different BNS mergers simulated here. The shock waves decelerate more slowly
than a single-shell model (dashed line; $\Gamma_{\rm sh}\propto R^{-3/2}$) due
to the radially structured ejecta refreshing the shock wave at late times.

####  5.5.2 Non-thermal radiation

We assume that a fraction $\epsilon_{e}$ of the thermal energy
$e=4(\Gamma-1)\Gamma nm_{p}c^{2}$ generated in the forward shock at time $t$
accelerates electrons into a power-law distribution in Lorentz factor
$\gamma_{e}$ in the frame comoving with the shock (Sari et al., 1998),

$Q(\gamma_{e},t)=Q_{0}\gamma^{-p}_{e}\Theta(\gamma_{e};\gamma^{\rm
min}_{e},\gamma^{\rm max}_{e}).$ (36)

Here, $Q_{0}$ is a time-dependent normalization factor, $p$ is the power-law
index, and $\Theta$ denotes the Heaviside function that restricts the
injection to an interval between the minimum energy given by normalization
constraints,

$\gamma^{\rm
min}_{e}=\epsilon_{e}\frac{(p-2)}{(p-1)}\frac{m_{p}}{m_{e}}(\Gamma-1),$ (37)

and a maximum energy given by balancing acceleration and synchrotron losses
(De Jager et al., 1996)

$\gamma^{\rm max}_{e}=\sqrt{\frac{6\pi e\epsilon_{\rm acc}}{\sigma_{T}B}}.$
(38)

Here, $e$ is the electron charge, $\sigma_{T}$ is the Thomson cross-section,
$B$ is the co-moving magnetic field strength, and $\epsilon_{\rm acc}$ is the
ratio of acceleration rate to the maximum possible particle energy-gain rate
that we fix to $0.35$ following Zhang et al. (2020). Furthermore, we assume a
fraction $\epsilon_{B}$ of the instantaneously generated post-shock thermal
energy is transformed into magnetic energy (Sari et al., 1998), which results
in a co-moving field field given by

$B=\sqrt{8\pi(\Gamma-1)(4nm_{p}\Gamma c^{2})\epsilon_{\rm B}}.$ (39)

The evolution of the electron distribution $n_{e}(\gamma_{e},t)$ in the shock
front is determined by the continuity equation for particle number. We
numerically solve the full resulting Fokker-Planck equation for the
distribution $n_{e}(\gamma_{e},t)$ in the comoving frame of the shock,

$\displaystyle\frac{\partial n_{e}(\gamma_{e},t)}{\partial t}$
$\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\gamma_{e}}[\dot{\gamma_{e}}(\gamma_{e},t)n_{e}(\gamma_{e},t)]$
(40) $\displaystyle+Q(\gamma_{e},t)-\frac{n_{e}(\gamma_{e},t)}{t_{\rm esc}}.$

Here, $t_{\rm esc}$ is the escape time of particles from the shock layer and
$\dot{\gamma_{e}}$ represents the energy gains/losses determined by various
radiative processes. The comoving time $t$ here is related to the evolution of
the shock as seen in the inertial frame of the merger site with time
$t^{\prime}$ by $t=\int_{0}^{\rm t^{\prime}}dt^{\prime}\Gamma^{-1}_{\rm
sh}(t^{\prime})=\int_{0}^{R}dR/(\Gamma^{-1}_{\rm sh}v_{\rm sh})$. To solve
this equation numerically and to compute the radiation fluxes we use the code
Paramo (Rueda-Becerril, 2021), taking into account synchrotron and inverse-
Compton cooling, together with synchrotron self-Compton/self-absorption, as
well as all relativistic effects required for computing fluxes in the observer
frame.

Most studies of non-thermal afterglows from BNS mergers (e.g., Piran et al.
2013; Hotokezaka & Piran 2015; Hotokezaka et al. 2018) employ analytic
approximations for the synchrotron spectrum, e.g., a simply connected power
law as a function of frequency (Sari et al., 1998). Here, we instead
numerically obtain the SED solving for the particle distribution, taking into
account self-consistently radiative cooling due to inverse Compton scattering,
and the full synchrotron emissivities (for details on the numerical methods,
see Rueda-Becerril et al. 2021; Rueda-Becerril 2021; Davis et al. 2022). While
providing more accurate predictions for non-thermal emission in general, this
approach, in particular, replaces the need to rely on order-of-magnitude
estimates for the cooling frequencies, which critically shape the synchrotron
light curves (Hotokezaka et al., 2018).

In order to take the angular dependence of the ejecta structure and viewing
angle effects into account, we follow the approach of Margalit & Piran (2015).
We assume an axisymmetric outflow and discretize the outflow into angular bins
according to polar angle $\Theta$, equidistantly spaced in $\cos\Theta$ , each
bin $i$ having its kinetic energy distribution $E_{{\rm K},i}(u,\Theta_{i})$
(cf. Fig. 7). We solve the shock and Fokker-Planck equations for each angular
bin separately, assuming no lateral spreading, and subsequently add the
emergent radiation from each annulus in solid angle in the observer frame.
Angular variation in the ejecta profiles can shift the peak time of the light
curves, although in our equal-mass models the ejecta is sufficiently isotropic
for this effect not to be significant (Margalit & Piran, 2015).

The free parameters of our model are

$\\{p,\epsilon_{e},\epsilon_{B},n,D_{L},z,\theta_{\rm obs},u_{\rm rad}\\},$
(41)

where $D_{L}$ is the luminosity distance to the merger site, $z$ the
corresponding cosmological redshift, $\theta_{\rm obs}$ the observer angle
relative to the orbital plane, and $u_{\rm rad}$ the external photon field
density relevant for inverse Compton scattering. As a fiducial photon field,
we use the thermal photons of the Cosmic Microwave Background, with $u_{\rm
rad}=0.26$ eV cm-3.

As a baseline test of our afterglow code, we consider a spherically symmetric
shock powered by ejecta following a power-law distribution in velocity
focusing on synchrotron emission only. Comparing to the public code
afterglowpy (Ryan et al., 2020)131313https://github.com/geoffryan/afterglowpy
we find very good agreement regarding the synchrotron emission.

####  5.5.3 Kilonova Afterglow Light Curves and GW170817

Figure 21: Representative kilonova non-thermal afterglow light curves
generated by the dynamical ejecta of our BNS simulation runs, labeled by the
corresponding EOSs employed. Also shown are X-ray (blue dots) and radio
(orange dot) observations reported by Hajela et al. (2022), who present
tentative evidence for X-ray and radio rebrightening following the GW170817
event. Our models are consistent with the observations for typical
microphysical parameters (see the text for details).

We apply our non-thermal afterglow model described above to the observations
of Hajela et al. (2022), who present evidence for a late-time
($\approx\\!1000$ days post-merger) excess of X-ray and radio emission above
the gamma-ray burst broad-band afterglow of GW170817. The existence of this
late-time rebrightening is still a matter of debate (Balasubramanian et al.,
2022; O’Connor & Troja, 2022) and, if present, could have other possible
origins as well (e.g., Metzger & Fernández 2021).

We fix $\epsilon_{e}$ to a standard value of $\epsilon_{e}=0.1$ and, as usual,
we vary the microphysical parameters $(p,n,\epsilon_{B})$. We find that our
models directly based on numerical relativity simulation data are compatible
with the observational data obtained so far over a range of typical parameter
values; Fig. 21 shows representative models for each EOS model. The stiffer
$\mathtt{LS220}$ simulation generates less energetic outflows and thus tends
to prefer a somewhat higher density of the ISM as well as higher magnetic
energy compared to the softer EOSs in order to reach the same observed fluxes
at $\approx\\!1230$ days. The properties of peak flux (timescale and flux
level) sensitively depend on the ISM density and the velocity of the fastest
(outermost) ejecta shell. We observe that the $\mathtt{APR}$ run with the
fastest ejecta profile tends to peak earliest, although all simulations
considered here produce overall similar results that are consistent with the
data.

Due to the mildly relativistic velocity structure of the outflow, the light
curves show long peak timescales of thousands of days and associated broad
peaks. The X-ray fluxes almost plateau around the peak timescale between
$\sim\\!200$ to $\sim\\!2000$ days, before decreasing again. As also evident
from Fig. 21, the X-ray light curves tend to have a faster rise and decay
timescale than the radio signal, which indicates that the cooling frequency
has crossed the X-ray band by $\sim\\!2000$ days. The radio emission plateaus
for several years.

Compared to Nedora et al. (2021a) who use a large set of numerical simulations
and a semi-analytical afterglow model, we find that our model is compatible
with similar ISM densities but tends to require higher magnetic energies. This
is because the extraction of asymptotic ejecta velocities in our simulations
leads to overall slower speeds (see Sec. 3.2). We also find that our model
favors a low spectral index, $p\leq 2.1$, in agreement with Hajela et al.
(2022). Finally, we note that at this early stage of what may represent excess
emission, the gamma-ray burst afterglow may still be contributing in part to
the observed light curve (not included in the emission model here), and thus
the derived microphysical parameters would likely need further adjustment if
the gamma-ray burst afterglow component was included self-consistently in this
analysis.

####  5.5.4 High-energy $\gamma$-ray emission

Given the mildly relativistic nature of the shock that the ejecta drives into
the ISM with Lorentz factor $\Gamma_{\rm sh}\approx 2$ (cf. Fig. 20), and the
low-density environment of photons and baryonic matter, gamma-ray emission
from inverse Compton and synchrotron self-Compton processes in these types of
afterglows is not expected to be detectable unless the merger event occurs
sufficiently nearby and/or the efficiency $\epsilon_{e}$ of converting kinetic
energy of the shock into accelerating electrons is very high (Takami et al.,
2014).

However, if the merger occurs within a medium of high radiation density of
soft photons $u_{\rm rad}\sim 10-10^{3}$ eV cm-3, such as in globular clusters
(Venter et al., 2009), for example, there may be noticeable emission of gamma-
rays generated by inverse Compton scattering. For $u_{\rm rad}\sim\\!10^{3}$
eV cm-3, we find fluxes of $\sim$ TeV gamma-rays of $\nu F_{\nu}<10^{-14}$ erg
s-1 cm-2 for $D_{L}>40$ Mpc. This still is too faint to be detected with the
Cherenkov Telescope Array (CTA) or $10$ years of Fermi Large Area Telescope
(LAT) data (Atwood et al., 2009). However, even in the absence of direct
detection of gamma-rays, inverse Compton processes may shape the observed non-
thermal afterglow emission by helping cool electrons at later times. We find
that the resulting decrease in X-ray emission is significant if $u_{\rm
rad}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}10^{2}$ eV cm-3 at timescales of
$t\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}1$ year.

Furthermore, at early times ($<\\!10-15$ d post-merger) the ejecta shock is
accompanied by thermal radiation from the kilonova with radiation density

$\displaystyle u_{\rm rad}$ $\displaystyle=$ $\displaystyle\frac{L_{\rm
bol}}{4\pi R^{2}c}\frac{1}{\Gamma_{\rm sh}^{2}}$
$\displaystyle\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}$ $\displaystyle 8.7\times
10^{5}\frac{\text{eV}}{\text{cm}^{3}}\,\left(\frac{L_{\rm bol}}{6\times
10^{40}\text{erg\,s}^{-1}}\right)\left(\frac{t}{10\,{\rm
d}}\right)^{-3.33}\mskip-40.0mu.$

Here, we have scaled to the bolometric luminosity of the GW170817 kilonova of
$\approx\\!10^{42}$ erg s-1 at $0.5$ d (Drout et al., 2017), rescaled
according to $\propto t^{-0.85}$ up to 5.5 d and $\propto t^{-1.33}$
thereafter (Drout et al., 2017), and have used a typical shock Lorentz factor
of $\Gamma_{\rm sh}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}2$ (cf. Fig. 20). In a similar manner as
in the case of the gamma-ray burst afterglow shock (Linial & Sari, 2019),
electrons in the kilonova ejecta shock interact with the low-energy kilonova
photon field and cool by inverse Compton scattering. We find that the X-ray
radiation of the ejecta shock may decrease by tens of percent at very early
times ($<10-15$ d), provided strong kilonova afterglow emission due to
sufficiently fast ejecta.

##  6 Conclusions

We have presented a comprehensive model of electromagnetic signatures arising
from the dynamical ejecta in BNS mergers. For this purpose, we performed GRMHD
simulations that incorporated, for the first time, tabulated three-parameter
nuclear EOS, magnetic fields, and approximate neutrino transport via a ray-by-
ray transport scheme. We provide a detailed analysis of the characteristics of
dynamical ejecta and its ejection mechanisms. We employ the dynamical ejecta
profiles extracted from these simulations, characteristics of the dynamical
ejecta as recorded by several families of tracer particles, and results of
detailed nucleosynthesis calculations with the nuclear network SkyNet to
compute the neutron-precursor thermal transient, kilonova emission, as well as
the broad-band non-thermal kilonova afterglow. The modeling of electromagnetic
counterparts presented here is directly based on ab-initio numerical
simulations of the BNS merger process and provides a comprehensive picture of
observable phenomena related to dynamical ejecta in a self-consistent
framework.

Our main results and conclusions can be summarized as follows:

  * •

We performed a detailed analysis of the dynamical ejecta properties and
ejection mechanisms of 1.35–1.35 $M_{\odot}$ binaries typical of the
distribution of Galactic BNS systems and consistent with the inferred system
parameters of GW170817 using three nuclear EOSs that span the range of NS
radii as allowed by current observational constraints (Fig. 2; Tab. 1). We
showed that matter is unbound during the decompression phase in quasi-radial
bounces of the remnant NS double-core structure. We find that for the soft EOS
cases, $\sim\\!75\%$ of dynamical ejecta is unbound in the first three
bounces, while for the stiffer $\mathtt{LS220}$ case, material is unbound more
secularly in a combination of torques and shock waves sourced by somewhat
weaker bounces.

  * •

All binaries generate a fast tail of material ($v>0.6c$) that can be traced
back to ejection during the first two bounces of the double-core structure
(Fig. 5). While the amount of such fast material (few $10^{-6}\,{M_{\odot}}$,
Tab. 1) is in good agreement with some grid-based GRHD simulations of ultra-
high resolution (Kiuchi et al., 2017; Hotokezaka et al., 2018), differences to
other grid-based GRHD simulations (Nedora et al., 2021a) can be understood in
terms of the criteria to extract unbounded mass and to compute the asymptotic
velocities (see Sec. 3.2). We conjecture that based on these results and
convergence studies (Dean et al., 2021), the amount of fast ejecta should be
roughly converged at our grid resolution and atmosphere level. Remaining
differences to Newtonian setups and SPH simulations (Dean et al., 2021;
Kullmann et al., 2022; Rosswog et al., 2022) remain an open question, but may
likely be attributable to differences in the overall setup, such as Newtonian
dynamics, velocity extraction, geometric effects due to assumed symmetries,
and not sufficiently converged ejecta tails in SPH simulations with
$\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}\\!\text{tens}$ of particles (see the
discussion in Sec. 3.2).

Although our present GRMHD setup includes neutrino absorption and employs a
lower atmosphere floor than previous grid-based GRHD simulations with
tabulated EOS, we are still not sensitive to a
$\sim\\!10^{-8}-10^{-7}M_{\odot}$ ultra-relativistic ejecta envelope as
suggested by Beloborodov et al. (2020), who employ this envelope as a breakout
medium for a relativistic jet to explain the prompt gamma-ray emission of GRB
170817A associated with the BNS merger event GW170817.

  * •

We showed that for soft EOSs, the second quasi-radial bounce after merger
produces a shock wave that reprocesses the cold, previously ejected neutron-
rich tidal tail material, resulting in an asymmetric, azimuthal pattern in
$Y_{\rm e}$, giving rise to ‘pockets’ of neutron-rich material while other
angular parts are shock-heated to higher $Y_{\rm e}$ values (Sec. 3.3, Figs. 4
and 8). For stiffer EOS, however, the action of multiple weaker shock waves
generates a more homogeneous $Y_{\rm e}$ pattern as a function of azimuthal
angle. Once r-process nucleosynthesis proceeds the low-$Y_{\rm e}$ regions in
the soft EOS cases translate into high-opacity ‘lanthanide/actinide’ pockets.
This is illustrated in Fig. 17.

  * •

Future work will need to explore the observational consequences of the
azimuthal opacity effect (Fig. 17) with radiative transfer calculations of the
early kilonova emission. If preserved to large radii, these “pockets” could
also significantly impact the nebular phase of the kilonova once the ejecta
becomes transparent at late times. 11Indeed, the radial velocities of these
pockets are larger than their azimuthal velocities by a factor of at least
${\sim}3-5$, translating into angular dispersion of $\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}(10-20)^{\circ}$ (see
Eq. (32)). This indicates that these structures could be preserved to large
distances and thus both to the early photospheric ($\sim$days) and late-time
nebular ($\sim$weeks) emission phases.. The dominant nebular lines are
determined by the elements that dominate the cooling (Hotokezaka et al.,
2021), i.e. lanthanide/actinide-rich pockets may produce different nebular
emission relative to other regions with equal mass but less lanthanide
content. Such nebular composition diagnostics may be performed with the James
Webb Space Telescope. 11This may provide an additional avenue to infer the
softness/stiffness of the EOS.

  * •

Our setup of several families of passive tracer particles injected into the
simulation (Sec. 2.5.2) allows us to track and characterize different ejecta
components and to perform detailed nucleosynthesis calculations of the
dynamical ejecta with the nuclear reaction network SkyNet (Sec. 4). We obtain
a mass-averaged abundance pattern in good agreement with solar abundances for
$A>80$, including, in particular, a robust pattern between and including the
2nd and 3rd r-process peak irrespective of the EOS (cf. Fig. 10). An
underproduction of first-peak elements relative to solar for $A<80$ evident
here is expected due to limited possibilities to increase $Y_{\rm e}$ in
dynamical ejecta via neutrino absorption and emission. Softer EOSs lead to
slightly higher abundance in the $A=50-80$ region due to somewhat stronger
shock heating and the resulting increase in $Y_{\rm e}$.

Directly post-processing with the nuclear reaction network, we find that in
all simulations a typical amount of $\approx\\!2\times 10^{-5}\,M_{\odot}$ of
the fast-ejecta tail leads to free neutrons (cf. Fig. 12; Tab. 1) whose decay
provides early heating on the free-neutron decay timescale of $\approx\\!900$
s in excess to that of the r-process (cf. Fig. 11). We find that although
stiffer EOS tend to generate less fast dynamical ejecta, a similar amount of
free neutrons can be generated due to relatively more neutron-rich outflows.

  * •

We present a novel kilonova model that incorporates a detailed method for
thermalization of radioactive decay products according to Hotokezaka & Nakar
(2020), heating due to free-neutron decay, and relativistic effects to
reconstruct the observer lightcurve (Sec. 5.1).

  * •

We find that the combination of free-neutron heating and relativistic effects
results in an enhancement of the kilonova emission by a factor $10-20$ at
$\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}\\!1$ h for all EOS cases (cf. Fig. 14),
with relativistic effects leading to noticeable enhancement up to
$\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}\\!10$ h. A moderate shift in the peak
timescale of the precursor emission due to relativistic effects to values
closer to the neutron decay timescale of $\approx\\!0.3$ h complicates
confusion with possible early cocoon heating (Gottlieb et al., 2018; Kasliwal
et al., 2017). We find that this modeling of early kilonova and neutron-
precursor emission based on ab-initio numerical simulations is compatible with
observations of GW170817 in that i) the main blue emission component of the
GW170817 kilonova cannot be explained with dynamical ejecta only as found here
and likely requires substantially more ejecta from the post-merger phase and
ii) earlier observations in UV and optical bands would likely have detected
the precursor emission (cf. Fig. 15). The boost in UV/optical brightness by a
factor of a few due to relativistic effects found here provides promising
prospects for detection of the neutron precursor following future merger
events with UV telescopes such as _Swift_ or with planned missions such as the
wide-field ULTRASAT satellite (Sagiv et al., 2014). Based on our precursor
modeling, we estimate a detection horizon of $\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}\\!250$ Mpc for
ULTRASAT.

  * •

The ejecta structure in our simulations suggests that in typical merger
systems such as those investigated here an opacity boost from highly ionized
lanthanide nuclei arising at temperatures $\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}\\!70000$ K as
recently reported by Banerjee et al. (2022) is unlikely to affect the early
UV/optical emission on timescales $\approx\\!1$ h (Sec. 5.3, Fig. 16).

  * •

Finally, we present a novel model to compute the non-thermal kilonova
afterglow emission generated by a shock wave that the fast dynamical ejecta
runs into the interstellar medium (Sec. 5.5). We numerically solve the 1D
hydrodynamics of a refreshed shock propagating into the interstellar medium
and compute the synchrotron and inverse-Compton radiation by numerically
solving the corresponding Fokker-Planck equation. We find that for typical
shock microphysics parameters and typical densities of the interstellar medium
inferred for GW170817, our model based on ab-initio numerical simulations is
compatible with the recent data by Hajela et al. (2022), who report tentative
evidence for a rebrightening of the broad-band GW170817 GRB afterglow emission
emerging at a timescale of $10^{3}$ days post-merger (Fig. 21). The presence
of this rebrightening, however, is still being debated (Balasubramanian et
al., 2021; O’Connor & Troja, 2022).

  * •

High-energy gamma-ray emission arising from inverse-Compton and synchrotron
self-Compton processes associated with the kilonova afterglow is unlikely
directly observable in future merger events. However, we find that in the
presence of intense photon fields (e.g., the kilonova at early times and/or in
environments such as globular clusters) indirect consequences such as
additional cooling of electrons leave observable imprints in the X-ray
radiation (leading to a decrease by $\sim\\!\text{tens}\,\%$ relative to the
radio band).

We thank Vassilios Mewes, Federico Lopez Armengol, Michael Müller, Aman
Agarwal, Dhruv Desai, Xinyu Li, Rodrigo Fernández, Brian Metzger, and Santiago
del Palacio for fruitful discussions and comments. We thank Jesús Rueda-
Becerril and Geoffrey Ryan for discussions and for providing help with Paramo
and afterglowpy. This research was enabled in part by support provided by
SciNet (www.scinethpc.ca) and Compute Canada (www.computecanada.ca). DMS
acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), funding reference number RGPIN-2019-04684. LC is a
CITA National fellow and acknowledges the support by the Natural Sciences and
Engineering Research Council of Canada (NSERC), funding reference
DIS-2022-568580. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.

##  Appendix A Convergence of ejecta

In BNS merger simulations it is challenging to obtain a satisfactory
convergence level of the total amount of dynamically ejected mass at finite
(typical) numerical resolutions employed. This is in part due to non-linear
effects involved in the ejection process (see, e.g., Radice et al. 2018) and
the requirement for and treatment of an artificial atmosphere floor in grid-
based simulations; another role is played by the methods of measuring the
amount of unbound ejecta in a given simulation due to finite numerical
resolution available for the numerical extraction techniques and ambiguities
in unboundedness criteria. In this appendix, we explore how the properties of
the dynamical ejecta change with resolution and other parameters such as the
extraction radius of outflow detectors. We focus on the $\mathtt{SFHo}$ case
and we compare results from simulations with a grid spacing of $\Delta
x=180,220,250$ m, which we label as high, medium, and low resolution,
respectively.

Figure 22: Convergence of total ejecta mass as a function of resolution and
extraction radius. Left: Cumulative unbound ejecta mass as computed from the
mass flux through a spherical detector surface at $\approx\\!300$ km using the
SFHo EOS simulations at different resolutions. Right: Cumulative unbound
ejecta mass according to the geodesic criterion (thick lines) and the
Bernoulli criterion (dashed lines) for $\mathtt{SFHo}$ at high resolution
($\Delta x=180$ m) and different detector radii.

The left panel of Fig. 22 shows a convergence test regarding the total amount
of dynamically ejected mass using the geodesic criterion at a detector
distance of 300 km. We observe convergence at the 10% level at the highest
resolution employed. A comparison of the corresponding detailed ejecta mass
distributions of the high and medium resolution runs according to asymptotic
velocity, electron fraction, and specific entropy are shown in Fig. 23. Except
for moderate differences in the $Y_{\rm e}$ distribution around $Y_{\rm
e}\approx 0.25$, which we attribute to better or worse resolved shock heating,
we find good agreement in the detailed ejecta distributions. 11The value
$Y_{\rm e}\approx 0.25$ being a threshold for lanthanide and actinide
production (e.g., Lippuner & Roberts 2015), these results emphasize the
importance of well-resolved shock dynamics during the merger process.

Finally, we compare the amount of unbound mass according to the Bernoulli and
geodesic criteria for different detector radii based on the high-resolution
run in the right panel of Fig. 22. As discussed in Sec. 3.2, the unbound mass
according to the geodesic criterion saturates at earlier times, because it
only captures the relatively faster fluid elements that are dynamically
unbound during the merger phase, while the Bernoulli criterion additionally
captures slower material of high specific enthalpy unbound through winds
during the post-merger phase, giving rise to a steadily growing unbound ejecta
mass in the early post-merger phase. Early on, both criteria lead to identical
ejecta masses, highlighting the dynamical nature of the underlying ejection
process. We find almost indistinguishable variations in the total ejecta mass
for detectors at radii larger than $\approx\\!450$ km using both the Bernoulli
and the geodesic criterion, once accounting for a shift in arrival times of
the outflow at the various detector locations.

Figure 23: Convergence of detailed dynamical ejecta distributions in terms of
asymptotic velocity (left), electron fraction (center), and specific entropy
(right) based on the SFHo EOS simulations at different resolutions.

##  Appendix B Tracer sampling

In Fig. 24, we compare the distributions of dynamical ejecta mass extracted
from our $\mathtt{SFHo}$ simulation using tracer particles (Sec. 2.5.2) and
computing the unbound mass flux through a spherical detector surface (Sec.
2.5.1). Overall, we find excellent agreement, specifically in thermodynamic
quantities such as electron fraction and specific entropy, which is key for
accurate nucleosynthesis calculations. We find a moderate oversampling of
high-velocity ejecta using the tracer approach, which is due to our method of
placing a large number of tracers in the high-entropy, under-dense regions of
shock heated ejecta during the merger (Sec. 2.5.2). We thus discard tracers
below a certain mass-threshold as ‘noise’ and exclude them from further
analyses. In doing so, we improve the velocity distributions shown in Fig. 24
and obtain an overall good agreement and sampling of ejecta according to
asymptotic velocity, with still more than 500 tracers resolving the high-
velocity tail $v_{\infty}>0.6c$. Other distributions, such as those in $Y_{\rm
e}$ are insensitive to such oversampling (compare the second and third panel
of Fig. 24).

Figure 24: Mass distribution of electron fraction for dynamical ejecta
material in our $\mathtt{SFHo}$ run, measured separately at $440$ km by
analyzing the mass flux on the Cartesian grid using a spherical detector
surface (‘Outflow’; Sec. 2.5.1) and by analyzing the properties recorded by
passive tracers advected with the flow (‘Tracers’; Sec. 2.5.2). The total
number of tracer particles with scale on the right is shown as dot-dashed
black lines. There is moderate oversampling of high-velocity ejecta (first
panel), which we fix in subsequent analyses by excluding low-mass tracers that
we consider sampling ‘noise’. Other distributions that are key to
nucleosynthesis analyses, such as $Y_{\rm e}$ and specific entropy, show
excellent agreement and are not affected by such oversampling, regardless of
whether the full (third panel) or reduced (second panel) set of tracers is
used. See the text for details.

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# RGB-D Neural Radiance Fields: Local Sampling for Faster Training

Arnab Dey
I3S-CNRS/Université Cote d’Azur
Sophia-Antipolis, France
<EMAIL_ADDRESS>
Andrew I. Comport
I3S-CNRS/Université Cote d’Azur
Sophia-Antipolis, France
<EMAIL_ADDRESS>

###### Abstract

Learning a 3D representation of a scene has been a challenging problem for
decades in computer vision. Recent advances in implicit neural representation
from images using neural radiance fields(NeRF) have shown promising results.
Some of the limitations of previous NeRF based methods include longer training
time, and inaccurate underlying geometry. The proposed method takes advantage
of RGB-D data to reduce training time by leveraging depth sensing to improve
local sampling. This paper proposes a depth-guided local sampling strategy and
a smaller neural network architecture to achieve faster training time without
compromising quality.

Figure 1: The RGB-D NeRF training process. (a) Depth guided sampling. (b) The
input to the network is 5D coordinate. (c) The network outputs volume density
and color for each sample. (d) The color and depth of a ray generated using
classic volume rendering. (e) The network is optimized using color and depth
loss.

##  1 Introduction and related work

Learning the 3D representation(shape and texture) of a scene is important for
novel view synthesis, 3D human modeling, virtual reality, etc. Recent
advancement in neural scene representations, more specifically NeRFMildenhall
et al. (2020), showed that neural networks can be used to encode high-quality
images of 3D scenes. NeRF-based methods use two multilayer perceptrons(coarse
and fine) to learn radiance and volume density from RGB images and their
corresponding camera poses.

Although NeRF-like methods can produce high-quality novel views of scenes,
they are too expensive to train. NeRF uses a classic volume rendering
technique to compute the color of the pixels by placing 256 samples along each
viewing ray. Each of those samples need a full network evaluation to compute a
color. Recently, Neff et al. (2021) proposed real-time rendering by limiting
the number of samples. They used an oracle network and ground-truth depth to
predict relevant sampling locations on rays. Their method is limited to
forward-facing scenes and poses belonging to a view cell. Alternatively, Deng
et al. (2021) uses sparse depth supervision generated by a Structure-from-
motion(SfM) algorithm to optimize the network using color and depth
information together, allowing them to use fewer input views. Sucar et al.
(2021) achieved real-time SLAM based on NeRF by using a smaller network, lower
resolution inputs and removed the viewing direction. The method proposed here
uses local sampling based on a depth sensor to reduce the number of samples
and replaces the coarse NeRF network. This study aims to prove that faster
training time can be achieved by local sampling without limiting scene
representation quality while using a single network.

##  2 Local sampling

NeRF-like methods estimate pixel color using classic volume rendering, alpha
composition Mildenhall et al. (2020) of sample volume density, and color to
generate the final rendering. Samples with a higher volume density have a
greater contribution to the final color. The proposed local sampling places
fewer samples only on the relevant part of the rays given depth information.

Figure 2: Visualization of three different sampling strategies. Black lines
represent rays coming from camera and circles are samples. Figure 3:
Qualitative results from simulated data. (a) Ground truth image;(b) Predicted
image; (c) Ground truth depth; (d) Predicted depth.

###  2.1 Stratified sampling

This approach is very similar to NeRFMildenhall et al. (2020) sampling, the
only difference is that the near and far bounds of the sampling are set using
depth information.

###  2.2 Gaussian sampling

Instead of placing the samples in a stratified manner, a Gaussian distribution
is used to distribute sample locations around the surface. The mean of the
distribution is the depth measurement, which ensures more samples are placed
close to the surface.

###  2.3 Adaptive sampling

A multiview depth error map $\\{e^{i}\\}_{i=1}^{N}$ is generated using all
depth maps of the training set (see the poster for the definition). The
standard deviation of the Gaussian distribution is computed from
$\\{e^{i}\\}_{i=1}^{N}$. It ensures that the spread of the samples are greater
when there is more uncertainty in the depth.

##  3 Preliminary Results

| Metrics
---|---
dataset |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  Abs Rel$\downarrow$ |  LPIPS$\downarrow$
Lego |  27.4 |  0.933 |  0.012 |  0.0009
Cube |  37.76 |  0.95 |  0.005 |  0.0001
Drums |  29.66 |  0.91 |  0.004 |  0.0008
Human |  38.83 |  0.98 |  0.003 |  0.00006
Table 1: The results of proposed method tested on 4 different simulated datasets. Underlying geometry is evaluated by Absolute relative distance(AbsRel). Photometric quality evaluated by PSNR(peak signal to noise ratio), SSIM(structural similarity index), and LPIPS(Learned Perceptual Image Patch Similarity). | Metrics
---|---
Strategy |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  AbsRel$\downarrow$ |  LPIPS$\downarrow$ |  Time$\downarrow$
Stratified |  21.81 |  0.891 |  0.003 |  0.002 |  30m
Gaussian |  24.17 |  0.912 |  0.017 |  0.002 |  22m
Adaptive |  23.40 |  0.910 |  0.018 |  0.002 |  22m
NeRF |  22.3 |  0.84 |  0.215 |  0.002 |  1h 42m
Table 2: The proposed local sampling strategies compared with baseline NeRF.
The dataset contains 8 training images. Experiments were performed with 16
sample points.

Qualitative results are shown in Table 1 and 2 . The Figure 3 shows
qualitative results of the proposed local sampling. The best method according
to the preliminary result is Gaussian sampling.

##  4 Conclusions

A preliminary study has been presented that shows that depth images can be
used to perform local sampling and that fewer samples can reduce training time
without compromising quality. The results suggest that surface information
about the scenes can provide additional supervision to achieve better
underlying geometry and photometry from fewer input views.

##  5 Acknowledgements

The project is funded by Horizon 2020 COFUND BoostUrCAreer under the Marie
Sklodowska-Curie grant agreement no. 847581 supported by the EU and the Region
Sud-Provence-Alpes-Cote d’Azur and IDEX ${UCA}^{{JEDI}}$. This work was
granted access to the HPC resources of IDRIS under the allocation
2021-AD011012578 made by GENCI.

## References

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