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Domain wall solution and variation of the fine structure constant in F ( R ) gravity Reference: K. Bamba, S. Nojiri and S. D. Odintsov, Phys. Rev. D 85, 044012 (2012) [arXiv:1107.2538 [hep-th]]. 2012 Asia Pacific School/Workshop on Cosmology and Gravitation 1st March, 2012 Yukawa Institute for Theoretical Physics, Kyoto University Presenter : Kazuharu Bamba ( KMI, Nagoya University ) Collaborators : Shin'ichi Nojiri ( KMI and Dep. of Phys., Nagoya University ) Sergei D. Odintsov ( ICREA and CSIC-IEEC )

I. Introduction No. 2 2011 Nobel Prize in Physics ・ Recent observations of Supernova (SN) Ia confirmed that the current expansion of the universe is accelerating. [Perlmutter et al . [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999)] [Riess et al . [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998)] [Astier et al . [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)] ・ There are two approaches to explain the current cosmic acceleration. [Copeland, Sami and Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)] [Tsujikawa, arXiv:1004.1493 [astro-ph.CO]] G ö÷ < Gravitational field equation > : Einstein tensor G ö÷ = ô 2 T ö÷ T ö÷ : Energy-momentum tensor Gravity Matter : Planck mass (1) General relativistic approach Dark Energy (2) Extension of gravitational theory

(1) General relativistic approach No. 3 ・ Cosmological constant ・ Scalar fields: X matter, Quintessence, Phantom, K-essence, Tachyon. R F ( R ) : Arbitrary function of the Ricci scalar ・ Fluid: Chaplygin gas [Capozziello, Cardone, Carloni and Troisi, Int. J. Mod. Phys. D (2) Extension of gravitational theory 12 , 1969 (2003)] ・ F ( R ) gravity [Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)] ・ Scalar-tensor theories [Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)] [Arkani-Hamed, Cheng, Luty and Mukohyama, ・ Ghost condensates JHEP 0405, 074 (2004)] G : Gauss-Bonnet term ・ Higher-order curvature term ・ f ( G ) T : torsion scalar gravity ・ DGP braneworld scenario [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] ・ Non-local gravity [Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)] ・ f ( T ) gravity [Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)] [Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (2010)]] ・ Galileon gravity [Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79, 064036 (2009)]

No. 4 ・ We construct a domain wall solution in F ( R ) gravity. ・ Static domain wall solution in a scalar field theory. ・ Explicit F ( R ) gravity model in which a static domain wall solution can be realized. ・ We show that there could exist an effective (gravitational) domain wall in F ( R ) gravity. ・ It is demonstrated that a logarithmic non-minimal gravitational coupling of the electromagnetic theory in F ( R ) gravity may produce time-variation of the fine structure constant which may increase with decrease of the curvature.

No. 5 II. Comparison of F ( R ) gravity with a scalar field theory having a runaway type potential g = det( g ö÷ ) < Action > : Metric tensor L M : Matter Lagrangian ・ We make a conformal transformation to the Einstein frame: A tilde represents quantities * in the Einstein frame. : Covariant d'Alembertian : Covariant derivative operator

・ No. 6 Action describing a runaway domain wall and a space-time varying fine structure constant : : Electromagnetic ø : Constant M : M ass scale field-strength tensor , : Constants : U(1) gauge field Discrete symmetry can be broken dynamically. : Bare fine structure constant A domain wall e : Charge of the electron can be formed. [Cho and Vilenkin, Phys. Rev. D 59, 021701 (1998)] [Olive, Peloso and Uzan, Phys. Rev. D 83, 043509 (2011)] [Chiba and Yamaguchi, JCAP 1103, 044 (2011)] < Flat Friedmann-Lema tre-Robertson-Walker (FLRW) space-time > a ( t ) : Scale factor

III. Reconstruction of a static domain wall solution No. 7 in a scalar field theory : Metric of the d - dimensional < Action > Einstein manifold: < dimensional warped metric > 1 /l 2 > 0 ・ de Sitter space: 1 /l 2 = 0 ・ Flat space: ϕ y We assume only depends on . 1 /l 2 < 0 ・ Anti-de Sitter space: < The Einstein equation > component: component: ・ The prime denotes the We may choose and take . * y derivative with respect to .

No. 8 ・ u 0 , y 0 : Constants Example: ú ( y ) y ø 0 is localized at and makes a domain wall.

IV. Reconstruction of an explicit F ( R ) gravity model No. 9 realizing a static domain wall solution < Gravitational field equation > : Energy-momentum tensor of matter ú M , P M : Energy density and pressure of matter : The Einstein tensor : Contribution to the energy-momentum tensor from the deviation of F ( R ) gravity from general relativity dimensional warped metric

Gravitational field equation No. 10 component: component: , ・ We derive an explicit form of F ( R ) realizing a domain wall solution. We consider the case in * which there is no matter.

No. 11 component: ・ We solve the equation of the scalar curvature R in terms of y . We define . For , we expand exponential terms and take the first leading terms in terms of Y .

: Arbitrary constant No. 12 F + , , A general solution Exponential model

V. Effective (gravitational) domain wall No. 13 P ( ψ ) , Q ( ψ ) Proper functions of the < Reconstruction method > : ψ auxiliary scalar field ψ The variation over By the variation of the metric, we find dimensional warped metric We have neglected the * contribution from the matter.

No. 14 component: ( i,j ) component: 1 /l 2 = 0 ψ = y By choosing , in case , we obtain

No. 15 U 0 , ψ 0 , ÿ : Constants By imposing the boundary condition that the universe becomes flat when , we find

No. 16 Since behaves non-trivially when , we may regard that there could be an effective (gravitational) domain wall y = 0 at . has a local maximum around . There could exist an effective (gravitational) y = 0 . domain wall at

< Reconstruction of an explicit form of F ( R ) > No. 17 ・ We derive an analytic relation . By substituting this relation into , we can obtain an explicit form of F ( R ). ・ We define . For , we expand each ö ö quantities in terms of and take leading terms in terms of . Y Y ・ For , we acquire an analytic solution: c 0 : Integration constant : Constant Power-law model

VI. Non-minimal Maxwell- F ( R ) gravity No. 18 < Action > R 0 : Current curvature : Bare fine structure constant ・ It has been found that such a logarithmic-type non-minimal gravitational coupling appears in the effective renormalization- group improved Lagrangian for an SU(2) gauge theory in matter sector for a de Sitter background. [Elizalde, Odintsov and Romeo, Phys. Rev. D 54, 4152 (1996)]

< Observations > No. 19 ・ Keck/HIRES (High Resolution Echelle Spectrometer) quasi-stellar object (QSO) absorption spectra over the redshift range : Time variation of ë EM ( significance level) [Murphy, Webb and Flambaum, Mon. Not. Roy. Astron. Soc. 345, 609 (2003)] ・ Combined dataset from the Keck telescope and the ESO Very Large Telescope (VLT) Spatial variation of ë EM c : Speed of light ( significance level) Θ : Angle on the sky : Look-back time at redshift z between sightline and best-fit dipole position [Webb, King, Murphy, Flambaum, Carswell and Bainbridge, Phys. Rev. Lett. 107, 191101 (2011)]

No. 20 < Theoretical estimation (in the Jordan frame) > Current value of H : [Freedman et al . [HST Collaboration], Astrophys. J. 553, 47 (2001)] ・ Naive model of a logarithmic non-minimal gravitational coupling of the electromagnetic field could not satisfy the constraints on the time variation of the fine structure constant from quasar absorption lines and therefore it would be ruled out.

No. 21 < Relation to a coupling between the electromagnetic field and a scalar field in the Einstein frame > þ ・ If the scalar curvature can be expressed by , J can be described þ J ( þ ) = B ( þ ) as a function of . ・ We can obtain the relation between a non-minimal gravitational coupling of the electromagnetic field in the Jordan frame and a coupling of the electromagnetic field to a scalar field in the Einstein frame.

No. 22 < Case for an exponential model >

No. 23 < Case for a power-law model >

VII. Summary No. 24 ・ We have studied a domain wall solution in F ( R ) gravity. ・ Static domain wall solution in a scalar field theory. ・ Explicit F ( R ) gravity model in which a static domain wall solution can be realized. ・ We have shown that there could exist an effective (gravitational) domain wall in the framework of F ( R ) gravity. ・ It has been demonstrated that a logarithmic non- minimal gravitational coupling of the electromagnetic theory in F ( R ) gravity may produce time-variation of the fine structure constant which may increase with decrease of the curvature.