Is Asymptotically Weyl-Invariant Gravity Viable?
IIs Asymptotically Weyl-Invariant Gravity Viable?
Daniel Coumbe
The Niels Bohr Institute, Copenhagen UniversityBlegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
Abstract
We explore the cosmological viability of a theory of gravity defined by the Lagrangian f ( R ) = R n ( R ) in the Palatini formalism, where n ( R ) is a dimensionless function of the Palatini scalar curvature R thatinterpolates between general relativity when n ( R ) = 1 and a locally scale-invariant and superficiallyrenormalizable theory when n ( R ) = 2 . We refer to this model as asymptotically Weyl-invariant gravity(AWIG).We analyse perhaps the simplest possible implementation of AWIG. A phase space analysis yieldsthree fixed points with effective equation of states corresponding to de Sitter, radiation and matter-dominated phases. An analysis of the deceleration parameter suggests our model is consistent withan early and late period of accelerated cosmic expansion, with an intermediate period of deceleratedexpansion. We show that the model contains no obvious curvature singularities. Therefore, AWIGappears to be cosmologically viable, at least for the simple implementation explored.PACS numbers: 04.60.-m, 04.60.Bc The general theory of relativity is currently our best description of gravity. One reason for this is itsexplanatory power: assuming little more than a single symmetry principle general relativity can explain atruly astonishing range of experimental phenomena [1].However, it is at best incomplete. It is often said that general relativity breaks down at high energies or small distances. Yet, it is more accurate to say high energies and small distances. This is an importantdistinction since it highlights the regime in which we must modify general relativity, namely for largeenergy densities, or equivalently for large spacetime curvatures. For example, general relativity predictsits own breakdown at curvature singularities, where scalar measures of curvature grow without bound.Furthermore, general relativity is known to become fundamentally incompatible with quantum field theoryat high curvature scales, a failure known as its non-renormalizability [2, 3]. Theoretical arguments aloneare enough to tell us that general relativity must be modified at high curvature scales.Experimental data also indicates that general relativity must either be augmented or replaced alto-gether if it is to agree with observation [4]. For example, general relativity by itself is unable to explainthe early phase of accelerated cosmic expansion, as evidenced by myriad high-precision measurements [4],and must be supplemented with unobserved exotic energy sources and scalar fields [5, 6]. Although thistop-down approach, as exemplified by the Λ CDM model, is currently our best description of observedcosmological dynamics [4], its ad hoc construction has driven attempts to replace general relativity fromthe bottom-up.
E-mail: [email protected] a r X i v : . [ g r- q c ] M a r inding a viable replacement of general relativity is challenging. Such a theory must at the very leastbe (i) equivalent to general relativity in the low-curvature limit, (ii) renormalizable in the high-curvaturelimit, (iii) unitary, (iv) stable, (v) contain no curvature singularities, (vi) consistent with observation.One attempt is that of higher-order gravity, in which the Lagrangian includes terms quadratic in thecurvature tensor. Although this approach is perturbatively renormalizable, and hence satisfies criterion(ii), such higher-order theories are not typically unitary or stable, thus failing to satisfy criteria (iii) and(iv). The only higher-order theories that are unitary and stable are so-called f ( R ) theories, in which theLagrangian is an arbitrary function of the Ricci scalar only [7].There are three types of f ( R ) theory: metric, Palatini and metric-affine variations [7]. Metric f ( R ) gravity assumes that the affine connection uniquely depends on the metric via the Levi-Civita connection,as in standard general relativity. The Palatini formalism generalises the metric formalism by relaxingthe assumption that the connection must depend on the metric. The metric-affine formalism is the mostgeneral approach since it even drops the implicit assumption that the matter action is independent of theconnection.Particular metric f ( R ) models have been shown to conflict with solar system tests [8], give an incorrectNewtonian limit [9], contradict observed cosmological dynamics [10, 11], be unable to satisfy big bangnucleosynthesis constraints [12] and contain fatal Ricci scalar instabilities [13]. Thus, metric f ( R ) theoriesdo not typically satisfy criteria (i), (iv) or (vi). As for the metric-affine formalism, it is not even a metrictheory in the usual sense, meaning diffeomorphism invariance is likely broken [7]. Thus, metric-affinetheories do not even seem to satisfy criterion (i). However, it has been shown that the Palatini variationis immune to any such Ricci scalar instability [14]. Palatini formulations also appear to pass solar systemtests and reproduce the correct Newtonian limit [15]. Remarkably, a Palatini action that is linear in thescalar curvature is identical to regular general relativity [7]. However, this equivalence does not hold forhigher-order theories [16, 7]. In particular, a Palatini action that is purely quadratic in the scalar curvatureis identical to normal general relativity plus a non-zero cosmological constant [17].In Ref. [18] we proposed the theory of asymptotically Weyl-invariant gravity (AWIG) within the Pala-tini formalism. By construction AWIG satisfies criteria (i)-(iv). The present work aims to test whetherthis theory also satisfies criteria (v) and (vi), and hence to determine if it may be a viable replacement ofgeneral relativity.In addition to satisfying criteria (i)-(iv), a major motivation for developing AWIG was finding a theorywith the symmetry of local scale invariance. The need for local scale invariance can be seen by recognisingthat all length measurements are local comparisons. For example, to measure the length of a rod requiresbringing it together with some standard unit of length, say a metre stick, at the same point in space andtime. In this way the local comparison yields a dimensionless ratio, for example, the rod might be longerthan the metre stick by a factor of two. Repeating this comparison at a different spacetime point mustyield the same result, even if the metric at this new point were rescaled by an arbitrary factor Ω ( x ) .This is because both the rod and metre stick would be equally rescaled, yielding the same dimensionlessratio. Such a direct comparison cannot be made for two rods with a non-zero space-like or time-likeseparation [21, 22]. Therefore, it has been argued that the laws of nature must be formulated in such away as to be invariant under local rescalings of the metric tensor g µν → Ω ( x ) g µν , or equivalently undera local change of units. Moreover, since scale-invariant theories of gravity are gauge theories [23, 24],unification with the other three fundamental interactions, which have all been successfully formulated aslocal gauge theories, becomes tractable. The theory analysed in this work is invariant with respect to localchanges of scale in the high-curvature limit.It is important to establish the standard against which we will judge whether the presented theoryis viable. Criterion (v) will be deemed to be satisfied if at least two different curvature invariants can In the low-curvature limit AWIG yields f ( R ) = R , which is identical to general relativity [7]. AWIG is at least superficiallyrenormalizable because the coupling constant of the theory becomes dimensionless in the high curvature limit, as shown in section 2.AWIG is likely to be unitary because states of negative norm (ghosts) that cause unitarity violations do not appear in f ( R ) theories [7, 19].AWIG appears stable since Ostragadsky’s instability is evaded by any f ( R ) theory [20], and the Dolgov-Kawasaki instability can notoccur in Palatini f ( R ) gravity [7] (see Ref. [18] for more details on the construction of AWIG).
2e shown to be divergence-free. To satisfy criterion (vi) we make the maximal demand that the theoryreproduces all four observed phases of cosmological evolution in the correct order [25], namely an earlyperiod of accelerated expansion, followed by radiation and matter-dominated phases, and finally a lateperiod of accelerated expansion [26].This paper is organised as follows. In section 2 we define the model of AWIG, including a detailedexploration of the dimensionless exponent n ( R ) . In section 3 we detail the methodology that will be usedto test the viability of our model. Results are presented in section 4 followed by a concluding discussionin section 5. The class of theories to which our model belongs is defined by the action S = 12 κ (cid:90) f ( R ) √− gd x, (1)where κ ≡ πG and G is the gravitational coupling. f ( R ) is an arbitrary function of the Palatini scalarcurvature R and g is the determinant of the metric tensor. Varying Eq. (1) with respect to the metricand taking the trace gives the field equations [7] f (cid:48) ( R ) R − f ( R ) = κT. (2)AWIG is defined by the specific case [18] f ( R ) = R n ( R ) , (3)where n ( R ) is a dimensionless function of R that interpolates between general relativity when n ( R ) = 1 and a locally scale-invariant and superficially renormalizable theory of gravity when n ( R ) = 2 . Bydefining n ( R ) in this way the Lagrangian density f ( R ) is purely a function of scalar curvature, and henceis guaranteed to be invariant under arbitrary differential coordinate transformations. In -dimensionalspacetime R n ( R ) has canonical mass dimension n ( R ) . Since √− g has mass dimension − , κ must havea mass dimension of n ( R ) − if Eq. (1) is to be dimensionless, which it must be since we are workingin units of (cid:126) = c = 1 . Thus, in the limit n ( R ) → the gravitational coupling becomes dimensionless,as demanded by scale-invariance. Superficially renormalizable field theories are those with dimensionlesscoupling constants [27].To complete the definition of this model we must specify the function n ( R ) . We begin by taking thefirst derivative of f ( R ) with respect to R , denoted by f (cid:48) ( R ) , finding f (cid:48) ( R ) = R n ( R ) − (cid:0) n ( R ) + R log ( R ) n (cid:48) ( R ) (cid:1) . (4)Substituting Eqs.(3) and (4) into Eq. (2) and rearranging yields n (cid:48) ( R ) = κT + R n ( R ) (2 − n ( R )) R n ( R )+1 log ( R ) . (5)We now use the fact that the symmetry of local scale invariance is signalled by the vanishing of thetraced energy tensor [28]. Thus, as n ( R ) → we must have T → . Applying the limits n ( R ) → and T → to Eq.(5) yields n (cid:48) ( R ) = 0 . Similarly, as n ( R ) → we must have κT → −R , and so Eq.(5) againyields n (cid:48) ( R ) = 0 . Therefore, the function we seek must satisfy the condition n (cid:48) ( R ) = 0 as n ( R ) → and n ( R ) → . If R = 1 when n ( R ) = 2 and κT = 0 then n (cid:48) ( R ) is undefined, since the numerator and denominator of Eq.(5) both equal zero.Likewise, if R = 0 when n ( R ) = 1 and κT = −R then n (cid:48) ( R ) is undefined. However, in the limiting cases R → and R → we have n (cid:48) ( R ) = 0 . n ( R ) at lower curvature scales. This is becausegeneral relativity agrees with experiment over a wide range of energy or curvature scales [29, 1], indicatingthat n ( R ) has at most a very weak dependence on R within the range of current experimental sensitivity.Similarly, the fact that in the high-curvature limit the theory becomes locally scale-invariant implies aconstant n ( R ) , since in this limit there can be no scale with respect to which n ( R ) can vary.We now proceed by assuming n ( R ) admits a series expansion in R of the form n ( R ∗ ) = ∞ (cid:88) m =0 c m R m ∗ , (6)where c m are dimensionless constants and R ∗ is defined by the dimensionless ratio R ∗ ≡ R / R , with R afinite constant of mass dimension two that represents the maximum value R can take. In this way, n ( R ∗ ) is a purely dimensionless function of the Palatini scalar curvature R . Truncating to a third-order functionwe have n ( R ∗ ) = c + c R ∗ + c R ∗ + c R ∗ . (7)Since the low-curvature limit corresponds to R ∗ → , the constraint n ( R ∗ →
0) = 1 immediately yields c = 1 . Similarly, since the high-curvature limit corresponds to R ∗ → , the constraint n ( R ∗ →
1) = 2 gives c + c + c = 2 , or equivalently c + c + c = 1 .The first derivative of n ( R ∗ ) with respect to R is n (cid:48) ( R ∗ ) = c R + 2 c R R + 3 c R R = c R R ∗ + 2 c R R ∗ + 3 c R R ∗ . (8)Since R ∗ ≡ R / R → in the low-curvature limit, Eq. (8) gives n (cid:48) ( R ∗ ) = c / R = 0 , which implies c = 0 since R is assumed to be finite. The high-curvature limit corresponds to R ∗ ≡ R / R → , andso Eq. (8) gives n (cid:48) ( R ∗ ) = c / R + 2 c / R + 3 c / R = 0 , which implies c + 3 c = 0 since c = 0 . Thepolynomial coefficients c and c can now be determined by solving the system of equations c + 3 c = 0 and c + c = 1 , with the result c = 3 , c = − . Therefore, n ( R ∗ ) = 1 + 3 R ∗ − R ∗ . (9)Eq. (9) is the lowest-order polynomial to satisfy our criteria, but there are potentially an infinitenumber of higher-order polynomial functions. Let n i ( R ∗ ) label this set of polynomial functions, where theorder of the polynomial is given by i + 1 . One can then generalise Eq. (9) to any higher-order using [30] n i ( R ∗ ) = 1 + R i +1 ∗ i (cid:88) j =0 (cid:18) i + jj (cid:19)(cid:18) i + 1 i − j (cid:19) ( −R ∗ ) j , i ∈ N . (10)The first thirteen functions generated by Eq. (10) are shown in Fig. 1 (left). The Lagrangian density inthis case is then f i ( R ) = R n i ( R ∗ ) = ( R R ∗ ) n i ( R ∗ ) . (11)For simplicity, we choose R to have the value of one when expressed in some particular unit of massdimension two. For example, one possibility is R = 1 m P , where m P is the Planck mass. The term R then only acts to set the dimensionality of f i ( R ) . The first thirteen functions f i ( R ) generated byapplying Eq. (10) to Eq. (11) are shown in Fig. 1 (middle), where we set R = 1 in some appropriate unit.Differentiating Eq. (11) with respect to R gives the set of first derivative functions f (cid:48) i ( R ) , with the first13 shown in Fig. 1 (right). It can be shown that first and second-order functions cannot produce the desired features [30]. .0 0.2 0.4 0.6 0.8 1.0 ℛ * n i ( ℛ * ) ℛ * f i ( ℛ ) ℛ * f i ' ( ℛ ) Figure 1: The first 13 exponents n i ( R ∗ ) (left), Lagrangian densities f i ( R ) (middle), and first derivativefunctions f (cid:48) i ( R ) (right) generated by Eq. (10) as a function of R ∗ .An important feature of Fig. 1 (right) is that the thirteenth function f (cid:48) ( R ) becomes negative forcertain values of R ∗ . A well-defined conformal transformation of the metric tensor ˜ g µν = f (cid:48) ( R ) g µν requiresthat f (cid:48) ( R ) > for all R . This condition is only satisfied if i ≤ . Thus, we can exclude Lagrangiandensities f i ( R ) with i ≥ . In this work, we shall focus on the simplest permitted Lagrangian density f ( R ) = R R ∗ − R ∗ . (12) In this section we detail the method used to test the cosmological viablility of the model defined byEq. (12). The methodology presented in this section follows the work of Refs. [25, 18].Since cosmological observations by the Planck satellite show that our universe is consistent with beingspatially flat at late times [4], we begin by assuming a flat Friedmann-Lemaıtre-Robertson-Walker (FLRW)metric ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) , (13)where a ( t ) is the scale factor of the universe, a function of cosmological time t . The evolution of aspatially homogenous and isotropic universe filled with a cosmological fluid composed of pressureless dustand radiation can be described by the modified Friedmann equation [7, 31] (cid:32) H + ˙ f (cid:48) ( R )2 f (cid:48) ( R ) (cid:33) = κ ( ρ m + 2 ρ r ) + f ( R )6 f (cid:48) ( R ) , (14)where the dot notation signifies a time derivative and H ≡ ˙ a/a is the Hubble parameter. ρ m and ρ r arethe energy density of matter and radiation, respectively, which satisfy the conservation conditions ˙ ρ m + 3 Hρ m = 0 , ˙ ρ r + 4 Hρ r = 0 . (15)Since the trace of the energy-momentum tensor for radiation is zero, we simply have T = − ρ m [25]. Byusing Eq. (14), combined with the conservation conditions of Eq. (15), we can express the time derivativeof the Palatini scalar curvature as [31, 25] ˙ R = − H ( f (cid:48) ( R ) R − f ( R )) f (cid:48)(cid:48) ( R ) R − f (cid:48) ( R ) . (16)Using Eq. (16) we can replace ˙ R in Eq. (14) to obtain [31, 7] H = (cid:115) κ ( ρ m + ρ r ) + f (cid:48) ( R ) R − f ( R )6 f (cid:48) ( R ) ξ , (17)5here ξ is defined by ξ = (cid:18) − f (cid:48)(cid:48) ( R ) ( f (cid:48) ( R ) R − f ( R )) f (cid:48) ( R ) ( f (cid:48)(cid:48) ( R ) R − f (cid:48) ( R )) (cid:19) . (18)If ρ r = 0 , it is possible to use T = − ρ m = ( f (cid:48) ( R ) R − f ( R )) /κ to obtain the simpler expression H = (cid:115) f ( R ) − f (cid:48) ( R ) R f (cid:48) ( R ) ξ . (19)In this work, we shall perform a detailed analysis of the phase space of the model defined by Eq. (12). Tofacilitate this analysis we establish an autonomous system of equations defined by the pair of dimensionlessvariables [25] y = f (cid:48) ( R ) R − f ( R )6 f (cid:48) ( R ) ξH , y = κρ r f (cid:48) ( R ) ξH . (20)Using Eqs. (20) and (17) it can be shown that ρ r can be expressed in terms of the variable y via ρ r = y − y (cid:18) ρ m + f (cid:48) ( R ) R κ − f ( R )2 κ (cid:19) . (21)The evolution of y and y as a function of the cosmic scale factor a are established by the differentialequations dy dN = y (cid:18) − y + y − f (cid:48) ( R ) R − f ( R )) f (cid:48)(cid:48) ( R ) R ( f (cid:48) ( R ) R − f ( R )) ( f (cid:48)(cid:48) ( R ) R − f (cid:48) ( R )) (1 − y ) (cid:19) (22)and dy dN = y (cid:18) − − y + y + 3 ( f (cid:48) ( R ) R − f ( R )) f (cid:48)(cid:48) ( R ) R ( f (cid:48) ( R ) R − f ( R )) ( f (cid:48)(cid:48) ( R ) R − f (cid:48) ( R )) y (cid:19) , (23)where N ≡ ln(a) . The fixed points of this system correspond to the values ( y , y ) that satisfy dy dN = dy dN = 0 . (24)Note that there is a direct relationship between R and the variables ( y , y ) given by [25] f (cid:48) ( R ) R − f ( R ) f (cid:48) ( R ) R − f ( R ) = − − y − y y . (25)By calculating the eigenvalues ( λ , λ ) of the Jacobian matrix at each point ( y , y ) the stability ofthe fixed points can be determined [25, 32]. The fixed point is stable when both eigenvalues are real andnegative, and unstable when both are real and positive. The fixed point is a saddle point when botheigenvalues are real and of opposite sign. The nature of the fixed point for different eigenvalues ( λ , λ ) issummarized in Tab. 1. Eigenvalues Fixed point λ (cid:54) = λ < Stable λ (cid:54) = λ > Unstable λ < < λ SaddleTable 1: Fixed point type based on eigenvalue pairs ( λ , λ ) .6he values ( y , y ) for each corresponding fixed point are then substituted into the effective equationof state w eff given by [25] w eff = − y + 13 y + ˙ f (cid:48) ( R )3 Hf (cid:48) ( R ) + ˙ ξ Hξ − ˙ f (cid:48) ( R ) R f (cid:48) ( R ) ξH , (26)where ˙ ξ is determined by taking the derivative of Eq. (18) with respect to time and using Eq. (16). ˙ f (cid:48) ( R ) is given by [25] ˙ f (cid:48) ( R ) = − H ( f (cid:48) ( R ) R − f ( R )) f (cid:48)(cid:48) ( R ) f (cid:48)(cid:48) ( R ) R − f (cid:48) ( R ) = ˙ R f (cid:48)(cid:48) ( R ) . (27)It will also prove useful to define the deceleration parameter q in terms of the effective equation ofstate w eff . Since the deceleration parameter is defined in terms of the Hubble parameter via q ≡ − (cid:32) ˙ HH + 1 (cid:33) , (28)and since [25] ˙ HH = −
32 (1 + w eff ) , (29)we then find q = 12 (1 + 3 w eff ) . (30)To further evaluate the viability criteria set out in the introduction, we must also test whether ourtheory contains scalar curvature singularities [18]. A local rescaling of the metric tensor by a conformalfactor Ω ( x ) is equivalent to the transformations [33, 34, 35] g µν → ˜ g µν = f (cid:48) ( R ) g µν , g µν → ˜ g µν = (cid:0) f (cid:48) ( R ) (cid:1) − g µν . (31)The Ricci scalar R defines the simplest possible curvature invariant. Thus, in the Palatini formalism, R raised to the power of any positive integer m transforms under (31) via R m → R m ( f (cid:48) ( R )) m . (32)The next simplest curvature invariant involves the Ricci tensor. Since our model is defined in thePalatini variation, the connection Γ νµσ is not assumed to depend on the metric g µν , and so the Ricci tensor R µν = ∂ ρ Γ ρνµ − ∂ ν Γ ρρµ + Γ ρρλ Γ λνµ − Γ ρνλ Γ λρµ (33)may remain invariant under the local rescaling transformation of Eq. (31). The Ricci tensor with upperindices, however, is given by R µν = g µρ g νσ R ρσ and so it does transform under Eq. (31) according to R µν → R µν ( f (cid:48) ( R )) − . Therefore, second order curvature invariants involving the Ricci tensor, namely R µν R µν , to any integer power m , will transform under Eq. (31) according to ( R µν R µν ) m → ( R µν R µν ) m ( f (cid:48) ( R )) m . (34)It is unclear whether the Kretchmann scalar is a scalar in the Palatini formalism [36], and so we omit thisfrom our analysis. As a cross-check of our methodology and computer code we verified that we are able to successfully reproduce the cosmologicaldynamics found in Ref. [25] for two different models. Results
We find that the model defined by the exponent of Eq. (12) contains three fixed points P , P and P .The eigenvalues and stability of these fixed points, defined by the roots ( y , y ) of Eqs. (22) and (23), aredisplayed in Tab. 2 in the low and high-curvature limits. Figure 2 displays how the eigenvalues ( λ , λ ) vary as a function of R ∗ for P (left), P (middle), and P (right).Fixed point ( y , y ) ( λ , λ ) ( R ∗ →
0) ( λ , λ ) ( R ∗ → P (1 ,
0) (6 , Unstable ( − , − Stable P (0 ,
1) (1 , − Saddle (1 , Unstable P (0 ,
0) ( − , − Stable ( − , SaddleTable 2: The dimensionless variables ( y , y ) , eigenvalues ( λ , λ ) in the low ( R ∗ → and high-curvature ( R ∗ → limits, and stability of the three fixed points P , P and P . λ λ ℛ * - - λ λ λ λ ℛ * - - λ λ λ λ ℛ * - - λ λ Figure 2: The eigenvalues ( λ , λ ) as a function of R ∗ for fixed points P (left), P (middle), and P (right).Figure 2 illustrates a potential advantage of AWIG. Unlike most other f ( R ) theories of gravity, thevariable power in the Lagrangian density of AWIG makes it possible for the eigenvalues and hence stabilityof each fixed point to vary with curvature scale, and hence to potentially vary with cosmological time. So,for example, the stability of the fixed point P can change from being stable in the high-curvature limitto being unstable at lower curvatures, as can be seen in Fig. 2 (left). This feature allows a richer set ofpossible cosmological dynamics.Inserting the obtained coordinate pairs ( y , y ) into Eq. (26) yields the effective equation of state w eff as a function of the Palatini scalar curvature. The results are displayed in Fig. 3 for the fixed points P and P . Since y = 1 for the fixed point P we can see from Eq. (21) that ρ r is undefined, and therefore viaEq. (17) H must also be undefined. Consequently, w eff for P is undefined, as is evident from Eq. (26).However, we know that as n ( R ∗ ) → AWIG is equivalent to general relativity plus a cosmologicalconstant [17]. Thus, w eff for P can be determined in the high-curvature limit by an equivalent analysisof the model f ( R ) = R − Λ , where Λ is the cosmological constant. We have repeated the methodologyoutlined in section 3 for the model f ( R ) = R − Λ finding eigenvalues ( λ , λ ) = (1 , , which agreeswith our result presented in Fig. 2 (middle) in the high-curvature limit, and an effective equation of state w eff = 1 / . Identical results are also found in Ref. [25]. Therefore, P corresponds to a radiation-likephase in the high-curvature limit. 8 .2 0.4 0.6 0.8 1.0 ℛ * - - - w eff ℛ * - - - w eff Figure 3: The effective equation of state parameter w eff as a function of R ∗ for the fixed point P (left)and P (right).The effective equation of state parameter w eff for the fixed points P , P and P in the low andhigh-curvature limits are summarised in Tab.3. Thus, we identify P as a de Sitter-like phase, P as aradiation-like phase, and P as a matter-like phase. Note that the unknown value of w eff for P in thelimit R ∗ → is denoted by − . Figures 2 and 3 suggest that if the matter-dominated phase P is totransition back to the de Sitter-like phase P , to account for the currently observed late period of cosmicacceleration, then this transition must occur at a curvature scale R ∗ (cid:38) . . This is because if R ∗ (cid:46) . then it is not possible to exit the stable matter-like phase.Fixed point w eff ( R ∗ → w eff ( R ∗ → Phase P -1 -1 De Sitter P - 1/3 Radiation P w eff ( R ∗ → , high-curvature limit w eff ( R ∗ → and the phase type for the fixed points P , P and P .To further analyse the cosmological evolution of our model we use Eq. (30) to investigate how thedeceleration parameter q varies as a function of R ∗ for the de Sitter-like phase. The results are shown inFig. 4. If q > then the universe is expanding but decelerating. If q < then the universe is expandingbut accelerating [37]. Figure 4, therefore, indicates that the de Sitter-like phase undergoes two periods ofaccelerated expansion, one in the high-curvature regime . (cid:46) R < and one in the low-curvature regime ≤ R (cid:46) . , mediated by a period of decelerated expansion for . (cid:46) R (cid:46) . (see Fig. 4). Assumingcurvature on cosmological scales decreases with cosmological time, this implies an early and late periodof accelerated cosmic expansion, with an intermediate period of decelerated expansion. In this sense, thedynamics appear consistent with cosmological observations, depending on the exact scale set by R .9 .2 0.4 0.6 0.8 1.0 ℛ * - - - q Figure 4: The deceleration parameter q as a function of scalar curvature R ∗ for the fixed point P .We now analyse the phase space of this model, with the results shown in Fig. 5. The phase space ofAWIG is 3-dimensional, with each point in the phase space uniquely specified by the set of coordinates ( y , y , R ∗ ) . Figure 5 shows the ( y , y ) plane for three different values of constant curvature. Onepossible route the system may take through the 3-dimensional phase space is depicted in the three plotsof Fig. 5, where the system evolves through the closed sequence of fixed points P → P → P → P with decreasing curvature scale R ∗ . Thus, the model presented is consistent with the sequence of anearly period of accelerated expansion, intermediate radiation and matter-dominated eras of deceleratedexpansion, followed by the return to a period of accelerated expansion at late times. P P P - - - - y y ℛ * = P P P - - - - y y ℛ * = P P P - - - - y y ℛ * = Figure 5: Slices of constant curvature through the 3-dimensional phase space of AWIG at R ∗ = 0 . (left), R ∗ = 0 . (middle) and R ∗ = 0 . (right). The red trajectory shows one possible way the systemmay evolve through the sequence of fixed points P → P → P → P .We now present results for various powers of the Ricci scalar curvature under the local rescaling ofEq. (31). Using Eqs. (4) and (32) we find R m → R m ( f (cid:48) ( R )) m = R ∗ → m . (35)The first three powers of the Ricci scalar curvature ( m = 1 , , ) are shown in Fig. 6. As can be seenfrom Fig. 6 each curvature invariant is divergence-free and approaches a constant in the limit R ∗ → .Similar results have been shown in Refs. [38, 39]. Likewise, Eqs. (4) and (34) can be used to show that thecurvature invariant ( R µν R µν ) m formed from the Ricci tensor asymptotically approaches / m as R ∗ → .Therefore, the model presented contains no curvature singularities in R or R µν R µν , at any order m .10 .0 0.2 0.4 0.6 0.8 1.0 ℛ * ( ℛ / f' ( ℛ )) m m = = = Figure 6: The first three powers ( m = 1 , , ) of the transformed Palatini scalar curvature as a functionof R ∗ . In this work we have shown that one of the simplest possible implementations of asymptotically Weyl-invariant gravity (AWIG) may be viable, as measured against criteria (i) − (vi) set out in the introduction.However, the model’s viability cannot yet be definitively established for several reasons. Firstly, AWIGis by construction superficially renormalizable, but establishing its renormalizability via explicit calculationremains an open problem. Secondly, the analysis performed in this work has raised some unansweredquestions. For example, the transition from the matter-dominated phase to the late phase of cosmicexpansion must occur at a curvature scale R (cid:38) . R . It is unknown whether this is consistent withcosmological observations since the dimensionful scale R is presently unknown. Furthermore, the effectiveequation of state parameter w eff for the fixed point P is negative for < R ∗ (cid:46) . , the meaning of whichis unclear. Finally, one of the three fixed points P has an undefined effective equation of state for ≤ R ∗ < , however, we can determine w eff for R ∗ → .Nevertheless, the model presented contains several encouraging features, such as the apparent absenceof curvature singularities and three fixed points with effective equation of states corresponding to deSitter, radiation and matter-like phases. The model also contains the correct sequence of early and lateperiods of accelerated cosmic expansion, with an intermediate period of decelerated expansion, somethingthat has proven difficult to achieve in other attempted modifications of general relativity [25]. Moreover,the early accelerating phase emerges from AWIG without adding a scalar field. This is because AWIGasymptotically approaches the Palatini formulation of pure R gravity in the high curvature limit, whichis equivalent to general relativity plus a non-zero cosmological constant and no massless scalar field [17].Another positive feature of AWIG is that the variable power in the Lagrangian density seems to permit aricher set of possible cosmological dynamics, as can be seen from the variable eigenvalues in Fig. (2).The dimensionless exponent n ( R ∗ ) explored in this work is among the simplest possible choices,but it is far from the only consistent choice. Going forward, we aim to narrow down, or perhaps evenuniquely determine, the functional form of n ( R ∗ ) and the value of R to more robustly test the viabilityof asymptotically Weyl-invariant gravity. I wish to thank Roberto Percacci and Subir Sarkar for their comments on the manuscript, and the anony-mous referee for invaluable corrections. 11 eferences [1] Clifford M. Will. 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