Abelian Symmetry Breaking and Modified Gravity

Transcription

1 Abelian Symmety Beaking and Modified Gavity Javie Chagoya Collaboations with Gustavo Niz and Gianmassimo Tasinato 1

2 Outline Context Abelian Galileon-Higgs votices Black holes in genealised Poca

3 GR & CDM Cuvatue, ξ (cm - ) Bake, Psaltis & Skodis, 014. BBN M Lambda Satellite SS Last scatteing Galaxies PSRs MS WD S stas CMB peaks MW M87 Clustes P(k) z=0 NS Potential, ε R SMBH BH Accn. scale

4 Lovelock s theoem The only second-ode, local gavitational field equations deivable fom an action containing solely the 4D metic tenso (plus elated tensos) ae the Einstein field equations with a cosmological constant. In ode to modify GR we must conside at least one of the following: New degees of feedom, Highe ode deivatives, Exta dimensions, Non-locality. 4

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6 Sceening Mechanisms Additional degees of feedom must be sceened: Chameleon The mass of the scala mode becomes lage in dense egions. Khouy & Weltman, Phys. Rev. Lett. 93 (004) M ( ) = g M pl Symmeton The mass of the scala mode becomes lage in dense egions. Hintebichle & Khouy, Phys. Rev. Lett. 104, m = g( ) M pl Vainshtein NL deivative self-inteactions become lage in dense egions. K( ) + m = g M pl Massive Gavity, DGP, Galileons... 6

7 Theoetical Consistency Highe ode opeatos ae negligible if E/ 1. Backgound-dependent functions in font of kinetic tems, couplings and potentials may lead to a stong coupling poblem. Ty to avoid fine tunings technical natualness. Take cae of the sign of the kinetic tem ghost fee. 7

8 Reviews of the popeties of MG models: S. Tsujikawa, Lect.Notes Phys. 800:99-145, 010 Clifton et al., Physics Repots 513, 1 (01) A. Baeia et al., axiv:

9 Some aspects of Galileons Scala Galileons in stong gavity. JC, K. Koyama, G. Niz, G. Tasinato, JCAP10(014)055 Votices in Higgs vecto Galileons. Black holes in genealised Poca action. JC, G. Tasinato, JHEP0(016)063 JC, G. Niz, G. Tasinato, axiv:

10 (No)Scalaisation Toy model Neglecting cuvatue and imposing int < V > V T = M 4 3 R3 1 V + 3 [...] = 4 M p T c Match F and F at R and v: "! 0 = 3 3 R 1/ 16 V 1 1/ V + 3 R s 30, # The diffeence between the cental and asymptotic values of the scala field is always suppessed. 10

11 (No)Scalaisation Binding enegy of a polytopic compact sta 0.14 e b M b êm ü 7 8 é é 0 Fixing an asymptotic value of the scala field, deviations fom GR inside a compact sta get smalle as the density inceases. No hai? Scala field solution beaks down nea the maximum obseved density of neuton stas. 11

12 Vecto Galileons Use new fields to dive acceleation. Vectos have been less exploed than scalas in DE models with deivative self-inteactions. Can be elevant in cosmology. Tasinato, CQG31(014)5004 Can be obtained fom spontaneous symmety beaking Hull, Koyama, Tasinato, JHEP03(015)154 1

13 Higgs Mechanism fo Vecto Galileons Hull, Koyama, Tasinato, JHEP03(015)154 Gauge invaiant action fo a complex scala field with highe ode deivative couplings: L = (D µ )(D µ ) 1 D µ µ iga µ µ Fµ V ( )+L(8) + L(1) + {4F L(16) Gauge invaiant opeatos of dimension 8, 1 and 16 that descibe deivative selfinteactions of the Higgs field. V = µ + ( ) v = µ 13

14 Higgs Mechanism fo Vecto Galileons Hull, Koyama, Tasinato, JHEP03(015)154 Demand that the Lagangian is invaiant unde a U(1) gauge symmety:! e i A µ! A µ + 1 µ 1 [(D µ ) (D )+(D ) (D µ )] L P µ 1 8 = 1! 4 " µ 1µ " D µ! e i [ D D + (D µ D ) 1 (8) L 1 µ 1 P µ + (8) L 1 µ 1 Q µ ] D L D µ D! e i Q µ i 1 = 1 8 " µ 1µ µ 3 " D µ D [ (D µd ) 1 3 (1) L 1 µ 1 P µ P 3 µ 3 + (1) L 1 µ 1 Q µ Q(D 3 µ 3µ D )] L 16 = 1 1 "µ 1µ µ 3 µ 4 " (16) L 1 µ 1 P µ P 3 µ 3 P 4 µ 4 + (16) L 1 µ 1 Q µ Q 3 µ 3 Q 4 µ 4 =) } Lµ Gauge invaiant combinations: 14

15 Higgs Mechanism fo Vecto Galileons Hull, Koyama, Tasinato, JHEP03(015)154 Extact the physical content: = 'e ig, Â µ A µ Gauge invaiants Covaiant deivatives can be expessed in tems of these fields, e.g. L, P and Q ae symmetic: D µ =(@ µ ' ig'âµ)e ig =) L µ µ '@ ' + g ' Â µ Â P µ = '@ ' g ' Â µ Â Q µ = g [@ µ(' Â )+@ (' Â µ )] L 0 s manifestly gauge invaiant: =) L 8 = g (8) 4 (@ µ'@ ' + g ' Â µ Â )@ (' Â )( µ µ ) 15

16 Higgs Mechanism fo Vecto Galileons Hull, Koyama, Tasinato, JHEP03(015)154 Aound the minimum of the potential: ' = v + h p L tot = 1 4 F µ F µ (gv)  3g 3 (8)v 4 4  µ   + gauge field inteactions + new inteactions between µ and h +... Same as the fist two Lagangians in genealised Poca action L. Heisenbeg, JCAP 1405 (014) 015 The emaining two Lagangians come fom L (1) and L (16) Effects yet to be analysed. Howeve they e suppessed by appopiate powes of m A = gv 16

17 Votices A simple way to study effects induced by vecto Galileon inteactions. Votices in the Abelian Higgs model L AH [,A a ]=(D a ) D a 1 4 F abf ab 4 (x )= X(x )e i (x ), A a (x )= 1 e Intoduce gauge inv. quantities, X and  hâa (x a (x )i c dops out fom the Lagangian L AH [X, Âa] a X@ a X + X  a  a 1 4e F abf ab 4 4 X 1 massive 17

18 Votices Votices in the Abelian Higgs model The physical degees of feedom obey X XÂaÂa + X 1 X a F ab +e X Â b =0 Static solution: Nielsen-Olesen votex, chaacteised by a winding numbe. At infinity: } X! 1 At =0: D a! 0 ) A a! N A a a (! 0) = 0, X(! 0) = 0 Flux = N It tuns out to be consistent Futhemoe: to set A = 0 eveywhee. = N 18

19 Votices Votices in the Abelian Higgs model The solution is uniquely detemined by: Rotational symmety  i dx i = Invaiance unde eflection accompanied by complex conjugation of F 1.0 h ij x j  ()+ x i A 1 (, ) = A 1 (, ) A (, ) =A (, ) i Â() dx i 0.8 X X 1 X  X + (X 0 + X 00 )=0 e  X  0 + Â00 =  N=1, l=e -- l=h=1.0 l=h=

20 Votices Now with Galileons Same gauge invaiant fields and asymptotic as befoe, (x )= X(x )e i (x ), A a (x )= 1 hâa (x a (x )i e Conside only new contibutions fom vecto field deivatives: L AHG a X@ a X + X  a  a 1 4e F abf ab 4 X 1 4 i + a X@ b X + X  a  b h c (X  c a (X  b ) X 3  " +  e  X 4 X 1 X  X +  Â0    0  0 +  e 4 X 4   X +6 4 X 4 " +  + X 0 + X 00  0 X 0 X 0 X 00 X X X " + Â0 + 4ÂX0 3X   0 4 X0 3X + X0 3X # # # =0, =0, =0. Setting  = 0 is inconsistent. L X   X... 0

21 Votices Now with Galileons  = 1 X 41 v u t 1 X 1 " ( Â0 X0 )+ 3X X 0 X 4Â!# 3 5 Only  q and X ae dynamical  fo a N-O and a Galileon votex y 0 y x x 1

22 Votices Now with Galileons X,  q and  b = b = 0.0 b = 0.50 Given b,  constaints the allowed values fo the voticity 1.4 b=0.10 b=0.0 Non-linea contibutions 1.0 b=0.40 b=0.50 Total enegy E 1 b= b=0.60 b=0.68 Enegy fom L

23 Votices Visual tick  = ea The non-tivial pofile of  can be attibuted eithe to A o to the phase Gauge choice: A =0, = N + () Lines of constant phase fo N-O, b=0.4 and b=0.5 3

24 Votices Minimal (and weak) coupling to gavity Is the geomety of the votex eflected on the spacetime? G ab =8 G(T (AH) ab + T (6) ab ) Contains -q component ds = e ( () ()) (dt d ) e dz () e d!()dd Futhe assumption: small b limit )Â O( ) Only T (AH) contibutes at the lowest ode, howeve it does contain Galileon effects. e p p p Asymptotically: X 1 x 0 p, Â a 0 e e N-O ) Â b 0 e p e,! 1 e p e 3/ New fields 4

25 Votices Minimal (and weak) coupling to gavity Diff. invaiance d! d! e d This edefinition effectively eats up the Galileon contibution that would souce an off-diagonal component of the EMT. The coodinate system adapts to the votex, and the deivative inteactions modulate the adial dependence of A q and X. The Galileon effects can be seen as a futhe contibution to the angula deficit of the cosmic sting. Howeve, in the end it is only seen in the cuvatue invaiants. 5

26 Some aspects of Galileons Scala Galileons in stong gavity. JC, K. Koyama, G. Niz, G. Tasinato, JCAP10(014)055 Votices in Higgs vecto Galileons. JC, G. Tasinato, JHEP0(016)063 Black holes in genealised Poca action. JC, G. Niz, G. Tasinato, axiv:

27 Genealised Poca L. Heisenbeg, JCAP 1405 (014) 015 L = 1 4 F µ + 5X n= nl n L =G (X) L 3 =G 3 (X)(D µ A µ ) X = 1 A µa µ L 4 =G 4 (X)R + G 4,X (Dµ A µ ) + c D A D A L 5 =... (1 + c )D A D A G s ae abitay functions, The non-minimal couplings to gavity keep the eqs. of motion nd ode, 3 degees of feedom (+ gavity). 7

28 Black holes in gen. Poca S = Z d 4 x p g ( M pl R 1 4 F + apple (D µ A µ ) D µ A D A µ 1 A R ) GR + Maxwell + c.c. + paticula choice G4 = X and c = 0 c would lead to a edefinition of the coupling to F. Total deivative in flat space S = Z d 4 x p g " M pl R 1 4 F + G µ A µ A # New contibutions excited only by the coupling to gavity 8

29 Black holes in gen. Poca M pl G µ = 1 apple F µ F 1 4 g µ F apple 1 g µ (D A ) A (µ D ) D A + g µ A D D A + 1 g µ D A D A D A (µ D ) A + D A ( D µ) A + A (µ D A ) A D (µ A ) 1 A G µ + A µ A R D µ D A + g µ A D µ F µ = G µ A µ Key to avoid Bekenstein s no Poca-hai theoem D µ F µ = m A At and Ai cannot be tuned on simultaneously without violating timeevesal symmety. Thee ae only puely electical and puely magnetic cases, fo which one can show that cetain integal identity is violate, concluding that the vecto field pofiles ae not suppoted. 9

30 Black holes in gen. Poca Asymptotically flat configuations without c.c. The coupling to G µn can switch off the tems that would violate time-evesal symmety. ds = f()dt + h() 1 d + d + sin d', A µ =(A 0 (), (), 0, 0). R = (f f0 )(f 0 + f 00 ) (f + f 0 ) Plug in the field equations Asymptotically, if f~ n, R~ - 30

31 Black holes in gen. Poca Asymptotically flat configuations without c.c. Asymptotic flatness can be ecoveed fo non-tivial pofiles only if = 1 4 Plug back to the exact field equations f =h =1 M A 0 = Q + P p Q +PQ+MP = M Unique solution Asymptotic flatness not imposed The new integation constant, P, contols the asymptotic pofile of p: 1 if P =0 1 p othewise 31

32 Black holes in gen. Poca Asymptotically flat configuations without c.c. Abelian symmety beaking tems completely sceen the geomety fom the vecto fields. (Simila to stealthy Schwazschild configuations in scala-tenso gavity, Babichev, Chamousis JHEP1408(014)) G µ =0 avoids Bekenstein s theoem. All cuvatue invaiants and the EMT ae well behaved fo > 0. does not contibute to F µ. No violation of no-hai conjectue. Howeve, an object coupled to A µ can pobe the longitudinal pofile. Stable unde spheical symmetic, but time dependent, petubations povided that the chages ae small. Moe geneal petubations not tested yet. 3

33 Black holes in gen. Poca Inclusion of c.c. Inteestingly, we can impose asymptotic flatness again with = 1 4 f =1 M + 4 P 3 h = f 1+ P, A 0 = Q + P 1+ 3 P, = s A 0 fh P (1 + P ) P, 8M pl P = 1+ P p f " 3Q + P 3+ P 9 f h P P 4Mpl P 6=4Mpl # 1/ P (1 + P )+8Mpl P 33

34 Black holes in gen. Poca Inclusion of c.c. The electic chage is still sceened fom the geomety. A0 and p do not vanish at infinity. If Q = P = 0 the electic field is tuned off, but the longitudinal pofile is still souced by the cosmological constant. Additional essential singulaity if LP < 0 R = 8 P (15M 5 +4 P 5 ) 5(1 + P ) 3 (Vanishes at infinity) G=0, this avoids Bekenstein s no-go aguments. f =1 M + 4 P 3 The hoizon coves the sing. if P ) 1 eal zeo ) Only 1 hoizon p P M>4/15 34

35 Black holes in gen. Poca Genealisations, conclusions, Without cosmological constant it is easy to include slow otation. The metic is the slow otation limit of Ke (not Ke-Newman). The time and angula components of the vecto field ae the slow otation limit of Ke-Newman. Also without L, genealisation to abitay dimensions is staightfowad ds = f() dt + h() 1 d + d (d ), A µ =(A 0 (), (), 0,...,0) f = h =1 A 0 = = Q d d M d d P d p Q d +M d Pd d 3 +P d Q d d 3 d 3 M d 35 = d 3 d 4 Diffeent asymptotic? (fix diffeent b) Moe geneal Lagangians Complete study of stability Astophysical consequences