When we look around us, at distant galaxies in any direction, we find that the spread of matter is quite the same. For example, you find 100,000 galaxies at 10 billion light-years within some solid angle in some direction. Now you look in the exact opposite direction at the same distance, within the same solid angle. Surprisingly, you will again find around 100,000 galaxies in that region! This property is called "Isotropy".

But what about moving to some other point. What if we shift to another galaxy and make the same observations? After all, there's nothing special about our position in this universe. Common sense tells us that the universe should maintain its isotropy at any point. This property is called "Homogeneity".

We conclude:

Isotropic = same in all directions

Homogeneous = same at all points

These are very important properties of our universe and together they form:

Cosmic Principle

We find that the expansion of our universe causes the light to have increased wavelength (equivalent to losing energy). So we find a relationship between the wavelength of scale factor at that time.

Using this relation, we construct a quantity called a redshift parameter "z". We define:

Redshift parameter (z) = (Change in wavelength)/(Original wavelength)

Which is further simplified as:

-------------------------->       1 + z = a(t2)/a(t1), such that t2 > t1

It is a very useful quantity and is helpful to relate observational quantities with theoretical ideas.

Now in the third chapter, we will construct a cosmological model using Newtonian mechanics. Before we dive into that, let's quickly discuss our favourite parameter. In the previous chapters, we defined the Hubble's parameter as:

H(t) = (da/dt)/a

Now if we consider an expanding grid and then take a ratio of the velocity and position of any galaxy, then we find that it is nothing but Hubble's parameter. We find that this ratio is independent of our coordinate system. Hence we say that Hubble's parameter is a constant in space. Usually, when we think about constants, we say they never change. At that time we consider 'in time constants will never change'. But for " H " we conclude:

Hubble's parameter is constant in space and not in time!

In this one, we derive the first Friedmann equation (Newtonian version). We consider a homogeneous universe and we choose a particular galaxy at "A". Now we know that there are billions of galaxies exerting force on this galaxy "A". But if we consider a frame with us at its centre and galaxy "A" at a distance D. Then the forces acting on "A" will only be due to galaxies with the sphere of radius D. Forces due to galaxies outside this sphere will simply cancel out.

Now we consider that the distance between us and "A" is changing according to a scale factor a(t). This gives rise to an equation, first Friedmann equation, that tells us about the dynamics of the scale factor.

In this one, we derive the second Friedmann equation (Newtonian version). We again consider a galaxy and all the other galaxies, in the given sphere, concentrated at the centre. We then use:

Total Energy ( E ) = Kinetic Energy + Potential Energy

If we take any object in a gravitational field with its kinetic part dominating, then that object can escape the gravitational field. Else, if the potential part is dominating, then that object may try to escape, but it will never escape the gravitational field. Similar is the case for our universe.

• Kinetic Energy > Potential Energy: Matter escapes, the universe keeps expanding

  
  

• Potential Energy > Kinetic Energy: Matter can not escape, the universe may expand up to some point but will collapse eventually

• Kinetic Energy = Potential Energy: Matter escapes after an infinite amount of time, the universe keeps expanding but at a slower & slower rate

For now, we consider a critical case where Kinetic energy = Potential energy i.e. E = 0. And then using the total energy equation, we find the second Friedmann equation.

Using the second Friedmann equation obtained in the last lecture, we understand the behaviour of a(t). We find that the a(t) grows with time as:

a(t) = c t^(2/3)

It is very important to know about the scale factor as we can use it to find the Hubble parameter, age of the universe and other concepts as well.

We consider a universe with only matter (no radiation, no dark energy) and then consider 2 cases of the energy equation. We know:

Kinetic Energy + Potential Energy = Total Energy

Case 1: Kinetic energy is dominant

In such a universe, the matter will have sufficient escape velocity to keep moving outwards. This gives us a forever expanding universe. The Hubble's parameter will be given as:

H^2 = (c1/a(t)^3) + (c2/a(t)^2)

Case 2: Potential energy is dominant

In such a universe, the matter will not have sufficient escape velocity. The universe grows initially and after a point, the scale factor (radius of universe) starts reducing, so the universe collapses into a singularity, i.e. we get a 'Big Crunch'. The Hubble's parameter will be given as:

H^2 = (c1/a(t)^3) - (c2/a(t)^2)

We consider a universe with only radiation. Using a result from the 'Redshift' lecture (wavelength of radiation is proportional to the scale factor), we replace the energy density parameter of our energy equation.

We arrive at:

H^2 = c/a(t)^4

This gives us the scale factor of radiation dominated universe as:

a(t)=t^(1/2)

In this lecture, we qualitatively analyze the critical Newtonian universe having radiation and matter. We discuss how the scale factor will grow at early and later times of the universe.

We find that:

H^2 = C_R/a(t)^4+C_M/a(t)^3

This gives us the scale factor in the early radiation-dominated universe as:

a(t)=t^(1/2)

And the scale factor of the late matter-dominated universe as:

a(t)=t^(2/3)

? Understanding the Free Particle Lagrangian: We start with the basics, introducing the Lagrangian formulation and its role in describing free particles' dynamics in a spacetime with a general metric.

? Defining the Christoffel Symbols: While exploring the Geodesic equation, we encounter the Christoffel symbols—a quantity that reveals the spacetime curvature. We define and explain these symbols concisely.

We derive the geodesic equation by starting with the action. Geodesic is a spacetime (4D) path along which any object moves under the influence of only gravity. An aeroplane does not follow a geodesic since other forces are acting along with the gravitation, ex. Atmospheric force.

We start with the action and recognize the Lagrangian. Then we make use of the Euler-Lagrange equation to find the geodesic equation.

In this lecture, you will see making very common Tensor Calculus mistakes and then fixing them. Do learn from them. The best way to learn anything is through mistakes!

In this video, we start with the regular form of the geodesic equation to write it in a very compact and smart way in terms of the covariant derivative. The geodesic equation gives us a spacetime path along which any freely falling object moves. Freely falling infers that there are no other forces except gravitational force acting on that object. For example, a rocket being launched from the surface of the Earth is not moving along a geodesic since there are forces other than the gravitational force, such as a force by the rocket engine and atmosphere. Any planet moving around the Sun moves along a geodesic. But remember that the orbit we will draw on paper for that planet will be a projection of 4D geodesic in 3D space.