The spectrum of gravitational waves in an model with a bounce.
We present an inflationary model preceded by a bounce in a metric theory. In this model, modified gravity affects only the early stages of the universe. We analyse the predicted spectrum of the gravitational waves in this scenario using the method of the Bogoliubov coefficients. We show that there are distinctive (oscillatory) signals on the spectrum for very low frequencies; i.e., corresponding to modes that are currently entering the horizon.
The bounce we will consider in our model is followed by an inflationary era which is asymptotically de Sitter where, in addition, the gravitational action approaches the Hilbert-Einstein action on that regime, such that the modification to Einstein’s General Relativity (GR) affects exclusively the very early universe, around the bounce and a few e-folds after that. We will constrain the model obtaining the spectrum of the stochastic gravitational fossil as would be measured today.
2 Model for the early universe
Inspired on the de Sitter solution for a closed space-time, we define the scale factor around the bounce as:
where is the scale factor, is a constant quantifying the size of the universe at the bounce. The parameter is related to the energy scale of inflation just after the bounce.
In a Friedmann-Lemaître-Robertson-Walker (FLRW) universe with a spatially flat metric and the scale factor defined as in Eq. (2), the minimization of the action (1) leads to a second order differential equation for the function . Solving this equation in conjunction with appropriate physical constraints gives BouhmadiLopez:2012qp ():
In the above equation .
3 Energy spectrum of the gravitational waves
The spectrum of the gravitational waves is determined using the method of the continuous Bogoliubov coefficients and , as in Ref. Parker1 (). The graviton density of the universe is given by , while the dimensionless logarithmic energy spectrum of the gravitational waves (GW) of angular frequency is defined at the present time as Allen:1987bk ():
To calculate the evolution of the gravitational waves, we express the the continuous Bogoliubov coefficients in terms of the variables and , see Refs. Sa1 (); Bouhmadi1 (), which obey the set of differential equations:
Here, is the wave-number, a prime indicates a derivative with respect to the conformal time () and . The differential equations (5) are integrated from an initial time , set before the bounce, until the present time. We describe the late time evolution of the universe in a GR setup, using the CDM model WMAP () complemented with a radiation phase and making the connection between the driven early inflation and the radiation phase with a model of a modified Generalized Chaplygin Gas suitable for the early universe Bouhmadi1 (). The results obtained for the GW spectra are shown in Fig. 2.
The existence of the bounce in the early universe affects the spectrum of GWs only in the low frequency range, Hz, where various peaks appear whose position and intensity depend on the parameters of the mode. The fact that the oscillatory structure appears in the spectra of (i) the GR treatment and (ii) the treatment suggests it is not a consequence of the effects of -gravity. Similar oscillations have been obtained in works of loop quantum cosmology first pointed out by Afonso et al Afonso1 (). Due to the low energy density of the cosmological GW’s, the results obtained in this work are hard to be detected in the near future (see Fig. 2 of Ref. Smith1 () and Fig. 6 of Ref. Kawasaki:2012rw ()). The detection of the B-mode polarization of the CMB radiation seems to be the best candidate to obtain information about the cosmological GW’s. Bourhrous:2012kr ()
Acknowledgements.M.B.L. is supported by the Spanish Agency “Consejo Superior de Investigaciones Científicas” through JAEDOC064. This work was supported by the Portuguese Agency “Fundção para a Ciência e Tecnologia” through PTDC/FIS/111032/2009.
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