PHYSICAL REVIEW D VOLUME 48, NUMBER 8 15 OCTOBER 1993 Amplification of gravitational waves in scalar-tensor theories of gravity John D. Barrow, Jose P. Mimoso, and Marcio R. de Garcia Maia Astronomy Centre, University of Sussex, Brighton BM 9', United Kingdom (Received 20 May 1993) The gravitational wave equation for a spatially flat Friedmann-Robertson-Walker universe is derived in the context of scalar-tensor theories of gravity, which have the Brans-Dicke theory as a particular case. This equation is solved for several cosmological scenarios, including the expansions governed by the Nariai as well as the Gurevich-Finkelstein-Ruban solutions of Brans-Dicke theory and for a new set of exact solutions of other scalar-tensor theories. The amplification of gravitational waves is studied in comparison to what happens in the general relativistic case. It is shown how the coupling with the sca- lar field changes the scales defining very large and very small wave numbers, and consequently the value of the amplification coefticient. It is found that very small values for the coupling parameter could lead to amplification of subhorizon waves. The creation of the corresponding high-frequency gravitons is ex- plained as a response to the rapid time variation of the gravitational "constant, " which can occur near the singularity in some models. It is also shown that there could be amplification of waves even in a radiation-dominated universe in some cases, because the wave equation is not conformally invariant, ex- cept for the case of Nariai's solution in the Brans-Dicke theory. PACS number(s): 04.30. + x, 04.50. + h, 12. 10.Gq, 98.80.Hw I. INTRODUCTION One of the most remarkable predictions of a metric theory of gravity, such as general relativity (GR), is that perturbations of a background spacetime generate gravity waves. The general results concerning gravitational waves have been investigated by numerous authors long ago [1 — 5]. In the context of GR cosmologies, Grishchuk has shown that the varying gravitational field of the ex- panding Universe would amplify zero-point fluctuations, and lead to the formation of a nonthermal, stochastic background of relic gravitons [6 — 9]. The advent of the infiationary models [10] (for an updated review see [11] and references therein) provides a new perspective on the cosmological features of gravitational radiation [12 — 14]. In fact, not only was an explanation for the origin of these perturbations put forward, but it also became possi- ble to study their contribution to the quadrupole aniso- tropies in the microwave background [15 — 17]. Thus, the idea of using gravitational radiation as probe of the early Universe has turned into a realistic possibility. Recently, the auspicious results of the Cosmic Background Explor- er (COBE) [18] seem to indicate that those expectations are now closer to being realized, and this fact has led to a reassessment of the predictions from inflation, so that a detailed understanding of the interplay between the data and possible theoretical models might be reached [19 — 27]. Among the various prescriptions for the early inflationary epoch, a framework was constructed which *On leave from Departamento de Fisica, F. C. Lisboa, Campo Grande 1700, Lisboa, Portugal. ~On leave from Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, RN, Brazil. seems to provide a simple way of circumventing the "graceful exit" problem, reviving the original idea of a first-order phase transition. We refer to the La- Steinhardt proposal of extended infiation [28], which con- siders Brans-Dicke (BD) [29] scalar-tensor theory of gravity instead of GR as the underlying gravitational theory. Notwithstanding the problems faced by the origi- nal proposal of La and Steinhardt, mainly due to the difhculty in matching the value of the coupling parameter required by the nucleation process with its post- Newtonian limits [30,31] and the restrictions imposed by the COBE data [19], the general idea of their prescription is endowed with appealing features. For instance, as pointed out by several authors, a coupling between a sca- lar field and gravity seems to be a generic outcome of the low-energy limit of string theories [32], and this in itself justifies further consideration of scalar-tensor theories of gravity. It is therefore a matter of great interest to address the question of cosmological gravitational waves within the context of the scalar-tensor theories characterized by the general action discussed by Bergmann [33], Wagoner [34], and Nordtvedt [35], which we write as found in Will [36]: S= — gd x ——— '"+2 U +S Here A is the Ricci curvature, P the scalar field, co(P) the coupling parameter, U(P) can be interpreted as a poten- tial associated with P, and SM represents the action for the matter fields. It is important to notice at this point that the usual condition (2) 0556-2821/93/48(8)/3630(11)/$06. 00 1993 The American Physical Society