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binbagsss
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Hi,
I'm looking at 'Lecture Notes on General Relativity' by Sean M.Carroll.
I have a question about p. 227, solving for ##a(t)## in the dark energy
case.
So for dust and radiation cases it was Friedmann equations you solve.
But in the case of a non-zero cosmological constant Eienstien equation, and consequently the Friedmann equations which are derived from Einstein's equation must differ.
Below I have on the top the steps to deriving Friedmann for Einstein equation without the cosmological constant, and on the steps I believe with a non-zero cosmological constant.
Please could someone let me know if this is correct? and my modified Friedmann equations are correct and what Carroll is referring to when solving here is this ?
Thanks very much in advance:
Zero Cosmological Constant:
Einsteins equation:
##8\pi G ( T_{ab}-\frac{Tg_{ab}}{2} ) = R_{ab}##
##R_{00}: \frac{-3\ddot{a}}{a}=4\pi G( \rho + 3p) ##
##R_{ij}: \frac{\ddot{a}}{a}+2\frac{\dot{a}}{a}^{2}+\frac{2k}{a^{2}}=4\pi G( \rho-p)##
Making ##\frac{\ddot{a}}{a} ## the subject in the 1st equation and plugging into the 2nd equation yields:
##\frac{\dot{a}}{a}^{2}=\frac{8\pi G}{3}(\rho)-\frac{k}{a^{2}}##
Non-zero cosmological constant:
Einsteins equation:
##8\pi G ( T_{ab}-\frac{Tg_{ab}}{2}+\Lambda g_{ab} ) =R_{ab}##
##R_{00}: \frac{-3\ddot{a}}{a}=4\pi G( \rho + 3p)-\Lambda ##
##R_{ij}: \frac{\ddot{a}}{a}+2\frac{\dot{a}}{a}^{2}+\frac{2k}{a^{2}}=4\pi G( \rho-p)+\Lambda ##
Making ##\frac{\ddot{a}}{a} ## the subject in the 1st equation and plugging into the 2nd equation yields:
##\frac{\dot{a}}{a}^{2}=\frac{8\pi G}{3}(\rho)-\frac{k}{a^{2}}+\Lambda a - \frac{\Lambda}{3} ##
I'm looking at 'Lecture Notes on General Relativity' by Sean M.Carroll.
I have a question about p. 227, solving for ##a(t)## in the dark energy
So for dust and radiation cases it was Friedmann equations you solve.
But in the case of a non-zero cosmological constant Eienstien equation, and consequently the Friedmann equations which are derived from Einstein's equation must differ.
Below I have on the top the steps to deriving Friedmann for Einstein equation without the cosmological constant, and on the steps I believe with a non-zero cosmological constant.
Please could someone let me know if this is correct? and my modified Friedmann equations are correct and what Carroll is referring to when solving here is this ?
Thanks very much in advance:
Zero Cosmological Constant:
Einsteins equation:
##8\pi G ( T_{ab}-\frac{Tg_{ab}}{2} ) = R_{ab}##
##R_{00}: \frac{-3\ddot{a}}{a}=4\pi G( \rho + 3p) ##
##R_{ij}: \frac{\ddot{a}}{a}+2\frac{\dot{a}}{a}^{2}+\frac{2k}{a^{2}}=4\pi G( \rho-p)##
Making ##\frac{\ddot{a}}{a} ## the subject in the 1st equation and plugging into the 2nd equation yields:
##\frac{\dot{a}}{a}^{2}=\frac{8\pi G}{3}(\rho)-\frac{k}{a^{2}}##
Non-zero cosmological constant:
Einsteins equation:
##8\pi G ( T_{ab}-\frac{Tg_{ab}}{2}+\Lambda g_{ab} ) =R_{ab}##
##R_{00}: \frac{-3\ddot{a}}{a}=4\pi G( \rho + 3p)-\Lambda ##
##R_{ij}: \frac{\ddot{a}}{a}+2\frac{\dot{a}}{a}^{2}+\frac{2k}{a^{2}}=4\pi G( \rho-p)+\Lambda ##
Making ##\frac{\ddot{a}}{a} ## the subject in the 1st equation and plugging into the 2nd equation yields:
##\frac{\dot{a}}{a}^{2}=\frac{8\pi G}{3}(\rho)-\frac{k}{a^{2}}+\Lambda a - \frac{\Lambda}{3} ##
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