Rewriting the Friedmann Equation

[SN_1987A]

Homework Statement

Consider a universe with Cold Matter, Radiation, and Dark Energy satisfying an equation of
state $p = w\rho.$

Variables: p is the pressure of the universe, w is a constant, and $\rho$ is the density of the universe.

a) Show that the Friedmann equation can be rewritten as
$H^2(t) = H_0^2(\Omega_{DE}(\frac{a_0}{a(t)})^{3(1+w)} + \Omega_K(\frac{a_0}{a(t)})^2 + \Omega_M(\frac{a_0}{a(t)})^3 + \Omega_R(\frac{a_0}{a(t)})^4)$

Variables: $H(t)= \frac{\dot{a}}{a}$ is the Hubble constant, $H_0$ is today's value of H, a(t) and $a_0$ are the scale factor at time t and today, respectively, and the Omegas are fractions of the critical density for Dark Energy (DE), Curvature (K), Matter(M), and Radiation (R).

b) For the case $w < \frac{-1}{3},$ which component of energy density is likely to dominate the early universe? How about the late universe?

Homework Equations

(1) $\dot{a}^2 + K = \frac{8 \pi G \rho a^2}{3}$
(2) $\dot{\rho} = \frac{-3\dot{a}}{a}(p+\rho)$
(3) $\Omega_K = \frac{-K}{a_0^2 H_0^2}$

The Attempt at a Solution

For Part a),

Putting in the given equation for pressure, (2) can be solved to obtain

(4) $\rho = \rho_0 \frac{a_0}{a}^{3(1+w)}$

where $\rho_0$ is the current density, which itself is just the sum of the densities of dark energy, matter, and radiation at today's time.

(5) $\rho_0 = \rho_{DE0} + \rho_{M0} + \rho_{R0}$

Plugging (5) into (4), (4) into (1), and dividing (1) by $a^2$ yields

(6) $H^2 = \frac{8 \pi G}{3} (\rho_{DE0} + \rho_{M0} + \rho_{R0} ) (\frac{a_0}{a})^{3(1+w)} - \frac{K}{a^2}$

The current densities are related to their fraction of the current critical densities by

(7) $\rho_{i0} = \frac{3H_0^2 \Omega_i}{8 \pi G}$
where i is Dark Energy, Matter, or Radiation

Putting (7) into each of the densities in (6), and pulling out a factor of $\frac{3H_0^2}{8 \pi G}$ gives

(8)$H^2 = H_0^2 (\Omega_{DE} + \Omega_{M} + \Omega_{R} ) (\frac{a_0}{a})^{3(1+w)} - \frac{K}{a^2}$

which, solving (3) for K and putting into (8) gives an answer very close to the solution

(9) $H^2 = H_0^2 (\Omega_{DE} + \Omega_{M} + \Omega_{R} ) (\frac{a_0}{a})^{3(1+w)} - \Omega_K (\frac{a_0}{a})^2$

Problem
This would get me the desired result if I could choose w as $\frac{1}{3}$ for the radiation term, and 0 for the matter term, which are the values of w that solve radiation and matter dominated universes, respectively, but I don't see how I would be allowed to just simply say "on this term, w is 0, and on this one, it's $\frac{1}{3}$, and for this one, I'll leave it as w".

My Solution to Part b)

When $w < \frac{-1}{3},$, using the solution in part a), in the early universe, when $\frac{a_0}{a}$ is large, the Radiation term (largest exponent) will dominate. When $\frac{a_0}{a}$ is approximately 1, the Dark Energy term (which has the largest value of the density fractions), should dominate.

Problem
Explicitly, I haven't been given values of the Omegas, which means I've just tailored an answer to what we know about our universe: radiation dominated while young, dark energy dominated now.

 

[Nate810]

But, without knowing the values of the Omegas, this answer might not be generalizable to all w < \frac{-1}{3}.