3.6 Cosmic Microwave Background

The CMB radiation is composed of photons emitted at the time of the recombination of hydrogen and helium when the universe was about 300,000 years old [see, e.g., [409Jump To The Next Citation Point] for details on the physics of the CMB]. This radiation is observed to be a black-body with a temperature T0 = 2.725 K with small anisotropies of order of the μK. The temperature fluctuation in a direction (πœ—,φ ) is usually decomposed on a basis of spherical harmonics as
δT ∑ m∑=+ β„“ --(πœ—, φ) = aβ„“mYβ„“m (πœ—, φ). (110 ) T β„“ m=− β„“
The angular power spectrum multipole 2 C β„“ = ⟨|alm| ⟩ is the coefficient of the decomposition of the angular correlation function on Legendre polynomials. Given a model of structure formation and a set of cosmological parameters, this angular power spectrum can be computed and compared to observational data in order to constrain this set of parameters.

The CMB temperature anisotropies mainly depend on three constants: G, αEM and me.

The gravitational constant enters in the Friedmann equation and in the evolution of the cosmological perturbations. It has mainly three effects [435Jump To The Next Citation Point] that are detailed in Section 4.4.1. αEM, me affect the dynamics of the recombination. Their influence is complex and must be computed numerically. However, we can trace their main effects since they mainly modify the CMB spectrum through the change in the differential optical depth of photons due to the Thomson scattering

Λ™τ = xenecσT, (111 )
which enters in the collision term of the Boltzmann equation describing the evolution of the photon distribution function and where xe is the ionization fraction (i.e., the number density of free electrons with respect to their total number density ne).

The first dependence arises from the Thomson scattering cross section given by

2 σ = 8π--ℏ---α2 (112 ) T 3 m2ec2 EM
and the scattering by free protons can be neglected since −4 me βˆ•mp ∼ 5 × 10.

The second, and more subtle dependence, comes from the ionization fraction. Recombination proceeds via 2-photon emission from the 2s level or via the Ly-α photons, which are redshifted out of the resonance line [405Jump To The Next Citation Point] because recombination to the ground state can be neglected since it leads to immediate re-ionization of another hydrogen atom by the emission of a Ly-α photons. Following [405, 338] and taking into account, for the sake of simplicity, only the recombination of hydrogen, the equation of evolution of the ionization fraction takes the form

[ ( ) ] dxe-= π’ž β (1 − xe)exp − B1-−-B2-- − β„›npx2 , (113 ) dt kBTM e
where TM is the temperature. At high redshift, TM is identical to the one of the photons T γ = T0(1 + z) but evolves according to
dTM-- 8σTaR-- 4--xe-- dt = − 3me TR 1 + xe(TM − Tγ) − 2HTM (114 )
where the radiation constant aR = 4σSBβˆ•c with 4 2 2 3 σSB = kBπ βˆ•(60πc ℏ ) the Stefan–Boltzmann constant. In Equation (113View Equation), Bn = − EI βˆ•n2 is the energy of the nth hydrogen atomic level, β is the ionization coefficient, β„› the recombination coefficient, π’ž the correction constant due to the redshift of Ly-α photons and to 2-photon decay and n = n p e is the number density of protons. β is related to β„› by the principle of detailed balance so that
( )3βˆ•2 ( ) β = β„› 2πmekBTM--- exp − -B2--- . (115 ) h2 kBTM
The recombination rate to all other excited levels is
8π ( k T )3βˆ•2∑∗ ∫ ∞ y2dy β„› = -2- --B--- (2l + 1)eBnβˆ•kBT σnl-y---- c 2πme n,l Bnβˆ•kBT e − 1
where σnl is the ionization cross section for the (n,l) excited level of hydrogen. The star indicates that the sum needs to be regularized and the α EM-, m e-dependence of the ionization cross section is complicated to extract. However, it can be shown to behave as −1 − 2 σnl ∝ α EMm e f (hνβˆ•B1 ). Finally, the factor π’ž is given by
π’ž = ---1 +-K-Λ2s(1 −-xe)--- (116 ) 1 + K (β + Λ2s)(1 − xe)
where Λ 2s is the rate of decay of the 2s excited level to the ground state via 2 photons; it scales as 8 me αEM. The constant K is given in terms of the Ly-α photon 2 λ α = 16πℏ βˆ•(3meα EMc) by 3 K = np λαβˆ•(8πH ) and scales as −3 − 6 m e αEM.

In summary, both the temperature of the decoupling and the residual ionization after recombination are modified by a variation of αEM or me. This was first discussed in [36Jump To The Next Citation Point, 277Jump To The Next Citation Point]. The last scattering surface can roughly be determined by the maximum of the visibility function g = Λ™τ exp(− τ), which measures the differential probability for a photon to be scattered at a given redshift. Increasing αEM shifts g to a higher redshift at which the expansion rate is faster so that the temperature and xe decrease more rapidly, resulting in a narrower g. This induces a shift of the C β„“ spectrum to higher multipoles and an increase of the values of the Cβ„“. The first effect can be understood by the fact that pushing the last scattering surface to a higher redshift leads to a smaller sound horizon at decoupling. The second effect results from a smaller Silk damping.

Most studies have introduced those modifications in the RECFAST code [454] including similar equations for the recombination of helium. Our previous analysis shows that the dependences in the fundamental constants have various origins, since the binding energies Bi scale has 2 me αEM, σT as 2 −2 α EMm e, K as −3 − 6 m e αEM, the ionisation coefficients β as 3 α EM, the transition frequencies as meα2EM, the Einstein’s coefficients as me α5EM, the decay rates Λ as me α8 EM and β„› has complicated dependence, which roughly reduces to α −1m −e 2 EM. Note that a change in the fine-structure constant and in the mass of the electron are degenerate according to Δ αEM ≈ 0.39Δme but this degeneracy is broken for multipoles higher than 1500 [36]. In earlier works [244Jump To The Next Citation Point, 277Jump To The Next Citation Point] it was approximated by the scaling 2(1+ξ) β„› ∝ α EM with ξ ∼ 0.7.

The first studies [244, 277] focused on the sensitivity that can be reached by WMAP7 and Planck8. They concluded that they should provide a constraint on α EM at recombination, i.e., at a redshift of about z ∼ 1,000, with a typical precision − 2 −3 |Δ αEM βˆ•αEM | ∼ 10 − 10.

The first attempt [21] to actually set a constraint was performed on the first release of the data by BOOMERanG and MAXIMA. It concluded that a value of αEM smaller by a few percents in the past was favored but no definite bound was obtained, mainly due to the degeneracies with other cosmological parameters. It was later improved [22Jump To The Next Citation Point] by a joint analysis of BBN and CMB data that assumes that only αEM varies and that included 4 cosmological parameters (Ωmat,Ωb, h,ns) assuming a universe with Euclidean spatial section, leading to − 0.09 < Δ αEM < 0.02 at 68% confidence level. A similar analysis [307Jump To The Next Citation Point], describing the dependence of a variation of the fine-structure constant as an effect on recombination the redshift of which was modeled to scale as z∗ = 1080[1 + 2Δ αEM βˆ•αEM ], set the constraint − 0.14 < ΔαEM < 0.02, at a 2σ level, assuming a spatially flat cosmological models with adiabatic primordial fluctuations that. The effect of re-ionisation was discussed in [350]. These works assume that only αEM is varying but, as can been seen from Eqs. (110View Equation116View Equation), assuming the electron mass constant.

With the WMAP first year data, the bound on the variation of αEM was sharpened [438Jump To The Next Citation Point] to − 0.05 < Δ αEM βˆ•αEM < 0.02, after marginalizing over the remaining cosmological parameters (Ωmath2, Ωbh2, Ωh2,ns, αs,τ) assuming a universe with Euclidean spatial sections. Restricting to a model with a vanishing running of the spectral index (α ≡ dn βˆ•d ln k = 0 s s), it gives − 0.06 < Δ αEM βˆ•αEM < 0.01, at a 95% confidence level. In particular it shows that a lower value of αEM makes αs = 0 more compatible with the data. These bounds were obtained without using other cosmological data sets. This constraint was confirmed by the analysis of [259Jump To The Next Citation Point], which got − 0.097 < Δ αEM αEM < 0.034, with the WMAP-1yr data alone and − 0.042 < Δ αEM βˆ•αEM < 0.026, at a 95% confidence level, when combined with constraints on the Hubble parameter from the HST Hubble Key project.

The analysis of the WMAP-3yr data allows to improve [476Jump To The Next Citation Point] this bound to − 0.039 < Δ αEM βˆ•αEM < 0.010, at a 95% confidence level, assuming (Ω ,Ω ,h,n ,z ,A mat b s re s) for the cosmological parameters (Ω Λ being derived from the assumption ΩK = 0, as well as τ from the re-ionisation redshift, zre) and using both temperature and polarization data (TT, TE, EE).

The WMAP 5-year data were analyzed, in combination with the 2dF galaxy redshift survey, assuming that both α EM and m e can vary and that the universe was spatially Euclidean. Letting 6 cosmological parameters [(2 2 Ωmath ,Ωbh ,Θ, τ,ns,As), Θ being the ratio between the sound horizon and the angular distance at decoupling] and 2 constants vary they, it was concluded [452Jump To The Next Citation Point, 453Jump To The Next Citation Point] − 0.012 < Δ αEM βˆ•αEM < 0.018 and − 0.068 < Δme βˆ•me < 0.044, the bounds fluctuating slightly depending on the choice of the recombination scenario. A similar analyis [381Jump To The Next Citation Point] not including me gave − 0.050 < Δ αEM βˆ•αEM < 0.042, which can be reduced by taking into account some further prior from the HST data. Including polarisation data data from ACBAR, QUAD and BICEP, it was also obtained [352Jump To The Next Citation Point] − 0.043 < Δ αEM βˆ•αEM < 0.038 at 95% C.L. and − 0.013 < Δ αEM βˆ•αEM < 0.015 including HST data, also at 95% C.L. Let us also emphasize the work by [351] trying to include the variation of the Newton constant by assuming that ΔαEM βˆ• αEM = QΔG βˆ•G, Q being a constant and the investigation of [380Jump To The Next Citation Point] taking into account αEM, me and μ, G being kept fixed. Considering (Ωmat,Ωb, h,ns,τ) for the cosmological parameters they concluded from WMAP-5 data (T T, T E, EE) that − 8.28 × 10 −3 < Δ α βˆ•α < 1.81 × 10−3 EM EM and − 0.52 < Δ μβˆ•μ < 0.17

The analysis of [452Jump To The Next Citation Point, 453Jump To The Next Citation Point] was updated [310Jump To The Next Citation Point] to the WMAP-7yr data, including polarisation and SDSS data. It leads to − 0.025 < Δ αEM βˆ•αEM < − 0.003 and 0.009 < Δme βˆ•me < 0.079 at a 1σ level.

The main limitation of these analyses lies in the fact that the CMB angular power spectrum depends on the evolution of both the background spacetime and the cosmological perturbations. It follows that it depends on the whole set of cosmological parameters as well as on initial conditions, that is on the shape of the initial power spectrum, so that the results will always be conditional to the model of structure formation. The constraints on α EM or m e can then be seen mostly as constraints on a delayed recombination. A strong constraint on the variation of αEM can be obtained from the CMB only if the cosmological parameters are independently known. [438Jump To The Next Citation Point] forecasts that CMB alone can determine αEM to a maximum accuracy of 0.1%.

Table 11: Summary of the latest constraints on the variation of fundamental constants obtained from the analysis of cosmological data and more particularly of CMB data. All assume Ω = 0 K.
Constraint Data Comment Ref.
(αEM × 102)
[–9, 2] BOOMERanG-DASI-COBE + BBN BBN with αEM only [22]
[–1.4, 2] COBE-BOOMERanG-MAXIMA (Ωmat,Ωb,h,ns) [307]
[–5, 2] WMAP-1 (Ωmath2,Ωbh2,ΩΛh2,τ,ns,αs) [438Jump To The Next Citation Point]
[–6, 1] WMAP-1 same + αs = 0 [438]
[–9.7, 3.4] WMAP-1 (Ωmat,Ωb,h,ns,τ,me) [259Jump To The Next Citation Point]
[–4.2, 2.6] WMAP-1 + HST same [259]
[–3.9, 1.0] WMAP-3 (TT,TE,EE) + HST (Ω ,Ω ,h,n ,z ,A mat b s re s) [476]
[–1.2, 1.8] WMAP-5 + ACBAR + CBI + 2df (Ωmath2,Ωbh2,Θ,τ,ns,As,me) [452]
[–1.9, 1.7] WMAP-5 + ACBAR + CBI + 2df (Ωmath2,Ωbh2,Θ,τ,ns,As,me) [453]
[–5.0, 4.2] WMAP-5 + HST (Ωmath2,Ωbh2,h,τ,ns,As) [381]
[–4.3, 3.8] WMAP-5 + ACBAR + QUAD + BICEP (Ωmath2,Ωbh2,h,τ,ns) [352Jump To The Next Citation Point]
[–1.3, 1.5] WMAP-5 + ACBAR + QUAD + BICEP+HST (Ωmath2,Ωbh2,h,τ,ns) [352]
[–0.83, 0.18] WMAP-5 (TT,TE,EE) (2 2 Ωmath ,Ωbh ,h,τ,ns,As,me, μ) [380]
[–2.5, –0.3] WMAP-7 + H0 + SDSS (2 2 Ωmath ,Ωbh ,Θ,τ,ns,As,me [310]

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