The Friedman-Lemaitre-Robertson-Walker (FLRW) equations can be derived in an elementary way from Newton’s laws, where the energy in the motion of a density of mass-energy in a region is determined by the gravitational potential. The FLRW energy (so called energy) equation for the evolution of a scale parameter of spatial distance $a$ derived from the metric is, $$ \Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi}{3}\rho~-~\frac{k}{a^2} $$ where $\rho$ is the energy density. The Hubble parameter or constant with space at each time is $H~=~{\dot a}/a$. We set $k~=~0$ for a flat space $R^3$ to match observations, and which recovers what is derived from Newton’s laws. Energy density for photons scales inversely with the length of the box. The box is thought of as a resonance cavity that is equivalent to a situation where the number of photons that leave is approximately equal to the number of photons that enter. During the radiation dominated period things were in a near equilibrium, so this is not out of line with some physical reasoning. In a stat-mech course an elementary problem of N-photons in a box uses the same logic, the energy of the photons scales inversely with the size of the box. So the energy of photons $E~=~hc/\lambda$, and the wave length scales with the scale factor a. So the density scales as $\rho~\sim~hc/a^4$.

So with this et up let us propose a time dependency on the scale factor a with time $a~\sim~t^n$. Put this into the "energy equation" and turn the crank and you find that $n~=~1/2$. The scale factor grows as the square root of time. This is an energy equation, and the balance tells us that the loss of energy in photons is equal to the gain in gravitational potential energy. This connects well with Newtonian analysis and the Pound-Rebka experiment.

We may continue further, for the photons in a box exert a pressure on the sides of the box $p~=~F/a^2$, and the force induces an increment of change in the size of the box $dE~=~Fdx$. The force is distributed on 3 different directions and so $p~=~ρ/3$. This may then be used in the equation $pV~=~NkT$ to find that for $p~\sim~a^{-4}$ and $V~\sim~a^3$ with the above $E~\sim~1/\lambda$ that $\lambda~\sim~1/T$, which is Wein's law for the wavelength as the peak of the BB curve. The proportionality of the energy density with scale factor and temperature also gives $E~\sim~T^4$. So this physics is remarkably in line with laboratory understanding of the basic thermodynamics of radiation.

The matter contribution scales as $a^{-3}$, which was smaller than the radiation contribution for a time. Around 380,000 years into the evolution of the universe the matter density surpassed the radiation density. The CMB demarks this transition in the mass-energy which dominated the universe. The above dynamics still apply for photons, but radiation is now a minor player in the spacetime structure of the universe.

The Friedman-Lemaitre-Robertson-Walker (FLRW) equations can be derived in an elementary way from Newton’s laws, where the energy in the motion of a density of mass-energy in a region is determined by the gravitational potential. The FLRW energy (so called energy) equation for the evolution of a scale parameter of spatial distance $a$ derived from the metric is, $$ \Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi}{3}\rho~-~\frac{k}{a^2} $$ where $\rho$ is the energy density. The Hubble parameter or constant with space at each time is $H~=~{\dot a}/a$. We set $k~=~0$ for a flat space $R^3$ to match observations, and which recovers what is derived from Newton’s laws. Energy density for photons scales inversely with the length of the box. The box is thought of as a resonance cavity that is equivalent to a situation where the number of photons that leave is approximately equal to the number of photons that enter. During the radiation dominated period things were in a near equilibrium, so this is not out of line with some physical reasoning. In a stat-mech course an elementary problem of N-photons in a box uses the same logic, the energy of the photons scales inversely with the size of the box. So the energy of photons $E~=~hc/\lambda$, and the wave length scales with the scale factor a. So the density scales as $\rho~\sim~hc/a^4$.

So with this et up let us propose a time dependency on the scale factor a with time $a~\sim~t^n$. Put this into the "energy equation" and turn the crank and you find that $n~=~1/2$. The scale factor grows as the square root of time. This is an energy equation, and the balance tells us that the loss of energy in photons is equal to the gain in gravitational potential energy. This connects well with Newtonian analysis and the Pound-Rebka experiment.

We may continue further, for the photons in a box exert a pressure on the sides of the box $p~=~F/a^2$, and the force induces an increment of change in the size of the box $dE~=~Fdx$. The force is distributed on 3 different directions and so $p~=~ρ/3$. This may then be used in the equation $pV~=~NkT$ to find that for $p~\sim~a^{-4}$ and $V~\sim~a^3$ with the above $E~\sim~1/\lambda$ that $\lambda~\sim~1/T$, which is Wein's law for the wavelength as the peak of the blackbody curve. The proportionality of the energy density with scale factor and temperature also gives $E~\sim~T^4$. So this physics is remarkably in line with laboratory understanding of the basic thermodynamics of radiation.

The matter contribution scales as $a^{-3}$, which was smaller than the radiation contribution for a time. Around 380,000 years into the evolution of the universe the matter density surpassed the radiation density. The CMB demarks this transition in the mass-energy which dominated the universe. The above dynamics still apply for photons, but radiation is now a minor player in the spacetime structure of the universe.