Cosmological parameters and critical density

The general form of the equation of Friedmann can be rewritten while posing $H = \frac{\dot a}{a }$ :

$$H^2(t) = \frac{8 \pi G}{3}\sum_i \rho_i - \frac{k}{a^2} + \frac{\Lambda}{3}$$ (1.14)

By dividing the two members by $H^2$, one obtains:

$$\frac{k}{H^2 a^2} = \frac{8 \pi G}{3 H^2}\sum_i \rho_i + \frac{\Lambda}{3 H^2} - 1.$$ (1.15)

As the first member is sign of K, whatever the T, it is thus second. One can pose:

$$\Omega = \frac{8\pi G }{3 H_0^2} \sum_i \rho_i + \frac{\Lambda}{3 H^2}$$ (1.16)

who defines a critical density without dimension, dependent on time.

This one separates the on-dense universes from the under-dense universes and corresponds to a universe spatially flat (k=0). In the cosmological absence of constant, the value of $\Omega _ \Lambda$ as being the reports/ratios of the terms of the member of right-hand side of the equation of Friedmann and the square of the growth rate:

$$\Omega_i = \frac{8\pi G}{3H^2} \rho_i$$ (1.17)

$$\Omega_{\rm k} = -\frac{k}{a^2H^2}$$ (1.18)

$$\Omega_\Lambda = \frac{\Lambda}{3H^2}$$ (1.19)

One can thus rewrite the equation of Friedmann-Lemaître in the extremely simplified form:

$$1 = \sum_i \Omega_i + \Omega_\Lambda + \Omega_{\rm k}$$ (1.20)

or in an equivalent way:

$$1 - \Omega_{\rm k}= \sum_i \Omega_i + \Omega_\Lambda$$ (1.21)

The sum of the energy components of the member of right-hand side makes it possible to measure the geometry of the universe directly. It will be noticed that this relation is true whatever $t$. The equation of state, i.e. , the relation which binds the density of these fluids to their pressure, of the three types of fluids introduced previously, makes it possible to describe the dynamic evolution their densities. It is the knowledge of the equations of state of each fluid present in the equation of Friedmann which will make it possible to solve it.

Figure 1.3 shows the typical evolution of these densities according to the scale factor. Typical evolutions of the three fluids: matter, radiation and cosmological constant are respectively proportional to $a^{-4}$ and constant (weinberg1972, chapter 15).

By combining these relationships to the equation 1.14 , one can reexpress the value of the growth rate according to the current values of the cosmological parameters:

$$\left(\frac{H}{H_0}\right)^2 = \Omega_{m_0} (a^{-3} - a^{-2}) + \Omega_{r_0} (a^{-4} - a^{-2}) + \Omega_{\Lambda_0} (1 - a^{-2}) + a^{-2}$$ (1.22)

where $\Omega_{m_0}$ are respectively the current values of the reduced densities for the components of the radiation type (relativistic) and matter (not relativist). This equation makes it possible to see the influence of the various parameters. The matter and the radiation tend to slow down the expansion. On the other hand, the term corresponding to the cosmological constant tends to accelerate the expansion.