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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 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.

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 : $$\tag{6} \frac{\lambda}{\lambda_{\text{e}}} = \frac{a(t_0)}{a(t_{\text{e}})}. $$ I'm not sure what to do next, with this relation, to define the distribution of all the $\lambda$'s received by the observer. I believe that I need to find a function $f(t_0, \, \lambda, \, \lambda_{\text{e}})$ such that $$\tag{7} \mathcal{I}(t_0) = \int_{0}^{^\infty} f(t_0, \, \lambda, \, \lambda_{\text{e}}) \; d\lambda. $$ So $f(t_0, \, \lambda, \, \lambda_{\text{e}}) = \; ?$. I'm not even sure that (7) should be equal to (5).

If the universe in non-expanding ; $a(t) = \textit{cste}$, then I guess that the distribution is a Dirac delta : $f(t_0, \, \lambda, \, \lambda_{\text{e}}) \propto \delta(\lambda - \lambda_{\text{e}})$.


EDIT : I'm tempted to differentiate equation (6) to get this : $$\tag{8} d\lambda = -\: \frac{a(t_0)}{a(t_{\text{e}})} \; H(t_{\text{e}}) \, \lambda_{\text{e}} \; dt_{\text{e}} = -\; \lambda \, H(t_{\text{e}}) \, dt_{\text{e}}. $$ Substituting this into (5) above gives $$\tag{9} \mathcal{I}(t_0) = \mathcal{P} \, n_0 \int_{?}^{?} \frac{\lambda_{\text{e}}}{H(t_{\text{e}}) \, \lambda^2} \; d\lambda \quad \Rightarrow \quad \frac{\mathcal{P} \, n_0}{\omega_{\text{e}}} \int_0^{\omega_{\text{e}}} \frac{1}{H(t_{\text{e}})} \; d\omega. $$ I'm not sure this is actually making sense. Not sure about the lower and upper limits, and what to do with $H(t_{\text{e}}) \equiv \frac{\dot{a}}{a}$ (it should be expressed as a function of $\lambda$ or the angular frequency $\omega \equiv 2 \pi / \lambda$). Any idea would be greatly appreciated !

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  • $\begingroup$ Equation (9) is interesting, because in the case of a deSitter universe, the expansion factor $H$ is just a constant, when $a(t) \propto e^{t/\ell_{\Lambda}}$. Then it implies that all the angular frequencies received by the observer are uniformly distributed on the intervall $[\, 0, \, \omega_{\text{e}}]$. $\endgroup$ – Cham Oct 21 '16 at 1:36
  • $\begingroup$ If equation (9) is right, then in a flat universe filled with dust the frequencies distribution is $$f(\omega, \, t_0) \, dω \propto t_0 \; \omega^{3/2} \, d\omega.$$ In the case of a flat universe filled with radiation, it gives $$f(\omega, \, t_0) \, d\omega \propto t_0 \; \omega^2 \, d\omega.$$ Apparently, the factor $H(t_{\text{e}})$ cannot be expressed as a function of $\omega$ in a general case. It can only be "inverted" on a case by case basis. Any comments on these would be appreciated. $\endgroup$ – Cham Oct 23 '16 at 18:31

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