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
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
In 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
WMAP7 and
Planck8.
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. |
(![]() |
|||
[–9, 2] | BOOMERanG-DASI-COBE + BBN | BBN with ![]() |
[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 + H0 + SDSS | (![]() |
[310] |
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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