Thursday, March 15, 2012

time-dependence of cosmological constants

in cosmology, we define all quantities by their values today. would we need to revise their numerical values in the future? if yes, on what time scale?

bonus question: what's larger, $|\pi^i|$ or $|i^\pi|$, with $i=\sqrt{-1}$ (principal values only)?


  1. they're both equal to 1.
    $$i^{\pi} = \exp(\ln i^{\pi}) = \exp(\pi \ln i) = \exp(\pi i \frac{\pi}{2})$$.
    But the modulus of $\exp(i \frac{\pi^2}[2])$ is 1.
    In a similar way,
    $$\pi^i = \exp (\ln \pi^i) = \exp (i \ln \pi)$$
    and $|\exp(i \ln \pi)| = 1$.

  2. almost all cosmological constants that somehow reflect the hubble expansion need to be adjusted in the future, of course on a time scale corresponding to $1/H_0$. everything related to the CDM spectrum of course stays constant, but the dark energy equation of state parameter can in principle vary much faster than on a time scale $1/H_0$.