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?


  1. the Hubble sphere is defined as the spherical shell around us at which the recession velocity,
    \frac{\upsilon_\mathrm{rec}}{c} = aH\int_a^1\:\frac{\mathrm{d}a}{a^2H}
    is equal to the speed of light. determining the derivative wrt the scale factor $a$ yields the condition
    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.

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