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 $x\geq-1$?

energy input into the CMB

the CMB is produced in thermal equilibrium.... would it be possible to notice if there was some extra energy put into it? what consequences could this have, depending on the way it was done?

maths bonus question: what are the asymptotes of $\exp(\tanh(x))$ for $x\rightarrow\pm\infty$ and what is the slope at $x=0$?

physics bonus question: can you derive the energy-time-uncertainty from the momentum-position-uncertainty in quantum mechanics?

inflationary perturbations

guest post by Shaun Hotchkiss (from the blog trenches of discovery):

if, through the CMB and large scale structure, we can study the primordial density perturbation over a range of scales corresponding to five orders of magnitude (i.e. $k_\mathrm{max} = 10^5 k_\mathrm{min}$), what fraction $\Delta \phi/\phi$ of the inflationary potential does this probe (if inflation is true and the inflationary potential is $V(\phi)=m^2\phi^2$)?

bonus question: where's the first minimum of $x^{\sinh(x)}$, $x>0$?