Detecting the massive bosonic zero-mode in expanding cosmological spacetimes
DDetecting the massive bosonic zero-mode inexpanding cosmological spacetimes
Vladimir Toussaint ∗ and Jorma Louko † School of Mathematical Sciences, University of Nottingham NingboChina,Ningbo 315100, PR China School of Mathematical Sciences, University of Nottingham,Nottingham NG7 2RD, UKFebruary 2021
Abstract
We examine a quantised massive scalar field in (1 + 1)-dimensional spatiallycompact cosmological spacetimes in which the early time and late time expansionlaws provide distinguished definitions of Fock “in” and “out” vacua, with thepossible exception of the spatially constant sector, which may become effectivelymassless at early or late times. We show, generalising the work of Ford andPathinayake, that when such a massive zero mode occurs, the freedom in therespective “in” and “out” states is a family with two real parameters. As anapplication, we consider massive untwisted and twisted scalar fields in the (1 + 1)-dimensional spatially compact Milne spacetime, where the untwisted field hasa massive “in” zero mode. We demonstrate, by a combination of analytic andnumerical methods, that the choice of the massive “in” zero mode state has asignificant effect on the response of an inertial Unruh-DeWitt detector, especiallyin the excitation part of the spectrum. The detector’s peculiar velocity withrespect to comoving cosmological observers has the strongest effect in the “in”vacuum of the untwisted field, where it shifts the excitation and de-excitationresonances towards higher values of the detector’s energy gap. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b Introduction
In quantum field theory on spatially compact spacetimes, it is well known that thewave equation may have normalisable zero frequency solutions, known as zero modes,which create an ambiguity in the choice of a vacuum state in Fock quantisation [1–8].It was demonstrated by Ford and Pathinayake [9] that a similar ambiguity arises incosmological spacetimes with compact spatial sections when a spatially constant modeof a massive field becomes asymptotically massless in an asymptotic “in” or “out” region,in such a way that the adiabatic evolution criterion cannot be invoked to define a uniquevacuum adapted to this “in” or “out” region. We refer to such field modes as massivezero modes.In this paper we examine how the choice of a massive zero mode quantum stateaffects the response of a localised quantum system that moves inertially in a cosmologicalspacetime. As a localised quantum system we consider an Unruh-DeWitt detector [10,11], which provides a simplified model for the interaction between atomic orbitals and thequantum electromagnetic field when angular momentum interchange is negligible [12,13].Unruh-DeWitt detectors have a long pedigree as a device for extracting local informationfrom quantum states defined by nonlocal criteria, including black hole spacetimes, andin recent years they have been much employed in entanglement extraction scenarios(see [14–16] for a sample). Our work fits in the context where the state of a quantumfield has been singled out by early universe cosmological considerations but the state isbeing probed by local observers in the late universe [17,18]. Specifically, we generalise tocosmological spacetimes previous work on observing zero modes in a static spacetime [7,8, 19].We work in two spacetime dimensions, for the technical reason that this allows usto analyse the response of an Unruh-DeWitt detector operating for a finite time in atime-dependent geometry without having to smear the detector in time or in space.We expect similar phenomena due to the choice of the quantum state and due to thedetector’s motion to be present also in higher spacetime dimensions, but there thesephenomena will be necessarily blurred by choices that will need to be made for smearingthe detector’s profile in time or in space [20, 21].We begin in Section 2 by discussing a quantised real massive scalar field in a (1 + 1)-dimensional spatially homogeneous cosmological spacetime whose spatial sections arecompactified to have the topology of a circle. The main points are to characterisesituations in which a massive zero mode exists, and to characterise the freedom inthe corresponding vacuum state. We show that the zero mode vacua form a familyparametrised by two real-valued parameters, as observed by Ford and Pathinayake [9]for a subset of our asymptotic conditions. On a par with the normal scalar field, we alsoinclude in this section the quantisation of scalar field that is antiperiodic on traversingthe spatial circle. As this field, called the “twisted” field, has no massive zero mode, itwill provide a point of contrast for the massive zero mode effects in the later sections.In Section 3 we recall the expression for the transition probability of an Unruh-DeWitt detector [10, 11], coupled linearly to the quantum field, treated to leading order2n perturbation theory, and operating for a finite time [22–25]. As we work in (1 + 1)spacetime dimensions, we can take the detector to operate at constant coupling strengthbetween a sharp switch-on moment and a sharp switch-off moment, without encounteringinfinities in the theory.In Section 4 we specialise to the spatially compactified Milne spacetime, in whichthe scale factor increases linearly in the cosmological time, and show that the untwistedscalar field has a massive zero mode in the asymptotic region near the initial singularity.In Section 5 we write out the response of a detector on inertial but not necessarilycomoving trajectories.Section 6 is a brief interlude in which we write out the response of an inertial finitetime detector in Minkowski spacetime, giving numerical plots. These plots will providea benchmark to which the corresponding plots in Milne will be seen to reduce in theappropriate limits.Section 7 presents the core results of the paper: numerical plots of the detector’sresponse in spatially compactified Milne, on inertial but not necessarily comoving de-tector trajectories, in vacua adapted to late time dynamics and to early time dynamics,and, for the untwisted field, with selected choices for the early time massive zero modestate. When the spatial period is small, or when the trajectory is at early cosmolog-ical times, we see significant deviations from the Minkowski vacuum response for thede-excitation part of the spectrum, whereas the choice of the massive zero mode statefor the untwisted field has a significant effect on the excitation part of the spectrum.The detector’s peculiar velocity with respect to comoving cosmological observers has thestrongest effect in the “in” vacuum of the untwisted field, where it shifts the excitationand de-excitation resonances towards higher values of the detector’s energy gap.Section 8 presents a summary and concluding remarks.We use units in which (cid:126) = c = 1. We work in the sign convention in which ds > In this section we quantise a real massive scalar field in a (1 + 1)-dimensional spatiallycompactified Friedmann-Lemaˆıtre-Robertson-Walker spacetime. We consider both afield that takes values in the trivial bundle, called an untwisted or periodic field, and afield that takes values in a nontrivial R / Z bundle, called a twisted or antiperiodic field. We consider a spacetime with the line element ds = dt − a ( t ) dx = C ( η )( dη − dx ) , (2.1)3here the scale factor a ( t ) is assumed positive, the cosmological time t and the conformaltime η are related by dη/dt = 1 /a ( t ), and C ( η ) = a ( t ( η )). C is by assumption positive,and we assume it to be a C ∞ function of η .We take x to be periodic with period L >
0, so that ( t, x ) ∼ ( t, x + L ), or ( η, x ) ∼ ( η, x + L ). The constant η surfaces are hence topologically circles.We write the range of η as η ∈ ( η in , η out ), where η in and η out may be finite orrespectively ∓∞ , depending on the cosmological model; the application to Milne inSection 4 will have η in = −∞ and η out = ∞ . We refer to the asymptotic region η → η in as the remote past, or the “in” region. Similarly, we refer to the asymptotic region η → η out as the remote future, or the “out” region.We consider a real scalar field φ of mass m >
0. In terms of the conformal time, theaction reads S = (cid:90) L dη dx , (2.2)where the Lagrangian density L is given by L = 12 (cid:2) ( ∂ η φ ) − ( ∂ x φ ) − µ ( η ) φ (cid:3) , (2.3a) µ ( η ) := m (cid:112) C ( η ) . (2.3b)Note that µ ( η ) appears as an effective time-dependent mass, and it is by assumptionpositive. The field equation is the Klein-Gordon equation, (cid:18) ∂ ∂η − ∂ ∂x + µ ( η ) (cid:19) φ ( η, x ) = 0 . (2.4)An untwisted field is periodic as ( η, x ) (cid:55)→ ( η, x + L ), while a twisted field is antiperiodicas ( η, x ) (cid:55)→ ( η, x + L ).We seek mode solutions to the field equation by the separation ansatz U n ( η, x ) = L − / χ n ( η ) exp( ik n x ) , (2.5)where n ∈ Z and k n := (cid:40) πn/L for untwisted field , π ( n + ) /L for twisted field . (2.6)The differential equation for χ n is χ (cid:48)(cid:48) n ( η ) + ω n ( η ) χ n ( η ) = 0 , (2.7)4here the prime denotes derivative with respect to η and ω n ( η ) := (cid:0) k n + µ ( η ) (cid:1) / . (2.8)To make the mode solutions U n (2.5) a positive norm orthonormal set in the Klein-Gordon inner product,( U n , U m ) := i (cid:90) L/ − L/ dx ( U ∗ n ∂ η U m − U m ∂ η U ∗ n ) = δ nm , (2.9)we require the mode functions χ n to be chosen so that they satisfy the Wronskiancondition W [ χ n , χ ∗ n ] := χ n χ (cid:48)∗ n − χ ∗ n χ (cid:48) n = i . (2.10)The complex conjugate mode solutions U ∗ n form then a negative norm orthonormal setin the Klein-Gordon inner product. The choice for the mode functions χ n is not unique. Suppose that one choice is made bycriteria that rely on the behaviour of the mode functions in the “in” region, and anotherchoice is made by criteria that rely on the behaviour of the mode functions in the “out”region. We denote these modes by respectively χ inn and χ outn . As the modes solve thesame differential equation, the modes must be related by the Bogoliubov transformation χ outn ( η ) = α n χ inn ( η ) + β n χ in ∗ n ( η ) , (2.11)where α n and β n are constants. The Wronskian condition (2.10) for each set implies | α n | − | β n | = 1 . (2.12)Denoting the corresponding positive frequency mode solutions by U inn and U outn , wemay expand the the quantised scalar field ˆ φ in either the “in” or “out” modes asˆ φ ( η, x ) = (cid:88) n (cid:16) U in/outn ( η, x )ˆ a in/outn + h.c. (cid:17) , (2.13)on a Hilbert space carrying a representation of the equal-time canonical commutationrelations (CCRs) for ˆ φ and the conjugate momentum ˆ π ( η, x ) = ∂ η ˆ φ ( η, x ), (cid:104) ˆ φ ( η, x ) , ˆ π ( η, x (cid:48) ) (cid:105) = iδ x,x (cid:48) . (2.14)The CCRs imply that the nonvanishing commutators for the creation and annihilation5perators are (cid:2) ˆ a in/outn , ˆ a in/out † m (cid:3) = δ n,m . (2.15)The in and out vacuum states | in/outL (cid:105) are defined, respectively, byˆ a in/outn | in/outL (cid:105) = 0 . (2.16)The Wightman functions in the two vacua are given by G in/out ( η, x ; η (cid:48) , x (cid:48) ) := (cid:104) in/outL | ˆ φ ( η, x ) ˆ φ ( η (cid:48) , x (cid:48) ) | in/outL (cid:105) = 1 L (cid:88) n χ in/outn ( η ) χ in/out ∗ n ( η (cid:48) ) e ik n ( x − x (cid:48) ) . (2.17) We now turn to the criteria for choosing the “in” and “out” modes.In many situations, it is possible to choose the “in” and “out” modes to have theasymptotic adiabatic form [26–28] χ in/outn ( η ) −−−−−−→ η → η in/out (cid:112) ω n ( η ) exp (cid:18) − i (cid:90) η ω n ( η (cid:48) ) dη (cid:48) (cid:19) , (2.18)which implements the physical requirement that the modes be of locally positive fre-quency with respect to the conformal Killing vector ∂ η . Such mode functions exist whenthe time dependence of the spacetime at η → η in/out is sufficiently slow; a technicalcondition that guarantees sufficient slowness is d p dη p C (cid:48) C −−−−−−→ η → η in/out , ∀ p ≥ . (2.19)In particular, if C ( η ) tends to a positive constant as η → η in/out , we may then think ofthe in/out region as asymptotically static.We are interested in a situation in which C ( η ) tends to zero as η → η in/out , in such away that (2.19) does not hold. In this case µ ( η ) = m (cid:112) C ( η ) tends to zero as η → η in/out ,and we see from (2.7) and (2.8) that the n = 0 mode of the untwisted field does nothave an adiabatic asymptotic form. Following Ford and Pathinayake [9], we call a modewith this property a massive zero mode.We assume the falloff of C ( η ) as η → η in/out to be such that the leading terms in thegeneral solution for the untwisted field’s spatially constant mode χ are χ ( η ) −−−−−−→ η → η in/out a + a η , a , a ∈ C . (2.20)6n example is when C ( η ) decays exponentially as η → η in/out = ∓∞ , so that (2.19)holds for p > p = 0. This is the out-region situation considered by Fordand Pathinayake [9] and the in-region situation that we shall encounter in Section 4.Another example is when C ( η ) is a multiple of η , 0 < η < ∞ , in which case a ( t ) isa multiple of t / , corresponding to a four-dimensional radiation-dominated expansionlaw: there is now a massive zero mode satisfying (2.20) in the in-region, η → + . In therest of this section we can however proceed assuming just (2.20), leaving the details ofthe falloff of C ( η ) unspecified.Given (2.20), the Wronskian condition (2.10) gives a a ∗ − a ∗ a = i , (2.21)which shows that neither a nor a can vanish. We may hence fix the overall phase of χ uniquely by taking a >
0, and then write the general solution of (2.21) as a = b + i b , (2.22a) a = b , (2.22b)where b ∈ R and b >
0. The choices for the mode function of the massive zero modeform hence a family with two real parameters.
We now write out a quantum theory of the untwisted field, assuming that the n = 0mode is a massive zero mode satisfying (2.20) in the past or in the future, while forall the other modes the positive norm mode functions can be chosen by the adiabaticcriterion (2.18), respectively in the past or in the future. The main issue is to identifythe consequences of the choice of the mode function for the massive zero mode.We decompose the quantum field ˆ φ asˆ φ ( η, x ) = ˆ φ ( η ) + ˆ φ osc ( η, x ) , (2.23a)ˆ φ osc ( η, x ) = (cid:88) n (cid:54) =0 (cid:16) U n ( η, x )ˆ a n + h.c. (cid:17) , (2.23b)ˆ φ ( η ) = 1 √ L χ ( η )ˆ a + h.c. , (2.23c)where the mode functions have been chosen as described above, and we have droppedthe superscripts specifying whether the choice of the modes refers to the “in” region orthe “out” region. We refer to the n (cid:54) = 0 modes as the oscillator modes.7he Wightman function (2.17) decomposes as G ( η, x ; η (cid:48) , x (cid:48) ) = G ( η ; η (cid:48) ) + G osc ( η, x ; η (cid:48) , x (cid:48) ) , (2.24a) G ( η ; η (cid:48) ) = 1 L χ ( η ) χ ∗ ( η (cid:48) ) , (2.24b) G osc ( η, x ; η (cid:48) , x (cid:48) ) = 1 L (cid:88) n (cid:54) =0 χ n ( η ) χ ∗ n ( η (cid:48) ) e ik n ( x − x (cid:48) ) . (2.24c)The part that depends on the choice of the mode functions of the massive zero modeis G ( η ; η (cid:48) ) (2.24b). In the asymptotic region, (2.20) and (2.22) show that G ( η ; η (cid:48) )redudes to G ( η ; η (cid:48) ) −→ L (cid:20)(cid:18) b + i b (cid:19) + b η (cid:21) (cid:20)(cid:18) b − i b (cid:19) + b η (cid:48) (cid:21) . (2.25)The expectation value of the stress-energy tensor requires a renormalisation, but themassive zero mode contribution to the expectation value of the energy density may befound by elementary considerations as follows. Let (cid:98) T µν denote the massive zero modecontribution to the stress-energy tensor operator. In the coordinates ( η, x ), we see from(2.2) and (2.3) that (cid:98) T ηη ( η ) = 12 (cid:104)(cid:0) ˆ φ (cid:48) ( η ) (cid:1) + m C ( η ) (cid:0) ˆ φ ( η ) (cid:1) (cid:105) , (2.26)where we recall that the prime denotes ddη . Using (2.23c), we have (cid:104) L | (cid:98) T ηη ( η ) | L (cid:105) = 12 L (cid:104) | χ (cid:48) ( η ) | + m C ( η ) | χ ( η ) | (cid:105) . (2.27)The contribution from the massive zero mode to the energy density seen by a comovingobserver is hence ρ ,comov = −(cid:104) L | (cid:98) T ηη ( η ) | L (cid:105) = 12 L (cid:34) | χ (cid:48) ( η ) | C ( η ) + m | χ ( η ) | (cid:35) . (2.28)In the asymptotic region, (2.20) and (2.22) give ρ ,comov −→ L (cid:20) β C ( η ) + m (cid:18) ( b + b η ) + 14 b (cid:19)(cid:21) . (2.29)As b >
0, and C ( η ) → ρ ,comov hence grows without bound in theasymptotic region, proportionally to β .We end this section with two comments.First, we note that the contributions to the CCRs (2.14) from the massive zero mode8nd the oscillator modes decompose as (cid:104) ˆ φ ( η ) , ˆ π ( η ) (cid:105) = iL , (2.30a) (cid:104) ˆ φ osc ( η, x ) , ˆ π osc ( η, x (cid:48) ) (cid:105) = i (cid:18) δ x,x (cid:48) − L (cid:19) , (2.30b)where ˆ π ( η ) = ddη ˆ φ ( η ) and ˆ π osc ( η, x ) = ∂∂η ˆ φ osc ( η, x ). The massive zero mode Hamilto-nian is ˆ H = L (cid:16) ˆ π + m C ( η ) ˆ φ (cid:17) . (2.31)The dynamics of the massive zero mode is therefore that of a nonrelativistic particle onthe real line in a quadratic potential with a time-dependent frequency. From (2.10) and(2.23c) we obtain χ (cid:48)∗ ( η ) ˆ φ ( η ) − χ ∗ ( η )ˆ π ( η ) = i ˆ a √ L . (2.32)In a “position” representation, in which ˆ π = − ( i/L ) ∂ φ by (2.30a), the wave functionof the massive zero mode Fock vacuum is henceΨ ( φ ) = N exp (cid:20) iLχ (cid:48)∗ ( η )2 χ ∗ ( η ) φ (cid:21) , (2.33)where η denotes a reference moment such that φ is the position representation of ˆ φ ( η ),and N is a normalisation constant. In the asymptotic region, where the nonrelativisticparticle becomes free, (2.20) and (2.22) giveΨ ( φ ) −→ (2 Lb /π ) / (cid:112) ib ( η + b /b ) exp (cid:20) − Lb ib ( η + b /b ) φ (cid:21) , (2.34)which is recognised as the Gaussian wave packet of a free nonrelativistic particle, with η specifying the moment of time. This offers another physical interpretation of theparameters b and b .Second, given that the massive zero mode Fock vacuum is not unique, it may be ofinterest to consider more general states for this mode. As an example, following [9],consider the coherent state | z (cid:105) , satisfying ˆ a | z (cid:105) = z | z (cid:105) , where z ∈ C is a parameter.The above analysis generalises in a straightforward fashion. The massive zero modecontribution to the Wightman function generalises from (2.24b) to G ( η ; η (cid:48) ) = 1 L (cid:104)(cid:0) zχ ( η ) + z ∗ χ ∗ ( η ) (cid:1)(cid:0) zχ ( η (cid:48) ) + z ∗ χ ∗ ( η (cid:48) ) (cid:1) + χ ( η ) χ ∗ ( η (cid:48) ) (cid:105) , (2.35)9nd the contribution to the comoving energy density generalises from (2.28) to ρ ,comov = 12 L (cid:40) | χ (cid:48) ( η ) | + (cid:0) zχ (cid:48) ( η ) + z ∗ χ (cid:48)∗ ( η ) (cid:1) C ( η )+ m (cid:104) | χ ( η ) | + (cid:0) zχ ( η ) + z ∗ χ ∗ ( η ) (cid:1) (cid:105)(cid:41) . (2.36) In this section we recall relevant properties of an Unruh-DeWitt detector coupled linearlyto the scalar field [10, 11], operating for a finite time [22–25].The detector is a spatially pointlike two-state system, moving on the timelike world-line x ( τ ), where τ is the proper time. The detector’s Hilbert space H D is spanned by theorthonormal states | (cid:105) D and | ω (cid:105) D , satisfying ˆ H D | (cid:105) D = 0 and ˆ H D | ω (cid:105) D = ω | ω (cid:105) D , whereˆ H D is the detector’s Hamiltonian and ω ∈ R . | (cid:105) D is the ground state if ω > ω < H D ⊗ H φ , where H φ is theHilbert space of the field φ . The total Hamiltonian is ˆ H = ˆ H D + ˆ H φ + ˆ H int , where ˆ H φ is the Hamiltonian of the free field andˆ H int := cχ ( τ )ˆ µ ( τ ) ˆ φ (cid:0) x ( τ ) (cid:1) , (3.1)where c ∈ R is a coupling constant, the real-valued switching function χ specifies how theinteraction is turned on and off, and ˆ µ ( τ ) is the detector’s monopole moment operator,evolving in the interaction picture asˆ µ ( τ ) = e i ˆ H D τ ˆ µ (0) e − i ˆ H D τ . (3.2)Suppose that the switching function χ has compact support, and the total systemis initially prepared in the product state | Ψ (cid:105) ⊗ | (cid:105) D , where the field state | Ψ (cid:105) satisfiesthe Hadamard condition [29]. After the interaction has ceased, the probability for thedetector to have made the transition to the state | ω (cid:105) D is, in first-order perturbationtheory, P = c | D (cid:104) | µ (0) | (cid:105) D | F ( ω ) , (3.3)where F ( ω ) := (cid:90) dτ dτ (cid:48) χ ( τ ) χ ( τ (cid:48) ) e − iω ( τ − τ (cid:48) ) G ( τ, τ (cid:48) ) (3.4)10nd G ( τ, τ (cid:48) ) := (cid:104) Ψ | ˆ φ (cid:0) x ( τ ) (cid:1) ˆ φ (cid:0) x ( τ (cid:48) ) (cid:1) | Ψ (cid:105) . (3.5) F is called the response function, and it encodes the dependence of P on the trajectory,the switching and ω , as the prefactor c | D (cid:104) | µ (0) | (cid:105) D | in (3.3) depends only on c andthe detector’s internal structure. With minor abuse of terminology, we refer to F as thetransition probability.As | Ψ (cid:105) is by assumption Hadamard, G ( τ, τ (cid:48) ) is a well-defined distribution [30, 31].If χ is smooth, F is hence well defined. In spacetime dimension 1 + 1, however, thecoincidence limit singularity of G ( τ, τ (cid:48) ) is only logarithmic [29], and F is then welldefined also for less regular χ . We shall take χ to have a sharp switch-on and switch-off, χ ( τ ) = Θ( τ − τ )Θ( τ − τ ) , (3.6)where τ denotes the switch-on moment and τ denotes the switch-off moment, and weassume τ < τ . The response function (3.4) then becomes F ( ω, τ , τ ) := (cid:90) τ τ dτ (cid:90) τ τ dτ (cid:48) e − iω ( τ − τ (cid:48) ) G ( τ, τ (cid:48) ) . (3.7) In this section we specialise to the expanding Milne spacetime with compactified spatialsections.
The expanding (1 + 1)-dimensional Milne universe is the special case of (2.1) for which[27] ds = dt − a t dx = e aη ( dη − dx ) , (4.1)where a is a positive constant, 0 < t < ∞ , −∞ < η < ∞ and t = a − e aη . In thenotation of (2.1), we have C ( η ) = e aη . t , x and η have the dimension of length and a has the dimension of inverse length.When x is not compactified, this spacetime is the future quadrant, y > | y | , ofMinkowski spacetime, with the metric ds = ( dy ) − ( dy ) , (4.2)11 igure 1: (1 + 1)-dimensional Milne spacetime. When x is not compactified, the ( t, x ) co-ordinates cover the future light-cone of Minkowski space, y > | y | . The lines x = constantare timelike geodesics emanating from the origin. Making x periodic means identifying thespacetime by a boost. as seen by the coordinate transformation y = a − e aη cosh( ax ) , (4.3a) y = a − e aη sinh( ax ) . (4.3b)The cosmological initial singularity t → + is a coordinate singularity on the future lightcone of the origin, y = | y | , as illustrated in Figure 1.We consider the case in which x is compactified, by the identification ( η, x ) (cid:55)→ ( η, x + L ), which is geometrically a boost of rapidity aL . In this case the initial singularityat t → + is a genuine singularity, known as the Misner space singularity [32].A selection of previous work on quantum field theory on the spacetime withoutspatial compactification (and on its four-dimensional counterpart) is [27, 33–37]. Weshall consider the spatially compactified spacetime. As the criterion (2.19) does nothold at η → −∞ for p = 0, we may expect the untwisted field to have a massive zeromode, and we shall see that this indeed happens.12 .2 Twisted field The mode equation (2.7) is solvable in terms of Bessel functions of imaginary order [38].For the twisted field, the criterion (2.18) selects the “in” and ”out” positive frequencymode functions χ inn ( η ) = [(2 a/π ) sinh( π | k n | /a )] − / J − i | k n | /a ( me aη /a ) , (4.4a) χ outn ( η ) = 12 ( π/a ) / e π | k n | / (2 a ) H (2) i | k n | /a ( me aη /a ) , (4.4b)where we recall that k n is given by the twisted field expression in (2.6). The “in” and“out” asymptotic behaviour is [38] χ inn ( η ) −−−−→ η →−∞ e iϕ n (2 | k n | ) − / exp ( − i | k n | η ) , (4.5a) χ outn ( η ) −−−−→ η → + ∞ e iπ/ (2 m e aη ) − / exp ( − im e aη /a ) , (4.5b)where the real-valued phase constant ϕ n has an expression in terms of Euler’s gamma-function.The Wightman functions for the “in” and “out” vacua are G in ( η, x, η (cid:48) , x (cid:48) ) = π aL (cid:88) n J − i | k n | /a ( s ) J i | k n | /a ( s (cid:48) ) e ik n ( x − x (cid:48) ) sinh( π | k n | /a ) , (4.6a) G out ( η, x, η (cid:48) , x (cid:48) ) = π aL (cid:88) n e π | k n | /a H (2) i | k n | /a ( s ) H (1) − i | k n | /a ( s (cid:48) ) e ik n ( x − x (cid:48) ) , (4.6b)where we have written s := me aη /a and s (cid:48) := me aη (cid:48) /a . For the response of the detectorin the two vacua, on the worldline x ( τ ) = (cid:0) t ( τ ) , x ( τ ) (cid:1) , (3.7) then gives F int ( ω, τ , τ ) = π aL ∞ (cid:88) n = −∞ | A n ( ω, τ , τ ) | sinh( π | k n | /a ) , (4.7a) F outt ( ω, τ , τ ) = π aL ∞ (cid:88) n = −∞ e π | k n | /a | B n ( ω, τ , τ ) | , (4.7b)where the subscript t stands for “twisted” and A n ( ω, τ , τ ) := (cid:90) τ τ dτ J − i | k n | /a ( mt ( τ )) e − i ( ωτ − k n x ( τ )) , (4.8a) B n ( ω, τ , τ ) := (cid:90) τ τ dτ H (2) i | k n | /a ( mt ( τ )) e − i ( ωτ − k n x ( τ )) . (4.8b)13 .3 Untwisted field For the untwisted field, there are no massive zero modes at η → ∞ , but the spatiallyconstant mode is a massive zero mode at η → −∞ . We shall therefore consider the“out” and “in” vacua separately. For the “out” vacuum, we may proceed as for the twisted field in subsection (4.2),with the exception that k n is now given by the untwisted field expression in (2.6). TheWightman function is G out ( η, x, η (cid:48) , x (cid:48) ) = π aL ∞ (cid:88) n = −∞ e π | k n | /a H (2) i | k n | /a ( s ) H (1) − i | k n | /a ( s (cid:48) ) e ik n ( x − x (cid:48) ) , (4.9)where again s = me aη/a and s (cid:48) = me aη (cid:48) /a . For the detector’s response, on the worldline x ( τ ) = (cid:0) t ( τ ) , x ( τ ) (cid:1) , we have F outu ( ω, τ , τ ) = π aL ∞ (cid:88) n = −∞ e π | k n | /a | B n ( ω, τ , τ ) | , (4.10)where B n is as in (4.8b), and the subscript u stands for untwisted. For the “in” vacuum, the n (cid:54) = 0 modes can be treated as in subsection 4.2. We callthese modes the oscillator modes. Their contribution to the Wightman function is G inosc ( η, x, η (cid:48) , x (cid:48) ) = π aL (cid:88) n (cid:54) =0 J − i | k n | /a ( s ) J i | k n | /a ( s (cid:48) ) e ik n ( x − x (cid:48) ) sinh( π | k n | /a ) , (4.11)and their contribution to the detector’s response is F inosc ( ω, τ , τ ) = π aL (cid:88) n (cid:54) =0 | A n ( ω, τ , τ ) | sinh( π | k n | /a ) . (4.12)The n = 0 mode functions are given by χ in ( η ) = c J ( me aη /a ) + c Y ( me aη /a ) , (4.13)where the coefficients c and c cannot be fixed by (2.18), but they are still constrained14y the Wronskian condition (2.10), which reads c c ∗ − c ∗ c = πi a . (4.14)Fixing the overall phase of χ in so that c >
0, we parametrise c and c as c = α + i π aβ , (4.15a) c = β , (4.15b)where α ∈ R and β > χ in is χ in ( η ) −−−−→ η →−∞ b + i b + b η , (4.16)where b = α + 2 βπ (cid:104) ln (cid:16) m a (cid:17) + γ (cid:105) , (4.17a) b = 2 aβπ , (4.17b)and γ is the Euler-Mascheroni constant [38]. The constants b ∈ R and b > α and β . This is the rationale for the “in” label for χ in .Note that in the special case α = 0 and β = ( π/a ) / , we have c = i ( π/a ) / and c = ( π/a ) / , so that χ in = iχ out . For all other values of α and β , χ in and χ out arelinearly independent.By (2.24b) and (4.13), the zero-momentum mode contribution to the Wightmanfunction is G in ( η, η (cid:48) ) = 1 L (cid:16) | c | J ( s ) J ( s (cid:48) ) + | c | Y ( s ) Y ( s (cid:48) )+ c c ∗ J ( s ) Y ( s (cid:48) ) + c ∗ c Y ( s ) J ( s (cid:48) ) (cid:17) , (4.18)where again s = me aη/a and s (cid:48) = me aη (cid:48) /a . By (3.7), the contribution to the detector’sresponse is F in ( ω, τ , τ ) = 1 L (cid:8) | c M ( ω, τ , τ ) | + | c N ( ω, τ , τ ) | + 2 Re [ c c ∗ M ( ω, τ , τ ) N ∗ ( ω, τ , τ )] (cid:9) , (4.19)15here M ( ω, τ , τ ) := (cid:90) τ τ dτ J ( mt ( τ )) e − iωτ , (4.20a) N ( ω, τ , τ ) := (cid:90) τ τ dτ Y ( mt ( τ )) e − iωτ . (4.20b)Collecting (4.12) and (4.19), the final formula for the response is F inu ( ω, τ , τ ) = F inosc ( ω, τ , τ ) + F in ( ω, τ , τ ) , (4.21)where the subscript u stands for “untwisted.” In this section we specialise to detector trajectories in Milne that are inertial but notnecessarily comoving with the cosmological expansion. We shall write the response as1 /m times a dimensionless function of dimensionless combinations of the parameters,suitable for numerical evaluation. We parametrise the detector’s worldline as t ( τ ) = (cid:113)(cid:2) t + ( τ − t ) cosh θ (cid:3) − (cid:2) ( τ − t ) sinh θ (cid:3) , (5.1a) x ( τ ) = 1 a arctanh (cid:18) ( τ − t ) sinh θt + ( τ − t ) cosh θ (cid:19) , (5.1b)where t > t at which the detector is switched on, θ ∈ R is the rapidity of the detector with the respect to the comoving worldline at theswitch-on moment, and the range of the proper time τ is chosen such that τ = τ := t at the switch-on moment. The comoving worldline is obtained as the special case θ = 0.Figure 2 shows a spacetime diagram with a comoving trajectory and a non-comovingtrajectory with the same value of t . For the twisted field, (4.7) and (5.1) give F int ( ω, τ , τ ) = m − Π int ( ω/m, mτ , mτ ) , (5.2a) F outt ( ω, τ , τ ) = m − Π outt ( ω/m, mτ , mτ ) , (5.2b)16 igure 2: Spacetime diagram of the worldline of a comoving inertial detector (on the y -axis,blue) and a non-comoving inertial detector (green). The switch-on event is at ( y , y ) = ( t , t, x ) = ( t , θ ∈ R . The proper time parameter τ on the trajectory is chosen tohave the value τ := t at the switch-on event. where Π int ( µ, ˜ τ , ˜ τ ) = π aL ∞ (cid:88) n = −∞ (cid:12)(cid:12) ˜ A n ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12) sinh (cid:0) π aL (cid:12)(cid:12) n + (cid:12)(cid:12)(cid:1) , (5.3a)Π outt ( µ, ˜ τ , ˜ τ ) = π aL ∞ (cid:88) n = −∞ exp (cid:18) π aL (cid:12)(cid:12)(cid:12)(cid:12) n + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:12)(cid:12) ˜ B n ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12) , (5.3b)with˜ A n ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du J − i πaL | n + | (cid:0) f θ, ˜ τ ( u ) (cid:1) exp (cid:20) − iµu + i πaL (cid:18) n + 12 (cid:19) g θ, ˜ τ ( u ) (cid:21) , (5.4a)˜ B n ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du H (2) i πaL | n + | (cid:0) f θ, ˜ τ ( u ) (cid:1) exp (cid:20) − iµu + i πaL (cid:18) n + 12 (cid:19) g θ, ˜ τ ( u ) (cid:21) , (5.4b)17nd f θ, ˜ τ ( u ) := (cid:113)(cid:2) ˜ τ + ( u − ˜ τ ) cosh θ (cid:3) − (cid:2) ( u − ˜ τ ) sinh θ (cid:3) , (5.5a) g θ, ˜ τ ( u ) := arctanh (cid:18) ( u − ˜ τ ) sinh θ ˜ τ + ( u − ˜ τ ) cosh θ (cid:19) . (5.5b)Apart from the overall dimensionful factor 1 /m , the response hence depends on theparameters only via the dimensionless combinations µ = ω/m , ˜ τ = mτ , ˜ τ = mτ and aL . The value of the cosmological time t at which the detector starts to operateis τ , and the detector operates for the total proper time τ − τ . For the untwisted field, we need to consider separately the “in” and “out” vacua.
For the “out” vacuum, we may proceed as with the twisted field. We find F outu ( ω, τ , τ ) = m − Π outu ( ω/m, mτ , mτ ) , (5.6)where Π outu ( µ, ˜ τ , ˜ τ ) = π aL ∞ (cid:88) n = −∞ exp (cid:18) π | n | aL (cid:19) (cid:12)(cid:12)(cid:12) ˜˜ B n ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12)(cid:12) , (5.7)with ˜˜ B n ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du H (2) i π | n | aL (cid:0) f θ, ˜ τ ( u ) (cid:1) exp (cid:20) − iµu + i πnaL g θ, ˜ τ ( u ) (cid:21) . (5.8) For the “in” vacuum, the oscillator modes may be treated as for the twisted field. Wefind that their contribution is F inosc ( ω, τ , τ ) = m − Π inosc ( ω/m, mτ , mτ ) , (5.9)where Π inosc ( µ, ˜ τ , ˜ τ ) = π aL (cid:88) n (cid:54) =0 (cid:12)(cid:12)(cid:12) ˜˜ A n ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12)(cid:12) sinh (cid:16) π | n | aL (cid:17) , (5.10)18ith ˜˜ A n ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du J − i π | n | aL (cid:0) f θ, ˜ τ ( u ) (cid:1) exp (cid:20) − iµu + i πnaL g θ, ˜ τ ( u ) (cid:21) . (5.11)For the zero momentum mode, we have, using (4.15), (4.19) and (4.20), F in ( ω, τ , τ ) = m − Π in ( ω/m, mτ , mτ ) , (5.12)where Π in ( µ, ˜ τ , ˜ τ ) = π aL (cid:26)(cid:16) ˜ α + ˜ β − (cid:17) (cid:12)(cid:12)(cid:12) ˜˜ M ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12)(cid:12) + ˜ β (cid:12)(cid:12)(cid:12) ˜˜ N ( µ, ˜ τ , ˜ τ ) (cid:12)(cid:12)(cid:12) + 2 Re (cid:104)(cid:16) ˜ α ˜ β + i (cid:17) ˜˜ M ( µ, ˜ τ , ˜ τ ) ˜˜ N ∗ ( µ, ˜ τ , ˜ τ ) (cid:105)(cid:27) , (5.13)with ˜˜ M ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du J (cid:0) f θ, ˜ τ ( u ) (cid:1) exp( − iµu ) , (5.14a)˜˜ N ( µ, ˜ τ , ˜ τ ) := (cid:90) ˜ τ ˜ τ du Y (cid:0) f θ, ˜ τ ( u ) (cid:1) exp( − iµu ) , (5.14b)and the dimensionless parameters ˜ α ∈ R and ˜ β > α and β in (4.15) by α = ( π/a ) / ˜ α , (5.15a) β = ( π/a ) / ˜ β . (5.15b)Collecting (5.10) and (5.13), we have F inu ( ω, τ , τ ) = m − Π inu ( ω/m, mτ , mτ ) , (5.16)where Π inu ( µ, ˜ τ , ˜ τ ) = Π inosc ( µ, ˜ τ , ˜ τ ) + Π in ( µ, ˜ τ , ˜ τ ) . (5.17) In Milne spacetime with noncompactified spatial sections, the “out” vacuum definedby the adiabatic criterion (2.18) coincides with the Minkowski vacuum [27, 36]. In thissection we give results for the response of a finite-time inertial detector in Minkowskivacuum. We shall see in Section 7 below that the response in spatially compactifiedMilne will duly reduce to the Minkowski vacuum response in appropriate limits.19ecall that in two-dimensional Minkowski spacetime, the pull-back of the Minkowskivacuum Wightman function of a massive scalar field to an inertial worldline is [39] G ( τ, τ (cid:48) ) = 12 π K (cid:0) m [ (cid:15) + i ( τ − τ (cid:48) )] (cid:1) , (6.1)where K is the modified Bessel function of the second kind and the limit (cid:15) → + isunderstood. While in higher spacetime dimensions the (cid:15) → + limit is distributional, intwo dimensions the coincidence singularity is so weak that the limit can be representedby an integrable function, as [38] G ( τ, τ (cid:48) ) = − i H (2)0 (cid:0) m ( τ − τ (cid:48) ) (cid:1) , for τ > τ (cid:48) ,i H (1)0 (cid:0) m ( τ (cid:48) − τ ) (cid:1) , for τ (cid:48) > τ , (6.2)where H (1)0 and H (2)0 are the Hankel functions. For a detector operating for the totalproper time ∆ τ , formula (3.7) hence gives the response function F Mink ( ω, ∆ τ ) = m − Π Mink ( ω/m, m ∆ τ ) , (6.3)where Π Mink ( µ, ∆˜ τ ) = − (cid:90) ∆˜ τ du (∆˜ τ − u ) (cid:2) J ( u ) sin( µu ) + Y ( u ) cos( µu ) (cid:3) . (6.4)Numerical plots of Π Mink ( µ, ∆˜ τ ) (6.4) are shown in Figure 3. At large ∆˜ τ , theprominent feature is a de-excitation peak near µ ≈ −
1, corresponding to ω ≈ − m . In this section we present the core numerical results of the paper: plots of the detector’sresponse on selected inertial trajectories in spatially compactified Milne, in the “in” and“out” vacua. The key aim is to see how the Milne response differs from the Minkowskivacuum response of Section 6.
Consider first the comoving detector.For the twisted field, the results in Figure 4 show that the prominent feature of thespectrum at late times or at large aL is still the de-excitation peak near ω = − m , forboth the “in” vacuum and the “out” vacuum, in close agreement with the Minkowskivacuum results of Section 6, as was to be expected. When aL is small, or when the20 a) Π Mink ( µ, ∆˜ τ )(b) Π Mink ( µ, Figure 3: Inertial detector’s response in Minkowski vacuum, evaluated from (6.4). Part (a)is a perspective plot, with the axix label τ in the plot denoting ∆˜ τ in (6.4). Part (b) is thecross-section at ∆˜ τ = 10. aL is small or when thedetector operates at early times.For the “in” vacuum of the untwisted field, effects of varying the parameters ˜ β and˜ α of the state are shown in respectively Figures 6 and 7. The prominent de-excitationpeak survives, but it is now accompanied by an excitation peak, near ω ≈ m . This isconsistent with the intuitive picture that changing the massive zero mode state puts inthe field a ‘particle’ that can be absorbed by the detector. Consider then a non-comoving detector.Figures 8–12 show results for the twisted and untwisted fields in the “in” and “out”vacua, with large and small values of aL , with the detector operating at early and latetimes, and with a selection of detector rapidities with respect to a comoving observer atthe turn-on moment. The “in” vacuum of the spatially constant mode of the untwistedfield is chosen to agree with that of the “out” vacuum, except in Figure 12, where the˜ α and ˜ β parameters of this mode are varied.The results show that in most cases within the parameter range probed, the effectof the rapidity is significant only when the detector operates at early times and aL is small: a representative example is the twisted field in the “out” vacuum, shown inFigure 8, where a significant effect appears only in Figure 8(a), as additional structurein the de-excitation probabilities.The exception to this pattern is the “in” vacuum of the untwisted field, for whichresults are shown in Figures 11 and 12. When the parameters ˜ α and ˜ β of the spatiallyconstant mode are chosen so that this mode agrees with the spatially constant modeof the “out” vacuum, Figure 11 shows a net shift of the de-excitation peaks to morenegative values of ω as the rapidity increases, and this shift persists even at late timesand for large aL , within the parameter range probed. When the parameters ˜ α and ˜ β arevaried, there appears also an excitation peak, and this excitation peak becomes shiftedto more positive values of ω as the rapidity increases, as shown in Figure 12.While we do not have an analytic explanation of why the spectral shift due to thedetector’s rapidity is most persistent for the “in” vacuum of the untwisted field, we haveverified that a similar shift appears for an untwisted field even in a spatially periodicstatic spacetime, in a Minkowski-like vacuum [7]. Taking the field on the static cylinderto be massless, taking the detector to be coupled to the proper time derivative of thefield (rather than to the value of the field), and taking the switching function to beGaussian (rather than sharp), we may evaluate the detector’s response from formulas(IV.12) in [7]. Choosing the state of the zero mode (which does not have a Fock vacuum)22 a) Π outt ( µ, ,
20) for selected aL . (b) Π outt ( µ, mτ , mτ ) for aL = 0 . mτ , mτ .(c) Π int ( µ, ,
20) for selected aL . (d) Π int ( µ, mτ , mτ ) for aL = 0 . mτ , mτ . Figure 4: Comoving detector’s response for the twisted field in Milne, as a function of µ = ω/m ,for the “in” and “out” vacua, and parameter values as indicated. The red curve masks theblue curve fully or almost fully. a) Π outu ( µ, ,
20) for selected aL . (b) Π outu ( µ, mτ , mτ ) for aL = 0 . mτ , mτ .(c) Π inu ( µ, ,
20) for selected aL , with ˜ α = 0 and˜ β = 1. (d) Π inu ( µ, mτ , mτ ) for aL = 0 . mτ , mτ , with ˜ α = 0 and ˜ β = 1. Figure 5: Comoving detector’s response for the untwisted field in Milne as a function of µ = ω/m , for the “in” and “out” vacua, and parameter values as indicated. The “in” vacuumspatially constant mode parameters are ˜ α = 0 and ˜ β = 1, so that this mode coincides withthat of the “out” vacuum. a) ˜ β = 2 / √ π (b) ˜ β = 10 / √ π (c) ˜ β = 20 / √ π (d) ˜ β = 30 / √ π Figure 6: Comoving detector’s response for the untwisted field in Milne with aL = 1 in the“in” vacuum, Π inu ( µ, , α = 0 but varying the parameter ˜ β as indicated. a) ˜ α = 4 / √ π (b) ˜ α = 6 / √ π (c) ˜ α = − / √ π (d) ˜ α = − / √ π Figure 7: Comoving detector’s response for the untwisted field in Milne with aL = 1 in the“in” vacuum, Π inu ( µ, , β = 1 but varying the parameter ˜ α as indicated. µ as the rapidity increases, inqualitative agreement with the shift seen in Figure 11. Note, however, that the heightsof the peaks in Figure 13 increase as the peaks shift, whereas the heights of the peaksin Figure 11 decrease as the peaks shift. We have investigated quantum field theory in spatially compact cosmological space-times where the early time or late time asymptotic behaviour makes a massive fieldasymptotically massless, without a distinguished Fock vacuum that could be singledout by adiabatic considerations. Focusing on a massive scalar field in 1 + 1 spacetimedimensions, we showed that the freedom in the choice of the vacuum state is a fam-ily with two real-valued parameters, in agreement with the observations of Ford andPathinayake [9] for a subset of our asymptotic conditions. Specialising to the expandingMilne spacetime with compactified spatial sections, where the ambiguity arises in theearly time vacuum, we examined how the ambiguity affects the response of an iner-tial Unruh-DeWitt detector coupled to the quantum field. In parallel, for contrast, weanalysed the Unruh-DeWitt’s detector’s response to a Z -twisted scalar field, for whichadiabatic considerations do single out unique “in” and “out” vacua.We found that the choice of the massive “in” zero mode state has a significant effecton the response of an inertial Unruh-DeWitt detector, especially in the excitation partof the spectrum. We also found that the inertial detector’s peculiar velocity with respectto the comoving cosmological observers affects the detector’s response mainly at earlytimes in spacetimes with a small spatial circumference, as could perhaps have been ex-pected, but with one notable exception: for an untwisted field in the “in” vacuum, thepeculiar velocity effect survives even for large circumferences and late times, within theparameter range of our numerical simulations, and produces a shift of the de-excitationand excitation resonances to larger detector energy gaps as the detector’s peculiar ve-locity increases. We verified that a qualitatively similar resonance shift occurs also fora static spacetime with compact spatial sections, but we do not have a quantitativeexplanation of why in Milne this effect is specific to the “in” vacuum of the untwistedfield.While Milne spacetime is flat, we expect similar phenomena to arise also in curvedspacetimes with compact spatial sections, including locally de Sitter and locally anti-de Sitter spacetimes. We leave investigation of these spacetimes subject to future work.27 a) Π outt ( µ, ,
20) for aL = 0 . θ . (b) Π outt ( µ, , aL = 0 . θ .(c) Π outt ( µ, ,
20) for aL = 1 and selected θ . (d) Π outt ( µ, , aL = 1 and selected θ . Figure 8: Non-comoving detector’s response for the twisted field in Milne, as a function of µ = ω/m , for the “out” vacuum, and parameter values as indicated. The sudden drop inFigure 8(d) at large negative µ appears to be a numerical artefact. a) Π int ( µ, ,
20) for aL = 0 . θ . (b) Π int ( µ, , aL = 0 . θ .(c) Π int ( µ, ,
20) for aL = 1 and selected θ . (d) Π int ( µ, , aL = 1 and selected θ . Figure 9: Non-comoving detector’s response for the twisted field in Milne, as a function of µ = ω/m , for the “in” vacuum, and parameter values as indicated. The sudden drop in Figure9(d) at large negative µ appears to be a numerical artefact. a) Π outt ( µ, ,
20) for aL = 0 . θ . (b) Π outu ( µ, , aL = 0 . θ .(c) Π outu ( µ, ,
20) for aL = 1 and selected θ . (d) Π outu ( µ, , aL = 1 and selected θ . Figure 10: Non-comoving detector’s response for the untwisted field in Milne, as a functionof µ = ω/m , for the “out” vacuum, and parameter values as indicated. The sudden drop inFigure 10(d) at large negative µ appears to be a numerical artefact. a) Π inu ( µ, ,
20) for aL = 0 . θ . (b) Π inu ( µ, , aL = 0 . θ .(c) Π inu ( µ, ,
20) for aL = 1 and selected θ . (d) Π inu ( µ, , aL = 1 and selected θ . Figure 11: Non-comoving detector’s response for the untwisted field in Milne, as a function of µ = ω/m , for the “in” vacuum, and parameter values as indicated. The “in” vacuum spatiallyconstant mode parameters are ˜ α = 0 and ˜ β = 1, so that this mode coincides with that of the“out” vacuum. a) ˜ α = 0 and ˜ β = 2 / √ π (b) ˜ α = 0 and ˜ β = 10 / √ π (c) ˜ α = 4 / √ π and ˜ β = 1 (d) ˜ α = 6 / √ π and ˜ β = 1 Figure 12: Non-comoving detector’s response for the untwisted field in Milne with aL = 1 inthe “in” vacuum, Π inu ( µ, , α and ˜ β as indicated. igure 13: Response of an inertial detector on a static (1+1)-dimensional cylinder, as a functionof the detector’s excitation energy, for selected values of the detector’s rapidity θ with respect tostatic observers, evaluated from (IV.12a) in [7]. The detector is coupled linearly to the propertime derivative of an untwisted massless scalar field, the field is prepared in the Minkowski-like vacuum, except for the zero mode, whose state is chosen so that the contribution tothe response, from (IV.12b) in [7], is negligible. The spatial circumference, the duration ofthe interaction and the normalisation of the horizontal scale are chosen to be comparable tothose in Figure 11(d), adjusted for the absence of a mass parameter. Note the shift of thede-excitation peak towards negative µ as the rapidity increases, in qualitative agreement withthe shift in Figure 11, but note also the increase in the height of the peak, in contrast withthe decrease of the height in Figure 11. cknowledgments We thank Larry Ford and Bei Lok Hu for helpful comments and discussions about thedominant behaviour of oscillator modes versus zero mode at early times. We also thankChris Fewster and Atsushi Higuchi for useful discussions. JL thanks Adam Magee fordiscussions on twisted and untwisted massive fields in Milne spacetime [40]. JL acknowl-edges partial support by United Kingdom Research and Innovation (UKRI) Science andTechnology Facilities Council (STFC) grant ST/S002227/1 “Quantum Sensors for Fun-damental Physics” and Theory Consolidated Grant ST/P000703/1.
References [1] R. Rajaraman,
Solitons and Instantons (North-Holland, Amsterdam 1982).[2] B. S. DeWitt, in
Les Houches 1983 , edited by C. DeWitt and B. S. DeWitt (Gordonand Breach, New York, 1983).[3] B. Allen and A. Folacci, “The massless minimally coupled scalar field in de Sitterspace,” Phys. Rev. D , 3771 (1987).[4] J. Garriga and A. Vilenkin, “Quantum fluctuations on domain walls, strings andvacuum bubbles,” Phys. Rev. D , 3469 (1992)[5] G. McCartor and D. G. Robertson, “Bosonic zero modes in discretized light conefield theory,” Z. Phys. C , 679 (1992).[6] K. Kirsten and J. Garriga, “Massless minimally coupled fields in de Sitter space:O(4) symmetric states versus de Sitter invariant vacuum,” Phys. Rev. D , 567(1993) [gr-qc/9305013].[7] E. Mart´ın-Mart´ınez and J. Louko, “Particle detectors and the zero mode of a quan-tum field,” Phys. Rev. D , 024015 (2014) [arXiv:1404.5621 [quant-ph]].[8] J. Louko and V. Toussaint, “Unruh-DeWitt detector’s response to fermions in flatspacetimes,” Phys. Rev. D , 064027 (2016) [gr-qc/1608.01002].[9] L. Ford and C. Pathinayake, “Bosonic zero frequency modes and initial conditions,”Phys. Rev. D , 3642 (1989).[10] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D , 870 (1976).[11] B. S. DeWitt, “Quantum gravity: the new synthesis”, in General Relativity: anEinstein centenary survey , edited by S. W. Hawking and W. Israel (CambridgeUniversity Press, Cambridge, 1979). 3412] E. Mart´ın-Mart´ınez, M. Montero and M. del Rey, “Wavepacket detection with theUnruh-DeWitt model,” Phys. Rev. D , 064038 (2013) [arXiv:1207.3248 [quant-ph]].[13] ´A. M. Alhambra, A. Kempf and E. Mart´ın-Mart´ınez, Phys. Rev. A , 033835(2014) [arXiv:1311.7619 [quant-ph]].[14] K. K. Ng, R. B. Mann and E. Mart´ın-Mart´ınez, “New techniques for entangle-ment harvesting in flat and curved spacetimes,” Phys. Rev. D , 125011 (2018)[arXiv:1805.01096 [quant-ph]].[15] P. Simidzija and E. Mart´ın-Mart´ınez, “Harvesting correlations from thermal andsqueezed coherent states,” Phys. Rev. D , 085007 (2018) [arXiv:1809.05547[quant-ph]].[16] K. K. Ng, R. B. Mann and E. Mart´ın-Mart´ınez, “Unruh-DeWitt detectors andentanglement: The anti–de Sitter space,” Phys. Rev. D , 125005 (2018)[arXiv:1809.06878 [quant-ph]].[17] G. L. Ver Steeg and N. C. Menicucci, “Entangling power of an expanding universe,”Phys. Rev. D , 044027 (2009) [arXiv:0711.3066 [quant-ph]].[18] L. J. Garay, M. Mart´ın-Benito and E. Mart´ın-Mart´ınez, “Echo of the quantumbounce,” Phys. Rev. D , 043510 (2014) [arXiv:1308.4348 [gr-qc]].[19] E. Tjoa and E. Mart´ın-Mart´ınez, “Vacuum entanglement harvesting with a zeromode,” Phys. Rev. D , 125020 (2020) [arXiv:2002.11790 [quant-ph]].[20] A. Sachs, R. B. Mann and E. Mart´ın-Mart´ınez, “Entanglement harvesting anddivergences in quadratic Unruh-DeWitt detector pairs,” Phys. Rev. D , 085012(2017) [arXiv:1704.08263 [quant-ph]].[21] E. Mart´ın-Mart´ınez, T. R. Perche and B. de S.L. Torres, “General RelativisticQuantum Optics: Finite-size particle detector models in curved spacetimes,” Phys.Rev. D , 045017 (2020) [arXiv:2001.10010 [quant-ph]].[22] B. F. Svaiter and N. F. Svaiter, “Inertial and noninertial particle detectors andvacuum fluctuations”, Phys. Rev. D , 5267 (1992).[23] A. Higuchi, G. E. A. Matsas and C. B. Peres, “Uniformly accelerated finite timedetectors,” Phys. Rev. D , 3731 (1993).[24] L. Sriramkumar and T. Padmanabhan, “Response of finite time particle detectorsin noninertial frames and curved space-time,” Class. Quant. Grav. , 2061 (1996)[arXiv:gr-qc/9408037]. 3525] C. J. Fewster, B. A. Ju´arez-Aubry and J. Louko, “Waiting for Unruh,” Class.Quant. Grav. , 165003 (2016) [arXiv:1605.01316 [gr-qc]].[26] L. Parker and S. Fulling, “Adiabatic regularization of the energy momentum tensorof a quantized field in homogeneous spaces,” Phys. Rev. D , 341 (1974).[27] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (CambridgeUniversity Press, Cambridge, 1982).[28] J. Audretsch, “Cosmological particle creation as above-barrier reflection: approxi-mation method and applications,” J. Phys. A , 1189 (1979).[29] Y. D´ecanini and A. Folacci, “Hadamard renormalization of the stress-energy tensorfor a quantized scalar field in a general spacetime of arbitrary dimension,” Phys.Rev. D , 044025 (2008) [arXiv:gr-qc/0512118].[30] L. H¨ormander, The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1986).[31] C. J. Fewster, “A general worldline quantum inequality,” Class. Quant. Grav. ,1897 (2000) [arXiv:gr-qc/9910060].[32] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time (Cam-bridge University Press, Cambridge, 1973).[33] A. diSessa, “Quantization on hyperboloids and full space-time field expansion,” J.Math. Phys. , 1892 (1974).[34] C. M. Sommerfield, “Quantization on spacetime hyperboloids,” Ann. Phys. (N.Y.) , 285 (1974).[35] D. Gromes, H. J. Rothe and B. Stech, “Field quantization on the surface x =constant,” Nucl. Phys. B , 313 (1974).[36] S. A. Fulling, L. Parker and B. L. Hu, “Conformal energy-momentum tensor incurved spacetime: Adiabatic regularization and renormalization,” Phys. Rev. D , 3905 (1974).[37] T. Padmanabhan, “Physical interpretation of quantum field theory in noninertialcoordinate systems,” Phys. Rev. Lett. , 2471 (1990).[38] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release1.1.0 of 2020-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl,and M. A. McClain, eds. 3639] S. Takagi, “Vacuum noise and stress induced by uniform acceleration: Hawking-Unruh effect in Rindler manifold of arbitrary dimension,” Prog. Theor. Phys. Suppl.88