## 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 $x\gg 1$?

$$\sinh_a(x)=\frac{a^x-a^{-x}}{2}=\frac{e^{x\ln{a}}-e^{-x\ln{a}}}{2}=sinh(x\ln(a))$$
$$\operatorname{arsinh}(x)=\ln(x+\sqrt{x^2+1})$$
inverting the above relation for $\sinh_a(x)$ leads to:
$$\operatorname{arsinh}_a(x)=\frac{\operatorname{arsinh}(x)}{\ln{a}}=\frac{\ln(x+\sqrt{x^2+1})}{\ln(a)}\\ =\log_a(x+\sqrt{x^2+1})\overset{x\gg1}{\approx}\log_a(2x)=\log_a(x)+\log_a(2)$$