Scale factor in standard cosmology

Physics Asked on May 8, 2021

In standard cosmology the density $rho$ is given by:


where $w=P/rho=$ const depends on the particle content ($P$ is the pressure). The Friedmann equation is

$left(frac{a’}{a}right)^2=frac{8pi G}{3}rho$,

where the prime is a cosmic time detivative and I have neglected curvature and the cosmological constant.

Now inserting the first equation into the Friedmann equation I get:

$left(frac{a’}{a}right)^2sim left(frac{a_0}{a}right)^{3(1+w)}Rightarrow a’sim a^{-3/2(1+w)+1}$

integrating this gives

$asim tau^{-3/2(1+w)+2}$

where $tau$ is the cosmic time. However in my book ( The Cosmic Microwave Background by Ruth Durrer, page 6, equation 1.25 ) they get

$asim tau^{2/3(1+w)}$

Can you spot my mistake?

One Answer

$da a^{3/2(1+w)-1}=dtauRightarrow a=tau^{2/(3(1+w))}$

Answered by jojo123456 on May 8, 2021

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