Optical Depth

The universal concept that may be fooling us about dark energy

If you’re interested in more of what I have to say about cosmology, in order of less to more technical: Brian Keating hosted me on his podcast; ICHEP has a youtube channel, which contains my talk on Monday (1hr:14min start here); and IHES sponsored a summer school a few weeks ago and posted the lectures.

Particles traveling through space, through any space, have a finite chance of hitting something and getting deflected. The optical depth is the expected total number of interactions a particle will experience over its full path length. Let’s take photons as an example. If the optical depth of a photon from a star to us is much smaller than one, then we will see almost all of the light emitted by the star so get a good view of it. On a clear night, the optical depth for light from the moon to the Earth is about 10%, so we see about 90% of the light coming from the moon. On a very cloudy night, the optical depth to the moon can be over 20. Mathematically, that means that the flux of light coming from the moon is exponentially suppressed by e^-20 . Physically, it means you can’t see the moon.

The optical depth is not just a thing in astronomy. Light has a huge optical depth when traveling through a brick wall or your garage door, so when you shine a flashlight at the inside of your garage door, your neighbor won’t see anything on the outside. The optical depth through glass is very small, so curious neighbors should turn their spyglasses on your windows. And there are some materials in between, like the shower door shown below.

This type of shower door has a relatively small optical depth, so you can see through it but the image will be slightly blurred.

The rods and cones in our eyes have interesting optical depths, carefully tuned by millions of years of evolution to help us see as well as possible on Earth. Back to astronomy, optical depth is a thing in gravitational lensing: a given star has a finite chance of being aligned with an intervening object for a short time, and during that short time, it appears much brighter to us. The chance is very small and is equal to the optical depth. This is called microlensing and it is a powerful technique fo searching for dark matter.

In cosmology, the optical depth is one of seven parameters in the fiducial cosmological model. What does it refer to? The photons that comprise the cosmic microwave background (CMB) travel to us over the course of 13.7 billion years without hitting almost anything. That is what makes the CMB so powerful: it is literally a picture of what the universe looked like so very long ago.

However. The photons in the CMB have a small chance of being deflected on their journey to us. Before getting into the why, let’s think about the consequences of those rare deflections. Suppose a given direction on the sky is a hot spot in the CMB: that means the temperature in that direction is slightly larger than in other directions, and there are more photons coming to us from that direction. If just a small fraction of them are deflected, then the hot spot will appear slightly less hot. Conversely, a cold spot has fewer photons than average if we neglect the rare deflections. But those deflections, being random, are likely to make the cold spots a bit hotter. To grok this, think about two heaps of rocks, one much larger than another. And think of randomly choosing one rock and moving it from one pile to another. Well, the rock is much more likely to come from the pile with lots of rocks (because it has lots of rocks) and be moved to the pile with very few rocks. In the same way, the hot spots in the CMB get slightly coolers and the cold spots slightly warmer due to these rare deflections, due to the small optical depth of the CMB.

What is responsible for these rare deflections? That turns out to open up a whole other field of astronomy/cosmology. Let’s quickly divert our attention to that, remembering that all we (as cosmologists) really care about is what the value of the optical depth is because at the end of the day, a large optical depth will mute the hot/cold spots, the anisotropies in the CMB. When we come to determine parameters, that matters.

The reason the photons can travel almost freely through the universe is that the electrons off of which they might scatter were almost all attracted by protons to form neutral hydrogen. The photons in the CMB do not scatter of of neutral hydrogen. However, at very late times, exactly when we’re not yet sure, the universe gets re-ionized. That is, stars (or something) form and spit out ionizing radiation that breaks up all the neutral hydrogen. The ensuing free electrons by this time are pretty dilute because the universe has expanded, but they can sometimes be targets for the photons in the CMB. And that leads to a non-zero optical depth for the CMB photons. Re-ionization is a mystery: how did it happen? When did it happen? Why? There are reams of astronomers and cosmologists with lots of different ways of attacking this question, and it is likely to be one of the more important issues in the field for the next decade or two.

Back to the CMB: we do have one very good way to infer the optical depth, so interesting that it may be worth exploring in the future. But for now, you should know that, using this technique, the Planck satellite team was able to constrain the optical depth to be about 5% with a small uncertainty. So it’s known fairly well.

OK, that was technical; what’s the teaser about the optical depth fooling us about dark energy? We’re not quite there yet. But we have set the stage: the optical depth is one of the 7 parameters in the fiducial model, and when you use the Planck data to determine it, the rest of the story unfolds, as I have outlined over the previous months: there is tension between different data sets that has put enormous pressure on the fiducial cosmological model. To give away the punchline, if you do not use the Planck data to determine the optical depth, then some folks have claimed that it moves to a higher value (about 10%); the tension between data sets goes away, and the fiducial cosmological model is restored. The claim is that if you do not use the Planck measurement of the optical depth, then you do not need to resort to a model in which dark energy evolves. You can stick with the cosmological constant.