What do we know about extra dimensions?

As far as we know, we live in a world with three spatial dimensions. There are some theories that it has more than that. Here, I will discuss the experimental searches for extra dimensions, how they work and what they have found.

Spoiler alert: no evidence of extra dimensions have been found.

But just because no evidence has been found, it doesn't mean we haven't learned anything. In physics, there is much to be learned from measuring zero, because it can tell you the largest value that something could have while still evading detection. I talk a bit more about measuring zero in my article on photon masses and lifetimes. 

The searches for extra dimensions do not involve trying to draw seven lines perpendicular to each other. All of them require making some sort of theoretical assumption involving extra dimensions, seeing what that theory implies, and looking for those implications. The strength of or constraint against those implications can give information about the extra dimensions that lead to them. However, whatever that theory is, it must reconcile the fact that we appear to live in a three dimensional universe. The reconciliation is usually that the extra dimensions are really small.

Why extra dimensions?

String theory and its relatives require that spacetime have 10 or 11 or 26 dimensions in order for certain calculations not to give infinite results. A lot of work goes into figuring out how these can be "compactified" so that it still seems like we live in three dimensions. String theory, being a theory of quantum gravity, generally has its extra dimensions on the order of the Planck length. The theories I'll be talking about are theory of Large Extra Dimensions, large being relative to the extremely small Planck length. Some of the problems these attempt to solve are the hierarchy problem, that gravity is so much weaker than other forces, and the vacuum catastrophy, that the measured energy density of the universe is 100 orders of magnitude smaller than the prediction from quantum field theory.

Our three dimensional universe as the surface of a higher dimensional universe. Source.

The best-known example of such a model is the Randall-Sundrum model. Its first author, Lisa Randall, is now hypothesizing that galactic dark matter distributions may lead to mass-extinction events on Earth. That is not really relevant but it's cool. This model posits we live in a three-dimensional surface in a four-dimensional universe (actually it's one higher in both cases, because of time), and gravity can propagate through the bulk of the universe while other forces are constrained to the surface. These types of models require that these extra dimensions have some characteristic size, compared to our three spatial dimensions which are infinite. How can a dimension have a size? Well, imagine if you lived on a really long narrow tube, so narrow that it seemed like you just lived on a one-dimensional line. The second dimension, that you don't notice, has a characteristic size that's the circumference of the tube. In fact, I found a picture demonstrating that on google images.


Tests of Newtonian Gravity at Short Distances

The fact that gravity follows an inverse square behaviour is a consequence of the fact that we live in a universe with three spatial dimensions. That is good for us, because an inverse square force is one of the only kinds that can give stable orbits. The inverse-squareness of gravity is attested by the elliptical orbits of planets around the sun, but it was first measured on a terrestrial scale by Cavendish in 1798, who observed the rotation of a torsional pendulum surrounded by massive spheres of lead, which acted as their own gravitational source. I once tried this experiment with my undergraduate lab partner Bon, and it was awful.

Within the Randall-Sundrum model of extra dimensions, it is expected that gravity will behave differently over distances smaller than the size of the extra dimensions. This was posited in order to explain the discrepancy between the observed density of dark energy, and the much much much larger prediction based on electromagnetic vacuum energy. So, Kapner and friends from the Eot-Wash group* simply measured Newtonian gravity with a Cavendish-type experiment to shorter and shorter distances, shorter than anyone had measured before, down to 44 microns separation between the source and test masses. The observed dark energy density has a characteristic length-scale of 85 microns, and they managed to get below that.

"We minimized electromagnetic torques by coating the entire detector with gold and surrounding it by a gold-coated shield."

The experiment functioned in such a way that if gravity was not following an inverse square law, there would be an extra torque on the pendulum, which they could then detect. As far as I know, this is the most precise lab-scale gravity experiment there is. They found no departure from the inverse square law down to that length, but were still able to learn some things. By fitting their results to a modified gravitational Yukawa potential (Newtonian gravity plus an exponentially decaying extra part), they could find experimental bounds on the strength and characteristic length scale of the non-observed deviation. They found, for a force with equal strength to gravity, it would have to be localized to dimensions smaller than 56 microns, or else it would have been detected. So, we know that mites crawling on human hairs are not subject to extra dimensional gravitational forces.

The Large Hadron Collider

The Large Hadron Collider (LHC) at CERN on the French-Swiss border was built to smash protons (and sometimes lead nuclei) together at almost the speed of light to see what comes out. The main thing they were looking for was the Higgs Boson, which was found in 2012. It also looks for other undiscovered particles, which I call splorks, and deviations from the predictions of the Standard Model of particle physics, which may be indicative of "new physics" going on in the background. One of these new physicses is Large Extra Dimensions. How are post-collision particle entrails related to extra dimensions?

A graviton penetrating the brane that is our universe. I'm not sure if this picture makes things more or less clear. Source.

In the paper I'm focusing on, they looked at the amount of monojets deteted. A jet in particle physics is a system of quarks that keeps creating more pairs and triplets of quarks as it is pulled apart. Quarks are weird. A monojet is...one jet. These are typically produced at the same time as Z bosons, and I gather they are less common than dijets. The ATLAS collaboration, one of the two bigger experiments at the LHC, under the alphabetic supremacy of Georges Aad, looked at how a model of large extra dimensions would lead to monojet production. In this model, we live on a three dimensional (mem)brane in a higher dimensional universe. Gravity can propagate through the bulk of the universe, while other interactions can only happen along the brane. In this scenario, quarks could be created in the collisions paired with gravitons, which are not detectable by ATLAS**. These single graviton-associated quarks would lead to monojet events, in greater number than predicted by the standard model. If this explanation seems incomplete, it is because I do not fully understand how this works.

So Georges Aad and his army of science friends looked at the monojet production data and scanned for excesses above the standard model, and tried to fit it to the extra dimensional brane quark model. They found that for two extra dimensions in this model, the upper experimental bound on their size was 28 microns. It surprised me that this is the same order of magnitude as the Newtonian gravity measurement. For larger numbers of dimension, their bound gets smaller.

There is a ridiculous amount of data produced at the LHC and a ridiculous number of ways to analyse it. For each of the many theories of extra dimensions out there, there may be multiple ways to probe it with LHC data, but I have just focused on one.

Pulsar Constraints

The Fermi Large Area Telescope is a space-borne gamma ray observatory. It gives a lot of data regarding gamma ray emission from various sources throughout the universe, including from pulsars. The collaboration wrote a paper trying to constrain a model that predicts how the gamma emission of a neutron star would be different if there were extra dimensions. In the model they consider, gravitons are massless only in the bulk universe, but gain mass in our brane. This allows them to be trapped in the gravitational potential of a neutron star, where they can decay into gamma rays, which could then be detected. For two extra dimensions, their analysis is sufficient to rule out large extra dimensions above 9 nanometers, and smaller for more dimensions. This is a much tighter bound than the LHC and gravity data.

Gravitational Radiation from Cosmic Strings

This one is premature, because cosmic strings have not been detected and may not exist, and we cannot yet detect gravitational radiation. They are not the same thing as string theory strings, they are more like the boundaries between different regions in the early universe that got shrunk as the universe homogenized, until what was left was an un-get-ridable*** string. They are analogous to grain boundaries  between different regions in a crystal.

O'Callaghan and Gregory computed the gravitational wave spectrum emitted by kinks in these strings. They did this assuming three spatial dimensions, and then assuming more. They found that these waves could exceed the detection threshold of future gravitational wave detectors, and that the signals would be weaker if there were more dimensions. This is a longshot, in regards to detection of both cosmic strings and extra dimensions, given that gravitational waves could be detected.

Summary

Even though extra dimensions have not been detected, we can still get information about how big they can be based on what we have not detected given our ability to detect things. However, the way the data is analysed depends on what model of extra dimensions is being considered.

*A combination of Lorand Eotvos and the University of Washington.
**...yet.
***topologically protected