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Particlelike solutions in modified gravity: the Higgs monopoles 5 1 0 2 Sandrine Schlögel∗ n UniversityofNamur,Namur,Belgium a UCLouvain,Louvain-la-Neuve,Belgium J E-mail: [email protected] 1 1 Theloreparadigmforsolvingso-calledhorizonandflatnessproblemsincosmologyistheprimor- ] c dialinflation.Plethoraofinflationarymodelshavebeenbuiltinlastdecadesandfirstexperimental q - probesseemtoappearinfavoroftheinflationaryparadigm. Wewillfocushereononeofthem, r g theHiggsinflation,andshowthecombinedconstraintrequiredforsuchamodelatcosmological [ aswellasgravitationalscales,i.e. forcompactobjects. WewillshowthatHiggsinflationmodel 1 givesrisetoparticlelikesolutionsaroundcompactobjects,dubbedHiggsmonopoles,character- v 5 izedbythenonminimalcouplingparameteraswellasthemassandthecompactnessoftheobject. 6 Forlargevaluesofthenonminimalcouplingconstantandspecificcompactness,theamplitudeof 4 2 theHiggsfieldinsidethematterdistributioncanbearbitrarilylarge. 0 . 1 0 5 1 : v i X r a FrontiersofFundamentalPhysics14 15-18July2014 AixMarseilleUniversity(AMU)Saint-CharlesCampus,Marseille,France ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Particlelikesolutionsinmodifiedgravity: theHiggsmonopoles SandrineSchlögel 1. Introduction If we look at the Universe history, we notice a series of puzzling mysteries like coincidence is- sues, darkmattereffectsandfine-tuningoftheinitialconditionsintheprimordialUniverse. Such features cannot be explained satisfactorily by general relativity (GR) and standard model of par- ticle physics (SM). On the other hand, GR and SM have never been faulted by observations and experimentsaswell. Inordertoencompasstheseissues, plethoraofmodifiedgravitymodelslike scalar-tensortheories, f(R),massivegravityandextradimensions,havebeenbuiltinlastdecades. Such GR modifications are a dangerous business since deviations from GR are rather well con- strained: in the solar-system, e.g. by Cassini probe, at astrophysical scales, e.g. through binary pulsars observations, and thanks to experimental tests like torsion balance. So, modified gravity models have to pass all these constraints. We will focus here on a scalar-tensor theory, the Higgs inflation [1], and we will show that Higgs distribution around compact objects predicted by this theoryisnontrivialandandthatitleadstoanamplificationmechanismofthescalarcharge. Higgs inflation is appealing since we know that the Higgs field is a fundamental scalar field ruling the inertial mass of elementary particles and as such it could be a partner of the metric in scalar-tensor theory of gravitation. This idea is not new: first inflationary models assumed that the Higgs field could be the inflaton. However, when introducing Higgs field in cosmology, some simplifications have been assumed so far. First, cosmologists always assume that Higgs field can bewrittenintheunitarygauge,i.e. (cid:32) (cid:33) 1 0 φ(x)= √ , 2 v+h(x) where v is the vacuum expectation value (vev) of the Higgs field φ and h is the excitation around the vev, while we cannot safely assume such a property in modified gravity models [2, 3]. Furthermore, up to now, no coupling have been considered between the Higgs and matter fields, like baryonic matter surrounding compact objects. Such a coupling, appearing in SM through the Yukawaterms,leadtoaweakequivalenceprinciplebreakinginHiggsinflation. In the following, we will focus on observational constraint of Higgs inflation in cosmology, thanks to CMB observations by Planck, and in astrophysics. We will show that Higgs inflation predicts particlelike solutions around compact objects, which we dubbed Higgs monopoles [4, 5] sinceitconsistsinisolatedcompactobjectschargedundertheHiggsfield. 2. CosmologicalconstraintsonHiggsinflation As mentioned previously, minimally coupled Higgs field to gravity was considered in very early inflationarymodel,theactionbeinggivenby (cid:90) √ (cid:20) R 1 (cid:21) S= d4x −g − (∂φ)2−V(φ) , (2.1) 2κ 2 where R is the Ricci scalar, φ is the Higgs field in unitary gauge, κ =8π/m2, m being the p p Planckmass,andV istheHiggspotentialcomingfromtheSM, 2 Particlelikesolutionsinmodifiedgravity: theHiggsmonopoles SandrineSchlögel V(φ)= λ (cid:0)φ2−v2(cid:1)2, (2.2) 4 where the vev v = 246 GeV and the self-coupling parameter λ ∼ 0.1 come from SM. The scalar field rolls slowly down its potential like for usual inflationary models. Unfortunately, this modelisnotviablesinceitdoesnotgiverisetoasufficiente-foldsnumberforsolvingthehorizon andflatnessproblems. Therefore,theideathatHiggsfieldcanbetheinflatonhasbeenconsidered intheframeworkofmodifiedgravity[1], (cid:34) (cid:32) (cid:33) (cid:35) (cid:90) √ R ξφ2 1 S= d4x −g 1+ − (∂φ)2−V(φ) , (2.3) 2κ m2 2 p ξ beingthenonminimalcouplingparameter. Thenonminimalcouplingleadstoaflatteningof theeffectivepotentialV definedby eff dV ξφ2R (cid:3)φ =− eff with V =−V+ , (2.4) eff dφ 16π so Higgs inflation is a viable model, provided that ξ > 104 for appropriate slow-rolling of the field during inflation. It appears to be favoured by Planck data [6] and is however ruled out by controversial BICEP2 experiment results since no tensor modes are generated [7]. In the next section,wewillstudythesamemodelatastrophysicalscalesinordertoconcludeifsuchadeviation fromGRpassesastrophysicaltestsofgravity. 3. Higgsmonopolesolutions InordertostudytheHiggsdistributionaroundcompactobjectsliketheSun,neutronstarsandblack holes,predictedbyHiggsinflation,westudythesamemodelinastaticandsphericallyspacetime, i.e. inSchwarzschildcoordinates. Intheminimallycoupledcase,theHiggsfieldissettledtoitsvev everywhere, a solution which is unrealistic since it would require strict homogeneity everywhere. Westartfromthesameaction(2.3) (cid:34) (cid:32) (cid:33) (cid:35) S=(cid:90) d4x√−g R 1+ξφ2 −1(∂φ)2−V(φ) +S (cid:2)ψ ,g (cid:3), (3.1) 2κ m2 2 mat m µν p whereweonlyaddedS whichdescribesbaryonicmatterfieldsψ whichconstitutethecom- mat m pactobject. Inthefollowing,wewillassumeatop-hatdensityprofileformatterfieldsforthesake of simplicity. We also introduce a convenient notation, φ(x)=m v˜h(x) with v˜the dimensionless p vevandh,thedeviationoftheHiggsfieldaroundthevev. Before deriving the numerical solution, let us take a look at the effective dynamics. During inflation, the Higgs field rolls slowly down its effective potential from high energy values and 3 Particlelikesolutionsinmodifiedgravity: theHiggsmonopoles SandrineSchlögel stabilizes at its vev during reheating. The scalar field dynamics is given by the Klein-Gordon equation(2.4)foraFriedmann-Lemaître-Robertson-Walkerspacetime,i.e. d2h 3dadh dV eff + = . (3.2) dt2 a dt dt dh with a(t), the scale factor. The role of the nonminimal coupling consists in flattening the ef- fectivepotential,sothee-foldsnumberisnowsufficienttosolvebothhorizonandflatnessproblem providedthatξ >104. Forastaticandsphericallysymmetricspacetime,theKlein-Gordonequationbecomes (cid:18) (cid:19) 2 dV h(cid:48)(cid:48)−h(cid:48) λ(cid:48)−ν(cid:48)− = − eff, (3.3) r dh where a prime denotes a derivative with respect to the radial coordinate, λ and ν being the metric fields. Because of the metric signature, the effective potential seems to be inverted (see the minus sign) with respect to the cosmological case. We plot the effective potential on Fig.1. We see that the effective potential differs in the interior and the exterior of the compact object becauseofthepresenceofmatter(cf. theRicciscalarRappearinginV ). TheHiggsfieldhasto eff stabilizeatitsvevatspatialinfinitysinceitisitsobservedvalueinvacuum, whichisamaximum oftheeffectivepotentialinthestaticcase. Actually, thevevistheonlyasymptoticpossiblevalue since it guarantees that the Higgs potential vanishes at spatial infinity and so, the spacetime is asymptoticallyflatandthesolutionoffiniteenergy. Wehavetofindasolutionwhichinterpolates between a value at the center of the compact object h and the vev at spatial infinity. If we now c look at the solution inside the compact object, we see that if h >hin, then the solution diverges c eq andthedistributioncannotinterpolatebetweenh andthevev. Ontheotherhand, ifh <hin, the c c eq Higgs field tends to oscillate around h=0, a solution which is not satisfactory since the potential does not vanish at spatial infinity. So, we have to find out a solution between both these regimes. Noticethatthepresenceofbaryonicmatterinsidethecompactobjectisessentialinthepresenceof thenonminimalcouplingsince,then,theeffectivepotentialdiffersinsideandoutsidethecompact objects. Intheabsenceofmatter,theonlysolutionistheGRone: h=1everywhere. On Fig.2, we plot the numerical Higgs profile around a compact object. This solution is the onlyonewhichisgloballyregular,offiniteenergyandasymptoticallyflatandwhichconvergesto thevevatspatialinfinity,i.e. aparticlelikesolution. ItismorerealisticthantheGRone,sinceitis nomorehomogeneous. Thefamilyofparticlelikesolutionislabelledbythenonminimalcoupling parameter, the compactness s, i.e. the ratio of the Schwarzschild radius and the body radius and the baryonic mass of the compact object m. We recover the dynamics predicted by the effective dynamics: if h(0)>h , the Higgs field diverges at spatial infinity while if h(0)<h , it oscillates c c aroundh=0. 4 Particlelikesolutionsinmodifiedgravity: theHiggsmonopoles SandrineSchlögel 1.5 1 v) e v d ( el 0.5 s fi g h=1 g Hi 0 −0.5 10−4 10−2 100 102 104 r/r S Figure 1: Effective dynamics associated to Figure 2: Higgs field distribution around a Higgs monopole: the effective potential dif- compactobjectofcompactnesss=0.75and fers inside (solid line) and outside (dashed massm=106kgforvariouscentralvaluesof line)thecompactobject. Equilibriumpoints theHiggsfield. Monopolesolutionisplotted arelabelledhienqandhoequtrespectively[5]. inblueandinterpolatesbetweenhcandv[5]. We notice that mass and compactness of monopole plotted on Fig.2 do not correspond to a physical object. Actually, we can show that it is not possible to find a combination of physical parameters which gives rise to a monopole solution [5] for SM potential parameters. Indeed, for astrophysical objects, h −→ 1, so we observe no deviation from GR even for large ξ. This is c due to a kind of "hierarchy" between the vev and the Planck scale. In conclusion, Higgs inflation predicts Higgs monopole solutions around compact object which are a unique solution depend- ing on parameters ξ, s and m, different than GR however compatible with current astrophysical observations. Monopole solutions exhibit also an amplification mechanism of h [5]. On Fig.3, we notice c that there exists a critical nonminimal coupling for which h becomes arbitrarily large for some c compactness - or equivalently for some body radii - the mass being fixed. This is due to the symmetryh =±1. Indeed,forthefirstbranchofthesolution,h >0inducesthath =+1while ∞ c ∞ forthesecondbranch,h <0⇒h =+1. Suchdivergencesinthesolutionleadtoaconstrainton c ∞ ξ: thereexistforbiddencompactnesses-orequivalentlybodyradii-wherenomonopolesolution exist,andso,nosolutionatall. Forlargerξ,moreandmoreforbiddencompactnessesappear. 4. Conclusion Insummary,wehighlightthatscalar-tensortheorieslikeHiggsinflationexhibitanewparticlelike solutionaroundcompactobjects. Themonopolesolutionrequiresthepresenceofapotentialforthe scalar field and the presence of baryonic matter. However, we observe negligible deviations from GR because of the difference between the Planck and the vev scale. Furthermore, we highlight a generalamplificationmechanismofthescalarfieldvalueatthecenterofthecompactobjectwhich leadstoforbiddenHiggsmonopolecompactnesses(orradii). 5 Particlelikesolutionsinmodifiedgravity: theHiggsmonopoles SandrineSchlögel Several open questions are still to be developed. Higgs field in scalar-tensor theories has been studied 20000 with some simplifications so far: we should consider an explicit coupling between the Higgs field and mat- 15000 terlikeYukawatermsofSMandgeneralizeourresults 10000 for other gauge than the unitary one. Another crucial hc question is the stability of the monopole. We know 5000 theonlypossiblesolutionsarethemonopoleonessince theyareoffiniteenergy,butwehavenoideaifthisso- 0 lutionisstablenoriftheycouldformduringagravita- tional collapse process. Eventually, we can generalize 0.44 0.46 0.48 0.5 0.52 s themonopoleresultstovariouspotentialsandnonmin- imalcouplingsappearingintheframeworkofmodified Figure 3: Higgs field values at the cen- gravity. ter of the compact object (m=103 kg) in function of the compactness. Divergences Acknowledgments appear upper a critical value of nonminimal coupling: ξ = 64.6 (solid line) and All computations were performed at the "Plate-forme ξ =64.7(dashedline)[5]. technologique en calcul intensif" of UNamur (Bel- gium)withthefinancialsupportoftheFRS-FNRSand S.S.isaFRIAResearchFellow. 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