Vainshtein mechanism Lecture 3
Screening mechanism Screening mechanisms The fifth force should act only on large scales and it should be hidden on small scales This implies that these mechanisms should work in environmental dependent ways Two well know examples Chameleon mechanism The mass of the scalar model depends on density thus changes according to environments Vainshtein mechanim
Massive gravity models Originally found in massive gravity theory (Fierz-Pauli 1939) L LEH m ( h h h ), h g This theory breaks diffeomorphism invariance h h x x,,, In GR there are two dynamical degrees of freedom (tensor) but in this theory there are five ( tensor, vector, 1 scalar) The scalar mode mediates the fifth force and the theory becomes the Brans-Dicke theory with BD 0, 1/ 5 excluded by solar system constraints 1 (.1.3) 10
Vainshtein mechanism van Dam-Veltman-Zakharov discontinuity (1970) Even in massless limit, the scalar mode does not disappear and GR is not recovered Vainshtein mechanism (197) This discontinuity disappears if we take into account non-linear interactions of the scalar mode Stückelberg approach restore diffeomorphism by introducing Stückelberg fields H h h h,,,
Vainshtein mechanism Scalar interactions Scalar interactions can be extracted from Stuckelberg fields A, A 0 After diagonalisng the kinetic terms and canonically normalise the field, we obtain action for the scalar 1 1 3 1 L T 5 5 M pl 4 mm 1/5 5 pl If we ignore the cubic interaction terms, the scalar mediates the unsuppressed fifth force spoiling the solar system constraints
Spherically symmetric solution Vainshtein radius Cubic interaction terms are important below the Vainshtein radius 4 1/5 r m r r, r GM V g g g m 1 Then GR is recovered below 4GM 3r GM 3r r r m v GM r O r r V GM r O r r V 1 rg r rv 5/ 5/ r g r V
Bolware-Deser ghost Bolware-Deser ghost The cubic interactions give rise to higher derivatives Ostrogradsky instability! A simple example 1 3 1 L T 5 5 M pl Equation of motion 3 ( ) T 5 5 M pl By introducing a new field, the action can be written as 1 1 L... The sixth degree of freedom is known as the BD ghost
Dvali-Gadadze-Porrati model 5D braneworld model Standard model particles are confined to a 4D brane in a 5D bulk spacetime 1 5 1 4 4 S d x g(5) R(5) d x gr d x gl 3Gr 16G c m cross over scale r c r r c r r c 4D gravity 5D gravity rc Infinite extra-dimension
Cosmology Friedmann equation H r c H 8 G 3 At early times, the Universe behaves as 4D. At late tiems the universe accelerates without the need of cosmological constant 1 H 0 r c Fine-tuning of the cross-over scale rc H 0
Quasi-static perturbations Brane bending mode ds N 1 dt A 1 dx (1 G) dy r dydx c, i represents the bending of the brane Quasi static perturbations 1 4 Ga, i i j c i j 3 ( t) r 8 Ga H 1 Hrc 1 3 H The equation of motion is second order No BD ghosts!
Linear theory Solution for linear perturbations 1 ( ) 1 ds dt a t dx k a k a 1 4 G 1, 3 1 4 G 1, 3 These solutions can be interpreted as those in the BD theory with BD 3 1 O(1) H 1 Hrc 1 3 H a
Non-linear solutions Brane bending mode spherically symmetric solution i j c i j 3 ( t) r 8 Ga d r g dr r r r 3 3 r r* ( r), ( r) 1 1 3 * r* 1 8rr 3 c g *, g 4 r r G M 9 rc 4D Einstein rg 1 r r, r g rc rg 1 r r r g rc 4D BD rg 1 1, r 3 rg 1 1 r 3 5D
DGP model Vainshtein mechanism A particular form of the cubic interaction gives rise to a second order differential equation realising the Vainshtein mechanism without introducing the Boulware-Deser ghost L DGP ghost Linear perturbations can be described by a Brans-Dicke theory with a negative BD parameter BD 3 1 3 H 1 Hrc 1 3 H The brane bending mode is a ghost
Galileon models The DGP cubic interaction L This is cubic interaction gives the second order e.o.m. Galilean symmetry 0 c L c 1 c this is a total derivative
Higher order terms (Nicolis, Rattazzi, Trincherini 09) Requirements The field equations are second order The terms are invariant up to total derivatives under the Galilean transformation c Schematically The Galileon terms also have a shift symmetry c The Neother theorem implies that the e.o.m is written as total derivatives L j 0, j ( ) In order for the e.o.m to be second order and satisfy the Galileon symmetry it is schematically given by F 0
Higher order terms F there is one and only one such term at the (n+1)-th order Total derivatives written as anti-symmetric and indices Noether current is given by The Galileon term is obtained by replacing by 1 1
Galileon terms In 4D there are only five Galileon terms
Spherically symmetric solutions Equation of motion r c c c M ( r) 3 1 3 ' ' ' (3) 3 4 r r r r r the fifth order term does not contribute The Vainshtein radius r c M / c, r M c / 8 c 3 6 3 * 3 * 4 1/ M ' r, ( r / r ) r r r 8 c3 1/3 1/ 3 * * * * M ', r ( r / r ) r 8 c4 3 * * *
Solar system constraints The fractional change in the gravitational potential The anomalous perihelion precession r r r r r The vainshtein radius is shorter for a smaller object 5 Lunar laser ranging: the Erath-moon distance re M4.110 km assume there is only the c3 term 1/ 3 3 M pl r EM 11.4 10, re M 4 8 cm 3 10 10 c 3 M DGP: c3 ( r M ) 10 r * 3 3 * 4 M pl pl 10 c pl H0 r cm
Covariantization (Deffyaet et.al. 10) Curved spacetime The galileon symmetry is lost on curved space A naïve covariantisation for 4 th and 5 th order Galileon terms leads to higher derivatives Changing the order of derivatives the curvature term and we non min need to cancel them L g G After integration by parts, 4 th and 5 th order action simplifies a lot 0 4 ( )( )
Generalisaiton (Deffayet et.al. 11, Kobayashi et.al. 11) The most general 4D action that gives nd order equation Standard kinetic term 1 1 V ( ) K X,, X K-inflation (K-essence) still gives the nd order equation of motion Generalisation of Galileon terms
Hordenski action In fact this action was derived in 1974 by Hordenski Generalised galileon terms are reproduced by
Strong coupling problem Strong coupling scale The cubic Galileon L 1 1 3 The strong coupling scale where non-linear interactions become important is naturally given by 1/3 HM 0 pl 1 ( ) 1000km Above this scale, quantum corrections are unsuppressed The Galileon models are effective theory up to
Strong coupling problem Two positive facts Galileon terms do not get renormalised their classical values can be trusted quantum mechanically Around non-trivial background, e.g. on the Earth 1 L Z 3, 1 Z cl cl cl the effective cut-off scale becomes higher Z 1/ 1cm 1 but not enough for the DGP case