Paper Explainer: Dark Radiation Isocurvature from Cosmological Phase Transitions
/I want to tell you about my most recent paper “Dark Radiation Isocurvature from Cosmological Phase Transitions,” which I wrote with my coauthors Mitch Weikert, Peizhi Du, and Nicolas Fernandez. Peizhi and Nico are postdocs here at Rutgers, and Mitch is my grad student. All three are great and fun to work with, and you should hire them.
As the title of the paper suggests, this is a project with a bunch of moving parts, and it’s not easy (even by the standards of theoretical physics research) to explain to outsiders. Which is unfortunate, because it has to do with what we know about the very early Universe, and how we know it. It’s one of those things that is really beautiful and tremendously informative, but complicated enough that its hard to convey how and why we know the things we know.
A Quick Primer on the Entire Universe
When you look on large enough scales (“large enough” being on the order of a 100 megaparsecs, with one parsec being a bit more than three light years), the Universe looks fairly uniform. Galaxies and even galaxy clusters sort of smooth out, and whichever way you look you sort of see the same thing. To all indications, the Universe on these scales are homogeneous and isotropic. Sure, there are variations, structures of galaxies and galaxy clusters and voids between them, but even the structure of that structure — the statistical properties of those clusters and voids — are the same everywhere we look. We promote the observation that the Universe is the same everywhere we look to a principle: the Cosmological or Copernican Principle. Though we should always remember that this is an assumption (backed by data) and we should continually test that assumption.
The Universe is also big, really big. Because it is so big, the future away we look the longer the light took to reach us, and so the older the objects we are seeing are. We don’t get to see what a galaxy a billion light years away looks like “today,” we only see what it looks like when the light was emitted. By looking at the structure of those distant objects, it is reasonable to assume that if you evolved them forward in time they’d look like the galaxies we see nearer to us, but we don’t really ever get to test that assumption. The Copernican Principle suggests that they are the same, but we should always push on that.
The Universe is also expanding. That is, the distance between distant galaxies is increasing. This isn’t because galaxies are running away from a central location (that would certainly violate the Copernican Principle) but because the distance measure itself — the thing we call the metric — is changing with time. This is the Hubble expansion. For things that are sufficiently close together, their gravitational interaction can overcome that expansion. For example, Andromeda and the Milky Way are gravitationally bound, they have “departed from Hubble flow” and they will merge together. But distant objects are not gravitationally bound, and the overall expansion of the Universe carries us apart. This expansion is an initial condition. I don’t know why the Universe started in this expanding phase, but given that it did, Einstein’s Theory of General Relativity allows us to calculate the dynamics of the stuff in the Universe and the metric itself.
Because the Universe’s metric is expanding, particles moving through them have their wavelength’s stretched (as the ruler itself it stretching). This means that light from distant stars is redshifted (losing energy as energy is inversely proportional to wavelength) as it travels through the expanding Universe to us.
If we look far enough away we stop seeing stars and galaxies. This is because the Universe has existed for only a finite time. When it started, there were no stars (again, an initial condition), and it took time for the stars and galaxies to evolve and form. JWST is beginning to probe that formation in new ways, but this most recent paper isn’t about that, let’s keep going back.
If you look past the Dark Ages, that stretch of cosmic time when there are no stars, you will see light again. Because of cosmic redshift, that light will be in the microwave wavelengths, but it used to be much higher energy. This light comes to us from every direction, and looks exactly like light from an object glowing with its own heat — what we call a blackbody. Today, this light looks like it comes from a very cold object, only 2.7 K. It’s cold because of redshift, but once, this light was emitted from plasma-hot gas.
What is this light? This cosmic microwave background (CMB)? It is completely consistent with being the light that was being emitted by the gas in the very early Universe. Remember, the Universe is expanding, so once in the past the distance between everything was smaller. If you compress gas, it heats up. Back when the Universe was 1/1100th the size it was today, the density and temperature would have been high enough that electrons couldn’t remain bound to atomic nuclei. Prior to this time, the Universe truly was a plasma, slowly cooling as it expanded.
When it cools enough, the electrons combine with nuclei. At this point, the Universe becomes electrically neutral and the photons flying between the particles in the plasma would suddenly stop scattering against free electrons and instead fly off in a straight line. Those that happen to be pointing right at us would, 13.7 billion years later, hit our radio detectors and be seen as the CMB. Since the physics of plasmas is something we can study right here on Earth, we can determine very exactly when the primordial plasma would have become electrically neutral. That allows us to benchmark this moment in the Universe’s history (this is how I know that the Universe was 1/1100th the size it is today when the CMB was released), and the CMB photons we see gives us a snapshot of the Universe at that moment.
Map of the Cosmic Microwave Background sky temperature perturbations, Planck (ESA)
The CMB is very uniform, it looks like a blackbody spectrum everywhere we look. But it isn’t exactly uniform: there are slight variations on the order of 1 part in 100,000 in the temperature of the CMB. The Planck space telescope measured these variations, and this is a map of what they look like on the sky.
What are we seeing? We’re seeing the local deviations from homogeneity and isotropy in that primordial plasma, spots where the gas was a tiny bit hotter or colder than it should be (technically, we’re seeing those deviations as projected through the aeons of travel-time of the photons, but that’s a detail you can ignore for now). We expect to see those variations, because those variations will, over time, grow and form the galaxy clusters and voids we see today. We are seeing the pattern of deviation from homogeneity in the early Universe. It is largely (with some interesting question marks) consistent with the variations in galaxies we see today, which is a non-trivial test of our models of cosmology. Studying the exact pattern of the CMB gives us tremendous knowledge of the content and evolution of the Universe, including its age and the best evidence for the existence of dark matter.
The key thing is not the exact pattern of hot and cold spots on the CMB, just as the interesting thing about the distribution of galaxy clusters isn’t the location of individual objects. It is the pattern of that pattern, the statistical properties, that we care about. What we do is decompose the CMB sky into hot and cold spots on different angular scales. You can think of this as counting the number and amplitude of waves of hotter and colder CMB photons, on all different scales. I often picture this like the surface of a lake in a storm: there would be big waves crossing the water, with smaller waves added inside that, and smaller and smaller waves within those. We are seeing a snapshot of that wildly fluctuating surface with waves on all different scales, projected on the sky.
But there’s a problem in this picture of the homogeneous Universe filled with little waves. How did the Universe arrange to be nearly the same everywhere? Remember, the CMB photon you see coming from one direction on the sky is (by definition) meeting the photons from the other side of the sky for the first time in the entire history of the cosmos. How did those two photons “know” to be the same temperature? Even though the Universe started very small (and so those two points on the opposite side of our CMB sky were once very close together), it didn’t stay small for very long. There was not time for light to traverse from one side of our CMB sky to another to bring the Universe we see today to thermal equilibrium. If you evolve the Universe as we see it back in time, the distance light could have traveled (and the distance over which the plasma could have come to the same temperature if it started with large variations) is much smaller than the entire CMB sky we see today. If you take this seriously, the problem is not “why are there variations in the CMB?” but “why are these variations so small?”
Again, this is an initial condition problem. Perhaps whatever started the Universe forced these regions that were out of contact with each other (because of the finite speed of light) to be the same temperature, because hey, that’s just the way it goes. And who am I to say otherwise? How many Universes have I started? Zero, that’s how many.
But we theoretical physicists don’t like just giving up like that. We’d like a mechanism that forces the Universe to be the same temperature in a causal way.
Which brings us to…
Inflation
The CMB is on very solid footing, both observationally and experimentally. The proposed mechanism to enforce the homogeneity of the CMB (solving the “Horizon Problem”) is a theory. I think it makes a lot of sense, but its important to remember we don’t know for sure it is true.
The idea is called “inflation.” The problem with the uniformity of the CMB is that when we evolve back the Universe as we see it, the way the Universe would expand up until the CMB is released doesn’t give enough time for light to have traveled from one side of the visible Universe to another. The solution is to introduce a new form of energy into the early Universe that changes that history. That might seem like a cheat, but it is the sort of cheat that theorists like to think about, and consider the consequences of.
So you introduce this new material, in the form of a new quantum field (imaginatively called the “inflaton” because that’s how theorists roll). If the inflaton has the right properties, it will act like a new form of “dark energy” or “cosmological constant” that will cause the metric of the Universe to expand exponentially. This will quickly blow up small regions of space into mind-bogglingly big volumes. This solves the Horizon Problem because you can imagine there was some tiny region (really tiny, like smaller than an atom small even) that all had the same initial conditions. Inflation blows that region up to be much much much larger than the entire visible Universe in a fraction of a fraction of fraction etc etc of a nanosecond. We see the Universe as homogeneous because we’re living in the middle of a tiny patch that was homogenous, just blown up past further than we can see.
It gets better. The inflaton is a quantum field. Quantum fields cannot stay perfectly constant. They fluctuate, in a way we can calculate. So imagine the inflaton as it drives the expansion of the Universe. At some moment in this expansion, parts of the field fluctuate, very slightly. The regions of space with that variation get expanded in size, baking in a variation in the Universe on enormous scales. Then, there is another fluctuation, which gets inflated as well, adding a new fluctuation on a smaller scale. These fluctuations are random because the quantum fluctuations are random, but since we understand the pattern by which quantum fields fluctuate, we can predict the statistical pattern of those random fluctuations.
When inflation ends (somehow), the Universe is super cold and empty. After all, you have made it larger by exponentially large factors — and not small exponentials. Minimum size of inflation to do some of the job we want it to do would be increases of $e^62 \sim 10^27$. But you have all this energy in the inflaton field, which didn’t dilute. The inflaton decays as inflation ends, and in that decay generates all the particles that fill the Universe (this process is called “reheating”).
This is all a completely hypothetical process. It solves some philosophical problems about the initial conditions of the Universe, but it should be emphasized that we don’t know for sure that this is what happened.
Two very interesting things should be noted. First, inflation generates a mechanism for those 1 part in 100,000 variations in the CMB (and for the initial seeds of the clustering that leads to galaxies). We can predict the statistics of those variations that the simplest models of inflation predict, and they match the CMB data. That’s a really interesting result, though some caution must be taken. I can as a theorist pretty easily change that result from inflation, and perhaps the statistical distribution of fluctuations in the Universe arose from some other mechanism and just happens to match the inflationary prediction. Nevertheless, interesting.
Second thing to note is that in the simplest model of inflation, you would have just a single inflaton field, which when it decays creates all the particles in the Universe. This would generate correlated fluctuations in photons, baryons, and even dark matter. That is, when the Universe starts evolving after inflation, wherever there are more photons (and a hotter plasma), there is more dark matter. These type of initial conditions for the fluctuations are called adiabatic. When we calculate the CMB fluctuation evolution, all the fluctuations interact with each other (in a really beautifully intricate set of interacting equations), and we can tell the difference between a CMB comes from correlated adiabatic initial conditions and ones where the regions of more photons don’t align with more dark matter (or baryons, or whatever). The CMB is very consistent with adiabatic initial conditions. Again, we get these from the simplest models of inflation. Again, interesting.
Finally, our work
Ok, so that was a lot of introductory material. What are we doing in this paper?
We are interested in thinking about modifications of the early Universe, that might leave imprints in current or near future experiments. In particular, we were thinking about ways to generate initial conditions that are not adiabatic — called isocurvature.
To get isocurvature, you need to have some other way of create random fluctuations in addition to those generated by the inflaton. One way to do this is through bubbles, in a process called a first order phase transition.
Think of opening a carbonated drink. Prior to unsealing the bottle, the CO2 gas is under pressure and as a result is in solution with the liquid. After the pressure drops (due to you opening the container), the gas wants to emerge from the solution — the system wants to transition from one phase to another. In this case, there’s what we call a potential energy barrier in the way. See, a bubble of gas represents a certain amount of energy which is proportional to the volume of the bubble. This energy is less than the energy contained in the same volume of gas/liquid phase. Systems try to go to lower energy, and so bubbles want to form and expand. However, the surface tension of the bubble also has energy, and the bigger the bubble, the more energy is in the surface. So there’s a problem: a bubble has to start small. When it is small, there is a relatively high surface area (which costs energy) to a relatively small volume (which gains energy). So you can’t nucleate bubbles, because you’d need to “jump” from a zero size bubble to a bubble large enough that the energy-from-volume outweighs the energy-from-surface tension. We call such phase transitions “first order” (second order transitions don’t have an energy barrier)
In a carbonated drink, you can see how this energy gap is bridged. Pour a coke into a glass and watch: the bubbles will tend to nucleate in the same spots on the glass wall. Why? Because there are microscopic defects in the wall there. Those physically deform the shape of the bubbles that would nucleate there, changing the ratio of surface area to volume and making that energy gap easier to bridge.
Schematic of our Bubble nucleation during inflation and after (From Buckley et al)
Quantum fields can be set up so that they have an energy barrier between the field value you start with and some other field value that has less energy. Unlike a bubble of CO2 in glass of coke, quantum fields can tunnel through barriers, and so the phase transition can occur completely spontaneously. Bubbles will just nucleate, and nucleate at random times and places. This gives us a mechanism to create random initial conditions that are distinct from those created by the random fluctuations of an inflaton.
People have considered such scenarios before. Our innovation is to consider a situation where the field that is undergoing the phase transition does so while inflation is also occurring, and doesn’t finish until after inflation itself ends. During inflation, bubbles in the field will be nucleated and immediately expand with the exponential increase in the metric. Later bubbles will expand less (because they exist during less time of the inflationary epoch), so you get a Universe filled with bubbles of various sizes, though there will be a characteristic scale set by the nucleation size of a bubble. After inflation finishes, there will be regions of field that haven’t transitioned, but they will quickly after reheating.
Simulation of Bubble nucleation during inflation. Earlier bubbles end up larger.
We assume that the field dumps its energy into a form of radiation that isn’t photons (this is generically called dark radiation, but in this context gravitational waves would be dark). We do this to make the imprint on the CMB more difficult to see — if it new field dumped energy into photons that would leave quite a mark.
We then calculate the statistical distribution of the bubbles and thus the new initial conditions that will result in the CMB — this is the power spectrum, which now has isocurvature in addition to the standard adiabatic perturbations. By evolving those perturbations up to the CMB, we can predict how the CMB would look if such first order phase transitions existed. This is done using a code called CLASS, suitably modified to account for our hypothetical additional physics.
We then test our model against the data, the actual statistical pattern in the CMB. We vary the amount of dark radiation our model dumps into the Universe, the characteristic scale of the bubbles, and a parameter that controls the rate of bubble nucleation during inflation. We find that, due to the isocurvature perturbations, we can typically set much stronger constraints on the presence of dark radiation than we get from adiabatic perturbations (at least, for bubbles whose characteristic size is large enough to be seen on the CMB sky and not too large that a single bubble takes up the entire sky). Again, this stronger constraint is because the CMB is largely consistent with adiabatic initial conditions. We do find that we fit the CMB data slightly better with a very small amount of such bubble nucleation at the right scale, but I wouldn’t read too much into that at the moment: when you add more parameters to a model, it is not surprising to find you can fit the data better. This paper serves as a bit of a jumping off point for us. The bubble nucleation model we put together can generate other possible imprints on the early Universe beyond the CMB power spectrum effects we calculated here, and we’re interested in following up on those as we think more about modifying the standard assumptions of how the initial conditions were set.
There was a lot of set up for a somewhat brief discussion of the work we actually did, but I hope you’ve found something interesting in learning more about what we know about the earliest moments of the Universe. The Cosmic Microwave Background tells us a tremendous amount of information, but we’re still puzzling out what is allowed by the data and what isn’t, and there is still room for a lot of interesting new physics beyond what we think we know right now.