Astron. Astrophys. 355, 1209-1213 (2000)

2. Imprint of the LSS in the astrophysical GWB

The angular fluctuations of intensity in any astrophysical GWB should correlate with those of the projected number of galaxies, so first we wish to remind the reader of the well-developed technique which is used for studies of angular properties of galactic counts on the sky. There is a lot of specially dedicated literature on this subject (see e.g. Peebles 1980, 1993). Before proceeding further, let us make a simple estimate. Suppose we have sources (galaxies) randomly distributed in space. What is the rms fluctuation of the number of galaxies within a small solid angle ? With the average space density of galaxies Mpc^-3 and neglecting cosmological effects for a while we would get

(here is the Hubble constant) while galactic counts demonstrate a much higher value for linear scales Mpc (i.e. for the most distant galaxies) (Peebles 1993). So clearly we should take into account the correlation properties of galaxies and galactic clusters.

To account for the non-Poissonian properties of galactic distribution in space, the correlation functions are used. For Gaussian fluctuations only two-point correlation function or its Fourier transform (power spectrum) would be sufficient. Taking as derived from LSS studies, we then can calculate the 2-dimensional correlation function of the projected distribution of galaxies on the sky or, equivalently, the 2-dimensional power spectrum . This is the last quantity that we actually need since the rms fluctuation of the energy flux per unit logarithmic frequency interval in a given direction is directly related to (Kashlinsky et al. 1999 and below).

2.1. Correlation functions and power spectrum

The brightness of a GW background can be characterized by the energy flux coming from a given direction within a solid angle (Thorne 1987)

where is the two dimensional coordinate across the sky. The integration over all sky yields the familiar value , the energy density per unitary logarithmic frequency interval in units of the critical energy density to close the Universe, which characterizes the isotropic stochastic GWB. To study angular properties of the noise we shall consider the flux from given direction itself. We shall assume some ideal GW detector with a beam-like sensitivity diagram, which is of course far from realistic ground-based LIGO-like or spaceborn LISA-like interferometers. However, in this paper we will not discuss the observability of GW backgrounds.

The fluctuation in the GW flux arrived at the detector from a given direction on the sky is , where denotes ensemble averaging. The Fourier transform of the fluctuation is .

The projected 2-dimensional correlation function represents the first non-trivial moment of the probability distribution function of : . The two dimensional power spectrum is by definition . We shall assume the phases to be random and the distribution of the flux field to be Gaussian so that the power spectrum is just the Fourier transform of the correlation function:

Here is the zero-order cylindrical Bessel function.

The mean square fluctuation of the flux on the detector within a finite solid angle subtended by the angle across the sky is zero-lag correlation signal

where W is the window function of the detector. For example, for a top-hat beam and the values of correspond to fluctuations of angular size .

The GWB flux and its angular properties measured in projection on the celestial sphere should reflect 3-dimensional structure of the Universe and the change of GW emission rate with redshift z. The LSS can be taken into account by the 3-dimensional correlation function or its 3-dimensional power spectrum , which for isotropic cases relates to through the equation

(here is the zero-order spherical Bessel function).

The projected correlation function of cosmic GWB is expressed through the two-point correlation function of the galaxy distribution and the rate of GW emission via the Limber equation (Limber 1953):

Substituting this equation in the limit of small angles for Friedman-Robertson-Walker metrics into Eq. (4), one gets (Kashlinsky et al. 1999)

where is the angular distance, is the metric distance. For degree angular scales of interest here the linear approximation of galactic clustering can be used , where describes the evolution of the clustering. On linear scales if (Peebles, 1980).

In our analysis we use the 3D spectrum as derived by Einasto et al. (1999) from a thorough analysis of different LSS studies. The mean spectrum of galaxies shows a power-law behaviour at small and large k with a maximum at .

The analysis of this formula (Kashlinsky et al. (1999) and references therein) shows that the relative fluctuations of the flux are and are weakly dependent on the cosmological model at angular scales of the order of one degree. However, strong dependence on redshift of the flux rate requires more accurate calculations.

2.2. GWB flux

The GW energy emitted by the population of some sources in a galaxy at frequency f per unit logarithmic frequency interval in the rest-frame of the galaxy can be calculated through the rate of GW-producing events (e.g. the rate of binary WD mergings or supernova explosions) in the galaxy. Under the stationary conditions we have (Kosenko and Postnov 1998)

where is the energy which is being carried away by gravitational waves from the typical source (e.g. for two point masses orbiting each other it is just the orbital binding energy). If and GW emission is the only dissipative mechanism

The comoving luminosity at proper frequency per unit logarithmic frequency interval produced by sources in galaxies at redshift z from the redshift interval dz from unit solid angle

where is the space density of galaxies, is the proper volume element. We assume no new galaxies have been created since their formation so that and  Mpc^-3 is the present-day density of galaxies normalized to the amount of baryons comprised in stars (in terms of the critical energy density ; (km/s/Mpc) is the Hubble constant). The strong star formation rate evolution is taken into account through the evolution of the event rate with redshift (Kosenko and Postnov 1998). We use the parametrization of Rowan-Robinson (1999) for the star formation rate history SFR as derived from optical, UV, and IR-observations, and normalized to unity at . The rate of particular events is thus where is the present-day event rate per galaxy, is the redshift dependence of the rate after a -function-like star formation burst. This is important especially for binary white dwarf coalescences because these are delayed typically by years since their formation. For events which relate to massive star evolution, like supernova explosions, we have and the change in the rate of events with redshift simply follows the star formation rate evolution .

The proper volume element is (Peebles 1993)

with the metric distance and being the functions of the cosmological model (Carroll et al. 1992). In our calculations we use the standard flat universe without cosmological constant and a -term dominated cosmological model with , .

Combining Eqs. (10), (11), and (12) together we arrive at

so the contribution to the total GW flux from the redshift interval dz from unit solid angle which is observed today by a detector with band-width df centered at frequency is

where we used the luminosity distance definition . Note that no additional factor accounting for the change of frequency interval appears in the numerator since we are working with unitary logarithmic frequency interval.

Now we are in the position to calculate relative fluctuations of any GWB produced by astrophysical sources associated with galaxies within a given solid angle . From Eqs. (8) and (14) we derive

Here the redshift corresponds to the beginning of star formation in the Universe. In our calculations we assumed (in fact, the exact value is of minor importance since sources at small and moderate redshifts mostly contribute to the flux).

For stochastic GWB the flux per unit logarithmic frequency interval is related to the dimensionless strain amplitude h as

so at a given frequency fluctuations in the strain amplitude .

Note that the value of fluctuations depends on the specific source type only through the dependence of energy carried away by gravitational waves on frequency (index for the power-law dependence) and the dimensionless change of event rate with redshift . The dependence on frequency can appear only through the change in spectral index with frequency. To ensure that the spectral shape has no effect on the relative fluctuations, observations should be performed at a frequency which is sufficiently (by an order of magnitude) lower than the high-frequency cut-off of the comoving power-law spectrum of the background.

2.3. Specific examples

As specific examples we consider GWB produced by extragalacting merging white dwarfs and rapidly rotating hot neutron stars. In the first case the energy carried away by gravitational radiation is the orbital energy of binary white dwarfs whose evolution is driven by gravitation wave emission. In the LISA sensitivity frequency band Hz we have , . A white dwarf binary system reaches the orbital frequency of the pre-merging stage  Hz typically years after the formation, so the dependence of the rate of events on redshift is smoother than star formation rate history SFR in the Universe. The isotropic stochastic GWB produced by extragalactic binary white dwarfs taking into account the star formation rate evolution in the Universe was calculated by Kosenko & Postnov (1998). At the frequency 0.01 Hz its level is .

For hot young neutron stars the emission of gravitational waves can be driven by r-mode instability (Lindblom et al. 1998, Owen et al. 1999). The energy carried away by gravitational waves is the rotational energy of the neutron star and (i.e. ) within the frequency band , where is the Kepler frequency at which mass shedding at the stellar equator makes the star unstable. The isotropic stochastic GWB produced by hot NS was studied by Ferrari et al. (1999b). Hot neutron stars are formed in the core collapse supernova explosion events in the end of evolution of massive stars, so no deviation from star formation dependence on redshift for their rate is expected. The isotropic background level at 100 Hz was estimated to be .

The relative fluctuations of these backgrounds at angular scales are shown in Fig. 1 as a function of wave numbers q. It is seen that at degree (), where the (quasi) linear regime of the galaxy clustering evolution with redshift is expected so that we can use , the fluctuations amount to 5% for coalescing white dwarfs and 10-20% for hot neutron stars. At larger scales (smaller q) the relative fluctuations slowly decrease. The calculations were performed for two cosmological models: a flat FRW universe with zero cosmological constant and with . In the case of hot neutron stars the level of fluctuations increases in the -dominated Universe at all scales considered, while in the case of coalescing white dwarfs the curves for two cosmological models intersect at (angular scales degrees).

Fig. 1. Relative rms fluctuation of the gravitational wave flux from unresolved extragalactic astrophysical sources at different angular scales . Shown are fluctuations of the backgrounds produced by hot rapidly rotating neutron stars at frequency 100 Hz (upper curves) and by coalescing binary white dwarfs at frequency Hz (lower curves). Calculations for flat FRW universes without cosmological constant () and with .

Note that at smaller angular scales (larger q) the non-linear evolution of LSS should be taken into account, so the present calculations cannot be applied.

© European Southern Observatory (ESO) 2000

Online publication: March 21, 2000
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