Quantum Gravity and Inflation

Haverford College Haverford Scholarship

Faculty Publications Astronomy

2004

Quantum gravity and inflation

Stephon Alexander Haverford College

Justin Malecki

Lee Smolin

Follow this and additional works at: https://scholarship.haverford.edu/astronomy_facpubs

Repository Citation Alexander, Stephon, Justin Malecki, and Lee Smolin. "Quantum Gravity and Inflation." Physical Review D 70.4 (2004): n. pag. Print.

This Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected]. PHYSICAL REVIEW D 70, 044025 ͑2004͒

Quantum gravity and inﬂation

Stephon Alexander Stanford Linear Accelerator Center and ITP Stanford University, Stanford, California 94309, USA

Justin Malecki Department of Physics, University of Waterloo, Ontario, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Canada

Lee Smolin Perimeter Institute for Theoretical Physics, Waterloo, Canada ͑Received 22 October 2003; published 20 August 2004͒ Using the Ashtekar-Sen variables of loop quantum gravity, a new class of exact solutions to the equations of quantum cosmology is found for gravity coupled to a scalar field that corresponds to inflating universes. The scalar field, which has an arbitrary potential, is treated as a time variable, reducing the Hamiltonian constraint to a time-dependent Schro¨dinger equation. When reduced to the homogeneous and isotropic case, this is solved exactly by a set of solutions that extend the Kodama state, taking into account the time dependence of the vacuum energy. Each quantum state corresponds to a classical solution of the Hamiltonian-Jacobi equation. The study of the latter shows evidence for an attractor, suggesting a universality in the phenomena of inflation. Finally, wave packets can be constructed by superposing solutions with different ratios of kinetic to potential scalar field energy, resolving, at least in this case, the issue of normalizability of the Kodama state.

DOI: 10.1103/PhysRevD.70.044025 PACS number͑s͒: 04.60.Ds

I. INTRODUCTION tum gravity.1 Therefore, a major goal of quantum gravity and cosmology is to find a quantum gravitational state which yields a consistent description of inflation. If this is accom- The inflationary scenario provides a framework for re- plished then one may be in a better position to make obser- solving the problems of the standard big bang ͑SBB͒ and, vational predictions for cosmic microwave background most importantly, provides a causal mechanism for generat- ͑CMB͒ experiments. ing structure in the universe. Currently, however, a com- The problem of inflation in quantum gravity has been pletely satisfactory realization of inflation is still lacking. much studied ͓3–5͔. However, in the past, the poor under- A hint for finding a concrete realization of inflation comes standing of quantum gravity necessitated that the study of from the trans-Plankian problem. Despite their successes inflation be restricted to the semiclassical approximation. in solving the formation of structure problem, most scalar This restriction makes it difficult to obtain reliable results field driven inflation models generically predict that the near about issues such as initial conditions and trans-Plankian ef- scale invariant spectrum of quantum fluctuations which fects that involve the regime in which quantum gravitational seeded structure were generated in the trans-Planckian effects will be significant. epoch. This, however, is inconsistent with the assumptions In this light it is worth noting that in recent years a great of a weakly coupled scalar field theory as well as the deal of progress has been made in a nonperturbative ap- assumption that quantum gravity effects can be ignored in proach to quantum gravity, called loop quantum gravity ͓7͔. the model ͓2͔. One can then view the trans-Plankian problem It is then appropriate to investigate whether these advances as an indication, and hence an opportunity, that the concrete allow us to treat the problem of inflation within cosmology derivation of inflation should be embedded in quantum more precisely. The recent results of Bojowald and others ͓8͔ gravity. indicate that in loop quantum gravity one can find exact There is also another interesting hint that suggests that quantum states that allow us to investigate more precisely the quantum gravity must play a role in our understanding of role of quantum gravitational effects on issues in cosmology, inflation. Inflation addresses the issue of initial conditions in including inflation and the fate of the initial singularity. Fur- the SBB, but solutions to scalar field theory driven inflation thermore, for constant cosmological constant, there is an ex- suffer from geodesic incompleteness. This is an indication act solution to the quantum constraints that define the full that inflation itself requires the specification of fine-tuned initial conditions ͓6͔. The issue of the robustness of the initial conditions necessary to start a phenomenologically ac- 1There are other motivations for expecting a quantum gravita- ceptable period of inflation, and their sensitivity to the pa- tional derivation of inflation but this is outside the scope of this rameters of the underlying field theory, remain open paper. A good discussion of this issue is nicely covered in a review questions which should in principle be addressed by quan- by Robert Brandenberger ͓1͔.

quantum general relativity, discovered by Kodama ͓9͔, which ratios of ␲2/V(␾), where ␲ is the canonical momenta of the has both an exact Planck scale description and a semiclassi- scalar field. It is then reasonable to superpose such solutions cal interpretation in terms of de Sitter spacetime. While there as there is no reason to believe that quantum state of the are open issues of interpretation concerning this state universe should at early times be an eigenstate of the ratio of ͓10,11,21͔, it is also true that it can be used as the basis of kinetic to potential energy. When we do this we find wave both nonperturbative and semiclassical calculations ͓12–14͔. packets which are exact normalizable solutions. This sug- Furthermore, exact results in the loop representation have gests that the problem of normalizability of the Kodama state made possible an understanding of the temperature and en- in the exact theory may be resolved similarly by adding mat- tropy of de Sitter spacetime ͓12,14͔ in terms of the kinemat- ter to the theory and then superposing extensions of the state ics of the quantum gravitational field. corresponding to different eigenvalues of the matter energy Thus, there appears to be no longer any reason to restrict momentum tensors. the study of quantum cosmology to the semiclassical ap- In the next section we describe the scalar field general proximation. In this paper we provide more evidence for this, relativity system in the formalism of Ashtekar ͓7͔ together by finding exact solutions to the equations of quantum cos- with the details of the procedure whereby the time gauge is mology that provide exact quantum mechanical descriptions fixed. Section III explains the reduction to homogeneous, of inflation. isotropic fields in these variables, while Sec. IV describes the In order to study the problem of inflation in quantum solutions to the resulting classical equations by means of a gravity we proceed by several steps: First, we couple general set of solutions to the Hamilton-Jacobi theory. The solutions relativity to a scalar field ␾ with an arbitrarily chosen poten- are studied numerically and evidence for an attractor is tial V(␾). We then choose a gauge for the Hamiltonian con- found. In Sec. V we quantize the homogeneous, isotropic straints in which this scalar field is constant on constant time system, discovering both semiclassical and exact solutions to hypersurfaces ͓16͔. This is appropriate for the study of infla- the Schro¨dinger equation. Our conclusions and some direction, because it has been shown that, in terms of the standard tions for further research are described in the final section. cosmological time coordinates, inflation cannot occur unless the fluctuations of the scalar field on constant time surfaces II. THE THEORY are small ͓15͔. There then always will exist a small, local rescaling of time that makes the scalar field constant.2 We consider general relativity coupled to a scalar field ␾ In this gauge the infinite number of Hamiltonian con- and additional fields ⌿, in the Ashtekar formulation of loop 3 straints are reduced to a single, time-dependent Shro¨dinger quantum gravity. Working in the canonical formalism, the equation ͓16͔. This is then solved, for homogeneous, isotro- Hamiltonian constraint is of the form pic fields, as follows. The corresponding classical Hamilton 1 1 ␾ϩH⌿, ͑1͒ ץ␾ ץevolution equations are solved exactly by a class of HϭHgravϩ ␲2ϩ EaiEb Hamilton-Jacobi functions. Each solution involves the nu- 2 2 i a b merical integration of an ordinary first order differential equation. These reduce, in the limit of vanishing slow role where ␲ is the conjugate momentum to ␾ and Eai is the ͑ ͒ parameter V˙ /V to the Chern-Simons invariant of the Ash- conjugate momentum to the complex SO 3 connection Aai . tekar connection. This is good, as the latter is known to be The latter couple satisfy the Poisson bracket relation the Hamilton-Jacobi function for de Sitter spacetime ͓9,12– ͕ ͑ ͒ bj͑ ͖͒ϭ ␦a␦ j␦3͑ ͒ ͑ ͒ 14͔. By exponentiating the actions of these solutions, one Aai x ,E y iG b i x,y , 2 obtains a semiclassical state that reduces in the same limit to the Kodama state. These new solutions are only good in the where G is Newton’s constant. ͑ ͒ H⌿ semiclassical approximation. However, in this case it is pos- In Eq. 1 we use to denote the Hamiltonian con- sible to find the corrections which make the wave functionals straint for all other matter fields. Unless otherwise noted, we into exact solutions of the time-dependent Schro¨dinger equa- adopt the convention that lowercase latin indices a, b, c,... tion. are spatial indices while lowercase latin indices i, j, k,... are ͑ ͒ The connection to the Kodama state allows us also to internal SO 3 indices. ␾ investigate issues regarding the physical interpretation of that We include the scalar field potential V( ) in the gravita- state such as the normalizability of the wave function tional term so that ͓10,11͔. In the case studied here, each exact quantum state 1 GV͑␾͒ we find is delta-function normalizable in the physical Hilbert Hgravϭ ⑀ ai bjͩ k ϩ ⑀ ckͪ ͑ ͒ 2 ijkE E Fab abcE , 3 space, corresponding to the reduced, homogeneous, isotropic l p 3 degrees of freedom. It is then interesting to ask whether fully k normalizable states can be constructed by superposing the where Fab is the curvature of the connection Aai . Note that different solutions. In fact, at a given time, defined by the any bare cosmological constant ⌳ is included in V(␾) and value of ␾, the different solutions correspond to different

3For an introduction to the Ashtekar formalism in the context of 2Technical subtleties regarding this choice of gauge are discussed cosmology, see Ref. ͓14͔. Other good, more general, and complete below. reviews are in Ref. ͓7͔.

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V(␾)ϭ⌳ gives the Hamiltonian constraint for general rela- where k is a constant. tivity sourced only by ⌳ and no scalar field. The gauge condition ͑7͒ is not good on the whole con- We also will impose the Gauss’s law constraint, which figuration space as there are solutions to Einstein’s equations enforces SO͑3͒ gauge invariance for which none of the constant ␾ surfaces are spacelike. Thus, Eq. ͑7͒ is more than a gauge choice, it is also a restric- Giϭ ai ͑ ͒ DaE 4 tion on the space of solutions. Nevertheless, it is a restriction which is appropriate to the study of inflation as there are and the diffeomorphism constraint, that imposes spatial dif- results that indicate that, in models where the metric is ap- feomorphism invariance proximately spatially homogeneous, inflation only takes ⌿ place for solutions in which ␾ is also, to a good approxima- ␾ϩD , ͑5͒ ץD ϭEbiF ϩ␲ a abi a a tion, spatially homogeneous. D⌿ However, for most initial data that satisfies Eq. ͑7͒,itis where a contains the matter fields. MϭSϫ known that the gauge condition will not be preserved for- We assume that spacetime has topology R, ␲ where S is the spatial manifold. As we are interested in cos- ever. The condition cannot be preserved if becomes zero at S any point on S. mology we assume has no boundary, so that the Hamil- ␲ tonian is given by Of course, is chosen on the initial data surface, and then evolves. Equation ͑9͒ tells us that an infinite lapse is required to preserve the gauge condition at points where ␲ vanishes. H͑N͒ϭ ͵ NH ͑6͒ So by fixing the gauge to Eq. ͑7͒ we will generally be able to S study only a finite period in the evolution of the universe. The extent of the period in which the gauge choice is good which is defined for any lapse N. We note that N has density ␲ weight minus 1. depends on the initial values taken for and the other fields. As we are interested in modeling inflation in which deviations from homogeneity must be assumed to be small, we A. Fixing the time gauge will assume that the gauge choice remains good for the entire In the Hamiltonian approach to general relativity, one is period of inflation. However, after we have built the quantum free to choose any slicing of space-time into a one parameter theory, we will have to be concerned with the extent to which family of spacelike surfaces, where that parameter can be these conditions are reliable. considered to be a time coordinate. All such slicings are As the Hamiltonian constraint does not commute with the physically equivalent and the Hamiltonian constraint gener- gauge condition, we have to solve all but one of the infinite ates gauge transformations that take us from any one spatial number of Hamiltonian constraints for the conjugate variable slice to any other. The choice of a slicing is then a gauge ␲. The one that is not solved is the constraint whose lapse is choice. inversely proportional to ␲, as that constraint commutes with In the usual treatments of inflation, the scalar field ␾ is the gauge condition. required to be homogeneous to a good approximation. As the We then find that deviations from homogeneity must be small for inflation to occur at all, in solutions to Einstein’s equations in which ␲ϭϮ͓Ϫ2HgravϪ2H⌿͔1/2. ͑10͒ inflation takes place we can assume that the surfaces of con- ␾ stant are spacelike. The scalar field also varies as the uni- There is one remaining Hamiltonian constraint which must verse expands, that is, it is ‘‘rolling down the hill.’’ It is then be imposed which is possible to measure time during inflation by the value of the ␾ field, keeping in mind that the forward progression of time k 1 corresponds to ␾ changing in the negative direction. 0ϭH͑Nϭk/␲͒ϭ ͵ ␲Ϫ H. ͑11͒ As a result, we will choose to gauge fix the action of the 2 S 2 Hamiltonian constraint so that ␾ is constant on constant time surfaces. We do this by imposing the gauge condition ͓16͔ To get the dimensions right, N should be dimensionless so we pick kϭ1/l2. Then H is the Hamiltonian for evolution in ␾ϭ ͑ ͒ p ץ a 0. 7 the gauge we have picked. It is given by

We need to ensure that this condition is maintained by evo- & lution generated by the Hamiltonian H(N). That is, we de- ϭϮ ͵ ͓ϪHgravϪH⌿͔1/2 ͑ ͒ H 2 . 12 mand l p S ␾ ץ d a N␲͒ ͑8͒ Finally, we deﬁne͑ ץ␾,H͑N͖͒ϭ ץ0ϭ ϭ͕ dt a a 1 ϭ ͵ ␲ ͑ ͒ which tells us that, to ensure the gauge condition is pre- P 2 13 served, we must use a lapse l p S

Nϭk/␲, ͑9͒ which has dimensions of energy.

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ϭ 2␾ 2 2 ͑ ͒ Thus, if we call the time T l p , where the factor of l p is 6 l pV T included so that T has dimensions of time, we have the Pois- Hgravϭ E2ͩ ϪA2ϩ Eͪ . ͑21͒ l2 3 son bracket p Given that S is not compact and our fields homogeneous, ͕T,P͖ϭ1. ͑14͒ we must give a well defined meaning to the integral over S. We then have from Eq. ͑11͒, As space is homogeneous, we can integrate over a compact region ⌺ʚS such that ϪPϩHϭ0, ͑15͒ ͵ ϭL3, ͑22͒ taking note of the fact that the Hamiltonian is time dependent ⌺ because the potential term in Hgrav depends on T. Herein, we will use V(T) to denote the value of the original potential where L is a fixed, nondynamical length scale. In this way, ⌺ ␾ ␾ϭ 2 V( ) evaluated at T/l p. We thus have reduced general is a finite representative of the entire homogeneous space. ϭ relativity to an ordinary Hamiltonian system with a time- We will use the dimensionless ratio R L/l p . R is a free dependent Hamiltonian. parameter in the homogeneous cosmological model that is not part of the full field theory, but arises from the reduction III. THE HOMOGENEOUS CASE from a field theory to a mechanical system. If there are no matter fields, we then have the Hamiltonian In this paper we will be concerned with the spatially homogeneous case, in order to be able to compare our approach 2 ͑ ͒ l pV T to the standard results in inflationary cosmology. Thus, we H͑A,E,T͒ϭϮR3ͱ12E2ͩ A2Ϫ E ͪ . ͑23͒ now turn to the reduction of the Hamiltonian system just 3 derived to the case of spatially homogeneous and isotropic universes. We will also consider from now on only the case In this way we have a finite dimensional Hamiltonian theory in which the scalar field is the sole matter field. of cosmology with a spatially homogeneous scalar field. The description of de Sitter spacetime in Ashtekar-Sen variables is described in Ref. ͓14͔. In a spatially flat slicing IV. SOLUTION OF THE HAMILTONIAN SYSTEM of de Sitter spacetime, the SO͑3͒ gauge can be chosen so that In order to find a solution for A(T) and E(T) we must the solution is given by diagonal and homogeneous fields first determine their symplectic relationship. Integrating bj ⌺ ͑ ͒4 A ϭi␦ A, Eaiϭ␦aiE, ͑16͒ E (y) over in Eq. 2 and substituting our homogeneous ai ai variables ͑16͒ gives the Poisson bracket where A and E are constant on each spatial slice S. ͕ 3 ͖ϭ ͑ ͒ de Sitter spacetime with cosmological constant ⌳ is given A,R l pE 3, 24 by 3 where we recognize (R l p/3)E as the conjugate momentum Aϭhf͑t͒, Eϭ f 2, f ͑t͒ϭeht, ͑17͒ to A. where the Hubble parameter is h2ϭG⌳/3 and t is the usual A. Derivation of the Hamilton-Jacobi function time coordinate defined so that the spacetime metric is given We now proceed to solve our model using Hamilton- by Jacobi ͑HJ͒ theory. We search for a Hamilton-Jacobi function 2ϭϪ 2ϩ 2 ͒2 ͑ ͒ S(A,T) such that5 ds dt f ͑ds3 , 18 Sץ Sץ S 3 2 where (ds3) is the flat metric on . ϭ ϭ ͑ ͒ 25 , ץ P , ץ E 3 We will consider the generalization of de Sitter spacetime R l p A T in which the homogeneous scalar field is used as the time coordinate, so that A and E are separately functions of T.In where the normalization of E is due to the relationship ͑24͒. these coordinates, the spacetime metric is Substituting these into Eq. ͑15͒ using the Hamiltonian ͑23͒ gives the HJ equation ͑with the positive root͒ 2ϭϪ 2 2ϩ ͒ ͒2 ͑ ͒ ds N dT E͑T ͑ds3 , 19 ץ ͒ 2 ץ ץ S 6 S l pV͑T S where N is the lapse ͑9͒. ϭ ͱ3ͩ ͪ ͩ A2Ϫ ͪ . ͑26͒ ץ 3 ץ ץ The gauge and diffeomorphism constraints are solved au- T l p A R A tomatically by the reduction to a homogeneous solution and the curvature is given by 4Ebj has spatial density weight 1 and so can be integrated over the ϭϪ 2⑀ ͑ ͒ Fabi A abi . 20 spatial manifold. 5Other approaches to inflation which involve solutions to the The gravitational part of the Hamiltonian constraint in this Hamilton-Jacobi equations, in the ‘‘old’’ canonical variables, are reduced model is given by described in Ref. ͓17͔.

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͑ ͒ A function Scs(A,T) will have zero energy if it satisfies We note that the A decouples, as each term in Eq. 26 is proportional to A3. The result is that for every solution to Eq. .S R3 ͑33͒ we get a cosmological modelץ cs ϭ 2 ͑ ͒ A 27 Before looking at the consequences of our solution, we ͒ ץ A l pV͑T first show that, for uÞ0, the spacetime deviates from ϭ ץ ץ so that H(A, Scs / A,T) 0. This implies de Sitter spacetime. Recall that de Sitter spacetime is the unique Lorentzian solution which satisfies the self-dual con- R3 dition ͑ ͒ϭ 3 ͑ ͒ Scs A,T ͑ ͒ A . 28 3l pV T 2⌳ l p Ji ϭFi ϩ ⑀ Eciϭ0, ͑34͒ However, this does not solve the Einstein equations because ab ab 3 abc where the proportionality can be taken to define ⌳.Inthe ˙S Vץ ϭ cs ϭϪ Þ ͑ ͒ homogeneous case we define J such that Ji ϭJ⑀i so the Scs 0 29 ab ab ץ P T V self-dual condition reads in general. 2⌳ l p We pause in our derivation to note that Eq. ͑28͒ is related JϭϪA2ϩ Eϭ0. ͑35͒ to the Kodama solution of the full quantum theory ͓9͔, be- 3 cause Scs is proportional to the Chern-Simons invariant For solutions generated by Eq. ͑32͒, taking V(T)ϵ⌳,we have 2 ͵ Y ͑A͒ϭ ͵ Trͩ A∧dAϩ A∧A∧Aͪ ϭiR3l3A3, CS 3 p JϭuA2 ͑36͒ ͑30͒ which, for nontrivial spacetimes, only vanishes if u(T)ϵ0. where the last equality comes from using the homogeneous variables ͑16͒. B. Analysis of the solutions We can understand why Scs is not a solution to our model. ⌳ϭ Given our HJ equation ͑32͒ we can calculate the lapse Were V(T) constant so that V were the cosmological ͑ ͒ constant, Eq. ͑28͒ would be the Hamilton-Jacobi function for function 9 to be de Sitter spacetime ͑see Ref. ͓14͔͒. Were this the case, we ␲ϭ Vlp would have to have 0 so that the scalar field contributed NϭϪi . ͑37͒ no kinetic energy, but only the constant potential energy. As 6A3͑1ϩu͒ͱ3u discussed above, this would violate our gauge condition ͑7͒. The deviation from de Sitter spacetime is then given by Introducing the integration constant ␣ that one obtains from terms proportional to the ratio rϭ␲2/2V. This is propor- integrating equation ͑33͒ and a further integration constant ␤ tional to the ‘‘slow roll parameter’’ ͑both dimensionless͒ we can derive a relationship between A and T by V˙ Sץ ␩ϭl . ͑31͒ p V u ϭ␤ ͑ ͒ 38 . ␣ץ When r and ␩ are small, the kinetic energy of the scalar field is small compared to its potential energy. This leads to the equation of motion Ϫ1 1/3 ץ ␤͒ To get a solution to our model, which requires ␲Þ0, we 3l pV͑T u need to modify de Sitter spacetime by terms proportional to A͑T͒ϭͫ ͩ ͪ ͬ . ͑39͒ ␣ץ r and ␩. To do this, we modify the HJ function ͑28͒, which R3 gives de Sitter spacetime, by adding a new dimensionless function of time u(T) so that In practice, the values of ␣ and ␤ are determined by the initial A(T0) and E(T0). Note that the partial derivative of u R3A3 will, in general, add further T dependence to A(T). We can S ͑A,T͒ϭ ͓1ϩu͑T͔͒. ͑32͒ ͑ ͒ u 3l V then use Eq. 39 to derive the T dependence of the conjugate p momentum to be Ϫ2/3 ץ ͒ We expect to find that u(T) scales with r and ␩. We plug this 5/3␤2/3͑ ϩ ͑ ͒ 3 1 u u into the Hamilton-Jacobi equation 26 and we find an equa- E͑T͒ϭ ͩ ͪ . ͑40͒ ␣ץ 1/3 4/3 2 tion for u(T) R l p V ͑ ͒ V˙ 18i Given the form of the metric in these coordinates 19 we u˙ ϭ ͑1ϩu͒ϩ ͑1ϩu͒ͱ3u. ͑33͒ see that the TT component of the metric is given by ϪN2.In V l p order for the metric to remain real and Lorentzian, we must

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Ͼ Unless otherwise noted, we always take T0 m so that ϭ 2␾ T l p proceeding in the negative direction corresponds to ␾ ‘‘rolling down the hill.’’ These solutions can be seen in ␭ϭ ϭ ϭ Fig. 1 for 1, m 2, and Vmin 5. These plots show the generic behavior of u for a wide range of parameters in Eq. ͑42͒. The most noticeable characteristic is the apparent attractor in Fig. 1. That is, all initial conditions satisfying our physical ͑ ͒ Ͻ condition 41 merge to the same solution for some T T0 . The significance of this apparent robustness to initial conditions suggests a universality in the phenomena of inflation, which should be further investigated. Furthermore, in Fig. 1 we see that, for TϾm, all of our solutions remain within the bounds Ϫ1ϽuϽ0. However, for TϽm, u quickly becomes positive and takes on an imaginary component and the metric becomes unphysical. Tϭm corresponds to the scalar field having its minimum potential energy and is traditionally the end of the inflationary period. Hence, it is not surprising that our gauge condition should break down at this point, as discussed in Sec. II A. This break-down of the gauge condition at the minimum of the inflaton potential is a common feature for all of the different parameters in Eq. ͑42͒ that were attempted. Of note is the ϭ ͑ ͒ case where Vmin 0 in which the evolution equation 33 becomes singular for Tϭm. While we defer a full numerical analysis of our model to a later time, we briefly discuss the behavior of u for ‘‘unphysical’’ initial conditions that do not satisfy Eq. ͑41͒.Itis ͑ ͒ Ͼ clear from Eq. 33 that an initial condition u(T0) 0 forces u to have an imaginary component and hence give an un- ϽϪ physical metric. For all of the initial conditions u(T0) 1 that were attempted we found that u rapidly diverged to Ϫϱ. Hence, the physical conditions imposed on u seem to be reflected in its functional behavior governed by Eq. ͑33͒. FIG. 1. Numerical solutions of u(T) using the potential ͑42͒ for ␭ϭ ϭ ϭ 1, m 2, and Vmin 5 and various initial conditions T0 , u(T0) Ͼ Ϫ Ͻ Ͻ V. QUANTUM MINISUPERSPACE satisfying T0 2 and 1 u(T0) 0. The real part of u is plotted on the top graph while the imaginary part is on the bottom graph We now proceed to build a quantum theory from our clas- and the different line types on the top and bottom correspond to the sical Hamiltonian theory and find a full solution to the re- same solution. The time variable T is in units of l p /c. Recall that sulting Schro¨dinger equation. First, we take quantum states forward time corresponds to T evolving in the negative direction. to be functions ⌿(A,T). We then define Aˆ to be a multipli- have that N2 be positive and real. Looking at Eq. ͑37͒ we see cative operator and define the operators that this requires that u(T) be negative and real. Further- ⌿ץ ⌿ Ϫ3iបץ more, for the metric to remain Lorentzian we require that EϾ0. Equation ͑40͒ then requires that uϾϪ1. Hence, our Pˆ ⌿ϭiប , Eˆ ⌿ϭ . ͑43͒ Aץ T l R3ץ gauge condition will break down unless p

Ϫ1Ͻu͑T͒Ͻ0. ͑41͒ Again, the normalization of Eˆ stems from the relation ͑24͒. Herein, we will work in units such that បϭ1. This then places a strong restriction on the value of u. The evolution equation becomes a time-dependent Schro¨- In order to get a sense of the behavior of u(T), we have dinger equation solved Eq. ͑33͒ numerically for the quartic potential ⌿ץ ͑ ͒ϭ␭͑ 2Ϫ 2͒2ϩ ͑ ͒ V T T m Vmin 42 i ϭHˆ ⌿, ͑44͒ Tץ for several initial conditions u(T0) consistent with the re- strictions discussed above. Potentials of this sort are renor- where we choose the ordering of the quantum Hamiltonian to malizable and satisﬁes the slow roll condition. be

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͑ ͱ ͒͐T ͱ ͑ ͒ ͱ 6 3i/lp u t dt ␹͑T͒ϭe T0 . ͑55͒ ץ 3i 6 Hˆ ϭ ͑AͱJˆ ͒ ͑45͒ ץ l p A This yields ͑note that we have taken the negative root͒ where we define ˆ ͑ ͱ ͒͐T ͱϪ ͑ ͒ ϩ ͑ 3 3 ͒͑ ϩ ͒ J as 6 3/lp u t dt i A R /3Vlp 1 u ⌿͑A,T͒ϭe T0 ͑56͒ ץ iVlp Jˆ 1ϩ . ͑46͒ which is a full solution to the Schro¨dinger equation ͑44͒.We Aץ ª R3A2 have factored the i into the square root term so that ͱϪu)is We then take a semiclassical quantum state real for solutions of u that satisfy the physical condition ͑41͒. To summarize, given a solution of Eq. ͑33͒ we have found a ϩ ͑ ͒ ϩ ͑ 3 3 ͒͑ ϩ ͒ complete solution ͑56͒ to the quantum theory of a homoge- ⌿ ͑A,T͒ϭe iSu A,T ϭe i A R /3Vlp 1 u ͑47͒ u neous cosmology coupled to a scalar field in a potential. and note that it is an eigenfunction of Jˆ with a time- dependent eigenvalue Ϫu(T) A. Wave packets and normalizable states Finally, we describe the physical inner product and show ˆ ⌿ ϭϪ ͒⌿ ͑ ͒ J u u͑T u 48 how to construct exact, normalizable, quantum states of the universe. In quantum gravity, the physical inner product is which implies determined by the reality conditions for physical observ- ables. In the present case, in which all gauge degrees of freedom are fixed by either gauge conditions or the reduction ͱJˆ ⌿ ϭiͱu͑T͒⌿ . ͑49͒ u u to homogeneous, isotropic solutions, A(T) and E(T) are Hence, the action of the Hamiltonian on our semiclassical physical degrees of freedom, and they are indeed real. In the state is representation we are using in which states are functionals of A and T and the latter is the time coordinate, the physical inner product is hence, 6iA3R3͑1ϩu͒ͱ3u 6ͱ3u ˆ ⌿ ϭϪͫ ϩ ͬ⌿ ͑ ͒ H u 2 u . 50 Vlp l p ͗⌿͑T͉͒⌿͑T͒͘ϭ ͵ dA͉⌿͑A,T͉͒2. ͑57͒ ⌿ We now show that u is, indeed, an approximate solution to the Schro¨dinger equation ͑44͒ by computing the time de- ͑ ͒ rivative Our exact solutions 56 are phases, so long as u(T)is real and negative, corresponding to real solutions to the clas- -⌿ 3 3 ϩ ͒ͱ sical Einstein equations. Hence, with this restriction the soץ u 6A R ͑1 u 3u ϭϪ ⌿ , ͑51͒ lutions are all delta function normalizable. u 2 ץ T Vlp Following the usual procedure in quantum theory, we can construct normalizable solutions by constructing wave pack- where we have used the fact that u(T) is a solution of Eq. ets. We may note that at a given T, different classical solu- ͑33͒. Comparing Eqs. ͑50͒ and ͑51͒ we see that, provided tions, with different values of u(T) correspond to cases in which the scalar field ␾ϷT is moving at different rates. A3R3͑1ϩu͒ Thus, while V(T) is fixed at the same T, ␲(T) can vary, ͯ ͯ ӷ1, ͑52͒ ␲ 2 Vl leading to different ratios of (T) /V(T). However, quan- p tum mechanically we would expect that this ratio would not be a sharp observable. Thus the quantum state of the expand- our semiclassical state ͑47͒ is indeed an approximate solu- ing universe should not physically be built from a single tion to the Schro¨dinger equation. To find a full solution to Eq. ͑44͒, we take the ansatz solution to the classical equations u(T). As there is no reason to expect that the initial conditions for u(T) are fixed ⌿͑A,T͒ϭ⌿ ͑A,T͒␹͑T͒, ͑53͒ classically in the very early universe, we should expect the u inflating universe to be described by a wave packet corresponding to superposing over solutions that differ in the where ␹ is an arbitrary function of T alone. Substitution of ͑ ͒ ¨ value of u(T) at fixed T. Eq. 53 into the Schrodinger equation yields the equation ϭ To do this, let us fix T T0 , and consider initial values for u(T ). To construct a wave packet we consider a central ␹ ͱ 0ץ 6i 3u value u and define vϭu(T )Ϫu . We then label ϭ ␹ ͑54͒ 0 0 0 ץ T l p

ϩ ͓ 3 3 ͑ ͒ ͔͑ ϩ ϩ ͒ ⌿͑ ͒ ϭ i A R /3V T0 lp 1 u0 v ͑ ͒ which can be immediately integrated to give A,T0 v e . 58

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Given a normalizable function f (v) we may then define lenging, but not impossible, to find exact results for the case an exact solution determined by the initial conditions of Hermitian ordering. If not, at least a semiclassical expansion could be constructed that would be reliable above the ⌿͑A,T ͒ ϭ ͵ dv f ͑v͒⌿͑A,T ͒ . ͑59͒ Planck scale, whose leading order terms would be given by 0 f 0 v the states found here. It will be also interesting to incorporate the inhomoge- By a suitable choice of f (v), with support in the physical ϭ ϩ ෈ Ϫ neous modes of the gravitational and matter fields by means interval u u0 v ( 1,0), the initial state is normalizable. of a perturbative expansion around the states constructed Let us then define E(T,v)ϭ͐T ͱϪu(t)dt with initial T0 here. The aim here will be first principles predictions for ϭ ϩ condition u(T0) u0 v and uv(T) to be the value of u at transplankian effects in the spectra of scalar and tensor per- time T with the same initial condition. Then the wave packet turbations, as well as polarizations, detectible in cosmologi- at later time is given by cal observations. Slow-roll inflation requires a sufficiently large initial ͑ ͱ ͒E͑ ͒ϩ ͑ 3 3 ͓͒ ϩ ͑ ͔͒ value of the scalar field in order to obtain the sufficient num- ⌿͑A,T͒ ϭ ͵ dv f ͑v͒e 6 3/lp T,v i A R /3Vlp 1 uv T . f ber of e foldings to match with observations. One mecha- ͑60͒ nism for attaining such initial conditions is that of eternal inflation ͓18͔ which proposes that quantum fluctuations al- VI. CONCLUSIONS AND DISCUSSION low the scalar field to diffuse up the potential well in certain regions of space. Once inhomogeneous modes have been in- The results of this paper represent a step towards a de- corporated into our model ͑as discussed in the previous tailed study of the very early universe beyond the semiclas- point͒ it will be interesting to see if such a mechanism exists sical approximation, in which quantum gravitational effects in a quantum gravity context. are treated in a nonperturbative and background-independent Recent results ͓19͔ have shown that loop quantum gravity manner. For each potential V(␾) and classical slow roll so- may provide a mechanism for driving the scalar field to high lution u(T) consistent with inflation, we have found a quan- values at early times. This was done in the context of Bo- tum state given by Eq. ͑56͒ which is an exact solution to the jowald’s cosmological formalism ͓8͔ so it will be interesting quantum equations of motion, but has a classical limit given to see if such a phenomenon can occur in our model as well. by that classical solution. Furthermore, we can construct nor- We note that the validity of our gauge condition ends malizable states which are wave packets around the initial roughly when inflation ends, as surfaces of constant ␾ no conditions that generate that classical solution. Thus, infla- longer track surfaces of constant scale factor, once the uni- tion is here described in terms of exact quantum states. verse enters the stage where, in terms of the latter, the scalar As a by-product, the simplicity of the Hamilton-Jacobi field oscillates around the minimum of V(␾). We see that at solutions to the coupled Einstein-scalar field problem, using this point in the classical dynamics, u(T) becomes complex, the Ashtekar formulation, may provide a new, simplified ap- leading to complex values of the spacetime metric. To study proach to studying inflation classically. In our first investiga- the problem of exiting from inflation it will then be neces- tion of the problem we found an attractor, suggesting, at least sary to choose another time parameter, which is good in this case, a possible universality in the dynamics of the throughout the exit from inflation, and use the wave function very early universe. This deserves more investigation, as it generated here as initial conditions for evolution in that pa- may provide an understanding of the hypothesis of chaotic rameter. inflation ͓5͔. There exist extensions of the Kodama state to supergrav- A number of very interesting questions remain, which ity with Nϭ1,2 ͓20͔. It is then likely that the results of the these results suggest can now be approached. present paper can be extended to supersymmetric models of It would be very interesting to understand the relationship inflation. of these results to those of Bojowald and others, in the con- It will be interesting to investigate whether the mecha- text of loop quantum cosmology ͓8͔. In that case a similar nism used here to construct normalizable states will work in reduction is used, and solutions to quantum cosmology are the full theory, perhaps resolving the issue of the normaliz- found which are exact and nonperturbative. However, Bo- ability of the Kodama state. jowald’s results obtained using a representation conjugate to that used here, which is roughly a dimensionally reduced spin network basis. A number of interesting results are ob- ACKNOWLEDGMENTS tained, including indications that the initial singularity is re- moved. It would then be very useful to establish a homoge- We are grateful to Robert Brandenberger, Laurent Freidel, neous version of the loop transform, to express the states John Moffat, Michael Peskin, and Hendryk Pfeiffer for con- studied here in Bojowald’s representation. versations during the course of this work. L.S. would like to The ordering of the Hamiltonian used here is not Hermit- thank SLAC and the Physics Department of Stanford Uni- ian. It is easy in the dimensional reduction to find Hermitian versity, and S.A. would like to thank the Perimeter Institute orderings for the Hamiltonian. The exact quantum states for hospitality during the course of this work. The work of found here will solve the Hermitian ordering of the Schro¨- S.A. was supported by the Department of Energy, Contract dinger equation at the semiclassical approximation. It is chal- No. DE-AC03-76SF00515.

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͓1͔ R. Brandenberger, ‘‘Inflation and the Theory of Cosmological ͓10͔ E. Witten, ‘‘A Note on the Chern-Simons and Kodama Wave- Perturbations,’’ astro-ph/971106. functions,’’ gr-qc/0306083. ͓2͔ J. Martin and R. H. Brandenberger, Phys. Rev. D 63, 123501 ͓11͔ L. Freidel and L. Smolin, hep-th/0310224. ͓ ͔ ͑ ͒ ͑2001͒. 12 L. Smolin and C. Soo, Nucl. Phys. B449, 289 1995 . ͓ ͔ ͓3͔ J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 ͑1993͒. 13 Chopin Soo and Lay Nam Chang, Int. J. Mod. Phys. D 3, 529 ͑1994͒; Chopin Soo, Class. Quantum Grav. 19, 105 ͑2002͒. ͓4͔ ‘‘A Bibliography of Papers on Quantum Cosmology,’’ ITP- ͓14͔ L. Smolin, ‘‘Quantum Gravity With a Positive Cosmological UCSB preprint No. NSF-ITP-88-132. Constant,’’ hep-th/0209079. ͓ ͔ ͑ ͒ 5 A. D. Linde, JETP 60,211 1984 ; A. Vilenkin, Phys. Rev. D ͓15͔ G. F. Mazenko, R. M. Wald, and W. G. Unruh, Phys. Rev. D ͑ ͒ 30, 509 1984 . 31, 273 ͑1985͒. ͓6͔ T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 ͓16͔ L. Smolin, ‘‘Time, measurement and information loss in quan- ͑2000͒. tum cosmology,’’ gr-qc/9301016; Carlo Rovelli and L. Smolin, ͓7͔ For current introductions and reviews see, C. Rovelli, Quan- Phys. Rev. Lett. 72, 446 ͑1994͒. tum Gravity ͑Cambridge University Press, Cambridge, in ͓17͔ William H. Kinney, Phys. Rev. D 56, 2002 ͑1997͒;D.S. press͒, draft in http://www.cpt.univ-mrs.fr/rovelli; T. Salopek, ‘‘Cosmological Inflation and the Nature of Time,’’ Thiemann, Lect. Notes Phys. 631,41͑2003͒; ‘‘Introduction to astro-ph/9512031; ‘‘The Role of Time in Physical Cosmology,’’ Modern Canonical Quantum General Relativity,’’ astro-ph/9510059. ͓18͔ A. D. Linde, D. Linde, and A. Mezhlumian, Phys. Rev. D 49, gr-qc/0110034; L. Smolin, ‘‘How far are we from the quantum 1783 ͑1994͒. theory of gravity,’’ hep-th/0303185; A. Perez, Class. Quantum ͓19͔ S. Tsujikawa, P. Singh, and R. Maartens, ‘‘Loop quantum grav- ͑ ͒ Grav. 20, R43 2003 ; D. Oriti, Rep. Prog. Phys. 64, 1489 ity effects on inflation and the CMB,’’ astro-ph/0311015. ͑ ͒ 2001 . ͓20͔ Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 ͑1993͒; ͓8͔ M. Bojowald and Hugo A. Morales-Tecotl, ‘‘Cosmological ap- ‘‘The Ashtekar Formalism and WKB Wave Functions of N plications of loop quantum gravity,’’ gr-qc/0306008, and refer- ϭ1,2 Supergravities,’’ hep-th/9211103. ences therein. ͓21͔ G. A. M. Marugan, ‘‘Is the Exponential of the Chern-Simons ͓9͔ H. Kodama, Prog. Theor. Phys. 80, 1024 ͑1988͒; Phys. Rev. D Action a Normalizable Physical State?’’ Class. Quant. Grav. 42, 2548 ͑1990͒. 12, 435 ͑1995͒.

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