A Color of Deep Space: Distribution of Light Wavelengths

Barnak
Messages
62
Reaction score
0
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:
Physics news on Phys.org
Barnak said:
Why the LaTeX codes aren't showing properly inside text lines, while it's working for larger equations ? What is the environment for LaTeX code here ?? For a Physics forum, this is really weird !
Use double #'s for inline code, just as you use double $'s for the rest.
 
So, no comments on this fascinating subject ?
 
OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox. If black holes "evaporate" via Hawking radiation, then they cannot exist forever. So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon. Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon? The...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
Back
Top