Wednesday, February 27, 2013

back to the future

imagine there hadn't been an inflationary epoch - how long would we need to wait until hydrogen atoms were formed and the cosmic microwave background was generated?

maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

physics bonus question: in quantum mechnics, can a perfect sphere rotate?


  1. maths bonus question: $\sqrt[x]{x}=\exp(\ln(x)/x)$, and therefore
    \frac{\mathrm{d}}{\mathrm{d}x}\sqrt[x]{x} = \frac{x^{1/x}}{x^2}\left(1-\ln(x)\right) = 0
    which has the solution $x=\exp(1)$.

  2. CQW thinks that the question is really tough... the answer would not depend a lot on the temperature before inflation (as it decreases a lot because inflation makes the scale-factor grow by $10^{30}$ and the temperature would decrease accordingly with $T\propto a^{-3(\gamma-1)}$ with the adiabatic index $\gamma$) but rather how much reheating is done if the inflaton field (or whatever field is responsible for inflation) decays and generates particles. after that, the photon temperature drops again from the end of inflation at $a=10^{-24}$ by $10^{24}$ ($\gamma=4/3$ for photons) until today.