The Fall of the Twin Prime Gap

The twin prime conjecture states that there are an infinite number of primes separated by a gap of two, e.g. 17 and 19 or 29 and 31. In May 2013, a relatively unknown mathematician named Yitang Zhang proved that there are infinitely primes separated by a gap of some number smaller than 70,000,000. This number is much bigger than the postulated two, but much smaller than infinity, which is where the number theory community was stuck at before. This finding heralded in a rush of work attempting to lower this bound, causing it to drop rapidly over the next several months, finally coming to a rest at 246 less than a year later, where it has sat. This post is about cataloguing and quantifying this drop.

Timeline of the twin prime gap, on log-linear and log-log axes.

Most of this work was carried out by the Polymath Project, which is a group of mathematicians that collaborate on unsolved math problems. The results have been catalogued here, which is where I got the data for those graphs. The Polymath Project can be really interesting to read through, because you can see chunks of the problem getting solved in real time on an online discussion forum, like reading the world's most high-brow youtube comments. The retrospective on the project  is a nice read even for non-mathematicians such as myself. Midway through this project, an independent mathematician (working in Montreal!) named James Maynard developed another method to shrink this gap, which was eventually worked into the project.

I will not say too much about the techniques that were used to prove these bounds; there are people who are more qualified to do so who have already done so, such as Terry Tao, who was part of the project. From what I gather, the  method involving generating a large list of numbers and then "sieving" them until the solution converges*. Two assumptions can be made to make the bound easier to shrink: the  Elliott-Halberstam conjecture, and Deligne's Theorem. While their validity in this situation is unproven, assuming the  Elliott-Halberstam can reduce the bound to 6. I have tried to take only data that does not rely on these conjectures (the proven bound, if you will), and have ignored data marked as conditional in the table.

Ok, so let's look at the falling bound. There are clearly three phases: the "let's get the ball rolling" phase, the wild-mathematical-rollercoaster phase, and the "honeymoon is over" phase. The first two weeks were just people starting to realize what could be done (Zhang kind of caught the mathematical community by surprise). In fact, the first post-Zhang paper was called "A poor man's improvement on Zhang's result" and subsequently there was a blog post called "I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart."  Then Terry Tao proposed the Polymath Project, and that's when it really started dropping. If you look through their website, the progress involves people posting tables of their iterated sieving data and discussing it; sometimes it works and sometimes it doesn't.

In my line of work, data is described by two types of functions: power laws and exponential curves, both of which look like straight lines if you have the right number of logarithmic axes (or the power is 1 or the x-variable covers a small range). Exponentials crop up more when something is varying with time (e.g. the length of the DNA molecule I'm looking at), while power laws come up when looking at how one quantity varies with another (e.g. how stretched a molecule is vs how confined it is). A big part of my thesis was demonstrating how we shouldn't just blindly look at power law fits when analyzing data, and that is exactly what I'm going to do now.

 When I fit the steep part of the bound dropping curve to a power law, the exponent is -9.6 with respect to time. That's pretty crazy; it falls 1000 times as much in a fortnight compared to a week. If instead I fit it to exponential decay, I find that the characteristic time for the gap to fall by a factor of e is about three days, although this doesn't fit the data as well as the power law. I have also ignored the slowdown that occurred around day 30 (although, within a month of Zhang's news, they had already dropped by bound by a factor of 300).

The results sort of petered out in the thousands, and then they were re-invigorated by Maynard's discoveries, which is what got them down to 246. If I do the same type of analysis for the slow phase, there is an inverse-square decay of the gap with respect to time, or a characteristic exponential decay of about 72 days. Can this teach us anything about the nature of collaborative number theory? Probably not.

The gap data with unjustified exponential (left) and power law (right) fits to the fast and slow portions of the drop.

I was watching this gap fall as it was happening, and I was curious as to why it stopped at the seemingly arbitrary 246. A googling came up with an excellent reddit comment explaining it from a user named BoundingBadger. It was pretty neat seeing an unsolved problem get hammered down by professional and sometimes famous mathematicians in real time and openly discussing their results, and I'm glad I caught that neat little bit of history. I was curious as to how the bound would end up behaving over time, and now I know!


*That is a really bad explanation but it's the limit of my understanding right now.

Perturbative Champions: Cohen and Hansen take it to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.

In 1999, in a preprint on arXiv.org, Thomas D. Cohen and James M. Hansen, physicists at the University of Maryland, claimed the following:

If one insists on an accuracy of ∼ 20%, one estimates contributions at their nominal order and Λ is taken to be 300 MeV, then one has to work to order (Q/Λ)$^7$ , this corresponds to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.

This octet is, to my knowledge, the largest string of next-to's ever to appear in scientific literature. As far as I can tell, the runner up is a recent paper by Eltern et al. in Physical Review C, with a paltry five next-to's, although this may be the champion of peer-reviewed literature. Below this, four next-to's is fairly common; it has its own notation, N4LO. I found a reference to N6LO in the literature, but it becomes hard to google these.

What does this actually mean, and why does it sound so silly? A powerful tool in physics is perturbation theory, where we approximate the solution to a problem with a power series, and then compute first the simplest terms, then terms of increasing complexity until we have a solution that is close enough to the exact solution that it's useful to us. There will be the zeroth-order solution that tells us the order of magnitude, the first-order solution with its basic dependences on system parameters, then second and higher order solutions for non-linear effects. In many cases, certain powers will be zero, so the "leading order" term might be the first non-zero term besides zeroth-order, although not necessarily first order. For example, the leading order term of something that is perturbed about equilibrium is second-order (which is one of the reasons why treating things as harmonic oscillators is so useful). Feynman diagrams are essentially a way to express perturbation theory in a graphic form, first drawing interactions with no loops, then one loop, then two loops, etc.

Anyway, it may have been the case that all of the even and odd powers of the perturbation series in Cohen and Hansen's paper were zero, they couldn't just say "eighth order," but thought they had to expand all those next-to's. Each term makes the approximation more accurate, and for their 20% desired accuracy, they needed to compute all these Feynman diagrams really far past leading order, but unfortunately, as they claim, "it is implausible that such calculations will ever prove tractable."

Another, similar situation involves describing sites on a lattice. For example, if you're sitting at a site on a square lattice (with lattice constant x), you have four "nearest neighbors" at distance x from you, four "next-nearest neighbors" across the diagonals at distance $\sqrt{2}$x, "next-next-nearest neighbors at distance 2x, etc. This paper uses the term "next next next next nearest neighbors" which is designates 4NN.

There was perhaps a better way to visualize this.

These record nextings are fairly insignificant but perhaps mildly interesting.

Measuring Newton's Constant with a Space-Borne Gravity Train

I recently came across a paper which was published in Classical and Quantum Gravity, a respected journal, after it had initially appeared on arXiv. It proposes a space mission consisting of a metal sphere with a cylindrical hole, floating through space as a smaller reflective object oscillates back and forth along the hole, pulled by the gravitational field of the sphere. The position of the the smaller object can be monitored by another space probe, and the period of oscillation can be used to measure Newton's gravitational constant, big G. I like this idea, it draws upon some of my recent work and I place it firmly in the "just crazy enough to work" paper category, which are my favourite papers to read.

Diagram of the proposed experiment, from the arXiv version of the paper.

The proposal is motivated by some recent analysis (by some of the current proposal's authors) of independent measurements of the gravitational constant, which showed that even though they are measuring the constant with smaller and smaller uncertainty, the different measurements are not in precise agreement of each other, sometimes deviating by 40 times the standard error on the measurements. The analysis makes the bolder claim that the difference between the measurements has the same periodicity as Earth's length-of-day variations, which are due to large-scale seismic effects. They conclude that there are systematic effects caused by the fact that all these experiments (which typically involve monitoring a rotating pendulum near large masses) take place on the Earth, and desire a way to measure this constant away from the Earth. The National Science Foundation has put out a call for proposals for more accurately measuring big G; I recommend reading its three paragraphs if you're wondering why anyone would bother caring about this.

The various non-agreeing measurements of G over the years, from Anderson et al. Pay more attention to the red than the black. The black is the "bold proposal" I mentioned.

To make this measurement, the authors, lead by independent researcher Michael Feldman, suggest sending a miniature gravity train into space. A gravity train, something I have written about in great detail, consists of a spherical mass (often taken to be the Earth, but not here) with a tunnel through it (often through its center), with a smaller object falling through the hole. It builds up speed due to gravitational attraction to the sphere, passes the halfway point, and then starts decelerating, coming to a rest on the other side. Inside the Earth, it would take 38 minutes to fall through this tunnel. Feldman and friends propose a small metal sphere, roughly 10 cm in diameter and 1.3 kg in mass, that would take about two hours for the small object to fall through.

How could G be measured from such a device? For a uniform density sphere, it can be shown that the period of a gravity train is $T=2\pi\sqrt{\frac{R^3}{MG}}$. If the mass and radius are known, and the time is measured, G can be extracted. In the proposed setup, the position of the small object will be monitored by a laser aimed at the tunnel from another nearby space probe, and from these periodic measurements of position, the time can be measured, and sent back to Earth by an antenna on the probe. The paper consists of more detailed derivations of the G-T measurement, unique to the proposed design (which consists of two layers of different materials).

Zoomed out schematic of the probe and sphere, from the arXiv version of the paper.

The authors are concerned with the precision of such a device, and which systematic errors contribute to the overall uncertainty in G. These include the metallurgy of the sphere and hole (uncertainties in R, M, and the uniformity thereof), the initial placement of the small reflecting object in the hole (which must be extremely gentle), the ideal place to position sphere with respect to the host probe so that the probe is close enough to block the sun but not so close that its own gravity affects the experiment, the radiation pressure from the probe laser on the device, deformations of the sphere due to the tidal influence of the sun, possible charging of the tunnel's interior due to cosmic rays, and more. They even calculate the change in the period if general relativity is taken into account, which is something I was curious about for my gravity tunnel research, but didn't have the tools to solve. The hypothetical uncertainty analysis was probably the most fun part of the paper to read.

They estimate that, given advances in metallurgy and aerospace deftness, they can get the precision of the G measurement down to an optimistic 63 parts per billion. The current record for Earth-based G measurements is 13,000 parts per billion. This would be a huge improvement if it actually worked, and would eliminate some of the systematic issues with measuring the strength of gravity in Earth's gravitational field.

The question, of course, is whether this thing will actually exist, and whether the budgetary will exists to make it so. The authors suggest that the experiment would not be the main mission payload of a space launch, but rather would piggyback on a larger, more important probe headed out of the solar system.

I was interested in this paper because I like crazy yet scientifically rigorous ideas, and it draws upon a system that is close to my scientific heart [disclaimer: the paper cites mine]. It was a pleasure to read about all the potential effects that could skew the time measurement, and how they planned to deal with them. I hope the available metallurgy, metrology, and money becomes sufficient to launch this thing into space.

PhysicsForums: The Interaction of Sound and Light

My last post about sound propagating through light inspired me to write about the more conventional interactions between sound and light. Read about it on physicsforums.com.

https://www.physicsforums.com/insights/interaction-sound-light/

The Speed of Sound in Light

This post will discuss something that I think is interesting, and wanted to think more about: the propagation of sound through light. One can show that the speed of sound in light is 57% the speed of light. I will talk about what the hell that actually means, and where that number comes from.

Sound in light?

Sound in light? Huh?

The first thing one must wrap their head around is: what does it mean for sound to propagate through light? I devised a simple thought experiment to help me understand that.
Imagine two mirrors with springs on their backs, facing each other with light propagating between them. You can either imagine this as photons bouncing back and forth, or a standing electromagnetic wave. The mirrors are in mechanical equilibrium: the radiation pressure from the light reflecting off them is balanced by compression of the springs holding the mirrors in place.

Two mirrors with a standing electromagnetic wave between them. Springs balance the radiation pressure, but if one mirror is moved, the induced Doppler shift will increase the radiation pressure on the opposite mirror, causing that spring to compress. This is my first foray into hand-drawn diagrams...sorry if it's terrible.

Now imagine you boop one of the mirrors, pushing it towards the other one. As light reflects off this mirror, it will be blue-shifted from the perspective of the other mirror. This blue-shifting will increase the magnitude of the radiation pressure, pushing the mirror back and compressing the spring a bit more. In a sense, pushing the first mirror caused an acoustic signal to propagate through the light to the other. The mirrors are both now in simple harmonic motion, coupled by the transfer of radiation overpressure and underpressure through blueshifting and redshifting as the mirrors move.

This scenario can be extended further, to an array of two-sided mirrors in an optical cavity, springs only on the mirrors on the end. Booping one end mirror would cause this blueshift-induced radiation overpressure to propagate down the line of mirrors. As the last mirror compresses the spring and rebounds, the wave will reverse direction.


The way I am picturing this scenario, the signal propagates through the light at the speed of light: as soon as you blueshift the first photon, it's already heading towards the other mirror at the speed of light.

The speed of sound in a photon gas

The scenario above envisioned light in mechanical equilibrium with mirrors. But what if it is also in thermal equilibrium? Now in this thought experiment, we have a box with mirrors on the inside, that is really hot (students of physics may recognize this as the canonical example of a blackbody). The mirrors are so hot that they start emitting blackbody radiation, which is reflected (or absorbed and re-emitted, depending on how you think about it) by the other mirrors in the box. Now what happens if you push on one side of the box?

Left. A really crappy depiction I made of a photon gas: a mirrored box with really hot walls, with radiation bouncing around in the box. Right. The best online depiction I found on google images, from photonics.com.

The way I am picturing the situation, basically the same thing happens but not deterministically: the blackbody radiation from the mirror you push on gets blueshifted, increasing the radiation pressure felt on the opposite side. However, the radiation is emitted isotropically from the surface, and some of it goes off on an angle and reaches the side walls, and the radiation pressure excess from the boop gradually propagates towards the other side. So instead of just one c-retarded propagation from side to the other, there is a distribution in the arrival time of the mechanical information. I suppose the same thing happens when we push on one side of a box full of gas, and because there are so many gas molecules, the sound propagation is effectively deterministic.

At this point I have exhausted my ability to picture what would happen, and will examine a more rigorous argument. The speed of sound in a photon gas can be derived from its thermodynamic equation of state and associated equipartition theorem, which is basically similar to the ideal gas law, except it is derived by assuming that all the particles are going so fast that they are effectively massless (in that the lion's share of their energy is kinetic). The main difference between the regular and relativistic results is that for slow particles like atoms in a gas, the equipartition tells us that the average kinetic energy is 3/2 kT per atom, whereas in the relativistic case is is 3kT. This is basically because you're writing down the energy as pc instead of 1/2mv$^{2}$=$\frac{p^2}{2m}$, so there's no factor of two. There are some more involved derivations to get the number of photons and calculate the pressure based on the temperature, which essentially involve integrating the Planck spectrum over all frequencies (similar to the derivation of the Stefan-Boltzmann law). This is not necessary here.

To find the speed of sound in a gas we want the ratio of pressure to density:

$v^{2}=\frac{P}{\rho}$.

And from the photon gas equation of state we have a relationship between pressure and internal energy.

U=3PV

With a bit of relativity we can relate the regular density to the energy density by dividing it by $c^2$:

$\rho=\frac{1}{c^2}\frac{U}{V}=\frac{3P}{c^2}$

Plugging this back into the equation or the velocity, we have:

$v^{2}=P\frac{c^2}{3P}=\frac{c^2}{3}$.

Thus, we find that the speed of sound through a photon gas is one-over-root-three, or 57%, the speed of light. Notice that the factor of three comes from the number of dimensions used in the equipartition theorem. In one dimension, which my initial mirror scenario took place in, the speed of sound in light is the same as the speed of light itself.

Photon-Photon Interactions

In the above discussions, the "sound waves" are essentially just signals of increased radiation pressure that are transmitted through Doppler shifts. However, at high-enough energy densities, you can start to have the light interacting with itself, via electron-positron pairs that are produced from the vacuum. This self-interaction of light causes a slight slowdown in the speed. It is derived from quantum electrodynamics by Partovi here, and he derives a correction to the speed that depends on the fourth power of the ratio of the temperature of the blackbody to the temperature-equivalent of the electron mass, 5.9 gigakelvin (511 keV divided by Boltzmann's constant). The correction to the velocity is of order 10$^-5$ even at gigakelvin temperatures:

This equation copied from Partovi's paper.

With this in mind, I can picture poking one side of a hot mirror box and expecting a small delay before the other side of the hot mirror box gets jolted.

The Relativistic Sound Speed in Astrophysics

This idea is neat, but is it important? This photon gas sound speed comes up in a few other places. The main one is baryon acoustic oscillations, sound waves through the very early universe that dictated the large-scale structure of the cosmos as the universe cooled down, still present in the periodic distribution of galaxy clusters. The very early universe was very hot, so there was a high energy photon gas just from thermal equilibrium, and all the massive particles zipzapping around were also highly relativistic. Thus, the speed associated with these baryon acoustic oscillations is that relativistic 0.57c.

A telescope attempting to measure Baryon Acoustic Oscillations, from BNL.gov

Another question that arises is whether this is the absolute maximum for sound speed propagation. As I demonstrated in my thought experiment, one-dimensional coherent systems can exceed it, but what about in thermodynamic systems? A paper in Physical Review Letters last year argues that certain models of  neutron star interiors may predict sound speeds that exceed this limit. There is a lot that is unknown about neutron star interiors, and many models put constraints on the maximum and minimum sizes of neutron stars. These models then get selectively ruled out by newer neutron star discoveries. Bedaque and Steiner argue that given the available models and certain observations of particularly small neutron stars, the speed of sound inside some neutron stars must exceed 0.57c. Unfortunately I don't really understand enough of their analysis to say anything for insightful. Alternately, if this 0.57c is the true upper limit, it will rule out a lot of these models.

To summarize, I think that the idea of a speed of sound in light is a cool idea (this is how most of my blog posts end). I make it make sense to me using a thought experiment involving radiation pressure and Doppler shifting. For a more complete understanding, we go to photon gas thermodynamics and QED. There is one point where my research experience touches upon this concept: my undergraduate thesis in general relativity, the first independent physics research I conducted, involved finding fluid mass distributions that solve Einstein's equations. These mass distributions then had to satisfy certain conditions to make them physically realistic: mass couldn't be negative, density had to always decrease with radius, I wanted to avoid singularities, etc, but I also required that the speed of sound not exceed the speed of light. This extended the previous work of my supervisor Kayll Lake, who showed that many existing solutions to Einstein's equations violated these conditions. I wonder if anything would change if we demanded that v<0.57 c instead of v<c.

Strange moments in science: The theft of the EBEX telescope

In early 2013, a balloon-born telescope, EBEX, was launched from Antarctica and circled the South Pole over the course of two weeks, attempting to observe the Cosmic Microwave Background and its polarization. A number of things went wrong with this mission, but it almost didn't happen because  the truck containing the telescope was stolen from a motel in Texas.

EBEX before and during launch. Images from the Ebex in Flight blog via space.com and NewScientist.

The EBEX (E and B EXperiment) telescope was assembled in Minnesota with input from an international team of collaborators. One group was at McGill where I was doing my PhD, and I played on a hockey team with a few people involved in the mission, including my friend Kevin who was the one of the main McGill guys on the collaboration. The telescope had a large number of bolometers (measuring incident radiation through a change in temperature) which, when floated above the lower atmosphere with a balloon would attempt to measure the polarization of the cosmic microwave background and look for B modes as a signature either of cosmic dust or primordial gravitational waves, something that was later somewhat controversially discovered by the BICEP2 telescope, also in Antarctica.

In May 2012, it was on its way from Minnesota to NASA's balloon centre in Palestine, Texas, being transported in the back of a truck. However, the truck failed to arrive, so the trucking company looked into it. According to its GPS, it was last seen at a truck stop Hutchins, Texas, and the trucking company attempted to contact the driver, but could not reach him. Several days later, he was found sleeping in the cab of his truck, but the trailer and the 8 million dollar cargo were still missing. The driver claimed the truck was stolen from the motel he was staying at. Eventually the trailer was found at a carwash in Dallas, with the telescope still inside and undamaged.

Artist's rendition.

According to a 2012 article, the driver had been fired, and the head of the trucking company commented "Why didn't he decide to do this on a load of potato chips?"

After it was recovered, the telescope was configured to the balloon and shipped to Antarctica, where it was launched. A number of things went wrong with EBEX: one of the orientation controllers overheated in the sun, causing the telescope to lose control of where it was pointing (I recall Kevin calling it "a dance party in space"), meaning it only imaged a donut-shaped swath of sky. It eventually landed out on the ice, and without sufficient time and resources to launch an immediate recovery the scientists had to wait until the next season to go access it. Not much has been heard from the collaboration since, although there may be plans to build and launch a new and improved version, hopefully one that will maintain orientation. To read more about EBEX and its launch and recovery (including the truck disappearance), I recommend this blog.

The flight of EBEX, from the EBEX in Flight blog.

As an experimental scientist I'm accustomed to things going wrong, but I can't imagine showing up to work one day to find my microscope missing.

Visit to the Roger Babson Anti-Gravity monument at Tufts

In the early 20th century lived an eccentric businessman named Roger Babson. He was a sort of Donald Trump of his time, writing books, founding a college, and running for president. Instead of hating Muslims and Mexicans, what Babson really hated was gravity, and declared it to be mankind's foremost enemy. 

He founded an organization devoted to defeating gravity's grip on humanity, part of which involved organizing the yearly Gravity Research Foundation essay contest, which has been won by the likes of Stephen Hawking and entered by the likes of me. He left a series of monuments around New England dedicated to his crusade. The closest one to me is at Tufts (yes, Tufts), and I visited it this evening. There used to be a yearly tradition of digging up the monument and carrying it to see if gravity had been defeated yet, but that was discontinued.

I plan to write a longer article about this history of his contest and his foundation, and for now you can find other articles about it online. I wrote an article about the history of the contest on PhysicsForums!

The text of the monument reads:

"This monument has been erected by the Gravity Research Foundation, Roger W. Babson founder. It is to remind students of the blessings forthcoming when a semi-insulator is discovered in order to harness gravity as a free power and reduce airplane accidents. 1961"