Finestructure constant constraints on dark energy
Abstract
We use astrophysical and atomic clock tests of the stability of the finestructure constant , together with Type Ia supernova and Hubble parameter data, to constrain the simplest class of dynamical dark energy models where the same degree of freedom is assumed to provide both the dark energy and (through a dimensionless coupling, , to the electromagnetic sector) the variation. We show how current data tightly constrains a combination of and the dark energy equation of state . At the confidence level and marginalizing over we find , with the atomic clock tests dominating the constraints. The forthcoming generation of highresolution ultrastable spectrographs will enable significantly tighter constraints.
pacs:
98.80.k, 04.50.KdI Introduction
The discovery of cosmic acceleration from luminosity distance measurements of Type Ia supernovas Riess et al. (1998); Perlmutter et al. (1999) led to wideranging theoretical and observational efforts trying to understand and characterize it. While a cosmological constant remains the simplest explanation consistent with most observational data, the wellknown finetuning problems associated with this solution imply that alternative scenarios should be sought and tested. The most natural alternative would involve scalar fields, an example of which is the recently discovered Higgs field Aad et al. (2012); Chatrchyan et al. (2012), which would lead to dynamical dark energy.
If dynamical scalar fields are present, one naturally expects them to couple to the rest of the model, unless a yetunknown symmetry is postulated to suppress these couplings. In particular, a coupling to the electromagnetic sector will lead to spacetime variations of the finestructure constant —see Uzan (2011); Martins (2014) for recent reviews. There are some recent indications of such a variation Webb et al. (2011), at the level of a few parts per million, which a dedicated Large Program at ESO’s Very Large Telescope (VLT) is aiming to test Molaro et al. (2013); Evans et al. (2014).
Here, in the same spirit of Calabrese et al. (2011, 2014), we discuss how astrophysical and local tests of the stability of can be used as additional tests of the underlying theories, in particular if one makes the ’minimal’ assumption that the same dynamical degree of freedom is responsible for the dark energy and the variations. In this case observational tests of the evolution of directly constrain dark energy. The future impact of these methods as a dark energy probe has recently been studied in detail in Amendola et al. (2012); Leite et al. (2014).(Earlier, more simplistic forecasts can also be found in Avelino et al. (2006); Nunes et al. (2009).) Here we show how current data already provides useful constraints.
We start by reviewing the relation between a varying and dynamical energy in the case (dubbed ’Class I models’ in Martins (2014)) where both are due to the same dynamical degree of freedom. We will then focus on the case of a constant dark energy equation of state, , constraining this model with a combination of cosmological data and local and astrophysical tests of . This choice is in the interest of simplicity, as it minimizes the number of free parameters and is frequently used as a fiducial model for forecasts; we leave a discussion of more general models for a subsequent, more detailed publication.
Ii Varying and dark energy
Dynamical scalar fields in an effective 4D field theory are naturally expected to couple to the rest of the theory, unless a (still unknown) symmetry suppresses this coupling Carroll (1998); Dvali and Zaldarriaga (2002). We assume this to be the case for the dynamical degree of freedom responsible for the dark energy. Specifically the coupling between the scalar field, , and the electromagnetic sector stems from a gauge kinetic function
(1) 
which one can assume to be linear,
(2) 
(with ): as pointed out in Dvali and Zaldarriaga (2002) the absence of such a term would require a symmetry, but such a symmetry must be broken throughout most of the cosmological evolution. Local tests of the Equivalence Principle lead to the conservative constraint on the dimensionless coupling parameter (see Uzan (2011) for an overview)
(3) 
while in Calabrese et al. (2011) an independent fewpercent constraint on this coupling was obtained using CMB and largescale structure data in combination with direct measurements of the expansion rate of the universe.
With these assumptions one can explicitly relate the evolution of to that of dark energy, as in Calabrese et al. (2011). The evolution of can be written
(4) 
and, since the evolution of the putative scalar field can be expressed in terms of the dark energy properties and as
(5) 
(with the prime denoting the derivative with respect to the logarithm of the scale factor), we finally obtain
(6) 
The is assumes a canonical scalar field, but the argument can be repeated for phantom fields Vielzeuf and Martins (2014), leading to
(7) 
the change of sign stems from the fact that one expects phantom filed to roll up the potential rather than down.
In the present work we’ll focus on models with a constant equation of state , and will constrain them using the following datasets

Laboratory data: the atomic clock constraint on the current drift of of Rosenband et al. Rosenband et al. (2008), which we can write as
(8) 
Astrophysical data: we will use both spectroscopic measurements of of Webb et al. Webb et al. (2011) (a large dataset of 293 archival data measurements) and the smaller and more recent dataset of 11 dedicated measurements listed in Table 1. The latter include the early results of the UVES Large Program for Testing Fundamental Physics Molaro et al. (2013); Evans et al. (2014), which is expected to be the one with a better control of possible systematics.
Object  z  Spectrograph  Ref.  

3 sources  1.08  HIRES  Songaila and Cowie (2014)  
HS15491919  1.14  UVES/HIRES/HDS  Evans et al. (2014)  
HE05154414  1.15  UVES  Molaro et al. (2008)  
HE05154414  1.15  HARPS/UVES  Chand et al. (2006)  
HS15491919  1.34  UVES/HIRES/HDS  Evans et al. (2014)  
HE00012340  1.58  UVES  Agafonova et al. (2011)  
HE11041805A  1.66  HIRES  Songaila and Cowie (2014)  
HE22172818  1.69  UVES  Molaro et al. (2013)  
HS19467658  1.74  HIRES  Songaila and Cowie (2014)  
HS15491919  1.80  UVES/HIRES/HDS  Evans et al. (2014)  
Q1101264  1.84  UVES  Molaro et al. (2008) 
Iii Observational constraints
We now use the above datasets to constrain the dynamical dark energy model with a constant equation of state . The behaviour of will be determined by Eq.(6) for and Eq.(7) for . Our main interest is in obtaining constraints on the – plane, and for this reason we will fix km/s/Mpc and (and assume a flat universe, so ). This choice of cosmological parameters is fully consistent with the supernova and Hubble parameter data we use. Moreover, we have explicitly verified that allowing , or the curvature parameter to vary (within observationally reasonable ranges) and marginalizing over these parameters does not significantly change our results: it is clear that the critical cosmological parameter here is itself.
We therefore consider a 2D grid of and values, and use standard chisquare techniques to compare the models to the aforementioned datasets. Figure 1 shows the results of this comparison for the Webb et al. data (top panel), and for the Table 1 data (middle panel)—in both cases, the constraints from the astrophysical data are shown by the thin red lines. The Webb data is not consistent with the null result Webb et al. (2011), and we correspondingly find a one sigma preference for a nonzero coupling (with a negative sign for a canonical field, or a positive sign for a phantom field). However, the data is compatible with the null result at two sigma. On the other hand, the Table 1 data is fully compatible with the null result. We note that in the former case the reduced chisquare of the bestfit model is , while in the latter case it is ; this may be an indication that some of the uncertainties in the Table 1 measurements are underestimated.
For comparison we also show in the bottom panel of Fig. 1, in the same scale as before (and also in thin red lines), the local atomic clock constraint of Rosenband et al. Rosenband et al. (2008). For the models under consideration this translates into
(9) 
(with the and signs respectively corresponding to the canonical and phantom field cases), and it is clear from the plot that this is currently more constraining than the astrophysical measurements.
The cosmological data we are considering is insensitive to . (Strictly speaking, a varying does affect the luminosity of Type Ia supernovas, but as recently shown in Calabrese et al. (2014) for partspermillion level variations the effect is too small to have an impact on current datasets, and we therefore neglect it in the present analysis.) Naturally, the cosmological data does constrain , effectively providing us with a prior on it. In Fig. 1 the cosmological data constraints are shown by the blue vertical lines, while the combined (cosmological plus astrophysical, or cosmological plus atomic clocks) constraints are shown by the thick black lines. Naturally, we can obtain tighter constraints by combining all datasets; this is straightforward to do since the Webb et al. and Table 1 measurements are independent. The results of this analysis are shown in Fig. 2.
Finally, in addition to constraints in the twodimensional – plane it is also interesting to obtain 1D constraints on the coupling by marginalizing over the dark energy equation of state . The results of this analysis are shown in Fig. 3. We confirm that in the case of the Webb et al. dataset there is a onesigma preference for a nonzero coupling, while in the other cases the null result provides the best fit. Significantly, the combination of all datasets allows us to obtain a nontrivial constraint on . At the twosigma () confidence level we find
(10) 
significantly improving upon previous constraints. As previously mentioned, the atomic clock measurement of Rosenband et al. currently provides tighter constraints than the astrophysical measurements. This new bound is the main result of our analysis. (Nevertheless, we also note that at threesigma is unconstrained.) We can similarly obtain the 1D likelihood for by marginalizing over . In this case we find at the threesigma () confidence level
(11) 
although this bound should be interpreted more cautiously given our assumptions on other cosmological parameters. We leave a more systematic exploration of the relevant parameter space for a subsequent publication.
Iv Conclusions and outlook
We have used a combination of astrophysical spectroscopy and local laboratory tests of the stability of the finestructure constant , complemented by background cosmological datesets, to constrain the simplest class of dynamical dark energy models where the same degree of freedom is responsible for both the dark energy and a variation of . In these models the redshift dependence of depends both on a fundamental physics parameter (the dimensionless coupling of the scalar field to the electromagnetic sector) and background ’dark cosmology’ parameters—for the simplest class of models we studied these are the dimensionless dark energy density and the dark energy equation of state .
We obtained new, tighter constraints on the dimensionless coupling of the scalar field to the electromagnetic sector. We note that these constraints are currently dominated by the atomic clock tests, which are only sensitive to the dark energy equation of state today. Thus a constant equation of state cosmological model is a reasonable assumption. Improvements in astrophysical measurements will allow more generic models to be constrained.
We have also pointed out how different currently available astrophysical measurements of (specifically the archival data of Webb et al. and the dedicated measurements of Table 1) lead to somewhat different constraints). This highlights the importance of obtaining improved astrophysical measurements of (both in terms of statistical uncertainty and in terms of control over possible systematics), not only for their own sake but also because there can have a strong impact on dark energy studies. The next generation of highresolution ultrastable spectrographs such as ESPRESSO and ELTHIRES will be ideal for this task. A roadmap for these studies is outlined in Martins (2014), and more detailed forecasts of the future impact of these measurements may be found in Leite et al. (2014).
Acknowledgements.
This work was done in the context of project PTDC/FIS/111725/2009 (FCT, Portugal). CJM is also supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE (EC).References
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