Cosmological perturbations in extended massive gravity
We study cosmological perturbations around self-accelerating solutions to two extensions of nonlinear massive gravity: the quasi-dilaton theory and the mass-varying theory. We examine stability of the cosmological solutions, and the extent to which the vanishing of the kinetic terms for scalar and vector perturbations of self-accelerating solutions in massive gravity is generic when the theory is extended. We find that these kinetic terms are in general non-vanishing in both extensions, though there are constraints on the parameters and background evolution from demanding that they have the correct sign. In particular, the self-accelerating solutions of the quasi-dilaton theory are always unstable to scalar perturbations with wavelength shorter than the Hubble length.
Recent years have seen the development of a non-linear theory propagating the five degrees of freedom of a massive graviton (dRGT theory deRham:2010ik (); deRham:2010kj (), see Hinterbichler:2011tt () for a review), without the Boulware-Deser ghost Boulware:1973my (); Hassan:2011hr (). This theory admits self-accelerating solutions deRham:2010tw (); Koyama:2011xz (); Nieuwenhuizen:2011sq (); Chamseddine:2011bu (); D’Amico:2011jj (); Gumrukcuoglu:2011ew (); Berezhiani:2011mt (), in which the universe is de Sitter without a cosmological constant in the action. The Hubble scale of these self-accelerating solutions is of order the mass of the graviton. Having a light graviton is technically natural ArkaniHamed:2002sp (); deRham:2012ew (), so these solutions are of great interest to account for cosmic acceleration in the late-time universe.
Given any non-trivial solution, it is natural to ask how the perturbations around it behave, and in particular whether there are interesting new effects in the propagation of the associated degrees of freedom. The perturbation theory for the self-accelerating solutions of dRGT has been studied in Gumrukcuoglu:2011zh (); D’Amico:2012pi (); Wyman:2012iw (); Khosravi:2012rk (); Fasiello:2012rw (). Freedom from the Boulware-Deser ghost means that around any background, at most five degrees of freedom propagate. Around a homogeneous and isotropic cosmology, these take the form of one transverse-traceless tensor, one transverse vector and one scalar. Even though the Boulware-Deser ghost is absent, the kinetic terms of these degrees of freedom can potentially have the wrong sign, in which case they are ghosts, signaling that that particular background is unstable.
In fact, around the self-accelerating solutions of dRGT theory, the scalar and vector degrees of freedom have vanishing kinetic terms Gumrukcuoglu:2011zh (); DeFelice:2012mx (). This result could have critical implications for the cosmology, and it is important to understand the extent to which it is a generic result in these types of models. There are several avenues one might consider. Quantum mechanically, kinetic terms may be generated by loops. Determining the sign of such terms, and hence whether the background propagates ghosts, is then a difficult question whose answer depends in general on the details of the matter propagating in the loops. If we wish to restore the kinetic terms at the classical level, one avenue is to move away from homogeneous and isotropic cosmologies Gumrukcuoglu:2012aa (); DeFelice:2013awa (). Another is to keep homogeneity and isotropy, but change the theory by adding more degrees of freedom. In this paper, we take the latter approach. We study cosmological perturbations in two extensions of dRGT theory: quasi-dilaton massive gravity, and mass-varying massive gravity.
Massive gravity admits an extension to a theory with a global scale symmetry through the inclusion of a specific scalar field, dubbed the quasi dilaton D’Amico:2012zv (). We examine self-accelerating background solutions to the quasi-dilaton model, and consider the behavior of cosmological perturbations (see also Haghani:2013eya ()). We find the conditions under which the backgrounds are free of ghosts. We find that there is always a ghost-like instability in the scalar sector, for fluctuations of physical wavelength shorter than the Hubble radius.
Ii Quasi-dilaton theory
We start with the quasi-dilaton theory, introduced in D’Amico:2012zv ().
The action governs a dynamical metric and a scalar field ,
The part of the action which provides the mass to the graviton is
where square brackets denote a trace. While these expressions are similar in form to the dRGT theory, in the case of the quasi-dilaton theory, the building block tensor is defined as
where is a non-dynamical fiducial metric. The theory is invariant under a global dilation of the space-time coordinates, accompanied by a shift of . This symmetry rules out a non-trivial potential for .
Throughout the analysis, we choose the fiducial metric to be Minkowski,
ii.1 Background equations of motion
For the physical background metric, we adopt the flat FRW ansatz
To obtain the background equations of motion, it is convenient to introduce time reparametrization invariance, so that we may write a mini-superspace action. We replace the fiducial metric with
where is the Stückelberg scalar ArkaniHamed:2002sp (), and unitary gauge corresponds to the choice .
The mini-superspace action is
where is the comoving volume and we have defined
In addition, to simplify expressions later on, we define
Varying with respect to and then choosing unitary gauge, we obtain a constraint equation:
The Friedmann equation is obtained by varying with respect to the lapse ,
and varying with respect to the scale factor, , yields the acceleration equation. This may be combined in a linear combination with (11)) to yield the simpler equation
Finally, the equation of motion for is
Since time reparametrization invariance was introduced with the Stückelberg field , there is a Bianchi identity which relates the four equations,
Therefore one equation is redundant with the others and may be dropped. In discussing solutions we will generally choose to drop the acceleration equation (LABEL:eqa), although we will use it to simplify expressions for the perturbations.
ii.2 Self-accelerating background solutions
We now discuss solutions, starting with the Stückelberg constraint (10). Integrating this equation gives
In an expanding universe, the right hand side of the above equation decays as . Thus after a sufficiently long time, saturates to a constant value , corresponding to a zero of the left hand side of (15).
These constant solutions lead to an effective energy density which acts like a cosmological constant. As pointed out in D’Amico:2012zv (), there are four such solutions for which is constant.
Of these, implies , and as in D’Amico:2012zv (), we drop this solution to avoid strong coupling.
For these solutions, we write the Friedmann equation (11) as
where the effective cosmological constant from the mass term is
The Friedmann equation (18) also provides a condition on the parameter ; on the self-accelerating solutions, one needs to have in order to keep the left hand side of the Friedmann equation (18) positive. This ensures that when ordinary matter is added to the right hand side, we will have standard cosmology during matter domination. (Although we do not include matter fields, the sign of the matter energy density can be determined by replacing the bare with .)
To find the action for quadratic perturbations, we expand the physical metric in small fluctuations around a solution ,
and keep terms to quadratic order in .
We break the perturbations into standard scalar, transverse vector and transverse-traceless tensor parts,
We then introduce the perturbation of the scalar via
We perform the entire analysis in unitary gauge, so that there are no issues of gauge invariance to worry about, and no need to form gauge invariant combinations. We write the actions expanded in Fourier plane waves, i.e. , . Raising and lowering of the spatial indices on perturbations is always carried out by and .
ii.4 Tensor perturbations
We begin by considering tensor perturbations around the background (5),
where . The tensor quadratic action reads
where the mass of the tensor modes is given by
To obtain this, we have first used the background acceleration equation to eliminate any terms with . Then we have used the self-accelerating branch of (15) (at late times when the right hand side is zero), the Friedman equation (11) evaluated on the self-accelerating solution (i.e. ), and the equation (LABEL:eqs) evaluated on the self-accelerating solution (i.e. ), to eliminate , and .
The tensor mode always has correct sign kinetic and gradient terms. However, it will be tachyonic if the mass term is negative: . The stability of long wavelength gravitational waves is thus ensured by the condition . Nevertheless, even if this condition is violated, the tachyonic mass is generically of order Hubble, so the instability would take the age of the universe to develop.
ii.5 Vector perturbations
We next turn to vector perturbations,
where . The field enters the action without time derivatives, so we may eliminate it as an auxiliary field using its own equation of motion (again we are using the equations of the background self-accelerating solution to eliminate , and )
Once this is inserted back into the action, what remains is a system of a single propagating vector,
and is the mass of the tensor modes as in (29).
From Eq.(33), we see that for , there exists a critical momentum scale, , above which the vector becomes a ghost. In the case , the kinetic terms of vector always has correct sign and thus there is no such critical momentum scale. In the first case, stability of the system requires that the physical critical momentum scale, , be above the ultraviolet cutoff scale of the effective field theory, , i.e.
To determine whether the vector modes suffer from other instabilities, we define canonically normalized fields,
in terms of which, the action (32) reads
where the dispersion relation of the modes is given by
and we have defined the dimensionless quantity
The second term in the dispersion relation (37), which originates from the time derivatives of , is always of order , provided that . Therefore, in this regime, this term does not introduce instabilities faster than the Hubble expansion rate. Moreover, if , the no-ghost condition (34) imposes . Thus, for any physical momenta sufficiently lower than the cutoff scale of the effective theory, the second term in (37) does not lead to any visible instability, i.e. the growth rate of any instability (if any exist) is at most of the cosmological scale.
The vector modes may potentially suffer from a gradient instability arising from the first term in (37), if and . The growth rate of this instability can be made lower than or at most of the order of the cosmological scale for all physical momenta below the UV cut-off , provided that
ii.6 Scalar perturbations
Finally we consider the action quadratic in scalar perturbations,
The scalar sector should consist of two dynamical degrees of freedom: the scalar field and the longitudinal mode of the massive graviton. The perturbations and stemming from and are free of time derivatives, and so we eliminate them as auxiliary fields using their equations of motion:
Inserting these back into the action, we obtain an action with three fields, , and . Since the “sixth” degree of freedom (which would come from the Boulware-Deser instability in generic massive theories) is removed by construction, there is another non-dynamical combination, which we determine to be
We also define an orthogonal combination,
With these field redefinitions, the action can be written in terms of and , with no time derivatives on . The latter is therefore auxiliary and can be eliminated via its equation:
Using this solution in the action, and introducing the notation , the scalar action can then be written as
where is a real anti-symmetric matrix, and and are real symmetric matrices. (Note that there is no loss of generality in taking anti-symmetric, since the symmetric part can be absorbed into by adding total derivatives). For now, we focus on the kinetic terms. The components of the matrix are
For the case at hand, it is sufficient to study the determinant of the kinetic matrix to determine the sign of the eigenvalues.
The sign of the determinant is determined by the sign of the quantity within the square brackets. First note that the determinant is always negative if . Along with the condition for a realistic cosmology obtained from (18), the range of allowed is thus
Generically, the right hand side of the inequality (51) is of order 1. This implies that for modes with physical wavelengths that are smaller than cosmological scales, one of the two degrees of freedom is a ghost. In other words, parametrically, there is an instability in the scalar sector at physical momenta above . As shown in Appendix A, both the physical momenta and the energies of those ghost modes near the threshold are not parametrically higher than and thus are below the UV cutoff scale of the effective field theory. This signals the presence of ghost instabilities in the regime of validity of the effective field theory.
We end this section by comparing our results to those found in D’Amico:2012zv (); Haghani:2013eya (). Noting that at the level of the kinetic matrix (48) the only scale other than is the momentum, the limit is equivalent to considering modes with wavelengths much shorter than the Hubble radius, i.e. , which is in contradiction with the no-ghost condition (51). In this limit, the kinetic matrix then becomes
which has one positive and one zero eigenvalue, as in D’Amico:2012zv () (See Appendix B for a more detailed comparison). In other words, the apparent stability of the self-accelerating solution is due to the loss of the dynamics of the ghost degree of freedom in the short wave-length limit, and so the decoupling limit is not sufficient to determine stability, in agreement with D’Amico:2012zv (). In Haghani:2013eya (), only the super-horizon limit is considered, so the instability which appears only for physical wavelengths Hubble is not visible in this limit.
ii.7 Higher derivative terms and UV sensitivity
The quasi-dilaton theory is governed by the global scaling symmetry described in D’Amico:2012zv (). The action (1) includes all possible terms compatible with the symmetry, with up to two derivatives, and we found there was no way to render the scalar perturbations of the self-accelerating solutions stable at all momenta.
However, beyond two derivative order there are many more terms
compatible with the symmetry. These higher derivative terms can be
thought of as encoding UV effects from whatever physics completes the
theory. Among the possible higher-derivative terms, we will focus here on two
classes of distinguished interaction terms which will not add new
degrees of freedom. There are the
Goldstone-like terms of the form , and the
three possible non-trivial covariantized Galileon terms, of the form
Deffayet:2009wt (); Goon:2011qf (); Goon:2011uw (); Burrage:2011bt (). The
strong coupling scale of the quasi-dilaton on flat space is D’Amico:2012zv (), so it is natural for the
Galileon-like terms to appear suppressed by this scale.
The Goldstone-like terms should carry the scale
The Friedmann equation now becomes:
where is the dimensionless coupling for the cubic covariant Galileon and we have chosen the form as the Goldstone-like term, where . On the other hand, the constraint equation and the value of remain the same. (We have omitted the quartic and quintic Galileon terms for simplicity.) There are still self-accelerating solutions with so the existence of these solutions appears insensitive to the UV effects encoded by the higher derivative operators. For a sensible cosmology, determined by the Friedmann equation should be an increasing function of the bare cosmological constant (which represents the matter energy density in our setup). Demanding this, we obtain the condition
The determinant of the kinetic matrix for scalar fluctuations changes by order one,
but the determinant is still always negative for sufficiently large momenta, provided that the condition (54) is satisfied.
The quartic and quintic covariant Galileon terms can render the determinant of the kinetic matrix for scalar perturbation positive for large momenta, depending on the values of the coupling constants. However, in this regime of parameters, the tensor and vector modes acquire negative kinetic terms. Moreover, after explicit diagonalization of the kinetic matrix one can show that there are two ghost modes in the scalar sector, provided that , determined by the Friedmann equation, is an increasing function of .
Thus, the form of the dispersion relations for the perturbations depends on and receives order one correction due to UV effects, but the presence of the ghost seems to be a robust feature.
Iii Varying mass theory
We now turn to the varying mass theory, obtained by introducing a scalar into dRGT theory and allowing the graviton mass to be a function of this scalar. This theory was first considered in D’Amico:2011jj (), and further studied in Huang:2012pe ().
The action is
We have further generalized to allow the dRGT parameters and to depend on the scalar . The part of the action which provides mass to the graviton takes the same form as in Eq.(2), but here the building block tensor is the same as in dRGT theory deRham:2010kj (), i.e.
One of our goals is to compare our results with the analogous analysis of perturbations in the dRGT theory. Since the original theory does not allow flat solutions for a Minkowski reference metric, it is necessary to adopt a more general form. We therefore extend the fiducial metric to be an arbitrary spatially flat homogeneous and isotropic metric,
iii.1 Cosmological Background Equations
We first study the cosmological background equations (see Huang:2012pe (); Saridakis:2012jy (); Cai:2012ag (); Hinterbichler:2013dv (); Wu:2013ii (); Leon:2013qh () for more on background cosmological solutions to mass-varying massive gravity.) For the physical background metric, we adopt the flat FRW ansatz
To write the mini-superspace action, we introduce time reparametrization via a Stückelberg field , by replacing the fiducial metric with
Unitary gauge corresponds to the choice .
The mini-superspace action is
where is the comoving volume and
In the following, for clarity, we will use the definitions
and we will omit the dependence of the functions , , and on the field value . (We also caution the reader that the above definitions of and are different than the ones in the quasi-dilaton theory, which we introduced in Section II.)
The equation of motion for the temporal Stückelberg field is
The Friedmann equation is obtained by varying the action with respect to ,
and by taking a variation with respect to , we obtain the dynamical equation which, after forming a linear combination with (66), can be expressed as
Finally, the equation of motion for is
where a prime denotes differentiation with respect to .