Wednesday, March 19, 2014

fat Hubble sphere

the Hubble sphere is a spherical surface around an observer in a FLRW-universe at which the recession velocity becomes larger than the speed of light. at what scale factor are the recession velocities stationary?


2 comments:

  1. the Hubble sphere is defined as the spherical shell around us at which the recession velocity,
    \begin{equation}
    \frac{\upsilon_\mathrm{rec}}{c} = aH\int_a^1\:\frac{\mathrm{d}a}{a^2H}
    \end{equation}
    is equal to the speed of light. determining the derivative wrt the scale factor $a$ yields the condition
    \begin{equation}
    \left(H+a\frac{\mathrm{d}H}{\mathrm{d}a}\right)\int_a^1\:\frac{\mathrm{d}a}{a^2H}=0
    \end{equation}
    implying that the logarithmic derivative $\mathrm{d}\ln H/\mathrm{d}\ln a$ must be $=-1$, because the Hubble function and all distances are positive. in $\Lambda$CDM this is met by $a=\sqrt[3]{\Omega_m/2/\Omega_\Lambda}$, i.e. roughly at $a=0.55$. there is necessarily a solution to this problem in $\Lambda$CDM, because the logarithmic derivative of the Hubble function changes from $-3/2$ in matter domination to $0$ in dark energy domination.

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    Replies
    1. of course this condition is related to the Universe switching from deceleration to acceleration!

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