On the optical transients from double white-dwarf mergers

Double white-dwarf (DWD) mergers are relevant astrophysical sources expected to produce massive, highly-magnetized WDs, supernovae (SNe) Ia, and neutron stars (NSs). Although they are expected to be numerous sources in the sky, their detection has evaded the most advanced transient surveys. This article characterizes the optical transient expected from DWD mergers in which the central remnant is a stable (sub-Chandrasekhar) WD. We show that the expansion and cooling of the merger's dynamical ejecta lead to an optical emission peaking at $1$-$10$ d post-merger, with luminosities of $10^{40}$-$10^{41}$ erg s$^{-1}$. We present simulations of the light-curves, spectra, and the color evolution of the transient. We show that these properties, together with the estimated rate of mergers, are consistent with the absence of detection, e.g., by The Zwicky Transient Facility (ZTF). More importantly, we show that the Legacy Survey of Space and Time (LSST) of the Vera C. Rubin Observatory will likely detect a few/several hundred per year, opening a new window to the physics of WDs, NSs, and SN Ia.

Three fates of the central remnant of a DWD merger can be envisaged: a fast-rotating (and possibly highlymagnetized) WD, a supernova (SN) of type Ia, or a neutron star (NS).The binary's component masses, the presence (or genesis) of high magnetic fields (García-Berro et al. 2012), and the rate of mass and angular momentum transfer from a surrounding debris disk are among the critical physical ingredients that determine the central object's fate (see, e.g., Becerra et al. 2018b, 2019, andreferences therein).Based on the above, the relevance of DWDs has been highlighted in various astrophysical scenarios, e.g.: • The double-degenerate scenario (Iben & Tutukov 1984;Webbink 1984) proposes that unstable thermonuclear fusion can be ignited in the central remnant of DWD mergers, leading to one of the most likely explanations of SNe Ia (see, e.g., Neopane et al. 2022, and references therein).Indeed, the DWD merger rate is sufficient to explain the rate of SNe Ia, which is about 5-8 times smaller (see, e.g., Ruiter et al. 2009;Maoz et al. 2018).
• DWD mergers have been, for a long time, thought to be the main channel leading to the observed WDs with high magnetic fields in the range 10 6 -10 9 G (Külebi et al. 2009;Ferrario et al. 2015;Kepler et al. 2016).
• A fraction of DWD mergers can explain the population of massive WDs of ∼ 1M ⊙ (see Maoz et al. 2018;Cheng et al. 2020;Kilic et al. 2023b, and references therein).See also section 4.
• Interestingly, most of those massive WDs are highly magnetic (see, e.g., Kepler et al. 2016).Additionally, LSST will observe more than 150 million WDs at the final depth of its stacked 10-year survey Fantin et al. (2020).
Despite the above theoretical and observational richness, additional physical phenomena in DWD mergers have remained unexplored.We aim to characterize them in this article.First of all, given that R DWD ∼ (5-8)R SN−Ia , we must conclude that there is a considerable population of DWD mergers that do not produce SNe Ia (see, also, Cheng et al. 2020).This article focuses on such systems, especially those where the central remnant is a massive WD (see section 2).Section 3 shows that the dynamical ejecta from DWD mergers produces a fast-rising and fast-declining optical emission, peaking at ∼ 1 d post-merger, from its cooling driven by the expansion.The energy injected by the central remnant (e.g., by accretion winds and/or pulsarlike emission) is considered.We exemplify such optical transient theoretically and observationally using fiducial model parameters.Section 4 discusses how our findings compare with the known optical transients population.We show the Bright Transient Survey (Perley et al. 2020) of The Zwicky Transient Facility (ZTF) has not detected/identified any of them.
Finally, we discuss our main conclusions in Section 5, including the consistency of our theoretical predictions with the lack of detections by the ZTF of DWD mergers optical transients.Furthermore, we provide an upper limit for the number of detections expected by the forthcoming Legacy Survey of Space and Time (LSST) of the Vera C. Rubin Observatory.Details on the theoretical modeling of the expected light-curves and spectra are given in the appendix.

MERGING BINARY AND POST-MERGER CONFIGURATION PROPERTIES
We are interested in DWD mergers leading to a central remnant that is a stable, sub-Chandrasekhar WD.Given the mass distribution of observed WDs, we expect that sub-Chandra mergers can lead to massive WDs in the 1.0 ≲ M ≲ 1.4M ⊙ range.In principle, such WDs might be fastly rotating with periods P ≳ 0.5 s (see, e.g., Boshkayev et al. 2013).Such post-merged WD can avoid exploding as an SN Ia if, during its evolution, its central density remains below some specific value estimated to be a few 10 9 g cm −3 (see, e.g., Becerra et al. 2018bBecerra et al. , 2019, and references therein for details).
Numerical simulations show that the merger of a DWD, in general, develops a rigidly rotating, central core surrounded by a hot, convective corona with differential rotation and a Keplerian disk that hosts nearly all the mass of the disrupted secondary star (Benz et al. 1990;Guerrero et al. 2004;Lorén-Aguilar et al. 2009;Longland et al. 2012;Raskin et al. 2012;Zhu et al. 2013;Dan et al. 2014;Becerra et al. 2018b).These compactobject mergers expel small amounts of mass in the dynamical phase of the merger.Dan et al. (2014) provided analytic functions that fit the results of their numerical simulations.Concerning the ejected mass, it can be estimated by where M = m 1 + m 2 is the total binary mass, and q ≡ m 2 /m 1 ≤ 1 is the binary mass ratio.Equation (1) tells us that, typically, DWD mergers eject m ej ∼ 10 −3 M ⊙ .Despite this amount of matter being negligible relative to the system mass, we will show that it is responsible Table 1.Parameter values used to model thermal and synchrotron radiation from the expansion of ejected material.

EXPECTED LIGHT-CURVES AND SPECTRA
We now turn to the results from modeling the emission of the expanding ejecta.As we have recalled, about 10 −3 M ⊙ are ejected from the system during the final dynamical phase of the merger.This ejecta expands nearly radially at about the escape velocity, namely, 10 8 -10 9 cm s −1 .In the early post-merger evolution, accretion winds further power the ejecta (see, e.g., Becerra et al. 2018a;Rueda et al. 2019).Magnetic braking and nuclear reactions can also contribute to the energy budget but to a much lesser extent.In Appendix A, we present our theoretical model to calculate the thermal evolution of the expanding ejecta subjected to the injection of energy from the central remnant.The model parameters are the ejecta mass (m ej ), the index defining the radial falloff of the density profile (m), the self-similar expansion index (n), the initial position and velocity of the innermost ejecta layer (R * ,0 and v * ,0 ), the parameters defining the power injected by the central remnant (H 0 , t c , and δ), and the optical opacity (κ).We refer the reader to Appendix A for technical details.
Table 1 lists the model parameters and the corresponding fiducial values we adopted to exemplify the model.Figure 1 shows the corresponding light-curves (luminosity as a function of time), predicted by the theoretical model in Appendix A, in the visible (r band) and the infrared (i and K s bands).
From the lightcurves in Fig. 1, we see that the thermal emission due to the expansion of the ejecta peak luminosity is ∼ 10 40 -10 41 erg s −1 , at about 11-12 d postmerger.The transparency time is t tr, * ≈ 1.55 × 10 5 s ≈ 1.79 d. Figure 2 shows the spectra νF (ν, t) at selected times, where F (ν, t) = J cool (ν, t) is the spectral density, as given by Eq. (A16).Although the electromagnetic detection of DWDs is a challenging observational task, the increasing quality, sensitivity, and capacity of performing accurate surveys by novel optical observational facilities (e.g., the SDSS, ZTF, Gaia) and the refinement of observational techniques have led to a ten-fold increase in the number of observed DWDs in the Milky Way in the last 20 years: from around 14 by 2000 (Nelemans et al. 2001) to about 150 by 2022 (Korol et al. 2022).That number has already increased (see, e.g., Kosakowski et al. 2023), also in view of the rapidly growing number of observed WDs in binaries in recent data from the Gaia Mission and ZTF, of which an appreciable percentage are expected to be DWDs (see, e.g., Brown et al. 2023;Kosakowski et al. 2023;Parsons et al. 2023;Jiménez-Esteban et al. 2023).
Using population synthesis models that matched the at-the-time number of observed DWDs, i.e., fourteen, in their pioneering work, Nelemans et al. (2001) estimated the Milky Way hosts about 2.5 × 10 8 DWDs, and a DWD merger rate ≈ 2.2 × 10 −2 yr −1 .Up-to-date analyses that match the increasing number of known DWDs have confirmed their estimate.We shall use the estimates by Maoz et al. (2018), to which we refer the reader for details.They estimated a DWD merger rate per WD of R DWD = (9.7 ± 1.1) × 10 −12 yr −1 .This estimate can be translated into a DWD merger rate per unit stellar mass by dividing it by the stellar mass to WD number ratio (15.5 ± 1.8)M ⊙ per WD, leading to R DWD ≈ (5-7) × 10 −13 yr −1 M −1 ⊙ .It is worth noticing that this number agrees with the initial estimate by Nelemans et al. (2001) when multiplied by the Milky Way stellar mass.Assuming a constant star-formation rate over the Milky Way lifetime, these figures imply that ∼ 10% of the Galactic WDs have merged with another WD.Therefore, as discussed in Maoz et al. (2018), this inferred fraction of already merged DWDs may explain the high-mass bump in the WD mass function (see also Kilic et al. 2023b).However, some massive WDs may have formed from different channels (see, e.g., the case of J004917.14-252556.81 in Kilic et al. 2023a).
The above result agrees with our basic assumption that a considerable fraction of DWD mergers do not lead to SNe Ia but to massive WDs (rapidly rotating and possibly highly magnetic).Therefore, attention must be given to the possibility of establishing the link between observed massive WDs and their possible DWD merger progenitors.The success of this task needs the observational determination of the WD parameters (e.g., mass, radius, rotation period, temperature, and magnetic field strength) and the accurate modeling of the merger and post-merger evolution of the system.Fortunately, there is a growing effort in both directions.Numerical simulations focusing on the merging phase of DWDs started in the 90s and have considerably improved over the years (see, e.g., Benz et al. 1990;Guerrero et al. 2004;Lorén-Aguilar et al. 2009;Longland et al. 2012;Raskin et al. 2012;Zhu et al. 2013;Dan et al. 2014;Becerra et al. 2018b).Theoretical analyses to constrain the physics of the post-merger remnant, to determine its possible fate either as a disrupting explosion (SN Ia), a stable massive WD or gravitational collapse to a NS, including magnetic fields, rotation, and general relativistic effects have also gained interest and been performed in the last decade (see, e.g., Schwab et al. 2012;Shen et al. 2012;Ji et al. 2013;Kashiyama et al. 2013;Beloborodov 2014;Schwab et al. 2016;Rueda et al. 2018;Becerra et al. 2018b;Shen et al. 2019;Neopane et al. 2022).Although there is still room for improvements in the merger and post-merger modeling, these works have already allowed us to test the viability of the connection between massive WDs and their possible DWD progenitors on a theoretical basis.For instance, Sousa et al. (2022) has positively assessed such a connection for the isolated, highly magnetic, rapidly rotating WDs ZTF J190132.9+145808.7 (Caiazzo et al. 2021) and SDSS J221141.80+113604.4 (Kilic et al. 2021), leading to the parameters of the possible DWD progenitor, which were found to agree with those of the DWD observed population.
Having set the theoretical and observational basis for the connection between DWD mergers and massive WDs, we next discuss the electromagnetic transient associated with such an astrophysical system, theoretically featured in section 2, from the observational viewpoint.

OBSERVED POPULATIONS OF FAST TRANSIENTS
In the last decades, the advent of wide-field, highcadence surveys led to the discovery of several classes of fast (t rise ≲ 10 d) transients, with luminosities spanning several decades (see Pastorello & Fraser 2019 for a review).The so-called 'Fast Blue Optical Transients' (FBOTs) are blue, fast-rising, with peak luminosities in the range −16 ≳ M g,peak ≳ −22 (e.g., Drout et al. 2014;Tanaka et al. 2016;Pursiainen et al. 2018;Tampo et al. 2020) and are also referred to as 'Rapidly Evolving Transients' or 'Fast-Evolving Luminous Transients'.The source AT2018cow (known as 'the cow'), at 60 Mpc, represents the best-studied case of this class.It exhibited some unprecedented characteristics: rise time of a few days; L p ∼ 4 × 10 44 erg s −1 ; mostly featureless spectra with blackbody temperatures above 10 4 K during the first 15 d with large expansion velocities (∼ 0.1 c); hard X-ray and variable soft X-ray emission; radio bright with L ν,p ∼ 4 × 10 28 erg s −1 Hz −1 at 8.5 GHz (Margutti et al. 2019;Perley et al. 2019;Ho et al. 2019; see also Coppejans et al. 2020;Ho et al. 2020;Perley et al. 2021;Ho et al. 2023;Matthews et al. 2023 for the few analogous cases yet observed).These properties suggest that a large amount of radioactive nickel cannot explain the high luminosity, and the relatively short effective diffusion timescale points to a low ejecta mass.In contrast, the long-lived X-ray variability suggests a compact and long-lived inner engine.Owing to their extreme peak luminosity from radio to hard X-rays, these FBOTs are hardly compatible with a DWD merger since the power injected from the central remnant at those times is lower than the observed luminosities.
In parallel, other transients sharing comparably fast rise times (∼ 12-15 d) but significantly less luminous have also been discovered, with peak luminosities in the gap between novae and supernovae.A class that raised interest is that of so-called calcium-rich transients (−13 ≳ M V ≳ −17; Perets et al. 2010;Kasliwal et al. 2012;De et al. 2020), which exhibit a strong [Ca II] emission in the nebular phase spectra with a high [Ca II ]/[O I] ratio.These share similar photospheric velocities with typical core-collapse Ib/c SNe.Still, their environment is strongly different from the latter since they prefer remote locations in the outskirts of earlytype galaxies, even more than type-Ia SNe and short gamma-ray bursts, indicative of a very old progenitor population (Lunnan et al. 2017).In this respect, a fraction of these transients could result from mergers of helium and oxygen/neon WDs (Shen et al. 2019).The local volumetric rate of Ca-rich, hydrogen-poor transients is estimated to be ≳ 15% of the type-Ia rate (De et al. 2020).The variety in peak luminosity and spectroscopic properties probably stems from a heterogeneous class of progenitors.
Some low-luminosity gap transients are still likely to be less energetic SNe.In particular, the so-called intermediate-luminosity red transients (ILRTs; Berger et al. 2009;Bond et al. 2009) have a peak luminosity in the range −12 ≳ M V ≳ −15, relatively long rise times and post-peak plateaus which resemble type II-L and II-P SNe.Although there is consensus that the progenitors are 8-15 M ⊙ stars in dusty cocoons, eruptive formation of a massive WD or eruptions from binary interactions could contribute to the observed population (Pastorello & Fraser 2019).
A few gap transients M V ≳ −13 mag, characterized by double or even triple-peaked light curves, have been proposed as a scaled-up version of red novae (typically less luminous than −10 mag) and, as such, are often referred to as luminous red novae (LRNe; see Kulkarni et al. 2007;Pastorello et al. 2019 and references therein).Their photometric evolution is reminiscent of eruptive variables such as V1309 Scorpii, whose final brightening was interpreted as the merger of a contact binary (Tylenda et al. 2011).
Figure 3 summarizes the zoo of the fast transients as observed with the Zwicky Transient Factory Bright Transient Survey (Perley et al. 2020) in the peak luminosity-duration plane.We show the region where our predictions on DWD mergers lie: despite the relatively high expected volumetric rate, this region is still poorly explored.Upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezić et al. 2019) are expected to boost the number of promising candidates for DWD mergers.

DISCUSSION AND CONCLUSIONS
We have estimated the optical transient from DWD mergers leading to stable, massive, fast-rotating WDs.The emission arises from the cooling down of the dynamical ejecta of the merger, about 10 −3 M ⊙ , that expands at 10 8 -10 9 cm s −1 .The ejecta is powered by the early activity of the central remnant, mainly fallback accretion (see, e.g., Rueda et al. 2019, and references therein, and Appendix A for a comparison of accretion power with nuclear energy and magnetic braking).Inspired by numerical simulations, we assumed spherical expansion.The theoretical model includes a power-law density profile and self-similar expansion.We solve the energy balance equation and determine the ejecta's thermal history (time evolution), estimating its photospheric emission and color evolution.
We have shown that the peak of the optical emission occurs at times 1-10 d, with a luminosity L p = 10 40 -10 41 erg s −1 , for typical parameters expected for these DWD mergers (see Table 1); see Figs. 1 and 2 for the light-curves and spectra, respectively.Although our model makes some approximations, we expect it to catch the main physics of these systems robustly.There-fore, further model refinements should not appreciably change the above qualitative and quantitative picture.
With this in mind, we turned to the observational considerations.Indeed, detecting the optical counterpart of DWD mergers would have several relevant consequences in physics and astrophysics.To mention some: -It will constrain the fraction of mergers producing SNe Ia, giving crucial hints for the SN Ia-associated physics, e.g., the unstable thermonuclear fusion and detonation (see Schwab et al. 2012;Shen et al. 2012;Ji et al. 2013;Schwab et al. 2016;Neopane et al. 2022, and references therein).
-If the rate of mergers leading to SN Ia will turn out lower than the SN Ia observed rate, it would imply the necessity of also having at work the single-degenerate scenario for their explanation (see, e.g., Han & Podsiadlowski 2004).
-It will alert facilities on ground and space to look for associated emissions at higher energies, e.g., in the Xand gamma-rays, constraining the physics of the central remnant such as magnetic fields and rotation (see, e.g., Ji et al. 2013;Kashiyama et al. 2013;Beloborodov 2014;Rueda et al. 2018;Becerra et al. 2018b).
-It will confirm DWD mergers as the formation channel of massive, fast-rotating WDs; -At late post-merger times, the central WD might be observed accompanied by a debris disk (see, e.g., Külebi et al. 2013;Rueda et al. 2013;Becerra et al. 2018b;Neopane et al. 2022, and references therein).Thus, it will be interesting to compare forthcoming advanced infrared-optical-UV survey estimates of the rate of WDs with debris disk (Fantin et al. 2020) and the DWD merger rate estimates.
-It will constrain the physics of the gravitational collapse of WDs into NSs while simultaneously possibly confirming DWD mergers as a formation channel of NSs.
Thus, in section 4, we checked whether current observational facilities could have observed such optical transients.We compare and contrast the model predictions with the emergence population of optical transients in the literature.Our analysis showed that the optical transients from DWD mergers presented here do not match the observed features of FBOTs, fast-evolving luminous transients (i.e., cow-like objects), and calciumrich transients.A plot of the peak absolute magnitude as a function of the rest-frame time duration for the transients detected by the ZTD Bright Transient Survey (see Fig. 3), highlighting the region the DWD merger optical transients should occupy, reveals overwhelmingly the above result.
Therefore, no optical transient from DWD mergers has ever been detected.Does this result agree with the model prediction?The limiting magnitude for detection by ZTF is m ZTF,lim = 19 mag (Perley et al. 2020).Assuming the peak luminosity L p = 10 40 erg s −1 , this turns into a detection horizon d ZTF,lim ∼ 11 Mpc.Using an expected volumetric rate for DWD mergers of 4 × 10 5 Gpc −3 yr −1 (see section 1), the upper limit to the expected number of events by ZTF is ∼ 2, considering the duty cycle of the survey.This result is consistent with our findings and the expectation that not all DWD mergers produce stable WDs: a fraction should lead to SNe Ia and another to NSs as central remnants.
We can apply the same kind of calculation to LSST, for which 5-σ limiting magnitudes for single exposures in filters g and r (the same considered for ZTF in Fig. 3) are 24.5 and 24.0, respectively 1 .Under these favorable conditions, the detection horizon becomes d LSST,lim ∼ 110-140 Mpc, corresponding to a gain by ∼ 10 3 in the expected detection rate.
The above analysis brings us to one of the main conclusions: in the transition from ZTF to LSST, the electromagnetic (optical) counterparts of DWD mergers will finally become observable, likely a few/several hundred per year, opening a new window to the physics of WDs, NSs, and SN Ia.
where σ is the Stefan-Boltzmann constant.The power per unit frequency, per unit area, is given by Planck's spectrum where ν is the radiation frequency, h and k B are the Planck and Boltzmann constants.Therefore, the spectral density (power per unit frequency) given by the thermal cooling at a frequency ν is J cool (ν, t) = 4πR 2 ph (t)B ν (ν, t), (A16) and the luminosity radiated in the frequency range [ν 1 , ν 2 ] can be then obtained as J cool (ν, t)dν. (A17) The parameter v max,0 has no appreciable effect in the evolution, so it can not be constrained from the data.This happens because most of the mass is concentrated in the innermost layers, so they dominate the thermal evolution.For self-consistency of the model, we have set v max,0 = 2v * ,0 (so R max,0 = 2R * ,0 ).As for the initial value of the internal energy of the shells, E i (t 0 ), we have set them to the initial kinetic energy of each layer, E i = (1/2)m i v i (t 0 ) 2 .

Figure 1 .
Figure1.Emission from the expanding, cooling ejecta at early times (solid lines) in the visible (r band) and in the infrared (i and Ks bands).We refer to Appendix A for details on the theoretical model.

Figure 2 .
Figure 2. Emission spectra from the expanding, cooling ejecta at selected post-merger times.We refer to Appendix A for details on the theoretical model

Figure 3 .
Figure 3. Different populations of fast transients observed with the Zwicky Transient Facility Bright Transient Survey (Perley et al. 2020).The shaded box highlights where we expect most DWD mergers should lie.Figure adapted from Perley et al. (2020).