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Regularizing cosmological singularities by varying physical constants.

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Konrad Marosek

Szczecin CosmologyGroup, University of Szczecin, Poland.

49th Winter School of TheoreticalPhysics

Lądek-Zdrój, Poland, February 10-16, 2013

The idea of variation of physicalconstantshasbeenestablishedwidelyinphysicsboththeoretically and experimentally. Fromthetheoreticalsidetheearlyideas of Weyl and Eddingtonwere most successfullyfollowed by Dirac’sLargeNumberHypothesisfromwhichit was concludedthatthegravitationalconstantshouldchangein time as . Thisled to thescalar-tensorgravitytheorydeveloped by Brans and Dickewhofollowedtheideas of Jordan.Theseideaswerefurtherembeddedintosuperstringtheoriesinwhichthecouplingconstant of gravitybecamerunningduringtheevolution of theearlyuniverse (Polchinski 1996).

The most popular theorieswhichadmitphysicalconstantsvariationarethevaryingspeed of lighttheoriesand varyingα alpha (finestructure) theories (Barrow et al. 1999). Ithasbeenshownthatboth of thesetheoriesallowthesolution of the standard cosmologicalproblemssuch as thehorizonproblem,flatnessproblem,and the−problem. Here, we will applythesetheories to solveyetanother problem - thesingularityproblem.

- Primordialnucleosynthesis(Accetta et al. 1990):
- Helioseismology (Guenther et al.1998):
- Lunar laser ranging(LLR) (Williams et al. 1996):

- Oklophenomenon (Shlyakhter 1976,Petrov et al. 2006):
- Meteorite dating (long-lived beta decays) (Olive et al. 2003):
- Quasarabsorption spectra withredshifts 2.33 < z < 3.08 (Murphy et al. 2001):
Where

Einstein-Friedman equations with varying fundamental constans can be expressed as:

and the energy-momentum conservation law is:

In contrast to many references dealing with non-standard singularities (e.g. sudden future singularities), which consider the scale factor:

Where , , , , are constants.

We propose a new form of the scale factor:

Where , , , are constants.

The first and second derivatives of the scale factor are as follows:

Einstein-Friedman equations for zero curvature (k=0) models now are expressed by:

For we dealwith Big-Bang singularity

Type 0 – Big-Bang (BB): , , , at .

For we have a Big-Ripsingularity

Type I - Big-Rip (BR): , , , at .

For we have a SuddenFuturesingularity

Type II - SuddenFutureSingularity(SFS): ,

, , at .

For we haveFiniteScalarFactorsingularity

Type III - FiniteScalarFactor(FSF): , ,

, at .

VARYING CONSTANTS MODELS

The plots of the scale factor , the energy density ,

and the pressure :

for theparameters , and , whichdescribe

a suddenfuturesingularity(SFS)

VARYING CONSTANTS MODELS

The plots of the scale factor , the energy density ,

and the pressure :

for theparameters , and , whichdescribe

a finitescalefactorsingularity(SFS)

We cansplitourscalefactorinthefollowingway:

where:

and

In a specialcase we get a standard big-bangscalefactorwith

Is possible to write down an equation of state inthe form of a barotropic perfect fluid as:

whereas in the standard notation.

The standard big-bang are decelerating for ,

and accelerating for .

Thepressureispositive for , and andnegative for ,

alsoin Big-Bang.

In a casewhere , ourscalefactorisreduced to an exoticsingularityscalefactor:

And nowdensity and pressureisexpressed as:

And also:

withthedecelerationparameter:

For energy density and pressurevanish ,

whilethe-indexblows-up to infinitywhichisexactlythecharacteristicsof a -singularity (Dąbrowski and Denkiewicz 2009).

Big-Bang singularity:

To avoidthe Big-Bang singularities a gravitationalconstantcould be inthe form:

whichis a fasterdecreasethatinthe standard Dirac’scase

Such a time-dependence of G wouldperhaps be less influenced by thegeophysicalconstraints on thetemperature of theEarth (Teller 1948).

Exoticsingularities

In order to regularize an SFS singularity by varyingspeed of light we suggestthatthetime-dependence of thespeed of lightisgiven by:

whichaftersubstitutingintopressureequationgives:

SFS singularityisregularized by varyingspeed of lightprovidedthat

However, thereis an interestingphysicalconsequence of thefunctionaldependence of thespeed of light. Namely, itgraduallydiminishesreaching zero atthesingularity. In otherwords, thelightslows and eventuallystopsat an SFS singularity. Such an effectispredictedwithintheframework of loop quantum cosmology (LQC), whereitiscalledtheanti-newtonian limit

for , withbeing a criticaldensity (Cailleteau, Mielczarek, Burrau, Grain 2012)

One of the standard assumptions on thevariation of thespeed of lightisthatitfollowstheevolution of thescalefactor:

The field equationsnowisexpressed by:

Withitispossible to remove a pressuresingularity

provided , , or , , .

Since does not depend on c(t) (for k = 0), thenitisimpossible to strengthen an SFS singularity to become an FSF singularity. Itispossibleonly, if we assumethatthegravitationalconstant G changesin time. Letusthenassumethat

where and

Nowdensity and pressureequationsare:

Itfollowsthat an SFS singularity(1 < n < 2) isregularized by varyinggravitationalconstantwhen .

An FSF singularity(0 < n < 1) isregularizedwhen .

Assumingthat we have an SFS singularity and that .

We getthatvarying G maychangean SFS singularityonto a stronger FSF singularitywhen

A physicalconsequence of thefunctionaldependence of thegravitationalconstantinisthatthestrength of gravitybecomesinfiniteatthesingularity. Thisisquitereasonableif we want to regularize an infinite (anti)tidalforceatthesingularity. Thisisalsoexactlywhathappensinthestrongcoupling limit of gravity.

A hybridcasewhichwould influence bothtypes of singularitiesis:

whichchangesdensity and pressureequationsinto:

Singularitiesin (anti-)Chaplygin gas cosmology

Since a couple of years ago, therehasbeen a proposalthatthedark energy can be simulated by a Chaplyginor an anti-Chaplygin gas model. One of theinterestsisthat an anti-Chaplygin gas model allowstheso-calledbig-brakesingularity( , ). whichis a specialcase of a suddenfuturesingularity and .

Theequation of state of the (anti-)Chaplygin gas reads as:

where A > 0 is a constantwiththe unit of the energy density

square.

Afterinsertingequation of state of the (anti-) Chaplygin gas intoenergy-momentumconservation law we get:

In order to find an exactsolution, we will first assumethatthespeed of lightisconstantand thatthegravitationalconstantchanges as:

Theenergy-momentumconservationnowis:

If we make an assumption we will get:

where

The more interesting solution in order to demonstrate regularization of singularities can be obtained in a general case of both varying G = G(t) and c = c(t) though with zero curvature k = 0 case.

Energy-momentum conservation law now is:

Solution of theenergy-momentumconservationequationisgiven by:

where .

Density and pressurenowisexpressed by:

Puttingthe standard big-bangscalefactor:

we nowhave

whichgive and , provided .Thesingularityat t=0 incan be regularized by taking . In ourcase we have a constantpressure (cosmological term) instead of zero pressure.

We haveshown by specifyingsomeexamplesthatitispossible to regularizecosmologicalsingularitiesdue to variation of thephysicalconstants. We haveconsideredthisphenomenoninthetheorieswithvaryingspeed of light (VSL) c(t), and withvaryinggravitationalconstant G = G(t).

Interestingly, in order to regularize an SFS by varying c(t), thelightshould stop propagatingat a singularity- thefactwhichappearsintheloop quantum cosmology. On theotherhand, to regularize an SFS by varyinggravitationalconstant - thestrength of gravityhas to becomeinfiniteat a singularity(as inthestrongcoupling limit of gravity(Isham 1976) whichisquitereasonablebecause of therequirement to overcometheinfinite (anti-)tidalforcesatsingularity.

Finally, we hopethatvariation of thephysicalconstantswhichleads to regularization of singularitiesmay be usefulinthediscussions of themultiverseconceptgivingthe link through a kind of “fake” singularities to various parts of theuniversewithdifferentphysics. Of courseourdiscussionispreliminary and should be continued by usingappropriatemathematicalformalism of both general relativity and particlephysics.