Fluctuations of the gravitational constant induced by primordial bubbles.

We consider the classical fluctuations of the gravitational constant generated by bubbles in the inflationary universe. For extended inflation, we demonstrate numerically how and how large fluctuations are produced during bubbleexpansion. The amplitude of the fluctuations depends on the Brans-Dicke parameter ω: if ω is of the order of unity, the amplitude becomes of the order of unity within one Hubble expansion time; if ω is large (say, ω = 1000), the growth rate of the fluctuations is small, but it keeps growing without freezing during inflation. We also discuss some astrophysical implications of our results. Electronic address: [email protected] 1 Particle physics predicts that the universe experienced many phase transitions in its early history. If any of these phase transitions is first-order, the universe changes its phase from false vacuum to true vacuum through creation, expansion and collision of bubbles. Old inflation [1] is based on a super-cooled first-order phase transition. In the inflationary scenario, the universe expands exponentially with time before the transition, thereby solving the horizon, flatness and monopole problems. However, it turns out that this exponential expansion is too rapid to permit a transition from false vacuum to true vacuum via percolation of true vacuum bubbles [2]. Extended inflation [3] revived the idea of old inflation by using the Brans-Dicke theory instead of the Einstein theory. The Brans-Dicke field decelerates the expansion of the universe so that true vacuum bubbles can coalesce, thus ending the phase transition that drives inflation. This model provides an interesting hypothesis that the large-scale structure of galaxy distribution is generated by primordial bubbles [4]. Many discussions about extended inflation have been made in past years. La et al. and Weinberg [5] estimated volume fraction of bubbles and found a difficulty: the constraint from the isotropy of the cosmic microwave background requires the Brans-Dicke parameter ω < 25, which contradicts the lower limit ω > 500 from the time-delay experiments [6]. This problem is called a ”bigbubble problem” because many big bubbles cause large-scale inhomogeneity. Thus interest shifted to other extended models based on general scalar-tensor theories [7][9]. In discussing density fluctuations in extended inflation or other extended models, several authors have estimated volume fraction of bubbles as we mentioned above, or quantum fluctuations of the gravitational constant (the Brans-Dicke field) [10]. In this paper, we investigate the classical fluctuations of the gravitational constant generated during bubble expansion. It was shown that, under the thin-wall approximation, the Brans-Dicke field must be inhomogeneous inside a bubble from the consistency of the junction conditions [11]. However, it is not clear how and how large fluctuations are really generated. If these fluctuations are not small, we should consider it for the constraint for the models, such as the structure formation process and the measurements of the gravitational constant. There is an alternative scenario of inflation which is accompanied by bubble nucleation: one-bubble inflation [12]. This model is distinguishable from extended inflation or its generalized version: the second slow-rollover inflation occurs inside a nucleated bubble, and our observable universe is entirely contained in one bubble. If we assume the O(4)-symmetric bubble, the interior of the bubble can be a homogeneous and isotropic open universe [13]; this model may account for the universe model with Ω0< ∼0.1, which is supported by increasing observations. Although the fluctuations of the gravitational constant may not be relevant to this model, it is interesting to study the fluctuations inside a bubble in a similar way.