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Modified Gravity
Outline


Motivation
The idea that we live in a Universe undergoing a period of acceleration is a new, yet strongly held, notion in cosmology. A wealth of data from supernovae, the cosmic microwave background radiation (CMB) and weak gravitational lensing, for example, has increasingly solidified this view as we have entered the age of precision cosmology. We are, therefore, currently faced with task of explaining this fascinating and intriguing problem.
A natural starting point might be to explain this phenomenon within the context of the currently accepted and successful theory of gravity: General Relativity. This is the generally accepted methodology given that the theory is not only elegant and physically motivated but well tested and verified by all local gravity tests. Within General Relativity one could invoke a cosmological constant, perhaps arising from some form of vacuum energy, by adding it to the Einstein field equations. Further still, one could introduce a dynamical scalar field which is either trapped within a false vacuum or slowly rolling down a potential. These models, which have an effective negative pressure and produce acceleration with the introduction of new energymatter to the stress energy tensor (), come under the heading of Dark Energy.
Alternatively, one could reason that Cosmology itself, a field performed within the formalism of a gravitational theory, is an experiment that our current gravitational theory is incapable of explaining. If this is true then we might need a more general General Relativity. This 'modified gravity' theory, whatever that may be, would then explain the latetime acceleration of our cosmos along with all other gravitational phenomena without the need to invoke a new unseen energy component. In this way changes would not occur to the stress energy tensor in the Einstein field equations (shown below) as in Dark Energy, but to the gravitational terms given on the left hand side. Although this is very much a radical idea considerable work has already been undertaken on this potentially daunting task.
It is worth noting that modified gravity also exists with a view to explain dark matter  another potential component of the Universe's energy budget that we currently have no explanation for. These models, such as MOdified Newtonian Dynamics (MOND) or TeVeS, are usually distinct and separate from those attempting to explain dark energy.
Modified Gravity Models
It is possible to obtain the field equations for General Relativity by varying the EinsteinHilbert action shown below, where g is the determinant of the metric, R is known as the Ricci scalar and the integral is over spacetime.
Given this natural starting point it has become increasingly popular to examine gravity models that act to generalise this particular action. That is, instead of having an action depending on just the Ricci scalar (R) one might argue that gravity could be some general function of R. This is commonly referred to as f(R) gravity. Early f(R) models, invoked to explain inflation, naturally added higher positive powers () to the Ricci scalar such that at early times the high curvature would cause it to dominate the action. For a resolution to acceleration in the late Universe inverse curvature terms (E.g. 1/R ) have been added such that they dominate today during a time of low curvature. Some of these early and more specific additions have recently been seen to fail solar system tests of gravity. However, in general, f(R) gravity is proving a rich and potentially viable course of action with regards to modifiying gravity or, at the very least, understanding its basic principles. Continuing on from f(R), one could potentially argue that there exist more general actions such as those containing Ricci tensor () or the curvature tensor () terms. While this seems like a reasonable generalisation these models tend to suffer from higher order instabilities (Ostrogradski's theorem).
Another modification to gravity which has previously proved popular in the literature is the DGP (DvaliGabadadzePorrati) braneworld model. Here the particles of the standard model and their interactions are confined to our everyday 4D spacetime  known as the brane. Gravity however is allowed to roam through 5 dimensions (called the bulk) which constitute the 4D brane and an extra infinitely large spatial dimension. Gravity thus slowly leaks into this extra dimension from the view of the brane and subsequently latetime acceleration is achieved through a long distance weakening effect. While a conceptionally interesting and much studied modified gravity model DGP potentially suffers from a strong coupling problem, a ghost and equally severely, tension with recent data.
Observational Signatures
Modified gravity models in general are expected to have a rich set of observational signatures. This is both interesting and useful in helping to detect, rule out or distinguish modified gravity from General Relativity. With regards to local gravity constraints, or solar system tests, there exists a well established formalism for understanding phenomena such as the deflection of light, the anomalous precession of Mercury and the Shapiro time delay, for example. This has culminated in the PPN (Parameterised PostNewtonian) formalism. The idea is that different gravity experiments, such as lunar ranging experiments for example, probe different gravitational characteristics and it is these characteristics that are parameterised and compared to data. Cosmology allows us yet further tests that are also capable of probing gravity over different distance, energy and temporal scales. It is therefore beneficial to see the characteristics that are likely to change under different gravity theories such that an equivalent parameterisation scheme can be developed for cosmology and its associated probes.
Given that we are attempting to explain the background latetime acceleration we expect different gravitational models to produce different expansion histories. Indeed this is a common observational signature that is often parameterised by the effective equation of state.
From this it is possible to reconstruct the expansion history and in turn measures of the angular diameter and luminosity distance. Using probes that are sensitive to this expansion such as weak lensing, Baryon Acoustic Oscillations (BAO) and supernovae, for example, one can then potentially constrain modified gravity.
As an example the DGP model briefly introduced above acts to modifiy the effective Friedmann equation for LCDM to that corresponding to DGP given by the equations below. From this it can be shown that DGP has an effective equal to 0.78 and a of 0.32, whereas LCDM corresponds to and .
It should be noted that it is possible to fine tune a general dark energy model to have any desired expansion history. Therefore, in principle, it is not possible to distinguish these models, within the formalism of General Relativity, from a modified gravity model with just expansion information: there exists a degeneracy. For smooth dark energy models, where there is no clustering of dark energy due to the presence of a high sound speed (), it is possible to break this degeneracy by examining the growth of structure. This is because a different force law resulting from a change in gravity will effectively modify the evolution of perturbations. Again using the DGP model as an example we can see how the growth of structure is altered from the General Relativity to the braneworld scenario in the two equations below. The extra factor in the source term follows from 5D gravitational effects. This change, and the subsequent break in degeneracy between smooth dark energy and modified gravity, is shown in Figure 1. It is worthwhile noting that a change in expansion also acts to alter the growth through the 'Hubble drag' seen in the second terms in the growth equations.
Figure 1  here.
In this way any probe that is sensitive to the growth of structure will be able to further constrain modified gravity and distinguish it from a nonclustering dark energy. By looking at the convergence power spectrum shown below (introduced in the Cosmic Lensing Section) we can see that weak gravitational lensing is in principle sensitive to this growth of structure through the matter power spectrum. In addition, it is also sensitive to the expansion history through the angular diameter distances (present in ) and the Hubble drag in the growth of structure.
Furthermore, we must remember (as shown in the From GR Section) that the deflection of light by mass is given by the tangential gradient of the sum of the metric potentials integrated along the light path.
Therefore, with weak lensing, we are fundamentally probing the power spectrum of the potentials which we must then relate to the matter power spectrum, such that we can deduce the lensing statistic as written above. Relating the potentials to the matter perturbation is performed by considering the Poisson equation below and any anisotropic stress that may exist  which acts to break the equivalence between the two potentials in General Relativity.
Then by using the definition of the matter power spectrum,
and similarly for the potential it is possible to relate the two quantities:
It is this relationship that has been assume in the convergence power spectrum above and for that found in the literature. If however we have a modification to gravity, or some very arbitrary dark energy, the usual Poisson equation will be modified and there will exist an anisotropic stress. Therefore, as seen in the methodology above the relation between the power spectra will be altered. Failure to include this change, along with changes to the expansion history and growth, will be inconsistent. Inclusion of this change allows a further sensitivity to a change in dark energy or modified gravity model.
Further Reading
Naturally we have only lightly touched the surface of what is a captivating yet challenging subject. The interested reader is therefore referred to the following reviews, papers and references therein:
Reviews
Durrer & Maartens (2007)  [Modified gravity: Theory Overview]
Sotiriou & Faraoni (2008)  [f(R)]
Lue (2005)  [DGP]
Will, C. M., (2006) The Confrontation between General Relativity and Experiment  [PPN]
Papers
Koyama & Maatens (2006)  [DGP growth of structure perturbations]
Kunz & Sapone (2007)  [Dark Energy vs Modified Gravity Degeneracy]
Jain & Zhang (2007)  [Observational tests of Modified Gravity]
Amendola, Kunz & Sapone (2007)  [Measuring the Dark Side with weak lensing]
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