Inflationary Paradigm in Modified Gravity

Transcription

1 Inflationary Paradigm in Modified Gravity Lavinia Heisenberg (ETH-ITS) Institute for Theoretical Studies, Zürich Jul. 21th, Nordita, Stockholm/Sweden

2 Theoretical modelling of the Universe General Relativity S = Z M 2 Pl 2 p gr + Lmatter varying the action geometry = matter Einstein tensor G µ = R µ 1 2 R = T 2 g (,p) Homogeneity & Isotropy Friedmann Lemaitre Robertson Walker (FLRW) solutions ds 2 = dt 2 + a(t) 2 ijdx i dx j

3 Challenges Cosmological and black hole singularities failure of GR for gravitational phenomena at UV Quantum theory of gravity UV completion Cosmological Constant Problem The necessity of Inflation

4 Modified Gravity GRAVITON } µ } f A µ

5 Modified Gravity A µ f µ MG A µ add new additional fields to the gravity sector scalar-tensor theories (Horndeski interactions) vector-tensor theories (generalized Proca interactions) tensor-tensor theories (massive gravity, bigravity) f µ A µ f µ A µ A µ

6 Modified Gravity MG Maybe not modifying that much!

7 Horndeski theory (scalar-tensor theory) } how do they couple

8 Horndeski theory (scalar-tensor theory) L 2 = K(,X) L 3 = G 3 (,X)[ ] µ = r X = 1 2 (@ )2 L 4 = G 4 (,X)R + G 4,X ([ ] 2 [ 2 ]) L 5 = G 5 (,X)G µ µ G 5,X 6 ([ ]3 3[ ][ 2 ] + 2[ 2 ]) G.W.Horndeski Int. J. Theo.Phys. 10, (1974) C.Deffayet, Esposito-Farese,Vikman Phys. Rev.D (79), (2009)

9 Horndeski theory (scalar-tensor theory) homogeneous & isotropic solutions ds 2 = dt 2 + a(t) 2 ijdx i dx j = (t) (quasi-)de Sitter solutions bouncing solutions

10 Vektor-Tensor Theories } how do they couple

11 Generalized Proca action Interactions on curved space-time requires the presence of non-minimal couplings to gravity L.H & J.Beltran, Phys.Lett.B757 (2016) , arxiv: L. H., JCAP 1405, 015 (2014), arxiv: G.Tasinato JHEP 1404 (2014)067 arxiv: Allys, Peter, Rodriguez, JCAP 1602 (2016) 02, 004 L 2 = G 2 (A µ,f µ, F µ ) L 3 = G 3 (Y )r µ A µ L 4 = G 4 (Y )R + G 4,Y (rµ A µ ) 2 r A r A L 5 = G 5 (Y )G µ r µ A 1 h 6 G 5,Y (r A) 3 +2r A r A r A 3(r A)r A r A i G 5 (Y ) F µ F µ r A L 6 = G 6 (Y )L µ r µ A r A + G 6,Y 2 F F µ r A µ r A

12 Cosmology with Vector Fields Flat Friedmann-Lemaitre- Robertson-Walker background ds 2 = dt 2 + a(t) 2 ijdx i dx j Homogeneity & Isotropy A µ =( (t), 0, 0, 0) A 0 A a i =0 i = A(t) i a A a µ

13 Cosmology in Multi-Proca Broken SU(2) symmetry A a µ L.H & J.Beltran, arxiv: L 2 =G 2 (A 2,F a µ ), L 4 =G 4 R + G 0 4 ab Sµ S a bµ Sµ aµ S b 4, L 6 =G 6 L µ F a µ F a + G0 6 2 F a F aµ S b µs b Homogeneity & Isotropy (New field configuration!) (Not so far considered in the literature!) A 0 A a i = i i = A(t) i a

14 Massive Gravity (tensor-tensor theory) (for theoretical consistency) C. de Rham, G. Gabadaze, A.J.Tolley, PRL106 (2011) S.F. Hassan, R.A. Rosen JHEP1107 (2011)

15 Massive Gravity (tensor-tensor theory) ds 2 g = N 2 g dt 2 + a 2 g ijdx i dx j ds 2 f = N 2 f dt 2 + a 2 f ijdx i dx j

16 Modified Gravity GRAVITON } } without adding additional degrees beyond Riemanian geometry

17 Relativistic Particle A free non-relativistic particle S NR = Z 1 2 mv2 dt ~v = d~x dt The free particle moves with constant velocity This action allows the particle to move with any constant velocity Principle of finiteness A free relativistic particle Z S R = mc 2 dt 1 r 1 v 2 c 2! The constraint of maximal velocity is implemented. In the limit of small velocities ~v << c we recover the non-rel. limit

18 Born-Infeld electromagnetism S Maxwell = 1 4 Z d 4 xf µ F µ Principle of finiteness Z 1 2 m2 dt v 2! mc 2 Z dt 1 r 1 v 2 c 2! M. Born and L. Infeld. Proc.Roy.Soc.Lond. A144.(1934) S BIE = l 4 Z d 4 x appleq det (h µn + l 2 F µn ) 1 For small electromagnetic fields it recovers Maxwell s theory: For large electromagnetic fields it prevents the unlimited growth of electric and magnetic fields! S BIE (F µn l 2 ) ' S Maxwell

19 Born-Infeld inspired gravity S DG = Z d 4 x q det (ag µn + br µn + cx µn ) S. Deser and G. Gibbons, CQG15 (1998) X µ = Higher order curvature terms to be tuned to avoid ghosts X µ contains terms of quadratic and higher orders in R µ There is a large freedom in the choice of clear immediate criterion. X µ and no

20 Born-Infeld inspired gravity S Ed = l 4 Z M. Ferraris and J. Kijowski, Letters in Mathematical Physics, , (1981) q d 4 x det R (µn) (G) Determinantal actions for gravity were considered by Eddington (1924) as a purely affine theory. Couplings to matter: The metric enters as an auxiliary field that can then be integrated out. Levi-Civita connection For Einstein-Hilbert metric Palatini G a µn = g a µn + L a µn(q)+k a µn(t) In general there are two independent traces of the Riemann tensor: r µ g ab = Q µab T a µn = G a [µn] R a aµn R a µan Non-metricity Torsion also independent from g ab R µ abn

21 Born-Infeld inspired gravity S BIP = l 4 Z d 4 x appleq det (g µn + l 2 R µn (G)) q det(g µn ) D. N. Vollick, PRD 69 (2004) In the Palatini formulation the ghost can be avoided without further corrections Existence of bouncing solutions... M. Bañados, P. G. Ferreira, PRL 105, (2010)

22 Extended Born-Infeld gravity We can rewrite the action as S = l 4 Z q d 4 x det (g µn + l 2 R µn )=l 4 Z d 4 x p g det q d µ n + l 2 g µa R an S = l 4 Z d 4 x p g det q ĝ 1 ˆq q an g an + l 2 R an (G) This reminds of the massive gravity potential: S MG = Z d 4 x p g 4 Â n=0 q b n n!(4 n)! e n( g 1 f ) C. de Rham, G. Gabadaze, A.J.Tolley, PRL106 (2011) S.F. Hassan, R.A. Rosen JHEP1107 (2011) elementary symmetric polynomials

23 Extended Born-Infeld gravity...and so, a natural generalization of BI inspired gravity is e 0 ( ˆM) = 1, e 1 ( ˆM) = [ ˆM], e 2 ( ˆM) = 1 [ ˆM] 2 [ ˆM 2 ], 2! e 3 ( ˆM) = 1 [ ˆM] 3 3[ ˆM][ ˆM 2 ]+2[ ˆM 3 ] 3! e 4 ( ˆM) = 1 4! S = l 4 Z d 4 x p [ ˆM] 4 6[ ˆM] 2 [ ˆM 2 ]+8[ ˆM][ ˆM 3 ]+3[ ˆM 2 ] 2 6[ ˆM 4 ], g 4 Â n=0 b n e n ( ˆM). J. Beltran J., L. H. and G.J. Olmo JCAP 11 (2014) 004 ˆM q + l 2 ĝ 1 ˆR(G) with matter minimally coupled. S ' Z d 4 x p g Low curvature limit apple l 4 (b 0 + 4b 1 + 6b 2 + 4b 3 + b 4 ) + l 4 2l 2 (b 1 + 3b 2 + 3b 3 + b 4 ) g µn R µn (G) Cosmological constant Newton s constant

24 Extended Born-Infeld gravity...and so, a natural generalization of BI inspired gravity is S = l 4 Z d 4 x p g 4 Â n=0 Field equations b n e n ( ˆM) J. Beltran J., L. H. and G.J. Olmo JCAP 11 (2014) 004 ˆM q + l 2 ĝ 1 ˆR(G) l 4 l 2 2 R al W l b + R bl W l a L G g ab = T ab r l p gw bn d n l r r p gw br + 2 p g T k lk Wbn d n l T k rkw br + T n lr Wbr = 0 Ŵ = f 1 ˆM 1 + f 2 + f 3 ˆM + f 4 ˆM 2 f 1 = b 1 e 0 + b 2 e 1 + b 3 e 2 + b 4 e 3 f 2 = (b 2 e 0 + b 3 e 1 + b 4 e 2 ) The equations have the same structure for all the terms. f 3 = b 3 e 0 + b 4 e 1 f 4 = b 4 e 0

25 Minimal Born-Infeld extension S min = l 2 Z d 4 x p gtr appleq + l 2 ĝ 1 ˆR J. Beltran J., L. H. and G.J. Olmo JCAP 11 (2014) 004 ˆM q + l 2 ĝ 1 ˆR(G) Metric field equations (M 1 ) a (µ R n)a Tr( ˆM )l 2 g µn = 1 T µn ˆR = l 2 ĝ( ˆM 2 ) 1 2 h i ĝ( ˆM ˆM 1 )+( ˆM ˆM 1 ) T ĝ Tr( ˆM )ĝ = 1 l 2 ˆT This equation allows to express content and the metric tensor. M a b as a function of the matter

26 Minimal Born-Infeld extension S min = l 2 Z d 4 x p gtr appleq + l 2 ĝ 1 ˆR J. Beltran J., L. H. and G.J. Olmo JCAP 11 (2014) 004 ˆM q + l 2 ĝ 1 ˆR(G) Connection field equations r l p gw bn d n l r r p gw br + 2 p g T k lk Wbn d n l T k rkw br + T n lr Wbr = 0 We will consider solutions without torsion T a µn = 0 Ŵ = ˆM 1 r l p r l p gg r(n W b) r gg r[n W b] r = 0 = 0 G = G( g) g µn = p det ˆMg aµ ( ˆM 1 ) n a We set torsion to zero a posteriori. This is a consistency equation for this Ansatz.

27 Cosmological solutions ds 2 = N(t) 2 dt 2 + a(t) 2 d ij dx i dx j d s 2 = N 2 (M 0 M 3 1 )1/2 dt 2 + a(t)2 p M0 M 1 d ij dx i dx j l 2 H 2 = 1 M M 3M 0M 0 1 M 1 h 6 1 3(r + p) r ln[(m 0 M 1 ) 1/4 ] J. Beltran J., L. H. and G.J. Olmo JCAP 11 (2014) 004 i 2 Λ 2 H w 1 GR w w Ρ

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