# The color of deep space background of an arbitrary universe

While writing my notes on cosmology in general relativity and the Olber's paradox, I was wondering about the color of the deep background of space. Our universe is mostly black because light didn't had time to reach us from everywhere (there was a Big Bang and there's a causal particle horizon that prevent the Olber's paradox). At the location of an arbitrary observer, the total luminosity of all the visible stars is finite (but increasing with time, since the particle horizon moves faster than light. More and more stars are visible with time). Also, the expansion of space produces a redshift of all wavelenghts, which helps in producing a dark background.

The following is a speculation of what other universes may look like, or what our universe may have been.

For simplicity, consider a deSitter universe, eternally inflating with the following metric (euclidian flat space, for simplicity here) : $$\tag{1} ds^2 = dt^2 - a^2(t)(dx^2 + dy^2 + dz^2).$$ The cosmological scale factor is $$\tag{2} a(t) = \textit{cste} \; e^{t / \ell_{\Lambda}},$$ where $\ell_{\Lambda} \equiv \sqrt{\frac{3}{\Lambda}}$ is just a constant. It can be shown that the total luminosity at the observer's time $t_0$ of all the visible "fake" stars of absolute power $\mathcal{P}$, uniformly distributed in that space, is defined by the following formula : $$\tag{3} \mathcal{I}(t_0) = \int I(t_0, t_e, x) \, n(t_e, t_0) \, a^3(t_e) \, d^3 x = \mathcal{P} \, n_0 \int_{t_{\text{min}}}^{t_0} \frac{a(t_e)}{a(t_0)} \; dt_e,$$ where $n(t, t_0) = n_0 \, a^3(t_0)/a^3(t)$ is the uniform density of stars, changing with time since space expands. $t_e$ is the time of emission of light. In the case of the scale factor (2) (with $t_{\text{min}} = -\, \infty$), this formula gives simply a constant : $$\tag{4} \mathcal{I}(t_0) = \mathcal{P} \, n_0 \, \ell_{\Lambda}.$$ In the case of most "standard" cosmologies, formula (3) may give a total luminosity increasing with the observer's time : $\mathcal{I}(t_0) = \kappa \, \mathcal{P} \, n_0 \, t_0$, where the constant $\kappa$ depends on the model.

So we have a deep background of some luminosity (not black, but not full white neither). The global color of that background depends on several things : stellar spectral types (red, orange, yellow, white, bue, etc), but depends also on the initial luminous material before the creation of stars (recombination of hydrogen, or of other substances).

So the question is this :

Is it physically possible to get (or define) an universe for which the deep space background "color" would be in the visible spectrum, as an uniform green color, or even rose (put in your favorite flavor here), visible to the naked eye, and still allow the existence of life on planets ? I would like to get some serious papers on this subject, if any (from arXiv ?).

Is it plausible to get/define/wathever "parallel" universe in which the night sky isn't black, but aurorae-like green, for example ?

Here are a few interesting references (the only one I've found yet, that are related to the question) :

https://en.wikipedia.org/wiki/Cosmic_latte

http://www.pha.jhu.edu/~kgb/cosspec/

http://www3.nd.edu/~techrev/Archive/Summer2002/a4.html

• – Sean E. Lake Oct 17 '16 at 18:41
• In the dust universe, the total luminosity is $$\mathcal{I} = \frac{3}{5}\; \mathcal{P} \, n_0 \, t_0.$$For stars of absolute luminosity like the Sun : $\mathcal{P} \approx 4 \times 10^{26} \; \text{W}/\text{m}^2$ and density like the Sun's galactic environment : $n_0 \approx 0.004 \; \text{stars}/\text{LY}^3$ (extrapolated to the whole universe !), and age $t_0 \approx 13.8 \times 10^9 \; \text{years}$, the night's luminosity would be $\mathcal{I} \approx 148 \; \text{W}/\text{m}^3$. For comparison, the Sun's luminosity on Earth is $\approx 1050 \; \text{W}/\text{m}^2$. Interesting, isn't ? – Cham Oct 20 '16 at 1:28
• I just found a very interesting paper on that subject. It's a very long paper (185 pages !), but its first part is worth reading : arxiv.org/abs/astro-ph/0407207 – Cham Oct 20 '16 at 14:56
• I think so or so you will get mainly a thermal black-body radiation. Its visible color is from deep red to white, depending on its temperature. It is the color of "arbitrary Universe". Note, the case of a free-eye visible CMB would most probably lead to such cosmic parameters where we can't exist in a biological form similar to our current one. The visible part of the CMB in our current Universe is many order below any free-eye visibility. – peterh - Reinstate Monica Nov 11 '16 at 16:54
• I think two important points are missing: 1. extinction. The interstellar medium absorbs light, so the light of stars is diminished, gets diffused and re-emitted in all direction at a longer wavelength. Stars needed to be very very hot and the interstellar medium quite dense to re-emit light in the visible range. 2. It is not the case that we will always see more stars. When the expansion is faster than light, objects we can see now might not be visible anymore. So we will see fewer stars than now at the "rim" of the observable universe. – kalle Dec 19 '18 at 19:18