Conformal Transformations in Cosmology of Modified Gravity: the Covariant Approach Perspective
The 1+3 covariant approach and the covariant gauge-invariant approach to perturbations are used to analyze in depth conformal transformations in cosmology. Such techniques allow us to obtain insights on the physical meaning of these transformations when applied to non-standard gravity. The results obtained lead to a number of general conclusions on the change of some key quantities describing any two conformally related cosmological models. For example, even if some of the geometrical properties of the cosmology are preserved (homogeneous and isotropic Universes are mapped into homogeneous and isotropic universes), it can happen that decelerating cosmologies can be mapped into accelerated ones. From the point of view of the cosmological perturbations it is shown how these fluctuation transform. We find that first-order vector and tensor perturbations equations are left unchanged in their structure by the conformal transformation, but this cannot be said of the scalar perturbations, which present differences in their evolutionary features. The results obtained are then explicitly interpreted and verified with the help of some clarifying examples based on -gravity cosmologies.
Keywords:conformal transformations modified gravity cosmology 1+3 covariant approach covariant gauge invariant theory of perturbations
Conformal or Weyl transformations have played for long time an important role in many fields, from geography (e.g. Mercator projection) to electromagnetism (see e.g. the work of Bateman Bateman ()) to quantum theories Kastrup:2008jn (). In relativity and cosmology conformal transformations are also widely used. For example they can be exploited to introduce the so-called conformally flat spacetimes, which are among the simplest possible nontrivial spacetimes compatible with General Relativity (GR) J.L.Synge:1960zz (), or to construct Penrose diagrams Hawking:1973uf (), which are one of the most important techniques for the study of black hole physics.
These transformations are also particularly important when one deals with non-standard theories of gravity. Such theories have recently been focus of much investigation because they are thought to offer a possible explanation for the problem of Dark Energy (for recent reviews see Nojiri:2006ri (); Capozziello:2007ec (); Sotiriou:2008rp () ). One of the main difficulties in dealing with these theories, however, is that the non-linearity of their structure makes it really hard to deal with them. Hence much effort has been put into developing new methods to analyze those models. Conformal transformations are particularly convenient in this respect. In fact, it is well known that with their use one is able to map non-standard theories of gravity into general relativity (GR) plus a scalar field () which is minimally coupled to the geometry Wands:1993uu (); Maeda:1988ab (). For many years the meaning and the physical interpretation of these maps have been debated and nowadays one cannot say that a complete agreement has been reached in the community on this issue (see e.g. Magnano:1993bd (); Flanagan:2004bz (); Cotsakis:2004zg (); Sotiriou:2007zu (); Capozziello:2006dj (); Dabrowski:2008kx () for some recent papers). In what follow we will not attempt to enter in this debate, but we will simply try to give a different perspective on the problem based on a new technique: the 1+3 covariant approach.
The 1+3 covariant approach has been developed in recent years thanks to pioneering work of Ehlers Ehlers1+3 (), subsequently developed by Ellis (see for example EllisCovariant ()). This formalism allows a treatment of any cosmological spacetime in a way that is, at the same time, mathematically rigorous and physically clear, and that can be easily adapted to non-standard theories of gravitation. The 1+3 covariant approach has been successful not only in the direct analysis of complicated cosmological models but also in other applications. One of them, the development of the dynamical system approach ellisbook (), has revealed itself very useful in shedding light on the dynamics of Bianchi Universes and the cosmology of scalar tensor and higher-order gravity (see ellisbook (); Carloni:2004kp (); Carloni:2007eu () for details).
It is a matter of fact that an important part of our knowledge on the cosmic evolution comes from the analysis of the perturbations of homogeneous and isotropic cosmological models. There are many different methods to describe the evolution of these perturbations, the most popular approach being Bardeen’s one bardeen (). Classical reviews of this approach can be found, e.g., in Mukhanov:1990me (); Kodama:1985bj (). In this paper, however, we will use the so-called Covariant Gauge Invariant (CoGI) approach BDE (); EB (); EBH (); DBE (); BED (). This technique, based on the 1+3 approach, preserves the most appealing properties of the 1+3 formalism and allows the description of the first order perturbation of any spacetime. It has revealed itself crucial in the development of a consistent theory of perturbations in -gravity, as well as in other extensions of General Relativity SantePert (); StructForm (). Differently from the other formulations Pogosian:2007sw (); Tsujikawa:2007tg (); Li:2008ai (); HuSawicki (); Bertschinger:2008zb (); HuSawicki2 (); Nesseris:2008mq (); delaCruzDombriz:2008cp (); Motohashi:2009qn (); DeFelice:2009ak (), the CoGI approach offers not only the simplest way to describe the evolution of the perturbations, but, as we will see, it also allows an easy comparison between the perturbations in the different frames.
The purpose of this paper is to look at conformal transformations from the point of view of the 1+3 approach and the CoGI approach. These formalism will allow to see the conformal transformation under an entire different view, which hopefully might be useful for further investigations of this topic. We will discover that conformal transformation can be associated to a passage from an inertial observer to an accelerated one plus a redefinition of fields, and that a real change in the physics can occur only at the stage of the field redefinition. Also we will give general formulas that relate the main quantities in cosmology in the different frames including the transformation of the perturbations and this will shed light on the difference in their behavior in the different frames. In order to show that in a concrete example, we will apply this transformation to some models of -gravity.
The paper is structured as follows. In Section II we give a brief review of the common procedure to perform a conformal transformation and we apply it to -gravity, with a specific focus on the distinction between change in the geometry and field redefinition, respectively. In Section III we introduce briefly the basics of the 1+3 covariant approach. In Section IV this formalism is applied to the conformal transformations and to the transformation of the covariant equations. Section V is dedicated to the corresponding transformation in the CoGI formalism. In Section VI the behavior of the scalar perturbations in the two frames is compared in detail for two simple models based on -gravity. Finally, Section VII is devoted to conclusions.
We now fix the notations. Unless otherwise specified, natural units () will be used throughout the paper, Latin indices running from 0 to 3. The symbol represents the usual covariant derivative and corresponds to partial differentiation. We use the signature and the Riemann tensor is defined by
where the are the Christoffel symbols (symmetric in the lower indices), defined by
The Ricci tensor is obtained by contracting the first and the third indices
Symmetrization and antisymmetrization over the indices of a tensor are defined as
Finally, the Hilbert–Einstein action in the presence of matter is given by
2 Conformal transformations in relativity
In this section we will give a basic introduction on the usual treatment of conformal transformations in Riemannian geometry, following mainly Wald:1984rg (); Faraoni:1998qx (); Magnano:1993bd (). Given a spacetime with dimension (), a conformal (or Weyl) transformation is defined as a transformation of the metric tensor given by
where is a regular, strictly positive function of the spacetime coordinates. This type of transformation does not affect the index structure and, as a consequence, preserves the angles between geometrical objects. In addition, since , it is clear that this transformation leaves the causal structure unchanged, i.e. null geodesics and light cones are preserved111For a rigorous proof of this fact see Wald:1984rg ()..
It is easy to derive the transformation rule for the determinant of the metric tensor
where we have used the fact that . This allows us to derive the transformation laws for the Christoffel symbols, the Riemann and Ricci tensors, the Ricci scalar, and the Weyl tensor J.L.Synge:1960zz (); birrell (); Wald:1984rg (); Faraoni:1998qx ():
An equation is said to be conformally invariant if there exists a number, , such that if is a solution of this equation then is a solution of its conformally transformed one Wald:1984rg ().
In relativity and cosmology, with the term conformal transformations one usually means two different transformations: (i) the transformation (6) on the metric and (ii) a rearrangement of the basic quantities in the theory. For example, a Friedmann–Lemaître-Robertson–Walker (FLRW) metric with flat spatial sections
can be written as
and, defining the conformal time as , one can write
We will see that exactly the same happens when we apply the conformal transformation to a theory of gravity: the transformation of the metric will be associated to a re-parametrization of the fields in the action. In fact the latter is a crucial point in understanding this application of the conformal transformations, because it is only in the last step that real changes in the model, which are unrelated to the geometrical conformal transformation (6), take place. In the following we will call Jordan frame (JF) the initial set of metric and fields present in the theory and Einstein frame (EF) the set of metric and fields obtained after this conformal transformation 222This definition is by no means commonly used in literature, where different aspect of the theories (like the presence and position of non minimal couplings in the action) or other arguments have been used to define the conformal frames. The reason behind our choice is that, in principle one can choose as starting point a theory with any feature, the structure depending on the observations and the data coming from experiments, and this choice is in fact irrelevant to the understanding of the features of the conformal transformations..
2.1 Conformal Transformations and Higher Order Gravity
Let us now see in detail what happens to the theory of gravity when we apply the conformal transformation (6) to the metric tensor. As said above, we will focus on the so-called -gravity, which is among the most studied models of modified gravity. Before starting, it is worth to point out that one can apply a conformal transformation at different levels, namely: (i) at the action level or (ii) at the level of the field equations. Of course, since these transformations are, in principle, different from each other, one has to prove their equivalence, but, as we will see, this can be verified directly.
2.1.1 Conformal transformations at the action level
Let us start at the action level. A general Lagrangian for -gravity is given by
being a generic matter field333It is worth to specify that here we are performing a conformal transformation of the metric only, leaving the matter fields (as well as any other additional field) untransformed. This assumption is widely used in the literature and, for the sake of simplicity, we will adopt it here too.. This action reduces to the Hilbert-Einstein one for . It is interesting to note that, performing the transformations (9-11) directly, with the action in this form, would result in a very complicated expression. Instead, using the tools of analytical mechanics Magnano:1993bd (); Magnano:1988tw (); Magnano:1990qu (), one manages to reduce the action to the so-called Helmoltz form, by means of defining a set of new fields associated with the higher-order momenta in the Lagrangian. A more pragmatic way to do that is to write the action as Faraoni:1998qx (); Nojiri:2003ft (); Wands:1993uu ()
where and are two auxiliary fields related to the canonical momenta of (16). As a consequence one can prove that, on the one side, variation upon the field implies , which means that this action is equivalent to (16), and on the other, that variation upon implies, instead, that and are related by Nojiri:2003ft ().
being and where we have used the transformation laws444Note that the conformal transformation performed here is limited to the metric tensor. Since the field defined in (17) is a generic field, one can set . (8)-(11).
At this point one proceeds to the reparametrization of the fields in the action. Setting555Note the absolute value in this definition. It is required in order to preserve the causal structure of the metric ( has to be positive) and implies that the conformal transformation can be only done on sections of the total history of the systems in which has a constant sign.
, and we have used the Gauss theorem to eliminate the term .
The Einstein frame can be then achieved by redefining as and considering a new (scalar) field in the theory. This step allows us to eliminate the non-minimal coupling in the kinetic term of (20) and to obtain:
once is chosen to be . In this form, this Lagrangian looks like a standard GR + minimally-coupled-scalar-field theory, and in vacuum this would be indeed the case. However, the matter Lagrangian in (22) appears to be coupled with the scalar field. This is the reason why in the literature it is often mentioned that in the Einstein frame matter is non-minimally coupled to the scalar field . The presence of such coupling is understandable if we bear in mind that is in fact a part of the gravitational interaction in the JF and, as such, it is bound to have a direct coupling with standard matter. To wit, a non-minimal coupling between gravity and matter is already present implicitly in (16), so that one can imagine that the conformal transformation separates fourth-order gravity into a tensorial part, which is minimally coupled with matter, and a scalar part, which carries the non-minimal coupling. In this sense the non-minimal coupling is an expression of the universality of the gravitational interaction in the Jordan frame.
2.1.2 Conformal transformations at the field equation level
Let us now look at the transformation (6) from the point of view of the gravitational field equations. Upon variation, the Lagrangian (16) gives rise to field equations that can be recast as Revnostra ():
where the term
can be considered to represent an effective fluid associated with the non-Einstein contributions to the gravitational interaction, the term represents the stress-energy tensor of standard matter, , and we have dropped the -dependence of and . Also these equations reduce to the standard Einstein field equations when .
At this point supposing Magnano:1988tw (); Magnano:1990qu (); Magnano:1993bd (), one can transform the energy momentum tensor , obtaining666Note that in the operation just mentioned we have left unchanged the Ricci scalar . This happens because one considers and is left unchanged by the transformation. This is analogous to the introduction of the field in the Lagrangian derivation.
In order to pass to the Einstein frame, we first need to set and then to introduce a scalar field such that . In this way
where the scalar field potential is defined as
which is equivalent to (21).
At this point, setting one obtains
Equations (31) describe Einstein gravity plus a scalar field minimally coupled with gravity and non minimally coupled with standard matter. This theory coincides with the one directly derived upon variation of (22). Such result shows that the conformal transformation (6), with
leads, both at the action and field equation levels, to the “same” theory in the Einstein frame.
Comparing (31) and (23) it is clear that the possibility to perform a conformal transformation has many advantages if one deals with a matter-less system. However, if matter is added the conformal transformation does not necessarily lead to an easier model. This is due mainly to the non-minimal coupling between standard matter and appearing in (31) and (22), which induces additional terms in the Bianchi identities. For example, the Klein Gordon equation for reads
and the energy-momentum conservation is given by
The above equations tell us that only a form of matter-energy for which the trace is null renders the above equation conformally invariant.
In spite of its usefulness, the conformal transformations of theories of gravity bring a serious problem: the possibility of changing the type and number of the fields in a theory by a simple change in the metric tensor implies that there is no reason, a priori, to choose a specific representation of the action among all possible ones. In other words, recognizing the freedom associated to the conformal mapping means, in fact, loosing the physics of the theory in an infinite set of representations. This fact, in itself, would not be a problem if those representations would describe the same physics but, as we will see later, this does not appear to be the case: they describe very different Universes. As a consequence we are left with the choice of either establish the existence of the particular “physical frame” (i.e. the specific field parametrization, that reflects the actual physical fields), or to prove that somehow all the frames are equivalent. There is a wide literature on this issue and we will not enter into the details of the debate referring the reader to some of the many papers and reviews on the topic (see e.g. Faraoni:1998qx (); Magnano:1993bd (); Flanagan:2004bz (); Sotiriou:2007zu ()). The purpose of this paper is to offer, using the covariant approaches, a new perspective on conformal transformation that might contribute to the clarification of this issue.
3 The 1+3 covariant approach to cosmology
In this section we will present a brief introduction to the covariant approach to cosmology. We will use this approach to understand better the physics behind the conformal transformations and for the construction of a theory of cosmological perturbations in the two frames.
Given a space-time associated to a cosmological model one can single out a family of preferred worldlines representing a certain class of observers (for example the ones comoving with standard matter). If we suppose that it is possible to define a unique 4-velocity vector field associated to these worldlines, then we can split the metric tensor as
i.e. the spacetime is foliated in hypersurfaces with metric orthogonal to the vector field . In this way any affine parameter on the worldlines associated to can be chosen to represent “time” and the tensor () determines the geometry of the instantaneous rest-spaces of the observers we have chosen. Using and , one can then define the projected volume form , the covariant time derivative () along the fundamental worldlines, and the fully orthogonally projected covariant derivative :
Performing a split of the first covariant derivative of into its irreducible parts, namely
one can define the basic kinematical quantities of this formalism EllisCovariant (). The trace is the rate of volume expansion scalar of the worldlines of (which is proportional to the standard Hubble parameter : ); is the trace-free symmetric rate of shear tensor (, , ) describing the rate of distortion of the observer flow; is the skew-symmetric vorticity tensor (, ) describing the rotation of the observers relative to a non-rotating (Fermi-propagated) frame, and is the acceleration vector, which describes the non-gravitational forces acting on the observers777For an introduction to relativistic fluid mechanics and more information on the meaning of these tensors we refer the reader to Ehlers1+3 (); Plebanski:2006sd ()..
A general matter energy-momentum tensor can also be decomposed locally using and . One has
where is the relativistic energy density relative to , () is the relativistic momentum density, which is also the energy flux relative to , is the isotropic pressure, and () is the trace-free anisotropic pressure.
The quantities presented above completely determine a cosmological model. Their evolution and constraint equations, also known as 1+3 covariant equations, are completely equivalent to the Einstein equations and characterize the full evolution of the cosmology. They are shown in Appendix A. The advantages in using these variables is that they allow for a treatment of cosmology that is both mathematically rigorous and physically meaningful and they are particularly useful in the construction of the theory of perturbations.
4 The 1+3 Conformal Transformation
As we have seen in the previous section, a conformal transformation in relativity and cosmology is, in fact, the combination of a geometric operation and a field redefinition. We will treat them separately.
4.1 The geometric part of the conformal transformation
The equations above show how the fact that the conformal factor is positive translates in the fact that the conformal observer velocity is always well defined and has to have the same direction of the Jordan observer. In addition, implies that the sign of the projector tensor remains the same, preserving the pseudo-Riemannian character of the manifold. The relations above also imply that in terms of the 1+3 formalism a conformal transformation888In the following we will consider this transformation as a passive transformation for . The reason for doing so is physical. If the transformations were active, we would basically change spacetime and the comparison of two observers in two different spacetimes would be less physically consistent. can be associated to a change from the Jordan observer , associated to , to a new one which we will call Conformal observer , associated to 999Consistently with the tradition in Relativity, here we call “observer” a reference frame in a specific state of motion.. In particular: (i) the conformal observer has a 4-velocity whose modulus depends on the spacetime coordinates (and as a consequence is accelerated), and (ii) the spatial metric of this observer is modified by the conformal factor. This tells us that (39) basically consists in switching from an inertial observer to an observer whose clock rate and rod length change continuously in spacetime. This can be seen clearly looking at the transformation of the derivative operators. For scalars, we have
and the ones for tensors follow accordingly101010Note that when one changes the position of the index of the corrections change their sign in the same way as for the Christoffel symbols..
Since the derivatives are changed, the basic quantities that one uses to describe the cosmology and the perturbations are also changed. The 1+3 kinematical quantities are transformed as follows
and the electric and magnetic parts of the Weyl tensor are transformed in themselves (one should beware of the position of the indices):
The transformations above describe clearly the differences between and . The conformal observer sees an expansion rate which is increased if the conformal factor grows in time and it might even observe the Universe undergoing cosmic acceleration when in the Jordan frame the expansion is decelerated111111Similarly, when the cosmology becomes singular in the Einstein frame the modified gravity description in the Jordan frame shows qualitatively a different behavior (e.g. it might become a complex theory BrisceseBamba ()). Capozziello:2006dj (). Instead, the vorticity and the shear are only changed by a multiplicative factor, so that under conformal transformation homogeneous, isotropic and irrotational universes do not loose their symmetries. Such effect can be traced back to the fact that the velocity of the conformal observer is always parallel to the one of the Jordan observer. Finally, the transformation of the acceleration vector shows that even if we start from Universes with zero acceleration (i.e. no additional forces other than gravity) the conformal observer perceives an acceleration which depends on the spatial dependence of the conformal factor.
The connection between conformal transformation and observers is also important when one looks at the thermodynamics. Since is moving with respect to with varying velocity, he/she will measure different thermodynamics. In fact, using the transformation (39) one can write the energy momentum tensor as
and the conformal observer will detect
Thus if we assume standard matter in the JF to be a perfect fluid in its rest frame, in the Einstein frame standard matter remains a perfect fluid in the conformal frame (i.e. , ). In fact, with these transformations, in general also the equation of state is preserved. However, since will measure only the barred quantities, the spacetime variation of all the thermodynamical quantities is different. The thermodynamics is further modified by the transformations in the derivatives which lead to further changes in the usual conservation laws. For example, in the homogeneous and isotropic cases one has
This is easy to understand in terms of the properties of the observer described above. Since the rods of the conformal observer change in time and space, the mass energy contained in a box at rest with this observer will change, and will measure a modification of the standard conservation laws.
At this point, using the transformation of the kinematics and the thermodynamics presented above, one can derive how the 1+3 equations transform under (6). The detailed set of equations is given in Appendix A. One can see that the conformal observer perceives many corrections to the standard cosmological equations. These equations show that for an accelerated observer, like , the expressions appear deeply modified in their structure. It is interesting to note, en passant, that such observer, if unaware of its acceleration, would conclude that some exotic physics or change in the gravitational interaction is taking place on cosmological scale.
4.2 The field-redefinition part of the conformal transformation
Let us now concentrate on the remaining part of the conformal transformation. As we have mentioned before, this consists basically in a field redefinition. In principle there is no standard prescription for the definition of a field in a theory, however the structure of the 1+3 equations and what we know about field theory (the correct form for a kinetic term, etc.) suggests the definition (32) should be taken. This specific choice (or any other whatsoever) leads to a tremendous change in the model, which the 1+3 formalism helps appreciate in detail.
As we have seen, the conformal observer uses clocks and rods that change with the spacetime coordinates. This means that such observer will perceive, for example, an object moving with a constant velocity with respect to the Jordan observer as if it was accelerating. Associating the conformal factor to a scalar field basically amounts to considering such effects as a result of the presence of a new interaction, rather than a kinetic effect. In a way this resembles Einstein’s lift Gedankenexperiment: the (accelerated) conformal observer becomes an inertial observer, which we will call Einstein observer , and a scalar field is introduced in the model which accounts for the additional kinematics.
What said above explains the fact that, if one performs a conformal transformation, even in pure General Relativity, one obtains Einstein’s gravity plus a scalar field. It also allows to clarify the nature of . Such field cannot be really considered a matter field, even if it behaves exactly like one: the best interpretation for is, in our view, to consider it as a kinematical effect promoted to interaction.
When one applies the conformal transformation to -gravity two additional operations are performed, namely the specification of the nature of the thermodynamical quantities and the connection of the scalar field with the term. Both steps require particular attention. Let us consider the first one. If we look at the 1+3 equations as perceived by , we can see that these equations are different from the ones we would obtain for the theory (22) even if we would write in terms of . The reason is that these equations miss a critical ingredient, i.e. the specification of the structure of which represent the total energy density as derived from in (24). In making this substitution one has to remember the presence of the non-minimal coupling between standard matter and the Ricci scalar and the fact that—differently to what happens with standard-matter thermodynamical variables—the effective variable associated to contain derivative terms. This implies that the effective thermodynamic quantities associated to do not follow strictly the transformations (51). In particular,
Once the correct transformations are introduced, one can substitute
to obtain the equations one would derive from (22). Again, the 1+3 approach helps shedding light on the physical meaning of this important step. The relation between and modifies the cosmological equations in such a way that all the higher-order terms are compensated, and one is just left with a linear theory of gravity and a scalar field minimally coupled to the geometry. In other words, one is thereby choosing a specific form of the conformal factor for which the kinematical terms compensate the non-Einstenian part of the equations. Thus, in practice, the Einstein observer moves in such a way to compensate the correction. Such compensation is complete in vacuum, but constraints matter to move non geodetically (at least with the choice (55)) as a footprint of the transformation we performed. Also this fact bears clear similarities with Einstein’s lift experiment: the only way in which the observer in the lift is able to infer the presence of an actual gravitational field is to study the geodesic deviation of matter.
5 Perturbations and conformal transformations
At this point we are ready to move our attention on how the conformal transformations affect the evolution of the cosmological perturbations. This will be done using the CoGI approach, which is based on the 1+3 equations mentioned in the previous section and listed in Appendix A.
The CoGI approach presents one main difference (which is at the same time a point of strength) with respect to other perturbation theory approaches in that it relies directly on the structure of the perturbed Universe, rather than on the concepts of background quantities and perturbations. In normal cases, the structure of the perturbed spacetime is trivial because one can just consider a completely generic spacetime. However, when we want to compare the perturbations of two conformally related spacetimes, the structure of the perturbed Universe in the “arrival” frame is not generic, but depends on the type of transformation chosen. In what follows we will assume that the conformal factor is a function of all the spacetime coordinates121212 In principle one could choose a conformal factor which depends on spatial or temporal coordinates only, but this would induce problems in the connection between the conformal factor and the scalar field made in Sect. 2.1. For example, performing a conformal transformation with a conformal factor that depends, say, only on time, would result in the disappearance of all the projected derivatives of in the 1+3 equations and would make impossible to characterize the perturbation of this field. If one would force the perturbations on these quantities, like one seems to be able to do in other perturbation formalisms, the perturbation of would represent a fluctuation of the conformal mapping, introducing something similar to gauge modes in the theory..
The next step in the construction of the CoGI formalism is the definition of the background. This is not done by assigning a metric, but rather by recognizing which 1+3 quantities are zero in the background and which are not. In what follows we will consider expanding () homogeneous and isotropic (, ) backgrounds. In this setting we will characterize the perturbations in terms the of 1+3 quantities seen in Sect. 3 and their projected gradients. For example, the key quantities relevant to the evolution of scalar perturbations in GR are
which represent the comoving normalized spatial gradient of the energy density, the comoving spatial gradient of the expansion, and the comoving spatial gradient of the 3-Ricci scalar, respectively. These variables are related by a constraint coming form the spatial derivative of the Gauss equation SantePert (); StructForm (). Moreover it can be proven that these variables, as well as any other quantity which vanish in the background, are gauge-invariant bi:stewart ().
A quick look to the Einstein frame 1+3 equation in App. A shows clearly that the tensor and vector perturbation equations are left unchanged in their structure, but, as we will see, the same cannot be said of the scalar perturbations. In the following we will focus on this last type of perturbations only and, specifically, on spherically symmetric collapse, which is associated to the cosmological density fluctuations. To extract this information from the variables (56), we use the local splitting
and we then single out the scalar parts of (56):
which are gauge invariant, for the same reasons that are. We will select (and deal) with variables of the type (59) only.
In order to describe the scalar fluctuations in -gravity (as in any other gravitational theory) we will use the variables (56), plus other ones that will take into account the additional degrees of freedom of the theory and are defined specifically for the theory itself. The results of bi:stewart () will guarantee that these new quantities are indeed gauge invariant.
At this point it is relatively easy to construct the perturbation equations. Starting from the 1+3 equations one obtains a set of propagation and constraint equations for the variables (59). Then, one chooses a background while recognizing which of these variables is zero in the background. These variables are then considered to be of order one. At this point the linearized equations can be obtained by dropping all the terms of order higher than one from the propagation and constraint equations.
Before analyzing in detail the transformation of the perturbation equations thus obtained, let us consider—as we already did for the kinematical quantities—what can we learn from the transformation of the perturbation variables (59) upon (6). We have
This clearly shows that the matter fluctuations in the Einstein frame are a combination of the matter fluctuations in the Jordan frame with the fluctuations of the conformal factor. This can be understood, intuitively, if one thinks that the conformal observer measures the matter fluctuations with clocks and rods which are also perturbed. It is useful to give the transformation for the scalar variables too, which read, to first order,
The differences between the Jordan and the Einstein frames appear clearly: even if the JF matter fluctuations are close to zero for some reason, can still be able to observe matter fluctuations, or, depending on the choice of the conformal factor, in spite of the presence of matter fluctuations in the Jordan frame the conformal observer could possibly see no matter fluctuations at all! In addition, because of the transformations above, it seems clear that does not necessarily imply that . This means that in the long wavelength limit the system of perturbation does not posses a conserved quantity, like it happens in GR. Such feature will have an important impact on the difference in the perturbation behaviors in the two frames.
The equations above also show that, in general, the perturbation equations are not conformally invariant in the sense of Wald:1984rg (). For example, given the structure of (60a) and (61a), one can see that it would be difficult to prove that there exists a number such that .
5.1 Scalar perturbations of -gravity in the Jordan frame
Let us now derive explicitly the perturbation equations for -gravity around an homogeneous and isotropic background in the presence of a barotropic fluid with equation of state . The zeroth order equations are given by
Now, in order to model the additional degrees of freedom of this theories one can add to (59) the following scalar quantities
where determine the fluctuations in the Ricci scalar and and the ones of its momentum 131313This choice of variables is by no means unique, but the ones we have chosen are definitely among the most convenient.. Again, since these quantities vanish in the background, we can say that, as in the case of and , they are gauge invariant. The set of variables completely characterizes the evolution of the density perturbations in -gravity. Their evolution equations constitute a system of first order partial differential equations SantePert (); StructForm (). In order to reduce it to a system of ordinary differential equations, one defines the eigenfunctions of the spatial Laplace-Beltrami operator:
where is the wave number and , and expands every first order quantity in the above equations:
where stands for both summation over discrete or integration over continuous indices. In this way, one obtains the equations describing the mode for scalar perturbations in gravity. They are SantePert (); StructForm ():