Observations and Screening mechanisms

All models of modified gravity presented in this section have in common the presence of at least one additional helicity-0 degree of freedom that is not an arbitrary scalar, but descends from a full-fledged spin-two field. As such it has no potential and enters the Lagrangian via very specific derivative terms fixed by symmetries. However, tests of gravity severely constrain the presence of additional scalar degrees of freedom. Interestingly this degree of freedom would severly affect the behavior of voids and could potentially help reducing the tension between Planck and supernovae. Euclid could detect such an effect at the 5σ confidence level [1120]. Outside voids, as it is well known, in theories of massive gravity the helicity-0 mode can evade fifth-force constraints in the vicinity of matter if the helicity-0 mode interactions are important enough to freeze out the field fluctuations [1186]. This Vainshtein mechanism is similar in spirit but different in practice to the chameleon and symmetron mechanisms presented in detail below. One key difference relies on the presence of derivative interactions rather than a specific potential. So, rather than becoming massive in dense regions, in the Vainshtein mechanism the helicity-0 mode becomes weakly coupled to matter (and light, i.e., sources in general) at high energy. This screening of scalar mode can yet have distinct signatures in cosmology and in particular for structure formation.

Different classes of screening. While quintessence introduces a new degree of freedom to

explain the late-time acceleration of the universe, the idea behind modified gravity is instead to tackle the core of the cosmological constant problem and its tuning issues as well as screening any fifth forces that would come from the introduction of extra degrees of freedom. As mentioned in Section 1.5.3.1, the strength with which these new degrees of freedom can couple to the fields of the standard model is very tightly constrained by searches for fifth forces and violations of the weak equivalence principle. Typically the strength of the scalar mediated interaction is required

to be orders of magnitude weaker than gravity. It is possible to tune this coupling to be as small as is required, leading however to additional naturalness problems. Here we discuss in more detail a number of ways in which new scalar degrees of freedom can naturally couple to standard model fields, whilst still being in agreement with observations, because a dynamical mechanism ensures that their effects are screened in laboratory and solar system tests of gravity. This is done by making some property of the field dependent on the background environment under consideration. These models typically fall into two classes; either the field becomes massive in a dense environment so that the scalar force is suppressed because the Compton wavelength of the interaction is small, or the coupling to matter becomes weaker in dense environments to ensure that the effects of the scalar are suppressed. Both types of behavior require the presence of nonlinearities.

Density dependent masses: The chameleon. The chameleon [664] is the archetypal model of

a scalar field with a mass that depends on its environment, becoming heavy in dense environments and light in diffuse ones. The ingredients for construction of a chameleon model are a conformal coupling between the scalar field and the matter fields of the standard model, and a potential for the scalar field, which includes relevant self-interaction terms.

In the presence of non-relativistic matter these two pieces conspire to give rise to an effective potential for the scalar field

Veff(φ) =V(φ) +ρA(φ), (1.5.69)

where V(φ) is the bare potential, ρ the local energy density and A(φ) the conformal coupling function. For suitable choices ofA(φ) and V(φ) the effective potential has a minimum and the position of the minimum depends on ρ. Self-interaction terms in V(φ) ensure that the mass of the field in this minimum also depends on ρ so that the field becomes more massive in denser environments.

The environmental dependence of the mass of the field allows the chameleon to avoid the constraints of fifth-force experiments through what is known as the thin-shell effect. If a dense object is embedded in a diffuse background the chameleon is massive inside the object. There, its Compton wavelength is small. If the Compton wavelength is smaller than the size of the object, then the scalar mediated force felt by an observer at infinity is sourced, not by the entire object, but instead only by a thin shell of matter (of depth the Compton wavelength) at the surface. This leads to a natural suppression of the force without the need to fine tune the coupling constant.

The Vainshtein Mechanism. In models such as DGP, the Galileon, Cascading gravity, mas-

sive gravity and bi- or multi-gravity, the effects of the scalar field(s) are screened by the Vainshtein mechanism [1186, 405], see also [96] for a recent review on the Vainshtein mechanism. This oc- curs when nonlinear, higher-derivative operators are present in the Lagrangian for a scalar field, arranged in such a way that the equations of motion for the field are still second order, such as the interactions presented in Eq. (1.5.57).

In the presence of a massive source the nonlinear terms force the suppression of the scalar force in the vicinity of a massive object. The radius within which the scalar force is suppressed is known as the Vainshtein radius. As an example in the DGP model the Vainshtein radius around a massive object of massM is r?∼ M 4πMPl 1/3 1 Λ, (1.5.70)

where Λ is the strong coupling scale introduced in section 1.5.5.2. For the Sun, ifm∼10−33eV, or in other words, Λ−1= 1000 km, then the Vainshtein radius is r?∼102 pc.

Inside the Vainshtein radius, when the nonlinear, higher-derivative terms become important they cause the kinetic terms for scalar fluctuations to become large. This can be interpreted as a

relative weakening of the coupling between the scalar field and matter. In this way the strength of the interaction is suppressed in the vicinity of massive objects.

Related to the Vainshtein mechanism but slight more general is the screening via a disformal coupling between the scalar field and the stress-energy tensor ∂µπ∂νπTµν [696] as is present in

DBI-braneworld types of models [401] and massive gravity [392].

The Symmetron. The symmetron model [589] is in many ways similar to the chameleon model

discussed above. It requires a conformal coupling between the scalar field and the standard model and a potential of a certain form. In the presence of non-relativistic matter this leads to an effective potential for the scalar field

Veff(φ) =− 1 2 ρ M2 −µ 2_φ2₊1 4λφ 4_{, (1.5.71)}

whereM,µandλare parameters of the model, andρis the local energy density.

In sufficiently dense environments, ρ > µ2M2, the field sits in a minimum at the origin. As the local density drops the symmetry of the field is spontaneously broken and the field falls into one of the two new minima with a non-zero vacuum expectation value. In high-density symmetry- restoring environments, the scalar field vacuum expectation value should be near zero and fluctu- ations of the field should not couple to matter. Thus, the symmetron force in the exterior of a massive object is suppressed because the field does not couple to the core of the object.

The Olive–Pospelov model. The Olive–Pospelov model [917] again uses a scalar conformally

coupled to matter. In this construction both the coupling function and the scalar field potential are chosen to have quadratic minima. If the background field takes the value that minimizes the coupling function, then fluctuations of the scalar field decouple from matter. In non-relativistic environments the scalar field feels an effective potential, which is a combinations of these two functions. In high-density environments the field is very close to the value that minimizes the form of the coupling function. In low-density environments the field relaxes to the minimum of the bare potential. Thus, the interactions of the scalar field are suppressed in dense environments.