Background of gravitational waves generated by astrophysical sources

Background of gravitational waves generated by astrophysical sources

José C. N. de Araujo; Oswaldo D. Miranda

Divisão de Astrofísica - Instituto Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758 - São José dos Campos - 12227-010 SP - Brazil

We have recently shown [1] that the dimensionless amplitude of the stochastic background of gravitational waves (GWs) produced by an ensemble of sources is given by

where h_single is the dimensionless amplitude produced by an event that generates a signal with observed frequency n_obs; and dR is the differential rate of production of GWs.

Eq.(1) is in fact a shortcut to the calculation of stochastic background of GWs. An interesting characteristic of this equation is that it is not necessary to know in detail the energy flux of the GWs produced at each frequency. If the characteristic values for the dimensionless amplitude and frequency are known and the event rate is given it is possible to calculate the stochastic background of GWs produced by an ensemble of sources.

In particular, for the case of a background produced by an ensemble of black holes, the differential rate reads

where

_*(z) is the star formation rate density (hereafter SFRD); in yr^-1 Mpc^-3), dV is the comoving volume element and f(m) is the initial mass function (IMF). In a few words, the f(m) dm represents the number of stars per unit mass in the interval [m,m + dm]. The normalization of the IMF is obtained through the relation

where it is usually considered that m_l = 0.1 and m_u = 125 (see, e.g., Ref. [1]).

We present here an alternative and in fact a more robust derivation of Eq.(1). We start from the specific flux received in GWs at the present epoch as (see, in particular, Eq. (15) in Ref.[2] and section 12.1 in Ref.[3])

where

is the comoving specific luminosity density (given, e.g, in erg s^-1 Hz^-1 Mpc^-3), which obviously refers to the source frame.

As discussed in Refs.[2, 3], the above equations are valid to estimate a stochastic background radiation received on Earth independent of its origin. In the present paper l_n can be written as follows

where dE_GW/dn is the specific energy of the source. Note that in the above equation _*(z) refers to the source frame, therefore, there is not any putative (1 + z) factor related to time dilation.

Thus, the flux F_n(n_obs) received on Earth reads

Using Eq.(2) it follows that

Note that in the above equation, what multiplies dR_BH is nothing but the specific energy flux per unity frequency (in, e.g., erg cm^-2 Hz^-1), i.e.,

(see, e.g., Ref.[4]).

On the other hand, the specific energy flux per unit frequency for GWs is given by Ref.[5]

Also, the spectral energy density, the flux of GWs, received on Earth, F_n, in erg cm^-2 s^-1 Hz^-1 can be written from Refs.[6, 7] as

From the above equations one obtains

which is nothing but the Eq. (1).

Acknowledgments

JCNA would like to thank the Brazilian agency CNPq for partial support (grant 303868/2004-0).

References

[1] J.C.N. de Araujo, O.D. Miranda, and O.D. Aguiar, 61, 124015 (2000).

[2] A. J. Farmer and E. S. Phinney, Mon. Not. R. Astron. Soc. 346, 1197 (2003).

[3] J. A. Peacock, in Cosmological Physics (Cambridge University Press, Cambridge, England, 1999).

[4] V. Ferrari, R. Schneider, and S. Matarrese, Mon. Not. R. Astron. Soc. 303, 258 (1999).

[5] B. J. Carr, Astron. Astrophys. 89, 6 (1980).

[6] D. H. Douglass and V. G. Braginsky, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press Cambridge, England, 1979), p. 90.

[7] D. Hils, P. L. Bender, and R. F. Webbink, Astrophys. J. 360, 75 (1990).

Received on 23 October, 2005

Publication Dates

• Publication in this collection
  16 Jan 2006
• Date of issue
  Dec 2005