Gravitational scalar-tensor theory

Transcription

1 Gravitational scalar-tensor theory Atsushi NARUKO (TiTech = To-Ko-Dai) with : Daisuke Yoshida (TiTech) Shinji Mukohyama (YITP)

2 plan of my talk 1. Introduction 2. Model 3. Generalisation 4. Summary & Discussion

3 Introduction

4 Accelerated expansion of the universe inflation (ancient) & dark energy (current) ä a = +3P 6 0 P apple 3 exotic matter?? change of gravity law?? (canonical) scalar field : = V ( ), P = 1 2 V ( ), 2 2 scalar - tensor theory P if 2 V

5 (G)R -> f (R) General relativity is a unique theory Landau & Lifshitz with 2nd-order EOMs for metric (R ddot{g}) f (R) theory 4th-order EOM : R + = 0 Under a Weyl (conformal) transformation, g µ! 2 g µ V (= p g)! 4 V f (R) theory Einstein + canonical scalar = functional degree of f(r) then, two theories are completely equivalent!!

6 correspondence metric + φ (scalar-tensor) metric (purely gravitational) canonical scalar f (R) theory

7 Horndeski (1974) Cédric et.al. (2011) Kobayashi et.al. (2011) k-essence Horndeski S-T theory with 2nd order EOM for g & φ : canonical : L = 1 2 r µ r µ V ( ) = X V ( ) k-essence : L = K(,X) X = (r ) 2 /2 KGB : L = K(,X)+G(,X) Horndeski : L 4 = G 4 (,X) L 5 = G 5 (,X) G µ r µ r 1 3 ( beyond Horndeski, GAO theory ) h(rr ) 2 ( ) h (rr ) 3 + i

8 my ambition metric + φ (scalar-tensor) metric (purely gravitational) canonical scalar f (R) theory k-essence conformal? KGB?? Horndeski (L4 B 2 )??? Horndeski (L5 B 3 )????!! a special Horndeski f (Gauss-Bonnet)!!

9 conformal disformal disformal transformation : Bekenstein (1993) g µ! A(,X) g µ + In φ = φ (t) gauge, only the lapse function is changed = change of time-coordinate N 2 dt 2! Ñ 2 dt 2 = N 2 d t 2 physics should not change!! checked invariance of action (matter & gravity) invariance of curvature pert. and GW multi-disformal tr. g µ! g µ + f IJ ( A,X AB µ J Y. Watanabe, AN, MS [EPL & editor s choice] & G. Domènech, AN, MS [JCAP]

10 my ambition metric + φ (scalar-tensor) metric (purely gravitational) canonical scalar f (R) theory k-essence conformal? KGB?? Horndeski (L4)??? Horndeski (L5) disformal????

11 Model

12 The model The action is given by R and its derivatives : f R,(rR) 2, R, c.f. f (Riemann) theory Nathalie et.al. (2009)

13 Ostrogradski theorem In applied mathematics, the Ostrogradsky instability is a consequence of a theorem of Mikhail Ostrogradsky in classical mechanics. A non-degenerate Lagrangian dependent on time derivatives of higher than the first corresponds to a linearly unstable Hamiltonian associated with the Lagrangian via a Legendre tr. L =(Q i, Q i, Q 2 ) H = P 1 Q 2 P 2 f(q 1,Q 2,P 2 ) unbounded below

14 The model The action is given by R and its derivatives : f R,(rR) 2, R, Since gravity system is a constrained system, that theorem cannot be directly applied to this theory -> constraint eq. can eliminate such instabilities e.g. Rio s talk : R! R +(rx) 2 + (constraint) In fact, we will show that this theory is healthy in the sense that there is no ghost instabilities once the functional form is properly chosen.

15 more on f(r) replacing R by a field, φ : f(r) =f( ) ( R) conformal transformation : f(r) = e R ( e r ) 2 f( ) constraint eq. from δφ (φ is non-dynamical) : f( ) # of degrees of freedom : > 1 + 2

16 proof of healthiness replacing R by a field, φ : f R,(rR) 2 = f, (r ) 2 ( R) conformal transformation : er ( r e ) 2 f, 2 ( r e ) 2 φ is dynamical = λ (canonical) + φ (k-essence) = k-essential multi-scalar fields # of degrees of freedom : ( 1 + 2)

17 Generalisations

18 Genralisations KGB-type term : K R,(rR) 2 + G R,(rR) 2 R L4-type term : G 4 R h ( R) 2 (rrr) 2i! G 4 R h ( ) 2 (rr ) 2i ( R)

19 Genralisations L5-type term G µ r µ r : definition of Riemann tensor : [r r r r ]r = R r contracting two indices (α, γ) : [r r ] = R r taking a derivative : r [r r ] = R r r +(r R )r

20 Genralisations L5-type term G µ r µ r : r µ 1 (r µ r µ )R + X R 2 R R = G µ r µ r R then the Einstein-tensor term can be also included Beyond Horndeski-type generalisations are also easy -> without the counter terms GAO-type generalisations will be also possible ->???

21 Summary & Discussion

22 summary We have considered a theory whose action is given by R and its derivatives. Despite the higher derivative nature of the action, the resultant system is healthy = no Ghost, no Ostrogradsky instabilities More higher derivative terms (KGB, L4, L5 ) can be also included.

23 discussion Is there any way to distinguish this model from a standard scalar field model (f(r) <-> canonical.)?? Is there any pure gravitational theory which ONLY possesses 1+2 degrees of freedom corresponding to Horndeski s theory (or relatives)??

24 Merci beaucoup!!!