Curvature of the Universe: Analysis

Introduction
1.1 Reviewing of General Relativity
1.1.1 Metric Tensor
The equation which describes the relationship between two given points is called metric and is given by

Where interval of space-time between two neighboring points, connects these two points and are the components of contra variant vector. Through the function, any displacement between two points is dependent on the position of them in coordinate system.
The displacement between two points in rectangular coordinates system is independent of their components due to homogeneity, so metric is given by

Where are the space-time coordinates, is speed of light and is metric for this case and is given by

Through the coordinates transformation from rectangular coordinates,, to curved coordinates system the components ofin a curved coordinates system can be found . For constructing rectangular coordinates system in a curved coordinates if space-time is locally flat then it is possible to that locally. From rectangular coordinates system defined locally in a point of a curved space-time to a curved coordinates system can be written as

So in this way we can find local values of metric tensor
Three important properties of metric tensor are:

is symmetric
so we have

metric tensors are used to lowering or raising indices

1.1.2 Riemann Tensor, Ricci Tensor, Ricci Scalar
The tool which plays an important role in identifying the geometric properties of spacetime is Riemann (Curvature) tensor. In terms of Christoffel symbols it is defined as:

Where .If the Riemann Tensor vanishes everywhere then the spacetime is considered to be flat. In term of spacetime metric Riemann Tensor can also be written as:

thus useful symmetries of the Riemann Tenser are:

so due to above symmetries, the Riemann tensor in four dimensional spacetime has only 20 independent components. Now simply contracting the Riemann Tensor over two of the indices we get Ricci Tensor as:

above equation is symmetric so it has at most 10 independent components. Now contracting over remaining two indices we get scalar known as Ricci Scalar.

Another important symmetry of Riemann Tensor is Bianchi identities

This after contracting leads to

1.1.3 Einstein Equation
The Einstein equation is the equation of motion for the metric in general theory of relativity is given by:

Where is stress energy momentum tensor and is Newton’s constant of Gravitation. Thus the left hand side of this equation measures the curvature of spacetime while the right hand side measures the energy and momentum contained in it.Taking trace of both sides of above equation we obtain

using this equation in eq. ( ), we get

In vacuum so for this case Einstein equation is

We define the Einstein tensor by

Taking divergence of above eq. we get

1.1.4 Conservation Equations for Energy momentum Tensor
In general relativity two types of momentum-energy tensor,are commonly used: dust and perfect fluid.
1.4.1 Dust: It is simplest possible energy-momentum tensor and is given by

The 4-velocity vector for commoving observer is given by, so energy momentum tensor is given by

It is an approximation,of the universe at later times when radiation is negligible
1.4.2 Perfect fluid: If there is no heat conduction and viscosity then such type of fluid is perfect fluid and parameterized by its mass density and pressure and is given by

It is an approximation of the universe at earlier times when radiation dominates so conservation equations for energy momentum tensor are given by

In Minkowski metric it becomes

1.1.5 Evolution of Energy-Momentum Tensor with Time
We can use eq. () to determine how components pf energy-momentum tensor evolved with time. The mixed energy-momentum tensor is given by:

and its conservation is given by

Consider component:

Now all non-diagonal terms of vanish because of isotropy so in the first term and in the second term so

For a flat, homogeneous and isotropic spacetime which is expanding in its spatial coordinate’s by a scale factor, the metric tensor is obtained from Minkowski metric is given by:

The Christoffel symbol by definition

Because
Because the only non-zero is so from eq. () conservation law in expanding universe becomes

after solving above equations we get

above equation is used to find out for both matter and radiation scale with expansion. In case of dust approximation we have so

So energy-density of matter scale varies as .Now the total amount of matter is conserved but volume of the universe goes as so
In case of radiation so from eq.() we obtain

Which implies that, science energy density is directly proportional to the energy per particle and inversely proportional to the volume, that is, because so the energy per particle decreases as the universe expands.
1.2 Cosmology
In physical cosmology, the cosmological rule is a suspicion, or living up to expectations theory, about the expansive scale structure of the universe. Throughout the time of Copernicus, much data were not accessible for the universe with the exception of Earth, few stars and planets so he expected that the universe might be same from all different planets likewise as it looked from the Earth. It suggests isotropy of the universe at all focuses. Once more, a space which is isotropic at all focuses, is likewise homogeneous. Copernicus rule and this result about homogeneity makes the Cosmological rule (CP) which states that, at a one-time, universe is homogeneous and isotropic. General covariance ensures validity of Cosmological Principle at other times also.
1.2.1 Cosmological metric:
Think about a 3D circle inserted in a 4d “hyperspace”:

where is the radius of the 3D sphere. The distance between two points in 4D space is given by

solving we get

now becomes

In spherical coordinates

Finally we obtain

We could also have a saddle with or a flat space. In literature shorthanded notation is adapted:

To isolate time-dependent term, make the following situation:

Then

where

If we introduce conformal time (arc parameter measure of time) as

then we can express the 4D line element in term of FRW metric:

1.2.2 Friedmann Equation:
We can now figure out Einstein field mathematical statement for perfect fluid. All the calculations are carried out in comoving frame where

and energy-momentum tensor is given by

Raising the index of the Einstein tensor equation

we get

After contracting over indices and we get

so Einstein’s Equation can be written as

It is easily found for perfect fluid

finally we obtain the components of Ricci tenser

The components are

and components are

To get a closed system of equations, we need a relationship of equation states which relates and so solving

At this point when we joined together with equation 62 comparisons in the connection of energy-momentum tensor and the equation of states, we get a closed frame work of Friedmann equations:

1.2.3 Solutions of Friedmann Equations:
We are going to comprehend Friedmann equation for the matter dominated and radiation dominated universe and get the manifestation of scale factor. From the definition of Hubble’s law

Matter Dominated Universe: : It is showed by dust approximation

As both and, for flat universe (), ( an) for . When combined with equation, this yields critical density

Currently it value is (we used).The quantity provide relationship between the density of the universe and the critical density so it is given by
Now the second Friedmann equation for matter dominated Universe becomes

so lastly

Radiation-dominated Universe: It is showed by perfect fluid approximation with
The second Friedmann Equation becomes

Flat Universe
Matter Dominated Universe (dust approximation)
The first Friedmann equation becomes

At the Big bang Using convention and universe flat condition we finally get

Now we can calculate the age of universe, which corresponds to the Hubble rate and scale factor to be:

Taking and we get
Years
Radiation-dominated:
The First Friedmann equation becomes

At the big bang and .Also we have

Closed Universe
Matter-dominated
The first Equation becomes

In term of conformal time we can rewrite the above integral as

After substituting and using equation

Then
but we have so we get

.
Now

but we have at sets. So we have now the dependence of scale factor in term of the time parameterized by the conformal time as

Radiation-dominated Universe:
The first Friedmann equation becomes

In term of conformal time we can re write the integral as

but we have conditions at sets so we get

and the requirement at sets , finally we have

Open Universe
Matter-dominated (dust approximation):
The first Friedmann equation

In term of conformal time we can rewrite the integral as

Take