Measuring the Duration of Last Scattering
Subject paper: Measuring the Duration of Last Scattering, by Boryana Hadzhiyska and David N. Spergel
General Background: A very common simplification one hears is that the cosmic microwave background (CMB) photons began freely streaming ~380k yrs after the big bang at redshift ~1090 when the universe cooled sufficiently for neutral hydrogen atoms to form and Thomson scattering no longer impeded the free flow of photons. What's rarely mentioned in science media articles is that what's called the surface of last scatter, when individual CMB photons decoupled and began freely traveling, has a thickness (or depth) layer to it. This thickness has been compared to the skin on an apple1. Though the term 'last scattering surface' is commonly used, it's more accurately characterized as a 'last scattering layer'2 due to its width. The layer is opaque and has been compared to a fog bank on Earth.3
The probability distribution for when an individual photon began freely streaming is well-understood theoretically (typically covered in cosmology textbooks) and given by the photon visibility function and is approximated by a Gaussian fit.
Since details about the width of the surface of last scatter are so often glossed over, I asked astrophysicst and science writer Ethan Siegel if he might cover that in one of his articles, which he did in What Was It Like When The Universe First Made Atoms? A summary comment from his article: the recombination process “takes us from a hot, plasma-filled Universe to an almost-equally-hot Universe filled with 100% neutral atoms. Although we say that the Universe formed these atoms 380,000 years after the Big Bang, this was actually a slow, gradual process that took about 100,000 years on either side of that figure to complete. Once the atoms are neutral, there is nothing left for the Big Bang's light to scatter off of. This is the origin of the CMB: the Cosmic Microwave Background.”
What's the Paper About? This paper by Hadzhiyska and Spergel constrains the thickness of that surface of last scatteriing layer to a high degree of precision. The CMB photons are slightly polarized during decoupling from matter, and this polarization is proportional to the width of the last scattering surface. Also, the polarization-to-temperature ratio is proportional to the square of the last scattering surface width. They used Planck 2015 data in their analysis resulting in a best fit to the visibility function (see Fig. 2 on pg3 of the paper). At one standard deviation, they constrain the width to be 19 +/– 0.065 Mpc, or ~6.2e+7 light years, or ∆z ~ 90.
Clarifying a Second Simplication: It's Not a Pure Spherical Shell Simplistically, the last scattering surface could be visualized as a thin spherical shell, as is sometimes portrayed in the literature, e.g. in this diagram from Ryden's textbook Introduction to Cosmology or in this image from lectures notes by Andrew H. Jaffe [source].
Addressing this simplification is one of the themes in the paper Distortions in the Surface of Last Scattering by Li, Dodelson, and Hu. Two effects involved in distorting the surface away from a pure spherical shell are time delay and deflection: “The surface of last scattering of the photons in the cosmic microwave background is not a spherical shell... each photon experiences a different gravitational potential along its journey to us, leading to different travel times in different directions...The theory of general relativity dictates that particles traveling through gravitational potential wells experience time delays. If two photons are emitted at the same time, then they will travel different distances depending upon the potential through which they travel... [Also] the photons in the CMB experience angular deflections as they traverse the inhomogeneous universe... Thus, the surface of last scattering is corrugated, a deformed spherical shell.” (bold italics added for emphasis) A more realistic graphic might be this one4 which depicts an unevenness in the surface.
Note: here is a very interesting graphic that gives a great illustrative explanation about the surface of last scattering.5
Other References: David Spergel gave a talk at the 2019 Simons Summer Workshop where he included discussion on the physics of the last scatter surface. The talk is titled Understanding CMB Fluctuations. It's best viewed at this YouTube link which provides a rather good transcript.
In-depth details of gravitational lensing of CMB photons is covered in a review paper Weak Gravitational Lensing of the CMB by Lewis and Challinor.
Reddit discussion thread on this paper.
___ 1 George Smoot's Nobel lecture, page 38 2 See Lecture 9 by M. Pettini, Recombination and the Cosmic Microwave Background, page 7. 3 Ibid. Note Figure 9.4 on page 8. 4The graphic image is from this article by the Smoot Group. 5Graphic is from Figure 9 in Inflation and the Cosmic Microwave Background by Charles Lineweaver.
