Abstract
We present up-to-date constraints on the value of the fine-structure constant at the epoch of decoupling from the recent observations made by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. In the framework of models we considered, a positive (negative) variation of the value of α at decoupling with respect to the present-day value is now bounded to be smaller than 2% (6%) at 95% confidence level. We point out that the existence of an early reionization epoch as suggested by the above measurements will, when more accurate cosmic microwave background polarization data is available, lead to considerably tighter constraints. The so-called 'reionization bump', in particular, will be extremely useful for this purpose. We find that the tightest possible constraint on α is about 0.1% using CMB data alone-tighter constraints will require further (non-CMB) priors. * Electronic address: ybenson@example.net
I. INTRODUCTION
Now that WMAP has validated beyond reasonable doubt the so-called concordance (or is it conspiracy?) model of cosmology [1, 2, 3, 4] , it is time to proceed to the next level of questioning and start asking what are the 'dark components' that make up about 96% of the energy budget of the universe. Most of this is in some non-baryonic form, for which there is at present no direct evidence or solid theoretical explanation. Understanding its nature will clearly require further progress in fundamental physics.
I. INTRODUCTION
Cosmology and astrophysics provide a laboratory with extreme conditions in which to test fundamental physics and search for new paradigms. Currently preferred unification theories [5, 6] predict the existence of additional space-time dimensions, which have a number of possibly observable consequences, including modifications in the gravitational laws on very large (or very small) scales [7] and space-time variations of the fundamental constants of nature [8, 9] . Recent evidence of a time variation of fundamental constants [10, 11, 12] offers an important opportunity to test such fundamental physics models. It should be noted that the issue is not if such theories predict such variations, but at what level they do so, and hence if there is any hope of detecting them in the near future.
I. INTRODUCTION
The most promising case is that of the fine-structure constant α, for which some evidence of time variation at redshifts z ∼ 2 − 3 already exists [10, 11] . Since one expects α to be a non-decreasing function of time [13, 14, 15] , it is particularly important to try to constrain it at earlier epochs, where any variations relative to the present-day value should be larger. (On the other hand, local laboratory tests can also provide useful constraints [16] .) The cosmic microwave background (CMB) anisotropies provide such a probe, being mostly sensitive to the epoch of decoupling, z ∼ 1100. Here we extend our previous work in this area [17, 18, 19] by analyzing the recently released WMAP first-year data and providing updated constraints on the value of α at decoupling. We emphasize that in previous work, constraints on α were obtained with the help of additional cosmological datasets. Here, by contrast, we will analyse the WMAP dataset alone. We also discuss how these constraints can be improved in the future, when more precise CMB polarization data will be available.
I. INTRODUCTION
In particular, we show that the existence of an early reionization epoch is of significant help in further constraining α. A more detailed analysis of this issue can be found in a companion paper [20] . 
II. WMAP CONSTRAINTS ON α
We compare the recent WMAP temperature and cross-polarization data with a set of flat cosmological models adopting the likelihood estimator method described in [4] . We don't consider gravity waves, running of the spectral index or iso-curvature modes since these further modifications are not required by the WMAP data.
II. WMAP CONSTRAINTS ON α
The likelihood distribution function for α dec /α 0 , obtained after marginalization over the remaining parameters, is plotted in Figure 1 . We found, at 95% C.L. that 0.94 < α dec /α 0 < 1.01, improving previous bounds (see [19] for the previous tighter bound, and [17, 18, 22] for earlier work) based on CMB and complementary datasets. Thus when it comes to constraining α in the primordial universe, the WMAP dataset alone is at least as good as everything else before it put together.
III. THE FUTURE: POLARIZATION AND REIONIZATION
In our previous work [19] , only the CMB temperature was taken into account. The precision with which cosmological parameters can be reconstructed using both CMB temperature and E-polarization measurements is now re-examined by means of the Fisher Matrix Analysis (FMA) technique. This technique was first applied in this context by [23] (see also [24] ). We consider the planned Planck satellite and an ideal experiment which would measure both temperature and polarization to the cosmic variance limit (in the following, 'CVL experiment').
III. THE FUTURE: POLARIZATION AND REIONIZATION
Cosmological models are characterized by the 8 dimensional parameter set
III. THE FUTURE: POLARIZATION AND REIONIZATION
where Ω m = Ω c + Ω b is the energy density in matter and Ω Λ the energy density due to a cosmological constant, and the Hubble parameter h is a dependent variable. The quantity inaccuracies. We refer to [19, 20] for a detailed description of the numerical technique used.
III. THE FUTURE: POLARIZATION AND REIONIZATION
The experimental parameters used for the Planck analysis are in Table I , and we use the first 3 channels of the Planck High Frequency Instrument (HFI) only. Note that we account for possible issues arising from point sources, foreground removal and galactic plane contamination by assuming that once these have been taken into account we are left with a 'clean' fraction of the sky given by f sky . For the cosmic variance limited (CVL) experiment, we set the experimental noise to zero, and we use a total sky coverage f sky = 1.00. Although this is never to be achieved in practice, the CVL experiment illustrates the precision which can be obtained in principle from CMB temperature and E-polarization measurements. to the present day's value. We emphasize that, for the reason pointed out above, we do not need specify a redshift dependence for the variation, although we could if we so choose. If the value of α at low redshift is different from that at decoupling, the peaks in the polarization power spectrum at small angular scales will be shifted sideways, while the reionization bump on large angular scales won't (lower right panel). It follows that by measuring the separation between the normal peaks and the bump, one can measure both α and τ , as illustrated in Fig. 3 . Thus we expect that the existence of an early reionization epoch will, when more accurate cosmic microwave background polarization data is available, lead to considerably tighter constraints on α. Table II cosmic variance limited for temperature but there will still be considerable room for improvement in polarization (Table II) . This therefore argues for a post-Planck polarization experiment, not least because polarization is, in itself, better at determining cosmological parameters than temperature. We conclude that Planck alone will be able to constrain variations of α at the epoch of decoupling within 0.34% (1σ, all other parameters marginalized), which corresponds to approximately a factor 5 improvement on the current upper bound.
III. THE FUTURE: POLARIZATION AND REIONIZATION
On the other hand, the CMB alone can only constrain variations of α up to O(10 −3 ) at z ∼ 1100. Going beyond this limit will require additional (non-CMB) priors on some of the other cosmological parameters. This result is to be contrasted with the variation measured in quasar absorption systems by Ref. [10] , δα/α 0 = O(10 −5 ) at z ∼ 2. Nevertheless, there are models where deviations from the present value could be detected using the CMB.
IV. CONCLUSIONS
We have tightened the CMB constraints on the value of the fine-structure constant at the epoch of decoupling, using the WMAP satellite data. We emphasize that this is the first constraint coming from the analysis of a single CMB measurement. Previous constraints on α used a combination of various CMB and other cosmological observables, such as supernovae, nucleosynthesis or the Hubble constant, in order to impose stronger priors on other key cosmological parameters. In this sense, the current WMAP data alone is at least as good as everything else before it put together. As in previous work [17, 18, 19] , the current data is consistent with no variation, though the likelihood is skewed towards smaller values at the epoch of decoupling.
IV. CONCLUSIONS
We have shown that CMB data alone will be able to constrain α up to the 0.1% level.
IV. CONCLUSIONS
Tighter constraints than this will require invoking further (non-CMB) priors. Nevertheless, the existence of an early reionization epoch as suggested by WMAP turns out to be a blessing for these measurements, once more accurate cosmic microwave background polarization data is available, and in particular we have proposed a novel way of exploiting it. These points are discussed in more detail in [20] . 

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