I am currently reading Baumann's Cambridge lecture slides on cosmology and am confused on the notion of the Hubble radius. On page 10 it is stated that for a perfect fluid with constant equation of state $w = P/\rho$, the comoving Hubble radius is $(aH)^{-1} \propto a^{0.5(1+3w)}$, but I haven't the faintest idea where this equation came from, as no derivation is given. Can anyone provide one?
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In Planck units, $M_P=(8\pi G)^{-1}=1$.
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$\begingroup$ Thank you very much! However, why do we assume a flat universe in this scenario? Most of the work I have done so far has not made this assumption $\endgroup$– wrb98Commented May 4, 2021 at 19:06
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$\begingroup$ @wrb98 observations suggest the universe is spatially flat $\endgroup$– KosmCommented May 4, 2021 at 19:07