Transcript Document 7134130

Why space-time behaves
homogeneously near the big
bang
Frans Pretorius
Princeton University
work with D. Garfinkle, W. Lim and P. Steinhardt
arXiv:0808.0542 [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:0808.0542 [hep-th]
Loop Qauntum Cosmology Workshop
Institute for Gravitation and the Cosmos
Penn State, 23 October, 2008
Overview

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



two contemporary, in-principle answers to the these problems

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 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
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

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
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 105, 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



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


energy densities approach the Planck scale, beyond which we cannot
reasonably trust the theories
to have a universe that is as flat as observed today requires that the postPlanck universe was flat to within ~ 1 part in 1060 … 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.
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

inflation (Guth 1981, Linde 1982, Albrecht & Steinhardt 1982)


ekpyrotic or cyclic models (Khoury et al. 2001, Steinhardt & Turok 2002)



a period of rapid expansion following 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
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/r=-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:
0
0
0
2



r
r
a
8

r
k
s


 m3  r4  3(1w w)   2  6
H2   
3 a
a
a
a
a
 a
2
where rm0, rr0, rw0 are the energy densities in a pressureless dust, radiation
and w-fluid respectively at some initial time, and k represents spatial curvature
and s 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


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

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)]
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
Formalism

We solve the Einstein field equations
Ga  8πTa
where the stress-energy tensor is sourced by a scalar field with a potential
of the form
V    V0e  k
where V0 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)

the metric is defined in terms of a set of four linearly independent 1-forms wa ,
which are dual to an orthonormal “tetrad” ea, with e0 being timelike and the 3 ea
spacelike:
ds2   abωaωb ,
  diag [1,1,1,1]
ab
Formalism - geometry

Choosing coordinates where there is no vorticity in the time-like
vector field e0 , and the spatial frame ea is non-rotating with no shift
e 0  N 1 t ,
ea  eai  i
we can decompose the commutators of the tetrad as
[e0 , ea ]  ua e 0  Ha  s a e 
[ea , e  ]  2a[a  ]   a n e
where N is the lapse; dua/dt is the acceleration, H the (Hubble)
expansion rate, and sa the shear of the time-like congruence; and
na and aa contain information about the spatial metric.

Hubble normalized (scale invariant) gravitational variables are
defined by
E
i
a
, a , Aa , Na ,1 eai ,sa ,aa , na , N 1/ H
Formalism - matter

Scale invariant matter quantities are define via
W  1 t
Sa  Ea i  i
V V / H2

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
W 2  12 Sa S a  V
w 
r
W 2  12 Sa S a  V
P
1
2
1
2
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
1
H  e t
3
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


Eai , a , Aa , Na  ...,...,...,...
t

The Klein-Gordon equation gives hyperbolic evolution equations for
the matter variables

 , Sa ,W   ...,...,...
t
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

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
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
York’s method for solving the constraints

Specifically we provide:

a conformally rescaled spatial metric ij
 ij   4hij

the scalar field  and its conformally rescaled velocity Q
Q   6W

and the divergence-free Xij part of the conformally rescaled shear Zij
(which symmetric and trace-free)
Zij   6 Eij
where
Z ij  X ij  Yij
 i X ij  0,
 iYij  Q k
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

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
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



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)
without loss of generality choose x to be periodic: x[0..2]
Initial data – solution procedure

A: at t=0, choose , Q, ij and Xij to be
 ij   ij
Q
f1
H
cosm1 x  d1 
  f 2 cosm2 x  d 2 
 b2

X ij   x
0

x
a1 cos( x )  b1
a2 cos( x )


a2 cos( x )

 a1 cos( x )  b1  b2 
0
where a1,a2,b1,b2,f1,f2,m1,m2,d1,d2,x are constants. In addition, recall that we
have the constants k and V0 in the potential V=-V0e-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.
Initial data – solution procedure

B: solve the divergence condition for Yij (which here reduces to a simple set of
algebraic equations) and reconstruct Zij
Z ij  X ij  Yij
 i X ij  0,

 iYij  Q k
C: solve the Hamiltonian constraint for the conformal factor
2  43 H 2  14 V  5  81     81 Q 2  Z ij Zij H 2 7
2
note : all our freely specifiable functions couple in here. 1 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 .

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

PAMR/AMRD can be downloaded from
ftp://laplace.physics.ubc.ca/pub/pamr/

the hyperbolic equations are integrated in time using an iterated CrankNicholson-like scheme

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 106 on the finest level in a simulation
Results

We have a rather large parameter space of initial conditions to explore,
choosing the initial shear (a1,a2,b1,b2,x ), scalar field (f1,f2,m1,m2,d1,d2) and
potential (k0,V0)

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
Wm  Wk  W s  1
W m  16 W 2  16 S a Sa  13 V
W s  16 a a
W k  16 N a Na  121 ( N   ) 2  Aa Aa  23 Eai  i Aa
where we identify
Wm as the matter contribution,
Ws the shear contribution, and
Wk the rest of the curvature contribution.

i.e., we choose parameters such that all 3 contributions Wm , Ws , Wk are nonnegligible 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),

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
the rest of the universe (or the entire universe if the potential is shallow) does
not smooth out, with both the matter Wm and shear Ws components being nonnegligible


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.


unlike mixmaster, there are only a finite number of transitions
isolated “spikes” also form, and here are the only places where Wk can be nonnegligible at late times
Results


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

it is for these reasons that we call the ekpyrotic smoothing
mechanism robust
Results – Example 1

choose , Q, ij , Xij and the potential to be
 ij   ij
Q  2 / H cos x  1.7 
  0.15 cos2 x  1
0.01
0
  0.15



X ij   0.01 0.70 cos( x )  1.80
0.10 cos( x )

 0
0.10 cos( x )
 0.70 cos( x )  1.65 

V  0.1e 10
Example 1: W at early times



1
0
Note: “t” is –t
Yellow --- Wm
Blue --- Ws
Pink --- Wk
Example 1: zoom-in of W at late times
change to DV one??



Yellow --- Wm
Blue --- Ws
Pink --- Wk
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
0
Note: “t” is –t
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.


A point on the circle is Kasnerlike (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
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

An easy way to get this information here is as follows. The proper volume
element is
S h
where h is the determinant of the spatial metric

For CMC slicing, the following holds
 t ln S  3

Thus, if  tends to a positive constant (as we will see it does), the volume
element along any world line shrinks as (recalling t  -)
S  e 3t
Example 1: 

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:
 ij   ij
Q0
 0
0.01
0
  0.15



5
5
X ij   0.01 0.70 cos( x  10 )  1.80
0.10 cos( x  10 )

 0
0.10 cos( x  105 )
 0.70 cos( x  105 )  1.65 

V  0.1e 5
Example 2: W
zoom-in to spike forming at left edge of universe
1
0
Note: “t” is –t



Yellow --- Wm
Blue --- Ws
Pink --- Wk
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
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/H2 remains non-zero and finite

in the anisotropic region V/H2 is negligible
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
t  
t  
lim   x, t   
lim W  k
lim V  3  k 2
2
t  
lim   2 k 2
t  
lim w  k 2 3  1
t  
w
lim   x, t   
lim W ( x, t )  W0 ( x )
t  

t  
lim V  0
W
t  
lim   1 3
t  
lim w  1
t  

V/H2
Note: “t” is –t
Analytic description - volume

Returning to the volume question, using this asymptotic behavior we have
in the matter dominated smooth region
Sm  e 6 t / k
2
while in the anisotropic region
Sa  et

Thus the ratio of smooth to non-smooth volumes tends to
S dx

R
e
 S dx
m

t 16 / k 2

a
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
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


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

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


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