Rami Ahmad El-Nabulsi and Waranont Anukool

1 Center of Excellence in Quantum Technology,Faculty of Engineering,Chiang Mai University,Chiang Mai 50200,Thailand

2 Quantum-Atom Optics Laboratory and Research Center for Quantum Technology,Faculty of Science,Chiang Mai University,Chiang Mai,50200,Thailand

3 Department of Physics and Materials Science,Faculty of Science,Chiang Mai University,50200,Thailand

Abstract In Moffat stochastic gravity arguments,the spacetime geometry is assumed to be a fluctuating background and the gravitational constant is a control parameter due to the presence of a timedependent Gaussian white noiseξ (t).In such a surrounding,both the singularities of gravitational collapse and the Big Bang have a zero probability of occurring.In this communication,we generalize Moffat’s arguments by adding a random temporal tiny variable for a smoothing purpose and creating a white Gaussian noise process with a short correlation time.The Universe accordingly is found to be non-singular and is dominated by an oscillating gravity.A connection with a quantum oscillator was established and analyzed.Surprisingly,the Hubble mass which emerges in extended supergravity may be quantized.

Keywords:Moffat stochastic gravity,Gaussian white noise,oscillating gravity,non-singular Universe,mass quantization

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 wellknown 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 thatG(t)=G0+σξ(t),G0being the gravitational constant,σis a parameter measuring the intensity of the geometrical fluctuations of the metric andξ(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].

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 parameterwhich 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.