5.1 Introducing new fields: generalities

5.1.1 The example of scalar-tensor theories

Let us start to remind how the standard general relativistic framework can be extended to make G dynamical on the example of scalar-tensor theories, in which gravity is mediated not only by a massless spin-2 graviton but also by a spin-0 scalar field that couples universally to matter fields (this ensures the universality of free fall). In the Jordan frame, the action of the theory takes the form

∫ 4 √ --- S = -d-x--- − g [F (φ)R − gμνZ(φ )φ,μφ,ν − 2U (φ )] + Smatter[ψ;gμν] (157 ) 16πG ∗
where G ∗ is the bare gravitational constant. This action involves three arbitrary functions (F, Z and U) but only two are physical since there is still the possibility to redefine the scalar field. F needs to be positive to ensure that the graviton carries positive energy. Smatter is the action of the matter fields that are coupled minimally to the metric gμν. In the Jordan frame, the matter is universally coupled to the metric so that the length and time as measured by laboratory apparatus are defined in this frame.

The variation of this action gives the following field equations

( ) [ ] 1- 1- 2 F (φ) R μν − 2gμνR = 8πG ∗Tμν + Z (φ ) ∂ μφ∂νφ − 2gμν(∂αφ ) +∇ μ∂ νF(φ ) − gμν□F (φ) − gμνU (φ) , (158 ) dF- dZ- 2 dU- 2Z(φ ) □ φ = − dφ R − dφ (∂ αφ) + 2dφ , (159 ) ∇ T μ= 0 , (160 ) μ ν
where T ≡ Tμμ is the trace of the matter energy-momentum tensor √ --- Tμν ≡ (2∕ − g) × δSm ∕δgμν. As expected [183Jump To The Next Citation Point], we have an equation, which reduces to the standard Einstein equation when φ is constant and a new equation to describe the dynamics of the new degree of freedom while the conservation equation of the matter fields is unchanged, as expected from the weak equivalence principle.

It is useful to define an Einstein frame action through a conformal transformation of the metric g∗ = F (φ)gμν μν. In the following all quantities labeled by a star (*) will refer to Einstein frame. Defining the field φ∗ and the two functions A(φ ∗) and V (φ∗) (see, e.g., [191]) by

( dφ )2 3( d lnF (φ))2 1 ---∗ = -- --------- + ------, A(φ ∗) = F −1∕2(φ), 2V (φ ∗) = U (φ )F −2(φ ), dφ 4 d φ 2F (φ)
the action (157View Equation) reads as
∫ ---- S = ---1--- d4x√ − g∗[R∗ − 2gμν∂ μφ∗∂νφ ∗ − 4V ] + Smatter[A2g∗ ;ψ ]. (161 ) 16πG ∗ ∗ μν
The kinetic terms have been diagonalized so that the spin-2 and spin-0 degrees of freedom of the theory are perturbations of ∗ gμν and φ ∗ respectively. In this frame the field equations are given by
∗ 1- ∗ ∗ ∗ ∗ αβ ∗ Rμν − 2 R gμν = 8πG ∗T μν + 2∂μφ∗∂νφ ∗ − gμν(g ∗ ∂αφ ∗∂ βφ∗) − 2V (φ)gμν , (162 ) □ φ = − 4πG α (φ ) T + dV(φ )∕dφ , (163 ) ∗∗ ∗μ ∗ ∗ ∗ ∗ ∇ μT ∗ν = α (φ∗) T∗∂νφ∗ , (164 )
with α ≡ dln A∕d φ∗ and β ≡ dα ∕dφ∗. In this version, the Einstein equations are not modified, but since the theory can now be seen as the theory in which all the mass are varying in the same way, there is a source term to the conservation equation. This shows that the same theory can be interpreted as a varying G theory or a universally varying mass theory, but remember that whatever its form the important parameter is the dimensionless quantity Gm2 ∕ℏc.

The action (157View Equation) defines an effective gravitational constant 2 Ge ff = G ∗∕F = G ∗A. This constant does not correspond to the gravitational constant effectively measured in a Cavendish experiment. The Newton constant measured in this experiment is

( 2 ) G = G A2(1 + α2) = G-∗ 1 + ---Fϕ----- (165 ) cav ∗ 0 0 F 2F + 3F 2ϕ
where the first term, G ∗A20 corresponds to the exchange of a graviton while the second term G ∗A20α20 is related to the long range scalar force, a subscript 0 referring to the quantity evaluated today. The gravitational constant depends on the scalar field and is thus dynamical.

This illustrates the main features that will appear in any such models: (i) new dynamical fields appear (here a scalar field), (ii) some constant will depend on the value of this scalar field (here G is a function of the scalar field). It follows that the Einstein equations will be modified and that there will exist a new equation dictating the propagation of the new degree of freedom.

In this particular example, the coupling of the scalar field is universal so that no violation of the universality of free fall is expected. The deviation from general relativity can be quantified in terms of the post-Newtonian parameters, which can be expressed in terms of the values of α and β today as

2 2 γPPN − 1 = − --2α0--, βPPN − 1 = 1---β0α-0--. (166 ) 1 + α20 2 (1 + α20)2
These expressions are valid only if the field is light on the solar system scales. If this is not the case then these conclusions may be changed [287Jump To The Next Citation Point]. The solar system constraints imply α0 to be very small, typically α20 < 10−5 while β0 can still be large. Binary pulsar observations [125, 189] impose that β0 > − 4.5. The time variation of G is then related to α0, β0 and the time variation of the scalar field today
G˙cav ( β0 ) ----- = 2α0 1 + -----2- φ˙∗0. (167 ) Gcav 1 + α 0
This example shows that the variation of the constant and the deviation from general relativity quantified in terms of the PPN parameters are of the same magnitude, because they are all driven by the same new scalar field.

The example of scalar-tensor theories is also very illustrative to show how deviation from general relativity can be fairly large in the early universe while still being compatible with solar system constraints. It relies on the attraction mechanism toward general relativity [130Jump To The Next Citation Point, 131].

Consider the simplest model of a massless dilaton (V (φ∗) = 0) with quadratic coupling (lnA = a = 12 βφ2∗). Note that the linear case correspond to a Brans–Dicke theory with a fixed deviation from general relativity. It follows that α0 = βφ0 ∗ and β0 = β. As long as V = 0, the Klein–Gordon equation can be rewritten in terms of the variable p = ln a as

2 ------′2 φ′′∗ + (1 − w )φ′∗ = − α (φ ∗)(1 − 3w). (168 ) 3 − φ ∗
As emphasized in [130], this is the equation of motion of a point particle with a velocity dependent inertial mass, m (φ ) = 2 ∕(3 − φ ′2) ∗ ∗ evolving in a potential α(φ )(1 − 3w ) ∗ and subject to a damping force, ′ − (1 − w)φ ∗. During the cosmological evolution the field is driven toward the minimum of the coupling function. If β > 0, it drives φ∗ toward 0, that is α → 0, so that the scalar-tensor theory becomes closer and closer to general relativity. When β < 0, the theory is driven way from general relativity and is likely to be incompatible with local tests unless φ ∗ was initially arbitrarily close from 0.

It follows that the deviation from general relativity remains constant during the radiation era (up to threshold effects in the early universe [108, 134Jump To The Next Citation Point] and quantum effects [85]) and the theory is then attracted toward general relativity during the matter era. Note that it implies that postulating a linear or inverse variation of G with cosmic time is actually not realistic in this class of models. Since the theory is fully defined, one can easily compute various cosmological observables (late time dynamics [348Jump To The Next Citation Point], CMB anisotropy [435Jump To The Next Citation Point], weak lensing [449], BBN [105, 106, 134]) in a consistent way and confront them with data.

5.1.2 Making other constants dynamical

Given this example, it seems a priori simple to cook up a theory that will describe a varying fine-structure constant by coupling a scalar field to the electromagnetic Faraday tensor as

∫ [ ] R 2 1 2 √ --- 4 S = 16πG--− 2(∂μϕ) − 4B (ϕ)F μν − gd x (169 )
so that the fine-structure will evolve according to α = B −1. However, such an simple implementation may have dramatic implications. In particular, the contribution of the electromagnetic binding energy to the mass of any nucleus can be estimated by the Bethe–Weizäcker formula so that
Z-(Z-−-1)- m (A,Z)(ϕ) ⊃ 98.25α (ϕ) A1 ∕3 MeV.
This implies that the sensitivity of the mass to a variation of the scalar field is expected to be of the order of
−2Z (Z − 1) ′ f(A,Z) = ∂ϕm (A,Z)(ϕ ) ∼ 10 ----4∕3---α (ϕ ). (170 ) A
It follows that the level of the violation of the universality of free fall is expected to be of the level of η12 ∼ 10− 9X (A1, Z1;A2, Z2)(∂ϕ ln B )20. Since the factor X (A1, Z1;A2, Z2) typically ranges as 𝒪 (0.1– 10), we deduce that (∂ϕ ln B )0 has to be very small for the solar system constraints to be satisfied. It follows that today the scalar field has to be very close to the minimum of the coupling function lnB. This led to the idea of the least coupling mechanism [135Jump To The Next Citation Point, 136Jump To The Next Citation Point] discussed in Section 5.4.1. This is indeed very simplistic because this example only takes into account the effect of the electromagnetic binding energy (see Section 6.3).

Let us also note that such a simple coupling cannot be eliminated by a conformal rescaling 2 ∗ gμν = A (ϕ)gμν since

∫ √ --- ∫ √ ---- B(ϕ )g μρg μνFνσFρσ − gd4x − → B (ϕ)AD −4(ϕ )g∗μρg μ∗νFνσFρσ − g∗d4x
so that the action is invariant in D = 4 dimensions.

This example shows that we cannot couple a field blindly to, e.g., the Faraday tensor to make the fine-structure constant dynamics and that some mechanism for reconciling this variation with local constraints, and in particular the university of free fall, will be needed.

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