The Gravitational Wave Background from Cosmological Compact Binaries

Abstract

We use a population synthesis approach to characterise, as a function of cosmic time, the extragalactic close binary population descended from stars of low to intermediate initial mass. The unresolved gravitational wave (GW) background due to these systems is calculated for the 0.1-10 mHz frequency band of the planned Laser Interferometer Space Antenna (LISA). This background is found to be dominated by emission from close white dwarf-white dwarf pairs. The spectral shape can be understood in terms of some simple analytic arguments. To quantify the astrophysical uncertainties, we construct a range of evolutionary models which produce populations consistent with Galactic observations of close WD-WD binaries. The models differ in binary evolution prescriptions as well as initial parameter distributions and cosmic star formation histories. We compare the resulting background spectra, whose shapes are found to be insensitive to the model chosen, and different to those found recently by Schneider et al. (2001). From this set of models, we constrain the amplitude of the extragalactic background to be 1E-12 < Omega(1 mHz) < 6E-12, in terms of Omega(f), the fraction of closure density received in gravitational waves in the logarithmic frequency interval around f.

Full text

Page 1

arXiv:astro-ph/0304393v2 8 Sep 2003
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 2 February 2008 (MN LATEX style file v2.2)
The Gravitational Wave Background from Cosmological
Compact Binaries
Alison J. Farmer⋆and E. S. Phinney
Theoretical Astrophysics, MC 130-33 Caltech, Pasadena, CA 91125, USA
2 February 2008
ABSTRACT
We use a population synthesis approach to characterise, as a function of cosmic time,
the extragalactic close binary population descended from stars of low to intermediate
initial mass. The unresolved gravitational wave (GW) background due to these systems
is calculated for the 0.1–10 mHz frequency band of the planned Laser Interferometer
Space Antenna (LISA). This background is found to be dominated by emission from
close white dwarf–white dwarf pairs. The spectral shape can be understood in terms
of some simple analytic arguments. To quantify the astrophysical uncertainties, we
construct a range of evolutionary models which produce populations consistent with
Galactic observations of close WD–WD binaries. The models differ in binary evolu-
tion prescriptions as well as initial parameter distributions and cosmic star formation
histories. We compare the resulting background spectra, whose shapes are found to
be insensitive to the model chosen, and different to those found recently by Schneider
et al. (2001). From this set of models, we constrain the amplitude of the extragalactic
background to be 1×10−12<
∼Ωgw(1 mHz)<
∼6×10−12, in terms of Ωgw(f), the fraction
of closure density received in gravitational waves in the logarithmic frequency interval
around f.
Key words: gravitational waves — binaries: close — diffuse radiation
1INTRODUCTION
Except at very low radio frequencies, most electromag-
netic telescopes have good angular rejection, so that faint
sources and backgrounds can be seen by looking between
bright sources. In contrast all currently implemented grav-
itational wave detectors, and most of those envisaged for
the future, simultaneously respond to sources all over
the sky, modified only by a beam pattern of typically
quadrupole form. It is therefore important to understand
the brightness of the gravitational wave sky, since this
will limit the ultimate sensitivity attainable in gravita-
tional wave astronomy. One immediate pressure to under-
stand this background comes from the need to set de-
sign requirements for the ESA/NASA Laser Interferometer
Space Antenna (LISA) mission (LISA mission documents
and status may be found at http://lisa.jpl.nasa.gov/ and
http://sci.esa.int/home/lisa/).
In this paper, we attempt to predict the gravitational
wave background produced by all the binary stars in the uni-
verse, excluding neutron stars and black holes. This is be-
lieved to be the principal source of gravitational wave back-
ground in the frequency range 10−5< f < 10−1Hz. Below
⋆e-mail: ajf@tapir.caltech.edu
10−5Hz, the background is probably dominated by merging
supermassive black holes, and above 10−1Hz, it is proba-
bly dominated by merging neutron stars and stellar mass
black holes (whose complicated and poorly understood for-
mation histories and birth velocities make predictions more
uncertain, cf. Belczynski, Kalogera & Bulik 2002).
Besides the extragalactic background, there is also a
Galactic background produced by the binary stars in our
Milky Way (Evans, Iben & Smarr 1987; Hils, Bender & Web-
bink 1990; Nelemans, Yungelson & Portegies Zwart 2001c).
Although the Galactic background is many times larger
in amplitude than both the extragalactic background and
LISA’s design sensitivity, the individual binaries contribut-
ing to it can be (spectrally) resolved and removed at fre-
quencies above ∼ 3 × 10−3Hz (Cornish & Larson 2002).
Below this frequency they cannot be removed (at least in
a mission of reasonable lifetime ∼ 3 years), but the unre-
solved Galactic background will be quite anisotropic. As the
detector beam pattern rotates about the sky, the Galactic
background will thus be modulated, while the isotropic (or
nearly so; see Kosenko & Postnov 2000) distant extragalac-
tic background will not. Modelling of the angular distribu-
tion of the Galactic background using both a priori models
and the observed distribution of higher frequency resolved
sources will thus allow the Galactic background to be sub-

Page 2

2A.J. Farmer and E.S. Phinney
tracted to some precision (Giampieri & Polnarev 1997). In
addition, there will be anisotropies due to the distribution
of local galaxies, at the level of 10 per cent of the distant ex-
tragalactic background from the LMC, and at the per cent
level from M31 or the Virgo cluster (see also Lipunov et al.
1995).
The immediate motivation for this work is a de-
signissueforLISA.One
goals (see the LISA Science Requirements document at
http://www.tapir.caltech.edu/listwg1/) is the detection of
gravitational waves from compact objects spiralling into su-
permassive black holes (Finn & Thorne 2000; Hils & Bender
1995), since these can provide precision tests of strong field
relativity and the no-hair theorem (Hughes 2001). However,
these signals are weak, and their templates not yet fully un-
derstood. It has thus been proposed that LISA should be
designed with somewhat greater sensitivity to increase the
probability that these signals are detected. However, this
would be pointless if the principal background were cosmo-
logical rather than instrumental. As we shall see (Fig. 16),
we find that this is most probably almost, but not quite the
case at the relevant frequencies (4−10 mHz). So there would
be a point to increasing LISA’s sensitivity in the 4−10 mHz
range, but not to increasing it by more than a factor of 3 in
gravitational wave amplitude h (9 in Ω ∝ f2h2).
A second motivation for this work comes from the
fact that this background is an astrophysical foreground to
searches (both with LISA and with future detectors with
extended frequency range and sensitivity) for backgrounds
produced in the very early universe. Gravitational waves
from bubble walls and turbulence following the electroweak
phase transition are expected to be in the LISA frequency
band, with amplitude that could be well above LISA instru-
mental sensitivity (Kamionkowski, Kosowski & Turner 1994;
Kosowsky, Mack & Kahniashvili 2002; Apreda et al. 2002).
Another potential source of isotropic gravitational waves in
the LISA band are those produced when dimensions beyond
the familiar four compactified, which occurred when the uni-
verse had temperature kT >TeV (Hogan 2000).
Note that detection of a gravitational wave background
can possibly be made even if it is considerably below the
noise limit of the LISA detectors shown in our Fig. 16.
This can be done by comparing the signals from Michel-
son beam combinations (sensitive to instrument noise and
gravitational waves) with Sagnac beam combinations (sen-
sitive to instrument noise, but insensitive to gravitational
waves), thus calibrating the instrumental noise —cf. Tinto,
Armstrong & Estabrook (2001), Hogan & Bender (2001).
Gravitational waves are the only directly detectable
relic of inflation in the early universe, and their detec-
tion over a range of frequencies would provide a valuable
test of models of inflation (Turner 1997). It has been pro-
posed that advanced space-based gravitational wave detec-
tors might search for the background of gravitational waves
from inflation. The gravitational waves from slow-roll infla-
tion models contribute to the critical density in the universe
Ωgw < 10−15per octave of frequency. We shall see (Fig.
8, 17) that the gravitational wave background from cosmo-
logical binaries makes such detection impractical except at
frequencies below 10−5Hz (where supermassive black holes
continue to make it impossible), or above 0.1 Hz.
A third motivation is that a detection of the extragalac-
ofLISA’smajorscience
tic binary background, e.g. by LISA, would set an indepen-
dent (and unaffected by dust extinction) constraint on a
combination of the star formation history of the universe
and binary star evolution.
There have been previous estimates of the extragalactic
binary background. Hils, Bender & Webbink (1990) made
detailed estimates of the Galactic binary background, and
estimated that the extragalactic background from close dou-
ble white dwarf pairs should be about 2 per cent (in flux or Ω
units) of the Galactic background. This estimate was refined,
using more modern star formation histories, by Kosenko &
Postnov (1998), who found instead a level of ∼10 per cent.
Schneider et al. (2001) used a descendant of the Utrecht pop-
ulation synthesis code to estimate the extragalactic binary
background as a function of frequency, and claimed that the
background should have a large peak at ∼ 3×10−5Hz, just
below the frequency at which typical binaries have a lifetime
that equals the age of the Universe.
We have followed the spirit of this previous work, but
with an independent binary population synthesis code. More
importantly, we have devoted much effort to the normalisa-
tion of the background, to understanding the contributions
of different types of binaries and their formation pathways
to the background, and to estimating the uncertainties in
all of these, so that we can have a better idea of the sources
and level of uncertainty in the predicted background.
The paper is organised as follows: In section 2, we de-
scribe the gravitational wave (GW) emission from a binary
system, then in section 3 we outline the main evolutionary
pathways to the close double degenerate (DD) stage, which
we shall see is the dominant source of GW background in
the LISA band. In section 4, we use the preceding sections to
make some simple analytic arguments about the nature of
DD inspiral spectra. We describe the use of the bse code
in our population synthesis, in section 5, then go on to
construct a set of synthesis models whose results we test
against the observed Galactic DD population. We also mo-
tivate some modifications made to the prescription for the
evolution of AM CVn stars in the bse code. In section 6, we
present the cosmological integrals used in the code, along
with the cosmic star formation history and overall normali-
sation chosen. Section 7 is devoted to a discussion of the GW
background spectra produced by our code, in terms of the
systems contributing to the background and the progenitors
of these sources. We also discuss the differences between our
population synthesis models. In section 8, we place limits
on the maximum and minimum expected background sig-
nals, and compare these with the LISA sensitivity and in
section 9 with previous work. In section 10 we summarise
and conclude.
2GRAVITATIONAL WAVES FROM A
BINARY SYSTEM
A binary system of stars in circular orbit with masses M1
and M2 and orbital separation a emits gravitational radia-
tion, at the expense of its orbital energy, at a rate given by
(Peters & Mathews 1963)
Lcirc
=
32
5
G4
c5
(M1M2)2(M1+ M2)
a5

Page 3

GW from cosmological binaries3
≃
1.0 × 1032(M′
1M′
2)2(M′
(a′)5
1+ M′
2)
erg s−1, (1)
where primes denote quantities expressed in solar units, i.e.
M/M⊙, a/R⊙. The gravitational radiation is emitted at
twice the orbital frequency ν of the binary, fcirc = 2ν = Ω/π.
If the binary is eccentric with eccentricity e, this ex-
pression must be generalised to include emission at all har-
monics n of the orbital frequency, fn = nν = nΩ/2π, where
Ω = (a−3G(M1+M2))1/2. The luminosity in each harmonic
is given by
L(n,e) = g(n,e)Lcirc,(2)
where Lcirc is the luminosity of a circular binary with sep-
aration a, as given in Eq. 1, where a is now the relative
semi-major axis of the eccentric orbit, and the g(n,e) are
defined in eq. (20) of Peters & Mathews (1963). The total
specific luminosity Lf = dL(f)/df of the system is then a
sum over all harmonics:
Lf(e) = Lcirc
∞
?
n=1
g(n,e)δ(f − nν).(3)
The total luminosity is
L = Lcirc
∞
?
n=1
g(n,e) =1 +73
24e2+37
(1 − e2)7/2
96e4
Lcirc
(4)
For eccentric orbits, the emission spectrum of Eq. 3,
g(n,e)Lcirc as a function of f = nν consists of points along
a skewed bell-shaped curve with maximum near the relative
angular velocity at pericentre, where the greatest acceler-
ations are experienced (2πν ∼ Ωp, where Ωp is the angu-
lar velocity of the relative orbit at pericentre, vp/rp). In
terms of harmonic number, a good approximation for all
e (becoming very good for e > 0.5) is that Lf peaks at
n = 1.63(1 − e)−3/2, and fLf peaks at n = 2.16(1 − e)−3/2.
3EVOLUTION TO THE DD STAGE
We shall see that the GW background is dominated by the
emission from close double degenerate (DD) binaries at fre-
quencies 10−4<
∼f<
refer to WD–WD pairs and loosely to WD–naked helium
star pairs, i.e. we exclude neutron stars from our definition.
In this section we describe the two main evolutionary path-
ways from the zero-age main sequence (ZAMS) to the close
DD stage. The route followed depends mainly on the initial
orbital separation of the ZAMS stars. Similar descriptions
can be found in e.g. Webbink & Han (1998).
We begin with an intermediate-mass ZAMS binary sys-
tem with primary mass M1, secondary mass M2 (< M1),
semi-major axis a and eccentricity e. The orbit may evolve
somewhat due to tidal interactions between the stars, partic-
ularly if they have convective envelopes. When the primary
evolves off the main sequence and swells in size, it may fill
its Roche lobe and start to transfer matter on to the sec-
ondary. The stability of this mass transfer determines which
of the two main pathways to the DD stage is commenced.
∼10−1Hz. In this work, the term DD will
3.1 CEE+CEE
If the primary fills its Roche lobe when it has a deep
convective envelope (i.e. on the red giant branch (RGB)
or asymptotic giant branch (AGB)), then for mass ratios
M1/M2>
∼0.6, the ensuing mass transfer is dynamically un-
stable (for conservative transfer). The envelope of the pri-
mary spills on to the secondary on a dynamical timescale,
leading to the formation of a common envelope, inside which
orbit the secondary and the core of the primary. The enve-
lope is frictionally heated at the expense of the stars’ orbital
energy, until eventually either they coalesce, or the envelope
is heated sufficiently that it is ejected from the system, leav-
ing the primary’s core (a hot subdwarf which will rapidly
cool to become a WD, or if the primary was on the RGB
and had mass M1>
∼2 M⊙, then a helium star which will
evolve to the WD stage). The basic idea of the common
envelope phase is well accepted and observationally moti-
vated, though not well simulated (see e.g. Livio & Soker
1988; Iben & Livio 1993; Taam & Sandquist 2000). Several
formalisms have been proposed to model it in population
synthesis studies. The evolution code used here (see section
5.1) follows closely the prescription of Tout et al. (1997)
(originally from Webbink 1984), in which
Ebind,i= α(Eorb,f− Eorb,i),(5)
where Ebind,i is the initial binding energy of the envelope
of the overflowing giant star (or the sum of both envelopes’
binding energies if both stars are giants), parametrized by
Ebind,i = −G/λ(M1Menv,1/R1), where λ is of order unity,
and is calculated in the bse code (see section 5.3). Eorb,i
and Eorb,f are respectively the initial and final orbital bind-
ing energies of the core-plus-secondary system, and α is the
so-called common envelope efficiency parameter, also of or-
der unity, usually taken to be a parameter to be fitted to
observations. Variations to this prescription will be consid-
ered in sections 5.2 and 5.3.
Continuing with the system’s evolution, the secondary
star later evolves off the main sequence, and a second com-
mon envelope phase is likely to occur, leading to further
orbital shrinkage. If once again the stars do not coalesce
then we will be left with a close(r) pair of remnants, one
or both of which may be helium stars, which in time will
evolve to the WD stage. (It is not uncommon for either he-
lium star to overflow its Roche lobe upon leaving the helium
main sequence; this can lead to either stable mass transfer
or to a futher common envelope phase.) In this picture, the
second-formed WD will be the less massive of the pair, since
the giant star from which it descended had a smaller core
mass when its core growth was halted as it lost its envelope.
3.2Stable RLOF+CEE
If Roche lobe overflow occurs when the primary is in the
Hertzsprung gap, that is after the primary has exhausted
its core hydrogen and before it has developed a deep con-
vective envelope and ascended the giant branch, then Roche
lobe overflow may be dynamically stable for moderate mass
ratios, and a phase of stable but rapid mass transfer can
occur. In this way, the primary transfers its envelope to the
secondary, leaving a compact remnant, and a common enve-
lope phase is avoided, since by the time the primary evolves
to the giant branch, the mass ratio has been sufficiently in-
verted that mass transfer remains dynamically stable. The
orbital separation will typically have increased during this
phase (for conservative mass transfer at least), since much of

Page 4

4A.J. Farmer and E.S. Phinney
the transfer was from the less-massive to the more-massive
star. When the secondary evolves off the main sequence, it
will most likely fill its Roche lobe on the RGB, so that a
common envelope phase ensues, and a close DD is born,
provided that the resulting orbital shrinkage does not lead
to coalescence. The second-formed WD will this time be the
more massive, since its progenitor was the more evolved at
the time of its overflow.
The initial conditions for this route occupy a smaller
range in initial orbital semimajor axis than the CEE+CEE
route, but as it results in the injection of DD systems only
at very short periods, we expect both pathways to be sig-
nificant contributors to the close DD population, i.e. those
systems contributing to the GW background in the LISA
waveband. We note also that both routes ought to lead to
the production of DDs with circular orbits, even if the ZAMS
eccentricity was non-zero, since tidal circularisation is rapid
when a system contains a near-Roche lobe-filling convective
star.
4ANALYTIC ARGUMENTS ABOUT
SPECTRAL SHAPE
Given only the above, we can make some predictions as to
the shape of the GW spectrum seen today. A somewhat
analogous treatment is given in Hils et al. (1990). We con-
sider the evolution under GW emission of a population of
DDs after creation as in Section 3, with circular orbits. We
deal here with detached systems; the spectral shape due to
interacting pairs is discussed in section 7.1.2.
Here and throughout, we use ν for orbital frequencies
and f for gravitational wave frequencies. For circular orbits,
f = 2ν.
The number density N(ν,t) of binary WDs per unit
orbital frequency interval at time t must obey the continuity
equation
∂N
∂t
+
∂
∂ν(˙ νN) =˙Nb(ν,t), (6)
where˙Nb(ν,t) is the birth rate (after nuclear evolution and
mass transfer) of WD–WD systems per unit frequency. Now
for a given source, we know that˙Eorb = −Lgw, and using
Eq. 1 along with Eorb = −GM1M2/(2a) and Kepler’s law,
we obtain
˙ ν=
96
5(2π)8/3
Kν11/3
(3.7 × 10−6s−2)(M/M⊙)5/3ν11/3,
?GM
c3
?5/3
ν11/3
≡
≃
(7)
where we have used the definition of the chirp mass M,
M ≡
M3/5
1
(M1+ M2)1/5.
M3/5
2
(8)
We solve Eq. 7 to give the evolution ν(t) for
δ(t− t′,ν − ν′), i.e. for a single source injected at frequency
ν′at time t′,
˙Nb =
ν(t)−8/3− ν′−8/3= 8K(t′− t)/3.(9)
The corresponding source number density (Green’s function
for Eq. 6) NG(ν,t;ν′,t′) as a function of time is given by
NGdν
∝
dt(ν)
∝
dν
M5/3ν11/3δ
?
t −
?
t′+
3
8K(ν′−8/3− ν−8/3)
??
(10)
since, as the system traces out a path in ν, it spends a time
at each point inversely proportional to its velocity ˙ ν through
frequency space.
We then consider a real injection spectrum ˙Nb(ν′,t′),
for νmin < ν′< νmax. The resulting number density N(ν,t)
is given by
N(ν,t) =
?νmax
νmin
?t
0
˙Nb(ν′,t′)NG(ν,t;ν′,t′)dt′dν′.(11)
Since Lgw ∝ ν10/3M10/3, we can then construct the GW
emission spectrum by taking Fgw(f,t) ∝ f10/3M10/3N(f,t).
The choice of DD injection spectrum is therefore in-
strumental in determining the shape of the GW emission
spectrum. We can estimate its shape as follows: we will later
choose to distribute ZAMS orbital semimajor axis uniformly
in loga, i.e. also uniformly in log ν, for given initial M1 and
M2. We suppose that, for at least the CEE+CEE route (see
section 3), the common envelope phases lead to some mean
orbital shrinkage factor, so that WD–WD pairs at their birth
are also distributed roughly uniformly in log ν. We then have
˙Nb(ν′) ∝ 1/ν′, from some νmin ≪ ν of interest, up to νmax
(see also fig. 1 of Webbink & Han 1998). This is the max-
imum orbital frequency at which a system can exit a com-
mon envelope phase and survive to become a WD–WD pair.
Upon CE exit, the newly exposed stellar core will be a hot
subdwarf, larger than the WD it will cool to become, or it
could be a naked helium star, which will eventually evolve to
the WD stage. The maximum injection frequency at WD–
WD birth is set by the minimum orbital separation that will
keep this object (and the first-formed WD) from overflowing
its Roche lobe on the way to the WD stage, whether this is
at the exit of common envelope or (applicable to the helium
star case) as its radius changes due to nuclear evolution.
For illustration, we compute the emergent spectrum for
a fiducial population of 0.5 M⊙ WD–WD pairs. The radius
of a 0.5 M⊙ naked helium star does not exceed ∼ 0.13 R⊙
on its way to the WD stage, which sets νmax ∼ 0.7 mHz.
If we then assign a constant pair formation rate, so that
˙Nb(ν′,t′) =˙Nb(ν′), and perform the integral in Eq. 11, we
obtain the spectral shape shown in Fig. 1. Note that the
spectrum is truncated at a frequency above which the inspi-
ralling WDs would undergo Roche lobe overflow and merge,
fmerge = 2νmerge ≃ 40 mHz.
If instead we only inject sources for 0 < t < τ, and look
at the spectrum obtained for t > τ = 1 Gyr, (Fig. 2), we see
that the basic spectral shape is little affected.
Because of the strong dependence of ˙ ν on ν, a given
system of specified age will either have merged or will have
remained at essentially constant separation. Thus there are
two clear physical regimes displayed in the spectra, sepa-
rated by the injection frequency from which a source could
have reached contact due to GW losses in the time t since
its birth, νcrit ≃ 0.03 − 0.04 mHz for t ∼ 5 − 10 Gyr. (In all
relevant situations for us, νcrit < νmax.)
At f < 2νcritlies a ‘static regime’, in which losses due to

Page 5

GW from cosmological binaries5
10−6
10−5
10−4
10−3
10−2
10−1
10−8
10−6
10−4
10−2
100
f/ Hz
f Ff / arbitrary units
Increasing time
Figure 1. Gravitational wave spectrum arising from constant
WD–WD formation rate, at times 2, 5, 10, 100 and 1000 Gyr,
increasing in the direction of the arrow shown.
10−6
10−5
10−4
10−3
10−2
10−1
10−10
10−8
10−6
10−4
10−2
f/ Hz
f Ff / arbitrary units
Increasing time
since burst
Figure 2. Gravitational wave spectrum arising from a burst of
WD–WD formation between 0 and 1 Gyr. Curves plotted are
spectra at times 2, 5, 10, 100 and 1000 Gyr, increasing in the
direction of the arrow shown.
GW are negligible in the time available, giving N(ν) ∝ ν−1
and hence fFgw ∝ f10/3M10/3. For f>
‘spiral-in’ regime. In the case of a burst of DD formation
(Fig. 2), sources simply sweep through this region on the
way to merger, so that we have N(ν) ∝ ν−11/3M−5/3, giving
fFgw ∝ f2/3M5/3. If we have a constant DD formation
rate (Fig. 1), then for 2νcrit < f < 2νmax, merging systems
are continually being injected, so that N(ν) is less steeply
decreasing than ν−11/3in this region. For f > 2νmax the
spectral slope is again 2/3. Reality will be some combination
of these histories.
We therefore expect the cosmological spectrum we cal-
culate later (section 6) to be composed of a superposition
of curves of these shapes, modified for chirp mass varia-
tions, redshift effects and time delay between progenitor star
formation and DD formation. The detailed calculations de-
scribed in following sections follow in detail the evolution
of all sources from ZAMS to merger, and do not rely upon
∼2νcrit, we are in the
approximate treatments of the kind given above. Simple es-
timates of the background amplitude are discussed in section
7.
5 MODEL CONSTRUCTION
5.1 The BSE code and population synthesis
The rapid evolution code bse (Hurley, Tout & Pols 2002)
is used throughout this work whenever a binary system
is evolved. This code is a fit to detailed models of stel-
lar evolution, and produces an evolutionary time-sequence
x(tj) of the properties x of any input ZAMS binary system.
The code’s time-resolution adapts to the shortest current
timescale for change of the system components and orbit,
due to e.g. nuclear evolution, angular momentum loss or
mass transfer, which are all treated iteratively and have fi-
nite duration. In this way, even the most fleeting of evolu-
tionary phases is captured in detail, without requiring ex-
cessive time resolution during long phases in which little
changes. This is especially useful in the study of gravita-
tional waves, since the majority of the GW emission from a
given system occurs over an inspiral timescale much shorter
than the nuclear timescales of the binary’s parent ZAMS
stars. Some of the most relevant features of the bse code
will be described in the following section; see Hurley et al.
(2002) for full details.
The output x(tj) from the code can be used to construct
a stellar population at time T as follows. This method is
similar to that used by Hurley et al. (2002) to characterise
the Galactic binary population.
We describe the ZAMS binary parameter space in terms
of the primary (larger) mass M1, secondary mass M2 (or
mass ratio q = M2/M1 ≤ 1), orbital semi-major axis a and
orbital eccentricity e. We divide this space into grid boxes,
and from each box k, we randomly choose a ZAMS system
to represent the evolution of all sources in that box.
The number Pk of sources born into box k per unit
binary system realised is determined by probability distri-
butions A(a), Ξ(e) and Φ(M1,M2) = Φ1(M1)f(q) in the
ZAMS system properties described above (see section 5.3).
Pk is obtained by integrating the product of these distribu-
tion functions over the extent of box k.
We wish to construct the population of sources present
at time T. For each output timestep tj, the system with
properties x(tj) can be viewed as a system born between
times (T − tj+1) and (T − tj). If at this point the star for-
mation rate was R = R(T − tj) (expressed as a number
of binary systems born per unit time), then the number of
systems with properties x(tj) we expect to see at time T is
given by
Nj,k(T) = (tj+1− tj)R(T − tj)Pk, (12)
so long as T > tj, so that stars were not born before time
began. We perform this calculation for all boxes k and all
timesteps j, so that the total population at time T is given
by the combination of all Nj,k(T).
This method of population synthesis ensures that
sources from even unlikely regions of ZAMS parameter space
are represented, weighted by their low formation probabil-
ity. Coupled with the adaptive time-resolution of the bse