Why space-time behaves Why space-time behaves homogeneously near the homogeneously near the

big bangbig bang

Frans PretoriusFrans PretoriusPrinceton University Princeton University

work with D. Garfinkle, W. Lim and P. Steinhardt work with D. Garfinkle, W. Lim and P. Steinhardt arXiv:0808.0542arXiv:0808.0542 [hep-th] [hep-th]

Loop Qauntum Cosmology WorkshopLoop Qauntum Cosmology WorkshopInstitute for Gravitation and the CosmosInstitute for Gravitation and the Cosmos

Penn State, 23 October, 2008Penn State, 23 October, 2008

Exploring the smoothing Exploring the smoothing power of the ekpyrotic power of the ekpyrotic

mechanismmechanism

Frans PretoriusFrans PretoriusPrinceton University Princeton University

work with D. Garfinkle, W. Lim and P. Steinhardt work with D. Garfinkle, W. Lim and P. Steinhardt arXiv:0808.0542arXiv:0808.0542 [hep-th] [hep-th]

Loop Qauntum Cosmology WorkshopLoop Qauntum Cosmology WorkshopInstitute for Gravitation and the CosmosInstitute for Gravitation and the Cosmos

Penn State, 23 October, 2008Penn State, 23 October, 2008

OverviewOverview Background and motivationBackground and motivation

seek “natural” solutions to cosmological puzzles, in particular why the pre-seek “natural” solutions to cosmological puzzles, in particular why the pre-CMB universe was in such a remarkably flat, homogenous and isotropic stateCMB universe was in such a remarkably flat, homogenous and isotropic state

two contemporary, in-principle answers to the these problemstwo contemporary, in-principle answers to the these problems inflation: a inflation: a period of rapid expansionperiod of rapid expansion of the universe of the universe after the big bangafter the big bang

the ekpyrotic/cyclic mechanism: a the ekpyrotic/cyclic mechanism: a period of slow contraction before the big bangperiod of slow contraction before the big bang

the main problem with either mechanism is lack of a compelling derivation coming from a the main problem with either mechanism is lack of a compelling derivation coming from a fundamental theoryfundamental theory

the rigorous content of this talk will be description of work investigating how the rigorous content of this talk will be description of work investigating how robust the ekpyrotic mechanism is in preparing the universe in a viable pre-robust the ekpyrotic mechanism is in preparing the universe in a viable pre-big bang statebig bang state

in relation to this conference, the underlying theme will be a reminder that in in relation to this conference, the underlying theme will be a reminder that in the quest to develop a fundamental theory of quantum gravity applicable to the quest to develop a fundamental theory of quantum gravity applicable to describing the early universe, that there are (at least) these 2 grails to describing the early universe, that there are (at least) these 2 grails to search for that would show the theory could provide a natural model for the search for that would show the theory could provide a natural model for the universe consistent with all present day observationsuniverse consistent with all present day observations

Formalism, Initial Data & ResultsFormalism, Initial Data & Results Conclusions and future workConclusions and future work

Horizon and Flatness ProblemsHorizon and Flatness Problems Established theory Established theory – – the standard model of particle physics and the standard model of particle physics and

general relativity general relativity –– (together with dark matter and dark energy), (together with dark matter and dark energy), provide a consistent picture of the evolution of the universe from provide a consistent picture of the evolution of the universe from ~100 seconds after the “big bang” until to today~100 seconds after the “big bang” until to today

This picture of the universe has been This picture of the universe has been assembled following the guidance of assembled following the guidance of remarkable observations over the remarkable observations over the past couple of decades that have past couple of decades that have shown that the universe was very shown that the universe was very close to flat, homogeneous and close to flat, homogeneous and isotropic at the time of recombinationisotropic at the time of recombination

Furthermore, the spectrum of the Furthermore, the spectrum of the CMB is that of a thermal black body to CMB is that of a thermal black body to within 1 part in 10within 1 part in 1055, better than , better than anything that can be produced in a anything that can be produced in a lab on earthlab on earth

Horizon and Flatness ProblemsHorizon and Flatness Problems Extrapolating the observations back in time using known theory Extrapolating the observations back in time using known theory

presents a couple of problemspresents a couple of problems energy densities approach the Planck scale, beyond which we cannot energy densities approach the Planck scale, beyond which we cannot

reasonably trust the theoriesreasonably trust the theories

going back from the era of the CMB to the Planck scale, regions of the going back from the era of the CMB to the Planck scale, regions of the CMB separated by more than roughly a degree where never in causal CMB separated by more than roughly a degree where never in causal contact, so whence came the black body spectrum? … contact, so whence came the black body spectrum? … horizon horizon problem problem

Envisioning an evolution forwards in time from just below the Envisioning an evolution forwards in time from just below the Planck scale also presents problemsPlanck scale also presents problems to have a universe that is as flat as observed today requires that the to have a universe that is as flat as observed today requires that the

post-Planck universe was flat to within ~ 1 part in 10post-Planck universe was flat to within ~ 1 part in 1060 60 … … flatness flatness problem problem

Without a fundamental theory of Planck scale physics there is no Without a fundamental theory of Planck scale physics there is no reasonable basis to suppose that a “solution” to these problems is reasonable basis to suppose that a “solution” to these problems is that the universe just happened to begin in an “un-natural”, fine-that the universe just happened to begin in an “un-natural”, fine-tuned state.tuned state.

Inflation & EkpyrosisInflation & Ekpyrosis At present there are 2 well-studied solutions to the horizon, flatness (& At present there are 2 well-studied solutions to the horizon, flatness (&

monopole) problems that are also consistent with current observations, monopole) problems that are also consistent with current observations, most notably with the near scale invariant power spectrum of the most notably with the near scale invariant power spectrum of the fluctuations of the CMBfluctuations of the CMB inflation (Guth 1981, Linde 1982, Albrecht & Steinhardt 1982) inflation (Guth 1981, Linde 1982, Albrecht & Steinhardt 1982)

a period of rapid expansion following the big-banga period of rapid expansion following the big-bang

ekpyrotic or cyclic models (Khoury et al. 2001, Steinhardt & Turok 2002)ekpyrotic or cyclic models (Khoury et al. 2001, Steinhardt & Turok 2002) a period of slow contraction preceding the big-banga period of slow contraction preceding the big-bang original inspiration based on the collision of two 4D branes in higher dimensional original inspiration based on the collision of two 4D branes in higher dimensional

spacetime, though here we will take the effective field theory modelspacetime, though here we will take the effective field theory model

In both models, the smoothing happens below the Planck scale, and the In both models, the smoothing happens below the Planck scale, and the usual assumptions made are that general relativity describes the usual assumptions made are that general relativity describes the evolution of spacetime, driven by some new kind of “exotic” matter or evolution of spacetime, driven by some new kind of “exotic” matter or effective mattereffective matter inflation: the matter has an effective equation of state parameter inflation: the matter has an effective equation of state parameter w= Pw= P//=-1=-1

ekpyrosis: the matter has an effective ekpyrosis: the matter has an effective w>>1w>>1

Why they smooth as they doWhy they smooth as they do Consider the Friedmann equation governing the evolution of the scale Consider the Friedmann equation governing the evolution of the scale

factor factor a(t)a(t) of the universe for a homogeneous, near FRW spacetime: of the universe for a homogeneous, near FRW spacetime:

where where mm00, , rr

00, , ww00 are the energy densities in a pressureless dust, are the energy densities in a pressureless dust,

radiation and radiation and ww-fluid respectively at some initial time, and -fluid respectively at some initial time, and kk represents represents spatial curvature and spatial curvature and the anisotropy the anisotropy

Inflation (Inflation (w=-1w=-1): ): a(t)a(t) growsgrows with time, hence the component with the with time, hence the component with the smallestsmallest power of power of aa in the denominator dominates the late-time evolution in the denominator dominates the late-time evolution of the universe … here, the of the universe … here, the ww-fluid, driving inflation-fluid, driving inflation without the without the ww-fluid it would be curvature (the flatness problem) -fluid it would be curvature (the flatness problem)

Ekpyrosis (Ekpyrosis (w>>1w>>1): ): a(t)a(t) shrinksshrinks with time, hence the component with the with time, hence the component with the largestlargest power of power of aa in the denominator dominates the approach to the big- in the denominator dominates the approach to the big-crunch … again, by construction, the crunch … again, by construction, the ww-fluid, but now this drives ekpyrosis-fluid, but now this drives ekpyrosis without the without the ww-fluid shear would dominate, resulting in chaotic mixmaster -fluid shear would dominate, resulting in chaotic mixmaster

behavior behavior

6

2

2)1(3

0

4

0

3

022

38

aak

aaaaaH w

wrm

How robust is the smoothing How robust is the smoothing mechanism?mechanism?

For inflation, several results have shown that the For inflation, several results have shown that the smoothing mechanism is robust in that even beginning smoothing mechanism is robust in that even beginning from large deviations from an FRW universe, if a from large deviations from an FRW universe, if a w=-1w=-1 matter component is present the spacetimes generically matter component is present the spacetimes generically evolve to de Sitter [Wald (1983), Jensen & Stein-Schabes evolve to de Sitter [Wald (1983), Jensen & Stein-Schabes (1986), …, Goldwirth (1991)](1986), …, Goldwirth (1991)]

Until now, the only comparable results for ekpyrosis Until now, the only comparable results for ekpyrosis showed that the mechanism worked for linear showed that the mechanism worked for linear perturbations about a contracting FRW spacetime [Erikson perturbations about a contracting FRW spacetime [Erikson et al (2004)]et al (2004)]

Here results are presented from numerical solution of the Here results are presented from numerical solution of the full Einstein equations coupled to a scalar field with a full Einstein equations coupled to a scalar field with a potential that can exhibit an ekpyrotic equation of state, potential that can exhibit an ekpyrotic equation of state, giving an example of a scenario where the ekpyrotic giving an example of a scenario where the ekpyrotic mechanism is robust even beginning from initial conditions mechanism is robust even beginning from initial conditions that are far from FRWthat are far from FRW

FormalismFormalism We solve the Einstein field equations We solve the Einstein field equations

where the stress-energy tensor is sourced by a scalar field with a where the stress-energy tensor is sourced by a scalar field with a potential of the formpotential of the form

where where VV00 and and kk are (positive) constants. Such potentials are are (positive) constants. Such potentials are common in compactified Brane-models of ekpyrosis, though there common in compactified Brane-models of ekpyrosis, though there at least two moduli fields are typically present: one representing at least two moduli fields are typically present: one representing the distance between the branes, the other the volume of the bulk the distance between the branes, the other the volume of the bulk spacetime spacetime

We expand the equations using the We expand the equations using the orthonormal-frame formalism orthonormal-frame formalism with Hubble-normalized variables with Hubble-normalized variables (Uggla et al, 2003) (Uggla et al, 2003) the metric is defined in terms of a set of four linearly independent 1-the metric is defined in terms of a set of four linearly independent 1-

forms forms a a , which are dual to an orthonormal “tetrad” , which are dual to an orthonormal “tetrad” eeaa, with , with ee00 being being timelike and the 3 timelike and the 3 ee spacelike: spacelike:

πTG 8

keVV 0

]1,1,1,1[diag,2 abab

bads ωω

Formalism - geometryFormalism - geometry Choosing coordinates where there is no vorticity in the time-Choosing coordinates where there is no vorticity in the time-

like vector field like vector field ee00 , and the spatial frame , and the spatial frame eeaa is non-rotating is non-rotating with no shiftwith no shift

we can decompose the commutators of the tetrad as we can decompose the commutators of the tetrad as

where where is the lapse; is the lapse; dudu/dt/dt is the acceleration, is the acceleration, HH the (Hubble) the (Hubble) expansion rate, and expansion rate, and the shear of the time-like congruence; the shear of the time-like congruence; and and nn and and aa contain information about the spatial metric. contain information about the spatial metric.

Hubble normalized (scale invariant) gravitational variables Hubble normalized (scale invariant) gravitational variables are defined byare defined by

HNnaeNAE ii /,,,,,, 1,

1,

eee

eeee

na

Hu

][

00

2],[

],[

ii

t eN ee ,1

0

Formalism - matterFormalism - matter Scale invariant matter quantities are define viaScale invariant matter quantities are define via

What will eventually be useful to characterize the region of What will eventually be useful to characterize the region of the universe that becomes smooth and matter dominated is the universe that becomes smooth and matter dominated is the effective equation of state parameter the effective equation of state parameter ww, defined as the , defined as the ratio of pressure to energy densityratio of pressure to energy density

VSSWVSSWPw

212

21

212

21

2

1

/ HVV

ES

W

ii

t

Formalism – evolution Formalism – evolution equationsequations

We will foliate spacetime such that each t=constant slice is We will foliate spacetime such that each t=constant slice is one of constant mean curvature, which is equivalent to the one of constant mean curvature, which is equivalent to the condition condition

where the big crunch is approached as where the big crunch is approached as t t - -. This condition . This condition results in an elliptic equation for the lapse functionresults in an elliptic equation for the lapse function

For the remaining geometric variables, the Einstein For the remaining geometric variables, the Einstein equations give a set of hyperbolic evolution equationsequations give a set of hyperbolic evolution equations

The Klein-Gordon equation gives hyperbolic evolution The Klein-Gordon equation gives hyperbolic evolution equations for the matter variablesequations for the matter variables

teH 31

.,......,...,..,, ,

NAEt

i

....,...,..,, WSt

Formalism – constraint Formalism – constraint equationsequations

In addition, the Einstein equations, Jacobi identities for the In addition, the Einstein equations, Jacobi identities for the commutators, and the introduction of auxiliary variables commutators, and the introduction of auxiliary variables gives several (mostly algebraic) constraint equations gives several (mostly algebraic) constraint equations amongst the variablesamongst the variables we only solve the constraints at the initial time, then use the we only solve the constraints at the initial time, then use the

evolution equations (plus elliptic slicing condition for the lapse) evolution equations (plus elliptic slicing condition for the lapse) to update the solution in time, a so-called to update the solution in time, a so-called free evolutionfree evolution

the structure of the equations guarantee that, to within truncation the structure of the equations guarantee that, to within truncation error, free evolution preserves the constraintserror, free evolution preserves the constraints

We will use the York procedure to provide self-consistent We will use the York procedure to provide self-consistent initial datainitial data separate the free from constrained degrees of freedom in the separate the free from constrained degrees of freedom in the

initial geometry via a conformal decomposition of the metric, initial geometry via a conformal decomposition of the metric, extrinsic curvature and matter variablesextrinsic curvature and matter variables

York’s method for solving the York’s method for solving the constraintsconstraints

Specifically we provide:Specifically we provide:

a conformally rescaled spatial metric a conformally rescaled spatial metric ijij

the scalar field the scalar field and its conformally rescaled velocity and its conformally rescaled velocity QQ

and the divergence-free and the divergence-free XXijij part of the conformally rescaled shear part of the conformally rescaled shear ZZijij(which symmetric and trace-free) (which symmetric and trace-free)

wherewhere

WQ 6

ijij h4

ijij EZ 6

kiji

iji

ijijij

QYX

YXZ

,0

Formalism and Initial DataFormalism and Initial Data To recap so far, the formalism we have described is general, though To recap so far, the formalism we have described is general, though

we have made several gauge choices to adapt the equations to the we have made several gauge choices to adapt the equations to the problem at handproblem at hand the temporal leg of the tetrad is vorticity free, and the spatial legs are the temporal leg of the tetrad is vorticity free, and the spatial legs are

Fermi propagated along itFermi propagated along it

we assume that a CMC foliation existswe assume that a CMC foliation exists

Due to limited computational resources we will now restrict to Due to limited computational resources we will now restrict to spacetimes with deviations in homogeneity in one spatial direction spacetimes with deviations in homogeneity in one spatial direction only (only (xx); thus we have two spatial Killing vectors. ); thus we have two spatial Killing vectors. This may seem like a serious restriction, howeverThis may seem like a serious restriction, however

as we will see, at late times even in regions where the spacetime is not as we will see, at late times even in regions where the spacetime is not homogeneous or isotropic, even the homogeneous or isotropic, even the xx-gradients of fields to not play any role in -gradients of fields to not play any role in the dynamics, expect at isolated spike pointsthe dynamics, expect at isolated spike points

in a study of a similar vacuum cosmology without any symmetries (Garfinkle in a study of a similar vacuum cosmology without any symmetries (Garfinkle 2004), the same behavior was found as with the 2-Killing field case there (except 2004), the same behavior was found as with the 2-Killing field case there (except possibly at isolated spike regions, which could not be resolved in that simulation)possibly at isolated spike regions, which could not be resolved in that simulation)

without loss of generality choose x to be periodic: without loss of generality choose x to be periodic: xx[0..2[0..2]]

Initial data – solution Initial data – solution procedureprocedure

A: at t=0, choose A: at t=0, choose Q, Q, ijij and and XXijij to be to be

where where aa11,a,a22,b,b11,b,b22,f,f11,f,f22,m,m11,m,m22,d,d11,d,d22,, are constants. In addition, recall that we are constants. In addition, recall that we have the constants have the constants kk and and VV00 in the potential in the potential V=-V=-VV00ee-k-kas free as free parameters parameters we believe this is a sufficiently general class of initial conditions to capture we believe this is a sufficiently general class of initial conditions to capture

generic behavior in these cosmologies, and is (modulo the scalar field) generic behavior in these cosmologies, and is (modulo the scalar field) similar to that used in Garfinkle (2004) that gave the same qualitative similar to that used in Garfinkle (2004) that gave the same qualitative conclusions in the 3D vs. 1D simulations away from spike points.conclusions in the 3D vs. 1D simulations away from spike points.

2112

211

2

222

11

)cos()cos(0)cos()cos(

0coscos1

bbxaxaxabxa

bX

dxmfdxmQ

ij

Hf

ijij

Initial data – solution Initial data – solution procedureprocedure

B: solve the divergence condition for B: solve the divergence condition for YYij ij (which here reduces to a (which here reduces to a simple set of algebraic equations) and reconstruct simple set of algebraic equations) and reconstruct ZZijij

C: solve the Hamiltonian constraint for the conformal factor C: solve the Hamiltonian constraint for the conformal factor

notenote : : allall our freely specifiable functions couple in here. our freely specifiable functions couple in here. is is notnot a a solution in general and thus we will solution in general and thus we will notnot have a flat physical metric at have a flat physical metric at t=0; all matter and geometric free data will contribute to the initial t=0; all matter and geometric free data will contribute to the initial curvature of the spacectimecurvature of the spacectime

D: now that we have the conformal factor, we can reconstruct all the D: now that we have the conformal factor, we can reconstruct all the initial physical geometric and matters variables, except the lapse initial physical geometric and matters variables, except the lapse ..

E: Solve the CMC slicing condition to arrive at the initial profile for E: Solve the CMC slicing condition to arrive at the initial profile for ..

kiji

iji

ijijij

QYX

YXZ

,0

722812

815

412

432 HZZQVH ij

ij

BriefBrief overview of numerical overview of numerical methodmethod

we solve all the differential equations using second order accurate we solve all the differential equations using second order accurate finite difference techniques with Berger and Oliger style adaptive finite difference techniques with Berger and Oliger style adaptive mesh refinement (AMR), as provided by the PAMR/AMRD packagemesh refinement (AMR), as provided by the PAMR/AMRD package PAMR/AMRD can be downloaded from PAMR/AMRD can be downloaded from

ftp://laplace.physics.ubc.ca/pub/pamr/ftp://laplace.physics.ubc.ca/pub/pamr/

the hyperbolic equations are integrated in time using an iterated the hyperbolic equations are integrated in time using an iterated Crank-Nicholson-like schemeCrank-Nicholson-like scheme

the elliptic equations are solved using a full approximation storage the elliptic equations are solved using a full approximation storage (FAS) multigrid algorithm(FAS) multigrid algorithm

surprisingly (though consistent with the BKL conjecture of locality surprisingly (though consistent with the BKL conjecture of locality approaching the singularity) the numerical evolution is stable even approaching the singularity) the numerical evolution is stable even with a spatial refinement ratio of 2 and time-sub-cycling turned off with a spatial refinement ratio of 2 and time-sub-cycling turned off (I.e., temporal refinement ratio of 1), and we have run simulations (I.e., temporal refinement ratio of 1), and we have run simulations where the CFL factor reaches 10where the CFL factor reaches 106 6 on the finest level in a simulationon the finest level in a simulation

ResultsResults We have a rather large parameter space of initial conditions to explore, We have a rather large parameter space of initial conditions to explore,

choosing the initial shear (choosing the initial shear (aa11,a,a22,b,b11,b,b22,,)), , scalar field (scalar field (ff11,f,f22,m,m11,m,m22,d,d11,d,d22) and ) and potential (potential (kk00,,VV00)) only focused on the subset of parameters that initially have non-negligible only focused on the subset of parameters that initially have non-negligible

contributions to the energy from both the gravity and matter sectors. contributions to the energy from both the gravity and matter sectors. Specifically, in Hubble normalized variables the Hamiltonian constraint takes Specifically, in Hubble normalized variables the Hamiltonian constraint takes the following formthe following form

where we identify where we identify m m as the matter contribution, as the matter contribution, s s the shear contribution, and the shear contribution, and k k the rest of the curvature contribution.the rest of the curvature contribution.

i.e., we choose parameters such that all 3 contributions i.e., we choose parameters such that all 3 contributions m m ,, s s ,, k k are non-are non-negligible in some region of the universe to begin withnegligible in some region of the universe to begin with

1 skm

AEAANNN

VSSW

ii

k

s

m

322

121

61

61

31

612

61

)(

ResultsResults Even with this restriction on parameters, we have certainly not Even with this restriction on parameters, we have certainly not

been able to explore parameter space exhaustively, but from the been able to explore parameter space exhaustively, but from the subset that we have studied we can draw the following subset that we have studied we can draw the following conclusions (will quantify some of these statements later), conclusions (will quantify some of these statements later), for a potential that is sufficiently steep some volume of the universe for a potential that is sufficiently steep some volume of the universe

will become homogeneous, isotropic, and matter dominatedwill become homogeneous, isotropic, and matter dominated in this region the scalar field behaves like a fluid with in this region the scalar field behaves like a fluid with w>>1w>>1

the rest of the universe (or the entire universe if the potential is the rest of the universe (or the entire universe if the potential is shallow) does not smooth out, with both the matter shallow) does not smooth out, with both the matter m m and shear and shear s s components being non-negligiblecomponents being non-negligible

here, the scalar field behaves like a fluid with here, the scalar field behaves like a fluid with w=1w=1

early behavior is similar to vacuum chaotic mixmaster, where each point in early behavior is similar to vacuum chaotic mixmaster, where each point in spacetime behaves similar to a Kasner solution for a while, then makes a spacetime behaves similar to a Kasner solution for a while, then makes a quick transition to a different Kasner-like solution.quick transition to a different Kasner-like solution.

unlike mixmaster, there are only a finite number of transitionsunlike mixmaster, there are only a finite number of transitions

isolated “spikes” also form, and here are the only places where isolated “spikes” also form, and here are the only places where k k can be can be non-negligible at late timesnon-negligible at late times

ResultsResults However, when a smooth matter dominated region However, when a smooth matter dominated region

does form, it very quickly grows to dominate the does form, it very quickly grows to dominate the volume of the universevolume of the universe

Thus, beginning even from highly in-homogeneous, Thus, beginning even from highly in-homogeneous, anistropic initial conditions that are not close to an anistropic initial conditions that are not close to an FRW universe, an open set of initial conditions will FRW universe, an open set of initial conditions will evolve to FRW, modulo isolated pockets of evolve to FRW, modulo isolated pockets of anisotropy that shrink to zero volume exponentially anisotropy that shrink to zero volume exponentially fast in the approach to singularityfast in the approach to singularity it is for these reasons that we call the ekpyrotic it is for these reasons that we call the ekpyrotic

smoothing mechanism smoothing mechanism robustrobust

Results – Example 1Results – Example 1 choose choose Q, Q, ij ij ,, XXijij and the potential to be and the potential to be

101.0

65.1)cos(70.0)cos(10.00)cos(10.080.1)cos(70.001.0

001.015.012cos15.07.1cos/2

eV

xxxxX

xxHQ

ij

ijij

Example 1: Example 1: at early timesat early times

Note: “t” is –tNote: “t” is –t

Yellow --- Yellow --- mm Blue --- Blue --- ss Pink --- Pink --- kk

00

11

Example 1: zoom-in of Example 1: zoom-in of at late at late times change to DV one??times change to DV one??

00

11

Note that spikes are Note that spikes are notnot being smoothed being smoothed

out – that they out – that they disappear after some disappear after some time is an artifact of time is an artifact of having converted the having converted the

data to a lo-res data to a lo-res uniform mesh for uniform mesh for

visualization purposesvisualization purposes

Note: “t” is –tNote: “t” is –t

Yellow --- Yellow --- mm Blue --- Blue --- ss Pink --- Pink --- kk

Example 1: effective equation Example 1: effective equation of state parameter of state parameter ww

Note: “t” is –tNote: “t” is –t

Example 1: state space orbitsExample 1: state space orbits Each frame of the Each frame of the

animation showsanimation shows

--= (= (1111--2222)/2/)/2/3 3 as a function ofas a function of++=1/2 (=1/2 (1111++2222))along an along an x=constantx=constant wordline, wordline, scanning from scanning from x=0 to x=2x=0 to x=2. .

A point on the circle is A point on the circle is Kasner-like (unstable), Kasner-like (unstable), points within an inner circle points within an inner circle of radius 1/of radius 1/3 (not shown) 3 (not shown) are stable Bianchi Type 1 are stable Bianchi Type 1 scalar field spacetimes, scalar field spacetimes, with the center a special with the center a special case of flat FRW.case of flat FRW.

A trajectory flowing to the A trajectory flowing to the center thus represents center thus represents evolution to a locally evolution to a locally smooth, isotropic geometrysmooth, isotropic geometry

Volume of smooth vs. non-smooth Volume of smooth vs. non-smooth regionsregions

In this example in certainly does not look like the smooth region In this example in certainly does not look like the smooth region is growing exponentially fast relative to the non-smooth region. is growing exponentially fast relative to the non-smooth region. However, the animations show However, the animations show coordinate volumecoordinate volume … we need to … we need to look at the look at the proper volumeproper volume

An easy way to get this information here is as follows. The proper An easy way to get this information here is as follows. The proper volume element isvolume element is

where where hh is the determinant of the spatial metric is the determinant of the spatial metric

For CMC slicing, the following holdsFor CMC slicing, the following holds

Thus, if Thus, if tends to a positive constant (as we will see it does), the tends to a positive constant (as we will see it does), the volume element along any world line shrinks as (recalling volume element along any world line shrinks as (recalling t t - -))

hS

3ln St

teS 3

Example 1: Example 1: Thus, a relatively large value of Thus, a relatively large value of of denotes a rapidly of denotes a rapidly

shrinking proper volumeshrinking proper volume

Note: “t” is –tNote: “t” is –t

Example 2Example 2 To show an example of a (temporary) spike, take similar To show an example of a (temporary) spike, take similar

initial conditions for the geometry as in example 1, but initial conditions for the geometry as in example 1, but now zero initial kinetic energy for the scalar field, and now zero initial kinetic energy for the scalar field, and we shift the domain by a small amount for visualization we shift the domain by a small amount for visualization purposes:purposes:

5

55

55

1.0

65.1)10cos(70.0)10cos(10.00)10cos(10.080.1)10cos(70.001.0

001.015.000

eV

xxxxX

Q

ij

ijij

Example 2: Example 2: zoom-in to spike forming at left edge of universezoom-in to spike forming at left edge of universe

Note: “t” is –tNote: “t” is –t Yellow --- Yellow --- mm Blue --- Blue --- ss Pink --- Pink --- kk

00

11

Analytic descriptionAnalytic description we can better understand the nature of the solution in the we can better understand the nature of the solution in the

limit approaching the big crunch if we assume spatial limit approaching the big crunch if we assume spatial derivative terms in the equations become negligiblederivative terms in the equations become negligible

dropping all terms from the equations involving spatial dropping all terms from the equations involving spatial derivatives, and variables that are defined as spatial derivatives, and variables that are defined as spatial derivatives, one can solve the equations exactly in the two derivatives, one can solve the equations exactly in the two regimes by further assumingregimes by further assuming in the smooth, matter dominated regime the Hubble in the smooth, matter dominated regime the Hubble

normalized potential normalized potential V/HV/H22 remains non-zero and finite remains non-zero and finite

in the anisotropic region in the anisotropic region V/HV/H22 is negligible is negligible

after-words we can compare to the numerical results to after-words we can compare to the numerical results to see if these assumptions were justifiedsee if these assumptions were justified

Analytic descriptionAnalytic description–– results & comparison to results & comparison to

numericsnumerics

smooth regionsmooth region non-smooth region non-smooth region

13lim

2lim

23lim

lim

,lim

2

2

2

kw

k

kV

kW

tx

t

t

t

t

t

1lim

31lim

0lim

)(),(lim

,lim

0

w

V

xWtxW

tx

t

t

t

t

t

Note: “t” is –tNote: “t” is –t

WW

V/HV/H22

ww

Analytic description - volumeAnalytic description - volume Returning to the volume question, using this asymptotic behavior we Returning to the volume question, using this asymptotic behavior we

have in the matter dominated smooth regionhave in the matter dominated smooth region

while in the anisotropic region while in the anisotropic region

Thus the ratio of smooth to non-smooth volumes tends toThus the ratio of smooth to non-smooth volumes tends to

assuming that the coordinate volumes of each region change assuming that the coordinate volumes of each region change negligibly in the limit, as suggested by the simulationsnegligibly in the limit, as suggested by the simulations

Thus, if Thus, if k>k>6 6 , and there is a region where the ekpyrotic mechanism , and there is a region where the ekpyrotic mechanism begins, it will eventually dominate the volume of the universe at late begins, it will eventually dominate the volume of the universe at late timestimes

ta eS

2/6 ktm eS

2/61 kt

a

m edxS

dxSR

Conclusions – future workConclusions – future work Demonstrated that the ekpyrotic mechanism works very well in preparing Demonstrated that the ekpyrotic mechanism works very well in preparing

a pre-big bang universe that is sufficiently smooth to conceivably be a pre-big bang universe that is sufficiently smooth to conceivably be consistent with present day observationsconsistent with present day observations obviates the need for a cyclic universe, as one of the original motivations in obviates the need for a cyclic universe, as one of the original motivations in

this context was to have a preceding phase of dark-energy dominated this context was to have a preceding phase of dark-energy dominated expansion to sufficiently smooth the universe prior to the next contracting expansion to sufficiently smooth the universe prior to the next contracting phase, to suppress chaotic mixmaster behaviorphase, to suppress chaotic mixmaster behavior

Even as a simple toy model, this one is not completeEven as a simple toy model, this one is not complete spacetime will evolve to a singularity, i.e., there is no bounce to a big-bangspacetime will evolve to a singularity, i.e., there is no bounce to a big-bang

if we want to model the bounce as 4D GR + an effective field theory, a new if we want to model the bounce as 4D GR + an effective field theory, a new matter field that violates the null energy condition needs to be addedmatter field that violates the null energy condition needs to be added

some suggestions on how to do this in the literature (Buchbinder et al 2007, some suggestions on how to do this in the literature (Buchbinder et al 2007, Creminelli & Senatore 2008)Creminelli & Senatore 2008)

if not, we need a quantum theory of gravity that resolves the big bang if not, we need a quantum theory of gravity that resolves the big bang

singularity, such as LQC (Bojowald 2001), and provides a mapping from the singularity, such as LQC (Bojowald 2001), and provides a mapping from the preceding contracting phase to the subsequent expansionpreceding contracting phase to the subsequent expansion