THE LISA GRAVITATIONAL WAVE FOREGROUND: A STUDY OF DOUBLE WHITE DWARFS

In this study, the combined Galactic population of RLOF and detached compact remnant binaries with orbital periods spanning the LISA band is considered. We use the StarTrack population synthesis code described in detail in Belczynski et al. (2008b) to evolve primordial binaries within the Galaxy from the zero-age main sequence (ZAMS) and calculate their properties. The code has undergone a number of important updates since Belczynski et al. (2002), including detailed mass-transfer calculations. Recent results on mass accumulation in the context of ultracompact X-ray binary formation are given in Belczynski & Taam (2004), while those for double WD binaries are given in Belczynski et al. (2005).

We evolve the field population of single and binary stars (50% binarity) with solar-like metallicity (Z = 0.02). Single star evolution is followed from the ZAMS employing modified analytic formulae and evolutionary tracks from Hurley et al. (2000). For the evolution of binary stars, the calculations also include full orbital evolution with tidal interactions, magnetic braking, supernova natal kicks, and GR emission (see Belczynski et al. 2008b, for formulae). ZAMS masses (M_ZAMS) span the mass range 0.08–100 M_☉. Single stars and binary primaries (M_a) are drawn from a three-component broken power law initial mass function (Kroupa et al. 1993), and secondary masses (M_b) are obtained from a flat mass ratio distribution q= secondary/primary (e.g., Mazeh et al. 1992). We note as well that binary interactions leading to mass-transfer events may alter the course of evolution for stars of a given initial mass. Initial orbital separations span a wide range up to 10⁵ R_☉ and are drawn from a distribution that is flat in the logarithm, while initial eccentricities are taken from a thermal-equilibrium eccentricity distribution (see Abt 1983; Duquennoy & Mayor 1991). Progenitors of all binaries are initially formed on eccentric orbits. Tidal interactions between binary components circularize orbits before the first RLOF in a given progenitor system occurs. However, for a small fraction (10%) of double WD progenitors that are initially found on very eccentric orbits, the first RLOF is encountered when the orbit is still eccentric. In such a case, we assume an instant circularization at periastron (a_new = a_old(1 − e_old) and e_new = 0), and follow with the RLOF calculation. At the time of double WD formation, all of the orbits are circular. Eccentric WD binaries are expected to arise from dynamical interactions in globular clusters (Benacquista 2001), and though we do not consider them here, could provide a unique opportunity for learning about WD structure with LISA (Willems et al. 2008).

For the population of Galactic disk binaries, we use a continuous star formation rate for 10 Gyr, while for the bulge population we use a constant star formation rate for the first Gyr, and no star formation thereafter (entire MW age is 10 Gyr). This translates to a present Galactic star formation rate of 4 M_☉ yr⁻¹ (disk mass =4 × 10¹⁰ M_☉ with a constant star formation rate for 10 Gyr =4 M_☉ yr⁻¹; see Section 2.2), which is within reasonable agreement with the current Galactic star formation rate estimate of 3.6 M_☉ yr⁻¹ (Cox 2000). We note however that it has been suggested that the star formation rate of the MW has been (mostly) decreasing exponentially with time, only reaching ∼3.6 M_☉ yr⁻¹ at the current epoch (Nelemans et al. 2001b, 2004, see Section 2.2), with the integrated mass in formed stars being closer to 8 × 10¹⁰ M_☉. Since the star formation history of the MW disk and bulge is not precisely known, we choose to model the Galaxy with simple star formation histories (1 Gyr-long "continuous burst" (bulge), and constant for 10 Gyr (disk)), which to first order is a reasonable representation of the global star formation history of the Galaxy.

The most uncertain phase in close binary evolution affecting the orbital separation for low- and intermediate-mass stars is the common envelope (CE) phase (Postnov & Yungelson 2006). Close binaries are expected to go through at least one CE event, and there are currently a few prescriptions of CE evolution in the literature (Webbink 1984; Nelemans & Tout 2005; van der Sluys et al. 2006; Beer et al. 2007). In our simulations, the CE phase is treated using energy balance as discussed in Webbink (1984), in which the orbital energy of the binary is diminished at the expense of the unbinding of the donor envelope. The post-CE separation is governed by the parameters λ (de Kool 1990), a function of the donor's structure, and the highly uncertain α, the efficiency with which the binary orbital energy is used to unbind the envelope. We have chosen to use α × λ = 1. The effects of using other CE efficiencies have not been fully explored here, though lower CE efficiencies result in closer post-CE binaries, and a higher number of stellar mergers.

At the current age of the Galaxy (10 Gyr), we extract all binaries containing two compact remnants: WDs, NSs, and BHs in the gravitational frequency range: 0.0001–1 Hz, which encompasses the LISA sensitivity range (orbital periods of 5.6 hr–2 s). Henceforth when we refer to "LISA binaries," we are referring to binary systems in our study within this GR frequency band. As previously stated, we consider only WD binaries when calculating the LISA signal. There are five types of WDs considered in the evolution: helium (He WD), formed by stripping the envelope off a Hertzsprung gap or red giant low-mass star; carbon–oxygen (CO WD), formed from progenitors with masses ∼0.8–6.3 M_☉; oxygen–neon (ONe WD), formed from progenitors with masses ∼6.3–8.0 M_☉; hybrid (Hyb WD), having a CO-core and an He-envelope, formed via the stripping of the envelope from a helium burning star, and hydrogen (H WD), formed by stripping the envelope from a very low mass (≲0.8 M_☉) main sequence (MS) star. In our LISA GR calculations, we only consider the first four types of WDs, as hydrogen WDs are more representative of brown dwarf (BD) like objects and thus we will from now on refer to them as BDs. For the current study, the BDs (formed through binary evolution) do not play a large role in contributing to the LISA GR signal due to their low mass (see Section 5). We note, however, that in older stellar populations, the fraction of binaries that contain H WDs becomes more significant (see Ruiter et al. 2009 for the calculation of the LISA signal from MW halo double WDs, which includes "hydrogen WDs").

In compact binaries, gravitational wave emission is a strong source of angular momentum loss. The average rate of orbital angular momentum loss due to GR for e = 0 binaries is calculated from Peters (1964):

$$\begin{equation} \frac{{\it dJ}_{\rm gr}}{dt}=\frac{-32}{5}\frac{G^{7/2}\,M_{\rm p}^{2}\,M_{\rm s}^{2}\,\sqrt{M_{\rm p} + M_{\rm s}}}{c^{5}\,a^{7/2}}, \end{equation} \tag{ 1 }$$

where G is the gravitational constant and c is the speed of light. At every time step in our calculations, the evolving system is checked for RLOF. Upon reaching contact (RLOF), we assume that mass-transfer is GR-driven and the donor (M_don) mass-transfer rate is calculated via

$$\begin{equation} \dot{M}_{\rm don} = M_{\rm don} D^{-1} {\,{\it d J}_{\rm gr}/\,d t \over J_{\rm orb}}, \end{equation} \tag{ 2 }$$

where RLOF stability is determined by the parameter D (see, e.g., King & Kolb 1995)

$$\begin{equation} D={5 \over 6}+ {1 \over 2} \zeta _{\rm don}-{1-f_{\rm a} \over 3 (1+q)}- { (1-f_{\rm a}) (1+q) \beta _{\rm mt}+f_{\rm a} \over q}. \end{equation} \tag{ 3 }$$

J_orb is the orbital angular momentum of the binary, f_a is the fraction of transferred mass accreted by the WD of mass M_acc (f_a = 1 for stable RLOF), q ≡ (M_acc/M_don), and β_mt = M²_don/(M_don + M_acc)². The radius mass exponent for the donor ζ_don is obtained from stellar models in each time step (see Belczynski et al. 2008b) and is ∼−0.35 for WD donors in a phase of stable mass transfer. Stable RLOF between two WDs leads to an increase in orbital period where dynamically unstable RLOF leads to a merger. See Belczynski et al. (2008b) for a more detailed description of treatment of mass transfer/accretion phases in StarTrack.

In our simulations, both the detached and RLOF systems have been distributed between the disk and bulge separately, assuming no correlation between the position and age of the system. For the current study, we have neglected the halo population of double WDs, but these systems have been investigated in another work (Ruiter et al. 2009). The density distribution of the Galactic disk is taken to have the following form

$$\begin{equation} \rho ({\bf R,z}) = \frac{N_{d}}{4\pi R_{0}^{2} z_{0}}e^{-R/R_{0}} e^{|-z|/z_{0}} \end{equation} \tag{ 4 }$$

in cylindrical coordinates, and the density distribution of the bulge has the form

$$\begin{equation} \rho ({\bf r}) = \frac{N_{b}}{4\pi r_{0}^3}e^{{-(r/r_{0})}^2} \end{equation} \tag{ 5 }$$

in spherical coordinates (bulge scale length $r_0 = \sqrt{x^{2} + y^{2} + z^{2}}$).

N_d and N_b represent the total number of simulated double WDs in the disk and bulge, respectively. We have calibrated our results using stellar Galactic disk and bulge masses from Klypin et al. (2002), with masses of 4 × 10¹⁰ M_☉ and 2 × 10¹⁰ M_☉, respectively. We have chosen R₀ = 2500 pc and z₀ = 200 pc for the disk scale length and scale height, while r₀ = 500 pc with a radial cutoff of 3500 pc for the bulge (Nelemans 2003a). While we note that even a scale height of 500 pc would be a reasonable choice for disk WDs (Majewski & Siegel 2002), we have chosen to use 200 pc so that our results are more readily comparable to those of previous studies (Nelemans et al. 2001b; Benacquista & Holley-Bockelmann 2006).

Our original LISA double WD binaries, which were born directly from StarTrack (8.4 × 10⁴), were the result of the evolution of 20 × 10⁶ ZAMS binaries. Since evolving the whole Galactic population of LISA binaries is time intensive, an interpolation scheme was developed in order to scale our number of StarTrack binaries to match those of the MW bulge and disk by stellar mass. Probability distribution functions (PDFs) were constructed for different evolutionary channels as a function of the primary and secondary masses and GR frequency. For RLOF binaries, it was only necessary to construct PDFs as a function of the masses since the orbital period, and hence GR frequency, was then uniquely determined by the masses. We chose to construct PDFs (instead of simply scaling the original binaries by a factor of ≳300) so that our systems would span a range of frequencies and masses pertinent to said formation channel. The PDFs were constructed using a kernel density estimator that best replicated the characteristics of the population (Williams 2008). The KDE package for Matlab was used to generate the specific PDFs. This technique was particularly successful for well-populated evolutionary channels, where we could be sure that the range of masses and frequencies was well covered. For some of the least-populated channels, the full range of binary masses and frequencies was not covered sufficiently to yield a smooth extrapolation of the data. Thus, apparent clumps of binaries that are comparable in number to our scaling factors are attributable to an undersampling of the frequency and mass range for these rare evolutionary channels.

The gravitational wave signal from the LISA data stream will be a time delay interferometry (TDI) variable in which the phases of the laser signals at each vertex of LISA are combined with the phases of other signals at delayed times in order to reduce the laser phase noise and accommodate the varying armlengths of the constellation of the spacecraft (Rubbo et al. 2004). At low frequencies, the signal at any given vertex can be very well approximated by the Michelson signal:

$$\begin{equation} h(t) = \frac{1}{2} h_{ab}\big(\ell _1^{a}\ell _1^{b} - \ell _2^{a}\ell _2^{b}\big), \end{equation} \tag{ 6 }$$

where h_ab is the wave metric and ℓ₁ and ℓ₂ are the unit vectors pointing along the two arms that join at the vertex. The transfer frequency is the frequency at which point the GR wavelength becomes comparable to the armlength of LISA. For high (e.g., above the transfer frequency) GR frequencies, the period of the gravitational wave is then less than the time it takes for light to propagate between the LISA spacecraft detectors, and the low-frequency approximation breaks down. The transfer frequency is given by f_* = c/(2πL) ≈ 0.01 Hz, where L is the LISA armlength taken to be L = 5 × 10⁹ m. At frequencies above or near f_*, a more accurate representation of the LISA signal is given by the rigid adiabatic approximation (Rubbo et al. 2004), but detailed analysis (Vecchio & Wickham 2004) has shown that the low-frequency approximation should be adequate for most tasks at frequencies below ∼30mHz. In this work, we have used the rigid adiabatic approximation to calculate the LISA signal for sources with frequencies above 0.003 Hz.

Space-borne gravitational wave observatories like LISA are constantly in motion, changing their relative speed and aspect with respect to astrophysical sources on the sky. The sensitivity of the observatory to different gravitational wave polarizations as a function of sky position is reflected in the antenna beam patterns, which have the highest gain in the direction perpendicular to the plane of the interferometer. As the observatory moves in its orbit, the gravitational wave signal is modulated in amplitude, frequency, and phase. Frequency (Doppler) modulation arises from the relative motion of the observatory and the source, amplitude modulation arises from the sweep of the anisotropic antenna pattern on the sky, and phase modulation results from the detector's changing response to the gravitational wave polarization state. The Michelson signal can be calculated using Equation (6) by either holding the source fixed and putting all the motion of the detector into ℓ₁ and ℓ₂ (Rubbo et al. 2004), or by holding the detector fixed and putting the motion into the source (Cutler 1998). Both approaches yield the same Michelson signal in the low-frequency limit:

$$\begin{equation} h(t) = \frac{\sqrt{3}}{2} A(t) \cos {\left[\int ^t2\pi f(t^{\prime }) dt^{\prime } + \varphi _{p}(t) + \varphi _D(t) + \varphi _0 \right]}, \end{equation} \tag{ 7 }$$

where A(t) is the amplitude modulation, f = 2/P_orb (where P_orb is the binary orbital period) is the gravitational wave frequency for systems with zero eccentricity, and φ₀ is the initial phase of the wave at t = 0. The polarization phase, φ_p(t), represents the phase modulation. The Doppler phase, φ_D(t), gives the frequency modulation and the amplitude modulation is described by $A(t) = \sqrt{(A_+F^+)^2+(A_{\times }F^{\times })^2}$. The sensitivity factors, F⁺ and F^× are complicated functions of the position angles and orientation of the binary, as well as the position of LISA in its orbit (see Cutler 1998; Rubbo et al. 2004 for specific forms of these functions). The plus and cross polarization amplitudes are given by

$$\begin{eqnarray} A_+ & = & 2 \frac{G^{5/3}}{c^4d}{\cal M}^{5/3} (\pi \,f)^{2/3}(1+\cos ^2{i}), \\ A_{\times } & = & -4 \frac{G^{5/3}}{c^4d}{\cal M}^{5/3} (\pi \,f)^{2/3} \cos {i}, \end{eqnarray} \tag{ 8 }$$

where d is the distance to the binary, i is the angle of inclination, and the chirp mass is ${\cal M} = (M_{\rm p}M_{\rm s})^{3/5}/(M_{\rm p}+M_{\rm s})^{1/5}$. We have used the approach of Rubbo et al. (2004) to calculate the timestreams, which are then added to produce the total observatory data stream, which is then Fourier transformed to produce the illustrated frequency domain representation of the double WD Galactic foreground.

The LISA instrument noise is simulated by assuming the power spectral density of the noise is made up of position (or shot) noise given by Cornish (2001):

$$\begin{equation} S_{\rm np} = 4.8 \times 10^{-42}\; {\rm Hz}^{-1}, \end{equation} \tag{ 10 }$$

and an acceleration noise (converted to strain) given by

$$\begin{equation} S_{\rm na} = 2.3\times 10^{-40}\left(\frac{10^{-3} \,{\rm Hz}}{f}\right)^4 {\rm Hz}^{-1}. \end{equation} \tag{ 11 }$$

These separate components are combined according to

$$\begin{equation} S_{\rm n} = 4 S_{\rm np} + 8S_{\rm na}(1+\cos ^2{(f/f_{*})}), \end{equation} \tag{ 12 }$$

where f_* is the transfer frequency. We roll off the acceleration below f_min = 10⁻⁵ Hz, so that S_na(f ⩽ f_min) = S_na(f_min). In reality, the LISA noise will probably not follow this simple power law all the way down to our choice of f_min, but will begin to rise at a higher frequency below 0.1 mHz.

We note that in our calculations, the total population of WD binaries with GR frequencies between 1 × 10⁻⁴and0.01 Hz comprises roughly 6% of the total StarTrack Galactic population of double WDs. While we include all RLOF double WD systems that currently exist in the MW in our study, only 2.3% of Galactic detached double WD systems are accounted for, given the orbital period cutoff of ∼5.6 hr. In Tables 1–4, we show the evolutionary history (formation channels) of the most common LISA double WDs. We indicate the contribution (percentages) of the said channel to the population of LISA WD binaries from that particular Galaxy component (disk, Tables 1 and 2, or bulge, Tables 3 and 4). The characters and numbers in parentheses in the central column of the tables represent the formation channel histories, and recount various stages of stellar evolution of the primary and secondary star progenitors (see Table 1 caption for description).

In our formation channel notation, for a "CO–He" binary the CO WD is the first WD formed, but which is not necessarily the more massive WD (and thus is not necessarily the primary star). In general, the detached systems have evolved from progenitors in which the stars initially have comparable masses,¹⁴ thus first RLOF mass transfer is more often dynamically stable and the binary does not undergo a CE at this stage (such as He–He detached double WDs). Typically, the detached double WD progenitors go through only one CE event, whereas in many cases, the RLOF binaries undergo two CE events.

We wish to point out that in the case of several RLOF binaries (e.g., those involving hybrid donors), there is a short-lived detached phase preceding the long-lived RLOF phase. For example, in the evolutionary history of a COHyb-R1 binary, the progenitor system only spends ∼10² Myr as a detached CO-Hyb WD after the second CE phase before being driven to contact via GR. Once contact is reached, however, the binary will spend a long time (Gyr) as a stable RLOF system. This explains why the COHyb-R1 channel shows up in Table 2, but its predecessor COHyb detached channel is absent in Table 1 (COHyb-D1 binaries only account for ∼1% of the disk formation channels for LISA double WDs).

In Figures 1–4, we show in (arbitrary) color the number densities of the most common formation channels for the detached and RLOF populations in the disk and bulge. We do not include the less-populated formation channels in the plot or it would be very difficult to distinguish between channels. We do, however, include some formation channels involving BDs, since even though these binaries do not significantly affect the overall GR signal (and we have ignored them in the signal calculation), they are relatively abundant, specifically in the bulge population.

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It is expected that the disk would host binaries that have evolved from a wider variety of formation channels than the bulge, since the age of the disk binaries extends over a large range: ∼few hundred Myr to 9 Gyr. Note that high-frequency systems only exist in the Galactic disk. There is a depletion of very high frequency (few minutes orbital period) RLOF systems in the bulge since the present bulge population only contains double WDs with long-lived (>9 Gyr) formation histories. All of the heavier (ONe+CO, CO+CO, and CO+hybrid) binaries originating from more massive progenitors that were born during star formation in the bulge have either long since merged, or in the case of RLOF binaries, the stars have long since reached contact and are now exchanging mass in RLOF on slowly expanding orbits.¹⁵ Once any system encounters stable RLOF, the binary's orbital period increases as a consequence of the conservation of angular momentum under continued mass exchange, since the less massive (larger) WD is losing matter to the more massive WD. For the Galactic disk, some WD binaries are still relatively young, and are in an earlier phase of stable RLOF and thus we "catch" these mass transferring systems at shorter orbital periods (GR frequencies above ∼4.5 mHz; P_orb ≲ 7–8 minutes).

Typical detached evolution. Detached double CO WDs are formed through a variety of evolutionary channels. For the most prominent CO–CO channel (COCO-D1; see Table 1), one particular example of evolution proceeds as follows: two MS stars (2.88 and 2.45 M_☉) start out with a P_orb of 14.2 days and an eccentricity of 0.54 when the MW is 9180 Myr old. At 9602 Myr, the more massive star (s1) begins to evolve off of the MS and becomes a Hertzsprung gap star. Shortly thereafter, s1 fills its Roche Lobe and begins to transfer matter to the companion (interaction between component stars aids in circularizing the orbit; P_orb ∼4.5 days). RLOF is dynamically stable, and since s1 quickly becomes the less massive star during mass transfer P_orb increases (to ∼143 days), and RLOF stops at 9607 Myr when s1 is a red giant (0.44 M_☉; core mass 0.43 M_☉) and s2 (3.67 M_☉) is still on the MS. s1 continues to evolve and shortly thereafter becomes a (naked) helium star (0.44 M_☉), as it has lost the remaining part of its depleted (by prior RLOF) envelope. At 9731 Myr, s2 has evolved off of the MS to become a Hertzsprung gap star. By 9777 Myr, s2 has now evolved into an early asymptotic giant branch (AGB) star (3.62 M_☉), and the orbital period has decreased to 81 days due to tidal interactions (expanding s2; s1 is still a helium star). s2 then fills its Roche Lobe, and since now the mass ratio is rather large (M_don/M_acc ≈ 8), the mass transfer is dynamically unstable, leading to a CE phase. After the CE event the donor (s2) has become a Hertzsprung gap helium star (0.82 M_☉) and P_orb has decreased by more than 2 orders of magnitude to 3 hr. The orbit then starts to decay slightly due to angular momentum losses associated with GR emission, and there is a brief RLOF phase in which the Hertzsprung gap helium star (s2) loses ∼0.1 M_☉, half of which is gained by s1 (still a helium star, now 0.49 M_☉). When the MW is 9779 Myr old, s2 finally becomes a CO WD (primary WD). s1 continues to burn helium and becomes a Hertzsprung gap helium star at 9838 Myr (P_orb = 2.7 hr). Within 20 Myr, s1 then becomes a low-mass CO WD (secondary WD), finishing its helium star evolution. At 10 Gyr (present), the system is found as a detached double CO WD with component masses of 0.70 and 0.49 M_☉, and P_orb = 2.28 hr (f = 0.24 mHz). The two WDs will be brought together in ∼300 Myr due to GR emission. The system is expected to merge due to dynamically unstable mass transfer with a combined mass of 1.19 M_☉ (pre-merger orbital period of ∼1.3 minutes). The total mass of the merged system does not exceed the canonical Chandrasekhar mass and so it is an unlikely Type Ia Supernova progenitor, though may result in the formation of a massive rapidly rotating WD (Saio & Nomoto 2004; Yoon & Langer 2005).

Typical RLOF evolution. In Figure 5, we show a representative evolution through RLOF for one of the Galactic double WDs, a COHe-R1 channel system (see Table 2). The primordial binary consists of two low-mass MS stars (2.60 and 1.40 M_☉) on a wide (P_orb = 31.0 yr) and eccentric (e = 0.9) orbit, born when the MW is 5410 Myr old. At 5966 Myr, the more massive star (s1) begins to evolve off of the MS. The orbit begins to shrink due to tidal interactions and by 6126 Myr, s1 is an early AGB star, the orbit has circularized (e = 0), and the period has decayed to 2.5 yr. Within 1 Myr, s1 becomes a late AGB star. Magnetic braking (convective envelope of s1) dominates angular momentum loss and the orbit decays slightly. s1 fills its Roche-Lobe (6127 Myr) and mass transfer is unstable, leading to a CE phase and a drastic decrease in orbital period (from 2.4 yr to 8.2 days). s1 then becomes a CO WD (primary WD, 0.64 M_☉). At 8782 Myr, s2 evolves off of the MS. The orbit decays due to magnetic braking (convective s2), and at 9085 Myr, s2 initiates a CE phase resulting in another prominent period decrease (6.1 days to 30 minutes) and the loss of its hydrogen-rich envelope. s2 then becomes an He WD (0.22 M_☉). GR causes the WDs to come into contact a few Myr later (P_orb ∼ 3 minutes), and the secondary (s2) fills its Roche Lobe. The mass ratio is only ∼0.34, thus RLOF is stable, and an AM CVn system is born. Initially, mass transfer is quite rapid (see Figure 5, top panel), but a slower mass-transfer phase follows. The binary is found at 10 Gyr with P_orb ∼ 1 hr (∼0.6 mHz) and with component masses of 0.83 and ∼0.01  M_☉ for the CO and He WDs, respectively. Many of the 25 observed AM CVn binaries' parameters are uncertain, and detection of these systems is biased against the typical, longer-period systems with lower mass-transfer rates. However, this particular simulated binary's properties are somewhat similar to the very few long-period AM CVn binaries that have been detected (e.g., SDSS J1552 and CE 315), though the nature of the donors is still unknown (Anderson et al. 2005; Roelofs et al. 2007b).

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Characteristic properties of the entire detached population are shown in Figure 6. Disk systems are found at all orbital periods (top panel) within the LISA sensitivity range, with a drop in number at shorter periods due to the fact that once systems reach contact, double WDs either fall into the RLOF population or otherwise merge and drop out of the LISA population altogether. The masses of the secondaries (M_s, less massive WDs, middle panel) lie predominantly in four regions: ∼0.1 M_☉ (He WDs with CO primaries), a prominent peak near 0.3 M_☉ (mostly He with some hybrid WDs with mostly CO or He companions), a small clump near 0.5 M_☉ (CO secondaries with CO primaries), and a short peak at ∼0.7 M_☉ (CO or sometimes ONe secondaries with either CO or ONe primaries). The He secondaries with very low mass (0.1 M_☉) derive from the same formation channels as cataclysmic variables (similar systems have been discussed in Podsiadlowski 2008), in which a CO WD accretes from a low-mass MS star for several Gyr (note the higher number of these systems in the Galactic bulge). Eventually, the donor becomes fairly exhausted of its hydrogen supply, and meanwhile has built up a significant helium core. The donor is eventually depleted of so much mass that it reaches the hydrogen burning limit (0.08 M_☉), and thus becomes degenerate (in this case, helium rich—a helium WD). Many of the systems whose secondary mass peaks near 0.3 M_☉ are progenitors of AM CVn stars (COHe), while most of the binaries involving more massive progenitors (CO and ONe) will likely merge once they reach contact. Primary WDs (M_p, more massive WDs; bottom panel) display a peak at 0.3–0.4 M_☉, though most primaries have masses 0.5–0.8 M_☉, and a smaller number of systems are quite heavy (0.8–1.4 M_☉). The low-mass peak is comprised of He WD primaries, with He secondaries. The majority of the rest are CO WDs, with either He or CO secondary companions. In both the middle and bottom panels, we note that there are relatively fewer heavy WDs in the bulge population. This is because all of the more massive progenitors (e.g., CO–CO) have had a sufficiently long enough time to evolve off of the MS, encounter 1 or 2 CE phases, reach contact (GR is also more efficient at bringing together two CO WDs than two He WDs) and have either merged out of the population altogether or in some cases, have moved into the RLOF population. A two-dimensional gray-shaded plot is presented in Figure 7, where we show the relationship between the secondary and primary masses of LISA binaries in the Galactic disk, presented in terms of relative total percentages of LISA double WDs.

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Characteristic properties of the entire RLOF population are shown in Figure 8. The RLOF systems with the maximum P_orb (lowest GR frequencies) appear at ∼75 minutes (log(f_gr)=−3.34). The shape of the period distribution (top panel) is due to the fact that most systems spend the majority of their time at periods of ∼1 hr during the dynamically stable RLOF phase. As indicated previously, 99% of RLOF systems are AM CVn (a WD accreting from an He or hybrid WD). For most WD binaries, at the onset of stable RLOF, orbital periods are on the order of only a few minutes, and mass-transfer rates are initially high (see Figure 5, top panel). The donor quickly becomes exhausted of most of its mass, the mass-transfer rate decreases, and the period continues to increase, though at a much slower rate. In the middle panel, we show WD secondary masses, which are strongly peaked at ∼0.01 M_☉, with a slight drop-off to 0.045 M_☉. These WDs are comprised of mostly He WDs, some hybrid WDs and a small number of CO WDs. The "pile-up" of WDs at 0.01 M_☉ stems from the fact that once an AM CVn WD–WD progenitor reaches contact, the mass-transfer rates are initially high (∼10⁻⁶ M_☉ yr⁻¹) and the donor is rapidly exhausted of mass, and thus it is rare to catch AM CVn secondaries that are relatively massive (>0.02 M_☉). In the bottom panel, we show the distribution of masses among the primary WDs, which are mostly CO WDs, with a small contribution of ONe WDs (masses >1.3 M_☉). We show in Figure 9 a two-dimensional gray-shaded plot of RLOF LISA binaries in the Galactic disk, presented in terms of relative total percentages of LISA double WDs.

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The "transition frequency" is said to correspond to the GR frequency at which the gravitational wave spectrum transitions from confusion limited to individually resolvable sources. The exact conditions that are necessary in order for this to occur (one possibility would be when the average number of binaries per frequency bin drops below 1 are not completely clear. The addition of RLOF systems to the detached population will increase the frequency at which the average number of binaries per bin drops below 1 simply by adding more sources to the population. This is particularly so since RLOF binaries are able to maintain short orbital periods (for longer than their detached counterparts) once contact has been established. Furthermore, as the transition frequency increases, the spread of the signal due to the motion of LISA also increases, thus the signals will continue to overlap at higher frequencies than might be inferred from a simple analysis of the number of binaries per resolvable frequency bin. For a simple detached system in a circular orbit whose orbital period is evolving solely due to the emission of GR, seven parameters are needed to fully characterize the signal. Basic estimates from information theory describe how the limited amount of information in the LISA data stream can be mapped onto an equivalent amount of information about detected binary systems. A common rule of thumb is the "three-bin rule": at least three (otherwise unpopulated) frequency bins of LISA data are needed to resolve a seven-parameter binary source (see Timpano et al. 2006 and references therein for a discussion). One can reasonably expect that a mass-transferring system will require more parameters to characterize the GR signal, and consequently more frequency bins to individually resolve the source. Therefore, the average number of binaries per frequency bin is not a particularly good indicator of how the transition frequency changes with the addition of the RLOF population.

For the sake of comparison with previous studies, we have estimated the transition frequency using the three-bin rule method. The point at which LISA binaries start to occupy one source per three frequency bins is found to be 1.5 mHz for the detached LISA binaries only, which is comparable to Nelemans et al. (2001b) and Nelemans (2003b), who found a transition frequency of 1.5–2 mHz (see also Nelemans et al. 2004 for a discussion of the inclusion of AM CVn binaries in the confusion limit calculation). However, we find that when we consider the entire LISA population of double WDs, the transition frequency is increased to 2.4 mHz. We would like to point out however that these frequencies do not represent the frequencies above which one expects to resolve all LISA binaries; there are still a number of populated bins at higher frequencies.

The three-bin rule method of estimating the frequency at which binaries start to become resolved is not a robust method, since it is expected that at these higher frequencies, the frequency evolution due to mass-transferring binaries can contribute a significant additional spreading of the signal. Further, this method does not take into account important criteria such as Doppler, phase, and amplitude modulation, and one should consider alternative (more technical) methods in computing the transition frequency region (e.g., chapter 2 of Ruiter 2009). In the following subsection, we discuss an approach other than the three-bin rule of thumb to identify the population of resolved sources. Instead of using a general estimator like the transition frequency, it is possible to use a predicted signal-to-noise ratio (S/N) for any particular system to determine whether it is potentially resolvable or not (see also Timpano et al. 2006).

For LISA observations of length T_obs, the sensitivity to a particular binary source can be expressed by constructing a simple estimator of the S/N. For circularized compact binaries, long observations will narrow the bandwidth of the source, and the binary will appear in the Fourier spectrum as a narrow spectral feature with root spectral density h_b(f), given in terms of the strain amplitude h_o as (Larson et al. 2000)

$$\begin{equation} h_{b}(f) = h_{o}\,\sqrt{T_{\rm obs}}. \end{equation} \tag{ 13 }$$

LISA will detect binaries in the presence of competing sources of noise. There will be two components to the noise in LISA analysis: instrumental noise in the detector itself and irreducible astrophysical noise from the total population of Galactic close binaries. The instrumental component of the noise is given by the shape of the LISA sensitivity curve (Larson et al. 2000). In this work, the Galactic foreground noise is computed from the synthesized Galaxy itself (i.e., with StarTrack). The gravitational wave strength of every binary in the synthesized Galaxy is estimated from its parameters, and a running median of the total signal is computed as a function of frequency. Since the total number of individually resolved binaries is expected to be small compared to the total population, the running median should be a good estimate of the residual astrophysical noise foreground that would result from a fully realized science analysis procedure. The foreground resulting from this procedure has been shown in Figure 10 (top panel). Additionally, we have calculated the median foreground signal that is expected to arise from the unresolvable sources alone (the "Reduced Galaxy"), which is likely a more realistic estimate of the true Galactic foreground. In Figure 11, we show the Galactic foreground alongside the Galactic foreground signal minus the signal arising from the sources that were determined to be resolvable using our signal-to-noise estimate technique, assuming an S/N >5 (see Section 4.3). Alongside our Galactic foregrounds, we show for comparison the standard LISA sensitivity curve as well as the average Michelson noise of LISA. The latter curve (see Section 2.3) is a more appropriate comparison of the LISA instrumental noise with our calculated gravitational wave signal from the Galactic binaries, as both curves (our signal and the Michelson noise) account for directional and frequency dependencies of the wave as measured by LISA (Rubbo et al. 2004). The standard LISA sensitivity curve (Larson 2000) is often presented in the literature, and is an approximation of the spectral amplitude h_f scaled from the barycenterd sensitivity amplitude h_o from the Online Sensitivity Curve Generator.

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The ultimate level of the confusion foreground (and as a result, the ultimate S/N of any given source) will depend strongly on the actual data analysis algorithm used to identify and subtract sources. To date, no fully realized implementation of such a procedure has been developed. The computed foreground shown here should be a good estimate of the output of a fully developed analysis procedure.

Though millions of double WDs are expected to be detected with LISA, a much smaller number of these systems are predicted to be resolved, depending upon the physical properties of each binary (see also Stroeer et al. 2005 for a discussion). Due to the small separations of these compact binaries, Cooray et al. (2004) predicted that nearly 1/3 of the resolved LISA binaries will be eclipsing in the optical regime and thus will offer an opportunity for these systems to be studied with follow-up observations, allowing for the determination of WD physical parameters (i.e., radii). It has been noted by Nelemans (2009), however, that the intrinsic brightness of WDs was overestimated in Cooray et al. (2004), and thus the expected number of LISA double WDs with electromagnetic counterparts should be lower. Nelemans (2009) predict a smaller, though still relatively optimistic, number of LISA systems that may have optical and/or near infrared counterparts—possibly as many as ∼2000 sources (see their Section 3.3).

The total effective noise in the detector (instrumental + foreground) is given by the spectral amplitude h_f(f), and the S/N is estimated as (Larson et al. 2000)

$$\begin{equation} {\rm S/N} \simeq \frac{h_{b}(f)}{h_{f}(f)} = \frac{h_{o}\sqrt{T_{\rm obs}}}{h_{f}(f)}. \end{equation} \tag{ 14 }$$

A binary is deemed "resolvable" (distinguishable from the confusion limited Galactic foreground) if its S/N is greater than some detection threshold (typical S/N ⩾5). Additionally, resolved sources in the population are sorted into monochromatic sources and chirping sources. Monochromatic sources do not evolve appreciably over the LISA observing time, T_obs. A binary is considered to be chirping if its frequency changes by more than the LISA frequency resolution, Δf ⩾ 1/T_obs. This criterion is a conservative one, as the expectation is that LISA will be able to detect frequency changes that are smaller than the frequency resolution Δf (Takahashi & Seto 2002). The detection of binaries with appreciable evolution in f (or "chirp") is useful, since this frequency evolution allows for the determination of ${\cal M}$, from which the distance to the source can be calculated (see, e.g., Nelemans et al. 2001b).

The estimate of the S/N in Equation (14) will not be precise in the regime where orbital periods change on timescales that are short compared to the observation time, or where the shape of the instrumental noise changes appreciably as a function of frequency. However, this simple estimator is a good tool to use as a first cut in a search over a large population of sources where a calculation of the S/N for each binary would be computationally prohibitive. For the purposes of this paper, this estimator should be perfectly adequate since in large part the binaries of interest are not chirping across a significant portion of the LISA band during T_obs. Once a set of Galactic binaries has been identified using the simple estimator in Equation (14), it is computationally plausible to consider a more robust estimate of the S/N for each system, which can then be constructed from the data analysis technique of choice. For example, with chirping binaries a good estimate of the S/N can be motivated from matched filtering techniques by integrating over a binary's spectral energy distribution, dE/df (Flanagan & Hughes 1998).

Using this simple S/N estimator to resolve LISA binaries, we find that a total of ∼11,300 double WDs will be resolvable in our Galactic sample: ∼600 chirping and ∼10,700 monochromatic. This is inferred under an optimistic assumption that all binaries with an S/N >5 have a chance to be resolved. Within the resolvable population, we find ∼4000 AM CVn binaries and ∼500 possible double degenerate scenario—DDS; Iben & Tutukov (1984); Webbink (1984)—SN Ia progenitors (detached COCO WD binaries that will merge within a Hubble time, whose total mass exceeds the Chandrasekhar mass limit). Out of these possible SN Ia progenitors, the majority are formed through the COCO-D4 channel, which involves the most massive primaries on the ZAMS. As for the AM CVn resolvable binaries, 7% are chirping and arise mostly from COHe-R1 and COHyb-R1 channels, in near-equal numbers. Both formation channels involve two CE events and thus if these systems will be resolved with LISA, they will serve as useful objects for understanding close binary evolution. Additionally, a number of progenitor AM CVn binaries are resolvable (∼3000; detached channels), but only ∼100 of these are chirping.

If we relax our criteria and assume that any binary with an S/N >1 is potentially resolvable, we find ∼43,000 resolvable double WDs (∼1000 chirping and ∼42,000 monochromatic). Additionally, we find ∼2800 LISA binaries with S/N >10, ∼300 of which are chirping sources. The binaries with large S/N (≳10) typically all have orbital periods <2000 s (f_gr>0.001 Hz) and have either just recently encountered RLOF, and thus they still have relatively large chirp masses, or they are close to reaching contact.

In Table 5, we show a breakdown of the 11,300 (2800) potentially resolvable LISA double WDs that have S/N > 5 (S/N >10), with percentages relative to the resolvable population in question. The properties of those resolved systems reflect some of the selection effects that are expected to arise from using LISA to identify double WDs. Systems with short orbital periods (and consequently high gravitational wave frequencies) are favored both because the amplitude of the signal scales as f^2/3 and because the number of binaries per bin drops with increasing frequency. Furthermore, systems with large chirp masses are favored simply because the amplitude of their GR scales as ${\cal M}^{5/3}$. Thus, we see that the resolvable population is biased toward high-mass, high-frequency (short period) systems. Realistically speaking, it is still unclear as to which method(s) of detecting resolvable binaries with LISA will be most successful, and so we have discussed the likelihood of potentially resolving individual binaries using a few different methods.

Detached resolved sources. A large number of the resolved sources are of the HeHe-D1 channel, where two initially low-mass (∼1 M_☉) evolved stars lose their envelopes during mass-transfer phases. HeHe-D1 is not only the most prolific formation channel of detached systems (Tables 1 and 3), but it also extends to high GR frequencies. Lumping all of the COCO-D formation channels together, COCO detached binaries make up ∼13%–18% of the resolved population, and potential DDS progenitors make up 5% of the total resolved population. Though HeCO-D1 double WDs are more numerous compared to COHe-D1 binaries, fewer of them are resolved. We note that COHe-D1 progenitors involve two CE phases (HeCO-D1 only one), and overall occupy closer orbits thus contributing to higher GR frequencies.

RLOF resolved sources. For RLOF systems, the majority are of AM CVn type. The most common RLOF channel is COHe-R1, followed by COCO-R1. COCO-R1 binaries are not common, but they lie on very close orbits, consisting of a massive (∼1 M_☉) CO WD accreting matter from a low-mass (<0.2 M_☉) CO WD. The resolved COHe-R1 AM CVn systems account for most of the total number of resolved AM CVn binaries. Since the formation of these systems involves two CE events, future LISA observations may put some very useful constraints on this rather uncertain evolutionary phase, which is crucial to the formation of close binaries. However, when comparing relative formation scenarios of AM CVn binaries, we must keep in mind that the precise behavior of the stars upon reaching contact—and the exact physical conditions of the donor at contact—are still unclear, as are the physical characteristics of the AM CVn progenitors themselves. Further observations and detailed modeling will be an important step toward understanding the donors in AM-CVn-type systems (Deloye et al. 2007). For example, deviations from the equilibrium evolution of AM CVn systems, e.g., brought on upon by changes in the mass-transfer rate following a nova explosion, result in a measurable frequency evolution, which can greatly differ from what is expected for the long-term, equilibrium evolution case. This was already shown by Stroeer & Nelemans (2009); if short-term changes in the evolution of AM CVn systems are neglected, it can lead to erroneous estimations of some of the binary parameters.