text: of the observer after which the younger version changes course and does not go back in time. After that, both versions live on into the unambiguous future. [8] It is the parallel worlds of the MSM that are connected by CTCs. This structure is not implicit in MWI itself, and is added by Deutsch as a solution to the paradoxes of time travel.6 Since Deutsch’s metaphysics plays such a central role in the formulation of his CTC model, it cannot be ignored. It is therefore problematic for quantum information theorists to adopt this model for analysis in a principle-theoretic context. Despite their claim to be neutral to questions of interpretation, the metaphysics is playing a fundamental role in the example. As a consequence, the ease of ﬁt between principle–theoretic considera- tions that reify the No Signaling principle in a context where the fundamen- tality of underlying metaphysical considerations is denied, with a model that relies crucially on such commitments to yield the predicted behavior under consideration. The D-CTC model requires a strong commitment to a particular ontology, and is therefore the product of a constructive version of quantum theory. On the constructive view, top–down principles are not considered to have any fundamental importance, and when they conﬂict with predictions based on the ontology and dynamics, they are discarded. The D-CTC model predicts signaling because of the commitments of the theoretical framework in which it was developed. Can we embed the mathematical features of the D-CTC model in a principle–theoretic framework without making the same metaphysical com- mitments? As argued in [10], denying the D-CTC model recourse to the explanatory resources provided by realism about the ontology of MSM will lead it to make diﬀerent predications in simple cases, undermining its ability to solve even the Grandfather Paradox. The D-CTC model includes elements of both the metaphysical approaches to quantum theory, and quantum information. Although these two ap- proaches come into conﬂict, I believe Deutsch had a consistent view in mind. His analysis of the power of quantum computation, for example, is based on parallelism. He considers the increased capabilities of a quantum computer 6There is a related result that suggests that Deutsch’s solution to a simple Grandfather Paradox circuit would not hold if he weren’t explicitly relying on the MSM structure. See [?]. 22 ##LTLine##

text: of

text: the

text: observer

text: after

text: which

text: the

text: younger

text: version

text: changes

text: course

text: and

text: does

text: not

text: go

text: back

text: in

text: time.

text: After

text: that,

text: both

text: versions

text: live

text: on

text: into

text: the

text: unambiguous

text: future.

text: It

text: is

text: the

text: parallel

text: worlds

text: of

text: the

text: MSM

text: that

text: are

text: connected

text: by

text: CTCs.

text: This

text: structure

text: is

text: not

text: implicit

text: in

text: MWI

text: itself,

text: and

text: is

text: added

text: by

text: Deutsch

text: as

text: a

text: solution

text: to

text: the

text: paradoxes

text: of

text: time

text: Since

text: Deutsch’s

text: metaphysics

text: plays

text: such

text: a

text: central

text: role

text: in

text: the

text: formulation

text: of

text: his

text: CTC

text: model,

text: it

text: cannot

text: be

text: ignored.

text: It

text: is

text: therefore

text: problematic

text: for

text: quantum

text: information

text: theorists

text: to

text: adopt

text: this

text: model

text: for

text: analysis

text: in

text: a

text: principle-theoretic

text: context.

text: Despite

text: their

text: claim

text: to

text: be

text: neutral

text: to

text: questions

text: of

text: interpretation,

text: the

text: metaphysics

text: is

text: playing

text: a

text: fundamental

text: role

text: in

text: the

text: example.

text: As

text: a

text: consequence,

text: the

text: ease

text: of

text: ﬁt

text: between

text: principle–theoretic

text: considera-

text: tions

text: that

text: reify

text: the

text: No

text: Signaling

text: principle

text: in

text: a

text: context

text: where

text: the

text: fundamen-

text: tality

text: of

text: underlying

text: metaphysical

text: considerations

text: is

text: denied,

text: with

text: a

text: model

text: that

text: relies

text: crucially

text: on

text: such

text: commitments

text: to

text: yield

text: the

text: predicted

text: behavior

text: under

text: consideration.

text: The

text: D-CTC

text: model

text: requires

text: a

text: strong

text: commitment

text: to

text: a

text: particular

text: ontology,

text: and

text: is

text: therefore

text: the

text: product

text: of

text: a

text: constructive

text: version

text: of

text: quantum

text: theory.

text: On

text: the

text: constructive

text: view,

text: top–down

text: principles

text: are

text: not

text: considered

text: to

text: have

text: any

text: fundamental

text: importance,

text: and

text: when

text: they

text: conﬂict

text: with

text: predictions

text: based

text: on

text: the

text: ontology

text: and

text: dynamics,

text: they

text: are

text: discarded.

text: The

text: D-CTC

text: model

text: predicts

text: signaling

text: because

text: of

text: the

text: commitments

text: of

text: the

text: theoretical

text: framework

text: in

text: which

text: it

text: was

text: developed.

text: Can

text: we

text: embed

text: the

text: mathematical

text: features

text: of

text: the

text: D-CTC

text: model

text: in

text: a

text: principle–theoretic

text: framework

text: without

text: making

text: the

text: same

text: metaphysical

text: com-

text: mitments?

text: As

text: argued

text: in

text: denying

text: the

text: D-CTC

text: model

text: recourse

text: to

text: the

text: explanatory

text: resources

text: provided

text: by

text: realism

text: about

text: the

text: ontology

text: of

text: MSM

text: will

text: lead

text: it

text: to

text: make

text: diﬀerent

text: predications

text: in

text: simple

text: cases,

text: undermining

text: its

text: ability

text: to

text: solve

text: even

text: the

text: Grandfather

text: Paradox.

text: The

text: D-CTC

text: model

text: includes

text: elements

text: of

text: both

text: the

text: metaphysical

text: approaches

text: to

text: quantum

text: theory,

text: and

text: quantum

text: information.

text: Although

text: these

text: two

text: ap-

text: proaches

text: come

text: into

text: conﬂict,

text: I

text: believe

text: Deutsch

text: had

text: a

text: consistent

text: view

text: in

text: mind.

text: His

text: analysis

text: of

text: the

text: power

text: of

text: quantum

text: computation,

text: for

text: example,

text: is

text: based

text: on

text: parallelism.

text: He

text: considers

text: the

text: increased

text: capabilities

text: of

text: a

text: quantum

text: computer

text: is

text: a

text: related

text: result

text: that

text: suggests

text: that

text: Deutsch’s

text: solution

text: to

text: a

text: simple

text: Grandfather

text: Paradox

text: circuit

text: would

text: not

text: hold

text: if

text: he

text: weren’t

text: explicitly

text: relying

text: on

text: the

text: MSM

text: structure.

text: See

text: ##LTLine##


text: (a) Mixture Model Subtypes (b) Proposed Model Subtypes 120 100 100 80 80 60 60 40 40 0 5 10 15 20 0 5 10 15 20 Years Since First Seen Years Since First Seen Figure 3: Comparison of sample subtypes learned by (a) a B-spline mixture model without a personalization component, and (b) the proposed model. To explain the remaining variation we use f ((cid:126)t ), which we know has a Gaussian process prior with OU co- i i variance kernel. Moreover, given f ((cid:126)t ) the residuals (cid:126)r are normally distributed. i i (cid:0) i (cid:1) ∼ N (cid:126)r f ((cid:126)t ), σ2I . (38) i i i (cid:48) To compute the value of the latent function at a new point t , we use the posterior predictive (cid:90) i (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:48) | (cid:48) | | P f (t ) (cid:126)r = P f (t ) f ((cid:126)t ) P f ((cid:126)t ) (cid:126)r df ((cid:126)t ), (39) i i i i i i i i i i i i RNi where (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) | ∝ N | | P fi((cid:126)ti) (cid:126)ri (cid:126)ri fi((cid:126)ti), σ2I GP f((cid:126)ti) 0, KOU((cid:126)ti,(cid:126)ti) . (40) The posterior predictive is itself a GP. The mean of this GP is used to predict new observations in GP regression (cid:48) [20]. Using standard results from [20] we have that the conditional expectation of f (t ) given z and(cid:126)b is (cid:2) (cid:3) i i i i KOU(t(cid:48)i,(cid:126)ti) KOU(cid:2)((cid:126)ti,(cid:126)ti) + σ2I −1 (cid:126)r(cid:3)i =(cid:16) (cid:17) (41) KOU(t(cid:48)i,(cid:126)ti) KOU((cid:126)ti,(cid:126)ti) + σ2I −1 (cid:126)yi − Φp((cid:126)ti)Λ(cid:126)xip − Φz((cid:126)ti)β(cid:126)zi − Φ(cid:96)((cid:126)ti)(cid:126)bi . (42) Just as with(cid:126)b , we need to take the expectation of this expression with respect to the posteriors over z and(cid:126)b to i i i obtain the unconditional posterior expectation. This is easy to do because the expression above is linear in both β(cid:126) and(cid:126)b so we can simply replace them with their expectations computed in Eq. 32 and Eq. 36 respectively: zi i (cid:2) (cid:3) (cid:16) (cid:17) f∗(t(cid:48)i) (cid:44) KOU(t(cid:48)i,(cid:126)ti) KOU((cid:126)ti,(cid:126)ti) + σ2I −1 (cid:126)yi − Φp((cid:126)ti)Λ(cid:126)xip − Φz((cid:126)ti)β(cid:126)i∗ − Φ(cid:96)((cid:126)ti)(cid:126)b∗i . (43) C Will a mixture model sufﬁce? A natural question one might ask is whether a B-spline mixture model would sufﬁciently explain the variability across individuals. In other words, is it necessary to include the individual-speciﬁc components that adjust for long-term and short-term deviations from the subtype trajectory in the proposed model? We ﬁt a B-spline 14 CVFP CVFP ##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##

text: (a)

label: paragraph

font: default

text: Mixture

text: Model

text: Subtypes

text: (b)

text: Proposed

text: Model

text: Subtypes

text: Years

text: Since

text: First

text: Seen

text: Years

text: Since

text: First

text: Seen

text: Figure

text: Comparison

label: caption

text: of

text: sample

text: subtypes

text: learned

text: by

text: (a)

text: a

text: B-spline

text: mixture

text: model

text: without

text: a

text: personalization

text: component,

text: and

text: (b)

text: the

text: proposed

text: model.

text: To

text: explain

text: the

text: remaining

text: variation

text: we

text: use

text: f

text: which

text: we

text: know

text: has

text: a

text: Gaussian

text: process

text: prior

text: with

text: OU

text: co-

text: i

text: i

text: variance

text: kernel.

text: Moreover,

text: given

text: f

text: the

text: residuals

text: are

text: normally

text: distributed.

text: i

text: i

text: i

text: N

text: f

text: i

text: i

text: i

text: To

text: compute

text: the

text: value

text: of

text: the

text: latent

text: function

text: at

text: a

text: new

text: point

text: t

text: we

text: use

text: the

text: posterior

text: predictive

text: i

text: P

text: f

text: (t

text: P

text: f

text: (t

text: f

text: P

text: f

text: df

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: RNi

text: where

text: N

text: P

text: GP

text: The

text: posterior

text: predictive

text: is

text: itself

text: a

text: GP.

text: The

text: mean

text: of

text: this

text: GP

text: is

text: used

text: to

text: predict

text: new

text: observations

text: in

text: GP

text: regression

text: Using

text: standard

text: results

text: from

text: we

text: have

text: that

text: the

text: conditional

text: expectation

text: of

text: f

text: (t

text: given

text: z

text: is

text: i

text: i

text: i

text: i

text: −1

text: −1

text: Just

text: as

text: we

text: need

text: to

text: take

text: the

text: expectation

text: of

text: this

text: expression

text: with

text: respect

text: to

text: the

text: posteriors

text: over

text: z

text: to

text: i

text: i

text: i

text: obtain

text: the

text: unconditional

text: posterior

text: expectation.

text: This

text: is

text: easy

text: to

text: do

text: because

text: the

text: expression

text: above

text: is

text: linear

text: in

text: both

text: so

text: we

text: can

text: simply

text: replace

text: them

text: with

text: their

text: expectations

text: computed

text: in

text: Eq.

text: and

text: Eq.

text: respectively:

text: zi

text: i

text: −1

text: C

text: Will

text: a

text: mixture

text: model

text: sufﬁce?

text: A

text: natural

text: question

text: one

text: might

text: ask

text: is

text: whether

text: a

text: B-spline

text: mixture

text: model

text: would

text: sufﬁciently

text: explain

text: the

text: variability

text: across

text: individuals.

text: In

text: other

text: words,

text: is

text: it

text: necessary

text: to

text: include

text: the

text: individual-speciﬁc

text: components

text: that

text: adjust

text: for

text: long-term

text: and

text: short-term

text: deviations

text: from

text: the

text: subtype

text: trajectory

text: in

text: the

text: proposed

text: model?

text: We

text: ﬁt

text: a

text: B-spline

text: CVFP

text: CVFP

text: ##LTFigure##

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text: (cid:88) (cid:90) (cid:90) (cid:104) (cid:105) (cid:16) (cid:17) yˆ(t(cid:48)) = G E(cid:124) y(cid:48) | z(cid:123),(cid:122)(cid:126)b , f , t(cid:48)(cid:125) P(cid:124) z ,(cid:126)b , f(cid:123)(cid:122)| (cid:126)y , X , Θ(cid:125) df d(cid:126)b (4) i (cid:104) i i i i i i i i i i (cid:105)i i zi=1 Rd(cid:96) RNi prediction given latent vars. posterior over latent vars. = E∗zi,(cid:126)bi,fi Φp (t(cid:48)i)(cid:62) Λ (cid:126)xip +(cid:122)Φz (cid:125)((cid:104)(cid:124)t(cid:48)i)(cid:62)(cid:105)(cid:123)β(cid:126)zi + Φ(cid:96) (t(cid:48)i(cid:122))(cid:62)(cid:125)(cid:126)b(cid:104)(cid:124)i +(cid:105)(cid:123)fi (t(cid:48)i)(cid:122) (cid:125)(cid:124) (cid:123) (5) ∗ ∗ (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122)β(cid:126)i (Eq. 32(cid:125)) (cid:124) (cid:123)(cid:122)(cid:126)bi (Eq. 36(cid:125)) f(cid:124)i∗(t(cid:48)i)(cid:123)((cid:122)Eq. 43(cid:125)) = Φ (t(cid:48))(cid:62) Λ (cid:126)x + Φ (t(cid:48))(cid:62) E∗ β(cid:126) + Φ (t(cid:48))(cid:62) E∗ (cid:126)b + E∗ [f (t(cid:48))] , (6) p i ip z i zi zi (cid:96) i (cid:126)bi i fi i i population prediction subpopulation prediction individual prediction structured noise prediction ∗ where E denotes an expectation conditioned on (cid:126)y , X , Θ. In moving from Eq. 29 to 30, we have i i written the integral as an expectation and substituted the inner expectation with the mean of the nor- mal distribution in Eq. 14. From Eq. 30 to 31, we use linearity of expectation. Eqs. 32, 36, and 43 below show how the expectations in Eq. 31 are computed. An expanded version of these steps are provided in the supplement. Computing the population prediction is straightforward as all quantities are observed. To compute the subpopulation prediction, we need to compute the marginal posterior over z , which we used in i the expectation step above (Eq. 19). The ex(cid:16)pected subtype c(cid:17)oefﬁcients are therefore (cid:80) ∗ (cid:44) ∗ β(cid:126) G π β(cid:126) . (7) i zi=1 izi zi To compute the individual prediction, note that by conditioning on z , the integral over the likelihood i with respect to f and the prior over(cid:126)b form the likelihood and prior of a Bayesian linear regression. i i Let K(cid:16)f = KOU((cid:126)ti,(cid:126)ti) +(cid:17) σ2I, (cid:16)then the po(cid:17)steri(cid:16)or over(cid:126)bi conditione(cid:0)d o(cid:1)n zi is: (cid:0) (cid:1) (cid:17) | ∝ N | N | P (cid:126)b z , (cid:126)y , X , Θ (cid:126)b 0, Σ (cid:126)y Φ Λ (cid:126)x + Φ (cid:126)t β(cid:126) + Φ (cid:126)t (cid:126)b , K . (8) i i i i i b i p ip z i zi (cid:96) i i f Just as in Eq. 16, we have integrated over f moving its effect from the mean of the normal distri- i bution to the covariance. Because the prior over (cid:126)b is conjugate to the likelihood on the right side i of Eq. 33, the posterior can be written in closed form as a normal distribution (see e.g. [10]). The mean of the left side of Eq. 33 is therefore (cid:104) (cid:105) (cid:104) (cid:16) (cid:17)(cid:105) − − (cid:62) − 1 (cid:62) − − − Σ 1 + Φ ((cid:126)t ) K 1Φ ((cid:126)t ) Φ ((cid:126)t ) K 1 (cid:126)y Φ ((cid:126)t )Λ (cid:126)x Φ ((cid:126)t )β(cid:126) , (9) b (cid:96) i f (cid:96) i (cid:96) i f i p i ip z i zi To compute the unconditional posterior mean of (cid:126)b we take the expectation of Eq. 9 with respect to i the poster(cid:104)ior over zi. Eq. 9 is linear in(cid:105)β(cid:126)zi(cid:104), so we can dire(cid:16)ctly replace β(cid:126)zi with its mean (cid:17)(E(cid:105)q. 32): − ∗ (cid:44) − (cid:62) − 1 (cid:62) − − − ∗ (cid:126)b Σ 1 + Φ ((cid:126)t ) K 1Φ ((cid:126)t ) Φ ((cid:126)t ) K 1 (cid:126)y Φ ((cid:126)t )Λ (cid:126)x Φ ((cid:126)t )β(cid:126) . (10) i b (cid:96) i f (cid:96) i (cid:96) i f i p i ip z i i Finally, to compute the structured noise prediction, note that conditioned on z and (cid:126)b , the GP prior i i and marker likelihood (Eq. 14) form a standard GP regression (see e.g. [20]). The conditional (cid:48) posterior of fi(ti) is t(cid:2)herefore a GP with(cid:3)mea(cid:16)n (cid:17) KOU(t(cid:48)i,(cid:126)ti) KOU((cid:126)ti,(cid:126)ti) + σ2I −1 (cid:126)yi − Φp((cid:126)ti)Λ (cid:126)xip − Φz((cid:126)ti)β(cid:126)zi − Φ(cid:96)((cid:126)ti)(cid:126)bi . (11) 6

label: equation

text: G

text: f

text: z

text: X

text: df

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: RNi

text: prediction

text: given

text: latent

text: vars.

text: posterior

text: over

text: latent

text: vars.

text: Φp

text: Λ

text: (Eq.

text: (Eq.

text: Φ

text: Λ

text: Φ

text: E∗

text: Φ

text: E∗

text: E∗

text: [f

text: p

text: i

text: ip

text: z

text: i

text: zi

text: zi

text: i

text: i

text: fi

text: i

text: i

text: population

text: prediction

text: subpopulation

text: prediction

text: individual

text: prediction

text: structured

text: noise

text: prediction

text: where

text: E

text: denotes

text: an

text: expectation

text: conditioned

text: on

text: X

text: Θ.

text: In

text: moving

text: from

text: Eq.

text: to

text: we

text: have

text: i

text: i

text: written

text: the

text: integral

text: as

text: an

text: expectation

text: and

text: substituted

text: the

text: inner

text: expectation

text: with

text: the

text: mean

text: of

text: the

text: nor-

text: mal

text: distribution

text: in

text: Eq.

text: From

text: Eq.

text: to

text: we

text: use

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text: 2 f 1 h=Hda(cid:144)dΤL(cid:144)a 1.5 rŽV 1.6 0.8 1.2 1.2 0.6 0.9 0.8 0.4 0.6 0.4 0.2 0.3 00 5 10 15 20 25Τ 00 5 10 15 20 25Τ 00 5 10 15 20 25Τ 20 Log@aD 1 h 1.5 rŽV 16 0.8 1.2 12 0.6 0.9 8 0.4 0.6 4 0.2 0.3 00 5 10 15 20 25Τ 01 2 3 4 5 6 Τ 01 2 3 4 5 6 Τ FIG. 1: Numerical solution for the evolution of the universe after a cosmic catastro- phe, neglecting dissipation. The model parameters of the ODEs (17) with auxiliary func- tions (18) are (w , γ , γ ) = (1/3, 0, 0) and the boundary conditions at τ = 1 are M f h { } { } a(1), h(1), h˙ (1), f(1), r (1) = 1, 0.55, 0, 0.2953, 0 . With these parameters and boundary M ≥ conditions, the matter energy density vanishes exactly, r (τ) = 0 for τ 1. M 2 f 1 h=Hda(cid:144)dΤL(cid:144)a 10 104 ´rŽV 1.6 0.8 8 1.2 0.6 6 0.8 0.4 4 0.4 0.2 2 00 5 10 15 20 25Τ 00 5 10 15 20 25Τ 00 50 100 150 200 250Τ 10 10-2 ´a 10 102 ´h 2 10´rM 8 8 1.6 6 6 1.2 4 4 0.8 2 2 0.4 00 50 100 150 200 250Τ 00 50 100 150 200 250Τ 00 5 10 15 20 25Τ FIG. 2: Numerical solution for the evolution of the universe after a cosmic catastrophe, taking into account dissipation, which leads to energy exchange between vacuum and matter. The model parameters of the ODEs (17) are (w , γ , γ ) = (1/3, 1, 1). The boundary conditions at τ = 1 M f h { } { } are the same as in Fig. 1, a(1), h(1), h˙ (1), f(1), r (1) = 1, 0.55, 0, 0.2953, 0 . M 2 f 1 h=Hda(cid:144)dΤL(cid:144)a 12 104 ´rŽV 1.6 0.8 8 1.2 0.6 4 0.8 0.4 0 0.4 0.2 -4 00 5 10 15 20 25Τ 00 5 10 15 20 25Τ -80 40 80 120 160 200Τ 2 10-2 ´a 12 102 ´h 2 10´rM 1.6 8 1.6 1.2 4 1.2 0.8 0 0.8 0.4 -4 0.4 00 40 80 120 160 200Τ -80 40 80 120 160 200Τ 00 5 10 15 20 25Τ FIG. 3: Numerical solution of the ODEs (17), taking into account dissipation. The model param- eters are the same as in Fig. 2, (w , γ , γ ) = (1/3, 1, 1), but the boundary conditions at τ = 1 M f h { } { } are diﬀerent, a(1), h(1), h˙ (1), f(1), r (1) = 1, 0.55, 0, 0.2952, 0 . 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text: with constants c and τ . There is a similar solution for τ > τ . With (A3a), it can h sing sing ¨ ˙ now be checked that the h term on the left-hand side of (A1b) dominates over the 4 h h − − − − − term, namely, the ﬁrst term goes as (τ τ) 2 and the second as (τ τ) 1 ln(τ τ). sing sing sing The scale parameter a(τ), deﬁned by a˙/a = h, is readily obtained from (A3a), (cid:104) (cid:105) − − − − a(τ) = C exp (c 1) τ (τ τ) ln(τ τ) , h sing sing (A4) for τ < τ and with an arbitrary constant C > 0. This scale parameter a(τ) does not sing ↑ vanish as τ approaches τ from below. Still, there is a physical singularity as τ τ sing sing with, for example, both the Ricci scalar R and the matter energy density r diverging as M − − (τ τ) 1. sing Incidentally, we also have another type of solution of the ODEs (A1), namely, the de-Sitter solution: f = f = const , (A5a) h = h = const , (A5b) r˙ = 0 , (A5c) M ˙ as discussed at the end of Sec. . The two types of solutions are distinguished by the h ˙ boundary condition for appropriate values of f: h(τ ) = 0 gives the de-Sitter solution (A5) in ˙ for τ > τ and h(τ ) < 0 gives the asymptotic singular solution (A3) for τ < τ < τ . in in in sing Expanding on the last statement, observe that the constraint (A2) allows for a heuristic understanding of the two diﬀerent types of solutions found numerically in Sec. . If the asymptotic value f is such that the right-hand side of (A2) is positive, then a de-Sitter vacuum is possible. But, if the asymptotic value f is such that the right-hand side of (A2) ˙ is negative, then a de-Sitter vacuum with h = 0 is strictly impossible and we are led to the asymptotic singular solution. In closing, we have two remarks on the relevance of the asymptotic singular solution as given by (A3). First, this singular analytic solution gives only a qualitative description of the numerical solution, because f (τ) appears to be not perfectly constant. Second, the num 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: constants

text: c

text: and

text: τ

text: There

text: is

text: a

text: similar

text: solution

text: for

text: τ

text: τ

text: With

text: it

text: can

text: h

text: sing

text: sing

text: now

text: be

text: checked

text: that

text: the

text: h

text: term

text: on

text: the

text: left-hand

text: side

text: of

text: dominates

text: over

text: the

text: h

text: h

text: term,

text: namely,

text: the

text: ﬁrst

text: term

text: goes

text: as

text: (τ

text: τ)

text: and

text: the

text: second

text: as

text: (τ

text: τ)

text: ln(τ

text: τ).

text: sing

text: sing

text: sing

text: The

text: scale

text: parameter

text: a(τ),

text: deﬁned

text: by

text: a˙/a

text: h,

text: is

text: readily

text: obtained

text: from

text: a(τ)

text: C

text: exp

text: (c

text: τ

text: (τ

text: τ)

text: ln(τ

text: τ)

text: h

text: sing

text: sing

text: for

text: τ

text: τ

text: and

text: with

text: an

text: arbitrary

text: constant

text: C

text: This

text: scale

text: parameter

text: a(τ)

text: does

text: not

text: sing

text: vanish

text: as

text: τ

text: approaches

text: τ

text: from

text: below.

text: Still,

text: there

text: is

text: a

text: physical

text: singularity

text: as

text: τ

text: τ

text: sing

text: sing

text: with,

text: for

text: example,

text: both

text: the

text: Ricci

text: scalar

text: R

text: and

text: the

text: matter

text: energy

text: density

text: r

text: diverging

text: as

text: M

text: (τ

text: τ)

text: sing

text: Incidentally,

text: we

text: also

text: have

text: another

text: type

text: of

text: solution

text: of

text: the

text: ODEs

text: namely,

text: the

text: de-Sitter

text: solution:

text: f

text: f

text: const

text: h

text: h

text: const

text: r˙

text: M

text: as

text: discussed

text: at

text: the

text: end

text: of

text: Sec.

text: The

text: two

text: types

text: of

text: solutions

text: are

text: distinguished

text: by

text: the

text: h

text: boundary

text: condition

text: for

text: appropriate

text: values

text: of

text: f:

text: h(τ

text: gives

text: the

text: de-Sitter

text: solution

text: in

text: for

text: τ

text: τ

text: and

text: h(τ

text: gives

text: the

text: asymptotic

text: singular

text: solution

text: for

text: τ

text: τ

text: τ

text: in

text: in

text: in

text: sing

text: Expanding

text: on

text: the

text: last

text: statement,

text: observe

text: that

text: the

text: constraint

text: allows

text: for

text: a

text: heuristic

text: understanding

text: of

text: the

text: two

text: diﬀerent

text: types

text: of

text: solutions

text: found

text: numerically

text: in

text: Sec.

text: If

text: the

text: asymptotic

text: value

text: f

text: is

text: such

text: that

text: the

text: right-hand

text: side

text: of

text: is

text: positive,

text: then

text: a

text: de-Sitter

text: vacuum

text: is

text: possible.

text: But,

text: if

text: the

text: asymptotic

text: value

text: f

text: is

text: such

text: that

text: the

text: right-hand

text: side

text: of

text: is

text: negative,

text: then

text: a

text: de-Sitter

text: vacuum

text: with

text: h

text: is

text: strictly

text: impossible

text: and

text: we

text: are

text: led

text: to

text: the

text: asymptotic

text: singular

text: solution.

text: In

text: closing,

text: we

text: have

text: two

text: remarks

text: on

text: the

text: relevance

text: of

text: the

text: asymptotic

text: singular

text: solution

text: as

text: given

text: by

text: First,

text: this

text: singular

text: analytic

text: solution

text: gives

text: only

text: a

text: qualitative

text: description

text: of

text: the

text: numerical

text: solution,

text: because

text: f

text: (τ)

text: appears

text: to

text: be

text: not

text: perfectly

text: constant.

text: Second,

text: the

text: num

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: of q-variable. The advantage of q-theory is that its equations are universal, in the sense that they essentially do not depend on the origin of the q–ﬁeld and that they naturally modify GR. Whatever the origin of q may be, the q-theory approach naturally solves the problem of the equilibrium energy. The diverging contribution of the zero-point-energies to the vac- uum energy density is automatically cancelled by the microscopic degrees of freedom. This cancellation follows from the Gibbs–Duhem identity, which is applicable to any equilibrium ground state, including the one of the physical vacuum. As a result, the proper vacuum energy density entering the Einstein gravitational equation is zero in full equilibrium at zero temperature. Let us consider the illustrative case where the matter sector is represented by a single real scalar ﬁeld with a non-zero absolute minimum of the potential. Without q-ﬁeld, the vacuum has a large energy density and a corresponding large cosmological con- stant. However, in equilibrium, the q-ﬁeld automatically compensates this contribution to the vacuum energy density without any ﬁne-tuning [15]. The only assumption is that the quantum vacuum is a self-sustained system with the energy being an extensive quantity. Out of equilibrium and/or at nonzero temperature, the vacuum energy density is determined in the infrared and, hence, is many orders of magnitude smaller than the value suggested by the non-zero absolute minimum of the potential or by the ultraviolet cut-oﬀ in RQFT. The CCP is then reduced to the problem of the relaxation of the vacuum to its thermodynamic equilibrium. Up till now, we have considered the classical version of q-theory [15], in which the quantum-dissipative energy exchange between vacuum and matter has been neglected. In the classical theory, the analog of the chemical potential µ – the variable thermodynamically conjugate to the variable q – becomes an integration constant. In a perfect equilibrium vac- uum, µ has the value µ determined by the microscopic parameters of the physical vacuum, 0 which gives a zero value for the cosmological constant. After a cosmic catastrophe, the energy density of the perturbed vacuum may be huge, of the order of the Planck energy scale. Still, if the cosmic catastrophe occurs in the original Minkowski vacuum (i.e., with µ = µ ), the state with a huge cosmological constant will relax back to the Minkowski state 0 with zero cosmological constant (see Figs. 1–5 in Ref. [16]). The drawback of the q-theory at the classical level is that, if the original chemical potential µ is unequal to the value µ , the vacuum does not relax to the Minkowski vacuum but to a 0 de-Sitter state(see Fig. 6 in Ref. [16]). This situation is similar to that of superconductors after a quench [7–12]: if dissipation is neglected, the power-law oscillatory attenuation does not necessarily lead to the equilibrium ground state. The next step is to extend q-theory in order to incorporate quantum-dissipation. This is why the next step should be to extend q-theory in order to incorporate quantum-dissipation and, thus, to allow µ to relax to its equilibrium value µ . In a full quantum theory, the 0 3

text: of

text: q-variable.

text: The

text: advantage

text: of

text: q-theory

text: is

text: that

text: its

text: equations

text: are

text: universal,

text: in

text: the

text: sense

text: that

text: they

text: essentially

text: do

text: not

text: depend

text: on

text: the

text: origin

text: of

text: the

text: q–ﬁeld

text: and

text: that

text: they

text: naturally

text: modify

text: GR.

text: Whatever

text: the

text: origin

text: of

text: q

text: may

text: be,

text: the

text: q-theory

text: approach

text: naturally

text: solves

text: the

text: problem

text: of

text: the

text: equilibrium

text: energy.

text: The

text: diverging

text: contribution

text: of

text: the

text: zero-point-energies

text: to

text: the

text: vac-

text: uum

text: energy

text: density

text: is

text: automatically

text: cancelled

text: by

text: the

text: microscopic

text: degrees

text: of

text: freedom.

text: This

text: cancellation

text: follows

text: from

text: the

text: Gibbs–Duhem

text: identity,

text: which

text: is

text: applicable

text: to

text: any

text: equilibrium

text: ground

text: state,

text: including

text: the

text: one

text: of

text: the

text: physical

text: vacuum.

text: As

text: a

text: result,

text: the

text: proper

text: vacuum

text: energy

text: density

text: entering

text: the

text: Einstein

text: gravitational

text: equation

text: is

text: zero

text: in

text: full

text: equilibrium

text: at

text: zero

text: temperature.

text: Let

text: us

text: consider

text: the

text: illustrative

text: case

text: where

text: the

text: matter

text: sector

text: is

text: represented

text: by

text: a

text: single

text: real

text: scalar

text: ﬁeld

text: with

text: a

text: non-zero

text: absolute

text: minimum

text: of

text: the

text: potential.

text: Without

text: q-ﬁeld,

text: the

text: vacuum

text: has

text: a

text: large

text: energy

text: density

text: and

text: a

text: corresponding

text: large

text: cosmological

text: con-

text: stant.

text: However,

text: in

text: equilibrium,

text: the

text: q-ﬁeld

text: automatically

text: compensates

text: this

text: contribution

text: to

text: the

text: vacuum

text: energy

text: density

text: without

text: any

text: ﬁne-tuning

text: The

text: only

text: assumption

text: is

text: that

text: the

text: quantum

text: vacuum

text: is

text: a

text: self-sustained

text: system

text: with

text: the

text: energy

text: being

text: an

text: extensive

text: quantity.

text: Out

text: of

text: equilibrium

text: and/or

text: at

text: nonzero

text: temperature,

text: the

text: vacuum

text: energy

text: density

text: is

text: determined

text: in

text: the

text: infrared

text: and,

text: hence,

text: is

text: many

text: orders

text: of

text: magnitude

text: smaller

text: than

text: the

text: value

text: suggested

text: by

text: the

text: non-zero

text: absolute

text: minimum

text: of

text: the

text: potential

text: or

text: by

text: the

text: ultraviolet

text: cut-oﬀ

text: in

text: RQFT.

text: The

text: CCP

text: is

text: then

text: reduced

text: to

text: the

text: problem

text: of

text: the

text: relaxation

text: of

text: the

text: vacuum

text: to

text: its

text: thermodynamic

text: equilibrium.

text: Up

text: till

text: now,

text: we

text: have

text: considered

text: the

text: classical

text: version

text: of

text: q-theory

text: in

text: which

text: the

text: quantum-dissipative

text: energy

text: exchange

text: between

text: vacuum

text: and

text: matter

text: has

text: been

text: neglected.

text: In

text: the

text: classical

text: theory,

text: the

text: analog

text: of

text: the

text: chemical

text: potential

text: µ

text: the

text: variable

text: thermodynamically

text: conjugate

text: to

text: the

text: variable

text: q

text: becomes

text: an

text: integration

text: constant.

text: In

text: a

text: perfect

text: equilibrium

text: vac-

text: uum,

text: µ

text: has

text: the

text: value

text: µ

text: determined

text: by

text: the

text: microscopic

text: parameters

text: of

text: the

text: physical

text: vacuum,

text: which

text: gives

text: a

text: zero

text: value

text: for

text: the

text: cosmological

text: constant.

text: After

text: a

text: cosmic

text: catastrophe,

text: the

text: energy

text: density

text: of

text: the

text: perturbed

text: vacuum

text: may

text: be

text: huge,

text: of

text: the

text: order

text: of

text: the

text: Planck

text: energy

text: scale.

text: Still,

text: if

text: the

text: cosmic

text: catastrophe

text: occurs

text: in

text: the

text: original

text: Minkowski

text: vacuum

text: (i.e.,

text: with

text: µ

text: µ

text: the

text: state

text: with

text: a

text: huge

text: cosmological

text: constant

text: will

text: relax

text: back

text: to

text: the

text: Minkowski

text: state

text: with

text: zero

text: cosmological

text: constant

text: (see

text: Figs.

text: in

text: Ref.

text: The

text: drawback

text: of

text: the

text: q-theory

text: at

text: the

text: classical

text: level

text: is

text: that,

text: if

text: the

text: original

text: chemical

text: potential

text: µ

text: is

text: unequal

text: to

text: the

text: value

text: µ

text: the

text: vacuum

text: does

text: not

text: relax

text: to

text: the

text: Minkowski

text: vacuum

text: but

text: to

text: a

text: de-Sitter

text: state(see

text: Fig.

text: in

text: Ref.

text: This

text: situation

text: is

text: similar

text: to

text: that

text: of

text: superconductors

text: after

text: a

text: quench

text: if

text: dissipation

text: is

text: neglected,

text: the

text: power-law

text: oscillatory

text: attenuation

text: does

text: not

text: necessarily

text: lead

text: to

text: the

text: equilibrium

text: ground

text: state.

text: The

text: next

text: step

text: is

text: to

text: extend

text: q-theory

text: in

text: order

text: to

text: incorporate

text: quantum-dissipation.

text: This

text: is

text: why

text: the

text: next

text: step

text: should

text: be

text: to

text: extend

text: q-theory

text: in

text: order

text: to

text: incorporate

text: quantum-dissipation

text: and,

text: thus,

text: to

text: allow

text: µ

text: to

text: relax

text: to

text: its

text: equilibrium

text: value

text: µ

text: In

text: a

text: full

text: quantum

text: theory,

text: the


text: Tidal debris dispersal in MW-sized halo 11 pancies. We found that the lack of dynamical symmetry the lumpiness of the halo would erase these streams, thus of the smooth halo can disperse tidal debris more than discouraging us from searching for more. As summarized subhalos do, similar to the conclusion of Siegal-Gaskins in Figures 5 and 7, our conclusion is that even though & Valluri (2008) which simulated the tidal disruptions streams inside realistic cases are more dispersed, many of dwarf galaxies in larger orbits. On the other hand, we of them can survive as long and thin structures over a found that subhalos with M > 107M(cid:12) can make some Hubble time. In fact, the presence of subhalos may even sub streams much denser, hence more easily detectable, by make some streams easier to detect. bunching up stream stars into clumps which were denser While our N-body streams are orbiting realistic halo than streams in the smooth halo. Meanwhile, subha- potentials, our simulations are still missing a few eﬀects. × los with M > 5 106M(cid:12) produced almost the same In addition to a dark matter halo, the Milky Way also has sub distribution of cell occupancies as the M > 107M(cid:12) a baryonic galaxy which was not included in our study. sub case. Therefore, even though subhalos with masses be- Previous studies such as Dehnen et al. (2004) and Brooks low 107M(cid:12) can produce gaps in streams as shown in & Zolotov (2014) (and references therein) found that the previous studies, these subhalos are not important for galactic disk can inﬂuence the mass loss rates of globular globally redistributing the material in a stream. clusters and dwarf galaxies, respectively. However, the We selected a few streams which produced the highest eﬀect of a galactic disk on the dispersal of tidal debris cell occupancies—hence, most easily detectable—in the has yet to be studied in detail. Furthermore, other than VL-2 halo in order to study their morphologies. In the the subhalos’ orbits, all our potentials are static since we (cid:46) ◦ smooth halo, these dense streams are long and 1 thin constructed them using only the redshift zero snapshot. as seen from the galactic center. This suggests that even In the hierarchical structure formation model, satellite though a realistic halo disperses tidal debris, long and systems are continuously accreted and merged into the thin streams can still survive for a Hubble time. In the main halo. Therefore, we expect both the main halo and lumpy halo, the streams with the highest cell occupancies the subhalos to be evolving in time. Finally, we selected were just as long and thin as those in the smooth halo; our stream orbits randomly in both positions and veloc- however, the densest parts of the streams were primarily ities, which likely resulted in an unrealistic distribution due to subhalo perturbations which caused stream parti- of orbits. Current surveys such as Gaia (Perryman et al. cles to bunch up. This increased the occupancies of the 2001) will be very valuable in providing detailed infor- cells near the point where the bunch up occurred. Com- mation of kinematics of globular clusters and streams in bined with the eﬀect of longitudinal compression at the the Milky Way for future studies. streams’ apocentric passages, streams in a lumpy halo The authors thank the referee for a very detailed and can be much denser than those in a smooth halo. constructive review. R.W., A.S., and B.B. (at JHU) are To put our results into context, the streams in our supported by NSF grant OIA-1124403, and P.M. by OIA- study were similar to (but not physical models of) glob- 1124452. Computations were performed on the GPC su- ular streams such as Pal-5 and GD-1. These streams percomputer at the SciNet HPC Consortium, as well as have been well modeled and studied, and are excellent Sunnyvale at the Canadian Institute of Theoretical As- sources of knowledge about the dark matter halo in our trophysics. SciNet is funded by: the Canada Foundation own galaxy. It would be very useful if more streams for Innovation under the auspices of Compute Canada; similar to them are detected in the future, should they the Government of Ontario; Ontario Research Fund - exist. The question we sought to answer was whether Research Excellence; and the University of Toronto. the lack of dynamical symmetry of the halo potential or REFERENCES Bernard, E. J., Ferguson, A. M. N., Schlaﬂy, E. F., et al. 2014, Grillmair, C. J., & Dionatos, O. 2006, ApJ, 643, L17 MNRAS, 443, L84 Hernquist, L. 1990, ApJ, 356, 359 Bonaca, A., Geha, M., & Kallivayalil, N. 2012, ApJ, 760, L6 Hernquist, L., & Ostriker, J. P. 1992, ApJ, 386, 375 Bonaca, A., Geha, M., Ku¨pper, A. H. W., et al. 2014, ApJ, 795, Ibata, R. A., Lewis, G. F., Irwin, M. J., & Quinn, T. 2002, 94 MNRAS, 332, 915 Brooks, A. M., & Zolotov, A. 2014, ApJ, 786, 87 Jing, Y. P., & Suto, Y. 2002, ApJ, 574, 538 Bullock, J. S., & Johnston, K. V. 2005, ApJ, 635, 931 Johnston, K. V. 1998, ApJ, 495, 297 Carlberg, R. G. 2009, ApJ, 705, L223 Johnston, K. V., Sackett, P. D., & Bullock, J. S. 2001, ApJ, 557, —. 2012, ApJ, 748, 20 137 —. 2013, ApJ, 775, 90 Johnston, K. V., Spergel, D. N., & Haydn, C. 2002, ApJ, 570, 656 Carlberg, R. G., & Grillmair, C. J. 2013, ApJ, 768, 171 Klypin, A., Kravtsov, A. V., Valenzuela, O., & Prada, F. 1999, Carlberg, R. G., Grillmair, C. J., & Hetherington, N. 2012, ApJ, ApJ, 522, 82 760, 75 Koposov, S. 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text: Tidal

text: debris

text: dispersal

text: in

text: MW-sized

text: halo

text: pancies.

text: We

text: found

text: that

text: the

text: lack

text: of

text: dynamical

text: symmetry

text: the

text: lumpiness

text: of

text: the

text: halo

text: would

text: erase

text: these

text: streams,

text: thus

text: of

text: the

text: smooth

text: halo

text: can

text: disperse

text: tidal

text: debris

text: more

text: than

text: discouraging

text: us

text: from

text: searching

text: for

text: more.

text: As

text: summarized

text: subhalos

text: do,

text: similar

text: to

text: the

text: conclusion

text: of

text: Siegal-Gaskins

text: in

text: Figures

text: and

text: our

text: conclusion

text: is

text: that

text: even

text: though

text: Valluri

text: which

text: simulated

text: the

text: tidal

text: disruptions

text: streams

text: inside

text: realistic

text: cases

text: are

text: more

text: dispersed,

text: many

text: of

text: dwarf

text: galaxies

text: in

text: larger

text: orbits.

text: On

text: the

text: other

text: hand,

text: we

text: of

text: them

text: can

text: survive

text: as

text: long

text: and

text: thin

text: structures

text: over

text: a

text: found

text: that

text: subhalos

text: with

text: M

text: can

text: make

text: some

text: Hubble

text: time.

text: In

text: fact,

text: the

text: presence

text: of

text: subhalos

text: may

text: even

text: sub

text: streams

text: much

text: denser,

text: hence

text: more

text: easily

text: detectable,

text: by

text: make

text: some

text: streams

text: easier

text: to

text: detect.

text: bunching

text: up

text: stream

text: stars

text: into

text: clumps

text: which

text: were

text: denser

text: While

text: our

text: N-body

text: streams

text: are

text: orbiting

text: realistic

text: halo

text: than

text: streams

text: in

text: the

text: smooth

text: halo.

text: Meanwhile,

text: subha-

text: potentials,

text: our

text: simulations

text: are

text: still

text: missing

text: a

text: few

text: eﬀects.

text: los

text: with

text: M

text: produced

text: almost

text: the

text: same

text: In

text: addition

text: to

text: a

text: dark

text: matter

text: halo,

text: the

text: Milky

text: Way

text: also

text: has

text: sub

text: distribution

text: of

text: cell

text: occupancies

text: as

text: the

text: M

text: a

text: baryonic

text: galaxy

text: which

text: was

text: not

text: included

text: in

text: our

text: study.

text: sub

text: case.

text: Therefore,

text: even

text: though

text: subhalos

text: with

text: masses

text: be-

text: Previous

text: studies

text: such

text: as

text: Dehnen

text: et

text: al.

text: and

text: Brooks

text: low

text: can

text: produce

text: gaps

text: in

text: streams

text: as

text: shown

text: in

text: Zolotov

text: (and

text: references

text: therein)

text: found

text: that

text: the

text: previous

text: studies,

text: these

text: subhalos

text: are

text: not

text: important

text: for

text: galactic

text: disk

text: can

text: inﬂuence

text: the

text: mass

text: loss

text: rates

text: of

text: globular

text: globally

text: redistributing

text: the

text: material

text: in

text: a

text: stream.

text: clusters

text: and

text: dwarf

text: galaxies,

text: respectively.

text: However,

text: the

text: We

text: selected

text: a

text: few

text: streams

text: which

text: produced

text: the

text: highest

text: eﬀect

text: of

text: a

text: galactic

text: disk

text: on

text: the

text: dispersal

text: of

text: tidal

text: debris

text: cell

text: occupancies—hence,

text: most

text: easily

text: detectable—in

text: the

text: has

text: yet

text: to

text: be

text: studied

text: in

text: detail.

text: Furthermore,

text: other

text: than

text: halo

text: in

text: order

text: to

text: study

text: their

text: morphologies.

text: In

text: the

text: the

text: subhalos’

text: orbits,

text: all

text: our

text: potentials

text: are

text: static

text: since

text: we

text: smooth

text: halo,

text: these

text: dense

text: streams

text: are

text: long

text: and

text: thin

text: constructed

text: them

text: using

text: only

text: the

text: redshift

text: zero

text: snapshot.

text: as

text: seen

text: from

text: the

text: galactic

text: center.

text: This

text: suggests

text: that

text: even

text: In

text: the

text: hierarchical

text: structure

text: formation

text: model,

text: satellite

text: though

text: a

text: realistic

text: halo

text: disperses

text: tidal

text: debris,

text: long

text: and

text: systems

text: are

text: continuously

text: accreted

text: and

text: merged

text: into

text: the

text: thin

text: streams

text: can

text: still

text: survive

text: for

text: a

text: Hubble

text: time.

text: In

text: the

text: main

text: halo.

text: Therefore,

text: we

text: expect

text: both

text: the

text: main

text: halo

text: and

text: lumpy

text: halo,

text: the

text: streams

text: with

text: the

text: highest

text: cell

text: occupancies

text: the

text: subhalos

text: to

text: be

text: evolving

text: in

text: time.

text: Finally,

text: we

text: selected

text: were

text: just

text: as

text: long

text: and

text: thin

text: as

text: those

text: in

text: the

text: smooth

text: halo;

text: our

text: stream

text: orbits

text: randomly

text: in

text: both

text: positions

text: and

text: veloc-

text: however,

text: the

text: densest

text: parts

text: of

text: the

text: streams

text: were

text: primarily

text: ities,

text: which

text: likely

text: resulted

text: in

text: an

text: unrealistic

text: distribution

text: due

text: to

text: subhalo

text: perturbations

text: which

text: caused

text: stream

text: parti-

text: of

text: orbits.

text: Current

text: surveys

text: such

text: as

text: Gaia

text: (Perryman

text: et

text: al.

text: cles

text: to

text: bunch

text: up.

text: This

text: increased

text: the

text: occupancies

text: of

text: the

text: will

text: be

text: very

text: valuable

text: in

text: providing

text: detailed

text: infor-

text: cells

text: near

text: the

text: point

text: where

text: the

text: bunch

text: up

text: occurred.

text: Com-

text: mation

text: of

text: kinematics

text: of

text: globular

text: clusters

text: and

text: streams

text: in

text: bined

text: with

text: the

text: eﬀect

text: of

text: longitudinal

text: compression

text: at

text: the

text: the

text: Milky

text: Way

text: for

text: future

text: studies.

text: streams’

text: apocentric

text: passages,

text: streams

text: in

text: a

text: lumpy

text: halo

text: The

text: authors

text: thank

text: the

text: referee

text: for

text: a

text: very

text: detailed

text: and

text: can

text: be

text: much

text: denser

text: than

text: those

text: in

text: a

text: smooth

text: halo.

text: constructive

text: review.

text: R.W.,

text: A.S.,

text: and

text: B.B.

text: (at

text: JHU)

text: are

text: To

text: put

text: our

text: results

text: into

text: context,

text: the

text: streams

text: in

text: our

text: supported

text: by

text: NSF

text: grant

text: and

text: P.M.

text: by

text: OIA-

text: study

text: were

text: similar

text: to

text: (but

text: not

text: physical

text: models

text: of)

text: glob-

text: Computations

text: were

text: performed

text: on

text: the

text: GPC

text: su-

text: ular

text: streams

text: such

text: as

text: and

text: These

text: streams

text: percomputer

text: at

text: the

text: SciNet

text: HPC

text: Consortium,

text: as

text: well

text: as

text: have

text: been

text: well

text: modeled

text: and

text: studied,

text: and

text: are

text: excellent

text: Sunnyvale

text: at

text: the

text: Canadian

text: Institute

text: of

text: Theoretical

text: As-

text: sources

text: of

text: knowledge

text: about

text: the

text: dark

text: matter

text: halo

text: in

text: our

text: trophysics.

text: SciNet

text: is

text: funded

text: by:

text: the

text: Canada

text: Foundation

text: own

text: galaxy.

text: It

text: would

text: be

text: very

text: useful

text: if

text: more

text: streams

text: for

text: Innovation

text: under

text: the

text: auspices

text: of

text: Compute

text: Canada;

text: similar

text: to

text: them

text: are

text: detected

text: in

text: the

text: future,

text: should

text: they

text: the

text: Government

text: of

text: Ontario;

text: Ontario

text: Research

text: Fund

text: exist.

text: The

text: question

text: we

text: sought

text: to

text: answer

text: was

text: whether

text: Research

text: Excellence;

text: and

text: the

text: University

text: of

text: Toronto.

text: the

text: lack

text: of

text: dynamical

text: symmetry

text: of

text: the

text: halo

text: potential

text: or

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