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in a surprising result , hartle and hawking twenty years ago showed @xcite how , at least in the semi - classical theory of quantum gravity , one could give an intuitively appealing definition for a preferred wave function for the universe . if one believes that nature selects this definition , then they argued that this wave function explains various features which we observe the universe to have . one of these features is that the universe is simple when it is small @xcite and becomes more and more complex as it grows in size . while hawking originally believed that the universe must always be simple when small @xcite , page and laflamme @xcite pointed out that this wave - function would be expected to also contain probablilities for the universe to be complex when small . this was interpreted as indicating a classical history of the universe which began simple when small , expanded , with the gravitational instability causing growth in the quantum fluctuations present when small . the universe would eventually re - collapse with the fluctuations continuing to grow . if one wrote the hh wave - function in terms of a basis which contained such semi - classical components , one would interpret them to say that the thermodynamic arrow of time would be driven by these fluctuations and that one should interpret those solutions such that the low fluctuation end should be interpreted as the beginning and the high as the end of the evolution . however , there is an alternative interpretation of the hh wave - function ( see for example jheeta and unruh@xcite ) . the prescription bears a strong resemblance to the prescription of a semi - classical wkb wave - function near a classical turning point of a potential . the classical solution for the hh models is that of a de sitter space with the the cosmological constant dominating the evolution ie the classical solution looks like de sitter space in spherical coordinates near the point where the spherical space reaches it minimum size . interpreted in this way there is another place where the universe could and does bounce ; namely if a closed universe has matter such that the energy density of the matter drops as the universe expands . in this case , the curvature terms in the frw equations eventually dominates and halts the expansion and the universe re - collapses . the question which we address is whether the hh prescription could be used to place boundary conditions at this bounce point rather than at the point where the universe achieves it minimum size ?    we will examine a simple model , with a homogeneous scalar field driving the dynamics of the universe . the space - time will be assumed to be a closed spherical homogeneous and isotropic frw cosmology with metric    ds^2= n^2 dt^2 - a(t)^2 ( dr^2 + sin(r)^2 ( d^2 + sin()^2 d^2 ) )    the mini - superspace action for this space - time becomes s= n a^3 ( -6a^2n^2 a^2 + 6a^2 + 12n^2 ^ 2 -v ( ) ) dt which has as semi - classical equations of motion    + na^3dv()d=0 + 12adndt dandt+6 dandt^2 + 6 + 3a^2(12n^2 ^2 - v)=0 + ( a ^22 n^2 a^2 + 6a^2 - 12n^2 ^ 2 -v ( ) ) = 0 where @xmath0 . ( the third is the constraint equation , and is an integral of motion of the other two ) . the semi - classical hh prescription is that the path integral will be dominated by a solution of the equations which has only a final boundary given by the classical state on which one wishes to evaluate the wave function , ( i.e. , the single boundary is defined by the values of @xmath1 and @xmath2 ( or @xmath3 ) which one wants on the final boundary ) . we must find solutions to these equations which have no initial boundary which have @xmath4 going to zero and @xmath5 regular as @xmath4 approaches zero . this is impossible if @xmath6 ( and @xmath1 ) are real . we can however find such solutions if @xmath6 is chosen to be complex , and in particular if we choose @xmath6 to become purely imaginary as @xmath1 approaches zero . ( for a generalisation to arbitrary complex @xmath6 see jheeta and unruh @xcite ) . one choice , if it works , is to take @xmath6 and @xmath1 to be purely real over some range of times ( the lorentzian regime ) , and then where @xmath4 has zero derivative , to take @xmath6 to be purely imaginary . this allows one to smoothly join @xmath4 across the surface where @xmath6 discontinuously . however , in general , both @xmath7 and @xmath3 can not be taken to be zero at the same time @xmath8 and still find a solution with only a single boundary . since the equations of motion for @xmath1 and @xmath2 are smooth second order equations , the derivatives @xmath9 must also be smooth . while we can keep @xmath4 real across this boundary , since going from real zero to imaginary zero is a continuous transformation , we can not do the same for @xmath2 . thus , @xmath2 must thus become complex as one crosses this boundary . we are going to try to model the universe such that at time @xmath10 we have @xmath11 , and for @xmath12 , the universe contracts from the value of @xmath1 at this time . ie , we take @xmath10 to be the time at which the universe achieves its maximum value in the lorentzian regime . examining the constraint equation for @xmath1 , let us examine this boundary where @xmath13 . since @xmath7 is , by assumption less , than zero ( the universe is contracting ) for larger values of @xmath8 , the potential term must be sub - dominant . the curvature term @xmath14 cancels with the kinetic energy . if @xmath15 were to dominate , then @xmath16 decreases as @xmath1 decreases , and @xmath17 would become imaginary . thus , in order that we have such a maximum in the radius of the universe , the term @xmath18 must be dominating the energy , and @xmath19 must be decreasing at least as fast as @xmath20 but for very small @xmath15 this is precisely how @xmath21 is doing . ( for small @xmath22 , @xmath2 obeys 0 or ie , in order that the universe re - collapse , the potential term can not dominate the evolution of the universe it must be dominated by the kinetic energy of the matter . now , @xmath23 must be continuous across that boundary where @xmath6 goes from real to imaginary . since we are taking n to be purely imaginary for @xmath24 , @xmath21 must also be imaginary for times just before zero . ie , @xmath2 must pick up an imaginary part . let us now examine the evolution of this model in this realm where @xmath6 is purely imaginary . let us assume that the potential @xmath25 is such that the potential remains real for @xmath26 , where @xmath27 is the value of @xmath2 at the point where @xmath11 . since @xmath6 is arbitrary , we will take @xmath28 for @xmath12 and @xmath29 for @xmath24 . the equation of motion for @xmath30 will be a^3 ddt = d vd and for @xmath1 , the constraint equation gives ( dadt)^2 = -1 + 16a^2(12ddt^2 + v ( ) )    the second order einstein equation for @xmath1 shows that through @xmath10 , @xmath31 reverses sign . thus , while the universe contracts toward the future in the lorentzian regime , it must expand into the past at this turning point .    thus @xmath32 is less than zero for @xmath24 near 0 , and the universe expands as go to negative @xmath8 . this raises the question of how one could get @xmath1 to go to zero in this n - imaginary region . the kinetic energy which dominates is driving the universe larger . the answer is that as @xmath1 increases , @xmath33 keeps decreasing . if the potential @xmath34 remains non - zero , eventually it will dominate once again . once it dominates , the potential term will keep increasing until it becomes of the same order as the curvature term ( the 1 in the equation for @xmath7 ) and @xmath1 has another turning point . the universe will then collapse as a function of @xmath8 , the kinetic term will again grow , and the danger is that one will hit another turning point . if however the potential is arranged so that during this collapse , the potential increases , and thus @xmath35 decreases , one can arrange the system so that @xmath1 can collapse all the way to zero smoothly . let us examine a specific model . we take the potential to be v()= v_0 + ^4 . we are going to assume that at t=0 , @xmath36 , @xmath37 is large , and @xmath38 . we will assume that @xmath39 is smaller than @xmath40 and thus @xmath41 .    if the solution is such that @xmath1 approaches zero smoothly at @xmath42 ( so that @xmath30 and @xmath33 are both finite there ) then near @xmath43 we have    ( t)(t_0 ) ( 1 - 8(t - t_0)^2 + ... ) + a(t)=(t - t_0)(1- ( v_0 - 4(t_0)^4)(t - t_0)^2 + ... )    in figure 1 we plot a solution with @xmath44 and @xmath45 . the lower axis is note that the field @xmath30 approaches a constant for most of the evolution . if the potential in the imaginary direction is sufficiently flat for a large enough value of @xmath30 , the universe can oscillate between the the kinetic @xmath33 dominated bounce at smaller values of @xmath1 and the potential dominated one at large values of @xmath1 for many bounces . in figure 2 we have a solution with @xmath46 and @xmath47 , where we have three bounces before the radius eventually goes to zero .    in the @xmath6 real regime , the potential @xmath1 will decrease , and the kinetic dominated term will grow roughly as @xmath48 . if the kinetic energy dominates , the solution for @xmath1 goes as @xmath49 , and @xmath2 grows logarithmically . if the potential in the real @xmath2 direction grows , it may eventually dominate the matter again , and the universe would suffer a de sitter like bounce at small values of @xmath1 . figure 2 is the solution for @xmath6 and @xmath2 real into the past from the large radius bounce which matches to the solution for figure 1 . in this particular case , because the potential eventually grows so rapidly with @xmath2 , the smaller bounce radius is not that much smaller than the largest size . however this depends entirely on the behaviour of the potential for large real @xmath2 . thus , if our potential were say @xmath50 , with @xmath51 very large , our potential along the real @xmath2 axis would remain bounded by @xmath27 , and the dynamics of @xmath1 would be kinetic dominated for all times to the past of the large bounce . in the imaginary direction , @xmath25 would have the required confining shape and would , once @xmath2 had grown to of order @xmath51 , allow for the radius to go smoothly to zero . in the real @xmath2 and @xmath6 regime , the solution would be kinetic dominated all the way to a singularity .    in the spirit of hh , we would interpret this universe as beginning in the euclidean regime with @xmath52 and @xmath53 and @xmath3 regular . the universe would `` expand '' and bounce between the potential and kinetic dominated bounce points for as long as desired , until it suddenly makes the transition to real @xmath6 and real @xmath2 at the lower kinetic dominated bounce point . the universe would then collapse into the future , until finally , either the potential dominated and the universe suffers a de sitter , inflationary type bounce , or until the the universe finally collapses to a singularity . since these solutions are possible , how would they enter into the path integral for the  wave function of the universe " ? there has of course been much heat expended on the various schools as to the sign one should take for the action in the euclidean regime . these amount to either a choice of sign for @xmath6 in the imaginary regime , or a choice in the sign of @xmath1 ( since @xmath1 enters only quadratically in all expressions of physical relevance ) . thus the contributions of these imaginary @xmath6 parts of the action will contribute either @xmath54 or @xmath55 where @xmath56 is the integrated action for the imaginary @xmath6 regime ( or is the imaginary part of the action in the case of a generic complex metric and field dominating the action ) . for our purposes , this will be the value of s_i= ( -6a + aa ^2 + a^3(12 ^ 2 + v(i ) ) ) dt during the bouncing , all the terms are of the same order as the first term , and we have s_ia dt by hypothesis both @xmath1 and @xmath8 are large , and furthermore , @xmath8 in general will be of the same order as @xmath57 , where @xmath58 is the number of bounces in the imaginary @xmath6 regime , making @xmath59 of order of @xmath60 . @xmath1 here will be of order its maximum where @xmath61 . ie , the action will be of order @xmath62 . for small @xmath39 , and for the positive choice of the @xmath56 in the action , this would completely dominate the path integral . the theory would predict with overwhelming probability that the universe began large and then collapsed down to a small size , rather than the more traditional way around . furthermore , the fluctuations would grow during the collapse phase . the universe would grow more and more chaotic as its size decreased , with the simplest being when the universe was at its largest diameter . this sign is the sign usually attributed to hartle and hawking @xcite .    the alternative sign , usually attributed to linde and vilenkin @xcite , @xcite ( although derivable as simply one of two possibilities in defining the hartle hawking wave function ) would on the other hand make the contributions of this particular solution to the semi - classical equations , completely sub - dominant . one of the interesting features of this analysis is the realization that the in order to maintain continuity of the scalar field equation through the transition from the real to imaginary @xmath6 , one must also allow @xmath2 to become complex . in particular , if the field and scale factor are real in the real @xmath6 regime , then the derivative of the field in general must be pure imaginary in the imaginary @xmath6 regime . thus , we note that the sign of the kinetic term @xmath63 does not change on the transition from real to imaginary @xmath2 . let us examine the action for the inhomogeneous minimally coupled scalar field @xmath64 in the geometry . the action for this field is s= na^3 ( 12n^2 ^ 2 - 1a^2()^2)dt d^3 where the @xmath65 is the spatial integral , and @xmath66 represents the spatial derivative part of the action . again at the surface where @xmath6 makes the transition from real to imaginary , and @xmath7 is zero , @xmath64 must again be complex to keep @xmath67 continuous . ie , again we can write @xmath68 . the action then becomes , taking @xmath29 , s_e= ia^3(12 ^2 + 1a^2()^2 ) dtd^3 taking the semi - classical approximation to the path integral with the vl sign we have @xmath69 as the contribution to the wave - function . all the terms in the action are positive , indicating that we want to minimize the action to find the dominant terms . using the apherical eigenstates for the spatial modes of the field we have the wave function factor as e^ - a^3 ( 12_k^2 + k^2a^2 _ k^2 ) dt near @xmath52 , a will again be linear in @xmath8 , and the solution for the field which is regular near @xmath43 is _ 0(1 + 18 t^2 + ...    the action exponent can be written as a^3 ( 12_k^2 + k^2a^2 _ k^2 ) dt = a^3_k(t)_k(t)_a=0^t the lower limit is zero . as long as @xmath70 @xmath71 will  slow roll " and _ k_k while in the other limit , _ k(ka)_k    thus , at the transition point , the exponent will be proportional to @xmath72 ( ie quadratic ) and will be concentrated near zero with a width proportional to either @xmath73 for short wavelengths , or @xmath74 for long wavelengths . hartle and s.w . hawking , rev d 28 , 2960 ( 1983 ) . j. halliwell and s.w . hawking , d 31 , 1777 ( 1985 ) . s.w . hawking , phys . d 32 , 2489 ( 1985 ) . d.n . page , phys . rev . d 32 , 2496 ( 1985 ) . m. jheeta , w. unruh http://arxiv.org : gr - qc/9812017 `` complex paths and the hartle hawking wave - function for slow roll cosmologies ''
using the extension of the standard hawking - hartle prescription for defining a wave function for the universe , we show that it is possible , given a suitable form for the scalar field potential , to have the universe begin at its largest size and thereafter contract , with the growth of perturbations proceeding from small , at the largest size , to largest when the universe is small . 
 this solution would dominate the wave function by an exponentially large amount if one chooses the hartle hawking prescription for the wave - function , but is exponentially sub - dominant for the linde - vilenkin prescription .    # 
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