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 $x\gg 1$?

2 comments:

  1. Bonus: I think it's the first guess solution
    $$\sinh_a(x)=\frac{a^x-a^{-x}}{2}=\frac{e^{x\ln{a}}-e^{-x\ln{a}}}{2}=sinh(x\ln(a))$$
    for the usual Arsinh the following relation holds:
    $$\operatorname{arsinh}(x)=\ln(x+\sqrt{x^2+1})$$
    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

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  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.

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