1. 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 2. Introduction 3. Introduction Much of my research (with Michael Haslam) is focused on applying dynamical systems methods to the Einstein eld equations to develop models of the early universe. 4. Introduction Much of my research (with Michael Haslam) is focused on applying dynamical systems methods to the Einstein eld equations to develop models of the early universe. 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. 5. Introduction Much of my research (with Michael Haslam) is focused on applying dynamical systems methods to the Einstein eld equations to develop models of the early universe. 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. We would like to understand how this is possible given that the early universe had a very high temperature and hence was unstable. 6. Introduction Much of my research (with Michael Haslam) is focused on applying dynamical systems methods to the Einstein eld equations to develop models of the early universe. 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. 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. 7. Introduction 8. 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 dene a time variable to get some idea of dynamics. 9. 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 dene 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 eld 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. 10. Spacetime Splitting 11. 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. 12. 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. The local region of M then can be made to look like R^n locally. 13. 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. 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. 14. Depiction of a manifold 15. Spacetime Splitting General Relativity is described by the Einstein eld equations: Rab 1 2 gabR + gab = kTab Rab : Ricci tensor gab : Metric tensor : Cosmological constant Tab : Energy-momentum tensor These are 10 hyperbolic, nonlinear, coupled PDEs! 16. Spacetime Splitting The solution of these equations is the metric tensor, typically written in the form: ds2 = gabdxa dxb One sees though, that the EFEs are not your typical eld equations, in the sense that the solution is a metric tensor, and not a time-varying eld, as would be the case with Maxwells equations, Navier-Stokes, etc 17. 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 18. (Baumgarte, Shapiro, 2010) Elliptic PDEs Hyperbolic PDEs 19. Energy-Momentum Tensor The energy-momentum tensor is given by: Tab = ( + p) uaub + gabp 3Hhab 2 ab, where , p, and ab denote the uids energy density, pressure, and shear respectively, while and denote the bulk and shear viscosity coe cients of the uid. Throughout this work, both coe cients are taken to be nonnegative constants. H denotes the Hubble parameter, and hab uaub + gab is the standard 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). 20. 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 1 6 + 1 2 w , (1) ab = 3H ab + 2 uv (a b)u v Sab 2 ab, (2) = 3H2 2 + 1 2 R, (3) 0 = 3 u a au uv a b unbv, (4) The Einstein eld equations: 21. where Sab and R are the three-dimensional spatial curvature and Ricci scalar and are dened as: Sab = bab 1 3 bu u ab 2 uv (a nb)uav, (1) R = 1 2 bu u 6auau , (2) where bab = 2nu anub (nu u) nab. We have also denoted by v the angular velocity of 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 b aab + uv a au v, (2) 0 = nb aab. (3) The contracted Bianchi identities give the evolution equation for as = 3H ( + p) b a a b + 2aaqa . (4) 22. In addition, we dene 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) 23. H0 = (1 H2 )q, 0 + = + H ( 2 + q) 6+ 0 S+, 0 = H ( 2 + q) 6 0 S , N0 1 = N1 H q 4+ , N0 2 = N2 H q + 2+ + 2 p 3 , N0 3 = N3 H q + 2+ 2 p 3 , 0 = H ( 1 + 2q 3w) + 9 H2 0 + 120 2 + + 2 where q = 2 2 + + 2 + 1 2 (1 + 3w) 9 2 0 H + 2 + V = 1 This is a constraint on the initial conditions 24. Note that: V = 1 12 N2 1 + N2 2 + N2 3 2 N1 N2 2 N2 N3 2 N3 N1 + 3 N1 N2 N3 2/3 S+ = 1 6 N2 N3 2 N1 2 N1 N2 N3 , S = 1 2 p 3 h N3 N2 N1 N2 N3 i . 25. Dynamical Analysis 26. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 27. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 28. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 2. Identify the invariant sets of the system 29. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 2. Identify the invariant sets of the system 3. Find all equilibrium points and analyze their local stability 30. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 2. Identify the invariant sets of the system 3. Find all equilibrium points and analyze their local stability 4. Find monotone functions in the various invariant sets 31. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 2. Identify the invariant sets of the system 3. Find all equilibrium points and analyze their local stability 4. Find monotone functions in the various invariant sets 5. Investigate bifurcations that occur as per the parameter space 32. Dynamical Analysis To understand the dynamics of such a universe, we typically proceed in the following manner: 1. Determine whether the state space is compact 2. Identify the invariant sets of the system 3. Find all equilibrium points and analyze their local stability 4. Find monotone functions in the various invariant sets 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. 33. Examples of Fixed Points Expanding Flat FLRW solution Expanding Bianchi II solution Closed Einstein Static Universe 34. Stability AnalysisEquilibrium Point EFE Solution Local Sink Local Source Saddle F+ Expanding k=0 FLRW solution Yes No Yes F- Contracting k=0 FLRW solution Yes Yes Yes P+(II) Expanding Bianchi Type II Solution No No Yes P_(II) (new discovery!) Contracting Bianchi Type II Solution No No Yes 35. 21 For nonlinear systems, we are bound by the Invariant Manifold theorem: Let x = 0 be an equilibrium point of the DE x0 = f(x) on Rn and let Es , Eu , and Ec denote the stable, unstable, and centre subspaces of the linearization at 0. Then there exists Ws tangent to Es at 0, Wu tangent to Eu at 0, Wc tangent 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. 36. Global Methods 22 37. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. 22 38. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: 22 39. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: Finding Lyapunov and Chetaev functions 22 40. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: Finding Lyapunov and Chetaev functions LaSalle invariance principle 22 41. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: Finding Lyapunov and Chetaev functions LaSalle invariance principle Monotonicity Principle 22 42. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: Finding Lyapunov and Chetaev functions LaSalle invariance principle Monotonicity Principle Numerical methods 22 43. Global Methods There are many such methods, although, the majority of the methods used only apply to planar dynamical systems. The methods that we have found useful are: Finding Lyapunov and Chetaev functions LaSalle invariance principle Monotonicity Principle Numerical methods The rst three give information on the alpha and omega limit sets of the dynamical system. 22 44. Numerical Experiments 45. 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. 46. 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. 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. 47. Numerical Experiments Parameter Space: Initial Conditions: + 2 + V = 1 Numerical Solver: ODE23s (MATLAB), ODE45 (MATLAB) Solution: H( ), ( ), N ( ), ( ) 1 w 1, 0 0, 0 0 48. 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 + N1 Chaotic Behaviour as 49. 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 + This gure shows the dynamical system behavior for 0 = 0, 0 = 0, and w = 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 gure. Numerical solutions were computed for 1000 1000. For clarity, we have displayed solutions for shorter timescales. 50. 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 + N1 This gure shows the dynamical system behaviour for 0 = 0, 0 = 1/2, and w = 1/3. The plus sign denotes the equilibrium point F+. The model also isotropizes as can be seen from the last gure, where 0 as . Numerical solutions were computed for 0 1000. For clarity, we have displayed solutions for shorter timescales. 51. 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 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 + N1 These gures show the dynamical system behavior for 0 = 0, 0 = 1/3, and w = 1. The plus sign denotes the equilibrium point F . The model also isotropizes as can be seen from the last gure, where 0 as . Numerical solutions were computed for 0 1000. For clarity, we have displayed solutions for shorter timescales. 52. 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 + 53. The at FLRW solution is clearly of primary importance with respect to modelling the present-day universe, which is observed to be very close to at. There are conditions in the parameter space for which this solution represents a saddle and a sink. 54. The at FLRW solution is clearly of primary importance with respect to modelling the present-day universe, which is observed to be very close to at. There are conditions in the parameter space for which this solution represents a saddle and 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. 55. The at FLRW solution is clearly of primary importance with respect to modelling the present-day universe, which is observed to be very close to at. There are conditions in the parameter space for which this solution represents a saddle and 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 congurations 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.