Talk given to Matt Johnson's Cosmology group at York University. The focus was on applying the orthonormal frame formalism and dynamical systems theory to cosmological models.

1. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods in Early-Universe
Cosmologies
Ikjyot Singh Kohli
Ph.D. Candidate
York University
Wednesday, February 12, 2014

2. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
1 Introduction
2 Description of the Matter
3 Deriving the Dynamical Equations
4 The Theory of Orthonormal Frames
5 Dynamical Systems Techniques

3. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.

4. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.

5. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.

6. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.
For a review of the justification of including viscosity
terms in early-universe cosmological models see the
articles by Barrow (1988,1982) [3] [2] and the article by
Belinskii and Khalatnikov (1976) [4].

7. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.
For a review of the justification of including viscosity
terms in early-universe cosmological models see the
articles by Barrow (1988,1982) [3] [2] and the article by
Belinskii and Khalatnikov (1976) [4].
As a result, we will keep our assumption of the spatial
homogeneity of the early universe, but now assume it is

8. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation I
In this section, we will derive the form of the required
energy-momentum tensor, namely, for that of a viscous fluid
without heat conduction. Recall that the energy-momentum
tensor for a perfect fluid takes the form
Tab
= (µ + p)ua
ub
− uc
ucgab
p. (1)
For the moment, letting µ + p = W, we obtain
Tab
= Wua
ub
− uc
ucgab
p, (2)
Denoting the viscous contributions by Vab, we seek a
modification of Eq. (2) such that
Tab = Wuaub − ucuc
gabp + Vab. (3)

9. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation II
To obtain the form of this additional tensor term, we note that
from classical fluid mechanics, the Euler equation is given as
(ρui ),t = −Πik,k, (4)
where Πik is the momentum flux tensor. Also, recall that for a
non-viscous fluid, one has the fundamental relationship
Πik = pδik + ρui uk. (5)
We simply add a term to Eq. (5) that represents the viscous
momentum flux, ˜Σik, to obtain
Πik = pδik + ρui uk − ˜Σik = −Sik + ρui uk. (6)
It is important to note that
Sik = −pδik + ˜Σik (7)
is the stress tensor, while, ˜Σik is the viscous stress tensor.

10. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation III
Note that, in what follows below, the viscous stress tensor,
˜Σik, is not to be confused with Σik, the Hubble-normalized
shear tensor.
The general form of the viscous stress tensor can be
formed by recalling that viscosity is formed when the fluid
particles move with respect to each other at different
velocities, so this stress tensor can only depend on spatial
components of the fluid velocity.
We assume that these gradients in the velocity are small,
so that the momentum tensor only depends on the first
derivatives of the velocity in some Taylor series expansion.

11. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation IV
Therefore, ˜Σik is some function of the ui,k. In addition,
when the fluid is in rotation, no internal motions of
particles can be occurring, so we consider linear
combinations of ui,k + uk,i , which clearly vanish for a fluid
in rotation with some angular velocity, Ωi . The most
general viscous tensor that can be formed is given by
˜Σik = η ui,k + uk,i −
2
3
δikul,l + ξδikul,l , (8)
where η and ξ are the coefficients of shear and
bulk/second viscosity, respectively [15] [13].

12. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation V
In Eq. (8), we note that δik ul,l is an expansion rate tensor,
and ui,k + uk,i − 2
3 δikul,l represents the shear rate
tensor. Since we would like to generalize this expression to
the general relativistic case, we replace the partial
derivatives above with covariant derivatives, and the
Kroenecker tensor with a more general metric tensor, that
is, δik → gik .
We thus have that
˜Σik = η ui;k + uk;i −
2
3
gikul;l + ξgikul;l . (9)
Denoting the shear rate tensor as σab, and the expansion
rate scalar as θ ≡ ua
;a, Eq. (9) becomes
Vab = −2ησab − ξθhab. (10)

13. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation VI
Since we are interested in the Hubble-normalized approach,
we will make use of the definition θ ≡ 3H, where H is the
Hubble parameter. This means that Eq. (9) becomes
Vab = −2ησab − 3ξHhab. (11)
Substituting Eq. (11) into Eq. (3) we finally obtain the
required form of the energy-momentum tensor as
Tab = (µ + p) uaub − ucuc
gabp − 2ησab − 3ξHhab. (12)
For simplicity, we shall let πab = −2ησab denote the
anisotropic stress tensor, and commit to the metric
signature (−1, +1, +1, +1) such that Eq. (12) takes the
form
Tab = (µ + p) uaub + gabp − 3ξHhab + πab. (13)

14. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion I
We will now derive the dynamical equations. Most of this
derivation follows from Ellis [6] and Grøn and Hervik [9], but I
have tried to make thing slightly easier to understand. Let us
first recall some basic properties of non-relativistic fluid
mechanics. In fluid mechanics, we typically measure the fluid
acceleration through the material derivative:
Dv
dt
≡ v,t + (v · ∇)v, (14)
where v,t is the local derivative, and (v · ∇)v is known as the
convective derivative. Note that
Dva
dt
= va
,t + va
,bxb
,t = va
,t + vb
va
,b. (15)

15. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion II
So we can write the convective term as Mv, where M = vi
,j is a
matrix, that is, it is the gradient of the velocity.
We can now decompose vi
,j into symmetric and anti-symmetric
parts:
vi,j = θij + ωij , (16)
where
θi
j =
1
2
vi
,j + vj
i (17)
is the expansion tensor and
ωi
j =
1
2
vi
,j − vj
i (18)
is the vorticity tensor.

16. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion III
I can now decompose the expansion tensor into trace and
trace-free parts:
θi
j =
1
3
θδi
j + σi
j , (19)
where
θ = vi
,i , σi
j =
1
2
vi
,j + vj
,i −
1
3
δi
j vi
,i . (20)
Note that σij is the trace-free tensor, and is known as the shear
tensor. The shear tensor essentially measures deformations of
the fluid.
Combining all of these, I can essentially write:
vi,j =
1
3
θδij + σij + ωij. (21)

17. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion IV
Switching to the relativistic form we obviously have
aα = uα;mum
= ˙uα. (22)
We want to project the four-dimensional quantities onto the
spatial slices and see how the evolve, these are our “dynamical”
quantities. That is,
uα;β =
1
3
θhαβ + σαβ + ωαβ − ˙uαuβ. (23)
Note that this last term is a consequence of purely relativistic
effects. It measures how much the fluid deviates from being
geodesic. In our work, we always assume the fluid is geodesic,
so the four-acceleration vanishes.

18. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion V
We can write the energy-momentum tensor of the previous
section in the more general form:
Tuv = µuuuv + phuv + πuv . (24)
Using the properties that
πu
u = 0, uu
πuv = 0, (25)
and the Bianchi identities Tv
u;v = 0 imply that
uu
Tv
u;v = 0
= uu
µ,v uuuv
+ µuu;v uv
+ µuuuv
;u + hv
u;v p + hv
up,v
+ πv
u;v
= − ˙µ − θµ − θp − σuv πuv
. (26)

19. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion VI
Therefore, the energy conservation equation is
˙µ + θ(µ + p) + σuv πuv
= 0, (27)

20. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
To close this system, we need dynamical equations for θ and
σuv . The idea for using kinematics to describe the dynamics of
universes obeying the Einstein field equations was first
introduced by Trumper as mentioned in Hawking’s 1966 paper
[10] and in Ellis’ Cargese Lectures [6].

21. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation I
The basic defining equation for General Relativity is the Ricci
identity. Let us start with:
−uu
uv
Raubv = uv
Rauvbuu
. (28)
I will write this as
−uu
uv
Raubv = ua;bv uv
+ ua;vuv
;b. (29)
If I now contract this equation, I obtain
− uu
uv
Ruv = ub
;bv uv
+ ua;v uv;a
. (30)
If I now substitute the Einstein field equations
Rab = Tab −
1
2
Tc
c gab. (31)

22. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation II
into the Ricci identity Eq. (30) and take the symmetric and
trace-free part, I obtain the shear-propagation equations:
ha
chb
d ˙σcd
= −2Hσab
− σa
c σbc
−
1
3
−2σ2
hab
+
1
2
πab
(32)
If I take the trace of Eq. (30), I get the Raychaudhuri equation:
˙θ +
1
3
θ2
+ σabσab
+
1
2
(µ + 3p) = 0. (33)
The assumption that ˙ua = 0 implies that ωuv = 0. In the 3 + 1
formalism, one can show that
ua;b = Kab, (34)

23. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation III
where Kab is the extrinsic curvature tensor. Gauss’ theorem
egregium (Page 171, [9]) states that
(n+1)
R =(n)
R + K2
− Kab
Kab + 2(−1)(n+1)
Rabua
ub
. (35)
Combining this with the Einstein field equations, we get:
Tab
uaub =
1
2
(3)
R − Kab
Kab + K2
. (36)
If we now use Eqs. (23) and (24), we get the Friedmann
equation:
1
3
θ2
=
1
2
σabσab
−
1
2
(3)
R + µ. (37)
This is a constraint equation on initial conditions as we will see
later.

24. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation IV
Note that, we still have a problem in that we really have not
simplified things much. Looking at the shear propagation
equation (32), for example, we have that
˙σab ≡ σab;mum
, (38)
which contains the covariant derivative, which contains the
Christoffel symbols, so we still need to work with the metric
tensor. This is where the grand theory of orthonormal frames
comes in!

25. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism I
The idea of orthonormal frames is based on the groundbreaking
1969 paper from Ellis and MacCallum [8], “A Class of
Homogeneous Cosmological Models”. First, what do we mean
when we say a space-time is spatially homogeneous? Let M be
a manifold with metric tensor g. The isometry group is defined
by
Isom(M = {φ : M → M|φ, isometry}, (39)
that is φ∗g = g, i.e., the metric is left unchanged after applying
an isometry. This isometry group is a Lie group, and the group
generators are Killing vectors. These Killing vectors span a
finite-dimensional space (the tangent space to the Lie group),
and we define the isotropy subgroup of a point p ∈ M by
fp(M) = {φ ∈ Isom(M)|φ(p) = p}. (40)

26. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism II
So a homogenous space is one that for each pair of points
a, b ∈ M, there exists a φ ∈ Isom(M) such that φ(p) = q.
For such a homogeneous space, there exists a set of Killing
vector fields ζi such that
[ζi , ζj ] = Dk
ij ζk. (41)
At a point p ∈ M, choose a basis set of vectors ei . It can be
shown that the frame ei span a Lie algebra:
[ei , ej ] = Ck
ij ek . (42)
Therefore, to construct a homogeneous space, one takes the
Ck
ij as the structure constants of the Lie algebra, and defines a

27. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism III
left-invariant frame by the above relation. For a dual basis, ωk
to ek , using the Cartan calculus, we have that
dwk
= −
1
2
Ck
ij ωi
∧ ωj
. (43)
Using these invariant one-forms, we can define the spatial
metric as
ds2
= gij ωi
⊗ ωj
, (44)
where the gij are constants. The amazing thing is that the
Killing vectors are ζi , and the basis vectors are ej , that is, we
can just read them off!
To proceed further, we also need to modify what we mean by a
connection on the manifold. First, consider a vector field u.
We have for any scalar function g that
u(g) = um
em(f ). (45)

28. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism IV
Now define
uv(g) = um
em(vn
en(g)), (46)
which is
uv(g) = um
em(vn
)en(g) + um
un
emen(g). (47)
This is clearly not a vector, since it has second-order partial
derivatives. However,
[u, v] = [um
em(vn
) − vm
em(un
)] en + um
un
[em, en] (48)
is indeed a vector. For an arbitrary basis,
[em, en] = cp
uv ep. (49)
These structure coefficients obviously vanish in a coordinate
basis, which is what most cosmologists usually work in.

29. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism V
We will also make use of a generalized connection ∇ that
associates ∇XY to any two vector fields X and Y. In a general
basis, we have that
∇v em = Γa
mnea, (50)
that is, the connection coefficients are defined as the
components of the directional derivative of the basis vectors.
For more information, see [9] and Abraham and Marsden’s
book [1]. For two vector fields, X = Xmem, and u = umem, we
have that
∇uX = (en(Xm
)un
+ Xa
Γm
anuv
) em. (51)
The commutator above therefore can now be written as:
[u, v] = ∇uv − ∇vu + (Γp
mn − Γp
nm + cp
mn) um
un
ep. (52)

30. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism VI
We defined the Torsion to be
ˆT(u ∧ v) ≡ ∇uv − ∇vu − [u, v] . (53)
We require spacetime in General Relativity to be torsion-free,
so we have
ca
mn = Γa
nm − Γa
mn. (54)
That is, all of the connection coefficients are now functions of
the structure constants!
In an orthonormal basis, gab = diag(−1, 1, 1, 1), so
Γamn =
1
2
gabcb
nm + gmbcb
an − gnbcb
ma . (55)
These are antisymmetric in the first two indices, so we can
write:
Γabt = −Γbat ≡ ϵabcΩc
, (56)

31. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism VII
where Ωc denotes the rotation of the spatial frame and is
defined by
Ωa
=
1
2
ϵabgd
ubeg · ˙ed . (57)
We can therefore write that
ca
tb = −θa
b + ϵa
bcΩc. (58)
The rest of the structure coefficients are all spatial in nature,
and we can write
ck
ij = ϵijl nlk
+ al δk
i δl
j − δk
j δl
i , (59)
where nlk and ai classify the Bianchi algebra, per the Behr
decomposition (See Stephani, Kramer, MacCallum,
Hoenselaers, and Herit [16]), Landau and Liftshitz [14], Grøn
and Hervik [9] for a further explanation:

32. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism VIII
Group Class Group Type n11 n22 n33 Contains FLRW?
A (ai = 0) I 0 0 0 k = 0
A II + 0 0
A VI0 0 + -
A VII0 0 + + k = 0
A VIII - + +
A IX + + + k = +1
B(ai ̸= 0) V 0 0 0 k = −1
B IV 0 0 +
B VIh 0 + -
B VIIh 0 + + k = −1
Table: The Bianchi classifications

33. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism IX
These classifications were due to Bianchi [5], but we follow the
conventions of Ellis, Maartens and MacCallum. [7].
One can find evolution equations for the nlk and the ai by
recognizing that the structure coefficients define a Lie algebra,
so that the Jacobi identity holds. That is, for the set of vectors
(u, ea, eb), we have that
0 = [u, [ea, eb]] + [ea, [eb, u]] + [eb, [u, ea]]. (60)
Go through some algebra, actually a lot of algebra (!), and find
that we get
u(ck
ab) + ck
td cd
ab + ck
ad cd
bt + ck
bd cd
ta = 0. (61)

34. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism X
Taking the trace of this equation, we get upon defining
u = ∂/∂t that
˙ai = −
1
3
θai − σij aj
− ϵijkaj
Ωk
, (62)
and the trace-free part yields
˙nab = −
1
3
θnab − 2nk
(aϵb)kl Ωl
+ 2nk(aσk
b). (63)
Returning to our propagation equations from before, we note
that for the shear propagation equation, we had
˙σab = um
∇mσab = u(σab) − Γm
anσmbun
− Γm
bnσamun
. (64)

35. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism XI
Using the orthonormal frame formalism, we can write this as
˙σab = u (σab) − 2σd
(aϵb)cd Ωc
. (65)
Setting a universal time gauge, u = ∂
∂t and using the
convention that H = 1
3θ, (where H is the Hubble parameter)
we have the Einstein field equations in an orthonormal frame:
˙H = −H2
−
2
3
σ2
− µ
1
6
+
1
2
w , (66)
˙σab = −3Hσab + 2ϵuv
(a σb)uΩv − Sab − 2ησab, (67)
µ = 3H2
− σ2
+
1
2
R, (68)
0 = 3σu
a au − ϵuv
a σb
unbv , (69)

36. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism XII
where Sab and R are the three-dimensional spatial curvature
and Ricci scalar and are defined as:
Sab = bab −
1
3
bu
uδab − 2ϵuv
(a nb)uav , (70)
R = −
1
2
bu
u − 6auau
, (71)
where bab = 2nu
a nub − (nu
u) nab. We have also denoted by Ωv
the angular velocity of the spatial frame.
˙nab = −Hnab + 2σu
(anb)u + 2ϵuv
(a nb)uΩv , (72)
˙aa = −Haaσb
a ab + ϵuv
a auΩv , (73)
0 = nb
a ab. (74)

37. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism XIII
The contracted Bianchi identities give the evolution equation
for µ as
˙µ = −3H (µ + p) − σb
a πa
b + 2aaqa
. (75)

38. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods I
With the propagation equations (66), (67), (75), and the
constraint equations (68) and (69) in hand, we can apply
techniques of dynamical systems theory to obtain a great deal
of information about the system’s future and past asymptotic
states. Let us introduce the following normalizations:
Σij =
σij
H
, Nij =
nij
H
, Ai =
ai
H
, Ri =
Ωi
H
, Sij =
(3)R⟨ij⟩
H2
,
(76)
Ω =
µ
3H2
, P =
p
3H2
, Qi =
qi
H2
, Πij =
πij
H2
, K = −
(3)R
6H2
.
(77)
The usefulness of this can be seen from the fact that, in
geometrized units, the Hubble parameter has units of inverse
length. Dividing each quantity above by an appropriate power
of H, makes the variables dimensionless.

39. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods II
We therefore obtain our dynamical system [11] [7] as:
Σ′
ij = −(2 − q)Σij + 2ϵkm
(i Σj)kRm − Sij + Πij
N′
ij = qNij + 2Σk
(i Nj)k + 2ϵkm
(i Nj)kRm
A′
i = qAi − Σj
i Aj + ϵkm
i AkRm
Ω′
= (2q − 1)Ω − 3P −
1
3
Σj
i Πi
j +
2
3
Ai Qi
Q′
i = 2(q − 1)Qi − Σj
i Qj − ϵkm
i RkQm + 3Aj
Πij + ϵkm
i Nj
kΠjm.(78)
These equations are subject to the constraints
Nj
i Aj = 0
Ω = 1 − Σ2
− K
Qi = 3Σk
i Ak − ϵkm
i Σj
kNjm. (79)

40. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods III
In the expansion-normalized approach, Σab denotes the
kinematic shear tensor, and describes the anisotropy in the
Hubble flow, Ai and Nij describe the spatial curvature, while
Ωi and Ri describe the relative orientation of the shear and
spatial curvature eigenframes and energy flux respectively.
Further the prime denotes differentiation with respect to a
dimensionless time variable τ such that
dt
dτ
=
1
H
. (80)
The remarkable thing about this method is as follows.The
system of equations (78) is a nonlinear, autonomous system of
ordinary differential equations, and can be written as
x′
= f(x), (81)

41. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods IV
where x = [Σij, Nij , Ai , Ω, Qi ] ∈ Rn, where the vector field f(x)
denotes the right-hand-side of the dynamical system.
Following [1], we first note that the vector field f(x) is clearly
at least C1 on M = Rn. We call a point m0 an equilibrium
point of f(x) if f(m0) = 0. Let (U, φ) be a chart on M with
φ(m0) = x0 ∈ Rn, and let x = [Σij , Nij , Ai , Ω, Qi ] denote
coordinates in Rn. Then, the linearization of f(x) at m0 in
these coordinates is given by
∂f(x)i
∂xj
x=x0
(82)

42. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods V
It is a remarkable fact of dynamical systems theory that if
the point m0 is hyperbolic, then there exists a
neighborhood N of m0 on which the flow of the system Ft
is topologically equivalent to the flow of the linearization
Eq. (82). This is the theorem of Hartman and Grobman
[17]. That is, in N, the orbits of the dynamical system can
be deformed continuously into the orbits of Eq. (82), and
the orbits are therefore topologically equivalent. We use
the following convention when discussing the stability
properties of the dynamical system. If all eigenvalues λi of
Eq. (82) satisfy Re(λi ) < 0(Re(λi ) > 0), m0 is local sink
(source) of the system. If the point m0 is neither a local
source or sink, we will call it a saddle point.
Given a linear DE x′ = Ax on Rn, we consider the
eigenvalues of A (complex in general, and not necessarily

43. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VI
distinct) and the associated generalized eigenvectors. We
define three subspaces of Rn,
stable subspace, Es
= span(s1, . . . , sns ),
unstable subspace, Eu
= span(u1, . . . , unu ),
centre subspace, Ec
= span(c1, . . . , cnc ),
where s1, . . . , sns are the generalized eigenvectors whose
eigenvalues have negative real parts, u1, . . . , unu are those
whose eigenvalues have positive real parts, and c1, . . . , cnc
are those whose eigenvalues have zero real parts. It is
well-known then that
Es
⊕ Eu
⊕ Ec
= Rn
,

44. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VII
and that
x ∈ Es
⇒ lim
t→+∞
eAt
x = 0,
x ∈ Eu
⇒ lim
t→−∞
eAt
x = 0.
These statements basically describe the asymptotic
behaviour of the system: All initial states in the stable
subspace are attracted to the equilibrium point 0, while all
initial states in the unstable subspace are repelled by 0. In
particular, if dim Es = n, all initial states are attracted to
0, which is referred to as a linear sink, while if
dim Eu = n, all initial states are repelled by 0, which is
referred to as a linear source.

45. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VIII
We will consider a DE x′ = f(x) on Rn, where f is of class
C1. If a is an equilibrium point (f(a) = 0), the linear
approximation of f at a becomes
f(x) ≈ Df(a)(x − a),
where
Df(a) =
∂fi
∂xj x=a
(83)
is the derivative matrix of f. Thus, with the given DE
x′ = f(x), we associate the linear DE
u′
= Df(a)u, (84)
where u = x − a, called the linearization of the DE at the
equilibrium point a.

46. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods IX
Theorem
(Invariant Manifolds) Let x = 0 be an equilibrium point of the
DE x′ = 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
W s
tangent to Es
at 0,
W u
tangent to Eu
at 0,
W c
tangent to Ec
at 0.
Further, each equilibrium point of the dynamical system
represents a solution to the Einstein field equations!. This
was discussed at length by Jantzen and Rosquist [12] and
Wainwright and Hsu [18].

47. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods X
Therefore, we have transformed the Einstein field
equations from the pseduo-Riemannian manifold to
n-dimensional phase space. Solutions to the EFE are
equilibrium points of the dynamical system, and the
stability analysis performed by analyzing the eigenvalues of
Eq. (82) tells us what the future and past asymptotic
state of a particular Bianchi cosmology will be. It should
be noted that it is often that one gets eigenvalues with
more than one zero eigenvalue, or that, the Jacobian
matrix in Eq. (82) is undefined. In this case, we must use
methods from topological dynamical systems theory to
analyze the future and past asymptotic states.
For examples of our work, see:

48. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XI
“Exploring Vacuum Energy in a Two-Fluid Bianchi Type I
Universe”, Ikjyot Singh Kohli and Michael C. Haslam -
arXiv:1402.1967 - Submitted to Phys. Rev. D - 2014
“On The Dynamics of a Closed Viscous Universe”, Ikjyot
Singh Kohli and Michael C. Haslam - arXiv:1311.0389 -
Phys. Rev. D 89, 043518 (2014)
“A Dynamical Systems Approach to a Bianchi Type I
Magnetohydrodynamic Model”, Ikjyot Singh Kohli and
Michael C. Haslam - arXiv:1304.8042 - Phys. Rev. D 88,
063518 (2013)
“The Future Asymptotic Behaviour of a Non-Tilted
Bianchi Type IV Viscous Model”, Ikjyot Singh Kohli and
Michael C. Haslam - arXiv:1207.6132 0 Phys. Rev. D 87,
063006 (2013)

49. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XII
In these works, we found two new solutions to the Einstein field
equations, and also showed that the early universe quite
possibly had large magnetic fields. In our newest work, we have
discussed a possible mechanism for the existence of vacuum
energy. So, dynamical systems methods are very powerful in
cosmology!
As an example, to be possibly discussed for next time, consider
our parameter space depiction of the Bianchi Type I MHD
model:

50. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XIII
Figure: A depiction of the different regions of sinks, sources, and
saddles of the dynamical system as defined by the aforementioned

51. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XIV
restrictions on the values of the expansion-normalized bulk and shear
viscosities, ξ0, η0 and equation of state parameter, w.

52. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources I
[1] Ralph Abraham and Jerrold E. Marsden.
Foundations of Mechanics.
AMS Chelsea Publishing, second edition, 1978.
[2] John D Barrow.
Dissipation and unification.
Monthly Notices of the Royal Astronomical Society,
199:45–48, 1982.
[3] John D Barrow.
String-driven inflationary and deflationary cosmological
models.
Nuclear Physics B, 310:743–763, 1988.

53. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources II
[4] V.A. Belinskii and I.M. Khalatnikov.
Influence of viscosity on the character of cosmological
evolution.
Soviet Physics JETP, 42:205, 1976.
[5] L. Bianchi.
Sugli Spazi I A Tre Dimensioni Che Ammettono Un Gruppo
Continuo Di Movimenti.
Soc. Ital. Sci. Mem. di Mat., 1898.
[6] George F.R. Ellis.
Cargese Lectures in Physics, volume Six.
Gordon and Breach, first edition, 1973.

54. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources III
[7] George F.R. Ellis, Roy Maartens, and Malcolm A.H.
MacCallum.
Relativistic Cosmology.
Cambridge University Press, first edition, 2012.
[8] G.F.R. Ellis and M.A.H. MacCallum.
A class of homogeneous cosmological models.
Comm. Math. Phys, 12:108–141, 1969.
[9] Øyvind Grøn and Sigbjørn Hervik.
Einstein’s General Theory of Relativity: With Modern
Applications in Cosmology.
Springer, first edition, 2007.

55. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources IV
[10] S.W. Hawking.
Perturbations of an expanding universe.
Astrophysical Journal, 145:544, 1966.
[11] C.G. Hewitt, R. Bridson, and J. Wainwright.
The asymptotic regimes of tilted bianchi ii cosmologies.
General Relativity and Gravitation, 33:65–94, 2001.
[12] R.T. Jantzen and K Rosquist.
Exact power law metrics in cosmology.
Classical and Quantum Gravity, 3:281, 1986.
[13] Pijush K. Kundu and Ira M. Cohen.
Fluid Mechanics.
Academic Press, fourth edition, 2008.

56. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources V
[14] L.D. Landau and E.M. Lifshitz.
Classical Theory of Fields.
Butterworth-Heinemann, fourth edition, 1980.
[15] L.D. Landau and E.M. Lifshitz.
Fluid Mechanics.
Butterworth-Heinman, second edition, 2011.
[16] Hans Stephani, Dietrich Kramer, Malcolm MacCallum,
Cornelius Hoenselaers, and Eduart Herlt.
Exact Solutions of Einstein’s Field Equations.
Cambridge University Press, second edition, 2009.
[17] J. Wainwright and G.F.R. Ellis.
Dynamical Systems in Cosmology.
Cambridge University Press, first edition, 1997.

57. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources VI
[18] J. Wainwright and L. Hsu.
A dynamical systems approach to bianchi cosmologies:
orthogonal models of class a.
Classical and Quantum Gravity, 6:1409–1431, 1989.