- #1
Barnak
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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 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 !
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 !
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