Cosmology of gravity
Abstract
Using dynamical system analysis, we explore the cosmology of theories of order up to eight order of the form . The phase space of these cosmology reveals that higherorder terms can have a dramatic influence on the evolution of the cosmology, avoiding the onset of finite time singularities. We also confirm and extend some of results which were obtained in the past for this class of theories.
I Introduction
General relativity deals with secondorder differential equations for the metric . Higherorder modifications of the gravitational interaction have been for long time the focus of intense investigation. They have been proposed for a number of reasons including the first attempts of unification of gravitation and other fundamental interactions. Nowadays, the main reason why one considers this kind of extension in general relativity is of quantum origin. Studies on the renormalisation of the stressenergy tensor of quantum fields in the framework of a semi classical approach to genral relativity, i.e., what we call quantum field theory in curved spacetime, shows that such corrections are needed to take into account the differences between the gravitation of quantum fields and the gravitation of classical fluids BOSbook ; Birrell:1982ix .
With the introduction of the paradigm of inflation and the requirement of a field able to drive it, it was natural, although not obvious, to consider these quantum corrections as the engine of the inflationary mechanism. Starobinski Starobinsky:1980te was able to show explicitly in the case of fourthorder corrections to general relativity that this was indeed the case: quantum corrections could induce an inflationary phase. Such result should not be surprising. Fourthorder gravity carries an additional scalar degree of freedom and this scalar degree of freedom can drive an inflationary phase. In the following years other researchers Amendola:1993bg ; Gottlober:1989ww ; Wands ; BerkinMaeda tried to look at the behaviour of sixthorder corrections, to see if in this case one could obtain a richer inflationary phase and more specifically a cosmology with multiple inflationary phases. However, it turned out that this is not the case: in spite of the presence of an additional scalar degree of freedom, multiple inflationary phases were not possible. The reason behind this result is still largely unknown.
The discovery of the dark energy offered yet another application for the additional degree of freedom of higherorder gravity. Like in the case of inflation, this perspective offered an elegant way to explain dark energy: higherorder corrections were a geometrical way to interpret the mysterious new component of the Universe GeomDE . Here an important point should be stressed: differently from the standard perturbative investigation of a physical system, in the case of higherorder gravity, the behaviour of the new theory cannot be deduced as a small perturbation of the original secondorder one. The reason is that, since the equations of motion switch order, the dynamics of the perturbed system are completely different from the nonperturbed one whatever the (nonzero) value of the smallness parameter. For this reason, the properties of higherorder gravity cannot be deduced from their lowerorder counterpart, even if the higherorder terms are suppressed by a small coupling constant. This fact calls for a complete reanalysis of the phenomenology of these theories. Such study should be performed with tools designed specifically for this task, which therefore contain no hidden assumptions or priors which might compromise the final result. One of these tools, which will be used in the following is the so called Dynamical System Approach (DSA) Collins ; DSA Book . DSA has been used now for long time to understand the dynamics of cosmologies of a number of different modifications of general relativity (see e.g. Refs. Bahamonde:2017ize ; Odintsov:2017tbc ; Carloni:2017ucm ; Carloni:2015jla ; Carloni:2014pha ; Carloni:2009jc ; Alho:2016gzi ; Carloni:2015lsa ; Carloni:2015bua ; Carloni:2013hna ; Bonanno:2011yx ; Roy:2011za ; Carloni:2009nc ; Carloni:2008jy ; Abdelwahab:2007jp ; Carloni:2007br ; Carloni:2007eu ; Carloni:2006mr ; Leach:2006br ; Carloni:2004kp ). It is based on the definition of a set of expansion normalised variables of clear physical meaning which help the physical interpretation of the orbits obtained. The first attempt to apply this technique to a theory of order six was made in Ref. BerkinMaeda . Recently a new version of DSA has been proposed Carloni:2015jla , which helped clarify the cosmological dynamics of fourthorder gravity, revealing new aspects of these theories. The technique is also extendable to consider higherorder theories and in the following we will propose a formalism able to treat a subclass of theories of gravity of order six and eight.
Among the many unresolved issues that are known to affect higherorder theories, it is worth to mention briefly the socalled Ostrogradski theorem Woodard:2006nt . The theorem shows that for a generic system with a higherorder Lagrangian, there exist a conserved quantity corresponding to time shift invariance. When this quantity is interpreted as a Hamiltonian, by the definition of a suitable Legendre transformation, it can be shown that such Hamiltonian, not being limited from below, leads to the presence of undesirable features of the theory upon quantisation, whereas the classical behaviour, which includes classical cosmology, has no problem. In view of this conclusion higherorder theories, with the notable exception of gravity, are deemed as unphysical. The most important issue for this work is then, why bother with higherorder gravity? We can give two arguments. The first is that, as mentioned above, the higherorder terms we will consider are terms of a series of corrections arising in a renormalisation procedure. In this perspective, therefore, there is no requirement that the truncated series had the same convergence property of its sum. A typical example is the Taylor series of . The truncated series is not bound, whereas its full sum is. In the same way the truncation of the original semiclassical model that gives rise to a higherorder theory might be fundamentally flawed on the quantum point of view. The problem only arises if one chooses the complete theory of quantum gravity to be given by a order truncation. The second is that a study of the behaviour of the truncation allows an understanding of the interplay between the different terms of the development and in particular if and how the pathologies of the theory at a certain order are changed by the terms of higherorder. This on one hand allows to give statements on the validity of the procedure of renormalisation in quantum field theory in curved spacetime and, on the other hand, it is interesting in the context of the cosmology of fourthorder gravity, as it is known that this class of theories can present a number of issues, i.e., scale factor can evolve towards a singularity at finite time Carloni:2015jla which is independent from the Ostrogradski instability. An analysis of the higherorder theories can therefore shed light on the real nature of these pathologies.
In this paper we will propose a DSA able to give a description of the dynamics of cosmological models based on a subclass of theories of gravity represented by the Lagrangian density . We will show that the higherorder terms in these theories act in an unexpected way on the cosmology: They can be dominant and prevent the appearance of finite time singularities. The calculations involved in this task are formidable so we will give the full expression only when strictly necessary.
The paper is organised as follows. In Section II, we give the general form of the field equations. In Section III, we construct a general formalism for theories of order six and we consider two specific examples. Section IV, we compare directly sixth order and fourth order theories. In Section V, the DSA formalism for theories of order eight is set up together with other three examples. In Section VI, we conclude.
Ii Basic equations
The general action for a relativistic theory of gravity of order six is given by Gottlober:1989ww ; Wands
(1) 
where is the metric determinant of the metric , is a generic function of the Ricci scalar and of its d’Alembertian , and is the standard matter Lagrangian. This theory is in general of order eight in the derivative of the metric. Since we consider the boundary terms as irrelevant, integrating by parts leads to a series of relevant properties in the theory above. First, it is important to note that, not differently from the case of the EinsteinHilbert action, if is linear in the theory is only of order six. In fact, any non linear term in appearing in can always be recast as a higherorder term. Thus, for example, can be written as
(2) 
Thus, terms of the type can be converted into terms of the form . In general, therefore, the class of theories of gravity with Lagrangian
(3) 
where the are functions of the Ricci scalar, will have the same equations of motion of a theory whose action contains terms of higher order like, e.g., . In this sense the analysis given in the following will extend also to this specific class of theories. In the following we will start describing the general theory and then, using the considerations above, we will present explicitly a dynamical systems formalism for Lagrangians of the type of Eq. (3).
Variation of Eq. (3) upon the metric tensor gives the gravitational field equations
(4) 
where is the standard stress energy tensor and
(5) 
We assume a FriedmannLemaîtreRobertsonWalker metric with an expansion factor denoted by and spatial curvature , and further assume the matter component to be an isotropic perfect fluid, i.e., , where is the energy density and is the pressure of the fluid. The cosmological equations are then usually written as
(6) 
(7) 
where
(8) 
is the Hubble parameter, a dot denotes a derivative with respect to time, and
(9) 
With an abuse of terminology, we will sometimes refer to Eq. (6) as the Friedmann equation and to Eq. (7) as the Raychaudhuri equation.
We introduce now the logarithmic time
(10) 
where is a constant with units of length that represents the value of the scale factor at the initial time . We also define a set of seven parameters as
(11) 
where represent the thderivative of with respect to . One can write the above equations in terms of these variables, but this is a long and rather tedious exercise which does not really add anything to the understanding of the problem. For this reason we will not show them here, giving directly the equations in terms of the dynamical variables in the following sections.
Iii Dynamical System Approach for the sixthorder case
iii.1 The basic equations
Let us start looking at the sixthorder case, i.e., . Recalling the argument of the previous section, all theories of order six that have the form can be written without loss of generality as
(12) 
where and are in general different functions of . Eq. (12) has the immediate consequence that and are not present in the cosmological equations and the analysis of this classes of modes is greatly simplified.
In order to apply the scheme presented in Ref. Carloni:2015jla the action will need to be written in a dimensionless way. We introduce therefore the constant , with , which has dimension of the inverse of a length squared and we will consider the function of the type , for some function . This implies the definition of an auxiliary dynamical variable related to . We then define the set of dynamical variables
(13) 
Note that in the above setting and are defined positive so that all fixed points with or should be excluded. The Jacobian of this variable definition reads
(14) 
which implies that the variables are always regular if and .
The requirement to have a closed system of equations demands the introduction of the auxiliary quantities
(15) 
where the prime represents the derivative with respect to the Ricci scalar . The dynamical equations can be written as
(16) 
To eliminate the equations for , , we have implemented in the equations above the Friedmann equation, Eq. (6), and the following constraints coming from the definition of and in Eqs. (9):
(17) 
As mentioned in the Introduction, for the sake of simplicity, we do not report here the full cosmological equations in terms of the variables in Eq. (11). They are very long and their full form does not add much to the understanding of the derivation of the fixed points and their properties. The reader can find some examples of the full form of these equations in the Appendix A.
The solutions associated to the fixed points can be derived writing Raychaudhuri equation, Eq. (7), in terms of the variables given in Eq. (11) and solving for . Since Eq. (7) is linear in via the term , this does not present any problem. From the definition of , in a fixed point we can write
(18) 
where here, and in the following, the asterisk indicates the value of a variable in a fixed point. The characteristic polynomial of Eq. (18) has one real and two pairs of complex roots. Hence, we can write an exact solution for :
(19) 
where , and are integration constants and and are given by
(20) 
i.e. are connected with the fifth root of unity. We are obviously interested in real solutions, which can be derived by a suitable redefinition of the integration constants. The solutions above are oscillating, however they do not correspond to oscillating scale factors. Indeed the scale factor is given by the equation
(21) 
which can be solved numerically. Notice that this solution, like Eq. (19) is parameterised only by the quantity and therefore . In the following we will characterise these solutions only by the value of .
In the case the equation to solve is
(22) 
which in terms of reads
(23) 
Eq. (22) can be solved by separation of variables and it has a solution that depends on the roots of the polynomial in on the right hand side. In particular, the scale factor can have a finite time singularity if any of the roots of the polynomial are complex, otherwise it evolves asymptotically towards a constant value of the scale factor, i.e., a static universe. Therefore, a fixed point with , will correspond to one of these two cosmic histories depending on the value of the constants . Considering that the solution given in Eq. (23) can be viewed as an approximation of the general integral of the cosmology, then the values of the constants should match the initial conditions of the orbit. This implies that the solution in the fixed point will depend on the initial condition of the orbit that reaches it. In Fig. 1 we show time dependence of the scale factor corresponding to this point.
In the following we will examine two specific examples. The first one will show the phase space of a theory in which only sixthorder terms are present other than the EinsteinHilbert one. This example will clarify the action of these terms. The second one will contain also fourthorder terms, so that the interaction between sixth and fourthorder corrections can be observed explicitly.
iii.2 Two Examples
iii.2.1 Case
In this case only the EinsteinHilbert plus sixthorder terms are present in the theory. It is an interesting example as it clarifies the interplay between these terms. In this case the action can be written as
(24) 
which implies and . Then the only nonzero auxiliary quantities in Eq. (15) are
(25) 
and the Friedmann and Raychaudhuri equations, Eqs. (6) and (7) respectively, can be found in Eq. (65) of Appendix A.
The dynamical system in Eq. (16) becomes
(26) 
The system presents three invariant submanifolds , and , therefore only points that belong to all of these three submanifolds can be true global attractors. The fixed points of the system can be found in Table 1, together with their associated solutions which are represented graphically in Fig. 2. Point has a solution of the type described by Eq. (23) and as such can indicate the occurrence of a finite time singularity.
Point  Coordinates  Solution  Stability  
NHS  


(23)  NHS  
S  
A  

The stability of fixed points for , and , can be deduced by the HartmannGrobmann theorem and it is also shown in Table 1. Points and are unstable, but is an attractor. Indeed this point is a global attractor for the cosmology as it lays on the intersection of the three invariant submanifolds of the phase space. The remaining points , for , and , are non hyperbolic, as they have a zero eigenvalue. Their stability can be analysed via the central manifold theorem Wiggins .
For point , for example, defining the variables
(27) 
and expressing the dynamical equations in the new variables, the equation of the center manifold is given by the system of equations
(28) 
where the vector has components . Solving the above system per series at thirdorder, i.e., setting
(29) 
gives the solutions
(30) 
Note that the center manifold coincides with the center space for the variables and . The equation for the central manifold is
(31) 
Using the Shoshitaishvili theorem we can conclude that the stability of is a complex combination of saddle nodes in each planes with and the center spaces for and . Looking at the coefficients of we can conclude that this point is in general unstable.
We can apply the same procedure to the other nonhyperbolic points. However, we can also evaluate the character of these points in a faster way. In fact, point has eigenvalues , i.e., with alternate signs. Therefore, regardless of the behaviour of the central manifold, this point is in fact always a saddle. This implies that in some cases we can evaluate the stability of a non hyperbolic fixed point without analysing in detail the central manifold. Clearly this is insufficient if the aim is to characterise the exact behaviour of the flow in the phase space. However, since we are mainly interested in the attractors in the phase space, such less precise analysis will be sufficient here.
iii.2.2 Case
In this case the EinsteinHilbert plus fourth and sixthorder correction terms are present in the theory and the interaction between them can be appreciated. Consider then the action
(32) 
which implies and . Hence the only non zero auxiliary quantities in Eq. (15) are
(33) 
and the cosmological equations can be decoupled to give an explicit equation for and . These are given in Eq. (66) of Appendix A.
The dynamical system Eq. (16) becomes
(34) 
The system above presents the same invariant submanifolds of Eq. (26) and therefore we can draw the same conclusions for the existence of global attractors. Table 2 summarises the fixed points for this system with the associated solution and their stability. All the solutions associated to the fixed points are characterised by with the exception of which is characterised by the solution Eq. (23).
Point  Coordinates  Solution  Existence/  Stability  
parameter  Phsyical  
always  NHS  
always 


S  
S  
Fig. 5  Fig. 5  
Fig. 5  Fig. 5  
Fig. 5  S  

Some of the fixed points exist only for specific values of the parameters and . For example, the existence of requires and more complex conditions hold for the points whose coordinates are determined by the equation
(35) 
In Fig. 5 we plot the region of existence of these points. With the exception of point all the other fixed points are hyperbolic, although their stability depends on the parameters and . This complex dependence makes very complicated to make general statements on the stability of points . We can conclude however that one of these points is always a saddle. As in the previous case, the stability of point can be determined by the analysis of the central manifold. However, from the sign of the other eigenvalues, we can conclude that the point is unstable. In Figs. 5 and 5 we also plot the stability, see Table 2.
Iv Sixth order terms vs. fourth order terms
It is useful to compare the results that we have obtained so far with an analysis of fourth order models made with the same approach (see also Ref. Carloni:2015jla for an equivalent, but slightly different choice of some of the dynamical variables). For simplicity we will consider here a fourth order theory of the form . For this choice of the cosmological equations read
(36) 
Defining the variables
(37) 
which are a subset of the variable in Eq. (13), the cosmological equations can be written as
(38) 
with the constraints
(39) 
The solutions associated to the fixed points can be obtained from the equation
(40) 
where is defined in Eq. (11) and its expression in the fixed point can be deduced by the second of Eqs. (36) as we have done for the higher order case. As in the previous sections the solution can be given in general noting that the characteristic polynomial for this equation has one real root and a pair of complex roots. Hence, we can write an exact solution for :
(41) 
where , and are integration constants. Naturally for we have the usual equation for the scale factor
(42) 
The fixed points for the system in Eq. (38) with their stability is presented in Table 3.
Point  Coordinates  Solution  Existence  Stability  
S  
R or S  

A if  
S  
A if 
Let us now repeat the same analysis for a theory that contains the fourth order term considered above plus a sixth order term. Consider then the action
(43) 
which implies and . The non zero auxiliary quantities in Eq. (15) are
(44) 
As before the cosmological equations can be decoupled to give an explicit equation for and and one can construct the dynamical system equations to have:
(45) 
In Table 4 we give the fixed points and their stability.
Point  Coordinates  Solution  Existence/  Stability  
parameter  Phsyical  
NHS  
always 



S  
S  
q=3, Fig. 5  Fig. 5  
q=3, Fig. 5  Fig. 5  
q=3, Fig. 5  S  
q<3  S  


Although fundamentally different the two phase spaces present some similarities. Points , and have exactly the same coordinates. In Points and Point , instead, the relation among the values of some of the coordinates is the same as the one of Points . The difference in the coordinates of these points is probably due to the additional contributions generated in the gravitational field equations by the correction. As one could expect, the same additional terms can change the stability of all the fixed points.
For our purposes, the most important result of this comparative analysis is the fact that both the phase spaces present the fixed point . As we have seen, such point is characterised by the vanishing of the quantity associated to both and , and it can represent a solution with a finite time singularity. Looking at Table 3 we see that the fourth order theory point for is an attractor. However, in the sixth order theory, it is possible to prove numerically that in the interval the point is always unstable, see Fig. 6. Therefore we can say that the introduction of the sixth order terms prevents the cosmology to evolve towards . Effectively, this amounts to “curing” the pathology of the fourth order model as the sixth order terms prevents the occurrence of a finite time singularity. In this sense, we can say that, as the time asymptotic state of sixth order cosmologies is never singular, these models are more “stable” with respect to the appearance of singularities. When we will consider eight order corrections, we will use in the results obtained in this section to reach the same conclusion.
V Going beyond sixthorder
v.1 The basic equations
Let us start extending the set of variables used in the previous section, i.e.,
(46) 
The Jacobian of this variable definition reads
(47) 
which implies that, as in the sixthorder case, the variables are always regular if and .
The requirement to have a closed systems of equations implies the introduction of the auxiliary quantities,
(48) 
(49) 
where, for simplicity, we indicate with the th derivative and the th derivative of .
The cosmological dynamics can be described by the autonomous system