Repulsive gravity induced by a conformally coupled scalar field implies a bouncing radiation-dominated universe

Abstract

In the present work we revisit a model consisting of a scalar field with a quartic self-interaction potential non-minimally (conformally) coupled to gravity (Novello in Phys Lett 90A:347 1980). When the scalar field vacuum is in a broken symmetry state, an effective gravitational constant emerges which, in certain regimes, can lead to gravitational repulsive effects when only ordinary radiation is coupled to gravity. In this case, a bouncing universe is shown to be the only cosmological solution admissible by the field equations when the scalar field is in such broken symmetry state.

Appendix A: Bouncing model generated by a vector field non-minimally coupled to gravity

We include here a short review of the bouncing cosmological model proposed in [12] for an easy comparison with the model presented above. The Lagrangian describing a vector field non-minimally coupled to gravity employed in [12] is

$$\begin{aligned} L = \sqrt{-g}\left[ \frac{1}{\kappa }R + \beta RA_{\mu }A^{\mu } - \frac{1}{4}F_{\mu \nu }F^{\mu \nu } \right] , \end{aligned}$$

(18)

where $\beta$ is a constant, and $F_{\mu \nu } = \nabla _{\mu }A_{\nu } - \nabla _{\nu }A_{\mu }$. The field equation are

$$\begin{aligned} R_{\alpha \beta } - \frac{1}{2}Rg_{\alpha \beta } =&-\frac{1}{{\frac{1}{\kappa } + \beta A_{\mu }A^{\mu }}}\tau _{\alpha \beta }(A^{\mu }) , \end{aligned}$$

(19a)

$$\begin{aligned} \nabla _{\nu }F^{\mu \nu } + \beta A^{\mu }&= 0, \end{aligned}$$

(19b)

where $\tau _{\alpha \beta }(A^{\mu })$ is the improved energy-momentum tensor of the vector field.

In a Friedmann geometry determined by the metric (14), making the choice $A_{\mu } = A(t)\delta ^0_{\mu }$, and defining $\Omega (t) = \frac{1}{\kappa } + \beta A^2(t)$, the set of field Eq. (19) assume the form

$$\begin{aligned}&3\frac{\ddot{a}}{a} = -\frac{\ddot{\Omega }}{\Omega }, \end{aligned}$$

(20a)

$$\begin{aligned}&\frac{\ddot{a}}{a} + 2\left( \frac{\dot{a}}{a} \right) ^2 + 2\frac{\epsilon }{a^2} = -\frac{\dot{a}}{a}\frac{\dot{\Omega }}{\Omega }, \end{aligned}$$

(20b)

$$\begin{aligned}&\Box \Omega = 0. \end{aligned}$$

(20c)

A particular solution for an open spatial section, $\epsilon = -1$, furnishes a scale factor with the form

$$\begin{aligned} a(t) = \sqrt{ t^2 + a_0^2}, \end{aligned}$$

(21)

where $a_0$ is the minimum value of the scale factor. Although this is the same solution (17 ) obtained for the scalar field, the two models differs in crucial points. First, the quantity $\Omega (t)$, which is analogous to the effective gravitational constant of the model discussed in the main text, is not a constant, but a function of the cosmic time (although a model similar to the one considered in the main text would arise in the special case $A_{\mu }A^{\mu } =$ constant). Second, here the non-minimally coupled vector field alone can be the source of the geometry, while in the bouncing model induced by the scalar field, in the absence of matter fields, and for the scalar field in the non-trivial (broken symmetry) ground state, the gravitational field equations reduce to the vacuum Einstein equations.