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Physics Research International
VolumeΒ 2012Β (2012), Article IDΒ 506285, 11 pages
http://dx.doi.org/10.1155/2012/506285
Research Article

On 𝑓(𝑅) Theories in Two-Dimensional Spacetime

Physics Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received 22 June 2011; Revised 26 October 2011; Accepted 3 December 2011

Academic Editor: AshokΒ Chatterjee

Copyright Β© 2012 M. A. Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 actionπΌξ…žπΊ1=βˆ’ξ€œπ‘‘2πœ…4π‘₯βˆšβˆ’π‘”π‘…,(1) where πœ…=8πœ‹πΊ, 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 become1𝐼′=βˆ’ξ€œπ‘‘2πœ…4π‘₯βˆšβˆ’π‘”π‘“(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 𝐼𝐺1=βˆ’2π‘”π‘ξ€œπ‘‘2π‘₯βˆšβˆ’π‘”π‘…,(3) where 𝑔𝑁=8πœ‹πΊ2, 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 π‘…πœ‡πœˆπœ†πœŒ 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]:π‘…πœ‡πœˆπœ†πœŒ=12π‘…ξ€·π‘”πœ‡πœ†π‘”πœˆπœŒβˆ’π‘”πœ‡πœŒπ‘”πœˆπœ†ξ€Έ,(4) so that 𝑅 alone completely characterizes the local geometry. Equation (4) implies thatπ‘…πœ‡πœˆ=12π‘”πœ‡πœˆπ‘…,(5) so that the Einstein tensor πΊπœ‡πœˆ=π‘…πœ‡πœˆβˆ’(1/2)π‘”πœ‡πœˆπ‘…, 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=𝐼𝐺+𝐼𝑀,(6) where𝐼𝐺1=βˆ’2π‘”π‘ξ€œπ‘‘2π‘₯βˆšβˆ’π‘”π‘“(𝑅)(7) is the gravitational action and 𝐼𝑀 is the matter action [15]. The field equations can be derived by varying the action with respect to the metric tensor π‘”πœ‡πœˆ. Upon noting that the stress-energy tensor is defined by𝛿𝐼𝑀=12ξ€œπ‘‘2π‘₯βˆšβˆ’π‘”π‘‡πœ‡πœˆπ›Ώπ‘”πœ‡πœˆ,(8) we derive the following field equation:π‘“ξ…ž(𝑅)π‘…πœ‡πœˆβˆ’12π‘”πœ‡πœˆπ‘“(𝑅)βˆ’π‘”πœ‡πœˆπ‘“ξ…ž(𝑅)+βˆ‡πœ‡βˆ‡πœ‡π‘“ξ…ž(𝑅)=βˆ’π‘”π‘π‘‡πœ‡πœˆ.(9) In (4) π‘…πœ‡πœˆ is the Ricci tensor, the prime denotes the differentiation with respect to 𝑅, and the operator is defined by π‘“ξ…ž1(𝑅)=βˆšπœ•βˆ’π‘”πœ‡ξ‚€βˆšβˆ’π‘”π‘”πœ‡πœˆπœ•πœˆπ‘“ξ…žξ‚.(𝑅)(10) Using (5) we can write (9) as12π‘”πœ‡πœˆξ€·π‘“ξ…žξ€Έ(𝑅)π‘…βˆ’π‘“(𝑅)βˆ’π‘”πœ‡πœˆπ‘“ξ…ž(𝑅)+βˆ‡πœ‡βˆ‡πœˆπ‘“ξ…ž(𝑅)=βˆ’π‘”π‘π‘‡πœ‡πœˆ.(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=βˆ’π‘‘π‘‘2+π‘Ž2(𝑑)1βˆ’π‘˜π‘₯2𝑑π‘₯2,(12) in terms of the comoving coordinates π‘₯ and 𝑑. The quantity π‘Ž(𝑑) is the usual time-dependent cosmic scale factor. A change of variable 𝑑π‘₯2/(1βˆ’π‘˜π‘₯2)→𝑑π‘₯2 leads to 𝑑𝑠2=βˆ’π‘‘π‘‘2+π‘Ž2(𝑑)𝑑π‘₯2.(13) Thus in two dimensions the time evolution of π‘Ž(𝑑) is not affected by the value of π‘˜=0,Β±1 corresponding to the three different cosmological models [16]. This is unlike the four-dimensional case. The values π‘˜=0,βˆ’1 still describe spatially open flat and hyperbolic universe respectively, while π‘˜=1 describes a closed universe. The stress-energy tensor of the homogeneous isotropic universe is taken to be that of a perfect fluid:π‘‡πœ‡πœˆ=π‘π‘”πœ‡πœˆ+(𝑝+𝜌)π‘ˆπœ‡π‘ˆπœˆ,(14) where 𝑝 is the pressure, 𝜌 is the energy density, and π‘ˆπœ‡ is the comoving velocity. Using (13) and (14) we obtain from (11) the following two independent cosmological field equations:12ξ€·π‘…π‘“ξ…žξ€Έ+(𝑅)βˆ’π‘“(𝑅)Μ‡π‘Žπ‘Žπœ•π‘‘π‘“ξ…ž(𝑅)=𝑔𝑁1𝜌,2ξ€·π‘…π‘“ξ…žξ€Έ(𝑅)βˆ’π‘“(𝑅)+πœ•2π‘‘π‘“ξ…ž(𝑅)=βˆ’π‘”π‘π‘,(15) where we use the dot as well as πœ•π‘‘ to indicate differentiation with respect to time. We note that if 𝑓(𝑅) is expressed as a sum of powers 𝑅𝑛 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:βˆ‡π›Όπ‘‡π›Όπ›½=0,(16) and this, for a perfect fluid, gives rise to the following two equations:π‘ˆπ›Όβˆ‡π›ΌπœŒ+(𝑝+𝜌)βˆ‡π›Όπ‘ˆπ›Ό=0,(17)(𝑝+𝜌)π‘ˆπ›Όβˆ‡π›Όπ‘ˆπ›½+𝑔𝛼𝛽+π‘ˆπ›Όπ‘ˆπ›½ξ‚βˆ‡π›Όπ‘=0.(18) For the FRW metric of (13) one readily obtains from (17) thatπ‘‘π‘‘π‘Ž(πœŒπ‘Ž)=βˆ’π‘.(19) Assuming an equation of state of the form 𝑝=π›ΎπœŒ, where 𝛾 is a constant, (19) immediately leads to𝜌=πΆπ‘Žβˆ’π›Ύβˆ’1,(20) where 𝐢 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 (πœŒπ‘šβ‰ 0,πœŒπ‘Ÿ=0,𝛾=0) we haveπœŒπ‘š=πΆπ‘šπ‘Žβˆ’1,(21) while for a pure radiation universe (πœŒπ‘š=0,πœŒπ‘Ÿβ‰ 0,𝛾=1), one hasπœŒπ‘Ÿ=πΆπ‘Ÿπ‘Žβˆ’2.(22) Denoting the present time by 𝑑0 and using the usual notation of π‘Ž0β‰‘π‘Ž(𝑑0) and 𝜌0β‰‘πœŒ(𝑑0) to denote present-day values of these quantities, we can write for a matter-dominated universeπ‘π‘š=0,πœŒπ‘š(𝑑)=πœŒπ‘š0π‘Ž0,π‘Ž(𝑑)(23) while for a radiation-dominated universe one hasπ‘π‘Ÿ=πœŒπ‘Ÿ(𝑑)=πœŒπ‘Ÿ0ξ‚΅π‘Ž0ξ‚Άπ‘Ž(𝑑)2.(24) Finally we wish to note that for the FRW metric the curvature scalar of this two-dimensional universe is given by𝑅=βˆ’2Μˆπ‘Žπ‘Ž,(𝑑)(25) where Μˆπ‘Ž=𝑑2π‘Ž/𝑑𝑑2.

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:12ξ€·π‘…π‘“ξ…žξ€Έ+(𝑅)βˆ’π‘“(𝑅)Μ‡π‘Žπ‘Žπœ•π‘‘π‘“ξ…ž(𝑅)=π‘”π‘πœŒπ‘š0π‘Ž0π‘Ž,12ξ€·π‘…π‘“ξ…žξ€Έ(𝑅)βˆ’π‘“(𝑅)+πœ•2π‘‘π‘“ξ…ž(𝑅)=0.(26)

For the radiation dominated epoch the corresponding equations read12ξ€·π‘…π‘“ξ…ž(ξ€Έ+𝑅)βˆ’π‘“(𝑅)Μ‡π‘Žπ‘Žπœ•π‘‘π‘“ξ…ž(𝑅)=π‘”π‘πœŒπ‘Ÿ0π‘Ž20π‘Ž2,(27)12ξ€·π‘…π‘“ξ…žξ€Έ(𝑅)βˆ’π‘“(𝑅)+πœ•2π‘‘π‘“ξ…ž(𝑅)=βˆ’π‘”π‘πœŒπ‘Ÿ0π‘Ž20π‘Ž2.(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+𝛼𝑅𝑛,(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(π‘›βˆ’1)𝛼𝑅𝑛+𝑛(π‘›βˆ’1)π›Όπ‘…π‘›βˆ’2Μ‡π‘…Μ‡π‘Žπ‘Ž=π‘”π‘πœŒπ‘š0π‘Ž0π‘Ž,(30)ΜˆΜ‡π‘…2𝑛𝑅𝑅+2𝑛(π‘›βˆ’2)2+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 bẏ𝑅𝑧(𝑅)=2,(32) the equation is transformed into the following form:𝑑𝑧+𝑑𝑅2(π‘›βˆ’2)𝑅𝑅𝑧+2𝑛=0.(33) This equation is easily solved and we obtain for 𝑛≠1/2̇𝑅𝑧(𝑅)=21=βˆ’π‘›π‘…(2π‘›βˆ’1)3+𝐢1𝑅4βˆ’2𝑛,(34) where 𝐢1 is a constant. Equation (34) then leads to the parametric solution:ξ€œξ‚Έβˆ’π‘…π‘‘=Β±3𝑛(2π‘›βˆ’1)+𝐢1𝑅4βˆ’2π‘›ξ‚Ήβˆ’1/2𝑑𝑅+𝐢2,(35) where 𝐢2 is a constant. For 𝑛=2 and 𝐢1β‰ 0 one can carry out the integration using the result [16]ξ€œπ‘‘π‘₯ξ€·πΎβˆ’π‘₯𝛼+2ξ€Έ1/2=π‘₯√𝐾2𝐹1ξ‚΅12,1,𝛼+2𝛼+3;π‘₯𝛼+2𝛼+2𝐾,(36) where 𝛼 and 𝐾 are constants and 2𝐹1 is the hypergeometric function. We obtainξ‚΅6𝑑=±𝐢1ξ‚Ά1/2𝑅2𝐹1ξ‚΅12,13,43;𝑅3𝐢1ξ‚Ά+𝐢2.(37)

Ideally one should solve (37) to obtain 𝑅 as a function of the cosmic time 𝑑 and plug that into (30) in order to solve for π‘Ž(𝑑) in the case of 𝑛=2, but that is a difficult task. Instead we consider solutions for which 𝐢1=0 in (34) and a general 𝑛≠1/2. One can then easily derive that𝑅=βˆ’4𝑛(2π‘›βˆ’1)ξ€·π‘‘βˆ’π‘‘π‘šξ€Έ2,(38) where we have renamed the integration constant 𝐢2 as π‘‘π‘š. 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 obtain𝐴2Μ‡π‘Ž+𝐴1ξ€·π‘‘βˆ’π‘‘π‘šξ€Έβˆ’1π‘Ž=πΎπ‘Ž0ξ€·π‘‘βˆ’π‘‘π‘šξ€Έ2π‘›βˆ’1,(39) where𝐴1=12(π‘›βˆ’1)𝑁𝑛,𝐴2=βˆ’2𝑛(π‘›βˆ’1)π‘π‘›βˆ’1,(40)𝑔𝑁=4𝑛(1βˆ’2𝑛),𝐾=π‘πœŒπ‘š0𝛼.(41) We readily solve (39) and getπ‘Ž(𝑑)=πΆξ€·π‘‘βˆ’π‘‘π‘šξ€Έ1βˆ’2𝑛+πΎξ€·π‘‘βˆ’π‘‘π‘šξ€Έ2𝑛,(42) where 𝐢 is a constant and𝐾=πΎπ‘Ž0(4π‘›βˆ’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 𝑑 dependence of 𝑅 is (π‘‘βˆ’π‘‘π‘š)βˆ’2 and thus independent of 𝑛, while that of π‘Ž(𝑑) does depend on 𝑛. We also note that the relation 𝑅=βˆ’2Μˆπ‘Ž/π‘Ž 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 𝐢 and 𝐾 must be such that π‘Ž(𝑑) is positive.

We now turn to the case of radiation. Upon adding (27) and (28) we obtain π‘…π‘“ξ…ž(𝑅)βˆ’π‘“(𝑅)+πœ•2π‘‘π‘“ξ…ž(𝑅)+Μ‡π‘Žπ‘Žπœ•π‘‘π‘“ξ…ž(𝑅)=0.(44) Employing in (44) the expression for 𝑓(𝑅) given in (29) above yieldsΜˆΜ‡π‘…π‘›π‘…π‘…+𝑛(π‘›βˆ’2)2+𝑅3̇𝑅+π‘›π‘…Μ‡π‘Žπ‘Ž=0.(45) Next we use (29) in (27) and obtain12(π‘›βˆ’1)𝛼𝑅𝑛+𝑛(π‘›βˆ’1)π›Όπ‘…π‘›βˆ’2Μ‡π‘…Μ‡π‘Žπ‘Ž=π‘”π‘πœŒπ‘Ÿ0π‘Ž20π‘Ž2.(46) Motivated by the structure of the solutions for the cosmological equations in the case of pure matter above, we seek solutions for 𝑅(𝑑) and π‘Ž(𝑑) of (45) and (46) in the form of powers in π‘‘βˆ’π‘‘π‘Ÿ where π‘‘π‘Ÿ is some reference time. We obtain the following results:𝑅(𝑑)=2𝑛(π‘›βˆ’1)π‘‘βˆ’π‘‘π‘Ÿξ€Έβˆ’2,(47)ξ€·π‘Ž(𝑑)=π΅π‘‘βˆ’π‘‘π‘Ÿξ€Έπ‘›,(48) where the constant 𝐡 is given by𝑔𝐡=π‘πœŒπ‘Ÿ0[]𝑛(π‘›βˆ’1)(1βˆ’3𝑛)2𝑛(1βˆ’π‘›)π‘›βˆ’1𝛼1/2π‘Ž0.(49) Note that, as in the case of matter, the 𝑑 dependence of 𝑅(𝑑) is independent of 𝑛, the only such dependence appears in the overall coefficient. We also note that the relation 𝑅=βˆ’2Μˆπ‘Ž/π‘Ž is satisfied by the solutions for 𝑅 given in (46) and (48). For an expanding universe one must have 𝑛>1 and 𝐡>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 𝑑𝑠 the proper distance 𝑑(𝑑𝑠) of the emitter is given by𝑑𝑑𝑠𝑑=π‘Žπ‘ ξ€Έξ€œπ‘‘π‘ π‘‘π‘’π‘‘π‘‘β€²π‘Ž(π‘‘ξ…ž),(50) where 𝑑𝑒 is the time of emission of the photon. Using (48) we obtain𝑑𝑑𝑠=ξ€·π‘‘π‘ βˆ’π‘‘π‘Ÿξ€Έπ‘›ξ‚ƒξ€·π‘‘1βˆ’π‘›π‘ βˆ’π‘‘π‘Ÿξ€Έ1βˆ’π‘›βˆ’ξ€·π‘‘π‘’βˆ’π‘‘π‘Ÿξ€Έ1βˆ’π‘›ξ‚„.(51) We can view π‘‘π‘Ÿ as signifying the onset of the radiation epoch. We see that as π‘‘π‘’β†’π‘‘π‘Ÿ, 𝑑(𝑑𝑠) is finite for 1βˆ’π‘›>0 and diverges for 1βˆ’π‘›<0. Hence no particle horizon problem will arise if 𝑛>1 which is the same condition required for an expanding universe. Reality of 𝐡 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 Μˆπ‘Ž(𝑑) which is given byξ€·Μˆπ‘Ž(𝑑)=𝑛(π‘›βˆ’1)π΅π‘‘βˆ’π‘‘π‘Ÿξ€Έπ‘›βˆ’2,(52) is positive for 𝑑>π‘‘π‘Ÿ since 𝑛>1 and is constant for 𝑛=2. Now in two-dimensional spacetime the radiation energy density is πœŒπ‘Ÿβˆπ‘‡2 where 𝑇 is the temperature [16] and it follows therefore from (24) thatπ‘Žβˆπ‘‡βˆ’1.(53) Since we have π‘Žβ†’0 as π‘‘β†’π‘‘π‘Ÿ, 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 𝐢=0 when the scale factor becomesπ‘Ž(𝑑)=πΎξ€·π‘‘βˆ’π‘‘π‘šξ€Έ2𝑛.(54) As we have stated earlier this is viable because it represents a solution of (39). The time π‘‘π‘š can be taken to signify the onset of matter dominance. The proper distance 𝑑(𝑑𝑠) is now given by𝑑𝑑𝑠=ξ€·π‘‘π‘ βˆ’π‘‘π‘šξ€Έ2𝑛𝑑1βˆ’2π‘›π‘ βˆ’π‘‘π‘šξ€Έ1βˆ’2π‘›βˆ’ξ€·π‘‘π‘’βˆ’π‘‘π‘šξ€Έ1βˆ’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 π‘Ž(𝑑)>0 for 𝑑>π‘‘π‘š 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 Μˆβ€Œπ‘Ž(𝑑) is positive. Finally we observe that for the pure matter universe we have π‘Ž(𝑑)β†’0 as π‘‘β†’π‘‘π‘š.

We now consider the case 𝐢≠0. Using (42) the proper distance is now given by𝑑𝑑𝑠𝑑=π‘Žπ‘ ξ€Έξ€œπ‘‘π‘ π‘‘π‘’ξ€·π‘‘βˆ’π‘‘π‘šξ€Έ2π‘›βˆ’1𝐢+πΎξ€·π‘‘βˆ’π‘‘π‘šξ€Έ4π‘›βˆ’1𝑑𝑑.(56)

It is clear that the integral converges for π‘‘π‘’β†’π‘‘π‘š and we do have a particle horizon. Performing the integral we determine the proper distance to the horizon to be𝑑𝑑𝑠=π‘Žξ€·π‘‘π‘ ξ€ΈπΆξ‚΅πΎπΆξ‚Ά2𝑛(4π‘›βˆ’1)Γ—ξƒ―βˆ’ξ€·ln1+πœ‰π‘ ξ€Έβˆ’14π‘›βˆ’14π‘›βˆ’12π‘›βˆ’1ξ“π‘˜=1ξ‚Έcos2π‘›πœ‹(2π‘˜βˆ’1)ξ‚Ήξ‚€4π‘›βˆ’1Γ—ln1βˆ’2πœ‰π‘ cos2π‘˜βˆ’14π‘›βˆ’1πœ‹+πœ‰2𝑠+24π‘›βˆ’12π‘›βˆ’1ξ“π‘˜=1ξ‚Έsin2π‘›πœ‹(2π‘˜βˆ’1)ξ‚ΉΓ—4π‘›βˆ’1arctgξ‚Έπœ‰π‘ βˆ’cos((2π‘˜βˆ’1)/(4π‘›βˆ’1))πœ‹ξ‚Ήβˆ’ξ€·πœ‰sin((2π‘˜βˆ’1)/(4π‘›βˆ’1))πœ‹π‘ βŸ·πœ‰π‘’ξ€Έξƒ°,(57) where πœ‰π‘—=𝐢𝐾4π‘›βˆ’1ξ€·π‘‘π‘—βˆ’π‘‘π‘šξ€Έ,𝑗=𝑠,𝑒.(58) Let us now study further properties of the solution given in (42). In the following we consider only values of 𝑑 such that 𝑑>π‘‘π‘š. Now it is evident that, except for values of 𝑛 in the interval 0<𝑛<1/2, the first term in (42) dominates for 𝑑 near π‘‘π‘š when 𝑛>1/2 while the second term dominates for 𝑛<0. Hence to ensure positivity of the scale factor we require that both 𝐢 and 𝐾 be positive. For 0<𝑛<1/2, 𝐢 and 𝐾 can have opposite signs but only in such a manner so as to keep π‘Ž>0. We shall for simplicity assume that 𝐢>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 Μˆπ‘Ž(𝑑) is given byξ€·Μˆπ‘Ž(𝑑)=2𝑛(2π‘›βˆ’1)π‘‘βˆ’π‘‘π‘šξ€Έβˆ’2π‘Ž(𝑑).(59) It is seen that Μˆπ‘Ž<0 for 0<𝑛<1/2, Μˆπ‘Ž=0 for 𝑛=1/2, and Μˆπ‘Ž>0 for 𝑛<0 or 𝑛>1/2.

Next we consider the behavior of π‘Ž(𝑑) as π‘‘β†’π‘‘π‘š for the case 𝐢≠0. We see from (42) that for 0<𝑛<1/2, π‘Ž(𝑑)β†’0 as π‘‘β†’π‘‘π‘š and accordingly the temperature π‘‡β†’βˆž in this limit. For 𝑛=1/2, we have π‘Ž(𝑑)→𝐢 as π‘‘β†’π‘‘π‘š and 𝑇 is finite. However for 𝑛 outside the interval 0≀𝑛≀1/2 the behavior of π‘Ž(𝑑) is very different as π‘‘β†’π‘‘π‘š. We see that π‘Ž(𝑑)β†’βˆž in this limit and energy density πœŒπ‘š and the temperature tend to zero. As 𝑑 increases beyond the value π‘‘π‘š, π‘Ž(𝑑) decreases to finite values and the density increases. However π‘Ž(𝑑) never reaches zero and attains a minimum value at 𝑑=𝑑𝑐 given by𝑑𝑐=π‘‘π‘š+(2π‘›βˆ’1)𝐢2𝑛𝐾1/4π‘›βˆ’1.(60) For 𝑑>𝑑𝑐, π‘Ž(𝑑) starts to increase. We also note from (38) that the curvature scalar is π‘…β†’βˆ’βˆž as π‘‘β†’π‘‘π‘š and then starts increasing through finite negative values as 𝑑 grows beyond π‘‘π‘š. 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 π‘Ž(𝑑):Μˆπ‘Ž(𝑑)>0.(61) Now as evident from the analysis of Section 4, Μˆπ‘Ž>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 π‘Ž(𝑑). Putting 𝐢1=0 enables the integration in (35) to be performed and leads to the solution given in (38) which we write as𝑅=βˆ’4𝑛(2π‘›βˆ’1)ξ€·π‘‘βˆ’π‘‘ξ€Έ2,(62) where 𝑛≠1/2, 1 and we have now denoted the integration constant by 𝑑. Using (62) in (30) with the R.H.S. set being equal to zero yields:Μ‡π‘Ž+2π‘›βˆ’1π‘‘βˆ’π‘‘π‘Ž=0,(63) the solution of which readsξ€·π‘Ž(𝑑)=π΄π‘‘βˆ’π‘‘ξ€Έ1βˆ’2𝑛,(64) where 𝐴>0 is a constant. We take the solution to hold for 𝑑>𝑑. The cosmic acceleration is given byξ€·Μˆπ‘Ž(𝑑)=2𝑛(2π‘›βˆ’1)π΄π‘‘βˆ’π‘‘ξ€Έβˆ’2π‘›βˆ’1.(65) The Hubble parameter is𝐻=Μ‡π‘Žπ‘Ž=1βˆ’2π‘›π‘‘βˆ’π‘‘,̇𝐻=2π‘›βˆ’1ξ€·π‘‘βˆ’π‘‘ξ€Έ2.(66) For 𝑛<βˆ’1 we can identify 𝑑 with the onset of inflation 𝑑=𝑑𝑖. Equation (64) then describes a universe that expands with positive acceleration for 𝑑>𝑑𝑖. We also have 𝐻>0 and ̇𝐻<0 for 𝑑>𝑑𝑖 which characterizes standard inflation. However if we make the identification 𝑑=𝑑𝑖 for 𝑛>1, we will have a situation in which π‘Ž(𝑑)β†’βˆž as 𝑑→𝑑𝑖 thus obtaining a universe that starts off already with an infinite size at the onset of inflation collapsing subsequently for 𝑑>𝑑𝑖 at an accelerated rate. Such a scenario can be avoided if 𝑑 is instead taken to have a relatively large value so that 𝑑<𝑑 during the inflationary epoch. We write π‘Ž(𝑑) now as||π‘Ž(𝑑)=π΄π‘‘βˆ’π‘‘||1βˆ’2𝑛.(67) The universe then starts off with a relatively small non-zero size at 𝑑=𝑑𝑖 and expands with positive acceleration as time progresses. We also have𝐻=2π‘›βˆ’1,̇𝐻=π‘‘βˆ’π‘‘1βˆ’2π‘›ξ€·ξ€Έπ‘‘βˆ’π‘‘2,(68) so that 𝐻>0 and ̇𝐻<0 and we again have standard inflation.

As in four spacetime dimensions we define the so-called slow-roll parameter πœ€ by [7]Μ‡π»πœ€=βˆ’π»2.(69) and in terms of which one hasΜˆπ‘Žπ‘Ž=𝐻2+̇𝐻=(1βˆ’πœ€)𝐻2.(70) Inflation can thus be attained only if πœ€<1. In our present context πœ€ is given by1πœ€=||||.2π‘›βˆ’1(71) For both cases of 𝑛<βˆ’1 and 𝑛>1 we clearly have πœ€<1. The slow-roll approximation corresponding to πœ€β‰ͺ1 then obtain when |2π‘›βˆ’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:Μ‡2π‘›π‘…Μ‡π‘Ž+𝑅2π‘Ž=0.(72) Now from (25) of Section 2 we obtaiṅ𝑅=βˆ’2βƒ›π‘Žπ‘Ž+2Μˆπ‘ŽΜ‡π‘Žπ‘Ž2.(73) Substituting (25) and (73) in (72) one obtainsβˆ’π‘›π‘ŽΜ‡π‘Žβƒ›π‘Ž+π‘ŽΜˆπ‘Ž2+π‘›Μ‡π‘Ž2Μˆπ‘Ž=0.(74) Next in terms of 𝐻, ̇𝐻, and ̈𝐻 we can express (74), after some manipulations, asΜˆΜ‡βˆ’π‘›π»π»βˆ’2(π‘›βˆ’1)𝐻𝐻2+̇𝐻2+𝐻4=0.(75) It is customary, in dealing with equations such as this, to invoke the slow-roll approximation |̇𝐻/𝐻2|β‰ͺ1 and |ΜˆΜ‡π»/𝐻𝐻|β‰ͺ1, [7, 17]. Applying this to (67) we obtain thatΜ‡βˆ’2(π‘›βˆ’1)𝐻+𝐻2=0.(76) The solution of (76) is𝐻(𝑑)=βˆ’2(π‘›βˆ’1)π‘‘βˆ’π‘‘β€²,(77) where π‘‘ξ…ž is a constant. Equation (77) in turn givesπ‘Ž(𝑑)=π΄β€²ξ‚€π‘‘βˆ’π‘‘ξ…žξ‚2(π‘›βˆ’1)(78) with π΄ξ…ž being another constant. Equation (78) for π‘Ž(𝑑) 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(𝑑) that holds for 𝑑 close to the instant 𝑑𝑖 that signifies the onset of inflation. Specifically we assume that 𝑑=𝑑𝑖 is a regular point of (31) and seek a solution for 𝑅(𝑑) in the form of a power series confining ourselves to small values of π‘‘βˆ’π‘‘π‘–. For simplicity we consider the case 𝑛=2 for which (31) becomes4Μˆπ‘…+𝑅2=0.(79)

We write𝑅(𝑑)=βˆžξ“π‘š=0π‘π‘šξ€·π‘‘βˆ’π‘‘π‘–ξ€Έπ‘š.(80)

Substituting (80) in (79) and solving we obtain𝑏21=βˆ’8𝑏20,𝑏31=βˆ’π‘120𝑏1,(81) and so forth. This leads to𝑅(𝑑)=𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έβˆ’18𝑏20ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έ2βˆ’1𝑏120𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έ3+β‹―.(82) We remark that if inflation lasts for a short period of time, then it is sensible to have a representation for 𝑅(𝑑) as given in (82). Moreover for sufficiently small π‘‘βˆ’π‘‘π‘– we can approximate 𝑅(𝑑) by the first two terms and substitute in (72) with 𝑛=2. Solving the resulting equation we obtainξƒ―βˆ’1π‘Ž(𝑑)β‰ˆπΆexp12𝑏21𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»3ξƒ°,(83) where 𝐢>0 is a constant. We can write (83) asπ‘Ž(𝑑)β‰ˆπ‘Žπ‘–ξƒ―βˆ’1exp12𝑏21𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»3βˆ’π‘30,(84) whereπ‘Žπ‘–ξ€·π‘‘=π‘Žπ‘–ξ€Έξƒ©βˆ’π‘=𝐢exp3012𝑏21ξƒͺ.(85) From (83) we obtain1Μ‡π‘Ž(𝑑)=βˆ’4𝑏1𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»2ξƒ―βˆ’1π‘Ž(𝑑),Μˆπ‘Ž(𝑑)=2𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–+1ξ€Έξ€»16𝑏21𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»4ξƒ°π‘Ž(𝑑).(86) From (84) we see that we must have 𝑏1<0 to ensure that Μ‡π‘Ž>0. We must also require π‘Ž(𝑑) to be increasing for 𝑑>𝑑𝑖. This can be achieved by having 𝑏0>0 for then 𝑏0+𝑏1(π‘‘βˆ’π‘‘π‘–) will start off at the value 𝑏0 and decreases reaching zero at π‘‘βˆ—βˆ’π‘‘π‘–=βˆ’π‘0/𝑏1. During the interval, 𝑑𝑖<𝑑<π‘‘βˆ—, π‘Ž(𝑑) will be increasing. We must also require the cosmic acceleration Μˆπ‘Ž(𝑑) to be positive during this interval and this leads to the following condition:18𝑏21𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»3>1.(87) This inequality will continue to hold until 𝑑=𝑑𝑓<π‘‘βˆ— when Μˆπ‘Ž(𝑑𝑓)=0. This implies that18𝑏21𝑏0+𝑏1ξ€·π‘‘π‘“βˆ’π‘‘π‘–ξ€Έξ€»3=1,(88) which yields𝑑𝑓=𝑑𝑖+𝑏0||𝑏1||βˆ’1||π‘βˆ’21||βˆ’1/3.(89) The time 𝑑𝑓 then signifies the end of inflation. Since 𝑅(𝑑𝑖)=𝑏0 and ̇𝑅(𝑑𝑖)=𝑏1, the conditions 𝑏0>0 and 𝑏1<0 can be expressed as𝑅𝑑𝑖̇𝑅𝑑>0,𝑖<0.(90) We can also express the duration of inflation asπ‘‘π‘“βˆ’π‘‘π‘–ξ€·π‘‘=𝑅𝑖||̇𝑅𝑑𝑖||βˆ’1||Μ‡π‘…ξ€·π‘‘βˆ’2𝑖||βˆ’1/3.(91)

The Hubble parameter is given by1𝐻=βˆ’4𝑏1𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘1ξ€Έξ€»2.(92)

It thus decreases from an initial value 𝐻𝑖 given by𝐻𝑖𝑑=𝐻𝑖𝑏=βˆ’204𝑏1=𝑅2𝑑𝑖4||̇𝑅𝑑𝑖||,(93)

to a value 𝐻𝑓 at the end of inflation where𝐻𝑓𝑑=𝐻𝑓=||𝑏1||1/3=||̇𝑅𝑑𝑖||1/3.(94) We note thatΜ‡1𝐻=βˆ’2𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»(95) is negative during 𝑑𝑖<𝑑<𝑑𝑓 and we thus have standard inflation. The slow-roll parameter is given byπœ€=8𝑏21𝑏0+𝑏1ξ€·π‘‘βˆ’π‘‘π‘–ξ€Έξ€»βˆ’3.(96) We recall that for inflation to proceed one must have πœ€<1 and this leads precisely to the condition expressed in (87) stated earlier.

The number of 𝑒-foldings from 𝑑=𝑑𝑖 to 𝑑=𝑑𝑓 is defined by [7, 18]ξ€œπ‘=𝑑𝑓𝑑𝑖𝐻𝑑𝑑,(97) which is evaluated to give2𝑁=3𝐻𝑖𝐻𝑓3/2ξƒ­.βˆ’1(98) In four dimensions, the solution of the horizon and flatness problems of big bang cosmology requires that 𝑁β‰₯70, [7, 19]. If we assume that we can use this value in our two-dimensional universe, we find that𝐻𝑖𝐻𝑓β‰₯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 as𝐼𝐺1=βˆ’2π‘”π‘ξ€œπ‘‘π‘‘π‘Ž(𝑑)𝑓(𝑅(𝑑)).(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 write𝐼𝐺1=βˆ’2π‘”π‘ξ€œπ‘‘π‘‘(π‘Žπ‘…βˆ’2π›ΌΜˆπ‘Žπ‘…).(101) We notice the appearance of the second derivative of π‘Ž in (101). The standard approach is to express the wave function in terms of π‘Ž and 𝑅 [20]. Hence integrating by parts in (101), we obtain𝐼𝐺=ξ€œβ„’ξ€·Μ‡π‘˜ξ€Έπ‘Ž,Μ‡π‘Ž,π‘˜,𝑑𝑑,(102) where1β„’=βˆ’2𝑔𝑁̇𝑅.π‘Žπ‘…+2π›ΌΜ‡π‘Ž(103) The canonical momenta are defined in the usual way:π‘ƒπ‘Ž=πœ•β„’π›Όπœ•Μ‡π‘Ž=βˆ’π‘”π‘Μ‡π‘ƒπ‘…,𝑅=πœ•β„’πœ•Μ‡π‘…π›Ό=βˆ’π‘”π‘Μ‡π‘Ž.(104) The Hamiltonian is then obtained asβ„‹=π‘ƒπ‘ŽΜ‡π‘Ž+π‘ƒπ‘…Μ‡π‘”π‘…βˆ’β„’=βˆ’π‘π›Όπ‘ƒπ‘Žπ‘ƒπ‘…+12π‘”π‘π‘Žπ‘….(105) Replacing π‘ƒπ‘Ž and 𝑃𝑅 by βˆ’π‘–(πœ•/πœ•π‘Ž) and βˆ’π‘–(πœ•/πœ•π‘…), respectively, in the Hamiltonian, we obtain the Wheeler-DeWitt equation for the wave function of the universe:ξ‚΅π‘”π‘π›Όπœ•2+1πœ•π‘…πœ•π‘Ž2π‘”π‘ξ‚Άπ‘Žπ‘…πœ“(π‘Ž,𝑅)=0.(106) Instead of π‘Ž and 𝑅 we shall work with the variables:πœ‰=𝑅+π‘Ž,πœ‚=π‘…βˆ’π‘Ž.(107) In terms of πœ‰ and πœ‚ the Wheeler-DeWitt equation becomesξƒ¬π‘”π‘π›Όξ‚΅πœ•2πœ•πœ‰2βˆ’πœ•2πœ•πœ‚2ξ‚Ά+1π‘”π‘ξ€·πœ‰2βˆ’πœ‚2ξ€Έξƒ­πœ“(πœ‰,πœ‚)=0.(108) We seek solutions of (108) in factorizable form:πœ“(πœ‰,πœ‚)=𝑋(πœ‰)π‘Œ(πœ‚)(109) and obtain the following equations for the functions 𝑋 and π‘Œ:𝑑2π‘‹π‘‘πœ‰2+𝛼8𝑔2π‘πœ‰2𝑋=𝐢𝛼𝑔𝑁𝑋,(110)𝑑2π‘Œπ‘‘πœ‚2+𝛼8𝑔2π‘πœ‚2π‘Œ=πΆπ›Όπ‘”π‘π‘Œ,(111) where 𝐢 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=𝛼8𝑔2𝑁,(112)𝐸=βˆ’πΆπ›Ό2𝑔𝑁.(113) In terms of 𝛾2 and 𝐸, (110) reads𝑑2π‘‹π‘‘πœ‰2+𝛾2πœ‰2𝑋+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 variableβˆšπ‘¦=2π›Ύπœ‰,(115) we cast (114) into the following form:𝑑2𝑋𝑑𝑦2+14𝑦2𝑋+πœ€π‘‹=0,(116) where πœ€=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 π‘Š(πœ€,𝑦) and π‘Š(πœ€,βˆ’π‘¦) [26]. For |𝑦|≫1 and |𝑦|≫|πœ€| these solutions display the following asymptotic behaviour:ξ‚™π‘Š(πœ€,π‘¦βŸΆβˆž)∼2π‘˜π‘¦ξ‚€1cos4𝑦21+πœ€ln𝑦+41πœ‹+2πœ™ξ‚,ξƒŽπ‘Š(πœ€,π‘¦βŸΆβˆ’βˆž)∼2π‘˜||𝑦||ξ‚€1sin4𝑦2||𝑦||+1+πœ€ln41πœ‹+2πœ™ξ‚,(117) whereξ€·π‘˜=1+π‘’βˆ’2πœ‹πœ€ξ€Έ1/2βˆ’π‘’βˆ’πœ‹πœ€,πœ™ξ‚€1=argΞ“2.βˆ’π‘–πœ€(118) The functions π‘Š(πœ€,𝑦) and π‘Š(πœ€,βˆ’π‘¦) satisfy the following normalization conditions [23]:ξ€œβˆžβˆ’βˆžξ‚€πœ€π‘Š(πœ€,𝑦)π‘Šβ€²ξ‚βŽ§βŽͺ⎨βŽͺ⎩0,βˆ’π‘¦π‘‘π‘¦=ifπœ€β‰ πœ€β€²,πœ‹π‘’βˆ’πœ‹πœ€ξ€·1+π‘’βˆ’2πœ‹πœ€ξ€Έ1/2ifπœ€=πœ€β€²,ξ€œβˆžβˆ’βˆžξ€·πœ€π‘Š(πœ€,𝑦)π‘Šξ…žξ€Έξ€·,𝑦𝑑𝑦=2πœ‹1+π‘’βˆ’2πœ‹πœ€ξ€Έ1/2π›Ώξ€·πœ€βˆ’πœ€ξ…žξ€Έ.(119) The parabolic cylinder functions can be expressed in several forms [26] and we can use the various relations between these forms to express π‘Š(π‘Ž,π‘₯) in terms of the more familiar function 𝐷𝑝(π‘₯) for some 𝑝. In

fact one can easily derive thatξ‚€π‘˜π‘Š(πœ€,𝑦)=21/2ξ€Ίπ‘’π‘–πœƒπ·π‘–πœ€βˆ’1/2ξ€·π‘¦π‘’βˆ’(𝑖/4)πœ‹ξ€Έ+π‘’βˆ’π‘–πœƒπ·βˆ’π‘–πœ€βˆ’1/2𝑦𝑒(𝑖/4)πœ‹,ξ€Έξ€»(120) where1πœƒ=2ξ‚€βˆ’12π‘–πœ‹πœ€+4.πœ‹+π‘–πœ™(121) Next we observe that the solutions to (111) are identical to those of (110) but expressed in terms of the vriable πœ‚. Hence we can write the following for the wavefunction πœ“:πœ“(πœ‰,πœ‚)=πœ“1(πœ‰)πœ“2,(πœ‚)(122) whereπœ“1(πœ‰)=𝐢1π‘Šξ‚΅πΈπ›Ύ,βˆšξ‚Ά2π›Ύπœ‰+𝐢2π‘Šξ‚΅πΈπ›Ύβˆš,βˆ’ξ‚Ά,πœ“2π›Ύπœ‰2(πœ‰)=πΆξ…ž1π‘Šξ‚΅πΈπ›Ύ,βˆšξ‚Ά2π›Ύπœ‚+πΆξ…ž2π‘Šξ‚΅πΈπ›Ύβˆš,βˆ’ξ‚Ά.2π›Ύπœ‚(123) We now consider the case in which the parameter 𝛼 is negative and write (110) and (111) as𝑑2π‘‹π‘‘πœ‰2βˆ’|𝛼|8𝑔2π‘πœ‰2𝑋=βˆ’πΆ|𝛼|𝑔𝑁𝑋,(124)𝑑2π‘Œπ‘‘πœ‚2βˆ’|𝛼|8𝑔2π‘πœ‚2π‘Œ=βˆ’πΆ|𝛼|π‘”π‘π‘Œ.(125) We define𝛾2=|𝛼|8𝑔2𝑁,𝐸=βˆ’πΆ|𝛼|2𝑔𝑁,(126) and thus they retain the same forms as in (112) and (113), respectively. Focussing on (124) we write it as𝑑2π‘‹π‘‘πœ‰2βˆ’π›Ύ2πœ‰2𝑋=2𝐸𝑋.(127) In terms of βˆšπ‘¦=2π›Ύπœ‰, (127) becomes𝑑2𝑋𝑑𝑦2+ξ‚€1𝜎+2βˆ’14𝑦2𝑋=0,(128) where1𝜎+2𝐸=βˆ’π›Ύ.(129)

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

n:𝑋1(𝑦)=𝐷𝜎(𝑦)=2𝜎/2+1/4π‘¦βˆ’1/2π‘ŠπœŽ/2+1/4,βˆ’1/4𝑦22ξ‚Ά.(130) In the above equation π‘Šπœ‡,𝜈 is the Whittaker function. Expressing π‘Šπœ‡,𝜈 in terms of the confluent hypergeometric function, we can write𝑋1Ξ“(𝑦)=(1/2)2𝜎/2𝑒Γ(1/2βˆ’πœŽ/2)βˆ’π‘¦2/4πΉξ‚΅βˆ’πœŽ2,12,𝑦22ξ‚Ά+Ξ“(βˆ’1/2)2𝜎/2βˆ’1/2Ξ“(βˆ’πœŽ/2)π‘¦π‘’βˆ’π‘¦2/4𝐹1βˆ’πœŽ2,32,𝑦22ξ‚Ά.(131) For the second solution of (128) we note that from (130) giving the relationship between 𝐷𝜎 and the Whittaker function, we know that π·βˆ’πœŽβˆ’1(±𝑖𝑦) are solutions linearly independent of 𝐷𝜎(𝑦) as π‘Šβˆ’(𝜎/2)βˆ’(1/4),βˆ’1/4(βˆ’π‘¦2/2) is linearly independent of π‘Š(𝜎/2)+(1/4),(βˆ’1/4)(𝑦2/2). From the asymptotic behaviour of the confluent hypergeometric function, we deduce that as π‘¦β†’βˆž,𝑋1(𝑦)βˆΌπ‘’βˆ’π‘¦2/4π‘¦πœŽ,(132) that is, 𝑋1β†’0. For π‘¦β†’βˆ’βˆž we have𝑋1(𝑦)βˆΌβˆ’(2πœ‹)1/2Γ𝑒(βˆ’πœŽ)πœŽπœ‹π‘–π‘’π‘¦2/4π‘¦βˆ’πœŽβˆ’1,(133) that is, 𝑋1β†’βˆž unless 𝜎 is a positive integer or zero in which case the R.H.S of (133) vanishes. In fact we have the relationship𝐷𝑛(𝑦)=2βˆ’π‘›/2π‘’βˆ’π‘¦2/4π»π‘›ξƒ©π‘¦βˆš2ξƒͺ,𝑛=0,1,2,…(134) that expresses the parabolic cylinder functions 𝐷𝑛 in terms of the Hermite polynomials 𝐻𝑛. Going back to (126) and (129) with 𝜎=𝑛, we obtain𝐢𝑛=𝑛+1/2√2|𝛼|1/2,(135) as the value of the separation constant. The functions 𝑋𝑛(πœ‰) that solve (127) are then precisely those that describe the one-dimensional quantum oscillator. We write𝑋𝑛𝛾(πœ‰)=πœ‹ξ‚1/41√2𝑛𝐻𝑛!π‘›ξ‚€βˆšξ‚π‘’π›Ύπœ‰βˆ’(1/2)π›Ύπœ‰2.(136) The solutions π‘Œπ‘›(πœ‚) are identical in form and we obtain for the normalized wave function the following:πœ“π‘›ξ‚€π›Ύ(πœ‰,πœ‚)=πœ‹ξ‚1/212𝑛𝐻𝑛!π‘›ξ‚€βˆšξ‚π»π›Ύπœ‰π‘›ξ‚€βˆšξ‚π‘’π›Ύπœ‚βˆ’(1/2)𝛾(πœ‰2+πœ‚2).(137) For πœŽβ‰ π‘› 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+𝛼𝑅𝑛. 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 π‘˜. 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 𝑑 near 𝑑𝑖, 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(𝑑) can be expanded in a power series about 𝑑=𝑑𝑖 with finite coefficients. In particular 𝑅(𝑑𝑖) and ̇𝑅(𝑑𝑖) are finite. On the other hand in the case of power law inflation these quantities exhibit singular behavior at 𝑑=𝑑. 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 π‘Ž, 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 πœ‰=𝑅+π‘Ž and πœ‚=π‘…βˆ’π‘Ž. 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.

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