This is a little thread where I will discuss the basic features of the Big Bang theory in a quantities fashion. I hope this will make discussions and debated on the Big Bang in other topic more well informed.
One of the key starting ideas in developing the theory of the universe is that on very large scales i. e. lengths spanning hundreds and thousands of Mega light years, the universe is roughly smooth in its average distribution of galaxies and other kinds of matter and Radiation. This feature has been verified by observations if galaxy distribution. What this means is that, on these large scales, any region of the universe looks the same as any other region of the universe and this is true whichever direction one looks.
The other key idea is based on observation that the universe is expanding. What is meant here is not that galaxies are moving through space away from each other... rather that space itself is blowing up (like a baking cake) taking the embedded galaxies (raisins) with it.
So let us begin with a simple model where we consider an observer in an expanding universe filled with a uniform mass density ρ. Since the universe looks the same everywhere and in every direction, we can take the location of this observer to be a center of the universe. In fact every location in this kind of uniform universe can function as a center equally well.
Now consider a small region of the universe at a distance r from the observer which contains a total mass of value m. By the theory of gravity, the gravitational force this mass m experiences as measured by the observer is only due to the total mass present in the spherical region of radius r centered around the observer. This mass M is the mass density multiplied by the volume of the sphere. So,
M = (4πr^3) /3 * ρ
Thus gravitational force on the mass m is,
F = GMm/r^2
The gravitational potential energy is,
V = - GMm/r
If the velocity of the mass m with respect to the observer is given by r', then kinetic energy of the mass is,
T = 0.5mr'^2
where r'= dr/dt
So, total energy of the mass m is
U =T + V = 0.5mr'^2 - ( 4πr^2/3)*ρGm
And this energy is conserved.
Now, this mass m is moving way from the observer because of the expansion of space. This is a bit like the zoom function when one is looking at a picture in the computer. If you zoom in, the entire picture expands so that each point is farther apart from the other. This feature of the expanding universe can be implemented in the model by implementing a coordinate system whose grids themselves expands with the space. Such a coordinate system is called a comoving coordinate system. A distance x between two points in the comoving coordinate system is related with the distance r between the same two points in the non-expanding physical coordinate frame by the relation,
r = a(t) * x
where a(t) is the scale factor that gives the rate of "zooming" or expansion rate of space with time. Since the comoving coordinate frame is also zooming with the universe, in terms of this new coordinates, the distance between observer and mass m does not change with time. Thus
x' = 0
So
r' = a'*x
where a' = da/dt
Using the new comoving coordinates, the total energy of mass m then becomes,
U = 0.5*m(a'*x) ^2 - ( 4π/3)*(a'*x)^2*ρGm
Defining the curvature of space
k= - 2U/(mc^2x^2)
the above expression can be written as,
(a'/a) ^2 = ( 8π/3)ρG - k*(c/a) ^2
This is the famous Friedmann equation that describes how our universe expands with time.
Next post will unpack important physical consequences of this equation.
One of the key starting ideas in developing the theory of the universe is that on very large scales i. e. lengths spanning hundreds and thousands of Mega light years, the universe is roughly smooth in its average distribution of galaxies and other kinds of matter and Radiation. This feature has been verified by observations if galaxy distribution. What this means is that, on these large scales, any region of the universe looks the same as any other region of the universe and this is true whichever direction one looks.
The other key idea is based on observation that the universe is expanding. What is meant here is not that galaxies are moving through space away from each other... rather that space itself is blowing up (like a baking cake) taking the embedded galaxies (raisins) with it.
So let us begin with a simple model where we consider an observer in an expanding universe filled with a uniform mass density ρ. Since the universe looks the same everywhere and in every direction, we can take the location of this observer to be a center of the universe. In fact every location in this kind of uniform universe can function as a center equally well.
Now consider a small region of the universe at a distance r from the observer which contains a total mass of value m. By the theory of gravity, the gravitational force this mass m experiences as measured by the observer is only due to the total mass present in the spherical region of radius r centered around the observer. This mass M is the mass density multiplied by the volume of the sphere. So,
M = (4πr^3) /3 * ρ
Thus gravitational force on the mass m is,
F = GMm/r^2
The gravitational potential energy is,
V = - GMm/r
If the velocity of the mass m with respect to the observer is given by r', then kinetic energy of the mass is,
T = 0.5mr'^2
where r'= dr/dt
So, total energy of the mass m is
U =T + V = 0.5mr'^2 - ( 4πr^2/3)*ρGm
And this energy is conserved.
Now, this mass m is moving way from the observer because of the expansion of space. This is a bit like the zoom function when one is looking at a picture in the computer. If you zoom in, the entire picture expands so that each point is farther apart from the other. This feature of the expanding universe can be implemented in the model by implementing a coordinate system whose grids themselves expands with the space. Such a coordinate system is called a comoving coordinate system. A distance x between two points in the comoving coordinate system is related with the distance r between the same two points in the non-expanding physical coordinate frame by the relation,
r = a(t) * x
where a(t) is the scale factor that gives the rate of "zooming" or expansion rate of space with time. Since the comoving coordinate frame is also zooming with the universe, in terms of this new coordinates, the distance between observer and mass m does not change with time. Thus
x' = 0
So
r' = a'*x
where a' = da/dt
Using the new comoving coordinates, the total energy of mass m then becomes,
U = 0.5*m(a'*x) ^2 - ( 4π/3)*(a'*x)^2*ρGm
Defining the curvature of space
k= - 2U/(mc^2x^2)
the above expression can be written as,
(a'/a) ^2 = ( 8π/3)ρG - k*(c/a) ^2
This is the famous Friedmann equation that describes how our universe expands with time.
Next post will unpack important physical consequences of this equation.
Always loved this particular study.
