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On the Stochastic Gravitational Radiation Background produced by an Ensemble of Single Neutron Stars

K.A. Postnov1   M.E. Prokhorov2

Received 13 January 1997 / Accepted 12 May 1997






Abstract: The possible stochastic background produced by single galactic NS is studied. The upper limit to this background independent of the NS ellipticities is derived. If ~ 0.1 of old NS population have a low surface magnetic field (<107 G; from the beginning or due to field decay), this background may be detected by the advanced LIGO interferometers with a sensitivity of ~ 10-25 at 100 Hz during 1-year integration.

gravitational waves --- stars: neutron


1   Introduction

Among possible sources of gravitational radiation (GR), neutron stars (NS), both single and entering binary systems, are considered as mostly promising (see Thorne 1987; Abramovici et al. 1992; Schutz 1996 for full review). NS are the end product of evolution of massive (¨ 8--10 M) stars, so their number in the Galaxy should amount to ~109 over the galactic lifetime of ~1010 years. The number of binary NS is more controversial. Simple estimate based upon the binary pulsar statistics (Phinney 1991; Curran & Lorimer 1995; van den Heuvel & Lorimer 1996) yield the coalescence rate of binary NS in the Galaxy of order 1 per 105 years, so to have an acceptable detection rate of these sources one must have a detector sensitivity of at least 10-21--10-22 at the frequency 100 Hz, which the initial laser inetrferometers currently under construction are aimed at. Theoretical estimates of the binary NS coalescence rate are typically an order of magnitude higher (1 per 104 years), and the possible solution of the discrepancy may be connected with the NS not being observed as a radiopulsar in a binary NS+NS system (see Lipunov, Postnov & Prokhorov (1997) for more detail).

The situation with single NS, however, differs from binary NS in that the latter are the most relaible sources of GR, which is confirmed by observations of the binary pulsar PSR 1913+16 orbit decay (Taylor 1992), whereas to be a noticable source of GR, the form of an isolating rotating NS must deviate from spherical symmetry. This deviation is usually described in terms of the relative difference of moment inertia along the different axis of the non-spherical body of the star, e = 1 - a2/a1 » D I/I, where a1 is the semimajor axis of the equatorial section, and a2 the semiminor axis.

In the last years, different mechanisms of symmetry bracking have been proposed for young NS (see, e.g., Lai & Shapiro 1995; Bonazzola, Frieben & Gourgoulhon 1996). It also has been suggested (Zimmermann 1978; Gal'tsov, Tsvetkov & Tsirulev 1984; Bonazzola & Gourgoulhon 1996) that an internal strong magnetic field (B~ 1015--1016 G) may cause the asymmetrical shape of the NS. It has been shown by Blair (1996) that the asymmetry of young NS may lead to appearance of a stochastic GR background at frequencies 1--300 Hz, and provided that the supernova explosions in the entire Universe are frequent enough, it can be marginally detected with the advanced LIGO interferometer. Recently, Giazotto, Bonazzola & Gourgoulhon (1996) studied the possibility of the detection of the GR background generated by all old NS in the Galaxy with only one GW interferometer using a quadratic detection technique. We are highly ignorant about old NS distribution in the Galaxy. However, old NS may populate an extensive halo around the galaxy (~100--300 kpc even without significant kick velocity; see e.g. calculations by Gurevich et al. (1993); Prokhorov & Postnov (1994)), so the diurnal modulation of the signal considered by Giazotto et al. (1996) could be smeared out, making quadratic detection by one GW interferometer questionable. In this paper, we consider the stochastic GW background produced by the old NS population in the Galaxy and beyond taking into account spin-down evolution of old NS. We show that in the limiting case of angular mometum loss only due to GR, the upper limit on the GR background formed by old NS is determined by NS production rate only. We also briefly discuss the possibility of its detection in future GR experiments.

2   Stochastic GR background produced by rotating single NS

Let us assume a stationary situation, i.e. that the number of NS in the Galaxy is determined by a constant formation rate R. Assuming the present star formation rate in the Galaxy 1 M/yr, Salpeter mass function f(M)µ M-2.35 (Salpeter 1955), and the minimal mass of the star Mmin=0.1 M (such a choice yields the total stellar mass in the Galaxy 1011 M during 1.5 â 1010 years), we find that the mean formation rate of massive stars (>10 M to produce NS) are about 1 per 30 years. Below we shall normalize all calcualtions to this rate, R30º 1/30 yr-1 » 10-9 s-1.

A rotating non-spherical NS with the ellipticity e loses energy in the form of GR at a rate (Shapiro & Teukolsky 1984)

EGR=
32 G
5 c5
I2e2w6     (1)
where G and c are the Newtonian gravity constant and speed of light, w=2pn is the NS rotation frequency, I is the moment of inertia.

If the NS were emitted GR at strictly twice rotational frequency and this frequency were not changed, in principle each star may be distinguished by an ideal detector provided its frequency band is sufficiently narrow. However, the rotating NS radiates both at w and 2 w (and possibly at other higher harmonics if its form is more complex) and its rotational frequency are constantly changing by the energy conservation law

Iw w = EGR+ Eem+ E
 
...
    (2)
where we have explicitly written down possible rotational energy losses -- electromagnetic ( Eem) and others. In the ideal case of GR being the only source of energy loss we would retain only first term in the expression above. We should note that the spin evolution of a magnetized NS becomes much more complicated when the NS is in a binary system (e.g. Lipunov 1992); however, their fraction among the total number of NS is hardly higher than 10%, and we will not consider them here.

As laser interferometers are broad-band detectors (D n~n), a large number of sources within the sensitivity band would produce a stochastic background. Long-term continuous observations, however, allow to make the sensitivity band efficiently narrower provided that the faweform of the signal is known (in fact, as D n ~ 1/T, where T is the integration time; this permits to increase signal-to-noise ratio for continous source observations as T using match filtering thechnique). For old NS, however, match filtering technique of data analysis would require enormous calculating time (Schutz 1996) (a priori we do not know the signal form, source location on the sky, etc.), so for the old NS GR background the interferometer always works as a broad-band detector.

Clearly, the condition that a stochastic signal appears within the detector band reads
D t  R ¨ 1     (3)
where D t is the time for a typical source to pass through the detector band dw. This time is determined by a particular mechanism of energy losses, and we calculate it separately for GR and electromagnetic losses.

1. Elecromagnetic losses. They are described by the law

w = A w3;     A=
2µ2
3c3I
    (4)
Here µ is the dipole magnetic moment of the NS. The solution to the equaiton (X) reads
D tem= 1/2A w-2((
w
w-dw
)2-1)     (5)
Assuming dw=1/2w we find the upper frequency of the stochastic background
n0em» 103(Hz)
R30I45
µ30
    (6)
with µ30=µ/(1030G cm3).

2. GR losses. These are

w = Bw5;     D tem= 1/4B w-4((
w
w-dw
)4-1)     (7)
and under the same assumption about dw we find
n0GR» 1.4â 104(Hz) R301/4I45-1/4e-6-1/2     (8)
with e-6=e/106.

Therefore, for plausible values of the NS magnetic fields (µ30=10-4--102) and ellipticities (e-6=10-3--102), at any frequency <103 Hz we deal with stochastic backgrounds from galactic NS. Physically, this is due to the inability of old NS to leave frequency interval Dw ~ w during the typical time between consecutive supernova explosions. This is not so for young NS (see, e.g., Lai & Shapiro 1995).

Now we ask the question: how many sources with changing frequency are to be simultaneously observed within a frequency interval Dw ~ w? The answer is immediate: Under stationary conditions the continuity equation implies

N(w)º
dN
dln w
=
dN
dt
â
w
w
= R â
w
w
    (9)
Here we assumed that all sources come into the interval through its upper boundary. This assumption is correct if the upper boundary of the interval lies sufficiently far from the initial frequency of NS (i.e. less than about 100 Hz). Now, if the number of sources within this interval is more than unity, the resulting GR signal at frequency w would read
h
 
S
(w)2=
N(w)
Å
i=1
hi2(w)     (10)
where dimensionless strain amplitude from one source relates to the energy flux F(w) at the frequency w as
h(w)2=
16 p G
c3
F(w)     (11)

Assuming each source to radiate identically at the given frequency, Eq. (X) may be rewritten in the form
h
 
S
(w)=h02(w)N(w)     (12)
where
h02(w)=
4 G
c3w2
EGR(n)
r2
    (13)
and r is the inverse-square average distance to the typical source. Using Eq. (X) and (X), we obtain
h
 
S
(w)=
1
r
4G I
c3
R
1+ Eem/ EGR
    (14)
where we omitted all but electromagnetic loss terms.

For purely GR-driven NS spin-down the resulting spectrum is independent of the unknown value of e in the NS population. (The independency of the resulting signal on the ellipticity when the pulsar spin-down is governed by GR losses only was noted by Thorne (1987) with the reference to private communication from R. Blandford in 1984). Note that any additional braking mechanism always lowers the resulting signal. For example, taking typical values I=1045 g cm2, R=1/30 yr-1 we obtain

hlim =
1
r
4GI R
c3
 
» 3â 10-24 Ô
Ã
Ã
Õ
10kpc
r
Æ
Â
Â
Ü		 R301/2I451/2     (15)
(here we assumed the characteristic distance to NS population of order 10 kpc). As we shall show, this is the upper limit to the stochastic GR background produced by old NS population at a typical distance of 10 kpc. The GR background of such strength could be detected by the advanced LIGO interferometers (Thorne 1987).

What kind of losses governs the NS spin evolution for realistic NS parameters? The ratio of electromagnetic losses to GR losses x= Eem/ EGR is
x=
A
B
w-2     (16)
with A and B determined as above, and for typical parameters µ and e we find
x» 4000 µ302e-6-2 Ô
Ã
Ã
Õ
100 Hz
n
Æ
Â
Â
Ü
2



 
    (17)
that is electormagnetic losses becomes insignificant only at frequencies
n >ncr» 6.3(kHz) 
µ30
e-6
    (18)
i.e. they dominate practically always. If we would take e-6=10-3 and µ30=10-4 as in millisecond pulsars, we would obtain ncr» 630 Hz, however millisecond pulsars are spun up by accretion in binary systems and are not considered here.

Therefore, for realistic NS we must consider the case x» 1. Using Eqs. (X) and (X) we derive that the stochastic background from old NS is
h
 
S
(n)» 5â 10-28 Ô
Ã
Ã
Õ
10kpc
r
Æ
Â
Â
Ü		 R301/2I451/2e-6µ30-1n     (19)
Note the frequency dependence appeared in this expression. If we take the estimate of magnetic field from observations of pulsar periods P and their change rate P : µ30» (P P-15)1/2 (here P-15º P/10-15) and assume maximum possible ellipticity allowed by P-- P observations: emax» 5.7â 10-3 (P3 P-15)1/2 , Eq. (X) immediately yields the same Eq. (X) for hlim as above.

3   Discussion

Now consider the contribution of old NS population from other galaxies. For distances described by Euclidean geometry (<100 Mpc) we may do a crude estimate as follows. For specific events, the rate within the volume V (Mpc-3) relates to the galactic event rate RG (e.g., Phinney 1991) as RV»0.01â RG h100, where h100=H0/100 km/s/Mpc is Hubble constant. Therefore, for a population of old NS within ~ 100 Mpc we obtain
hlim~ 10-23 Ô
Ã
Ã
Õ
100 Mpc
r
Æ
Â
Â
Ü		 Ô
Ã
Ã
Õ
RG
1/30 yr
Æ
Â
Â
Ü
1/2



 
I451/2     (20)
a few times larger than from galactic NS.

Going further away, however, cosmological effects become significant. Old NS population from other galaxies may fairly well be considered isotropic and of probably not strongly varying comoving density. Then we should use mean photometric distance in (X) which is in the standard flat Friedman Universe is <rph>=20/3(c/H0) (if zlim>>1). For limiting redshifts zlim=5 we find <rph>» 10 Gpc. The supernova rate even with strong evolutionary effets is <109 per year (for baryonic content in stars Wb» 0.005; see JÜrgensen et al. (1997) for more detail), so we obtain hlim < 10-25.

We have shown that if the NS form ellipticity is present, the stochastic GR background produced by old NS population is naturally formed due to NS rotation braking. In the limiting case when only GR angular momentum loss causes NS spin-down, this background is independent on both exact value of the NS form ellipticity e and frequency and can be detected by advanced LIGO interferometers. In reality, the magnetic field of NS causes more effective electromagnetic energy loss: to be insignificant, magnetic field of NS should be less than (see Eq. (X))
µ < 1.5â 1026 (G cm3) e-6n     (21)

According to Urpin & Muslimov (1992), the magnetic field can decay very fastly provided that the field was initially concentrated in the outer crust layers with the density <1010- 1011 g cm-3, and such very low magnetic field for old NS may be possible. In the limiting case that the NS magnetic field does not decay at all (for example, if only accretion-induced field decay is possible in binary systems (Bisnovatyi--Kogan & Komberg 1974)), old NS should lose their energy through electromagnetic losses and be very slow rotators with periods of about a few seconds. Then the initial magnetic field distribution becomes crucial. If it is centered at ~ 1012 G (as implied by radipulsar P-- P measurements), we have little chances to detect the old NS population at 10--100 Hz frequency band unless close mean distances (<10 kpc) are assumed (Giazotto et al. 1996). However, if nature prefers scale-free law (like f(µ)µ 1/µ), the fraction of low-field NS could amount to a few 10% and they can contribute to the frequency-independent GR background. Then Eq. (X) implies that such a background can be detected by the advanced LIGO interferometer in the frequency band 10--1000 Hz in one-year integration even if the formation rate of such NS is as small as 1 per 300 years and the characteristic distance to them is 100 kpc.

KAP acknowledge the staff of Cosmic Radiation Laboratory of RIKEN for hospitality and JISTEC through STA Fellowship Grant No 496057 for financial support. Both authors partially supported by RBRF(?) Grant No 95-02-06053-a and INTAS Grant No 93-3364.

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2
On leave from Moscow State University, Russia
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