PHYSICAL REVIEW D 71, 063513 (2005) Cosmology of generalized modified gravity models

Sean M. Carroll,1,* Antonio De Felice,2,† Vikram Duvvuri,1,‡ Damien A. Easson,2,x Mark Trodden,2,k and Michael S. Turner1,3,4,{ 1Enrico Fermi Institute, Department of Physics, and Kavli Institute for Cosmological Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637-1433, USA 2Department of Physics, Syracuse University, Syracuse, New York 13244-1130, USA 3Department of Astronomy & Astrophysics, University of Chicago, Chicago, Illinois 60637-1433, USA 4NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, USA (Received 3 December 2004; published 18 March 2005) We consider general curvature-invariant modifications of the Einstein-Hilbert action that become important only in regions of extremely low space-time curvature. We investigate the far future evolution of the Universe in such models, examining the possibilities for cosmic acceleration and other ultimate destinies. The models generically possess de Sitter space as an unstable solution and exhibit an interesting set of attractor solutions which, in some cases, provide alternatives to dark energy models.

DOI: 10.1103/PhysRevD.71.063513 PACS numbers: 98.80.–k, 04.50.+h

expansion rate inevitably slows. Our question is can this I. INTRODUCTION be avoided within the gravitational sector of the theory, as The acceleration of the Universe presents one of the opposed to adding new energy sources. One way is by greatest problems in theoretical physics today. The increas- adding a cosmological constant. In this paper we explore ingly accurate observations of type Ia supernovae light- a wider class of modifications that share the feature of late- curves, coupled with exquisite measurements of cosmic time accelerating behavior, thus fitting cosmological microwave background (CMB) anisotropies and large observations. scale structure data [1–6] have forced this issue to the Consider a simple correction to the Einstein-Hilbert forefront of those facing particle physicists, cosmologists, action, Z   Z and gravitational physicists alike. M2 p 4 p This problem has been attacked head on, but no com- S ˆ P d4x ÿg R ÿ ‡ d4x ÿgL ; (1) 2 R M pelling, well-developed, and well-motivated solutions have yet emerged. While much work has focused on the search where  is a new parameter with units of [mass] and LM is for new matter sources that yield accelerating solutions to the Lagrangian density for matter. This action gives rise to general relativity, more recently some authors have turned fourth-order equations of motion. In the case of an action to the complementary approach of examining whether new depending purely on the Ricci scalar (and on its deriva- gravitational physics might be responsible for cosmic tives) it is possible to transform from the frame used in (1), acceleration. which we call the matter frame,toanEinstein frame,in There have been a number of different attempts [7–15] which the gravitational Lagrangian takes the Einstein- to modify gravity to yield accelerating cosmologies at late Hilbert form and the additional degrees of freedom (H times. The path we are concerned with in this paper is the and H_ ) are represented by a scalar field . The details of direct addition of higher order curvature invariants to the this can be found in [7]. The scalar field is minimally Einstein-Hilbert action. The first example of this was pro- coupled to Einstein gravity, nonminimally coupled to mat- vided by the model of Carroll, Duvvuri, Trodden, and ter, and has a potential given by v Turner (CDTT) [7] (see also [8]). For subsequent work  s u s  tu on various aspects and extensions of this model, see [16– 2 2 2  2  V †ˆ MP exp ÿ2 exp ÿ 1: (2) 32]. In particular, the simplest model has been shown to 3 MP 3 MP conflict with solar system tests of gravity [19,22,25]. Our Consider vacuum cosmological solutions. We must approach is purely phenomenological. The evidence for specify the initial values of  and 0, denoted as  and cosmic acceleration is very sound. In pure Einstein gravity, i 0. For simplicity we take   M . There are three as matter dilutes away in the expanding Universe, the i i P qualitatively distinct outcomes, depending on the value 0 of i. *[email protected] 1. Eternal de Sitter. There is a critical value of 0  0 † i C [email protected] for which  just reaches the maximum of the potential ‡[email protected] [email protected] V † and comes to rest. In this case the Universe asymp- [email protected] totically evolves to a de Sitter solution. This solution {[email protected] requires tuning and is unstable, since any perturbation

1550-7998=2005=71(6)=063513(11)$23.00 063513-1  2005 The American Physical Society SEAN M. CARROLL et al. PHYSICAL REVIEW D 71, 063513 (2005) will induce the field to roll away from the maximum of its where an overdot denotes differentiation with respect to potential. cosmic time, and H ˆ a=a_ . 0 0 2. Power-Law Acceleration. For i >C, the field over- To perform a phase-space analysis of such equations we shoots the maximum of V † and the Universe evolves to write H_ as a function of H. Let late-time power-law inflation, with observational conse- _ quences similar to dark energy with equation-of-state pa- x ˆÿH t†;yˆ H t†; (6) rameter wDE ˆÿ2=3. so that 0 0 3. Future Singularity. For i <C,  does not reach the maximum of its potential and rolls back down to  ˆ 0. dH_ dH_ dH dy H ˆ ˆ ˆÿy : (7) This yields a future curvature singularity. dt dH dt dx In the more interesting case in which the Universe In this way we may write (5), a third-order equation in contains matter, it is possible to show that the three pos- a t†, as a second-order equation in H t†, and hence as a sible cosmic futures identified in the vacuum case remain first-order equation in y x†. The fact that the Friedmann- in the presence of matter. Robertson-Walker (FRW) equation for any f R† theory can By choosing   10ÿ33 eV, the corrections to the stan- be reduced to the first-order equation for the spatially flat dard cosmology only become important at the present case (second order in case of a nonzero spatial curvature) epoch, making this theory a candidate to explain the ob- was first introduced in [35,36]. served acceleration of the Universe without recourse to Many cosmologically interesting solutions, including dark energy. Since we have no particular reason for choos- accelerating ones, are power-law solutions of the form ing this value of , such a tuning is certainly not attractive. a t†/tp. In such cases H_ ˆÿH2=p (i.e. y ˆÿx2=p). However, it is worth commenting that this small correction In anticipation of finding such solutions as asymptotic to the action, unlike most small corrections in physics, is solutions to our equations, we define a new function v x† destined to be important as the Universe evolves. by Clearly our choice of correction to the gravitational action can be generalized. Terms of the form x2 ‡ † v x†ˆÿ ; (8) ÿ2 n 1 =Rn, with n>1, lead to similar late-time self y acceleration, with behavior similar to a dark energy com- ponent with equation of state parameter with v Þ 0. Power-law solutions in the asymptotic future are then easily identified as v x†!p ˆ constant 2 n ‡ 2† w ˆÿ1 ‡ : (3) as H ! 0. eff 3 2n ‡ 1† n ‡ 1† Furthermore, if Therefore, such modifications can easily accommodate jH_ jˆjyj!1; then jvj!0ifxÞ 0: (9) current observational bounds [33,34] on the equation of state parameter ÿ1:45

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FIG. 1 (color online). Two phase portraits for the modified gravity model proposed in [7]. The left portrait is in the coordinates x; v†, for which an attractor at constant v ˆ p corresponds to a power-law solution with a t†/tp. The right portrait is for the same theory in the H;_ H† plane, with the unstable de Sitter solution at (0, 1). study of the asymptotically late-time behavior of Eq. (10). body of the paper and Appendix B consists of a proof of the In order to have a constant v0 as a solution, we need stability of the vacuum solutions under the addition of matter. 2 2v0 ÿ 5v0 ‡ 2 ˆ 0; (11) which has two real solutions: v0 ˆ 1=2, 2, as expected. II. A GENERAL NEW GRAVITATIONAL ACTION The singularity at p ˆ 1=2 occurs because the Ricci We now generalize the action of [7] to include other scalar vanishes for this particular power. One might worry curvature invariants. There are, of course, any number of that this is problematic for describing the radiation- terms that we could consider. We have chosen to consider dominated phase in standard cosmology, since nucleosyn- those invariants of lowest mass dimension that are also thesis occurs during this epoch and provides a particularly parity-conserving strong constraint on deviations from the standard   Friedmann equation at that time. However, as we shall P  RR ;Q R R : (12) see later, this is not a problem when matter sources are included explicitly, since even during radiation domination Since we are interested in adding terms to the action that the Ricci scalar does not vanish exactly, but rather, has a explicitly forbid flat space as a solution, we will, in a small contribution from nonrelativistic matter. similar way as in [7], consider inverse powers of the above Nevertheless, the singularity is a new feature that was invariants. not found in the Einstein-frame analysis [7]. This is pre- It is likely that such terms introduce ghost degrees of sumably because R ˆ 0 is a singular point of the confor- freedom. We shall not address this problem here, since it is mal transformation used to reach the Einstein frame. We beyond the scope of this paper. Rather, if ghosts arise we shall see similar singularities for some of the more general shall require that some as yet unknown mechanism (for actions we consider in this paper. example, extra-dimensional effects) cut off the theory in Another way to visualize the solutions to our models is such a way that the associated instabilities do not appear on to use a more traditional phase portrait in the H;_ H† plane. cosmological time scales [37] (see [14] for an example of a The outline of this paper is as follows. In the next section concrete model where ghosts are brought under control by we shall introduce the general class of actions we are higher-derivative terms). We therefore consider actions of interested in. In section III we analyze the vacuum equa- the form tions, describing the singularity and attractor structure in Z  Z  4 p 4 p detail before moving on to some simple special cases. In S ˆ d x ÿg‰R ‡ f R; P; Q†Š ‡ d x ÿgLM; section IV we introduce matter into the equations, demon- strating briefly that the late-time attractor solutions of the (13) system remain unchanged and setting up the formalism where f R; P; Q† is a general function describing devia- used in the appendices to establish stability of the system. tions from general relativity. In section V we summarize our findings and comment on It is convenient to define the status of these models as origins of cosmic accelera- tion. The paper contains two appendices. Appendix A @f @f @f fR  ;fP  ;fQ  ; (14) contains definitions of a number of functions used in the @R @P @Q

063513-3 SEAN M. CARROLL et al. PHYSICAL REVIEW D 71, 063513 (2005) in terms of which the equations of motion are 1 1 ᮀ R ÿ 2gR ÿ 2gf ‡ fRR ‡ 2fPRR  ‡ 2fQR R  ‡ g fR ÿrrfR ÿ 2r r ‰fPR  †Š ᮀ ‡ fPR†‡gr r fPR †ÿ4r r ‰fQR † Šˆ8GT: (15)

It is straightforward to check that these equations reduce As mentioned in the previous section, we will consider to those of the simple model of [7] for f R†ˆÿ4=R. inverse powers of our curvature invariants and, for sim- We would like to obtain constant curvature vacuum plicity, we will specialize to a class of actions with solutions to these field equations. To do so, we take the 4n‡2 trace of (15) and substitute Q ˆ R2=4 and P ˆ R2=6 f R; P; Q†ˆÿ 2 n ; (20) (which are identities satisfied by constant curvature aR ‡ bP ‡ cQ† space-times) into the resulting equation to obtain the alge- where n is a positive integer,  has dimensions of mass, braic equation: and a, b, and c are dimensionless constants.1 We will focus 2 on the case n ˆ 1 for most of the paper, because the 2fQ ‡ 3fP†R ‡ 6 fR ÿ 1†R ÿ 12f ˆ 0: (16) analysis is less involved for that case. For general n the Solving this equation for the Ricci scalar yields the con- qualitative features of the system are as for n ˆ 1 and we stant curvature vacuum solutions. discuss the quantitative differences in our conclusions. Evidently, actions of the form (13) generically admit a maximally-symmetric solution: R ˆ a nonzero constant. A. Distinguished Points of the Action (and Equations) However, an equally generic feature of such models is that this de Sitter solution is unstable. In the CDTT model the In general, the analogue of the Friedmann equation may instability is to an accelerating power-law attractor. This is be written in the following convenient form a possibility that we will also see in many of the more A ‡ BH ˆ M; (21) general models under consideration here. C Before we leave this section, note that, in analyzing the _ _ _ cosmology of these general models, it is useful to have at where A ˆ A H; H†, B ˆ B H; H†, and C ˆ C H; H† hand the following expressions, which hold in a flat FRW arise from the gravitational part of the action and M ˆ background M a† describes the possible inclusion of matter.   It is also convenient to write this schematically in terms a_ 2 a of our variables of the previous section as R ˆ 6 ‡ ˆ 6 H_ ‡ 2H2† (17) a2 a dv x6f v†‡6v2g v†   x ˆ 6 ; (22) a_ 4 a2 a_ 2 a dx 2 vh v† P ˆ 12 ‡ ‡ a4 a2 a2 a where f v†, g v†, and h v† are 6th, 4th, and 2nd degree ˆ 12‰ H_ ‡ H2†2 ‡ H4 ‡ H2 H_ ‡ H2†Š (18) polynomials, respectively, in the variable v, whose explicit form is given in Appendix A. We are often interested in a   particular subset of the phase space, the region where v> a_ 4 a2 _ 2 2 4 1=2. This is because from nucleosynthesis to the matter- Q ˆ 12 4 ‡ 2 ˆ 12‰ H ‡ H † ‡ H Š: (19) a a dominated epoch, we expect Einstein’s equations to pro- We have provided these both in terms of the scale factor vide a good approximation to the dynamics, and therefore, a t† and in terms of the Hubble parameter H t†ˆ when matter becomes subdominant we should have 1=2 < a_ t†=a t†, since they will be separately important in this v<2=3. paper. There are three types of special points in the phase-space plots of these equations: III. VACUUM SOLUTIONS (1) Singular Points of the Friedmann Equation—In our introduction we reconsidered the model of [7] and In this section we study cosmological solutions to the discovered a singular point of both the action and field equations (15) in the absence of matter sources. the equations of motion, corresponding to a power- Physically, this is important because we are hoping to find law evolution with exponent p ˆ 1=2. In this par- novel cosmological consequences arising purely from the gravitational sector of the theory. Mathematically, this 1Another potentially interesting possibility is a correction of provides us with valuable insight into the structure of the 4R form f R; P; Q†ˆ P . This term has the same mass dimension equations, which take a significantly simplified form as the 4=R term of CDTT. However, it turns out that this model wherein the Hubble parameter may be treated as the inde- does not possess any accelerating attractors. Specifically, the pendent variable. scale factor asymptotically approaches a t†/t0:37.

063513-4 COSMOLOGY OF GENERALIZED MODIFIED GRAVITY MODELS PHYSICAL REVIEW D 71, 063513 (2005) ticular case the singularity occurred because R  0 constant (distinct from our previously mentioned for p ˆ 1=2. In our more general models, analogous singular values). Taking an asymptotic limit of the singularities occur whenever the denominator of the equations of motion yields C Friedmann equation blows up; i.e., at zeros of , 2 2 2 4 where C is defined by (21). For the flat cosmological v‰ 288a ‡ 18b ‡ 8c ‡ 144ab ‡ 96ac ‡ 24bc†v ansatz this occurs when ÿ 1512a2 ‡ 90b2 ‡ 36c2 ‡ 738ab ‡ 468ac   H_ 2 3 2 2 2 H_ ‡ H2 ˆÿ ; (23) ‡ 114bc†v ‡ 1836a ‡ 123b ‡ 62c ‡ 951ab 4 H ‡ 678ac ‡ 175bc†v2 ÿ 846a2 ‡ 69b2 ‡ 44c2 where we have defined ‡ 489ab ‡ 414ac ‡ 113bc†v ‡ 135a2 ‡ 15b2 12a ‡ 4b ‡ 4c  : (24) 2 12a ‡ 3b ‡ 2c ‡ 15c ‡ 90ab ‡ 90ac ‡ 30bcŠˆ0; (30) Recall that a, b, and c are parameters in the with solutions Lagrangian (20). In our variables of the previous  s section this becomes 20 ÿ 3 120  s1;2 ˆ 1  1 ÿ ; 1 p 8 20 ÿ 3 †2 (31) v ˆ p  1  1 ÿ †; (25) 1;2 2 s3 ˆ p1; each zero having multiplicity 3. These singularities only exist if p are real, i.e. 1;2 s ˆ p ; (32)  1 and therefore there are many invariants that 4 2 do not admit this type of singularity. In the simple case of [7], corresponding to b ˆ c ˆ 0, note that s5 ˆ 0; (33) we have ˆ 1 and recover pc ˆ 1=2 as expected. and s >s . If  1 2 1 It is clear that only s1 and s2 can be late-time power- p1  1=2;p2  1=2: (26) law solutions. The other ones represent singular- _ ities; v ˆ 0 implies that y ˆ H !1, whereas s3 (2) Singular Points at which H !1—Points at which and s4 are the singular points we discussed earlier. H !1occur at zeros of B and, in our variables of the previous section, correspond to B. Summary of Possibilities

dv Here we summarize the different vacuum possibilities. It !‡1 (27) is useful to define the following two constants dx p at finite x and v. We denote the zeros by v x†ˆ 280 ‡ 60 3 1 ˆ p  1:78 (34) v1;v2, where the vi are (in general complex) con- 111 ‡ 60 3 stants constructed from a, b, and c. p If g v† Þ 0 one singular point is at x ˆ 0. 280 ÿ 60 3 2 ˆ p  24:9: (35) Otherwise the singularities occur at solutions of 111 ÿ 60 3 vh v†ˆ0, i.e. v‰ 108a2 ‡ 51ab ‡ 7bc ‡ 30ac ‡ 2c2 ‡ 6b2†v2 (1) <1. In this case vi are complex, whereas pi and ÿ 63ab ‡ 6c2 ‡ 9b2 ‡ 108a2 ‡ 15bc ‡ 54ac†v si are real. It is straightforward to show that s2 > p2 > 1=2 and s2 > 1. ‡ 18ab ‡ 3c2 ‡ 6bc ‡ 27a2 ‡ 18ac ‡ 3b2Šˆ0; (a) 0 < <1. In this case 1=2 >s1 >p1 > 0 (28) and 1 >p2 > 1=2. Solutions close to p2 are repelled from it, whereas s1 is an attractor. which are given by This leads to decelerating late-time behavior. s   (b) 0 > >ÿ160=9. Here s1 1. (29) (c) <ÿ160=9. In this case p1 1. (3) Late-time Stable Points.—Finally, we look for late- (2) >1. In this case vi are real, whereas pi are time power-law attractors. This happens if v x†! complex.

063513-5 SEAN M. CARROLL et al. PHYSICAL REVIEW D 71, 063513 (2005)  (a) 1 < <4=3. Here v1 < 1=2 and 1=2 1. Both s1 and s2 Let us begin by dealing only with actions containing are attractors, but s1 describes a decelerating  modifications involving P  RR . Our prototype is to phase. consider f P†ˆÿm6=P, with m a parameter with dimen- (b) 4=3 < <1. Here 1=2 sions of mass. ˆ ˆ 1. Both (x 0, v 0) and s1;2 are attractors, Using (16) we can see that there is a constant curvature with the separatrix being at v1;2. vacuum solution to this action given by (c) 1 < <4. In this case 1=2 1. Again, both (x ˆ 0, v ˆ 0) and v2 Rconst ˆ 16† m : (38) are attractors and the separatrix is at v1. However, we would like to investigate other cosmological There are no real solutions for s1;2. solutions and analyze their stability. (d) 4 < <2. Here there are no real late-time From (15), with the flat cosmological ansatz, the ana- attractors since the si are complex. We also logue of the Friedmann equation becomes have v2 < 0 and v1 > 1. Solutions either evolve to (x ˆ 0, v ˆ 0)ortov ˆ‡1, m6 with separatrix at v . 3H2 ÿ ‰H_ 4 ‡ 11H2H_ 3 ‡ 2HH_ 2H 1 8 3H4 ‡ 3H2H_ ‡ H_ 2†3 (e) >2. In this case v2 < 0, v1 > 1, and s1 < ‡ 4 _ 2 ‡ 6 _ ‡ 3 _  ‡ 8 ‡ 5  Šˆ s2 < 0. Evolution is to a decelerating 33H H 30H H 6H H H 6H 4H H 0: attractor. (39) (3) ˆ 1. This yields a promising class of solutions. We have We analyze this equation using the same technique, with the same definitions, as in the example of the previous 1 p ˆ p ˆ v ˆ v ˆ s ˆ : (36) section. The relevant equation is 1 2 1 2 1 2 dv 1 All singularities occur as v x†!1=2. However, 2 x ˆ 2v ‡ 6 2 from nucleosynthesis onwards we never encounter dx 2m v 2v ÿ 3v ‡ 1† this point. It is simple to show that solutions evolve ‰ÿx6 24 ÿ 216v ‡ 864v2 ÿ 1944v3 ‡ 2592v4 to a late-time power-law attractor describing an ÿ 1944v5 ‡ 648v6†‡m6 v2 ÿ 11v3 ‡ 33v4 accelerating phase, with ÿ 30v5 ‡ 6v6†Š: (40) 15 s ˆ ˆ 3:75; (37) 2 4 The solution to this equation is displayed graphically in which is otherwise independent of a, b, c. Fig. 3. We identify four fixed points of the system; two The results of this section may be summarized in Fig. 2, attractors at v ’ 0:77 and v ’ 3:22 and two repellers at v ’ showing the values of the various distinguished points as 0:5 and v ˆ 1. Clearly, in order to obtain a late-time is varied. accelerating solution (p>1), it is necessary to give accel- erating initial conditions (a> 0), otherwise the system is in the basin of attraction of the nonaccelerating attractor at p ’ 0:77. 5 The exact exponents of the two late-time attractors of the system are obtained by studying the asymptotic behavior of

4 s2 (40). Substituting in a power-law ansatz and taking the late-time limit we find that, in order to have a constant v ˆ v0 as a solution, the exponent must satisfy 3 v2 4 3 2 6v0 ÿ 30v0 ‡ 41v0 ÿ 23v0 ‡ 5 ˆ 0: (41)

2 This equation has two real solutions (andp two complex ones). Thep real solutions are: v0 ˆ 2 ÿ 6=2 ’ 0:77 and v ˆ 2 ‡ 6=2 ’ 3:22. p 0 1 2 s1 Even the nonaccelerating attractor is of some interest in

v1 this model. In order for structure to form in the Universe, p1 0 there must be a sufficiently long epoch of matter domina- 0 0.5 1 1.5 2 2.5 2=3 α tion, for which a t†/t . As matter redshifts away, how- ever, since the Universe is decelerating, we expect the FIG. 2. The values of the various distinguished points as is Universe to approach the attractor at p ’ 0:77. This corre- varied. sponds to an effective equation of state weff ’ÿ0:13; i.e.,

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FIG. 3 (color online). Two phase portraits for the f R; P; Q†ˆÿm6=P modification. The left portrait is in the coordinates x; v†, for p which an attractor at constant v ˆ p corresponds to a power-law solution with a t†/t . Thep right portrait is for the same theory in the H;_ H† plane. There are two late-time power-law attractors corresponding to p ˆ 4  6†=2. The ( ÿ ) branch, represented by the solid lines, is nonaccelerating, while the ( ‡ ) branch, represented by the dash-dotted line, is accelerating. negative pressure, although not negative enough to provide sions of mass, and the analysis follows much the same as in a good fit to the supernova observations. the previous subsection, albeit with different results. Again, (16) demonstrates that there is a constant curva- D. Inverse Powers of Q  RR ture vacuum solution to this action given by

Now let us move on to actions containing modifications Q† 1=3 2  Rconst ˆ 24† M : (42) involving only Q ˆ R R . Our prototype example is f Q†ˆÿM6=Q, with M another parameter with dimen- What about other possible cosmological solutions? From (15), with the flat cosmological ansatz, the analogue of the Friedmann equation becomes 2.5 M6 3H2 ÿ 24‰H_ 2 ‡ 2H2H_ ‡ 2H4Š3 ‰8H8 ‡ 36H6H_ ‡ 54H4H_ 2 ‡ 20H2H_ 3 ‡ 3H_ 4 ‡ 4H5H 2 ‡ 12H3H_ H ‡6HH_ 2H Šˆ0: (43)

Employing our phase-space technique once more, in the 1.5 variables best-suited for analyzing power-law behavior, the relevant equation is v dv 1 x ˆ 2v ‡ 1 dx 2M6v‰2v2 ÿ 6v ‡ 3Š ‰ÿx6 576v6 ÿ 1728v5 ‡ 2592v4 ÿ 2304v3 ‡ 1296v2 ÿ 432v ‡ 72†‡M6 8v6 ÿ 36v5 0.5 ‡ 54v4 ÿ 20v3 ‡ 3v2†Š: (44)

It is clear from this that our action consisting of 0 f R; P; Q†ˆÿM6=Q does not admit any late-time -0.04-0.03 -0.02 -0.01 0 power-law attractors (see Fig. 4). x This is consistent with a study of the late-time asymp- FIG. 4 (color online). Phase portrait for the f R; P; Q†ˆ totics of (44). As in our previous analysis, late-time power- ÿM6=Q modification in the coordinates x; v†, for which an law solutions correspond to real solutions to the equation attractor at constant v ˆ p corresponds to a power-law solution 4 3 2 with a t†/tp. 8v0 ÿ 36v0 ‡ 62v0 ÿ 44v0 ‡ 15 ˆ 0: (45)

063513-7 SEAN M. CARROLL et al. PHYSICAL REVIEW D 71, 063513 (2005) However, no such real solutions exist, confirming our @F F ˆ phase-space analysis. v @v v2g ‡ 2vg ÿ 2xhs ÿ 2xvh s ˆ 6 v v IV. INCLUDING MATTER d3 (53) v2g ÿ 2xvhs We now show that the late-time behavior of our vacuum ÿ 36 d ; solutions remains unaltered upon the inclusion of matter. d4 v We begin by rewriting the equation of motion in a more @F 26xvh convenient form. Let  be the tensor defined by the left- Fs ˆ ˆÿ  @s d3 hand side of (15). Then, the generalized Friedmann equa- tion takes the form and a prime denotes differentiation with respect to x. Seeking late-time power-law behavior, we take the lim- its x ! 0 and s ! 0 in (51). This yields the condition 00 ˆ 8G; (46) g v0†ˆ0, which is solved by the same power-law fixed points as those obtained in vacuum. This is as one might where  is the energy density of a perfect fluid with expect, since matter redshifts away in the asymptotic fu- equation of state p ˆ w. Now, since x ˆÿH, y ˆÿx_ , ture. However, the above description proves useful for  / aÿ3 1‡w† d ln =0† ˆ ‡ w† v and , we have dx 3 1 x . dealing with the issue of stability. Since this is somewhat Combining this relation with (46) yields the equation of technical, we relegate the details to Appendix B and motion merely assert here that these fixed points do remain stable in the presence of matter sources. d00 x ˆ 3 1 ‡ w†v00: (47) dx V. COMMENTS AND CONCLUSIONS Thus far, we have not imposed any dynamics. In the extreme low curvature regime, our only tests of Specializing to a theory with f R; P; Q†ˆ6= aR2 ‡ general relativity are cosmological. The discovery of new bP ‡ cQ† gives phenomena at these scales may point to new matter sources, but alternatively may hint at hitherto undetected 1 modifications of gravity.  ˆ F x; v; s†; (48) 00 24x4 The acceleration of the Universe provides a particular challenge to modifications of gravity. Unlike the known with perturbative corrections to the Einstein-Hilbert action aris- ing from string theory, late-time acceleration requires mod- ifications that become important at extremely low energies, v2g v†ÿ2xvh v†s 6 6 so low that only today, at the largest scales in the Universe, F ˆ 72x ‡  3 ; (49) ‰d v†Š is the resulting curvature low enough to lead to measurable deviations from general relativity. dv where s ˆ dx . The explicit forms of the functions h v†, These considerations led some of the current authors and g v†, and d v†, defined in Appendix A, are not needed in others to consider new terms in the action for gravity that this section. Substituting (48) and (49) into (47) gives the consist of inverse powers of the Ricci scalar. It is easy to equation governing the dynamics of our theory in the show that such an approach introduces a de Sitter solution. presence of matter: However, this solution is unstable to a late-time accelerat- ing power-law attractor. For appropriate choices of pa- dF rameters, this theory is a candidate to explain cosmic x ˆ v†F; (50) dx acceleration without the need for dark energy, although the simplest such theories are in conflict with solar system where v†‰3 1 ‡ w†v ‡ 4Š. Using the chain rule this tests. For Lagrangians that are functions of only the Ricci becomes scalar, there exists a map to an Einstein frame, in which the new degrees of freedom are represented by a scalar field. As a result, such modified gravity theories share many x F ‡ F s ‡ F s0†ˆ v†F; (51) x v s features in common with some dark energy models. In this paper we have introduced a much more general where class of modifications to the Einstein-Hilbert action, becoming relevant at extremely low curvatures. @F vhs F ˆ ˆ 432x5 ÿ 26 ; (52) Specifically, we have considered inverse powers of arbi- x @x d3 trary linear combinations of the curvature invariants R2,

063513-8 COSMOLOGY OF GENERALIZED MODIFIED GRAVITY MODELS PHYSICAL REVIEW D 71, 063513 (2005)   P  RR , and Q  R R . Such modifications For n ˆ 1, all the above expressions agree with the Q† are not simply equivalent to Einstein gravity plus scalar values found earlier. It is interesting to note that v1;2 are matter sources. imaginary for n ˆ 1, but are real for all n>1. We have performed a general analysis of the late-time Of course, much remains to be done. We have not evolution of cosmological solutions to these theories. addressed solar system tests of these theories since this is Many of the theories exhibit late-time attractors of the a complicated analysis that is beyond the scope of our form a t†/tp, with p some constant power. Indeed, there current paper. We have not focused on detailed compari- are often multiple such attractors. For a large class of sons between our models and the supernova data and it theories there exists at least one attractor satisfying p> is possible that there are specific signatures of this new 1, corresponding to cosmic acceleration. physics in such data. For example, the modified Friedmann The detailed structure of cosmological solutions to these equations arising from the theories presented here should theories turns out to be quite rich and varied, depending on directly provide information about the jerk parameter. We the dimensionless parameters entering the particular linear intend to consider such effects, not only in the present combination. Two distinct types of singularities may exist, models, but in those proposed by us and other authors, in as well as the late-time power-law attractors. We have a separate paper focused on the connections of these mod- identified all those theories for which the late-time behav- els with observations. In this context, as with the case of the ior is consistent with the observed acceleration of the simple modifications introduced in [7], it may also be Universe, providing a whole new class of theories—gen- interesting to study more complicated functions of the eralized modified gravity theories—which are alternatives curvature invariants we have considered. to dark energy. The results we have found for the modification f ˆ ACKNOWLEDGMENTS 6= aR2 ‡ bP ‡ cQ† may be generalized to modifica- tions M. T. thanks Gia Dvali, Renata Kallosh, Burt Ovrut, and Richard Woodard for helpful discussions. The work of 4n‡2 S. M. C. is supported in part by the Department of Energy f ˆ ; (54) aR2 ‡ bP ‡ cQ†n (DOE), the NSF, and the Packard Foundation. V.D. is supported in part by the NSF and the DOE. A. D. F., using exactly the same calculational techniques. D. A. E., and M. T. are supported in part by the NSF under In general there are power-law attractors with the fol- Grant No. PHY-0094122. D. A. E. is also supported in part lowing exponents, whenever they are real by funds from Syracuse University. M. S. T. is supported in p part by the DOE (at Chicago), the NASA (at Fermilab), and 8n2 ‡ 10n ‡ 2 ÿ 3  vgen ˆ ; (55) the NSF (at Chicago). 1;2 4 n ‡ 1† where APPENDIX A: SOME DEFINITIONS

ˆ 9n2 2 ÿ 80n3 ‡ 116n2 ‡ 40n ‡ 4† ‡ 64n4 In Section III we defined ‡ 160n3 ‡ 132n2 ‡ 40n ‡ 4: (56) 4 3a ‡ b ‡ c† ˆ ; (A1) Clearly, as n !1, the smaller attractor tends to 0, whereas the larger one increases linearly as 4n. where Two special cases are ˆ 12a ‡ 3b ‡ 2c: (A2) m4n‡2 M4n‡2 f ˆÿ n ;fˆÿ n ; (57) P Q We should distinguish between two cases: for which the power-law attractors, v1;2 are 1. ˆ 0 P† v1;2 In this case diverges, but we may define p 12n2 ‡ 9n ‡ 3  144n2 ‡ 120n3 ÿ 15n2 ÿ 30n ÿ 3 ˆ g v†ˆ 3a ‡ b ‡ c† 282a ‡ 69b ‡ 44c†v 2 3 ‡ 3n† ‡ 15 3a ‡ b ‡ c†2 (A3) (58) 2 p h v†ˆ3 3a ‡ b ‡ c† (A4) 4n2 ‡ 2n ‡ 1  16n4 ÿ 16n2 ÿ 10n ÿ 1 v Q† ˆ : (59) 1;2 2n ‡ 2 d v†ˆ3a ‡ b ‡ c: (A5)

063513-9 SEAN M. CARROLL et al. PHYSICAL REVIEW D 71, 063513 (2005)

Note that both , and 3a ‡ b ‡ c, cannot vanish simulta- 1E-6 neously, as this causes the Friedmann equation to be sin- gular for all values of x and v.

2. Þ 0 5E-7 In this case we have

2 g v†ˆ2 v ÿ s2† v ÿ s1† v ÿ p2† v ÿ p1† (A6)

2 v 0 h v†ˆ 1 ÿ † v ÿ v2† v ÿ v1† (A7) 4 -0.1 -0.05 0 0.05 0.1 s d v†ˆ v ÿ p2† v ÿ p1†; (A8) ˆ where either v1;2 or p1;2 are complex unless 1,in -5E-7 which case, v1;2 ˆ p1;2 ˆ s1 ˆ 1=2 and s2 ˆ 3:75. Also s1;2 in general may be complex. As we saw in Section III, si, vi, and pi are all functions of only.

-1E-6 APPENDIX B: STABILITY OF FIXED POINTS IN THE PRESENCE OF MATTER FIG. 5 (color online). Phase plot for the linearized equations Let us rewrite (51) as a system of first-order ordinary close to the power-law solution in the coordinates x; v†, for differential equations which an attractor at constant v ˆ p corresponds to a power-law solution with a t†/tp. x0 ˆ 1 (B1)

v†F F F Now, introducing equilibrium coordinates s0 ˆ ÿ v s ÿ x (B2) xFs Fs Fs u1 ˆ x ÿ x0 (B7)

v0 ˆ s: (B3) u2 ˆ s (B8) Treating x as a dependent variable makes the system autonomous, facilitating a phase-space analysis. u3 ˆ v ÿ v0; (B9) To obtain the phase portrait of the system we define a we finally obtain the linearized equations vector field by W~ T  x0;s0;v0† and plot this in the vicinity 0 of the fixed point. In Fig. 5 we show a 2D slice of this phase u1 ˆ 1 (B10) portrait by choosing a section at constant x. Note that at the 0† fixed point, W~ ˆ 1; 0; 0†. 0 0 ‡ 0 ÿ 1 00 u2 ˆ u2 ÿ 2 u3 (B11) To analyze the stability of the system, we first linearize x0 x0 the system about the fixed point v ˆ v0, s ˆ 0, and x ˆ 0 x0  1 (all expressions evaluated at these values of v, s, u3 ˆ u2; (B12) 0† and x will carry the superscript ). v0gv;0 with 0 ˆ . We write the linearized system of equations as 2h0 X Notice that the first equation has decoupled from the 0† 0† 0† Wi ˆ ‰MŠi  ÿ  †‡Wi ; (B4) other two. Hence, it suffices to study the behavior of the subsystem u2;u3†. We will use standard results from the matrix M being defined by the theory of dynamical systems to do so (see [38]).   In the vicinity of the fixed point, the behavior of the 0† 0† @Wi system can be classified by the eigenvalues of the subma- ‰MŠi ˆ ; (B5) @ trix Mij, with i; j ˆ 2, 3, with characteristic equation and 2 ÿ M 0† ÿ M 0† ˆ 0: (B13) 0 1 2s 2v x @ A Since we are only interested in values of v and w (the ~  s : (B6) equation of state parameter) which satisfy v>1=2 and v ÿ1

063513-10 COSMOLOGY OF GENERALIZED MODIFIED GRAVITY MODELS PHYSICAL REVIEW D 71, 063513 (2005)

2 sensible choice, and [42,43] for how one may be tricked 0 ‡ 0 ÿ 1† ÿ 4 00 < 0; (B15) into inferring values outside this range when considering gravitational theories other than General Relativity), we which holds for all x0  1. Applying the last condition to the case ˆ 1 and w ˆ 0 have that 0 > 1. Furthermore if we choose v0 ˆ s2 (see Section III), the value of the bigger power law, typically (dust), we have that (see Appendix A) 0† 0† 0 > 0. This implies that both M2s and M2v are negative, 4 0 ˆ s2 s2 ÿ 0:5†ˆ16:25 (B16) so that s2 is either a stable node or a stable spiraling helix. 3 The latter case is tantamount to stability. and We have a stable spiralizing helix if 0 ˆ 3s2 ‡ 4 ˆ 15:25: (B17) M 0††2 ‡ 4M 0† < 0: (B14) 2s 2v For these values of  and , (B15) is easily satisfied, so that If not, we get a stable node. Relation (B14) leads to the the power-law solution s2 ˆ 3:75 is a stable spiraling condition helix.

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