On 𝑓(𝑅) Theories in Two-Dimensional Spacetime

Abstract

In recent years, theories in which the Einstein-Hilbert Lagrangian is replaced by a function 𝑓(𝑅) of the Ricci Scalar have been extensively studied in four-dimensional spacetime. In this paper we carry out an analysis of such theories in two-dimensional spacetime with focus on cosmological implications. Solutions to the cosmological field equations are obtained and their properties are analysed. Inflationary solutions are also obtained and discussed. Quantization is then carried out, the Wheeler-DeWitt equation is set up, and its exact solutions are obtained.

1. Introduction

Attempts to modify the theory of general relativity, by including higher-order invariants in the action, started not too long after its inception [1, 2]. Later the nonrenormalizability of general relativity gave impetus to the inclusion of higher-order terms in the action [3, 4]. More recently it was shown that when quantum corrections are taken into consideration, higher order curvature invariants need to be added to the low-energy gravitational action [5, 6]. Such considerations further increased the interest in constructing theories in which the Einstein-Hilbert action is extended by the inclusion of higher-order curvature invariants with respect to the Ricci Scalar. Our interest here is in the so-called 𝑓(𝑅) theories of gravity. In these theories the Lagrangian in the Einstein-Hilbert action𝐼^𝐺=−12𝜅𝑑⁴𝑥√−𝑔𝑅,(1) where 𝜅=8𝜋𝐺, 𝐺 is the gravitational constant, 𝑔 is the determinant of the metric tensor and 𝑅 is the Ricci scalar (in units 𝑐=ℏ=1), is generalized to become𝐼′=−12𝜅𝑑⁴𝑥√−𝑔𝑓(𝑅).(2) In (2) 𝑓(𝑅) is a general function of 𝑅 [7]. Our focus here is on the cosmological aspects of 𝑓(𝑅) theories.

Now in another direction, the quest for quantum theory of gravity has led to the study of the simpler case of gravitational theory in two-dimensional spacetime. Such a spacetime provides an interesting arena in which to explore some fundamental aspects of both classical and quantum gravity. The reduction in the degrees of freedom greatly simplifies the analysis of the field equations. This leads to appreciable understanding of several problems in gravity theory. In two-dimensional spacetime, the two-dimensional gravitational constant 𝐺₂ is dimensionless and formally the theory with the bare action 𝐼_𝐺=−12𝑔_𝑁𝑑²𝑥√−𝑔𝑅,(3) where 𝑔_𝑁=8𝜋𝐺₂, is power counting renormalizable in perturbation theory. However the Einstein-Hilbert action term is purely topological in two dimensions. In fact in two spactime dimensions, the curvature tensor 𝑅_{𝜇𝜈𝜆𝜌} has only one independent component since all nonzero components may be obtained by symmetry from 𝑅₀₁₀₁. Equivalently the curvature tensor may be written in terms of the curvature scalar [8]:𝑅_{𝜇𝜈𝜆𝜌}=12𝑅𝑔^𝜇𝜆𝑔_𝜈𝜌−𝑔_𝜇𝜌𝑔_𝜈𝜆,(4) so that 𝑅 alone completely characterizes the local geometry. Equation (4) implies that𝑅_𝜇𝜈=12𝑔^𝜇𝜈𝑅,(5) so that the Einstein tensor 𝐺_𝜇𝜈=𝑅_𝜇𝜈−(1/2)𝑔_𝜇𝜈𝑅, vanishes identically and the usual Einstein equations are meaningless in two dimensions. This led to various models for gravity in two-dimensional spacetime being proposed [9]. Of special interest are those models that involve a scalar field, the dilaton, in the action [9–12]. We have previously studied some aspects of classical and quantum cosmology in two-dimensional dilaton gravity models [13, 14]. In the present work we study 𝑓(𝑅) theories as an alternative way to formulate gravitational theory in two-dimensional spactime and explore some of their cosmological implications.

In Section 2 we set up the 𝑓(𝑅) gravity theory in two-dimensional spacetime and derive the general field equations. We then specialize to the case of the Friedmann-Robertson-Walker metric and obtain the field equations with matter treated as a perfect fluid. Section 3 is devoted to obtaining solutions to the cosmological field equations under various conditions of matter or radiation dominance. Properties of these solutions are discussed in Section 4. In particular, conditions for ensuring cosmic acceleration and solving the horizon problem are elucidated. Inflation is discussed in Section 5 and solutions to the field equations in the absence of matter or radiation are obtained and their properties are discussed. In Section 6 we carry out the quantization. We establish the Wheeler-DeWitt equation and obtain its solutions. In Section 7 we offer some concluding remarks.

2. Field Equations

We write the two-dimensional action for 𝑓(𝑅) gravity as𝐼=𝐼_𝐺+𝐼_𝑀,(6) where𝐼_𝐺=−12𝑔_𝑁𝑑²𝑥√−𝑔𝑓(𝑅)(7) is the gravitational action and 𝐼_𝑀 is the matter action [15]. The field equations can be derived by varying the action with respect to the metric tensor 𝑔_𝜇𝜈. Upon noting that the stress-energy tensor is defined by𝛿𝐼_𝑀=12𝑑²𝑥√−𝑔𝑇^𝜇𝜈𝛿𝑔_𝜇𝜈,(8) we derive the following field equation:𝑓^(𝑅)𝑅_𝜇𝜈−12𝑔^𝜇𝜈𝑓(𝑅)−𝑔_𝜇𝜈𝑓^(𝑅)+∇_𝜇∇_𝜇𝑓^(𝑅)=−𝑔_𝑁𝑇_𝜇𝜈.(9) In (4) 𝑅_𝜇𝜈 is the Ricci tensor, the prime denotes the differentiation with respect to 𝑅, and the operator is defined by 𝑓^(𝑅)=1√−𝑔𝜕^𝜇√−𝑔𝑔^𝜇𝜈𝜕_𝜈𝑓^(𝑅).(10) Using (5) we can write (9) as12𝑔^𝜇𝜈𝑓^(𝑅)𝑅−𝑓(𝑅)−𝑔_𝜇𝜈𝑓^(𝑅)+∇_𝜇∇_𝜈𝑓^(𝑅)=−𝑔_𝑁𝑇_𝜇𝜈.(11) In the following we will concern with cosmological implications of (11). For this purpose will adopt the Friedman-Robertson-Walker (FRW) metric which in two-dimensional spacetime reads (𝑐=1)𝑑𝑠²=−𝑑𝑡²+𝑎²(𝑡)1−𝑘𝑥²𝑑𝑥²,(12) in terms of the comoving coordinates 𝑥 and 𝑡. The quantity 𝑎(𝑡) is the usual time-dependent cosmic scale factor. A change of variable 𝑑𝑥²/(1−𝑘𝑥²)→𝑑𝑥² leads to 𝑑𝑠²=−𝑑𝑡²+𝑎²(𝑡)𝑑𝑥².(13) Thus in two dimensions the time evolution of 𝑎(𝑡) is not affected by the value of 𝑘=0,±1 corresponding to the three different cosmological models [16]. This is unlike the four-dimensional case. The values 𝑘=0,−1 still describe spatially open flat and hyperbolic universe respectively, while 𝑘=1 describes a closed universe. The stress-energy tensor of the homogeneous isotropic universe is taken to be that of a perfect fluid:𝑇_𝜇𝜈=𝑝𝑔_𝜇𝜈+(𝑝+𝜌)𝑈_𝜇𝑈_𝜈,(14) where 𝑝 is the pressure, 𝜌 is the energy density, and 𝑈^𝜇 is the comoving velocity. Using (13) and (14) we obtain from (11) the following two independent cosmological field equations:12𝑅𝑓^(𝑅)−𝑓(𝑅)+𝑎̇𝑎𝜕^𝑡1𝑓^(𝑅)=𝑔_𝑁𝜌,2𝑅𝑓^(𝑅)−𝑓(𝑅)+𝜕^2𝑡𝑓^(𝑅)=−𝑔_𝑁𝑝,(15) where we use the dot as well as 𝜕_𝑡 to indicate differentiation with respect to time. We note that if 𝑓(𝑅) is expressed as a sum of powers 𝑅^𝑛 of 𝑅, then a term linear in 𝑅 would cancel out in the bracketed terms in (15) and would not contribute to the derivative terms either. Hence it has no effect on the dynamics. The stress-energy tensor obeys the conservation law:∇^𝛼𝑇_𝛼𝛽=0,(16) and this, for a perfect fluid, gives rise to the following two equations:𝑈^𝛼∇_𝛼𝜌+(𝑝+𝜌)∇^𝛼𝑈_𝛼=0,(17)(𝑝+𝜌)𝑈^𝛼∇_𝛼𝑈_𝛽+𝑔_𝛼𝛽+𝑈_𝛼𝑈_𝛽∇^𝛼𝑝=0.(18) For the FRW metric of (13) one readily obtains from (17) that𝑑𝑑𝑎(𝜌𝑎)=−𝑝.(19) Assuming an equation of state of the form 𝑝=𝛾𝜌, where 𝛾 is a constant, (19) immediately leads to𝜌=𝐶𝑎^−𝛾−1,(20) where 𝐶 is a constant. Equation (18) is seen to be identically satisfied and does not give rise to anything new. For a pressureless (dust) pure matter universe (𝜌_𝑚≠0,𝜌_𝑟=0,𝛾=0) we have𝜌_𝑚=𝐶_𝑚𝑎⁻¹,(21) while for a pure radiation universe (𝜌_𝑚=0,𝜌_𝑟≠0,𝛾=1), one has𝜌_𝑟=𝐶_𝑟𝑎⁻².(22) Denoting the present time by 𝑡₀ and using the usual notation of 𝑎₀≡𝑎(𝑡₀) and 𝜌₀≡𝜌(𝑡₀) to denote present-day values of these quantities, we can write for a matter-dominated universe𝑝_𝑚=0,𝜌_𝑚(𝑡)=𝜌_𝑚0𝑎₀𝑎(𝑡),(23) while for a radiation-dominated universe one has𝑝_𝑟=𝜌_𝑟(𝑡)=𝜌_𝑟0𝑎𝑎(⁰𝑡)².(24) Finally we wish to note that for the FRW metric the curvature scalar of this two-dimensional universe is given by𝑅=−2𝑎̈𝑎(𝑡),(25) where 𝑎̈=𝑑²𝑎/𝑑𝑡².

3. Solutions of the Cosmological Field Equations

In this section we seek solutions of the cosmological field equation (15) with the energy density and pressure given by (23) and (24) for each component of the cosmological fluid thus obtaining two sets of equations. For the matter dominated epoch we obtain the following:12𝑅𝑓^(𝑅)−𝑓(𝑅)+𝑎̇𝑎𝜕^𝑡𝑓^(𝑅)=𝑔_𝑁𝜌_𝑚0𝑎₀1𝑎,2𝑅𝑓^(𝑅)−𝑓(𝑅)+𝜕^2𝑡𝑓^(𝑅)=0.(26)

For the radiation dominated epoch the corresponding equations read12𝑅𝑓^(𝑅)−𝑓(𝑅)+𝑎̇𝑎𝜕^𝑡𝑓^(𝑅)=𝑔_𝑁𝜌_𝑟0𝑎²⁰𝑎²,(27)12𝑅𝑓^(𝑅)−𝑓(𝑅)+𝜕^2𝑡𝑓^(𝑅)=−𝑔_𝑁𝜌_𝑟0𝑎²⁰𝑎².(28) To proceed further we need to specify the function 𝑓(𝑅). Similar to the procedure followed in the four-dimensional case [7] we take for 𝑓(𝑅) the following expression:𝑓(𝑅)=𝑅+𝛼𝑅^𝑛,(29) where the real constants 𝛼 and 𝑛 are, at this stage, only restricted by 𝛼≠0 and 𝑛≠1. Upon substitution of (29) into (26) we obtain 12(𝑛−1)𝛼𝑅^𝑛+𝑛(𝑛−1)𝛼𝑅^𝑛−2𝑅̇𝑎̇𝑎=𝑔^𝑁𝜌_𝑚0𝑎₀𝑎,(30)2𝑛𝑅𝑅̈+2𝑛(𝑛−2)𝑅̇²+𝑅³=0.(31) Equations (30) and (31) describe the matter dominated epoch and we shall attempt to find solutions for them now. We start with (31) and note that in terms of the function 𝑧(𝑅) defined by𝑧(𝑅)=𝑅̇²,(32) the equation is transformed into the following form:𝑑𝑧𝑑𝑅+2(𝑛𝑅−2)𝑧+𝑅𝑛²=0.(33) This equation is easily solved and we obtain for 𝑛≠1/2𝑧(𝑅)=𝑅̇²=−1𝑛(2𝑛−1)𝑅³+𝐶₁𝑅^4−2𝑛,(34) where 𝐶₁ is a constant. Equation (34) then leads to the parametric solution:𝑡=±−𝑅³𝑛(2𝑛−1)+𝐶¹𝑅^4−2𝑛^−1/2𝑑𝑅+𝐶₂,(35) where 𝐶₂ is a constant. For 𝑛=2 and 𝐶₁≠0 one can carry out the integration using the result [16]𝑑𝑥𝐾−𝑥^𝛼+2^1/2=𝑥√𝐾²𝐹₁12,𝛼+12,𝛼𝛼++32;𝑥𝐾^𝛼+2,(36) where 𝛼 and 𝐾 are constants and ₂𝐹₁ is the hypergeometric function. We obtain𝑡=±6𝐶₁^1/2𝑅₂𝐹₁12,13,34;𝑅𝐶³₁+𝐶².(37)

Ideally one should solve (37) to obtain 𝑅 as a function of the cosmic time 𝑡 and plug that into (30) in order to solve for 𝑎(𝑡) in the case of 𝑛=2, but that is a difficult task. Instead we consider solutions for which 𝐶₁=0 in (34) and a general 𝑛≠1/2. One can then easily derive that𝑅=−4𝑛(2𝑛−1)𝑡−𝑡_𝑚²,(38) where we have renamed the integration constant 𝐶₂ as 𝑡_𝑚. In fact one can verify directly by substitution that the expression for 𝑅 in (38) is a solution of (31).

Next we substitute (38) into (30) and obtain𝐴₂𝑎̇+𝐴₁𝑡−𝑡_𝑚⁻¹𝑎=𝐾𝑎₀𝑡−𝑡_𝑚^2𝑛−1,(39) where𝐴₁=12(𝑛−1)𝑁^𝑛,𝐴₂=−2𝑛(𝑛−1)𝑁^𝑛−1,(40)𝑁=4𝑛(1−2𝑛),𝐾=𝑔^𝑁𝜌_𝑚0𝛼.(41) We readily solve (39) and get𝑎(𝑡)=𝐶𝑡−𝑡_𝑚^1−2𝑛+𝐾𝑡−𝑡_𝑚^2𝑛,(42) where 𝐶 is a constant and𝐾=𝐾𝑎⁰(4𝑛−1)𝐴₂.(43) Clearly 𝑛 must be such that 𝐴₁ and 𝐴₂ are real and 𝐾 is finite. We will return to this issue later. It is interesting to note that the 𝑡 dependence of 𝑅 is (𝑡−𝑡_𝑚)⁻² and thus independent of 𝑛, while that of 𝑎(𝑡) does depend on 𝑛. We also note that the relation 𝑅=−2𝑎/̈𝑎 is satisfied by the solutions for 𝑅 given in (38) and (42), respectively. We further note that the second term in (42) is a solution of (39) in its own right. On the other hand the first term in (42) is a solution of the homogeneous form of (39). Furthermore the constants 𝐶 and 𝐾 must be such that 𝑎(𝑡) is positive.

We now turn to the case of radiation. Upon adding (27) and (28) we obtain 𝑅𝑓^(𝑅)−𝑓(𝑅)+𝜕^2𝑡𝑓^(𝑅)+𝑎̇𝑎𝜕^𝑡𝑓^(𝑅)=0.(44) Employing in (44) the expression for 𝑓(𝑅) given in (29) above yields𝑛𝑅𝑅̈+𝑛(𝑛−2)𝑅̇²+𝑅³+𝑛𝑅𝑅̇𝑎̇𝑎=0.(45) Next we use (29) in (27) and obtain12(𝑛−1)𝛼𝑅^𝑛+𝑛(𝑛−1)𝛼𝑅^𝑛−2𝑅̇𝑎̇𝑎=𝑔^𝑁𝜌_𝑟0𝑎²⁰𝑎².(46) Motivated by the structure of the solutions for the cosmological equations in the case of pure matter above, we seek solutions for 𝑅(𝑡) and 𝑎(𝑡) of (45) and (46) in the form of powers in 𝑡−𝑡_𝑟 where 𝑡_𝑟 is some reference time. We obtain the following results:𝑅(𝑡)=2𝑛(𝑛−1)𝑡−𝑡_𝑟⁻²,(47)𝑎(𝑡)=𝐵𝑡−𝑡_𝑟^𝑛,(48) where the constant 𝐵 is given by𝐵=𝑔^𝑁𝜌_𝑟0𝑛(𝑛−1)(1−3𝑛)[2𝑛(1−𝑛)]^𝑛−1𝛼^1/2𝑎₀.(49) Note that, as in the case of matter, the 𝑡 dependence of 𝑅(𝑡) is independent of 𝑛, the only such dependence appears in the overall coefficient. We also note that the relation 𝑅=−2𝑎/̈𝑎 is satisfied by the solutions for 𝑅 given in (46) and (48). For an expanding universe one must have 𝑛>1 and 𝐵>0. Furthermore the value of 𝑛 must ensure that the bracketed term in (49) is finite and real.

4. Properties of the Solutions

We now discuss some properties of the solutions of the cosmological field equations found in the previous section. Let us first look at the radiation dominated case and determine whether our vision of the universe is limited by a particle horizon. At a given cosmic time 𝑡_𝑠 the proper distance 𝑑(𝑡_𝑠) of the emitter is given by𝑑𝑡_𝑠=𝑎𝑡_𝑠^𝑡𝑠_𝑡𝑒𝑑𝑡′𝑎(𝑡^),(50) where 𝑡_𝑒 is the time of emission of the photon. Using (48) we obtain𝑑𝑡_𝑠=𝑡^𝑠−𝑡_𝑟^𝑛1−𝑛𝑡^𝑠−𝑡_𝑟^1−𝑛−𝑡_𝑒−𝑡_𝑟^1−𝑛.(51) We can view 𝑡_𝑟 as signifying the onset of the radiation epoch. We see that as 𝑡_𝑒→𝑡_𝑟, 𝑑(𝑡_𝑠) is finite for 1−𝑛>0 and diverges for 1−𝑛<0. Hence no particle horizon problem will arise if 𝑛>1 which is the same condition required for an expanding universe. Reality of 𝐵 also requires 𝑛 to be an integer. For 𝑛 an even integer, the parameter 𝛼 must be positive while for 𝑛 odd, 𝛼 should be negative. Thus we take 𝑛 to be a positive integer greater than one. Next we note that the cosmic acceleration 𝑎(̈𝑡) which is given by𝑎̈(𝑡)=𝑛(𝑛−1)𝐵𝑡−𝑡_𝑟^𝑛−2,(52) is positive for 𝑡>𝑡_𝑟 since 𝑛>1 and is constant for 𝑛=2. Now in two-dimensional spacetime the radiation energy density is 𝜌_𝑟∝𝑇² where 𝑇 is the temperature [16] and it follows therefore from (24) that𝑎∝𝑇⁻¹.(53) Since we have 𝑎→0 as 𝑡→𝑡_𝑟, we conclude that this radiation universe has a hot big bang origin.

Next we turn to the case of the matter dominated universe described by (38) and (42). First let us consider the case 𝐶=0 when the scale factor becomes𝑎(𝑡)=𝐾𝑡−𝑡_𝑚^2𝑛.(54) As we have stated earlier this is viable because it represents a solution of (39). The time 𝑡_𝑚 can be taken to signify the onset of matter dominance. The proper distance 𝑑(𝑡_𝑠) is now given by𝑑𝑡_𝑠=𝑡^𝑠−𝑡_𝑚^2𝑛1−2𝑛𝑡^𝑠−𝑡_𝑚^1−2𝑛−𝑡_𝑒−𝑡_𝑚^1−2𝑛.(55) Hence no particle horizon will arise if 2𝑛>1. Also as we stated following (43), the parameter 𝑛 must be such that the constants 𝐴₁ and 𝐴₂ given by (40) are real. Since for 2𝑛>1 the number 𝑁 of (41) is negative, it follows that 𝑛 has to be a positive integer. Now the requirement that 𝑎(𝑡)>0 for 𝑡>𝑡_𝑚 implies that 𝐾>0. For 𝑛 even we have 𝐴₂>0 and hence 𝛼 should be positive to ensure 𝐾>0 while for 𝑛 odd one has 𝐴₂<0 and 𝛼 should be negative. Since we exclude 𝑛=1, the smallest permissible value is 𝑛=2. For such values of 𝑛 it is evident that the cosmic acceleration 𝑎̈‌(𝑡) is positive. Finally we observe that for the pure matter universe we have 𝑎(𝑡)→0 as 𝑡→𝑡_𝑚.

We now consider the case 𝐶≠0. Using (42) the proper distance is now given by𝑑𝑡_𝑠=𝑎𝑡_𝑠^𝑡𝑠_𝑡𝑒𝑡−𝑡_𝑚^2𝑛−1𝐶+𝐾𝑡−𝑡_𝑚^4𝑛−1𝑑𝑡.(56)

It is clear that the integral converges for 𝑡_𝑒→𝑡_𝑚 and we do have a particle horizon. Performing the integral we determine the proper distance to the horizon to be𝑑𝑡_𝑠=𝑎𝑡^𝑠𝐶𝐾𝐶^{2𝑛(4𝑛−1)}×−ln1+𝜉^𝑠4𝑛−1−4𝑛1−1^2𝑛−1_𝑘=1cos2𝑛𝜋(2𝑘−1)×ln1−2𝜉_𝑠4𝑛−1cos2𝑘−1+24𝑛−1𝜋+𝜉^2𝑠4𝑛−1^2𝑛−1_𝑘=1sin2𝑛𝜋(2𝑘−1)×4𝑛−1arctg𝜉^𝑠−cos((2𝑘−1)/(4𝑛−1))𝜋−𝜉sin((2𝑘−1)/(4𝑛−1))𝜋_𝑠⟷𝜉_𝑒,(57) where 𝜉_𝑗=𝐶𝐾^4𝑛−1𝑡_𝑗−𝑡_𝑚,𝑗=𝑠,𝑒.(58) Let us now study further properties of the solution given in (42). In the following we consider only values of 𝑡 such that 𝑡>𝑡_𝑚. Now it is evident that, except for values of 𝑛 in the interval 0<𝑛<1/2, the first term in (42) dominates for 𝑡 near 𝑡_𝑚 when 𝑛>1/2 while the second term dominates for 𝑛<0. Hence to ensure positivity of the scale factor we require that both 𝐶 and 𝐾 be positive. For 0<𝑛<1/2, 𝐶 and 𝐾 can have opposite signs but only in such a manner so as to keep 𝑎>0. We shall for simplicity assume that 𝐶>0 and 𝐾>0 for all values of 𝑛. Next we observe that outside the interval 0<𝑛<1/2, the number 𝑁 of (41) is negative, and to ensure the reality of 𝐴₂ given by (40), the number 𝑛 has to be an integer. We readily deduce that for 𝛼>0, 𝑛 can be a positive even integer or a negative odd integer. On the other hand for 𝛼<0, 𝑛 can be a positive odd integer or a negative even integer. The cosmic acceleration 𝑎(̈𝑡) is given by𝑎̈(𝑡)=2𝑛(2𝑛−1)𝑡−𝑡_𝑚⁻²𝑎(𝑡).(59) It is seen that 𝑎̈<0 for 0<𝑛<1/2, 𝑎̈=0 for 𝑛=1/2, and 𝑎̈>0 for 𝑛<0 or 𝑛>1/2.

Next we consider the behavior of 𝑎(𝑡) as 𝑡→𝑡_𝑚 for the case 𝐶≠0. We see from (42) that for 0<𝑛<1/2, 𝑎(𝑡)→0 as 𝑡→𝑡_𝑚 and accordingly the temperature 𝑇→∞ in this limit. For 𝑛=1/2, we have 𝑎(𝑡)→𝐶 as 𝑡→𝑡_𝑚 and 𝑇 is finite. However for 𝑛 outside the interval 0≤𝑛≤1/2 the behavior of 𝑎(𝑡) is very different as 𝑡→𝑡_𝑚. We see that 𝑎(𝑡)→∞ in this limit and energy density 𝜌_𝑚 and the temperature tend to zero. As 𝑡 increases beyond the value 𝑡_𝑚, 𝑎(𝑡) decreases to finite values and the density increases. However 𝑎(𝑡) never reaches zero and attains a minimum value at 𝑡=𝑡_𝑐 given by𝑡_𝑐=𝑡_𝑚+(2𝑛−1)𝐶2𝑛𝐾^1/4𝑛−1.(60) For 𝑡>𝑡_𝑐, 𝑎(𝑡) starts to increase. We also note from (38) that the curvature scalar is 𝑅→−∞ as 𝑡→𝑡_𝑚 and then starts increasing through finite negative values as 𝑡 grows beyond 𝑡_𝑚. The singular behavior of the scale factor noted here should be contrasted with that of the FRW cosmological models in four-dimensional general relativity where the scale factor and energy density go to zero and infinity, respectively, as the initial moment is approached.

5. Inflation

The horizon problem in four-dimensional standard FRW cosmology is a consequence of deceleration in the expansion of the universe. The problem can be solved by postulating a phase of the universe, prior to the decelerating phase, in which the expansion is accelerating and such a phase is called a period of inflation. Hence inflation is characterized by the following property for the scale factor 𝑎(𝑡):𝑎̈(𝑡)>0.(61) Now as evident from the analysis of Section 4, 𝑎̈>0 is readily achieved in our 𝑓(𝑅) theory in two-dimensional spacetime and the universe is accelerating. The solutions obtained for the scale factor displayed power dependence on time akin to that of power-law inflation. It would seem that there is no need to require an inflationary phase since matter or radiation dominated epochs yield an accelerating universe. Here we are not seeking to introduce scalar fields to propel acceleration as in the usual inflationary cosmology. We recall that one of the motivations for introducing modified or 𝑓(𝑅) theories of gravity in four-dimensional spacetime is the desire to explain acceleration of the universe as an alternative to using scalar fields. For this purpose solutions for the cosmological field equations are sought in the absence of the matter fluid [7]. We carry out such an analysis in our case by considering solutions to (30) and (31) of Section 3 with the R.H.S set being equal to zero. We have earlier obtained a general solution for (31) given by (35) of Section 3. However the parametric nature of that solution makes it difficult to use in (30) in order to solve for 𝑎(𝑡). Putting 𝐶₁=0 enables the integration in (35) to be performed and leads to the solution given in (38) which we write as𝑅=−4𝑛(2𝑛−1)𝑡−𝑡²,(62) where 𝑛≠1/2, 1 and we have now denoted the integration constant by 𝑡. Using (62) in (30) with the R.H.S. set being equal to zero yields:𝑎̇+2𝑛−1𝑡−𝑡𝑎=0,(63) the solution of which reads𝑎(𝑡)=𝐴𝑡−𝑡^1−2𝑛,(64) where 𝐴>0 is a constant. We take the solution to hold for 𝑡>𝑡. The cosmic acceleration is given by𝑎̈(𝑡)=2𝑛(2𝑛−1)𝐴𝑡−𝑡^−2𝑛−1.(65) The Hubble parameter is𝐻=𝑎̇𝐻̇=𝑎=2𝑛1−𝑡−−12𝑡𝑛,𝑡−𝑡².(66) For 𝑛<−1 we can identify 𝑡 with the onset of inflation 𝑡=𝑡_𝑖. Equation (64) then describes a universe that expands with positive acceleration for 𝑡>𝑡_𝑖. We also have 𝐻>0 and 𝐻̇<0 for 𝑡>𝑡_𝑖 which characterizes standard inflation. However if we make the identification 𝑡=𝑡_𝑖 for 𝑛>1, we will have a situation in which 𝑎(𝑡)→∞ as 𝑡→𝑡_𝑖 thus obtaining a universe that starts off already with an infinite size at the onset of inflation collapsing subsequently for 𝑡>𝑡_𝑖 at an accelerated rate. Such a scenario can be avoided if 𝑡 is instead taken to have a relatively large value so that 𝑡<𝑡 during the inflationary epoch. We write 𝑎(𝑡) now as𝑎(𝑡)=𝐴||𝑡−𝑡||^1−2𝑛.(67) The universe then starts off with a relatively small non-zero size at 𝑡=𝑡_𝑖 and expands with positive acceleration as time progresses. We also have𝐻=2𝑛−1𝐻̇=1𝑡−−2𝑡𝑛,𝑡−𝑡²,(68) so that 𝐻>0 and 𝐻̇<0 and we again have standard inflation.

As in four spacetime dimensions we define the so-called slow-roll parameter 𝜀 by [7]𝜀=−𝐻̇𝐻².(69) and in terms of which one has𝑎̈𝑎=𝐻²+𝐻̇=(1−𝜀)𝐻².(70) Inflation can thus be attained only if 𝜀<1. In our present context 𝜀 is given by𝜀=1||2𝑛−1||.(71) For both cases of 𝑛<−1 and 𝑛>1 we clearly have 𝜀<1. The slow-roll approximation corresponding to 𝜀≪1 then obtain when |2𝑛−1|≫1. As we have stated previously the solution for 𝑅 given in (62) arises as a special case of the general solution given in (35). As an alternative to solving (30) and (31) one can derive an equation for the Hubble parameter [7, 17]. We write (30) with the R.H.S set being equal to zero:2𝑛𝑅̇𝑎̇+𝑅²𝑎=0.(72) Now from (25) of Section 2 we obtain𝑅̇=−2𝑎⃛𝑎+2𝑎𝑎²𝑎̈̇.(73) Substituting (25) and (73) in (72) one obtains−𝑛𝑎𝑎𝑎̇⃛+𝑎𝑎̈²+𝑛𝑎̇²𝑎̈=0.(74) Next in terms of 𝐻, 𝐻̇, and 𝐻̈ we can express (74), after some manipulations, as−𝑛𝐻𝐻̈−2(𝑛−1)𝐻̇𝐻²+𝐻̇²+𝐻⁴=0.(75) It is customary, in dealing with equations such as this, to invoke the slow-roll approximation |𝐻̇/𝐻²|≪1 and |𝐻̈/𝐻𝐻̇|≪1, [7, 17]. Applying this to (67) we obtain that−2(𝑛−1)𝐻̇+𝐻²=0.(76) The solution of (76) is𝐻(𝑡)=−2(𝑛−1)𝑡−𝑡^′,(77) where 𝑡^ is a constant. Equation (77) in turn gives𝑎(𝑡)=𝐴′𝑡−𝑡^^2(𝑛−1)(78) with 𝐴^ being another constant. Equation (78) for 𝑎(𝑡) is similar in structure to (67) and the properties of the solution are therefore similar to what we discussed before and hence will not be considered any further.

We shall next seek a general solution to (31) for 𝑅(𝑡) that holds for 𝑡 close to the instant 𝑡_𝑖 that signifies the onset of inflation. Specifically we assume that 𝑡=𝑡_𝑖 is a regular point of (31) and seek a solution for 𝑅(𝑡) in the form of a power series confining ourselves to small values of 𝑡−𝑡_𝑖. For simplicity we consider the case 𝑛=2 for which (31) becomes4𝑅̈+𝑅²=0.(79)

We write𝑅(𝑡)=^∞_𝑚₌₀𝑏_𝑚𝑡−𝑡_𝑖^𝑚.(80)

Substituting (80) in (79) and solving we obtain𝑏₂=−18𝑏²⁰𝑏₃,=−112𝑏⁰𝑏₁,(81) and so forth. This leads to𝑅(𝑡)=𝑏₀+𝑏₁𝑡−𝑡_𝑖−18𝑏²⁰𝑡−𝑡_𝑖²−112𝑏⁰𝑏₁𝑡−𝑡_𝑖³+⋯.(82) We remark that if inflation lasts for a short period of time, then it is sensible to have a representation for 𝑅(𝑡) as given in (82). Moreover for sufficiently small 𝑡−𝑡_𝑖 we can approximate 𝑅(𝑡) by the first two terms and substitute in (72) with 𝑛=2. Solving the resulting equation we obtain𝑎(𝑡)≈𝐶exp−112𝑏²¹𝑏₀+𝑏₁𝑡−𝑡_𝑖³,(83) where 𝐶>0 is a constant. We can write (83) as𝑎(𝑡)≈𝑎_𝑖exp−112𝑏²¹𝑏⁰+𝑏₁𝑡−𝑡_𝑖³−𝑏³⁰,(84) where𝑎_𝑖=𝑎𝑡_𝑖=𝐶exp−𝑏³⁰12𝑏²¹.(85) From (83) we obtain𝑎̇(𝑡)=−14𝑏₁𝑏₀𝑎̈(𝑡)=−1+𝑏₁𝑡−𝑡_𝑖²𝑎(𝑡),2𝑏⁰+𝑏₁𝑡−𝑡_𝑖+116𝑏²¹𝑏₀+𝑏₁𝑡−𝑡_𝑖⁴𝑎(𝑡).(86) From (84) we see that we must have 𝑏₁<0 to ensure that 𝑎̇>0. We must also require 𝑎(𝑡) to be increasing for 𝑡>𝑡_𝑖. This can be achieved by having 𝑏₀>0 for then 𝑏₀+𝑏₁(𝑡−𝑡_𝑖) will start off at the value 𝑏₀ and decreases reaching zero at 𝑡^∗−𝑡_𝑖=−𝑏₀/𝑏₁. During the interval, 𝑡_𝑖<𝑡<𝑡^∗, 𝑎(𝑡) will be increasing. We must also require the cosmic acceleration 𝑎(̈𝑡) to be positive during this interval and this leads to the following condition:18𝑏²¹𝑏₀+𝑏₁𝑡−𝑡_𝑖³>1.(87) This inequality will continue to hold until 𝑡=𝑡_𝑓<𝑡^∗ when 𝑎(̈𝑡_𝑓)=0. This implies that18𝑏²¹𝑏₀+𝑏₁𝑡_𝑓−𝑡_𝑖³=1,(88) which yields𝑡_𝑓=𝑡_𝑖+𝑏₀||𝑏₁||⁻¹−2||𝑏₁||^−1/3.(89) The time 𝑡_𝑓 then signifies the end of inflation. Since 𝑅(𝑡_𝑖)=𝑏₀ and 𝑅̇(𝑡_𝑖)=𝑏₁, the conditions 𝑏₀>0 and 𝑏₁<0 can be expressed as𝑅𝑅̇𝑡𝑡_𝑖>0,_𝑖<0.(90) We can also express the duration of inflation as𝑡_𝑓−𝑡_𝑖=𝑅𝑡_𝑖||𝑅̇𝑡_𝑖||⁻¹−2||𝑅̇𝑡_𝑖||^−1/3.(91)

The Hubble parameter is given by𝐻=−14𝑏₁𝑏₀+𝑏₁𝑡−𝑡₁².(92)

It thus decreases from an initial value 𝐻_𝑖 given by𝐻_𝑖=𝐻𝑡_𝑖=−𝑏²⁰4𝑏₁=𝑅²𝑡_𝑖4||𝑅̇𝑡_𝑖||,(93)

to a value 𝐻_𝑓 at the end of inflation where𝐻_𝑓=𝐻𝑡_𝑓=||𝑏₁||^1/3=||𝑅̇𝑡_𝑖||^1/3.(94) We note that𝐻̇=−12𝑏⁰+𝑏₁𝑡−𝑡_𝑖(95) is negative during 𝑡_𝑖<𝑡<𝑡_𝑓 and we thus have standard inflation. The slow-roll parameter is given by𝜀=8𝑏²¹𝑏₀+𝑏₁𝑡−𝑡_𝑖⁻³.(96) We recall that for inflation to proceed one must have 𝜀<1 and this leads precisely to the condition expressed in (87) stated earlier.

The number of 𝑒-foldings from 𝑡=𝑡_𝑖 to 𝑡=𝑡_𝑓 is defined by [7, 18]𝑁=^𝑡𝑓_𝑡𝑖𝐻𝑑𝑡,(97) which is evaluated to give𝑁=23𝐻𝐻^𝑖_𝑓^3/2−1.(98) In four dimensions, the solution of the horizon and flatness problems of big bang cosmology requires that 𝑁≥70, [7, 19]. If we assume that we can use this value in our two-dimensional universe, we find that𝐻_𝑖𝐻_𝑓≥22,(99) that is, the Hubble parameter decreases to about 4.5% of its initial value by the time inflation ends.

6. Quantization

As we stated in the introduction two-dimensional spacetime models of gravity provide an arena where issues like quantization are studied since in such a setting they prove to be more tractable than in four-dimensional spacetime. In this section we thus consider quantization of the 𝑓(𝑅) gravity theory defined by the action of (3). Our objective is to derive the Wheeler-DeWitt equation for the wave function of the universe and obtain its solutions. Since we are considering a spatially homogeneous and isotropic universe, we drop the spatial integral and write the action as𝐼_𝐺=−12𝑔_𝑁𝑑𝑡𝑎(𝑡)𝑓(𝑅(𝑡)).(100) We take for 𝑓(𝑅) the expression given in (29) and put 𝑛=2. We use (25) that expresses the scalar curvature in terms of the scale factor and write𝐼_𝐺=−12𝑔_𝑁𝑑𝑡(𝑎𝑅−2𝛼𝑎𝑅̈).(101) We notice the appearance of the second derivative of 𝑎 in (101). The standard approach is to express the wave function in terms of 𝑎 and 𝑅 [20]. Hence integrating by parts in (101), we obtain𝐼_𝐺=ℒ𝑎,𝑎,̇𝑘,𝑘̇𝑑𝑡,(102) whereℒ=−12𝑔_𝑁𝑎𝑅+2𝛼𝑎𝑅̇̇.(103) The canonical momenta are defined in the usual way:𝑃_𝑎=𝜕ℒ𝜕𝑎̇=−𝑔𝛼_𝑁𝑃𝑅̇,_𝑅=𝜕ℒ𝜕𝑅̇=−𝑔𝛼_𝑁𝑎.̇(104) The Hamiltonian is then obtained asℋ=𝑃_𝑎𝑎̇+𝑃_𝑅𝑅̇−ℒ=−𝑔^𝑁𝛼𝑃^𝑎𝑃_𝑅+12𝑔_𝑁𝑎𝑅.(105) Replacing 𝑃_𝑎 and 𝑃_𝑅 by −𝑖(𝜕/𝜕𝑎) and −𝑖(𝜕/𝜕𝑅), respectively, in the Hamiltonian, we obtain the Wheeler-DeWitt equation for the wave function of the universe:𝑔𝛼^𝑁𝜕𝑅𝜕²𝜕𝑎+2𝑔1_𝑁𝑎𝑅𝜓(𝑎,𝑅)=0.(106) Instead of 𝑎 and 𝑅 we shall work with the variables:𝜉=𝑅+𝑎,𝜂=𝑅−𝑎.(107) In terms of 𝜉 and 𝜂 the Wheeler-DeWitt equation becomes𝑔𝛼^𝑁𝜕𝜕𝜉²²−𝜕²𝜕𝜂²+𝑔1_𝑁𝜉²−𝜂²𝜓(𝜉,𝜂)=0.(108) We seek solutions of (108) in factorizable form:𝜓(𝜉,𝜂)=𝑋(𝜉)𝑌(𝜂)(109) and obtain the following equations for the functions 𝑋 and 𝑌:𝑑²𝑋𝑑𝜉²+𝛼8𝑔²_𝑁𝜉²𝑋=𝐶𝛼𝑔_𝑁𝑋,(110)𝑑²𝑌𝑑𝜂²+𝛼8𝑔²_𝑁𝜂²𝑌=𝐶𝛼𝑔_𝑁𝑌,(111) where 𝐶 is the separation constant. The two equations are identical and hence it is enough to consider one of them. We first

take 𝛼>0 and define𝛾²=𝛼8𝑔²_𝑁,(112)𝐸=−𝐶𝛼2𝑔_𝑁.(113) In terms of 𝛾² and 𝐸, (110) reads𝑑²𝑋𝑑𝜉²+𝛾²𝜉²𝑋+2𝐸𝑋=0.(114) It is interesting to note that (114) is identical to that describing the inverted or reversed oscillator discussed by several authors in a number of contexts [21–25]. By performing the change of variable𝑦=√2𝛾𝜉,(115) we cast (114) into the following form:𝑑²𝑋𝑑𝑦²+14𝑦²𝑋+𝜀𝑋=0,(116) where 𝜀=𝐸/𝛾. Equation (116) is one of the standard forms of the equation for the parabolic cylinder functions. Two linearly independent solutions are given by the real functions 𝑊(𝜀,𝑦) and 𝑊(𝜀,−𝑦) [26]. For |𝑦|≫1 and |𝑦|≫|𝜀| these solutions display the following asymptotic behaviour:𝑊(𝜀,𝑦⟶∞)∼2𝑘𝑦cos41𝑦²+𝜀ln𝑦+1𝑊(𝜀,𝑦⟶−∞)∼4𝜋+21𝜙,2𝑘||𝑦||sin14𝑦²+𝜀ln||𝑦||+14𝜋+21𝜙,(117) where𝑘=𝜙=1+ar𝑒gΓ^−2𝜋𝜀1^1/2−𝑒^−𝜋𝜀,2−𝑖𝜀.(118) The functions 𝑊(𝜀,𝑦) and 𝑊(𝜀,−𝑦) satisfy the following normalization conditions [23]:^∞−∞𝑊(𝜀,𝑦)𝑊𝜀^′,−𝑦𝑑𝑦=⎧⎪⎨⎪⎩0𝜋𝑒_−𝜋𝜀if𝜀≠𝜀^′,1+𝑒^−2𝜋𝜀^1/2_∞−∞if𝜀=𝜀^′,𝑊(𝜀,𝑦)𝑊𝜀^,𝑦𝑑𝑦=2𝜋1+𝑒^−2𝜋𝜀^1/2𝛿𝜀−𝜀^.(119) The parabolic cylinder functions can be expressed in several forms [26] and we can use the various relations between these forms to express 𝑊(𝑎,𝑥) in terms of the more familiar function 𝐷_𝑝(𝑥) for some 𝑝. In

fact one can easily derive that𝑊(𝜀,𝑦)=𝑘2^1/2𝑒^𝑖𝜃+𝑒_−𝑖𝜃𝐷_{𝑖𝜀−1/2}𝑦𝑒^{−(𝑖/4)𝜋}𝐷_{−𝑖𝜀−1/2}𝑦𝑒^(𝑖/4)𝜋,(120) where𝜃=12−12𝜋𝜀+4𝑖𝜋+𝑖𝜙.(121) Next we observe that the solutions to (111) are identical to those of (110) but expressed in terms of the vriable 𝜂. Hence we can write the following for the wavefunction 𝜓:𝜓(𝜉,𝜂)=𝜓₁(𝜉)𝜓₂(𝜂),(122) where𝜓₁(𝜉)=𝐶₁𝑊𝐸𝛾,√2𝛾𝜉+𝐶₂𝑊𝐸𝛾,−√𝜓2𝛾𝜉₂,(𝜉)=𝐶^1𝑊𝐸𝛾,√2𝛾𝜂+𝐶^2𝑊𝐸𝛾,−√2𝛾𝜂.(123) We now consider the case in which the parameter 𝛼 is negative and write (110) and (111) as𝑑²𝑋𝑑𝜉²−|𝛼|8𝑔²_𝑁𝜉²𝑋=−𝐶|𝛼|𝑔_𝑁𝑋,(124)𝑑²𝑌𝑑𝜂²−|𝛼|8𝑔²_𝑁𝜂²𝑌=−𝐶|𝛼|𝑔_𝑁𝑌.(125) We define𝛾²=|𝛼|8𝑔²_𝑁𝐸=−𝐶|𝛼,|2𝑔_𝑁,(126) and thus they retain the same forms as in (112) and (113), respectively. Focussing on (124) we write it as𝑑²𝑋𝑑𝜉²−𝛾²𝜉²𝑋=2𝐸𝑋.(127) In terms of 𝑦=√2𝛾𝜉, (127) becomes𝑑²𝑋𝑑𝑦²+𝜎+12−14𝑦²𝑋=0,(128) where𝜎+12=−𝐸𝛾.(129)

Equation (128) has the form of Weber’s equation [27] and possesses the following solutio

n:𝑋₁(𝑦)=𝐷_𝜎(𝑦)=2^𝜎/2+1/4𝑦^−1/2𝑊_{𝜎/2+1/4,−1/4}𝑦2².(130) In the above equation 𝑊_𝜇,𝜈 is the Whittaker function. Expressing 𝑊_𝜇,𝜈 in terms of the confluent hypergeometric function, we can write𝑋₁(𝑦)=Γ(1/2)2^𝜎/2Γ(1/2−𝜎/2)𝑒^−𝑦2/4𝐹−𝜎+Γ(−1/2)2_𝜎/2−1/22,12,𝑦2²Γ(−𝜎/2)𝑦𝑒^−𝑦2/4𝐹1−𝜎2,32,𝑦2².(131) For the second solution of (128) we note that from (130) giving the relationship between 𝐷_𝜎 and the Whittaker function, we know that 𝐷_−𝜎−1(±𝑖𝑦) are solutions linearly independent of 𝐷_𝜎(𝑦) as 𝑊_{−(𝜎/2)−(1/4),−1/4}(−𝑦²/2) is linearly independent of 𝑊_{(𝜎/2)+(1/4),(−1/4)}(𝑦²/2). From the asymptotic behaviour of the confluent hypergeometric function, we deduce that as 𝑦→∞,𝑋₁(𝑦)∼𝑒^−𝑦2/4𝑦^𝜎,(132) that is, 𝑋₁→0. For 𝑦→−∞ we have𝑋₁(𝑦)∼−(2𝜋)^1/2Γ(−𝜎)𝑒^𝜎𝜋𝑖𝑒^𝑦2/4𝑦^−𝜎−1,(133) that is, 𝑋₁→∞ unless 𝜎 is a positive integer or zero in which case the R.H.S of (133) vanishes. In fact we have the relationship𝐷_𝑛(𝑦)=2^−𝑛/2𝑒^−𝑦2/4𝐻_𝑛√𝑦2,𝑛=0,1,2,…(134) that expresses the parabolic cylinder functions 𝐷_𝑛 in terms of the Hermite polynomials 𝐻_𝑛. Going back to (126) and (129) with 𝜎=𝑛, we obtain𝐶_𝑛=𝑛+1/2√2|𝛼|^1/2,(135) as the value of the separation constant. The functions 𝑋_𝑛(𝜉) that solve (127) are then precisely those that describe the one-dimensional quantum oscillator. We write𝑋_𝑛(𝜉)=𝛾𝜋^1/41√2^𝑛𝑛!𝐻^𝑛√𝛾𝜉𝑒^{−(1/2)𝛾𝜉2}.(136) The solutions 𝑌_𝑛(𝜂) are identical in form and we obtain for the normalized wave function the following:𝜓_𝑛(𝜉,𝜂)=𝛾𝜋^1/212^𝑛𝑛!𝐻^𝑛√𝛾𝜉𝐻_𝑛√𝛾𝜂𝑒^{−(1/2)𝛾(𝜉2+𝜂2)}.(137) For 𝜎≠𝑛 the wavefunctions will not have finite norm and solutions of (124) and (125) of the type given in (131) would have to be superposed, just as wave packets are constructed in quantum mechanics, in order to obtain wave functions capable of describing physical states.

7. Conclusions

In this work we studied 𝑓(𝑅) theories of gravity in two-dimensional spacetime with focus on applications to cosmology. With the metric taken to have to the FRW form we were able to obtain solutions for the cosmological field equations in the case of pure matter or radiation-dominated universe when 𝑓(𝑅)=𝑅+𝛼𝑅^𝑛. The remarkable feature of these solutions is that they readily describe an accelerating universe in contrast to the standard FRW cosmology of four-dimensional general relativity. The horizon problem is also readily solved. As we have stated in Section 2, the time evolution of the scale factor is not affected by the value of the curvature constant 𝑘. We have also seen that the solution for the radiation-dominated universe and one solution for the case of pure matter-domination, describe a hot big bang. However an interesting solution in a matter dominated universe, given in (42), describes a universe that kicks off with an infinite size and zero temperature at the start of matter dominance. It subsequently collapses to a finite size and then begins to expand.

Now as we mentioned before, the interest behind the pursuit of 𝑓(𝑅) theories is partially due to the desire to obtain a description of inflation without the introduction of scalar fields. This is done by seeking solutions to the cosmological field equations with the energy-momentum tensor set equal to zero, [7]. In Section 5 we obtained such solutions that characterize power law inflation. Furthermore, with inflation presumed to last for a short period of time, we obtained for the case 𝑛=2 a solution for 𝑡 near 𝑡_𝑖, the instant of onset of inflation. This solution displayed exponential dependence on time. For this case we computed the duration of inflation and the number of 𝑒-foldings as well as an estimate for the change that ensues in the value of the Hubble parameter from the start to the end of inflation. The basic distinguishing feature between power law and exponential inflation appears to be in the behavior of the Ricci scalar. Exponential inflation is obtained when we assumed that 𝑅(𝑡) can be expanded in a power series about 𝑡=𝑡_𝑖 with finite coefficients. In particular 𝑅(𝑡_𝑖) and 𝑅̇(𝑡_𝑖) are finite. On the other hand in the case of power law inflation these quantities exhibit singular behavior at 𝑡=𝑡. Another characteristic of our inflationary solutions is that they do not depend on the parameter that appears in (29) for 𝑓(𝑅). This is in contrast to the inflationary solution in four-dimensional 𝑓(𝑅) theories where 𝑛=2 describes the Starobinsky model [28]. In that case with 𝛼 being written as 𝛼=1/6𝑀², where the constant 𝑀 has the dimension of mass, exponential inflation is obtained with 𝑎, 𝐻, and 𝑅 all depending on 𝑀, [7].

Interest in two-dimensional theories stems partially from the desire to investigate the quantum theory in a simple setting. Hence we carried out quantization of the theory in the case of 𝑛=2. The Wheeler-DeWitt equation was derived and its solutions were obtained. We were able to solve the equation exactly in the entire domain of the variables, unlike the situation in the four-dimensional case [17, 20]. Interestingly we found that for 𝛼>0 the equation for the wave function coincided with that of the inverted oscillator. For 𝛼<0 the wave function, under certain conditions, turned out to be a product of two quantum harmonic oscillator wave functions in the variables 𝜉=𝑅+𝑎 and 𝜂=𝑅−𝑎. In conclusion we have studied some aspects of classical and quantum cosmology in two-dimensional 𝑓(𝑅) theories. Clearly a lot more issues need to be investigated and we hope to return to them in the near future.