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text: 3 (cid:90) (cid:90) The fact that the ﬁeld is in the (Minkowski) vacuum does not necessarily imply that a detector will fail to be- ∞ − τ (cid:48) (cid:48) (cid:48) come excited when interacting with it [14, 32]. This is due X = dτ dτ η(τ)η(τ )eiΩ(τ+τ ) to counter-rotating terms (such as σˆ−aˆ and σˆ aˆ†) in the −∞ −∞ + × (cid:48) (cid:48) expansion of Eq. (2.2). In fact, most window functions [D+(x (τ); x (τ )) + D+(x (τ); x (τ ))], (2.7) a b b a and trajectories will produce a nonzero response (exci- tation probability) in an Unruh-DeWitt detector due to the presence of these terms. Their contribution—and where the Wightman functions are taken with respect thus the probability of excitation—will tend to 0, how- to the two trajectories xa and xb (cf. Eq. (2.4), which only ever, when the energy gap and coupling strength (during involves one trajectory at a time), and Ω is the energy gap detection) are constant, the trajectory is inertial, and the of the detector. The amount of entanglement harvested detection time tends to inﬁnity [32]. can be quantiﬁed by the negativity [37] of the ﬁnal state In general, then, each detector (initially in the ground of the detectors, which in this case is simply state) is prone to excitation through its traveling interac- tion with the ﬁeld. To lowest nontrivial order (i.e., second {| | − } N = max X A, 0 . (2.8) order) in the detector coupling strength η , the probabil- 0 ity A that exactly one of the detectors will be excited is given(cid:90)by [32](cid:90) A two-qubit state has nonzero negativity iﬀ it is en- ∞ ∞ tangled [36]. (cid:48) (cid:48) − − (cid:48) (cid:48) Entanglement is therefore seen as a competition be- A = dτ dτ η(τ)η(τ )e iΩ(τ τ )D+(x (τ); x (τ )) −∞ −∞ a a tween the probability of actual detector excitation (A) (2.4) and the magnitude of an oﬀ-diagonal term in the den- sity matrix representing the amplitude for particle ex- (cid:48) where Ω is the detector energy gap, and D+(x; x ) = change (X) [4]. When the detection events are entirely (cid:104) (cid:48) (cid:105) φˆ(x)φˆ(x ) is the Wightman function for the ﬁeld φˆ, spacelike separated, the exchange term can only be inter- which, for the case of the Minkowski vacuum, is given preted as swapping preexisting entanglement out of the by Eq. (3.59) of Ref. [32]: ﬁeld and into the detectors—rather than simply using the ﬁeld as a transmission medium by which to exchange − real excitations. When the detection events are only ap- (cid:48) (cid:48) 1 D+(xa(τ); xa(τ )) = 4π2[(t − t(cid:48) − i(cid:15))2 − |x − x(cid:48) |2]. proximately spacelike separated, as is the case for some a a choices of parameters here (since η(τ) is Gaussian), there (2.5) are additional subtleties, which we address later. D. Accelerating trajectories and comparison with Note that, due to symmetry, we could equally well · (cid:55)→ · other scenarios make the trajectory substitution x ( ) x ( ) and get a b the same value for A. Also note that all quantities are → The canonical formalism for studying accelerated ob- evaluated with (cid:15) 0+ at the end. servers in Minkowski spacetime [31, 32, 38] involves deﬁn- C. Quantifying the harvested entanglement ing a Rindler wedge corresponding to an accelerating trajectory and using that wedge to deﬁne three addi- To quantify the harvested entanglement, we follow the tional wedges that separate Minkowski spacetime into standard procedure ﬁrst proposed by Reznik [34] and left, right, future, and past regions with lightsheet bound- used also by later authors [4, 15, 17, 18]: We calculate the aries. A key feature of this partitioning is that all four elements of the reduced density matrix of the detectors wedges share a common apex. This suggests a natural set and then use the partial transpose criterion [35], which of accelerating trajectories: those hyperbolas that share is necessary and suﬃcient for two qubits [36], to detect the bounding lightsheets as asymptotes. the presence or absence of entanglement after interaction We take a more operational approach to the problem with the ﬁeld. by focusing on the physical objects: two detectors accel- As in previous work [4, 15, 17, 18, 34], the partial trans- erating uniformly through space. There is no physical pose criterion for our setup reduces to a simple compar- reason why the Rindler wedges deﬁned by each observer ison between two quantities. Entanglement exists be- should be constrained to meet at the origin, and in fact tween the detectors if and only if we argue that this approach is restrictive. Forcing the wedges to meet at the origin is tantamount to forcing | | the distance between the detectors at closest approach, X > A, (2.6) L, to be constrained by the acceleration parameter κ: − speciﬁcally, L = 2κ 1. We relax the condition of wedges where A is deﬁned in Eq. (2.4), and X can be inter- with overlapping apexes and study two uniformly accel- preted as an amplitude for (virtual) particle exchange erated detectors with an arbitrary distance at closest ap- between the detectors (to second order in η ): proach. We also consider both anti-parallel and parallel 0 ##LTLine##

label: paragraph

label: equation

text: The

text: fact

text: that

text: the

text: ﬁeld

text: is

text: in

text: the

text: (Minkowski)

text: vacuum

text: does

text: not

text: necessarily

text: imply

text: that

text: a

text: detector

text: will

text: fail

text: to

text: be-

text: τ

text: come

text: excited

text: when

text: interacting

text: with

text: it

text: This

text: is

text: due

text: X

text: dτ

text: dτ

text: η(τ)η(τ

text: )eiΩ(τ+τ

text: to

text: counter-rotating

text: terms

text: (such

text: as

text: σˆ−aˆ

text: and

text: σˆ

text: aˆ†)

text: in

text: the

text: expansion

text: of

text: Eq.

text: In

text: fact,

text: most

text: window

text: functions

text: [D+(x

text: (τ);

text: x

text: (τ

text: D+(x

text: (τ);

text: x

text: (τ

text: a

text: b

text: b

text: a

text: and

text: trajectories

text: will

text: produce

text: a

text: nonzero

text: response

text: (exci-

text: tation

text: probability)

text: in

text: an

text: Unruh-DeWitt

text: detector

text: due

text: to

text: the

text: presence

text: of

text: these

text: terms.

text: Their

text: contribution—and

text: where

text: the

text: Wightman

text: functions

text: are

text: taken

text: with

text: respect

text: thus

text: the

text: probability

text: of

text: excitation—will

text: tend

text: to

text: how-

text: to

text: the

text: two

text: trajectories

text: xa

text: and

text: xb

text: (cf.

text: Eq.

text: which

text: only

text: ever,

text: when

text: the

text: energy

text: gap

text: and

text: coupling

text: strength

text: (during

text: involves

text: one

text: trajectory

text: at

text: a

text: time),

text: and

text: Ω

text: is

text: the

text: energy

text: gap

text: detection)

text: are

text: constant,

text: the

text: trajectory

text: is

text: inertial,

text: and

text: the

text: of

text: the

text: detector.

text: The

text: amount

text: of

text: entanglement

text: harvested

text: detection

text: time

text: tends

text: to

text: inﬁnity

text: can

text: be

text: quantiﬁed

text: by

text: the

text: negativity

text: of

text: the

text: ﬁnal

text: state

text: In

text: general,

text: then,

text: each

text: detector

text: (initially

text: in

text: the

text: ground

text: of

text: the

text: detectors,

text: which

text: in

text: this

text: case

text: is

text: simply

text: state)

text: is

text: prone

text: to

text: excitation

text: through

text: its

text: traveling

text: interac-

text: tion

text: with

text: the

text: ﬁeld.

text: To

text: lowest

text: nontrivial

text: order

text: (i.e.,

text: second

text: N

text: max

text: X

text: A,

text: order)

text: in

text: the

text: detector

text: coupling

text: strength

text: η

text: the

text: probabil-

text: ity

text: A

text: that

text: exactly

text: one

text: of

text: the

text: detectors

text: will

text: be

text: excited

text: is

text: A

text: two-qubit

text: state

text: has

text: nonzero

text: negativity

text: iﬀ

text: it

text: is

text: en-

text: tangled

text: Entanglement

text: is

text: therefore

text: seen

text: as

text: a

text: competition

text: be-

text: A

text: dτ

text: dτ

text: η(τ)η(τ

text: )e

text: iΩ(τ

text: τ

text: )D+(x

text: (τ);

text: x

text: (τ

text: a

text: a

text: tween

text: the

text: probability

text: of

text: actual

text: detector

text: excitation

text: (A)

text: and

text: the

text: magnitude

text: of

text: an

text: oﬀ-diagonal

text: term

text: in

text: the

text: den-

text: sity

text: matrix

text: representing

text: the

text: amplitude

text: for

text: particle

text: ex-

text: where

text: Ω

text: is

text: the

text: detector

text: energy

text: gap,

text: and

text: D+(x;

text: x

text: change

text: (X)

text: When

text: the

text: detection

text: events

text: are

text: entirely

text: φˆ(x)φˆ(x

text: is

text: the

text: Wightman

text: function

text: for

text: the

text: ﬁeld

text: φˆ,

text: spacelike

text: separated,

text: the

text: exchange

text: term

text: can

text: only

text: be

text: inter-

text: which,

text: for

text: the

text: case

text: of

text: the

text: Minkowski

text: vacuum,

text: is

text: given

text: preted

text: as

text: swapping

text: preexisting

text: entanglement

text: out

text: of

text: the

text: by

text: Eq.

text: of

text: Ref.

text: ﬁeld

text: and

text: into

text: the

text: detectors—rather

text: than

text: simply

text: using

text: the

text: ﬁeld

text: as

text: a

text: transmission

text: medium

text: by

text: which

text: to

text: exchange

text: real

text: excitations.

text: When

text: the

text: detection

text: events

text: are

text: only

text: ap-

text: D+(xa(τ);

text: xa(τ

text: |x

text: proximately

text: spacelike

text: separated,

text: as

text: is

text: the

text: case

text: for

text: some

text: a

text: a

text: choices

text: of

text: parameters

text: here

text: (since

text: η(τ)

text: is

text: Gaussian),

text: there

text: are

text: additional

text: subtleties,

text: which

text: we

text: address

text: later.

text: D.

text: Accelerating

text: trajectories

text: and

text: comparison

text: with

text: Note

text: that,

text: due

text: to

text: symmetry,

text: we

text: could

text: equally

text: well

text: other

text: scenarios

text: make

text: the

text: trajectory

text: substitution

text: x

text: x

text: and

text: get

text: a

text: b

text: the

text: same

text: value

text: for

text: A.

text: Also

text: note

text: that

text: all

text: quantities

text: are

text: The

text: canonical

text: formalism

text: for

text: studying

text: accelerated

text: ob-

text: evaluated

text: with

text: at

text: the

text: end.

text: servers

text: in

text: Minkowski

text: spacetime

text: involves

text: deﬁn-

text: C.

text: Quantifying

text: the

text: harvested

text: entanglement

text: ing

text: a

text: Rindler

text: wedge

text: corresponding

text: to

text: an

text: accelerating

text: trajectory

text: and

text: using

text: that

text: wedge

text: to

text: deﬁne

text: three

text: addi-

text: To

text: quantify

text: the

text: harvested

text: entanglement,

text: we

text: follow

text: the

text: tional

text: wedges

text: that

text: separate

text: Minkowski

text: spacetime

text: into

text: standard

text: procedure

text: ﬁrst

text: proposed

text: by

text: Reznik

text: and

text: left,

text: right,

text: future,

text: and

text: past

text: regions

text: with

text: lightsheet

text: bound-

text: used

text: also

text: by

text: later

text: authors

text: We

text: calculate

text: the

text: aries.

text: A

text: key

text: feature

text: of

text: this

text: partitioning

text: is

text: that

text: all

text: four

text: elements

text: of

text: the

text: reduced

text: density

text: matrix

text: of

text: the

text: detectors

text: wedges

text: share

text: a

text: common

text: apex.

text: This

text: suggests

text: a

text: natural

text: set

text: and

text: then

text: use

text: the

text: partial

text: transpose

text: criterion

text: which

text: of

text: accelerating

text: trajectories:

text: those

text: hyperbolas

text: that

text: share

text: is

text: necessary

text: and

text: suﬃcient

text: for

text: two

text: qubits

text: to

text: detect

text: the

text: bounding

text: lightsheets

text: as

text: asymptotes.

text: the

text: presence

text: or

text: absence

text: of

text: entanglement

text: after

text: interaction

text: We

text: take

text: a

text: more

text: operational

text: approach

text: to

text: the

text: problem

text: with

text: the

text: ﬁeld.

text: by

text: focusing

text: on

text: the

text: physical

text: objects:

text: two

text: detectors

text: accel-

text: As

text: in

text: previous

text: work

text: the

text: partial

text: trans-

text: erating

text: uniformly

text: through

text: space.

text: There

text: is

text: no

text: physical

text: pose

text: criterion

text: for

text: our

text: setup

text: reduces

text: to

text: a

text: simple

text: compar-

text: reason

text: why

text: the

text: Rindler

text: wedges

text: deﬁned

text: by

text: each

text: observer

text: ison

text: between

text: two

text: quantities.

text: Entanglement

text: exists

text: be-

text: should

text: be

text: constrained

text: to

text: meet

text: at

text: the

text: origin,

text: and

text: in

text: fact

text: tween

text: the

text: detectors

text: if

text: and

text: only

text: if

text: we

text: argue

text: that

text: this

text: approach

text: is

text: restrictive.

text: Forcing

text: the

text: wedges

text: to

text: meet

text: at

text: the

text: origin

text: is

text: tantamount

text: to

text: forcing

text: the

text: distance

text: between

text: the

text: detectors

text: at

text: closest

text: approach,

text: X

text: A,

text: L,

text: to

text: be

text: constrained

text: by

text: the

text: acceleration

text: parameter

text: κ:

text: speciﬁcally,

text: L

text: 2κ

text: We

text: relax

text: the

text: condition

text: of

text: wedges

text: where

text: A

text: is

text: deﬁned

text: in

text: Eq.

text: and

text: X

text: can

text: be

text: inter-

text: with

text: overlapping

text: apexes

text: and

text: study

text: two

text: uniformly

text: accel-

text: preted

text: as

text: an

text: amplitude

text: for

text: (virtual)

text: particle

text: exchange

text: erated

text: detectors

text: with

text: an

text: arbitrary

text: distance

text: at

text: closest

text: ap-

text: between

text: the

text: detectors

text: (to

text: second

text: order

text: in

text: η

text: proach.

text: We

text: also

text: consider

text: both

text: anti-parallel

text: and

text: parallel

text: ##LTLine##


text: 9 − Notice that L = 2κ 1 is not necessary to see this reso- t nance eﬀect. For the special case κσ2Ω = π, of course, we 2 get resonance at that distance, but this is an additional − requirement. Therefore, L = 2κ 1 alone is neither a nec- essary nor a suﬃcient condition for resonance. This is contrary to the intuition provided by the Rindler-wedge picture often used to derive and discuss the Unruh ef- fect [31, 32, 38]. C. Causal residue contribution x The entanglement harvested by detectors in diﬀerent L L regions of parameter space seems to arise from distinct (cid:45) 2 2 physical mechanisms. For Lκ < 2, the detectors are time- like separated, and real particles can, in principle, be ex- changed. There is a residue contribution to X that is largest for small Lκ and vanishes when Lκ > 2. Due to this behavior, we hypothesize that the residue contribu- tion in this region may arise from causal dynamics (see Sec. ). FIG. 4: (Color online.) Interpretation of the residue contri- For detectors on the hyperbolic trajectories shown in bution from causally connected detectors (Lκ < 2). While the right panel of Fig. 1, the Gaussian window functions the analytic form of the Gaussian was essential for calculat- cause the detectors to spend a very long time traveling ing the coherence term, X, the inﬁnite tails give rise to very near the speed of light and interacting with the under- long-time interactions with ﬁeld modes (shown in green) while lying ﬁeld. Although the interaction is very weak, we the detector is traveling near the speed of light. Eﬀects due conjecture that the long interaction time gives rise to to interaction between the ﬁrst detector and the ﬁeld over a a nontrivial contribution to X comparable to (or even long period in the inﬁnite past can build up and be felt by the second detector in ﬁnite time as it crosses the lightlike exten- greater than) the residue-free portion. sion of the ﬁrst detector’s past asymptote (and vice versa). Speciﬁcally, we hypothesize that the ﬁrst detector’s in- Similarly, small ﬂuctuations caused by short-time interaction teraction with the ﬁeld for a long time in the inﬁnite past from the ﬁrst detector can be ampliﬁed through long inter- may produce a large eﬀect concentrated near the lightlike actions in the inﬁnite future of the second detector (and vice past asymptote of the ﬁrst detector’s trajectory that is versa). We hypothesize that these eﬀects together give rise to felt by the other detector (in the coherence term X) as the nontrivial contribution to X discussed in Sec. and shown the second detector crosses the future extension of that on the left side of Fig. 2 (right). asymptote. (The same eﬀect also happens from the sec- ond to the ﬁrst detector.) Similarly, the second detector will spend a long time near the lightlike future asymp- feeling the eﬀects of entangled Rindler modes [31, 32, 38]. tote of its own trajectory, thus amplifying any eﬀect due To understand this reasoning, consider dividing space- to the presence of the ﬁrst detector near the past exten- time into four wedges (using the usual Rindler prescrip- sion of that asymptote (and vice versa). The net result tion [31]) whose common origin is at (x = 0, t = 0). If of these four (symmetric) processes is a nontrivial con- the actual detector trajectories have asymptotes that are tribution to the coherence term X, which indicates that close to the boundaries of these origin-centered Rindler (cid:38) the detectors “feel each other’s presence” due to the long wedges (i.e., Lκ 2), then the detectors will be nearly tails of the Gaussian and overlapping wedges, as shown resonant with the Rindler modes that are two-mode → in Fig. 4. squeezed (see Sec. ). This eﬀect only increases as Lκ D. Noncausal residue contribution 2+. However, the fact that it cuts oﬀ sharply as soon as Lκ < 2 also suggests that the inﬁnite tails may play a role Even as the causal residue contribution discussed in in this eﬀect. Perhaps conﬁnement of the inﬁnite tails to Sec. vanishes as Lκ > 2, a second residue contribution the left and right origin-centered wedges is required to see takes over in that region when, in addition, κσ2Ω > π. the eﬀect. If the detectors are too close, then the long 2 This means that there exists a nontrivial residue contri- tails of each detector’s trajectory will leak out into the bution for spacelike separated detectors (Lκ > 2) but origin-centered future and past wedges. This eﬀect would only for certain choices of the other parameters. Because then be traded out for the causal eﬀect described in Sec. , (cid:46) the detectors are spacelike separated for the entirety of which is tiny for Lκ 2. Further work is needed to ex- their trajectories in this case, this contribution is more plore the eﬀect of using switching functions with compact mysterious. There are some clues that we can use to support, but this requires new calculational methods and hypothesize about its physical origin, however. is beyond the scope of this work. We therefore leave the The most important clue is that the eﬀect is strongest above as a working hypothesis. Further justiﬁcation may (cid:38) near Lκ 2. This suggests that the detectors may be be found in Sec. . ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Notice

text: that

text: L

text: 2κ

text: is

text: not

text: necessary

text: to

text: see

text: this

text: reso-

text: t

font: default

text: nance

text: eﬀect.

text: For

text: the

text: special

text: case

text: κσ2Ω

text: π,

text: of

text: course,

text: we

text: get

text: resonance

text: at

text: that

text: distance,

text: but

text: this

text: is

text: an

text: additional

text: requirement.

text: Therefore,

text: L

text: 2κ

text: alone

text: is

text: neither

text: a

text: nec-

text: essary

text: nor

text: a

text: suﬃcient

text: condition

text: for

text: resonance.

text: This

text: is

text: contrary

text: to

text: the

text: intuition

text: provided

text: by

text: the

text: Rindler-wedge

text: picture

text: often

text: used

text: to

text: derive

text: and

text: discuss

text: the

text: Unruh

text: ef-

text: fect

text: C.

label: section

text: Causal

text: residue

text: contribution

text: x

text: The

text: entanglement

text: harvested

text: by

text: detectors

text: in

text: diﬀerent

text: L

text: L

text: regions

text: of

text: parameter

text: space

text: seems

text: to

text: arise

text: from

text: distinct

text: physical

text: mechanisms.

text: For

text: Lκ

text: the

text: detectors

text: are

text: time-

text: like

text: separated,

text: and

text: real

text: particles

text: can,

text: in

text: principle,

text: be

text: ex-

text: changed.

text: There

text: is

text: a

text: residue

text: contribution

text: to

text: X

text: that

text: is

text: largest

text: for

text: small

text: Lκ

text: and

text: vanishes

text: when

text: Lκ

text: Due

text: to

text: this

text: behavior,

text: we

text: hypothesize

text: that

text: the

text: residue

text: contribu-

text: tion

text: in

text: this

text: region

text: may

text: arise

text: from

text: causal

text: dynamics

text: (see

text: Sec.

text: FIG.

text: (Color

text: online.)

text: Interpretation

text: of

text: the

text: residue

text: contri-

text: For

text: detectors

text: on

text: the

text: hyperbolic

text: trajectories

text: shown

text: in

text: bution

text: from

text: causally

text: connected

text: detectors

text: (Lκ

text: While

text: the

text: right

text: panel

text: of

text: Fig.

text: the

text: Gaussian

text: window

text: functions

text: the

text: analytic

text: form

text: of

text: the

text: Gaussian

text: was

text: essential

text: for

text: calculat-

text: cause

text: the

text: detectors

text: to

text: spend

text: a

text: very

text: long

text: time

text: traveling

text: ing

text: the

text: coherence

text: term,

text: X,

text: the

text: inﬁnite

text: tails

text: give

text: rise

text: to

text: very

text: near

text: the

text: speed

text: of

text: light

text: and

text: interacting

text: with

text: the

text: under-

text: long-time

text: interactions

text: with

text: ﬁeld

text: modes

text: (shown

text: in

text: green)

text: while

text: lying

text: ﬁeld.

text: Although

text: the

text: interaction

text: is

text: very

text: weak,

text: we

text: the

text: detector

text: is

text: traveling

text: near

text: the

text: speed

text: of

text: light.

text: Eﬀects

text: due

text: conjecture

text: that

text: the

text: long

text: interaction

text: time

text: gives

text: rise

text: to

text: to

text: interaction

text: between

text: the

text: ﬁrst

text: detector

text: and

text: the

text: ﬁeld

text: over

text: a

text: a

text: nontrivial

text: contribution

text: to

text: X

text: comparable

text: to

text: (or

text: even

text: long

text: period

text: in

text: the

text: inﬁnite

text: past

text: can

text: build

text: up

text: and

text: be

text: felt

text: by

text: the

text: second

text: detector

text: in

text: ﬁnite

text: time

text: as

text: it

text: crosses

text: the

text: lightlike

text: exten-

text: greater

text: than)

text: the

text: residue-free

text: portion.

text: sion

text: of

text: the

text: ﬁrst

text: detector’s

text: past

text: asymptote

text: (and

text: vice

text: versa).

text: Speciﬁcally,

text: we

text: hypothesize

text: that

text: the

text: ﬁrst

text: detector’s

text: in-

text: Similarly,

text: small

text: ﬂuctuations

text: caused

text: by

text: short-time

text: interaction

text: teraction

text: with

text: the

text: ﬁeld

text: for

text: a

text: long

text: time

text: in

text: the

text: inﬁnite

text: past

text: from

text: the

text: ﬁrst

text: detector

text: can

text: be

text: ampliﬁed

text: through

text: long

text: inter-

text: may

text: produce

text: a

text: large

text: eﬀect

text: concentrated

text: near

text: the

text: lightlike

text: actions

text: in

text: the

text: inﬁnite

text: future

text: of

text: the

text: second

text: detector

text: (and

text: vice

text: past

text: asymptote

text: of

text: the

text: ﬁrst

text: detector’s

text: trajectory

text: that

text: is

text: versa).

text: We

text: hypothesize

text: that

text: these

text: eﬀects

text: together

text: give

text: rise

text: to

text: felt

text: by

text: the

text: other

text: detector

text: (in

text: the

text: coherence

text: term

text: X)

text: as

text: the

text: nontrivial

text: contribution

text: to

text: X

text: discussed

text: in

text: Sec.

text: and

text: shown

text: the

text: second

text: detector

text: crosses

text: the

text: future

text: extension

text: of

text: that

text: on

text: the

text: left

text: side

text: of

text: Fig.

text: (right).

text: asymptote.

text: (The

text: same

text: eﬀect

text: also

text: happens

text: from

text: the

text: sec-

text: ond

text: to

text: the

text: ﬁrst

text: detector.)

text: Similarly,

text: the

text: second

text: detector

text: will

text: spend

text: a

text: long

text: time

text: near

text: the

text: lightlike

text: future

text: asymp-

text: feeling

text: the

text: eﬀects

text: of

text: entangled

text: Rindler

text: modes

text: tote

text: of

text: its

text: own

text: trajectory,

text: thus

text: amplifying

text: any

text: eﬀect

text: due

text: To

text: understand

text: this

text: reasoning,

text: consider

text: dividing

text: space-

text: to

text: the

text: presence

text: of

text: the

text: ﬁrst

text: detector

text: near

text: the

text: past

text: exten-

text: time

text: into

text: four

text: wedges

text: (using

text: the

text: usual

text: Rindler

text: prescrip-

text: sion

text: of

text: that

text: asymptote

text: (and

text: vice

text: versa).

text: The

text: net

text: result

text: tion

text: whose

text: common

text: origin

text: is

text: at

text: (x

text: t

text: If

text: of

text: these

text: four

text: (symmetric)

text: processes

text: is

text: a

text: nontrivial

text: con-

text: the

text: actual

text: detector

text: trajectories

text: have

text: asymptotes

text: that

text: are

text: tribution

text: to

text: the

text: coherence

text: term

text: X,

text: which

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text: that

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text: the

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text: of

text: these

text: origin-centered

text: Rindler

text: the

text: detectors

text: “feel

text: each

text: other’s

text: presence”

text: due

text: to

text: the

text: long

text: wedges

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text: Lκ

text: then

text: the

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text: Gaussian

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text: as

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text: modes

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text: If

text: the

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text: ##LTFigure##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: MIT-CTP 4571 A THEORY OF SELF-RESONANCE AFTER INFLATION ∗ Part 1: Adiabatic and Isocurvature Goldstone Modes ∗ † Mark P. Hertzberg , Johanna Karouby , William G. Spitzer, Juana C. Becerra, Lanqing Li † ‡ (Dated: May 26, 2020) We develop a theory of self-resonance after inﬂation. We study a large class of models involving multiple scalar ﬁelds with an internal symmetry. For illustration, we often specialize to dimension 4 potentials, but we derive results for general potentials. This is the ﬁrst part of a two part series of papers. Here in Part 1 we especially focus on the behavior of long wavelengths modes, which are found to govern most of the important physics. Since the inﬂaton background spontaneously breaks the time translation symmetry and the internal symmetry, we obtain Goldstone modes; these are the adiabatic and isocurvature modes. We ﬁnd general conditions on the potential for when a large instability band exists for these modes at long wavelengths. For the adiabatic mode, this is determined by a sound speed derived from the time averaged potential. While for the isocurvature mode, this is determined by a speed derived from a time averaged auxiliary potential. Interestingly, we ﬁnd that this instability band usually exists for one of these classes of modes, rather than both simultaneously. We focus on backgrounds that evolve radially in ﬁeld space, as setup by inﬂation, and also mention circular orbits, as relevant to Q-balls. In Part 2 [1] we derive the central behavior from the underlying description of many particle quantum mechanics, and introduce a weak breaking of the symmetry to study corrections to particle-antiparticle production from preheating. CONTENTS 5A. pplication to Power Law Potentials 11 CI.socurvature Modes δn 11 I.Introduction 1 1E. quation of Motion 12 2T. ime Average with Auxiliary Potential 12 IIS.ymmetric Theories 2 3I.socurvature Speed c 13 I AB. ackground Evolution 2 4A. pplication to Dim 4 Potentials 15 BL.inearized Perturbations 3 5A. pplication to Power Law Potentials 15 CF. loquet Theory for Self-Resonance 3 VCI.ircular Motion for Background 15 IIMI.otivation from Dimension 4 Potentials 4 AB. ackground Evolution 15 AP.ositive vs Negative Quartic Behavior 4 BF.ull Floquet Result 15 BA. diabatic vs Isocurvature Behavior 4 CL.ong Wavelength Limit – Sound Speed c 16 CN. egative Mass Squared Behavior 6 S VCIIo.nclusions 18 IVA.uxiliary Potential for Isocurvature Modes 6 AC. orrespondence between Modes 6 VAIIpIp. endix – Non-Canonical Kinetic Terms 19 BI.ntegral Form for General Potentials 7 CA. pplication to Dim 4 Potentials 7 References 19 VG. eneral Analysis for Long Wavelength Modes 8 AG. oldstone Modes 8 BA. diabatic Mode δε 8 I. INTRODUCTION 1E. quation of Motion 8 2T. ime Average 9 Cosmological inﬂation is a successful theory of the 3S. ound Speed cS 10 early universe [2–4], which accounts for the approxi- 4A. pplication to Dim 4 Potentials 10 mately scale-invariant distribution of structures on large scales. The structures arises from quantum ﬂuctuations in the inﬂaton scalar ﬁeld/s φ by stretching modes to large scales due to inﬂation’s exponential expansion. ∗ †Electronic address: mphertz@mit.edu Evidence for inﬂation is increasing with detailed mea- ∗ Electronic address: karouby@mit.edu surement of the cosmic microwave background radiation † [5–8], and this has motivated the construction of many ‡ interesting theoretical models [9–17]. ##LTLine##

text: MIT-CTP

text: A

label: title

text: THEORY

text: OF

text: SELF-RESONANCE

text: AFTER

text: INFLATION

text: Part

text: Adiabatic

text: and

text: Isocurvature

text: Goldstone

text: Modes

label: author

text: Mark

text: P.

text: Hertzberg

text: Johanna

text: Karouby

text: William

text: G.

text: Spitzer,

text: Juana

text: C.

text: Becerra,

text: Lanqing

text: Li

text: (Dated:

text: May

label: date

text: We

label: abstract

text: develop

text: a

text: theory

text: of

text: self-resonance

text: after

text: inﬂation.

text: We

text: study

text: a

text: large

text: class

text: of

text: models

text: involving

text: multiple

text: scalar

text: ﬁelds

text: with

text: an

text: internal

text: symmetry.

text: For

text: illustration,

text: we

text: often

text: specialize

text: to

text: dimension

text: potentials,

text: but

text: we

text: derive

text: results

text: for

text: general

text: potentials.

text: This

text: is

text: the

text: ﬁrst

text: part

text: of

text: a

text: two

text: part

text: series

text: of

text: papers.

text: Here

text: in

text: Part

text: we

text: especially

text: focus

text: on

text: the

text: behavior

text: of

text: long

text: wavelengths

text: modes,

text: which

text: are

text: found

text: to

text: govern

text: most

text: of

text: the

text: important

text: physics.

text: Since

text: the

text: inﬂaton

text: background

text: spontaneously

text: breaks

text: the

text: time

text: translation

text: symmetry

text: and

text: the

text: internal

text: symmetry,

text: we

text: obtain

text: Goldstone

text: modes;

text: these

text: are

text: the

text: adiabatic

text: and

text: isocurvature

text: modes.

text: We

text: ﬁnd

text: general

text: conditions

text: on

text: the

text: potential

text: for

text: when

text: a

text: large

text: instability

text: band

text: exists

text: for

text: these

text: modes

text: at

text: long

text: wavelengths.

text: For

text: the

text: adiabatic

text: mode,

text: this

text: is

text: determined

text: by

text: a

text: sound

text: speed

text: derived

text: from

text: the

text: time

text: averaged

text: potential.

text: While

text: for

text: the

text: isocurvature

text: mode,

text: this

text: is

text: determined

text: by

text: a

text: speed

text: derived

text: from

text: a

text: time

text: averaged

text: auxiliary

text: potential.

text: Interestingly,

text: we

text: ﬁnd

text: that

text: this

text: instability

text: band

text: usually

text: exists

text: for

text: one

text: of

text: these

text: classes

text: of

text: modes,

text: rather

text: than

text: both

text: simultaneously.

text: We

text: focus

text: on

text: backgrounds

text: that

text: evolve

text: radially

text: in

text: ﬁeld

text: space,

text: as

text: setup

text: by

text: inﬂation,

text: and

text: also

text: mention

text: circular

text: orbits,

text: as

text: relevant

text: to

text: Q-balls.

text: In

text: Part

text: we

text: derive

text: the

text: central

text: behavior

text: from

text: the

text: underlying

text: description

text: of

text: many

text: particle

text: quantum

text: mechanics,

text: and

text: introduce

text: a

text: weak

text: breaking

text: of

text: the

text: symmetry

text: to

text: study

text: corrections

text: to

text: particle-antiparticle

text: production

text: from

text: preheating.

text: CONTENTS

text: pplication

text: to

text: Power

text: Law

text: Potentials

text: CI.socurvature

text: Modes

text: δn

text: I.Introduction

text: quation

text: of

text: Motion

text: ime

text: Average

text: with

text: Auxiliary

text: Potential

text: IIS.ymmetric

text: Theories

text: Speed

text: c

text: I

text: AB.

text: ackground

text: Evolution

text: pplication

text: to

text: Dim

text: Potentials

text: BL.inearized

text: Perturbations

text: pplication

text: to

text: Power

text: Law

text: Potentials

text: CF.

text: loquet

text: Theory

text: for

text: Self-Resonance

text: VCI.ircular

text: Motion

text: for

text: Background

text: IIMI.otivation

text: from

text: Dimension

text: Potentials

text: AB.

text: ackground

text: Evolution

text: AP.ositive

text: vs

text: Negative

text: Quartic

text: Behavior

text: BF.ull

text: Floquet

text: Result

text: BA.

text: diabatic

text: vs

text: Isocurvature

text: Behavior

text: CL.ong

text: Wavelength

text: Limit

text: Sound

text: Speed

text: c

text: CN.

text: egative

text: Mass

text: Squared

text: Behavior

text: S

text: VCIIo.nclusions

text: IVA.uxiliary

text: Potential

text: for

text: Isocurvature

text: Modes

text: AC.

text: orrespondence

text: between

text: Modes

text: VAIIpIp.

text: endix

text: Non-Canonical

text: Kinetic

text: Terms

text: BI.ntegral

text: Form

text: for

text: General

text: Potentials

text: CA.

text: pplication

text: to

text: Dim

text: Potentials

text: References

text: VG.

text: eneral

text: Analysis

text: for

text: Long

text: Wavelength

text: Modes

text: AG.

text: oldstone

text: Modes

text: BA.

text: diabatic

text: Mode

text: δε

text: I.

text: INTRODUCTION

text: quation

text: of

text: Motion

text: ime

text: Average

text: Cosmological

text: inﬂation

text: is

text: a

text: successful

text: theory

text: of

text: the

text: ound

text: Speed

text: cS

text: early

text: universe

text: which

text: accounts

text: for

text: the

text: approxi-

text: pplication

text: to

text: Dim

text: Potentials

text: mately

text: scale-invariant

text: distribution

text: of

text: structures

text: on

text: large

text: scales.

text: The

text: structures

text: arises

text: from

text: quantum

text: ﬂuctuations

text: in

text: the

text: inﬂaton

text: scalar

text: ﬁeld/s

text: φ

text: by

text: stretching

text: modes

text: to

text: large

text: scales

text: due

text: to

text: inﬂation’s

text: exponential

text: expansion.

text: †Electronic

text: address:

text: mphertz@mit.edu

text: Evidence

text: for

text: inﬂation

text: is

text: increasing

text: with

text: detailed

text: mea-

text: Electronic

text: address:

text: karouby@mit.edu

text: surement

text: of

text: the

text: cosmic

text: microwave

text: background

text: radiation

text: and

text: this

text: has

text: motivated

text: the

text: construction

text: of

text: many

text: interesting

text: theoretical

text: models

text: ##LTLine##


text: 14 FIG. 7. Left panel: Contour plot of the real part of Floquet exponent µ for dimension 4 potentials as a function of physical k wavenumber k and background amplitude φ for δφ⊥, with m2 > 0 and λ > 0 (zoomed in region of Fig. 1’s lower left p a panel). The dashed green line indicates the redshifti√ng physical wavenumber and amplitude of a ﬁxed √comoving wavenumber ≡ k = 0.4 a m, where a is the√scale factor at φ = m/ λ. We have plotted µ in units of rH where r λ M /m, k in units 0 0 a k Pl p of m, and φa in units of m/ λ. Right panel: Growth in power Pδφ⊥ as a function of time for the same comoving wavenumber k = 0.4 a0 m with r = 20. We have plotted Pδφ⊥ in units of 1/(a3 m) and t in units of 1/m. For λ < 0 (not shown here) qualitatively similar behavior occurs for δφ(cid:107). (cid:90) (cid:104) (cid:105) ator for parallel ﬂuctuations as initial condition − δˆφ(cid:107)(x, t) = (2dπ3k)3 v(cid:107),k(t) aˆk eik·x + v(cid:107)∗,k(t) aˆ†k e−ik·x v(cid:107),k, v⊥,k → √(cid:112)e2 ωiωkkat3 (84) (cid:90) (cid:104) (7(cid:105)8) where the frequency is ω = m2 + k2/a2. In fact at and the operator for orthogonal ﬂuctuations as k → early times it is suﬃcient to use ω k/a. k d3k · ∗ † − · We can go further and quantize the perturbations in δˆφ⊥(x, t) = v⊥ (t)ˆb eik x + v⊥ (t)ˆb e ik x (2π)3 ,k k ,k k energy densit(cid:90)y δε and(cid:104) number density δn. We writ(cid:105)e the (79) operator for energy density (adiabatic) ﬂuctuations as where v(cid:107) and v⊥ are the respective mode functions. The d3k · ∗ † − · creation and an(cid:104)nihilat(cid:105)ion operators satisfy the standard δˆε(x, t) = z (t) aˆ eik x + z (t) aˆ e ik x quantization condition (2π)3 k k k k (cid:104) (cid:105) (85) † − (cid:48) aˆk, aˆk(cid:48) = (2π)3δ3(k k ) (80) and the opera(cid:90)tor for (cid:104)number density (isocurvature) (cid:105)ﬂuc- tuations as † − (cid:48) ˆbk,ˆbk(cid:48) = (2π)3δ3(k k ) (81) d3k · ∗ † − · δˆn(x, t) = w (t)ˆb eik x + w (t)ˆb e ik x (2π)3 k k k k As this is a free theory, the mode functions satisfy the (86) classical equations of motion that we previously discussed We can relate these density ﬂuctuations to the ﬁeld ﬂuc- for δφ in eqs. (11, 12). S(cid:18)o, allowing for(cid:19)Hubble expansion, tuations by using the quantized versions of the linearized we have energy density and number densities, as we deﬁned in (cid:18)k2 (cid:48)(cid:48) (cid:19) Part 1 [1]. This leads to the following relationship be- v¨(cid:107) + 3Hv˙(cid:107) + + V (φ ) v(cid:107) = 0 (82) a2 0 tween the density m(cid:18)ode functions z (cid:19)and w and the ﬁeld (cid:48) mode functions v(cid:107) and v⊥ k2 V (φ ) v¨⊥ + 3Hv˙⊥ + + 0 v⊥ = 0 (83) a2 φ0 z (t) = φ(cid:18)˙ ∂ + V (cid:48)(φ(cid:19)) v(cid:107) (t) (87) k 0 ∂t 0 ,k We assume that at early times, the mode functions are in their Minkowski vacua, and then evolve; this is the − ∂ − w (t) = φ φ˙ v⊥ (t) (88) Bunch-Davies vacuum. So at early times we require the k 0 ∂t 0 ,k ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Left

label: caption

text: panel:

text: Contour

text: plot

text: of

text: the

text: real

text: part

text: of

text: Floquet

text: exponent

text: µ

text: for

text: dimension

text: potentials

text: as

text: a

text: function

text: of

text: physical

text: k

text: wavenumber

text: k

text: and

text: background

text: amplitude

text: φ

text: for

text: δφ⊥,

text: with

text: and

text: λ

text: (zoomed

text: in

text: region

text: of

text: Fig.

text: lower

text: left

text: p

text: a

text: panel).

text: The

text: dashed

text: green

text: line

text: indicates

text: the

text: redshifti√ng

text: physical

text: wavenumber

text: and

text: amplitude

text: of

text: a

text: ﬁxed

text: √comoving

text: wavenumber

text: k

text: a

text: m,

text: where

text: a

text: is

text: the√scale

text: factor

text: at

text: φ

text: m/

text: λ.

text: We

text: have

text: plotted

text: µ

text: in

text: units

text: of

text: rH

text: where

text: r

text: λ

text: M

text: /m,

text: k

text: in

text: units

text: a

text: k

text: Pl

text: p

text: of

text: m,

text: and

text: φa

text: in

text: units

text: of

text: m/

text: λ.

text: Right

text: panel:

text: Growth

text: in

text: power

text: Pδφ⊥

text: as

text: a

text: function

text: of

text: time

text: for

text: the

text: same

text: comoving

text: wavenumber

text: k

text: m

text: with

text: r

text: We

text: have

text: plotted

text: Pδφ⊥

text: in

text: units

text: of

text: m)

text: and

text: t

text: in

text: units

text: of

text: For

text: λ

text: (not

text: shown

text: here)

text: qualitatively

text: similar

text: behavior

text: occurs

text: for

text: ator

text: for

text: parallel

text: ﬂuctuations

text: as

text: initial

text: condition

text: t)

text: aˆk

text: eik·x

text: aˆ†k

text: e−ik·x

text: v⊥,k

text: where

text: the

text: frequency

text: is

text: ω

text: In

text: fact

text: at

text: and

text: the

text: operator

text: for

text: orthogonal

text: ﬂuctuations

text: as

text: k

text: early

text: times

text: it

text: is

text: suﬃcient

text: to

text: use

text: ω

text: k/a.

text: k

text: We

text: can

text: go

text: further

text: and

text: quantize

text: the

text: perturbations

text: in

text: δˆφ⊥(x,

text: t)

text: v⊥

text: (t)ˆb

text: eik

text: x

text: v⊥

text: (t)ˆb

text: e

text: ik

text: x

text: ,k

text: k

text: ,k

text: k

text: energy

text: δε

text: number

text: density

text: δn.

text: We

text: the

text: operator

text: for

text: energy

text: density

text: (adiabatic)

text: ﬂuctuations

text: as

text: where

text: and

text: v⊥

text: are

text: the

text: respective

text: mode

text: functions.

text: The

text: creation

text: and

text: operators

text: satisfy

text: the

text: standard

text: δˆε(x,

text: t)

text: z

text: (t)

text: aˆ

text: eik

text: x

text: z

text: (t)

text: aˆ

text: e

text: ik

text: x

text: quantization

text: condition

text: k

text: k

text: k

text: k

text: aˆk,

text: k

text: and

text: the

text: for

text: density

text: (isocurvature)

text: tuations

text: as

text: k

text: δˆn(x,

text: t)

text: w

text: (t)ˆb

text: eik

text: x

text: w

text: (t)ˆb

text: e

text: ik

text: x

text: k

text: k

text: k

text: k

text: As

text: this

text: is

text: a

text: free

text: theory,

text: the

text: mode

text: functions

text: satisfy

text: the

text: classical

text: equations

text: of

text: motion

text: that

text: we

text: previously

text: discussed

text: We

text: can

text: relate

text: these

text: density

text: ﬂuctuations

text: to

text: the

text: ﬁeld

text: ﬂuc-

text: for

text: δφ

text: in

text: eqs.

text: allowing

text: expansion,

text: tuations

text: by

text: using

text: the

text: quantized

text: versions

text: of

text: the

text: linearized

text: we

text: have

text: energy

text: density

text: and

text: number

text: densities,

text: as

text: we

text: deﬁned

text: in

text: Part

text: This

text: leads

text: to

text: the

text: following

text: relationship

text: be-

text: V

text: (φ

text: tween

text: the

text: density

text: functions

text: z

text: w

text: and

text: the

text: ﬁeld

text: mode

text: functions

text: and

text: v⊥

text: V

text: (φ

text: v¨⊥

text: v⊥

text: φ0

text: z

text: (t)

text: V

text: (t)

text: k

text: ∂t

text: ,k

text: We

text: assume

text: that

text: at

text: early

text: times,

text: the

text: mode

text: functions

text: are

text: in

text: their

text: Minkowski

text: vacua,

text: and

text: then

text: evolve;

text: this

text: is

text: the

text: w

text: (t)

text: φ

text: φ˙

text: v⊥

text: (t)

text: Bunch-Davies

text: vacuum.

text: So

text: at

text: early

text: times

text: we

text: require

text: the

text: k

text: ∂t

text: ,k

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 Dimension 4 Potential the adiabatic modes. In particular, for λ > 0 the isocur- vature mode leads to an instability, despite the repulsion among the particles (and antiparticles). The reason is that Bose-Einstein statistics favor the particles to clump with particles and the antiparticles to clump with an- tiparticles. We also developed a small amplitude analysis, which captured not only the long wavelength behavior, but also the higher band structure, and we explained this in terms of Feynman diagrams of annihilation and decay, where appropriate. We then performed the quantization of our perturba- tions. As an example we computed the distribution of the number density of particles minus the density of antipar- ticles for λ > 0. For strong resonance, we showed that the inﬂaton fragments into separate regions of particles and antiparticles. Finally, we applied the quantization of the inﬂaton ﬁelds to the case of particle-antiparticle asymmetry, which is relevant to some models of baryogenesis [2, 3]. We showed that the symmetric theory can be used to Toy Potential compute the leading order behavior of the asymmetric theory in the limit of weak symmetry breaking. We computed the corrections to the homogeneous theory from the inhomogeneous theory due to self-resonance. The result involves an integral over the diﬀerence in the power between the adiabatic modes and the isocurvature modes. This diﬀerence is ﬁnite and is dominated by the resonant modes at relatively long wavelengths. Altogether, along with Part 1 [1], our work gives a detailed theory of self-resonance after inﬂation in single and multi-ﬁeld models. In Part 1 [1] we understood the long wavelength behavior using the Goldstone theorem. Here we have provided the deep underlying physical un- derstanding of the entire resonance structure, with the main structure determined by the attraction/repulsion of particles. An interesting direction for future work is to incorpo- rate corrections from possibly large ﬂuctuations of the metric, and explore gravitational waves production. An- other direction is to extend this theory by including cou- FIG. 12. The relative correction to the baryon asymmetry plings to other ﬁelds beyond the inﬂaton, such as Stan- ∆N /∆N (quantum ﬂuctuations relative to classical back- δ 0 dard Model ﬁelds. ground) as a function of the dimensionless coupling r for two diﬀerent kinds of inﬂationary potentials with n = 4. Up- per panel is for the dimension 4 potential (with m2 > 0 and Acknowledgments λ > 0). Lower panel is for the toy potential of eq. (118) (an m2 > 0 and λ < 0 model). We have plotted the relative correction in units of λ and absorbed an overall minus sign. We thank Jacovie Rodriguez for help in the early stage of this project. We also thank Alan Guth and David Kaiser for useful discussions. We would like to acknowl- particles (and antiparticles) exhibit repulsion, so the ho- edge support by the Center for Theoretical Physics and mogeneous conﬁguration established by inﬂation is stable the Undergraduate Research Opportunities Program at against adiabatic perturbations. While for λ < 0 there MIT. This work is supported by the U.S. Department of is mutual attraction leading to breakup of the homoge- Energy under cooperative research agreement Contract neous conﬁguration and instability. On the other hand, Number DE-FG02-05ER41360. JK is supported by an the isocurvature modes have very diﬀerent behavior from NSERC PDF fellowship. ##LTFigure## ##LTFigure##

text: Dimension

text: Potential

text: the

text: adiabatic

text: modes.

text: In

text: particular,

text: for

text: λ

text: the

text: isocur-

text: vature

text: mode

text: leads

text: to

text: an

text: instability,

text: despite

text: the

text: repulsion

text: among

text: the

text: particles

text: (and

text: antiparticles).

text: The

text: reason

text: is

text: that

text: Bose-Einstein

text: statistics

text: favor

text: the

text: particles

text: to

text: clump

text: with

text: particles

text: and

text: the

text: antiparticles

text: to

text: clump

text: with

text: an-

text: tiparticles.

text: We

text: also

text: developed

text: a

text: small

text: amplitude

text: analysis,

text: which

text: captured

text: not

text: only

text: the

text: long

text: wavelength

text: behavior,

text: but

text: also

text: the

text: higher

text: band

text: structure,

text: and

text: we

text: explained

text: this

text: in

text: terms

text: of

text: Feynman

text: diagrams

text: of

text: annihilation

text: and

text: decay,

text: where

text: appropriate.

text: We

text: then

text: performed

text: the

text: quantization

text: of

text: our

text: perturba-

text: tions.

text: As

text: an

text: example

text: we

text: computed

text: the

text: distribution

text: of

text: the

text: number

text: density

text: of

text: particles

text: minus

text: the

text: density

text: of

text: antipar-

text: ticles

text: for

text: λ

text: For

text: strong

text: resonance,

text: we

text: showed

text: that

text: the

text: inﬂaton

text: fragments

text: into

text: separate

text: regions

text: of

text: particles

text: and

text: antiparticles.

text: Finally,

text: we

text: applied

text: the

text: quantization

text: of

text: the

text: inﬂaton

text: ﬁelds

text: to

text: the

text: case

text: of

text: particle-antiparticle

text: asymmetry,

text: which

text: is

text: relevant

text: to

text: some

text: models

text: of

text: baryogenesis

text: We

text: showed

text: that

text: the

text: symmetric

text: theory

text: can

text: be

text: used

text: to

text: Toy

text: Potential

text: compute

text: the

text: leading

text: order

text: behavior

text: of

text: the

text: asymmetric

text: theory

text: in

text: the

text: limit

text: of

text: weak

text: symmetry

text: breaking.

text: We

text: computed

text: the

text: corrections

text: to

text: the

text: homogeneous

text: theory

text: from

text: the

text: inhomogeneous

text: theory

text: due

text: to

text: self-resonance.

text: The

text: result

text: involves

text: an

text: integral

text: over

text: the

text: diﬀerence

text: in

text: the

text: power

text: between

text: the

text: adiabatic

text: modes

text: and

text: the

text: isocurvature

text: modes.

text: This

text: diﬀerence

text: is

text: ﬁnite

text: and

text: is

text: dominated

text: by

text: the

text: resonant

text: modes

text: at

text: relatively

text: long

text: wavelengths.

text: Altogether,

text: along

text: with

text: Part

text: our

text: work

text: gives

text: a

text: detailed

text: theory

text: of

text: self-resonance

text: after

text: inﬂation

text: in

text: single

text: and

text: multi-ﬁeld

text: models.

text: In

text: Part

text: we

text: understood

text: the

text: long

text: wavelength

text: behavior

text: using

text: the

text: Goldstone

text: theorem.

text: Here

text: we

text: have

text: provided

text: the

text: deep

text: underlying

text: physical

text: un-

text: derstanding

text: of

text: the

text: entire

text: resonance

text: structure,

text: with

text: the

text: main

text: structure

text: determined

text: by

text: the

text: attraction/repulsion

text: of

text: particles.

text: An

text: interesting

text: direction

text: for

text: future

text: work

text: is

text: to

text: incorpo-

text: rate

text: corrections

text: from

text: possibly

text: large

text: ﬂuctuations

text: of

text: the

text: metric,

text: and

text: explore

text: gravitational

text: waves

text: production.

text: An-

text: other

text: direction

text: is

text: to

text: extend

text: this

text: theory

text: by

text: including

text: cou-

text: FIG.

text: The

text: relative

text: correction

text: to

text: the

text: baryon

text: asymmetry

text: plings

text: to

text: other

text: ﬁelds

text: beyond

text: the

text: inﬂaton,

text: such

text: as

text: Stan-

text: ∆N

text: /∆N

text: (quantum

text: ﬂuctuations

text: relative

text: to

text: classical

text: back-

text: δ

text: dard

text: Model

text: ﬁelds.

text: ground)

text: as

text: a

text: function

text: of

text: the

text: dimensionless

text: coupling

text: r

text: for

text: two

text: diﬀerent

text: kinds

text: of

text: inﬂationary

text: potentials

text: with

text: n

text: Up-

text: per

text: panel

text: is

text: for

text: the

text: dimension

text: potential

text: (with

text: and

text: Acknowledgments

text: λ

text: Lower

text: panel

text: is

text: for

text: the

text: toy

text: potential

text: of

text: eq.

text: (an

text: and

text: λ

text: model).

text: We

text: have

text: plotted

text: the

text: relative

text: correction

text: in

text: units

text: of

text: λ

text: and

text: absorbed

text: an

text: overall

text: minus

text: sign.

text: We

text: thank

text: Jacovie

text: Rodriguez

text: for

text: help

text: in

text: the

text: early

text: stage

text: of

text: this

text: project.

text: We

text: also

text: thank

text: Alan

text: Guth

text: and

text: David

text: Kaiser

text: for

text: useful

text: discussions.

text: We

text: would

text: like

text: to

text: acknowl-

text: particles

text: (and

text: antiparticles)

text: exhibit

text: repulsion,

text: so

text: the

text: ho-

text: edge

text: support

text: by

text: the

text: Center

text: for

text: Theoretical

text: Physics

text: and

text: mogeneous

text: conﬁguration

text: established

text: by

text: inﬂation

text: is

text: stable

text: the

text: Undergraduate

text: Research

text: Opportunities

text: Program

text: at

text: against

text: adiabatic

text: perturbations.

text: While

text: for

text: λ

text: there

text: MIT.

text: This

text: work

text: is

text: supported

text: by

text: the

text: U.S.

text: Department

text: of

text: is

text: mutual

text: attraction

text: leading

text: to

text: breakup

text: of

text: the

text: homoge-

text: Energy

text: under

text: cooperative

text: research

text: agreement

text: Contract

text: neous

text: conﬁguration

text: and

text: instability.

text: On

text: the

text: other

text: hand,

text: Number

text: JK

text: is

text: supported

text: by

text: an

text: the

text: isocurvature

text: modes

text: have

text: very

text: diﬀerent

text: behavior

text: from

text: NSERC

text: PDF

text: fellowship.

text: ##LTFigure##

text: ##LTFigure##


text: 3 are possible). So here we assume, for simplicity, that to erase angular momentum, even in ﬁeld space. This all higher order derivative corrections to the Einstein- leads to the multi-ﬁeld inﬂaton moving radially in ﬁelds Hilbert action are small. Furthermore, we specialize to space. As in Part 1 [1], the purely radial motion for the canonical kinetic energy in the Einstein frame. Since background shall be denoted by the ﬁeld φ (t). From 0 inﬂaton couplings are usually small in order to achieve varying the action, we obtain the standard equation of small ﬂuctuations, this is technically natural. So we take motion for a homogeneous scalar ﬁeld N − the ac(cid:90)tion for (cid:20)scalar ﬁelds to be (signature +(cid:21) ++, (cid:48) units (cid:126) = c = 1) φ¨0 + 3Hφ˙0 + V (φ0) = 0 (7) √ − M2 R − 1 − where H = a˙/a is the Hubble parameter. During slow- S = d4x g Pl δ ∂ φi∂µφj V (φ(cid:126)) (3) 2 2 ij µ roll inﬂation, the second and third terms here dominate. √ After inﬂation, as is the focus of this work, the ﬁrst and ≡ where M 1 8πG is the reduced Planck mass. In third terms dominate and the second “friction” term is Pl N the Appendix of Part 1 [1] we developed some results for sub-dominant. more general potentials, including higher derivative cor- In Section we will properly track the behavior for φ , 0 rections and non-trivial metrics Gij on ﬁeld space. But where we self consistently s(cid:18)olve for the Hu(cid:19)bble parameter these generalizations are not important for our analysis H from the Friedmann equation here. Ignoring higher order corrections, the residual freedom H2 = 1 1φ˙2 + V (φ ) (8) is in the choice of the potential V (φ(cid:126)). In Part 1 [1] we 3M2 2 0 s 0 Pl exclusively studied symmetric potentials that carry the This leads to a redshifting of the background (classical) internal rotational symmetry ﬁelds that will inﬂuence the self-resonance of the pertur- → φi Ri φj (4) bations in an important fashion. j B. Quantal Evolution (Preliminary) where R is a rotation matrix acting on ﬁeld space. For- N mally this implied an O( ) symmetry and the potential | | Due to quantum mechanics the ﬁeld cannot have a well may be written as V (φ(cid:126)) = V ( φ(cid:126) ). Here we often focus deﬁned value, so there are necessarily quantum ﬂuctua- on these symmetric potentials, but we also allow for a tions. Focussing then on the symmetric case, we can de- breakly of the symmetry. So we decompose the potential compose these ﬂuctuations into those that are parallel to as the radial motion of the background δφ(cid:107), and those that | | are orthogonal to the background δφ⊥. Later in Section V (φ(cid:126)) = V ( φ(cid:126) ) + V (φ(cid:126)) (5) s b we will be precise about the quantization of these ﬂuctu- ations. But for now it suﬃces to treat them as any form The term V is a symmetric potential that carries the in- s of ﬂuctuation, either classical or quantum, even though ternal rotational symmetry, while V does not. For most b its origin is necessarily quantum. We expand the ﬁeld of this paper we ignore the breaking term and utilize the around the background as symmetry to simplify the analysis. We then make use of the breaking term in Section to provide an asymme- φ(cid:126)(x, t) = φ(cid:126) (t) + δφ(cid:126)(x, t) (9) try between particles and antiparticles. This breaking of 0 a symmetry means that particle number is not exactly where conserved, potentially leading to the matter-antimatter { } asymmetry. δφ(cid:126)(x, t) = δφ⊥ (x, t), . . . , δφ⊥N− (x, t), δφ(cid:107)(x, t) (10) 1 1 A. Classical Evolution (Preliminary) N and we have put the background motion in the th direc- We begin by discussing the evolution of the classical tion, without loss of generality, in the symmetric theory. background. The metric is established by inﬂation to be Later in Section we will allow for a general direction for the standard ﬂat FRW metric the asymmetry theory. Expanding the pertu(cid:18)rbations (clas(cid:19)sical or quanta) to − ds2 = dt2 + a(t)2dx2 (6) ﬁrst order we have where a(t) is the scale factor. δφ¨(cid:107) + 3Hδφ˙ (cid:107) + k(cid:18)2 + V (cid:48)(cid:48)(φ ) δ(cid:19)φ(cid:107) = 0 (11) The evolution of the classical ﬁeld is, in general, a2 0 (cid:48) cisomprpelsiecnatte,dt.he Wﬁehldentetnhdes styomgmetetkriyckebdreaarkoinugndteinrmﬁeVldb δφ¨⊥i + 3Hδφ˙⊥i + ka22 + V φ(φ0) δφ⊥i = 0 (12) space. If the breaking is small, then it is roughy a kind 0 of elliptic behavior. This will play a role later in our where we have Fourier transformed to k-space. For the Section on baryogenesis. orthogonal components, we have included an “i” index, N − On the other hand, if the breaking term is negligible, where i runs over i = 1, . . . , 1; each equation carries then the motion simpliﬁes considerably. Inﬂation tends the same structure due to the symmetry. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: are

text: possible).

text: So

text: here

text: we

text: assume,

text: for

text: simplicity,

text: that

text: to

text: erase

text: angular

text: momentum,

text: even

text: in

text: ﬁeld

text: space.

text: This

text: all

text: higher

text: order

text: derivative

text: corrections

text: to

text: the

text: Einstein-

text: leads

text: to

text: the

text: multi-ﬁeld

text: inﬂaton

text: moving

text: radially

text: in

text: ﬁelds

text: Hilbert

text: action

text: are

text: small.

text: Furthermore,

text: we

text: specialize

text: to

text: space.

text: As

text: in

text: Part

text: the

text: purely

text: radial

text: motion

text: for

text: the

text: canonical

text: kinetic

text: energy

text: in

text: the

text: Einstein

text: frame.

text: Since

text: background

text: shall

text: be

text: denoted

text: by

text: the

text: ﬁeld

text: φ

text: (t).

text: From

text: inﬂaton

text: couplings

text: are

text: usually

text: small

text: in

text: order

text: to

text: achieve

text: varying

text: the

text: action,

text: we

text: obtain

text: the

text: standard

text: equation

text: of

text: small

text: ﬂuctuations,

text: this

text: is

text: technically

text: natural.

text: So

text: we

text: take

text: motion

text: for

text: a

text: homogeneous

text: scalar

text: ﬁeld

text: N

text: the

text: for

text: ﬁelds

text: to

text: be

text: (signature

text: units

text: c

text: φ¨0

text: V

text: R

text: where

text: H

text: a˙/a

text: is

text: the

text: Hubble

text: parameter.

text: During

text: slow-

text: S

text: g

text: Pl

text: δ

text: φi∂µφj

text: V

text: ij

text: µ

text: roll

text: inﬂation,

text: the

text: second

text: and

text: third

text: terms

text: here

text: dominate.

text: After

text: inﬂation,

text: as

text: is

text: the

text: focus

text: of

text: this

text: work,

text: the

text: ﬁrst

text: and

text: where

text: M

text: is

text: the

text: reduced

text: Planck

text: mass.

text: In

text: third

text: terms

text: dominate

text: and

text: the

text: second

text: “friction”

text: term

text: is

text: Pl

text: N

text: the

text: Appendix

text: of

text: Part

text: we

text: developed

text: some

text: results

text: for

text: sub-dominant.

text: more

text: general

text: potentials,

text: including

text: higher

text: derivative

text: cor-

text: In

text: Section

text: we

text: will

text: properly

text: track

text: the

text: behavior

text: for

text: φ

text: rections

text: and

text: non-trivial

text: metrics

text: Gij

text: on

text: ﬁeld

text: space.

text: But

text: where

text: we

text: self

text: consistently

text: for

text: the

text: parameter

text: these

text: generalizations

text: are

text: not

text: important

text: for

text: our

text: analysis

text: H

text: from

text: the

text: Friedmann

text: equation

text: here.

text: Ignoring

text: higher

text: order

text: corrections,

text: the

text: residual

text: freedom

text: 1φ˙2

text: V

text: (φ

text: is

text: in

text: the

text: choice

text: of

text: the

text: potential

text: V

text: In

text: Part

text: we

text: s

text: Pl

text: exclusively

text: studied

text: symmetric

text: potentials

text: that

text: carry

text: the

text: This

text: leads

text: to

text: a

text: redshifting

text: of

text: the

text: background

text: (classical)

text: internal

text: rotational

text: symmetry

text: ﬁelds

text: that

text: will

text: inﬂuence

text: the

text: self-resonance

text: of

text: the

text: pertur-

text: φi

text: Ri

text: φj

text: bations

text: in

text: an

text: important

text: fashion.

text: j

text: B.

text: Quantal

text: Evolution

text: (Preliminary)

text: where

text: R

text: is

text: a

text: rotation

text: matrix

text: acting

text: on

text: ﬁeld

text: space.

text: For-

text: N

text: mally

text: this

text: implied

text: an

text: O(

text: symmetry

text: and

text: the

text: potential

text: Due

text: to

text: quantum

text: mechanics

text: the

text: ﬁeld

text: cannot

text: have

text: a

text: well

text: may

text: be

text: written

text: as

text: V

text: V

text: Here

text: we

text: often

text: focus

text: deﬁned

text: value,

text: so

text: there

text: are

text: necessarily

text: quantum

text: ﬂuctua-

text: on

text: these

text: symmetric

text: potentials,

text: but

text: we

text: also

text: allow

text: for

text: a

text: tions.

text: Focussing

text: then

text: on

text: the

text: symmetric

text: case,

text: we

text: can

text: de-

text: breakly

text: of

text: the

text: symmetry.

text: So

text: we

text: decompose

text: the

text: potential

text: compose

text: these

text: ﬂuctuations

text: into

text: those

text: that

text: are

text: parallel

text: to

text: as

text: the

text: radial

text: motion

text: of

text: the

text: background

text: and

text: those

text: that

text: are

text: orthogonal

text: to

text: the

text: background

text: δφ⊥.

text: Later

text: in

text: Section

text: V

text: V

text: V

text: s

text: b

text: we

text: will

text: be

text: precise

text: about

text: the

text: quantization

text: of

text: these

text: ﬂuctu-

text: ations.

text: But

text: for

text: now

text: it

text: suﬃces

text: to

text: treat

text: them

text: as

text: any

text: form

text: The

text: term

text: V

text: is

text: a

text: symmetric

text: potential

text: that

text: carries

text: the

text: in-

text: s

text: of

text: ﬂuctuation,

text: either

text: classical

text: or

text: quantum,

text: even

text: though

text: ternal

text: rotational

text: symmetry,

text: while

text: V

text: does

text: not.

text: For

text: most

text: b

text: its

text: origin

text: is

text: necessarily

text: quantum.

text: We

text: expand

text: the

text: ﬁeld

text: of

text: this

text: paper

text: we

text: ignore

text: the

text: breaking

text: term

text: and

text: utilize

text: the

text: around

text: the

text: background

text: as

text: symmetry

text: to

text: simplify

text: the

text: analysis.

text: We

text: then

text: make

text: use

text: of

text: the

text: breaking

text: term

text: in

text: Section

text: to

text: provide

text: an

text: asymme-

text: t)

text: (t)

text: t)

text: try

text: between

text: particles

text: and

text: antiparticles.

text: This

text: breaking

text: of

text: a

text: symmetry

text: means

text: that

text: particle

text: number

text: is

text: not

text: exactly

text: where

text: conserved,

text: potentially

text: leading

text: to

text: the

text: matter-antimatter

text: asymmetry.

text: t)

text: δφ⊥

text: (x,

text: t),

text: δφ⊥N−

text: (x,

text: t),

text: t)

text: A.

text: Classical

text: Evolution

text: (Preliminary)

text: N

text: and

text: we

text: have

text: put

text: the

text: background

text: motion

text: in

text: the

text: th

text: direc-

text: We

text: begin

text: by

text: discussing

text: the

text: evolution

text: of

text: the

text: classical

text: tion,

text: without

text: loss

text: of

text: generality,

text: in

text: the

text: symmetric

text: theory.

text: background.

text: The

text: metric

text: is

text: established

text: by

text: inﬂation

text: to

text: be

text: Later

text: in

text: Section

text: we

text: will

text: allow

text: for

text: a

text: general

text: direction

text: for

text: the

text: standard

text: ﬂat

text: FRW

text: metric

text: the

text: asymmetry

text: theory.

text: Expanding

text: the

text: or

text: quanta)

text: to

text: ﬁrst

text: order

text: we

text: have

text: where

text: a(t)

text: is

text: the

text: scale

text: factor.

text: 3Hδφ˙

text: V

text: The

text: evolution

text: of

text: the

text: classical

text: ﬁeld

text: is,

text: in

text: general,

text: cisomprpelsiecnatte,dt.he

text: Wﬁehldentetnhdes

text: δφ¨⊥i

text: 3Hδφ˙⊥i

text: V

text: δφ⊥i

text: space.

text: If

text: the

text: breaking

text: is

text: small,

text: then

text: it

text: is

text: roughy

text: a

text: kind

text: of

text: elliptic

text: behavior.

text: This

text: will

text: play

text: a

text: role

text: later

text: in

text: our

text: where

text: we

text: have

text: Fourier

text: transformed

text: to

text: k-space.

text: For

text: the

text: Section

text: on

text: baryogenesis.

text: orthogonal

text: components,

text: we

text: have

text: included

text: an

text: “i”

text: index,

text: N

text: On

text: the

text: other

text: hand,

text: if

text: the

text: breaking

text: term

text: is

text: negligible,

text: where

text: i

text: runs

text: over

text: i

text: each

text: equation

text: carries

text: then

text: the

text: motion

text: simpliﬁes

text: considerably.

text: Inﬂation

text: tends

text: the

text: same

text: structure

text: due

text: to

text: the

text: symmetry.

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