Fine-structure constant constraints on dark energy
aa r X i v : . [ a s t r o - ph . C O ] M a y Fine-structure constant constraints on dark energy
C. J. A. P. Martins
1, 2, ∗ and A. M. M. Pinho
1, 3, † Centro de Astrof´ısica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Instituto de Astrof´ısica e Ciˆencias do Espa¸co, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal Faculdade de Ciˆencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal (Dated: 12 March 2015)We use astrophysical and atomic clock tests of the stability of the fine-structure constant α ,together with Type Ia supernova and Hubble parameter data, to constrain the simplest class ofdynamical dark energy models where the same degree of freedom is assumed to provide both thedark energy and (through a dimensionless coupling, ζ , to the electromagnetic sector) the α variation.We show how current data tightly constrains a combination of ζ and the dark energy equation ofstate w . At the 95% confidence level and marginalizing over w we find | ζ | < × − , withthe atomic clock tests dominating the constraints. The forthcoming generation of high-resolutionultra-stable spectrographs will enable significantly tighter constraints. I. INTRODUCTION
The discovery of cosmic acceleration from luminositydistance measurements of Type Ia supernovas [1, 2] led towide-ranging theoretical and observational efforts tryingto understand and characterize it. While a cosmologicalconstant Λ remains the simplest explanation consistentwith most observational data, the well-known fine-tuningproblems associated with this solution imply that alter-native scenarios should be sought and tested. The mostnatural alternative would involve scalar fields, an exam-ple of which is the recently discovered Higgs field [3, 4],which would lead to dynamical dark energy.If dynamical scalar fields are present, one naturally ex-pects them to couple to the rest of the model, unless ayet-unknown symmetry is postulated to suppress thesecouplings. In particular, a coupling to the electromag-netic sector will lead to spacetime variations of the fine-structure constant α —see [5, 6] for recent reviews. Thereare some recent indications of such a variation [7], at thelevel of a few parts per million, which a dedicated LargeProgram at ESO’s Very Large Telescope (VLT) is aimingto test [8, 9].Here, in the same spirit of [10, 11], we discuss how as-trophysical and local tests of the stability of α can beused as additional tests of the underlying theories, inparticular if one makes the ’minimal’ assumption thatthe same dynamical degree of freedom is responsible forthe dark energy and the α variations. In this case ob-servational tests of the evolution of α directly constraindark energy. The future impact of these methods as adark energy probe has recently been studied in detailin [12, 13].(Earlier, more simplistic forecasts can also befound in [14, 15].) Here we show how current data al-ready provides useful constraints.We start by reviewing the relation between a varying α and dynamical energy in the case (dubbed ’Class I ∗ [email protected] † [email protected] models’ in [6]) where both are due to the same dynamicaldegree of freedom. We will then focus on the case of aconstant dark energy equation of state, w , constrainingthis model with a combination of cosmological data andlocal and astrophysical tests of α . This choice is in theinterest of simplicity, as it minimizes the number of freeparameters and is frequently used as a fiducial model forforecasts; we leave a discussion of more general modelsfor a subsequent, more detailed publication. II. VARYING α AND DARK ENERGY
Dynamical scalar fields in an effective 4D field theoryare naturally expected to couple to the rest of the the-ory, unless a (still unknown) symmetry suppresses thiscoupling [16, 17]. We assume this to be the case for thedynamical degree of freedom responsible for the dark en-ergy. Specifically the coupling between the scalar field, φ , and the electromagnetic sector stems from a gaugekinetic function B F ( φ ) L φF = − B F ( φ ) F µν F µν (1)which one can assume to be linear, B F ( φ ) = 1 − ζκ ( φ − φ ) , (2)(with κ = 8 πG ): as pointed out in [17] the absenceof such a term would require a φ → − φ symmetry, butsuch a symmetry must be broken throughout most of thecosmological evolution. Local tests of the EquivalencePrinciple lead to the conservative constraint on the di-mensionless coupling parameter (see [5] for an overview) | ζ local | < − , (3)while in [10] an independent few-percent constraint onthis coupling was obtained using CMB and large-scalestructure data in combination with direct measurementsof the expansion rate of the universe.With these assumptions one can explicitly relate theevolution of α to that of dark energy, as in [10]. Theevolution of α can be written∆ αα ≡ α − α α = ζκ ( φ − φ ) , (4)and, since the evolution of the putative scalar field canbe expressed in terms of the dark energy properties Ω φ and w as 1 + w φ = ( κφ ′ ) φ (5)(with the prime denoting the derivative with respect tothe logarithm of the scale factor), we finally obtain∆ αα ( z ) = ζ Z z q φ ( z ) (1 + w φ ( z )) dz ′ z ′ . (6)The is assumes a canonical scalar field, but the argumentcan be repeated for phantom fields [18], leading to∆ αα ( z ) = − ζ Z z q φ ( z ) | w φ ( z ) | dz ′ z ′ ; (7)the change of sign stems from the fact that one expectsphantom filed to roll up the potential rather than down.In the present work we’ll focus on models with a con-stant equation of state w , and will constrain them usingthe following datasets • Cosmological data: the Union2.1 Type Ia super-nova dataset [19] and the compilation of Hubble pa-rameter measurements from Farooq & Ratra [20]. • Laboratory data: the atomic clock constraint onthe current drift of α of Rosenband et al. [21],which we can write as1 H ˙ αα = ( − . ± . × − . (8) • Astrophysical data: we will use both spectroscopicmeasurements of α of Webb et al. [7] (a largedataset of 293 archival data measurements) andthe smaller and more recent dataset of 11 dedi-cated measurements listed in Table I. The latterinclude the early results of the UVES Large Pro-gram for Testing Fundamental Physics [8, 9], whichis expected to be the one with a better control ofpossible systematics. III. OBSERVATIONAL CONSTRAINTS
We now use the above datasets to constrain the dynam-ical dark energy model with a constant equation of state w . The behaviour of α will be determined by Eq.(6) for w > − w < −
1. Our main interest is
Object z ∆ α/α
Spectrograph Ref.3 sources 1.08 4 . ± . − . ± . − − . ± . − . ± . − . ± . − − . ± . − − . ± . − . ± . − . ± . − . ± . −
264 1.84 5 . ± . α . Listed are,respectively, the object along each line of sight, the redshiftof the measurement, the measurement itself (in parts per mil-lion), the spectrograph, and the original reference. The firstmeasurement is the weighted average from 8 absorbers in theredshift range 0 . < z < .
53 along the lines of sight ofHE1104-1805A, HS1700+6416 and HS1946+7658, reportedin [22] without the values for individual systems. The UVES,HARPS, HIRES and HDS spectrographs are respectively inthe VLT, ESO 3.6m, Keck and Subaru telescopes. in obtaining constraints on the ζ – w plane, and for thisreason we will fix H = 70 km/s/Mpc and Ω m = 0 . φ = 0 . H , Ω m or thecurvature parameter to vary (within observationally rea-sonable ranges) and marginalizing over these parametersdoes not significantly change our results: it is clear thatthe critical cosmological parameter here is w itself.We therefore consider a 2D grid of ζ and w values,and use standard chi-square techniques to compare themodels to the aforementioned datasets. Figure 1 showsthe results of this comparison for the Webb et al. data(top panel), and for the Table I data (middle panel)—inboth cases, the constraints from the astrophysical dataare shown by the thin red lines. The Webb data is notconsistent with the null result [7], and we correspond-ingly find a one sigma preference for a non-zero coupling ζ (with a negative sign for a canonical field, or a positivesign for a phantom field). However, the data is com-patible with the null result at two sigma. On the otherhand, the Table I data is fully compatible with the nullresult. We note that in the former case the reduced chi-square of the best-fit model is χ min,W ebb = 1 .
04, whilein the latter case it is χ min,T able = 1 .
29; this may be anindication that some of the uncertainties in the Table Imeasurements are underestimated.For comparison we also show in the bottom panel ofFig. 1, in the same scale as before (and also in thin redlines), the local atomic clock constraint of Rosenband et −1.1 −1.05 −1 −0.95 −0.9 −0.85−3−2−10123 x 10 −5 w ζ −1.1 −1.05 −1 −0.95 −0.9 −0.85−3−2−10123 x 10 −5 w ζ −1.1 −1.05 −1 −0.95 −0.9 −0.85−3−2−10123 x 10 −5 w ζ FIG. 1. One, two and three sigma constraints on the ζ − w plane from Webb et al. data (top panel), Table I data (mid-dle) and the atomic clock bound (bottom). In each panel thethin red lines correspond to the constraints from the astro-physical or clock data alone, the blue vertical ones correspondto the cosmological data (which constrain w but are insensi-tive to ζ ) and the black thick lines correspond to the combineddatasets. −1.1 −1.05 −1 −0.95 −0.9 −0.85−3−2−10123 x 10 −5 w ζ FIG. 2. One, two and three sigma constraints on the ζ − w plane from the full dataset considered in our analysis: Webb et al. data plus Table I data plus atomic clock bound plus cos-mological (Type Ia supernova and Hubble parameter) data.The reduced chi-square of the best fit is χ min,full = 0 . al. [21]. For the models under consideration this trans-lates into 1 H ˙ αα = ∓ ζ q φ | w | (9)(with the − and + signs respectively corresponding to thecanonical and phantom field cases), and it is clear fromthe plot that this is currently more constraining than theastrophysical measurements.The cosmological data we are considering is insensitiveto ζ . (Strictly speaking, a varying α does affect the lu-minosity of Type Ia supernovas, but as recently shown in[11] for parts-per-million level α variations the effect istoo small to have an impact on current datasets, and wetherefore neglect it in the present analysis.) Naturally,the cosmological data does constrain w , effectively pro-viding us with a prior on it. In Fig. 1 the cosmologi-cal data constraints are shown by the blue vertical lines,while the combined (cosmological plus astrophysical, orcosmological plus atomic clocks) constraints are shownby the thick black lines. Naturally, we can obtain tighterconstraints by combining all datasets; this is straightfor-ward to do since the Webb et al. and Table I measure-ments are independent. The results of this analysis areshown in Fig. 2.Finally, in addition to constraints in the two-dimensional ζ – w plane it is also interesting to obtain1D constraints on the coupling ζ by marginalizing overthe dark energy equation of state w . The results of thisanalysis are shown in Fig. 3. We confirm that in the caseof the Webb et al. dataset there is a one-sigma prefer-ence for a non-zero coupling, while in the other cases thenull result provides the best fit. Significantly, the com-bination of all datasets allows us to obtain a non-trivial −3 −2 −1 0 1 2 3x 10 −5 ζ ∆ χ FIG. 3. 1D likelihood for ζ , marginalizing over w . Plottedis the value of ∆ χ = χ − χ min , for cosmological + Webbdata (blue dashed), cosmological + Table I data (blue dash-dotted), cosmological + atomic clock data (red dotted) andthe combination of all datasets (black solid). constraint on ζ . At the two-sigma (95 . | ζ | < × − , (10)significantly improving upon previous constraints. Aspreviously mentioned, the atomic clock measurement ofRosenband et al. currently provides tighter constraintsthan the astrophysical measurements. This new boundis the main result of our analysis. (Nevertheless, we alsonote that at three-sigma ζ is unconstrained.) We cansimilarly obtain the 1D likelihood for w by marginaliz-ing over ζ . In this case we find at the three-sigma (99 . − . < w < − . , (11)although this bound should be interpreted more cau-tiously given our assumptions on other cosmological pa-rameters. We leave a more systematic exploration of therelevant parameter space for a subsequent publication. IV. CONCLUSIONS AND OUTLOOK
We have used a combination of astrophysical spec-troscopy and local laboratory tests of the stability of thefine-structure constant α , complemented by backgroundcosmological datesets, to constrain the simplest class ofdynamical dark energy models where the same degree offreedom is responsible for both the dark energy and avariation of α . In these models the redshift dependenceof α depends both on a fundamental physics parame-ter (the dimensionless coupling ζ of the scalar field tothe electromagnetic sector) and background ’dark cos-mology’ parameters—for the simplest class of models westudied these are the dimensionless dark energy densityΩ φ and the dark energy equation of state w .We obtained new, tighter constraints on the dimen-sionless coupling ζ of the scalar field to the electromag-netic sector. We note that these constraints are cur-rently dominated by the atomic clock tests, which areonly sensitive to the dark energy equation of state today.Thus a constant equation of state cosmological model isa reasonable assumption. Improvements in astrophysi-cal measurements will allow more generic models to beconstrained.We have also pointed out how different currently avail-able astrophysical measurements of α (specifically thearchival data of Webb et al. and the dedicated measure-ments of Table I) lead to somewhat different constraints).This highlights the importance of obtaining improved as-trophysical measurements of α (both in terms of statisti-cal uncertainty and in terms of control over possible sys-tematics), not only for their own sake but also becausethere can have a strong impact on dark energy studies.The next generation of high-resolution ultra-stable spec-trographs such as ESPRESSO and ELT-HIRES will beideal for this task. A roadmap for these studies is out-lined in [6], and more detailed forecasts of the futureimpact of these measurements may be found in [13]. ACKNOWLEDGMENTS
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