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

The gravitational constant enters in the Friedmann equation and in the evolution of the cosmological perturbations. It has mainly three effects [435] that are detailed in Section 4.4.1. , 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

which enters in the collision term of the Boltzmann equation describing the evolution of the photon distribution function and where is the ionization fraction (i.e., the number density of free electrons with respect to their total number density ).The first dependence arises from the Thomson scattering cross section given by

and the scattering by free protons can be neglected since .The second, and more subtle dependence, comes from the ionization fraction. Recombination proceeds via 2-photon emission from the level or via the Ly- photons, which are redshifted out of the resonance line [405] 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- 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

where is the temperature. At high redshift, is identical to the one of the photons but evolves according to where the radiation constant with the Stefan–Boltzmann constant. In Equation (113), is the energy of the th hydrogen atomic level, is the ionization coefficient, the recombination coefficient, the correction constant due to the redshift of Ly- photons and to 2-photon decay and is the number density of protons. is related to by the principle of detailed balance so that The recombination rate to all other excited levels isIn summary, both the temperature of the decoupling and the residual ionization after recombination are modified by a variation of or . This was first discussed in [36, 277]. The last scattering surface can roughly be determined by the maximum of the visibility function , which measures the differential probability for a photon to be scattered at a given redshift. Increasing shifts to a higher redshift at which the expansion rate is faster so that the temperature and decrease more rapidly, resulting in a narrower . This induces a shift of the spectrum to higher multipoles and an increase of the values of the . 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 scale has , as , as , the ionisation coefficients as , the transition frequencies as , the Einstein’s coefficients as , the decay rates as and has complicated dependence, which roughly reduces to . Note that a change in the fine-structure constant and in the mass of the electron are degenerate according to but this degeneracy is broken for multipoles higher than 1500 [36]. In earlier works [244, 277] it was approximated by the scaling with .

The first studies [244, 277] focused on the sensitivity that can be reached by
WMAP^{7} and
Planck^{8}.
They concluded that they should provide a constraint on at recombination, i.e., at a redshift of
about , with a typical precision .

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 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 [22] by a joint analysis of BBN and CMB data that assumes that only varies and that included 4 cosmological parameters ( assuming a universe with Euclidean spatial section, leading to at 68% confidence level. A similar analysis [307], 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 , set the constraint , at a 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 is varying but, as can been seen from Eqs. (110–116), assuming the electron mass constant.

With the WMAP first year data, the bound on the variation of was sharpened [438] to , after marginalizing over the remaining cosmological parameters ( assuming a universe with Euclidean spatial sections. Restricting to a model with a vanishing running of the spectral index (), it gives , at a 95% confidence level. In particular it shows that a lower value of makes more compatible with the data. These bounds were obtained without using other cosmological data sets. This constraint was confirmed by the analysis of [259], which got , with the WMAP-1yr data alone and , 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 [476] this bound to , at a 95% confidence level, assuming () for the cosmological parameters ( being derived from the assumption , as well as from the re-ionisation redshift, ) and using both temperature and polarization data (, , ).

The WMAP 5-year data were analyzed, in combination with the 2dF galaxy redshift survey, assuming that both and can vary and that the universe was spatially Euclidean. Letting 6 cosmological parameters [(), being the ratio between the sound horizon and the angular distance at decoupling] and 2 constants vary they, it was concluded [452, 453] and , the bounds fluctuating slightly depending on the choice of the recombination scenario. A similar analyis [381] not including gave , 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 [352] at 95% C.L. and 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 , being a constant and the investigation of [380] taking into account , and , being kept fixed. Considering () for the cosmological parameters they concluded from WMAP-5 data (, , ) that and

The analysis of [452, 453] was updated [310] to the WMAP-7yr data, including polarisation and SDSS data. It leads to and at a 1 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 or can then be seen mostly as constraints on a delayed recombination. A strong constraint on the variation of can be obtained from the CMB only if the cosmological parameters are independently known. [438] forecasts that CMB alone can determine to a maximum accuracy of 0.1%.

Constraint | Data | Comment | Ref. |

( × 10^{2}) |
|||

[–9, 2] | BOOMERanG-DASI-COBE + BBN | BBN with only | [22] |

() | |||

[–1.4, 2] | COBE-BOOMERanG-MAXIMA | () | [307] |

[–5, 2] | WMAP-1 | () | [438] |

[–6, 1] | WMAP-1 | same + | [438] |

[–9.7, 3.4] | WMAP-1 | () | [259] |

[–4.2, 2.6] | WMAP-1 + HST | same | [259] |

[–3.9, 1.0] | WMAP-3 (TT,TE,EE) + HST | () | [476] |

[–1.2, 1.8] | WMAP-5 + ACBAR + CBI + 2df | () | [452] |

[–1.9, 1.7] | WMAP-5 + ACBAR + CBI + 2df | () | [453] |

[–5.0, 4.2] | WMAP-5 + HST | () | [381] |

[–4.3, 3.8] | WMAP-5 + ACBAR + QUAD + BICEP | () | [352] |

[–1.3, 1.5] | WMAP-5 + ACBAR + QUAD + BICEP+HST | () | [352] |

[–0.83, 0.18] | WMAP-5 (TT,TE,EE) | () | [380] |

[–2.5, –0.3] | WMAP-7 + H_{0} + SDSS |
( | [310] |

Living Rev. Relativity 14, (2011), 2
http://www.livingreviews.org/lrr-2011-2 |
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