# Prospects of gravitational-waves detections from common-envelope evolution
with LISA

M. Renzo Department of Physics, Columbia University, New York, NY 10027, USA
Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New
York, NY 10010, USA  T. Callister Center for Computational Astrophysics,
Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  K. Chatziioannou
Department of Physics, California Institute of Technology, Pasadena,
California 91125, USA  LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA  Center for Computational Astrophysics, Flatiron
Institute, 162 5th Ave, New York, NY 10010, USA  L. A. C. van Son Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  C. M. F. Mingarelli Department of Physics, University of Connecticut, 196
Auditorium Road, U-3046, Storrs, CT 06269-3046, USA  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  M.
Cantiello Center for Computational Astrophysics, Flatiron Institute, 162 5th
Ave, New York, NY 10010, USA  Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08544, USA  K. E. S. Ford Dept. of Science, Borough
of Manhattan Community College, City University of New York, New York, NY
10007 USA  Dept. of Astrophysics, American Museum of Natural History, New
York, NY 10024 USA  Physics Program, CUNY Graduate Center, City University of
New York, New York, NY 10016 USA  Center for Computational Astrophysics,
Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  B. McKernan Dept. of
Science, Borough of Manhattan Community College, City University of New York,
New York, NY 10007 USA  Dept. of Astrophysics, American Museum of Natural
History, New York, NY 10024 USA  Physics Program, CUNY Graduate Center, City
University of New York, New York, NY 10016 USA  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  G.
Ashton Royal Holloway University of London, Egham, Surrey, TW20 0EX, U.K.
School of Physics and Astronomy, Monash University, VIC 3800, Australia

###### Abstract

Understanding common envelope (CE) evolution is an outstanding problem in
binary evolution. Although the CE phase is not driven by gravitational-wave
(GW) emission, the in-spiraling binary emits GWs that passively trace the CE
dynamics. Detecting this GW signal would provide direct insight on the gas-
driven physics. Even a non-detection might offer invaluable constraints. We
investigate the prospects of detection of a Galactic CE by LISA. While the
dynamical phase of the CE is likely sufficiently loud for detection, it is
short and thus rare. We focus instead on the self-regulated phase that
proceeds on a thermal timescale. Based on population synthesis calculations
and the (unknown) signal duration in the LISA band, we expect $\sim 0.1-100$
sources in the Galaxy during the mission duration. We map the GW observable
parameter space of frequency $f_{\mathrm{GW}}$ and its derivative
$\dot{f}_{\mathrm{GW}}$ remaining agnostic on the specifics of the inspiral,
and find that signals with $\mathrm{SNR}>10$ are possible if the CE stalls at
separations such that $f_{\mathrm{GW}}\gtrsim 2\times 10^{-3}\,\mathrm{Hz}$.
We investigate the possibility of mis-identifying the signal with other known
sources. If the second derivative $\ddot{f}_{\mathrm{GW}}$ can also be
measured, the signal can be distinguished from other sources using a GW
braking-index. Alternatively, coupling LISA with electromagnetic observations
of peculiar red giant stars and/or infrared and optical transients, might
allow for the disentangling of a Galactic CE from other Galactic and extra-
galactic GW sources.

Common-envelope evolution — Common-envelope binary stars — Gravitational-wave
astronomy

††journal: ApJ††software:  COSMIC (Breivik et al., 2020), ipython/jupyter
(Pérez & Granger, 2007), matplotlib (Hunter, 2007), NumPy (van der Walt et
al., 2011).

##  1 Introduction

Common envelope (CE) evolution is one of the most challenging processes to
study in the evolution of binary stars. CE evolution was originally proposed
by Paczyński (1976) to account for the short orbital separations observed in
cataclysmic variables. Despite its importance for many categories of stellar
objects, the details of this process remain elusive.

A CE is typically assumed to occur when the evolution of a binary system leads
to dynamically unstable mass transfer, causing the accretor to be engulfed in
the donor’s envelope. This results in a CE binary, in which the core of the
donor star and the accretor star coexist within a shared envelope. Drag forces
then drive the inspiral of the binary, injecting orbital energy (and angular
momentum, Nelemans et al., 2000) into the CE. Depending on the complex
coupling between the orbit and the envelope structure, the CE can ultimately
be ejected, or, alternatively, the two stars can merge. We expand on the
current understanding of CE evolution in Sec. 1.1.

Evolution through a CE phase is a key step in the formation of almost any
compact binary system from an isolated binary, including cataclysmic variables
(e.g., Paczyński, 1976), subdwarf B-stars (e.g., Han et al., 2003; Igoshev et
al., 2020) and stripped stars (Götberg et al., 2020), double white dwarfs (WD,
e.g., Nandez et al., 2015), X-ray binaries (e.g., Podsiadlowski et al., 2002;
Chen et al., 2020a), binary neutron-stars (e.g., Tauris et al., 2017; Kruckow
et al., 2018; Vigna-Gómez et al., 2020), black-hole neutron-star binaries
(e.g., Kruckow et al., 2018; Chattopadhyay et al., 2020; Broekgaarden et al.,
2021), black-hole WD binaries (e.g., Sberna et al., 2020), and binary black-
holes (e.g., Belczynski et al., 2016; Kruckow et al., 2018). CE evolution also
produces the tightest binaries which can in turn make the fastest runaway
stars from isolated systems if disrupted by the explosion of one star (e.g.,
Justham et al., 2009; Renzo et al., 2019; Evans et al., 2020; Neunteufel,
2020). Finally, CE preceeds the majority of stellar mergers111We neglect here
the possibility of (close to) head-on collisions between stars in dense
stellar environments which can also lead to mergers (e.g. Glebbeek et al.,
2009; Di Carlo et al., 2020; Renzo et al., 2020)., which are common across the
entire mass range of stars (e.g., de Mink et al., 2013; Zapartas et al., 2017;
Temmink et al., 2020) and may possibly lead to the formation of Thorne-Zytkow
objects (e.g., Thorne & Zytkow, 1977).

The final phases of a CE – the envelope ejection or the merger event – can
produce visible electromagnetic (EM) transients called luminous red novae
(LRN). The detection of the pre-transient period decay of V1309Sco (Tylenda et
al., 2011) unambiguously connected this type of transient to CE. LRNe exhibit
a variety of morphologies (e.g., Pastorello et al., 2020), possibly related to
the variety of CE progenitors and outcomes possible.

Numerical simulations of CE evolution remain challenging (e.g., Ricker & Taam
2008; Ohlmann et al. 2016; MacLeod et al. 2018; Chamandy et al. 2020; Cruz-
Osorio & Rezzolla 2020; Sand et al. 2020, see also Ivanova et al. 2013b, 2020
for reviews). The main difficulties relate to the large range of temporal and
spatial scales that need to be resolved in CE evolution. Moreover, there are
several possible energy sources other than the orbit that need to be
considered during CE evolution, such as recombination energy (e.g., Ivanova et
al., 2013a; Ivanova, 2018; Sand et al., 2020), accretion energy (e.g., Voss &
Tauris, 2003; MacLeod & Ramirez-Ruiz, 2014), and energy released by nuclear
burning (e.g., Ivanova et al., 2002). Both where the energy is released, and
how it is transported through the CE also have a crucial role in determining
the outcome (e.g., Sand et al., 2020; Wilson & Nordhaus, 2020). Therefore, CE
evolution remains one of the major uncertainties in the modeling of binary
evolution and stellar populations.

The inspiral within a CE is not driven by gravitational-wave (GW) emission.
Nevertheless, the binary inside of a CE, consisting of the core of the donor
and the companion star, has a time-varying mass quadrupole moment and hence
radiates GWs. If detectable, GWs from the binary inside of a CE will passively
trace the gas-drag driven inspiral, and could provide invaluable insight into
the CE process. Conversely, because of the low density of the envelope of
giant stars (which is possibly further lowered by the energy injection during
the CE), any GW signal from the envelope itself is unlikely to be detectable.

The GWs radiating from a Galactic CE binary would be in the mHz band, and thus
could be detectable by space-based GW-detectors, such as the Laser
Interferometer Space Antenna (LISA, Baker et al. 2019), and TianQin (Huang et
al., 2020). This can be expected because of the existence of double-WD
verification binaries for LISA (e.g., Stroeer & Vecchio, 2006), which likely
went through at least one CE phase (e.g., Korol et al., 2017). LISA is
currently scheduled for launch in 2034 and will commence operations in the mHz
GW frequency range shortly thereafter. Its projected mission lifetime is
$5-10$ years, during which time it is expected to detect a wide variety of GW
signals in the mHz band, from supermassive black hole binary mergers to
Galactic compact binaries with a fraction of a solar mass.

Here, we examine the prospects for observing the inspiral of two Galactic
stellar cores within a CE with LISA. Such signals are not accessible to
ground-based detectors (e.g., LIGO – Aasi et al. 2015, Virgo – Acernese et al.
2015, and KAGRA – Somiya 2012) as they are restricted to frequencies above a
few to tens of Hz. At these frequencies, CE binaries are expected to either
have merged or have ejected the envelope. In fact, Ginat et al. (2020) showed
that LISA could detect the portion of the CE inspiral proceeding on a
dynamical timescale ($\sim$ days).

However, not all CE events are expected to rapidly complete their evolution
over a dynamical timescale (Ivanova et al., 2013b). A significant fraction of
the envelope may be ejected on thermal timescales much longer than the
dynamical timescale (e.g., Meyer & Meyer-Hofmeister, 1979; Glanz & Perets,
2018; Michaely & Perets, 2019). Currently, the duration and occurrence rate of
this so-called ‘self-regulated inspiral’ is very uncertain (e.g., Meyer &
Meyer-Hofmeister, 1979; Fragos et al., 2019; Sand et al., 2020; Igoshev et
al., 2020), but if it is of the order of years or longer, this phase is likely
to constitute the most promising CE target for LISA.

Constraints placed by low-frequency GW on CE evolution will be from direct
detection/non-detection. These constraints will be complementary to those
obtained by ground-based detectors which observe the aftermath of successful
CE events in massive binaries where the envelope has been ejected and the
binary merged under the influence of GWs. The properties and rate of such
detections can be used to place constraints on the CE properties, for example
the CE efficiency (e.g., Wong et al., 2020; Zevin et al., 2020). Space-based
detectors, on the other hand, offer the possibility of studying binaries while
they undergo CE, possibly including systems that might fail to eject the
envelope. Provided the signal can be distinguished from other potential mHz GW
sources, space-based detectors will give us direct insight into the gas-driven
dynamics.

In particular, the GW frequency of the signal ($f_{\mathrm{GW}}$), can
constrain the orbital separation at which the self-regulated inspiral stalls,
and the frequency change ($\dot{f}_{\mathrm{GW}}$) and event duration can
constrain the rapidity of the inspiral and thus the internal gas density of
the CE (and potentially its radial distribution). Even a non-detection would
offer invaluable constraints on the occurrence rate, the orbital separation
during the self-regulated inspiral, and the duration of this phase, which are
not accessible by any other means.

###  1.1 Phases of common envelope evolution

Figure 1: Schematic representaion of all possible phases during a common
envelope event. We focus on the most promising source of GW in the LISA band,
i.e. the thermal-timescale self-regulated phase at step d. However, dynamical
phases such as the plunge-in at step c or the final ejection or merger might
also be detectable (e.g., Ginat et al., 2020).

Because of the aforementioned complexity of CE evolution and the diversity of
binary systems that can evolve through this phase (both in terms of masses and
evolutionary stages at the onset of the CE), a complete and exhaustive picture
of this phenomenon does not yet exist. Nevertheless, several phases which
might subsequently occur during the CE have been identified (e.g.,
Podsiadlowski, 2001; Ivanova et al., 2013b; Fragos et al., 2019; Ivanova et
al., 2020), and are outlined in Figure 1. Below we emphasize which are
potentially interesting targets for GW observations. It is possible that not
every binary evolving through a CE will experience all of these phases
(Ivanova et al., 2013b; Law-Smith et al., 2020).

#### a. Onset of the common envelope

Mass transfer through a Roche lobe overflow (RLOF) can become dynamically
unstable and result in the initiation of a CE for several reasons:

  * (i)

an increasing rate of overflow – either because of the Roche lobe shrinks or
because it expands slower than the donor’s star radius;

  * (ii)

because the mass transfer timescale is much shorter than the accretor thermal
timescale, the accretor is pushed out of thermal equilibrium. This leads to
its radial expansion until the accretor too fills its Roche lobe (e.g., Nariai
& Sugimoto, 1976);

  * (iii)

because of the Darwin instability that redistributes angular momentum between
the stellar spins and the orbit (Darwin, 1879), causing the orbit to shrink;

  * (iv)

because of mass loss from the L2 Lagrangian point, causing large orbital
energy loss. This accelerates the evolution to proceed on a dynamical
timescale (e.g., Pejcha et al., 2017).

These processes can cause the accretor to enter the envelope of the donor,
starting the CE on a timescale that is either thermal or dynamical. Possible
pre-CE mass ejection can also occur, depending on what drives the CE
initiation (e.g., Pejcha et al., 2017; MacLeod et al., 2017; MacLeod & Loeb,
2020). When RLOF becomes dynamically unstable, the separation, $a$, between
the core of the donor and the accretor is still wide and of the order of the
Roche radius of the dono,r $a\gtrsim R_{\mathrm{RL,1}}$. Except for binaries
already involving two compact objects (e.g., AM CVn systems, Liu et al.,
2021), this is likely to correspond to a GW signal of frequency
$f_{\mathrm{GW}}$ too low for the LISA band.

#### b. Loss of co-rotation

Soon after the initiation of the CE, the envelope will rapidly stop co-
rotating with the binary (even under the optimistic assumption that the
envelope was initially co-rotating with the donor’s core). The timescale for
the loss of co-rotation depends on how the CE was initiated at step (_a_), but
it varies between the tidal timescale up to a thermal timescale (e.g.,
Podsiadlowski, 2001; Fragos et al., 2019). The loss of co-rotation still
happens at separations wider than could be detectable in the LISA band, but it
allows for the onset of strong gas-drag which drives the subsequent CE
evolution.

#### c. Dynamical plunge-in

After the loss of co-rotation, with the envelope density still only slightly
perturbed (Fragos et al., 2019), a phase of dynamical plunge-in happens. This
rapid inspiral can take (a fraction of) an orbital period and possibly
decrease the orbital separation $a$ to zero (i.e., a prompt merger).
Alternatively it can lead to a prompt ejection of the CE (e.g., Law-Smith et
al., 2020). However, this is not always the case. The orbital energy liberated
in this very fast plunge-in could expand the CE, leading to a decrease in gas-
drag and thus stalling the inspiral, setting up the conditions for phase (_d_)
(see below). At which separation $a_{\mathrm{post-plunge}}$ the plunge-in
stalls is likely not to be universal, and is not yet fully understood.
Podsiadlowski (2001) suggested that $a_{\mathrm{post-plunge}}$ corresponds to
the location where the orbital energy equates the envelope binding energy,
however De Marco et al. (2011) caution about the subtleties of how the binding
energy is defined. Depending on the pre- and post-plunge-in separations, this
dynamical phase could be detectable in GW for Galactic binaries (e.g., Ginat
et al., 2020). However, because of its short duration ($\sim$ hours – days),
it is not the most promising source (see Sec. 4). Should one happen within the
Galaxy during the LISA mission, it would produce a frequency change
$\dot{f}_{\mathrm{GW}}$ much larger than expected from GW emission, and
possibly be associated to a detectable EM transient. The plunge-in phase can
also produce an eccentric binary within the CE (Ivanova et al., 2013b; Sand et
al., 2020), with potential implications for the GW signal they might produce.

#### d. Self-regulated inspiral

Assuming the plunge-in did not produce a prompt merger, what follows is a
relatively slow phase of self-regulated inspiral, that can last several
thermal time-scales (10-$10^{5}$ years, e.g., Meyer & Meyer-Hofmeister 1979;
Clayton et al. 2017; Fragos et al. 2019; Chamandy et al. 2020; Igoshev et al.
2020). During the self-regulated inspiral, the quasi-Keplerian motion of the
binary slowed by gas-drag progressively injects orbital energy and angular
momentum in the CE. Other energy sources can intervene and radiative transport
becomes relevant during this CE phase (e.g., hydrogen recombination can occur
if it happens in a sufficiently optically thick layer, Fragos et al. 2019).
Depending on the separation $a_{\mathrm{post-plunge}}$ (and the masses of the
binary), this phase might be the most promising for detecting a GW signal,
since it could allow for a slowly222We still expect $\dot{f}_{\mathrm{GW}}$ to
exceed the value predicted by general relativity for GW-driven inspiral in
vacuum. varying $f_{\mathrm{GW}}$ that can be integrated over several years to
build up signal-to-noise ratio (SNR). However, we emphasize that this self-
regulated inspiral might not happen for all CE events (see for example the
short timescale for subdwarf B-type progenitors inferred by Igoshev et al.
2020, or the numerical results for neutron star progenitors by Law-Smith et
al. 2020).

#### e. CE ejection or merger

Finally, the CE will end with one last dynamical phase during which either the
envelope is ejected (possibly because of the intervention of energy sources
other than the orbit) or one of the stars of the binary inside the CE fills
the equipotential surface distinguishing it from the other (improperly, its
Roche lobe) leading to the final merger (e.g, Podsiadlowski, 2001; Ivanova et
al., 2002). In the cases where the CE is successfully ejected, the surviving
binary is left with the final $a_{\mathrm{post-CE}}$ orbital separation
smaller than the initial radius of the donor. What governs the binary post-CE
orbital evolution depends on the specific details of the system (magnetic
braking, tidal evolution, wind mass loss, or possibly GW emission). The short
duration of phase (_c_), phase makes it an unpromising target for GW searches.
However, it would result in a large $\dot{f}_{\mathrm{GW}}$ for a brief
moment, and likely be accompanied by a detectable EM transient.

In the rest of this study, we will focus on potential signals from a long
lived, self-regulated inspiral (step _d_) unless otherwise stated. In Sec. 2
we show that CE binaries of various mass are expected to cross or end within
the LISA GW frequency band. We then consider the issue of whether a typical
binary crossing the LISA band can be detected in Sec. 3, and use population
synthesis simulations to estimate the number of sources expected at any given
time in the Galaxy in Sec. 4. Sec. 5 deals with the possible “stealth bias”,
that is the possibility of misinterpreting the detection of GWs from a CE as
another kind of GW source, and we consider how EM observation can help
resolving this bias in Sec. 6. We further discuss our results in Sec. 7 before
concluding in Sec. 8.

##  2 Common-envelope evolution across the LISA band

For a CE event to cross the LISA frequency band and produce a detectable GW
signal, the mass of the binary inside the CE needs to be sufficiently large.
This favors systems where both stars are evolved and have dense and well
defined cores (or are only what is left of their original core, like WDs). On
top of this, the separation of the binary inside the CE needs to correspond to
GW frequencies roughly $10^{-4}\,\mathrm{Hz}\lesssim f_{\mathrm{GW}}\lesssim
0.1\,\mathrm{Hz}$ (gray band in Fig. 2), bound at the low end by the
population of unresolved Galactic WD and at the high end by the instrument
response function (e.g., Robson et al., 2019). The existence of verification
WD binaries in the LISA band, which are likely to have formed through a CE
event, suggests that CE binaries crossing the LISA band may exist.

Figure 2: Estimates of the post-CE separations using the $\alpha\lambda$
energy formalism. Given a pre-CE separation (different colors in plot) and the
GWs emitting mass (x-axis), the post-CE separation can be determined. We show
a range of GW-emitting masses, consisting of the core mass of the donor star
plus the mass of the companion WD (0.3$\,M_{\odot}$ in this plot). We assume
$\alpha=1$ and $\lambda=0.3$ for illustrative purposes. The gray band shows
where the corresponding GW signal approximately be in the LISA frequency range
($10^{-4}\,\mathrm{Hz}\lesssim f_{\mathrm{GW}}\lesssim 0.1\,\mathrm{Hz}$).
Systems with post-CE separations within or below this band therefore
potentially experience a phase during which their GW emission could be
detectable by LISA. The hatched area corresponds to post-CE separation smaller
than the core radius, that is presumably merging systems.

To illustrate that some binaries will indeed cross the LISA band during a CE
we use a population-synthesis approach (see also Sec. 4). In rapid population
synthesis calculations, CE is typically treated as an instantaneous event.
This is usually a sufficient approximation since the duration of a CE is at
most of order of the thermal timescale (depending on the mass, luminosity and
radius of the CE this ranges from several hundreds to several thousand years),
that is much shorter than the entire evolution of a binary system. To relate
the pre-CE initial conditions to the estimated post-CE configuration,
conservation of energy is usually used ($\alpha\lambda$-algorithms, see e.g.,
Webbink, 1984; De Marco et al., 2011), although algorithms considering
conservation of angular momentum also exist (e.g., Nelemans et al., 2000).

Figure 2 shows the post-CE separation ($a_{\mathrm{post-CE}}$) as a function
of the mass that would be contributing to the GW emission ($M_{\mathrm{GW\
emitting}}$). In a rapid population synthesis calculation, each binary system
would drop instantaneously from its pre-CE separation to the post-CE
separation (as indicated by the vertical arrows). For illustrative purposes,
we assume fully efficient use of the orbital energy to eject the envelope and
no extra energy source (i.e., $\alpha=1$), and $\lambda=0.3$ as a typical
value for the binding energy parameter of giant donors (e.g., De Marco et al.,
2011). Increasing either value shifts the curves to larger post-CE
separations.

In Fig. 2 we assume $M_{\mathrm{GW\ emitting}}$ to be the (helium) core mass
of the donor star at the end of the main sequence plus $0.3\,M_{\odot}$
assumed to be the mass of a WD companion. We chose this value to reflect the
highest rate of initiation of Galactic CE events from the simulations
described in Sec. 4, which is expected for systems with
$M_{\mathrm{core}}\simeq 0.5\,M_{\odot}$ and $M_{\mathrm{WD}}\simeq
0.3\,M_{\odot}$. Although the combination of core masses with this particular
WD mass might not be realistic throughout the range shown, we have also
performed the same calculation assuming a more massive companion of
$1.4\,M_{\odot}$ (mimicking a Chandrasekhar mass WD or a neutron star). The
resulting curves are shown in Fig. 6 in Appendix C, each curve moves up to
larger post-CE separations, but most curves still intersect the LISA band
(gray area). For large core-radii the final dynamical phase (e. in Fig. 1
above) might start at frequencies in the LISA band (hatched area in Fig. 2).
The systems ending at separation smaller than the He core radii could result
in mergers rather than successful common envelope ejections. Whether this
leads to a detectable GW signal or not requires further investigation.

We use the solar metallicity models from Pols et al. (1998) to obtain the
(helium) core mass of the donor from their total masses. Our core masses are a
lower bound since we neglect the core-growth due to the ashes of (hydrogen)
shell burning. The gray shaded area indicates core separations corresponding
to a GW signal of frequency $10^{-4}\,\mathrm{Hz}\lesssim
f_{\mathrm{GW}}\lesssim 0.1\,\mathrm{Hz}$, roughly within the LISA band.
Binaries crossing this gray area or ending within it (e.g., verification WD
binaries) are potential candidates for GW detection of the CE evolution with
LISA, depending on the duration of the CE event. The duration cannot be
obtained from a population synthesis approach, it either needs to be modeled
with challenging multi-dimensional hydrodynamic simulations or, as we will
argue below, it might be directly constrained with observation of the GW
signal.

We return to a full population synthesis calculation of the rate of events in
Sec. 4, after discussing the observability of GW from a CE in Sec. 3.

##  3 Detectability of common envelope in LISA

Figure 3: LISA signal-to-noise ratio of a $M_{\mathrm{core}}=0.5\,M_{\odot}$
and $M_{\mathrm{WD}}=0.3\,M_{\odot}$ binary as a function of its initial
observed GW frequency $f_{\mathrm{GW}}$ and frequency derivative
$\dot{f}_{\mathrm{GW}}$ (blue shading and black solid lines). For the duration
of the observations, we assume the shortest between 5 years or the time to
merger at constant $\dot{f}_{\mathrm{GW}}$, and adopt 3 kpc as the source
distance. The top axis shows the orbital separation corresponding to
$f_{\mathrm{GW}}$ for the masses considered. Although the exact
$f_{\mathrm{GW}}$ and $\dot{f}_{\mathrm{GW}}$ values for a binary undergoing
CE are not known, the dashed and solid orange lines illustrate lower and upper
bounds on the expected values of $\dot{f}$ due to CE evolution. The former
assumes pure GW emission (see Appendix A), and the latter assumes pure gas
drag with no feedback (see Appendix B). Finally, dotted and dot-dashed black
lines mark different regimes in which we may or may not expect to successfully
distinguish a CE event from other sources of gravitational radiation (see Sec.
5). Frequency derivatives above the dotted line are likely detectable after
five years of observation, and systems above the dot-dashed line are further
expected to have measurable second derivatives $\ddot{f}_{\rm GW}$, enabling
measurement of braking indices. Systems lying below these thresholds,
meanwhile, are most likely to suffer from stealth bias or mis-identification.
The red crosses indicate known verification WD binary.

We now explore in what regimes we expect LISA to successfully detect GW
signals from binaries undergoing CE. Since the details of CE are not yet fully
understood, we do not know the precise frequency evolution of a GW source
undergoing CE. Instead, in Fig. 3 we agnostically quantify the expected
signal-to-noise ratios (SNRs) with which LISA will observe a $M_{\mathrm{GW\
emitting}}=0.5+0.3\,M_{\odot}$ binary with an initial GW frequency $f_{\rm
GW}$ and frequency derivatives $\dot{f}_{\rm GW}$ at a distance of 3 kpc. We
chose a value for the typical mass of the donor star core ($0.5\,M_{\odot}$)
and the WD companion ($0.3\,M_{\odot}$) from our population synthesis (see
Sec. 4). Expected SNRs have been averaged over possible sky locations and
binary inclinations. Throughout, we assume the standard quadrupole radiation
such that $f_{\rm GW}=2f_{\mathrm{orb}}$, where $f_{\mathrm{orb}}$ is the
(quasi-) Keplerian orbital frequency of the CE binary. We further assume the
observation to persist for five years333The SNR is proportional to the square
root of the observing time., or until the given binary’s frequency reaches
$f_{\rm GW}=10^{-1.5}\,{\rm Hz}$. Beyond this frequency, the He core would
fill its Roche lobe within the CE, initiating the dynamical phase that leads
to a prompt merger (bottom right-hand side of Fig. 1) and termination of the
GW signal. Additional details of the SNR calculation are given in Appendix A.

Although the true rate $\dot{f}_{\mathrm{GW}}$ at which a binary’s GW
frequency evolves under CE is not known, we can nevertheless place sensible
bounds on frequency evolution of a CE binary. In Fig. 3, the orange dashed
line shows the frequency derivative $\dot{f}_{\mathrm{GW}}(f_{\mathrm{GW}})$
due solely to the emission of GWs. For a given total mass, this is a strict
lower bound on the evolution rate of a binary undergoing CE. Meanwhile, the
orange solid line shows the value of $\dot{f}_{\mathrm{GW}}(f_{\mathrm{GW}})$
expected in the case of pure gas drag, neglecting the reaction of the envelope
and assuming $\rho=10^{-6}\,\mathrm{g\ cm^{-3}}$, $T=9\times
10^{5}\,\mathrm{K}$ to represent the density and temperature deep inside the
shared CE (see Appendix B). This provides a strict upper-limit, since in
nature we expect the envelope to expand due to the energy injected into the
envelope by the inspiral, lowering the gas drag on the binary inside the CE.
Generally, we expect CE binaries to lie between these two bounds. Under these
constraints, we see that a systems occupying a large portion of the remaining
$f_{\mathrm{GW}}-\dot{f}_{\mathrm{GW}}$ phase space are detectable with LISA.
In particular, systems whose gravitational wave frequency is
$f_{\mathrm{GW}}\gtrsim 10^{-3}\,{\rm Hz}$ (corresponding to orbital
separations less than $\sim{}0.3\,R_{\odot}$ for our fiducial masses) yield
${\rm SNR}\gtrsim 10$.

Although galactic binaries undergoing self-regulated CE evolution may be
detectable by LISA, they are not necessary identifiable as such. Without
additional information or careful consideration, we run the risk of
misidentifying such a system as a purely GW-driven compact binary, incorrectly
attributing an anomalously high $\dot{f}_{\mathrm{GW}}$ to large component
masses rather than accelerated evolution due to CE. This issue of
identification will be discussed further in Sec. 5 below.

##  4 Estimate of the number of Galactic common-envelope binaries

To determine the number of CE binaries in the Galaxy that LISA (and future GW
observatories) might be able to detect, we use the population synthesis code
COSMIC444https://cosmic-popsynth.github.io/. Our input files and results are
available at https://zenodo.org/record/4490011. version 3.2 (Breivik et al.,
2020). As in all rapid binary population synthesis codes, COSMIC treats CE as
an instantaneous event, therefore, we obtain from COSMIC the rate at which CE
event are initiated ($R_{\mathrm{CE,init}}$) by scaling the number of binaries
initiating a CE with the total mass of the population and multiplying by a
star formation rate (SFR) of $3.5\,M_{\odot}\ \mathrm{yr^{-1}}$ (commonly used
for Milky-way equivalent galaxies in population synthesis calculations of GW
sources, e.g., Dominik et al., 2012). The value of $R_{\mathrm{CE,init}}$ can
be rescaled for an arbitrary SFR, which anyway does not constitute the
dominant uncertainty in the determination of the number of Galactic sources.

Population synthesis calculations assume a CE event occurs if one of the
following condition is met: (i) both stars simultaneously fill their Roche
lobe (e.g., for initially equal mass ratio systems), (ii) one of the stellar
radii exceeds the periastron point – for eccentric binaries, or (iii) when the
mass transfer is deemed to be unstable. Case (iii) is most common. Mass
transfer stability is decided in COSMIC using critical mass ratio values
$q_{c}$ (Breivik et al. 2020, see also Hurley et al. 2002, Pavlovskii et al.
2017, Vigna-Gómez et al. 2020). Whenever the mass ratio between the donor star
and the accretor is larger than a given threshold
$M_{\mathrm{donor}}/M_{\mathrm{accretor}}\geq q_{c}$, mass transfer is defined
to be unstable and a CE is initiated. Different $q_{c}$ values can be used for
each evolutionary phase of the donor star, to reflect the changes in thermal
timescales as stars evolve.

In our fiducial run – labeled as “Clayes et al. 14 $q_{c}$” in Fig. 4, we
assume the donor-type dependent $q_{c}$ values555However, Claeys et al. (2014)
define the critical mass ratio as $M_{\mathrm{accretor}}/M_{\mathrm{donor}}$.
from Tab. 2 of Claeys et al. (2014), a metallicity $Z=0.02$, and the
correlated initial distributions in primary mass, mass ratio, period, and
eccentricity from Moe & Di Stefano (2017). We explore several variations of
our fiducial run, keeping the same setup except one of the following:

  * •

constant $q_{c}$ for all evolutionary phases of the donor star. We consider
values from $q_{c}=0.1$ (more likely to enter CE) to $q_{c}=2$ (less likely to
enter CE);

  * •

lower metallicity $Z=0.002$;

  * •

independent and uncorrelated initial distributions – with a Kroupa et al. 1993
initial mass function for the primary stars, a flat mass-ratio distribution, a
log-normal period distribution (e.g., Duchêne & Kraus, 2013), and a thermal
eccentricity distribution.

Figure 4: Rate at which CE events are initiated $R_{\mathrm{CE,init}}$ as a
function of the minimum pre-CE separation (left) and minimum total mass
emitting GWs (right). The solid lines include all CE events, while the dashed
line only include CE where both stars are beyond the main sequence, or in
other words, where two dense stellar cores are involved. Different colors
correspond to different assumptions on the stability of mass transfer
($q_{c}$, see also text) and initial distribution of binaries (yellow).

COSMIC is a Monte Carlo population synthesis code (Breivik et al., 2020), and
for each population described above, we stop drawing systems when the sampling
uncertainty in the initial distributions is less than $10^{-5}$, that is
drawing more systems has an effect smaller than this threshold. More
specifically, we check the initial primary and secondary mass distributions in
the range [0,80] $M_{\odot}$, and initial orbital period below 1000 days. The
number of systems simulated to reach this threshold ultimately determines the
total mass of the simulated population, which we use to determine
$R_{\mathrm{CE,init}}$.

These parameter variations do not exhaust the entire range of uncertainties.
For example, the fraction of mass that is accreted during stable mass transfer
can influence the orbital evolution of the binary, and thus modify the chances
of a subsequent reverse CE from the initially less massive star to the
initially more massive (e.g., Zapartas et al., 2017). However, we expect the
impact of other parameters to be smaller than the impact of varying $q_{c}$
and the initial distributions for binaries.

We find a rate of CE initiation regardless of the masses involved of
$R_{\mathrm{CE,init}}=0.18^{+0.02}_{-0.09}\,\mathrm{yr^{-1}}$ for all CE
events, and $0.06^{+0.03}_{-0.02}\,\mathrm{yr^{-1}}$ when considering only CE
between two evolved stars (see below). The uncertainties on
$R_{\mathrm{CE,init}}$ are significantly smaller than the uncertainty on the
duration of the CE (cf. Sec. 1.1).

Figure 4 shows the rate of CE initiation that we obtain as a function of the
minimum pre-CE separation (left) and the minimum mass emitting GWs (right),
i.e. the rate of events with pre-CE separation (or GW emitting mass) indicated
by the abscissa or larger. For example, for our fiducial simulation, about
$1.6\times 10^{-2}$ CE events are initiated per year with a pre-CE separation
$a_{\mathrm{pre-CE}}\geq 10^{3}\,\mathrm{R_{\odot}}$. The solid lines in Fig.
4 show all CE events initiated, while the dashed lines focus on CE events
where both stars are evolved beyond the main sequence, meaning both have a
well defined, dense core. The latter (dashed lines) represent systems that are
more likely to be detectable in GWs.

When considering all CE events, higher $q_{\mathrm{crit}}$ correspond to lower
$R_{\mathrm{CE,\ init}}$. Conversely, CE with both stars post-MS, i.e. with a
(semi-) degenerate core, might not be the first mass transfer episode in a
given binary, and their occurrence and rate depend also on what is assumed for
previous mass transfer phases. The CE initiation rate is rather flat for pre-
CE separations as large as $a_{\mathrm{pre-CE}}\lesssim 700\,R_{\odot}$, while
it drops very quickly with increasing $M_{\mathrm{GW\ emitting}}$ because of
the initial mass function.

To determine the mass emitting GWs in each binary, we assume the following: if
a star (donor or accretor) is on the main sequence (MS), that is, it lacks a
well defined core, we use its total mass in the mass emitting GWs. If a star
has evolved beyond the MS, only the core mass adds to the GW emitting mass. If
a star has become a compact object (WD, neutron star or black hole) again the
total mass of the object adds to the GW emitting mass. There are two edge
cases where our assumptions might overestimate the amount of mass involved in
GW-emission as the total mass of the binary: CE between two MS stars, and
between two compact stars (e.g., a double-degenerate SNIa progenitor, Dan et
al. 2011). However, this has very little effect on our conclusions. With our
initial period distribution, CE between two MS stars are relatively rare, and
they are unlikely to be the most interesting target for LISA, since the lack
of dense cores limits the mass involved in the generation of GWs and thus the
signal amplitude. Setting instead the total mass emitting GWs to zero for
these systems has a negligible effect on the right panel of figure 4. CE
between two compact stars do not occur with our fiducial setup.

Assuming the stellar population of the Galaxy is in equilibrium, an estimate
of the current number of Galactic CE can then be obtained with

$N_{\mathrm{CE}}=R_{\mathrm{CE,init}}\times\Delta t_{\mathrm{CE}}\ \ ,$ (1)

where $\Delta t_{\mathrm{CE}}$ is the typical duration of a CE event, which is
not known. Moreover, what matters for GW detection is not the actual duration
$\Delta t_{\mathrm{CE}}$, but the fraction of it spent at a separation
corresponding to a GW frequency in the detector bandpass.

Depending on the evolutionary stage of the CE binary and the model assumed,
estimates of the CE duration $\Delta t_{\mathrm{CE}}$ can range from dynamical
timescale ($\sim$ days, e.g., Ginat et al. 2020; Law-Smith et al. 2020) to
thermal timescale ($\sim 10^{5}\,\mathrm{years}$), although the radial
expansion and luminosity increase of the envelope during the CE are likely to
reduce the relevant thermal timescale to $10^{3}-10^{4}$ years (e.g., Meyer &
Meyer-Hofmeister, 1979; Clayton et al., 2017; Igoshev et al., 2020). Recent
hydrodynamical simulations suggest a $\sim$$10$ years duration (e.g., Fragos
et al., 2019; Chamandy et al., 2020).

Based on these previous estimates, assuming a short duration of the CE of
$\sim$ years, the rate of CE at the low mass end ($R_{\mathrm{CE,init}}\approx
0.1$ yr-1) is sufficiently large that at least one source might be detectable
during a 10-year LISA mission. Considering only CE with evolved stars (dashed
lines), $R_{\mathrm{CE,init}}$ drops to $\sim 0.5\times 10^{-2}$ yr, making
the expected number of sources only marginally non-zero.

If the self-regulated phase of the CE instead lasts much longer than the
planned LISA mission, there might be upwards of hundreds of detectable sources
in the Galaxy ($R_{\mathrm{CE,init}}\approx 0.1$ yr-1 $\times 10^{4}$ yr
$\simeq 1000$). Thus, under the most pessimistic assumptions, there might be
only a slim chance of detecting a CE event through LISA, while under more
optimistic, but not extreme, assumptions there could be a significant number
of target sources available. We return on the possible implications of a non-
detection in Sec. 7.2.

##  5 Stealth bias

There is an ever growing catalog of potential systems that LISA might be
sensitive to, both galactic and extragalactic. Examples include exoplanets
(e.g., Tamanini & Danielski, 2019), WD binaries in various stages of their
evolution (e.g., Kupfer et al., 2020; Liu et al., 2021), and binary black
holes (BBH) in vacuum or embedded in gas (e.g., Chen et al., 2020b). Our study
suggests that CE events are also potentially detectable by LISA, raising the
question: how can we distinguish GWs from a CE from these other possibly
confounding sources? In this section we discuss GW-based ways that could be
used to conclude that a detected signal is a CE event and the expected bias if
we fail to do so. We discuss further EM-based ways of resolving this stealth
bias in Sec. 6.

Depending on the exact frequency evolution of the CE event, there may exist an
observational degeneracy between the CE inspiral and purely GW-driven binaries
of higher mass hosting black holes and/or neutron stars. In standard GW
analysis that assumes a binary in vacuum is described by general relativity
(GR), a measurement of a binary’s $f_{\rm GW}$ and $\dot{f}_{\rm GW}$ can be
used to estimate its mass and distance. If a particular binary is actually
undergoing CE, the assumption of vacuum breaks down and the resulting
estimates will be biased compared to their true values (see also Chen et al.,
2020b, for an extended discussion).

This bias is quantified in Fig. 5, which shows the chirp mass

$M_{c}=\frac{(M_{1}M_{2})^{3/5}}{(M_{1}+M_{2})^{1/5}}\ \ ,$ (2)

(color in top) and distance ($D$, color in bottom) that we would incorrectly
assign to a $0.5+0.3\,M_{\odot}$ binary undergoing CE at $3$ kpc, if we have
measured only its gravitational-wave frequency and frequency derivative (and
the signal amplitude). We note that for our example binary, the correct value
of the chirp mass would be $M_{c}=0.34\,M_{\odot}$ (pink line in the top
panel). Depending on the exact values of ($f_{\mathrm{GW}}$,
$\dot{f}_{\mathrm{GW}}$), we find the possibility of large biases in the
inferred parameters, some of them being consistent with BBHs at cosmological
distances.

We identify three possible means of breaking this degeneracy between BBHs and
CE events:

  * (i)

Increased observation time could result in a measurement of
$\ddot{f}_{\mathrm{GW}}$ (and higher derivatives) in addition to
$f_{\mathrm{GW}}$ and $\dot{f}_{\mathrm{GW}}$, enabling a measurement of the
binary’s _braking index_. This scenario is further discussed in Sec. 5.1.

  * (ii)

The (incorrectly) inferred chirp mass may be high enough that the system is
consistent with a BBH. In this case, one would expect to continue to see the
signal evolve, and eventually observe the merger in the LIGO band. Failure to
observe the final merger, or a signal that is different from a BBH merger
signal in vacuum GR would indicate that the binary may have been
misidentified.

  * (iii)

The simultaneous measurement of an apparent high chirp mass and a low distance
might indicate the presence of a BBH close to Earth, potentially even in the
Galaxy for certain combinations of $f_{\mathrm{GW}}$ and
$\dot{f}_{\mathrm{GW}}$. Such a system might be inconsistent with independent
information about the BBH inspiral and merger rates obtained from ground-based
detectors. This possibility is further discussed in Sec. 5.2.

If these outcomes fail, the possibility of a _stealth_ bias on the system
parameters remains present, it may be impossible to conclusively identify the
true nature of the given binary.

Figure 5:  Inferred chirp mass (top) and distance (bottom) for each measured
$f_{\mathrm{GW}}$ and $\dot{f}_{\mathrm{GW}}$ for a system with
$M_{\mathrm{core}}=0.5\,M_{\odot},M_{\mathrm{WD}}=0.3\,M_{\odot}$ at $3$ kpc.
Similar to Figure 3, we use a a dot-dashed line to show the border above which
$\ddot{f}_{\mathrm{GW}}$ can be measured and the nature of the binary can be
determined through the braking index measurement (see Sec. 5.1). Below that
line there is a possible stealth bias, which these two plots quantify. Above
the dotted line, frequency derivatives are likely detectable after five years
of observation. Systems lying below the dotted line would be observed as
stationary sources. The magenta line corresponds to the correct value of the
chirp mass and distance, namely the GR relation between $f_{\mathrm{GW}}$ and
$\dot{f}_{\mathrm{GW}}$ for these masses. Hence the bias vanishes.

###  5.1 Measuring the braking index

Stellar binaries evolving purely via GW radiation have GW frequencies that
evolve in time as $\dot{f}_{\mathrm{GW}}\propto f_{\mathrm{GW}}^{11/3}$. In
general, given some source evolving as $\dot{f}\propto f^{n}$, a measurement
of the braking index $n$ given by

$n=\frac{f\ddot{f}}{\dot{f}^{2}}\ \ ,$ (3)

is possible if the second time derivative $\ddot{f}$ is measurable. We
emphasize the analogy with the pulsar braking index, where $f$ is the pulsar
spin frequency rather than the frequency of GW radiated by the CE binary.

If LISA can successfully measure $f_{\mathrm{GW}}$, $\dot{f}_{\mathrm{GW}}$,
and $\ddot{f}_{\mathrm{GW}}$ for a given GW source, inference on the source’s
braking index offers a means of identifying the physical processes driving its
evolution. In particular, we can seek to verify if $n$ is consistent with
$n_{\mathrm{GW}}=\frac{11}{3}$, as expected for pure gravitational radiation
in vacuum, or inconsistent with $n_{\mathrm{GW}}$, possibly indicating a
binary in the midst of CE or subject to other environmental influences (e.g.,
Fedrow et al., 2017).

While LISA is expected to measure $\dot{f}_{\mathrm{GW}}$ for a considerable
number of binaries, $\ddot{f}_{\mathrm{GW}}$ is likely measurable only in a
much smaller subset of cases. Although determining when
$\dot{f}_{\mathrm{GW}}$ and $\ddot{f}_{\mathrm{GW}}$ are measurable requires
expensive simulations (Seto, 2002; Littenberg et al., 2020), we can make an
order-of-magnitude estimate by considering how LISA’s frequency resolution
scales with time. As LISA accumulates more data, its frequency resolution
scales with the observing time $T$ as $\Delta f_{\mathrm{GW}}\approx 1/T$.
Accordingly, we assume that $\dot{f}_{\mathrm{GW}}$ and
$\ddot{f}_{\mathrm{GW}}$ are measurable when they cause a binary’s frequency
to change enough such that it moves from one frequency bin to another over the
course of LISA’s lifetime. Then, lower limits on the minimum detectable
$\dot{f}_{\mathrm{GW}}$ and $\ddot{f}_{\mathrm{GW}}$ are

$\displaystyle\dot{f}_{\rm GW,min}T$ $\displaystyle=\Delta f_{\rm
GW}\quad\Rightarrow\quad\dot{f}_{\rm GW,min}\approx\frac{1}{T^{2}}\ \ ,$ (4)
$\displaystyle\frac{1}{2}\ddot{f}_{\rm GW,min}T^{2}$ $\displaystyle=\Delta
f_{\rm GW}\quad\Rightarrow\quad\ddot{f}_{\rm GW,min}\approx\frac{2}{T^{3}}\ \
.$ (5)

We refer the readers to Takahashi & Seto (2002) and Robson et al. (2018) for
more accurate estimates accounting for the SNR dependence.

On the $\dot{f}_{\mathrm{GW}}-f_{\mathrm{GW}}$ plane, the first condition is a
straight line depicted by the black dotted horizontal line in Fig. 3 and 5. We
can also identify the region of this plane in which $\ddot{f}_{\rm GW}$ is
measurable for a system evolving in vacuum GR. Using the definition of braking
index (Eq. 3) to express $\dot{f}_{\mathrm{GW}}$ as a function of
$f_{\mathrm{GW}}$ and $\ddot{f}_{\mathrm{GW}}$, a binary evolving purely via
GW emission with the minimum detectable $\ddot{f}_{\rm GW}$ (Eq. 5) has a
frequency derivative

$\dot{f}^{2}_{\rm GW}=\frac{3}{11}\ddot{f}_{\rm GW,min}f_{\rm GW}\ \ .$ (6)

This relation is plotted in Figs. 3 and 5 with black dot-dashed line. Consider
a system lying above this line. Such a system _should_ have a measurable
$\ddot{f}_{\rm GW}$ (and hence a measurable braking index) if it evolves via
GW emission. Therefore, if this system’s $\ddot{f}_{\rm GW}$ or $n$ is not
measured, or is measured to be different than the GR prediction, then the non-
vacuum or non-GR nature of the binary can be inferred. Next, we consider
systems lying below the line defined by Eq. (6). For such systems, GW
evolution alone cannot lead to a detectable $\ddot{f}_{\rm GW}$. If
$\ddot{f}_{\rm GW}$ is nevertheless measured, one would infer that the binary
does not evolve solely under vacuum GR effects. If $\ddot{f}_{\rm GW}$ is not
measured, then it is not possible to determine the nature of the binary.

###  5.2 Distinction from other sources bases on rates

One potential way to infer the nature of sources that cannot be unambiguously
identified using only their GW signature is through their rate of occurrence.
For example, if the rate of CE detections in LISA is higher than the
uncertainty in the rate of BBH mergers, one can statistically filter out the
over-abundance of BBH detections in LISA when comparing to the BBH merger rate
from ground-based detectors.

As of LIGO/Virgo O3a, the rate of BBH mergers
(29.9${}^{+14.9}_{-8.6}$Gpc-3yr-1, Abbott et al., 2020a, b) is relatively
poorly constrained due to the small sample size of BBH events and the
dependence on the assumed BH mass distribution. However, when LISA will become
operational (around the year 2035, Amaro-Seoane et al., 2017) the BBH catalog
size from ground based detectors will have increased to approximately
$10^{4.5}$–$10^{5}$ BBH detections per year (e.g. Baibhav et al., 2019). This
means that the constraints on the BBH merger rate will have improved
drastically.

The LISA mission will result in an independent measurement of the rate of BBH
mergers, which will be based on the GW signals from wide (in-spiraling) BBH
systems. If the BBH merger rate as inferred from LISA is significantly higher
than the, by then, well constrained BBH merger rate from ground based
detectors, one could argue that this overabundance could originate from, for
example, CE-events.

Moreover, the BBH merger rate from ground based detectors will be sufficiently
constrained to allow for a split based on event parameters such as BH mass and
spin. This will enable us to constrain, for example, the rate of massive BHs
($M_{\rm{BH}}\sim 40$ M⊙) in the local Universe. CE-events might appear as
more massive BBH events (see Fig. 5)

Whether or not the discrepancy between the BBH merger rates inferred from LISA
with respect to the ground based detectors will be significant, will depend
both on the size of the discrepancy and the uncertainty in the rate
measurement from LISA. If we assume that such an overabundance in the rate
comes from CE-events, then the size of the discrepancy is directly related to
the rate of CE events. Based on our discussion in Sec. 4 we expect the number
of CE events detectable by LISA, to lie between 0.1 and 1000. Although this
estimate spans several orders of magnitude, we will be able to constrain the
upper limit of this estimate (corresponding to long lasting CE-events with
$\Delta t_{\mathrm{CE}}\propto 10^{4}$yr), given the uncertainty in the
stellar mass BBH-merger rate in LISA.

##  6 Sky localization and electromagnetic counterparts

EM transients could provide interesting triggers for GW searches, and in this
case the probability of misinterpreting the signal might be lower. In fact,
the self-regulated inspiral phase is expected to follow the dynamical plunge-
in phase (step (_c_) in Fig. 1), which might cause optical (e.g., LRNe Soker &
Tylenda, 2003; Kulkarni et al., 2007; Ivanova et al., 2013a; Pastorello et
al., 2019) and infrared (IR) transients (e.g., SPRITES Kasliwal et al., 2017).

An example of what a possible EM precursor could look like is V1309Sco
(Tylenda et al., 2011), and given that the initial parameters of the binary
stars are known fairly well, it would be interesting to monitor this object
for possible GWs once LISA is operational. Concerning the characterization and
discovery of EM precursors, the spectral evolution and late-time observations
of V1309Sco suggest that LRNe could produce large amounts of dust and be
particularly bright in the infrared (IR, Pejcha et al., 2017; Metzger &
Pejcha, 2017; Iaconi et al., 2020; Nicholls et al., 2013; Jencson et al.,
2019; Blagorodnova et al., 2020). Therefore, it is conceivable that some of
the transients associated with CE events might only be observable in the IR.
The SPitzer InfraRed Intensive Transients Survey (SPIRITS) detected unusual
infrared transients with no optical counterparts (Kasliwal et al., 2017).
These transients, dubbed eSPecially Red Intermediate-luminosity Transient
Events (SPRITEs), are in the infrared luminosity gap between novae and
supernovae, and could be associated with stellar mergers and CE events.

Another way to identify possible interesting targets where the CE event might
have been initiated too long ago to detect its beginning is through peculiar
circumstellar environments. An example could be TYC 2597-735-1 which was
interpreted as a merger product based on its “blue-ring nebula” but still
exhibits variability (Hoadley et al., 2020, although its variability is not
necessarily related to a still ongoing merger).

Conversely, in the case of a GW detection, the interpretation of the signal
might be helped by EM followup. This might be necessary to distinguish a CE
signal from other potential mHz GW sources (e.g., AM CVns, sdO+WD binaries).
As mentioned before, the most promising GW signal for LISA does not come from
the dynamical phase of a CE associated to the brightest EM transients (e.g.,
LRNe). Instead, the long-lasting self-regulated inspiral produces a possibly
slowly-evolving GW source. The LISA sky localization is strongly dependent on
the signal and its SNR, for a monochromatic source typical uncertainty
ellipses are of order tens of degrees squared (Cutler, 1998).

During the self-regulated phase, the CE binary might look like a red giant
with the following possible features. The surrounding circumstellar material
might be peculiar: a CE is initiated by an unstable phase of RLOF (step _a_ in
Fig. 1), which can be non-conservative and spill over in the surroundings of
the system (e.g., Pejcha et al., 2017; MacLeod & Loeb, 2020). One-dimensional
CE simulations suggest that periodic outbursts of mass ejection during the
self-regulated phase are possible (e.g., Clayton et al., 2017). Thus, a CE
binary might appear as a red giant with an excess of circumstellar material
and peculiar time-variability.

##  7 Discussion

###  7.1 Comparison to other common envelope rate estimates

Assuming a steady-state stellar population in the Galaxy, the number of CE
binaries viable as GW sources depends on the initiation rate
$R_{\mathrm{CE,init}}$ and the time duration when the system might be
detectable. While presently the latter needs to be estimated from numerical
simulations (although, see also Igoshev et al. 2020), the former can be
constrained using observations of EM transients associated with CE events.

The most relevant EM transient to constrain the rate of CE are LRNe, which
have been conclusively linked to a CE event via observations of the period
decay preceding the outburst event in V1309Sco (Tylenda et al., 2011).
Kochanek et al. (2014) estimate the Galactic rate of these events with $V$
($I$) band magnitude brighter than $-3$ ($-4$) to be $\sim
0.5\,(0.3)\,\mathrm{yr^{-1}}$, in reasonable agreement with our
$R_{\mathrm{CE,init}}=0.18^{+0.02}_{-0.09}\,\mathrm{yr^{-1}}$ (cf. Sec. 4).
Their estimate was based on the then available sample of 4 events in 25 years
and correction for observational biases. Kochanek et al. (2014) also noted the
good agreement with population synthesis results obtained with the StarTrack
code. More recently, Howitt et al. (2020) used the COMPAS population synthesis
code to suggest a Galactic rate of LRNe of 0.2$\,\mathrm{yr^{-1}}$, which is
in agreement with our estimates obtained with COSMIC.

###  7.2 Consequences of non-detection

Even a non-detection of GWs from a CE event throughout the LISA mission might
provide unique and direct constraints on CE evolution. In this case,
regardless of the rate of CE initiation, the first possibility is that the
self-regulated thermal-timescale phase happens at too wide separations,
corresponding to $f_{\mathrm{GW}}\lesssim 10^{-4}$ Hz below the LISA band
pass. For our representative CE binary with $M_{\mathrm{core}}=0.5\,M_{\odot}$
and $M_{2}=0.3\,M_{\odot}$, this would mean $a_{\mathrm{post-plunge}}\gtrsim
1.5\,R_{\odot}$, although from Fig. 3 we expect to build up significant SNR
only for $a_{\mathrm{post-plunge}}\lesssim 0.5\,R_{\odot}$.

Alternatively, direct constraints on the duration in the LISA band ($\Delta
t_{\mathrm{CE}}$) can be derived with some informed assumptions on the rate of
initiation ($R_{\mathrm{CE,init}}$). Considering all CE events, the latter can
be estimated with the observed rate of LRNe at $\approx 0.1$ yr-1 (consistent
with our population synthesis calculations, see also Sec. 7.1). Restricting to
CE between two evolved stars with dense cores, our simulations suggest a
decrease of about a factor of $\sim$3 (cf. Sec. 4). Therefore, despite the
many uncertainties in binary mass transfer stability and CE evolution, both EM
transients observations and population synthesis estimates suggest a value of
about one CE initiation every decade in the Galaxy. Assuming most of these
event will cross the LISA band at some point (cf. Fig. 2) and a mission
duration of $T\approx 10$ yr, a non-detection would be at odds with models for
which $\Delta t_{\mathrm{CE}}\gg 1$ yr, which we predict might result in a
significant number of detectable sources. The longer the time $T$ during which
a GW signal might be observed is, the less dependent on uncertainties in
$R_{\mathrm{CE,init}}$ and more stringent this constrain will be.

###  7.3 Further caveats

There are several effects which we have not explored in detail in this study
which could impact the LISA detectability of a CE event. The dynamical plunge-
in could make the orbit inside the CE eccentric, modifying the GW emission in
two ways: make it more “bursty”, with stronger emission at each pericenter
passage, and make it more “directional”. If the rate of CE events is on the
optimistic side, LISA can still expect to detect eccentric CE events, and will
unveil a great deal about the physics of the dynamical plunge-in.

We have also neglected triple stellar systems. Triples can enhance the rate of
mass transfer in binaries, either because the third star fills its Roche lobe,
or because of the Kozai-Lidov oscillations inducing high eccentricity in the
inner binary (e.g., Toonen et al., 2020). This can in principle enhance the
rate of CE and produce events which are qualitatively different in EM and GWs,
see for example Glanz & Perets (2021). Hence, our most pessimistic rate
calculations may be regarded as a lower limit.

##  8 Summary and conclusions

Common-envelope evolution remains one of the largest uncertainties in stellar
physics. Here, we have investigated the possibility of detecting
gravitational-wave emission from a binary made of the core of the donor star
and the companion star inspiraling within a shared common envelope.

Although common-envelope evolution is never _driven_ by the loss of energy to
gravitational radiation, the binary inside the shared envelope does have a
time-dependent mass quadrupole moment and can thus emit gravitational
radiation. If detectable, such gravitational radiation offers invaluable
insight on the process by passively tracing the motion of the binary.

While the final dynamical phase of a common envelope might emit detectable
signals (Ginat et al., 2020), its short duration makes it an extremely rare
target. Conversely, we focus on a longer duration phase that might occur in
some common-envelope events: the self-regulating thermal-timescale phase.
Detections of this phase would constrain the stalling radius (from the
gravitational-wave frequency) and the duration of the stalled phase (from the
number of sources detected and/or the signal duration). Even non-detections
could put upper-limits on the duration or lower limits on the radius at which
common envelopes might stall (see Sec. 7.2).

The gravitational-wave frequency range of interest is likely in the mHz range.
This is suggested by the existence of verification white-dwarf binary for the
LISA mission, which are thought to be the outcome of successful common-
envelope ejections.

Based on rapid population synthesis calculations, we estimate that about one
common-envelope event is initiated per decade in the Galaxy (cf. Fig. 4), in
agreement with previous observational and theoretical determinations (Sec. 4
and Sec. 7.1). The largest uncertainty in converting this to a number of
Galactic sources is the duration of the common envelope, which is predicted to
lie between a dynamical ($\sim$ days) and a thermal (decades to $10^{5}$
years) timescale, and is likely to depend on the masses and evolutionary
stages of the binary considered. Nevertheless, assuming that some common-
envelope events go through a self-regulated thermal-timescale phase, it is
realistic to expect at least one source within the LISA mission, and possibly
more than hundreds.

While we remained agnostic on the details of the physics governing the common-
envelope dynamics, we can bracket the range of possibilities by requiring the
common envelope to proceed faster than if driven by pure gravitational-wave
emission (corresponding to the unrealistic assumption of neglecting the gas
drag), and slower than if the gas drag was constant (corresponding to the
unrealistic assumption of neglecting the envelope reaction to the inspiraling
binary). We found that a system representative of the most common kind of
Galactic common envelope – a $0.5\,M_{\odot}$ core with a $0.3\,M_{\odot}$
companion embedded in a shared envelope – might be (marginally) detectable
during the stalled phase within these limits (cf. Fig. 3). More specifically,
if the separation at which the inspiral stalls is of order of $0.1\,R_{\odot}$
and lasts $\gtrsim 5$ years, a SNR $>$10 is possible. The shared envelope is
likely too low density to contribute directly to the gravitational-wave
signal.

We have further investigated the risk of misinterpreting a future detection of
a Galactic common-envelope binary if analyzed as a compact binary in vacuum
evolving according to general relativity (cf. Sec. 5). Although the
possibility of misinterpretation exist (cf. Fig. 5), there are several ways in
which it could be avoided. To break the degeneracies and recognize a signal as
the product of a Galactic common envelope, one could couple electromagnetic
followup or using electromagnetic observations to attempt directional searches
in the gravitational-wave data (Sec. 6); measure deviations from the “braking
index” predicted by general relativity (for fast evolving signals); or compare
the rate of detections with the double compact-object-merger rate from ground-
based detectors.

Space-based gravitational-wave detectors might improve our understanding the
inner dynamics of common-envelope evolution by using gravitational waves to
probe in a direct way a phenomenon that is _not_ gravitational wave driven.

Portions of this study were performed during the LISA Sprint at the Center for
Computational Astrophysics of the Flatiron Institute, supported by the Simons
Foundation. MR thanks S. Justham and Y. F. Jiang for helpful discussions early
on during this project, and K. Breivik for guidance in using COSMIC and
helpful feedback. We thank T. Littenberg for useful discussion on LISA
parameter estimation.

##  Appendix A Calculation of LISA Signal-to-Noise ratios

We calculate expected LISA signal-to-noise ratios following the discussion in
Robson et al. (2019), our code is available at
https://github.com/tcallister/LISA-and-CE-Evolution/. The gravitational-wave
signal from a quasicircular binary can be generically described via

$\displaystyle h_{+}(t)$
$\displaystyle=A(t)\frac{1+\cos^{2}\iota}{2}\cos\Psi(t)\ \ ,$ (A1)
$\displaystyle h_{\times}(t)$ $\displaystyle=A(t)\cos\iota\sin\Psi(t).$

Here, $A(t)$ is the gravitational wave’s amplitude, $\iota$ is the inclination
of the binary’s orbital plane with respect to our line of sight, and $\Psi(t)$
is the gravitational wave’s phase (the corresponding frequency of
gravitational-wave emission is $f_{\rm GW}=\frac{1}{2\pi}\dot{\Psi}$). In the
frequency domain (denoted by a tilde), the resulting signal that is measured
by LISA is the linear combination

$\tilde{h}(f_{\rm GW})=F_{+}(f_{\rm GW},\hat{n})\,\tilde{h}_{+}(f_{\rm
GW})+F_{\times}(f_{\rm GW},\hat{n})\,\tilde{h}_{\times}(f_{\rm GW})\ \ ,$ (A2)

assuming a signal incident on LISA from direction $\hat{n}$, and where
$F_{+}(f_{\rm GW},\hat{n})$ and $F_{\times}(f_{\rm GW},\hat{n})$ are the LISA
antenna response functions. Note that, unlike ground-based detectors, LISA
observes signals whose wavelengths are comparable to the size of the
instrument itself, and therefore its antenna patterns are strongly frequency-
dependent. Also, here we explicitly use $f_{\rm GW}$ to refer to
gravitational-wave frequencies in order to minimize confusion with orbital
frequencies $f_{\mathrm{orb}}=f_{\rm GW}/2$.

The resulting signal-to-noise ratio (SNR) with which this signal is observed
in LISA is given by

${\rm SNR}^{2}=4\int_{0}^{\infty}\frac{|h(f_{\rm GW})|^{2}}{S_{n}(f_{\rm
GW})}df_{\rm GW}\ \ ,$ (A3)

$S_{n}(f_{\rm GW})$ is the LISA strain sensitivity curve; we adopt the
analytic expression in Eq. 1 of Robson et al. (2019). As we do not know in
advance the sky location or orientation of possible CE sources, Eq. (A3) can
be averaged over all sky positions and binary inclination angles, yielding an
expected signal-to-noise ratio

$\langle{\rm
SNR}^{2}\rangle=\frac{16}{5}\int_{0}^{\infty}\frac{|\tilde{A}(f_{\rm
GW})|^{2}}{S_{n}(f_{\rm GW})}df_{\rm GW}\ \ .$ (A4)

We use Eq. (A4) to obtain the results shown with the colors in Fig. 3.

To compute the Fourier domain amplitudes $\tilde{A}(f_{\rm GW})$, we first
note that the time-domain amplitude of a quasi-circular binary with chirp mass
$\mathcal{M}_{c}$, at distance $D$, and gravitational-wave frequency $f_{\rm
GW}$ is (Maggiore, 2008)

$A(t)=\frac{4}{D}\left(\frac{G\mathcal{M}_{c}}{c^{2}}\right)^{5/3}\left(\frac{\pi
f_{\rm GW}}{c}\right)^{2/3}\ \ .$ (A5)

The corresponding Fourier-domain amplitude is (Buonanno et al., 2009)

$\tilde{A}(f_{\rm GW})=\frac{A\left(t(f_{\rm GW})\right)}{\sqrt{\dot{f}_{\rm
GW}\left(t(f_{\rm GW})\right)}}\ \ ,$ (A6)

where $\dot{f}_{\rm GW}$ is the time derivative of the gravitational-wave
frequency and we now regard time as a function of $f_{\rm GW}$.

##  Appendix B Calculation of gas-drag neglecting the envelope reaction

Any massive object moving linearly through gas with a constant density
interacts with its own gravitational wake and loses momentum via dynamical
friction. An object of mass $M$ and radius $R$ travelling with velocity $v$
through gas of uniform density $\rho$ and sound speed $c_{s}$ generates an
enhanced density tail. This wake is confined to the Mach cone at supersonic
speeds ($\mathcal{M}=v/c_{s}\gg 1$) and at sub-sonic speeds the over-dense
wake lies within a sphere of radius $c_{s}t$ a distance $vt$ behind the mass.
If the wake is low amplitude, linear perturbation theory yields a drag force
of the form $F_{\rm DF}=-C4\pi(GM)^{2}\rho/v^{2}$ (Ostriker, 1999) where $C$
is a term that approaches $C\rightarrow\mathcal{M}^{3}/3$ as $\mathcal{M}$
$\ll$ $1$ and $C\rightarrow\ln(vt/R)$ as $\mathcal{M}$ $\gg$ $1$. For a
massive object on a circular orbit, the path of the wake bends and is
therefore asymmetric, leading to both azimuthal and radial components of drag.
The azimuthal component dominates and is generally well approximated by
$F_{\rm DF}$ (Kim & Kim, 2007). Where the perturber is more massive, the wake
is larger amplitude and the non-linear regime modifies $F_{\rm DF}$ by an
additional factor $(\eta/2)^{-0.45}$ where $\eta$ is a non-linear factor given
by $\eta=A/(\mathcal{M}^{2}-1)$ and $A=GM/c_{s}^{2}R_{s}$ with $R_{s}$ a
characteristic softening scale (Kim & Kim, 2009). Further modifications to the
Kim & Kim (2007) picture of drag can occur when the embedded mass is a binary
and the binary orbit or its orbit around its own center of mass become
supersonic (Sánchez-Salcedo & Chametla, 2014).

To calculate the solid orange line in Fig. 3, we assumed
$\rho=10^{-6}\,\mathrm{g\ cm^{-3}}$, roughly corresponding to the average
envelope density for a red supergiant, and a temperature $T=9\times
10^{5}\,\mathrm{K}$, corresponding to the temperature somewhere inside the
envelope. To obtain a (conservative) upper-limit on the gas drag force, we
consider these parameters constant throughout the inspiral, that is we neglect
the reaction of the envelope to the injection of energy and angular momentum
by the binary spiraling-in. With these parameters, we calculated $F_{\rm DF}$
and assumed it to be the only source of energy loss and inspiral.

##  Appendix C Post-common-envelope separation with more massive companion

Figure 6: Same as Fig. 2 but assuming a compact companion of $1.4\,M_{\odot}$
representing a Chandrasekhar mass WD or a neutron star.

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