Qiaochu Dai

University of Alberta

Spacetime Curvature drives the Big Bounce

Qiaochu Dai

University of Alberta

Instead of the hypothesis of the dark matter and dark energy,curvature of a four dimensional Lorentzian manifold can be used to explain the accelerating expansion of the universe today and the shrink of that in the future,as well as two critical points for each period life of the universe.Therefore,the Big Bang can be treated as the beginning point of the process that the total energy changes to to?tal mass along the positive direction of entropy on the cosmic scales, where the curvature is positive and,conversely,the Big Crunch can be treated as the beginning point of the process that total mass chang?es to total energy along the negative direction of entropy,where the curvature is negative.By derivative,sinusoidal functions obtained to describe the change rate of the scale and that of the curvature along the periodic life of the universe.

1 Background

The question about the gravitational influence over the universe and its dynamics has been attributed to the presence of dark matter in the universe.The discovery of dark matter seemed to answer the ques?tion of infinite expansion towards destruction（Khoury,Ovrut,Seiberg, Steinhardt,&Turok,2002）.To this extent,theoretical physicists sur?mised the existence of dark energy as a form of vacuum energy to ex?plain the acceleration of our universe.Nevertheless,the hypothesis of dark matter and dark energy has come under close scrutiny in recent timesObservations,likePan-STARRS,indicatedarkenergy's strength increases over time so that the ratio ω between the dark ener?gy pressure and its energy density is no longer a constant,which al?most appears as a mythical and supernatural force operating in the cos?mos.If the amount of time for accelerating expansion lasts longer than the time of the Standard Big Bang cosmology,fluctuation will be caused with wavelengths of cosmological scales originating from Planckian values at the beginning of inflation.Furthermore,infinite density at the beginning of the Big Bang runs counter to the uncertain?ty principle.In addition,second order Friedmann equations from the quantum corrected Raychaudhuri equation（Ahmed&Saurya,2015）shown below contain the radiation term in the early universe,which eliminates the singularity of the Big Bang，

where（1）is the Friedmann equation withsignifying the current epoch and the scale factor of the universeais isolate timedependent.Therefore,

which predicts an everlasting universe with the approximation of

F(H)=F(n)(Hp)(H-Hp)n

near the fixed point whereH˙=F() H.

To avoid the Planckian problem and fix the inconsistency of the Big Bang,the model of bouncing cosmology with curvature of a four di?mensional Lorentzian manifold can used to explain the accelerating ex?pansion of the universe as well as the shrink of it in the future.

The cosmological model of Big bounce provides a phase of an os?cillatory universe interpreting the Big Bang as the result of the col?lapse of a previous universe.When mass density approaches infinity at the end of the expansion,the behavior of the quantum foam will change,which means fundamental physical constants are unnecessary to be constant during one period of life of the universe,including the speed of light in a vacuum.The condition of the bouncing universe can be expressed by the equations,

whereΩΛ+Ωr+Ωm+Ωk=1andΩmis the present value of the density in the universe.In this equation,cossiscosif the total matter density is bigger than0.5andcoshif it is smaller than0.5.

2 Introduction

Moving towards a cosmological constant is a heavily contested means of approaching the interpretation of cosmological events since the universe is not stable.This has been witnessed through the colli?sion of galaxies.Therefore,the universe appears to moving towards higher instability.

Within seconds of the big bang,the temperature of the cosmos standing at the excesses of 1032Kelvin cooled to 2.725K,the normal temperature of the cosmos.Two theories exist for the explanation of where the heat went:（1）it was swallowed by black hole singularities, or（2）there was an external energy source,which gave out the cold in exchange for the heat.A case example of this occurrence could be seen in VIRGOHI21,which swallowed adjacent stars and pulled up to 2000 galaxies within it,creating a galaxy-in-mass singularity（Joseph, 2017）.Eradinus is also another phenomenon in outer space bearing the gravity of millions of galaxies.Having swallowed entire surrounding galaxies,the singularity sits in nothingness and has swallowed all gas, light and radiation of surrounding galaxies（Shahalam,Pacif,&Myrza?kulov,2016）.The gravitation pull of the Eradinus is so strong thateven light cannot escape,thereby bringing out the image of a supermassive black hole.With these phenomena in mind,one considers this against the theory of an ever-expanding cosmos pulled out by dark matter,but these are not consistent（Joseph,2017）.However,the ob?servations are amicable with the views of a collapsing universe.

Without the need of dark energy,second order Friedmann equa?tion indicates the big bang without singularity and everlasting universe. According to the assumption of the uniformity on large-scale structure that four-dimensional Lorentzian manifold is homogeneous and isotro?pic,the Friedman-Lema??tre-Robertson-Walker metric is written as,

whereηis conformal time,xare the comoving spatial coordinates anda(t)is the isolate time-dependent scale factor thata(t0)=1.

Therefore,in a gauge where the scalar metric f l uctuations are di?agonal,the action with linear f l uctuations takes the form,

where Φ（x,t）and Ψ（x,t）are scalar metric f l uctuations,and the transverse traceless tensorhijdepends on spacetime,which represents the gravitational waves.（Robert&Patrick,2016）

Then consider a closed universe that the expansion is reversible, Hubble expansion rate is given by

where the total mass is modelled in terms of a scalar fieldφlin?earized about its background thatφ(x,t)=φ0(t)+σφ(x,t).In gener?al relativity,gravitation can be seen as curvature of a four dimension?al Lorentzian manifold.Curvature of spacetime is distinct with k=0, 1,-1 at any local points,however；the sum for rate change of the total number of curvature in large scale can be seen as continuous.

In these cases,expansion will cease and all matter will collapse in on itself.The universe retracts from infinite large space then bounc?es back to re-expand as the result that the spacetime extremely curved.Therefore,the Big Bang can be treated as the beginning point of the process that the total energy changes to total mass along the pos?itive direction of entropy on the cosmic scales,where the curvature reaches negative minimum and,conversely,the Big Crunch can be treated as the beginning point of the process that total mass changes to total energy along the negative direction of entropy,where the curva?ture reaches positive maximum.

Scientists successfully use curvature perturbations that form a scale-invariant primordial power spectrum to explain the cosmic mi?crowavebackground observations.Themodelofnonsingularbig bounce is constructed within the frame of standard gravitation theory.（Yi-Fu,Damien&Robert,2012）Furthermore,the theory of Big Bounce provides a possible solution to the horizon problem in Big Bang.

3 Calculation

To obtain a nonsingular bouncing model,introduce a free scalar f i eldψwith opposite sign kinetic energy term whose density is dominat?

.2ed byψand scales asa-6.The total mass of the universe can be de?scribed through a scalar φ fi eld whose time-averaged energy density scales asa-3.The action is

The Polyakov Action with embedding coordinatesxμin a back?ground space-time with metricgμνis,

wheree2=-x˙μx˙νgμνis the constraint.

For the case of p=1 in Minkowski space-time,gμν=ημν, γαβ=xμ?βxvgμν.

It can be used to describe the transition from a big crunch to a big bang corresponds to the collision of two empty orbifold planes ap?proaching each other.

Using the metric of a Kerr black hole in any large mass that,

According to the analytic series representation of the solution of the Teukolsky equation,the gravitational perturbations is,

whereCαβγδis the Weyl tensor.Put equation（16）into Fourier-har?monic components,

From equation（11）and（12）the potential V（r）is given by

wheresSlm(θ)is the spin-weighted spheroidal harmonic angular function and the coordinates of time and space will switch within large mass groups.Soθbecomes the measure of periodic life of our universe andsinθcan be treated as the change rate in scale.By derivation,cur?vature changes in one period life is the function ofcosθ.Graphs of the expanding and shrinking rate of the scale can be obtained as below.（如圖所示）

4 Discussion

（Figure1.The expanding and shrinking rate of the scale）

（Figure2.Curvature changes in one period life）

Based on the graphs of the total curvature change,it is believed that the universe may not continually expand forever.During the pro?cess of expansion,The effect of curvature of spacetime increases from negative maximum to total mass of the universe plays the main role as needed to go from expected slowing down the expansion and leading to the Big Crunch–a situation in which the effect of the big bang is re?versed.Predictions show that this event is likely to happen in more than 13.8 billion years.With the assumption that the average massdensity of the universe is sufficiently high as well as the curvature of spacetime is finitely large at the critical point of the Big Crunch,the reversal of the metric expansion of space predictably cause the uni?verse to fall on itself,as indicated by the Analytic solution to the Fried?mann equation,until the cosmic scale factor to reach negative maxi?mum and restart of the Big Bang.

Since the universe is under accelerating expansion now,θ＜θ1is obtained from Figure（1）in a quarter of the period where a maximum redshift given by,

which can be obtained from Figure（2）that the total energy is still dominant.This agrees with astronomical observations recently.

5 Conclusion

It is in this spirit that due to curvature of a four dimension?al Lorentzian manifold,the universe has been expanding as the domi?nation of energy is changing to that of matter.Based on the presump?tion that the universe gains negative curvature to hold itself together, the expansion may predictably get slower and diminish as the domina?tion of matter returns to that of energy,which leads the universe to fall on itself until the curvature reaches its positive maximum.Thus, the Big Bang is actually the result of the collapse of a previous uni?verse,which will go through the same process–expand then collapse–thereby recreating itself until infinity.

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