Having read a number of books on cosmology and particle physics, I found my-self raking through 5 or 6 books or looking on the web as I tried to remember some tangible fact that had interested me. In the end, I decided to gather this info and post it under various headings as blogs on MySpace. With the introduction of LaTeX at Physics Forums, I decided to move a couple of them over here. Some are a year old, some are more recent. MySpace blogs
Friedmann Acceleration Equation
Continued from Critical Density
Equations that demonstrate a flat and accelerating universe-
Friedmann equation-
[tex]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}[/tex]
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 [itex]P[/itex] is pressure and the dots indicate derivatives by proper time.
Both equations 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].
Friedmann equation becomes-
[tex]H^2=\frac{8\pi G}{3}\rho'-\frac{kc^2}{a^2}[/tex]
which reduces to the critical density equation when [itex]k=0[/itex].
Friedmann acceleration equation becomes-
[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) means the universe accelerates.
[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 in this case [itex]w=P'/(\rho'c^2)[/itex], the equation of state of the universe, which in the case of a universe which is virtually flat, can be rewritten as-
[tex]q=\frac{1}{2} (1+3w)[/tex]
This implies that the universe is decelerating for any cosmic fluid with equation of state [itex]w[/itex] greater than -1/3.
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
http://www.answers.com/topic/deceleration-parameter
Equations that demonstrate a flat and accelerating universe-
Friedmann equation-
[tex]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}[/tex]
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 [itex]P[/itex] is pressure and the dots indicate derivatives by proper time.
Both equations 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].
Friedmann equation becomes-
[tex]H^2=\frac{8\pi G}{3}\rho'-\frac{kc^2}{a^2}[/tex]
which reduces to the critical density equation when [itex]k=0[/itex].
Friedmann acceleration equation becomes-
[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) means the universe accelerates.
[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 in this case [itex]w=P'/(\rho'c^2)[/itex], the equation of state of the universe, which in the case of a universe which is virtually flat, can be rewritten as-
[tex]q=\frac{1}{2} (1+3w)[/tex]
This implies that the universe is decelerating for any cosmic fluid with equation of state [itex]w[/itex] 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.
http://en.wikipedia.org/wiki/FLRW#Solutions
http://en.wikipedia.org/wiki/Deceleration_parameter
http://scienceworld.wolfram.com/phys...Parameter.html
http://www.answers.com/topic/deceleration-parameter
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