## 2. Imprint of the LSS in the astrophysical GWBThe 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 (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 spectrumThe 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 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 (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 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 fluxThe GW energy emitted by the population of some sources in a galaxy
at frequency 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 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 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
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 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 examplesAs 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
Note that at smaller angular scales (larger © European Southern Observatory (ESO) 2000 Online publication: March 21, 2000 |