# A The color of deep space

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1. Oct 28, 2016

### Barnak

I'm looking for the distribution of all wavelengths (or frequencies) of light that a stationary observer would receive at his location (at $r = 0$ and time $t_0$), from all light sources emitting a single wavelength $\lambda_{\text{e}}$ (or angular frequency $\omega_{\text{e}}$). The light sources are uniformly distributed in a general expanding FLRW universe, and comoving with the cosmic fluid. The spectral distribution of frequencies would tell something about the "color of deep space" (which is dark micro-waves "reddish" in our universe).

Because of the expansion of space with time, the light received by the observer will not have a single wavelength, it will have a blur instead (i.e. a dispersion). What is the distribution of wavelengths ?

More specifically, consider a universe with the following standard Robertson-Walker metric :
$$\tag{1} ds^2 = dt^2 - a^2(t)\Big( \, \frac{1}{1 - k \, r^2} \; dr^2 + r^2 \, (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2) \Big),$$
where $k = -1, \, 0, \, 1$, and $a(t)$ is the cosmological scale factor (arbitrary function). The apparent luminosity at an observer's location, at time $t_0$, of a punctual light source of proper absolute power $\mathcal{P}$, located at coordinate $r_{\text{e}}$ and emitting light at time $t_{\text{e}}$, is defined as the emitted energy per unit time per unit area (this is in Weinberg's book) :
$$\tag{2} I = \frac{\mathcal{P} \, a^2(t_{\text{e}})}{4 \pi \, a^4(t_0) \, r^2}.$$
The sources density (number of stars per unit volume) is
$$\tag{3} n(t) = \frac{a^3(t_0)}{a^3(t)} \; n_0,$$
and the volume of a spherical shell of radius $r_{\text{e}}$ is
$$\tag{4} d\mathcal{V} = 4 \pi \, a^3(t) \frac{r_{\text{e}}^2}{\sqrt{1 - k \, r_{\text{e}}^2}} \; dr_{\text{e}}.$$
Thus, the total luminosity at the observer's location at time $t_0$, of all the sources is the following (using metric (1) to change the variable of integration. We assume that $\mathcal{P}$ and $n_0$ are constants) :
$$\tag{5} \mathcal{I}(t_0) = \int_{\mathcal{V}} I \, n \; d\mathcal{V} = \mathcal{P} \, n_0 \int_{t_{\text{min}}}^{t_0} \frac{a(t_{\text{e}})}{a(t_0)} \; dt_{\text{e}}.$$
Usually $t_{\text{min}} = 0$ (Big Bang) or $t_{\text{min}} = -\, \infty$ in some universe models.

Now, the light's wavelength is a fixed constant at emission time : $\lambda_{\text{e}}$ (at time $t_{\text{e}}$), and stretches to $\lambda$ at time $t_0$ during propagation to the observer :
$$\tag{6} \frac{\lambda}{\lambda_{\text{e}}} = \frac{a(t_0)}{a(t_{\text{e}})}.$$
The differential of this equation is
$$\tag{8} d\lambda = -\: \frac{a(t_0)}{a(t_{\text{e}})} \; H(t_{\text{e}}) \, \lambda_{\text{e}} \; dt_{\text{e}} = -\; \lambda \, H(t_{\text{e}}) \, dt_{\text{e}}.$$
Substituting this into (5) above gives (changing to angular frequencies) :
$$\tag{9} \mathcal{I}(t_0) = \mathcal{P} \, n_0 \int \frac{\lambda_{\text{e}}}{H(t_{\text{e}}) \, \lambda^2} \; d\lambda \quad \Rightarrow \quad \frac{\mathcal{P} \, n_0}{\omega_{\text{e}}} \int_0^{\omega_{\text{e}}} \frac{1}{H(t_{\text{e}})} \; d\omega.$$
Now, $H(t_{\text{e}}) \equiv \frac{\dot{a}}{a}$ should be expressed as a function of $\lambda$ or the angular frequency $\omega \equiv 2 \pi / \lambda$. This way, we can get the spectral distribution $f(\omega)$ of light, which is now "blurred" by the expansion of space.

This is interesting since for a deSitter space, we have a constant expansion rate ; $H = \textit{cste}$ (when the scale factor is $a(t) \propto e^{t \,/\, \ell_{\Lambda}}$), so the frequencies received by the observer are all uniformly distributed on the intervall $0 \le \omega \le \omega_{\text{e}}$.

For a dust universe ; $a(t) \propto t^{2/3}$, we get a frequency distribution $f(\omega) \, d\omega \propto \omega^{3/2} \, d\omega$.

The problem is that I never saw this analysis anywhere, in any book of General Relativity. Someone has references for this ?

Any idea would be greatly appreciated !

Last edited: Oct 28, 2016
2. Oct 28, 2016

### Bandersnatch

Use double #'s for inline code, just as you use double \$'s for the rest.

3. Nov 2, 2016

### Barnak

So, no comments on this fascinating subject ?