Variations of the fine-structure constant α in exotic singularity models
Mariusz P. Dabrowski, Tomasz Denkiewicz, C. J. A. P. Martins, P.E. Vielzeuf
aa r X i v : . [ a s t r o - ph . C O ] J un Variations of the fine-structure constant α in exotic singularity models Mariusz P. D¸abrowski,
1, 2, ∗ Tomasz Denkiewicz,
1, 2, † C. J. A. P. Martins, ‡ and P. E. Vielzeuf
3, 4, § Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Copernicus Center for Interdisciplinary Studies, S lawkowska 17, 31-016 Krak´ow, Poland Centro de Astrofisica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Faculdade de Ciˆencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Various classes of exotic singularity models have been studied as possible mimic models for theobserved recent acceleration of the universe. Here we further study one of these classes and, un-der the assumption that they are phenomenological toy models for the behavior of an underlyingscalar field which also couples to the electromagnetic sector of the theory, obtain the correspondingbehavior of the fine-structure constant α for particular choices of model parameters that have beenpreviously shown to be in reasonable agreement with cosmological observations. We then comparethis predicted behavior with available measurements of α , thus constraining this putative couplingto electromagnetism. We find that values of the coupling which would provide a good fit to spec-troscopic measurements of α are in more than three-sigma tension with local atomic clock bounds.Future measurements by ESPRESSO and ELT-HIRES will provide a definitive test of these models. I. INTRODUCTION
The discovery of cosmic acceleration from supernovaobservations [1, 2], unveiled the presence of an unknownsource of energy which can be modeled in the easiestapproach by a cosmological constant Λ, resulting in thestandard ΛCDM model. Despite the fact that a rangeof observational tests appears to be in good agreementwith this model, the physical interpretation of Λ remainsambiguous. Thus a range of alternative scenarios grad-ually emerged, the most natural of which ascribes darkenergy to the presence of a dynamical scalar field. Thesealternatives have to be tested by the local and globalcosmological observations.One specific class of models aiming to mimic the ob-served dark energy behavior are the so-called exotic sin-gularity models [3, 4]. In fact, the emergence of exoticsingularities is related to some physical fields which phe-nomenologically are mimicked in the form of a specificparametrization of the evolution of the scale factor. Inother words, exotic singularity models may be seen as atoy-model parametrization of the evolution of a physicaldegree of freedom, such as a dynamical scalar field andits coupling to gravity and other fields.The issue of exotic singularities in cosmology was in-vestigated more intensively soon after the discovery ofcosmic acceleration and the first example of such a singu-larity was a big-rip due to the non-canonical scalar fieldknown as phantom [5]. Then, other options such as asudden future singularity (SFS) [6, 7], finite scale factorsingularity (FSFS) [3, 8], a big separation [3], and w-singularity [9] and many others have been proposed (fora recent review see [10]). These singularities are weak in ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] the sense that both particles and extended objects canpass through them [11, 12]. It also emerged that mod-els which contain these singularities can, with suitableparameter choices, fit current observations [13–19].Whenever dynamical scalar fields are present, one nat-urally expects them to couple to the rest of the model, un-less a yet-unknown symmetry suppresses these couplings.In particular, a coupling to the electromagnetic sectorwill lead to spacetime variations of the fine-structureconstant—see [20] for a recent review. In fact there issome recent evidence for such a variation [21], which adedicated VLT/UVES Large Program is aiming to test[22]. In any case, these spectroscopic measurements canbe used 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 [23, 24]. This is theapproach we will take here, though note that alternativesalso exist, as discussed in [25, 26].Thus if one envisages exotic singularity models astoy model parametrizations for an underlying dynami-cal scalar field, one may ask what variations of α willensue. As shown in [25, 27], with the above minimalassumption this question can be answered without ex-plicit knowledge of the field dynamics—the evolution ofthe dark energy equation of state and density are suffi-cient. (Additionally, there will be a parameter describingthe strength of the coupling to electromagnetism, that isthe evolution of the gauge kinetic function.) Thus wewill consider some representative exotic singularity mod-els which were shown (in our recent [28] and referencestherein) to be in reasonable agreement with current back-ground cosmological data, and study the behavior of α therein, under the assumptions stated above.The paper is organized as follows. In section II wepresent a brief review of the exotic singularity modelsuseful for our further study. In section III we discussthe physics behind the variation of the fundamental con-stants and our specific assumptions regarding this classof models. The result of applying these to our study-caseexotic singularity models will be exposed in section IV.Our conclusions are given in section V. II. EXOTIC SINGULARITY MODELPHENOMENOLOGY
In this section we will briefly review the phenomenol-ogy of some previously studied exotic singularity modelsthat are in reasonable agreement with cosmological ob-servations. While several classes of such singularities canbe studied, we will be focusing here on SFS models. Wewill also briefly contrast these with a related alternative(FSFS models) which turn out not to provide observa-tionally viable α models.In these models one assumes the standard Einstein-Friedmann standard field equations for the energy den-sity and pressure: ρ ( t ) = 38 πG (cid:18) ˙ a a + kc a (cid:19) (II.1) p ( t ) = − c πG (cid:18) aa + ˙ a a + kc a (cid:19) (II.2)appended by the continuity equation:˙ ρ ( t ) = − aa (cid:20) ρ ( t ) + p ( t ) c (cid:21) , (II.3)where a ≡ a ( t ) is the scale factor, the dot means thederivative with respect to physical time t , G is the grav-itational constant, c is the speed of light, and the cur-vature index k = 0 , ±
1. For further analysis we will set k = 0, in agreement with observational results. The mainassumption of these models resides in the scale factorwhich is parametrized differently than for the standardmodel and can be expressed as a function of the fourparameters: δ , m , n , t s , namely a ( t ) = a s (cid:20) δ + (1 − δ ) (cid:18) tt s (cid:19) m − δ (cid:18) − tt s (cid:19) n (cid:21) . (II.4)The parameter m characterizes the evolution of the uni-verse near the initial big-bang singularity at t = 0, theparameter δ gives the standard Friedmann limit δ → n characterizes an exotic singularity (anSFS singularity appears for 1 < n < < n < t s tellsus the moment of an exotic singularity to appear duringthe evolution, and a s ≡ a ( t s ). The ansatz (II.4) is fullyequivalent to the one applied in Ref. [7] but differs fromthat one proposed in Ref. [29] which uses an exponen-tial function of time. From the relation (II.4) one definesthe redshift of an object being at radial distance r atthe moment t with respect to an observer receiving thesignal at t :1 + z = a ( t ) a ( t ) = δ + (1 − δ ) (cid:16) t t s (cid:17) m − δ (cid:16) − t t s (cid:17) n δ + (1 − δ ) (cid:16) t t s (cid:17) m − δ (cid:16) − t t s (cid:17) n , (II.5) as well as the Hubble function H ( t ( z )) = 1 t s m (1 − δ ) (cid:16) tt s (cid:17) m − + δn (cid:16) − tt s (cid:17) n − δ + (1 − δ ) (cid:16) tt s (cid:17) m − δ (cid:16) − tt s (cid:17) n , (II.6)for which eq. (II.5) has to be applied.We consider the scenario in which the universe containstwo fluid components, namely non-relativistic matter andthe scalar field which drives an exotic singularity. Thesefluids obey independently their conservation laws. Weassume the standard behaviour for the non-relativistic(dust) matter component ρ m = Ω m ρ (cid:16) a a (cid:17) (II.7)and the evolution of the other fluid, which we name here ρ Φ , can be determined by taking the difference betweenwhole energy density, ρ as given in Friedmann eq. (II.1)and ρ m , i.e. ρ Φ = ρ − ρ m . (II.8)In fact, it is just the ρ Φ component of the Universe whichis responsible for the appearance of an exotic singularityat t → t s . Using this we can rewrite the Friedmann eq.(II.1) as ρ = 3 H πG (cid:20) Ω m (cid:16) a a (cid:17) + Ω Φ (cid:21) (II.9)so that the dark energy density is given byΩ Φ = 1 − Ω m H H (cid:16) a a (cid:17) = 1 − Ω m . (II.10)The barotropic index of the equation of state for the darkenergy given by the canonical scalar field φ is defined as w Φ = p Φ /ρ Φ , where p Φ = (1 /
2) ˙Φ − V (Φ) and ρ Φ =(1 /
2) ˙Φ + V (Φ) ( V (Φ) is the potential). In the phantomregime which has negative kinetic energy [5] one has p Φ = − (1 /
2) ˙Φ − V (Φ) and ρ Φ = − (1 /
2) ˙Φ + V (Φ). On theother hand, the effective barotropic index of the equationof state is w eff = p/ρ . In the case in which we considerthe times when the radiation can be neglected p = p Φ .The model parameters used here will be the same asthe ones taken in our previous paper [28]; they are listedin Table I.Note that SFS2 and FSFS2 correspond to the dustlimit of these models. Clearly they are amply ruled out,but they provide pedagogically useful fiducial compar-isons for some of the discussion that follows.Using these parameters for the redshift function (II.5)one can check whether our models are consistent withcurrent observations of the Hubble parameter as a func-tion of redshift (II.6) and the plots for our choices of SFSand FSFS parameters given in the Table I are shown inFig. (1), with the observational data taken from the re-cent compilation [30]. These illustrate the point that thebackground evolution of the dust models is highly dis-crepant. Model m n δ y SFS1 2/3 1.9999 -0.43 0.99SFS2 2/3 1.9999 0 0.99SFS3 0.749 1.99 -0.45 0.77FSFS1 0.56 0.8 0.42 0.96FSFS2 2/3 0.7 0.0 0.79FSFS3 2/3 0.7 0.24 0.96
TABLE I: The sets of parameters for the scale factor(II.4) which are used for SFS and FSFS models. See[28] and references therein for further discussion onthese choices.
Object z ∆ α/α
Spectrograph Ref.HE0515 − − . ± . − . ± . − − . ± . − . ± . −
264 1.84 5 . ± . TABLE II: Currently specific measurements of α . Thecolumns respectively contain the object along each lineof sight, the redshift of the absorber, the measuredvariation of fine structure constant α (in parts permillion), the name of the spectrograph, and thereference reporting the measurement. The fourth entrycorresponds to the recent Large Program measurement. III. VARYING FINE-STRUCTURE CONSTANT
High-resolution spectroscopic observations of absorp-tion clouds along the line of sight of quasars have pro-vided indications of spacetime variations of the fine-structure constant α at the level of a few parts per mil-lion, in the approximate redshift range 1 < z <
4, themost recent one being that of Ref. [21]. A possiblecause for concern is that these measurements come fromarchival data, and thus several efforts have been made toindependently confirm this result through dedicated mea-surements. A summary list of some of these new measure-ments is provided in Table II; the latest of these effortsis the ongoing Large Program at the VLT UVES (VeryLarge Telescope Ultraviolet and Visual Echelle Spectro-graph) [31]. We will use both the data in the Table II andthat of Ref. [21] as sets of data to constrain our mod-els. Note that the former has fewer data points (and asmaller redshift sampling) but smaller uncertainties, thereverse being true for the latter.Any dynamical scalar field providing the dark energy isnaturally expected to couple to the rest of the model andin particular to lead to spacetime variations of fundamen-tal couplings [35]. The coupling between the scalar-fieldand the electromagnetic field can be described by L Φ ,F = − B F (Φ) F µν F µν , (III.11)where as usual the gauge kinetic function is such that FIG. 1: The redshift evolution of the Hubble function(II.6) for SFS (top) and FSFS (bottom) type ofsingularities with the set of parameters shown in TableI, plotted against the observational data of Ref. [30]. B F (Φ) = α /α (Φ). To a good approximation (at leastfor the low redshifts of interest in the present work) wemay assume a linearized gauge kinetic function B F (Φ) = 1 − ξκ (Φ − Φ ) , (III.12)where κ = 8 πG/c and ξ parametrizes the coupling be-tween the scalar field and the electromagnetic sector. Itthen follows that the evolution of α can be written as∆ αα ≡ α − α α = B − F (Φ) − ξκ (Φ − Φ ) . (III.13)If one assumes that the same degree of freedom pro-vides all of the dark energy and the variation of α , thenthe dark energy equation of state can be inferred fromthe dynamics of the field, as first discussed in [24]. Usingthe fact that for a canonical scalar field ˙Φ = p Φ + ρ Φ and changing the derivative with respect to time into thederivative with respect to logarithm of the scale factori.e. that ( ... ) ′ ≡ d/d ln a = H − d/dt we have for thedynamics of the scalar field w Φ + 1 = ˙Φ ρ Φ = ( κ Φ ′ ) Φ , (III.14)where Ω Φ if the fraction of the universe’s energy in thescalar field componentΩ Φ = ρ Φ ρ Φ + ρ m = ρ Φ a ρ Ω m + ρ Φ a . (III.15)The equation for the field can easily be integrated withrespect to the scale factor [25, 27], and changing variablesusing dz/ (1 + z ) = da/a we finally find, in terms of theredshfit∆ αα ( z ) = ξ Z z p Φ (ˆ z ) | (1 + w (ˆ z )) | d ˆ z (1 + ˆ z ) . (III.16)Notice that the above expression is only valid forcanonical (quintessence-type) scalar fields. On the otherhand, in the phantom regime w < − w + 1 = − ( κ Φ ′ ) Φ (III.17)and this now leads to∆ αα ( z ) = − ξ Z z p Φ (ˆ z ) | w (ˆ z ) | d ˆ z (1 + ˆ z ) ; (III.18)the extra minus sign comes from the fact that in thecanonical case one physically expects the field to berolling down the potential, while in the phantom caseit should be nominally climbing up.In the above formulas Ω Φ ( z ) and w ( z ) are the frac-tion of the universe’s energy in the form of dark energyand its equation of state respectively. We thus see thatknowledge of these parameters is sufficient (up to a nor-malization provided by the coupling ξ ) to determine theevolution of α . Thus with the above assumptions we caneasily determine this evolution in the exotic singularitymodels under consideration.Note that in some of these models w ( z ) can cross the w = − α need not be monotonic, but may have inflectionpoints and change sign. On the other hand, this cannothappen in the dust case. This is one reason for keepingthis model in the analysis, as a simple comparison point.In particular the above equations apply at redshift z =0, for which atomic clock measurements provide a verytight limit [37] on the current drift rate of α , namely (cid:18) ˙ αα (cid:19) = ( − . ± . × − yr − . (III.19)This bound was later refined (under plausible theoreticalassumptions) in [38], but we use the direct (and moreconservative) bound in our analysis. Similarly we do not FIG. 2: The present-day drift rate of the fine-structureconstant α as a function of the coupling ξ , for the threeSFS models under consideration, compared to theone-sigma experimental bound of [37].use Oklo bound [39] at z = 0 .
14: although nominallyquite strong, it is subject to much larger theoretical andsystematic uncertainties than the spectroscopic measure-ments we are considering. With the assumptions we aremaking for this class of models we therefore have from(III.16) that [25] (cid:12)(cid:12)(cid:12)(cid:12) ˙ αα (cid:12)(cid:12)(cid:12)(cid:12) = | ξ | H p Φ0 | w Φ0 | , (III.20)where the modulus signs allow for the fact that the mod-els can be at either side of the phantom divide and thesign of the coupling in the gauge kinetic function is nota priori defined. Using the current value of the Hubbleconstant (say the H = (67 . ± .
4) km . s − Mpc − Planckvalue) one gets to the following conservative (3 σ ) bound | ξ | p Φ0 | w Φ0 | < − . (III.21)(Obviously the choice of a different value of H —say fromlocal measurements—has a negligible effect on the abovebound.) Therefore the different models being consideredwill be subject to different bounds on ξ , since they willhave different values of Ω Φ0 and w Φ0 . These bounds aresummarized in Table III, together with the maximumvariation of α allowed in each model, up to a redshift z = 5, when the ξ bound is saturated. The choice ofa maximum redshift of z = 5 is meant to represent therange over which future measurements may be expected,in particular from the European Extremely Large tele-scope (E-ELT) [40].Note that in the case of the FSFS models the ex-tremely negative present-day equation of state leads to avery tight bound on the coupling (coming from atomicclock measurements). The result of this is that the al-lowed variations of α in these models are extremely small:about two orders of magnitude smaller than would beneeded to explain the results of [21], and even difficultto detect at all with the next generation of observationalFIG. 3: The top panel shows the redshift dependence of α for the values of the coupling that saturate the redshift z = 0 constraints; the bottom panels illustrate the range of allowed variations for each of the models, compared tothe dedicated measurements of Table II and the data of [21] respectively. The thin black rectangle in the redshiftrange 2 < z < α is monotonic (and therefore themaximum variation occurs for the highest redshift con-sidered), this is not the case for SFS1 and SFS3. Thereason for this is the previously mentioned fact that thedark energy equation of state of these models crosses thephantom divide at some points, the precise redshift ofwhich depends on the choice of model parameters. Thiscan be seen in Fig. 3, which shows the range of allowedvariations of α in these models. The bottom two panelsof this figure also provide (through the thin black rectan-gle) a simple visual illustration of the expected sensitivityand redshift span of E-ELT measurements (through the Model Ω Φ0 w Φ0 | ξ | max × z | α max | ∆ α/α | max × SFS1 0 . − .
06 2 .
76 1 . . .
685 0 . .
70 5 . . . − .
92 2 .
42 2 . . . − .
49 0 .
44 0 . . .
685 0 . .
70 5 . . . − .
68 0 .
43 0 . . TABLE III: Bounds on the coupling ξ , coming from theatomic clock measurements of [37], for the differentmodels under consideration. Also listed is the maximumallowed variation of α , in the redshift range 0 < z ≤ z = 5 since the evolution of α is monotonic.ELT-HIRES instrument [41]) as compared to currentlyavailable measurements. IV. RESULTS
We can now compare the SFS models with the spec-troscopic measurements of the fine-structure constant α discussed in the previous section, using the standard chi-square statistic. Figure 4 summarizes the results of thiscomparison, for various choices of dataset: we consideredboth the dedicated measurements listed in Table II andthe larger archival dataset of Webb et al. [21], separatelyincluding the two subsets of the latter (correspondingto measurements with the Keck and VLT telescopes).For each model we only explore tha range of couplingsallowed by the (conservative) bound coming from localatomic clock measurements.For the dust (SFS2) model, where the α evolution ismonotonic, one recovers as expected that the Keck data(which contains predominantly negative measurements)prefers a negative coupling ξ , while the VLT data (andalso that of Table II) prefers a positive coupling. Thetrend is opposite for the SFS1 and SFS3 models, since(as in clear from Fig. 3) the putative underlying scalarfield is in the phantom regime for at least part of theredshift range under consideration.More importantly, one also notices that there is nominimum of the reduced chi-square for this range of cou-plings. In other words, tha value of coupling that wouldprovide the best fit to any of these spectroscopic datasetswould be incompatible with the local atomic clock boundat least at the three sigma level.A similar analysis can be done considering all the avail-able fine-structure constant measurements as well as theHubble parameter measurements in [30]; these results areshown in Fig. 5. Naturally the dust (SFS2) model pro-vides an extremely poor fit, while the status of the SFS1and SFS3 models remains as before (as these are in rea-sonable agreement with the H ( z ) data). Thus with thechosen values of the cosmological parameters (which wehaven’t allowed to vary, as they had been found in previ-ous works to provide the best fits to this class of models),and under the previously discussed assumptions, we findthat these models do not provide good fits to availablespectroscopic measurements of the fine-structure con-stant. V. CONCLUSIONS
The so-called exotic singularity models have been re-cently suggested as possible mimic models for the ob-served recent acceleration of the universe. Here we havetreated them as toy models for the behavior of an under- lying scalar field and, assuming that this also couples tothe electromagnetic sector of the theory (which a scalarfield would naturally do, unless a new symmetry is pos-tulated to suppress the coupling), calculated the ensuingbehavior of the fine-structure constant α .We have shown that with the above assumptions thisquestion can be answered without explicit knowledge ofthe dynamics of the putative scalar field: the evolutionof the dark energy equation of state and density are suf-ficient, since exotic singularity models assume that thedynamical effects of the field are phenomenologically en-coded in the behavior of the scale factor a ( t ), given by(II.4). We focused on specific choices of SFS and FSFSmodel parameters, previously shown to be in reason-able agreement with cosmological observations, and usedavailable laboratory and astrophysical tests of the stabil-ity of α to further constrain these models.Our results highlight the importance of local atomicclock measurements such as those of [37], in constrain-ing these cosmological models. Specifically, for the FSFSmodels we considered, the local constraints on the cou-pling of the putative scalar field to the electromagneticsector of the theory are so tight that the allowed varia-tions of α at the redshifts probe by optical/UV measure-ments would be too low to be detected, not only withcurrent spectroscopic facilities but possibly even with fu-ture ones. For the SFS class the allowed variations arelarger, but nevertheless the values of the coupling ξ thatwould provide the best fit to currently available spectro-scopic measurements of α are in more than three-sigmatensions with the local atomic clock bound.Nevertheless, at the phenomenological level the SFSmodels do have one interesting feature: since they cancross the phantom divide (and often do so more thanonce, at redshifts determined by the model parametersthemselves), they will usually lead to a non-monotonicredshift dependence of α . This is in contrast with mostother single-field, dilaton-type models where its evolu-tion tends to be monotonic—again the dust model we in-cluded in the analysis is a simple example of this. Forth-coming more precise measurements with high-resolutionultra-stable spectrographs such as ESPRESSO and ELT-HIRES will allow a detailed mapping of the allowed red-shift dependence of α and provide a definitive test ofthese models. VI. ACKNOWLEDGEMENTS
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