My research focuses on modified theories of gravity, particularly (but not limited to) viable modified gravity alternatives to dark energy. I have been particularly interested in massive gravity and its recent extension to massive bigravity. Recently I have started working on scalar-tensor theories of gravity and non-gravitational alternatives to dark energy.

# Dark Energy and Modified Gravity

We have known since the 1920s that the Universe is expanding, but it wasn’t until the very end of the 20th century that observations showed that the rate of expansion is accelerating. This is *weird!* On cosmological scales, gravity should be the only relevant force, and we all know that gravity is supposed to be attractive. This means that the expansion should slow down, for the simple reason that all the galaxies (and dark matter) will pull on each other.

When a theory makes the wrong prediction — in this case, that the Universe should be decelerating rather than accelerating — there are two possibilities: either we’ve put the wrong ingredients into it, or the theory is wrong. The former is the possibility that there is a *dark energy* for which we haven’t accounted, which dominates the Universe’s mass-energy. We don’t know what this mysterious dark energy is, but it would have the property that its gravity is repulsive rather than attractive. The dominant model is ΛCDM, where the acceleration is driven by a cosmological constant. This is favored because it only has one free parameter. However, it has a very unnaturally small value from a quantum point of view.

The other possibility is that our premier theory of gravity, general relativity, is incorrect on the largest distance scales. If gravity itself turns repulsive at distances close to the size of the observable Universe, this might explain the accelerated expansion that we observe.

I am interested in these modified theories of gravity, both on the conceptual side and the observational side. Conceptually, we should ask: what should a viable modified gravity theory look like, especially compared to the nearly-impossible simplicity of general relativity? Can the small late-time acceleration be stable against quantum corrections? How do we avoid dangerous fifth forces which are ruled out by solar system tests?

# Structure Formation

It is not especially difficult to construct a modified gravity theory which gives the right self-acceleration at late times, *i.e.*, which gives a consistent cosmology at the *background* level. In order to distinguish these models from ΛCDM we need to use more refined tests. One powerful probe is to look at the structure which forms on top of the expanding Universe. Many modified gravity models which predict the same expansion history as ΛCDM give very different results for the growth of structure. Large-scale structure probes like Euclid will be able to distinguish these growth histories and rule out many modified gravity models (or perhaps general relativity!). To do this means analyzing the linearized Einstein equations around a cosmological background — rarely a simple task — in different limits.

# Massive (Bi-)Gravity

Earlier we asked: how do we construct a compelling theory of modified gravity? Many theories in the past have tried to add mathematically simple terms, such as higher-order curvature invariants, to the Einstein-Hilbert action of general relativity. However, these models are often complicated from a field theory point of view, involving a massless spin-2 graviton coupled non-minimally to one or more scalar degrees of freedom.

Another approach would be to modify general relativity’s degrees of freedom in the most obvious way: by giving the graviton a mass. Mathematically this is quite challenging, and for several decades was thought to be impossible, without generating dangerous ghost instabilities. However, in 2010 a breakthrough was made and a ghost-free non-linear theory of massive gravity was found. In addition to fulfilling the long-term vision of finding a theory of a massive graviton, the mass of the graviton is thought to be stable under quantum corrections. It holds promise, therefore, as a technically natural explanation for the accelerating Universe.

This non-linear massive gravity invokes a second metric, or a reference metric, which must be put in by hand. It is then natural to give this metric its own dynamics, so that massive gravity leads to massive *bigravity*. I am interested in this new ghost-free massive bigravity and its theoretical and cosmological implications. This theory has realistic self-accelerating cosmological solutions. These solutions can be tested, and with my collaborators we have been studying the evolution of structures in these theories. Moreover, interesting conceptual issues arise when both metrics are allowed to couple to matter, because then there is no simple notion of a single physical spacetime. How are distances and time measured in such a theory? Importantly, how do we make measurements and compare the theory to observations?

With a group of collaborators based around Oslo, Stockholm, and Heidelberg, I am currently studying both observational and conceptual issues in this intriguing and still mysterious new theory. On the observational side, we have studied the growth of cosmic structure in the version of the theory where matter couples to a single metric, finding that there are deviations from GR structure formation which should be detectable by near-future large-scale structure experiments such as the Euclid satellite. We are working on extending these predictions to other régimes and sectors of the theory.

We are also studying the conceptual implications of the doubly-coupled version, in which the entire theory, including the matter couplings, is symmetric under the interchange of the two metrics, placing both on entirely equal footing. We are working on understanding the underlying structure of a theory with two metrics, which resembles physics in a Finsler (non-metric) geometry. This is not only conceptually interesting, but moreover these steps need to be taken before we can properly compare this theory with observations.

# Violation of Lorentz Invariance

Lorentz symmetry is one of the most fundamental ingredients of both general relativity and the standard model of particle physics. For this reason it demands to be studied and probed as far as we can. Indeed, while Lorentz symmetry is quite well-tested in the standard model, the constraints are much weaker for gravity, as well as for more exotic and speculative forms of matter and energy. To model boost violation (a subset of Lorentz violation) in the gravitational sector, we can use Einstein-aether theory, a modified gravity theory where the metric couples non-minimally to a fixed-norm, timelike vector, or “aether.” By coupling this aether to other forms of matter we can describe Lorentz violation in those sectors as well. With John Barrow, we have determined the strongest bounds to date on a coupling between the aether and a slowly-rolling scalar field driving a period of early-universe inflation.