Wednesday, May 15, 2013

tests for FLRW-cosmologies

is there an observational test of verifying that we live in a FLRW-universe, i.e. of all (which?) symmetry assumptions behind choosing the Robertson-Walker-metric? is this a sufficient condition?

physics bonus question: can we test if the cosmological fluids are ideal?

maths bonus question: can you prove (by induction) the Bernoulli inequality, namely that
\begin{equation}
(1+x)^n\geq 1+nx
\end{equation}
for all real numbers $x\geq-1$?

2 comments:

  1. the Ehlers-Geren-Sachs theorem states that it suffices to observe an isotropic background radiation as the CMB to infer homogeneity and isotropy of the metric, i.e. the Robertson-Walker ansatz. surprisingly, this even holds for small anisotropies, as Stoeger, Maartens and Ellis found out.

    ReplyDelete
  2. bonus question: the Bernoulli-inequality is certainly true for $n=0$ or $n=1$. assuming that it is valid for arbitrary $n$, we need to show for induction that it is then also valid for $n+1$:
    \begin{equation}
    (1+x)^{n+1} = (1+x)(1+x)^n \geq (1+x)(1+nx)
    \end{equation}
    with the assumption that the inequality is valid for $n$. continuing the argumentation yields:
    \begin{equation}
    \ldots = 1+nx+x+nx^2 \geq 1+nx+x = 1+(n+1)x
    \end{equation}
    using that $x^2$ is always positive in the last step.

    ReplyDelete