4.4 Galileons
4.5.3 Dark energy from curvature corrections
A proposal for IR modifications of gravity has been put forward by Piazza [1036, 1037]. The starting point for this is the usual semi-classical gravity, where matter fields are quantised on a curved background manifold. The operators of the matter field theory are then modified in the IR in a way we will now describe. Schematically, in a cosmological setup, operators corresponding to Fourier modes of physical momentum k are corrected by terms of order H2/k2, where H is the Hubble parameter. These modifications lead
to the apparent existence of Dark Energy, but without introducing a new scale in the problem.
To illustrate this idea consider the vacuum expectation value of the local energy density of a massless field42:
hT0 0(t, ~x)ibare = Z d3k " k + fquad(t) k + flog(t) k3 + . . . # (575) = local terms + non-local terms,
42see e.g.[1298, 516] for the explicit expression of a massive scalar on flat FLRW background. 169
where spatial homogeneity has been assumed for simplicity. The local terms can be re- moved by local gravitational counter-terms, while the non-local pieces represent the gen- uine particle/energy content of the chosen “vacuum” state. The first term contributes to the cosmological constant, and in flat space-time can be removed by the usual procedure of normal-ordering (the f ’s vanish in flat space-time). In curved space-time, however, the presence of the time-dependent f ’s makes the normal-ordering procedure meaningless. The conjecture of [1036, 1037] is that there exists a theory that resembles semi-classical GR on small scales, but that has an IR-completion that prohibits the time dependent pieces in (575). If that is the case then one can still deal with the cosmological constant term by the usual procedure of normal-ordering, as in flat space-time.
To try to construct such a theory [1036, 1037] propose what they call the Ultra Strong Equivalence Principle: For each matter field or sector sufficiently decoupled from all other matter fields, there exists a state (the “vacuum”) for which the expectation value of the (bare) energy-momentum tensor is the same as in flat space, regardless of the configuration of the gravitational field.
What this principle aims to achieve is to remove the time-dependent terms in Eq. (575) by appropriate modifications of semi-classical gravity that manifest themselves when the Fourier modes have wavelengths comparable to the inverse extrinsic curvature (i.e. the inverse Hubble radius H−1). At the present, a complete theory that imple-
ments this idea is lacking, but a toy-model with massive scalar fields has been considered in [1036, 1037]. Letting ~n be the comoving momentum that labels operators in Fourier space (related to physical momentum as ~n/a), the modification to O(H2a2/n2) is given
by the modified commutation relation h A(1)~n , A(1)†~n0 i = δ(3)(~n − ~n0) 1−H 2a2 2n2 + . . . , (576)
where A(1)~n is the annihilation operator. This prescription is equivalent to using the standard commutation relation hA(0)~n , A~(0)†n0
i
= δ(3)(~n
− ~n0) for the standard operator
A(0)~n , but with a modified comoving momentum given by ~k = ~n1−H2a2
2n2
that locally defines the infinitesimal translations. In a local neighbourhood (smaller than a Hubble patch) the above prescription can be shown to cancel the quadratically divergent piece fquad(t) in Eq. (575). Note that the momentum ~k is not conserved, but ~n is.
To extend the above to the global picture one can use the translation operator e−iλP(1) , where ~P(1) =Rd3n ~n A(1)† ~ n A (1) ~ n = ~P(1) = R d3n ~k A(0)† ~ n A (0) ~ n is the momentum
operator constructed with the modified Fourier modes, and λ is the comoving proper distance to a point far away from the origin. In GR the comoving distance λ = d(t)/a(t) is a constant given by the ratio of the physical distance, d(t), to the scale factor, a(t). However, in the present theory one finds instead that
˙λ = 1 4λ
3d
dt(a
2H2) + higher orders. (577)
Comoving distances obeying Eq. (577) are, in fact, already strongly disfavoured by ob- servations [958]. One may, however, try to explore further whether the dynamical Hubble scale H(t) itself could provide the scale required by cosmic acceleration by considering
the more general expansion ˙λ = A1λH + A2(λH)2+ . . . +B1λ2 d dt(aH) + B2λ 3 d dt(a 2H2) + . . . , (578)
where Ai and Bi are a set of constants. The authors find that certain regions of the
resulting parameter space can fit the data as well as ΛCDM.
5. Higher Dimensional Theories of Gravity
The first systematic studies of higher dimensional geometry date back to the likes of Riemann, Cayley and Grassmann in the mid nineteenth century. It lies at the heart of General Relativity, where space and time form part of a curved 3 + 1 dimensional manifold, as described in Section 2. Of course, Riemannian geometry is not restricted to 3+1 dimensions, so we have the tools to study gravitational theories in higher dimensions. Indeed, this is more than just a theoretical curiosity. Superstring theory, arguably our best candidate for a quantum theory of gravity, can only be formulated consistently in 10 dimensions.
The problem now is a phenomenological one: Gravity does not behave like a 10 dimensional force in our experiments and observations. Perhaps the simplest observation along these lines is the stability of earth’s orbit. In D dimensions of space-time, the Newtonian potential due to a point source will typically go like 1/rD−3. For D 6= 4, it
follows that we cannot have stable planetary orbits, and so it is clear that gravity should not appear 10 dimensional on solar system scales. We use the word appear, because there exist gravitational models where the extra dimensions are hidden from experiment, but which open up at shorter and/or larger distances.
In this section we will review various models of higher dimensional gravity that have been proposed. We will only discuss the case of extra spatial dimensions, although extra temporal dimensions have been studied (see eg [1141]). One might worry that extra temporal dimensions lead to problems with causality, as they permit closed time-like curves in the form of circles in the plane of the two temporal directions.
5.1. Kaluza-Klein Theories of Gravity
Kaluza-Klein (KK) theory grew out of an attempt to unify gravity and electrody- namics [985, 673, 701, 702]. The basic idea was to consider General Relativity on a 4 + 1 dimensional manifold where one of the spatial dimensions was taken to be small and compact. One can perform a harmonic expansion of all fields along the extra dimension, and compute an effective 3 + 1 dimensional theory by integrating out the heavy modes. This idea has been embraced by string theorists who compactify 10 dimensional string theories and 11 dimensional supergravity/M-theory on compact manifolds of 6 or 7 di- mensions respectively, often switching on fluxes and wrapping branes on the compact space (see [552] for a review). Each different compactification gives a different effective 4-dimensional theory, so much so that we now talk about an entire landscape of effective theories [1194].
Assuming that the extra dimensions have been stabilised, the late-time dynamics of KK theories is most easily understood at the level of the 4D effective theory. As we will show, this will generically correspond to a 4D gravity theory with extra fields, examples of which are studied in detail in Section 3. At early times, when the 3 dimensional space is comparable in size to the extra dimensions, the effective description clearly breaks down. This forms the basis of KK cosmology where one can ask the deeply profound question of why and how the 3 extended dimensions of space were able to grow large, while the extra dimensions remained microscopically small. It seems fair to say that a fully satisfactory answer to this question has yet to emerge.
We will now discuss some aspects of KK theory, starting with an overview of dimen- sional reduction and effective theory before moving on to a discussion of KK cosmology at early times. For a more detailed review of KK theory see [76, 1005].