## Wednesday, March 7, 2012

### precision cosmology

a question for the era of precision cosmology... future surveys such as EUCLID and PLANCK are designed to yield almost $10^{-4}$ precision on the cosmological parameters $\Omega_m$ and $H_0$ (in a reasonably simple model). at the same time table-top experiments constrain the gravitational constant $G$ to a similar precision (it's a bit scary to realise that $G$ is the least known constant of Nature with only 4 significant digits!). would it make sense to improve cosmological measurements without improving on the gravitational constant?

bonus question: what's the asymptotic behaviour of $\frac{d}{dx}\left(\frac{1}{x}\right)^\frac{1}{x}$ for large $x$?

1. The answer to the bonus question: Since $\left(\frac{1}{x}\right)^{\frac{1}{x}}=\mathrm{e}^{-\frac{\ln x}{x}}$, the derivative is given by $\mathrm{e}^{-\frac{\ln x}{x}}\frac{\ln x-1}{x^2}$. Since $\frac{\ln x}{x}\rightarrow 0$ for $x\rightarrow\infty$, we have $\mathrm{e}^{-\frac{\ln x}{x}}\rightarrow 1$. Additionally, $\frac{\ln x-1}{x^2}\rightarrow 0$ and thus $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{1}{x}\right)^{\frac{1}{x}}\rightarrow 0$ for $x\rightarrow\infty$.
2. At least for $\Omega_m$ and the other densities it can't make sense. It is defined with respect to critical density and therefore $G$