text: 4 From the detected root position of the tree structure, authors C. From 2D Parts to 3D Pose Estimation in [1] employ a backtracking algorithm to ﬁx all the body part The Gaussian process regressor is one of the most widely positions. It starts from the root of the tree structure and ﬁx used regression model for learning 2D to 3D mapping in the its child node by picking the maximum response from all the pose estimation since it has been proved to be an effective ap- child nodes. This method causes the double counting problem. proach for the nonlinear 2D to 3D pose mapping problem [8], Since each body part is ﬁxed only considering the response of [9], [10]. Gaussian Process Regression (GPR) is considered the test image with the trained templates, when sibling body a model-free framework. GPR deﬁned as a distribution over parts (the same body part, but on different sides of the human functions, extends statistics from data points to functions. With body, like a left hand and a right hand) resemble each other, kernel trick, we can even get rid of the function deﬁnition, and the same image patch might be picked repetitively. In this case, only concentrate on kernel matrix instead. Once we normalize the estimated pose is occluded while in the real case it is not. the training input to have a zero mean, we only need to To solve this problem, we use global optimization to ﬁx deﬁne a covariance matrix, that is ,the kernel matrix, for each body part positions. In the original MoP model, where GPR. Frequently used covariance matrices include squared there is only HOG feature, optimizing body part position is exponential covariance function, Mate´rn covariance function very time consuming. For example, if the model uses 26 body and so on. In the following subsections, we will explain parts, and each body parts use 5 or 6 mixtures, the minimum detailed representations and settings for the Gaussian process number of possible combination for all body parts is 525. This regressor used here. is a huge amount of possible guesses. In our case, where we consider human blob as another feature cues. The possible According to [11], Gaussian process is deﬁned as: a col- number of mixtures for each body parts is great reduced due to lection of random variables, any ﬁnite number of which have the constraint. So we can optimize the tree structure globally. (consistent) joint Gaussian distribution. A Gaussian process The body part positions are optimized to maximize the score is completely speciﬁed by its mean function and a covariance o(cid:88)f the proposed model fo(cid:88)r combining multiple cues SMC: function. Integrating with our problem, we denote the mean (cid:48) function as m(s) and the covariance function as k(s, s ), so a ∈ ωiti · φ(I,pi) · F(pi) + ∈ ωitji,tj · ψ(pi − pj) · F(pi) · F(pj), (7) Gaussian process is represented as: i V ij E ∼ GP (cid:48) ζ(s) j(m(s),k(s,s )), (8) where F(p ) is the ratio of overlap between the body part i p and the foreground model deﬁne in equation(3). i where overlap ratio = 7.261652e−001, bbsize = 4.525483e+001 m(s) = E[ζ(s)], (cid:48) − (cid:48) − (cid:48) k(s,s ) = E[(ζ(s) m(s))(ζ(s ) m(s ))], (9) We assume prediction noise as a Gaussian distribution and formulate ﬁnding the optimal hyperparameters as an optimiza- tion problem. We seek the optimal solution of hyperparameters by maximizing the log marginal likelihood (see [11] for Fig. 2. Detected body part localization with the Mixture of Parts (MoP) details): model. The left ﬁgure shows the localizations of all body parts with MoP. The right ﬁgure shows the overlap between extracted human blobs and all bounding boxes. Color variations in human blobs denotes different number log p(Ψ(cid:48)|s,θ) = −1Ψ(cid:48)TK−(cid:48)1Ψ(cid:48) − 1 log |K (cid:48)| − n log 2π, (10) of bounding boxes that are overlapped. 2 Ψ 2 Ψ 2 where K (cid:48) is the calculated covariance matrix of the target Ψ vector. overlap ratio = 8.982789e−001, bbsize = 4.525483e+001 With the optimal hyperparameters, the prediction distribu- tion is represented as: (cid:48)∗| ∗ (cid:48) ∼ N ∗ − (cid:48) Ψ s ,s,Ψ (k(s ,s)T[K + σ2 I] 1Ψ , ∗ ∗ − ∗ n−oise ∗ k(s ,s ) + σ2 k(s ,s)T[K + σ2 I] 1k(s ,s)), (11) noise noise where K is the calculated covariance matrix from training 2D image features s and σ is the covariance of Gaussian Fig. 3. Body part localization after optimization (left) and ﬁnal overlap map noise noise. between all bounding boxes and the extracted human blob. Equation11 for referencing test data is deducted from marginal and conditional properties of Gaussian distributions. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: From

text: the

text: detected

text: root

text: position

text: of

text: the

text: tree

text: structure,

text: authors

text: C.

text: From

text: Parts

text: to

text: Pose

text: Estimation

text: in

text: employ

text: a

text: backtracking

text: algorithm

text: to

text: ﬁx

text: all

text: the

text: body

text: part

text: The

text: Gaussian

text: process

text: regressor

text: is

text: one

text: of

text: the

text: most

text: widely

text: positions.

text: It

text: starts

text: from

text: the

text: root

text: of

text: the

text: tree

text: structure

text: and

text: ﬁx

text: used

text: regression

text: model

text: for

text: learning

text: to

text: mapping

text: in

text: the

text: its

text: child

text: node

text: by

text: picking

text: the

text: maximum

text: response

text: from

text: all

text: the

text: pose

text: estimation

text: since

text: it

text: has

text: been

text: proved

text: to

text: be

text: an

text: effective

text: ap-

text: child

text: nodes.

text: This

text: method

text: causes

text: the

text: double

text: counting

text: problem.

text: proach

text: for

text: the

text: nonlinear

text: to

text: pose

text: mapping

text: problem

text: Since

text: each

text: body

text: part

text: is

text: ﬁxed

text: only

text: considering

text: the

text: response

text: of

text: Gaussian

text: Process

text: Regression

text: (GPR)

text: is

text: considered

text: the

text: test

text: image

text: with

text: the

text: trained

text: templates,

text: when

text: sibling

text: body

text: a

text: model-free

text: framework.

text: GPR

text: deﬁned

text: as

text: a

text: distribution

text: over

text: parts

text: (the

text: same

text: body

text: part,

text: but

text: on

text: different

text: sides

text: of

text: the

text: human

text: functions,

text: extends

text: statistics

text: from

text: data

text: points

text: to

text: functions.

text: With

text: body,

text: like

text: a

text: left

text: hand

text: and

text: a

text: right

text: hand)

text: resemble

text: each

text: other,

text: kernel

text: trick,

text: we

text: can

text: even

text: get

text: rid

text: of

text: the

text: function

text: deﬁnition,

text: and

text: the

text: same

text: image

text: patch

text: might

text: be

text: picked

text: repetitively.

text: In

text: this

text: case,

text: only

text: concentrate

text: on

text: kernel

text: matrix

text: instead.

text: Once

text: we

text: normalize

text: the

text: estimated

text: pose

text: is

text: occluded

text: while

text: in

text: the

text: real

text: case

text: it

text: is

text: not.

text: the

text: training

text: input

text: to

text: have

text: a

text: zero

text: mean,

text: we

text: only

text: need

text: to

text: To

text: solve

text: this

text: problem,

text: we

text: use

text: global

text: optimization

text: to

text: ﬁx

text: deﬁne

text: a

text: covariance

text: matrix,

text: that

text: is

text: ,the

text: kernel

text: matrix,

text: for

text: each

text: body

text: part

text: positions.

text: In

text: the

text: original

text: MoP

text: model,

text: where

text: GPR.

text: Frequently

text: used

text: covariance

text: matrices

text: include

text: squared

text: there

text: is

text: only

text: HOG

text: feature,

text: optimizing

text: body

text: part

text: position

text: is

text: exponential

text: covariance

text: function,

text: Mate´rn

text: covariance

text: function

text: very

text: time

text: consuming.

text: For

text: example,

text: if

text: the

text: model

text: uses

text: body

text: and

text: so

text: on.

text: In

text: the

text: following

text: subsections,

text: we

text: will

text: explain

text: parts,

text: and

text: each

text: body

text: parts

text: use

text: or

text: mixtures,

text: the

text: minimum

text: detailed

text: representations

text: and

text: settings

text: for

text: the

text: Gaussian

text: process

text: number

text: of

text: possible

text: combination

text: for

text: all

text: body

text: parts

text: is

text: This

text: regressor

text: used

text: here.

text: is

text: a

text: huge

text: amount

text: of

text: possible

text: guesses.

text: In

text: our

text: case,

text: where

text: we

text: consider

text: human

text: blob

text: as

text: another

text: feature

text: cues.

text: The

text: possible

text: According

text: to

text: Gaussian

text: process

text: is

text: deﬁned

text: as:

text: a

text: col-

text: number

text: of

text: mixtures

text: for

text: each

text: body

text: parts

text: is

text: great

text: reduced

text: due

text: to

text: lection

text: of

text: random

text: variables,

text: any

text: ﬁnite

text: number

text: of

text: which

text: have

text: the

text: constraint.

text: So

text: we

text: can

text: optimize

text: the

text: tree

text: structure

text: globally.

text: (consistent)

text: joint

text: Gaussian

text: distribution.

text: A

text: Gaussian

text: process

text: The

text: body

text: part

text: positions

text: are

text: optimized

text: to

text: maximize

text: the

text: score

text: is

text: completely

text: speciﬁed

text: by

text: its

text: mean

text: function

text: and

text: a

text: covariance

text: the

text: proposed

text: model

text: combining

text: multiple

text: cues

text: SMC:

text: function.

text: Integrating

text: with

text: our

text: problem,

text: we

text: denote

text: the

text: mean

text: function

text: as

text: m(s)

text: and

text: the

text: covariance

text: function

text: as

text: k(s,

text: s

text: so

text: a

text: ωiti

text: φ(I,pi)

text: F(pi)

text: ωitji,tj

text: ψ(pi

text: pj)

text: F(pi)

text: F(pj),

text: Gaussian

text: process

text: is

text: represented

text: as:

text: i

text: V

text: ij

text: E

text: GP

text: ζ(s)

text: j(m(s),k(s,s

text: where

text: F(p

text: is

text: the

text: ratio

text: of

text: overlap

text: between

text: the

text: body

text: part

text: i

text: p

text: and

text: the

text: foreground

text: model

text: deﬁne

text: in

text: i

text: where

text: overlap

text: ratio

text: bbsize

text: m(s)

text: E[ζ(s)],

text: k(s,s

text: E[(ζ(s)

text: m(s))(ζ(s

text: m(s

text: We

text: assume

text: prediction

text: noise

text: as

text: a

text: Gaussian

text: distribution

text: and

text: formulate

text: ﬁnding

text: the

text: optimal

text: hyperparameters

text: as

text: an

text: optimiza-

text: tion

text: problem.

text: We

text: seek

text: the

text: optimal

text: solution

text: of

text: hyperparameters

text: by

text: maximizing

text: the

text: log

text: marginal

text: likelihood

text: (see

text: for

text: Fig.

text: Detected

text: body

text: part

text: localization

text: with

text: the

text: Mixture

text: of

text: Parts

text: (MoP)

text: details):

text: model.

text: The

text: left

text: ﬁgure

text: shows

text: the

text: localizations

text: of

text: all

text: body

text: parts

text: with

text: MoP.

text: The

text: right

text: ﬁgure

text: shows

text: the

text: overlap

text: between

text: extracted

text: human

text: blobs

text: and

text: all

text: bounding

text: boxes.

text: Color

text: variations

text: in

text: human

text: blobs

text: denotes

text: different

text: number

text: log

text: log

text: |K

text: n

text: log

text: of

text: bounding

text: boxes

text: that

text: are

text: overlapped.

text: Ψ

text: Ψ

text: where

text: K

text: is

text: the

text: calculated

text: covariance

text: matrix

text: of

text: the

text: target

text: Ψ

text: vector.

text: overlap

text: ratio

text: bbsize

text: With

text: the

text: optimal

text: hyperparameters,

text: the

text: prediction

text: distribu-

text: tion

text: is

text: represented

text: as:

text: N

text: Ψ

text: s

text: ,s,Ψ

text: (k(s

text: ,s)T[K

text: σ2

text: I]

text: 1Ψ

text: n−oise

text: k(s

text: ,s

text: σ2

text: k(s

text: ,s)T[K

text: σ2

text: I]

text: ,s)),

text: noise

text: noise

text: where

text: K

text: is

text: the

text: calculated

text: covariance

text: matrix

text: from

text: training

text: image

text: features

text: s

text: and

text: σ

text: is

text: the

text: covariance

text: of

text: Gaussian

text: Fig.

text: Body

text: part

text: localization

text: after

text: optimization

text: (left)

text: and

text: ﬁnal

text: overlap

text: map

text: noise

text: noise.

text: between

text: all

text: bounding

text: boxes

text: and

text: the

text: extracted

text: human

text: blob.

text: for

text: referencing

text: test

text: data

text: is

text: deducted

text: from

text: marginal

text: and

text: conditional

text: properties

text: of

text: Gaussian

text: distributions.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: z 16 4 2 Y 0 - 2 - 4 - - 4 2 0 2 4 | | | | x FIG. 13: The numerical error in BP (as seen in ∆B / B ) being generated at subdomain boundaries (signiﬁed by dense concentrations of black lines). The gray frames highlight the FIG. 14: Example congruence of geodesics in the equatorial locations of the error on such boundaries. Note that the warp- plane of the Schwarzschild black hole, with a nonvanishing P ing scale and the colour map are diﬀerent from Fig. 12, but impact parameter b translating into = 2/27M2 (not the the plane shown is the same as those appearing in Fig. 12. value we use for the simulations, but instead chosen to ac- centuate the perturbation). The red circle is the location of the event horizon. Angle Y is the angle at which the dashed geodesic strikes the origin. the initial data we see in Fig. 10 (b), which is absent from the analytical initial data used for the convergence tests. The fact that the convergence behaviour improves after modes). we apply spectral ﬁltering to the initial data before start- However, the wave component (the φ piece of the ing the evolutions provides evidence for this conclusion. 0 Faraday tensor) of the simple perturbation studied in In particular, the SpEC code utilizes a penalty method Sec. IV A still follows the GPNDs initially as we have [85–88] to enforce consistency across internal boundaries, retained the use of the Kinnersley tetrad when construct- which allows for discontinuities to exist temporarily. It ing the perturbed Faraday tensor (φ is by deﬁnition the has been noted in several previous studies [80, 91] that 0 wave propagating along the tetrad’s ingoing null direc- high-frequency noise tends to induce large discontinuities tion, which for the Kinnersley tetrad is in a doubly degen- at the internal boundaries, thereby creating errors in B. erate GPND direction) and is therefore (at least initially) The situation is the same for our FFE evolution system, not necessarily backscattered by the spacetime curvature as can be seen in Fig. 13. The high-frequency noise also (see Sec. V). In other words, the perturbation may be destroys the convergence at the highest resolutions (see restricted to a rather speciﬁc submanifold in the solu- Fig. 11), because lower resolution (smaller k) acts as an tion space. In principle, it is possible that even though eﬀective spectral ﬁlter, shielding the lower resolution sim- null+ solutions are stable on this submanifold, they may ulations from high-frequency noise. We expect this com- still be unstable under perturbations oﬀ of it. Although plication to disappear as we ascertain the source of the we would expect numerical truncation errors to induce high-frequency noise (one possibility is the boundary con- perturbations away from this submanifold regardless of dition of Φ = 0 being too simplistic) and further improve our procedure of solving Eq. (35). B. Perturbation in the propagation direction With numerical experiments, we can only state that there are no diverging modes being excited by the par- ticular perturbations that we introduce into the initial data. Therefore, in order to provide as strong evidence as possible for stability (meaning the nonexistence of di- verging modes in general), it is important that we con- sider a set of initial perturbations that is as general as possible. In the last section, although we started with speciﬁc modiﬁcations to φ , our initial data construction 0 procedure nevertheless has to go through the blending FIG. 15: Similar to Fig. 10, but for initial data with a per- and constraint solving stages, which means the resulting turbed propagation direction. Left: The right-hand side of Eq. (35). Right: The left-hand side of Eq. (35). perturbation is in fact rather general (containing many ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: z

label: paragraph

font: default

text: Y

label: caption

text: x

text: FIG.

text: The

text: numerical

text: error

text: in

text: BP

text: (as

text: seen

text: in

text: ∆B

text: B

text: being

text: generated

text: at

text: subdomain

text: boundaries

text: (signiﬁed

text: by

text: dense

text: concentrations

text: of

text: black

text: lines).

text: The

text: gray

text: frames

text: highlight

text: the

text: FIG.

text: Example

text: congruence

text: of

text: geodesics

text: in

text: the

text: equatorial

text: locations

text: of

text: the

text: error

text: on

text: such

text: boundaries.

text: Note

text: that

text: the

text: warp-

text: plane

text: of

text: the

text: Schwarzschild

text: black

text: hole,

text: with

text: a

text: nonvanishing

text: P

text: ing

text: scale

text: and

text: the

text: colour

text: map

text: are

text: diﬀerent

text: from

text: Fig.

text: but

text: impact

text: parameter

text: b

text: translating

text: into

text: (not

text: the

text: the

text: plane

text: shown

text: is

text: the

text: same

text: as

text: those

text: appearing

text: in

text: Fig.

text: value

text: we

text: use

text: for

text: the

text: simulations,

text: but

text: instead

text: chosen

text: to

text: ac-

text: centuate

text: the

text: perturbation).

text: The

text: red

text: circle

text: is

text: the

text: location

text: of

text: the

text: event

text: horizon.

text: Angle

text: Y

text: is

text: the

text: angle

text: at

text: which

text: the

text: dashed

text: geodesic

text: strikes

text: the

text: origin.

text: the

text: initial

text: data

text: we

text: see

text: in

text: Fig.

text: (b),

text: which

text: is

text: absent

text: from

text: the

text: analytical

text: initial

text: data

text: used

text: for

text: the

text: convergence

text: tests.

text: The

text: fact

text: that

text: the

text: convergence

text: behaviour

text: improves

text: after

text: modes).

text: we

text: apply

text: spectral

text: ﬁltering

text: to

text: the

text: initial

text: data

text: before

text: start-

text: However,

text: the

text: wave

text: component

text: (the

text: φ

text: piece

text: of

text: the

text: ing

text: the

text: evolutions

text: provides

text: evidence

text: for

text: this

text: conclusion.

text: Faraday

text: tensor)

text: of

text: the

text: simple

text: perturbation

text: studied

text: in

text: In

text: particular,

text: the

text: SpEC

text: code

text: utilizes

text: a

text: penalty

text: method

text: Sec.

text: IV

text: A

text: still

text: follows

text: the

text: GPNDs

text: initially

text: as

text: we

text: have

text: to

text: enforce

text: consistency

text: across

text: internal

text: boundaries,

text: retained

text: the

text: use

text: of

text: the

text: Kinnersley

text: tetrad

text: when

text: construct-

text: which

text: allows

text: for

text: discontinuities

text: to

text: exist

text: temporarily.

text: It

text: ing

text: the

text: perturbed

text: Faraday

text: tensor

text: (φ

text: is

text: by

text: deﬁnition

text: the

text: has

text: been

text: noted

text: in

text: several

text: previous

text: studies

text: that

text: wave

text: propagating

text: along

text: the

text: tetrad’s

text: ingoing

text: direc-

text: high-frequency

text: noise

text: tends

text: to

text: induce

text: large

text: discontinuities

text: tion,

text: which

text: for

text: the

text: Kinnersley

text: tetrad

text: is

text: in

text: a

text: doubly

text: degen-

text: at

text: the

text: internal

text: boundaries,

text: thereby

text: creating

text: errors

text: in

text: B.

text: erate

text: GPND

text: direction)

text: and

text: is

text: therefore

text: (at

text: least

text: initially)

text: The

text: situation

text: is

text: the

text: same

text: for

text: our

text: FFE

text: evolution

text: system,

text: not

text: necessarily

text: backscattered

text: by

text: the

text: spacetime

text: curvature

text: as

text: can

text: be

text: seen

text: in

text: Fig.

text: The

text: high-frequency

text: noise

text: also

text: (see

text: Sec.

text: V).

text: In

text: other

text: words,

text: the

text: perturbation

text: may

text: be

text: destroys

text: the

text: convergence

text: at

text: the

text: highest

text: resolutions

text: (see

text: restricted

text: to

text: a

text: rather

text: speciﬁc

text: submanifold

text: in

text: the

text: solu-

text: Fig.

text: because

text: lower

text: resolution

text: (smaller

text: k)

text: acts

text: as

text: an

text: tion

text: space.

text: In

text: principle,

text: it

text: is

text: possible

text: that

text: even

text: though

text: eﬀective

text: spectral

text: ﬁlter,

text: shielding

text: the

text: lower

text: resolution

text: sim-

text: null+

text: solutions

text: are

text: stable

text: on

text: this

text: submanifold,

text: they

text: may

text: ulations

text: from

text: high-frequency

text: noise.

text: We

text: expect

text: this

text: com-

text: still

text: be

text: unstable

text: under

text: perturbations

text: oﬀ

text: of

text: it.

text: Although

text: plication

text: to

text: disappear

text: as

text: we

text: ascertain

text: the

text: source

text: of

text: the

text: we

text: would

text: expect

text: numerical

text: truncation

text: errors

text: to

text: induce

text: high-frequency

text: noise

text: (one

text: possibility

text: is

text: the

text: boundary

text: con-

text: perturbations

text: away

text: from

text: this

text: submanifold

text: regardless

text: of

text: dition

text: of

text: Φ

text: being

text: too

text: simplistic)

text: and

text: further

text: improve

text: our

text: procedure

text: of

text: solving

text: Eq.

text: B.

text: Perturbation

text: in

text: the

text: propagation

text: direction

text: With

text: numerical

text: experiments,

text: we

text: can

text: only

text: state

text: that

text: there

text: are

text: no

text: diverging

text: modes

text: being

text: excited

text: by

text: the

text: par-

text: ticular

text: perturbations

text: that

text: we

text: introduce

text: into

text: the

text: initial

text: data.

text: Therefore,

text: in

text: order

text: to

text: provide

text: as

text: strong

text: evidence

text: as

text: possible

text: for

text: stability

text: (meaning

text: the

text: nonexistence

text: of

text: di-

text: verging

text: modes

text: in

text: general),

text: it

text: is

text: important

text: that

text: we

text: con-

text: sider

text: a

text: set

text: of

text: initial

text: perturbations

text: that

text: is

text: as

text: general

text: as

text: possible.

text: In

text: the

text: last

text: section,

text: although

text: we

text: started

text: with

text: speciﬁc

text: modiﬁcations

text: to

text: φ

text: our

text: initial

text: data

text: construction

text: procedure

text: nevertheless

text: has

text: to

text: go

text: through

text: the

text: blending

text: FIG.

text: Similar

text: to

text: Fig.

text: but

text: for

text: initial

text: data

text: with

text: a

text: per-

text: and

text: constraint

text: solving

text: stages,

text: which

text: means

text: the

text: resulting

text: turbed

text: propagation

text: direction.

text: Left:

text: The

text: right-hand

text: side

text: of

text: Eq.

text: Right:

text: The

text: left-hand

text: side

text: of

text: Eq.

text: perturbation

text: is

text: in

text: fact

text: rather

text: general

text: (containing

text: many

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text: 27 × × · −Bj × (E B)j (E nˆ)j E B (A )j = (E nˆ) + nˆ + α B + α (cid:15)ji nˆ , (B13) EE k B2 k B2 k 1 B2 k 2 B2 k i × · − Bj × (E nˆ)j − E B × (A )j = (cid:15)ji nˆ + (B nˆ) + α E 2α (E nˆ)jB , (B14) EB k k i B2 k 1 B2 k 3 B4 k × (B nˆ)j (A )j = (cid:15)ji nˆ + α B , (B15) BE k k i 5 B2 k × × · · (E B)j (B nˆ)j E B − E B × (A )j = α nˆ + α E + α (cid:15)ji nˆ 2α (B nˆ)jB . (B16) BB k 4 B2 k 5 B2 k 6 B2 k i 7 B4 k Multiplying with symmetrizer S as given by Eq. (B8) with ζ = 1/2 and an extra overall factor of 2 for convenience, we obtain the components in Eq. (B10): − × × − × · × × (SA ) = B (E nˆ) + 2(E B) nˆ 2(E nˆ) B + B (E nˆ)B B + B (E nˆ) EE ik i k · ×i k i k i k i k − × E (B nˆ) 2E (B nˆ) + B B , (B17) i k B2 i k · · × − × × × − E B × − B (E nˆ) (SA ) = E (B nˆ) + 2B (E nˆ) + E (B nˆ) 2 B (B nˆ) E E BB ik i k i k j k B2 i k B2 i k · · × E B · × × × − E (B nˆ) + B (E nˆ)E B + 2(E B) n + 2(B nˆ) E E E B4 i k i k i k B2 i k · · · × − E B × E B E (B nˆ) 2 (B nˆ) B + E B , (B18) B2 i k B2 B2 i k · − × E B × × (SA ) = 2B2(cid:15) nˆl 2(E nˆ) E + 2 (E nˆ) B + 2B (B nˆ) , (B19) EB ik ikl i k B2 i k i k · − − × E B × × (SA ) = 2B2(cid:15) nˆl 2(E nˆ) E + 2 (E nˆ) B + 2B (B nˆ) , (B20) BE ik ikl k i B2 k i k i (B21) − where we have specialized to the AU2 system of Eq. (B4). We then have (SA ) (SA ) = 0, and it is EB ik BE ki straightforward to show that the antisymmetric part of the diagonal blocks are × × × × × × (SA ) (cid:15)jik = (SA ) (cid:15)jik = 2(B (E nˆ) + nˆ (B E) + E (nˆ B))j = 0 (B22) EE ik BB ik For completeness, we explicitly write out the charac- right eigenvectors are teristic modes and speeds for the AU2 system. The right − × × eα = ((cid:16) PE + nˆ B, PB + nˆ E) , (B30) eigenvalues satisfying the equation ˆ1 − − × − × eα = ( PE nˆ B, PB nˆ E) , (B31) ˆ2 (cid:17) − × − eα = PB + ν nˆ E + (1 ν2 )B, (niAi)αβeβαˆ = ναˆeααˆ (B23) ˆ3,ˆ4 (cid:0) − − ˆ3,ˆ4 × ˆ3,ˆ4 (cid:1) PE ν nˆ B , (B32) ˆ3,ˆ4 · − · − × eα = E(B nˆ) B(E nˆ), B2nˆ E Bν ,(B33) are ˆ5 eα = (B, 0) . (B34) ˆ6 − We note that the unphysical modes for the AU2 system as ν = 1 , (B24) ˆ1 given by Eqs. (B33) and (B34) are much less complicated νˆ2 = 1 ,− (B25) than for the AU system as given in Ref. [79]. This is ben- ν = ν ω , (B26) eﬁcial for inverting the characteristic modes in order to ˆ3 obtain the fundamental variables E and B, which is nec- ν = ν + ω , (B27) ˆ4 essary for some pseudospectral implementations such as ν = ν , (B28) ˆ5 ours, where both the internal (between the adjacent sub- νˆ6 = 2ν , (B29) domains) [85–88] and the external boundary conditions [116–118] are imposed on the characteristic modes, and so need to be translated into the fundamental evolution where ν and ω are given in Eqs. (A11) and (A12). The variables before they become useful. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −Bj

text: (E

text: B)j

text: (E

text: nˆ)j

text: E

text: B

text: (A

text: )j

text: (E

text: nˆ)

text: nˆ

text: α

text: B

text: α

text: nˆ

text: EE

text: k

text: k

text: k

text: k

text: k

text: i

text: Bj

text: (E

text: nˆ)j

text: E

text: B

text: (A

text: )j

text: nˆ

text: (B

text: nˆ)

text: α

text: E

text: 2α

text: (E

text: nˆ)jB

text: EB

text: k

text: k

text: i

text: k

text: k

text: k

text: (B

text: nˆ)j

text: (A

text: )j

text: nˆ

text: α

text: B

text: BE

text: k

text: k

text: i

text: k

text: (E

text: B)j

text: (B

text: nˆ)j

text: E

text: B

text: E

text: B

text: (A

text: )j

text: α

text: nˆ

text: α

text: E

text: α

text: nˆ

text: 2α

text: (B

text: nˆ)jB

text: BB

text: k

text: k

text: k

text: k

text: i

text: k

text: Multiplying

text: with

text: symmetrizer

text: S

text: as

text: given

text: by

text: Eq.

text: with

text: ζ

text: and

text: an

text: extra

text: overall

text: factor

text: of

text: for

text: convenience,

text: we

text: obtain

text: the

text: components

text: in

text: Eq.

text: (SA

text: B

text: (E

text: nˆ)

text: B)

text: nˆ

text: nˆ)

text: B

text: B

text: (E

text: nˆ)B

text: B

text: B

text: (E

text: nˆ)

text: EE

text: ik

text: i

text: k

text: ×i

text: k

text: i

text: k

text: i

text: k

text: i

text: k

text: E

text: (B

text: nˆ)

text: (B

text: nˆ)

text: B

text: B

text: i

text: k

text: i

text: k

text: E

text: B

text: B

text: (E

text: nˆ)

text: (SA

text: E

text: (B

text: nˆ)

text: (E

text: nˆ)

text: E

text: (B

text: nˆ)

text: B

text: (B

text: nˆ)

text: E

text: E

text: BB

text: ik

text: i

text: k

text: i

text: k

text: j

text: k

text: i

text: k

text: i

text: k

text: E

text: B

text: E

text: (B

text: nˆ)

text: B

text: (E

text: nˆ)E

text: B

text: B)

text: n

text: nˆ)

text: E

text: E

text: E

text: i

text: k

text: i

text: k

text: i

text: k

text: i

text: k

text: E

text: B

text: E

text: B

text: E

text: (B

text: nˆ)

text: (B

text: nˆ)

text: B

text: E

text: B

text: i

text: k

text: i

text: k

text: E

text: B

text: (SA

text: nˆl

text: nˆ)

text: E

text: (E

text: nˆ)

text: B

text: (B

text: nˆ)

text: EB

text: ik

text: ikl

text: i

text: k

text: i

text: k

text: i

text: k

text: E

text: B

text: (SA

text: nˆl

text: nˆ)

text: E

text: (E

text: nˆ)

text: B

text: (B

text: nˆ)

text: BE

text: ik

text: ikl

text: k

text: i

text: k

text: i

text: k

text: i

text: where

text: we

text: have

text: specialized

text: to

text: the

text: system

text: of

text: Eq.

text: We

text: then

text: have

text: (SA

text: (SA

text: and

text: it

text: is

text: EB

text: ik

text: BE

text: ki

text: straightforward

text: to

text: show

text: that

text: the

text: antisymmetric

text: part

text: of

text: the

text: diagonal

text: blocks

text: are

text: (SA

text: (SA

text: (E

text: nˆ)

text: nˆ

text: (B

text: E)

text: E

text: (nˆ

text: B))j

text: EE

text: ik

text: BB

text: ik

text: For

text: completeness,

text: we

text: explicitly

text: write

text: out

text: the

text: charac-

text: right

text: eigenvectors

text: are

text: teristic

text: modes

text: and

text: speeds

text: for

text: the

text: system.

text: The

text: right

text: eα

text: PE

text: nˆ

text: B,

text: PB

text: nˆ

text: E)

text: eigenvalues

text: satisfying

text: the

text: equation

text: ˆ1

text: eα

text: PE

text: nˆ

text: B,

text: PB

text: nˆ

text: E)

text: ˆ2

text: eα

text: PB

text: ν

text: nˆ

text: E

text: ν2

text: )B,

text: (niAi)αβeβαˆ

text: ναˆeααˆ

text: PE

text: ν

text: nˆ

text: B

text: eα

text: E(B

text: nˆ)

text: B(E

text: nˆ),

text: E

text: Bν

text: are

text: ˆ5

text: eα

text: (B,

text: ˆ6

text: We

text: note

text: that

text: the

text: unphysical

text: modes

text: for

text: the

text: system

text: as

text: ν

text: ˆ1

text: given

text: by

text: Eqs.

text: and

text: are

text: much

text: less

text: complicated

text: νˆ2

text: than

text: for

text: the

text: AU

text: system

text: as

text: given

text: in

text: Ref.

text: This

text: is

text: ben-

text: ν

text: ν

text: ω

text: eﬁcial

text: for

text: inverting

text: the

text: characteristic

text: modes

text: in

text: order

text: to

text: ˆ3

text: obtain

text: the

text: fundamental

text: variables

text: E

text: and

text: B,

text: which

text: is

text: nec-

text: ν

text: ν

text: ω

text: ˆ4

text: essary

text: for

text: some

text: pseudospectral

text: implementations

text: such

text: as

text: ν

text: ν

text: ˆ5

text: ours,

text: where

text: both

text: the

text: internal

text: (between

text: the

text: adjacent

text: sub-

text: νˆ6

text: 2ν

text: domains)

text: and

text: the

text: external

text: boundary

text: conditions

text: are

text: imposed

text: on

text: the

text: characteristic

text: modes,

text: and

text: so

text: need

text: to

text: be

text: translated

text: into

text: the

text: fundamental

text: evolution

text: where

text: ν

text: and

text: ω

text: are

text: given

text: in

text: Eqs.

text: and

text: The

text: variables

text: before

text: they

text: become

text: useful.

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text: 4 product between Eq. (10) and B which gives us form to Ref. [27]. Speciﬁcally, we add a damping term − · × Nδ(E B)B/B2 to ∂ E, so that the full set of evolution t · B E B equations becomes 4πNJ = 4πN(B J) + 4πNρ , (11) B2 B2 × − L ∇ × − E B ∇ · where the second term above can be seen as the charge (∂ )E = NKE + (NB) N E t β B2 density moving∇at·the plasma drift velocity, and we can − B · ∇ × − · ∇ × (cid:1) replace ρ with E/4π using one of the Maxwell co·n- N (B B E E straint(cid:112)equations. To further work out the current B J − B2 · · along the B ﬁeld, we note that 2KijEiBj + 2KE B + δE B (,15) − L − ∇ × (B / B Bb)FabJ(4) = 0 ⇒ E · B = 0 or ρ = 0 ,(12) (∂t β)B = NKB (NE) . (16) a b b The damping term is proportional to the constraint, and and for nonvacuum solutions (vacuum here refers to will not aﬀect the physical constraint-satisfying solutions. J(4) = 0, with these solutions satisfying the force-free · However, it modiﬁes the constraint evolution equation to condition trivially) we would like to enforce the E B = 0 a damped form condition, which should be preserved along the timelike normal to the spatial hypersurfaces, and so − L · − · (∂ )E B = δNE B . (17) t β − L · (∂ )E B = 0 . (13) t β We note that our damping strategy diﬀers from that of Using the deﬁnition of the extrinsic curvature tensor Ref. [27], in that we have kept the original current terms Eq. (2), and substituting in the Maxwell equations, we from Eq. (14) and treated the new damping term as an · obtain an equation for B J that reads addition instead of a replacement. In contrast, Ref. [27] removed all of the original current terms from the evo- · − · ∇ × · ∇ × 4πNB J = E (NE) + B (NB) lution equations, replacing them with only the damping − · 2NK EiBj + 2NKE B , (14) term. With their strategy, the evolution equations be- ij come simpler, but those current terms forcibly removed (note that the extrinsic curvature terms on the second from the evolution equations will resurface in the con- line appear to be missing in some of the existing liter- straint evolution equation, speciﬁcally on the right-hand ature). Substituting Eq. (14) into Eq. (11) yields J in side of Eq. (17). Therefore Eq. (17) won’t reduce to − L · · terms of E and B. Substituting this expression back into (∂ )E B = 0 when E B = 0, so that constraint- t β the Maxwell-equations yields the desired minimalist FFE satisfying FFE solutions at some instance of time can not evolution system. stay constraint-satisfying as the simulation progresses. There is also a set of constraints that needs to be satis- However, this does not invalidate the results of Ref. [27], ﬁed, which comes from both the Maxwell equations and as the size of the constraint violation would be negatively the force-free condition. When deriving the current J, correlated with δ (and positively correlated with the mag- ∇ · we have explicitly used q = E/4π as the deﬁnition of nitudes of the derivatives of B and E), as the damping · charge density, so there is no need to enforce this con- will be activated when E B grows too large. Indeed, ∇ · straint. The nontrivial constraints are B = 0 and the Ref. [27] adopted a δ greater than the inverse of the time · · force-free constraint E B = 0. These two constraints step size and observed a well controlled E B. Such a are preserved automatically by the evolution equations strategy however leads to a stiﬀ term in the evolution ∇ · [31]. Speciﬁcally, the B = 0 constraint is preserved equations that has to be treated with an implicit-explicit by the original Maxwell equations and inherited by the (IMEX) evolution scheme [27]. As the analytical solu- · force-free specialization. The E B = 0 condition is tions we examine are exactly constraint-satisfying at all also preserved, as we have explicitly used the condition times, and since we do not use an IMEX scheme, we will − L · (∂ )E B = 0 to derive the current. Physically, this use the damping term as an addition with a moderate t β condition ﬁxes the magnitude of the conduction current coeﬃcient δ = 100, allowing us to enjoy its constraint along the B direction [17], which would short out the E cleaning beneﬁts but avoid the aforementioned compli- ﬁeld along B by redistributing charge to eliminate the cations. Finally, note that this damping term does not potential diﬀerence associated with that E component, introduce magnetic monopoles, see Eq. (B51). · thus enforcing E B = 0. We note that the introduction of the additional damp- ∇ · · Although the B = 0 and E B = 0 constraints ing term replaces another alternative constraint cleaning are preserved by the evolution equations when they were strategy of removing the component of E along B after satisﬁed initially, numerical noise inevitably creates some taking each time step, which has been widely utilized seed constraint violation that may grow further under (e.g. Refs. [13, 15, 17, 77]). Such an alteration of the minimal evolution system. It is therefore beneﬁcial the evolving ﬁelds at a discrete set of times (which de- to modify the evolution equations so as to be able to pend on resolution) will result in a failure of the system clean up the constraint violations as they emerge. For to achieve the usual convergence behaviour that would · the E B = 0 constraint, we adopt a strategy similar in be expected if the scheme were just applied to a set of ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: product

text: between

text: Eq.

text: and

text: B

text: which

text: gives

text: us

text: form

text: to

text: Ref.

text: Speciﬁcally,

text: we

text: add

text: a

text: damping

text: term

text: Nδ(E

text: to

text: E,

text: so

text: that

text: the

text: full

text: set

text: of

text: evolution

text: t

text: B

text: E

text: B

text: equations

text: becomes

text: J)

text: 4πNρ

text: L

text: E

text: B

text: where

text: the

text: second

text: term

text: above

text: can

text: be

text: seen

text: as

text: the

text: charge

text: )E

text: NKE

text: (NB)

text: N

text: E

text: t

text: β

text: density

text: moving∇at·the

text: plasma

text: drift

text: velocity,

text: and

text: we

text: can

text: B

text: replace

text: ρ

text: with

text: using

text: one

text: of

text: the

text: Maxwell

text: co·n-

text: N

text: (B

text: B

text: E

text: E

text: To

text: further

text: work

text: out

text: the

text: current

text: B

text: J

text: along

text: the

text: B

text: ﬁeld,

text: we

text: note

text: that

text: B

text: δE

text: B

text: L

text: (B

text: B

text: E

text: B

text: or

text: ρ

text: (∂t

text: β)B

text: NKB

text: (NE)

text: a

text: b

text: b

text: The

text: damping

text: term

text: is

text: proportional

text: to

text: the

text: constraint,

text: and

text: and

text: for

text: nonvacuum

text: solutions

text: (vacuum

text: here

text: refers

text: to

text: will

text: not

text: aﬀect

text: the

text: physical

text: constraint-satisfying

text: solutions.

text: with

text: these

text: solutions

text: satisfying

text: the

text: force-free

text: However,

text: it

text: modiﬁes

text: the

text: constraint

text: evolution

text: equation

text: to

text: condition

text: trivially)

text: we

text: would

text: like

text: to

text: enforce

text: the

text: E

text: B

text: a

text: damped

text: form

text: condition,

text: which

text: should

text: be

text: preserved

text: along

text: the

text: timelike

text: normal

text: to

text: the

text: spatial

text: hypersurfaces,

text: and

text: so

text: L

text: )E

text: B

text: δNE

text: B

text: t

text: β

text: L

text: )E

text: B

text: t

text: β

text: We

text: note

text: that

text: our

text: damping

text: strategy

text: diﬀers

text: from

text: that

text: of

text: Using

text: the

text: deﬁnition

text: of

text: the

text: extrinsic

text: curvature

text: tensor

text: Ref.

text: in

text: that

text: we

text: have

text: kept

text: the

text: original

text: current

text: terms

text: Eq.

text: and

text: substituting

text: in

text: the

text: Maxwell

text: equations,

text: we

text: from

text: Eq.

text: and

text: treated

text: the

text: new

text: damping

text: term

text: as

text: an

text: obtain

text: an

text: equation

text: for

text: B

text: J

text: that

text: reads

text: addition

text: instead

text: of

text: a

text: replacement.

text: In

text: contrast,

text: Ref.

text: removed

text: all

text: of

text: the

text: original

text: current

text: terms

text: from

text: the

text: evo-

text: J

text: E

text: (NE)

text: B

text: (NB)

text: lution

text: equations,

text: replacing

text: them

text: with

text: only

text: the

text: damping

text: EiBj

text: B

text: term.

text: With

text: their

text: strategy,

text: the

text: evolution

text: equations

text: be-

text: ij

text: come

text: simpler,

text: but

text: those

text: current

text: terms

text: forcibly

text: removed

text: (note

text: that

text: the

text: extrinsic

text: curvature

text: terms

text: on

text: the

text: second

text: from

text: the

text: evolution

text: equations

text: will

text: resurface

text: in

text: the

text: con-

text: line

text: appear

text: to

text: be

text: missing

text: in

text: some

text: of

text: the

text: existing

text: liter-

text: straint

text: evolution

text: equation,

text: speciﬁcally

text: on

text: the

text: right-hand

text: ature).

text: Substituting

text: Eq.

text: into

text: Eq.

text: yields

text: J

text: in

text: side

text: of

text: Eq.

text: Therefore

text: Eq.

text: won’t

text: reduce

text: to

text: L

text: terms

text: of

text: E

text: and

text: B.

text: Substituting

text: this

text: expression

text: back

text: into

text: )E

text: B

text: when

text: E

text: B

text: so

text: that

text: constraint-

text: t

text: β

text: the

text: Maxwell-equations

text: yields

text: the

text: desired

text: minimalist

text: FFE

text: satisfying

text: FFE

text: solutions

text: at

text: some

text: instance

text: of

text: time

text: can

text: not

text: evolution

text: system.

text: stay

text: constraint-satisfying

text: as

text: the

text: simulation

text: progresses.

text: There

text: is

text: also

text: a

text: set

text: of

text: constraints

text: that

text: needs

text: to

text: be

text: satis-

text: However,

text: this

text: does

text: not

text: invalidate

text: the

text: results

text: of

text: Ref.

text: ﬁed,

text: which

text: comes

text: from

text: both

text: the

text: Maxwell

text: equations

text: and

text: as

text: the

text: size

text: of

text: the

text: constraint

text: violation

text: would

text: be

text: negatively

text: the

text: force-free

text: condition.

text: When

text: deriving

text: the

text: current

text: J,

text: correlated

text: with

text: δ

text: (and

text: positively

text: correlated

text: with

text: the

text: mag-

text: we

text: have

text: explicitly

text: used

text: q

text: as

text: the

text: deﬁnition

text: of

text: nitudes

text: of

text: the

text: derivatives

text: of

text: B

text: and

text: E),

text: as

text: the

text: damping

text: charge

text: density,

text: so

text: there

text: is

text: no

text: need

text: to

text: enforce

text: this

text: con-

text: will

text: be

text: activated

text: when

text: E

text: B

text: grows

text: too

text: large.

text: Indeed,

text: straint.

text: The

text: nontrivial

text: constraints

text: are

text: B

text: and

text: the

text: Ref.

text: adopted

text: a

text: δ

text: greater

text: than

text: the

text: inverse

text: of

text: the

text: time

text: force-free

text: constraint

text: E

text: B

text: These

text: two

text: constraints

text: step

text: size

text: and

text: observed

text: a

text: well

text: controlled

text: E

text: B.

text: Such

text: a

text: are

text: preserved

text: automatically

text: by

text: the

text: evolution

text: equations

text: strategy

text: however

text: leads

text: to

text: a

text: stiﬀ

text: term

text: in

text: the

text: evolution

text: Speciﬁcally,

text: the

text: B

text: constraint

text: is

text: preserved

text: equations

text: that

text: has

text: to

text: be

text: treated

text: with

text: an

text: implicit-explicit

text: by

text: the

text: original

text: Maxwell

text: equations

text: and

text: inherited

text: by

text: the

text: (IMEX)

text: evolution

text: scheme

text: As

text: the

text: analytical

text: solu-

text: force-free

text: specialization.

text: The

text: E

text: B

text: condition

text: is

text: tions

text: we

text: examine

text: are

text: exactly

text: constraint-satisfying

text: at

text: all

text: also

text: preserved,

text: as

text: we

text: have

text: explicitly

text: used

text: the

text: condition

text: times,

text: and

text: since

text: we

text: do

text: not

text: use

text: an

text: IMEX

text: scheme,

text: we

text: will

text: L

text: )E

text: B

text: to

text: derive

text: the

text: current.

text: Physically,

text: this

text: use

text: the

text: damping

text: term

text: as

text: an

text: addition

text: with

text: a

text: moderate

text: t

text: β

text: condition

text: ﬁxes

text: the

text: magnitude

text: of

text: the

text: conduction

text: current

text: coeﬃcient

text: δ

text: allowing

text: us

text: to

text: enjoy

text: its

text: constraint

text: along

text: the

text: B

text: direction

text: which

text: would

text: short

text: out

text: the

text: E

text: cleaning

text: beneﬁts

text: but

text: avoid

text: the

text: aforementioned

text: compli-

text: ﬁeld

text: along

text: B

text: by

text: redistributing

text: charge

text: to

text: eliminate

text: the

text: cations.

text: Finally,

text: note

text: that

text: this

text: damping

text: term

text: does

text: not

text: potential

text: diﬀerence

text: associated

text: with

text: that

text: E

text: component,

text: introduce

text: magnetic

text: monopoles,

text: see

text: Eq.

text: thus

text: enforcing

text: E

text: B

text: We

text: note

text: that

text: the

text: introduction

text: of

text: the

text: additional

text: damp-

text: Although

text: the

text: B

text: and

text: E

text: B

text: constraints

text: ing

text: term

text: replaces

text: another

text: alternative

text: constraint

text: cleaning

text: are

text: preserved

text: by

text: the

text: evolution

text: equations

text: when

text: they

text: were

text: strategy

text: of

text: removing

text: the

text: component

text: of

text: E

text: along

text: B

text: after

text: satisﬁed

text: initially,

text: numerical

text: noise

text: inevitably

text: creates

text: some

text: taking

text: each

text: time

text: step,

text: which

text: has

text: been

text: widely

text: utilized

text: seed

text: constraint

text: violation

text: that

text: may

text: grow

text: further

text: under

text: (e.g.

text: Refs.

text: Such

text: an

text: alteration

text: of

text: the

text: minimal

text: evolution

text: system.

text: It

text: is

text: therefore

text: beneﬁcial

text: the

text: evolving

text: ﬁelds

text: at

text: a

text: discrete

text: set

text: of

text: times

text: (which

text: de-

text: to

text: modify

text: the

text: evolution

text: equations

text: so

text: as

text: to

text: be

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text: to

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text: on

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text: will

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text: For

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text: E

text: B

text: constraint,

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text: adopt

text: a

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text: be

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text: if

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text: scheme

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text: applied

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text: ##LTLine##

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text: 31 whereby the decreasing number of times that øA ap- Ψ0 = Cabcdlamblcmd , (E12a) pears in the deﬁnitions establishes the hierarchy of decay Ψ = C lanblcmd , (E12b) rates of these scalars under the peeling theorem [80, 97], 1 abcd a feature not as visible in the tensorial expressions in Ψ = C lambm¯ cnd , (E12c) 2 abcd Eq. (E12). Lastly, to carry out some calculations in the Ψ = C lanbm¯ cnd , (E12d) main text, we note the fact that under the dyad basis, 3 abcd (cid:15) can be written as Ψ = C nam¯ bncm¯ d , (E12e) AB 4 abcd − (cid:15) = ø ι ι ø . (E14) AB A B A B can be rewritten as Ψ = Ψ øAøBøCøD , (E13a) 0 ABCD Ψ = Ψ øAøBøCιD , (E13b) 1 ABCD Ψ = Ψ øAøBιCιD , (E13c) 2 ABCD Ψ = Ψ øAιBιCιD , (E13d) 3 ABCD Ψ = Ψ ιAιBιCιD , (E13e) 4 ABCD [1] R. D. Blandford and R. L. Znajek, Mon. Not. Roy. As- ph/0408318. tron. Soc. 179, 433 (1977). [21] J. C. McKinney, Mon.Not.Roy.Astron.Soc. 367, 1797 [2] P. Goldreich and W. H. Julian, Astrophys.J. 157, 869 (2006), astro-ph/0601410. (1969). [22] C. Yu, Mon.Not.Roy.Astron.Soc. 411, 2461 (2011), [3] A. Gruzinov (2007), 0710.1875. 1010.3592. [4] M. Ruderman and P. Sutherland, Astrophys.J. 196, 51 [23] I. Contopoulos, D. Kazanas, and C. Fendt, Astrophys.J. (1975). 511, 351 (1999), astro-ph/9903049. [5] D. MacDonald and K. Thorne, [24] D. A. Uzdensky, Astrophys.J. 620, 889 (2005), astro- Mon.Not.Roy.Astron.Soc. 198, 345 (1982). ph/0410715. [6] R. D. Blandford, in Lighthouses of the Universe: The [25] A. N. Timokhin, Astrophys.Space Sci. 308, 575 (2007), Most Luminous Celestial Objects and Their Use for astro-ph/0607165. Cosmology: Proceedings of the MPA/ESO/MPE/USM [26] C. Palenzuela, L. Lehner, and S. L. Liebling, Science Joint Astronomy Conference Held in Garching, edited 329, 927 (2010), 1005.1067. by M. Gilfanov, R. Sunyaev, and E. Churazov (2002), [27] D. Alic, P. Mosta, L. Rezzolla, O. Zanotti, and J. L. p. 381. Jaramillo, Astrophys.J. 754, 36 (2012), 1204.2226. [7] R. D. Blandford, mnras 176, 465 (1976). [28] C. Kalapotharakos, I. Contopoulos, and D. Kazanas, [8] P. Meszaros and M. Rees, Astrophys.J. 482, L29 (1997), Mon.Not.Roy.Astron.Soc. 420, 2793–2798 (2012), astro-ph/9609065. 1109.5122. [9] G. Menon and C. D. Dermer (2011), 1105.4139. [29] F. C. Michel, Physical Review Letters 23, 247 (1969). [10] F. C. Michel, Astrophys.J. 180, L133 (1973). [30] T. D. Brennan and S. E. Gralla (2013), 1311.0752. [11] M. Lyutikov, Phys.Rev. D83, 124035 (2011), 1104.1091. [31] S. E. Gralla and T. Jacobson (2014), 1401.6159. [12] T. D. Brennan, S. E. Gralla, and T. Jacobson, [32] A. Lupsasca, M. J. Rodriguez, and A. Strominger Class.Quant.Grav. 30, 195012 (2013), 1305.6890. (2014), 1406.4133. [13] A. Spitkovsky, Astrophys.J. 648, L51 (2006), astro- [33] A. Lupsasca and M. J. Rodriguez (2014), 1412.4124. ph/0603147. [34] F. Zhang, H. Yang, and L. Lehner, Phys.Rev. D90, [14] C. Kalapotharakos and I. Contopoulos, As- 124009 (2014), 1409.0345. tron.Astrophys. 496, 495 (2009), 0811.2863. [35] H. Li, C. Yu, J. Wang, and Z. Xu (2014), 1403.6959. [15] C. Palenzuela, T. Garrett, L. Lehner, and S. L. Liebling, [36] S. Appl, in Energy Transport in Radio Galaxies and Phys. Rev. D82, 044045 (2010), 1007.1198. Quasars, edited by P. E. Hardee, A. H. Bridle, and J. A. [16] J. Petri (2012), 1205.0889. Zensus (1996), vol. 100 of Astronomical Society of the [17] K. Parfrey, A. M. Beloborodov, and L. Hui (2011), Paciﬁc Conference Series, p. 129. 1110.6669. [37] S. Appl, Astrophysical Letters and Communications 34, [18] S. Komissarov, Mon.Not.Roy.Astron.Soc. 336, 759 395 (1996). (2002), astro-ph/0202447. [38] S. Appl and M. Camenzind, Astron. Astrophys. 270, 71 [19] E. Asano, T. Uchida, and R. Matsumoto, (1993). Publ.Astron.Soc.Jap. (2005), astro-ph/0502371. [39] S. Appl and M. Camenzind, Astron. Astrophys. 274, [20] J. Cho, Astrophys.J. 621, 324 (2005), astro- 699 (1993). ##LTLine## ##LTLine## ##LTLine##

text: whereby

text: the

text: decreasing

text: number

text: of

text: times

text: that

text: øA

text: ap-

text: Ψ0

label: equation

text: Cabcdlamblcmd

text: pears

text: in

text: the

text: deﬁnitions

text: establishes

text: the

text: hierarchy

text: of

text: decay

text: Ψ

text: C

text: lanblcmd

text: rates

text: of

text: these

text: scalars

text: under

text: the

text: peeling

text: theorem

text: abcd

text: a

text: feature

text: not

text: as

text: visible

text: in

text: the

text: tensorial

text: expressions

text: in

text: Ψ

text: C

text: lambm¯

text: cnd

text: abcd

text: Eq.

text: Lastly,

text: to

text: carry

text: out

text: some

text: calculations

text: in

text: the

text: Ψ

text: C

text: lanbm¯

text: cnd

text: main

text: text,

text: we

text: note

text: the

text: fact

text: that

text: under

text: the

text: dyad

text: basis,

text: abcd

text: can

text: be

text: written

text: as

text: Ψ

text: C

text: nam¯

text: bncm¯

text: d

text: AB

text: abcd

text: ø

text: ι

text: ι

text: ø

text: AB

text: A

text: B

text: A

text: B

text: can

text: be

text: rewritten

text: as

text: Ψ

text: Ψ

text: øAøBøCøD

text: ABCD

text: Ψ

text: Ψ

text: øAøBøCιD

text: ABCD

text: Ψ

text: Ψ

text: øAøBιCιD

text: ABCD

text: Ψ

text: Ψ

text: øAιBιCιD

text: ABCD

text: Ψ

text: Ψ

text: ιAιBιCιD

text: ABCD

text: R.

text: D.

text: Blandford

text: and

text: R.

text: L.

text: Znajek,

text: Mon.

text: Not.

text: Roy.

text: As-

text: tron.

text: Soc.

text: J.

text: C.

text: McKinney,

text: Mon.Not.Roy.Astron.Soc.

text: P.

text: Goldreich

text: and

text: W.

text: H.

text: Julian,

text: Astrophys.J.

text: C.

text: Yu,

text: Mon.Not.Roy.Astron.Soc.

text: A.

text: Gruzinov

text: M.

text: Ruderman

text: and

text: P.

text: Sutherland,

text: Astrophys.J.

text: I.

text: Contopoulos,

text: D.

text: Kazanas,

text: and

text: C.

text: Fendt,

text: Astrophys.J.

text: D.

text: MacDonald

text: and

text: K.

text: Thorne,

text: D.

text: A.

text: Uzdensky,

text: Astrophys.J.

text: astro-

text: Mon.Not.Roy.Astron.Soc.

text: R.

text: D.

text: Blandford,

text: in

text: Lighthouses

text: of

text: the

text: Universe:

text: The

text: A.

text: N.

text: Timokhin,

text: Astrophys.Space

text: Sci.

text: Most

text: Luminous

text: Celestial

text: Objects

text: and

text: Their

text: Use

text: for

text: Cosmology:

text: Proceedings

text: of

text: the

text: MPA/ESO/MPE/USM

text: C.

text: Palenzuela,

text: L.

text: Lehner,

text: and

text: S.

text: L.

text: Liebling,

text: Science

text: Joint

text: Astronomy

text: Conference

text: Held

text: in

text: Garching,

text: edited

text: by

text: M.

text: Gilfanov,

text: R.

text: Sunyaev,

text: and

text: E.

text: Churazov

text: D.

text: Alic,

text: P.

text: Mosta,

text: L.

text: Rezzolla,

text: O.

text: Zanotti,

text: and

text: J.

text: L.

text: p.

text: Jaramillo,

text: Astrophys.J.

text: R.

text: D.

text: Blandford,

text: mnras

text: C.

text: Kalapotharakos,

text: I.

text: Contopoulos,

text: and

text: D.

text: Kazanas,

text: P.

text: Meszaros

text: and

text: M.

text: Rees,

text: Astrophys.J.

text: Mon.Not.Roy.Astron.Soc.

text: G.

text: Menon

text: and

text: C.

text: D.

text: Dermer

text: F.

text: C.

text: Michel,

text: Physical

text: Review

text: Letters

text: F.

text: C.

text: Michel,

text: Astrophys.J.

text: T.

text: D.

text: Brennan

text: and

text: S.

text: E.

text: Gralla

text: M.

text: Lyutikov,

text: Phys.Rev.

text: S.

text: E.

text: Gralla

text: and

text: T.

text: Jacobson

text: T.

text: D.

text: Brennan,

text: S.

text: E.

text: Gralla,

text: and

text: T.

text: Jacobson,

text: A.

text: Lupsasca,

text: M.

text: J.

text: Rodriguez,

text: and

text: A.

text: Strominger

text: Class.Quant.Grav.

text: A.

text: Spitkovsky,

text: Astrophys.J.

text: astro-

text: A.

text: Lupsasca

text: and

text: M.

text: J.

text: Rodriguez

text: F.

text: Zhang,

text: H.

text: Yang,

text: and

text: L.

text: Lehner,

text: Phys.Rev.

text: C.

text: Kalapotharakos

text: and

text: I.

text: Contopoulos,

text: As-

text: tron.Astrophys.

text: H.

text: Li,

text: C.

text: Yu,

text: J.

text: Wang,

text: and

text: Z.

text: Xu

text: C.

text: Palenzuela,

text: T.

text: Garrett,

text: L.

text: Lehner,

text: and

text: S.

text: L.

text: Liebling,

text: S.

text: Appl,

text: in

text: Energy

text: Transport

text: in

text: Radio

text: Galaxies

text: and

text: Phys.

text: Rev.

text: Quasars,

text: edited

text: by

text: P.

text: E.

text: Hardee,

text: A.

text: H.

text: Bridle,

text: and

text: J.

text: A.

text: J.

text: Petri

text: Zensus

text: vol.

text: of

text: Astronomical

text: Society

text: of

text: the

text: K.

text: Parfrey,

text: A.

text: M.

text: Beloborodov,

text: and

text: L.

text: Hui

text: Paciﬁc

text: Conference

text: Series,

text: p.

text: S.

text: Appl,

text: Astrophysical

text: Letters

text: and

text: Communications

text: S.

text: Komissarov,

text: Mon.Not.Roy.Astron.Soc.

text: S.

text: Appl

text: and

text: M.

text: Camenzind,

text: Astron.

text: Astrophys.

text: E.

text: Asano,

text: T.

text: Uchida,

text: and

text: R.

text: Matsumoto,

text: Publ.Astron.Soc.Jap.

text: S.

text: Appl

text: and

text: M.

text: Camenzind,

text: Astron.

text: Astrophys.

text: J.

text: Cho,

text: Astrophys.J.

text: astro-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Ξm (cid:68)Φ Ξm 2 (cid:68)Φ 2 Ξm → ∞ FIG. 3: The shape of strings at τ projected on the Poincar´e disk. We plotted the shape of 3 diﬀerent kinds of loops and labeled the corresponding ξ and ∆φ. The parameters (η , χ ) are m n n − − − ( π/4, π/2) for the blue loop, ( π/8, π/4) for the red loop and ( 5π/8, π/4) for the green loop. and mark the corresponding distances ξ . The asymptotic shape of the loop depends on m the location of its nucleation center (η , χ ). n n For a loop nucleated in green or blue regions in Fig. 1, the string will appear to have inﬁnite length, with its two asymptotes becoming straight and spanning a ﬁxed angle from (cid:20) (cid:21) (cid:20) (cid:21) → ∞ the origin. This angle can be found by sending ξ , − − cos η (1 + r2) e 2Htn ∆φ = 2 arccos n = 2 arccos n . (30) sin χ 2r n n For a nucleation center in the red region of Fig. 1, the loop initially surrounds the origin, then shrinks and ﬁnally freezes in comoving coordinates, but physically always expands. At (cid:20) (cid:21) (cid:20) (cid:21) → ∞ τ , we can ﬁnd the maximal distance from the origin to the loop: − − − cos χ + sin η − eHtn (1 r ) 1 n n n ξ = log = log . (31) a cos ηn + sin χn eHtn (1 + rn) + 1 Some examples of loop worldsheets with diﬀerent nucleation centers are shown in Figs. 4-65. 5 We use a 1 + 1 dimensional version of the conformal diagram for de Sitter in these ﬁgures in order to be 12 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ξm

text: Ξm

text: Ξm

text: FIG.

text: The

text: shape

text: of

text: strings

text: at

text: τ

text: projected

text: on

text: the

text: Poincar´e

text: disk.

text: We

text: plotted

text: the

text: shape

text: of

text: diﬀerent

text: kinds

text: of

text: loops

text: and

text: labeled

text: the

text: corresponding

text: ξ

text: and

text: ∆φ.

text: The

text: parameters

text: (η

text: χ

text: are

text: m

text: n

text: n

text: for

text: the

text: blue

text: loop,

text: for

text: the

text: red

text: loop

text: and

text: for

text: the

text: green

text: loop.

text: and

text: mark

text: the

text: corresponding

text: distances

text: ξ

text: The

text: asymptotic

text: shape

text: of

text: the

text: loop

text: depends

text: on

text: m

text: the

text: location

text: of

text: its

text: nucleation

text: center

text: (η

text: χ

text: n

text: n

text: For

text: a

text: loop

text: nucleated

text: in

text: green

text: or

text: blue

text: regions

text: in

text: Fig.

text: the

text: string

text: will

text: appear

text: to

text: have

text: inﬁnite

text: length,

text: with

text: its

text: two

text: asymptotes

text: becoming

text: straight

text: and

text: spanning

text: a

text: ﬁxed

text: angle

text: from

text: the

text: origin.

text: This

text: angle

text: can

text: be

text: found

text: by

text: sending

text: ξ

text: cos

text: η

text: e

text: ∆φ

text: arccos

text: n

text: arccos

text: n

text: sin

text: χ

text: n

text: n

text: For

text: a

text: nucleation

text: center

text: in

text: the

text: red

text: region

text: of

text: Fig.

text: the

text: loop

text: initially

text: surrounds

text: the

text: origin,

text: then

text: shrinks

text: and

text: ﬁnally

text: freezes

text: in

text: comoving

text: coordinates,

text: but

text: physically

text: always

text: expands.

text: At

text: τ

text: we

text: can

text: ﬁnd

text: the

text: maximal

text: distance

text: from

text: the

text: origin

text: to

text: the

text: loop:

text: cos

text: χ

text: sin

text: η

text: eHtn

text: r

text: n

text: n

text: n

text: ξ

text: log

text: log

text: a

text: cos

text: ηn

text: sin

text: χn

text: eHtn

text: rn)

text: Some

text: examples

text: of

text: loop

text: worldsheets

text: with

text: diﬀerent

text: nucleation

text: centers

text: are

text: shown

text: in

text: Figs.

text: We

text: use

text: a

text: dimensional

text: version

text: of

text: the

text: conformal

text: diagram

text: for

text: de

text: Sitter

text: in

text: these

text: ﬁgures

text: in

text: order

text: to

text: be

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: enter the bubble can be characterized by the parameters ξ and ∆φ. In the case of strings, m there is also a third parameter, α, which is the angle between the plane of the loop and the line connecting the loop and bubble nucleation centers. We shall ﬁrst consider the simpler case of domain walls. Since the walls get frozen in comoving coordinates shortly after they enter the bubble, their distribution at a late time τ, e.g., at the end of inﬂation, is well → ∞ approximated by the distribution at τ . We start with the distribution of domain walls with nucleation centers at (t , r ). Then, n n using Eqs. (28) and (30) with H = H , we can ﬁnd the distribution with respect to (ξ , ∆φ). T m The calculation is somewhat tedious but straightforward. The number of domain walls with nucleation centers in spacetime volume element dΩ = e3HFtn4πr2dr dt (39) n n n n is dN = ΓdΩ , (40) n where Γ is the nucleation rate per unit spacetime volume. Then we have − dN = ΓHF 3e3H(cid:12)Ftn4πrn2drnd(cid:12) tn (cid:12) (cid:12) (cid:12) (cid:12) = 4πλc2r2d(cid:12)r dc (cid:12) (41) n n ∂ (r , c) = 4πλc2r2 n dξ d∆φ, (42) n ∂ (ξ , ∆φ) m m where, as before we have deﬁned ≡ − λ ΓH 4. (43) F (cid:0) (cid:1) Using the Jacobian calculated in Appendix , we ﬁnally obtain (cid:104) (cid:105) (cid:0) (cid:1) − dN γ4 2 cosh ξ sin ∆φ sin ∆φ sinh ξ = 32πλ m 2 m . (44) dξmd∆φ 2 cos ∆φ cosh ξ − 2 sinh ξ 2 + 2γ2(1 − cos ∆φ) 5/2 2 m m (cid:113) This distribution is plotted in Fig. 11. It is deﬁned only in the region − − γ cos ∆φ + 1 (1 γ2) cos2 ∆φ 2 − 2 > eξm, (45) 1 cos ∆φ 2 which is below the dashed red line in the ﬁgure. This is because the ﬂat chart of de Sitter − ≤ space covers only the region of χ η π. n n 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: enter

text: the

text: bubble

text: can

text: be

text: characterized

text: by

text: the

text: parameters

text: ξ

text: and

text: ∆φ.

text: In

text: the

text: case

text: of

text: strings,

text: m

text: there

text: is

text: also

text: a

text: third

text: parameter,

text: α,

text: which

text: is

text: the

text: angle

text: between

text: the

text: plane

text: of

text: the

text: loop

text: and

text: the

text: line

text: connecting

text: the

text: loop

text: and

text: bubble

text: nucleation

text: centers.

text: We

text: shall

text: ﬁrst

text: consider

text: the

text: simpler

text: case

text: of

text: domain

text: walls.

text: Since

text: the

text: walls

text: get

text: frozen

text: in

text: comoving

text: coordinates

text: shortly

text: after

text: they

text: enter

text: the

text: bubble,

text: their

text: distribution

text: at

text: a

text: late

text: time

text: τ,

text: e.g.,

text: at

text: the

text: end

text: of

text: inﬂation,

text: is

text: well

text: approximated

text: by

text: the

text: distribution

text: at

text: τ

text: We

text: start

text: with

text: the

text: distribution

text: of

text: domain

text: walls

text: with

text: nucleation

text: centers

text: at

text: (t

text: r

text: Then,

text: n

text: n

text: using

text: Eqs.

text: and

text: with

text: H

text: H

text: we

text: can

text: ﬁnd

text: the

text: distribution

text: with

text: respect

text: to

text: (ξ

text: ∆φ).

text: T

text: m

text: The

text: calculation

text: is

text: somewhat

text: tedious

text: but

text: straightforward.

text: The

text: number

text: of

text: domain

text: walls

text: with

text: nucleation

text: centers

text: in

text: spacetime

text: volume

text: element

text: dΩ

text: dt

text: n

text: n

text: n

text: n

text: is

text: dN

text: ΓdΩ

text: n

text: where

text: Γ

text: is

text: the

text: nucleation

text: rate

text: per

text: unit

text: spacetime

text: volume.

text: Then

text: we

text: have

text: dN

text: ΓHF

text: tn

text: dc

text: n

text: n

text: (r

text: c)

text: n

text: dξ

text: d∆φ,

text: n

text: (ξ

text: ∆φ)

text: m

text: m

text: where,

text: as

text: before

text: we

text: have

text: deﬁned

text: λ

text: ΓH

text: F

text: Using

text: the

text: Jacobian

text: calculated

text: in

text: Appendix

text: we

text: ﬁnally

text: obtain

text: dN

text: γ4

text: cosh

text: ξ

text: sin

text: ∆φ

text: sin

text: ∆φ

text: sinh

text: ξ

text: 32πλ

text: m

text: m

text: dξmd∆φ

text: cos

text: ∆φ

text: cosh

text: ξ

text: sinh

text: ξ

text: cos

text: ∆φ)

text: m

text: m

text: This

text: distribution

text: is

text: plotted

text: in

text: Fig.

text: It

text: is

text: deﬁned

text: only

text: in

text: the

text: region

text: γ

text: cos

text: ∆φ

text: ∆φ

text: eξm,

text: cos

text: ∆φ

text: which

text: is

text: below

text: the

text: dashed

text: red

text: line

text: in

text: the

text: ﬁgure.

text: This

text: is

text: because

text: the

text: ﬂat

text: chart

text: of

text: de

text: Sitter

text: space

text: covers

text: only

text: the

text: region

text: of

text: χ

text: η

text: π.

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:101) (cid:101) We can also deﬁne the comoving size as R = R/R and the comoving volume dV = 0 − dV sinh 3 H τ. Then (cid:101) (cid:101) T (cid:101) (cid:101) (cid:39) RdR dN λγ4dV , (66) (R2 + 1)3 which simply says that the defects are formed at early times τ and are then stretched and diluted by the expansion, leaving the comoving distribution unchanged. The distribution (64) shows that the number of defects is strongly suppressed by γ4. This suppression is in qualitative agreement with the shrinking of the red region in Fig. 7. Hence, we should not expect to see any closed loops or walls from the multiverse within our horizon, ∼ unless γ 1. Closed strings and walls can also nucleate in the true vacuum inside the bubble. The corresponding nucleation rates are given by Eq. (6) with the instanton action (7) or (12) and with H replaced by H . This rate, however, will be exponentially suppressed for H < H , F T T F assuming that the string or wall tension is the same inside and outside the bubble. If the true vacuum admits lighter defects, which do not exist in the false vacuum, they could have a higher nucleation rate and could produce some observable eﬀects (see e.g. [17]). However, observation of such defect would not provide evidence for a multiverse, so we do not consider them here. V. THE NUMBER OF COLLISION EVENTS We shall now estimate the number of collision events within the observer’s past light cone. We consider a central observer, located at ξ = 0 inside the bubble. The past light cone of such an observer, originating at time τ , is given by O tanh(H τ /2) eξ = T O . (67) tanh(H τ/2) T (cid:20) (cid:21) Its physical radius at time τ < τ is O − − − cosh2(H τ/2) R(τ) = H 1 sinh ξ sinh H τ = H 1 coth(H τ /2) 1 T . (68) T T T T O cosh2(H τ /2) T O The bubble wall in these coordinates is at τ = 0; it intersects the past light cone on a sphere S of radius − R(0) = H 1 tanh(H τ /2). (69) T T O 24 ##LTLine## ##LTLine## ##LTLine##

text: We

text: can

text: also

text: deﬁne

text: the

text: comoving

text: size

text: as

text: R

text: R/R

text: and

text: the

text: comoving

text: volume

text: dV

text: dV

text: sinh

text: H

text: τ.

text: Then

text: T

text: RdR

text: dN

text: which

text: simply

text: says

text: that

text: the

text: defects

text: are

text: formed

text: at

text: early

text: times

text: τ

text: and

text: are

text: then

text: stretched

text: and

text: diluted

text: by

text: the

text: expansion,

text: leaving

text: the

text: comoving

text: distribution

text: unchanged.

text: The

text: distribution

text: shows

text: that

text: the

text: number

text: of

text: defects

text: is

text: strongly

text: suppressed

text: by

text: This

text: suppression

text: is

text: in

text: qualitative

text: agreement

text: with

text: the

text: shrinking

text: of

text: the

text: red

text: region

text: in

text: Fig.

text: Hence,

text: we

text: should

text: not

text: expect

text: to

text: see

text: any

text: closed

text: loops

text: or

text: walls

text: from

text: the

text: multiverse

text: within

text: our

text: horizon,

text: unless

text: γ

text: Closed

text: strings

text: and

text: walls

text: can

text: also

text: nucleate

text: in

text: the

text: true

text: vacuum

text: inside

text: the

text: bubble.

text: The

text: corresponding

text: nucleation

text: rates

text: are

text: given

text: by

text: Eq.

text: with

text: the

text: instanton

text: action

text: or

text: and

text: with

text: H

text: replaced

text: by

text: H

text: This

text: rate,

text: however,

text: will

text: be

text: exponentially

text: suppressed

text: for

text: H

text: H

text: F

text: T

text: T

text: F

text: assuming

text: that

text: the

text: string

text: or

text: wall

text: tension

text: is

text: the

text: same

text: inside

text: and

text: outside

text: the

text: bubble.

text: If

text: the

text: true

text: vacuum

text: admits

text: lighter

text: defects,

text: which

text: do

text: not

text: exist

text: in

text: the

text: false

text: vacuum,

text: they

text: could

text: have

text: a

text: higher

text: nucleation

text: rate

text: and

text: could

text: produce

text: some

text: observable

text: eﬀects

text: (see

text: e.g.

text: However,

text: observation

text: of

text: such

text: defect

text: would

text: not

text: provide

text: evidence

text: for

text: a

text: multiverse,

text: so

text: we

text: do

text: not

text: consider

text: them

text: here.

text: V.

text: THE

text: NUMBER

text: OF

text: COLLISION

text: EVENTS

text: We

text: shall

text: now

text: estimate

text: the

text: number

text: of

text: collision

text: events

text: within

text: the

text: observer’s

text: past

text: light

text: cone.

text: We

text: consider

text: a

text: central

text: observer,

text: located

text: at

text: ξ

text: inside

text: the

text: bubble.

text: The

text: past

text: light

text: cone

text: of

text: such

text: an

text: observer,

text: originating

text: at

text: time

text: τ

text: is

text: given

text: by

text: O

text: tanh(H

text: τ

text: eξ

text: T

text: O

text: tanh(H

text: T

text: Its

text: physical

text: radius

text: at

text: time

text: τ

text: τ

text: is

text: O

text: R(τ)

text: H

text: sinh

text: ξ

text: sinh

text: H

text: τ

text: H

text: coth(H

text: τ

text: T

text: T

text: T

text: T

text: T

text: O

text: τ

text: T

text: O

text: The

text: bubble

text: wall

text: in

text: these

text: coordinates

text: is

text: at

text: τ

text: it

text: intersects

text: the

text: past

text: light

text: cone

text: on

text: a

text: sphere

text: S

text: of

text: radius

text: H

text: tanh(H

text: τ

text: T

text: T

text: O

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: It simply states that the domain wall should be close enough to overlap with the bubble. The total number of intersections can now be found by counting all domain walls in the (cid:90) (cid:90) distribution (11) that satisfy the conditions (80), (81), ∼ − − N (cid:38) γ 1 dR γ 1+R ≈ 2π − λ 4πr2 dr λγ 2, (82) R4 − phy phy 3 1 γ 1+1 (cid:28) where in the last step we assumed that γ 1. We see that the number of potentially − observable collisions is enhanced by a factor of γ 2 compared to the number of walls (48) that are physically present within the visible universe. A similar enhancement factor was (cid:28) N (cid:29) found in Ref. [7] for observable bubble collisions. For γ 1 it is possible to have 1, provided that the wall nucleation rate λ is not very strongly suppressed. N The total number of intersections for cosmic strings can be found by a calculation s similar to that in Sec. IV.B. One obtains an integral similar to Eq. (82), but with a weighting factor sin α (R, r , γ), due to the variable inclination angle. α is the maximal inclination m phy (cid:20) (cid:21) m angle for given R and r , phy − − r2 + R2 γ 2 α = arccos phy . (83) m 2r R phy For small γ, we ﬁnd N ≈ 15π2 − λγ 2. (84) s 16 In the above analysis we disregarded defects that have their nucleation centers inside the bubble. Such defects encircle the bubble at t = 0, and we shall assume as before that they ∼ can only be formed at t < 0. Then their nucleation centers must be at t < t∗ ln S . f n E The number of such defects can be estimated as N (cid:48) ∼ ΓΩ , (85) 4 (cid:90) (cid:90) where Ω is the 4-volume of the relevant spacetime region, 4 r∗ · t∗ Ω = dr 4πr2 dte3t. (86) 4 0 t (r) b Here, − − t (r) = ln(1 r) (87) b is the worldsheet of the bubble wall and ∼ − 1 r∗ 1 (88) S E 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: simply

text: states

text: that

text: the

text: domain

text: wall

text: should

text: be

text: close

text: enough

text: to

text: overlap

text: with

text: the

text: bubble.

text: The

text: total

text: number

text: of

text: intersections

text: can

text: now

text: be

text: found

text: by

text: counting

text: all

text: domain

text: walls

text: in

text: the

text: distribution

text: that

text: satisfy

text: the

text: conditions

text: N

text: γ

text: dR

text: γ

text: 2π

text: λ

text: dr

text: λγ

text: phy

text: phy

text: γ

text: where

text: in

text: the

text: last

text: step

text: we

text: assumed

text: that

text: γ

text: We

text: see

text: that

text: the

text: number

text: of

text: potentially

text: observable

text: collisions

text: is

text: enhanced

text: by

text: a

text: factor

text: of

text: γ

text: compared

text: to

text: the

text: number

text: of

text: walls

text: that

text: are

text: physically

text: present

text: within

text: the

text: visible

text: universe.

text: A

text: similar

text: enhancement

text: factor

text: was

text: N

text: found

text: in

text: Ref.

text: for

text: observable

text: bubble

text: collisions.

text: For

text: γ

text: it

text: is

text: possible

text: to

text: have

text: provided

text: that

text: the

text: wall

text: nucleation

text: rate

text: λ

text: is

text: not

text: very

text: strongly

text: suppressed.

text: N

text: The

text: total

text: number

text: of

text: intersections

text: for

text: cosmic

text: strings

text: can

text: be

text: found

text: by

text: a

text: calculation

text: s

text: similar

text: to

text: that

text: in

text: Sec.

text: IV.B.

text: One

text: obtains

text: an

text: integral

text: similar

text: to

text: Eq.

text: but

text: with

text: a

text: weighting

text: factor

text: sin

text: α

text: (R,

text: r

text: γ),

text: due

text: to

text: the

text: variable

text: inclination

text: angle.

text: α

text: is

text: the

text: maximal

text: inclination

text: m

text: phy

text: m

text: angle

text: for

text: given

text: R

text: and

text: r

text: phy

text: γ

text: α

text: arccos

text: phy

text: m

text: R

text: phy

text: For

text: small

text: γ,

text: we

text: ﬁnd

text: N

text: λγ

text: s

text: In

text: the

text: above

text: analysis

text: we

text: disregarded

text: defects

text: that

text: have

text: their

text: nucleation

text: centers

text: inside

text: the

text: bubble.

text: Such

text: defects

text: encircle

text: the

text: bubble

text: at

text: t

text: and

text: we

text: shall

text: assume

text: as

text: before

text: that

text: they

text: can

text: only

text: be

text: formed

text: at

text: t

text: Then

text: their

text: nucleation

text: centers

text: must

text: be

text: at

text: t

text: t∗

text: ln

text: S

text: f

text: n

text: E

text: The

text: number

text: of

text: such

text: defects

text: can

text: be

text: estimated

text: as

text: N

text: ΓΩ

text: where

text: Ω

text: is

text: the

text: of

text: the

text: relevant

text: spacetime

text: region,

text: r∗

text: t∗

text: Ω

text: dr

text: t

text: (r)

text: b

text: Here,

text: t

text: (r)

text: r)

text: b

text: is

text: the

text: worldsheet

text: of

text: the

text: bubble

text: wall

text: and

text: r∗

text: S

text: E

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 0.1 S 3 N 20 0.01 S 3 N 100 Γ(cid:61) E(cid:61) (cid:61) Γ(cid:61) E(cid:61) (cid:61) FIG. 14: Collision marks of cosmic strings on the sky. The numbers of collision events, the values of γ and the color code are the same as in Fig. 13. Note that all strings in these simulations nucleate in the blue region of the parameter space. centers in the blue and green areas of the parameter space of Fig. 7. The green collision marks correspond to domain walls that formed prior to the bubble formation and initially enclosed the bubble. A typical bubble can be expected to collide with such walls only if the wall nucleation rate is relatively high, as in the left panel of Fig. 13. Blue collision marks are due to walls that formed outside the bubble. For blue curves the area of impact on the sky is enclosed by the curve; in other words, the aﬀected area is always less than half of the sky. For green curves the aﬀected area can either be inside or outside, but we ﬁnd that it is predominantly outside. For example, all of the green curves in the left panel of Fig. 13 have the aﬀected region outside. For a head-on (α = 0) collision with a string loop, the mark is just a straight segment, (cid:54) while for a collision with α = 0 the mark segment is curved. Two simulated distributions of string collision marks in the sky are shown in Fig. 14. As for the walls, the distributions were generated for a ﬁxed number of collisions, but now the nucleation centers and loop inclination angles α had to be chosen at random. The details of the simulations are given in Appendix D. The distribution of angular sizes of domain wall collision marks on the sky can be found analytically by calculating the 4-volume of the appropriate nucleation region. The result is shown in Fig. 15 by the red solid line. The calculation involves some tedious integrations, and we do not reproduce it here. Instead, we shall ﬁnd the form of the distribution for relatively large angles, which can be calculated as follows. As in the preceding section, it will be convenient to use units in which H = 1 and F 29 ##LTFigure## ##LTFigure##

text: S

text: N

text: S

text: N

text: FIG.

text: Collision

text: marks

text: of

text: cosmic

text: strings

text: on

text: the

text: sky.

text: The

text: numbers

text: of

text: collision

text: events,

text: the

text: values

text: of

text: γ

text: and

text: the

text: color

text: code

text: are

text: the

text: same

text: as

text: in

text: Fig.

text: Note

text: that

text: all

text: strings

text: in

text: these

text: simulations

text: nucleate

text: in

text: the

text: blue

text: region

text: of

text: the

text: parameter

text: space.

text: centers

text: in

text: the

text: blue

text: and

text: green

text: areas

text: of

text: the

text: parameter

text: space

text: of

text: Fig.

text: The

text: green

text: collision

text: marks

text: correspond

text: to

text: domain

text: walls

text: that

text: formed

text: prior

text: to

text: the

text: bubble

text: formation

text: and

text: initially

text: enclosed

text: the

text: bubble.

text: A

text: typical

text: bubble

text: can

text: be

text: expected

text: to

text: collide

text: with

text: such

text: walls

text: only

text: if

text: the

text: wall

text: nucleation

text: rate

text: is

text: relatively

text: high,

text: as

text: in

text: the

text: left

text: panel

text: of

text: Fig.

text: Blue

text: collision

text: marks

text: are

text: due

text: to

text: walls

text: that

text: formed

text: outside

text: the

text: bubble.

text: For

text: blue

text: curves

text: the

text: area

text: of

text: impact

text: on

text: the

text: sky

text: is

text: enclosed

text: by

text: the

text: curve;

text: in

text: other

text: words,

text: the

text: aﬀected

text: area

text: is

text: always

text: less

text: than

text: half

text: of

text: the

text: sky.

text: For

text: green

text: curves

text: the

text: aﬀected

text: area

text: can

text: either

text: be

text: inside

text: or

text: outside,

text: but

text: we

text: ﬁnd

text: that

text: it

text: is

text: predominantly

text: outside.

text: For

text: example,

text: all

text: of

text: the

text: green

text: curves

text: in

text: the

text: left

text: panel

text: of

text: Fig.

text: have

text: the

text: aﬀected

text: region

text: outside.

text: For

text: a

text: head-on

text: (α

text: collision

text: with

text: a

text: string

text: loop,

text: the

text: mark

text: is

text: just

text: a

text: straight

text: segment,

text: while

text: for

text: a

text: collision

text: with

text: α

text: the

text: mark

text: segment

text: is

text: curved.

text: Two

text: simulated

text: distributions

text: of

text: string

text: collision

text: marks

text: in

text: the

text: sky

text: are

text: shown

text: in

text: Fig.

text: As

text: for

text: the

text: walls,

text: the

text: distributions

text: were

text: generated

text: for

text: a

text: ﬁxed

text: number

text: of

text: collisions,

text: but

text: now

text: the

text: nucleation

text: centers

text: and

text: loop

text: inclination

text: angles

text: α

text: had

text: to

text: be

text: chosen

text: at

text: random.

text: The

text: details

text: of

text: the

text: simulations

text: are

text: given

text: in

text: Appendix

text: D.

text: The

text: distribution

text: of

text: angular

text: sizes

text: of

text: domain

text: wall

text: collision

text: marks

text: on

text: the

text: sky

text: can

text: be

text: found

text: analytically

text: by

text: calculating

text: the

text: of

text: the

text: appropriate

text: nucleation

text: region.

text: The

text: result

text: is

text: shown

text: in

text: Fig.

text: by

text: the

text: red

text: solid

text: line.

text: The

text: calculation

text: involves

text: some

text: tedious

text: integrations,

text: and

text: we

text: do

text: not

text: reproduce

text: it

text: here.

text: Instead,

text: we

text: shall

text: ﬁnd

text: the

text: form

text: of

text: the

text: distribution

text: for

text: relatively

text: large

text: angles,

text: which

text: can

text: be

text: calculated

text: as

text: follows.

text: As

text: in

text: the

text: preceding

text: section,

text: it

text: will

text: be

text: convenient

text: to

text: use

text: units

text: in

text: which

text: H

text: and

text: F

text: ##LTFigure##

text: ##LTFigure##


text: collision time. The same applies to collisions with nucleating defects. This is demonstrated by an explicit calculation in Appendix . VII. DISCUSSION We investigated various aspects of collisions between strings and domain walls nucleating in an inﬂating false vacuum and our bubble, which we assume to be expanding in that vacuum. These collisions have some similarity to collisions with other bubbles, but in some respects they are very diﬀerent. The character of collision between the bubbles crucially depends on whether the vacuum energy density of the other bubble is higher or lower than that in our bubble. If it is lower, then the invading bubble will expand into our bubble at nearly the speed of light, and all objects it encounters on its way, including observers, will be turned into some alien forms of matter. If it is higher, then our bubble will expand into the other bubble, and only a peripheral region of our bubble will be aﬀected by the collision. In the case of defects, the collisions are always rather benign. Defects that do not collapse shortly after they enter the bubble, eventually come to rest relative to the comoving observers and can be detected through their gravitational eﬀects. We found, however, that there can be no more than a few defects from the multiverse in our observable region, and only if ∼ their nucleation rate is very high, λ 1. Because of their small number, such defects are not subject to observational bounds that have been derived for “regular” defects, formed in cosmological phase transitions. The tension µ of “regular” cosmic strings is bounded by (cid:46) − [18] Gµ 10 8. Strings from the multiverse, on the other hand, could have, for example, (cid:38) − Gµ 10 5 without conﬂict with observational data, provided that the angular extent of these strings on the sky is suﬃciently small. “Regular” domain walls are essentially ruled (cid:46) out by observations, unless their tension is extremely small, σ (1 MeV)3 [19]. A wall with a larger tension crossing our observable region would introduce a very large gravitational perturbation, which would be in conﬂict with the isotropy of the CMB and of the galaxy distribution. The tension of walls from the multiverse can be greater by many orders of magnitude, if these walls are located at suﬃciently high redshifts. The eﬀect of massive defects on the CMB power spectrum has been discussed in [20] for a straight cosmic string and in [21] for a planar domain wall. In particular, it has been argued in [21] that a massive domain wall at about twice the horizon distance from us could account for the hemispherical 32

text: collision

text: time.

text: The

text: same

text: applies

text: to

text: collisions

text: with

text: nucleating

text: defects.

text: This

text: is

text: demonstrated

text: by

text: an

text: explicit

text: calculation

text: in

text: Appendix

text: VII.

label: section

text: DISCUSSION

text: We

text: investigated

text: various

text: aspects

text: of

text: collisions

text: between

text: strings

text: and

text: domain

text: walls

text: nucleating

text: in

text: an

text: inﬂating

text: false

text: vacuum

text: and

text: our

text: bubble,

text: which

text: we

text: assume

text: to

text: be

text: expanding

text: in

text: that

text: vacuum.

text: These

text: collisions

text: have

text: some

text: similarity

text: to

text: collisions

text: with

text: other

text: bubbles,

text: but

text: in

text: some

text: respects

text: they

text: are

text: very

text: diﬀerent.

text: The

text: character

text: of

text: collision

text: between

text: the

text: bubbles

text: crucially

text: depends

text: on

text: whether

text: the

text: vacuum

text: energy

text: density

text: of

text: the

text: other

text: bubble

text: is

text: higher

text: or

text: lower

text: than

text: that

text: in

text: our

text: bubble.

text: If

text: it

text: is

text: lower,

text: then

text: the

text: invading

text: bubble

text: will

text: expand

text: into

text: our

text: bubble

text: at

text: nearly

text: the

text: speed

text: of

text: light,

text: and

text: all

text: objects

text: it

text: encounters

text: on

text: its

text: way,

text: including

text: observers,

text: will

text: be

text: turned

text: into

text: some

text: alien

text: forms

text: of

text: matter.

text: If

text: it

text: is

text: higher,

text: then

text: our

text: bubble

text: will

text: expand

text: into

text: the

text: other

text: bubble,

text: and

text: only

text: a

text: peripheral

text: region

text: of

text: our

text: bubble

text: will

text: be

text: aﬀected

text: by

text: the

text: collision.

text: In

text: the

text: case

text: of

text: defects,

text: the

text: collisions

text: are

text: always

text: rather

text: benign.

text: Defects

text: that

text: do

text: not

text: collapse

text: shortly

text: after

text: they

text: enter

text: the

text: bubble,

text: eventually

text: come

text: to

text: rest

text: relative

text: to

text: the

text: comoving

text: observers

text: and

text: can

text: be

text: detected

text: through

text: their

text: gravitational

text: eﬀects.

text: We

text: found,

text: however,

text: that

text: there

text: can

text: be

text: no

text: more

text: than

text: a

text: few

text: defects

text: from

text: the

text: multiverse

text: in

text: our

text: observable

text: region,

text: and

text: only

text: if

text: their

text: nucleation

text: rate

text: is

text: very

text: high,

text: λ

text: Because

text: of

text: their

text: small

text: number,

text: such

text: defects

text: are

text: not

text: subject

text: to

text: observational

text: bounds

text: that

text: have

text: been

text: derived

text: for

text: “regular”

text: defects,

text: formed

text: in

text: cosmological

text: phase

text: transitions.

text: The

text: tension

text: µ

text: of

text: “regular”

text: cosmic

text: strings

text: is

text: bounded

text: by

text: Gµ

text: Strings

text: from

text: the

text: multiverse,

text: on

text: the

text: other

text: hand,

text: could

text: have,

text: for

text: example,

text: Gµ

text: without

text: conﬂict

text: with

text: observational

text: data,

text: provided

text: that

text: the

text: angular

text: extent

text: of

text: these

text: strings

text: on

text: the

text: sky

text: is

text: suﬃciently

text: small.

text: “Regular”

text: domain

text: walls

text: are

text: essentially

text: ruled

text: out

text: by

text: observations,

text: unless

text: their

text: tension

text: is

text: extremely

text: small,

text: σ

text: A

text: wall

text: with

text: a

text: larger

text: tension

text: crossing

text: our

text: observable

text: region

text: would

text: introduce

text: a

text: very

text: large

text: gravitational

text: perturbation,

text: which

text: would

text: be

text: in

text: conﬂict

text: with

text: the

text: isotropy

text: of

text: the

text: CMB

text: and

text: of

text: the

text: galaxy

text: distribution.

text: The

text: tension

text: of

text: walls

text: from

text: the

text: multiverse

text: can

text: be

text: greater

text: by

text: many

text: orders

text: of

text: magnitude,

text: if

text: these

text: walls

text: are

text: located

text: at

text: suﬃciently

text: high

text: redshifts.

text: The

text: eﬀect

text: of

text: massive

text: defects

text: on

text: the

text: CMB

text: power

text: spectrum

text: has

text: been

text: discussed

text: in

text: for

text: a

text: straight

text: cosmic

text: string

text: and

text: in

text: for

text: a

text: planar

text: domain

text: wall.

text: In

text: particular,

text: it

text: has

text: been

text: argued

text: in

text: that

text: a

text: massive

text: domain

text: wall

text: at

text: about

text: twice

text: the

text: horizon

text: distance

text: from

text: us

text: could

text: account

text: for

text: the

text: hemispherical