6.2 Cosmological scales

During inflation, any light scalar field develop super-Hubble fluctuations of quantum origin, with an almost scale invariant power spectrum (see chapter 8 of [409Jump To The Next Citation Point]). It follows that if the fundamental constants depend on such a field, their value must fluctuate on cosmological scales and have a non-vanishing correlation function. More important these fluctuations can be correlated with the metric perturbations.

In such a case, the fine-structure constant will behave as α = α (t) + δα (x,t) EM EM EM, the fluctuations being a stochastic variable. As we have seen earlier, αEM enters the dynamics of recombination, which would then become patchy. This has several consequences for the CMB anisotropies. In particular, similarly to weak gravitational lensing, it will modify the mean power spectra (this is a negligible effect) and induce a curl component (B mode) to the polarization [466Jump To The Next Citation Point]. Such spatial fluctuations also induce non-Gaussian temperature and polarization correlations in the CMB [466, 417]. Such correlations have not allowed to set observational constraints yet but they need to be included foe consistency, see e.g., the example of CMB computation in scalar-tensor theories [435]. The effect on large the scale structure was also studied in [30, 363] and the Keck/HIRES QSO absorption spectra showed [377] that the correlation function of the fine-structure constant is consistent on scales ranging between 0.2 and 13 Gpc.

Recently, it has been claimed [50, 523] that the fine structure constant may have a dipolar variation that would explain consistently the data from the Southern and Northern hemispheres (see Section 3.4.3). Let assume a constant, X say, depend on the local value of a dynamical scalar field ϕ. The value of X at the observation point is compared to its value here and today,

ΔX ∕X0 ≡ X (ϕ )∕X (ϕ0) − 1.
Decomposing the scalar field as ϕ = ϕ0 + Δ ϕ, one gets that ΔX ∕X0 = sX(ϕ )Δ ϕ, with sX defined in Equation (233View Equation). Now the scalar field can be decomposed into a background and perturbations as ¯ ϕ = ϕ(t) + δϕ (x, t) where the background value depends only on t because of the Copernican hypothesis. It follows that
ΔX (x, t) ---------= sX(¯ϕ)[¯ϕ(t) − ϕ0] + {sX (¯ϕ) + s′X(¯ϕ )[¯ϕ(t) − ϕ0]}δϕ(x,t) X0 ≡ sX(¯ϕ)Δ ¯ϕ + 𝒮X (¯ϕ)δϕ (x, t). (223 )
The first term of the r.h.s. depends only on time while the second is space-time dependent. It is also expected that the second term in the curly brackets is negligible with respect to the first, i.e., 𝒮X (¯ϕ) ∼ sX (¯ϕ). It follows that one needs δϕ (x,t) not to be small compared to the background evolution term Δ ¯ϕ for the spatial dependence to dominate over the large scale time dependence. This can be achieved for instance if ϕ is a seed field whose mean value is frozen. Because of statistical isotropy, and in the same way as for CMB anisotropies (see, e.g., [409Jump To The Next Citation Point]), one can express the equal-time angular power spectrum of ΔX ∕X0 for two events on our past lightcone as
⟨ ⟩ ΔX--(n1,-r,t)-ΔX--(n2,r,t) ∑ 2-ℓ +-1 (XX ) X X = 4π Cℓ (z )P ℓ(n1 ⋅ n2). (224 ) 0 0 ℓ
If δ ϕ is a stochastic field characterized by its power spectrum, ⟨δ ϕ(k1,t)δϕ(k2,t)⟩ = Pϕ(k,t)δ(k1 + k2) in Fourier space, then
∫ (XX ) 2 2 2 Cℓ (z) = --𝒮X [¯ϕ(z)] Pϕ (k, z)jℓ[k(η0 − η)]k dk, (225 ) π
jℓ being a spherical Bessel function. For instance, if P ϕ ∝ kns−1 where ns is a spectral index, ns = 1 corresponding to scale invariance, one gets that ℓ(ℓ + 1)C (XX ) ∝ ℓns−1 ℓ on large angular scales. The comparison of the amplitude of the angular correlation and the isotropic (time) variation is model-dependent and has not yet been investigated. It was suggested that a spatial variation of αEM would induce a dipolar modulation of CMB anisotropies [362], but at a level incompatible with existing constraints [424].

This has lead to the idea [396] of the existence of a low energy domain wall produced in the spontaneous symmetry breaking involving a dilaton-like scalar field coupled to electromagnetism. Domains on either side of the wall exhibit slight differences in their respective values of αEM. If such a wall is present within our Hubble volume, absorption spectra at large redshifts may or may not provide a variation in αEM relative to the terrestrial value, depending on our relative position with respect to the wall.

Another possibility would be that the Copernican principle is not fully satisfied, such as in various void models. Then the background value of ϕ would depend, e.g., on r and t for a spherically symmetric spacetime (such as a Lemaître–Tolman–Bondi spacetime). This could give rise to a dipolar modulation of the constant if the observer (us) is not located at the center of the universe. Note, however, that such a cosmological dipole would also reflect itself, e.g., on CMB anisotropies. Similar possibilities are also offered within the chameleon mechanism where the value of the scalar field depends on the local matter density (see Section 5.4.2).

More speculative, is the effect that such fluctuations can have during preheating after inflation since the decay rate of the inflaton in particles may fluctuate on large scales [293, 294].

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