What is the Friedmann acceleration equation

[Greg Bernhardt]

Definition/Summary

While the Friedmann equation can demonstrate a flat universe, the Friedmann acceleration equation, in conjunction, can demonstrate a flat yet accelerating universe.

Equations

Friedmann acceleration equation-

[tex]\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}[/tex]

where H is a function of time (in this case, the inverse of Hubble time), a is the time-scale factor (0 to 1, now being 1), G is the gravitaional constant, [itex]\rho[/itex] is density, P is pressure, [itex]\Lambda[/itex] is the cosmological constant and the dots indicate derivatives by proper time. (G, [itex]\Lambda[/itex] and c are universal constants and H, [itex]\rho[/itex], P and a are a function of time. a is established by [itex]a=1/(1+z)[/itex] where z is the redshift.

[tex]\frac{\dot{H}}{H^2}=-(1+q)[/tex]

where q is the deceleration parameter-

[tex]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) [/tex]

where [itex]\Omega[/itex] is the density parameter ([itex]\Omega[/itex]=actual density/critical density)

Extended explanation

The Friedmann acceleration equation can be rewritten where-

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

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

where the equation of state for dark energy is [itex]w=-1[/itex] and for ordinary and dark matter, [itex]w=0[/itex].

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

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) which has negative pressure means the universe accelerates.

In the case of a universe which is virtually flat (i.e. [itex]\Omega=\rho'/\rho_c=1[/itex]) the equation for q can be rewritten as simply-

[tex]q=\frac{1}{2} (1+3w)[/tex]

where [itex]w=P'/(\rho'c^2)[/itex] is the equation of state of the universe.

This implies that the universe is decelerating for any cosmic fluid with equation of state [itex]w[/itex] greater than -1/3 (with current predictions, the EOS of our universe is ~-3/4 and q=~-0.625 which means it is accelerating).

..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.

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[Aaron Paulo]

To provide a brief summary of the Friedmann acceleration equation, it is used to demonstrate that a flat universe can still experience an accelerated expansion due to the inclusion of dark energy with negative pressure. The equation for the deceleration parameter (q) can be rewritten as q=1/2(1+3w) where w is the equation of state of the universe. If q is greater than 0.5, the universe will eventually collapse, whereas if q is less than 0.5, the universe will continue to expand forever.