10 Gravitational Waves from Compact Binaries with White-Dwarf Components

It was expected initially that contact W UMa binaries will dominate the gravitational wave spectrum at low frequencies [258]. However, it was shown in [41725416293Jump To The Next Citation Point224Jump To The Next Citation Point] that it is, most probably, totally dominated by detached and semidetached double white dwarfs.

As soon as it was recognized that the birth rate of Galactic close double white dwarfs may be rather high and even before the first close DD was detected, Evans, Iben, and Smarr in 1987 [93] accomplished an analytical study of the detectability of the signal from the Galactic ensemble of DDs, assuming certain average parameters for DDs. Their main findings may be formulated as follows. Let us assume that there exists a certain distribution of DDs over frequency of the signal f and strain amplitude h: n(f,h ). The weakest signal is hw. For the time span of observations τint, the elementary resolution bin of the detector is Δfint ≈ 1∕τint. Then, integration of n(f, h) over amplitude down to a certain limiting h and over Δf gives the mean number of sources per unit resolution bin for a volume defined by h. If for a certain hn

∫ ∞ ( ) --dn- Δ ν = 1, (66 ) hn df dh int
then all sources with hn > h > hw overlap. If in a certain resolution bin hn > hw does not exist, individual sources may be resolved in this bin for a given integration time (if they are above the detector’s noise level). In the bins where binaries overlap they produce so-called “confusion noise”: an incoherent sum of signals; the frequency, above which the resolution of individual sources becomes possible got the name of “confusion limit”, f c. Evans et al. found f ≈ 10 mHz c and 3 mHz for integration times 106 s and 108 s, respectively.

Independently, the effect of confusion of Galactic binaries was demonstrated by Lipunov, Postnov, and Prokhorov [225] who used simple analytical estimates of the GW confusion background produced by unresolved binaries whose evolution is driven by GWs only; in this approximation, the expected level of the background depends solely on the Galactic merger rate of binary WDs (see [124Jump To The Next Citation Point] for more details). Later, more involved analytic studies of the GW background produced by binary stars at low frequencies were continued in [97145330144Jump To The Next Citation Point].

A more detailed approach to the estimate of the GW background is possible using population synthesis models [224445Jump To The Next Citation Point286Jump To The Next Citation Point287Jump To The Next Citation Point].

Nelemans et al. [286Jump To The Next Citation Point] constructed a model of the gravitational wave signal from the Galactic disk population of binaries containing two compact objects. The model included detached DDs, semidetached DDs, detached systems of NSs and BHs with WD companions, binary NSs and BHs. For the details of the model we refer the reader to the original paper and references therein. Table 6 shows the number of systems with different combinations of components in the Nelemans et al. model15. Note that these numbers strongly depend on the assumptions in the population synthesis code, especially on the normalization of stellar birth rate, star formation history, distributions of binaries over initial masses of components and their orbital separations, treatment of stellar evolution, common envelope formalism, etc. For binaries with relativistic components (i.e. descending from massive stars) an additional uncertainty is brought in by assumptions on stellar wind mass loss and natal kicks. The factor of uncertainty in the estimated number of systems of a specific type may be up to a factor ∼ 10 (cf. [133286Jump To The Next Citation Point423157]). Thus these numbers have to be taken with some caution; we will show the effect of changing some of approximations below. Table 6 immediately shows that detached DDs, as expected, dominate the population of compact binaries.

Population synthesis computations yield the ensemble of Galactic binaries at a given epoch with their specific parameters M1, M2, and a. Figure 10View Image shows examples of the relation between frequency of emitted radiation and amplitude of the signals from a “typical” double degenerate system that evolves into contact and merges, for an initially detached double degenerate system that stably exchanges matter after contact, i.e. an AM CVn-type star and its progenitor, and for an UCXB and its progenitor. For the AM CVn system effective spin-orbital coupling is assumed [281251Jump To The Next Citation Point]. For the system with a NS, the mass exchange rate is limited by the Eddington one and excess of the matter is “re-ejected” from the system” (see Section 3.2.3 and [472]). Note that for an AM CVn-type star it takes only ∼ 300 Myr after contact to evolve to log f = − 3 which explains their accumulation at lower f. For UCXBs this time span is only ∼ 20 Myr. In the discussed model, the systems are distributed randomly in the Galactic disk according to

ρ(R, z) = ρ e−R∕H sech(z∕β )2 pc −3, (67 ) 0
where H = 2.5 kpc [362] and β = 200 pc. The Sun is located at R ⊙ = 8.5 kpc and z⊙ = − 30 pc.


Table 6: Current birth rates and merger rates per year for Galactic disk binaries containing two compact objects and their total number in the Galactic disk [286Jump To The Next Citation Point].








Type Birth rate Merger rate Number




Detached DD 2.5 × 10–2 1.1 × 10–2 1.1 × 108
Semidetached DD 3.3 × 10–3 4.2 × 107
NS + WD 2.4 × 10–4 1.4 × 10–4 2.2 × 106
NS + NS 5.7 × 10–5 2.4 × 10–5 7.5 × 105
BH + WD 8.2 × 10–5 1.9 × 10–6 1.4 × 106
BH + NS 2.6 × 10–5 2.9 × 10–6 4.7 × 105
BH + BH 1.6 × 10–4 2.8 × 106









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Figure 10: Dependence of the dimensionless strain amplitude for a WD + WD detached system with initial masses of the components of 0.6M ⊙ + 0.6 M ⊙ (red line), a WD + WD system with 0.6M ⊙ + 0.2M ⊙ (blue line) and a WD + NS system with 1.4M ⊙ + 0.2 M ⊙ (green line). All systems have an initial separation of components 1 R ⊙ and are assumed to be at a distance of 1 kpc (i.e. the actual strength of the signal has to be scaled with factor 1 ∕d, with d in kpc). For the DD system the line shows an evolution into contact, while for the two other systems the upper branches show pre-contact evolution and lower branches – a post-contact evolution with mass exchange. The total time-span of evolution covered by the tracks is 13.5 Gyr. Red dots mark the positions of systems with mass-ratio of components q = 0.02 below which the conventional picture of evolution with a mass exchange may be not valid. The red dashed line marks the position of the confusion limit as determined in [287Jump To The Next Citation Point].
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Figure 11: GWR background produced by detached and semidetached double white dwarfs as it would be detected at the Earth. The assumed integration time is 1 yr. The ‘noisy’ black line gives the total power spectrum, the white line the average. The dashed lines show the expected LISA sensitivity for a S ∕N of 1 and 5 [212Jump To The Next Citation Point]. Semidetached double white dwarfs contribute to the peak between logf ≃ − 3.4 and − 3.0. (Figure from [286Jump To The Next Citation Point].)
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Figure 12: The number of systems per bin on a logarithmic scale. Semidetached double white dwarfs contribute to the peak between log f ≃ − 3.4 and − 3.0. (Figure from [286Jump To The Next Citation Point].)
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Figure 13: Fraction of bins that contain exactly one system (solid line), empty bins (dashed line), and bins that contain more than one system (dotted line) as function of the frequency of the signals. (Figure from [286Jump To The Next Citation Point].)

Then it is possible to compute strain amplitude for each system. The power spectrum of the signal from the population of binaries as it would be detected by a gravitational wave detector, may be simulated by computation of the distribution of binaries over Δf = 1∕T wide bins, with T being the total integration time. Figure 11View Image shows the resulting confusion limited background signal. In Figure 12View Image the number of systems per bin is plotted. The assumed integration time is T = 1 yr. Semidetached double white dwarfs, which are less numerous than their detached cousins and have lower strain amplitude dominate the number of systems per bin in the frequency interval − 3.4 ≲ logf (Hz ) ≲ − 3.0 producing a peak there, as explained in the comment to Figure 10View Image.

Figure 11View Image shows that there are many systems with a signal amplitude much higher than the average in the bins with f < fc, suggesting that even in the frequency range seized by confusion noise some systems may be detectable above the noise.

Population synthesis also shows that the notion of a unique “confusion limit” is an artifact of the assumption of a continuous distribution of systems over their parameters. For a discrete population of sources it appears that for a given integration time there is a range of frequencies where there are both empty resolution bins and bins containing more than one system (see Figure 13View Image). For this “statistical” notion of fc, Nelemans et al. [286Jump To The Next Citation Point] get the first bin containing exactly one system at log f (Hz ) ≈ − 2.84, while up to log f (Hz) ≈ − 2.3 there are bins containing more than one system.

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Figure 14: Gravitational waves background formed by the Galactic population of white dwarfs and signal amplitudes produced by some of the most compact binaries with white dwarf components ([273]). The green line presents results from [286Jump To The Next Citation Point], the black one from [287Jump To The Next Citation Point], while the red one presents a model with all assumptions similar to [287Jump To The Next Citation Point], but with the γ-formalism for the treatment of common envelopes [285284] (see also Section 3.5). The blue line shows the background derived in [28]. (Figure from [275].)

As we noted above, predictions of the population synthesis models are sensitive to the assumptions of the model; one of the most important is the treatment of common envelopes (see Section 3.5). Figure 14View Image compares the average gravitational waves background formed by Galactic population of white dwarfs under different assumptions on star formation history, IMF, assumed Hubble time, and treatment of some details of stellar evolution (cf. [286Jump To The Next Citation Point287Jump To The Next Citation Point]). The comparison with the work of other authors [143445375Jump To The Next Citation Point] shows that both the frequency of the confusion limit and the level of confusion noise are uncertain within a factor of ∼ 4. This uncertainty is clearly high enough to influence seriously the estimates of the possibilities for a detection of compact binaries. Note, however, that there are systems expected to be detected above the noise in all models (see below).

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Figure 15: Strain amplitude h as a function of the frequency for the model populations of resolved DDs (left panel, ≃ 10,700 objects) and AM CVn systems (right panel, ≃ 11,000 objects.). The gray shades give the density distribution of the resolved systems normalized to the maximum density in each panel (1,548 and 1,823 per “cell” for the double white dwarfs and AM CVn-stars panels, respectively). The 200 strongest sources in each sample are shown as dots to enhance their visibility. In the AM CVn panel the periods of several observed short period systems are indicated by the vertical dotted lines. The solid line shows the average background noise due to detached white dwarfs. The LISA sensitivities for an integration time of one year and a signal-to-noise ratio of 1 and 5 are indicated by the dashed lines [212]. (Figure from [287Jump To The Next Citation Point].)

Within the model of Nelemans et al. [286Jump To The Next Citation Point] there are about 12,100 detached DD systems that can be resolved above fc and ≈ 6,100 systems with f < fc that are detectable above the noise. This result was confirmed in a follow-up paper [287Jump To The Next Citation Point] which used a more up-to-date SFH and Galaxy model (this resulted in a slight decrease of the number of “detectable” systems – to ∼ 11,000). The frequency – strain amplitude diagram for DD systems is plotted in the left panel of Figure 15View Image. In the latter study the following was noted. Previous studies of GW emission of the AM CVn systems [144Jump To The Next Citation Point286Jump To The Next Citation Point] have found that they hardly contribute to the GW background noise, even despite at f = (0.3 – 1.0) mHz they outnumber the detached DDs. This happens because at these f their chirp mass ℳ is well below that of a typical detached system. But it was overlooked before that at higher f, where the number of AM CVn systems is much smaller, their ℳ is similar to that of the detached systems from which they descend. It was shown that, out of the total population of ∼ 140,000 AM CVn-stars with P ≤ 1500 s, for Tobs = 1 yr, LISA may be expected to resolve ∼ 11,000 AM CVn systems at S∕N ≥ 1 (or ∼ 3,000 at S∕N ≥ 5), i.e. the numbers of “resolvable” detached DDs and interacting DDs are similar. Given all uncertainties in the input data, these numbers are in reasonable agreement with the estimates of the numbers of potentially resolved detached DDs obtained by other authors, e.g., 3,000 to 6,000 [37563]. These numbers may be compared with expectation that ∼ 10 Galactic NS + WD binaries will be detected [62].

The population of potentially resolved AM CVn-type stars is plotted in the right panel of Figure 15View Image. Peculiarly enough, as a comparison of Figures 14View Image and 15View Image shows, the AM CVn-type systems appear, in fact, dominant among so-called “verification binaries” for LISA: binaries that are well known from electromagnetic observations and whose radiation is estimated to be sufficiently strong to be detected; see the list of 30 promising candidates in [390] and references therein, and the permanently updated (more rigorous) list of these binaries supported by G. Nelemans, G. Ramsay, and T. Marsh at [276].

Systems RXJ0806.3+1527, V407 Vul, ES Cet, and AM CVn are currently considered as the best candidates. We must note that the most severe “astronomical” problems concerning “verification binaries” are their distances, which for most systems are only estimates, and poorly constrained component masses.


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