Continued from first post.
Last time I provided a rough derivation of the Friedmann equation for our expanding universe as,
(a'/a) ^2 = ( 8π/3)ρG - k*(c/a) ^2
Where a(t) is the magnification factor or scale factor for the expansion of space between two points in the universe and
a' = da/dt.
So a'/a is the Rate of increase or decrease of expansion of space divided by the current expansion factor value.
ρ is the uniform mass density of the universe.
And a more rigorous derivation from General Relativity shows that the parameter k,
k= - 2U/(mc^2x^2)
Is the curvature of space, with U being the total energy of a small region with mass m at a distance x from the observer in the comoving frame that expands with space. Note that since energy is conserved and the distance x between two points in space remains the same in the comoving frame that expands with space, k is a constant.
Thus the Friedmann equation relation the expansion rate with the curvature and mass density of the universe.
Consider now that you are an observer and you are looking at a spherical region of space with unit radius in the comoving frame centered around you. Then the radius in the physical frame (which does not expand with expansion of the universe) increases with expansion of space as,
r= a(t) * 1 = a(t)
Look at the spherical region of space below and imagine you are in the middle. As space expands, this region blows up, so actual radius is increasing along with the expansion scale factor a(t).
This expansion is just like the expansion of an ideal frictionless gas. Hence for a small increase in volume dV of this spherical region of space the 1st law of thermodynamics energy conservation equation gives
dW + dE =0
Where dW = p*dV is the work done due to expansion of the spherical volume of space by an amount dV and p is the pressure inside this volume. dE is the change in the internal energy content of this volume due to expansion. In terms of rates with respect to time, we get,
dW/dt + dE/dt =0
Now volume of sphere is
V = (4/3)*πa^3
So
dW/dt = p*dV/dt = p*(4πa^2)*(da/dt)
By Einstein,
E =mc^2
The total mass inside the spherical volume is given by volume of the sphere multiplied by the mass density. Hence,
m= ρV
So
dE/dt = (ρc^2)*dV/dt + (dρ/dt) * c^2*V
Thus the 1st law equation
dE/dt + dW/dt = 0
Gives
ρ' + 3*(a'/a)*(ρ + p/c^2) = 0
where ρ' = dρ/dt is the rate of change of Mass density due to expansion
a' = da/dt is the expansion rate of space
p = pressure of matter, radiation etc. in the universe.
This equation is called the fluid equation for the universe and is the second most important relationship governing the behavior of the expanding cosmos after the Friedmann equation. It relates the mass density, expansion rate and pressure together.
Last time I provided a rough derivation of the Friedmann equation for our expanding universe as,
(a'/a) ^2 = ( 8π/3)ρG - k*(c/a) ^2
Where a(t) is the magnification factor or scale factor for the expansion of space between two points in the universe and
a' = da/dt.
So a'/a is the Rate of increase or decrease of expansion of space divided by the current expansion factor value.
ρ is the uniform mass density of the universe.
And a more rigorous derivation from General Relativity shows that the parameter k,
k= - 2U/(mc^2x^2)
Is the curvature of space, with U being the total energy of a small region with mass m at a distance x from the observer in the comoving frame that expands with space. Note that since energy is conserved and the distance x between two points in space remains the same in the comoving frame that expands with space, k is a constant.
Thus the Friedmann equation relation the expansion rate with the curvature and mass density of the universe.
Consider now that you are an observer and you are looking at a spherical region of space with unit radius in the comoving frame centered around you. Then the radius in the physical frame (which does not expand with expansion of the universe) increases with expansion of space as,
r= a(t) * 1 = a(t)
Look at the spherical region of space below and imagine you are in the middle. As space expands, this region blows up, so actual radius is increasing along with the expansion scale factor a(t).
This expansion is just like the expansion of an ideal frictionless gas. Hence for a small increase in volume dV of this spherical region of space the 1st law of thermodynamics energy conservation equation gives
dW + dE =0
Where dW = p*dV is the work done due to expansion of the spherical volume of space by an amount dV and p is the pressure inside this volume. dE is the change in the internal energy content of this volume due to expansion. In terms of rates with respect to time, we get,
dW/dt + dE/dt =0
Now volume of sphere is
V = (4/3)*πa^3
So
dW/dt = p*dV/dt = p*(4πa^2)*(da/dt)
By Einstein,
E =mc^2
The total mass inside the spherical volume is given by volume of the sphere multiplied by the mass density. Hence,
m= ρV
So
dE/dt = (ρc^2)*dV/dt + (dρ/dt) * c^2*V
Thus the 1st law equation
dE/dt + dW/dt = 0
Gives
ρ' + 3*(a'/a)*(ρ + p/c^2) = 0
where ρ' = dρ/dt is the rate of change of Mass density due to expansion
a' = da/dt is the expansion rate of space
p = pressure of matter, radiation etc. in the universe.
This equation is called the fluid equation for the universe and is the second most important relationship governing the behavior of the expanding cosmos after the Friedmann equation. It relates the mass density, expansion rate and pressure together.
