We believe that the topology of spacetime changes with time at all scales ranging from microscopic to very large scales. At microscopic scales, the modification of the spacetime topology is due to huge quantum effects occurring near the initial singularity. Although there exist in literature several well-known phenomenological theories addressing this problem such as string theory or loop quantum gravity, no complete or convincing framework exists till the present moment [1–3]. We believe that a complete theory of quantum gravity will be characterized by a new set of field equations where Einstein's field equations are considered as a particular case. Some of these theories include canonical Hamiltonian formalism, string theory and loop quantum gravity. One more interesting attempt is based on the concept of stochastic gravity in which gravity is treated as a stochastic phenomenon based on fluctuations of the metric tensor of general relativity [4]. In this context, modifications of the macroscopic behavior of the dynamical system emerge due to a nonlinear coupling of the gravitational system to the fluctuating spacetime. The spacetime metric is comparable to a fluctuating surrounding and a probabilistic interpretation of spacetime is then adapted. Its microscopic structure is unidentified whereas its subsystems undergo stochastic fluctuations when it couple to matter at a certain length scale, generating accordingly a number of macroscopic correlation lengths and self-organized behavior. The gravitational constant is considered as a control parameter such that $G(t)={G}_{0}+\sigma \xi (t),$ ${G}_{0}$ being the gravitational constant, $\sigma $ is a parameter measuring the intensity of the geometrical fluctuations of the metric and $\xi (t)$ is a Gaussian white noise with zero expectation value. The gravitational constant is therefore Gaussian distributed due to the central limit theorem. Several approaches to stochastic gravity have been addressed in literature proving the importance of stochasticity in the geometry of spacetime [5, 6].
The main aim of this study is to extend the stochastic gravity approach introduced by Moffat in [4] by introducing in the theory a random variable $\tau $ considered to be too small added for a smoothing purpose and created, by sampling, a white Gaussian noise process with a short correlation time. We assume that the gradient of the Gaussian white noise $\xi (t)$ with respect to time is given by ${{\rm{\nabla }}}_{t}\xi (t)=\tfrac{1}{2}({{\rm{\nabla }}}_{t}\xi (t+\tau )+{{\rm{\nabla }}}_{t}\xi (t-\tau )),$ i.e. $\xi (t)=\tfrac{1}{2}(\xi (t+\tau )+\xi (t-\tau )).$ Its Taylor series expansion generates higher-order derivatives and hence a new form of the time-dependent gravitational constant will arise. In fact, the variability of fundamental constants in nature is not new and goes back to Dirac [7]. Although it leads to a violation of the equivalence principle [8], it may direct us toward a new physics that gives a complete understanding of quantum gravity. It is noteworthy that in the framework of Brans–Dicke gravity, the variation of the gravitational coupling constant is determined by the vacuum expectation value of the dilaton scalar field and the measurement of its variations in time is quite delicate [9]. It should be stressed that gravity theories with time-varying gravitational constant are still far from complete and additional Solar system gravitation tests and binary pulsar astronomical observations are required [8].
The Taylor series expansion of $\xi (t)=\tfrac{1}{2}(\xi (t+\tau )\,+\xi (t-\tau ))$ permits to write the gravitational constant as:2 ) gives:6 ) is reduced to:6 ) takes the subsequent form:7 ) is reduced to:9 ) is given by:9 ). Such a feature emerges in brane cosmology in which the observed Universe is realized as a four-dimensional thin brane embedded in a higher dimensional spacetime (bulk). In these theories, at high energies an extra term quadratic in the energy density of matter on the brane emerges in the Friedmann equation [12, 13]. We may discuss two energy limits:
$\begin{eqnarray}\begin{array}{l}G(t)={G}_{0}+\sigma \left(\xi (t)+\displaystyle \frac{1}{2}\displaystyle \sum _{k=1}^{n}\displaystyle \frac{1+{\left(-1\right)}^{k}}{k!}{\tau }^{k}{\left(\displaystyle \frac{\partial }{\partial t}\right)}^{k}\xi (t)\right)\\ \,=\,{G}_{0}+\sigma \left(\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+\displaystyle \frac{1}{24}{\tau }^{4}{\xi }^{(4)}+{\rm{O}}\left({\tau }^{6}\right)\right).\end{array}\end{eqnarray}$
In what follows, we will assume that the form of $G(t)$ is unknown and it will be determined from cosmological arguments. We consider the case of the Friedmann–Robertson–Walker (FRW) spacetime metric ${\rm{d}}{s}^{2}=-{\rm{d}}{t}^{2}+{a}^{2}(t)({\rm{d}}{r}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}))$ where $a(t)$ is the scale factor. We neglect at the moment the spatial variations of the gravitational constant. We assume that the Universe is filled with perfect fluid with density $\rho $ and pressure $p$ obeying the equation of state (EoS) $p=(\gamma -1)\rho $ where $\gamma $ is a real constant. Besides, the cosmological constant is considered to be time-dependent which is considered largely in literature since it is consistent with observations (see [10] and references therein). Using the approach of Moffat, the nonlocal field equations up to 2nd order on $\tau $ are given by: $\begin{eqnarray}{H}^{2}=\displaystyle \frac{8\pi }{3}\left({G}_{0}+\sigma \xi +\displaystyle \frac{\sigma }{2}{\tau }^{2}\ddot{\xi }+{\rm{O}}({\tau }^{4})\right)\rho +\displaystyle \frac{{\rm{\Lambda }}}{3}.\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\dot{H}+{H}^{2}=-\displaystyle \frac{4\pi }{3}\left({G}_{0}+\sigma \left(\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+{\rm{O}}({\tau }^{4})\right)\right)\left(\rho +3p\right)\\ \,+\displaystyle \frac{{\rm{\Lambda }}}{3},\end{array}\end{eqnarray}$
where $H=\dot{a}/a$ is the Hubble parameter. The divergence of the Einstein tensor, i.e. ${G}_{\mu ;\nu }^{\nu }=0$ leads to: $\begin{eqnarray}8\pi \dot{G}\rho +\dot{{\rm{\Lambda }}}+8\pi G\left(\dot{\rho }+3\gamma H\rho \right)=0.\end{eqnarray}$
Using the usual conservation equation ${{\rm{\nabla }}}^{\mu }{T}_{\mu \nu }=0$ gives: $\dot{\rho }+3\gamma H\rho =0$ and $8\pi \dot{G}\rho +\dot{{\rm{\Lambda }}}=0.$ The first equation gives after simple integration $\rho ={\rho }_{0}{a}^{-3\gamma }$ where ${\rho }_{0}$ is an integration constant whereas the second equation may be written as: $8\pi \rho \left(\sigma \dot{\xi }+\tfrac{\sigma }{2}{\tau }^{2}\dddot{\xi }+{\rm{O}}({\tau }^{4})\right)+\dot{{\rm{\Lambda }}}=0.$ In order to solve these differential equations, we follow the arguments of [11] and we conjecture that ${\rm{\Lambda }}={{\rm{\Lambda }}}_{0}{\rho }^{-\varepsilon },$ $\varepsilon $ is a constant different from unity and ${{\rm{\Lambda }}}_{0}$ is a real parameter. We obtain accordingly: $\begin{eqnarray}\sigma \left(\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+{\rm{O}}({\tau }^{4})\right)\approx -\displaystyle \frac{{{\rm{\Lambda }}}_{0}\varepsilon }{8\pi (\varepsilon +1)}{\rho }^{-\varepsilon -1}.\end{eqnarray}$
Inserting this equation into equation ( $\begin{eqnarray}{\dot{a}}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{2-3\gamma }+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}^{-\varepsilon }}{3(\varepsilon +1)}{a}^{2+3\varepsilon \gamma }.\end{eqnarray}$
Evidently for $\gamma =0,$ the Universe is expanding exponentially with time which corresponds for the inflationary regime dominated by vacuum energy. For $\gamma =4/3$ which corresponds for the radiation-dominated era, equation ( $\begin{eqnarray}{\dot{a}}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{-2}+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}^{-\varepsilon }}{3(\varepsilon +1)}{a}^{2+4\varepsilon },\end{eqnarray}$
whereas for $\gamma =1$ which corresponds to the matter-dominated era, equation ( $\begin{eqnarray}{\dot{a}}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{-1}+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}^{-\varepsilon }}{3(\varepsilon +1)}{a}^{2+3\varepsilon }.\end{eqnarray}$
One can check that for $\varepsilon =-2,$ equation ( $\begin{eqnarray}\begin{array}{l}{\dot{a}}^{2}=\left(\displaystyle \frac{8\pi {G}_{0}\rho }{3}-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }^{2}}{3}\right){a}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{2-3\gamma }\\ \,-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}^{2}}{3}{a}^{2-6\gamma },\end{array}\end{eqnarray}$
and its solution is given by: $\begin{eqnarray}a(t)=\sqrt[4]{\displaystyle \frac{32\pi \alpha {G}_{0}{\rho }_{0}}{3}{t}^{2}+{c}_{1}t+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{8\pi \alpha {G}_{0}}},\end{eqnarray}$
whereas the solution of equation ( $\begin{eqnarray}a(t)=\displaystyle \frac{1}{\sqrt[3]{4}}\sqrt[3]{24\pi \alpha {G}_{0}{\rho }_{0}{t}^{2}+{c}_{2}t+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{8\pi \alpha {G}_{0}}},\end{eqnarray}$
${c}_{i},i=1,2,\mathrm{..}.$ are constants of integration. The main difference between these solutions and the standard FRW model concerns the absence of singularity. We observe the emergence of the square of matter density in the Friedmann equation (1-In the low energy limit, $\rho \gg {\rho }^{2}$ as it is naturally expected and hence equation (9 ) is reduced to:14 ) is approximated by:
$\begin{eqnarray}{\dot{a}}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{2-3\gamma },\end{eqnarray}$
where its solution is given by: $a(t)\propto {t}^{2/3\gamma }$ and hence accelerated expansion occurs for $0\lt \gamma \lt 2/3.$ This corresponds to an EoS parameter $w$ such that $-1\lt w=\gamma \,-1\lt -1/3$ which is within the observed value $w\,=-{1.01}_{-0.40}^{+0.37}$ [14]. Given that the Universe is dominated by a power-law expansion, the present value of the Hubble constant ${H}_{0}$ and the deceleration parameter $q=-\ddot{a}a/{\dot{a}}^{2}$ play a significant role. The recent joint test using $H(z)$ at $1\sigma $ level yields the constraints $q=-{0.18}_{-0.12}^{+0.12}$ and ${H}_{0}\,={64.18}_{-0.54}^{+0.55}{{\rm{Kms}}}^{-1}{{\rm{Mpc}}}^{-1}$[15]. Since $q=(3\gamma -2)/2,$ then we find $0.46\lt \gamma \lt 0.62$ or $-0.54\lt w=\gamma -1\lt -0.38$ which is within the observational limit. The Universe is therefore dominated by dark energy and a decaying energy density. Both the energy density and the cosmological constant decrease in time whereas the Gaussian white noise is governed for $\varepsilon =-2$ by the following differential equation: $\begin{eqnarray}\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+{\rm{O}}({\tau }^{4})=-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{4\pi \beta }{a}^{-3\gamma }=-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{4\pi \beta }{t}^{-2},\end{eqnarray}$
where its solution is given by: $\begin{eqnarray}\begin{array}{l}\xi (t)\approx {c}_{3}\,\sin \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+{c}_{4}\,\cos \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{2\pi \beta {\tau }^{2}}\\ \,\times \,\left({\rm{Ci}}\left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\cos \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+{\rm{Si}}\left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\sin \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\right),\end{array}\end{eqnarray}$
where ${\rm{Ci}}$ and ${\rm{Si}}$ are respectively the cosine and sine integral. This shows that the Gaussian white noise oscillates with time. For very large time, equation ( $\begin{eqnarray}\xi (t)\approx \frac{\xi (\tau )\sin \left(\displaystyle \frac{\sqrt{2}}{\tau }\left(t-{t}_{0}\right)\right)+\xi ({t}_{0})\sin \left(\sqrt{2}\left(1-\displaystyle \frac{t}{\tau }\right)\right)}{\sin \left(\sqrt{2}\left(1-\displaystyle \frac{{t}_{0}}{\tau }\right)\right)},\end{eqnarray}$
where $\xi ({t}_{0})=\xi (t={t}_{0})$ and $\xi (\tau )=\xi (t=\tau ),$ ${t}_{0}$ is the present time. Since $\xi (\tau )\gg \xi ({t}_{0}),$ i.e. the gravitational constant at high energy limit is expected to be much larger than the gravitational constant at low energy limit, then $\xi (t)\,\propto \sin (\sqrt{2}(t-{t}_{0})/\tau )$ and hence the period of oscillations is $T=2\pi \tau /\sqrt{2}.$2-In the high energy limit, $\rho \ll {\rho }^{2}$ and hence equation (10 ) is reduced to:14 ) is given by $a(t)\propto {t}^{1/3\gamma }$ and hence accelerated expansion occurs for $0\lt \gamma \lt 1/3.$ The deceleration parameter is $q=3\gamma -1$ and gives $-0.77\lt w=\gamma -1\lt -0.69$ which lies within the observational limit. The Gaussian white noise is governed, in that case, by the subsequent differential equation:
$\begin{eqnarray}{\dot{a}}^{2}=-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}^{2}}{3}{a}^{2-6\gamma },\end{eqnarray}$
and a plausible solution is obtained for ${{\rm{\Lambda }}}_{0}\lt 0$ which corresponds to a negative cosmological constant. In fact, a negative cosmological constant was reconsidered in literature based on high-redshift and low-redshift cosmological observations which suggest that the dark energy part of the Universe may be dominated by a negative lambda [16]. It was also proved that it may play an important role in gravitational theories with higher-order curvature terms [17] and it may be consistent with low-energy observations [18–20]. A plausible solution of equation ( $\begin{eqnarray}\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+{\rm{O}}({\tau }^{4})=-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{4\pi \beta }{a}^{-3\gamma }=-\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{4\pi \beta }{t}^{-1},\end{eqnarray}$
where its solution is given by: $\begin{eqnarray}\begin{array}{l}\xi (t)={c}_{5}\,\sin \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+{c}_{6}\,\cos \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+\displaystyle \frac{{{\rm{\Lambda }}}_{0}{\rho }_{0}}{2\pi \beta {\tau }^{2}}\\ \,\times \,\left({\rm{Ci}}\left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\cos \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+{\rm{Si}}\left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\sin \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)\right),\\ \,\approx \,\left({c}_{5}+\displaystyle \frac{\pi {{\rm{\Lambda }}}_{0}{\rho }_{0}}{4\pi \beta {\tau }^{2}}\right)\sin \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right)+{c}_{6}\,\cos \left(\displaystyle \frac{\sqrt{2}t}{\tau }\right),\end{array}\end{eqnarray}$
for very large time. Setting $\xi ({t}_{0})=\xi (t={t}_{0})$ and $\xi (\tau )\,=\xi (t=\tau ),$ we find $\xi (t)\propto \,\sin (\sqrt{2}(t-{t}_{0})/\tau ),$ i.e. $G(t)\,\approx {G}_{0}+\sigma \,\sin (\sqrt{2}(t-{t}_{0})/\tau ).$ Periodic oscillations of $G$ was discussed in several research works. In [21], a correlation between measurements of the gravitational constant and the length of day is observed. The ground state measurements of $G$ oscillates between $6.672\times {10}^{-11}N{({\rm{m}}/{\rm{Kg}})}^{2}$ and $6.675\,\times {10}^{-11}N{({\rm{m}}/{\rm{Kg}})}^{2}$ with a period $T=5.9$ years and an amplitude of $A=0.0016\times {10}^{-11}\,{{\rm{m}}}^{3}\,{{\rm{Kg}}}^{-1}\,{{\rm{s}}}^{-1}.$ This measurement is subject to a number of studies in order to better constrain the measure of the gravitational constant [22–25]. Besides, an oscillation $G$ is able to explain some dynamical anomalies in galaxies. In [26], the time dependence proposed for the gravitational constant is $G(t)={G}_{0}+{G}_{1}\,\cos (\lambda (t-{t}_{0})+\psi ),$ $\lambda $ is a constant and $\psi $ is the phase. This law leads to an apparent variation in the isotropic density of galaxies and has peaks lying on concentric spherical shells at periodically spaced radii. Periodic gravity was also discussed in [27–34] through different frameworks.More generally, if lambda is constant, i.e. $\varepsilon =0,$ equation (7 ) is reduced to:20 ) is converted into:
$\begin{eqnarray}{\dot{a}}^{2}=\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{3}{a}^{2-3\gamma }+\displaystyle \frac{{{\rm{\Lambda }}}_{0}}{3}{a}^{2},\end{eqnarray}$
and the solution is given by: $\begin{eqnarray}\begin{array}{l}a(t)={a}_{0}\left(\cos \left(\displaystyle \frac{3\gamma -2}{2}\sqrt{-\displaystyle \frac{{{\rm{\Lambda }}}_{0}}{3}}\left(t-{t}_{0}\right)\right)\right.\\ \,+\sqrt{-\displaystyle \frac{8\pi {G}_{0}{\rho }_{0}}{{{\rm{\Lambda }}}_{0}}{a}_{0}^{2-3\gamma }-1}\\ \,\times {\left.\sin \left(\displaystyle \frac{3\gamma -2}{2}\sqrt{-\displaystyle \frac{{{\rm{\Lambda }}}_{0}}{3}}\left(t-{t}_{0}\right)\right)\right)}^{\displaystyle \frac{2}{3\gamma -2}},\end{array}\end{eqnarray}$
provided that ${{\rm{\Lambda }}}_{0}\lt 0$ and $-8\pi \alpha {G}_{0}{\rho }_{0}{a}_{0}^{2-3\gamma }\gt {{\rm{\Lambda }}}_{0}.$ Here ${a}_{0}=a(t={t}_{0}).$ Obviously, if $\gamma \lt 2/3,$ then equation ( $\begin{eqnarray}\begin{array}{l}a(t)={a}_{0}\left(\cos \left(\left|\displaystyle \frac{3\gamma -2}{2}\right|\sqrt{-\displaystyle \frac{{{\rm{\Lambda }}}_{0}}{3}}\left(t-{t}_{0}\right)\right)\right.\\ \,-\sqrt{-\displaystyle \frac{8\pi \alpha {G}_{0}{\rho }_{0}}{{{\rm{\Lambda }}}_{0}}{a}_{0}^{2-3\gamma }-1}\\ \,\times {\left.\sin \left(\left|\displaystyle \frac{3\gamma -2}{2}\right|\sqrt{-\displaystyle \frac{{{\rm{\Lambda }}}_{0}}{3}}\left(t-{t}_{0}\right)\right)\right)}^{\displaystyle \frac{2}{3\gamma -2}},\end{array}\end{eqnarray}$
and the scale factor diverges an infinite number of times where each divergence describes a Big Rip that occurs at: $\begin{eqnarray}\begin{array}{l}{t}_{BR}={t}_{0}+\displaystyle \frac{2}{3\gamma -2}\sqrt{-\displaystyle \frac{3}{{{\rm{\Lambda }}}_{0}}}\\ \,\times \arctan \left(\displaystyle \frac{1}{\sqrt{-\displaystyle \frac{8\pi \alpha {G}_{0}{\rho }_{0}}{{{\rm{\Lambda }}}_{0}}{a}_{0}^{2-3\gamma }-1}}\right)\\ \,+\displaystyle \frac{2\pi n}{3\gamma -2}\sqrt{-\displaystyle \frac{3}{{{\rm{\Lambda }}}_{0}}},\end{array}\end{eqnarray}$
where $n$ is a natural number [35, 36]. Since $\gamma \lt 2/3,$ the EoS parameter is $w=\gamma -1\lt -1/3$ and accordingly we have infinite Big Rip singularities without necessarily crossing the phantom-divide line [37]. For constant lambda, one may check that the Gaussian white noise obeys the higher-order differential equation: $\begin{eqnarray}\xi +\displaystyle \frac{1}{2}{\tau }^{2}\ddot{\xi }+\displaystyle \frac{{\tau }^{4}}{24}{\xi }^{(4)}+\mathrm{...}.=\displaystyle \frac{\xi (t+\tau )+\xi (t-\tau )}{2}=0,\end{eqnarray}$
assuming a zero integration constant. Since $\xi (t)\propto \,\sin (\omega t)$ we get: $\begin{eqnarray}\left(1-\displaystyle \frac{1}{2}{\omega }^{2}{\tau }^{2}+\displaystyle \frac{1}{4!}{\omega }^{2}{\tau }^{2}-\cdot \cdot \cdot \right)\sin (\omega t)=0,\end{eqnarray}$
which is a finite approximation of $\cos (\omega \tau )\sin (\omega t)=0.$ This equation holds for ${\omega }_{l}=\tfrac{2n+1}{2\tau }\pi ,n\in {\mathbb{Z}}$ and the solution is therefore expressed by a Fourier series and is given by: $\begin{eqnarray}\xi (t)=\displaystyle \sum _{n}{k}_{n}\,\sin \left({\omega }_{n}t\right),\end{eqnarray}$
where ${k}_{l}$ is an unknown coefficient. This solution may have a quantum analogy with the quantum harmonic oscillator if we define the energy ${E}_{l}^{+}=\tfrac{2n+1}{2}\hslash \bar{\omega }$ where $\bar{\omega }=\tfrac{\pi \sqrt{2}}{\tau }$ and $n\in {{\mathbb{N}}}^{0}.$ The quantum mechanical energy level is given by ${\rm{\Delta }}E={E}_{n}^{+}-{E}_{0}^{+}={E}_{0}^{-}-{E}_{n}^{-}=n\hslash \bar{\omega }$ where ${E}_{n}^{-}=\tfrac{2n-1}{2}\hslash \bar{\omega },$ ${E}_{0}^{+}=\tfrac{1}{2}\hslash \bar{\omega }$ and ${E}_{0}^{-}=-\tfrac{1}{2}\hslash \bar{\omega }.$ Of course for ${\omega }_{n}\lt 0$ instabilities occur. Since ${\omega }_{n}=\tfrac{2n+1}{2\tau }\pi ,n\in {\mathbb{Z}},$ then quantization of $\tau $ may be realized. If we assume that ${\omega }_{0}=\tfrac{\pi }{2\tau }=\tfrac{2\pi }{{T}_{0}}$ where ${T}_{0}$ is the period of oscillations at the ground state $l=0$ and if we set $\tau \approx {H}_{0}^{-1}$[38, 39], then ${E}_{0}^{+}=\tfrac{\pi \hslash {H}_{0}\sqrt{2}}{2}=m{c}^{2}$ and hence $m\approx \hslash {H}_{0}/{c}^{2}.$ Such a mass is known as the ‘Hubble mass' and it emerges in different theories including supersymmetry and extended supergravity [40–44]. In general, ${\rm{\Delta }}E=n\hslash \bar{\omega }\,=n\hslash \tfrac{\pi \sqrt{2}}{\tau }=m{c}^{2}$ and hence ${\rm{\Delta }}E\approx \pi \sqrt{2}n\hslash {H}_{0}=m{c}^{2}$ and hence the Hubble mass is quantized in agreement with some phenomenological studies on mass quantization [45–55].To conclude, in this letter we have tried to generalize Moffat stochastic gravity arguments where the gravitational constant is assumed to be controlled by a time-dependent Gaussian white noise parameter due to the geometrical fluctuations of the spacetime metric. The Gaussian white noise is very irregular but is a practical model for speedily fluctuating phenomena, in particular metric fluctuations assumed to be comparable to a Markov process, i.e. any supplementary information on its earlier period history is considered irrelevant for the prediction of its future evolution. Recall that Einstein's general relativity is a nonlinear theory where spacetime topology is fluctuating comparable to an elastic manifold. Hence, the spacetime manifold may be described by a probabilistic theory where the spacetime metric tensor is defined as a random stochastic variable (see [56–60] regarding some interesting studies on Gaussian white noise) fluctuating randomly at some length scale. These fluctuations are expected to be huge during the Planck era, and then they move from being close together in a group to being in different places across a larger area of the Universe. This will lead to an instantaneous spreading of information throughout the Universe. Such a stochastic scenario is motivating since the horizon problem will be solved and a plausible explanation of the high degree of isotropy and homogeneity of the present universe is offered. Within the framework of FRW cosmology, we have observed that the Universe is governed by two periods of accelerated expansion: the inflation followed by the radiation and matter-dominated epoch, then by a period of accelerated expansion dominated by dark energy. The Friedmann equation is characterized by the presence of a new square density term comparable to the term obtained in brane cosmology suggesting that the Universe is dominated by low and high energy limits. In both cases, the Universe is found to obey a power-law evolution of the scale factor and is dominated by dark energy. Nevertheless, the Universe is governed by an oscillating gravitational constant and a decaying cosmological constant which is in agreement with recent astrophysical observations. For the case of a constant lambda, the scale factor of the Universe diverges an infinite number of times where each divergence describes a Big Rip without necessarily crossing the phantom-divide line. In fact, the Big Rip is predicted by several phantom cosmological models that could describe the future evolution of our Universe. It is generally believed that quantum gravity effects may smooth or even shun these singularities. In the present model, the Universe could reach this singularity in a finite time from the present epoch yet its scale factor diverge an infinite number of times and as a result, the size of the apparent Universe, the Hubble rate and its cosmic time derivative would diverge, i.e. the scale factor diverges to infinity and the size of the cosmic horizon goes to zero. That would have drastic effects on galaxy formation in the Universe. Although the Big Rip is associated in phantom dark energy models, in this study, it was observed that infinite Big Rip singularities may occur without necessarily crossing the phantom-divide line. It is noteworthy that this scenario may be associated with the conformal cyclic cosmology model where the energy density of matter relative to radiation will become irrelevant at the Big Rip resulting on a conformally invariant spacetime metric [61]. The late-time acceleration of the Universe characterized by such a cosmological singularity has been discussed in literature within the framework of modified or extended gravity theories (see [62, 63] and references therein) and all frameworks are in fact characterized by a violation of strong energy conditions. A connection to quantization was observed suggesting that the parameter $\tau \approx {H}_{0}^{-1}$ which leads to the emergence of the Hubble mass and its quantization. This shows that it is possible to explain certain quantum gravity phenomena of quantization and discrete nature [64–66]. We were further motivated by the spontaneous conviction that quantum theory may be applied to large scales and there should be detectable effects at cosmological distances. These features may have motivating impacts in high energy physics and may open some possible roads toward quantum gravity.
Acknowledgments
The authors would like to thank Chiang Mai University for funding this research and the anonymous referee for useful comments and valuable suggestions.
Disclosure of conflicts of interest
The author declares that he has no conflicts of interest.
Funding
The author received no direct funding for this work.