It was expected initially that contact W UMa binaries will dominate the gravitational wave spectrum at
low frequencies [258]. However, it was shown in [417, 254, 162, 93, 224
] 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
and strain amplitude
:
. The weakest signal is
. For the time span of
observations
, the elementary resolution bin of the detector is
. Then,
integration of
over amplitude down to a certain limiting
and over
gives the mean
number of sources per unit resolution bin for a volume defined by
. If for a certain
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 [124] for more details). Later,
more involved analytic studies of the GW background produced by binary stars at low frequencies were
continued in [97, 145, 330, 144
].
A more detailed approach to the estimate of the GW background is possible using population synthesis
models [224, 445, 286
, 287
].
Nelemans et al. [286] 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. [133, 286
, 423, 157]). 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 ,
, and
. Figure 10
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 [281, 251
]. 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
which
explains their accumulation at lower
. For UCXBs this time span is only
20 Myr. In
the discussed model, the systems are distributed randomly in the Galactic disk according to
|
|
|
|
|
|
|
|
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 |
|
|
|
|
|
|
|
|
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 wide bins, with
being the total integration time. Figure 11
shows the resulting confusion limited background
signal. In Figure 12
the number of systems per bin is plotted. The assumed integration time is
. 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
producing a peak there, as explained in the comment to
Figure 10
.
Figure 11 shows that there are many systems with a signal amplitude much higher than the average in
the bins with
, 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 13). For this
“statistical” notion of
, Nelemans et al. [286
] get the first bin containing exactly one system at
, while up to
there are bins containing more than one
system.
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 14
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. [286
, 287
]). The comparison with the work of other
authors [143, 445, 375
] 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).
Within the model of Nelemans et al. [286] there are about 12,100 detached DD systems that can be
resolved above
and
6,100 systems with
that are detectable above the noise. This result
was confirmed in a follow-up paper [287
] 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 15
. In the latter study the following
was noted. Previous studies of GW emission of the AM CVn systems [144
, 286
] have found that they
hardly contribute to the GW background noise, even despite at
= (0.3 – 1.0) mHz they
outnumber the detached DDs. This happens because at these
their chirp mass
is
well below that of a typical detached system. But it was overlooked before that at higher
,
where the number of AM CVn systems is much smaller, their
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
, for
, LISA may be expected to resolve
11,000 AM CVn systems at
(or
3,000 at
), 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 [375, 63]. 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 15.
Peculiarly enough, as a comparison of Figures 14
and 15
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.
http://www.livingreviews.org/lrr-2006-6 |
© Max Planck Society and the author(s)
Problems/comments to |