Thermodynamics Of The Universe

The thermodynamics of the universe is dictated by which form of energy dominates it - Relativistic Particle s which are referred to as Radiation , or non-relativistic particles which are referred to as matter. The former are particles whose Rest Mass is zero or negligible compared to their energy, and therefore move at the speed of light or very close to it; The latter are particles whose Kinetic Energy is much lower than their Rest Mass and therefore move much slower than the speed of light. The intermediate case is not treated well Analytically .

ENERGY DENSITY IN THE EXPANDING UNIVERSE

If the universe is not undergoing a Phase Transition , one can approximate its Thermodynamics by neglecting interactions between particles, and assuming all the Energy is in the form of Heat . Then by the First Law Of Thermodynamics :

0 = dQ = dU + P dV

Where

Q

is the total heat which is assumed to be constant,

U

is the internal energy of the matter and radiation in the universe,

P

is the pressure and

V

the volume.

One then finds an equation for the Energy Density

u\equiv U/V

du = d({U\over V})={dU\over V}-U{dV\over V^2}=-(p+u){dV\over V} = -3(p+u){da\over a}

where in the last equality we used the fact that the total volume of the universe is proportional to

a^3

a

being the Scale Factor of the universe.

In fact this equaion can be directly obtained from the euqations of motions governing the .

In the Comoving Coordinates ,

u

is equal to the Mass Density

ho

. For radiation,

p=u/3

whereas for matter

p< and the pressure can be neglected. Thus we get:

For radiation

du = -4u {da\over a}

thus

u

is proportional to

a^{-4}

For matter

du = -3u {da\over a}

thus

u

is proportional to

a^{-3}

This can be understood as follows: For matter, the Energy Density is equal (in our approximation) to the Rest Mass density. This is inversly proportional to the volume, and is therefore proportional to

a^{-3}

.
For Radiation , the Energy Density depends on the Temperature

T

as well, and is therefore proportional to

T a^{-3}

. As the universe expands it cools down, so

T

depends on

a

as well. In fact, since the Energy of a Relativistic Particle is inversely proportional to its Wavelength , which is proportional to

a

, the Energy Density of the Radiation must be proportional to

a^{-4}

.

From this discussion it is also obvious that the Temperature of radiation is inversly proportional to the Scale Factor

a

.

RATE OF EXPANSION OF THE UNIVERSE

Plugging this information to the Friedmann-Lemaître-Robertson-Walker Equations Of Motion and neglecting both the Cosmological Constant

\Lambda

and the curvatue parameter

k

, which is justified for the early universe (

a<<1

), one gets the following equation:

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Thermodynamics Of The Universe