Introduction

The plane optical wavefronts of a distant background light source become rippled when they cross a perturbation. For a distant perturbation, the focusing of light at the observer changes with the curvature of the ripples. It is the usual geometrical scintillation effect. It accounts, e.g., for the twinkling of stars under the atmospheric turbulence.

The question whether gravitational waves can cause the light emitted by a distant source to scintillate is an old problem. In general relativity, it is well known from the early works by Zipoy (1966), Zipoy & Bertotti (1968) and Bertotti & Trevese (1972) that gravitational waves have no focusing property to the first order in their amplitude.

However, it has been recently pointed out by Faraoni (1996) that a first-order scintillation effect can be expected in scalar-tensor theories of gravity . Furthermore, some actual improvements of the observational techniques renew the interest in the search of gravitational scintillation (Labeyrie 1993) and related effects (Fakir 1995).

The aim of the present work is to make a detailed analysis of the scintillation effect in monoscalar-tensor theories for a monochromatic electromagnetic wave propagating in a weak gravitational field. We adopt the point of view that the physical metric is the metrical tensor $g_{\mu \nu}$ to which matter is universally coupled. This basic assumption defines the usual "Jordan-Fierz" frame. We find a scintillation effect proportional to the value of the scalar field perturbation at the observer.

Our result contrasts with the zero effect found by Faraoni & Gunzig (1998) by using the "Einstein" conformal frame, in which the original physical metric $g_{\mu \nu}$ is replaced by a conformal one . However, their negative conclusion is seemingly due to the fact that the authors do not take into account the changes in areas and other physical variables induced by the conformal transformation (Damour & Esposito-Farèse 1998).

The paper is organized as follows. In Sect.2, we give the notations and we recall the fundamental definitions. In Sect.3, we construct the theory of gravitational scintillation for a very distant light source emitting quasi plane electromagnetic waves. Our calculations are valid for any metric theory of gravity in the limit of the geometrical optics approximation. We obtain the variation with respect to time of the photon flux received by a freely falling observer as a sum of two contributions: a change in the scalar amplitude of the electromagnetic waves, that we call a geometrical scintillation, and a change in the spectral shift. We express each of these contributions in the form of an integral over the light ray arriving to the observer. In Sect.4, we study the scintillation within the linearized weak-field approximation. We show that the geometrical scintillation is related to the Ricci tensor only. Thus we recover as a particular case the conclusions previously drawn by Zipoy and Zipoy & Bertotti for gravitational waves in general relativity. Moreover, we show that the contribution due to the change in the spectral shift is entirely determined by the curvature tensor. In Sect.5, we apply the results of Sect.4 to the scalar-tensor theories of gravity. We prove that these theories predict a scintillation effect of the first order, proportional to the amplitude of the scalar perturbation. Furthermore, we find that this effect has a local character: it depends only on the value of the scalar field at the observer. Finally, we briefly examine the possibility of observational tests in Sect.6.