Olbers' paradox - Poisson model

[bpet]

Olbers' paradox states that if the universe is infinite, static and homogeneous then why is the night sky dark. Of course it's been resolved but it brings up an interesting probability question:

If we model the universe with a spatial Poisson model (probability that a small element is occupied is proportional to the volume) and ignoring decay, variations in star brightness and relativistic effects etc we get

$$L \propto \int_r^R \tfrac{1}{s^2}dN(s)$$

as the amount of light reaching your eye originating from stars between r and R units of distance away, where N(s) is a Poisson process with rate at time s proportional to s^2. Does L go to infinity as R increases?

 

[pmsrw3]

If $N(s)$ is a Poisson process with intensity $\rho s^2$, the compensated process whose differential is $d M(s)=d N(s)-(\rho s^2)d s$ is a martingale, $\mathbb{E}\left[d M(s)\right]=0$.

$$\begin{eqnarray*} 0 & = & \mathbb{E}\left[{\int_0^R \frac{d M(s)}{s^2}}\right] \\ & = & \mathbb{E}\left[L\right] - \rho R \\ \\ \mathbb{E}\left[L\right] & = & \rho R \end{eqnarray*}$$
The mean of $L(R)$ increases without bound. To show that L itself almost surely goes to infinity, consider the variance. The spherical shell $s \leq r < s+ds$ contains on average $\rho s^2 ds$ stars, with variance $\rho s^2 ds$. This shell contributes $\frac{\rho s^2 ds}{s^4}=\frac{\rho ds}{s^2}$ to the variance of L. All the shells are independent, so their variances add.

$$Var(L) = \int_0^R \frac{\rho ds}{s^2}$$

which is bounded. Since $\mathbb{E}\left[L\right]$ increases without bound while its variance is bounded, $L$ is almost surely unbounded. (I.e., the probability that $L \leq b$ for any finite b goes to 0 as R goes to infinity.)

 

[bpet]

That's neat! I didn't think to check that the variance is finite or at least slowly growing. For the variance to be bounded the lower integration limit would have to be r>0, but that's fine if the observer is not inside a star.

It looks like the argument works in >3 dimensions as well. To work in 1 or 2 dimensions apply Chebyshev's inequality:

Pr[L<=C] <= Pr[|X-M|>=|M-C|] <= V/|M-C|^2

where M and V are the mean and variance and C < M is arbitrary. We have M=k.(R-r) for all dimensions d and V=k.(R^(2-d)-r^(2-d))/(2-d) for d<>2 or V=k.log(R/r) so the RHS is O(1/R) or less in all cases.

 

[pmsrw3]

For the variance to be bounded the lower integration limit would have to be r>0, ...

Oops! You're right. What saves you, of course, is that stars have finite size, so it's not Poisson at low r.

 

[blue_raver22]

I can say that Olbers' paradox has been a topic of discussion among astronomers and cosmologists for centuries. It raises important questions about the nature of the universe and our understanding of it.

The Poisson model is a useful tool for understanding the distribution of stars in the universe. However, as mentioned in the paradox, there are other factors that must be taken into account such as the decay of light and relativistic effects. These factors can greatly affect the amount of light reaching our eyes from distant stars.

In regards to the question of whether L goes to infinity as R increases, the answer is not straightforward. While the Poisson model may suggest that the amount of light increases with distance, we must also consider the expansion of the universe and the redshift of light. As the universe expands, the wavelength of light increases, resulting in a decrease in its energy and brightness. This is known as cosmological redshift.

Additionally, there are other factors at play such as the finite age of the universe and the fact that not all matter in the universe has had enough time to emit light. These factors also contribute to the dark night sky.

In conclusion, while the Poisson model can provide some insights into the distribution of stars in the universe, it is not the only factor to consider when trying to understand Olbers' paradox. There are still many unanswered questions about the nature of the universe and its infinite or finite nature. As scientists, it is our duty to continue exploring and expanding our understanding of the universe.