Gravitational Waves from Double White Dwarfs as probes of the Milky Way

Maria Georgousi^1,2, Nikolaos Karnesis¹, Valeriya Korol^3,4, Mauro Pieroni⁵ and Nikolaos Stergioulas¹
¹Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
²Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, PO Box 1527, 6R-71110 Heraklion, Greece
³School of Physics and Astronomy & Institute for Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
⁴Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany
⁵Blackett Laboratory, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK E-mail: [email protected]

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

Abstract

Future gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA), will be able to resolve a significant number of the ultra compact stellar-mass binaries in our own Galaxy and its neighborhood. These will be mostly double white dwarf (DWD) binaries, and their underlying population characteristics can be directly correlated to the different properties of the Galaxy. In particular, with LISA we will be able to resolve $\sim\mathcal{O}(10^{4})$ binaries, while the rest will generate a confusion foreground signal. Analogously to how the total electromagnetic radiation emitted by a galaxy can be related to the underlying total stellar mass, in this work we propose a framework to infer the same quantity by investigating the spectral shape and amplitude of the confusion foreground signal. For a fixed DWD evolution model, and thus a fixed binary fraction, we retrieve percentage-level relative errors on the total stellar mass, which improves for increasing values of the mass. At the same time, we find that variations in the Milky Way shape, at a fixed mass and at scale heights smaller than 500 pc, are not distinguishable based on the shape of stochastic signal alone. We perform this analysis on simulations of the LISA data, estimating the resolvable sources based on signal-to-noise criteria. Finally, we utilize the catalogue of resolvable sources to probe the characteristics of the underlying population of DWD binaries. We show that the DWD frequency, coalescence time and chirp mass (up to $<0.7\,$ M_⊙) distributions can be reconstructed from LISA data with no bias.

keywords:

gravitational waves – white dwarfs – binaries:close – Galaxy:structure

^†^†pubyear: 2021^†^†pagerange: Gravitational Waves from Double White Dwarfs as probes of the Milky Way–LABEL:lastpage

1 Introduction

The Milky Way harbours a large variety of double compact objects composed of white dwarfs (WDs), neutron stars and black holes (for a review see amaro22). Theoretical studies forecast hundreds of millions of double WDs (DWDs), and millions of double neutron stars and double black holes (e.g. Nelemans01a; rui10; yu10; nis12; lam18; bre20; vig20; wag21). Although numerous, these are very challenging to detect through electromagnetic radiation because they either are too dim or do not emit light. The future Laser Interferometer Space Antenna (LISA) (LISAwhitepaper) will survey these populations at shortest orbital periods delivering complete samples for periods of less than 17 min within the Galaxy (lam19; KorolHallakoun2021). LISA will measure the ensemble signal from these objects. However, out of millions, LISA will be able to extract $\sim\mathcal{O}(10^{4})$ DWDs, which makes them the most numerous type among LISA sources (e.g. Korol2017; lam19; bre20; zen20; Wilhelm2021; thi21; tianquin). The rest of compact Galactic binaries will blend together to form a confusion-limited foreground that is expected to affect the LISA data at frequencies below a few mHz (e.g. LISAwhitepaper; Bender1997hs; nis12; rui10; Robson2017; KorolHallakoun2021).

The Galactic confusion signal will have an overall spectral shape that depends on the properties of the actual DWD population. From earlier theoretical works, the spectrum is predicted to have two distinct attributes: the lower frequency tail ( $\lesssim~{}\mathrm{mHz}$ ) and the higher frequency ‘knee’ (at around few $\mathrm{mHz}$ ). Under the assumption of orbital evolution depending solely on gravitational-wave (GW) emission, the low-frequency tail is expected to follow a power-law with the index that should be $\sim 2/3$ in energy density $\Omega_{\mathrm{gw}}$ units (see for example Phinney2001). The form of the high-frequency ‘knee’ structure depends on assumptions about the detectability of these sources with LISA. For example, it heavily depends on the observation duration, and on the signal-to-noise (SNR) threshold criteria to classify the DWD sources as resolvable (Karnesis2021tsh; nis12; Timpano2006; crowder2007; Digman2022jmp). Variations of intrinsic properties of the DWD population such as the chirp mass distribution can influence the detectability of binaries at few mHz, and, hence, the shape of the knee as well. For example, if DWDs are on average more massive than predicted by binary population synthesis studies and follow the observed mass function of single white dwarfs, the confusion foreground sharply drops at 2 mHz instead of 3 mHz (KorolHallakoun2021). This may seem like a small difference, however it bares important implications for the detectability of other LISA sources, including merging massive black-hole binaries ( $\sim 10^{4}-10^{7}$ M_⊙, e.g. Sesana2009; Klein2016; Dayal2019), extreme-mass-ratio inspirals (e.g. Babak2017; moo17; bon20), and backgrounds of cosmological origin (e.g. Caprini2016; Tamanini2016; Caprini2018mtu). In addition, changes to initial binary fractions have been shown to affect the level of the Galactic confusion signal up to a factor of 2 (thi21).

Several studies investigated the use of the resolved binaries for constraining the properties of the Milky Way (adams2012; Korol2019; Wilhelm2021). Only a few have exploited the confusion foreground to address similar constraints (Benacquista2006; Breivik2020). Using different strategies, both studies explored the effect of changing the disc scale height on the shape of confusion foreground only. Benacquista2006 demonstrated that at a fixed stellar space density, an increase in the scale height (leading to an increase in the total number of DWDs) raises the overall confusion level at frequencies lower than a few mHz and shifts the transfer frequency – i.e. the frequency above which LISA can resolve all DWDs – to slightly higher values. They also showed, which is confirmed with our work here, that at a fixed total number of binaries, changing the scale heights does not influence the shape of the confusion foreground. On the other hand, in Breivik2020, the scale height was probed by decomposing the foreground signal on a basis of spherical harmonics. This methodology allows us to constrain the scale height of the Galaxy, even so with limited sensitivity compared to other methods, i.e. in adams2012 and Korol2019 using resolved DWDs.

In this work, we fix a DWD population model and investigate different methods to characterize the properties of the Galaxy, by using two approaches. First, we describe the adopted fiducial population model in section 2, and describe the methodology to estimate the residual foreground signal after subtracting the loudest sources in LABEL:sec:methodology. Then, in LABEL:sec:Properties_impact, as proposed in the earlier studies summarised above, we investigate the capabilities of LISA to probe the scale height of the Galaxy ( $Z_{\mathrm{d}}$ ), by measuring the confusion foreground signal. We do that by simulating catalogues with different $Z_{\mathrm{d}}$ , and estimating the confusion signal by evaluating the number of resolvable sources given the observation time of LISA. By examining the SNRs and the waveform parameter errors of the recovered sources, we confirm that for small variations of $Z_{\mathrm{d}}$ , it is challenging to measure the disc scale height from the confusion signal alone. We then go one step further to design a framework in order to assess the overall number of Galactic DWDs from the confusion signal, which can be linked to the total Galactic stellar mass. Interestingly, we find that number of sources can be reasonably well constrained by measuring the Galactic confusion noise. In LABEL:sec:hierarchical, we perform the complementary analysis with the sub-set population of the resolvable sources. We use them to probe the probability distributions of three parameters of interest, namely the chirp mass, time of coalescence, and emission frequency. Finally, we draw our conclusions and dicuss our results in LABEL:sec:conclusions.

This work is based on the Bsc thesis of marythesis.

2 Mock double white dwarf population

To investigate the shape of the Galactic confusion noise we construct a fiducial Galactic DWD population based on the (publicly available) population synthesis code SeBa (PZ96; Nelemans01a; Toonen12). The detailed description of this model is given in Toonen12, to which we refer for further details. Below, we outline its most relevant features for this work. Our choice is motivated by the fact that this DWD evolution model yields the space density of DWDs in agreement with observations of the local white dwarf population and reproduces the general trend of the observed DWD mass ratio distribution (Toonen12; Toonen2017). We consider only one DWD evolution model and explore how the shape of the confusion noise changes when varying global properties of the Milky Way: its total stellar mass and shape (e.g. through the scale height parameter).

2.1 Initial population

The progenitor zero age main sequence population is assembled through a Monte Carlo technique. The mass of the primary stars – the most massive in each pair – is sampled from the initial mass function of KroupaIMF between 0.95 - 10 M_⊙, where the lower limit represents the minimum mass for an (isolated) star to reach the white dwarf stage in a Hubble time. The mass of the secondary star is drawn uniformly between 0.08 M_⊙ and the mass of the primary (Raghavan2010; Duchene2013). Initial binaries’ semi-major axes are drawn from a log-uniform distribution extending up to $10^{6}\,$ R_⊙ (Abt1983; Raghavan2010; Duchene2013). Orbit eccentricities are sampled from a thermal distribution (Heggie1975). The metallicity of the progenitor population is set to the Solar value, while the initial binary fraction is assumed to be of 50 per cent. Here we neglect a potential correlation between metallicity and initial binary fraction. Electromagnetic observations of close ( $\lesssim$ 10 au) low-mass binaries in the Solar neighborhood hint at a possible anti-correlation of binary fraction with metallicity (Bad18; Elb19; Moe19, and references therein). thi21 showed that, when implementing this anti-correlation in the binary population synthesis models for LISA, the size of the population may decrease noticeably. We note that, in this work, we will neglect the degeneracy effects of the initial binary fraction to the total Galactic stellar mass (see LABEL:sec:Mass_impact). However, in practice, the initial binary fraction represents a normalization factor and, if desired, the results can be re-scaled accordingly in the post-processing, or considered as a prior from electromagnetic observations (Duchene2013; Bad18; belokurov2020; Korol2020; maozhallakoun; napi).

2.2 Binary evolution

Next, SeBa evolves the progenitor population until both stars turn into white dwarfs following prescriptions for processes involved in the binary evolution, such as mass and angular momentum transfer, common envelope evolution, magnetic braking, and gravitational radiation (PZ96; Toonen12, and references therein). In SeBa, binaries starting on wide orbits ( $a\gtrsim$ 10 au) end up evolving independently, as they would if they were born in isolation. However, to form a close binary system that falls in LISA’s frequency window, at least one common envelope phase is required (Paczynski1976; Webbink1984). Specifically, to form a close DWD pair, the binary typically experiences at least two mass-transfer phases, of which at least one should be a common envelope (e.g. Nelemans01a).

Despite the importance of the common envelope evolution in the formation of double compact objects, there is no clear consensus regarding details governing this phase. Based on the observed DWDs, Nelemans2000 determined the possible masses and radii of DWD progenitor stars, which were used to reconstruct binaries’ past mass-transfer phases. Their work revealed that the DWD evolution models using the standard common-envelope formalism alone (i.e. the $\alpha$ -formalism, see Ivanova2013 for details, equating the energy balance in the system and implicitly assuming angular momentum conservation) could not reproduce the observed mass ratios of DWDs. Therefore, they concluded that an alternative formalism for the first phase of the mass-transfer is required and proposed a formalism that parametrizes the angular momentum balance equation—instead of the energy balance equation—through a $\gamma$ parameter. The result originally found by Nelemans2000 was later confirmed by studies using larger DWD samples (nel05; slu06) and by the binary population synthesis studies (Nelemans01a; Toonen12).

We adopt a DWD evolution model based on the results prescribed above: a model combining the two aforementioned common envelope parametrizations, such that the first common envelope phase is typically described by the $\gamma$ formalism, while the second by the $\alpha$ formalism. In the standard SeBa setting, $\gamma=1.75$ and $\alpha\lambda=2$ (he