Constraints on variation in and from WMAP 7year data
S. J. Landau^{1,}^{}  G. Scóccola^{2,}^{}
1  Departamento de Física, FCEyN, Universidad de Buenos Aires,
Ciudad Universitaria  Pab. 1, 1428 Buenos Aires, Argentina
2  MaxPlanckInstitut für Astrophysik, KarlSchwarzschild Str. 1, Postfach 1317,
85741 Garching, Germany
Received 8 February 2010 / Accepted 4 May 2010
Abstract
Aims. We update the constraints on the time variation of the fine structure constant
and the electron mass ,
using the latest CMB data, including the 7yr release of WMAP.
Methods. We made statistical analyses of the variation of each
one of the constants and of their joint variation, together with the
basic set of cosmological parameters. We used a modified version of
CAMB and COSMOMC to account for these possible variations.
Results. We present bounds on the variation of the constants for
different data sets, and show how results depend on them. When using
the latest CMB data plus the power spectrum from Sloan Digital Sky
Survey LRG, we find that
at 1
level, when the 6 basic cosmological parameters were fitted, and only variation in
was allowed. The constraints in the case of variation of both constants are
and
.
In the case of only variation in ,
the bound is
.
Key words: cosmic background radiation  cosmological parameters  early Universe
1 Introduction
The variation of fundamental constants over cosmological time scales is a prediction of theories that attempt to unify the four interactions in nature, like string derived field theories, related braneworld theories and KaluzaKlein theories (see Uzan 2003; GarcíaBerro et al. 2007, and references therein). Many observational and experimental efforts have been made to put constraints on such variations. Most of the reported data are consistent with null variation of fundamental constants. Although there have been recent claims for time variation of the fine structure constant () and of the proton to electron mass ratio ( ) (Reinhold et al. 2006; Murphy et al. 2003), independent analyses of similar data give null results (Malec et al. 2010; King et al. 2008; Srianand et al. 2004; Thompson et al. 2009). On the other hand, a recent analysis of ammonia spectra in the Milky Way suggests a spatial variation of (Molaro et al. 2009; Levshakov et al. 2010).
Unifying theories predict variation of all coupling constants, being all variations related in general to the rolling of a scalar field. Therefore, the relationship between variations of coupling constants depends on the unifying model. In this paper we adopt a phenomenological approach and analyse the possible variation of and/or at the time of the formation of neutral hydrogen without assuming any theoretical model. Nakashima et al. (2010) have considered also the variation in the proton mass (). This quantity affects mainly the baryon mass density and the baryon number density. Their results confirm the strong degeneracy with the baryon density. Therefore, we will not consider the variation in in this work.
Cosmic microwave background radiation (CMB) is one of the most powerful tools to study the early universe and in particular, to put bounds on possible variations in the fundamental constants between early times and the present. Changing or at recombination affects the differential optical depth of the photons due to Thompson scattering, changing therefore Thompson scattering cross section and the ionization fraction. The signatures on the CMB angular power spectrum due to varying fundamental constants are similar to those produced by changes in the cosmological parameters, i.e. changes in the relative amplitudes of the Doppler peaks and a shift in their positions. Moreover, an increment in or decreases the high diffusion damping, which is due to the finite thickness of the lastscattering surface, and thus, increases the power on very small scales (Hannestad 1999; Kaplinghat et al. 1999).
Recent analyses of CMB data (earlier than the WMAP sevenyear release) including a possible variation in have been performed by Martins et al. (2010); Menegoni et al. (2009); Scóccola et al. (2009,2008); Nakashima et al. (2010), and including a possible variation in have been performed by Scóccola et al. (2009); Nakashima et al. (2010); Scóccola et al. (2008).
In our previous works, we have also analyzed the dependence of the updated recombination scenario (that includes the recombination of helium, and was implemented in R ECFAST following Wong et al. 2008) on and , and show that these dependencies are not relevant for WMAP data.
In this paper we adopt a phenomenological approach and analyse the possible variation in and/or without assuming any theoretical model. We use WMAP sevenyear release, together with other recent CMB data. We also combine CMB data with other cosmological data sets: i) the power spectrum of the Sloan Digital Sky Survery DR7 LRG; ii) a recent constraint of the Hubble constant H_{0} with data from the Hubble Space Telescope. In Sect. 2 we describe the method and data sets we used in the statistical analysis. We present and discuss our results in Sect. 3. We conclude in Sect. 4.
2 Statistical analysis
We performed our statistical analysis by exploring the parameter space with Monte Carlo Markov chains generated with the CosmoMC code (Lewis & Bridle 2002) which uses the Boltzmann code CAMB (Lewis et al. 2000) and R ECFAST to compute the CMB power spectra. We modified them in order to include the possible variation in and at recombination.
We use data from the WMAP 7year temperature and temperaturepolarization power spectrum (Larson et al. 2010), and other CMB experiments such as CBI (Readhead et al. 2004), ACBAR (Kuo et al. 2004), BOOMERANG (Jones et al. 2006; Piacentini et al. 2006), BICEP (Chiang et al. 2010) and QUAD (Brown et al. 2009). In order to reduce degeneracies of the cosmological parameters, we combine the CMB data sets with other cosmological data: i) the power spectrum of the Sloan Digital Sky Survey LRG (Reid et al. 2010) and ii) the recent constraint on the Hubble constant, km s^{1} Mpc^{1}, presented by Riess et al. (2009).
We have considered a spatiallyflat
cosmological model with adiabatic density fluctuations, and the
following parameters:
where is the baryon density and is the dark matter density, both in units of the critical density; gives the ratio of the comoving sound horizon at decoupling to the angular diameter distance to the surface of last scattering (and is related to the Huble constant H_{0}); is the reionization optical depth; the scalar spectral index; and is the amplitude of the density fluctuations.
We have performed statistical analyses using the data mentioned above and considering variation of only one constant ( or ) and variation of both constants. We present our results in the next section.
3 Results and discussion
Results for the variation of the constants in the case when only one constant is allowed to vary are shown in Table 1 and for the case when both are allowed to vary, are presented in Table 2. The obtained values are consistent with no variation of or at recombination. The obtained errors are at the same percent level than those obtained by Martins et al. (2010); Menegoni et al. (2009); Scóccola et al. (2009,2008) using WMAP5 year release. The parameter space has higher dimension when both constants are allowed to vary. Therefore, limits on and are more stringent in the case were only one constant is allowed to vary. Results for the cosmological parameters have similar values for all of the analyses. Therefore, we only report the values obtained in the case where both and were allowed to vary and the data from CMB and the power spectrum of the SDSS DR7 were considered (see Table 3). The mean values and errors for the cosmological parameters are in agreement within 1 with those obtained by the WMAP collaboration (Larson et al. 2010) with no variation of fundamental constants.
Table 1: Mean values and 1 errors for the analysis with variation of only , and only .
Table 2: Mean values and 1 errors for the analysis with the joint variation of and .
Table 3: Mean values and 1 errors for the cosmological parameters using all CMB data and the SDSS DR7 power spectrum.
In Fig. 1 we show the 68% and 95% c.l. constraints for versus H_{0}, for the analysis of the variation of alone. The results correspond to different data sets: all the CMB data alone; all the CMB data plus the H_{0} prior taken from Riess et al. (2009); and all the CMB data plus the power spectrum from Sloan Digital Sky Survery DR7 LRG (Reid et al. 2010). The large degeneracy between and H_{0} from CMB data is reduced when another data set is added. However, since the value of H_{0} obtained from the extra data sets are different, the obtained constraint on depends strongly on the data chosen for the analysis. Nevertheless, the results are consistent within 1.
Figure 1: 68% and 95% c.l. constraints for versus H_{0}, for the analysis of the variation of alone. Results from different data sets. 

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In Fig. 2 we present the constraints for versus and in Fig. 3 we present the constraints for versus . There are degeneracies among these parameters. The contours change because of the different mean value of obtained with different data sets.
Figure 2: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

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Figure 3: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

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In Fig. 4 we present the result for the case where only was allowed to vary. The degeneracy between and H_{0} is larger than between and H_{0}, making impossible to find reliable constraints using CMB data alone. When another data set is added, the bounds result tighter, but the mean value for depends strongly on which data set was added. Results are marginally consistent at 1.
Figure 4: 68% and 95% c.l. constraints for versus H_{0}, for the analysis of the variation of alone. Results from different data sets. 

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The constraints on versus are shown in Fig. 5, and on versus are shown in Fig. 6. In both cases, the results depend on the data set added to CMB data in the statistical analysis.
Figure 5: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

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Figure 6: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

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Figure 7: 68% and 95% c.l. constraints for the joint variation of and from different data sets. 

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In Fig. 7 we show the posterior distribution for and , for the case of joint variation of these quantities, marginalized over the cosmological parameters. The results correspond to different data sets. The difference in the contours is mainly due to the large degeneracy of and H_{0}, and the different H_{0} values derived from the Sloan power spectrum and from the H_{0} prior. We see that the mean value of is more affected than the mean value of . These results can also be seen in Table 2.
A variation of or affects the recombination scenario (see Scóccola et al. 2009, for example). As a consequence, the angular diameter distance at recombination is modified if any of these constants varies. This results in a change in the Doppler peak positions and heights (see Kaplinghat et al. 1999, for example). This explains the degeneracy between and shown in Fig. 7 and confirmed by the correlation coefficient. On the other hand, the degeneracy between or with the baryon mass density or the Hubble constant can be explained since these effects are similar to a change in the cosmological parameters. A variation in and/or at recombination, affects mainly the binding energy of hydrogen. This quantity is proportional to . When only one constant is allowed to vary, its influence on the parameter estimation is similar, regardless of the constant. However, when a joint variation analysis is performed, the results are different for and , due to the power with which they enter the hydrogen binding energy. In particular, in Fig. 7, we note that the bounds on are not affected when including additional data sets to the CMB data. This is due to the fact that is no longer correlated with H_{0}, as it was shown previously (see, for example, Landau et al. 2008).
4 Conclusions
In this paper we have updated the constraints on the time variation of the fine structure constant and the electron mass during recombination epoch, using the latest CMB data, including the 7yr release of WMAP. We perform several statistical analyses adding two different data sets; the H_{0} prior taken from Riess et al. (2009); and the power spectrum from Sloan Digital Sky Survery DR7 LRG (Reid et al. 2010). The bounds on the variation of the constants are tighter than previous results because of the higher precision of the new data used in this work.
Our results show no variation of the constants at recombination time. We also emphasize that the constraints depend strongly on which data set we choose in the analysis, due to the large degeneracy between or and H_{0}. Yet, the results are consistent within 1.
AcknowledgementsThe research leading to these results has received funding from the European Community's Seventh Framework Programme ([FP7/20072013] under grant agreement No. 237739), and PICT 200702184 from Agencia Nacional de Promoción Científica y Tecnológica, Argentina. The authors would like to thank H. C. Chiang for help with compiling the BICEP dataset in COSMOMC.
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Footnotes
 ...^{}
 Member of the Carrera del Investigador Científico y Tecnológico, CONICET.
 ...^{}
 Marie Curie fellow.
All Tables
Table 1: Mean values and 1 errors for the analysis with variation of only , and only .
Table 2: Mean values and 1 errors for the analysis with the joint variation of and .
Table 3: Mean values and 1 errors for the cosmological parameters using all CMB data and the SDSS DR7 power spectrum.
All Figures
Figure 1: 68% and 95% c.l. constraints for versus H_{0}, for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 2: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 3: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 4: 68% and 95% c.l. constraints for versus H_{0}, for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 5: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 6: 68% and 95% c.l. constraints for versus , for the analysis of the variation of alone. Results from different data sets. 

Open with DEXTER  
In the text 
Figure 7: 68% and 95% c.l. constraints for the joint variation of and from different data sets. 

Open with DEXTER  
In the text 
Copyright ESO 2010