A straightforward non-anthropocentric approach

In the main Friedman Equation page, we saw that the solution t(R) of the Friedman equation depends on the following parameters:

1. A characteristic mass M_* that we can define from conservation of mass:
   M_* = $${4 \pi R^3 \over 3}$$$$\rho_{m}^{}$$

2. The cosmological constant
   $$\Lambda$$ = $${8 \pi G \rho_\Lambda \over c^2}$$

3. The curvature constant k = - 1, 0, or +1.

However, we can reduce the number of interesting continuous parameters by one by making the solution dimensionless: the second continuous parameter then just specifies the overall scale.

In fact there are two characteristic lengths in the problem, determined by M_* and $\Lambda$ respectively:

$${3 G M_* \over c^2} \Lambda \Lambda$$ (1)

If $\Lambda$ = 0, the maximum size reached by a closed universe is 2R_m/3, and in an open universe gravitational deceleration becomes negligible for R $\gg$ R_m (i.e. we approach the Milne solution). R_$\Lambda$ defines the scale on which the cosmological constant becomes important: if $\Lambda$ is negative, R_$\Lambda$ is of order the maximum size reached before recollapse begins; for positive $\Lambda$ it gives the size of the universe when it begins to grow exponentially.

We will use R_m to define scaled versions of radius and time:

$$\tau$$ (2)

In a critical universe R is arbitrary and hence so are M_* and R_m; one can use R_$\Lambda$ for scaling in this case, or else consider this as the limit when R_m $\rightarrow$ $\infty$.

The cosmological constant has dimensions inverse area, so we can make it dimensionless by defining

$$\lambda \Lambda \Lambda \left(\vphantom{{R_m \over R_\Lambda}}\right. {R_m \over R_\Lambda} \left.\vphantom{{R_m \over R_\Lambda}}\right)^{2}_{}$$ (3)

Using these parameters the solution of the Friedman equation can be written

$$\tau \int^{r}_{r_0} \sqrt{3 r \over 2 - 3kr + \lambda r^3}$$ (4)

The lower limit of the integral, r₀ , is the radius at our chosen t = 0; it makes sense to choose this as the time when r is a minimum, usually zero.

Typically the integral only exists over a finite range of r, limited by the denominator becoming zero. These limiting sizes correspond to turnarounds (if there is an upper limit to r) or bounces (for lower limits). After turnaround the size evolution reverses, corresponding to taking the negative square root; similarly before the bounce (and so at negative times with t = 0 defined as suggested above).

As promised, the only continuous parameter that affects the shape of the $\tau$(r) curve is $\lambda$. R_m only affects the overall scaling. In addition, of course, there are the three possibilities for k. Note that $\lambda$, k and R_m are constant throughout the history of each universe; in particular $\lambda$ is constant along each track in the $\Omega_{m}^{}$- $\Omega_{\Lambda}^{}$ plane.

$\lambda$ parameterizes how important the cosmological constant is when the universe has size of order R_m. For $\lambda$ $\ll$ 1 and k = - 1, there is a range of scales between R_m and R_$\Lambda$ when neither gravity nor cosmological constant have much effect, and the universe expands linearly, close to the Milne solution (see below). If k = + 1 and $\lambda$ $\ll$ 1, the universe recollapses under gravity before it ever reaches size R_$\Lambda$, and the cosmological constant has little effect. For $\lambda$ $\gg$ 1, the cosmological constant takes over while the universe is still in its initial Einstein-de Sitter phase (see below), and the scale R_m is no longer important. If $\lambda$ $\approx$ 1 and k = + 1, gravity and cosmological repulsion nearly balance as r approaches unity: eventually the universe either turns around to recollapse or begins to expand exponentially, but the times for this to occur can be much longer than either R_m/c or R_$\Lambda$/c.

If 0 < $\lambda$ < 1 and k = + 1, the integral solution is valid for two separate ranges of r: from zero to a turnaround radius which ranges from 2/3 (when $\lambda$ = 0) to 1 (when $\lambda$ = 1); and from a bounce radius to infinity. $\lambda$ = 1 is the loitering case, when the bounce radius is coincident with the turnaround radius; otherwise it is larger. Thus the two distinct regions of the $\Omega_{m}^{}$- $\Omega_{\Lambda}^{}$ plot with positive $\Lambda$ and k = + 1, which are outside the loitering trajectory, correspond to these two branches of the solution for the same values of $\lambda$.

The solutions for positive, negative and zero $\lambda$ have rather different properties; furthermore, there are the two branches for 0 < $\lambda$ < 1, k = + 1. This seems to give ten different families of solutions; however we will see that cases with k or $\lambda$ zero are actually single solutions, not families, and they can be treated as special cases of the five main families: furthermore we will find a way to continuously link the families with k = - 1 and +1, so that really there are only three families of curves to worry about, i.e. the bang-crunch solutions, the bounce solutions, and the continuous expansion solutions.

In terms of the more familiar cosmological parameters, using

$$\Omega_{k}^{}$$ = 1 - $$\Omega$$ = 1 - $$\Omega_{m}^{}$$ - $$\Omega_{\Lambda}^{}$$

we have

$${2 \vert \Omega_k\vert \over 3 \Omega_m} \left(\vphantom{{R \over R_0}}\right. {R \over R_0} \left.\vphantom{{R \over R_0}}\right) \tau {2 \Omega_m \over 3 \vert\Omega_k\vert^{3/2}} \left(\vphantom{ { t \over t_H } }\right. {t \over t_H} \left.\vphantom{ { t \over t_H } }\right)$$ r = (5)

$${3\over2} {c \over H_0} \Omega_{m}^{} \Omega_{k}^{} \Omega_{k}^{} \lambda {\Omega_\Lambda (\Omega_m/2)^2 \over (\vert\Omega_k\vert/3)^3}$$ R_m = (6)

We can convert back to $\Omega_{m}^{}$, $\Omega_{\Lambda}^{}$ and H using:

$$\Omega_{m}^{} {2 \over 2 - 3 k r + \lambda r^3} \Omega_{\Lambda}^{} {\lambda r^3 \over 2} \Omega_{m}^{}$$ (7)

$$\left(\vphantom{{c \over R_m}}\right. {c \over R_m} \left.\vphantom{{c \over R_m}}\right)^{2}_{} {\lambda \over 3 \Omega_\Lambda(r)} \left(\vphantom{{c \over R_m}}\right. {c \over R_m} \left.\vphantom{{c \over R_m}}\right)^{2}_{} {2 \over 3 r^3 \Omega_m(r)}$$ (8)

Notice that only H depends on the dimensional parameter R_m.