Cheetox said:
Is the equation your referring to that could help us further the FRW equation? How would the effect of dark energy manifest itself in that?
Yes! I am going to use a rather simplified form of the equation.
There's a simple intuitive derivation of a universe model with matter and no dark energy in the PF library, using Newtonian reasoning rather than relativity. It happens to get the right answer, however, so it is often use to help explain this. See
Friedmann Equation.
I am going to use equations with Ω factors for the different energy terms. These are fractions of critical density. Equations similar to the form I am using are in the appendix of
Expanding Confusion by Lineweaver and Davis (arXiv:astro-ph/0310808v2).
Added in edit. I have just seen ([post=2529194]msg #2[/post] in thread "light geodesics in FLWR") an excellent pedagogical reference which explains these equations even better. See Distance measures in cosmology, by David Hogg, at arXiv:astro-ph/9905116v4.
Here's a simple equation using matter and dark energy, and for completeness a curvature term as well, though this term is thought to be very close to zero, for an essentially flat universe.
\frac{\dot{a}}{a} = H_0 \sqrt{\Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_\Lambda}
In this equation, a is the "scale factor" of the universe, defined to be 1 in the present, H
0 is the Hubble constant in the present, and the Ω terms are constants required to add up to 1. The Ω
m is the amount of matter, as a fraction of critical density, and the Ω
Λ term is the dark energy fraction, at the time when a=1. The current best estimate has values close to
- Ωm = 0.27
- ΩΛ = 0.73
- Ωk = 1 - Ωm - ΩΛ = 0
Hence:
\begin{align*}<br />
\frac{\dot{a}}{a} & = H_0 \sqrt{\Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_\Lambda} \\<br />
\dot{a} = \frac{da}{dt} & = H_0 \sqrt{\Omega_m a^{-1} + \Omega_k + \Omega_\Lambda a^2} \\<br />
& = H_0 \sqrt{1 + \Omega_m (a^{-1} -1) + \Omega_\Lambda (a^2-1)}<br />
\end{align*}
(1) Age of the universe
(I have edited this to make t=0 the origin of the universe, and t=T the present epoch.)[/size]
Let’s assume t=0 at the origin of the universe. Hence if T is the age of the universe, it is the value of t now. From the scale factor definition, we have a(0) = 0 and a(T) = 1.
\begin{align*}<br />
\frac{da}{dt} &= H_0 \sqrt{1 + \Omega_m (a^{-1} -1) + \Omega_\Lambda (a^2-1)} \\<br />
T = \int_0^T dt & = \frac{1}{H_0} \int_0^1 \frac{da}{\sqrt{1 + \Omega_m (a^{-1} -1) + \Omega_\Lambda (a^2-1)}}<br />
\end{align*}
Caveat: in some cases we cannot satisfy the boundary condition a(0) = 0; this means there is no initial singularity in the model. Ω
m = 0, Ω
Λ = 1 is the main example. In such cases there is no age to the universe, and I shall apply the boundary condition a(0) = 1, to make 0 the time co-ordinate of the present. But let's not worry about that case just yet. I'll presume a(0) = 0 and a(T) = 1 for the time being.
(2) Distance to particle horizon
(Added in edit: I have corrected these formulae some hours after posting to use the time co-ordinate consistently.)[/size]
As for distance to the particle horizon, I can use d to represent the "co-moving" distance to a photon, and this differential equation
\frac{dd}{dt} = c/a
Hence, for a photon we observe in the present, when the time co-ordinate is T, the co-moving distance it has traveled since time t is
\begin{align*}<br />
d(t) & = c \int_t^T \frac{dt}{a} \\<br />
& = c \int_{a(t)}^1 \frac{da}{\dot{a}a} \\<br />
& = \frac{c}{H_0} \int_{a(t)}^1 \frac{da}{a \sqrt{1 + \Omega_m (a^{-1} -1) + \Omega_\Lambda (a^2-1)}} \end{align*}
This gives co-moving distance from our present location to co-moving surface that emitted the photon back at time t. The "proper distance" now is just co-moving distance d times the present scale factor, and so this same value is used as the current "proper" distance to whatever surface emitted the photon. The particle horizon D
ph is the largest possible value of this distance, given by starting the photon at t=0. The distance to the surface of last scattering is given by starting the photon when a(t) = 1/1100, because this is the scale change since last scattering.
D_{ph} = \frac{c}{H_0} \int_0^1 \frac{da}{a \sqrt{1 + \Omega_m (a^{-1} -1) + \Omega_\Lambda (a^2-1)}} \end{align*}
If everyone is still on board, I'll go ahead next with some solutions to these equations.
Cheers -- sylas