No-go theorem for inflation in an extended Ricci-inverse gravity model
NNo-go theorem for inflation in an extended Ricci-inverse gravity model
Tuan Q. Do ∗ Phenikaa Institute for Advanced Study, Phenikaa University, Hanoi 12116, Vietnam andFaculty of Basic Sciences, Phenikaa University, Hanoi 12116, Vietnam (Dated: January 28, 2021)In this paper, we propose an extension of the Ricci-inverse gravity, which has been proposedrecently as a very novel type of fourth-order gravity, by introducing a second order term of theso-called anticurvature scalar as a correction. The main purpose of this paper is that we wouldlike to see whether the extended Ricci-inverse gravity model admits the homogeneous and isotropicFriedmann-Lemaitre-Robertson-Walker metric as its stable inflationary solution. However, a no-gotheorem for inflation in this extended Ricci-inverse gravity is shown to appear through a stabilityanalysis based on the dynamical system method. As a result, this no-go theorem implies that it isimpossible to have such stable inflation in this extended Ricci-inverse gravity model. ∗ [email protected] a r X i v : . [ g r- q c ] J a n I. INTRODUCTION
Cosmic inflation [1–3] has been regarded as one of the leading paradigms for modern cosmology. Remarkably, manytheoretical predictions derived within the context of cosmic inflation have been shown to be highly consistent withthe observed data of the leading cosmic microwave background radiation (CMB) detectors such as the WilkinsonMicrowave Anisotropy Probe satellite (WMAP) [4] and the Planck one [5].It is widely believed that the inflationary phase of our universe, which happens very shortly after the Big Bang,is driven by the so-called inflaton field, which is a hypothesis scalar field [2, 3]. The nature of this scalar field,however, has been a great mystery of modern cosmology. It appears that many inflation models have been proposedto realize the origin of the inflaton field, e.g., see Ref. [6]. In addition, many inflation models have been examinedtheir cosmological viability through comparing with the observational data of the Planck satellite, e.g., see Ref. [7].It is worth noting that among the well-known inflation models the Starobinsky model [1], one of the first inflationmodels, involving the R correction term, has remained as one of the most favorable models in the light of the Planckobservation [5]. Although the Starobinsky model originally contains no scalar field, but it has been shown that theStarobinsky model can be conformally transformed into an effective model of scalar field [8–12]. More interestingly,according to Ref. [13] the Starobinsky model can become, under a suitable Weyl transformation, an effective gravitymodel involving a more general matter sector, which contains not only scalar field but also other fields. It is notedthat the Starobinsky model is one of the simplest fourth-order gravity models (a.k.a. quadratic gravity), where thecosmic inflation can be found [14–23]. The other types of the fourth-order gravity such as R µν R µν and R µναβ R µναβ can be seen in Refs. [17–19]. It is also noted that the fourth-order gravity has been one of the leading alternativeapproaches to solve the so-called accelerated expansion problem of the current universe [24–29]. More interestingly,inspired by the fourth-order gravity a number of interesting gravity models have been proposed to unify both theearly and late time phases of our universe into a single scenario, see, e.g., Refs. [25, 29]. All these facts indicate thatthe fourth-order gravity has been one of the most attractive frameworks for studying both the early and late timephases of our universe. For an interesting review on the rich history and cosmological implications of the fourth-ordergravity, see Ref. [30]. In addition to the cosmological aspects, it is worth noting that gravitational actions includingterms quadratic in the curvature tensor such as R and R µν R µν have been shown to be renormalizable by Stelle [31].This important result indicates that the fourth-order gravity is a promising approach to quantum gravity despite thefact that it could admit the Ostrogradsky ghost [32] due to the existence of higher derivatives. It is noted that theStarobinsky model turns out to be free of the Ostrogradsky ghost [32]. For detailed discussions on the interestingissues related to the quantum scenario of the fourth-order gravity such as the renormalizability, ghost problem, Landaupoles, etc., see an interesting review [13].Recently, a very novel fourth-order gravity model, which is called the Ricci-inverse gravity, has been proposed byAmendola, Giani, and Laverda in Ref. [33]. This model is constructed by introducing a very novel geometrical objectcalled an anticurvature scalar denoted by the capital letter A . More specific, the anticurvature scalar A is nothingbut the trace of anticurvature tensor denoted as A µν , which is defined to be equal to the inverse Ricci tensor, i.e., A µν = R − µν . Very interestingly, this model has been shown to admit a no-go theorem stating that both deceleratedand accelerated expansions cannot exist together in this model. Consequently, it is impossible for the Ricci-inversegravity model to be a dark energy candidate [33]. One therefore might think of a possibility that the Ricci-inversegravity may be suitable for describing the early time inflationary phase rather than the late time accelerated expansionphase of the universe [33]. However, our follow-up study in Ref. [34] has indicated that we also have another no-gotheorem, not for accelerated expansion but for inflation, of the Ricci-inverse gravity model. In particular, we haveshown that although the original Ricci-inverse gravity model admits both the Friedmann-Lemaitre-Robertson-Walker(FLRW) and Bianchi type I solutions but all of these solutions turn out to be unstable against perturbations duringthe inflationary phase. All of these results have raised doubts about the cosmological viability of the Ricci-inversegravity. Hence, non-trivial extensions such as the f ( R, A ) seem to be necessary to cure the Ricci-inverse gravity [33].In fact, some simple extensions of the Ricci-inverse gravity have been proposed in Ref. [33] in order to overcome thefirst no-go theorem. Some of them turn out to be promising and of course need to be verified by further investigations.In this paper, motivated by the Starobinsky model [1] we would like to propose a simple extension of the Ricci-inversegravity by introducing a second order term A . Note that the introduction of A is not for violating the first no-gotheorem for the accelerated expansion. Therefore, it has not been proposed in Ref. [33]. By doing this, we expectto have the corresponding isotropic inflation solution, which would be stable against perturbations. Unfortunately, ano-go theorem based on the stability analysis will be shown to hold for the isotropic inflation. This result together withour previous investigation [34] raise more doubt about the implications of the Ricci-inverse gravity for the inflationaryphase of our universe.As a result, this paper will be organized as follows: (i) A brief introduction of the present study has been writtenin the Sec. I. (ii) Basic setup of the extended Ricci-inverse gravity model will be presented in Sec. II. (iii) Isotropicinflationary solution to this model will be solved analytically in Sec. III. (iv) The proof of the no-go theorem for thisinflationary solution will be shown in Sec. IV. (v) Finally, concluding remarks will be written in Sec. V. II. BASIC SETUP
As a result, an action of extended Ricci-inverse gravity has been proposed in Ref. [33] as follows S = (cid:90) √− gd x [ R + f ( A ) − , (2.1)where the reduced Planck mass, M p , has been set to be one for convenience, while Λ > f ( A ) is an arbitrary function of the so-called anticurvature scalar, A , which isnothing but the trace of the so-called anticurvature tensor, A µν . By construction, the anticurvature tensor is nothingbut the Ricci-inverse tensor [33], i.e., A µν = R − µν . (2.2)It is noted that this relation does not lead to A = R − . As a result, varying the action (2.1) with respect to g µν willyield the corresponding Einstein field equation [33], R µν − Rg µν + Λ g µν = f A A µν + 12 f g µν − (cid:2) g ρµ ∇ α ∇ ρ f A A ασ A νσ − ∇ ( f A A µσ A νσ ) − g µν ∇ α ∇ ρ ( f A A ασ A ρσ ) (cid:3) , (2.3)where f A = ∂f /∂A and ∇ µ is understood as the covariant derivative.It turns out that the right hand side of Eq. (2.3) looks very complicated which addresses a very lengthy calculation,even for the simplest metrics such as the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric [33]. For convenience,therefore, we decide to use another approach based on the Euler-Lagrange equations, which will turn out to be veryeffective in computing explicitly the non-vanishing components of the Einstein field equation. Of course, this methodhas also been used in our previous study [34]. As a result, the calculation process is as follows: (i) given a specificmetric we will first define the corresponding Lagrangian of an extended Ricci-inverse gravity model, i.e., L = √− g [ R + f ( A ) − , (2.4)(ii) then we will define the corresponding Euler-Lagrange equations of scale factors, which are nothing but the desiredfield equations.In this paper, motivated by the Starobinsky gravity [1], we propose to study a simple generalization of the Ricci-inverse gravity, in which f ( A ) contains not only A but also the second order term A , i.e., f ( A ) = α A + α A , (2.5)where α and α are free parameters. We expect that the inclusion of the higher order term A would lead to stabilityregions for the existing of isotropic inflationary solutions.In this paper, we will consider the spatially homogeneous and isotropic FLRW spacetime described by the followingmetric, ds = − N ( t ) dt + exp[2 β ( t )] (cid:0) dx + dy + dz (cid:1) , (2.6)where N ( t ) is the lapse function, whose existence is necessary to derive the corresponding Friedmann equation from itsEuler-Lagrange equation [21, 35]. In particular, after deriving the corresponding Euler-Lagrange equation equation, N will be set as one in order to recover the well-known Friedmann equation [21, 35]. In addition, β ( t ) is an isotropicscale factor, which is assumed to be a function of cosmic time t due to the homogeneity of the FLRW spacetime. Asa result, the corresponding non-vanishing components of the Ricci tensor, R µν ≡ R ρµρν , can be defined to be R = 3 (cid:32) ˙ NN ˙ β − Φ (cid:33) , (2.7) R = R = R = − g N (cid:32) ˙ NN ˙ β − Π (cid:33) , (2.8)respectively. Here, two additional variables Φ and Π have been defined asΦ = ¨ β + ˙ β , (2.9)Π = ¨ β + 3 ˙ β , (2.10)respectively, for convenience. Note that ˙ β ≡ dβ/dt , ¨ β ≡ d β/dt , β (3) ≡ d β/dt , and so on. Thanks to these usefulresults, the corresponding Ricci scalar, R ≡ g µν R µν , and anticurvature scalar, A ≡ g µν A µν , turn out to be R = − N (cid:34) ˙ NN ˙ β − (cid:16) ¨ β + 2 ˙ β (cid:17)(cid:35) , (2.11) A = − N (cid:32) ˙ NN ˙ β − Φ (cid:33) − − N (cid:32) ˙ NN ˙ β − Π (cid:33) − , (2.12)respectively. This result confirms the above claim that A (cid:54) = R − . As a result, the Lagrangian, L = exp[3 β ] N (cid:0) R + α A + α A − (cid:1) , (2.13)can now be defined explicitly. It appears that this Lagrangian explicitly contains two independent variables, N ( t )and β ( t ), and their time derivatives. Next, we are going to figure out the corresponding Euler-Lagrange equations inorder to figure out cosmological solutions. First, the following Euler-Lagrange equation for the lapse function N isdefined as follows ∂ L ∂N − ddt (cid:18) ∂ L ∂ ˙ N (cid:19) = 0 , (2.14)which will become, after setting N = 1, the corresponding Friedmann equation, α (cid:20) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 2Φ ˙ β (cid:16) β (3) + 2 ˙ β ¨ β (cid:17)(cid:21) + 3 α Π (cid:20) ˙ β (cid:16) β (3) + 6 ˙ β ¨ β (cid:17)(cid:21) + α (cid:20) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 6Φ ˙ β (cid:16) β (3) + 2 ˙ β ¨ β (cid:17)(cid:21) + 9 α Π (cid:20) ˙ β (cid:16) β (3) + 6 ˙ β ¨ β (cid:17)(cid:21) + α ΠΦ (cid:20) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 4Φ ˙ β (cid:16) β (3) + 2 ˙ β ¨ β (cid:17) + 2ΠΦ ˙ β (cid:16) β (3) + 4 ˙ β ¨ β (cid:17)(cid:21) + α ΦΠ (cid:20) ˙ β (cid:16) β (3) + 6 ˙ β ¨ β (cid:17) + 2ΦΠ ˙ β (cid:16) β (3) + 4 ˙ β ¨ β (cid:17)(cid:21) + 2 (cid:16) β − Λ (cid:17) = 0 . (2.15)On the other hand, the corresponding Euler-Lagrange equation of β turns out to be ∂ L ∂β − ddt (cid:18) ∂ L ∂ ˙ β (cid:19) + d dt (cid:18) ∂ L ∂ ¨ β (cid:19) = 0 , (2.16)which will be reduced, after setting N = 1, to α (cid:26) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 2Φ (cid:104) β (4) + 6 ˙ ββ (3) + 2 ¨ β (cid:16) ¨ β + 4 ˙ β (cid:17)(cid:105) − (cid:16) β (3) + 2 ˙ β ¨ β (cid:17) (cid:27) + 3 α Π (cid:20) (cid:16) β (4) + 6 ˙ ββ (3) + 6 ¨ β (cid:17) − (cid:16) β (3) + 6 ˙ β ¨ β (cid:17) (cid:21) + α (cid:26) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 6Φ (cid:104) β (4) + 6 ˙ ββ (3) + 2 ¨ β (cid:16) ¨ β + 4 ˙ β (cid:17)(cid:105) − (cid:16) β (3) + 2 ˙ β ¨ β (cid:17) (cid:27) + 27 α Π (cid:20) (cid:16) β (4) + 6 ˙ ββ (3) + 6 ¨ β (cid:17) − (cid:16) β (3) + 6 ˙ β ¨ β (cid:17) (cid:21) + α ΠΦ (cid:26) − (cid:16) ¨ β + 3 ˙ β (cid:17) + 4Φ (cid:104) β (4) + 6 ˙ ββ (3) + 2 ¨ β (cid:16) ¨ β + 4 ˙ β (cid:17)(cid:105) − (cid:16) β (3) + 2 ˙ β ¨ β (cid:17) + 2ΠΦ (cid:104) β (4) + 6 ˙ ββ (3) + 4 ¨ β (cid:16) ¨ β + 3 ˙ β (cid:17)(cid:105) − (cid:16) β (3) + 2 ˙ β ¨ β (cid:17) (cid:16) β (3) + 14 ˙ β ¨ β (cid:17)(cid:27) + α ΦΠ (cid:26) (cid:16) β (4) + 6 ˙ ββ (3) + 6 ¨ β (cid:17) − (cid:16) β (3) + 6 ˙ β ¨ β (cid:17) + 2ΦΠ (cid:104) β (4) + 6 ˙ ββ (3) + 4 ¨ β (cid:16) ¨ β + 3 ˙ β (cid:17)(cid:105) − (cid:16) β (3) + 6 ˙ β ¨ β (cid:17) (cid:16) β (3) + 10 ˙ β ¨ β (cid:17)(cid:27) + 6 (cid:16) β + 3 ˙ β − Λ (cid:17) = 0 . (2.17)It is noted that the existence of the third term in the left hand side of the Euler-Lagrange equation (2.16) is dueto the fact that L contains not only β and ˙ β but also ¨ β . Up to now, the desired field equations (2.15) and (2.17)have been worked out thanks to the effective Euler-Lagrange equation approach. It is apparent that all of these fieldequations are higher order nonlinear ordinary differential equations. In particular, the Friedmann equation (2.15) isthe third-order differential equation of β , while the other field equation (2.17) is the fourth-order differential equationof β . Therefore, these field equations seems to be very difficult to be solved analytically. Fortunately, the studiesdone in Refs. [17, 18], which are also about inflation in the fourth-order gravity, provide us a useful hint to figure outanalytical solution to these field equations. As a result, detailed inflationary solutions will be presented in the nextsections. III. INFLATIONARY SOLUTIONS
Following the Barrow-Hervik’s papers in Refs. [17, 18] as well as our previous paper [34], we will assume thefollowing ansatz for the scale factor as β ( t ) = ζt, (3.1)where ζ is a constant, whose value will be determined after solving the field equations. Consequently, the correspondinganticurvature scalar A turns out to be A = 43 ζ . (3.2)It is clear that A is always singularity-free during an inflationary phase with ζ (cid:29)
1. It is noted that the correspondingvalue of the Ricci scalar is given by R = 12 ζ . (3.3)Hence, it is straightforward to have a relation AR = 16 . (3.4)As a result, the field equations (2.15) and (2.17) both reduce to the corresponding algebraic equation of ζ ,27 ζ − ζ + 9 α ζ + 16 α = 0 . (3.5)Setting ˆ ζ ≡ ζ > ζ − ζ + 9 α ˆ ζ + 16 α = 0 . (3.6) A. Case 1: Vanishing α It is noted that when setting α = 0, i.e., ignoring the contribution of the second order term A , we will arrive atan equation of ˆ ζ , 3ˆ ζ − Λˆ ζ + α = 0 , (3.7)which is nothing but that derived in our previous paper [34]. Furthermore, as pointed out in Ref. [34], for theexistence of an inflationary solution with ζ (cid:29)
1, or equivalently ˆ ζ (cid:29) α must be negative definite along with | α | (cid:29) Λ ∼ O (1). Indeed, we can have an approximated solution for this equation such asˆ ζ (cid:39) (cid:114) − α (cid:29) . (3.8) B. Case 2: Vanishing α In this case, Eq. (3.6) simply becomes as 27ˆ ζ − ζ + 16 α = 0 . (3.9)It is clear to see that if α < ζ >
0. On the other hand,it turns out that if α < − (cid:2) α − Λ α (cid:3) < . (3.10)In conclusion, in this case α should be negative, similar to α in the case 1, in order to have inflation. And theabsolute value of α should be much larger than Λ ∼ O (1) to have the following solution asˆ ζ (cid:39)
23 ( − α ) / (cid:29) . (3.11) C. Case 3: Non-vanishing α and α Now, we would like to consider a geneal scenario, in which α and α are both non-vanishing. In particular, wewould like to see whether Eq. (3.6) with additional parameter α admits an inflationary solution to the present model.It is clear to see that if α < ζ >
0, no matter the signof α . Furthermore, if one need only one real, positive root to this equation, which could represent an inflationarysolution, an additional constraint should be mathematically satisfied,∆ = − (cid:2) (cid:0) α + 64 α (cid:1) + 864Λ α α − α − α (cid:3) < , (3.12)where ∆ is the following discriminant of this cubic equation. See Fig. 1 for a constraint region of both α and α given that Λ = 1, in which the cubic equation admits only one real, positive root ˆ ζ . According to this plot, we observethat there exists a wide region of positive α , in which the corresponding ˆ ζ will be positive definite, possible to be adesired inflationary solution. As shown in Fig. 2, the contribution of the second order term A will increase a littlebit the value of scale factor ˆ ζ when α < | α | (cid:29)
1. However, there is an interesting point that it is possible tohave an inflationary for positive α . Furthermore, given a fixed negative value of α , whose absolute value is assumedto be much larger than one, the smaller positive value of α is, the larger value ζ will be. For convenience, we classifyall possibilities to have inflation into three sets of inequalities (I)-(III) for Λ = 1:(I) : α < , α < , | α | (cid:29) | α | ∼ O (1) , (3.13)(II) : α < , α < , | α | (cid:29) | α | (cid:29) , (3.14)(III) : α > , α < , | α | (cid:29) α ∼ O (1) . (3.15)In other words, one of these sets needs to be fulfilled if we would like to have inflation. It seems that these inequalitiesare quite flexible to be fulfilled. Hence, we now face to a very important point that the inclusion of A term wouldlead to the stability of the corresponding inflationary solutions or not. This issue will be investigated in detailed inthe next section. - -
20 0 20 40 - - - - - α α FIG. 1. The purple region of both α and α for Λ = 1, in which the cubic equation admits only one real, positive root ˆ ζ . -
10 000 - α ζ FIG. 2. Behavior of the positive root ˆ ζ for Λ = 1 and several values of α . In particular, the black, red, green, and blue curvescorrespond to α = 0, − − IV. NO-GO THEOREM FOR INFLATIONARY SOLUTIONS
In this section, we would like to investigate the stability of the isotropic inflationary solution within the extendedRicci-inverse gravity. As a result, we will finally reach to a no-go theorem for inflationary solutions of this proposedmodel due to the result that all these inflationary solutions will be shown to be unstable unexpectedly.
A. Dynamical system
In cosmology, the stability analysis based on the dynamical system has been widely used [18, 36]. Therefore, wewill construct the corresponding dynamical system, which is nothing but a set of the first order differential equationscalled the autonomous equations, from the higher order field equations [18]. Note that this method has been usedin our previous study on the original Ricci-inverse gravity model [34]. As a result, we have shown using this methodthat the original Ricci-inverse gravity admits unstable isotropic inflation. First, we will introduce dynamical variablesas follows [18, 34] B = 1˙ β , (4.1) Q = ¨ β ˙ β , (4.2) Q = β (3) ˙ β , (4.3)Ω Λ = Λ3 ˙ β , (4.4)here the Hubble constant is given by H = ˙ β . Note that we no longer have the shear terms like Σ = ˙ σ/ ˙ β since theparameter σ ( t ) representing spatial anisotropies vanishes for isotropic metrics. As a result, the corresponding set ofautonomous equations of dynamical variables turn out to be B (cid:48) = − QB, (4.5)Ω (cid:48) Λ = − Q Ω Λ , (4.6) Q (cid:48) = Q − Q , (4.7) Q (cid:48) = β (4) ˙ β − QQ , (4.8)where (cid:48) ≡ d/dτ with τ = (cid:82) ˙ βdt is nothing but the dynamical time variable. It is noted that the term β (4) / ˙ β in Eq.(4.11) can be figured out from the field equation (2.17), which can be written in terms of the dynamical variables asfollows α (cid:26) B −
1Φ ( Q + 3) + 2 B Φ (cid:20) β (4) ˙ β + 6 Q + 2 Q ( Q + 4) (cid:21) − B Φ ( Q + 2 Q ) (cid:27) + 3 α Π (cid:20) B + 2 B Π (cid:18) β (4) ˙ β + 6 Q + 6 Q (cid:19) − B Π ( Q + 6 Q ) (cid:21) + α (cid:26) B −
2Φ ( Q + 3) + 6 B Φ (cid:20) β (4) ˙ β + 6 Q + 2 Q ( Q + 4) (cid:21) − B Φ ( Q + 2 Q ) (cid:27) + 27 α Π (cid:20) B + 2 B Π (cid:18) β (4) ˙ β + 6 Q + 6 Q (cid:19) − B Π ( Q + 6 Q ) (cid:21) + α ΠΦ (cid:26) B −
2Φ ( Q + 3) + 4 B Φ (cid:20) β (4) ˙ β + 6 Q + 2 Q ( Q + 4) (cid:21) − B Φ ( Q + 2 Q ) + 2 B ΠΦ (cid:20) β (4) ˙ β + 6 Q + 4 Q ( Q + 3) (cid:21) − B ΠΦ ( Q + 2 Q ) (3 Q + 14 Q ) (cid:27) + α ΦΠ (cid:26) B + 4 B Π (cid:18) β (4) ˙ β + 6 Q + 6 Q (cid:19) − B Π ( Q + 6 Q ) + 2 B ΦΠ (cid:20) β (4) ˙ β + 6 Q + 4 Q ( Q + 3) (cid:21) − B ΦΠ ( Q + 6 Q ) (3 Q + 10 Q ) (cid:27) + 6 (2 Q + 3 − Λ ) = 0 . (4.9)where Φ and Π are now functions of the dynamical variables such asΦ = 1 B ( Q + 1) , Π = 1 B ( Q + 3) . (4.10)Before going to solve fixed point to the dynamical system, we would like to note that there is a constraint equation,which is nothing but the Friedmann equation (2.15) written in terms of the dynamical variables as α (cid:20) B −
1Φ ( Q + 3) + 2 B Φ ( Q + 2 Q ) (cid:21) + 3 α Π (cid:20) B + 2 B Π ( Q + 6 Q ) (cid:21) + α (cid:20) B −
2Φ ( Q + 3) + 6 B Φ ( Q + 2 Q ) (cid:21) + 9 α Π (cid:20) B + 6 B Π ( Q + 6 Q ) (cid:21) + α ΠΦ (cid:20) B −
2Φ ( Q + 3) + 4 B Φ ( Q + 2 Q ) + 2 B ΠΦ ( Q + 4 Q ) (cid:21) + α ΦΠ (cid:20) B + 4 B Π ( Q + 6 Q ) + 2 B ΦΠ ( Q + 4 Q ) (cid:21) + 2 (3 − Λ ) = 0 . (4.11)Hence, all found fixed point solutions should satisfy this important constraint equation. B. Fixed point
Following the previous works [18, 34], we are going to figure out the corresponding fixed point of the dynamicalsystem described by the autonomous equations (4.5), (4.6), (4.7), (4.8), (4.9), and (4.11). Mathematically, the fixedpoint is a solution of the following equations B (cid:48) = Ω (cid:48) Λ = Q (cid:48) = Q (cid:48) = 0 . (4.12)As a result, these equations lead to Q = Q = 0 (4.13)along with an equation of B , 16 α B + 9 α B −
27 (Ω Λ −
1) = 0 . (4.14)More interestingly, this cubic equation can be reduced to27 ˙ β −
9Λ ˙ β + 9 α ˙ β + 16 α = 0 , (4.15)which is nothing but Eq. (3.5) given that ˙ β = ζ . This result implies that the exponential solution found in theprevious section is equivalent with the isotropic fixed point found in this section. Therefore, investigating the stabilityof the fixed point will yield the stability of the exponential solution.It is appearance that B = ˙ β − and Ω Λ = Λ ˙ β − / β (cid:29)
1. Consequently, Eq. (4.14) indicates that16 α B + 9 α B + 27 = 27Ω Λ (cid:28) , (4.16)or equivalently, 16 α B + 9 α B (cid:39) − < . (4.17)This result implies that at least either α or α is negative definite and has absolute value much larger than one.This requirement can be fulfilled by one of three sets of inequalities (I)-(III) shown in Eqs. (3.13), (3.14), and (3.15),respectively. To be more specific, we will plot in the Fig. 3 the green region of α B and α B with 27Ω Λ is chosento be 10 − , where any real solution of Eq. (4.14) should belong to.0 C. Stability analysis of fixed point
Similar to the previous investigations [18, 34], we are going to perturb the autonomous equations around the fixedpoint to see whether or not the unstable mode(s) exists. As a result, a set of perturbed equations is given by δB (cid:48) = − BδQ, (4.18) δ Ω (cid:48) Λ = − Λ δQ, (4.19) δQ (cid:48) = δQ , (4.20) δQ (cid:48) = δ (cid:18) β (4) ˙ β (cid:19) , (4.21)where δ (cid:16) β (4) / ˙ β (cid:17) will be figured out from the following perturbed equation, α (cid:26) δB + 3Φ δ Φ − δQ + 2 B Φ (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ + 8 δQ (cid:21)(cid:27) + 3 α Π (cid:26) δB − δ Π + 2 B Π (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ (cid:21)(cid:27) + α (cid:26) δB + 12Φ δ Φ − δQ + 6 B Φ (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ + 8 δQ (cid:21)(cid:27) + 27 α Π (cid:26) δB − δ Π + 2 B Π (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ (cid:21)(cid:27) + α ΠΦ (cid:26) δB + 9Φ δ Φ + 3ΠΦ δ Π − δQ + 4 B Φ (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ + 8 δQ (cid:21) + 2 B ΠΦ (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ + 12 δQ (cid:21)(cid:27) + α ΦΠ (cid:26) δB − δ Φ − δ Π + 4 B Π (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ (cid:21) + 2 B ΦΠ (cid:20) δ (cid:18) β (4) ˙ β (cid:19) + 6 δQ + 12 δQ (cid:21)(cid:27) + 6 (2 δQ − δ Ω Λ ) = 0 , (4.22)where δ Φ = − B δB + 1 B δQ, (4.23) δ Π = − B δB + 1 B δQ, (4.24)along with Φ = 1
B ,
Π = 3
B . (4.25)This perturbed equation is derived from the field equation (4.9). It is noted that the perturbed Friedmann equationturns out to be α (cid:20) δB + 3Φ δ Φ − δQ + 2 B Φ ( δQ + 2 δQ ) (cid:21) + 3 α Π (cid:20) δB − δ Π + 2 B Π ( δQ + 6 δQ ) (cid:21) + α (cid:20) δB + 8Φ δ Φ − δQ + 6 B Φ ( δQ + 2 δQ ) (cid:21) + 9 α Π (cid:20) δB − δ Π + 6 B Π ( δQ + 6 δQ ) (cid:21) + α ΠΦ (cid:20) δB + 7Φ δ Φ + 1ΠΦ δ Π − δQ + 4 B Φ ( δQ + 2 δQ ) + 2 B ΠΦ ( δQ + 4 δQ ) (cid:21) + α ΦΠ (cid:20) δB − δ Φ − δ Π + 4 B Π ( δQ + 6 δQ ) + 2 B ΦΠ ( δQ + 4 δQ ) (cid:21) − δ Ω Λ = 0 . (4.26)Thanks to the useful relations shown in Eqs. (4.23), (4.24), and (4.25), Eq. (4.26) can be solved to give δ Ω Λ = 2 B
81 [9 (8 α B + 3 α ) δB + 2 B (11 α B + 3 α ) (3 δQ + δQ )] . (4.27)By inserting this solution into Eq. (4.22) we are able to obtain the following value of δ (cid:16) β (4) / ˙ β (cid:17) as δ (cid:18) β (4) ˙ β (cid:19) = 3 (cid:0) α B + 9 α B − (cid:1) B (11 α B + 3 α ) δQ − δQ . (4.28)1By taking exponential perturbations, δB = A exp [ µτ ] , (4.29) δ Ω Λ = A exp [ µτ ] , (4.30) δQ = A exp [ µτ ] , (4.31) δQ = A exp [ µτ ] , (4.32)it is possible to write all perturbation equations (4.18), (4.19), (4.20), and (4.21) as a matrix equation, M A A A A ≡ µ B µ Λ
00 0 µ −
10 0 − ( α B +9 α B − ) B (11 α B +3 α ) µ + 3 A A A A = 0 . (4.33)Mathematically, this matrix equation admits non-trivial solutions if and only ifdet M = 0 , (4.34)which can be determined to be µ (cid:2) B (11 α B + 3 α ) µ + 6 B (11 α B + 3 α ) µ − α B − α B + 81 (cid:3) = 0 . (4.35)As a result, besides two trivial eigenvalues, µ , = 0, this equation admits two non-trivial ones given by µ , = − (cid:34) ∓ (cid:115) α B + 27 α B − B (11 α B + 3 α ) (cid:35) . (4.36)Note that a stable inflationary solution happens only when all obtained µ , or their real part (if they are complexnumber) turn out to be non-positive definite. Otherwise, unstable mode(s) to the inflationary solution will ariseaccordingly. It is straightforward to see that if we set α = 0, or equivalently neglecting the contribution of A , theeigenvalues µ , all reduce to µ , → ¯ µ , = − (cid:20) ∓ (cid:114) − α B (cid:21) , (4.37)which are nothing but µ , found in our previous works [34] provided a replacement that α → α . It is clear in thiscase that ¯ µ is always positive for α <
0, making the inflationary solution unstable [34].One can ask what will happen if α = 0 and α (cid:54) = 0. It turns out that the eigenvalues µ , now become as µ , → ˆ µ , = − (cid:20) ∓ (cid:114) − α B (cid:21) . (4.38)It is clear that ˆ µ is always positive for α < α A in terms of α B to see whether the eigenvalues µ , defined in Eq. (4.36) with both non-vanishing α and α act as stable modes. We expect that the appearance of α in Eq. (4.36) would leave extra space for the existence of stable modes of the inflationary solution. As a result, thenon-positivity of µ , addresses the following inequality97 α B + 27 α B − B (11 α B + 3 α ) ≤ . (4.39)It turns out that for a stable inflationary solution it should satisfy not only the the inequality (4.39) but also one ofthree sets of inequalities (I), (II), and (III) described by Eqs. (3.13), (3.14), and (3.15), respectively. Therefore, wewill numerically examine whether the inequality (4.39) is satisfied or not in one of three different regions (I), (II),and (III). It turns out that only the region (III) is suitable for the existence of stable modes. To be more specific, weplot in Fig. 3 the blue region for the inequality (4.39), where any real fixed point solution will be stable. However,one can wonder that is there any inflationary solution existing in this blue region. To address this question, we alsoplot in Fig. 3 the green region for the existence of inflationary solution, following the inequality shown in Eq. (4.16)with 27Ω Λ is chosen to be 10 − . It appears that these two colored regions do not have common points, meaning thatthe inflationary solution found in this extended Ricci-inverse gravity model will no longer be stable as expected. Thisresult unexpectedly raises more doubt about the cosmological validity of the Ricci-inverse gravity.2 - - - - α B α B FIG. 3. The stability region is colored as blue, while the existence region of real solution of Eq. (4.14) is colored as green. Itis clear that these two colored regions do not overlap each other, meaning no stable isotropic inflation exists in the extendedmodel.
V. CONCLUSIONS
We have proposed an extension of the Ricci-inverse gravity model [33] by introducing the second order term A asa correction. As a result, we have been able to derive the homogeneous and isotropic inflation within this extension.Unfortunately, the no-go theorem based on the stability analysis has been achieved for this isotropic inflation. Asa result, this theorem implies that it is impossible to have a stable isotropic inflation in this extended Ricci-inversegravity model. This result together with our previous investigation [34] raise more doubt about the implication of theRicci-inverse gravity for inflationary phase of our universe. This result also indicates that the isotropic inflation mightbe not suitable for inflationary phase of our universe within the context of the Ricci-inverse gravity. An investigationon anisotropic inflation within this extended Ricci-inverse gravity model is therefore necessary [34]. Of course, otherpossible types of f ( A ) or f ( R, A ) of the Ricci-inverse gravity like that proposed in Ref. [33] should also be examined.We will leave these issues for our further studies. We hope that our present study would be useful for other studiesof the cosmological implications of the Ricci-inverse gravity.
ACKNOWLEDGMENTS
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