We adopt here a working definition of the universe as the

Edwin Hubble around 1930 discovered that the distant galaxies are moving away from us. The velocity of recession follows

Hubble discovered the

According to

Consider any two galaxies 𝐴 and 𝐵 which are taking part in the general notion of expansion of the universe. The distance between them can be written as 𝑓

Imagine a spherical balloon like

(1.1) 𝑑

The speed 𝑣

(1.2) 𝑣

Because the expansion of balloon is uniform and the angle 𝜃

From the rate at which galaxies are receiving from each other, it can be deduced that all galaxies must have been very close each other at the same time in the past. For the universe it is believed that at the initial moment (some time between 10 and 20 billion years ago) there was a universal explosion, at every point of the universe. This was the

The expansion of the universe may continue forever, as in the open models, or it may halt at some future time and contraction set in, as in closed models. These possibilities are illustrated in

There is an important a piece of evidence apart from the recession of the galaxies that the contents of the universe in the past must have been in the highly compressed form. This is the

Hubble's law implies arbitrarily large velocities of the galaxies as the distance increases indefinitely. The red-shift 𝑧 is defined as 𝑧 = (𝜆

There is a certain

(1.3) 𝛆

Here 𝐺 is Newton's gravitational constant and R is the scale factor and represents the size of the universe in a sense. If 𝑡

Thus the

There are several other ways of determining if the universe will expanding forever. One is to measure the rate by the

Another way is to determine the precise age of the universe and compare it the

(1.5) tan 𝛼 = 𝑃𝑁/𝑁𝑇 = Ṙ(𝑡

(1.6) 𝑁𝑇 = 𝑃𝑁/Ṙ(𝑡

Thus 𝑁𝑇, the Hubble time is the reciprocal of Hubble's constant. For the value of 50 km/s per million parsecs of Hubble's Constant, the Hubble time is about 20 billion years. Again in the simple models, if the universe is older than two-thirds of Hubble time it will expand forever, and otherwise, the opposite being the case.

Here we shall use the term

The

To deal with these problems Alan Guth in 1981 proposed the

The period referred to as

**2. Introduction to general relativity**^{*} _{}

**2.1 Summary of general relativity** _{.}

The Robertson-Walker metric or line-element is fundamental in the standard models of cosmology. The mathematical framework of the metric is that of general relativity. Here we will summarize general relativity as a reminder of main result and for the sake of completeness. Because [Dirac 1975]*, which is the main source in this chapter, was studied previously here and some contents which are different from the book will be described chiefly.

General relativity is formulated in a *four-dimensional Riemannian space* in which points are labelled by a general coordinate system (𝑥^{0}, 𝑥^{1}, 𝑥^{2}, 𝑥^{3}), often written as 𝑥^{𝜇} (𝜇 = 0, 1, 2, 3). In the whole of space-time coordinates, 𝑥^{0} = 𝑐𝑡 (𝑐: the speed of light, 𝑡: time), 𝑥^{1} which has distance dimension, 𝑥^{2} and 𝑥^{3} which have same dimension and are corresponding to spatial coordinates of three dimensional Euclidean space. According to the law of special relativity, the interaction of material particles takes time of speed of light among them and the law was experimentally proved [Landau-Lifshitz 1975 pp.1~]. If we take a event or a point close to the point of 𝑥^{𝜇} as 𝑥^{𝜇} + 𝑑𝑥^{𝜇}, the *interval* between the two evens or points 𝑑𝑠, which is *invariant* as follows:

(2.1) 𝑥^{0} = 𝑐𝑡, 𝑥^{1} = 𝑥, 𝑥^{2} = 𝑦, 𝑥^{3} = 𝑧 (𝑐 = 2.998 x 10^{10}cm/sec); 𝑑𝑠^{2} = (𝑑𝑥^{0})^{2} - (𝑑𝑥^{1})^{2} - (𝑑𝑥^{2})^{2} - (𝑑𝑥^{3})^{2}

Any 𝐴^{𝜇} that transforms under a change of coordinates like as above called a *contravariant vector* as a *four-vector*. we define *covariant vector* 𝐴_{𝜇} as a four-vector and thus have the scalar product invariant as follows:

(2.2) (𝐴^{0})^{2} - (𝐴^{1})^{2} - (𝐴^{2})^{2} - (𝐴^{3}) ^{2} = (𝐴, 𝐴); 𝐴_{0} = 𝐴^{0}, 𝐴_{1} = -𝐴^{1}, 𝐴_{2} = -𝐴^{2}, 𝐴_{3} = -𝐴^{3}; 𝐴^{𝜇}𝐵_{𝜇} = 𝐴^{0}𝐵_{0} + 𝐴^{1}𝐵_{1} + 𝐴^{2}𝐵_{2} + 𝐴^{3}𝐵_{3} = (𝐴, 𝐵),

where 𝐴^{𝜇}𝐵_{𝜇} are sums of a pair of dummy indices at upper and lower indices separately.according to *Einstein summation notation* enable to simplify complicate tensor equations.

Under a coordinate transformation from 𝑥^{𝜇} to 𝑥'^{𝜇} a contravariant vector field 𝐴^{𝜇}, a covariant vector field 𝐵_{𝜇} and a tensor such as 𝑇^{𝜇}_{𝜈𝜆} transform as follows:

(2.3) 𝐴'^{𝜇} = (∂𝑥'^{𝜇}/∂𝑥^{𝜈}) 𝐴^{𝜈}, 𝐵'_{𝜇} = (∂𝑥^{𝜈}/∂𝑥'^{𝜇}) 𝐵_{𝜈}, 𝑇'^{𝜇}_{𝜈𝜆} = (∂𝑥'^{𝜇}/∂𝑥^{𝜌}) (∂𝑥^{𝜎}/∂𝑥'^{𝜈}) (∂𝑥^{𝜏}/∂𝑥'^{𝜆}) 𝑇^{𝜌}_{𝜎𝜏}.

All the information about gravitational field is contained in the second rank tensor 𝑔_{𝜇𝜈} called *fundamental tensor or metric tensor* which determines the square of space-time intervals 𝑑𝑠^{2} between infinitesimally separated events or points 𝑥^{𝜇} and 𝑥^{𝜇} + 𝑑𝑥^{𝜇} as follows:

(2.4) 𝑑𝑠^{2} = 𝑔_{𝜇𝜈}𝑑𝑥^{𝜇}𝑑𝑥^{𝜈}, 𝑔_{𝜇𝜈} = 𝑔_{𝜈𝜇}. 𝑔_{𝜇𝜈}𝑔^{𝜈𝜆} = 𝛿^{𝜆}_{𝜇}, 𝛿^{𝜆}_{𝜇} = 1 for 𝜆 = 𝜇, 𝛿^{𝜆}_{𝜇} = 0 for 𝜆 ≠ 𝜇,

The contravariant tensor corresponding to 𝑔_{𝜇𝜈} called *inverse metric tensor* is denoted by 𝑔^{𝜇𝜈} and is defined by as above and where the 𝛿^{𝜆}_{𝜇} is the Kronecker delta, which equals unity if 𝜆 = 𝜇 and zero otherwise. Indices can be raised or lowered or changed by using the metric tensor and Kronecker delta as follows:

(2.5) 𝐴^{𝜇} = 𝑔^{𝜇𝜈}𝐴_{𝜈}, 𝐴_{𝜇} = 𝑔_{𝜇𝜈}𝐴^{𝜈}, 𝐴_{𝜇} = 𝛿_{𝜇}^{𝜌}𝐴_{𝜌}.

The generalization of partial differentiation to Riemannian space is given by covariant differentiation denoted by semi-colon and defined as follows:

(2.6 a, b, c) 𝐴^{𝜇}_{;𝜈} = ∂𝐴^{𝜇}/∂𝑥^{𝜈} + 𝛤^{𝜇}_{𝜈𝜆}𝐴^{𝜆}; 𝐴_{𝜇}_{;𝜈} = ∂𝐴_{𝜇}/∂𝑥^{𝜈} - 𝛤^{𝜆}_{𝜇𝜈}𝐴_{𝜆}. 𝐴^{𝜇}_{𝜈𝜆;𝜎} = ∂𝐴_{𝜇}_{𝜈𝜆}/∂𝑥^{𝜎} + 𝛤^{𝜇}_{𝜎𝜌}𝐴^{𝜌}_{𝜈𝜆} - 𝛤^{𝜌}_{𝜈𝜎}𝐴^{𝜇}_{𝜌𝜆} - 𝛤^{𝜌}_{𝜆𝜎}𝐴^{𝜇}_{𝜈𝜌}.
Here the 𝛤^{𝜇}_{𝜈𝜆} are called Christoffel symbols; their lower indices are interchangeable and are given in terms of metric tensor as follows:

(2.7) 𝛤^{𝜇}_{𝜈𝜆} = 1/2 𝑔^{𝜇𝜎}(𝑔_{𝜎𝜇,𝜆} + 𝑔_{𝜎𝜆,𝜇} - 𝑔_{𝜈𝜆,𝜎}), 𝑔_{𝜎𝜇,𝜆} ≡ ∂𝑔_{𝜎𝜇}/∂𝑥^{𝜆}; 𝛤^{𝜇}_{𝜈𝜆} = 𝛤^{𝜇}_{𝜆𝜈}.

Equation (2.7) has the consequence that the covariant derivative of the metric tensor or inverse metric tensor vanishes:

(2.8) 𝑔_{𝜇𝜈;𝜆} = 0, 𝑔^{𝜇𝜈}_{;𝜆} = 0.

With metric tensors the indices of vectors in covariant differentiation can be lowered and raised also as folllows:

(2.9) 𝑔_{𝜎𝜇}𝐴^{𝜇}_{;𝜈} = 𝐴_{𝜎;𝜈}, 𝑔^{𝜎𝜇}𝐴_{𝜇;𝜈} = 𝐴^{𝜎}_{;𝜈}

Under a coordinate transformation from 𝑥^{𝜇} to 𝑥'^{𝜇} the 𝛤^{𝜇}_{𝜈𝜆} transform as folllows:

(2.10) 𝛤^{𝜇}_{𝜈𝜆} = (∂𝑥'^{𝜇}/∂𝑥^{𝜌}) (∂𝑥^{𝜎}/∂𝑥'^{𝜈}) (∂𝑥^{𝜏}/∂𝑥'^{𝜆})𝛤^{𝜌}_{𝜎𝜏} + (∂^{2}𝑥^{𝜎}/∂𝑥'^{𝜈}𝑥'^{𝜆}) (∂𝑥'^{𝜇}/∂𝑥^{𝜎}),

so the 𝛤^{𝜇}_{𝜈𝜆} is not a tensor. Therefore at any specific point a coordinate system always be chosen so that the 𝛤^{𝜇}_{𝜈𝜆} vanish there. From (2,7) it follows that the first derivatives of metric tensor also vanish at the point. This is one form of the *equivalence of principle*, according to which the gravitational field can be 'transformed away' at any point bt choosing a suitable frame of reference. At this point one can carry out a further transformation of the coordinates to reduce the metric to that of *flat (Minkowski) space* as (2.1);

(2.11) 𝑑𝑠^{2} = (𝑑𝑥^{0})^{2} - (𝑑𝑥^{1})^{2} - (𝑑𝑥^{2})^{2} - (𝑑𝑥^{3})^{2}.

For a covariant vector 𝐴_{𝜇}, in general, two covariant differentiation in succession their order does matter and we get

(2.12) 𝐴_{𝜇;𝜈;𝜆} - 𝐴_{𝜇;𝜆;𝜈} = 𝑅^{𝜎}_{𝜇𝜈𝜆},
where 𝑅^{𝜎}_{𝜇𝜈𝜆} is called the *Riemann(-Christoffel) curvature tensor* and defined by

(2.13) 𝑅^{𝜎}_{𝜇𝜈𝜆} = 𝛤^{𝜎}_{𝜇𝜆,𝜈} - 𝛤^{𝜎}_{𝜇𝜈,𝜆} + 𝛤^{𝜎}_{𝛼𝜈}𝛤^{𝛼}_{𝜇𝜆} - 𝛤^{𝜎}_{𝛼𝜆}𝛤^{𝛼}_{𝜇𝜈}.
The Riemann curvature tensor has the following properties:

(2.14 a, b, c) 𝑅_{𝜎𝜇𝜈𝜆} = 𝑔_{𝜎𝛼}𝑅^{𝜎𝛼}_{𝜇𝜈𝜆}, 𝑅_{𝜎𝜇𝜈𝜆} = -𝑅_{𝜇𝜎𝜈𝜆} = -𝑅_{𝜎𝜇𝜆𝜈}, 𝑅_{𝜎𝜇𝜈𝜆} = 𝑅_{𝜈𝜆𝜎𝜇}, 𝑅_{𝜎𝜇𝜈𝜆} + 𝑅_{𝜎𝜆𝜇𝜈} + 𝑅_{𝜎𝜈𝜆𝜇} = 0,

and satisfies the *Bianchi identity* :

(2.15) 𝑅^{𝜎}_{𝜇𝜈𝜆;𝜌} + 𝑅^{𝜎}_{𝜇𝜌𝜈;𝜆} + 𝑅^{𝜎}_{𝜇𝜆𝜌;𝜈} = 0.

The *Ricci tensor* 𝑅_{𝜇𝜈} is defined by

(2.16) 𝑅_{𝜇𝜈} = 𝑔^{𝜆𝜎}𝑅_{𝜆𝜇𝜎𝜈} = 𝑅^{𝜎}_{𝜇𝜌𝜈},
From (2.13) and (2.16) 𝑅_{𝜇𝜈} is given as follows:

(2.17) 𝑅_{𝜇𝜈} = 𝛤^{𝜆}_{𝜇𝜈,𝜆} - 𝛤^{𝜆}_{𝜇𝜆,𝜈} + 𝛤^{𝜆}_{𝜇𝜈}𝛤^{𝜎}_{𝜆𝜎} - 𝛤^{𝜎}_{𝜇𝜆}𝛤^{𝜆}_{𝜈𝜎}.
Let the determinant of 𝑔_{𝜇𝜈} considered as a matrix be denoted by 𝑔 which is negative quantity. Then another expression for 𝑅_{𝜇𝜈} is given by the following:

(2.18) 𝑅_{𝜇𝜈} = 1/(-𝑔)^{1/2} [𝛤^{𝜆}_{𝜇𝜈}(-𝑔)^{1/2}]_{,𝜆} - [log(-𝑔)^{1/2}]_{,𝜇𝜈} - 𝛤^{𝜎}_{𝜇𝜆}𝛤^{𝜆}_{𝜈𝜎}.

This follows from (2.7) and the next properties which is proved at [Dirac 1975 p.25]

(2.19) 𝛤^{𝜆}_{𝜇𝜆} = [log(-𝑔)^{1/2}]_{,𝜇}.

From (2.18) it follows that 𝑅_{𝜇𝜈} = 𝑅_{𝜈𝜇}. Some authors define Riemann curvature tensor and Ricci tensor with opposite signs. The *Ricci scalar* 𝑅 is defined by

(2.20) 𝑅 = 𝑔^{𝜇𝜈}𝑅_{𝜇𝜈}.

By contracting the Biancchi identity on the pair of indices 𝜇𝜈 and 𝜎𝜌 (that is, multiplying by 𝑔^{𝜇𝜈} and 𝑔^{𝜎𝜌}). One can deduce the identity

(2.21) (𝑅^{𝜇𝜈} - 1/2 𝑔^{𝜇𝜈}𝑅)_{;𝜈}.

The tensor 𝐺^{𝜇𝜈} = 𝑅^{𝜇𝜈} - 1/2 𝑔^{𝜇𝜈}𝑅 is sometimes called the *Einstein tensor*.

We are now in a position to write down the fundamental equations of general relativity. These are Einstein's equations given by [Landau-Lifshitz 1975 § 95.5]:

(2.22) 𝑅_{𝜇𝜈} - 1/2 𝑔_{𝜇𝜈}𝑅 = 8𝜋𝐺/𝑐^{4} 𝑇_{𝜇𝜈},

where 𝑇_{𝜇𝜈} is the *energy-momentum tensor* of the source producing the gravitational field and 𝐺 is Newton's gravitational constant. For a perfect fluid, 𝑇_{𝜇𝜈} takes a following form [Landau-Lifshitz 1975 §94.9]:

(2.23) 𝑇^{𝜇𝜈} = (𝜌 + 𝑝)𝑢^{𝜇}𝑢^{𝜈} - 𝑝𝑔^{𝜇𝜈} or 𝑇_{𝜇𝜈} = (𝜌 + 𝑝)𝑢_{𝜇}𝑢_{𝜈} - 𝑝𝑔_{𝜇𝜈},

where 𝜌 is the *mass-energy dnsity*, 𝑝 is the pressure and 𝑢^{𝜇} is the four-velocity of the matter given by

(2.24) 𝑢^{𝜇} = 𝑑𝑥^{𝜇}/𝑑𝑠,

where 𝑥^{𝜇}(𝑠) describes the worldline of matter in terms of the proper time 𝜏 = 𝑠/𝑐 along the worldline. We will consider some other forms of the energy-momentum tensor than (2.23). Refer to [Dirac 1975 p.45; 𝑇^{𝜇𝜈} = 𝜌𝜐^{𝜇}𝜐^{𝜈}]. From (2.21) Einstein's equations are compatible with the following equations

(2.25) 𝑇^{𝜇𝜈}_{;𝜈} = 0,

which is the equation for the conservation of mass-energy and momentum.

The equations of motion of a particle in a gravitational field are

(2.26) 𝑑^{2}𝑥^{𝜇}/𝑑𝑠^{2} + 𝛤^{𝜇}_{𝜆𝜈} (𝑑𝑥^{𝜆}/𝑑𝑠) (𝑑𝑥^{𝜈}/𝑑𝑠) = 0.

Geodesics can also be introduced through the concept of parallel transfer. Consider a curve 𝑥^{𝜇}(𝜆), where 𝑥^{𝜇} are suitably differentiable functions of the real parameter 𝜆, varying over some interval of real time. For an arbitrary vector field 𝑌^{𝜇} its covariant derivative along the curve is 𝑌^{𝜇}_{;𝜈}(𝑑𝑥^{𝜈}/𝑑𝜆). The 𝑌^{𝜇} is said to be parallelly transported along the curve if

(2.27) 𝑌^{𝜇}_{;𝜈}(𝑑𝑥^{𝜈}/𝑑𝜆) = 𝑌^{𝜇}_{,𝜈} (𝑑𝑥^{𝜈}/𝑑𝜆) + 𝛤^{𝜇}_{𝜈𝜎}𝑌^{𝜎} (𝑑𝑥^{𝜈}/𝑑𝜆) = 𝑑𝑌^{𝜇}/𝑑𝜆 + 𝛤^{𝜇}_{𝜈𝜎}𝑌^{𝜎} (𝑑𝑥^{𝜈}/𝑑𝜆) = 0.

The curve is said to be a geodesic curve if the tangent vector is transported parallelly, that is, putting (𝑌^{𝜇} = 𝑑𝑥^{𝜇}/𝑑𝜆), if

(2.28) 𝑑^{2}𝑥^{𝜇}/𝑑𝜆^{2} + 𝛤^{𝜇}_{𝜈𝜎} (𝑑𝑥^{𝜈}/𝑑𝜆) (𝑑𝑥^{𝜎}/𝑑𝜆) = 0.

The curve, or a portion of it, is time-like, light-like or space-like according to whether 𝑔_{𝜇𝜈} (𝑑𝑥^{𝜇}/𝑑𝜆) (𝑑𝑥^{𝜈}/𝑑𝜆) > 0, = 0, or < 0.

**2.2 Killing vectors**

Einstein's exterior equation 𝑅

In the following we will sometimes we write 𝑥, 𝑦, 𝑥' for 𝑥

(2.29) 𝑔'

Therefore

(2.30 a, b, c) 𝑔'

(2.31) 𝑥'

with 𝛼 constant and ∣𝛼∣ ≪ 1. Substituting in (2.30 c) and expanding by Taylor's formula and neglecting terms involving 𝛼

(2.32) 𝑔

(2.33) 𝜉

Equation (2.33) is Killing's equation and a vector field 𝜉

We now derives a property of Killing vectors which we might use later. Let 𝜉

(2.34) 𝜁

We will show that 𝜁

(2.35) 𝜁

(2.36) 𝜁

(2.37) 𝜉

(2.38 a, b, c) 𝜉

(23.8 a) - (23.8 b) = 0.

The other terms can be canceled by using Killing's equation. For example

(2.39) 𝜉

which cancels the second term and last term in (2.36), and so on. Thus 𝜁

(2.40) [𝜉

* The main textbook: Islam. J.N. *An introduction to Mathematical Cosmology* (2nd edition, Cambridge University Press 2002)

For General Relativity, refer to P.A.M. Dirac *General Theory of Relativity* [Dirac 1975], which is at [88] here and thus described briefly.

Also refer to L.D. Landau and E.M. Lifshitz *The Classical Theory of Fields* (4th edition) [Landau Lifshitz 1975], a classic reference.

** Refer to Steven Weinberg *Gravitation and Cosmology* [Weinberg 1972], which is one of the best reference for GR with cosmology

Also refer to H.C. Ohanian and R. Ruffini *Gravitation and Spacetime* [Ohanian Ruffini 1994] describing Killing vectors.

^{※} attention: some rigorous derivation might be required