Quantum field theory in de Sitter and quasi-de Sitter spacetimes: Revisited
aa r X i v : . [ g r- q c ] M a y Quantum field theory in de Sitter and quasi-de Sitter spacetimes:Revisited
Suprit Singh a ∗ , Chandrima Ganguly b † and T. Padmanabhan a ‡ a IUCAA, Ganeshkhind, Pune 411007, INDIA. b Department of Physics, IIT Hyderabad, Yeddumailaram 502 205, INDIA.
Abstract
It is possible to associate temperatures with the non-extremal horizons of a large class ofspherically symmetric spacetimes using periodicity in the Euclidean sector and this procedureworks for the de Sitter spacetime as well. But, unlike e.g., the black hole spacetimes, the deSitter spacetime also allows a description in Friedmann coordinates. This raises the questionof whether the thermality of the de Sitter horizon can be obtained, working entirely in theFriedmann coordinates, without reference to the static coordinates or using the symmetries ofde Sitter spacetime. We discuss several aspects of this issue for de Sitter and approximatelyde Sitter spacetimes, in the Friedmann coordinates (with a time-dependent background and theassociated ambiguities in defining the vacuum states). The different choices for the vacuum states,behaviour of the mode functions and the detector response are studied in both (1+1) and (1+3)dimensions. We compare and contrast the differences brought about by the different choices. Inthe last part of the paper, we also describe a general procedure for studying quantum field theoryin spacetimes which are approximately de Sitter and, as an example, derive the corrections tothermal spectrum due to the presence of pressure-free matter.
There exists an extensive literature on quantum field theory in de Sitter spacetimes (for a nonexhaustive sample, see Ref. [1, 2]). From a theoretical point of view, the high level of symmetryexhibited by the de Sitter geometry makes it an important and tractable example. On the otherhand, observations suggest that the evolution of our universe is described by a (near) de Sittergeometry both during the early inflationary phase as well as during the current accelerated phaseof expansion. While quantum effects are not expected to play a serious role in the current phaseof the expansion (see, however, [3]), they play an important role during the inflationary phase andpossibly seeds the cosmic structure we see today. This was part of the motivation to study quantumfield theory in de Sitter background. ∗ [email protected] † [email protected] ‡ [email protected] exact de Sitter spacetime, many of these techniques will fail when the spacetime is only approximately deSitter. It is interesting to ask how much progress one can make in studying such (approximatelyde Sitter) spacetimes and how much of the results valid in exact de Sitter will continue to hold(in an approximate sense) in such spacetimes. For example, it is fairly complicated to work withan analogue of approximately static coordinate system when the universe is not strictly de Sitter.Many of the techniques used to define the vacuum states in exactly de Sitter spacetimes will also beinapplicable when the manifold has no de Sitter symmetry.It is thus clear that, given the special features possessed by de Sitter spacetime, one can approachthe problem of quantum field theory in de Sitter spacetime from many different perspectives, notall of which will be easily generalizable to an approximately de Sitter spacetime. This motivatesus to examine closely several aspects of quantum field theory in de Sitter spacetime delineating theproperties which arise (in one way or the other) from the symmetry of de Sitter spacetime from thosewhich are of more general nature. Such a study also reveals some significant differences between deSitter spacetime in (1+1) dimension and de Sitter spacetime in (1+3) dimensions.We will now briefly describe some of these issues which will be discussed in detail in the paper.As we mentioned earlier, there is a very standard procedure for obtaining the thermality of thehorizon in the static coordinate system. This procedure works for a very wide class of spacetimes2nd can, for example, handle Rindler, black hole and de Sitter spacetimes at one go. But whenthe cosmological spacetime is not exactly de Sitter, no static coordinate system will exist. One canstill define an “approximately static” coordinate system but this proves to be difficult to handlemathematically.If one decides to work with a Friedmann coordinate system, then the mathematics simplifiesconsiderably because we will be dealing with a quantum mechanical problem rather than quantumfield theory. But conceptually, we now have to tackle the issue of defining the vacuum state in atime dependent background. This turns out to be reasonably straight forward in (1+1) dimensionin which conformal invariance of a massless scalar field helps the analysis. But in (1+3) dimensionit is not possible to have sensible limits for the mode functions in the infinite past if one works withthe massless scalar field Φ( t, x ) as the primary variable. The usual trick is to work instead with thevariable χ ( t, x ) ≡ a ( t )Φ( t, x ) and define a vacuum state in the asymptotic past for χ ( t, x ). When t → −∞ , a ( t ) → a ( t ) and there is no naturalanalogue of this vacuum state for non-de Sitter spacetimes.An alternative to the above procedure is to define a vacuum state at some fixed time t = t , say,by choosing the modes which behave as close to the positive frequency modes as possible at thisinstant. We will call this the co-moving vacuum since it is based on the co-moving time coordinateof Friedmann spacetime. In general, this vacuum state differs from the Bunch-Davies vacuum butit has the advantage that the evolution of a ( t ) for t < t becomes irrelevant for its definition. It is,therefore, well suited to study spacetimes which are de Sitter at late times with deviations from deSitter geometry in the early epochs.Once the vacuum state is defined, in the asymptotic past or at some other chosen moment, onecould study the mixing of positive and negative frequency modes due to the time dependence of thebackground expansion. In particular, one would be interested in knowing whether the mixing leadsto a thermal nature for the state at later times. It does happen in the case of (1+1) dimension butthe spectrum is not strictly Planckian in the case of (1+3) dimensions. There are some interestingpeculiarities which arise in this context when we try to obtain thermality working entirely in theFriedmann coordinates.Finally, one can also study the inter-relationship between the mode functions defined in Fried-mann coordinate system and those defined with static coordinates. This is an exercise in evaluatingthe Bogolioubov coefficients and we do find that one recovers standard Planck spectrum without anydeviation. This allows us to establish a correspondence between the vacuum states defined usingthe two coordinate systems but — since static coordinate systems do not exist for approximately deSitter spacetimes — the approach does not allow an easy generalization to more realistic cases.In the last part of the paper we study the mode functions in approximately de Sitter spacetimesin Friedmann coordinates. We find an explicit solution to the wave equation, correct to the necessaryorder of approximation, and use it to describe the deviations from the exact de Sitter spacetime.This approach is quite general and is capable of handling a wide variety of cases when the evolutionis approximately de Sitter.The plan of the paper is as follows: We briefly review thermal aspects of horizons in staticcoordinate system in Section 2. In Section 3, we solve for the modes of a massless scalar field inspatially flat de Sitter spacetime in (1+1) and (1+3) dimensions and define the Bunch-Davies and3omoving vacuum states in the Friedmann coordinate patch. These modes evolve in time and thephysical content of the modes at later times is determined by evaluating the mixing coefficients inSection 4, working entirely in the Friedmann coordinates. In Section 5, we study the response of adetector coupled with the field as a way to provide an operational meaning to the mixing coefficients.We next compare, in Section 6, the mode functions defined in the Friedmann patch with those definedin the static patch of the de Sitter spacetime to reproduce some standard results. Finally, we studythe corresponding effects in the quasi-de Sitter geometry. We consider a small perturbation to thede Sitter metric and develop the perturbative framework to find the corrections to the field modesand the corresponding power spectrum in the quasi-de Sitter case. This procedure is illustrated bytaking the model of the universe containing pressure-free matter and cosmological constant whichbehaves like quasi-de Sitter at late times. Section 8 describes the conclusions. To set the stage, we shall begin by briefly reviewing some well known results (see e.g., chapter 14of Ref. [9] for more details) related to the temperature of horizons in static coordinates. Severalspacetimes of interest including the Schwarzschild, de Sitter, Rindler, can be described by a lineelement of the form ds = f ( r ) dt − dr f ( r ) − dL ⊥ (1)where dL ⊥ is the transverse metric and f ( r ) vanishes at the horizon, r = r with f ′ ( r ) ≡ κ = 0.Then, using a Taylor series expansion near the horizon, we can write f ≈ κl where l = ( r − r ) andthe metric near the horizon takes the form: ds = 2 κl dt − dr κl − dL ⊥ . (2)In the case of Rindler spacetime, this is exact and κ denotes the acceleration of the Rindler observer.In other cases, the metric reduces to this form close to the horizon with κ denoting the surfacegravity.This (Rindler) form of the metric makes it obvious that the singular behaviour of the metric near l = 0 is a coordinate artefact. It is possible to introduce several, different, sets of coordinates whichwill cover the entire manifold without any pathology at the horizon. One such choice, ( T, X ), whichwe will call Kruskal-like coordinates is obtained by the transformations: κX = e κr ∗ cosh κt ; κT = e κr ∗ sinh κt ; r ∗ ≡ Z d rf ( r ) (3)which lead to the metric ds = fκ ( X − T ) ( dT − dX ) + dL ⊥ (4)that covers the full manifold. Here f should be treated as a function of ( T, X ). The horizon at r = r is now mapped to T = X but with the factor f / ( X − T ) remaining finite at the horizon.It is now possible to show that the vacuum state of a quantum field defined on the T = 0hypersurface appears as a thermal state to observers confined on the right wedge X > | T | . This is4ost easily done by making an analytic continuation to the imaginary time coordinates by T E = − iT and t E = − it . The time evolution of the system in terms of T E will take the field configurationfrom T E = 0 to T E → ∞ and will be governed by a global Hamiltonian H gl . One can equivalentlydescribe the same evolution in terms of t E , which behaves like an angular coordinate from t E = 0to t E = 2 π/κ when we use the Hamiltonian H st which determines time evolution in the static timecoordinates. The entire upper half-plane T > T E or in terms of the evolution in t E . In ( T E , X ) coordinates, wevary X in the range ( −∞ , ∞ ) for each T E and vary T E in the range (0 , ∞ ). In ( t E , x ) coordinates, x varies in the range (0 , ∞ ) for each t E which varies in the range (0 , π/κ ) like an angular variable.This allows us to prove, using standard path integral techniques [9, 10] that, h vac | φ L , φ R i ∝ h φ L | e − πH st /κ | φ R i . (5)where φ L and φ R are the field configurations in left and right parts of the plane on the T = 0hypersurface. One can find the density matrix for observations confined to the right wedge bytracing out the field configuration φ L on the left wedge. This computation gives: ρ ( φ R , φ ′ R ) = h φ R | e − πH st /κ | φ ′ R i T r ( e − πH st /κ ) (6)which is thermal with the temperature β − = κ/ π . Thus, the vacuum state of the field defined onthe T = 0 hypersurface leads to a thermal density matrix with temperature κ/ π as far as staticobservers in the right hand wedge are concerned.In the case de Sitter the metrics in the static and Kruskal-like coordinates ds = (1 − H r ) dt − dr (1 − H r ) − dL ⊥ = 4[ H ( X − T ) + 1] ( dT − dX ) − dL ⊥ (7)are connected by the coordinate transformations: X = 1 H (cid:18) Hr − Hr (cid:19) / cosh Ht, T = 1 H (cid:18) Hr − Hr (cid:19) / sinh Ht ; (8)From the form of the metric in in the two coordinate systems, it is obvious that the ( T, X ) coordinatesystem is not static because the metric depends on T . On the other hand, the coordinate system( t, x ) which covers the right wedge has a static time coordinate t . It is well known that defininga vacuum state in a time dependent background is non-trivial and often ambiguous. In the aboveanalysis we have chosen to define a vacuum state at a particular space-like hypersurface T = 0 andexamine its properties in terms of the static coordinates. A different definition for the vacuum state,in general, will lead to a different description in static coordinates. We will see that similar issuesarise later on when we study de Sitter universe in the Friedmann coordinates as well.An alternative procedure to determine the thermal nature of the horizon is based on the cal-culation of relevant Bogoliubov coefficients. Since we have two coordinate systems — Kruskal-likeand static — covering part of the manifold, one can obtain, in principle, the relation between the5eld modes which are natural to these coordinate systems and compute the Bogoliubov coefficientsbetween them. Let the field modes be given by some functions φ ( T, X ) and χ ( t, r ) in the Kruskal-likeand static coordinates respectively in the region of the manifold where both are well defined. (Forsimplicity, we have ignored the dependence on the transverse coordinates which play no role in thediscussion, as we shall see.) It is often not possible to obtain closed expressions for the field modesdue to mathematical complexity. However, it is possible to evaluate the Bogoliubov coefficientsusing a simple trick: Since the Bogoliubov coefficients that relate the two sets of field modes areindependent of the hypersurface which is used to evaluate the Klein-Gordon inner product, we canchoose this hypersurface to be arbitrarily close to the horizon. The field equations reduce to a two-dimensional wave equation near the horizon making the dependence in the transverse coordinates(and the mass of the field) irrelevant. Conformal invariance then allows us to determine the fieldmodes near the future horizon which take the form of plane waves in the relevant coordinates. Thatis, χ ω ( t, r ) = 1 √ ω e − iωu (9)and φ k ( T, X ) = 1 √ k e − ikU . (10)where u = t − r ∗ and U = T − X respectively. These are related by κU = − e − κu which signifies anexponential redshift near the horizon. As is well known, the relevant Bogoliubov coefficient (whichwe will have the occasion to evaluate explicitly later on) will now lead to a thermal spectrum ofparticles.This discussion is, of course, applicable to the de Sitter universe described by the metric in Eq. (1)with f ( r ) = 1 − H r and will lead to a temperature H/ π . More precisely, if we introduce Kruskal-like coordinates in the de Sitter manifold and define a vacuum state on the T = 0 hypersurface,then such a vacuum state will lead to a density matrix with temperature H/ π for the observersconfined to the region r < H − . Once again, it should be stressed that the de Sitter metric in theKruskal-like coordinates is not static and the vacuum state is defined using the T = 0 hypersurface.This analysis is completely in tune with what could be done in black hole spacetimes as wellas in the case of Rindler spacetime. But in the case of de Sitter we have an alternative coordinatesystem available to us, viz.., the standard Friedmann coordinate system. This allows us to study thedynamics of a quantum field entirely in the Friedmann coordinate system and explore whether wecan recover the thermality of the horizons and other features. In such a study we necessarily haveto work with a time dependent background but — as we have emphasized above — this is implicit even when we use Kruskal-like coordinates and relate them to static coordinates. We can, therefore,adopt a similar strategy in the Friedmann coordinate system by defining a vacuum state at somesuitable hypersurface and studying its particle content as the evolution proceeds.This approach has one extra advantage. The static coordinate system exists only for the exact deSitter universe. When there are deviations from de Sitter nature, we can still describe the universein a very natural fashion using Friedmann coordinate system. But in this case, we will not have theluxury of an alternative static coordinate system to describe the physics. Therefore, a formalismwhich addresses issues like thermality working entirely in Friedmann coordinate system, withoutusing any of the symmetries of the de Sitter universe, is well suited for the study of near de Sittergeometry. We will find that obtaining thermal nature of the horizon working entirely in Friedmann6oordinates is — surprisingly — not an easy task. In fact we could not find any previous workin published literature which discusses quantum field theory in de Sitter spacetime from such anapproach and obtains thermal nature of the horizon. We shall now turn to this study. Throughout the paper, we will confine ourselves to massless, minimally coupled scalar field in deSitter spacetime. The action for the field Φ( t, x ) is given by: S [Φ] = 12 Z d n x √− g ∂ a Φ ∂ a Φ (11)It turns out that the dynamics is somewhat different in (1+1) dimensional spacetime compared to(1+3) dimensional spacetime. We will first study the behaviour in (1 + 1) and then follow the sameprocedure for the (1 + 3) case. This will bring out the similarities and some curious differencesbetween the two cases. spacetime We will describe the de Sitter spacetime in Friedmann coordinates with k = 0. Then, the (1+1)dimensional metric is given by, ds = dt − a ( t ) dx (12)and the field equation reads, ∂ t Φ + ˙ aa ∂ t Φ − a ∂ x Φ = 0 . (13)We decompose Φ in terms of a complete set of orthonormal functions f k in the formΦ( t, x ) = Z ∞−∞ dk π h ˆ a k f k + ˆ a † k f ∗ k i . (14)Spatial homogeneity allows us to separate out the x dependence and write: f k ( x, t ) = e ikx ψ | k | ( t ) (15)Substituting in Eq. (13) and solving the resulting equation, we find that ψ k ( t ) = A k s k ( t ) + B k s ∗ k ( t ); s k ( t ) = 1 √ k exp (cid:18) − ik Z d ta ( t ) (cid:19) (16)(The k in these expressions actually stand for | k | ; we will not explicitly show the modulus signhereafter for notational simplicity.) The result is obvious from the fact that in (1+1) dimension thescalar field action is conformally invariant and any Friedmann spacetime is conformally flat with theconformal time coordinate η defined through dη = dt/a ( t ). For dS with the scale factor a ( t ) = e Ht ,the solution is, s k ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (17)7here the phase ensures that, in the H → A k and B k in Eq. (16) are determined using the appropriateboundary conditions and thus decide the choice of the vacuum state for the field. For example, when H →
0, the choice A k = 1 and B k = 0 gives the positive frequency mode and selects the standardinertial vacuum for the flat spacetime.In the presence of an expanding background, it is difficult to define a unique choice for thevacuum and we need to study different choices and their physical properties. One possible choicewould be to define the vacuum state at the asymptotic past by choosing the field modes such thatthey reduce to positive frequency modes in this limit. It is, however, clear that the mode functionin Eq. (17) does not have a well defined phase when t → −∞ . (This is related to the fact thatin the asymptotic past a → t coordinate and instead usethe conformal time η which, for the de Sitter universe, can be taken to be η ≡ (1 − e − Ht ) /H . (Theintegration constant is chosen to give the correct limit of η → t when H → s k ( η ) = 1 √ k e − ikη (18)is indeed a positive frequency solution with respect to η (at all times) and therefore the choice A k = 1and B k = 0 gives a natural choice for the vacuum. This is the conventional Bunch-Davies vacuumdefined with respect to conformal time by the choice of mode functions ψ ( BD ) k ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (19)While the Bunch-Davies vacuum is the preferred choice in the literature, it is clear that it ismore in tune with the conformal time coordinate η than with the co-moving time coordinate t . Inthe Friedmann metric, the co-moving time t has a direct physical significance as the proper time ofthe co-moving, geodesic clocks. This motivates us to look at the possibility of defining a co-movingvacuum with mode functions which behave as close as possible to the positive frequency modes withrespect to co-moving time coordinate t . We can take a cue from the discussion in the last sectionwhere we saw that, even in the Kruskal-like coordinates for the de Sitter spacetime, the metric istime dependent and the vacuum state is defined on a particular hypersurface T = 0. In a similarfashion, we can choose the modes in Eq. (16) by demanding that at some time t = t they behavelike positive frequency modes. Because of the time translational invariance, we can take t = 0,without the loss of generality, as long as t is finite. That is, we impose the conditions: ψ k (0) = 1 √ k e − ikt | t =0 ; ˙ ψ k (0) = − ik √ k e − ikt | t =0 . (20)(The same physics is obtained if we take t = 0 with the replacement of k by ke − Ht which ensuresthat k is the co-moving wave number defined at t = 0). These conditions imply that at t = t (= 0)the mode function and its derivative behave like a positive frequency mode.We can now determine the coefficients A k and B k using this condition and — somewhat curiously— we will again find that A k = 1 and B k = 0. That is, the mode function ψ ( CM ) k ( t ), evolved from the co-moving vacuum defined at t = t (= 0) is same as Bunch-Davies state ψ ( BD ) k ( t ) defined earlier in8S . This result is independent of the choice for t thereby showing that the Bunch-Davies vacuumcan also be interpreted as a co-moving vacuum state defined using the conditions in Eq. (20).We will see later, this equivalence is a special feature of (1+1) dimension and does not hold in(1+3) dimensions where the co-moving and Bunch-Davies vacua are different. spacetime We shall now follow the same procedure as above in the (1 + 3) dimensions. The metric is now givenby ds = dt − exp(2 Ht ) d x (21)where H is the Hubble constant and sets the only length-scale (or time-scale) in the problem to be1 /H . The field equation for Φ( t, x ) in this metric reads: ∂ t Φ + 3 H∂ t Φ − exp( − Ht ) ∂ x Φ = 0 (22)As usual we expand the field in terms of a complete set of orthonormal functions f k and write:Φ( x , t ) = Z d k (2 π ) n ˆ a k f k ( t, x ) + ˆ a † k f ∗ k ( t, x ) o (23)where spatial homogeneity allows us to express the field modes in the form: f k ( t, x ) = e i k · x ψ k ( t ) (24)where k = | k | . The equation in ψ k ( t ) then becomes,¨ ψ k + 3 H ˙ ψ k + exp( − Ht ) k ψ k = 0 . (25)with the solution, ψ k ( t ) = A k s k ( t ) + B k s ∗ k ( t ) (26)where s k ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (cid:18) iHk + e − Ht (cid:19) (27)(Because of the existence of A k and B k , the normalization of s k is not unique; we choose it insuch a way that, when A k = 1 and B k = 0, the functions s k satisfy the standard orthonormalityconditions with respect to the Klein-Gordon inner product.) Again, the constants A k and B k areto be determined using the appropriate boundary conditions which makes a choice for the vacuumstate for the field. In (1+3) dimensions also, we see that when H → A k = 1 and B k = 0leads to the standard positive frequency mode in flat spacetime and selects the inertial vacuum. Ourinterest is to explore the different choices in the presence of expanding background.As in dS case, let us first study the behaviour of the modes in the asymptotic past. We seethat, in the t → −∞ limit, the expression in Eq. (27) goes to: s k ( t ) → √ k exp (cid:18) ikH e − Ht (cid:19) e − Ht . (28)9his does not have a well-defined limit and hence cannot be used to define a vacuum state for thefield. In this respect, both (1+1) and (1+3) dimensional results are similar.We found that, in the (1+1) dimensional case we could use the conformal time coordinate η todefine a natural vacuum state in the asymptotic past. In the present case, however, the situation isdifferent. In terms of the conformal time η , the mode function becomes (in the asymptotic past): s k ( η ) → a ( η ) e − ikη √ k (29)and we see now the crucial difference from the (1 + 1) dimensional case. There is an extra a ( η ) inthis case which prevents us from treating it as the standard positive frequency mode.The result also suggests a possible way-out which is usually adopted in the literature. Instead ofquantising Φ we may choose to quantise ¯Φ ≡ a ( t )Φ. This is a (time dependent) point transformationof the dynamical variable which is permissible in the classical description. We then see that thechoice A k = 1 and B k = 0 will give the modes exp( − ikη ) which has the standard form for ¯Φ, whentreated as a quantum field. This is the usual procedure in the literature and this choice leads tothe conventional Bunch-Davies vacuum. But note the the situation was not as straightforward as inthe case of (1 + 1) dimensions and we needed to remove a factor a ( t ) to define the vacuum state in(1+3) dimensions.The difference is more acute when we try to define a co-moving vacuum. As in the case of (1 + 1)dimension one can define the co-moving vacuum by imposing the conditions given in Eq. (20) andthus determining A k and B k . Because of the time translation invariance, we can again define theco-moving vacuum at at t = 0 and the result for any other time, t can be obtained by a finite shift.Hence the conditions we impose on the modes are ψ k (0) = 1 √ k e − ikt | t =0 ; ˙ ψ k (0) = − ik √ k e − ikt | t =0 . (30)These allow us to determine the constants A k and B k as: A k = H + 2 ik ik ; B k = H ik (31)which define the mode function, ψ ( CM ) k ( t ), evolved from the co-moving vacuum choice defined at t = 0.When we did this in (1+1) dimension, we found that A k = 1 and B k = 0 - instead of theexpressions in Eq. (31) — thereby showing the equivalence of co-moving and Bunch-Davies vacuum.But in (1+3) dimensions we get a different result, viz. that the co-moving vacuum is different fromthe Bunch-Davies vacuum. The difference can be traced, algebraically, to the existence of the a ( η )factor in Eq. (29).To summarise, we can define the vacuum states by imposing suitable boundary conditions on themode functions and thus determining the constants A k and B k . If we work in the asymptotic past,then one can choose the modes to be exp( − ikη ) in (1+1) dimension, thereby defining the Bunch-Davies vacuum. In (1+3) dimensions this is not possible with the original scalar field. But if wework with a ( t )Φ, instead of Φ, one can again define the modes such that they behave as exp( − ikη )in the asymptotic past. Alternatively, one can attempt to define a co-moving vacuum by imposing10he condition that the modes must behave as close to positive frequency solutions as possible, withrespect to the co-moving time coordinate t , at some time t = t . Because of time translationinvariance, we can choose t = 0 without loss of generality. We then find that, in (1+1) dimension,the co-moving vacuum is equivalent to the Bunch-Davies vacuum. But in (1+3) dimensions, thesetwo mode functions (and hence the vacua are different). We shall now explore the properties of thesevacuum states. The Bunch-Davies and the co-moving vacua are defined by the condition that the mode function ispurely positive frequency at a given moment of time t = t . In the case of Bunch-Davies vacuum,this is done in the asymptotic past ( t → −∞ ) while in the case of co-moving vacuum we choosethis to be t = 0. Once this initial condition is set, expansion of the universe will evolve the modefunctions to a mixture of positive and negative frequency modes, with respect to the co-moving timecoordinate, at any later time. This mixing can be analysed in terms of two mixing coefficients, α ν and β ν in the expansion: ψ k ( t ) = Z ∞ d ν π (cid:0) α ν e − iνt + β ν e iνt (cid:1) (32)It is slightly more convenient to let the frequency vary over both positive and negative values andwrite: ψ k ( t ) = Z ∞−∞ d ν π f ( ν ) e − iνt (33)so that α ν = f ( ν ) , β ν = f ( − ν ); ν > not the same.We stress that in Eq. (32) ψ k ( t ) is expanded in terms of the complete set of orthonormal functionsexp( ± iνt ) which are not the solutions to scalar field wave equation in the de Sitter background.Physically, one can think of these functions exp( ± iνt ) as defining the instantaneous positive andnegative frequency mode functions with respect to the co-moving time. But as we shall see, thesemixing coefficients have interesting properties and in fact play a direct role in the response of detec-tors. We shall say more about it later on.The task of determining the mixing coefficients is thus reduced to calculating the the Fouriertransform of ψ k ( t ), f ( ν ) = Z ∞−∞ d t e iνt ψ k ( t ) . (35)Often we will be interested in | α ν | and | β ν | which can be obtained from the power spectrum | f ( ν ) | .We shall now compute these for the different cases. We will begin with the (1+1) dimensional case for which the modes are ψ k ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (36)11o, the Fourier transform is: f ( ν ) = e − ik/H √ k Z ∞−∞ d t e iνt e ik/H e − Ht = e − ik/H √ k (cid:18) H (cid:19) (cid:18) kH (cid:19) iν/H Γ (cid:18) − iνH (cid:19) e πν/ H . (37)Similarly, f ( − ν ) = e ik/H √ k (cid:18) H (cid:19) (cid:18) kH (cid:19) iν/H Γ (cid:18) − iνH (cid:19) e − πν/ H . (38)so that the modulus square of the coefficients of mixing in Eq. (34) are given by : | α ν | = 12 kν βe βν e βν − | β ν | = 12 kν βe βν − β = 2 πH . (39)That is the power spectrum per logarithmic band at negative frequencies (given by | β ν | ) is Planckianat temperature, H/ π . At first sight this might look like the familiar result, well known in literature.However, there are some peculiar features which need to be commented on.Note that we have started with a solution to the wave equation in de Sitter background (givenby Eq. (36)) and expanded it using the complete set of functions exp( ± iνt ). These functions haveno “legality” in the de Sitter spacetime since they are not the solutions of the wave equation. Wecould have, for example, used any other complete set of orthonormal functions in place of exp( ± iνt )and could have defined the mixing coefficients through an equation like Eq. (32). The two propertieswhich favour our choice are: (a) they were precisely the mode functions used to define the co-movingvacuum and (b) they are instantaneous, monochromatic, plane waves with respect to the co-movingtime t . It is therefore interesting that the overlap between positive and negative frequencies in suchan expansion gives rise to the thermal spectrum.If we had used the conformal time instead of co-moving time, then the result would have beenvery different — and very trivial. In terms of the conformal time, the modes are just exp( ± ikη ) at all η and there is no mixing of the positive and negative frequencies defined with respect to η . So if wehad defined another set of mixing coefficients with an equation like Eq. (32) but with conformal time η , then we would have got the trivial result β ν = 0. So if we define the vacuum state with respect toconformal time and work entirely in terms of conformal time we will see no trace of thermal spectrumin the de Sitter universe. This is, of course, obvious from the fact that the metric is conformallyflat in ( η, x ) coordinates and the scalar field theory is conformally invariant in (1+1) dimension; sowe are back to the evolution of inertial vacuum in flat spacetime. On the other hand, the modesundergo exponential redshift when frequencies are defined with respect to co-moving time and —as we had already mentioned — the exponentially redshifted wave will lead to a thermal mixingcoefficient. We will next see that the situation is somewhat different in the (1+3) dimensional case. In this case, which we want to study in detail, it is convenient to work with a general mode function,having arbitrary coefficients A k and B k . From Eq. (26), we have, ψ k ( t ) = A k s k ( t ) + B k s ∗ k ( t ) = ψ (1) k ( t ) + ψ (2) k ( t ) (40)12here s k ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (cid:18) iHk + e − Ht (cid:19) (41)Recall that taking A k = 1 and B k = 0 gives Bunch-Davies state and for the co-moving statethe corresponding values are provided by Eq. (31). We again choose the normalization such that f k ≡ s k exp( i k · x ) satisfies the standard orthonormality conditions ( f k , f k ′ ) = δ ( k − k ′ ) with respectto the Klein-Gordon inner product:(Φ , Φ ) ≡ − i Z t d x a ( t )[Φ ∂ t Φ ∗ − Φ ∗ ∂ t Φ ] (42)where the integral is evaluated over a hypersurface of constant t .Some amount of algebra yields the following results for the Fourier transform of the respectiveparts, ψ (1) k ( t ) and ψ (2) k ( t ): f (1) ( ν ) = A k e − ik/H e πν/ H (2 k ) / (cid:18) kH (cid:19) iνH Γ (cid:18) − iνH (cid:19) (cid:16) i + νH (cid:17) f (2) ( ν ) = B k e ik/H e − πν/ H (2 k ) / (cid:18) kH (cid:19) iνH Γ (cid:18) − iνH (cid:19) (cid:16) − i − νH (cid:17) . (43)Taking the square of the modulus of the above expressions we get, ν | f (1) ( ν ) | = H k |A k | βe βν e βν − (cid:18) ν H (cid:19) ν | f (2) ( ν ) | = H k |B k | βe βν − (cid:18) ν H (cid:19) (44)where β = 2 π/H . For negative frequencies, the forms of power spectrum are: ν | f (1) ( − ν ) | = H k |A k | βe βν − (cid:18) ν H (cid:19) ν | f (2) ( − ν ) | = H k |B k | βe βν e βν − (cid:18) ν H (cid:19) (45)Let us first consider the Bunch-Davies state, with A k = 1 and B k = 0. This gives α ν = f (1) ( ν ) and β ν = f (1) ( − ν ) so that we find: | α ν | = H k ν βe βν e βν − (cid:18) ν H (cid:19) (46) | β ν | = H k ν βe βν − (cid:18) ν H (cid:19) (47)In contrast to the (1+1) dimensional case, these are not thermal, due to the extra factor (1+ ν /H ).So the expansion of the universe leads to a mixing of positive and negative frequencies but the13esulting mixing coefficients do not have a thermal form. It should be noted, however, that the ratioof the mixing coefficients is | β ν | | α ν | = | f (1) ( − ν ) | | f (1) ( ν ) | = e − βν (48)When the field ψ k ( t ) couples linearly to a detector, the rate of upward and downward transitionsbetween any two levels of the detector will be determined by the mixing coefficients. Therefore,when the condition in Eq. (48) holds, one is led to a level population in the detector at thermalequilibrium with the temperature β − = H/ π . Any multiplicative function h ( ν ) with α ν and β ν drops off in the ratio. [Usually, one works with Bogoliubov coefficients which satisfy the constraint | α | − | β | = 1; in that case, if Eq. (48) holds, then | β | must be thermal. In the case of mixing coefficients we have defined, the condition | α | − | β | = 1 does not hold which allows extra factorslike (1 + ν /H ).]Let us next consider the co-moving vacuum which holds more surprises. We now require thesquare of the modulus of complete f ( − ν ) which is combination of individual quantities, | f (1) ( − ν ) | , | f (2) ( − ν ) | evaluated above and a cross-term given by,2 ν | f (1) ( − ν ) || f (2) ( − ν ) | cos θ = H k |A k ||B k | (cid:18) ν H (cid:19) β e βν/ e βν − θ (49)where θ = arg( f (1) , f (2) ) . (50)The complete expression becomes: ν | f ( − ν ) | = H β k (cid:18) ν H (cid:19)h(cid:16) |A k | + |B k | e βν (cid:17) N + 2 |A k ||B k | p N ( N + 1) cos θ i (51)where N = 1 e βν − A k , B k given by Eq. (31). This result shows that for a massless scalarfield prepared in the co-moving vacuum state, we obtain an expression having the Planckian factorwith the temperature H/ π . In addition, we obtain an interference term involving p N ( N + 1)which can be thought of as the fluctuation in the occupation number in thermal equilibrium. Thisfactor has been noticed earlier [11] in the case of horizon thermodynamics though no clear physicalexplanation is available. As far as we know, this has not been noticed earlier in the case of de Sitterspacetime in any context. The mixing coefficients defined through Eq. (32) are directly related to the response of a co-movinggeodesic detector in Friedmann universe. Since the clock carried by such a detector will measure the14o-moving time t , the rate of transition between the levels of the detector will involve the factorsexp( ± it ∆ E ) where ∆ E is the energy difference between the two levels. This gives an operationalmeaning to the mixing coefficients and we will show that the response of a co-moving , geodesicdetector shows features very similar to what we obtained in the last section.Consider a stationary detector, located at the spatial origin in a de Sitter spacetime and coupledto the massless scalar field by monopole interaction. The amplitude for excitation of this detectorduring the time interval ( − T, + T ) due to its interaction with the scalar field can be computed, infirst order perturbation theory as: A k = M Z T − T dτ e iν τ h k | Φ( x [ τ ]) | i (53)where M = iλ h E | ˆ m (0) | E i is amplitude of transition in the internal levels of the detector with λ asthe coupling constant and ˆ m (0) is the detector’s monopole operator. (In the above expression, weare confining our attention to a final field state containing a particle with a specified momentum k .The total excitation probability for the detector is obtained by integrating | A k | over all k .) Thedetector interacts with the field only during the period − T to T and x a ( τ ) = x a ( t ) = ( t, , ,
0) isthe trajectory of the detector. Expanding Φ( x [ τ ]) as in Eq. (14), we find that the only term thatsurvives in the T → ∞ limit is the negative frequency term. The amplitude arising from this termis given by, A k = M Z T − T dt e iνt (cid:20) A ∗ k √ k e ik/H (cid:18) − iHk + e − Ht (cid:19) e − ik/H e − Ht + B ∗ k √ k e − ik/H (cid:18) iHk + e − Ht (cid:19) e ik/H e − Ht (cid:21) . (54)This can be recast as A k = MA ∗ k √ k e ik/H (cid:18) − iHk (cid:19) lim µ → (1 − ∂ µ ) I µ ( ν )+ MB ∗ k √ k e − ik/H (cid:18) iHk (cid:19) lim µ → (1 − ∂ µ ) I ∗ µ ( − ν ) (55)where I µ ( ν ) = Z T − T dt e iνt e − ikµ/H e − Ht = (cid:18)Z ∞−∞ dt − Z − T −∞ dt − Z ∞ T dt (cid:19) e iνt e − ikµ/H e − Ht . (56)The above integral can be evaluated to give, I µ = 1 H (cid:18) kH (cid:19) iν/H e − πν/ H e iνH ln µ (cid:20) Γ (cid:18) − iνH , i kµH e − HT (cid:19) − Γ (cid:18) − iνH , i kµH e HT (cid:19)(cid:21) (57)where Γ( a, b ) is an incomplete gamma function. With this the amplitude becomes, A k = M e ik/H √ k (cid:18) − iHk (cid:19) A ∗ k (cid:18) − iνH (cid:19) I ( ν ) + M e − ik/H √ k (cid:18) iHk (cid:19) B ∗ k (cid:18) − iνH (cid:19) I ∗ ( − ν ) (58)15here we have ignored terms coming from differentiating the gamma functions since those are purelyoscillatory and can be made to vanish in the large ν limit by using the standard iǫ prescription. Theprobability P k for the transition is now given by, P k = | A k | = M H k (cid:18) ν H (cid:19) (cid:0) |A k | | I ( ν ) | + |B k | | I ( − ν ) | +2 |A k ||B k || I ( ν ) || I ∗ ( − ν ) | cos θ ) (59)To avoid the transients arising due to finite T , we will take the limit of HT ≫
1. In this case, anelementary computation gives: I ≈ e − πν/ H H (cid:18) kH (cid:19) iν/H Γ (cid:18) − iνH (cid:19) − ik e − HT e − i kH e HT e − iνT . (60)so that | I ( ν ) | = βNν − e − HT k r βNν cos θ ′ + O ( e − HT ) | I ∗ ( − ν ) | = βe βν Nν + 2 e − HT e βν/ k r βNν cos θ ′′ + O ( e − HT ) (61)where N = ( e βν − − and β = 2 π/H . Therefore, when HT ≫
1, we get the transition probabilityto be, P k = M H k (cid:18) ν H (cid:19) βν h(cid:16) |A k | + |B k | e βν (cid:17) N + 2 |A k ||B k | p N ( N + 1) cos θ i (62)A comparison with Eq. (51) shows that the detector response is triggered by essentially | f ( − ν ) | which should be obvious from the fact that the amplitude in Eq. (53) picks out the negative frequencycomponent of the field when the time integration is extended over the range ( −∞ , ∞ ). This resultshows that our mixing coefficients have a direct connection with the operational definition of particlecontent, as determined by the detector response.The above result is general and is valid for arbitrary A k , B k . By taking specific values we candetermine the detector response in Bunch-Davies and co-moving vacuum. In the Bunch-Davies case,we have A k = 1 and B k = 0 giving P k = M H k (cid:18) ν H (cid:19) βν N (63)This result shows that the detector response does pick up the extra factor (1 + ν /H ) just as themixing coefficients do. (The same factor has been noticed earlier in ref. [12]). The correspondingresult for co-moving vacuum can be obtained by substituting Eq. (31) into Eq. (62) but the result hasno special features worth mentioning. The above results arise because, by definition, the geodesicdetector measures the co-moving time t . 16 Relation to the results in static coordinate system
In Sec. 2, we briefly described how thermal nature of the de Sitter horizon arises in the staticcoordinate system and in the last few sections we studied the field theory in Friedmann coordinatesystem. Since both coordinate systems coexist in part of the de Sitter manifold, one can make anexplicit comparison of the quantum states defined in these two coordinate systems. (This is similar tocomparing the states in inertial coordinate system and Rindler coordinate system in flat spacetime.)For this, we need to compute the relevant Bogoulibov coefficients on a spacelike hypersurface betweenthe relevant mode functions by using the Klein-Gordon inner product. As we shall see, this is fairlystraightforward in (1+1) but somewhat complicated in (1+3). We begin by noting that the metric, ds = dt − e Ht dx (64)in ( t, x ) coordinates can be written in the static coordinates (˜ t, ˜ x ) as ds = (cid:0) − H ˜ x (cid:1) d ˜ t − (cid:0) − H ˜ x (cid:1) − d ˜ x = (cid:0) − H ˜ x ( x ∗ ) (cid:1) ( d ˜ t − d ˜ x ∗ ) (65)where ˜ x = e Ht x ; ˜ t = t − / H ln (cid:0) − H ˜ x (cid:1) (66)and ˜ x ∗ = Z d˜ x (1 − H ˜ x ) . (67)is the tortoise coordinate. Using these transformations we can express the field modes in the time-dependent dS coordinates in terms of the static coordinates. We will focus on a fixed ( k >
0) modeso that the mode function f k ( t, x ) = 1 √ k e − ik/H e ikx e ik/H e − Ht (68)becomes f k (˜ t, ˜ x ) = 1 √ k exp ike − H ˜ t (1 − H ˜ x ) / + ike − H ˜ t ˜ xH (1 − H ˜ x ) / ! = 1 √ k e i ( k/H ) e − Hu (69)in static coordinates where u ≡ ˜ t − ˜ x ∗ . In the static deSitter patch, conformal flatness of the metricin Eq. (65) allows us to write down the solution to the field equation as exp( ± iωu ). This allows theexpansion Φ Rω = 1 √ ω (cid:16) ˆ b ω e − iωu + ˆ b † ω e iωu (cid:17) (70)etc. which is valid on the complete manifold. We now need to determine the Bogoliubov coefficientsthat relate the above two sets of field modes. These are given by the standard Klein-Gordon innerproduct. β ωk = − i Z ˜ t d˜ x ∗ (cid:0) Φ Rω ∂ ˜ t f k − f k ∂ ˜ t Φ Rω (cid:1) (71)17here the integral is over any spacelike hypersurface. Choosing ˜ t = 0 surface, the above integralover ˜ x ∗ can be recast as: β ωk = − i √ ω Z ∞−∞ d u (cid:0) e − iωu ∂ u f k − f k ∂ u e − iωu (cid:1) (72)Integrating the first term by parts gives, β ωk = √ ω Z ∞−∞ d u e − iωu f k ( u ) + f k e − iωu | ∞−∞ = √ ω Z ∞−∞ d u e − iωu f k ( u ) (73)since the second term vanishes. Thus β ωk = r ωk Z ∞−∞ d ue − iωu e ik/H e − Hu = r ωk (cid:18) H (cid:19) (cid:18) kH (cid:19) iω/H Γ (cid:18) − iωH (cid:19) e − πω/ H . (74)We find that modulus | β ωk | is again Planckian at temperature, H/ π : | β ωk | = βk ( e βω −
1) ; β = 2 πH . (75)This shows that the Bunch-Davies vacuum (which is the same as the co-moving vacuum in (1+1)dimension) has a thermal character in the static patch bounded by the horizon. The transformation from the Friedmann coordinates to static coordinates goes through in dS exactlyin the same way as dS . The metric ds = dt − e Ht (cid:0) dr + r d Ω (cid:1) (76)in ( t, r, Ω) system can be written in the static coordinates (˜ t, ˜ r, Ω) as ds = (cid:0) − H ˜ r (cid:1) d ˜ t − (cid:0) − H ˜ r (cid:1) − d ˜ r − ˜ r d Ω = (cid:0) − H ˜ r ( r ∗ ) (cid:1) ( d ˜ t − dr ∗ ) − ˜ r ( r ∗ ) d Ω (77)with the same transformations as before˜ r = e Ht r ; ˜ t = t − / H ln (cid:0) − H ˜ r (cid:1) (78)and defining the tortoise coordinate r ∗ = Z d˜ r (1 − H ˜ r ) . (79)18n the static coordinates, the field equation reads " ∂ ∂ ˜ t − f (˜ r )˜ r ∂∂ ˜ r (cid:18) ˜ r f (˜ r ) ∂∂ ˜ r (cid:19) − f (˜ r ) ˆ L r Φ(˜ t, ˜ r, Ω) = 0 (80)where f (˜ r ) = (1 − H ˜ r ) and ˆ L is the standard angular Laplacian operator. TakingΦ = φ l (˜ r ) Y lm (Ω) e − iω ˜ t / ˜ r , we find that φ l (˜ r ) satisfies the equation: − ω φ l − f ˜ r dd ˜ r (cid:18) ˜ r f dd ˜ r (cid:18) φ l ˜ r (cid:19)(cid:19) − l ( l + 1) f ˜ r φ l = 0 . (81)Since f (˜ r ) vanishes at the horizon ˜ r = 1 /H , only the s -mode makes a dominant contribution nearthe horizon and hence we will focus on l = 0 mode. For this mode, the wave equation becomes d φdr ∗ + (cid:18) ω − f f ′ ˜ r (cid:19) φ = 0 (82)where the prime denotes derivative with respect to ˜ r . Clearly, in the near horizon limit ( f → ± iωr ∗ ). Thus near the past horizon, ˜ r → /H and ˜ t → −∞ , the modes inthe static coordinate system behave as exp( ± iωv ).On the other hand, the modes describing the Bunch-Davies vacuum can be expressed in sphericalcoordinates by the standard plane wave expansionΦ BDk = e − ik/H √ k ∞ X l =0 i l (2 l + 1) j l ( kr ) P l (cos θ ) e ik/He − Ht (cid:18) iHk + e − Ht (cid:19) (83)Using the transformations in Eq. (78) we can express this in (˜ t, ˜ r ) coordinates. Concentrating onthe s -wave contribution we obtainΦ ( BD ) k = e − ik/H √ k (cid:16) e ik/He − Hu − e ik/He − Hv (cid:17) (cid:18) iHk e H ˜ t (cid:0) − H ˜ r (cid:1) + 1 (cid:19) . (84)which, near the past horizon, ˜ r → /H and ˜ t → −∞ , behaves asΦ ( BD ) k → √ k e i ( k/H ) e − Hv . (85)We now use the fact that the Klein-Gordon inner product between the field modes is independentof the surface over which it is evaluated. It is, therefore, convenient to evaluate the Bogoliubovcoefficients on a space-like surface very close to the horizon. Since the Bunch-Davies mode behaves as e i ( k/H ) e − Hv while the static modes behave as exp( ± iωv ), it is obvious that the Bogoliubov coefficientsdefined in Eq. (73) will give β ωk = √ ω Z ∞−∞ d v e − iωv Φ ( BD ) k ( v ) (86)which has a thermal character: | β ωk | = βk ( e βω −
1) ; β = 2 πH . (87)19e again see that the Bunch-Davies vacuum has a thermal property when viewed in the static patchin (1+3) dimensions as well. In this sense, (1+1) and (1+3) dimensions behave identically. It isalso straightforward to show that a detector at rest in the static coordinates will perceive a thermalradiation in the Bunch-Davies vacuum state. On the other hand, we saw earlier that a freely-falling detector will also see the modified thermal spectrum [see Eq. (63)] in the same vacuum state. Itshould be noted that this is somewhat contrary to the results in black hole spacetime.Finally, we quote the result for the co-moving vacuum transformed to static coordinates. Theanalysis is again straightforward when we use the fact that the co-moving modes can be expressedin terms of the Bunch-Davies modes by the relation ψ ( CM ) k ( t ) = A k ψ ( BD ) ( t ) + B k ψ ( BD ) ∗ ( t ) (88)Therefore, Φ ( CM ) k → √ k (cid:16) A k e ik/He − Hv + B k e − ik/He − Hv (cid:17) . (89)on the past horizon. It follows that the spectrum is now given by k | β ωk | = (cid:16) |A k | + |B k | e βν (cid:17) βN + |A k ||B k | β p N ( N + 1) cos θ (90)where N = 1 e βν − p N ( N + 1) even when we compare the modes between Friedmann description and static descriptionsuggesting that there must be some physical explanation for the origin of this factor. We hope toaddress this question in a future publication. So far, we have been concentrating on the features which are special to de Sitter spacetime. However,in the evolution of the real universe, it is impossible to obtain a pure de Sitter evolution due to thepresence of external matter. Both, during the inflationary phase as well as during the late timeacceleration phase, we only have a quasi-de Sitter phase rather than a pure de Sitter universe.In this section we will extend the formalism described earlier to a quasi-de Sitter spacetime bydetermining an approximate solution to the wave equation. This approach is quite general and cantake into account any first order deviation from the pure de Sitter universe. After developing theformalism we will apply it to a specific example to illustrate its utility.
Consider a Friedmann spacetime with the scale factor given by: a ( t ) = e ( Ht + ǫλ ( t )) ≈ e Ht (1 + ǫλ ( t )) = a + ǫλa (91)which can be treated as quasi-de Sitter if the condition ¨ λ ≪ ˙ λH is satisfied. In the above expansion,we have retained the perturbation to first order as indicated by the bookkeeping parameter ǫ (which20ill be set to unity at the end of the computation). Correspondingly, the mode functions, which arethe solutions to the wave equation in the perturbed metric, will differ from those in the de Sitterspacetime by a small amount: ψ ( t ) = ψ ( t ) + ǫ δψ ( t ) (92)where ψ ( t ) is the unperturbed mode function and we have omitted the subscript k for notationalsimplicity. Substituting the above expressions for a ( t ) and ψ ( t ) into the time dependent part of thewave equation written in the form: d ψdt + 3 (cid:18) ˙ aa (cid:19) dψdt + k a ψ = 0 . (93)we get, d ( δψ ) dt + 3 H d ( δψ ) dt + 3 ˙ λ dψ dt − k a λψ + k a δψ = 0 . (94)This equation can be solved by writing δψ = ψ s . The function s ( t ) then satisfies the equation, d sdt + ψ ψ + 3 H ! dsdt = µ ( t ) (95)where µ ( t ) ≡ k a λ − λ ˙ ψ ψ (96)acts like a source term. Eq. (95) is first order in ds/dt and hence can be immediately integrated.(This result holds for a generic class of second-order homogeneous linear differential equation; seeAppendix B for details). The solution for s ( t ) is s ( t ) = C Z t d t ′ ψ − e − Ht ′ + Z t d t ′ ψ − ( t ′ ) e − Ht ′ Z t ′ d t ′′ ψ ( t ′′ ) e Ht ′′ µ ( t ′′ ) (97)where C is a constant of integration. Thus, given a model for λ ( t ) and appropriate boundaryconditions we can, in principle, solve for the perturbation δψ by this method. As an illustration of the above method, let us consider the late time accelerated phase of the universecontaining dust-like matter and a cosmological constant. The expansion factor of such a universe isgiven by a ( t ) = 2 / (cid:18) sinh 32 Ht (cid:19) / (98)In the spirit of the above discussion, we will treat this as a perturbation to an exact de Sitter universeand write a ( t ) ≈ e Ht (cid:18) − e − Ht (cid:19) (99)21here λ ( t ) = − (2 /
3) exp( − Ht ) which vanishes as t goes to infinity. This behaviour suggests thatwe use the boundary conditions s ( ∞ ) = 0 and ˙ s ( ∞ ) = 0 in our general solution given by Eq. (97).In the pure de Sitter case, the mode functions can be taken to be ψ ( t ) = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (cid:18) iHk + e − Ht (cid:19) (100)which amounts to taking A k = 1 and B k = 0 in the Eq. (26) i.e., we have chosen to work withBunch-Davies state. Calculating the integrals in Eq. (97) is straightforward (see Appendix C fordetails) and we obtain, s ( t ) = 7 k H e − Ht (101)Therefore, the first-order change in the mode function is given by: δψ ( t ) = ψ ( t ) s ( t ) = 7 ik e − ik/H H √ k (cid:18) − ikH e − Ht (cid:19) e − Ht e ikH e − Ht . (102)We can now compute the Fourier transform of this expression to determine the first order correctionin Fourier space: δf ( ν ) = Z ∞−∞ d t δψ ( t ) e iνt = 7 ik H √ k e − ik/H Z ∞−∞ d t (cid:18) − ikH e − Ht (cid:19) e − Ht e ikH e − Ht e iνt = 7 ik H √ k e − ik/H lim µ → (1 − ∂ µ ) Z ∞−∞ d t e − Ht + iνt e ikµH e − Ht = − H e − ik/H k √ k (cid:18) kH (cid:19) iν/H e πν/ H (cid:18) − iνH (cid:19) Γ (cid:18) − iνH (cid:19) (103)The resulting power spectrum, to the lowest order, is given by: ν | F ( ν ) | ≈ ν | f ( ν ) | + ǫ ν Re[2 f ∗ ( ν ) δf ( ν )] (104)where we have reintroduced ǫ for bookkeeping and ν Re[2 f ∗ ( ν ) δf ( ν )] = 7 H k βe βν e βν − (cid:18) νH + 64 ν H − ν H + ν H (cid:19) (105)is the correction to the power spectrum in the case of quasi-de Sitter phase arising from the mattercontribution in the late-time acceleration. The periodicity in the Euclidean time allows us to attribute a temperature H/ π using the staticcoordinates on the de Sitter manifold. In this sense, de Sitter spacetime behaves just like other static22pacetimes with a horizon. However, in such an analysis, one has to define a vacuum state on a T = 0 hypersurface in the Kruskal-like coordinate system which is not static. In this particular case,thermal nature of the de Sitter horizon arises because the vacuum state in Kruskal-type coordinatesystem leads to a thermal density matrix for the observers bounded by the de Sitter horizons.The de Sitter spacetime is unique in the sense that it also allows introducing Friedmann coordi-nates in which the metric is homogeneous. This, in turn, reduces the field theoretic problem to thatof a quantum oscillator with a time dependent frequency. In such a, time dependent, backgroundthere is no unique definition for the vacuum state and the best one could do is to introduce wellmotivated vacuum states and study their physical properties. Quite generically, such states can beintroduced by giving a suitable boundary condition for the mode functions at some time t = t . Thequestion arises as to whether one can understand the thermality of de Sitter universe working en-tirely within the Friedmann coordinates i.e., without comparing the results between Friedmann andstatic coordinates. (We have not seen such a derivation in the literature for a massless scalar field.)We investigated several aspects of this question both in (1+1) dimension and in (1+3) dimensionsin this paper.Two natural vacuum states one can introduce are the Bunch-Davies and co-moving vacuum statesin this spacetime. In (1+1) dimension, the Bunch-Davies vacuum state corresponds to choosing themodes to be positive frequency with respect to the conformal time η in the asymptotic past while theco-moving vacuum state corresponds to imposing the positive frequency condition at some arbitraryinstant of time t = t . It turns out that both these states are identical in (1+1) dimension. To studythe time evolution of this state, we expand the mode function in terms of positive and negativefrequency modes defined with respect to the co-moving time. The mixing of positive and negativefrequency modes then reveals a thermal character with temperature H/ π . This, of course, doesnot happen during the time evolution in the conformal time; the positive frequency mode remains apositive frequency mode at all times.The situation in (1+3) dimensions is quite different. To begin with, co-moving vacuum state andthe Bunch-Davies vacuum state do not coincide in (1+3) dimensions. Further, the mixing coefficientbetween positive and negative frequency modes does not have a pure thermal character (and ismodified by an extra frequency dependent factor) in the case of Bunch-Davies vacuum. The resultfor the case of co-moving vacuum is more complicated and involves an interference term containing p N ( N + 1) factor which is reminiscent of the fluctuations in the occupation numbers of masslessthermal radiation.The physical meaning of the mixing coefficients introduced to analyse the above phenomenacan be understood by studying the response of particle detectors in the de Sitter spacetime. Wecomputed the rate of excitation of a geodesic detector evolving in co-moving time. This rate exactlymatches with the particle content of the state as determined by the mixing coefficients in bothBunch-Davies vacuum and co-moving vacuum.We also compared the states defined using Friedmann coordinate system with those definedusing the static coordinate system. This requires evaluating the necessary Bogoliubov coefficientbetween the mode functions defined in the static patch and Friedmann patch in the region of themanifold where they coexist. We found that the Bunch-Davies vacuum appears to be a thermalstate for static observers bounded by the horizon, both in (1+1) and (1+3) dimensions. This is incontrast with the results obtained within the Friedmann coordinate system where the results for(1+3) dimensions differ from the results for (1+1). On the other hand, the co-moving vacuum in231+3) dimension, defined in Friedmann coordinates, does not have a simple thermal interpretationin the static coordinates.In the last part of the paper, we studied the effects of small deviations from de Sitter evolutionand the resulting corrections to the mode functions. This formalism is sufficiently general to handleany functional form of the deviation in the lowest order of perturbation theory. As an illustrationof this formalism, we studied the deviations in the power spectrum arising due to the existence ofpressure-free matter during the late time accelerated phase of the universe. This formalism mighthave applications for studying the spectral deviations in the case of inflationary universe as well. Acknowledgements
SS is supported by a fellowship from the Council of Scientific and Industrial Research (CSIR), India.CG would like to thank IUCAA for hosting her in the VSP program. TP’s research is partiallysupported by the J.C.Bose Research Grant of DST, India.
A Calculation of the Fourier transform in Eq. (33)
To evaluate I = Z ∞−∞ dt e iνt e ikµH e − Ht , we define u = e − Ht and b = kµ/H . This gives, I = 1 H Z ∞ du u − − iνH e ibu = 1 H exp (cid:20) iνH ln (cid:12)(cid:12)(cid:12)(cid:12) kµH (cid:12)(cid:12)(cid:12)(cid:12) + πν H sign (cid:18) kµH (cid:19)(cid:21) Γ (cid:18) − iνH (cid:19) = 1 H (cid:18) kH (cid:19) iν/H e πν/ H Γ (cid:18) − iνH (cid:19) e iνH ln µ . (106) B A result in perturbation theory
Consider a generic second-order homogeneous linear differential equation a ( t )¨ x ( t ) + b ( t ) ˙ x ( t ) + c ( t ) x ( t ) = 0Let x be the solution of above equation for some functions a ( t ), b ( t ) and c ( t ). We are nowinterested in the corresponding solution of the equation when the parameter functions a , b and c areperturbed about their original forms. Then, to the first order in perturbation, we have a δ ¨ x + ¨ x δa + b δ ˙ x + ˙ x δb + c δx + x δc = 0 . (107)Scaling the perturbation δx with the unperturbed solution as, δx ≡ x s , gives for s ( t ) the equation¨ s + A ( t ) ˙ s = B ( t ) (108)24ith A ( t ) = 2 ˙ x x + b a B ( t ) = − ¨ x x δaa − ˙ x x δba − δca (109)which is a first order differential equation in ˙ s and can be solved immediately. C Solution of Eq. (97) for late-time accelerated phase
The basic ingredients that go in are ψ = 1 √ k exp (cid:20) − ikH (cid:0) − e − Ht (cid:1)(cid:21) (cid:18) iHk + e − Ht (cid:19) , which gives ˙ ψ ψ = − ik e − Ht H (cid:0) i + kH e − Ht (cid:1) and µ ( t ) = 2 k e − Ht λ − λ ˙ ψ ψ = − k e − Ht + 6 ik e − Ht ( i + k/H e − Ht )so that under the conditions s ( ∞ ) = ˙ s ( ∞ ) = 0, we can set C , the constant of integration in thehomogeneous part to be zero and obtain: s ( t ) = (cid:18) H k (cid:19) e − ikH Z ∞ t d t ′ e − Ht ′ ψ ( t ′ ) Z ∞ t ′ d t ′′ e ikH e − Ht ′′ e − Ht ′′ (cid:20) − − k H e − Ht ′′ + 10 ikH e − Ht ′′ (cid:21) = (cid:18) H (cid:19) Z ∞ t d t ′ e − ikH e − Ht ′ e − Ht ′ (cid:20) − γ (cid:18) , − i kH e − Ht ′ (cid:19) + γ (cid:18) , − i kH e − Ht ′ (cid:19) − γ (cid:18) , − i kH e − Ht ′ (cid:19)(cid:21) Noting that, − γ (2 , x ) + γ (4 , x ) − γ (3 , x ) = −
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