Modified Gravity

Another way to produce accelerated cosmic expansion is to modify the equations describing gravity themselves (for a review, seeJain & Khoury, 2010). One possible method of achieving this is to modify the Ricci scalar, R, in the action giving rise to Einstein’s field equations (Einstein-Hilbert action), to f(R), a function of the Ricci scalar R. This formalism is referred to as f(R) gravity, and incorporates a family of models each defined by the specific form of f(R). These models were first introduced by Buchdahl (Buchdahl,

1970), as a generalisation of the Einstein-Hilbert action, and later gained traction after they were used in the context of inflationary cosmology as a possible driver of inflationary expansion (Starobinsky, 1980). As it turns out, quantum corrections to GR naturally produce f(R) models (e.g. Birrell & Davies, 1984; Buchbinder, Odintsov & Shapiro,

1992;Vilkovisky,1992), whose solution when curvatures are large involves an effective cosmological constant that drives inflation.

The generalisation to f(R) introduces additional degrees of freedom that also produce an effective cosmological constant, giving rise to accelerated expansion at late times with-

out invoking a form of dark energy. Some f(R) models also modify the Poisson equation, thus altering the relation between the energy density and the associated gravitational poten- tial to include some dependence on local curvature and offering an alternative explanation for structure formation without the need for some form of dark matter. However, not all models are viable cosmological models for our Universe, either because they do not match observations or they produce theoretical problems. For example, the very first f(R) model proposed to explain late-time cosmic acceleration (of the form f(R) = R − µ4_{/R, where}

µ ∼10−33_{eV) was ruled out because it violates tests of GR (Chiba,2003) and contains}

catastrophic instabilities (Dolgov & Kawasaki,2003). Furthermore, some f(R) models exhibit curvature singularities (Frolov,2008), while others offer no physical solutions for the field equations beyond the special case of GR (Barausse, Sotiriou & Miller,2008). Constraining viable models can be difficult due to the large number of possible forms of f(R), the complexity of finding solutions to the modified field equations, and because deviations from GR can be made arbitrarily small for some models in order to match observations.

Other modified gravity theories include Galileons, massive gravity theories and scalar- tensor theories. Galileons are a class of modified gravity theories that contain a self- interacting scalar field and whose Lagrangian is invariant under Galileon symmetry, which removes instability and other problems that plague other modified gravities (Nicolis, Rattazzi & Trincherini,2009). Massive gravity and its extensions modify GR at large scales, by introducing a non-zero mass to the graviton, in order to produce accelerated cosmic expansion without introducing dark energy (seede Rham, Gabadadze & Tolley,

2011). This also forces gravitational waves to travel at less than the speed of light. Scalar- tensor theories offer a natural framework in which a massless scalar field is linked to the gravitational field, while also preserving the coupling of matter and gravity present in GR (e.g. Brans-Dicke theory;Brans & Dicke,1961). These models are referred to as scalar-tensor, because GR is a tensor theory (the metric is a spin-2 tensor), while the addition of a linked scalar field provides the scalar component. They are of particular interest because they also arise naturally out of unification theories that include gravity (e.g. Jordan’s Projective Relativity theory;Jordan,1955, and string theory;Damour & Vilenkin,

1996). Many of these models also tend to approach GR at the current time, making them consistent with observations that support it, while some also offer an alternative to dark matter, instead explaining cluster and rotation dynamics through modified gravitational interactions.

The additional scalar degrees of freedom introduced as part of these modifications are expected to couple to matter and thereby produce an observable fifth force. Such a force has never been observed in local tests of gravity (e.g. tests of the Strong Equivalence Principle within the Solar System;Williams, Turyshev & Boggs,2009,2012), so many modified gravity theories require a ‘screening mechanism’ in their formalism to achieve consistency with local observations (for example, the Vainshtein mechanism, which includes non-linear derivative self-interactions to suppress the fifth force;Vainshtein,1972, and the chameleon mechanism, which increases the mass of the scalar field in regions of high density;Khoury

& Weltman,2004). The screening allows the fifth force introduced by additional degrees of freedom (e.g. scalar fields coupling to matter with gravitational strength) to produce accelerated cosmic expansion on cosmological scales, while suppressing its effects on small scales. These mechanisms apply to f(R) gravity, Galileons, massive gravity theories, scalar-tensor theories, and other modified gravity theories.

It is worth noting that the interpretation of the scalar fields in modified gravity theories is not the same as the interpretation of the scalar field in quintessence models. In quintessence, the scalar field appears as a source of energy in the stress-energy tensor, rather than as an alteration to the behaviour of the gravitational force in modified gravity. However, the distinction can be considered inconsequential, as will be discussed in greater detail in Chapter5. It is also worth highlighting that some f(R) models behave like quintessence models, where the particular alteration to the Einstein-Hilbert action produces an expansion history and a modified Poisson equation that mimics the effect of a quintessence scalar field. This similarity in behaviour will also be further explored in Chapter5.