case than the
$\alpha\lambda=0.1$ case. The reason is that a larger $\alpha\lambda$ means it
is easier to unbind the envelope during a CE phase than a smaller
$\alpha\lambda$. On the other hand, there are still a few Type II SNe whose
progenitor ZAMS masses are more than $24M_{\odot}$, due to the mass accretion
by a mass transfer or the remaining envelope after a CE phase. In this case,
they often lose the envelope by stellar wind mass loss. Thus the number of
such SNe is small.

Table 2 shows the fractions of SN types for each of our population synthesis
realizations. The fraction of SN type strongly depends on the combined CE
parameter $\alpha\lambda$. In the case of $\alpha\lambda=0.1$, fractions of SN
types are II:Ibc $\sim$ 75:25. On the other hand, in the case of
$\alpha\lambda=1$, fractions of SN types are $\sim$ 60:40. If $\alpha\lambda$
is small, a large orbital energy is need to unbind the envelope. It is easier
to merge during the CE phase and the merged remnants tend to maintain large
envelopes. Thus, in the case of small $\alpha\lambda$, the fraction of merged
type II SNe are high (Fig. 4). The fraction of SN types from a volume-limited
sample of low-redshift SNe is II:Ibc=75:25 Li  _et al._ (2011), consistent
with our models using $\alpha\lambda=0.1$.

| Ibc | II
---|---|---
$\alpha\lambda$=1 Fiducial | 41.8% | 58.2%
$\alpha\lambda$=1 Extrapolated | 33.6% | 66.4%
$\alpha\lambda$=1 NoRotation | 42.1% | 57.9%
$\alpha\lambda$=0.1 Fiducial | 23.1% | 76.8%
$\alpha\lambda$=0.1 Extrapolated | 25.9% | 74.1%
$\alpha\lambda$=0.1 NoRotation | 26.0% | 74.0%
Table 2: Fractions of SN types for each of our population synthesis
realizations.

##  III Supernova neutrino predictions including binary treatment

We now quantify the impact of binary interactions on the DSNB. Binaries cause
three important changes relative single stellar populations: (1) due to mass
transfer and mergers, stars initially below the core-collapse threshold can
become a core-collapse progenitor, (2) mergers can cause two core-collapse
progenitors to reduce to a single core-collapse progenitor, and (3) due to
mass transfer, the distribution of final CO core masses is changed. Here, we
include these effects in a new prediction of the DSNB.

###  III.1 DSNB formulation

The DSNB is predicted by integrating over time the comoving volumetric
occurrence rate of core collapses, $R_{\rm CC}$, weighted by the mean neutrino
spectrum for a population of core-collapse progenitors, $dN_{\nu}/dE$,
appropriately redshifted (see, e.g., reviews Beacom (2010); Lunardini (2016)),

$\frac{d\phi}{dE}=c\int R_{\rm
CC}(z)\frac{dN_{\nu}}{dE^{\prime}}(1+z)\left|\frac{dt}{dz}\right|dz,$ (10)

where $E^{\prime}=E(1+z)$ and
$|dz/dt|=H_{0}(1+z)[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$. In what
follows, we perform the integration out to $z_{\rm max}=5$, which is
sufficiently large for flux predictions in the detectable range of neutrinos.

We compute the cosmic core-collapse rate $R_{CC}(z)$ from the cosmic star
formation rate (SFR) history, $\dot{\rho}_{*}(z)$. This is because while the
uncertainties of the SFR are fairly large Hopkins and Beacom (2006); Mathews
_et al._ (2014); Madau and Dickinson (2014), they are smaller than those for
direct CCSN rate measurements Horiuchi  _et al._ (2013); Strolger  _et al._
(2015); Mattila  _et al._ (2012); Botticella  _et al._ (2012); Taylor  _et
al._ (2014); Cappellaro  _et al._ (2015). Furthermore, not all core collapse
may yield CCSNe. We adopt the piece-wise linear fit to $\dot{\rho}_{*}(z)$,
i.e., Eq. (5) of Ref. Yuksel  _et al._ (2008), with shape parameters $a=3.4$,
$b=-0.3$, and $c=-3.5$; redshift breaks at $z=1$ and $4$; and smoothing
parameter $\eta=10$. We convert the SFR to a core-collapse rate by the
conversion

$R_{\rm CC}(z)=\dot{\rho}_{*}(z)\frac{\int_{M_{\rm
min}=8}^{140}\psi(M)dM}{\int_{0.1}^{140}M\psi(M)dM},$ (11)

where $\psi(M)=dN/dM$ is the IMF, defined so that $\psi(M)dM$ gives the number
of stars in the mass range $M$ to $M+dM$, and $M_{\rm min}$ is the minimum
mass for a star to undergo core collapse. We adopt measurements of
$\dot{\rho}_{*}(z)$ assuming a Baldry-Glazebrook IMF Baldry and Glazebrook
(2003), and consistently use the same IMF for computing Eq. (11). The Baldry-
Glazebrook IMF has a shallow high-mass slope of $-2.15$ and produces better
agreements with other astronomical observations such as the stellar mass
density buildup Hopkins and Beacom (2006) and the extragalactic background
light Horiuchi  _et al._ (2009). Importantly, the Baldry-Glazebrook high-mass
slope is also close to the effective IMF of our population synthesis setup (a
power-law index of approximately $-2.2$).

Fortunately, the IMF impacts the core-collapse rate only at the percent level.
For example, Ref. Hopkins and Beacom (2006) explored the impacts of different
IMFs, including the Baldry-Glazebrook and Salpeter IMFs. Since the same
massive stars acting as tracers of star-formation activity are also core-
collapse progenitors, the IMF is not needed to sample the entire stellar
population mass range (see, also, Horiuchi  _et al._ (2009)). As a result, our
CCSN rate normalization at z=0 is $R_{CC}(0)=(1.3\pm 0.26)\times 10^{-4}{\rm
Mpc^{-3}\,yr^{-1}}$, consistent with values obtained with other IMFs (e.g.,
Hopkins and Beacom (2006); Horiuchi  _et al._ (2009); Møller  _et al._
(2018)).

We do not model the potential impact of binary interactions on the SFR
calibration factors. While binary interactions can dramatically impact stellar
evolution, their effects are strongly concentrated on the late evolutionary
epochs. The main sequence, which dominate as SFR tracers, are relatively
unaffected, so we do not expect the SFR calibrations to be strongly
influenced.

###  III.2 Including binary effects

We account for the binary effects on the mean neutrino emission by using the
modifications to the distribution of CO core masses (Figure 2). To this end,
we first establish the connection between the CO core mass and the neutrino
emission during core collapse. Then, we derive the mean neutrino emission
based on the distribution of CO core mass as predicted by our population
synthesis calculations.

In order to obtain the neutrino emission from a wide range of massive stars,
we adopt the systematic set of two-dimensional (2D) core-collapse simulations
of Summa et al (2016) Summa  _et al._ (2016), hereafter S16, and augment them
by additional simulations. The simulations of S16 include a total of 18
progenitor models in the ZAMS mass range $11.2$–$28M_{\odot}$. Self-gravity is
computed using the general relativistic monopole corrections as described in
Marek  _et al._ (2006), and neutrino transport is solved with a ray-by-ray
approximation along radial rays using a variable Eddington factor method
(e.g., Buras  _et al._ (2006)) for all neutrinos: $\nu_{e}$, $\bar{\nu}_{e}$,
and $\nu_{x}$ ($=\nu_{\mu}$, $\nu_{\tau}$, $\bar{\nu}_{\mu}$,
$\bar{\nu}_{\tau}$).

To the set of 18 simulations of S16 we add two core-collapse simulations: (i)
a $9.6M_{\odot}$ progenitor of Heger and Woosley (2010) and (ii) an
$8.8M_{\odot}$ ONeMg progenitor of Nomoto (1984, 1987). For the $9.6M_{\odot}$
progenitor, the hydrodynamic simulation has been performed by the 3DnSNe code
(see the references (Nakamura  _et al._ , 2019; Kotake  _et al._ , 2018;
Takiwaki  _et al._ , 2016) for recent applications). The method for the 2D
supernova model is summarized in previous works Zaizen  _et al._ (2020a);
Sasaki  _et al._ (2020). The code provides consistent results on neutrino
luminosities and average energies with more sophisticated schemes (see Ref.
O’Connor  _et al._ (2018) for detailed comparison), whereas the code employs
an idealized neutrino transport scheme of IDSA (Isotropic Diffusion Source
Approximation) Liebendörfer  _et al._ (2009). The simulation of the
$8.8M_{\odot}$ progenitor is the same as adopted in the DSNB study of Horiuchi
_et al._ (2018) and is based on simulations of Hudepohl  _et al._ (2010). Core
collapse of the $8.8M_{\odot}$ progenitor of Nomoto (1984, 1987) is simulated
in 1D with the PROMETHEUS-VERTEX code long-term, until $\sim 25$ seconds. A
variable Eddington-factor closure scheme is used to model Boltzmann transport
Rampp and Janka (2002).

At each simulation time snapshot, the neutrino energy spectrum is well-
described by a pinched Fermi-Dirac form described by three parameters,

$\displaystyle f(E)$ $\displaystyle=$
$\displaystyle\frac{(1+\langle\alpha\rangle)^{(1+\langle\alpha\rangle)}}{\Gamma(1+\langle\alpha\rangle)}$
$\displaystyle\quad\times\frac{E^{\rm
tot}_{\nu}E^{\langle\alpha\rangle}}{\langle
E_{\nu}\rangle^{2+\langle\alpha\rangle}}{\rm
exp}\left[{-(1+\langle\alpha\rangle)\frac{E}{\langle E_{\nu}\rangle}}\right],$

where $\langle E\rangle$ is the mean neutrino energy, $\alpha$ is the pinching
parameter, and $E_{\nu}^{\rm tot}$ is the total neutrino energy in the time
bin. Since the models of S16 and the $9.6M_{\odot}$ simulation were terminated
typically at $\sim 0.5$–1 seconds post bounce, we estimate the late-time
neutrino emission using the following assumptions: (1) the protoneutron star
radius contracts from the radius at the final time- step to a final radius of
15 km; (2) the protoneutron star mass grows according to
$M(r)=M_{0}+M_{1}(1-{\rm e}^{-t/\tau_{M}})$, where $M_{0}$, $M_{1}$, and
$\tau_{M}$ are found by fitting this function to the time evolution of the
protoneutron mass during the simulated epoch; (3) the gravitational binding
energy released after the final simulation time step is equipartitioned
between all neutrino flavours; and (4) the average neutrino energies and
pinching factors are fixed to the final value of each simulation. The
simulation of the $8.8M_{\odot}$ progenitor includes the late-time cooling
phase and does not need such extrapolations.

Figure 5: Time-integrated neutrino spectral parameters: total neutrino
energetic (top panel), mean energy (middle panel), and pinching parameter
(bottom panel), shown separately for $\nu_{e}$ (red circles), $\bar{\nu}_{e}$
(blue squares), and $\nu_{x}$ (gray triangles). These are based on the 2D
simulations of S16 augmented by simulations of a $9.6M_{\odot}$ star and a
$8.8M_{\odot}$ ONeMg star. Note the axis change between the left and right, in
logarithmic and linear, respectively. The dashed lines indicate our
phenomenological fits through the simulations.

The neutrino emission is then summed over time, adopting the set of
$(L,\langle E\rangle,\alpha)$ at each time step. The summed emission is then
fitted to a functional form identical to Eq. (III.2), except now the
normalization $E_{\nu}^{\rm tot}$ represents the neutrino energetics over the
entire duration of the CCSN. The results of the fits are shown in Fig. 5,
where for each progenitor the neutrino emission parameters $E_{\nu}^{\rm
tot}$, $\langle E\rangle$, and $\alpha$ are plotted against the compactness of
the progenitor. Here, the compactness is simple parameter to capture the size
of the core at the moment of core collapse,

$\xi_{M}=\left.\frac{M/M_{\odot}}{R(M_{\rm bary}=M)/1000\,{\rm
km}}\right|_{t},$ (13)

where $R(M_{\rm bary}=M)$ is the radial coordinate that encloses a baryonic
mass $M$ at epoch $t$. First described in O’Connor and Ott (2011) as a proxy
for collapse to black holes, it has been shown to be also a versatile
indicator including the neutrino emission Horiuchi  _et al._ (2017); Horiuchi
and Kneller (2018). We adopt $t$ as the onset of collapse as was done in Ref.
Sukhbold and Woosley (2014), rather than at core bounce as in Ref. O’Connor
and Ott (2011). Figure 5 reveals a growing trend of higher compactness stars
emitting more neutrinos with higher mean energies, consistent with previous
findings in the literature.

Figure 6: The core compactness (defined by $M=2.5M_{\odot}$) at moment of
core-collapse, as a function of the CO core mass. The dashed lines separate
the ONeMg regime and the Fe core regime in both compactness and CO mass
planes.

We fill in the gaps between our 20 progenitors by performing a linear fit
through the simulation outputs versus compactness. When performing a global
fit through both ONeMg and Fe core progenitors combined, we find a
systematically lower neutrino output in the ONeMg case. Therefore, we perform
a fit only on the iron core progenitors (19 in total), and maintain the ONeMg
case separately. We put the separation at a compactness value of
$\xi_{2.5}=5\times 10^{-5}$, which is between the $8.8M_{\odot}$ ONeMg core
and the lowest compactness Fe core, $11.2M_{\odot}$ progenitor. The final fit
results are shown by the dashed lines in Fig. 5.

To incorporate our new CO mass distributions, we map the core compactness of
our adopted core-collapse progenitors to the CO core mass, as shown in Fig. 6.
In general, the relation is non-monotonic, and recently has been shown to
depend on also the ratio of C and O masses Patton and Sukhbold (2020).
However, for typical solar mass progenitors they should trace a trajectory
within the large C/O mass plane Patton and Sukhbold (2020). We interpolate
between core-collapse progenitors, but for CO mass $<2M_{\odot}$ and
$>7M_{\odot}$ we use a linear fit through the last few progenitors, as shown
by the black curve. The dashed lines in Fig. 6 illustrates the transition
between ONeMg and Fe core regions, in both compactness and CO mass. The
separation in CO mass is $M_{\rm CO}=1.35M_{\odot}$ Suwa  _et al._ (2018);
Schwab  _et al._ (2016).

Finally, we compute the mean neutrino emissions by summing the neutrino
emission for each CO mass bin, appropriate weighted by the CO core mass
distribution. To account for changes in the number of core-collapse
progenitors, we apply a re-normalization factor $f_{b}$ (see final column of
Table 1) to our predictions. As discussed in Section II.3, this physically
arises from the increased mass range for core-collapse progenitors, i.e., a
smaller $M_{\rm min}$. While direct measurements of $R_{CC}$ in principle
allows a data-driven approach to constrain $f_{b}$, in reality measurements
are strongly affected by systematic uncertainties due to corrections for dust
and missing CCSNe. Furthermore, older direct measurements of $R_{CC}$ and
estimates from the SFR show hints of a systematic normalization discrepancy at
a factor $\sim 2$ level Horiuchi  _et al._ (2011). This discrepancy may be
explained by new $R_{CC}$ measurements (e.g., Dahlen  _et al._ (2012); Perley
_et al._ (2020)) and/or new SFR (e.g., Mathews  _et al._ (2014); Madau and
Dickinson (2014)). We therefore do not attempt to constrain large binary
enhancements based on rates alone.

Figure 7: The predicted DSNB flux of $\bar{\nu}_{e}$ for NH, for 6 different
binary population synthesis models, compared with a single stellar evolution
model. The binary population synthesis models include different treatments for
the post-merger rapidly rotating stars and different CE modeling. We find that
the minimal estimate, which neglects post-merger rotation effects (blue solid
and dashed), yields a DSNB flux indistinguishable from a single stellar
evolution model (red dot-dashed), while our fiducial scheme (green solid and
dashed) and the extrapolated scheme (black solid and dashed) yield higher DSNB
fluxes.

###  III.3 DSNB predictions

The neutrinos that are emitted by the collapsing cores of massive stars
undergo flavor transformations during their propagation to Earth. We implement
the well-established matter-induced Mikheyev-Smirnov-Wolfenstein (MSW) effect
which occurs during the neutrino’s propagation through the progenitor Mirizzi
_et al._ (2016). The mixing of neutrino flavors depends on the neutrino mass
hierarchy. For example, the $\bar{\nu}_{e}$ survival probability is
$\cos^{2}\theta_{12}\approx 0.7$ in the normal mass hierarchy (NH), and
$\approx 0$ in the inverted mass hierarchy (IH). The terrestrial flux, $F^{\rm
obs}_{\bar{\nu}_{e}}$, as a mixture of fluxes at core collapse, $F_{i}$, is
then,

$\displaystyle{\rm(NH)}\quad F^{\rm obs}_{\bar{\nu}_{e}}$
$\displaystyle\simeq$
$\displaystyle\cos^{2}\theta_{12}F_{\bar{\nu}_{e}}+\sin^{2}\theta_{12}F_{{\nu}_{x}}~{},$
(14) $\displaystyle{\rm(IH)}\,\,\quad F^{\rm obs}_{\bar{\nu}_{e}}$
$\displaystyle\simeq$ $\displaystyle F_{{\nu}_{x}},$ (15)

Additional flavour mixing can be induced by the coherent neutrino–neutrino
forward scattering potential. However, this so-called collective oscillations
are complex and the detailed oscillation predictions and their time-dependence
are far from known (see, e.g., review Tamborra and Shalgar (2020)). Since the
DSNB samples the entire core-collapse population, it may be expected that any
particular features (e.g., spectral splits caused by swaps) may be washed out,
resulting in an overall minimal effect. Also, the differences between neutrino
flavors are not large during the cooling phase. The effect for the DSNB has
even been estimated to be less than $\sim 10$% Lunardini and Tamborra (2012).
However, it is not clear to what extent this holds in light of recently
discovered flavor instabilities, e.g., due to spontaneously broken symmetries
Raffelt  _et al._ (2013); Abbar  _et al._ (2015); Chakraborty  _et al._
(2014); Mirizzi  _et al._ (2015); Zaizen  _et al._ (2020b) or so-called fast
conversion Sawyer (2005, 2009). Therefore, we do not consider mixing effects
beyond MSW.

| SK-Gd [/yr] | HK [/yr]
---|---|---
| Normal | Inverted | Normal | Inverted
No binary evolution | 2.3 | 2.4 | 5.5 | 6.2
Binary $\alpha\lambda=0.1$ Extrapolated | 4.7 | 4.6 | 11.4 | 12.0
Binary $\alpha\lambda=0.1$ Fiducial | 2.7 | 2.7 | 6.4 | 7.1
Binary $\alpha\lambda=0.1$ No rotation | 2.3 | 2.4 | 5.5 | 6.2
Binary $\alpha\lambda=1$ Extrapolated | 3.8 | 3.8 | 9.1 | 9.9
Binary $\alpha\lambda=1$ Fiducial | 2.7 | 2.7 | 6.3 | 7.0
Binary $\alpha\lambda=1$ No rotation | 2.3 | 2.5 | 5.5 | 6.4
Table 3: Annual DSNB event rates at Super-K (Hyper-K) employing 22.5 kton (187
kton) fiducial volume. Signals are computed over an energy range of
$E_{e}=[10,28]$ MeV ($[18,28]$ MeV) for gadolinium-doped Super-K (pure-water
Hyper-K). Perfect detection efficiencies are used and only MSW oscillations
are included shown separately for normal/inverted mass hierarchies.

Figure 7 shows the DSNB predictions for our three population synthesis models:
fiducial, extrapolated, and no rotation, each for our two different choice of
CE parameter $\alpha\lambda=0.1$ and 1. We also add the prediction of the
single stellar evolution for comparison. For clarify, only the NH case is
considered, and only the $\bar{\nu}_{e}$ flux is shown; results for the IH and
other flavors show similar trends. We see that in our pessimistic no rotation
scheme, where post-merger rotation effects are neglected, the DSNB flux
remains to good accuracy the same as the no binary case. However, for both our
fiducial and extrapolated schemes, the DSNB flux is noticeably increased.

To estimate the DSNB event rates, we consider separately the NH and IH, and
compute the event rates at two detector setups: Super-K assuming a fiducial
volume of 22.5 kton containing gadolinium doped water for a signal energy
range of 10–28 MeV, and Hyper-Kamiokande (Hyper-K) assuming a fiducial volume
of 187 kton containing pure water for a signal energy range of 18–28 MeV. The
event rates for Hyper-K filled with gadolinium-doped water can be
approximately made by scaling the Super-K-Gd numbers by the fiducial volume.
We assume perfect detection efficiency for simplicity. The predicted DSNB
event rates are summarized in Table 3.

In the absence of binary interactions, we predict annual event rates of
2.3–2.4 yr-1 at Super-K-Gd and 5.5–6.2 yr-1 at Hyper-K. While Hyper-K is $\sim
8$ times the volume of Super-K, its predicted signal rate is only $\sim 2.4$
that of Super-K. This is because we assume a narrower signal energy window for
Hyper-K due to larger backgrounds.

When binary interactions are included, we observe increases in the predicted
DSNB event rates. The largest increase is seen in our extrapolated scheme,
based on extrapolating the Limongi models, where the predicted rates at Super-
K-Gd and Hyper-K are 4.6–4.7 yr-1 and 11.4–12.0 yr-1, respectively, i.e., a
factor $\sim 2$ increase. The increase is more modest in our fiducial scheme,
generally in the 15–20% range compared to the no binary case. In our most
conservative case where we neglect the effect of increased rotation in
mergers, the change is at the percent level.

The rate changes can be understood by the balance of two competing effects. On
the one hand, the mergers of stars whose individual masses were high enough to
each undergo core collapse reduce the DSNB. While the merged star will be more
massive than the individual stars and therefore will have a stronger neutrino
emission, it cannot compensate for the reduction from two to one progenitor.
On the other hand, the mergers of light stars whose individual masses are
below the core collapse threshold can increase the DSNB if the combined mass
is above the core collapse threshold. The net effect can be seen in the counts
in Table 1. In the no rotation case, we find that these effects nearly cancel
each other out, with a net change in the total number of core-collapse
progenitors of less than percent (for $\alpha\lambda=0.1$) and $+5$% (for
$\alpha\lambda=1.0$) compared with the no binary case. More impressively, both
in our fiducial and extrapolated schemes, the mergers of even light stars can
form a core-collapse progenitor owing to rotational effects, reducing the
minimal core mass for core collapse. The increase through this additional
population results in net changes in the total number of core-collapse
progenitors of $+24$% and $+53$–$75$% for the fiducial and extrapolated
schemes, respectively. These largely explain the major trends seen in the DSNB
rates.

A secondary effect occurs due to mass transfer in binary systems. Since our
mass transfer prescription conserves mass, any mass lost by a star is accreted
onto its companion. We can quantify the impact of this by comparing the
_shape_ of CO mass among different population syntheses, i.e., we re-normalize
the non-merge populations to match the no binary counts and compare the DSNB
event rates. We find that the DSNB event rate increases by 2–3% for
$\alpha\lambda=0.1$, while the increase is smaller for $\alpha\lambda=1$. We
therefore conclude that the effect of mass stripping is more or less
compensated by the effect of mass gained by its companion.

##  IV Discussion and conclusions

In this paper, we explored the impacts of binary interactions on the predicted
DSNB detection rates. We incorporate the key effects of binary
interactions—mass transfer and mergers—through binary population synthesis
calculations to predict the final distributions of CO core masses of
progenitors that undergo core collapse at the end of stellar evolution. Using
a suite of 20 core-collapse supernova simulations, we compute the neutrino
emissions from such progenitor distributions and predict the DSNB detection
rate at present and future neutrino detectors.

Figure 8: Reverse cumulative plot showing the fraction of progenitors with
higher CO mass. Shown are the results for no binary treatment (solid red), the
extrapolated scheme (black dot-dashed-dashed), our fiducial scheme (green dot-
dashed), and a scheme neglecting post-merger rotation (dashed blue). Binary
populations have higher fractions of CO cores massive enough to likely
collapse to black holes ($M_{\rm CO}\gtrsim 10M_{\odot}$) than a no binary
population. CE parameter is $\alpha\lambda=0.1$.

We find that binary effects can be large. Compared to predictions without
binary considerations, we find increases of 15–20% for our fiducial scheme.
However, it is as small as 1% in our estimate ignoring rotation effects, or as
large as 70% in our extrapolated scheme. Binary effects can be separated into
three effects: change in mass due to mass transfer, reduced progenitor counts
due to mergers of stars originally massive enough to core collapse, and
increased progenitor counts due to mergers of stars originally not massive
enough to core collapse (and massive enough after merger). Of these, the
latter has the strongest effect. However, its contribution to the DSNB depends
on the evolution of the post-merger stars which would be rapidly rotating as a
result of the merger. Completely ignoring the rotation or extrapolating the
results of Limongi et al. Limongi (2017) yields increases of 1–70%. Therefore,
it is important to further investigate the evolution of such post-merger,
rapidly rotating stars near the white dwarf / core collapse threshold.

Our results differ from those of Ref. Kresse  _et al._ (2020) who conclude
that binary interactions yield a reduction of the DSNB. However, their study
uses the suite of hydrogen-stripped progenitors of Ref. Woosley (2019) as a
proxy for stars undergoing close-binary interactions. Therefore, it includes
the effect of mass stripping from stars, but no consideration for mass gained
by its binary companion. The study also did not include the impacts of stellar
mergers. Our study differs in that we perform binary population syntheses
incorporating mass transfer and mergers. We find that mergers can cause a
sizable increase in the core-collapse progenitor population numbers and hence
the DSNB rates.

In this study, we have not included other contributions to the DSNB, most
notably the collapse to black holes. This is because our intent is to focus on
the impacts of binary interactions on the primary population: collapse to
neutron stars. Binary interactions would also affect the fraction of massive
stars that collapse to black holes. Indeed, our binary population synthesis is
already being applied to interpret compact object merger rates (e.g., Refs.
Kinugawa  _et al._ (2014); Yu and Jeffery (2015); Ablimit and Maeda (2018)).
However, the emission of neutrinos from black holes comes with additional
uncertainties including the equation of state of hot dense matter and the
fraction of massive stars which collapse to black hole. We can still speculate
what the impact may be, based on the relative fractions of potential black
hole progenitors. Figure 8 shows the reverse cumulative plot of CO mass, i.e.,
the fraction of stars with higher CO mass. It is evident that all our binary
population syntheses have a long tail of high CO mass progenitors, but these
remain too rare to make substantial impact on the DSNB. More important would
be the stars near the neutron star/black hole boundary, around CO core mass of
$\sim 10M_{\odot}$. Compared to the no binary model, our binary models have
larger fractions, implying any consideration of collapse to black holes will
likely enhance our results due to their harder neutrino spectra.

It should be emphasized that there are various sources of uncertainties
affecting predictions of the DSNB which we have neglected, e.g., from the
cosmic core-collapse rate, collective neutrino oscillation effects, as well as
uncertainties arising from different simulation suites. Our intent here is not
to make the most accurate or robust error estimates of the DSNB. Rather, we
aim to highlight the potential impacts of binary interactions on the DSNB
flux. Thorough discussions of the DSNB uncertainties be found in the
literature, e.g., Kresse  _et al._ (2020).

While the DSNB has not yet been detected, this is expected to change in the
coming decade. Super-K has completed its gadolinium upgrade, and next-
generation detectors such as Hyper-K and DUNE are under construction.
Interpreting the results of DSNB searches will require a commensurate effort
in theoretical modeling of the DSNB, including binary effects which massive
stars are subject to. As simulations advance, DSNB predictions will continue
to improve and allow future DSNB detections to reveal the diversity in massive
stellar core collapse and their neutrino emissions.

###### Acknowledgements.

We thank Shin’ichiro Ando, Takami Kuroda, and Masamichi Zaizen for useful
discussions, and Hans-Thomas Janka for sharing simulations data of S16. We
also thank the hospitality of the organizers of the workshop “Workshop on
core-collapse supernova explosions and related physics” where parts of the
discussion took place. S.H. is supported by the U.S. Department of Energy
Office of Science under award number DE-SC0020262 and NSF Grants Nos.
AST-1908960 and PHY-1914409. This study was supported in part by the Grants-
in-Aid for the Scientific Research of Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant Number (JP17H05206, JP17K14306, and JP17H01130,
JP17H06364, JP18H01212 (T.T. and K.K.), by the Central Research Institute of
Explosive Stellar Phenomena (REISEP) of Fukuoka University and the associated
project (No. 207002), and by JICFuS as a priority issue to be tackled by using
the Post ‘K’ Computer. The numerical simulations were done using XC50 at
Center for Computational Astrophysics at National Astronomical Observatory of
Japan. T.K. acknowledges support from University of Tokyo Young Excellent
Researcher program.

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