Wednesday, March 6, 2013

thinking outside the box

we can observe the cosmos on the past light cone and naturally, we see finite amounts of objects and of fluctuations which sets statistical limits on inferences from cosmological data (aka cosmic variance). are there possibilities of looking outside the past light cone and what type of information would that provide?

bonus question: can you estimate the value of
\begin{equation}
\log\int_0^\infty\mathrm{d}t\:t^x\exp(-t)
\end{equation}
for large $x$?

3 comments:

  1. I know at least one such possibility: If the homogeneity of the cosmos would end shortly behind the present CMB "wall", the CMB would not be as homogeneous as we see it - and most of all, it would not have such a perfect Planck spectrum. Thus, the universe as we know it must "go on" for quite a distance (technically: quite an optical depth at CMB release) behind the "wall".

    Uli Bastian

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  2. The answer to the bonus question is: x(log x-1)

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    Replies
    1. yea! in fact the integral is the definition of the $\Gamma$-function $\Gamma(x+1)$, which of course is the generalisation of the factorial $x!$ - and can be approximated with the Stirling formula $\log(x!)\simeq x(\log x-1)$ or even $x\log x$ for large $x$.

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