Gravity and Cosmology 2018 Massive and Kyoto, 2018 march 1st. Partially Massless (PM) Cédric Deffayet graviton (IAP and IHÉS, CNRS Paris) on curved space-times 1. DOF counting for a Massive FP7/2007-2013 graviton ( à la Fierz-Pauli) « NIRG » project no. 307934 on an Einstein spacetime 2. Consistent massive graviton on arbitrary spacetime. L. Bernard, C.D., M. von Strauss + A. Schmidt-May (2015-2016, PR D , JCAP) 3. PM graviton on non Einstein spacetimes. L. Bernard, C.D., K. Hinterbichler and M. von Strauss arXiv:1703.02538 (PR D )
1. DOF counting for a massive graviton on an Einstein space-time (Fierz-Pauli linear theory). Consider an Einstein space-time obeying
1. DOF counting for a massive graviton on an Einstein space-time (Fierz-Pauli linear theory). Consider an Einstein space-time obeying Fierz-Pauli theory can be defined by Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001)
1. DOF counting for a massive graviton on an Einstein space-time (Fierz-Pauli linear theory). Consider an Einstein space-time obeying Fierz-Pauli theory can be defined by Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001) Field equations on shell
1. DOF counting for a massive graviton on an Einstein space-time (Fierz-Pauli linear theory). Consider an Einstein space-time obeying Fierz-Pauli theory can be defined by Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001) Field equations on shell with Cosmological Kinetic Mass constant operator term
Cosmological Kinetic Mass constant operator term Comes from expanding the Einstein-Hilbert action
Cosmological Kinetic Mass constant operator term Only consistent mass term among
DOF counting The fierz Pauli theory for a massive graviton of mass m propagates Massless graviton 2 DOF if m = 0 •
DOF counting The fierz Pauli theory for a massive graviton of mass m propagates Massless graviton 2 DOF if m = 0 • Generic massive graviton 5 DOF if m 0 and m 2 2 ¤ /3 •
DOF counting The fierz Pauli theory for a massive graviton of mass m propagates Massless graviton 2 DOF if m = 0 • Generic massive graviton 5 DOF if m 0 and m 2 2 ¤ /3 • Partially Massless 4 DOF if m 2 = 2 ¤ /3 • graviton
How to count DOF ?
How to count DOF ? Cosmological Kinetic Mass constant operator term Comes from expanding the Einstein-Hilbert action This implies the (Bianchi) offshell identities
Results in the off-shell identity
Results in the off-shell identity i.e.
NB: in this talk on shell Equals off-shell up to undifferentiated or (only) once differentiated h ¹ º but NOT twice differentiated h ¹ º
Results in an the off-shell identity i.e. And the on-shell relation 4 vector constraints Kills 4 out of 10 DOF of
Taking an extra derivative of the field equation operator yields (off shell)
Taking an extra derivative of the field equation operator yields (off shell) While tracing it with the metric gives
Taking an extra derivative of the field equation operator yields (off shell) While tracing it with the metric gives Hence we have the identity Yielding on shell
Generically: yields i.e. a “scalar constraint” reducing from 6 to 5 the number of propagating DOF
However, if Then this vanishes identically
However, if Then this vanishes identically As is
However, if Then this vanishes identically As is Shows the existence of a gauge symmetry
Hence, if one has 6 - 2 = 4 DOF (and a gauge symmetry) The massive graviton is said to be “Partially massless” (PM)
Two questions: Can a fully non linear PM theory exist ? Can a PM graviton exist on non Einstein space-times ?
Two questions: Can a fully non linear PM theory exist ? Can a PM graviton exist on non Einstein space-times ? Here we address the second one…
First, we need to introduce the theory of a massive graviton on arbitrary backgrounds 2. Consistent massive graviton h ¹ º º on an arbitrary background specified by some given (non dynamical) metric g ¹ º
First, we need to introduce the theory of a massive graviton on arbitrary backgrounds 2. Consistent massive graviton h ¹ º º on an arbitrary background specified by some given (non dynamical) metric g ¹ º Einstein-Hilbert kinetic operator Mass term
The theory has been obtained in L.Bernard, CD, M. von Strauss 1410.8302 + 1504.04382 + 1512.03620 (with A. Schmidt-May) out of the dRGT theory de Rahm, Gabadadze; de Rham, Gababadze, Tolley, 2010, 2011
Our massive graviton theory is defined by
Our massive graviton theory is defined by 1. A symmetric tensor obtained from the background curvature solving with ¯ 0 , ¯ 1 and ¯ 2 dimensionless parameters and m the graviton mass, e n the symmetric polynomials
2. The following (linear) field equations
2. The following (linear) field equations Linearized Einstein operator
2. The following (linear) field equations
With
How we got it out of dRGT theory ? Where Uses two metrics: g and f with Has been shown to propagate 5 DOF in a fully non linear way (dRGT, Hassan, Rosen ) … … evading Boulware -Deser no-go « theorem »
Idea: expand dRGT theory around arbitrary backgrounds However The « square root » tensor S ¹ º • makes things unpleasant ! Two metrics in the game ! •
Linearize around some background geometry (e.g. here for the « ¯ 1 » models) Expand the dynamical metric g ¹ º around Easy to get in terms of the arbitrary backgrounds for f ¹ º and g ¹ º linear perturbation h ¹ º of g ¹ º A technical difficulty: expand the matrix square root S ¹ º from One has Sylvester (Matrix) equation (A X + X B = C)
Solving the Sylvester equation (which is possible iff the spectra of S and - S do not intersect) we get with
One thus obtains the linearized field equations Linearized Einstein tensor (curvature dependent) Mass term I.e. the field equations shown previously (where previous E ¹ º is denoted above as ± E ¹ º ):
In these field equations … Using the background equations of motion: • We got rid of the second metric f ¹ º and … … expressed everything in terms of g ¹ º and its curvature
In these field equations … Expressions have been simplified using Cayley- • Hamilton theorem stating that S ¹ º obeys 4th power of S in the matricial sense Allowing to replace any power of S with i ¸ 4 by linear combinations of lower powers of S
This provides consistent (as we now show) field equations (and action) for a massive graviton on a background simply defined by just one arbitrary metric g ¹ º NB: We cheked that (in the special cases of diagonal metrics g and f) our equations (before getting rid of the non dynamical metric f) match those obtained by Guarato & Durrer 2014 (which are not fully general and not explicitly covariant)
DOF counting (consistency check) Start from the (linear) field equations
DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e.
DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e. Yield the (off-shell) identities
DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e. Yield the (off-shell) identities Yielding the four on shell vector constraints
Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ?
Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ? Need to trace over the field equations and their second derivatives and look for a linear combination of these traces yielding a constraints
Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ? Need to trace over the field equations and their second derivatives and look for a linear combination of these traces yielding a constraints However, we can trace both with the metric g ¹ º and with its Ricci curvature R ¹ º (or equivalently S ¹ º )
So we look for linear combinations of the scalars
So we look for linear combinations of the scalars Thanks again to Cayley Hamilton theorem, only 8 of them are independent … … I.e. we look for 4 u i and 4 v i such that
After some algebra we find : 26 « irreducible » scalars made by contracting with powers of S , e.g. … . …
After some algebra we find Are scalar functions of
After some algebra we find Are scalar functions of
After some algebra we find Are scalar functions of That should all vanish … … . ….i.e. 26 equations for the 8 unknown
After some algebra we find Are scalar functions of That should all vanish … … . ….i.e. 26 equations for the 8 unknown
Recommend
More recommend
