The effective dynamics of loop quantum R 2 cosmology
TThe e ff ective dynamics of loop quantum R cosmology Long Chen ∗ College of Physics and Electrical Engineering, Xinyang Normal University, Xinyang, 464000, Henan, China (Dated: November 21, 2018)The e ff ective dynamics of loop quantum f ( R ) cosmology in Jordan frame is considered by using the dynam-ical system method and numerical method. To make the analyze in detail, we focus on R model since it issimple and favored from observations. In classical theory, ( φ = , ˙ φ =
0) is the unique fixed point in both con-tracting and expanding states, and all solutions are either starting from the fixed point or evolving to the fixedpoint; while in loop theory, there exists a new fixed point (saddle point) at ( φ (cid:39) / , ˙ φ =
0) in contracting state.We find the two critical solutions starting from the saddle point control the evolution of the solutions startingfrom the fixed point ( φ = , ˙ φ =
0) to bounce at small values of scalar field in 0 < φ <
1. Other solutions,including the large field inflation solutions, all have the history with φ < ff ective theory of loop quantum f ( R ) theory. Another di ff erent thing from loop quantum cosmology withEinstein-Hilbert action is that there exist many solutions do not have bouncing behavior. PACS numbers: 04.60.Pp, 04.50.Kd,98.80.Qc
I. INTRODUCTION
Since the development of observational cosmology and as-trophysics in recent years, such as Cosmic Microwave Back-ground(CMB) [1, 2] and Supernovae surveys[3], some modi-fied theories of gravity, such as Brans-Dicke theory[4], brane-world[5], Einstein-Aether theory[6],etc, have be studied invariant ways in the literature. The f ( R ) theory[7] is a rel-atively simple theory among these modified theories, and ithas been shown that Starobinsky model[8], a special model of f ( R ) theory, is a candidate inflation model to fit the observa-tions of CMB[2].Although f ( R ) theories may give a unified description ofearly-time inflation with late-time cosmic acceleration, theyalso inherit the big-bang singularity as in classical general rel-ativity, which means the dyanmics of these modified theoriesof gravity are not complete. One the other hand, it is widelybelieved that the quantum e ff ect of gravity can avoid the big-bang singularity, and loop quantum gravity (LQG)[9, 10] issuch a promising theory of quantum gravity which is non-perturbative and background independent. It has been shownthat loop quantum cosmology (LQC)[11–13], the cosmologi-cal application of LQG, replaces the big-bang singularity bya big-bounce thus solves the initial singularity of general rel-ativity. So trying to insert loop quantum correction into f ( R )cosmology is an interesting work.Loop quantum gravity is based on connection-triad vari-ables, and since f ( R ) theory is a higher-derivative theory, oneneeds to introduce other variables besides the connection-triadvariables. It has been shown that[7] f ( R ) theory can be trans-formed into a special theory of Brans-Dicke theory with a po-tential for the e ff ective scalar field where the scalar field andits conjugate momentum could be viewed as the other vari-ables. The connection-triad formulation of Brans-Dicke the-ory has been derived, and the full theory of loop quantum f ( R )theory also has been constructed in the work of Zhang and his ∗ Electronic address: chen˙[email protected] collaborations[14, 15]. In this paper, we want to study the cos-mological part of loop quantum f ( R ) gravity, and for explic-ity, we will consider the Starobinsky model, i.e., the R -modelwith f ( R ) = R + R µ in detail.There exists a conformal transformation can cast Brans-Dicke theory into an theory of a scalar field minimally cou-pled with geometry, i.e., Einstein theory with a scalar field,and thus the theory can be loop quantized directly. We callthe theory quantized in this frame the Einstein frame theory,and call the loop quantum theory without conformal trans-formation the Jordan frame theory. Works on loop quantum f ( R ) cosmology based on Einstein frame[16, 17] or on Jor-dan frame[18–20] both can be seen in the literature. But inthe viewpoint of loop quantum theory, geometry is quantizedand whether making a conformal transformation before quan-tization or not would give di ff erent quantum theories. In thecurrent work we use Jordan frame since Jordan frame is closerto the original f ( R ) theory. Although some works on Jordanframe have been given in the literature[18–20], we find thedynamics of cosmos before bouncing is not studied in detailespecially there exists a sign problem of the variable cos b thathas not been addressed.The structure of the paper is as follows: In Sec II, we firstgives some general aspects of classical f ( R ) cosmology, andfor compariasion with loop theory, we also consider the dyan-mics of R theory in detail. In SecIII, we give the e ff ec-tive equations of loop f ( R ) theory, and analyse the dynamicsof R theory by the dynamical system method and numeri-cal method. In SecIV, some discussions and conclusions aregiven. II. CLASSICAL THEORY OF f ( R ) COSMOLOGY
In this section, we first present some results of connection-triad form of f ( R ) gravity, and the cosmological dynamics of f ( R ) cosmology in classical theory, then focus on an importantexample, R model in detail.Note that in this paper we take c = = π G . a r X i v : . [ g r- q c ] N ov A. general considerations
To get the connection-triad form of f ( R ) theory whose ac-tion is S [ g ] = (cid:82) d x √− g f ( R ), one can transform the theoryinto a special theory of Brans-Dicke theory with zero ω andthe action of f ( R ) theory is transformed into S [ g , φ ] = (cid:90) d x √− g (cid:34) φ R − V ( φ ) (cid:35) , (1)where the potential of the scalar field φ is defined as V ( φ ) = ( φ s − f ( s )) and s is the solution of φ = f (cid:48) ( s ) (assuming f (cid:48)(cid:48) ( s ) (cid:44) f ( R ) theory, where the basicvariables are ( A ia , E bj ) and ( φ, π ) and the Hamilton’s constraintdensity is (other constriants are not given here since their triv-iality in cosmology): H = φ (cid:34) F jab − (cid:32) γ + φ (cid:33) (cid:15) jmn ˜ K ma ˜ K nb (cid:35) (cid:15) jkl E ak E bl √ h +
13 ( ˜ K ia E ai ) √ h φ +
2( ˜ K ia E ai ) π √ h + π φ √ h + √ hV + √ hD a D a φ, (2)where γ ˜ K ia : = A ia − Γ ia .In the spatially flat isotropic model of cosmology, as in theliterature, one first introduces an elementary cell V and re-stricts all integrations to this cell. Then by fixing a fiducialflat metric o q ab and denoting the volume the elementary cell V by V o , the basic variables of gravity are of the forms A ia = cV − / oo ω ia and E ai = pV − / oo e ai , (3)and the variables of scalar field are φ = φ and π = π φ V − o . (4)where ( o e ai , o ω ia ) are a set of orthogonal triads and co-trialscompatible with o q ab . The Possion brackets become { c , p } = γ { φ, π } = . (5)In cosmological part, the first three terms in the constraintdensity (2) are canceled by each other, and the total Hamil-tonian constraint becomes C cl = γ c sgn( p ) π φ (cid:112) | p | + π φ φ | p | / + | p | / V ( φ ) . (6)By the canonical equations and the constraint equation, onecan get the Friedmann equation in f ( R ) cosmology, (cid:32) H +
12 ˙ φφ (cid:33) = (cid:32) ˙ φφ (cid:33) + V ( φ ) φ = : φρ , (7)where H : = ˙ p p is the Hubble constant. The dynamical equation of scalar field also can be derivedas ¨ φ + H ˙ φ + φ V (cid:48) ( φ ) − V ( φ ) = . (8)Note that we will restrict the scalar field’s region to φ > φ appears in the denominator ofthe Hamilton’s constraint (2), and thus in above Eq.(7), weintroduced an “energy density”: ρ : = φ (cid:32) ˙ φφ (cid:33) + V ( φ ) φ , (9)which is inspired by the theory in the Einstein frame and theevolution of ρ is d ρ dt = − φ φ (cid:32) H +
12 ˙ φφ (cid:33) . (10)From Eq.(7) and since V ( φ ) is generally positive, we will findthis expression is generally sign fixed which will help analyz-ing the dynamics of the system. B. R model In this paper we focus on the model with f ( R ) = R + R µ ,called Starobinsky theory or R -model, so in this subsection,we first consider the classical dyanamics of this model.It is easy to find the potential V ( φ ) of the scalar field inBrans-Dicke theory of R model is V ( φ ) = µ ( φ − , (11)and then the Friedmann’s equation and the dynamical equa-tion of scalar field are (cid:32) H + ˙ φ φ (cid:33) = (cid:32) ˙ φφ (cid:33) + µ ( φ − φ , (12)¨ φ + H ˙ φ + µ ( φ − = . (13)The right hand of Friedmann equation is positive except thepoint ( φ = , ˙ φ = H = H + forever or H = H − forever,where H ± : = −
12 ˙ φφ ± (cid:115)(cid:32) ˙ φφ (cid:33) + µ ( φ − φ . (14)Moreover, one can find (i) the solution space of the system istime-inverse invariant: if φ ( t ) is a solution, then ˜ φ ( t ) : = φ ( − t )is also a solution; (ii) and if φ ( t ) has H = H + (cid:62)
0, then itstime-inversion ˜ φ ( t ) has H = H − (cid:54)
0. So in the following, weonly consider the solutions having the property H = H + , andother solutions can be derived by time inversion.To analyse the dynamics, we rescale the time variable andintroduce new variables: τ : = µ t , x : = − φ ∈ ( −∞ , , y : = µ ˙ φφ / . (15)The dynamicial equations of x , y are x (cid:48) = y √ − x , y (cid:48) = − (cid:112) x + y y √ − x − x √ − x , (16)where the prime (cid:48) is for dd τ . It is interesting to note that theseequations are independent on the model parameter µ , thus weknow the dynamics of all R models with di ff erent µ s are sim-ilar. Note that we will find loop theory does not have thisproperty.The energy density becomes ρ = µ ( x + y ), and its evo-lution is d ρ d τ = − µ y (cid:114) x + y − x (cid:54) . (17)From the dynamical equations (16), one can find the fixedpoint ( φ = , ˙ φ = x , y ) = (0 , ρ decreases as time flows and the minimumof ρ is taken at the unique fixed point, we know ( x , y ) = (0 , | y | (cid:28) x ) and (ii) ˙ φ varies slow making y (cid:48) small compared to the other two termsin the second equation of Eq.(16). The slow-roll approxima-tions transform the dynamics equations(16) into the slow-rollequation y = − (1 − x ). Recall that if one wants to get thesolutions with H = H − (cid:54)
0, just takes the transformation: x − ( τ ) : = x + ( − τ ) , y − ( τ ) : = − y + ( − τ ). III. EFFECTIVE THEORY OF LOOP f ( R ) COSMOLOGYA. general considerations
In this section, we consider the LQC correction to f ( R )cosmology. In loop quantum theory, one uses holonomiesof edges and the fluxes on some 2-dim surfaces, instead ofconnections and triads. The loop quantum construction of f ( R ) cosmology or Brans-Dicke cosmology has been derivedin Ref.[18, 19], and in this paper we consider the e ff ectivetheory with holonomy corrections. E ff ectively, the holonomycorrection is equivalent to do the following replacement in theconstraint (6): c → sin( ¯ µ c )¯ µ , (18) Figure 1: (color online) Phase potrait with H = H + for Starobinskymodel, where x ≡ − φ , y = µ ˙ φφ / . where ¯ µ = (cid:113) ∆ | p | . Thus we get the constraint of loop quantumcosmology of f ( R ) cosmology: C e ff = γ sin( ¯ µ c )¯ µ π φ (cid:112) | p | + π φ φ | p | / + | p | / V ( φ ) . (19)As in the literature, it is convenient to introduce ρ c = γ ∆ (cid:39) . ρ P , b : = ¯ µ c and v : = p / . Then the Possion bracket of b and v is { b , v } = (cid:113) ρ c from Eq.(5), and the constraint C e ff inEq.(19) becomes C e ff = (cid:114) ρ c b ) π φ + φπ φ v + vV ( φ ) . (20)By the canonical equations and some direct calculations, onecan simplify the constraint equation assin b = ˙ φ + φ V ( φ ) ρ c = : ρ e ff ρ c , (21)where we introduced the energy density: ρ e ff : =
34 ˙ φ + φ V ( φ ) , (22)which is di ff erent from the classical theory’s energy density ρ ,and the relation of the two is ρ e ff = φ ρ . From the Eq.(21),one can get ρ e ff (cid:54) ρ c , and one may think ρ e ff = ρ c is justthe bouncing point like in LQC with ordinary scalar field, butwe will find it is a necessary condition but not a su ffi cientcondition.The Hubble variable H = ˙ v v in the e ff ective theory satisfies H = cos b φ (cid:32) (cid:114) ρ c b − ˙ φ (cid:33) . (23)To compare the Hubble variable’s equation(23) with the Fried-mann equation(7) in classical theory, one can get (cid:32) H + cos b φφ (cid:33) = φρ (cid:32) − ρ e ff ρ c (cid:33) . (24)Since cos b = ± (cid:113) − ρ e ff ρ c , we can find when ρ e ff (cid:28) ρ c andcos b >
0, the Friedmann equation of e ff ective theory closesto classical theory(cf.Eq.(7)), but those cases with cos b < φ + H ˙ φ + φ V (cid:48) + V (1 − b ) = . (25)From the Friedmann equation and the equation of scalar field,we have known loop quantum f ( R ) cosmology needs moreconditions to get classical theory than the LQC with Einstein-Hilbert action, where the energy condition ρ (cid:28) ρ c is enoughto get classical theory.Since the potential function of scalar field is not given ex-plicitly, the dynamics of the theory can not be analysed indetail, and we consider the R model in the next subsection. B. R model The potential of scalar field in R model is given by Eq.(11),and we also make some rescalings: τ = µ t and y = ˙ φµ ≡ d φ d τ ≡ φ (cid:48) . (26)Then the constraint equation (21) becomessin b = µ ρ c [ y + φ ( φ − ] = : (cid:15) ˜ ρ ( φ, y ) , (27)where ˜ ρ : = y + φ ( φ − , and (cid:15) = µ ρ c ∼ − (cid:28) H = µ √ (cid:15) cos b φ (cid:16) sin b − √ (cid:15) y (cid:17) , (28)and the dynamical equations of scalar fields are d φ d τ = y , (29) dyd τ = − H µ y − ( φ − (cid:34) φ + φ −
12 (1 − b ) (cid:35) . (30)To solve the system, we view ( φ, y ) as the dynamical vari-ables, and b is not viewed as a dynamical variable since theEq.(27) almost determines b except the sign of sin b and cos b .And for sin b , we find the sign of sin b never changes since ˜ ρ (cid:62) = ’ only happens at the fixed point ( φ = , y = φ ( τ ) , y ( τ ) , b ( τ )) → ( φ ( − τ ) , − y ( − τ ) , − b ( − τ )). Thus wecan safely only consider the subset of the solution space whereevery solution (except the fixed point) hassin b = (cid:112) (cid:15) ˜ ρ > , (31)which makes b ∈ (0 , π ). After this, we can split the range of b into (cid:16) , π (cid:17) and (cid:16) π , π (cid:17) , then consider each part separately thusthe sign of cos b = ± (cid:112) − (cid:15) ˜ ρ is also solved.In the following, we first consider the behavior near b = π since it is related to bouncing states, then we try to find allfixed points and analyze the dynamics near those fixed points.The singular edge φ = φ =
0. Afterthese analytical analyses, we confer to the numerical method.It may be noted that the analytical analyses will help to thenumerical analyse since the complexity of the system.
1. The behavior near b = π The range of b is (0 , π ), and by the equation of Hubble vari-able (28), it would have H = b = π . But not allthe points with b = π are the bouncing points. A bouncingpoint is a point where the collapsing world is transforminginto expanding world, so the Hubble variable H should evolvefrom negative to positive at the point. Since sin b − √ (cid:15) y = √ (cid:15) ( √ ˜ ρ − y ) > φ =
1) and by the equa-tion of Hubble variable(28), we have sign( H ) = sign(cos b ),which means a bouncing solution has a decreasing b as timeflows. Then from the canonical equation of b , dbd τ = y φ (cid:16) sin b − √ (cid:15) y (cid:17) , (32)we know the points with y < b = π are the bouncingpoints. Fig.3 gives solutions of this case. But for the solutionswith y > b = π , they evolve from expanding states tocollapsing states. The case(b1) in Fig.5 gives such a case.
2. The fixed points and the dynamics near them
The fixed points and the behavior near the fixed points canhelp to understand the global behavior of a dynamical system.In this part, we consider related problems.Fixed points satisfy the equations (cid:40) y = , ( φ − (cid:104) φ + φ − (1 − b ) (cid:105) = . (33)There are three solutions of these equations. Two of themare easy to find, which also appear in classical theory:( φ = , y =
0) with b = π . There is a new fixedpoint satisfying y = φ + φ − (1 − b ) =
0, i.e., (cid:16) φ = ˆ φ (cid:39) − (cid:15) + o ( (cid:15) ) , y = (cid:17) with b = arccos (cid:18) φ − φ − (cid:19) ∈ (cid:16) π , π (cid:17) . [To solve φ = ˆ φ , let x = φ − b = ± (cid:112) − (cid:15) ˜ ρ , then x satisfies x = − − (cid:15) ( x + x who has a unique solution approximated by x = − − (cid:15) + o ( (cid:15) )as (cid:15) ∼ − (cid:28) φ = , y =
0) with b =
0. Inspired fromclassical theory, there is a Lyapunov function ρ : = ρ e ff φ whichis positive except the fixed point. It can be shown that nearthis fixed point, ρ decreases as time flows (by Eq.(27),(32)): d ρ d τ = ρ c y φ sin( b ) (cid:104) cos b (cid:16) − √ (cid:15) y + sin b (cid:17) − sin b (cid:105) = ρ c y φ sin( b ) (cid:104) − √ (cid:15) y + O (sin b ) (cid:16) sin b − √ (cid:15) y (cid:17)(cid:105) (cid:39) − ρ c √ (cid:15) y φ sin b (cid:54) , so this fixed point is asymptotically stable and a sink of thesystem, which is the same with classical theory.(ii) 2nd fixed point: ( φ = , y =
0) with b = π . The func-tion ρ = ρ e ff φ is not a Lyapunov function since the dynamicsof loop theory is quite di ff erent from classical theory whencos b <
0. Thus we need to consider the dynamical equationsnear the fixed point. Let x = φ − x = r cos θ, y = r sin θ , then the equations of r , θ are approximated as (cid:40) d θ d τ = − + O ( r ) − drd θ = r (sin θ + sin θ − θ ) + O ( r ) , The first equation gives that θ decreases as time flows. Thenby the second equation, one can find after a period of θ , i.e., θ evolves from θ to θ − π , r would increase, r ( θ − π ) − r ( θ ) = (cid:90) θ − πθ drd θ d θ (cid:39) r ( θ ) (cid:90) θ θ − π [sin θ + sin θ − θ ] d θ = π r ( θ ) > . This proves that the fixed point is asymptotically unstable anda source of the system. Note that we also can use this methodto analyze the stability of the 1st fixed point which gives thesame result as above.(iii) 3rd fixed point: (cid:16) φ = ˆ φ (cid:39) − (cid:15) + o ( (cid:15) ) , y = (cid:17) with b = arccos (cid:18) φ − φ − (cid:19) ∈ (cid:16) π , π (cid:17) . Let ξ = φ − ˆ φ, η = y , and thedynamics near the fixed point to linear order is d ξ d τ = η d η d τ = ξ + √ η , whose general solution is (cid:34) ξη (cid:35) = c + g + ( τ ) + c − g − ( τ ) , where g ± ( τ ) : = e τλ ± (cid:34) λ ± (cid:35) , λ ± = √ (cid:18) ± (cid:113) (cid:19) and c ± ∈ R .Since λ + > , λ − <
0, we know this fixed point is a saddlepoint.Since there are only one source (in collapsing world) andonly one sink (in expanding world), one may think if there ex-ists no limit cycle, then all solutions, except the fixed pointsand the critical solutions ± g ± (here and later, what we say ± g ± is to mean the global solutions of φ = ξ + ˆ φ, y = η where[ ξ, η ] T (cid:39) ± g ± when the points are near the saddle point.),would start from the source and evolve to the sink, like the dy-namics in the LQC of Einstein-Hilbert action [23]. However,we will find this is not the case. The problem is the region φ > φ < φ = φ >
3. The dynamics near the singular edge φ = In this part, we consider the behavior of the system near thesingular edge φ =
0. Let y (cid:44) φ ∼ + , then by theconstraint equation (27), one can get sin b (cid:39) √ (cid:15) | y | (cid:16) + φ y (cid:17) .Substitude this expression into H in Eq.(28), one can find theHubble variable satisfies H (cid:39) (cid:40) µ y cos b , if y > µ y cos b φ , if y < . Then the dynamics of y in Eq.(30) becomes dyd τ = − + cos b + O ( φ ) , if y > , − y cos b φ + O ( φ ) , if y < . In the case of y <
0, we have dyd τ → ±∞ when φ → + ,so no solution can cross the edge φ =
0. However for thesolutions with y > dyd τ is finite when φ → + which meansthere exists solutions can cross the edge φ = φ < R theory but common for the general f ( R ) theory where d ˙ φ dt (cid:39)− V (1 − b ) when ˙ φ > φ → + . Note that φ = φ = ρ = ∞ .Since φ = φ =
4. Numerical analyze
The critical solutions ± g ± related with the saddle point cangrab the global behavior of the phase portrait, so we first con-sider the behaviors of these four solutions, which are given inFig.2, where for pretty plotting we choose σ = µ √ ρ c = . (cid:15) = σ = . (cid:28) (cid:15) ∼ − will be given in the last of this subsubsection whose resultsare not quite di ff erent. Figure 2: (color online) ( σ ≡ µ √ ρ c = .
3) The global behavior ofcritical solutions ± g ± from or to the saddle point ( φ (cid:39) , φ (cid:48) = ± g + restrict the solutions starting from source (1 ,
0) canonly bouncing at 0 < φ <
1. We also show the initial points (at φ = From Fig.2, we know: (1) the solution − g − starts at ( φ = , y = . b <
0; (2) g − starts from the source(1 ,
0) with b = π ; (3) g + evolves over the source (1 ,
0) to(1 . ,
0) and bounces at φ = . <
1, then evolves to thesink (1 , − g + bounces at φ = .
192 and also evolves tothe sink (1 , ± g + together with the boundary ρ = ρ c make a closed regionwhere the source (1 ,
0) sits in which means all solutions start-ing from the source will bounce at φ ∈ (0 . , . ⊂ (0 , , φ = , φ (cid:48) ≡ y > .
793 with cos b < φ = , < φ (cid:48) < .
793 with cos b < φ ∼ , φ (cid:48) ∼ b ∼ π .Each a typical solution of these three cases are shown inFig.3.These solutions don’t cover all the bouncing points on theboundary ρ = ρ c , since in case (a1), even we let φ (cid:48) (0) bethe maximum φ (cid:48) max , it only bounces at φ (cid:39) . < φ max (see Fig.4). We find for those solutions bouncing at φ ∈ (2 . , φ max ), the initial values would have cos b > φ =
0. But these solutions are also not all cases and only coverinitial values φ (cid:48) > .
745 with φ =
0, and the solution of initialvalue ( φ = , φ (cid:48) = . b >
0, bounces at φ max .So we find the initial values with cos b > φ = , φ (cid:48) > .
745 with cos b > φ = , < φ (cid:48) < .
745 with cos b > Figure 3: (color online) ( σ ≡ µ √ ρ c = .
3) Three examples of solutions(a1),(a2) and (a3) whose initial values have cos b <
0. The solutionsin case (a1) bounce at large field φ > . < φ < . For a solution in case (b1) (cf. Fig.5 and Eq.(28)(32)), itfirst expands with b increasing, and when b = π , it begins tocollapse but b still increases until y =
0, and when b decreasesto π , it bounces at φ ∈ (2 . , φ max ), finally it approaches thesink (1 ,
0) with b =
0. On the other hand, for the solutions incase (b2), b also increases initially but never reaches b = π ,and when y = b begins to decrease to zero. These solutionshave no bouncing point. Figure 4: (color online) ( σ ≡ µ √ ρ c = .
3) Two critical solutions: onestarts at φ (cid:48) = φ (cid:48) max and bounces at φ = . φ (cid:48) = .
745 and bounces at φ = φ max . We also show the initial points(at φ =
0) of the two classes of solutions: (b1) and (b2).
Things are qualitatively same for the real model where σ ≡ µ √ ρ c ∼ − with (cid:15) ∼ − , and we only gives the behaviorsof the critical solutions in Fig.6 and Fig.7. In Fig.6, for prettyplotting, we use the Poincare’s sphere method[21, 22] to map Figure 5: (color online) ( σ ≡ µ √ ρ c = .
3) Two examples of the so-lutions of classes (b1) and (b2). The solution of (b1) approaches to b = π twice (the first one is not bouncing point and the second oneis) while the solution of (b2) never bounces. ( x ≡ φ − , y ) into ( u , v ): u : = x (cid:112) + ( x + y ) and v : = y (cid:112) + ( x + y ) , (34)since the values of φ and φ (cid:48) can be quite large with order of10 and 10 while the fixed points are of order 1. Quantita-tively, the solution g + evolves over the source and bounces at φ = .
004 which is much smaller than the previous case, sothe solutions from source are also not inflation solutions whileother solutions, including inflation solutions, have the historywith φ <
0. From the critical solutions in Fig.7 and the ana-lyze of the case σ = .
3, we know that for the solutions whoseinitial values are φ = , φ (cid:48) > . × with b < π , they willbounce at φ ∈ (9523 , φ max ), and for the solutions with initialvalues of φ = , φ (cid:48) < . × , no bouncing phenomenonhappens.Let’s make some comments to end this section: There arefive classes of solutions of this system (not including the crit-ical solutions). Only case (a3) has the similar property withloop theory in Einstein-Hilbert action, i.e., the universe startsfrom low energy density, bounces at some point, and then goesback to low energy density state. But these solutions do notbelong to large field inflation models, since they bounce at lowvalues of the field φ . On the other hand, those large field infla-tion solutions are in the cases of (a1), (b1) or (b2) where all so-lutions come from φ (cid:54) IV. CONCLUSIONS
The f ( R ) theory of gravity is a class of modified theories,which can explain some phenomena from cosmology and as-trophysics, and has got much attention in the literature. In Figure 6: (color online) ( σ ≡ µ √ ρ c = − ) Critical solutions in ( u , v )-space: ± g ± , which give the classes of solutions (a1), (a2) and (a3). u , v are defined in Eq.(34).Figure 7: (color online) ( σ ≡ µ √ ρ c = − ) Critical solutions whichgive the classes of solutions (b1) and (b2). the paper, we considered the dynamics of f ( R ) theory in loopquantum cosmology in Jordan frame. We first considered theclassical dynamics of f ( R ) cosmology, and then consideredthe e ff ective dynamics of loop quantum cosmology of f ( R ).We focus on an important model of f ( R ) theory, i.e., R the-ory. The classical theory is easy to analyze since there exists aglobal inequality (17) of energy density to control the dynam-ics and only one fixed point in collapsing world or expandingworld. However the e ff ective LQC of the theory is some com-plicate.Firstly, we find a new fixed point in collapsing world whichis a saddle point. Then by analyzing the critical solutionsof the saddle point and considering the range of the value ofscalar field of bouncing points, we get to know there are fiveclasses of solutions of the system. Only one case of solutionshas the property that the universe starting from low energydensity states, bouncing at some point and then evolving backto the low energy density states, but these solutions are notlarge field inflationary solutions since they bounce at smallfields. Other cases of solutions all have a history with φ < f ( R ) theory φ could not be zero, and the potential partof energy density defined in Eq.(22) should be positive. Wealso note that this phenomenon is not only a special case of R model, but also a phenomenon of general f ( R ) theory. An-other quite di ff erent thing is that not all solutions have thebouncing behavior, i.e., the solutions in case(b2). This meansputting initial values at bouncing points used in the loop the-ory of Einstein-Hilbert action [24–26] may not be reasonablewhen considering the e-foldings and the probabilities of hap-pening of inflation in R model. And the initial values given at φ = φ = ω > φ ∼ + ,one can find the kinetic energy of the energy density wouldmake ˙ φ → ∞ , but the full dynamics of such model should beconsidered in detail which is not the work of the current workand may be considered in the future work. Acknowledgments
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