Thursday, March 29, 2012

CMB temperature

if the ionisation temperature of hydrogen is $T=10^4$K, the CMB decoupled at a redshift of $z=10^3$, and if for photons the temperature drops proportional to $1/a$, shouldn't the CMB have a temperature of $10$K instead of $3$K?

bonus question: would it be possible to generalise the hyperbolic functions to a basis $a\neq e=\exp(1)$ in a way that $\mathrm{asinh}(x)$ behaves asymptotically as $\log_{a}(x)$ for $x\gg 1$?


  1. Bonus: I think it's the first guess solution
    for the usual Arsinh the following relation holds:
    inverting the above relation for $\sinh_a(x)$ leads to:
    $$\operatorname{arsinh}_a(x)=\frac{\operatorname{arsinh}(x)}{\ln{a}}=\frac{\ln(x+\sqrt{x^2+1})}{\ln(a)}\\ =\log_a(x+\sqrt{x^2+1})\overset{x\gg1}{\approx}\log_a(2x)=\log_a(x)+\log_a(2)$$
    which is the desired property

  2. yes, and the CMB temperature today is so low because the hydrogen recombined at lower temperatures compared to the (naive) ionisation temperature. this is because there's a large number of photons per baryon, which keeps the hydrogen ionised. hydrogen can slowly (re)combine by a two-photon process, which is suppressed. it's funny enough to realised that the overwhelming number of photons in the Universe originates from a forbidden transition.