Aller guten Dinge sind drei: Cosmology with three interacting spin-2 fields

Aller guten Dinge sind drei: Cosmology with three interacting spin-2 fields

Marvin Lüben lueben@stud.uni-heidelberg.de Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg
Philosophenweg 16, 69120 Heidelberg, Germany
   Yashar Akrami akrami@thphys.uni-heidelberg.de Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg
Philosophenweg 16, 69120 Heidelberg, Germany
   Luca Amendola amendola@thphys.uni-heidelberg.de Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg
Philosophenweg 16, 69120 Heidelberg, Germany
   Adam R. Solomon adamsol@physics.upenn.edu Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg
Philosophenweg 16, 69120 Heidelberg, Germany
Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania
209 S. 33rd St., Philadelphia, PA 19104, USA
July 20, 2019
Abstract

Theories of massive gravity with one or two dynamical metrics generically lack stable and observationally-viable cosmological solutions that are distinguishable from CDM. We consider an extension to trimetric gravity, with three interacting spin-2 fields which are not plagued by the Boulware-Deser ghost. We systematically explore every combination with two free parameters in search of background cosmologies that are competitive with CDM. For each case we determine whether the expansion history satisfies viability criteria, and whether or not it contains beyond-CDM phenomenology. Among the many models we consider, there are only three cases that seem to be both viable and distinguishable from standard cosmology. One of the models has only one free parameter and displays a crossing from above to below the phantom divide. The other two provide scaling behavior, although they contain future singularities that need to be studied in more detail. These models possess interesting features that make them compelling targets for a full comparison to observations of both cosmological expansion history and structure formation.

modified gravity, massive gravity, trimetric gravity, trigravity, background cosmology, cosmic acceleration, dark energy

I Introduction

The past half-decade has borne witness to a revolution in our understanding of the physics of spin-2 fields. While it has been known for decades that the unique theory describing a massless spin-2 field is general relativity Gupta (1954); Weinberg (1965); Deser (1970); Boulware and Deser (1975); Feynman (1996), it had similarly been a long-standing belief that massive and interacting spin-2 fields were generically plagued by the nonlinear Boulware-Deser ghost Boulware and Deser (1972), despite admitting a healthy linear formulation Fierz and Pauli (1939). This story was turned on its head when, building on earlier work in Refs. Arkani-Hamed et al. (2003); Creminelli et al. (2005), de Rham, Gabadadze, and Tolley (dRGT) constructed a theory of a massive graviton de Rham and Gabadadze (2010); de Rham et al. (2011a) which has been shown through a variety of methods to be free of the Boulware-Deser mode Hassan and Rosen (2011, 2012a); de Rham et al. (2012, 2011b); Hassan et al. (2012a); Hassan and Rosen (2012b); Hassan et al. (2012b); Hinterbichler and Rosen (2012).

The breakthrough in massive gravity led to a corresponding advance in theories of multiple interacting gravitons, or, equivalently, multiple metrics. The dRGT construction contains two metrics, a spacetime metric and a fixed reference metric which must be inserted by hand (typically chosen to be that of Minkowski space). By promoting this fixed metric to a dynamical one, one arrives at a theory of bimetric gravity (or bigravity) which is also ghost-free Hassan and Rosen (2012c).111Both massive gravity and bimetric gravity have deep histories describing rich physics; for further details we refer the reader to the reviews in Refs. de Rham (2014); Hinterbichler (2012) on massive gravity, and Refs. Schmidt-May and von Strauss (2016); Solomon (2015) on bigravity. Theories describing multiple metrics can be trivially constructed from here by coupling various pairs of metrics in the same manner as in bigravity, using the ghost-free potential. These theories also avoid the Boulware-Deser ghost Hinterbichler and Rosen (2012), up to certain conditions on which we elaborate below Nomura and Soda (2012); Scargill et al. (2014); de Rham and Tolley (2015); de Rham et al. (2015a).222In addition to these terms containing two metrics each, interactions directly involving three or four different spin-2 fields can be written using vielbeins Hinterbichler and Rosen (2012), but have no counterpart in terms of metric interactions. Such interactions turn out to contain the Boulware-Deser ghost precisely because they lack a metric-language formulation de Rham and Tolley (2015); de Rham et al. (2015a).

With theoretically-consistent theories in hand, the next step is to search for physical solutions. Given that multimetric theories are fundamentally theories of massive gravitons in addition to a massless one—generically a theory of metrics contains massive gravitons and one massless one, of which matter couples to some combination333One might consider coupling matter to the massless graviton exclusively, but this turns out to reintroduce the Boulware-Deser ghost Hassan et al. (2013).—they modify general relativity predominantly at large distances, i.e., they are infrared modifications to gravity. As it turns out, general relativity has a well-known and significant problem in reconciling theory and observation at cosmological distances (see, e.g., Ref. Bull et al. (2016)): the accelerating Universe Riess et al. (1998); Perlmutter et al. (1999), which naturally lends itself to solutions involving modifying gravity on large scales Clifton et al. (2012). It is therefore entirely natural to ask whether massive gravity or its multimetric generalizations can solve this problem.

There are two immediately necessary (though not sufficient) criteria for a modified-gravity theory to successfully address the accelerating Universe. First, it needs to have cosmological solutions which self-accelerate, i.e., which possess late-time acceleration in the absence of dark energy. Second, it needs to have stable fluctuations about these self-accelerating solutions. Unfortunately, this has proven rather difficult to achieve in massive gravity and bigravity. In the simplest massive gravity case, in which the reference metric is flat space, spatially-flat and closed Friedmann-Lemaître-Robertson-Walker (FLRW) solutions do not exist D’Amico et al. (2011). Solutions can be obtained by considering open FLRW or more general reference metrics, but these solutions seem to generically contain instabilities Gümrükçüoğlu et al. (2011, 2012); Vakili and Khosravi (2012); De Felice et al. (2012); Fasiello and Tolley (2012); De Felice et al. (2013). In bigravity, the situation is slightly improved, as it is not difficult to find FLRW solutions that agree with observations of the cosmic expansion history Volkov (2012); Comelli et al. (2012a); von Strauss et al. (2012); Akrami et al. (2013a, b); Könnig et al. (2014a); Enander et al. (2015a). However, linear perturbations, studied extensively in Refs. Comelli et al. (2012b); Khosravi et al. (2012a); Berg et al. (2012); Könnig and Amendola (2014); Solomon et al. (2014); Könnig et al. (2014b); Lagos and Ferreira (2014); Cusin et al. (2015); Yamashita and Tanaka (2014); De Felice et al. (2014); Fasiello and Tolley (2013); Enander et al. (2015b); Amendola et al. (2015); Johnson and Terrana (2015); Könnig (2015), tend to contain either ghost or gradient instabilities. In each of these cases there are potential ways out. In massive gravity, one might consider large-scale inhomogeneities D’Amico et al. (2011). In bigravity, cosmological solutions can be made stable back to arbitrarily early times by taking one Planck mass to be much smaller than the other Akrami et al. (2015), or by reintroducing a cosmological constant which is much larger than the bimetric interaction parameter Könnig and Amendola (2014). It is also possible that the gradient instability in bigravity is cured at the nonlinear level Mortsell and Enander (2015) due to a version of the Vainshtein screening mechanism Vainshtein (1972); Babichev and Deffayet (2013). However, there remains strong motivation to find a massive gravity or multigravity theory with self-accelerating solutions that are linearly stable at all times.

One logical step in this direction is to inquire what happens cosmologically if we have three, rather than two, interacting spin-2 fields. This generalization has been discussed before in Refs. Khosravi et al. (2012b); Tamanini et al. (2014); Scargill et al. (2014), but all the cases studied in those references are pathological. Ref. Khosravi et al. (2012b) studied a theory of massive trimetric gravity where all three metrics interact with each other directly, making a cycle of interactions; as we will discuss in the next section, such theories are plagued by the Boulware-Deser ghost Nomura and Soda (2012). Ref. Tamanini et al. (2014), on the other hand, studied a multigravitational theory in terms of vierbeins with no metric formulation. Such theories turned out later to also suffer from the Boulware-Deser ghost. Ref. Scargill et al. (2014) compares multimetric models in the metric and vierbein formalism, but does not look at cosmological solutions, while Ref. Baldacchino and Schmidt-May (2016) examines maximally-symmetric solutions in multigravity.

In this paper, we consider the cosmologies of healthy theories of trigravity and scan various models by examining a large number of combinations of parameters in search for interesting background cosmological solutions. Here, “interesting” means cosmological solutions that are both viable and qualitatively different from those occurring in bigravity. Appropriate perturbative analyses of the interesting models will then be required to see whether any of them could be free from instabilities; we leave this for future work.

We note that this paper examines a number of trimetric models in detail and is therefore fairly lengthy. The especially busy reader is directed to section V for a summary of “positive” results. An overview with an additional level of detail can be found in tables 2 and 1, where we present various cosmological viability criteria for all of the models studied.

Ii The theory of massive trigravity

We begin by presenting the theory of massive trimetric gravity, or trigravity, describing three interacting spin-2 fields in four dimensions. We will work in the metric formulation of trigravity, in which the spin-2 fields are described by three metric-like tensors.

How should the three metrics couple to each other? When metrics interact, the Boulware-Deser ghost looms as a nearly-inevitable pitfall, and extreme caution must be taken to avoid it. Fortunately, as mentioned in the previous secion, the interaction potentials which avoid this ghost are known and have been formulated in the context of massive gravity and bigravity. This provides an immediate avenue for trimetric gravity: we could couple each pair of metrics via the ghost-free potential, leading to a cycle of interactions. However, such cycles turn out to be plagued by the Boulware-Deser ghost Nomura and Soda (2012); Scargill et al. (2014); de Rham and Tolley (2015). Therefore we are forced to consider breaking the cycle into a line, i.e., there must be one pair of metrics which do not directly interact with each other.

Next we must consider how these metrics couple to matter. In the simpler cases of massive gravity and bigravity, where there are two metrics rather than three, the question of how matter couples was the source of much discussion and debate Hassan and Rosen (2012c); Hassan et al. (2013); Akrami et al. (2013c); Tamanini et al. (2014); Akrami et al. (2014); Yamashita et al. (2014); de Rham et al. (2015b, b); Hassan et al. (2014); Enander et al. (2015a); Solomon et al. (2015); Schmidt-May (2015); de Rham et al. (2014); Gümrükçüoğlu et al. (2015a); Heisenberg (2015a); Gümrükçüoğlu et al. (2015b); Hinterbichler and Rosen (2015); Heisenberg (2015b, c); Lagos and Noller (2016); Melville and Noller (2016), leading to the conclusion that the Boulware-Deser ghost almost always re-emerges if any matter field couples to more than one metric, or if matter coupled to one metric interacts with matter coupled to another.444This conclusion can be partially avoided by coupling matter to a composite metric of the form , although the Boulware-Deser ghost is present at high energies where the decoupling limit is no longer a valid effective field theory de Rham et al. (2015b). We will therefore take all matter to couple minimally to a single metric, which we will call . Because matter moves on geodesics of this metric, we can interpret it as the physical metric describing the geometry of spacetime, exactly like in general relativity. The other two metrics, which we will denote as and (or and ),555In this paper, commas do not denote spacetime derivatives. couple only to each other or to , and thus are responsible for modifying gravity.

This leaves us with two different classes of ghost-free trimetric theory. In the first, the metrics and both couple to the physical metric , but not to each other. We will call this star trigravity. The other possibility is to couple one of the additional metrics, without loss of generality , to each of the other metrics, and . In this theory, which we call path trigravity, there is no coupling between and . The two theories are depicted schematically in fig. 1. In the rest of this section, we proceed with discussing both classes of trigravity in full detail and generality, before moving on with studying the background cosmology of the two theories in the next sections.

Figure 1: Visual depiction of star trigravity (left) and path trigravity (right). The white circles and the single lines between them represent, respectively, the three metrics and the interaction terms mixing them. The shaded circles and the double lines represent the matter fields and their couplings to the physical metric, , respectively.

ii.1 Star trigravity

In star trigravity, couples to , , as well as all matter fields, . The action is given by

(1)

where is the matrix square root of , the are the elementary symmetric polynomials of the eigenvalues of the square-root matrix, as presented in, e.g., Ref. Hassan and Rosen (2012c), and are the dimensionless coupling constants for the interactions between and . The first index corresponds to the metric involved in the interaction with the physical metric , while the second index specifies the order of the interaction and can take the values . and are the Planck masses and and are the Ricci scalars for the metrics and , respectively. This theory is symmetric under the interchange of the metrics and , along with their Planck masses and interaction parameters. The two mass parameters can be absorbed into the , so that the will have dimensions of mass squared.

The two Planck masses of , , are redundant parameters and can be set equal to .666Though this rescaling does not change the physical solutions, one has to be careful when considering certain limits of this theory. This was demonstrated explicitly for bigravity in Ref. Akrami et al. (2015), and shown to be quite important for cosmological applications. We expect an analogous story to hold in trigravity; this should be explored in future work. To see this, consider the rescaling . The Ricci scalars transform as , so the corresponding Einstein-Hilbert terms in the action become

(2)

In addition to the Einstein-Hilbert terms, the interaction terms in the action also depend on . These transform as

(3)

where in the last equality we used the scaling properties of the elementary polynomials . Redefining the interaction couplings as , we end up with the original star trigravity action, but with .

Variation of the action (1) with respect to and yields the modified Einstein equations for the metrics (after absorbing into and seting ),

(4)
(5)

where and are the Einstein tensors of and , respectively, and is the stress-energy tensor defined with respect to as . The matrices for a matrix are defined as

(6)

where is the identity matrix and is the trace operator.

Let us now consider the divergence of the Einstein equations (4) and (5). The Einstein tensors satisfy the Bianchi identities and . General covariance of the matter sector implies conservation of the stress energy tensor, . Thus we are left with the Bianchi constraints,777The sum of the three equations will vanish, i.e., one of the equations is redundant. Thus, this set of equations really gives only two constraints.

(7)
(8)

where is the -metric covariant derivative raised with respect to , and are the corresponding operators for the metrics. These constraints arise from the fact that the ghost-free potentials are invariant under combined diffeomorphisms of the two metrics involved. They will be important in reducing some freedom in the cosmological solutions.

ii.2 Path trigravity

In path trigravity, couples directly to matter and to one of the reference metrics, which we choose to be . The latter couples in turn to . The action is therefore given by

(9)

with the same notations as in star trigravity, up to different definitions of the interaction parameters. Here the parameters describe the interactions between the physical metric and the metric , while the describe the interactions between and . In what follows, the two mass parameters will again be absorbed into the .

Let us take a closer look at the -metric Planck masses, , which, as discussed in the context of star trigravity, are redundant parameters. Under the rescaling , the Ricci scalars for and transform as above. Therefore, the Einstein-Hilbert terms transform as in eq. 2. However, the mass terms transform differently,

(10)
(11)

where we have again used the scaling properties of the elementary symmetric polynomials . By redefining the interaction parameters and we end up with the original path trigravity action, but with . Therefore, we set from now on.

Variation of the action (9) with respect to and yields the modified Einstein equations for the metrics,

(12)
(13)
(14)

where and are the Einstein tensors for and , respectively. The matrices are given by eq. 6 and is the stress-energy tensor defined with respect to the physical metric .

Let us take the covariant derivative of the Einstein equations (12)–(14). The Bianchi identities for , , and , and the covariant conservation of the stress-energy tensor lead to the Bianchi constraints888See footnote 7.

(15)
(16)
(17)

As in star trigravity, these constraints will allow us to fix some otherwise-free variables.

Iii The background cosmology of trigravity

After having introduced the theories of star and path trigravity, we now turn to their cosmological solutions. We want to describe an isotropic and homogeneous universe, so we choose all our metrics to be of the FLRW form.999We follow the standard recipe as in bigravity, where both metrics are usually taken to be of an FLRW form. One could in principle consider cosmologies with some of the metrics being anisotropic or inhomogeneous. In those cases, it is important to first investigate the consistency of such choices. This has been done in, e.g., Ref. Nersisyan et al. (2015) for bigravity. We leave a similar study for trigravity to future work. This allows us to derive Friedmann equations for all the metrics. After further massaging, we can analyze the solutions to the Friedmann equations and find expressions for the matter density parameter and the effective equation of state.

We first study the background equations of star trigravity and then turn to the case of path trigravity, where we repeat the same procedure. The results of this section are general and hold for any choices of parameters.

iii.1 Star trigravity

We assume that at the background level, the Universe is described by spatially-flat FLRW metrics for , , and ,

(18)
(19)

where is conformal time. The scale factor of and the scale factors and lapses of are functions of conformal time only. Since is the physical metric that minimally couples to matter, its scale factor is the observable scale factor, and similarly the cosmic time measured by observers is given by . Plugging these ansätze in the Bianchi constraints (7) and (8) gives

for : (20)
for : (21)

where an overdot denotes a derivative with respect to conformal time . The ratios of the scale factors of the physical and reference metrics,

(22)

will be of major importance in the cosmological solutions. We will use the Bianchi constraints to fix the -metric lapses as101010Recall from above that out of the three Bianchi constraints, two are independent. In each case we can choose either the dynamical branch, fixing one of the lapses, or the algebraic branch, fixing one of the . These correspond to setting to zero either the first term in the parentheses of eqs. 21 and 20 or the second, respectively. In general, there are four possibilities to solve the Bianchi constraints in star trigravity: taking the dynamical branch for both constraints, the algebraic branch for both constraints, or mixing the dynamical branch for one and the algebraic branch for the other. This is a novel feature of trigravity; in bigravity such mixed branches are not possible. In bigravity, the algebraic branch reproduces general relativity with a cosmological constant at the background level, as we have a fixed solution for , which, when plugged back into the Friedmann equations, generates a constant term von Strauss et al. (2012); Comelli et al. (2012a). However, these solutions possess perturbations with vanishing kinetic terms Comelli et al. (2012b), signalling an infinitely strong coupling, and moreover, are plagued by instabilities in the tensor sector Cusin et al. (2016). Whether this is the case also in trigravity needs investigation, and we leave it for future work.

(23)

We additionally define the conformal-time Hubble parameter for each metric as and . These quantities are related via

(24)

Let us now turn to the Einstein field equations (4) and (5). Inserting our ansätze for and into the - components of the equations, we obtain the three Friedmann equations,

(25)
(26)

where we have assumed a perfect fluid source with . Using eq. 23 we can write the -metric lapses as , and the Friedmann equations for therefore become

(27)

The spatial components of the -metric Einstein equation yield

(28)

where for a perfect fluid. We rewrite the lapses as , where denotes a derivative with respect to the number of -foldings .111111From now on, we will work only in terms of as our time variable. A conformal-time derivative of a quantity can be transformed into a derivative with respect to as (29) as long as . That yields

(30)

As the Friedmann eqs. 30 and 27 suggest, the dynamics of trimetric cosmology are captured by the scale-factor ratios and . Thus, we need to find an expression for in order to be able to analyze the background cosmology of star trigravity. We start by subtracting eq. 27 with from eq. 27 with to obtain

(31)

With this equation, it is possible to relate the two ratios of the scale factors and . It is a cubic polynomial in and , and therefore always has analytic solutions for as a function of , and vice versa, though of course there is more than one solution in general.121212The only exception is the model with , i.e., with only and being nonzero. In that case eq. 31 reduces to , but does not give a relation between and . For every solution, one has to therefore check whether it leads to viable cosmologies. For the star trigravity models discussed in this paper it turns out that the different solutions are redundant at the level of the Friedmann equations and the models’ phase space.

Taking the derivative of eq. 31 with respect to and rearranging the whole expression give

(32)

where we use as a short-hand notation. With these two equations, it is possible to reduce the dimension of the phase space from to , which simplifies the analysis significantly.131313One can use eq. 32 only when the denominator does not vanish. If it vanishes, then the relation between the derivatives of the two scale factor ratios does not hold anymore. However, this situation does not occur in the -parameter models of star trigravity discussed in this paper. Combining the Friedmann eq. 27 with and eq. 30 gives an algebraic equation for and ,

(33)

and the same for exchanged. Taking the derivative with respect to , specializing to pressureless dust with obeying the continuity equation

(34)

and using eq. 32 to rewrite in terms of , yields a differential equation for ,

(35)

Since exchanging in this equation yields the same result, we need an expression for the density that is symmetric under . In order to find such an expression, we add eq. 27 for and the one for , and combine the resulting equation with eq. 30. We obtain

(36)

With these equations we can analyze the phase space.

In order to check the cosmological viability of a model, we will make use of the matter density parameter defined as

(37)

where the matter density follows . Using eqs. 36 and 27 to rewrite and in terms of we obtain

(38)

We can also define the modified-gravity energy density parameter as since we are working in flat space without curvature terms. Note that we additionally do not consider radiation here as we are interested in observations at low redshifts. However, we could easily add a radiation component to the pressureless matter and it would qualitatively not change any of the conclusions below.

The effective equation of state of a fluid consisting of different constituents is defined as

(39)

with the total pressure and the total energy density. We can then rewrite the Friedmann eq. 30 as , and the acceleration eq. 30 as , yielding

(40)

Using eqs. 30 and 27, the effective equation of state in star trigravity reads

(41)

iii.2 Path trigravity

Let us now repeat the procedure of the previous subsection for path trigravity. We assume the metrics , , and to be of the spatially-flat FLRW form

(42)
(43)

where the scale factors and of and , respectively, as well as the lapses of , are all functions of conformal time only. As is the physical metric that couples to matter, its scale factor plays the same role as in general relativity, and in particular is the same scale factor as usually deduced from observations. The path trigravity Bianchi constraints (15)–(17) simplify to

for : (44)
for : (45)
for : (46)

where overdot again denotes a derivative with respect to conformal time . The quantities

(47)

are the ratios of the scale factors of and , and and , respectively. Note the different definition here compared to star trigravity. We use these Bianchi constraints to fix the lapses as

(48)

but we note that other solutions are also possible, in principle.141414See footnote 10 for star trigravity. The same statements are true for path trigravity. Similarly to star trigravity, we define the conformal-time Hubble parameter for the metrics as and . These quantities are related via

(49)

Let us turn back to the Einstein field equations (12)–(14) and insert the ansätze for the metrics into the - components of the equations. We arrive at the Friedmann equations for the metrics,

(50)
(51)
(52)

where we have assumed a perfect fluid source with . The Bianchi constraints on the lapses can be rewritten as and . The -metric Friedmann equations then become

(53)
(54)

If we plug in the ansätze for the metrics into the - components of the -metric Einstein field equations, we obtain

(55)

where we have assumed . Rewriting the lapse as , the equation reads