Accelerating universe with decreasing gravitational constant

1 Introduction

Since the discovery of an accelerated expansion of the universe from astrophysical observations (Riess et al., 1998; Perlmutter et al., 1999), cosmologists think that beside the standard matter there must be an unknown fluid with a negative pressure $p_{\mathit{de}}<-(1/3){\rho }_{\mathit{de}}$ , that drives the universe into acceleration. Such a component, dubbed “dark energy” was completely negligible in most of the past and nowadays dominates entirely the universe. The origin and the exact nature of dark energy are still unknown. Subsequently, the Einstein cosmological constant has been commonly considered as the best candidate to play the role of dark energy; however it faces fine-tuning and coincidence puzzles. The late-time cosmic acceleration may alternatively be driven by a dynamic dark energy which could be a time evolving and spatially dependent scalar field. Lots of such dynamic dark energy models have been proposed and are roughly classified into three categories: quintessence (Peebles and Ratra, 1988; Ratra and Peebles, 1988; Caldwell et al., 1998; Copeland et al., 2006), phantom (Caldwell, 2002) and quintom (Feng et al., 2005; Guo et al., 2005). In quintessence models, a scalar field $\varphi$ with a canonical kinetic energy and a self-interaction potential energy $V\left(\varphi \right)$ is supposed to be minimally coupled to Einstein gravity. In a flat Robertson-Walker background the quintessence scalar behaves as a perfect fluid with an evolving equation-of-state (EoS) parameter $\omega =p/\rho$ lying in the range $-1\le \omega \le 1$ . In phantom models the quintessence is replaced by a ghost scalar with negative kinetic energy yielding $\omega <-1$ . The quintom models generally involve two scalar fields, one is a quintessence scalar and another a phantom. Quintom models characterize themselves by the property that their effective EoS parameter can cross the $\omega =-1$ barrier, which enables them to evade the divergence of linear perturbations and makes them to fit the observational data better (Cai et al., 2010). Having $\omega =-1$ crossing in a dynamic dark energy model is bewitching. However, the emergence of a phantom scalar with negative kinetic energy in quintom models brings about great embarrassment on physics. It is worthwhile to study the mechanism of removing the phantom field from the quintom models. In fact, there has lots of attempts to investigate the possibility of $\omega =-1$ crossing in quintessence like models. It has been empirically realized that to have $\omega =-1$ crossing and remove ghost mode at the same time, the model building should be involved in either modifying the general theory of Einstein’s relativity or introducing some higher derivative terms for the scalar fields (Cai et al., 2010). For example, in the so-called Galileon cosmology (Deffayet et al., 2009; Nicolis et al., 2009) of a scalar field, the higher derivatives of operators are introduced into the Lagrangian but the equation of motion of the scalar remains of the second order. The Galileon models can have $\omega =-1$ crossing without ghost modes. It goes without saying, however, that these models are very complicated.

Aimed at finding the explanation for the cosmic acceleration, some authors (Sivaram et al., 1975; Pande et al., 2000; Tsagas and Maartens, 2000; Vishwakarma, 2001; Rane and Katore, 2014) have constructed homogeneous isotropic cosmological models with variable cosmological constant and time-dependent gravitational constant satisfying the present day observational data. A time variation of the gravitational constant, which characterizes the strength of the gravitational interaction, has been first suggested by Dirac (1937a,b, 1938) and extensively described in literature (Dicke, 1961; Wu and Wang, 1986; Carroll et al., 1992; Kordi, 2009), where the gravitational constant is considered in methods that include studies of the evolutions of clusters of galaxies and of the Sun, observations of lunar occultations, planetary radar-ranging measurements and laboratory experiments. Many other extensions of Einstein theory with variable gravitational constant have been proposed in order to achieve a possible unification of gravitation and elementary particles physics or incorporate Mach’s principle in general relativity (Canuto et al., 1977; Uehara and Kim, 1982; Alfonso-Faus, 1986). Results on the time variation of the gravitational constant $G$ usually yield experimental bounds $\left|\dot{G}/G\right|\le 4.0\times {10}^{-11}{\mathrm{y}\mathrm{r}}^{-1}$ (Benvenuto et al., 2004). The variability of $G$ and Λ becomes then one of the most striking and unsettled problems in cosmology.

In this paper, we study the $\omega =-1$ crossing possibility in a cosmological model in the framework of the extended theories of gravity (ETG) in Lyra’s geometry (Lyra, 1951; Scheibe, 1952; Sen, 1957, 1960; Halford, 1970; Sen and Dunn, 1971; Manoukian, 1972; Hudgin, 1973; Soleng, 1987; Hova, 2013, 2014, Hova et al., 2019) and the references therein. Despite the impossibility for the standard ΛCDM model to have $\omega =-1$ crossing property in Einstein’s gravity in pseudo-Riemannian geometry, the existence of a displacement field interacting with the cold dark matter (CDM) in the ETG in Lyra’s geometry leads to an effective time-dependent gravitational constant and does probably modify the effective distribution of the cosmic fluids so that the effective EoS parameter may cross the $\omega =-$ 1 barrier as the effective time-dependent gravitational constant decreases with time.

In Section II, we describe a model, where an effective gravitational constant is derived, and compute the effective equation of stage parameter that depends exclusively on the effective time-dependent gravitational constant. Section III is devoted to the discussions and conclusions are given in Section 4.

2 Model

In Planckian units $c=\hslash ={\kappa }^2=1$ , the Einstein-Hilbert action invariant under both coordinate and gauge transformations in Lyra’s geometry which is a modification of Riemannian geometry by the introduction of a nonzero gauge function $\psi$ into the structureless manifold, is described by (Sen, 1957)

In the so-called normal-gauge Lyra’s geometry, defined by $\psi =1$ , the gravitational Eq. (2) reduce to (Sen, 1957)

Due to the interaction of matter field with the displacement field during the evolution of the universe, the matter energy ${\rho }_m$ does not conserve separately; there may be transfer of energy between matter fields and the displacement field. In (Hova, 2013) an interaction was built between $\varphi$ and the matter energy ${\rho }_m$ by encoding Eq. (10) into two conservation equations. Here, we are to consider the first Friedmann equation (8) to find the connection between $\varphi$ , $H$ and ${\rho }_m$ . Assuming therefore the geometrical quantities $H$ and $\varphi$ are both real, we can recast Eq. (8) in the form

Combining Eqs. (17) and (18) yields a relationship between $H$ and $\varphi$ :

energy ${\rho }_m$ in the form

To find the effective EoS parameter, ${\omega }_{\mathit{eff}}$ , that controls the different phases of the expanding universe, let us rewrite the Friedmann equation (8) (or equivalently Eq. (17)) in the form

3 Discussions

The EoS parameters ${\widehat{\omega }}_m\left(x\right)$ and ${\omega }_{\mathit{eff}}\left(x\right)$ coincide when the function $\zeta$ is constant, in particular ${\omega }_{\mathit{eff}}={\widehat{\omega }}_m=-1+2{\omega }_m$ for $\zeta =\pi /4$ , leading to an accelerating universe when ${\omega }_m<1/3$ . Moreover, a pressureless matter will yield the cosmological constant boundary since ${\omega }_{\mathit{eff}}={\widehat{\omega }}_m=-1$ . The difference between ${\widehat{\omega }}_m\left(x\right)$ and ${\omega }_{\mathit{eff}}\left(x\right)$ is due to the variation of the effective gravitational constant that will be responsible for the late-time accelerated expansion of the universe. Equation (31) shows that both ${\omega }_{\mathit{eff}}\left(x\right)$ and $\widehat{g}\left(x\right)$ either increase or decrease. As observations indicate that the dark energy EoS parameter is decreasing with time, the possible accelerated expansion of the universe will therefore be the result of the time-dependent gravitational constant that decreases following the expansion of the universe. In view of (29) and (31) we do have ${\omega }_{\mathit{eff}}\left(x\right)\le {\omega }_m$ , especially ${\omega }_{\mathit{eff}}\left(x\right)={\omega }_m$ when $\widehat{g}\left(x\right)=1$ . When the background fluid is a stiff matter ( ${\omega }_m=1$ ), the total energy fluid will also behave as a stiff matter during the evolution of the universe since ${\omega }_{\mathit{eff}}\left(x\right)={\omega }_m=1$ whatever the variation of $\widehat{g}\left(x\right)$ : the universe will be permanently decelerating. And ${\omega }_{\mathit{eff}}\left(x\right)$ vanishes when $\widehat{g}\left(x\right)=1-{\omega }_m$ . As shown in Tables 1 and 2, when $3\left(1-{\omega }_m\right)/4<\widehat{g}\left(x\right)\le 1,$ the expansion is decelerated, and starts accelerating when $\widehat{g}\left(x\right)=3\left(1-{\omega }_m\right)/4$ . The universe then crosses the cosmological constant boundary (quintom scenario) for $\widehat{g}\left(x\right)=\left(1-{\omega }_m\right)/2$ and evolves therefore into a phantom regime. For a positive definite Hubble parameter and to avoid a permanently decelerating expanding universe or a static universe, we need to have $\cos \zeta \left(x\right)>0$ with $\zeta \left(x\right)\ne k\pi$ for $k\in \mathbb{Z}$ , and ${\omega }_m\ne 1$ ( $0\le {\omega }_m<1$ ).

Now we will be interesting in the geometrical diagnostic of the model under consideration, we thus introduce the so-called statefinder pair $\left\{r,s\right\}$ , defined by (Alam et al., 2003; Sahni et al., 2003):

As our model is based on a modification of general relativity, it is important to constrain the different functions so that general relativity Solar System tests remain valid. All the cosmological quantities depending on time via the function $\zeta \left(x\right)$ or equivalently $\widehat{\mathrm{g}}\left(x\right)$ , a necessary condition to have a realistic gravitational theory is

4 Conclusions

In this work we have constructed within Lyra’s geometry a cosmological model involving a matter energy, with equation of state $0\le {\omega }_m\le 1$ , interacting with the displacement field. It has been shown that the late-time accelerated expansion of the universe could be explained for ${\omega }_m\ne 1$ by a nonzero effective time-dependent gravitational constant, resulting from the displacement field. A quintom scenario could even appear without a ghost mode involved in the model when the effective gravitational constant takes some particular value.