Why space-time behaves homogeneously near the big bang Frans Pretorius Princeton University work with D. Garfinkle, W. Lim and P. Steinhardt arXiv: [hep-th] Loop Qauntum Cosmology Workshop Institute for Gravitation and the Cosmos Penn State, 23 October, 2008

Exploring the smoothing power of the ekpyrotic mechanism Frans Pretorius Princeton University work with D. Garfinkle, W. Lim and P. Steinhardt arXiv: [hep-th] Loop Qauntum Cosmology Workshop Institute for Gravitation and the Cosmos Penn State, 23 October, 2008

Overview Background and motivation Background and motivation seek “natural” solutions to cosmological puzzles, in particular why the pre-CMB universe was in such a remarkably flat, homogenous and isotropic state seek “natural” solutions to cosmological puzzles, in particular why the pre-CMB universe was in such a remarkably flat, homogenous and isotropic state two contemporary, in-principle answers to the these problems two contemporary, in-principle answers to the these problems inflation: a period of rapid expansion of the universe after the big bang inflation: a period of rapid expansion of the universe after the big bang the ekpyrotic/cyclic mechanism: a period of slow contraction before the big bang the ekpyrotic/cyclic mechanism: a period of slow contraction before the big bang the main problem with either mechanism is lack of a compelling derivation coming from a fundamental theory the main problem with either mechanism is lack of a compelling derivation coming from a fundamental theory 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-big bang state 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-big bang state 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 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 universe consistent with all present day observations 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 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 universe consistent with all present day observations Formalism, Initial Data & Results Formalism, Initial Data & Results Conclusions and future work Conclusions and future work

Horizon and Flatness Problems Established theory – the standard model of particle physics and general relativity – (together with dark matter and dark energy), provide a consistent picture of the evolution of the universe from ~100 seconds after the “big bang” until to today Established theory – the standard model of particle physics and general relativity – (together with dark matter and dark energy), provide a consistent picture of the evolution of the universe from ~100 seconds after the “big bang” until to today This picture of the universe has been assembled following the guidance of remarkable observations over the past couple of decades that have shown that the universe was very close to flat, homogeneous and isotropic at the time of recombination This picture of the universe has been assembled following the guidance of remarkable observations over the past couple of decades that have shown that the universe was very close to flat, homogeneous and isotropic at the time of recombination Furthermore, the spectrum of the CMB is that of a thermal black body to within 1 part in 10 5, better than anything that can be produced in a lab on earth Furthermore, the spectrum of the CMB is that of a thermal black body to within 1 part in 10 5, better than anything that can be produced in a lab on earth

Horizon and Flatness Problems Extrapolating the observations back in time using known theory presents a couple of problems Extrapolating the observations back in time using known theory presents a couple of problems energy densities approach the Planck scale, beyond which we cannot reasonably trust the theories energy densities approach the Planck scale, beyond which we cannot reasonably trust the theories 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 contact, so whence came the black body spectrum? … horizon problem 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 contact, so whence came the black body spectrum? … horizon problem Envisioning an evolution forwards in time from just below the Planck scale also presents problems Envisioning an evolution forwards in time from just below the Planck scale also presents problems to have a universe that is as flat as observed today requires that the post- Planck universe was flat to within ~ 1 part in … flatness problem to have a universe that is as flat as observed today requires that the post- Planck universe was flat to within ~ 1 part in … flatness problem Without a fundamental theory of Planck scale physics there is no reasonable basis to suppose that a “solution” to these problems is that the universe just happened to begin in an “un-natural”, fine-tuned state. Without a fundamental theory of Planck scale physics there is no reasonable basis to suppose that a “solution” to these problems is that the universe just happened to begin in an “un-natural”, fine-tuned state.

Inflation & Ekpyrosis At present there are 2 well-studied solutions to the horizon, flatness (& monopole) problems that are also consistent with current observations, most notably with the near scale invariant power spectrum of the fluctuations of the CMB At present there are 2 well-studied solutions to the horizon, flatness (& monopole) problems that are also consistent with current observations, most notably with the near scale invariant power spectrum of the fluctuations 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-bang a 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-bang a period of slow contraction preceding the big-bang original inspiration based on the collision of two 4D branes in higher dimensional spacetime, though here we will take the effective field theory model original inspiration based on the collision of two 4D branes in higher dimensional spacetime, though here we will take the effective field theory model In both models, the smoothing happens below the Planck scale, and the usual assumptions made are that general relativity describes the evolution of spacetime, driven by some new kind of “exotic” matter or effective matter In both models, the smoothing happens below the Planck scale, and the usual assumptions made are that general relativity describes the evolution of spacetime, driven by some new kind of “exotic” matter or effective matter inflation: the matter has an effective equation of state parameter w= P /  =-1 inflation: the matter has an effective equation of state parameter w= P /  =-1 ekpyrosis: the matter has an effective w>>1 ekpyrosis: the matter has an effective w>>1

Why they smooth as they do Consider the Friedmann equation governing the evolution of the scale factor a(t) of the universe for a homogeneous, near FRW spacetime: where  m 0,  r 0,  w 0 are the energy densities in a pressureless dust, radiation and w -fluid respectively at some initial time, and k represents spatial curvature and  the anisotropy Consider the Friedmann equation governing the evolution of the scale factor a(t) of the universe for a homogeneous, near FRW spacetime: where  m 0,  r 0,  w 0 are the energy densities in a pressureless dust, radiation and w -fluid respectively at some initial time, and k represents spatial curvature and  the anisotropy Inflation ( w=-1 ): a(t) grows with time, hence the component with the smallest power of a in the denominator dominates the late-time evolution of the universe … here, the w -fluid, driving inflation Inflation ( w=-1 ): a(t) grows with time, hence the component with the smallest power of a in the denominator dominates the late-time evolution of the universe … here, the w -fluid, driving inflation without the w -fluid it would be curvature (the flatness problem) without the w -fluid it would be curvature (the flatness problem) Ekpyrosis ( w>>1 ): a(t) shrinks with time, hence the component with the largest power of a in the denominator dominates the approach to the big-crunch … again, by construction, the w -fluid, but now this drives ekpyrosis Ekpyrosis ( w>>1 ): a(t) shrinks with time, hence the component with the largest power of a in the denominator dominates the approach to the big-crunch … again, by construction, the w -fluid, but now this drives ekpyrosis without the w -fluid shear would dominate, resulting in chaotic mixmaster behavior without the w -fluid shear would dominate, resulting in chaotic mixmaster behavior

How robust is the smoothing mechanism? For inflation, several results have shown that the smoothing mechanism is robust in that even beginning from large deviations from an FRW universe, if a w=-1 matter component is present the spacetimes generically evolve to de Sitter [Wald (1983), Jensen & Stein-Schabes (1986), …, Goldwirth (1991)] For inflation, several results have shown that the smoothing mechanism is robust in that even beginning from large deviations from an FRW universe, if a w=-1 matter component is present the spacetimes generically evolve to de Sitter [Wald (1983), Jensen & Stein-Schabes (1986), …, Goldwirth (1991)] Until now, the only comparable results for ekpyrosis showed that the mechanism worked for linear perturbations about a contracting FRW spacetime [Erikson et al (2004)] Until now, the only comparable results for ekpyrosis showed that the mechanism worked for linear perturbations about a contracting FRW spacetime [Erikson et al (2004)] Here results are presented from numerical solution of the full Einstein equations coupled to a scalar field with a potential that can exhibit an ekpyrotic equation of state, giving an example of a scenario where the ekpyrotic mechanism is robust even beginning from initial conditions that are far from FRW Here results are presented from numerical solution of the full Einstein equations coupled to a scalar field with a potential that can exhibit an ekpyrotic equation of state, giving an example of a scenario where the ekpyrotic mechanism is robust even beginning from initial conditions that are far from FRW

Formalism We solve the Einstein field equations where the stress-energy tensor is sourced by a scalar field with a potential of the form where V 0 and k are (positive) constants. Such potentials are common in compactified Brane-models of ekpyrosis, though there at least two moduli fields are typically present: one representing the distance between the branes, the other the volume of the bulk spacetime We solve the Einstein field equations where the stress-energy tensor is sourced by a scalar field with a potential of the form where V 0 and k are (positive) constants. Such potentials are common in compactified Brane-models of ekpyrosis, though there at least two moduli fields are typically present: one representing the distance between the branes, the other the volume of the bulk spacetime We expand the equations using the orthonormal-frame formalism with Hubble-normalized variables (Uggla et al, 2003) We expand the equations using the orthonormal-frame formalism with Hubble-normalized variables (Uggla et al, 2003) the metric is defined in terms of a set of four linearly independent 1-forms  a, which are dual to an orthonormal “tetrad” e a, with e 0 being timelike and the 3 e  spacelike: the metric is defined in terms of a set of four linearly independent 1-forms  a, which are dual to an orthonormal “tetrad” e a, with e 0 being timelike and the 3 e  spacelike:

Formalism - geometry Choosing coordinates where there is no vorticity in the time-like vector field e 0, and the spatial frame e a is non-rotating with no shift we can decompose the commutators of the tetrad as where  is the lapse; du  /dt is the acceleration, H the (Hubble) expansion rate, and   the shear of the time-like congruence; and n  and a  contain information about the spatial metric. Choosing coordinates where there is no vorticity in the time-like vector field e 0, and the spatial frame e a is non-rotating with no shift we can decompose the commutators of the tetrad as where  is the lapse; du  /dt is the acceleration, H the (Hubble) expansion rate, and   the shear of the time-like congruence; and n  and a  contain information about the spatial metric. Hubble normalized (scale invariant) gravitational variables are defined by Hubble normalized (scale invariant) gravitational variables are defined by

Formalism - matter Scale invariant matter quantities are define via Scale invariant matter quantities are define via What will eventually be useful to characterize the region of the universe that becomes smooth and matter dominated is the effective equation of state parameter w, defined as the ratio of pressure to energy density What will eventually be useful to characterize the region of the universe that becomes smooth and matter dominated is the effective equation of state parameter w, defined as the ratio of pressure to energy density

Formalism – evolution equations We will foliate spacetime such that each t=constant slice is one of constant mean curvature, which is equivalent to the condition where the big crunch is approached as t  - . This condition results in an elliptic equation for the lapse function We will foliate spacetime such that each t=constant slice is one of constant mean curvature, which is equivalent to the condition where the big crunch is approached as t  - . This condition results in an elliptic equation for the lapse function For the remaining geometric variables, the Einstein equations give a set of hyperbolic evolution equations For the remaining geometric variables, the Einstein equations give a set of hyperbolic evolution equations The Klein-Gordon equation gives hyperbolic evolution equations for the matter variables The Klein-Gordon equation gives hyperbolic evolution equations for the matter variables

Formalism – constraint equations In addition, the Einstein equations, Jacobi identities for the commutators, and the introduction of auxiliary variables gives several (mostly algebraic) constraint equations amongst the variables In addition, the Einstein equations, Jacobi identities for the commutators, and the introduction of auxiliary variables gives several (mostly algebraic) constraint equations amongst the variables we only solve the constraints at the initial time, then use the evolution equations (plus elliptic slicing condition for the lapse) to update the solution in time, a so-called free evolution we only solve the constraints at the initial time, then use the evolution equations (plus elliptic slicing condition for the lapse) to update the solution in time, a so-called free evolution the structure of the equations guarantee that, to within truncation error, free evolution preserves the constraints the structure of the equations guarantee that, to within truncation error, free evolution preserves the constraints We will use the York procedure to provide self-consistent initial data We will use the York procedure to provide self-consistent initial data separate the free from constrained degrees of freedom in the initial geometry via a conformal decomposition of the metric, extrinsic curvature and matter variables separate the free from constrained degrees of freedom in the initial geometry via a conformal decomposition of the metric, extrinsic curvature and matter variables

York’s method for solving the constraints Specifically we provide: Specifically we provide: a conformally rescaled spatial metric  ij a conformally rescaled spatial metric  ij the scalar field  and its conformally rescaled velocity Q the scalar field  and its conformally rescaled velocity Q and the divergence-free X ij part of the conformally rescaled shear Z ij (which symmetric and trace-free) where and the divergence-free X ij part of the conformally rescaled shear Z ij (which symmetric and trace-free) where

Formalism and Initial Data To recap so far, the formalism we have described is general, though we have made several gauge choices to adapt the equations to the problem at hand To recap so far, the formalism we have described is general, though we have made several gauge choices to adapt the equations to the problem at hand the temporal leg of the tetrad is vorticity free, and the spatial legs are Fermi propagated along it the temporal leg of the tetrad is vorticity free, and the spatial legs are Fermi propagated along it we assume that a CMC foliation exists we assume that a CMC foliation exists Due to limited computational resources we will now restrict to spacetimes with deviations in homogeneity in one spatial direction only ( x ); thus we have two spatial Killing vectors. Due to limited computational resources we will now restrict to spacetimes with deviations in homogeneity in one spatial direction only ( x ); thus we have two spatial Killing vectors. This may seem like a serious restriction, however This may seem like a serious restriction, however as we will see, at late times even in regions where the spacetime is not homogeneous or isotropic, even the x -gradients of fields to not play any role in the dynamics, expect at isolated spike points as we will see, at late times even in regions where the spacetime is not homogeneous or isotropic, even the x -gradients of fields to not play any role in the dynamics, expect at isolated spike points 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 possibly at isolated spike regions, which could not be resolved in that simulation) 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 possibly at isolated spike regions, which could not be resolved in that simulation) without loss of generality choose x to be periodic: x  [0..2  ] without loss of generality choose x to be periodic: x  [0..2  ]

Initial data – solution procedure A: at t=0, choose  Q,  ij and X ij to be where a 1,a 2,b 1,b 2,f 1,f 2,m 1,m 2,d 1,d 2,  are constants. In addition, recall that we have the constants k and V 0 in the potential V=- V 0 e -k  as free parameters A: at t=0, choose  Q,  ij and X ij to be where a 1,a 2,b 1,b 2,f 1,f 2,m 1,m 2,d 1,d 2,  are constants. In addition, recall that we have the constants k and V 0 in the potential V=- V 0 e -k  as free parameters we believe this is a sufficiently general class of initial conditions to capture generic behavior in these cosmologies, and is (modulo the scalar field) similar to that used in Garfinkle (2004) that gave the same qualitative conclusions in the 3D vs. 1D simulations away from spike points. we believe this is a sufficiently general class of initial conditions to capture generic behavior in these cosmologies, and is (modulo the scalar field) similar to that used in Garfinkle (2004) that gave the same qualitative conclusions in the 3D vs. 1D simulations away from spike points.

Initial data – solution procedure B: solve the divergence condition for Y ij (which here reduces to a simple set of algebraic equations) and reconstruct Z ij B: solve the divergence condition for Y ij (which here reduces to a simple set of algebraic equations) and reconstruct Z ij C: solve the Hamiltonian constraint for the conformal factor C: solve the Hamiltonian constraint for the conformal factor note : all our freely specifiable functions couple in here.  is not a solution in general and thus we will not have a flat physical metric at t=0; all matter and geometric free data will contribute to the initial curvature of the spacectime D: now that we have the conformal factor, we can reconstruct all the initial physical geometric and matters variables, except the lapse . D: now that we have the conformal factor, we can reconstruct all the 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 .

Brief overview of numerical method we solve all the differential equations using second order accurate finite difference techniques with Berger and Oliger style adaptive mesh refinement (AMR), as provided by the PAMR/AMRD package we solve all the differential equations using second order accurate finite difference techniques with Berger and Oliger style adaptive mesh refinement (AMR), as provided by the PAMR/AMRD package PAMR/AMRD can be downloaded from ftp://laplace.physics.ubc.ca/pub/pamr/ PAMR/AMRD can be downloaded from ftp://laplace.physics.ubc.ca/pub/pamr/ the hyperbolic equations are integrated in time using an iterated Crank- Nicholson-like scheme the hyperbolic equations are integrated in time using an iterated Crank- Nicholson-like scheme the elliptic equations are solved using a full approximation storage (FAS) multigrid algorithm the elliptic equations are solved using a full approximation storage (FAS) multigrid algorithm surprisingly (though consistent with the BKL conjecture of locality approaching the singularity) the numerical evolution is stable even 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 where the CFL factor reaches 10 6 on the finest level in a simulation surprisingly (though consistent with the BKL conjecture of locality approaching the singularity) the numerical evolution is stable even 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 where the CFL factor reaches 10 6 on the finest level in a simulation

Results We have a rather large parameter space of initial conditions to explore, choosing the initial shear ( a 1,a 2,b 1,b 2,  ), scalar field ( f 1,f 2,m 1,m 2,d 1,d 2 ) and potential ( k 0, V 0 ) We have a rather large parameter space of initial conditions to explore, choosing the initial shear ( a 1,a 2,b 1,b 2,  ), scalar field ( f 1,f 2,m 1,m 2,d 1,d 2 ) and potential ( k 0, V 0 ) only focused on the subset of parameters that initially have non-negligible contributions to the energy from both the gravity and matter sectors. Specifically, in Hubble normalized variables the Hamiltonian constraint takes the following form where we identify  m as the matter contribution,  s the shear contribution, and  k the rest of the curvature contribution. only focused on the subset of parameters that initially have non-negligible contributions to the energy from both the gravity and matter sectors. Specifically, in Hubble normalized variables the Hamiltonian constraint takes the following form where we identify  m as the matter contribution,  s the shear contribution, and  k the rest of the curvature contribution. i.e., we choose parameters such that all 3 contributions  m,  s,  k are non- negligible in some region of the universe to begin with i.e., we choose parameters such that all 3 contributions  m,  s,  k are non- negligible in some region of the universe to begin with

Results Even with this restriction on parameters, we have certainly not been able to explore parameter space exhaustively, but from the subset that we have studied we can draw the following conclusions (will quantify some of these statements later), Even with this restriction on parameters, we have certainly not been able to explore parameter space exhaustively, but from the subset that we have studied we can draw the following conclusions (will quantify some of these statements later), for a potential that is sufficiently steep some volume of the universe will become homogeneous, isotropic, and matter dominated for a potential that is sufficiently steep some volume of the universe will become homogeneous, isotropic, and matter dominated in this region the scalar field behaves like a fluid with w>>1 in this region the scalar field behaves like a fluid with w>>1 the rest of the universe (or the entire universe if the potential is shallow) does not smooth out, with both the matter  m and shear  s components being non- negligible the rest of the universe (or the entire universe if the potential is shallow) does not smooth out, with both the matter  m and shear  s components being non- negligible here, the scalar field behaves like a fluid with w=1 here, the scalar field behaves like a fluid with w=1 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 quick transition to a different Kasner-like solution. 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 quick transition to a different Kasner-like solution. unlike mixmaster, there are only a finite number of transitions unlike mixmaster, there are only a finite number of transitions isolated “spikes” also form, and here are the only places where  k can be non- negligible at late times isolated “spikes” also form, and here are the only places where  k can be non- negligible at late times

Results However, when a smooth matter dominated region does form, it very quickly grows to dominate the volume of the universe However, when a smooth matter dominated region does form, it very quickly grows to dominate the volume of the universe Thus, beginning even from highly in-homogeneous, anistropic initial conditions that are not close to an FRW universe, an open set of initial conditions will evolve to FRW, modulo isolated pockets of anisotropy that shrink to zero volume exponentially fast in the approach to singularity Thus, beginning even from highly in-homogeneous, anistropic initial conditions that are not close to an FRW universe, an open set of initial conditions will evolve to FRW, modulo isolated pockets of anisotropy that shrink to zero volume exponentially fast in the approach to singularity it is for these reasons that we call the ekpyrotic smoothing mechanism robust it is for these reasons that we call the ekpyrotic smoothing mechanism robust

Results – Example 1 choose  Q,  ij, X ij and the potential to be choose  Q,  ij, X ij and the potential to be

Example 1:  at early times Note: “t” is –t Yellow ---  m Yellow ---  m Blue ---  s Blue ---  s Pink ---  k Pink ---  k 0 1

Example 1: zoom-in of  at late times change to DV one?? 0 1 Note that spikes are not being smoothed out – that they disappear after some time is an artifact of having converted the data to a lo-res uniform mesh for visualization purposes Note: “t” is –t Yellow ---  m Yellow ---  m Blue ---  s Blue ---  s Pink ---  k Pink ---  k

Example 1: effective equation of state parameter w Note: “t” is –t

Example 1: state space orbits Each frame of the animation shows  - = (  11 -  22 )/2/  3 as a function of  + =1/2 (  11 +  22 ) along an x=constant wordline, scanning from x=0 to x=2 . Each frame of the animation shows  - = (  11 -  22 )/2/  3 as a function of  + =1/2 (  11 +  22 ) along an x=constant wordline, scanning from x=0 to x=2 . A point on the circle is Kasner- like (unstable), points within an inner circle of radius 1/  3 (not shown) are stable Bianchi Type 1 scalar field spacetimes, with the center a special case of flat FRW. A point on the circle is Kasner- like (unstable), points within an inner circle of radius 1/  3 (not shown) are stable Bianchi Type 1 scalar field spacetimes, with the center a special case of flat FRW. A trajectory flowing to the center thus represents evolution to a locally smooth, isotropic geometry A trajectory flowing to the center thus represents evolution to a locally smooth, isotropic geometry

Volume of smooth vs. non-smooth regions In this example in certainly does not look like the smooth region is growing exponentially fast relative to the non-smooth region. However, the animations show coordinate volume … we need to look at the proper volume In this example in certainly does not look like the smooth region is growing exponentially fast relative to the non-smooth region. However, the animations show coordinate volume … we need to look at the proper volume An easy way to get this information here is as follows. The proper volume element is where h is the determinant of the spatial metric An easy way to get this information here is as follows. The proper volume element is where h is the determinant of the spatial metric For CMC slicing, the following holds For CMC slicing, the following holds Thus, if  tends to a positive constant (as we will see it does), the volume element along any world line shrinks as (recalling t  -  ) Thus, if  tends to a positive constant (as we will see it does), the volume element along any world line shrinks as (recalling t  -  )

Example 1:  Thus, a relatively large value of  of denotes a rapidly shrinking proper volume Thus, a relatively large value of  of denotes a rapidly shrinking proper volume Note: “t” is –t

Example 2 To show an example of a (temporary) spike, take similar initial conditions for the geometry as in example 1, but now zero initial kinetic energy for the scalar field, and we shift the domain by a small amount for visualization purposes: To show an example of a (temporary) spike, take similar initial conditions for the geometry as in example 1, but now zero initial kinetic energy for the scalar field, and we shift the domain by a small amount for visualization purposes:

Example 2:  zoom-in to spike forming at left edge of universe Note: “t” is –t Yellow ---  m Yellow ---  m Blue ---  s Blue ---  s Pink ---  k Pink ---  k 0 1

Analytic description we can better understand the nature of the solution in the limit approaching the big crunch if we assume spatial derivative terms in the equations become negligible we can better understand the nature of the solution in the limit approaching the big crunch if we assume spatial derivative terms in the equations become negligible dropping all terms from the equations involving spatial derivatives, and variables that are defined as spatial derivatives, one can solve the equations exactly in the two regimes by further assuming dropping all terms from the equations involving spatial derivatives, and variables that are defined as spatial derivatives, one can solve the equations exactly in the two regimes by further assuming in the smooth, matter dominated regime the Hubble normalized potential V/H 2 remains non-zero and finite in the smooth, matter dominated regime the Hubble normalized potential V/H 2 remains non-zero and finite in the anisotropic region V/H 2 is negligible in the anisotropic region V/H 2 is negligible after-words we can compare to the numerical results to see if these assumptions were justified after-words we can compare to the numerical results to see if these assumptions were justified

Analytic description – results & comparison to numerics smooth region non-smooth region Note: “t” is –t  W V/H 2 w 

Analytic description - volume Returning to the volume question, using this asymptotic behavior we have in the matter dominated smooth region while in the anisotropic region Returning to the volume question, using this asymptotic behavior we have in the matter dominated smooth region while in the anisotropic region Thus the ratio of smooth to non-smooth volumes tends to assuming that the coordinate volumes of each region change negligibly in the limit, as suggested by the simulations Thus the ratio of smooth to non-smooth volumes tends to assuming that the coordinate volumes of each region change negligibly in the limit, as suggested by the simulations Thus, if k>  6, and there is a region where the ekpyrotic mechanism begins, it will eventually dominate the volume of the universe at late times Thus, if k>  6, and there is a region where the ekpyrotic mechanism begins, it will eventually dominate the volume of the universe at late times

Conclusions – future work Demonstrated that the ekpyrotic mechanism works very well in preparing a pre-big bang universe that is sufficiently smooth to conceivably be consistent with present day observations Demonstrated that the ekpyrotic mechanism works very well in preparing a pre-big bang universe that is sufficiently smooth to conceivably be consistent with present day observations 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 expansion to sufficiently smooth the universe prior to the next contracting phase, to suppress chaotic mixmaster behavior 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 expansion to sufficiently smooth the universe prior to the next contracting phase, to suppress chaotic mixmaster behavior Even as a simple toy model, this one is not complete Even 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-bang spacetime 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 matter field that violates the null energy condition needs to be added 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 added some suggestions on how to do this in the literature (Buchbinder et al 2007, Creminelli & Senatore 2008) some suggestions on how to do this in the literature (Buchbinder et al 2007, Creminelli & Senatore 2008) 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 preceding contracting phase to the subsequent expansion 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 preceding contracting phase to the subsequent expansion