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Alexey Golovnev
St. Petersburg State University

Hamiltonian analysis of modified gravity models

The XXI International Workshop High Energy Physics and Quantum Field Theory Repino, 23 ­30 June 2013


Modifying the gravity?
The Einstein theory of gravity is extremely successful But: 1. Dark Matter is somewhat intriguing 2. Dark Energy points at a technically very unnatural value of the cosmological constant 3. The underlying fundamental physics of inflation is not known 4. Signatures of statistical anisotropy in the CMB radiation Hence, people are looking for IR modifications of GR One of the most straight-forward IR modifications amounts to introducing a mass term However it is not that easy in GR (ghosts etc) Only very recently a potentially healthy modification has been obtained: de Rham ­ Gabadadze ­ Tolley non-linear massive gravity There are still some problems despite the healthy number (five) of degrees of freedom: viable cosmological regimes are not easy to obtain, superluminal propagation is ubiquitous (Deser, Waldron) Nevertheless, it is very remarkable that a reasonable candidate is at hand; and the structure of the model is worth studying Other possible modifications are also of a great interest



The Hamiltonian analysis of modified gravity models
It is more comfortable to calculate Riemann tensor components for the metric connection due to its symmetries. After that, any connection might be presented as a sum of the metric part and the variation which, in turn, consists of the contortion tensor (to represent the effects of torsion) and the contribution from the non-metricity tensor. And exact relation for the Riemann tensor would take the form
R= - + -

Suppose the connection is generated by some modification of the physical metric In this case we get The inverse metric can be computed in ADM-type variables, or in perturbation theory:


Computing the Riemann components 1
Now we reasonable they might trade for nee to be the d the Riemann tensor (ADM) components. It is calculate the connection components first. Anyway, needed during the further steps. We will thoroughly extrinsic curvatures using

In what follows we need:


Computing the Riemann components 2


Computing the Riemann components 3
Finally, in the standard GR we use and, using curvatures again, we get and trading velocities for extrinsic

For the Einstein-Hilbert density, we use and get

-g = N

,

However, we have all the Riemann tensor components at hand, and more general models can be studied.


Linearised GR
We use the "mostly plus" (-,+,+,+) sign convention, and the EinsteinHilbert action. At quadratic level around Minkowski, the action reads



-g R-

1 (( h )( h )-2 ( h )( h )+ 2 ( h )( h )-( h )( h ) 4

)

In the usual variables h 00=2 , h 0i =i b+ s i where i s i 0, and 2 (TT ) (TT ) h ik =2 ik + 2 ik +i v k +k v i+ hik where i v i0 , i hik 0 and h(iiTT ) 0 (and, of course, g +h ), we get up to surface terms
1 ( - ( h i 4
TT ) k

)( h



(TT ) ik

)+

1 ( k (v i - si 2

))

2

2 Ё 2 -3 +( i ) + 4 ( -b +

)

The transverse-traceless tensor sector is physical (gravitons, two degrees of freedom), there are four Lagrange multipliers (two scalars and one transverse vector), and the constraints are actually first class making four more variables unphysical (pure gauge). We see the wrong sign kinetic term for , however everything apart from h( TT ) is not genuinely dynamical due to the gauge symmetry. But once we break it, we are to expect some problems to come about! And indeed, breaking the four gauge invariances, generically we get six degrees of freedom, five of the normal spin-2 field, and one of the additional scalar which is actually the Boulware-Deser ghost.


Fierz-Pauli action


In massive gravity, we always need an auxiliary metric, even if a mere Minkowski one. This is just because otherwise there exist no nonderivative invariant to construct the potential from. Generically, a theory of massive gravity has six degrees of freedom: five of massive spin-two, and one of a scalar ghost Linearly, there is the Fierz-Pauli mass term with only five of them:
m ( h h -h h 4
2





)




In Stueckelberg trick this is the action with Maxwellian kinetic term for the vector. Therefore, the scalar is healthy. h h + + There is a well-known problem of vDVZ-discontinuity It might very well be cured by Vainstein mechanism But, anyway, background) it of freedom), be true for generic non-linearly (or, equivalently, around a curved develops the Boulware-Deser ghost (the sixth degree there any Vainstein mechanism or not. And it remains non-linear completions of the potential.
+








dRGT-gravity


It is an amusing fact that a family of completely BD-ghost-free massive gravity models does exist
(Claudia de Rham, Gregory Gabadadze, Andrew Tolley)



dRGT have worked with the matrix H defined by
g I - H
2

-1

and
1 /3 Pl

K I - I - H
4 1 /5 5 Pl

and used the Stueckelberg trick in perturbation theory. The EFT cutoff is raised to ( m M ) from the Vainshteinian ( m M ) due to elimination of leading (kinetic) self-interactions of the scalar.
3



The potential can be put into the form

V = 2m

2

(

(

g ) - 3
-1



)



It is just the first symmetric polynomial (trace) of the eigenvalues of the basic matrix g-1 Non-perturbatively, it works if a real matrix square root exists. This is not always the case, and there is also uniqueness problem.



((

1 3 -4 5 -4 -3

)) ( )
=1 0 01

2


Immediate generalisations


The other possible mass terms are the second and the third order symmetric polynomials of the eigenvalues (the fourth one is just the determinant)
V 1 = Tr



V 2 = ( Tr V 3 =( Tr


)

2

- Tr

( )

2

)

3

-3 ( Tr ) Tr (

( )

2

)

+ 2 Tr

()


3

One can use an arbitrary background metric Minkowski one

f

instead of the



Generalisation to bimetric version is fairly straightforward since the symmetric polynomials of g-1 f multiplied by -g can be expressed in terms of those for f -1 g multiplied by - f


The ADM formalism


The standard way to perform a non-perturbative analysis in gravity is to invoke the ADM decomposition
ds -( N - N k N ) dt + 2 N i dx dt + ij dx dx
2 2 k 2 i i j



In standard GR the lapse and shift are Lagrange multipliers, and the associated constraints decrease the number of degrees of freedom in from 6 to 2.
11 3 (3) H =- d x N R + ( 2

((

(

j2 j

)

- ik

ik

))

+2 N i k

( 3)

ik

)
ik



In massive gravity with potential V, the Hamiltonian is non-linear in them, thus leaving generically the full set of 6 independent variables
11 H =- d x N R + ( 2
3 (3)

((

(

j2 j

)

- ik

ik

))

-V +2 N i k

( 3)

)

where the indices are handled with respect to the metric , and ik are canonical momenta of ik


Hassan-Rosen analysis 1


In dRGT gravity one has to work with the square root of
g






1 N2 = N - N

(

j 2

N N2 Ni N ij - N2

i

j

)
j

where we have assumed the Minkowski auxiliary metric f =


The approach of Hassan and Rosen is to take the root It would have been fairly simple if it were not for the since



(


1 -a

j

a i -a a

i

j

)

2

=( 1- a a
k

k

)

(

1 -a

a i -a a

i

j

)

How to proceed with the actual case at hand?
Hassan, Rosen; 1106.3344 1109.3515 1111.2070


Hassan-Rosen analysis 2


The main idea is to make a clever redefinition of shits such that
N i =( ij + ND ij ( n , ) ) n
j




1 g = N 1- nk nk -n
-1

1

(

j

ni 0 + -ni n j 0

)(

0 X ij ( , n )

)

We see that, after this redefinition, the quantity -g V in the action is linear in the lapse, and therefore one of the constraints remains, leaving 5 degrees of freedom One can show that the secondary constraint is also there (it is a second class couple), and the analysis can be extended to arbitrary background metrics, and to bimetric versions too.
1 1 In higher potentials the N and N terms drop out due to specific properties of the first matrix in the expression for g f
2 3





-1

? Some particular type of the square root is (partially) chosen ?


Other approaches (without explicit square roots)


As a simple way to perform this analysis, we have proposed to introduce the square root implicitly Golovnev, Phys.Lett.B 707 (2012), 404-408
N V = 2m2 + ( - N 2 g

)

or, after integrating out the Lagrange multiplier,
N V = m 2 +(


(

-1

)

N 2g







)

Now the constraint analysis can be done without explicitly taking any square roots; and it was used in Stueckelberg formalism (Hassan et al.). There is also a very interesting vierbein (vielbein) formulation of the model (Hinterbichler, Rosen; 1203.5783). Moreover, this approach allows to tackle the multi-metric theories. The ghost free potentials are antisymmetrised wedge products of the vielbeins in the model ­ a very nice and beautiful result! The theory can also be generalised by allowing the mass to depend on a scalar field ­ improvements in cosmological model building.






Another class of bimetric theories


Recently, the actions of the form
(Amendola, Enqvist, Koivisto; 1010.4776)

-g g



R ( g )

have been proposed



Unfortunately, the ghosts appear. For fluctuations of g (denote them by h) and the difference between the two metrics (denote it by f) we have at the quadratic level EH(h) ­ EH(f). (Beltran Jimenez, Golovnev, Ё Karciauskas, Koivisto; 1201.4018) Not very surprising! Consider dt ( x + x ) y
2



At non-linear level, there are also dynamical vectors and scalars. The conformal mode of the metric difference was shown to be healthy
(Koivisto; 1103.2743)



The tensor ghosts might be cured by combining the new term with the standard GR (1201.4018), but then it flips the sign of conformal mode too. The ghosts are ubiquitous!
Can we use some f ( g R ( g ) , g g R ( g ) R ( g )) model? Hard question in general. May be, one might try to look for a function with one degree of freedom less? ADM analysis might again be useful!





Conclusions and outlook
The ADM type of analysis can be successfully applied to modified gravity theories. It gives very powerful tools for understanding a model at hand, e.g. to count the degrees of freedom in a completely non-perturbative manner. Beyond the perturbative approach, the dRGT massive gravity exhibits some problems with the existence and uniqueness of the square root matrix It is interesting to further investigate the relations among the metric formulation, the vielbein approach, auxiliary fields method, etc. At the same time, other modifications and bimetric models are worth studying In the simplest version of the metric-affine Amendola-EnqvistKoivisto theory, there are ghosts around the (double) Minkowski background One way to study more general set-ups of this class is to perform a suitable ADM-type analysis It seems feasible to do so.....