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# Friedmann Acceleration Equation

Posted Jul4-11 at 05:52 AM by stevebd1
Updated Jul7-11 at 02:47 AM by stevebd1

Continued from Critical Density

Equations that demonstrate a flat and accelerating universe-

Friedmann equation-

$$H^2=\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}$$

Friedmann acceleration equation-

$$\dot{H}+H^2=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)+\frac{ \Lambda c^2}{3}$$

where $P$ is pressure and the dots indicate derivatives by proper time.

Both equations can be rewritten where-

$$\rho'\Rightarrow \rho_m+\frac{\Lambda c^2}{8\pi G}=(\rho_m+\rho_\Lambda)$$

$$P'\Rightarrow P_m-\frac{\Lambda c^4}{8\pi G}=(P_m-P_\Lambda)$$

where the equation of state for dark energy is $w=-1$ and for ordinary and dark matter, $w=0$.

Friedmann equation becomes-

$$H^2=\frac{8\pi G}{3}\rho'-\frac{kc^2}{a^2}$$

which reduces to the critical density equation when $k=0$.

Friedmann acceleration equation becomes-

$$\dot{H}+H^2=-\frac{4\pi G}{3}\left(\rho'+\frac{3P'}{c^2}\right)$$

which would normally show that both energy density and pressure would cause a deceleration in the expansion of the universe though the inclusion of the cosmological constant (or dark energy or vacuum energy) means the universe accelerates.

$$\frac{\dot{H}}{H^2}=-(1+q)$$

where q is the deceleration parameter-

$$q= -\frac{\ddot{a}}{\dot{a}^2}a = \frac{1}{2\rho_c} \left(\rho'+\frac{3P'}{c^2} \right)=\frac{1}{2} \Omega (1+3w)$$

where in this case $w=P'/(\rho'c^2)$, the equation of state of the universe, which in the case of a universe which is virtually flat, can be rewritten as-

$$q=\frac{1}{2} (1+3w)$$

This implies that the universe is decelerating for any cosmic fluid with equation of state $w$ greater than -1/3.

Quote:
..A value of q greater than 0.5 indicates that the expansion is decelerating quickly enough for the universe eventually to collapse. A value less than 0.5 indicates that the expansion will continue for ever. In models with a cosmological constant, q can even be negative, indicating an accelerated expansion, as in the inflationary universe.
For calculations for Lambda, see Dark Energy

http://en.wikipedia.org/wiki/FLRW#Solutions
http://en.wikipedia.org/wiki/Deceleration_parameter
http://scienceworld.wolfram.com/phys...Parameter.html