Dynamical Systems Methods in Early-Universe Cosmologymath.yorku.ca/~isk/Site/Welcome_files/sonad2014talknewnewest.pdf · dynamical systems methods to the Einstein ﬁeld equations

Dynamical Systems Methods in Early-Universe Cosmology

!SONAD 2014 Talk - May 15, 2014

Ikjyot Singh Kohli, Ph.D. Candidate - Dept. of Physics and Astronomy - York Universitywww.yorku.ca/kohli

http://www.yorku.ca/kohli

Introduction

Introduction

• Much of my research (with Michael Haslam) is focused on applying dynamical systems methods to the Einstein field equations to develop models of the early universe.

Introduction


• Our present-day universe is isotropic (to a very high degree) and spatially homogeneous. The universe is then described by the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric.

Introduction



• We would like to understand how this is possible given that the early universe had a very high temperature and hence was unstable.

Introduction



• We would like to understand how this is possible given that the early universe had a very high temperature and hence was unstable.

• In our work, we relax the condition of isotropy to obtain cosmological models that admit Killing vectors that describe spatial translations only, and as a result are spatially homogeneous and anisotropic in general. Such models are known as the Bianchi models.

Introduction

Introduction

• This task is non-trivial, as in General Relativity, space and time are one entity, spacetime, so it is not immediately clear how one should essentially “define” a time variable to get some idea of dynamics.

Introduction

• This task is non-trivial, as in General Relativity, space and time are one entity, spacetime, so it is not immediately clear how one should essentially “define” a time variable to get some idea of dynamics.

• I will describe in this talk the method of orthonormal frames (Ellis and MacCallum, Comm. Math. Phys, 12, 108-141,1969), which turn the Einstein field equations into a coupled system of nonlinear ordinary differential equations, and then describe the importance of numerical methods in understanding the global behaviour of the system.

Spacetime Splitting

Spacetime Splitting

• M is a manifold if for every point of M there is an open neighbourhood of that point for which there is a continuous one-to-one and onto map f to an open neighbourhood of R^n for some integer n.

Spacetime Splitting


• The local region of M then can be made to look like R^n locally.

Spacetime Splitting


• The local region of M then can be made to look like R^n locally.

• In general relativity, n = 4, for three spatial coordinates, and one time coordinate.

Depiction of a manifold

Spacetime Splitting

General Relativity is described by the Einstein field equations:

Rab � 1

2gabR + �gab = kTab

Rab : Ricci tensor

gab : Metric tensor

� : Cosmological constant

Tab : Energy-momentum tensor

These are 10 hyperbolic, nonlinear, coupled PDEs!

Spacetime Splitting

The solution of these equations is the metric tensor, typically written in the form:

ds2 = gabdxadxb

One sees though, that the EFEs are not your “typical” field equations, in the sense that the solution is a metric tensor, and not a time-varying field, as would be the case with Maxwell’s equations, Navier-Stokes, etc…

Spacetime Splitting

To get a dynamical interpretation, then, we must split our spacetime into space and time. We accomplish this by foliating M into a collection of spacelike hypersurfaces and allow a timelike vector to thread through them.

na

t1

t2

t3

t4

(Baumgarte, Shapiro, 2010)

Elliptic PDEs

Hyperbolic PDEs

Energy-Momentum Tensor

The energy-momentum tensor is given by:Tab = (µ+ p)uaub + gabp� 3⇠Hhab � 2⌘�ab,

where µ, p, and �ab denote the fluid’s energy density, pressure, and shearrespectively, while � and � denote the bulk and shear viscosity coe�cients ofthe fluid. Throughout this work, both coe�cients are taken to be nonnegativeconstants. H denotes the Hubble parameter, and hab � uaub + gab is thestandard projection tensor corresponding to the metric signature (�, +, +, +).

See: (Kohli and Haslam, Phys. Rev. D. 87, 063006 (2013) arXiv: 1207.6132) for further details.

In principle, other terms too, involving heat conduction, but these are acausal, (parabolic and elliptic PDEs do not

occur in the real, physical universe, except in the sense of spacetime symmetries).

Evolution and Constraint Equations

We will choose an orthonormal frame: {n, e�}That is, n is tangent to a hypersurface-orthogonal congruence of geodesics, and we obtain:

H = �H2 � 2

3�2 � µ

�1

6+

1

2w

�, (1)

�ab = �3H�ab + 2�uv(a �b)u�v � Sab � 2��ab, (2)

µ = 3H2 � �2 +1

2R, (3)

0 = 3�uaau � �uv

a �bunbv, (4)

The Einstein field equations:

where Sab and R are the three-dimensional spatial curvature and Ricci scalarand are defined as:

Sab = bab � 1

3buu�ab � 2�uv

(a nb)uav, (1)

R = �1

2buu � 6auau, (2)

where bab = 2nuanub�(nu

u) nab. We have also denoted by �v the angular velocityof the spatial frame.

Using the Jacobi identities, one obtains evolution equations for these vari-ables as well:

nab = �Hnab + 2�u(anb)u + 2�uv

(a nb)u�v, (1)

aa = �Haa�baab + �uv

a au�v, (2)

0 = nbaab. (3)

The contracted Bianchi identities give the evolution equation for µ as

µ = �3H (µ + p) � �ba�a

b + 2aaqa. (4)

In addition, we define a dimensionless “time” variable:dt

d⌧=

1

D

In our research, we are particularly interested in a closed universe, that is, of topology S^3. !The EFEs for such a universe take the form: (Kohli and Haslam, Phys. Rev. D. 89, 043518 (2014), arXiv: 1311.0389)

H 0 = �(1� H2)q,

⌃0+ = ⌃+H (�2 + q)� 6⌃+⌘0 � S+,

⌃0� = ⌃�H (�2 + q)� 6⌃�⌘0 � S�,

N 01 = N1

⇣Hq � 4⌃+

⌘,

N 02 = N2

⇣Hq + 2⌃+ + 2

p3⌃�

⌘,

N 03 = N3

⇣Hq + 2⌃+ � 2

p3⌃�

⌘,

⌦0 = ⌦H (�1 + 2q � 3w) + 9H2⇠0 + 12⌘0⇣⌃2

+ + ⌃2�

⌘

where q = 2⇣⌃2

+ + ⌃2�

⌘+

1

2⌦ (1 + 3w)� 9

2⇠0H

⌦+ ⌃2 + V = 1This is a constraint on the initial

conditions

Note that:V =

1

12

N2

1 + N22 + N2

3 � 2N1N2 � 2N2N3 � 2N3N1 + 3⇣N1N2N3

⌘2/3�

S+ =1

6

⇣N2 � N3

⌘2� N1

⇣2N1 � N2 � N3

⌘�,

S� =1

2p3

h⇣N3 � N2

⌘⇣N1 � N2 � N3

⌘i.

Dynamical Analysis

Dynamical Analysis

• To understand the dynamics of such a universe, we typically proceed in the following manner:

Dynamical Analysis


1. Determine whether the state space is compact

Dynamical Analysis



2. Identify the invariant sets of the system

Dynamical Analysis




3. Find all equilibrium points and analyze their local stability

Dynamical Analysis





4. Find monotone functions in the various invariant sets

Dynamical Analysis






5. Investigate bifurcations that occur as per the parameter space

Dynamical Analysis






5. Investigate bifurcations that occur as per the parameter space

6. Knowing all this information allows one to state precisely information about the asymptotic past and future evolution of the universe model under consideration.

Examples of Fixed PointsExpanding Flat FLRW

solution

Expanding Bianchi II solution

Closed Einstein Static Universe

Stability AnalysisEquilibrium Point EFE Solution Local Sink Local Source Saddle

F+Expanding k=0 FLRW

solutionYes No Yes

F-Contracting k=0 FLRW

solutionYes Yes Yes

P+(II)Expanding

Bianchi Type II Solution

No No Yes

P_(II) (new discovery!)

Contracting Bianchi Type II

SolutionNo No Yes

21

For nonlinear systems, we are bound by the Invariant Manifold theorem:

Let x = 0 be an equilibrium point of the DE x

0= f(x) on Rn

and let Es, Eu,

and Ecdenote the stable, unstable, and centre subspaces of the linearization at

0. Then there exists

• W stangent to Es

at 0,

• Wutangent to Eu

at 0,

• W ctangent to Ec

at 0.

So, in nonlinear systems, linearization techniques only tell you the orbits of the dynamical system that belong to the

stable, unstable, or centre manifolds.

So, we need global methods that will determine asymptotic stability of the various equilibrium points.

Global Methods

22

Global Methods• There are many such methods, although, the majority of the

methods used only apply to planar dynamical systems.

22



• The methods that we have found useful are:

22




• Finding Lyapunov and Chetaev functions

22





• LaSalle invariance principle

22






• Monotonicity Principle

22







• Numerical methods

22







• Numerical methods

• The first three give information on the alpha and omega limit sets of the dynamical system. 22

Numerical Experiments


• Numerical methods are key, and without them, we would be somewhat limited in how far we can take the analysis of the dynamical system.


• Numerical methods are key, and without them, we would be somewhat limited in how far we can take the analysis of the dynamical system.

• In general, we need numerical methods to verify and check our topology-based work as outlined before, but, also, because of the dimension and nonlinearity of the dynamical system, there are situations in which a stability analysis will not yield any information, and where the global methods outline above will only give limited information.


Parameter Space: Initial Conditions:

⌦+ ⌃2 + V = 1

Numerical Solver: ODE23s (MATLAB), ODE45 (MATLAB)

Solution: H(�), �±(�), N�(�), �(�)

�1 � w � 1, �0 � 0, �0 � 0

−1

−0.5

0

0.5

1

−1−0.8

−0.6−0.4

−0.20

0.20.4

0.60.8

1

0

0.5

1

1.5

2

2.5

3

Σ +

Σ −

N1

Chaotic Behaviour as � � ��

https://www.youtube.com/watch?v=0-whlK8arN4

−1.5

−1

−0.5

0

0.5

1

1.5

−1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0.5

1

1.5

H

Σ +

Σ−

This figure shows the dynamical system behavior for �0 = 0, �0 = 0, andw = 1/3. In particular, it displays the heteroclinic orbits joining K+ to K�,where K+ is located at H = 1, and K� is located at H = �1 in the figure.Numerical solutions were computed for �1000 � � � 1000. For clarity, we havedisplayed solutions for shorter timescales.

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Σ +

Σ −

N1

This figure shows the dynamical system behaviour for �0 = 0, �0 = 1/2,and w = �1/3. The plus sign denotes the equilibrium point F+. The modelalso isotropizes as can be seen from the last figure, where �± � 0 as � � �.Numerical solutions were computed for 0 � � � 1000. For clarity, we havedisplayed solutions for shorter timescales.

−0.4−0.3

−0.2−0.1

00.1

0.20.3

0.40.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5

0

0.5

1

1.5

2

2.5

3

Σ +

Σ −

N1

These figures show the dynamical system behavior for �0 = 0, �0 = 1/3,and w = 1. The plus sign denotes the equilibrium point F�. The model alsoisotropizes as can be seen from the last figure, where �± � 0 as � � �.Numerical solutions were computed for 0 � � � 1000. For clarity, we havedisplayed solutions for shorter timescales.

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

o

Σ +

Σ −

• The flat FLRW solution is clearly of primary importance with respect to modelling the present-day universe, which is

observed to be very close to flat. There are conditions in the parameter space for which this solution represents a saddle and

a sink.



a sink. • When it is a saddle, the equilibrium point attracts along its stable

manifold and repels along its unstable manifold. Therefore, some orbits will have an initial attraction to this point, but will eventually

be repelled by it.



a sink. • When it is a saddle, the equilibrium point attracts along its stable

manifold and repels along its unstable manifold. Therefore, some orbits will have an initial attraction to this point, but will eventually

be repelled by it. • In the case when it was found to be a sink, all orbits approach

the equilibrium point in the future. Therefore, there exists a time period and two separate configurations for which our

cosmological model will isotropize and be compatible with present-day observations of a high degree of isotropy in the

cosmic microwave background.

Documents

Dynamical Systems Methods in Early-Universe Cosmologymath.yorku.ca/~isk/Site/Welcome_files/sonad2014talknewnewest.pdf · dynamical systems methods to the Einstein ﬁeld equations