# Observational Signatures of Carbon-Oxygen White Dwarf Merger Remnants

Philippe Z. Yao ,1 Eliot Quataert ,1 Andy Goulding 1
1Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA E-mail<EMAIL_ADDRESS>0000-0003-3024-7218 0000-0001-9185-5044 0000-0003-4700-663X

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Many double white dwarf (WD) mergers likely do not lead to a prompt
thermonuclear explosion. We investigate the prospects for observationally
detecting the surviving remnants of such mergers, focusing on the case of
mergers of double Carbon-Oxygen WDs. For $\sim 10^{4}$ yr, the merger remnant
is observationally similar to an extreme AGB star evolving to become a massive
WD. Identifying merger remnants is thus easiest in galaxies with high stellar
masses (high WD merger rate) and low star formation rates (low birth rate of
$\sim 6-10\,{\rm M_{\odot}}$ stars). Photometrically identifying merger
remnants is challenging even in these cases because the merger remnants appear
similar to He stars and post-outburst classical novae. We propose that the
most promising technique for discovering WD merger remnants is through their
unusual surrounding photoionized nebulae. We use CLOUDY photoionization
calculations to investigate their unique spectral features. Merger remnants
should produce weak hydrogen lines and strong carbon and oxygen recombination
and fine-structure lines in the UV, optical and IR. With narrow-band imaging
or integral field spectrographs, we predict that multiple candidates are
detectable in the bulge of M31, the outskirts of M87 and other nearby massive
galaxies, and the Milky Way. Our models roughly reproduce the WISE nebula
surrounding the Galactic WD merger candidate IRAS 00500+6713; we predict
detectable [Ne VI] and [Mg VII] lines with JWST but that the mid-IR WISE
emission is dominated by dust not fine-structure lines.

###### keywords:

white dwarfs – supernovae: general – planetary nebulae: general – stars:
evolution

††pubyear: 2015††pagerange: Observational Signatures of Carbon-Oxygen White
Dwarf Merger Remnants–Observational Signatures of Carbon-Oxygen White Dwarf
Merger Remnants

##  1 Introduction

The merger of two white dwarfs (WDs) can result in a wide range of outcomes
depending on the total mass, mass ratio, and chemical composition of the
system. Although some double WD mergers likely lead to a thermonuclear
explosion and a Type Ia supernova (Webbink, 1984; Iben & Tutukov, 1984), many
probably do not and instead leave behind a long-lived remnant. If the total
mass of the merger remnant remains above the Chandrasekhar mass, the merger
remnant ultimately develops an iron core and collapses to form a neutron star
(Schwab et al., 2016). Lower mass merger remnants can evolve to become a
cooling WD, an R Coronae Borealis star, or a hot subdwarf (Hollands et al.,
2020; Staff et al., 2012; Schwab, 2018, respectively). In this paper, we will
focus on the merger of two C/O WDs or a C/O and O/Ne/Mg WD, which will not
result in R Coronae Borealis stars or hot subdwarfs. In WD merger models, the
former is thought to originate from He+C/O WD mergers (e.g. Webbink, 1984;
Saio & Jeffery, 2002; Munson et al., 2021), and the latter from double He WD
mergers (e.g. Webbink, 1984; Iben & Tutukov, 1986).

The timescale for the merger remnant to evolve towards its final fate is $\sim
10^{3-4}$ yr, set by the thermal time needed to radiate away the excess
thermal energy created during the merger (Shen et al., 2012). During this
thermal timescale evolution, the remnant initially evolves to become a giant
and then contracts and heats up to become a hot proto-WD (Shen et al., 2012;
Schwab et al., 2016; Schwab, 2021). The resulting evolution is similar to that
of a $\sim 6-10{\rm M_{\odot}}$ star as it evolves from the AGB to become a
cooling WD. To highlight this similarity, Figure 1 shows the post-merger
evolution of double WD merger remnants predicted by Schwab (2021) compared to
the evolution of $1,3$, and $6{\rm M_{\odot}}$ stars (we also show post
outburst classical novae for reasons discussed later in the paper). There is
considerable overlap in the luminosity and effective temperatures of the WD
merger remnants and the 6 $M_{\odot}$ star (the similarity would be even
stronger for stars with $7-10{\rm M_{\odot}}$, but those models are very
difficult to evolve to late stages because of off-center nuclear burning;
Jones et al. 2013).

Figure 1: C/O WD merger remnants and potential stellar contaminants in the H-R
diagram. The left panel shows models from Schwab (2021) with fixed total mass
($1.5{\rm M_{\odot}}$) and different mass ratios q. The right panel shows
models from Schwab (2021) with fixed mass ratios (q = 0.9) and different total
masses. The orange line is the same in both panels. MESA stellar tracks of
typical post-main-sequence stars evolving into cooling WDs with a range of
masses from Farag et al. (2020) are plotted with dotted lines. The grey solid
line shows the trajectory of post-outburst classical novae from Wolf et al.
(2013); markers are separated by equal time portions of 1 year and 30 years
for $1{\rm M_{\odot}}$ and $0.6{\rm M_{\odot}}$ progenitors respectively.
Post-outburst novae and $\sim 6-10{\rm M_{\odot}}$ post-AGB stars are similar
in $L-T_{\rm eff}$ space to WD merger remnants.

As shown in Figure 1, theoretical models predict that surviving WD merger
remnants should evolve through a giant phase to become a hot, bright proto-WD
(Shen et al., 2012; Schwab et al., 2016; Schwab, 2021). Mass loss during the
giant phase would likely produce a H- and He-free nebula analogous to a
standard planetary nebula (PN) around young WDs. The central object might be
dust-obscured during some of this evolution, particularly because of the
carbon-enriched composition of the mass lost during the giant phase.
Gvaramadze et al. (2019) reported the first detection of a possible WD merger
remnant (IRAS 00500+6713) matching the above criteria: a H-free nebula
detected in WISE surrounding a H- and He-free star with a surface temperature
around $2\times 10^{5}$K and luminosity around $10^{4.5}L_{\odot}$. Gvaramadze
et al. (2019) showed that IRAS 00500+6713 also had a remarkably high-speed
wind suggestive of a strongly magnetized and rapidly rotating central object.
Simulations of WD mergers indeed find that they can produce very strongly
magnetized remnants (Ji et al., 2013). Ritter et al. (2021) later found high
speed $\sim 1100$ km/s expansion of the Pa 30 nebula from the [S II] doublet.
Strong mid-IR (Gvaramadze et al., 2019) and X-ray emissions (Oskinova et al.,
2020), moderate [O III] 5007 Å emissions, weak [Ar III] 7136 Å features, and
filamentary [S II] 6716 and 6731 Å features (Ritter et al., 2021; Fesen et
al., 2023) from this nebula are also observed.

The first technique that we consider is whether merger candidates can be
photometrically identified given their predicted luminosity and effective
temperatures. This turns out to be challenging. The second technique we
consider is more promising: merger candidates can be spectroscopically
identified by their unusual H-free photoionized nebulae. This is analogous to
how the dusty circumstellar nebulae around R Coronae Borealis stars have
proven valuable at elucidating the nature of these enigmatic stars (De Marco
et al., 2002; Clayton et al., 2011a, b; Montiel et al., 2018). In the case of
more massive C/O merger remnants, however, we propose to use the surrounding
nebula to infer the presence of a central merger remnant and we focus on the
much hotter phase of the central remnant’s evolution when the surrounding
nebula is analogous to a photoionized planetary nebula.

To orient the reader, we note that the number of active C/O-C/O merger
remnants in a galaxy can be estimated as follows:

${\rm N_{active}}\approx 10\times\left(\frac{\dot{N}}{10^{-14}{\rm
yr^{-1}}{\rm M_{\odot}}^{-1}}\frac{M_{\star}}{10^{11}{\rm
M_{\odot}}}\frac{t_{\rm remnant}}{\rm 10^{4}years}\right),$ (1)

where $\dot{N}$ is the WD merger rate, and $t_{\rm remnant}$ is the lifetime
of a merger remnant. Assuming the super-Chandrasekhar double WD merger rate to
be $1.0^{+1.6}_{-0.6}\times 10^{-14}{\rm yr^{-1}}{\rm M_{\odot}}^{-1}$
(Badenes & Maoz, 2012), the number of white dwarf merger remnants in the
‘thermal’ phase of evolution highlighted in Figure 1 is around 20 in the Milky
Way or M31, and around 100 in M87. The total WD merger rate is probably $\sim
5-10\times$ higher than the super-Chandrasekhar rate, though it remains
uncertain what fraction of those mergers will explode as type Ia supernovae.

One general challenge in identifying WD merger remnants highlighted in Figure
1 is that the remnants are similar in luminosity and effective temperature to
intermediate-mass stars evolving to become WDs. In the Milky Way, sufficiently
detailed observations may be possible to distinguish between these
possibilities (e.g., by environment or spectroscopy). In other galaxies, the
natural way to minimize the impact of the contaminants in Figure 1 is to focus
on galaxies with high stellar masses (high WD merger rates) and low star
formation rates (low birth rates of intermediate-mass stars), i.e., low
specific star formation rates. To be more quantitative, the birth rate of
intermediate-mass stars that are the most likely to overlap in $L,T_{\rm eff}$
space with WD merger remnants is $\sim 10^{-2}\,{\rm
yr^{-1}}\,(\dot{M}_{\star}/1{\rm{\rm M_{\odot}}\,yr^{-1}})$. Including sub-
Chandrasekhar mass systems the WD merger rate is $\sim 5\times 10^{-3}\,{\rm
yr^{-1}}\,(M_{\star}/10^{11}\,\rm{\rm M_{\odot}})$. Thus galaxies with
specific star formation rates $\dot{M}_{\star}/M_{\star}\ll 5\times 10^{-12}$
yr-1 are likely to have low contaminants from intermediate-mass stars. The
focus of our work will thus be on massive, quiescent galaxies such as M87 or
the bulge of M31.

The remainder of this paper is organized as follows. In §2 we present our
methods for constructing a composite stellar population for an old stellar
population with both single and binary stellar isochrones as well as WD merger
remnants. We also summarize our methods for calculating photo-ionized nebulae
surrounding WD merger remnants for comparison to planetary nebulae. In §3, we
summarize the main observational challenges in identifying merger remnants
based on photometric properties such as the luminosity and effective
temperature of the central star. In §4, we show that spectroscopic signatures
in the surrounding photoionized nebula are a more promising way of identifying
WD merger remnants. Finally, in §5 we summarize our results and discuss
optimal strategies for identifying the unique population of WD merger
remnants.

##  2 Methods

In §2.1 we describe how we construct synthetic color-magnitude diagrams (CMDs)
to assess the photometric detectability of WD merger remnants. In §2.2 we
discuss how we calculate the spectrum of the photoionized nebula surrounding
WD merger remnants.

###  2.1 Simulated Composite Color-Magnitude Diagram

In order to compare the theoretically predicted population of WD merger
remnants with the rest of a galaxy’s stellar population, we construct
synthetic CMDs. We do so separately for single star and binary star models
using the MESA Isochrones and Stellar Tracks
code111https://waps.cfa.harvard.edu/MIST/(MIST, Dotter, 2016; Choi et al.,
2016) and the Binary Population and Spectral Synthesis
code222https://bpass.auckland.ac.nz(BPASS, Eldridge & Stanway, 2009; Eldridge
et al., 2008; Eldridge et al., 2017), respectively.

MIST self-consistently computes stellar evolutionary tracks using Module for
Experiments in Stellar Astrophysics (MESA, Paxton et al., 2011; Paxton et al.,
2013, 2015, 2018, 2019), covering a large parameter space in age, mass, and
metallicity. We use the python-Flexible Stellar Population Synthesis
code333https://dfm.io/python-fsps/(python-fsps, Conroy et al., 2009; Conroy &
Gunn, 2010) with MIST to construct a composite CMD. We assume a Salpeter
initial mass function, because strong lensing suggests there is a significant
population of low-mass stars in elliptical galaxies (Leier et al., 2016).

The MESA models in Figure 1 show that the primary single star contaminants
come from the mass range between $6<$ M $\rm<10\,{\rm M_{\odot}}$. This
indicates that only single stars born in the past $\lesssim 300$ Myr will be
in a similar part of the HR diagram to our merger remnants. As a result, we
take isochrones with $3<$ log(age) $<9$ years with age bins of 0.5 in log.
This increases the efficiency of generating our composite CMDs by eliminating
the need to compute and plot numerous low-mass stars that are always fainter
than our models of interest. Then, at each age, we interpolate the luminosity,
effective temperature, fsps weights, and AB magnitudes at relevant wavebands
to a linear grid of $10^{6}$ points plus an additional linear grid of $10^{7}$
points at luminosity $L>10^{4}\rm L_{\odot}$ to cover the evolutionary track
with enough resolution, including where stellar evolution is relatively rapid.
To obtain the composite stellar population, we randomly sample from the
interpolated luminosity and effective temperature with the probability
distribution given by fsps. Since the contaminants here are from relatively
massive stars, only the recent star formation rate affects the size of this
population. Assuming a constant star formation rate in the past is thus a fine
approximation. Hence, at each age bin, we sample a total of $N=\Delta m/m_{\rm
mean}\simeq(\dot{M}_{\star}/0.1{\rm M_{\odot}})\times\Delta t$ stars, where
$m_{\rm mean}$ is the mean stellar mass of the IMF, which we set to 0.1${\rm
M_{\odot}}$ here.

It is well known that mass transfer and stellar mergers in binary systems can
make an older stellar population appear bluer than predicted by single stars
alone. This is believed to be the origin of the UV upturn in elliptical
galaxies (Yi & Yoon, 2004; Han et al., 2007). Binary evolution is thus likely
to introduce additional sources that appear similar to WD merger remnants on
top of those associated with single intermediate-mass stars. Since the MIST
isochrones do not evolve binary systems, we alternatively adopt BPASS models
(Eldridge & Stanway, 2009; Eldridge et al., 2008; Eldridge et al., 2017) to
quantify the impact of binarity on our model CMDs. With binary evolution,
stars with the same mass and age can be inherently different, so using python-
fsps is no longer an option for sampling a CMD as it only has access to the
integrated SSP spectra from BPASS. Therefore, we use the Hoki code to access
and sample BPASS models444https://heloises.github.io/hoki/ (Stevance et al.,
2020).

With Hoki, we start with a BPASS model and sample stellar populations in a
range of log ages $6<$ log(age [yr]) $<9$, leaving out low-mass stars like
before. For each value of luminosity and effective temperature in a $i\times
j$ grid, Hoki provides a weight $W_{i,j}$ that indicates the number of systems
formed in a single 1 million ${\rm M_{\odot}}$ population. As a result, the
probability of a system being present in a population is given by
$\mathcal{P}_{i,j}=W_{i,j}/\sum W_{i,j}$. As with MIST, we sample a total of
$N\simeq(\dot{M}_{\star}/0.1{\rm M_{\odot}})\times\Delta t$ number of stars
based on this probability to construct the composite stellar population. Each
sample includes a system’s luminosity and effective temperature on the grid,
but lacks information about the initial stellar mass of the system. We also
note that BPASS reports the properties of the primary and secondary stars
separately and independently; the connection between the two stars in a binary
system has not yet been implemented.

Figure 2: Simulated H-R diagram of a massive galaxy with a low star formation
rate and constant star formation history sampled from MIST (left) and BPASS
(right) isochrones. The luminosity of the WD merger remnants from Schwab et
al. (2016) are plotted in star-shaped points; the population is estimated to
be around 70 for a galaxy with the stellar mass of M87 ($\rm 6.8\times
10^{11}{\rm M_{\odot}}$ in Forte et al. 2012a). The color in the left panel
indicates the stars’ initial masses, but we do not have that information for
BPASS models, as systems with different masses can appear in the same part of
the HR diagram due to binary interaction.

###  2.2 Simulated Nebula Spectrum

Mass loss may play an important role in the thermal evolution of the merger
remnant, particularly during the giant phase when the surface gravity of the
star is low, reaching an escape velocity of $\rm\lesssim 60km/s$. Such ejected
mass could extend to $\rm\sim 2\times 10^{18}cm$ as the remnant evolves over
its $\sim 10^{4}$ years of thermal evolution. The winds are likely to be dusty
due to the remnants’ C/O-dominated composition. They may appear bright in the
infrared because of dust reprocessing (Schwab, 2021). Here we focus on using
the CLOUDY spectral synthesis code555https://gitlab.nublado.org/cloudy/cloudy
(v17.02; Ferland et al., 1998; Ferland et al., 2017) to calculate the spectrum
of the surrounding nebula accounting for photoionization and dust.

We use CLOUDY to perform a suite of photoionization simulations both for our
WD merger remnant with a C/O-dominated composition, and for normal planetary
nebulae with a solar composition. We assume a total ejecta mass $M_{\rm
ejecta}=0.1{\rm M_{\odot}}$ based on Schwab (2021), although we note that the
exact ejecta mass is uncertain (calculations with ejecta masses of $0.05{\rm
M_{\odot}}$ and $0.20{\rm M_{\odot}}$ were very similar to the models shown
here). We assume blackbody emission from the central source and consider a
range of ejecta sizes and stellar effective temperatures. Since the luminosity
of the merger remnant remains relatively constant between $\rm
10^{4}L_{\odot}$ and $\rm 3\times 10^{4}L_{\odot}$ (see Figure 1), we use the
same luminosity $\rm\sim 5\times 10^{37}ergs^{-1}$ in all our simulations.
Tests using $\sim 2\times$ higher luminosity produce similar results.

To obtain the abundance of elements for the WD merger remnant, we use a MESA
test suite which produces a $\rm 0.6{\rm M_{\odot}}$ carbon-oxygen WD, roughly
the expected mass of one WD in the binary system, from a $\rm 3{\rm
M_{\odot}}$ progenitor. We take the final mass fraction of elements in this
white dwarf as the composition of the ejecta, which includes C, N, O, and Ne,
and trace amounts of H and He. Additionally, we assume solar abundance for
elements heavier than neon, including Mg, Si, S, Cl, Ar, and Fe. Based on the
calculations of Schwab et al. (2016) (their Figure 10), we expect that this
assumption is valid throughout the majority of the thermal evolution of the
merger remnant; only at the latest stages, when a convectively bounded Ne
flame develops, does the surface abundance change significantly (at least
according to these spherically symmetric models).

Since the composition of the ejecta is C/O-rich, a large amount of dust is
expected to be produced during the giant phase when the photosphere is cool.
As a result, modeling the effect of dust in CLOUDY is crucial to understanding
the appearance of the merger remnant. Our default models with dust assume that
10% of the ejecta by mass is graphite grains. CLOUDY performs the heating and
cooling of dust grains self-consistently, and models the size distribution
using 10 bins. Such dust can survive outside the sublimation radius

$R_{\rm sub}=\left(\frac{L}{4\pi\sigma T_{\rm sub}^{4}}\right)^{1/2}$ (2)

where $R_{\rm sub}\simeq 10^{14}\rm cm$ when $T_{\rm sub}\rm\sim 1500K$. Since
$R_{\rm sub}$ is orders of magnitudes smaller than the outer radius of the
ejecta, dust grains will very likely be present in the ejecta.

##  3 Photometric Detectability of Merger Remnants

We begin by discussing the detectability of merger remnants in theoretical
$L-T_{\rm eff}$ space and then consider the more observationally relevant case
of optical and UV CMDs. We adopt properties of an M87-like elliptical galaxy
with $\rm SFR=0.01{\rm M_{\odot}}/yr$ (Davis et al., 2014). In between $3<$
log(age) $<9$ years, we sample a total of $10^{8}$ stars that follows a
Salpeter-like initial mass function. For BPASS, we sample the same number of
stars in each corresponding age bin. We use the stellar mass of M87 of $\rm
6.8\pm 1.1\times 10^{11}{\rm M_{\odot}}$ (Forte et al., 2012b) to estimate a
total of $70$ active super-Chandrasekhar remnants; we thus sample a total of
70 systems from Schwab et al. (2016)’s MESA model with even spacing for the
remnant’s age during its evolutionary track. We note that our results are
converged with respect to taking a wider age range for sampling the isochrones
because the older stars included by doing so only appear in the lower
luminosity parts of the HR diagram. We also reiterate that sub-Chandrasekhar
merger remnants are initially similar in luminosity and effective temperature
to super-Chandrasekhar merger remnants (Figure 1), which likely increases the
number of expected remnants relative to the 70 sampled here and shown in
Figures 2 and 3.

###  3.1 Theoretical Distinction in $L-T_{\rm eff}$ Space

Figure 2 shows mock H-R diagrams for our M87-like galaxy from MIST and BPASS.
In both cases, the general distribution of stars is very similar in the
$T_{\rm eff}\lesssim 10^{4.7}$ K part of the H-R diagrams, which represent the
main sequence and giant phases. On the MIST plot, we see a small number of
stars in the proto-WD/planetary nebula phase. These are the main sources of
contamination in $L-T_{\rm eff}$ space. However, the white dwarf merger
remnants are typically a few times more luminous and significantly more
abundant for a galaxy like M87. However, the number of standard single stars
in this portion of the HR diagram will scale up linearly with the star
formation rate. Hence, in a Milky Way-like star-forming galaxy with $\rm
SFR\sim 1{\rm M_{\odot}}/yr$, these contaminants will significantly outnumber
the white dwarf merger remnants (which will also be less numerous because of
the lower total stellar mass of the Milky Way). In this case, additional
information is needed to distinguish between the photometrically similar
objects, such as the spectroscopic features described in the next section. For
the bulge of the Milky Way or M31, on the other hand, the results in Figure 2
will remain qualitatively applicable, though the total number of WD mergers
will be smaller because of the lower total stellar mass of the system (eq. 1).

In the BPASS models in the right panel of Figure 2 there is a significant
population of stars on the hotter part of the H-R diagram, which is absent in
the single-star evolutionary tracks. These are stars whose hydrogen envelopes
have been stripped during binary evolution. The lower luminosity stars are
helium stars while the higher luminosity stars are primarily Wolf-Rayet stars.
The existence of this hotter population of stripped stars somewhat complicates
the identification of white dwarf merger remnants in L-$\rm T_{eff}$ space,
but the merger remnants still stand out at $T_{\rm eff}\gtrsim\rm 10^{5}K$.

###  3.2 Observational Distinguishability

Figure 3: CMDs derived from the H-R diagrams in Figure 2 for MIST (top) and
BPASS (bottom) isochrones. Stars (blue dots) are sampled from the isochrones
and WD merger remnants are from Schwab et al. (2016). The simulated plots are
in FUV (1565 Å) and NUV (2740 Å) filters (left), or the U (3640 Å) and V (5400
Å) filters (right). The dotted line highlights the observational limiting
sensitivity of The Ultraviolet Explorer (UVEX, orange) and the Hubble Space
Telescope’s ultraviolet and visible light imager (HST UVIS, brown) when
placing the galaxy at two different distances (16.7 & 0.78 Mpc). The bottom
panel highlights that existing and planned UV filters are not blue enough to
distinguish He stars (the blue ridge of BPASS points) and WD merger remnants.

We now consider how single & binary stars and WD merger remnants would appear
in observationally accessible ultraviolet (FUV & NUV) and optical (U & V)
bands. Python-fsps computes the AB magnitude of stars for a large number of
commonly used filters. We choose the FUV and NUV filters centered on 1565 Å
and 2740 Å, and the U and V filters centered on 3640 Å and 5400 Å
respectively. For the white dwarf merger remnant or stellar systems sampled
from BPASS, we assume blackbody emission to compute their AB magnitudes. Note
that for the WD merger remnants, this assumption neglects the impact of the
surrounding ejecta, which will be considered in the following section.

The CMDs from MIST and BPASS in both the optical and UV filters are plotted in
Figure 3. For the MIST CMDs, we notice that the merger remnants easily stand
out from other stars at later stages of evolution, particularly in the UV
bands. However, the limiting sensitivity of the most advanced near-term UV
instrument, the Ultraviolet Explorer (UVEX), is 25.8 mags (Kulkarni et al.,
2021). In this case, Figure 3 shows that merger remnants cannot be detected in
the UV at the distance of M87 (16.4 Mpc). A limiting sensitivity closer to 28
mags or, equivalently, observing a closer galaxy at a distance of 10.9 Mpc
such as NGC 1023, is required for WD merger remnants to be detectable in the
UV in the left part of the CMD. Likewise, the right top panel of Figure 3
suggests that deep HST imaging of a galaxy significantly closer than M87 (such
as Andromeda at 0.78 Mpc indicated by the lower dotted lines) might be
sufficient to detect some of the merger remnants at high effective
temperature.

The bottom panels of Figure 3 show the analogous UV and optical CMDs for BPASS
models. The He stars are now a major contaminant at essentially all accessible
UV or optical luminosity. The reason is simply that an FUV filter of 1565 Å is
still not short enough wavelength to be sensitive to the higher effective
temperature of the WD merger remnants relative to He stars. Overall, given
realistic binary stellar populations, Figure 3 strongly suggests that deep
photometric imaging is unlikely to be an effective route to identifying
candidate WD merger remnants even in the case of nearby galaxies such as the
bulge of M31.

A caveat in the analysis presented here is that Hoki provides only the
characteristics of primary or secondary stars separately, but there is no
information about which secondary and primary stars go together. Hence, all
the models plotted here are the primary stars of each binary system, which
corresponds to the hotter star. This is true for the CMDs in Figure 3 as well
as the BPASS models in the H-R diagram (Figure 2). Since the He stars are
likely much hotter than any main sequence companion, the UV CMD should be
accurate. However, it is possible that in the optical the secondary will
dominate the system’s luminosity in which case the BPASS optical CMD would
likely be more similar to the MIST CMDs. Even in this case, the WD merger
remnants are still very difficult to detect. Additionally, we will discuss how
dust will affect the transmitted stellar continuum, and hence the UV and
optical luminosity, in §4.2.

Figure 4: Output spectra from CLOUDY for C/O (left) and solar (right)
composition nebula, with the former representing our predictions for WD merger
remnants. The bottom panel highlights the ratio between the two spectra. We
fix the inner radii of the clouds here to be $\rm 10^{17}cm$, while varying
the effective temperature of the central star between $\rm\sim 2\times
10^{4}-2\times 10^{5}K$. The grey shaded region represents the blackbody
spectrum from the range of temperature in the legend. No dust is included in
these models; see Figure 7 for models with dust.

###  3.3 Novae as Contaminants

Classical novae can also appear in the same region of the H-R diagram as WD
merger remnants and are likely an even more significant contaminant than He
stars. Classical novae are binary systems hosting a WD accreting from a non-
degenerate stellar companion. The accretion leads to a thermonuclear runaway
on the surface of the WD which causes the WD to brighten to near Eddington and
then cool and contract over time (see Chomiuk et al. 2021 for a review). The
process repeats with a recurrence time that is a function of the mass of the
WD and the accretion rate.

Figure 1 shows this post-outburst nova cooling phase for $0.6$ and $\rm 1{\rm
M_{\odot}}$ WDs (using models from Wolf et al. 2013). The time between points
is 1 yr and 30 yr for the $\rm 1{\rm M_{\odot}}$ and $\rm 0.6{\rm M_{\odot}}$
models, respectively. Figure 1 shows that post-outburst novae appear very
similar to WD merger remnants. Moreover, the cooling timescale for the post-
outburst novae is sufficiently long that secular cooling is unlikely to
distinguish them from WD merger remnants in all cases. The most promising part
of parameter space for distinguishing post-outburst novae and WD merger
remnants photometrically is the hottest most luminous systems because the
post-outburst novae evolve on $\sim$ year timescales.

We can estimate the number of potential novae contaminants using the estimated
novae rate of $26\pm 5$ in the Milky Way (Kawash et al., 2022) and
$363^{+33}_{-45}$ in M87 (Shafter et al., 2017). The year- to decade-long
evolution timescale in Figure 1 for novae translates to $\sim 10-100$ years as
contaminants. Hence, there are likely $\sim 10^{4}-10^{5}$ post-outburst novae
in M87 and $\sim 10^{3}-10^{4}$ in the Milky Way, $\sim 10-100$ times more
abundant than WD merger remnants.

Despite the photometric resemblance between post-outburst novae and WD merger
remnants, the spectroscopic differences between them should be significant. In
particular, novae eject a $\sim 10^{-4-5}{\rm M_{\odot}}$ envelope per
eruption (with significant hydrogen), while WD merger remnants are likely
surrounded by $\sim 0.1{\rm M_{\odot}}$ of ejecta with a very different
composition. In the next section, we quantify the spectroscopic signatures of
the photoionized nebulae surrounding WD merger remnants.

##  4 Spectroscopic Signatures of WD Merger Remnants

###  4.1 CLOUDY spectra

We ran a 2D logarithmically spaced grid (20x10) of simulations across a range
of stellar effective temperatures and ejecta radii expected for the merger
remnants: from $\sim 5\times 10^{3}$K to $\sim 2\times 10^{5}$K, and inner
radii range between $10^{16}$cm and $10^{19}$cm. We consider both C/O and
solar compositions in order to compare merger remnant nebulae and standard
planetary nebulae. To begin the models do not include dust but we return to
models with dust in §4.2. It is worth noting that the first ionization energy
of oxygen is nearly identical to that of hydrogen so that the ionization of
oxygen and hydrogen set in at very similar stellar effective temperatures.

CLOUDY outputs the incident and transmitted spectrum as in Figure 4, allowing
us to identify spectral signatures of nebulae produced by WD merger remnants.
The shaded region in Figure 4 reflects the blackbody incident continuum for
the range of effective temperatures in the legend. The output continuum
emission at long wavelengths in Figure 4 is free-free emission from the photo-
ionized nebula. In the soft X-ray ($\sim 10^{-3}-10^{-2}$ microns), the C/O
remnants produce more prominent emission features. In the extreme UV ($\sim
10^{-2}-10^{-1}$ microns), the cloud is more opaque to incident radiation for
the C/O remnant until the central star becomes hotter than $\rm\sim 10^{5}K$.

Simulated CLOUDY spectra also enable us to take the ratio of line strengths
between individual spectral lines produced by the two photo-ionized clouds.
For all parameters, almost all hydrogen lines in the C/O-rich remnant are more
than three orders of magnitudes weaker than in the solar remnant. This is of
course due to the lack of hydrogen in the merger remnant ejecta; hence the
typical PNe $\rm H\alpha$ and $\rm H\beta$ line features should be extremely
weak or nonexistent in WD merger remnants.

Since our target objects are C/O-rich, we then focus on carbon and oxygen
emission lines in the output spectra. Planetary nebulae are known for bright
[O III] 5007 Å lines where oxygen is doubly ionized above stellar effective
temperatures of 30,000 K. Hence, this line is often used in planetary nebula
surveys. The significantly larger abundance of oxygen in the merger remnant
does not necessarily guarantee a much brighter [O III] 5007 Å line. Unlike
standard planetary nebulae in which oxygen is the dominant coolant, the WD
merger remnants effectively have much higher metallicity, allowing the ejecta
to cool more rapidly. Figure 5 shows that the C/O dominated nebula has
significantly lower electron temperatures than solar composition nebula in
almost the whole parameter space we have explored. Since collisionally excited
lines are weaker at lower temperatures, even though the WD merger remnant is
rich in oxygen, the [O III] 5007 Å line can be similar in strength, or even
weaker than, in a typical solar composition PNe. This is shown explicitly in
Figure 6 which compares the strength of the [O III] 5007 Å line, the C IV 7727
Å line, and the [O IV] 25.9 $\rm\mu m$ line in solar composition and CO-
dominated nebulae. We note that this carbon recombination line is rarely
observed from photoionized gas, which could be a unique tracer for WD merger
remnants. It is, however, also faint and so likely only detectable in the MW.
In the bottom row, the lines show radii vs. stellar effective temperature
given the post-merger models of $T_{\rm eff}$ vs. time in Figure 1 and
assuming outflow velocities of 20, 40, and 60 km/s.

The left panels of Figure 6 show that in the later stages of the merger
remnant, the [O III] line-strengths in solar and C/O composition ejecta are
typically within a factor of $\sim 10$ of each other. Note that these larger
ejecta radii are where the system spends most of its time and where dust has
the least impact. Hence, an initial search for potential WD merger remnant
candidates can identify planetary nebulae with strong [O III] 5007 Å lines and
weak hydrogen lines. Table 1 provides a list of additional spectral lines that
are the strongest for the merger remnant, when compared to typical solar PNe.
Spectral features that will potentially help narrow down the number of
candidate WD merger remnant nebulae include bright carbon and oxygen
recombination lines, as well as oxygen fine structure lines. The middle and
right panels of Figure 6 show, e.g., that the C IV 7727 Å line is typically
$\sim 40$ times stronger in the C/O-rich nebula and the [O IV] 25.8832 $\rm\mu
m$ line is $\sim 10$ times stronger when the effective temperature of the
C/O-rich merger remnant is larger than $10^{5}$ K. Other oxygen fine-structure
lines including [O III] 51.81 $\rm\mu m$ and [O III] 88.36 $\rm\mu m$ are also
significantly stronger, but lie outside the range of most current
instrumentation, notably JWST. Many other oxygen and carbon lines in the UV or
optical are also suitable for the same purpose, especially carbon and oxygen
recombination lines such as C IV 4657 Å and O IV 4631 Å.

Figure 5: Electron temperature of the photoionized nebula from CLOUDY for the
C/O (left) and solar (middle) composition clouds. The right panel highlights
the temperature ratio showing that the effectively high metallicity C/O
nebulae are significantly cooler than standard planetary nebulae; overplotted
is the expected trajectory in the evolution of WD merger remnants for outflow
velocities of 20, 40, and 60 km/s.

Figure 6: Select line luminosity for solar (top) and C/O (middle) composition nebulae in a grid of simulations across a range of stellar effective temperatures and cloud sizes ($R_{17}$ is cloud inner radii in units of $10^{17}$cm). The bottom panel highlights the ratio of the line luminosity overplotted with the expected trajectory in the evolution of WD merger remnants for outflow velocities of 20, 40, and 60 km/s. We predict that WD merger remnants are bright in [O III] 5007 Å, C IV 7727 Å and [O IV] 25.8832 $\rm\mu m$ but faint in hydrogen lines. Line | Wavelength | $F_{\rm WD}/F_{\rm PN}$ | $F_{\rm WD,d}/F_{\rm PN,d}$
---|---|---|---
UV
C IV | 2528.83 Å | 34.2 | 40.2
O IV | 2450.01 Å | 22.2 | 26.0
O IV | 3037.66 Å | 20.9 | 25.5
O V | 2941.99 Å | 31.3 | 29.6
Optical
C IV | 4658.74 Å | 34.9 | 41.0
C IV | 7727.12 Å | 35.5 | 41.7
O IV | 4632.41 Å | 22.9 | 26.8
O IV | 7714.25 Å | 23.6 | 27.4
O V | 4930.35 Å | 32.6 | 30.4
O V | 7612.08 Å | 34.1 | 32.4
O V | 6486.60 Å | 25.6 | 25.7
O VI | 5291.36 Å | 78.1 | 76.7
IR
[O III] | 88.3585 $\rm\mu m$ | 2.2 | 7.5
$\left[\rm O\,III\right]$ | 51.8184 $\rm\mu m$ | 1.2 | 4.1
$\left[\rm O\,IV\right]$ | 25.8832 $\rm\mu m$ | 5.7 | 12.2
$\left[\rm O\,V\right]$ | 32.6035 $\rm\mu m$ | 5.0 | 6.4
Table 1: Notable emission lines from CLOUDY photoionization calculations for
the C/O WD merger remnant. The flux ratios ($F_{\rm WD}/F_{\rm PN}$) are for
comparison with a typical solar-composition PNe with the same stellar and
ejecta attributes: a stellar effective temperature of 150,000 K, and an ejecta
shell with a mass of $\rm 0.1{\rm M_{\odot}}$ and an inner radius of $10^{17}$
cm, where both models are dust-free. $F_{\rm WD,d}$ represents a model merger
remnant nebula with a D/G mass ratio of 10%, while $F_{\rm PN,d}$ has a D/G
mass ratio of 1%.

###  4.2 Models With Dust

Figure 7: WD merger remnants from CLOUDY models with graphite grains and a D/G
mass ratio of 10%. The left panel shows spectra of merger remnants with a
central stellar effective temperature of 150,000K, and a range of cloud inner
radii from $10^{16}$cm to $2.2\times 10^{18}$cm. The right panel shows the
line luminosity of [O III] 5007 Å as a function of cloud size (inner radius)
from CLOUDY, assuming a D/G mass ratio of 10%. An example model without dust
at $1.1\times 10^{5}$K is overplotted with the cyan dashed line. Different
stellar effective temperatures are plotted to cover the thermal evolution of
the remnant in the near-constant luminosity ($\sim 10^{4}L_{\odot}$) phase
(Figure 1). Dust somewhat suppresses the optical line emission when the nebula
is small but has little effect at larger sizes.

Dust may play an important role in modifying the appearance of the merger
remnant due to the carbon and oxygen-rich composition of WD merger remnant
ejecta. To assess the impact of dust, we carried out the same grid of
photoionization calculations as in §4.1 but for dust-to-gas (D/G) mass ratio
set to 10%. Although the exact D/G mass ratio in merger remnant ejecta is
uncertain, we assume a value that is larger than a typical planetary nebula
value of $\sim$0.1% - 1.0% (Stasińska & Szczerba, 1999) because of the higher
metallicity of the merger remnant ejecta.

The dust composition in the case of C/O WD mergers depends on the relative
mass in carbon and oxygen. When the central star is a giant, the absence of UV
radiation permits the formation of CO molecules in the outflow. If carbon
dominates by mass, the remaining carbon not in the form of CO will form
graphite dust. On the other hand, if oxygen dominates by mass, there is
remaining oxygen which forms silicate grains. Here we assume graphite grains
for concreteness, but in the next section we will separately show results for
silicate grains in models of the Milky Way merger candidate IRAS 00500+6713.

In the presence of dust, the short-wavelength transmitted spectrum depends on
the UV optical depth to dust which is $\tau\simeq M_{\rm dust}\kappa_{\rm
dust}/(4R_{\rm out}^{2})$ where $M_{\rm dust}$ is the total dust mass in the
cloud of outer radius $R_{\rm out}$ and $\kappa_{\rm dust}$ is the UV optical
depth per unit mass of dust. Requiring $\tau\lesssim 1$ implies

$R_{\rm out}\gtrsim 2\times 10^{17}{\rm cm}\,\left(\frac{M_{\rm dust}}{\rm
0.01{\rm M_{\odot}}}\,\frac{\kappa_{\rm dust}}{10^{4}\,{\rm
cm^{2}\,g^{-1}}}\right)^{1/2}$ (3)

The left panel of Figure 7 indeed shows that when the cloud radius is
$\lesssim 10^{17}$cm for the C/O remnant, the grains absorb nearly all of the
incident photons and re-emit them in an infrared continuum.666Recall that our
nebula models assume $R_{\rm out}=3r_{\rm in}$ and $r_{\rm in}$ is the size-
parameter shown in Figure 7. At these stages, dust would significantly modify
the appearance of the merger remnants in the CMDs in Figure 3. For larger
nebula radii, however, when the surrounding gas is photoionized and the dust
is optically thin, the UV and optical continuum becomes detectable. This
likely applies during most of the time of the merger remnant evolution, and
thus to most of the sources in the model CMDs in §2.1.

However, when the ejecta expands to $\gtrsim 10^{17}$cm, the optical depth to
dust drops below unity. In this case, the transmitted spectrum of the cloud
resembles that of the dust-free scenario with the UV/optical lines becoming
slightly weaker. It is important to stress that strong lines such as the [O
III] 5007 line are still present even when the UV continuum is absent. This is
because the photoionization cross-section is much larger than the dust
absorption cross-section so that the (carbon and oxygen) ionizing photons are
initially absorbed by photoionization leading to strong optical lines even
when dust ultimately absorbs and re-radiates much of the input radiation.

The long-wavelength infrared continuum due to dust in Figure 7 is much
stronger than the long-wavelength continuum due to free-free emission in the
dust-free models in Figure 4. Weaker lines are no longer detectable relative
to this stronger continuum, but strong lines such as the [O IV] 25.8832
$\rm\mu m$ should still be observable.

The right panel of Figure 7 shows the [O III] 5007 Å line luminosity in our
models with dust as a function of the inner radius of the ejecta for different
stellar effective temperatures; we also show an example without dust for
comparison (dashed cyan line). The peak line luminosity in the case with dust
is a factor of $\sim 5$ lower than without dust; nonetheless, a relatively
strong line is still present even when the cloud is small and the UV dust
optical depth is large (this is again due to the photoionization cross-section
being much larger than the dust cross-section so that an optical-IR emitting
photoionized nebula is present even when the UV dust optical depth is $\gtrsim
1$). The peak line luminosity occurs when the cloud radius is $\sim 10^{16}$
cm both with and without dust and is predominantly set by photoionization
physics. We also note that the results in Figure 6 and Table 1 for the most
prominent lines in merger remnant PNe relative to solar composition PNe remain
reasonably applicable in our models with dust, particularly the infrared
lines.

###  4.3 Detectability in External Galaxies

The primary challenges in detecting the ionized nebulae predicted here in
external galaxies lie in (1) the large underlying stellar continuum that
dilutes the detectability of the line emission, and (2) the low surface
density of merger remnants which requires surveying a relatively large area.
Here we briefly quantify these effects.

We consider an external galaxy with stellar surface mass density $\Sigma_{*}$
and stellar mass to light ratio in the optical of $\Upsilon$. If observations
are made with pixels (or seeing-limited resolution) of area $A_{\rm pixel}$
the total stellar luminosity per pixel is $\Sigma_{*}\Upsilon^{-1}A_{\rm
pix}$. For observations with spectral resolution $R=3000R_{3000}$ the stellar
continuum background per spectral resolution element is then
$\Sigma_{*}\Upsilon A_{\rm pix}/R$ (we scale $R$ here to the rough resolution
of MUSE on the VLT, but also discuss the case of narrow-band imaging below).
In order to detect, say, the OIII line luminosity relative to the background
stellar continuum we thus require $L_{\rm OIII}\gtrsim\Sigma_{*}\Upsilon
A_{\rm pix}/R$, i.e.,

$\Sigma_{*}\lesssim\,3\times 10^{5}\,{\rm
M_{\odot}\,arcsec^{-2}}\left(\frac{L_{\rm
OIII}}{10L_{\odot}}\right)\left(\frac{A_{\rm pix}}{\rm
0.1arcsec^{2}}\right)^{-1}{R_{3000}\Upsilon}$ (4)

The stellar surface density limit in equation 4 is comparable to the surface
brightness of M31’s bulge or that of M87 exterior to $\simeq 10$ kpc (note
that the latter region contains $\gtrsim 3\times 10^{11}M_{\odot}$; Gebhardt &
Thomas 2009).

For a given stellar mass surface density the expected number of merger
remnants per arcsec2 $\Sigma_{\rm remnants}$ is given by

$\Sigma_{\rm remnants}\simeq
10^{-10}\,\Sigma_{*}\left(\frac{\dot{N}}{10^{-14}\,{\rm yr^{-1}}{\rm
M_{\odot}}^{-1}}\frac{t_{\rm remnant}}{\rm 10^{4}\,yr}\right).$ (5)

where $\Sigma_{*}$ is in units of $M_{\odot}\,{\rm arcsec^{-2}}$ and where we
have normalized the merger rate and remnant lifetime as in equation 1. The
constraint in equation 4 together with the expected surface density of
remnants in equation 5 thus translates into a lower limit on the area $A$ that
must be surveyed to detect of order N merger remnant nebulae:

$\displaystyle A\gtrsim$ $\displaystyle 10\,N\,{\rm
arcmin^{2}}\,\left(\frac{\dot{N}}{10^{-14}\,{\rm yr^{-1}}{\rm
M_{\odot}}^{-1}}\frac{\rm t_{remnant}}{\rm 10^{4}\,yr}\right)^{-1}\times$
$\displaystyle\left(\frac{L_{\rm
OIII}}{10L_{\odot}}\right)^{-1}\left(\frac{A_{\rm pix}}{\rm
0.1arcsec^{2}}\right)\Upsilon^{-1}R_{3000}^{-1}$ (6)

Note that equation 4.3 assumes that the stellar mass satisfying equation 4 is
$\gtrsim 10^{10}M_{\odot}$ so that at least one remnant is present in the
galaxy of interest (eq. 1). Equation 4.3 is a non-trivial constraint given
that, e.g., the field of view of MUSE of 1 arcmin2. We note, however, that
Sarzi et al. (2018) surveyed 33 galaxies for a total area of order $60\,{\rm
arcmin^{2}}$ in Fornax searching for planetary nebulae, with 23 of them being
ellipticals. It is thus possible that WD merger nebulae are present in this
data set (eq. 4.3). Narrow band imagers with $R\sim 100$ can have a field of
view of $\simeq$ 1 square degree (e.g., the WIYN ODI) and so are significantly
more efficient at surveying a large area than IFUs. The downside is that the
lower spectral resolution restricts the observations to lower stellar surface
densities (eq. 4) which have less stellar mass and thus a smaller fraction of
the expected WD merger remnants.

As a final point of comparison, we note that the expected total number of PNe
in a galaxy would scale with the host-galaxy bolometric luminosity and the
specific PN number density, which is correlated with the age and metallicity
of the parent stellar population. For late-type and early-type galaxies, the
total number of PNe could be $\sim 1000\times$ and $\sim 100\times$ higher
respectively than the number of WD merger remnant nebulae present (Buzzoni et
al., 2006; Longobardi et al., 2013). The WD merger remnant nebulae are
distinguishable by their absence of strong Hydrogen and Helium lines. In
addition, for galaxies with old stellar populations, the WD merger remnant
nebulae may preferentially populate the more readily detectable high
luminosity end of the planetary nebula luminosity function. This is because WD
merger remnants are significantly brighter than the WDs formed from solar-mass
stars (Fig. 1); the dust-to-gas ratio in WD merger nebulae is uncertain,
however, and also influences the exact optical line luminosity from the
resulting nebulae (Fig. 7).

###  4.4 IRAS 00500+6713

Parameters of the Central Star
---
$\log_{10}[L_{\rm\star}(\rm L_{\odot})]$ | $4.60\pm 0.14$
$T_{\rm eff}(\rm K)$ | $211,000^{+40,000}_{-23,000}$
Distance (kpc) | $2.30\pm 0.14$
Shell Radius (pc) | $1.1$
Halo Radius (pc) | $1.6$
O mass fraction | $0.8\pm 0.1$
C mass fraction | $0.2\pm 0.1$
Ne mass fraction | $0.01$
Nebula Luminosity (erg/s)
$\nu L_{\nu,W3}$ | $1.7\times 10^{34}$
$\nu L_{\nu,{\rm W4,1.1pc}}$ | $5.7\times 10^{34}$
$\nu L_{\nu,{\rm W4,1.6pc}}$ | $1.6\times 10^{35}$
Line Luminosity (erg/s)
$L_{\rm[Ne\,VI]}$ | $3.1\times 10^{33}$
$L_{\rm[Mg\,VII]}$ | $2.2\times 10^{32}$
Table 2: Parameters of IRAS 00500+6713, a potential WD merger remnant,
obtained from Gvaramadze et al. (2019); distances updated based on Gaia Early
Data Release 3 (Bailer-Jones et al., 2021); monochromatic luminosity obtained
from WISE images in W3 ($\lambda_{\rm ref}=22.2\mu\rm m$) and W4
($\lambda_{\rm ref}=12.1\mu\rm m$) bands, also plotted in Figure 8; and
predicted line luminosity for lines [Ne VI] 7.64$\mu\rm m$ and [Mg VII]
9.01$\mu\rm m$ within JWST MIRI’s range, assuming a D/G mass ratio of 10%. Two
luminosity are provided for the W4 band depending on if halo subtraction is
performed, and is discussed further in §4.4. Figure 8: CLOUDY spectra for the
potential WD merger remnant in the Milky Way (IRAS 00500+6713) calculated
using parameters listed in Table 2 with a $0.1{\rm M_{\odot}}$ ejecta mass and
a range of D/G ratios. The nebula luminosity and filter response functions for
WISE W3 and W4 bands are plotted in orange and red; the circular and star-
shaped points represent the nebula luminosity from reduced WISE images in
these bands. The lower opacity star-shaped WISE point results from a different
data reduction choice for the stellar background, and is discussed more in-
depth in §4.4. Our models predict that the WISE detections are primarily dust
continuum, not infrared spectral lines.

Gvaramadze et al. (2019) reported the detection of a hot luminous Galactic
star surrounded by a hydrogen- and helium-free nebula (IRAS 00500+6713) that
was bright in the WISE W3 ($\lambda_{\rm ref}=22.2\mu\rm m$) and W4
($\lambda_{\rm ref}=12.1\mu\rm m$) bands. The star was also H and He-free with
a remarkable $\sim 16,000$ km s-1 wind. Table 2 summarizes some of the
properties of the star and nebula reported by Gvaramadze et al. 2019. They
interpreted this source as a pre-collapse super-Chandrasekhar WD merger
remnant with the unusual wind powered by strong magnetic fields and rotation
generated during the merger.

We use the properties listed in Table 2 together with an assumed nebula mass
of $0.1{\rm M_{\odot}}$ and several assumed D/G ratios to compute CLOUDY
models of the surrounding nebula. Since the derived stellar abundance is 80%
oxygen and 20% carbon, we assume the dust composition is primarily silicate.
From X-ray spectroscopy Oskinova et al. (2020) measures a different
composition in the nebula relative to Gvaramadze et al. (2019)’s optical
measurements for the star (higher C and Ne fractions). The origin of this
difference is unclear but introduces some systematic uncertainty in our
predictions of the nebula’s properties.

For our assumed parameters of IRAS 00500+6713, we find that the infrared
emission is dominated by dust with only a small contribution from lines. The
properties of the infrared emission thus depend primarily on the total dust
mass ($m_{\rm ej}\times$D/G ratio) and assumptions about the grain composition
and size distribution. Spectral lines between $1\mu\rm m<\lambda<10\mu\rm m$
such as [Ne VI] 7.64 $\mu\rm m$ are present in all of our models, but do not
contribute significantly to the WISE photometric data. This is a property of
the relatively low gas densities associated with our assumed ejecta mass of
$0.1{\rm M_{\odot}}$ and the large measured nebula size. More line features in
the optical and on top of the dust continuum start to appear in models with
$m_{\rm ej}\gtrsim 1.0{\rm M_{\odot}}$ but even then dust dominates the total
luminosity in the WISE bandpasses unless $m_{\rm ej}\gg 1{\rm M_{\odot}}$ or
there is significant clumping. Gvaramadze et al. (2019) suggested that the
observed nebula emission is dominated by [O IV] 25.89 $\mu\rm m$ and [Ne V]
14.32 $\mu\rm m$ and 24.32 $\mu\rm m$ lines. However, our models show that
this is not the case unless the remnant is dust-free and has $m_{\rm ej}\gg
0.1{\rm M_{\odot}}$, in which case it is unlikely to be the result of a double
WD merger. Follow-up mid and far-infrared spectroscopy with JWST could help
clarify the nature of this unusual source. In particular, confirmation that
the emission in the WISE bands is dust dominated (not line-dominated) would
provide additional support for the interpretation that this object is a WD
merger remnant.

Gvaramadze et al. (2019) do not report luminosity for the nebula surrounding
IRAS 00500+6713. We thus reduced the WISE data surrounding IRAS 00500+6713 by
masking the foreground and background star that overlaps with the nebula and
performing sky subtraction. The resulting luminosity in the WISE bands are
shown in Figure 8 and Table 2. The halo structure outside the nebula (see
Gvaramadze et al. 2019 Figure 1) in the WISE data can only be seen in W4, not
in W3. Hence, subtracting the stellar background in W4 with or without the
halo results in different nebula luminosity illustrated by the two different
points in Figure 8. The W4 luminosity we obtain matches our model for dust-
dominated WD merger remnant spectra quite well, but the W3 luminosity is a few
times higher. Overall, given the large uncertainty in dust properties
including total dust mass, grain size distribution, dust clumping, etc, our
model provides a reasonable explanation for the WISE observations.

One striking feature of the central star IRAS 00500+6713 is its remarkably
high speed wind with $\dot{M}\simeq 3\times 10^{-6}\,{\rm M_{\odot}}\,{\rm
yr^{-1}}$ and $v_{wind}\simeq 16,000\,{\rm km\,s^{-1}}$ (Gvaramadze et al.,
2019). This may account for one slightly puzzling feature of the WISE nebula:
the size of 1.6 pc corresponds to a wind speed of $\gtrsim 150{\rm
km\,s^{-1}}$ if the age of the system is $\lesssim 10^{4}$ yr, as expected for
a WD merger remnant. This is larger than the wind speeds observed for red
giants and asymptotic giant branch stars (Netzer & Elitzur, 1993). The net
momentum supplied by the current wind of IRAS 00500+6713 over a time $t$ is
$\sim 50{\rm M_{\odot}}{\rm km\,s^{-1}}\,(t/10^{3}{\rm yr})$. This is
sufficient to accelerate $0.1{\rm M_{\odot}}$ of ejecta to $500\,{\rm
km\,s^{-1}}$ (if the interaction is non-radiative and conserves energy not
momentum, the speed the ejecta can be accelerated to will be even higher). It
is thus possible that the mass of the nebula around IRAS 00500+6713 was set
during an earlier giant phase but that its kinematics is due to interaction
with the unusually powerful current wind of the central star. This could also
account for the high expansion velocity of $\sim$ 1100 km/s measured in Fesen
et al. (2023)’s [S II] optical spectra. It is likewise worth noting that the
interaction between the current high-speed wind and a previous cooler, slower,
denser wind could modify the thermal and ionization state of the denser gas,
an effect that is not accounted for in our photoionization models. This could
explain the filamentary structure seen in the [S II] 6716 and 6731 Å spectra
in Fesen et al. (2023).

##  5 Summary & Discussion

It is plausible that many WD mergers do not lead to prompt thermonuclear
supernovae, but instead lead to a long-lived stellar remnant. This is
particularly true for mergers of two C/O WDs or a C/O and O/Ne/Mg WD because
the temperature required to fuse Carbon is $\sim 10$ times higher than that to
fuse He and so dynamical burning is less likely during the merger. In this
paper, we have investigated the best ways of observationally detecting
surviving C/O-C/O or C/O-O/Ne/Mg merger remnants, utilizing the models of
Schwab et al. (2016); Schwab (2021) for the post-merger evolution of the
remnant. These models predict that the merger remnant inflates to become a
giant before subsequently contracting and evolving to higher $T_{\rm eff}$
(see Figure 1). The overall evolution of the merger remnant is similar to that
of a $6-10{\rm M_{\odot}}$ star as it evolves from the AGB through to the WD
cooling sequence. As a result, we highlight the value of searching for WD
merger remnants in galaxies with low specific star formation rates since this
implies a higher ratio of WD mergers to intermediate-mass stars. This
includes, e.g., the bulge of the MW and M31 and the outskirts of M87.

To quantify the detectability of WD merger remnants in photometric data, we
created composite stellar populations to simulate H-R diagrams of an M87-like
late-type galaxy, comparing the population of WD merger remnants to those of
single and binary stellar evolution models using MIST and BPASS, respectively
(see Figure 2). In principle, WD merger remnants appear to stand out as an
unusually high $T_{\rm eff}\sim 10^{5.5}$ K population relative to single and
binary stellar population synthesis models. However, in practice, this
technique for finding WD merger remnants has several challenges. First,
existing and planned UV instrumentation is not sufficiently blue to
distinguish WD merger remnants from He stars formed by binary evolution (see
Figure 3). Second, and probably more severely, post-outburst classical novae
undergo a very similar evolution to WD merger remnants (see Figure 1).
Moreover, the thermal evolution time of post-outburst novae is years-decades
and they significantly outnumber WD merger remnants even in old stellar
populations. Spectroscopic information to distinguish classical novae (with
hydrogen) from WD merger remnants (without hydrogen) is thus likely needed.

Just like AGB stars generate strong dusty winds, it is likely that WD merger
remnants during their giant phase generate significant outflows (Schwab et
al., 2016). We argue that the resulting photoionized nebula (analogous to a
planetary nebula) is one of the most promising ways of searching for WD merger
remnants. Our CLOUDY models show that WD merger remnant nebulae have unique
traits such as a high [O III]/$\rm H\alpha$ ratio and strong carbon and oxygen
recombination lines and oxygen fine structure lines (see Figure 6). The
merger-remnant wind is likely even more dust-enriched than AGB winds,
producing a strong mid-far-infrared continuum. Although dust suppresses the
strength of optical emission lines by a factor of $\simeq 5$ when the nebula
is small (and the dust optical depth is high), the optical lines are still
detectable even in this phase (Figure 7). In addition, dust has less of an
effect on the optical emission when the nebulae are larger, which is also
where they spend most of their time. Photo-absorption and extinction in the UV
suppress the short wavelength continuum emission when the cloud radius is
$\lesssim 10^{17}$ cm, which likely precludes photometrically finding the
merger remnants in these bands in the early phase of their thermal evolution.
At later times, however, the stellar continuum emission is detectable through
the surrounding fully ionized nebula.

Overall, we predict that M87 should have $\sim 100$ WD merger remnants with
detectable [O III] lines (and large [O III]/$\rm H\alpha$ line ratios) in
integral field unit or narrow-band imaging data. For other old stellar
populations, the number of luminous hot merger remnants detectable at a given
time should scale roughly with the stellar mass of the galaxy. The primary
observational challenge in detecting these ionized nebulae in external
galaxies is that the background stellar continuum dilutes the line emission
making it only detectable in lower stellar density regions with
$\Sigma_{*}\lesssim 3\times 10^{5}M_{\odot}\,{\rm arcsec}^{-2}$ (eq. 4) even
with integral field unit spectrographs (with narrow-band imaging the
observations are restricted to even lower stellar surface densities). This
implies that the bulge of the MW and M31 and the outskirts of nearby massive
galaxies such as M87 are the most promising targets. Observations will need to
survey a large area $\gtrsim 10\,{\rm arcmin^{2}}$ to detect even a single
merger remnant given the low surface density of such objects (eq. 4.3.) Sarzi
et al. (2018) have already covered such an area in Fornax searching for
planetary nebulae.

In the Milky Way disk, the challenge of detecting WD merger remnants is that
the population of photometrically similar objects due to ongoing star
formation greatly outnumbers the modest number $\sim 10$ of expected merger
remnants. Nonetheless, the latter should stand out spectroscopically by their
H and He-free ionized nebulae. Indeed, Gvaramadze et al. (2019) proposed that
the Galactic star IRAS 00500+6713 is a super-Chandrasekhar WD merger remnant
due to its C/O-dominated composition, luminosity and effective temperature
similar to the Schwab et al. (2016) merger remnant models, bright H-free WISE
nebula, and remarkably high-speed stellar wind suggestive of strong magnetic
fields generated during a stellar merger. Our photoionization models explain
the WISE nebula surrounding IRAS 00500+6713 as dust emission; although some
mid-infrared lines should be present (e.g., [Ne VI] 7.64 $\mu\rm m$) they do
not contribute significantly to the WISE luminosity (see Figure 8). Additional
infrared spectroscopy with JWST would be very valuable for testing this
prediction and using line diagnostics to constrain the density and electron
temperature of the nebula. In particular, our fiducial $0.1{\rm M_{\odot}}$
nebula model predicts that the [Ne VI] line at 7.64 $\mu\rm m$ should have a
luminosity $\sim 3\times 10^{33}\,{\rm erg\,s^{-1}}$ and that the [Mg VII]
line at 9.01 $\mu\rm m$ should have a luminosity of $\sim 2\times
10^{32}\,{\rm erg\,s^{-1}}$. Slight variations to the dust mass within a
factor of a few have little effect on the line luminosity. There are, however,
uncertainties in the composition of the nebula: Oskinova et al. (2020) infer a
higher C and Ne fraction in the nebula using X-ray spectroscopy, relative to
the abundances inferred from the optical stellar spectrum. Additional mid-IR
spectroscopic data would help clarify the origin of these differences and
better constrain the physical conditions in the nebula.

The theoretical models and stellar population estimates in this paper could be
significantly improved in future work, although some of the theoretical
uncertainties are sufficiently complex that progress will likely require the
detection of (or strong limits on) a sample of WD merger remnants. In
particular, our predictions of the properties of merger remnant nebulae are
uncertain because of the lack of good models for (or observations of) the
C/O-dominated cool winds which we believe produce the nebula (during the
merger remnants’ earlier giant phase). This leads to uncertainty in the mass,
dust content, and speed of the ejecta, all of which factor into the
detectability of the photoionized nebula. Separately, when estimating the
number of merger remnants detectable at any time, we focused on super-
Chandrasekhar mergers. However, lower mass merger remnants have a similar
evolutionary track and cover almost the same effective temperature range as
their more massive counterparts (see Figure 1). Hence, the actual number of
active remnants could be up to $\sim 10$ times higher than the $\sim 100$ we
estimated for M87 if the total formation rate of long-lived merger remnants is
of order the Type Ia supernova rate (Maoz & Mannucci, 2012).

## Acknowledgements

We thank Max Briel, Shany Danieli, Bruce Draine, Jan Eldridge, Götz Gräfener,
Jenny Greene, Zachary Hemler, Shri Kulkarni, Norbert Langer, Ben Margalit, Ken
Shen, Héloïse Stevance for helpful conversations and/or correspondence. We are
particularly grateful to Nadia Zakamska for useful conversations and comments
on an initial draft of the paper. EQ was supported in part by a Simons
Investigator award from the Simons Foundation. This research benefited from
interactions at workshops funded by the Gordon and Betty Moore Foundation
through Grant GBMF5076.

## Data Availability

The data underlying this paper will be shared on reasonable request to the
corresponding author. The WISE data can be accessed via the IPAC Infrared
Science Archive https://irsa.ipac.caltech.edu/applications/wise/.

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# Local and uniform moduli of continuity of chi–square processes

Michael B. Marcus Jay Rosen Research of Jay Rosen was partially supported by
grants from the Simons Foundation.

00footnotetext:  Key words and phrases: chi–square processes, moduli of
continuity 00footnotetext:  AMS 2020 subject classification: 60F15, 60G15,
60G17

##  1 Introduction

Let $\eta=\\{\eta(t);t\in[0,1]\\}$ be a mean zero continuous Gaussian process
with covariance $U=\\{U(s,t),s,t\in[0,1]\\},$ with $U(0,0)>0$. Let
$\\{\eta_{i};i=1,\ldots,k\\}$ be independent copies of $\eta$ and set,

$Y_{k}(t)=\sum_{i=1}^{k}\eta^{2}_{i}(t),\qquad t\in[0,1].$ (1.1)

The stochastic process $Y_{k}=\\{Y_{k}(t),t\in[0,1]\\}$ is referred to as a
chi–square process of order $k$ with kernel $U$.

###### Theorem 1.1

Let $\phi(t)$ be a positive function on $[0,\delta]$ for some $\delta>0$. If

$\limsup_{t\to 0}\frac{\eta(t)-\eta(0)}{\phi(t)}=1\qquad a.s.,$ (1.2)

then for all integers $k\geq 1$,

$\limsup_{t\to 0}\frac{Y_{k}(t)-Y_{k}(0)}{\phi(t)}=2Y^{1/2}_{k}(0)\qquad a.s.$
(1.3)

When $k=1$ this is particularly simple. Since $\eta$ is symmetric it follows
from (1.2) that,

$\liminf_{t\to 0}\frac{\eta(t)-\eta(0)}{\phi(t)}=-1\qquad a.s.$ (1.4)

Therefore, writing $Y_{1}(t)-Y_{1}(0)=(\eta(t)-\eta(0))(\eta(t)+\eta(0))$ and
using the continuity of $\eta$, we see that

$\limsup_{t\to
0}\frac{Y_{1}(t)-Y_{1}(0)}{\phi(t)}=2\left(\eta(0)\vee-\eta(0)\right)\qquad
a.s.,$ (1.5)

which is (1.3).

A result similar to Theorem 1.1 for the limiting behavior of chi–square
sequences at infinity is given in [2, Lemma 6.5].

Set

$\sigma^{2}(u,v)=E(\eta(u)-\eta(v))^{2}\quad\text{and}\quad\widetilde{\sigma}^{2}(x)=\sup_{|u-v|\leq
x}\sigma^{2}(u,v).$ (1.6)

###### Theorem 1.2

Assume that $\inf_{t\in[0,1]}U(t,t)>0$ and,

$\lim_{x\to 0}\widetilde{\sigma}^{2}(x)\log 1/x=0.$ (1.7)

Let $\varphi(t)$ be a positive function on $[0,1]$. Then if

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\eta(u)-\eta(v)}{\varphi(|u-v|)}=1\qquad a.s.,$
(1.8)

for all intervals $\Delta\subset[0,1]$, it follows that for all intervals
$\Delta\subset[0,1]$ and all integers $k\geq 1$,

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}=2\sup_{u\in\Delta}Y_{k}^{1/2}(u),\hskip
14.45377pta.s.$ (1.9)

When $\eta$ is a continuous Gaussian process with stationary increments,
$\sigma^{2}(u,v)$ in (1.6) can be written as $\sigma^{2}(u-v,0)$. In this case
if $\widetilde{\sigma}^{2}(x)$ is asymptotic to an increasing function at 0,
then (1.8) implies (1.7). We discuss this further in Remark 2.1.

An extensive treatment of Gaussian processes satisfying (1.2) and (1.8) is
given in [1, Chapter 7].

##  2 Proofs

Proof of Theorem 1.1 Let $\eta_{i}(t)$, $i=1,\ldots,k$, be independent copies
of $\eta(t)$. We write,

$\displaystyle\eta_{i}^{2}(t)-\eta_{i}^{2}(0)$ $\displaystyle=$
$\displaystyle(\eta_{i}(t)-\eta_{i}(0))(\eta_{i}(t)+\eta_{i}(0))$
$\displaystyle=$
$\displaystyle(\eta_{i}(t)-\eta_{i}(0))(2\eta_{i}(0)+(\eta_{i}(t)-\eta_{i}(0)))$
$\displaystyle=$ $\displaystyle
2(\eta_{i}(t)-\eta_{i}(0))\eta_{i}(0)+(\eta_{i}(t)-\eta_{i}(0))^{2}.$

By (1.2)

$\displaystyle\limsup_{t\to
0}\frac{\sum_{i=1}^{k}(\eta_{i}(t)-\eta_{i}(0))^{2}}{\phi(t)}$ (2.2)
$\displaystyle\qquad\leq\sum_{i=1}^{k}\limsup_{t\to
0}\frac{|\eta_{i}(t)-\eta_{i}(0)|}{\phi(t)}\lim_{t\to
0}|\eta_{i}(t)-\eta_{i}(0)|=0.$

Consequently, using (1.1) we see that,

$\limsup_{t\to 0}\frac{Y_{k}(t)-Y_{k}(0)}{\phi(t)}=\limsup_{t\to
0}\frac{2\sum_{i=1}^{k}(\eta_{i}(t)-\eta_{i}(0))\eta_{i}(0)}{\phi(t)}.$ (2.3)

Write,

$\displaystyle(\eta_{i}(t)-\eta_{i}(0))\eta_{i}(0)$ (2.4)
$\displaystyle\qquad=\left(\eta_{i}(t)-\frac{U(0,t)}{U(0,0)}\eta_{i}(0)\right)\eta_{i}(0)-\left(\frac{U(0,0)-U(0,t)}{U(0,0)}\right)\eta^{2}_{i}(0).$

We show below that

$\limsup_{t\to 0}\frac{|U(0,0)-U(0,t)|}{U(0,0)\phi(t)}=0.$ (2.5)

Consequently,

$\limsup_{t\to 0}\frac{Y_{k}(t)-Y_{k}(0)}{\phi(t)}=\limsup_{t\to
0}\frac{2\sum_{i=1}^{k}\left(\eta_{i}(t)-\frac{U(0,t)}{U(0,0)}\eta_{i}(0)\right)\eta_{i}(0)}{\phi(t)}.$
(2.6)

Let $\\{\xi_{i}(t),t\in[0,1]\\}$, $i=1,\ldots,k$, be independent copies of a
mean zero Gaussian process $\\{\xi(t),t\in[0,1]\\}$, and set
$\vec{\xi}(t)=(\xi_{1}(t),\ldots,\xi_{k}(t))$. Let $\vec{v}\in R^{k}$ with
$\|\vec{v}\|_{2}=1$. Computing the covariances we see that,

$\\{(\vec{v}\cdot\vec{\xi}(t)),t\in[0,1]\\}\stackrel{{\scriptstyle
law}}{{=}}\\{\xi(t),t\in[0,1]\\}.$ (2.7)

(This relationship is used by P. Revesz in [3, Theorem 18.1] to obtain LILs
for Brownian motion in $R^{k}$.)

Therefore, since $\left(\eta_{i}(t)-(U(0,t)/U(0,0))\eta_{i}(0)\right)$ and
$\eta_{i}(0)$ are independent for $i=1,\ldots,k$, we see that,

$\displaystyle\Big{\\{}\left(\vec{\eta}(t)-\frac{U(0,t)}{U(0,0)}\vec{\eta}(0)\right)\cdot\frac{\vec{\eta}(0)}{\|\vec{\eta}(0)\|_{2}},t\in[0,1]\Big{\\}}$
(2.8) $\displaystyle\qquad\stackrel{{\scriptstyle
law}}{{=}}\Big{\\{}\left(\eta(t)-\frac{U(0,t)}{U(0,0)}\eta(0)\right),t\in[0,1]\Big{\\}},$

where $\vec{\eta}(t)=(\eta_{1}(t),\ldots,\eta_{k}(t))$. Consequently, (2.6)
implies that

$\limsup_{t\to
0}\frac{Y_{k}(t)-Y_{k}(0)}{\phi(t)\|\vec{\eta}(0)\|_{2}}\stackrel{{\scriptstyle
law}}{{=}}\limsup_{t\to
0}\frac{2\left(\eta(t)-\frac{U(0,t)}{U(0,0)}\eta(0)\right)}{\phi(t)}.$ (2.9)

Using (2.4) again and (2.5) we see that this implies that,

$\limsup_{t\to
0}\frac{Y_{k}(t)-Y_{k}(0)}{\phi(t)\|\vec{\eta}(0)\|_{2}}\stackrel{{\scriptstyle
law}}{{=}}\limsup_{t\to 0}\frac{2(\eta(t)-\eta(0))}{\phi(t)}=2,$ (2.10)

where the last equality uses (1.2). Since,

$\|\vec{\eta}(0)\|_{2}=\left(\sum_{i=1}^{k}\eta^{2}_{i}(0)\right)^{1/2}=Y_{k}^{1/2}(0),$
(2.11)

we obtain (1.3).

To obtain (2.5) we first note that it follows from (1.2) that,

$\phi(t)=(E(\eta(t)-\eta(0))^{2})^{1/2}h(t),$ (2.12)

for some function $h$ such that $\lim_{t\downarrow 0}h(t)=\infty$. To see
this, suppose that it is false. Then there exists a sequence $\\{t_{k}\\}$,
with $\lim_{k\to\infty}t_{k}=0$, such that $\sup_{k}h(t_{k})\leq M$.
Therefore, if (1.2) holds, we would have,

$\sup_{k}\frac{\eta(t_{k})-\eta(0)}{(E(\eta(t_{k})-\eta(0))^{2})^{1/2}}\leq
M\qquad a.s.$ (2.13)

This is not possible because
$\\{{\eta(t_{k})-\eta(0)}/{(E(\eta(t_{k})-\eta(0))^{2})^{1/2}}\\}$ is a
sequence of standard normal random variables.

Since,

$U(0,0)-U(0,t)=E\left(\left(\eta(t)-\eta(0)\right)\eta(0)\right)\leq
E\left(\left(\eta(t)-\eta(0)\right)^{2}\right)^{1/2}U^{1/2}(0,0),$ (2.14)

we have,

$\frac{U(0,0)-U(0,t)}{\phi(t)}\leq\frac{U^{1/2}(0,0)}{h(t)}.$ (2.15)

Using the fact that $\lim_{t\downarrow 0}h(t)=\infty$ we get (2.5).

Proof of Theorem 1.2 Note that (1.8) implies that $\\{\eta(t),t\in[0,1]\\}$
and therefore $\\{\eta^{2}(t),t\in[0,1]\\}$ are uniformly continuous on
$[0,1]$, which in turn implies that for all $k\geq 1,$
$\\{Y_{k}(t);t\in[0,1]\\}$ is uniformly continuous on $[0,1]$.

To show,

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}\geq
2\sup_{t\in\Delta}Y_{k}^{1/2}(t),\hskip 14.45377pta.s.$ (2.16)

it suffices to show that for any $d\in\Delta$,

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}\geq
2Y_{k}^{1/2}(d),\hskip 14.45377pta.s.$ (2.17)

This is because, (2.17) holding almost surely implies that for any countable
dense set $\Delta^{\prime}\subset\Delta$,

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}\geq
2\sup_{d\in\Delta^{\prime}}Y_{k}^{1/2}(d),\hskip 14.45377pta.s.$ (2.18)

which implies (2.16).

Let $u,v,d\in\Delta$. We write,

$\displaystyle\eta_{i}^{2}(u)-\eta_{i}^{2}(v)$ $\displaystyle=$
$\displaystyle(\eta_{i}(u)-\eta_{i}(v))(\eta_{i}(u)+\eta_{i}(v))$
$\displaystyle=$
$\displaystyle(\eta_{i}(u)-\eta_{i}(v))(2\eta_{i}(d)+(\eta_{i}(u)-\eta_{i}(d))+(\eta_{i}(v)-\eta_{i}(d))).$

It follows from (1.8) that,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}(\eta_{i}(u)-\eta_{i}(v))(\eta_{i}(u)-\eta_{i}(d))}{\varphi(|u-v|)}$
$\displaystyle\qquad\leq\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}|\eta_{i}(u)-\eta_{i}(v)|\sup_{u\in\Delta}|\eta_{i}(u)-\eta_{i}(d)|}{\varphi(|u-v|)}$
$\displaystyle\qquad\leq\sum_{i=1}^{k}\sup_{u\in\Delta}|\eta_{i}(u)-\eta_{i}(d)|:=\Delta^{*}.$

Therefore, using (2.16)–(2) and (2.11) with 0 replaced by $d$ we see that,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)Y^{1/2}_{k}(d)}$
(2.21) $\displaystyle\qquad\geq\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{2\sum_{i=1}^{k}(\eta_{i}(u)-\eta_{i}(v))\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}-\frac{2\Delta^{*}}{\|\vec{\eta}(d)\|_{2}}.$

To simplify the notation we take $V(u,v)={U(u,v)}/{U(d,d)}$. Write

$\eta_{i}(u)-\eta_{i}(v)=V(v,d)\eta_{i}(u)-V(u,d)\eta_{i}(v)+G_{i}(u,v),$
(2.22)

where,

$G_{i}(u,v)=(1-V(v,d))\eta_{i}(u)-(1-V(u,d))\eta_{i}(v).$ (2.23)

In this notation,

$\displaystyle\frac{\sum_{i=1}^{k}(\eta_{i}(u)-\eta_{i}(v))\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}-\frac{\sum_{i=1}^{k}G_{i}(u,v)\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}$
$\displaystyle\qquad=\frac{\sum_{i=1}^{k}\left(V(v,d)\eta_{i}(u)-V(u,d)\eta_{i}(v)\right)\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}.$
(2.24)

Note that for all $u,v\in[0,1],$

$E\left((V(v,d)\eta_{i}(u)-V(u,d)\eta_{i}(v))\eta_{i}(d)\right)=0.$

This shows that $\eta_{i}(d)$ is independent of
$\\{V(v,d)\eta_{i}(u)-V(u,d)\eta_{i}(v);u,v\in[0,1]\\}$. Therefore by (2.7),

$\displaystyle\Big{\\{}\sum_{i=1}^{k}(V(v,d)\eta_{i}(u)-V(u,d)\eta_{i}(v))\frac{\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}};\quad
u,v\in[0,1]\Big{\\}}$ (2.25) $\displaystyle\qquad\stackrel{{\scriptstyle
law}}{{=}}\Big{\\{}V(v,d)\eta(u)-V(u,d)\eta(v);\quad u,v\in[0,1]\Big{\\}}.$

It follows that,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}(\eta_{i}(u)-\eta_{i}(v))\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}$
(2.26) $\displaystyle\qquad\qquad+\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}|G_{i}(u,v)||\eta_{i}(d)|}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}$
$\displaystyle\qquad\stackrel{{\scriptstyle law}}{{\geq}}\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{V(v,d)\eta(u)-V(u,d)\eta(v)}{\varphi(|u-v|)}.$

Using (2.22) we can write,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{V(v,d)\eta(u)-V(u,d)\eta(v)}{\varphi(|u-v|)}$ (2.27)
$\displaystyle\qquad\geq\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{(\eta(u)-\eta(v))}{\varphi(|u-v|)}-\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|G(u,v)|}{\varphi(|u-v|)}$
$\displaystyle\qquad=1-\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|G(u,v)|}{\varphi(|u-v|)},$

where

$G(u,v)=(1-V(v,d))\eta(u)-(1-V(u,d))\eta(v)$ (2.28)

and we use (1.8) for the last line in (2.27).

It follows from (2.26) and (2.27) that,

$\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}(\eta_{i}(u)-\eta_{i}(v))\eta_{i}(d)}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}\geq
1-{\cal H},$ (2.29)

where,

${\cal H}=\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sum_{i=1}^{k}|G_{i}(u,v)||\eta_{i}(d)|}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}+\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|G(u,v)|}{\varphi(|u-v|)}$

Using the Schwartz inequality followed by the triangle inequality we note
that,

$\frac{\sum_{i=1}^{k}|G_{i}(u,v)||\eta_{i}(d)|}{\|\vec{\eta}(d)\|_{2}\,\varphi(|u-v|)}\leq\frac{\sum_{i=1}^{k}|G_{i}(u,v)|}{\varphi(|u-v|)}.$

Therefore,

${\cal H}\leq\sum_{i=0}^{k}\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|G_{i}(u,v)|}{\varphi(|u-v|)},\qquad a.s.,$ (2.30)

where for notational convenience we have set $G_{0}=G$.

Set,

$\sigma^{2}(u,v)=E(\eta(u)-\eta(v))^{2},$ (2.31)

and

$\widetilde{\sigma}^{2}(x)=\sup_{|u-v|\leq x}\sigma^{2}(u,v).$ (2.32)

Then it follows from (1.8) that we can write,

$\varphi(h)=\widetilde{\sigma}(h)g(h),\mbox{ where necessarily, $\lim_{h\to
0}g(h)=\infty$}.$ (2.33)

This follows by a minor modification of the argument used to prove (2.12).
Note that for any sequence $h_{k}\to 0$ we can find sequences $\\{u_{k}\\}$,
$\\{v_{k}\\}$ in $\Delta$, with $|u_{k}-v_{k}|\leq h_{k}$ such that
$\widetilde{\sigma}(h_{k})\leq 2\sigma(u_{k},v_{k})$. Now, suppose that
$\limsup_{h\to 0}g(h)=M$. Then by (1.8) we would have,

$\sup_{k\to\infty}\frac{\eta(u_{k})-\eta(v_{k})}{\sigma(u_{k},v_{k})}\leq
4M\qquad a.s.$

This is not possible because each term
$(\eta(u_{k})-\eta(v_{k}))/\sigma(u_{k},v_{k})$ is a standard normal random
variable.

We show in Lemma 2.1 below that,

$|G(u,v)|\leq\frac{\sigma(d,v)}{U^{1/2}(d,d)}|\eta(u)-\eta(v)|+\frac{\sigma(u,v)}{U^{1/2}(d,d)}|\eta(v)|.$
(2.34)

Therefore, using (1.8) and (2.33) we have,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|G(u,v)|}{\varphi(|u-v|)}\leq\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{|\eta(u)-\eta(v)|\sigma(d,v)}{U^{1/2}(d,d)\,\varphi(|u-v|)}\,\,\,$
(2.35) $\displaystyle\hskip 144.54pt+\lim_{h\to
0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{\sigma(u,v)}{U^{1/2}(d,d)\,\varphi(|u-v|)}\eta(v)$
$\displaystyle\qquad\qquad\leq\sup_{d,v\in\Delta}\frac{\sigma(d,v)}{U^{1/2}(d,d)}+\lim_{h\to
0}\frac{1}{g(h)}\sup_{v\in\Delta}\frac{|\eta\left(v\right)|}{U^{1/2}(d,d)}=\frac{\widetilde{\sigma}(|\Delta|)}{U^{1/2}(d,d)}.$

where $\widetilde{\sigma}(|\Delta|)$ is defined in (2.32). We now use (2.21),
(2.29) and (2.35) to see that,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}$ (2.36)
$\displaystyle\qquad\geq
2Y^{1/2}_{k}(d)\left(1-(k+1)\frac{\widetilde{\sigma}(|\Delta|)}{U^{1/2}(d,d)}\right)-2\Delta^{*},\qquad
a.s.,$

where $d\in\Delta$ and $\Delta^{*}$ is defined in (2).

Suppose that $|\Delta|=1/n$. Then it follows from Lemma 2.2 that,

$P\left(\Delta^{*}\geq k((1+2C)\widetilde{\sigma}^{2}(1/n)\log
n)^{1/2}\right)\leq\frac{2k}{n^{C}}.$ (2.37)

Now let $\Delta(d,n)\subseteq\Delta$, be an interval of size $1/n$ that
contains $d$. It follows from (2.36) and (2.37) applied to $\Delta(d,n)$ and
$\Delta^{*}(d,n)$ that ,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}$ (2.38)
$\displaystyle\qquad\geq\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta(d,n)}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}$
$\displaystyle\qquad\geq
2Y^{1/2}_{k}(d)\left(1-(k+1)\frac{\widetilde{\sigma}(|\Delta|)}{U^{1/2}(d,d)}\right)-2k((1+2C)\widetilde{\sigma}^{2}(1/n)\log
n)^{1/2},$

except, possibly, on a set of measure $2k/n^{C}$. Taking $n\to\infty$, and
using (1.7), gives (2.17) and consequently (2.16), which is the lower bound in
(1.9).

We now obtain the upper bound in (1.9). Let
$\Delta_{m,n}=\Delta\cap[\frac{m-1}{n},\frac{m+1}{n}]$. Analogous to (2.38) we
have,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta_{m,n}}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}$ (2.39)
$\displaystyle\qquad\leq
2Y^{1/2}_{k}(m/n)\left(1+(k+1)\frac{\widetilde{\sigma}(2/n)}{U^{1/2}(\frac{m}{n},\frac{m}{n})}\right)+k((1+2C)\widetilde{\sigma}^{2}(2/n)\log
n)^{1/2}$ $\displaystyle\qquad:=2Y^{1/2}_{k}(m/n){\cal
A}\left(\widetilde{\sigma}(2/n),k,n,C\right),$

except, possibly, on a set of measure $2k/n^{C}$. The proof of (2.39) proceeds
in essentially the same way as the proof of (2.38). In the proof of the lower
bound in (2.38) we subtract several terms. In proving the upper bound in
(2.39) we add these terms. It then follows that,

$\displaystyle\lim_{h\to 0}\sup_{\stackrel{{\scriptstyle|u-v|\leq
h}}{{u,v\in\Delta}}}\frac{Y_{k}(u)-Y_{k}(v)}{\varphi(|u-v|)}$
$\displaystyle\leq 2\sup_{m=1,\ldots,n-1}Y^{1/2}_{k}(m/n){\cal
A}\left(\widetilde{\sigma}(2/n),k,n,C\right)$ $\displaystyle\leq
2\sup_{v\in\Delta}Y^{1/2}_{k}(v){\cal
A}\left(\widetilde{\sigma}(2/n),k,n,C\right),$

except possibly on a set of measure $2k/n^{C-1}$. Taking the limit as
$n\to\infty$ gives the upper bound in (1.9).

###### Lemma 2.1

$|G(u,v)|\leq\frac{\sigma(d,v)}{U^{1/2}(d,d)}|\eta(u)-\eta(v)|+\frac{\sigma(u,v)}{U^{1/2}(d,d)}|\eta(v)|.$
(2.41)

Proof:  We write,

$G(u,v)=(1-V(v,d))(\eta(u)-\eta(v))+(V(u,d)-V(v,d))\eta(v).$

Note that,

$\displaystyle|(V(u,d)-V(v,d))|$ $\displaystyle=$
$\displaystyle\frac{|\left(E(\eta(u)-\eta(v))\eta(d)\right)|}{U(d,d)}$
$\displaystyle\leq$
$\displaystyle\frac{\left(E(\eta(u)-\eta(v))^{2}E\eta^{2}(d)\right)^{1/2}}{U(d,d)}$
$\displaystyle\leq$ $\displaystyle\frac{\sigma(u,v)}{U^{1/2}(d,d)},$

and,

$\displaystyle(1-V(v,d))$ $\displaystyle=$
$\displaystyle\frac{\left(E(\eta(d)-\eta(v))\eta(d)\right)}{U(d,d)}$
$\displaystyle\leq$
$\displaystyle\frac{\left(E(\eta(d)-\eta(v))^{2}E\eta^{2}(d)\right)^{1/2}}{U(d,d)}$
$\displaystyle\leq$ $\displaystyle\frac{\sigma(d,v)}{U^{1/2}(d,d)}.$

###### Lemma 2.2

$P\left(\sum_{i=1}^{k}\sup_{u\in\Delta}|\eta_{i}(u)-\eta_{i}(d)|\geq
k((1+2C)\widetilde{\sigma}^{2}(|\Delta|)\log 1/|\Delta|)^{1/2}\right)\leq
2k|\Delta|^{C}.$ (2.42)

Proof:  Let a be the median of $\sup_{u\in\Delta}(\eta_{1}(u)-\eta_{1}(d))$.
It follows from [1, Lemma 5.4.3] that,

$P\left(\sup_{u\in\Delta}|\eta_{1}(u)-\eta_{1}(d)|\geq\mathbf{a}+\widetilde{\sigma}(|\Delta|)t\right)\leq
2e^{-t^{2}/2}.$ (2.43)

Since by [1, (7.113)],

$\mathbf{a}=o(\widetilde{\sigma}^{2}(|\Delta|)\log 1/|\Delta|)^{1/2},$ (2.44)

we see that

$P\left(\sup_{u\in\Delta}|\eta_{1}(u)-\eta_{1}(d)|\geq((1+2C)\widetilde{\sigma}^{2}(|\Delta|)\log
1/|\Delta|)^{1/2}\right)\leq 2|\Delta|^{C},$ (2.45)

which gives (2.42).

###### Remark 2.1

When $\eta$ has stationary increments, $\sigma^{2}(u,v)$ in (1.6) can be
written as $\sigma^{2}(u-v)$. For $x\in[0,\delta]$ define,

$\overline{\sigma}^{2}(x)=\mu\left(s:\sigma^{2}(x)\leq x\right),$

where $\mu$ is Lebesgue measure. Clearly, $\overline{\sigma}^{2}(x)$ is an
increasing function. It is called the increasing rearrangement of
$\sigma^{2}(x)$. (See [1, Section 6,4] for more details.) We show in [1,
(6.138)] that when $\eta$ is continuous a.s.,

$\lim_{x\to 0}\overline{\sigma}^{2}(x)\log(1/x)=0.$

This shows that if $\widetilde{\sigma}^{2}(x)$ is asymptotic to an increasing
function at 0, then (1.8), which implies that $\eta$ is continuous a.s.,
implies (1.7).

## References

  * [1] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes and Local Times, Cambridge University Press, New York, (2006).
  * [2] M. B. Marcus and J. Rosen, Asymptotic Properties of Permanental Sequences, Springer Briefs in Prob. and Math. Stat., Switzerland, (2021).
  * [3] P. Revesz, Random Walk in Random and Non-random Environments, Cambridge University Press, New York, (1990).

|  Michael B. Marcus |  Jay Rosen
---|---|---
|  253 West 73rd. St., Apt. 2E |  Department of Mathematics
|  New York, NY 10023, USA |  College of Staten Island, CUNY
|<EMAIL_ADDRESS>|  Staten Island, NY 10314, USA
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