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# Critical density

Posted Jan26-11 at 05:57 AM by stevebd1

Einstein field equations (EFE)-

$$G_{\mu\nu}+g_{\mu\nu}\Lambda=\frac{8\pi G}{c^4} T_{\mu\nu}$$

Gμν is the Einstein tensor of curvature (spacetime), gμν is the metric tensor, Λ is the cosmological constant, $8\pi$ is the concentration factor and Tμν is the energy tensor of matter (matter energy)

c and G are introduced to convert the quantity (which is expressed in physical units) to geometric units (G/c4 is used to convert units of energy into geometric units while G/c2 is used to convert units of mass, when mass is used instead of energy, the c4 is replaced with c2).

Using the EFE to establish Λ (presuming that gμν=1)

$$\Lambda =\frac{8\pi G}{c^4}u_{\Lambda}\ \equiv\ \ \frac{8 \pi G}{c^2}\rho_{\Lambda}$$

where uΛ is vacuum energy and ρΛ is vacuum density (basically, uΛ=ρΛ·c2)

The equation is more commonly known as-

$$\rho_{vac}=\frac{\Lambda c^2}{8\pi G}$$

where ρvac is essentially ρΛ

Critical density (ρc)-

The critical density is derived from the Friedmann equations which govern the homogeneous and isotropic expansion of space-

$$H^2=\frac{8 \pi G}{3} \rho - \frac{k c^2}{a^2}+\frac{\Lambda c^2}{3}$$

where H is a function of time (in this case, the inverse of Hubble time), G is the gravitaional constant, ρ is density in kg/m3, k is the spatial curvature parameter (-1 to +1, -k=hyperbolic, 0=flat, +k=hyperspherical), a is the time-scale factor (0 to 1, now=1) (k/a2 being the spatial curvature in any time-slice of the universe) and Λ is the cosmological constant. (G, Λ and c are universal constants, k is a constant throughout a solution and H, ρ, and a are a function of time.
Source- Friedmann equations

substituting for Λ, the equation can be rewritten-

\begin{flalign} H^2&=\frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2}+\frac{8 \pi G}{3}\rho_{\Lambda}\\[6mm] &=\frac{8 \pi G}{3} (\rho+\rho_{\Lambda}) - \frac{kc^2}{a^2} \end{flalign}

where $\rho_c=(\rho+\rho_{\Lambda})$, ρ representing baryonic and dark matter, ρΛ representing dark energy.

if we considered a flat universe, then k=0 and the equation can be reduced to-

$$H^2=\frac{8 \pi G}{3} \rho_c$$

rewriting the equation relative to ρc and based on a Hubble constant of ~70 (km/s)/Mpc, the critical density for a flat universe is-

$$\rho_{c}=\frac{3H^2}{8\pi G}=\frac{3\ \times\ (2.26\times10^{-18})^2}{8\ \times\ 3.14159\ \times\ 6.6742\times10^{-11}} = 0.918\times10^{-26}\ \text{kg/m}^3$$

The critical density of 0.918x10-26 kg/m3 is equivalent to 0.825 joules per km3.
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