text: 4 (cid:29) (cid:29) as well as the continuum limit L w 1 which makes the derivations analytically tractable. The potential is a c “variational formulation” of the message passing algorithm on coding systems. It is a functional whose stationary points are the ﬁxed points of the density evolution equations described by this algorithm, and has been used to prove threshold saturation in [3] for general BMS channels. A signiﬁcant part of the formalism in [3] is used in the present work. We also note that potential formulations have been used to characterize the ﬁxed point(s) of general scalar systems at the MAP threshold using displacement convexity in [23], [24]. An extension of the ideas in these works could shed some light on the question of the independence of the soliton’s shape from the initial conditions. Section introduces a few preliminary notions that we will need and reviews the phenomenology of the solitonic wave. In Section , we formulate the continuum limit and state our main formula for the velocity of the soliton on general BMS channels; the derivation is presented in Section . Comparisons with numerical experiments are presented in Section . These concern transmission over the BEC as well as general BMS channels in the so- called Gaussian approximation. We also discuss a possible application of our formula to scaling laws for ﬁnite-size ensembles in Section . The case of general scalar spatially coupled systems is treated in Section , and illustrated for the examples of generalized LDPC codes (on the BEC or BSC channels) and compressive sensing. We present concluding remarks and propose further directions in Section . A summary of this work has appeared in [25], [26]. II. PRELIMINARIES We consider (almost) the same setting as in [3] and adopt most of the notation introduced in that work. For more information about the formalism in these preliminaries, one can consult [27]. (cid:82) We denote by M(R¯) the space of probability measures x on the extended real numbers α ∈ R¯ = R ∪ {∞}. (cid:82) Here α ∈ R¯ should be interpreted as a “log-likelihood variable”. We call the measure x symmetric if dx(α) = E − e−αdx(α) for all measurable sets E ⊂ R¯. E We deﬁne an entropy functional H : M → R that maps a ﬁnite probability measure from M(R¯) to a real number. It is deﬁned as (cid:90) − H(x) = dx(α) log (1 + e α). (1) 2 Note that this is a linear functional. Linearity is used in an important way to compute the entropy of convex combinations of measures (which also yield probability measures). But we will also compute the “entropy” associated − ≡ − to differences of measures by setting H(x1 x2) H(x1) H(x2). In other words, the entropy functional is extended in an obvious way to the space of signed measures. In the remainder of this work, we will use the Dirac masses ∆ (α) and ∆∞(α) at zero and inﬁnite log-likelihood, 0 that have entropies H(∆ ) = 1 and H(∆∞) = 0, respectively. (cid:16) (cid:24) 0 We will also use the standard variable-node and check-node convolution operators and for log-lik(cid:16)elihood ratio message distributions involved in DE equations [27]. For x1, x2 ∈ M(R¯), the usual convolution x1 x2 is May 29, 2020 DRAFT

text: as

text: well

text: as

text: the

text: continuum

text: limit

text: L

text: w

text: which

text: makes

text: the

text: derivations

text: analytically

text: tractable.

text: The

text: potential

text: is

text: a

text: c

text: “variational

text: formulation”

text: of

text: the

text: message

text: passing

text: algorithm

text: on

text: coding

text: systems.

text: It

text: is

text: a

text: functional

text: whose

text: stationary

text: points

text: are

text: the

text: ﬁxed

text: points

text: of

text: the

text: density

text: evolution

text: equations

text: described

text: by

text: this

text: algorithm,

text: and

text: has

text: been

text: used

text: to

text: prove

text: threshold

text: saturation

text: in

text: for

text: general

text: BMS

text: channels.

text: A

text: signiﬁcant

text: part

text: of

text: the

text: formalism

text: in

text: is

text: used

text: in

text: the

text: present

text: work.

text: We

text: also

text: note

text: that

text: potential

text: formulations

text: have

text: been

text: used

text: to

text: characterize

text: the

text: ﬁxed

text: point(s)

text: of

text: general

text: scalar

text: systems

text: at

text: the

text: MAP

text: threshold

text: using

text: displacement

text: convexity

text: in

text: An

text: extension

text: of

text: the

text: ideas

text: in

text: these

text: works

text: could

text: shed

text: some

text: light

text: on

text: the

text: question

text: of

text: the

text: independence

text: of

text: the

text: soliton’s

text: shape

text: from

text: the

text: initial

text: conditions.

text: Section

text: introduces

text: a

text: few

text: preliminary

text: notions

text: that

text: we

text: will

text: need

text: and

text: reviews

text: the

text: phenomenology

text: of

text: the

text: solitonic

text: wave.

text: In

text: Section

text: we

text: formulate

text: the

text: continuum

text: limit

text: and

text: state

text: our

text: main

text: formula

text: for

text: the

text: velocity

text: of

text: the

text: soliton

text: on

text: general

text: BMS

text: channels;

text: the

text: derivation

text: is

text: presented

text: in

text: Section

text: Comparisons

text: with

text: numerical

text: experiments

text: are

text: presented

text: in

text: Section

text: These

text: concern

text: transmission

text: over

text: the

text: BEC

text: as

text: well

text: as

text: general

text: BMS

text: channels

text: in

text: the

text: so-

text: called

text: Gaussian

text: approximation.

text: We

text: also

text: discuss

text: a

text: possible

text: application

text: of

text: our

text: formula

text: to

text: scaling

text: laws

text: for

text: ﬁnite-size

text: ensembles

text: in

text: Section

text: The

text: case

text: of

text: general

text: scalar

text: spatially

text: coupled

text: systems

text: is

text: treated

text: in

text: Section

text: and

text: illustrated

text: for

text: the

text: examples

text: of

text: generalized

text: LDPC

text: codes

text: (on

text: the

text: BEC

text: or

text: BSC

text: channels)

text: and

text: compressive

text: sensing.

text: We

text: present

text: concluding

text: remarks

text: and

text: propose

text: further

text: directions

text: in

text: Section

text: A

text: summary

text: of

text: this

text: work

text: has

text: appeared

text: in

text: II.

text: PRELIMINARIES

text: We

text: consider

text: (almost)

text: the

text: same

text: setting

text: as

text: in

text: and

text: adopt

text: most

text: of

text: the

text: notation

text: introduced

text: in

text: that

text: work.

text: For

text: more

text: information

text: about

text: the

text: formalism

text: in

text: these

text: preliminaries,

text: one

text: can

text: consult

text: We

text: denote

text: by

text: M(R¯)

text: the

text: space

text: of

text: probability

text: measures

text: x

text: on

text: the

text: extended

text: real

text: numbers

text: α

text: R¯

text: R

text: Here

text: α

text: R¯

text: should

text: be

text: interpreted

text: as

text: a

text: “log-likelihood

text: variable”.

text: We

text: call

text: the

text: measure

text: x

text: symmetric

text: if

text: dx(α)

text: E

text: e−αdx(α)

text: for

text: all

text: measurable

text: sets

text: E

text: R¯.

text: E

text: We

text: deﬁne

text: an

text: entropy

text: functional

text: H

text: M

text: R

text: that

text: maps

text: a

text: ﬁnite

text: probability

text: measure

text: from

text: M(R¯)

text: to

text: a

text: real

text: number.

text: It

text: is

text: deﬁned

text: as

text: H(x)

text: dx(α)

text: log

text: e

text: α).

text: Note

text: that

text: this

text: is

text: a

text: linear

text: functional.

text: Linearity

text: is

text: used

text: in

text: an

text: important

text: way

text: to

text: compute

text: the

text: entropy

text: of

text: convex

text: combinations

text: of

text: measures

text: (which

text: also

text: yield

text: probability

text: measures).

text: But

text: we

text: will

text: also

text: compute

text: the

text: “entropy”

text: associated

text: to

text: differences

text: of

text: measures

text: by

text: setting

text: In

text: other

text: words,

text: the

text: entropy

text: functional

text: is

text: extended

text: in

text: an

text: obvious

text: way

text: to

text: the

text: space

text: of

text: signed

text: measures.

text: In

text: the

text: remainder

text: of

text: this

text: work,

text: we

text: will

text: use

text: the

text: Dirac

text: masses

text: (α)

text: and

text: ∆∞(α)

text: at

text: zero

text: and

text: inﬁnite

text: log-likelihood,

text: that

text: have

text: entropies

text: H(∆

text: and

text: H(∆∞)

text: respectively.

text: We

text: will

text: also

text: use

text: the

text: standard

text: variable-node

text: and

text: check-node

text: convolution

text: operators

text: and

text: for

text: ratio

text: message

text: distributions

text: involved

text: in

text: DE

text: equations

text: For

text: M(R¯),

text: the

text: usual

text: convolution

text: is

text: May

text: DRAFT


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Zdeborova´, “Probabilistic reconstruction in compressed sensing: algorithms, phase diagrams, and threshold achieving matrices,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2012, no. 08, p. P08009, 2012. [10] D. L. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” in International Symposium on Information Theory Proceedings (ISIT). IEEE, 2012, pp. 1231–1235. [11] S. H. Hassani, N. Macris, and R. Urbanke, “Coupled graphical models and their thresholds,” in Proceedings of Information Theory Workshop IEEE (Dublin), 2010, pp. 1–5. [12] ——, “Threshold saturation in spatially coupled constraint satisfaction problems,” Journal of Statistical Physics, vol. 150, no. 5, pp. 807–850, 2013. [13] D. Achlioptas, S. H. Hassani, N. Macris, and R. Urbanke, “Bounds for random constraint satisfaction problems via spatial coupling,” in ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2016. [14] S. H. Hassani, N. Macris, and R. Urbanke, “Chains of mean ﬁeld models,” Journal of Statistical Mechanics: Theory and Experiment, vol. P02011, 2012. [15] F. Caltagirone, S. Franz, R. G. Morris, and L. Zdeborova´, “Dynamics and termination cost of spatially coupled mean-ﬁeld models,” Physical Review E, vol. 89, no. 1, p. 012102, 2014. [16] S. Kudekar, T. Richardson, and R. Urbanke, “Wave-like solutions of general one-dimensional spatially coupled systems,” preprint arXiv:1208.5273, 2012. [17] V. Aref, L. Schmalen, and S. ten Brink, “On the convergence speed of spatially coupled LDPC ensembles,” in 51st Annual Allerton Conference on Comm., Control, and Comp. IEEE, 2013, pp. 342–349. [18] S.-Y. Chung and G. D. J. Forney, “On the capacity of low-density parity-check codes,” in International Symposium on Information Theory Proceedings (ISIT). IEEE, 2001, p. 320. [19] S.-Y. Chung, “On the construction of some capacity-approaching coding schemes,” Ph.D. dissertation, MIT, 2000. [20] P. Olmos and R. Urbanke, “A scaling law to predict the ﬁnite-length performance of spatially-coupled LDPC codes,” IEEE Transactions on Information Theory, vol. 61, no. 6, pp. 3164–3184, 2014. [21] A. Yedla, Y.-Y. Jian, P. S. Nguyen, and H. D. Pﬁster, “A simple proof of Maxwell saturation for coupled scalar recursions,” IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 6943–6965, 2014. [22] K. Takeuchi, T. Tanaka, and T. Kawabata, “A phenomenological study on threshold improvement via spatial coupling,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 95, no. 5, pp. 974–977, 2012. [23] R. El-Khatib, N. Macris, and R. Urbanke, “Displacement convexity, a useful framework for the study of spatially coupled codes,” in Information Theory Workshop (ITW), Sevilla, Spain. IEEE, 2013, pp. 1–5. [24] R. El-Khatib, N. Macris, T. Richardson, and R. Urbanke, “Analysis of coupled scalar systems by displacement convexity,” in International Symposium on Information Theory Proceedings (ISIT), Honolulu, Hawaii. IEEE, 2014, pp. 2321–2325. [25] R. El-Khatib and N. Macris, “The velocity of the decoding wave for spatially coupled codes on bms channels,” in Information Theory Proceedings (ISIT), 2016 IEEE International Symposium on. IEEE, 2016, pp. 2119–2123. [26] ——, “The velocity of the propagating wave for general coupled scalar systems,” in Information Theory Workshop, Cambridge, UK, 2016. [27] T. Richardson and R. Urbanke, Modern coding theory. Cambridge University Press, 2008. [28] A. Giurgiu, N. Macris, and R. Urbanke, “Spatial coupling as a proof technique and three applications,” IEEE Trans. Inform. 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text: analysis

text: for

text: LDPC

text: convolutional

text: codes,”

text: IEEE

text: Transactions

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text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: V.

text: Aref,

text: N.

text: Macris,

text: R.

text: Urbanke,

text: and

text: M.

text: Vuffray,

text: “Lossy

text: source

text: coding

text: via

text: spatially

text: coupled

text: LDGM

text: ensembles,”

text: in

text: Information

text: Theory

text: Proceedings

text: (ISIT),

text: IEEE

text: International

text: Symposium

text: on.

text: IEEE,

text: pp.

text: V.

text: Aref,

text: N.

text: Macris,

text: and

text: M.

text: Vuffray,

text: “Approaching

text: the

text: Rate-Distortion

text: Limit

text: With

text: Spatial

text: Coupling,

text: Belief

text: Propagation,

text: and

text: Decimation,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory

text: vol.

text: no.

text: pp.

text: S.

text: Kudekar

text: and

text: H.

text: D.

text: Pﬁster,

text: “The

text: effect

text: of

text: spatial

text: coupling

text: on

text: compressive

text: sensing,”

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text: Allerton

text: Conference

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text: Control,

text: and

text: Computing

text: (Allerton).

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text: Krzakala,

text: M.

text: Me´zard,

text: F.

text: Sausset,

text: Y.

text: Sun,

text: and

text: L.

text: Zdeborova´,

text: “Probabilistic

text: reconstruction

text: in

text: compressed

text: sensing:

text: algorithms,

text: phase

text: diagrams,

text: and

text: threshold

text: achieving

text: matrices,”

text: Journal

text: of

text: Statistical

text: Mechanics:

text: Theory

text: and

text: Experiment,

text: vol.

text: no.

text: p.

text: D.

text: L.

text: Donoho,

text: A.

text: Javanmard,

text: and

text: A.

text: Montanari,

text: “Information-theoretically

text: optimal

text: compressed

text: sensing

text: via

text: spatial

text: coupling

text: and

text: approximate

text: message

text: passing,”

text: in

text: International

text: Symposium

text: on

text: Information

text: Theory

text: Proceedings

text: (ISIT).

text: IEEE,

text: pp.

text: S.

text: H.

text: Hassani,

text: N.

text: Macris,

text: and

text: R.

text: Urbanke,

text: “Coupled

text: graphical

text: models

text: and

text: their

text: thresholds,”

text: in

text: Proceedings

text: of

text: Information

text: Theory

text: Workshop

text: IEEE

text: (Dublin),

text: pp.

text: “Threshold

text: saturation

text: in

text: spatially

text: coupled

text: constraint

text: satisfaction

text: problems,”

text: Journal

text: of

text: Statistical

text: Physics,

text: vol.

text: no.

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text: S.

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text: Hassani,

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text: and

text: R.

text: Urbanke,

text: “Bounds

text: for

text: random

text: constraint

text: satisfaction

text: problems

text: via

text: spatial

text: coupling,”

text: in

text: ACM-SIAM

text: Symposium

text: on

text: Discrete

text: Algorithms

text: (SODA),

text: January

text: S.

text: H.

text: Hassani,

text: N.

text: Macris,

text: and

text: R.

text: Urbanke,

text: “Chains

text: of

text: mean

text: ﬁeld

text: models,”

text: Journal

text: of

text: Statistical

text: Mechanics:

text: Theory

text: and

text: Experiment,

text: vol.

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text: Caltagirone,

text: S.

text: Franz,

text: R.

text: G.

text: Morris,

text: and

text: L.

text: Zdeborova´,

text: “Dynamics

text: and

text: termination

text: cost

text: of

text: spatially

text: coupled

text: mean-ﬁeld

text: models,”

text: Physical

text: Review

text: E,

text: vol.

text: no.

text: p.

text: S.

text: Kudekar,

text: T.

text: Richardson,

text: and

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text: solutions

text: of

text: general

text: one-dimensional

text: spatially

text: coupled

text: systems,”

text: preprint

text: V.

text: Aref,

text: L.

text: Schmalen,

text: and

text: S.

text: ten

text: Brink,

text: “On

text: the

text: convergence

text: speed

text: of

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text: ensembles,”

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text: Conference

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text: Control,

text: and

text: Comp.

text: IEEE,

text: pp.

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text: Chung

text: and

text: G.

text: D.

text: J.

text: Forney,

text: “On

text: the

text: capacity

text: of

text: low-density

text: parity-check

text: codes,”

text: in

text: International

text: Symposium

text: on

text: Information

text: Theory

text: Proceedings

text: (ISIT).

text: IEEE,

text: p.

text: S.-Y.

text: Chung,

text: “On

text: the

text: construction

text: of

text: some

text: capacity-approaching

text: coding

text: schemes,”

text: Ph.D.

text: dissertation,

text: MIT,

text: P.

text: Olmos

text: and

text: R.

text: Urbanke,

text: “A

text: scaling

text: law

text: to

text: predict

text: the

text: ﬁnite-length

text: performance

text: of

text: spatially-coupled

text: LDPC

text: codes,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: A.

text: Yedla,

text: Y.-Y.

text: Jian,

text: P.

text: S.

text: Nguyen,

text: and

text: H.

text: D.

text: Pﬁster,

text: “A

text: simple

text: proof

text: of

text: Maxwell

text: saturation

text: for

text: coupled

text: scalar

text: recursions,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: K.

text: Takeuchi,

text: T.

text: Tanaka,

text: and

text: T.

text: Kawabata,

text: “A

text: phenomenological

text: study

text: on

text: threshold

text: improvement

text: via

text: spatial

text: coupling,”

text: IEICE

text: Transactions

text: on

text: Fundamentals

text: of

text: Electronics,

text: Communications

text: and

text: Computer

text: Sciences,

text: vol.

text: no.

text: pp.

text: R.

text: El-Khatib,

text: N.

text: Macris,

text: and

text: R.

text: Urbanke,

text: “Displacement

text: convexity,

text: a

text: useful

text: framework

text: for

text: the

text: study

text: of

text: spatially

text: coupled

text: codes,”

text: in

text: Information

text: Theory

text: Workshop

text: (ITW),

text: Sevilla,

text: Spain.

text: IEEE,

text: pp.

text: R.

text: El-Khatib,

text: N.

text: Macris,

text: T.

text: Richardson,

text: and

text: R.

text: Urbanke,

text: “Analysis

text: of

text: coupled

text: scalar

text: systems

text: by

text: displacement

text: convexity,”

text: in

text: International

text: Symposium

text: on

text: Information

text: Theory

text: Proceedings

text: (ISIT),

text: Honolulu,

text: Hawaii.

text: IEEE,

text: pp.

text: R.

text: El-Khatib

text: and

text: N.

text: Macris,

text: “The

text: velocity

text: of

text: the

text: decoding

text: wave

text: for

text: spatially

text: coupled

text: codes

text: on

text: bms

text: channels,”

text: in

text: Information

text: Theory

text: Proceedings

text: (ISIT),

text: IEEE

text: International

text: Symposium

text: on.

text: IEEE,

text: pp.

text: “The

text: velocity

text: of

text: the

text: propagating

text: wave

text: for

text: general

text: coupled

text: scalar

text: systems,”

text: in

text: Information

text: Theory

text: Workshop,

text: Cambridge,

text: UK,

text: T.

text: Richardson

text: and

text: R.

text: Urbanke,

text: Modern

text: coding

text: theory.

text: Cambridge

text: University

text: Press,

text: A.

text: Giurgiu,

text: N.

text: Macris,

text: and

text: R.

text: Urbanke,

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text: proof

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text: Theory,

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text: “How

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text: proof

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text: “And

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text: proof

text: technique,”

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text: { } × { } { } × where µ and ν are probability distributions on 0, 1 n 0,√1 n and 0, 1 { } ≈ ≈ 0, 1 , respectively. Here, 2 log 3 3.1699 and 2 log(1 + 2/ 3) 2.2150. Since randomized/distributional communication complexity of the equal- ity function is constant [10], these improvements are meaningful. If the fac- tor 2 is missing, the above two lower bounds for the equality function are smaller than 2 for ρ = 1. In this sense, this improvement is signiﬁcant. By applying an argument for generalized discrepancy theory [4, 11], we can further improve this bound, and obtain a class of lower bounds for ran- domized communication complexity including an exact Linial and Shraib- man’s lower bound as a special case. { } × Theorem 2 (Linial and Shraibman’s lower bound [14, 11]). For any f : 0, 1 n { } → { } ∈ 0, 1 n 0, 1 and (cid:15) [0, 1/2], − E − ≥ (1 (cid:15)) ∼ [f(x, y)h(x, y)] (cid:15) (x,y) µ R (f) 2 log max ∗ (1) (cid:15) h,µ β (h, µ) where R (f) is the randomized communication complexity of f with an (cid:15) { } × { } → { } error probability at most (cid:15), where h: 0, 1 n 0, 1 n 0, 1 , and where { } × { } µ is a probability distribution on 0, 1 n 0, 1 n. While the original proof of Theorem 2 uses factorization norms of matri- ces, the proof in this paper is based on Buhrman’s argument, which derived Theorem 2 in an intuitive way without the constant factor 2 [11, 4]. For obtaining the constant factor 2, we use a concept of nonlocal boxes and Paw(cid:32)lowski et al.’s idea, which showed the violation of information causality in superquantum theories [18]. In the generalized lower bounds, the con- ∗ stant factor 2 is replaced by larger constants while β (h, µ) is also replaced by larger quantities. Although any concrete lower bound by the generalized lower bound is not obtained in this paper, the generalized lower bound may improve Linial and Shraibman’s lower bound at most a constant factor. 2 Preliminaries 2.1 Nonlocal box The nonlocal box is an abstract device with two input ports and two ∈ { } ∈ { } output ports. When x 0, 1 and y 0, 1 are given to a nonlocal box, ∈ { } ∈ { } a nonlocal box randomly outputs a 0, 1 and b 0, 1 . A nonlocal box | is speciﬁed by a conditional probability distribution p(a, b x, y). Here, a 3 ##LTLine## ##LTLine##

text: where

text: µ

text: and

text: ν

text: are

text: probability

text: distributions

text: on

text: n

text: n

text: and

text: respectively.

text: Here,

text: log

text: and

text: Since

text: randomized/distributional

text: communication

text: complexity

text: of

text: the

text: equal-

text: ity

text: function

text: is

text: constant

text: these

text: improvements

text: are

text: meaningful.

text: If

text: the

text: fac-

text: tor

text: is

text: missing,

text: the

text: above

text: two

text: lower

text: bounds

text: for

text: the

text: equality

text: function

text: are

text: smaller

text: than

text: for

text: ρ

text: In

text: this

text: sense,

text: this

text: improvement

text: is

text: signiﬁcant.

text: By

text: applying

text: an

text: argument

text: for

text: generalized

text: discrepancy

text: theory

text: we

text: can

text: further

text: improve

text: this

text: bound,

text: and

text: obtain

text: a

text: class

text: of

text: lower

text: bounds

text: for

text: ran-

text: domized

text: communication

text: complexity

text: including

text: an

text: exact

text: Linial

text: and

text: Shraib-

text: man’s

text: lower

text: bound

text: as

text: a

text: special

text: case.

text: Theorem

text: (Linial

text: and

text: Shraibman’s

text: lower

text: bound

text: For

text: any

text: f

text: n

text: n

text: and

text: E

text: [f(x,

text: y)h(x,

text: y)]

text: (x,y)

text: µ

text: R

text: (f)

text: log

text: max

text: h,µ

text: β

text: (h,

text: µ)

text: where

text: R

text: (f)

text: is

text: the

text: randomized

text: communication

text: complexity

text: of

text: f

text: with

text: an

text: error

text: probability

text: at

text: most

text: where

text: h:

text: n

text: n

text: and

text: where

text: µ

text: is

text: a

text: probability

text: distribution

text: on

text: n

text: n.

text: While

text: the

text: original

text: proof

text: of

text: Theorem

text: uses

text: factorization

text: norms

text: of

text: matri-

text: ces,

text: the

text: proof

text: in

text: this

text: paper

text: is

text: based

text: on

text: Buhrman’s

text: argument,

text: which

text: derived

text: Theorem

text: in

text: an

text: intuitive

text: way

text: without

text: the

text: constant

text: factor

text: For

text: obtaining

text: the

text: constant

text: factor

text: we

text: use

text: a

text: concept

text: of

text: nonlocal

text: boxes

text: and

text: et

text: al.’s

text: idea,

text: which

text: showed

text: the

text: violation

text: of

text: information

text: causality

text: in

text: superquantum

text: theories

text: In

text: the

text: generalized

text: lower

text: bounds,

text: the

text: con-

text: stant

text: factor

text: is

text: replaced

text: by

text: larger

text: constants

text: while

text: β

text: (h,

text: µ)

text: is

text: also

text: replaced

text: by

text: larger

text: quantities.

text: Although

text: any

text: concrete

text: lower

text: bound

text: by

text: the

text: generalized

text: lower

text: bound

text: is

text: not

text: obtained

text: in

text: this

text: paper,

text: the

text: generalized

text: lower

text: bound

text: may

text: improve

text: Linial

text: and

text: Shraibman’s

text: lower

text: bound

text: at

text: most

text: a

text: constant

text: factor.

text: Preliminaries

text: Nonlocal

text: box

text: The

text: nonlocal

text: box

text: is

text: an

text: abstract

text: device

text: with

text: two

text: input

text: ports

text: and

text: two

text: output

text: ports.

text: When

text: x

text: and

text: y

text: are

text: given

text: to

text: a

text: nonlocal

text: box,

text: a

text: nonlocal

text: box

text: randomly

text: outputs

text: a

text: and

text: b

text: A

text: nonlocal

text: box

text: is

text: speciﬁed

text: by

text: a

text: conditional

text: probability

text: distribution

text: p(a,

text: b

text: x,

text: y).

text: Here,

text: a

text: ##LTLine##

text: ##LTLine##


text: ˙ 18 M. M. Serim, S¸. S¸ahiner, D. C¸erri–Serim, S. C¸. Inam, and A. Baykal ˙ 2003; Inam & Baykal 2005) and LMC X-4 (Naik & Paul 2003). This correlation might be an indication of the fact that the iron lines originates from the cool matter near the neutron star (Makishima 1986). Pulse phase resolved spectra of the source are also studied. Using the same spectral model used for the analysis of time resolved spectra, it is found that the spectral bin corresponding to the pulse minimum has signiﬁcantly higher cutoﬀ energy, lower photon index and lower e-folding energy indicating a harder spectrum for pulse minimum. Moreover, the same pulse phase shows an increase in equivalent width of Fe line whithout any signiﬁcant change in line ﬂux. A constant line ﬂux over all the pulse phases might be an indication of the fact that the line emitting region surrounds the pulsar, whereas broadening of Fe line on continuum of minimum phase might be a consequence of the blending of Kα and Kβ lines at 6.4 and 7.05 keV, while Kβ at 7.05 keV is smothered due to enhanced emission on the continuum of other phases. ACKNOWLEDGMENT We acknowledge support from TU¨BI˙TAK, the Scientiﬁc and Technological Research Council of Turkey through the research project MFAG 114F345. We would like to thank the anonymous referee for valuable comments which helped to improve the manuscript. REFERENCES Acuner Z., I˙nam SC¸, S¸ahiner S¸., Serim M.M., Baykal A., Swank J., 2014, MNRAS, 444, 457 Baykal A., O¨gelman H., 1993, A&A, 267, 1 Baykal A., 1997, A&A, 319, 515 Baykal A., 2000, MNRAS, 313, 637 Baykal A., I˙nam C¸., Alpar M.A., in’t Zand J., Strohmayer T., 2001, MNRAS, 327, 1269 Baykal A., I˙nam S.C¸., Beklen E., 2006, MNRAS, 369, 176 Baykal A., I˙nam S.C¸., Stark M. J., Heﬀner C. M., Erkoca A. E., Swank J. H., 2007, MNRAS, 374, 1108 Bildsten L., Chakrabarty D., Chiu J., Finger M.H., Koh D.T., Nelson R.B., Prince T.A, Rubin B.C, Scott, D.M., Stollberg M., Vaughan, B.A., Wilson C.A., Wilson R.B., 1997, ApJ, 113, 367 Braga J., Pereira M.G., Jablonski F.J, 2000, AIPC, 510, 188 Chakrabarty D., Roche P., 1997a, ApJ, 489, 254 Chakrabarty D., Bildsten L., Finger M.H., Grunsfeld J.M., Koh, D.T., Nelson R.W., Prince T.A., Vaughan B.A., Wilson R.B., 1997b, ApJL, 481, 101 Corbet R.H.D., Sokoloski J.L., Mukai K., Markwardt C.B., Tueller J., 2008, ApJ, 675, 1424 Cordes J.M., 1980, ApJ, 237, 216 Cui W., 1997, ApJL, 482, 163 Cui W., Smith B., 2004, ApJ, 602, 320 Cutler E.P, Dennis B.R., Dolan J.F., 1986, ApJ, 300, 551 Deeter J. E., 1984, ApJ, 281, 482 MNRAS 000, 1–19 (2017)

text: M.

text: M.

text: Serim,

text: S¸.

text: S¸ahiner,

text: D.

text: C¸erri–Serim,

text: S.

text: C¸.

text: Inam,

text: and

text: A.

text: Baykal

text: Inam

text: Baykal

text: and

text: LMC

text: (Naik

text: Paul

text: This

text: correlation

text: might

text: be

text: an

text: indication

text: of

text: the

text: fact

text: that

text: the

text: iron

text: lines

text: originates

text: from

text: the

text: cool

text: matter

text: near

text: the

text: neutron

text: star

text: (Makishima

text: Pulse

text: phase

text: resolved

text: spectra

text: of

text: the

text: source

text: are

text: also

text: studied.

text: Using

text: the

text: same

text: spectral

text: model

text: used

text: for

text: the

text: analysis

text: of

text: time

text: resolved

text: spectra,

text: it

text: is

text: found

text: that

text: the

text: spectral

text: bin

text: corresponding

text: to

text: the

text: pulse

text: minimum

text: has

text: signiﬁcantly

text: higher

text: cutoﬀ

text: energy,

text: lower

text: photon

text: index

text: and

text: lower

text: e-folding

text: energy

text: indicating

text: a

text: harder

text: spectrum

text: for

text: pulse

text: minimum.

text: Moreover,

text: the

text: same

text: pulse

text: phase

text: shows

text: an

text: increase

text: in

text: equivalent

text: width

text: of

text: Fe

text: line

text: whithout

text: any

text: signiﬁcant

text: change

text: in

text: line

text: ﬂux.

text: A

text: constant

text: line

text: ﬂux

text: over

text: all

text: the

text: pulse

text: phases

text: might

text: be

text: an

text: indication

text: of

text: the

text: fact

text: that

text: the

text: line

text: emitting

text: region

text: surrounds

text: the

text: pulsar,

text: whereas

text: broadening

text: of

text: Fe

text: line

text: on

text: continuum

text: of

text: minimum

text: phase

text: might

text: be

text: a

text: consequence

text: of

text: the

text: blending

text: of

text: Kα

text: and

text: Kβ

text: lines

text: at

text: and

text: keV,

text: while

text: Kβ

text: at

text: keV

text: is

text: smothered

text: due

text: to

text: enhanced

text: emission

text: on

text: the

text: continuum

text: of

text: other

text: phases.

text: ACKNOWLEDGMENT

text: We

text: acknowledge

text: support

text: from

text: TU¨BI˙TAK,

text: the

text: Scientiﬁc

text: and

text: Technological

text: Research

text: Council

text: of

text: Turkey

text: through

text: the

text: research

text: project

text: MFAG

text: We

text: would

text: like

text: to

text: thank

text: the

text: anonymous

text: referee

text: for

text: valuable

text: comments

text: which

text: helped

text: to

text: improve

text: the

text: manuscript.

text: REFERENCES

text: Acuner

text: Z.,

text: I˙nam

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text: S¸ahiner

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text: ApJ,

text: MNRAS


text: × both on a two-dimensional rectangular domain [0, 1] [0, 1]. For the second × case, ε has a value of 100 in the lower left quadrant [0, 0.25] [0, 0.25], and a value of 1 in the rest of the domain. In both cases, we pick the following solution for u | − | | − | u(r) = exp( (cid:126)r (cid:126)r /σ) + exp( (cid:126)r (cid:126)r /σ), (11) 1 2 where r(cid:126) = (0.25, 0.25), r(cid:126) = (0.75, 0.75) and σ = 0.04. An analytic ex- 1 2 pression for the source term ρ is obtained by plugging the solution in equations (9) and (10) (note that jumps in ε also contribute to the source term ρ). The solution is used to set Dirichlet boundary conditions. For these examples, we have used N = 2, N = 2 and N = 4 smoothing steps, and boxes with down up base 82 cells. | | The reﬁnement criterion is based on the source term ρ: reﬁne if ∆x2 ρ /ε > − 10 3, where ε is one for the ﬁrst problem. The resulting mesh spacing, which is the same for both problems, is shown in ﬁgure 6a. Figure 6b shows that in both cases, one FMG (full multigrid) cycle is enough to achieve convergence up to the discretization error. Consecutive FMG cycles further reduce the residual r = −∇ ρ 2u. The convergence behaviour is similar for both cases, with each iteration reducing the residual by a factor of about 0.07. This factor decreases when more smoothing steps are taken and when a higher order prolongation or restriction method is used. For this example we have used ﬁrst order prolongation and simple averaging for restriction, as discussed in section c. The oﬀset between the lines is caused by the ε = 100 region, which locally ampliﬁes the source term by a factor of 100. 4.2. Multigrid performance and scaling Here we brieﬂy investigate the performance and scaling of the multigrid routines. Although the numbers presented here depend on the particular system and compiler used, they can be used to estimate feasible problem sizes with Aﬁvo. As a performance test we use a 3D Poisson problem ∇ 2φ = 1, on a domain [0, 1]3 with φ = 0 at the boundaries. For simplicity (and for com- parison with other methods), the domain is uniformly reﬁned up to a resolution of 5123 cells. Figure 7 shows the duration of a single FMG cycle versus the number of processors, again using N = 2, N = 2 and N = 4 smoothing steps. down up base Curves are shown for box sizes of 83, 163 and 323, which aﬀect the overhead of the adaptive octree mesh. The runs were performed on a node with Xeon E5-2680 processors (2.8 GHz, 20 cores per node). The maximal speedups are about 4.9 (83 case), 5.4 (163 case) and 5.7 (323 case). The performance of the geometric multigrid algorithm, which performs only a few additions and multiplications per cell during each smoothing step, is 17

text: both

text: on

text: a

text: two-dimensional

text: rectangular

text: domain

text: For

text: the

text: second

text: case,

text: ε

text: has

text: a

text: value

text: of

text: in

text: the

text: lower

text: left

text: quadrant

text: and

text: a

text: value

text: of

text: in

text: the

text: rest

text: of

text: the

text: domain.

text: In

text: both

text: cases,

text: we

text: pick

text: the

text: following

text: solution

text: for

text: u

label: equation

text: u(r)

text: exp(

text: /σ)

text: exp(

text: /σ),

text: where

text: and

text: σ

text: An

text: analytic

text: ex-

text: pression

text: for

text: the

text: source

text: term

text: ρ

text: is

text: obtained

text: by

text: plugging

text: the

text: solution

text: in

text: equations

text: and

text: (note

text: that

text: jumps

text: in

text: ε

text: also

text: contribute

text: to

text: the

text: source

text: term

text: ρ).

text: The

text: solution

text: is

text: used

text: to

text: set

text: Dirichlet

text: boundary

text: conditions.

text: For

text: these

text: examples,

text: we

text: have

text: used

text: N

text: N

text: and

text: N

text: smoothing

text: steps,

text: and

text: boxes

text: with

text: down

text: up

text: base

text: cells.

text: The

text: reﬁnement

text: criterion

text: is

text: based

text: on

text: the

text: source

text: term

text: ρ:

text: reﬁne

text: if

text: ρ

text: /ε

text: where

text: ε

text: is

text: one

text: for

text: the

text: ﬁrst

text: problem.

text: The

text: resulting

text: mesh

text: spacing,

text: which

text: is

text: the

text: same

text: for

text: both

text: problems,

text: is

text: shown

text: in

text: ﬁgure

text: Figure

text: shows

text: that

text: in

text: both

text: cases,

text: one

text: FMG

text: (full

text: multigrid)

text: cycle

text: is

text: enough

text: to

text: achieve

text: convergence

text: up

text: to

text: the

text: discretization

text: error.

text: Consecutive

text: FMG

text: cycles

text: further

text: reduce

text: the

text: residual

text: r

text: ρ

text: The

text: convergence

text: behaviour

text: is

text: similar

text: for

text: both

text: cases,

text: with

text: each

text: iteration

text: reducing

text: the

text: residual

text: by

text: a

text: factor

text: of

text: about

text: This

text: factor

text: decreases

text: when

text: more

text: smoothing

text: steps

text: are

text: taken

text: and

text: when

text: a

text: higher

text: order

text: prolongation

text: or

text: restriction

text: method

text: is

text: used.

text: For

text: this

text: example

text: we

text: have

text: used

text: ﬁrst

text: order

text: prolongation

text: and

text: simple

text: averaging

text: for

text: restriction,

text: as

text: discussed

text: in

text: section

text: c.

text: The

text: oﬀset

text: between

text: the

text: lines

text: is

text: caused

text: by

text: the

text: ε

text: region,

text: which

text: locally

text: ampliﬁes

text: the

text: source

text: term

text: by

text: a

text: factor

text: of

text: Multigrid

text: performance

text: and

text: scaling

text: Here

text: we

text: brieﬂy

text: investigate

text: the

text: performance

text: and

text: scaling

text: of

text: the

text: multigrid

text: routines.

text: Although

text: the

text: numbers

text: presented

text: here

text: depend

text: on

text: the

text: particular

text: system

text: and

text: compiler

text: used,

text: they

text: can

text: be

text: used

text: to

text: estimate

text: feasible

text: problem

text: sizes

text: with

text: Aﬁvo.

text: As

text: a

text: performance

text: test

text: we

text: use

text: a

text: Poisson

text: problem

text: 2φ

text: on

text: a

text: domain

text: with

text: φ

text: at

text: the

text: boundaries.

text: For

text: simplicity

text: (and

text: for

text: com-

text: parison

text: with

text: other

text: methods),

text: the

text: domain

text: is

text: uniformly

text: reﬁned

text: up

text: to

text: a

text: resolution

text: of

text: cells.

text: Figure

text: shows

text: the

text: duration

text: of

text: a

text: single

text: FMG

text: cycle

text: versus

text: the

text: number

text: of

text: processors,

text: again

text: using

text: N

text: N

text: and

text: N

text: smoothing

text: steps.

text: down

text: up

text: base

text: Curves

text: are

text: shown

text: for

text: box

text: sizes

text: of

text: and

text: which

text: aﬀect

text: the

text: overhead

text: of

text: the

text: adaptive

text: octree

text: mesh.

text: The

text: runs

text: were

text: performed

text: on

text: a

text: node

text: with

text: Xeon

text: processors

text: GHz,

text: cores

text: per

text: node).

text: The

text: maximal

text: speedups

text: are

text: about

text: case),

text: case)

text: and

text: case).

text: The

text: performance

text: of

text: the

text: geometric

text: multigrid

text: algorithm,

text: which

text: performs

text: only

text: a

text: few

text: additions

text: and

text: multiplications

text: per

text: cell

text: during

text: each

text: smoothing

text: step,

text: is


text: 1e20 1e18 1e16 1e14 y 1e12 x electrons (cm-3) 2.0e7 1.5e7 1 cm 1.0e7 0.5e7 0 E (V/m) 2 ns 4 ns 6 ns 8 ns Figure 8: The evolution of the electron density (top) and electric ﬁeld (bottom) in a 2D electric discharge simulation in nitrogen at standard temperature and pressure. The discharge started from a pre-ionized layer, which destabilizes into streamer channels. A zoom-in of the mesh around a streamer head at t = 8ns is shown in ﬁgure 9. for dereﬁnement when ∆x < 25 µm and α∆x < 0.1. The quadtree mesh for this example was constructed from boxes containing 82 cells. The model described above is used to simulate discharges in a domain of − (1 cm)2, see ﬁgure 8. Initially, a density n of approximately 1015 cm 3 electrons 0 and ions is present between y = 9 mm and y = 9.5 mm, elsewhere the density is zero. The precise density in each cell is drawn using random numbers, by taking samples from a normal distribution with mean and variance n ∆x3, with ≈ (cid:29) 0 ∆x 9.8µm in the region of the initial condition. For n ∆x3 1, as we have 0 here, this approximates the Poisson distribution of physical particle noise (when the simulation would be truly 3D). At y = 1 cm the domain is grounded, and at y = 0 a background ﬁeld of 8 MV/m is applied through a Neumann condition for the electric potential; therefore the electrons drift downward in the ﬁeld. The electron and ion density at the y-boundaries are set to zero, and the domain has periodic boundary conditions in the x-direction. Figure 8 shows how the electron density and the electric ﬁeld evolve in time. At ﬁrst, the pre-ionized layer grows rather homogeneously downwards due to electron drift and creation of more charges through impact ionization, but because small inhomogeneities locally enhance the electric ﬁeld [30], the layer quickly destabilizes into streamer channels. The faster channels electrically screen the slower ones, reducing the number of active channels over time. Figure 9 shows a zoom of the adaptively reﬁned mesh at t = 8 ns. 5. Conclusion & outlook This paper describes Aﬁvo, a framework for parallel simulations on adap- tively reﬁned quadtree/octree grids with a geometric multigrid solver. We have 20 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

font: default

text: y

text: x

text: electrons

text: cm

text: E

text: (V/m)

text: ns

text: ns

text: ns

text: ns

text: Figure

text: The

label: caption

text: evolution

text: of

text: the

text: electron

text: density

text: (top)

text: and

text: electric

text: ﬁeld

text: (bottom)

text: in

text: a

text: electric

text: discharge

text: simulation

text: in

text: nitrogen

text: at

text: standard

text: temperature

text: and

text: pressure.

text: The

text: discharge

text: started

text: from

text: a

text: pre-ionized

text: layer,

text: which

text: destabilizes

text: into

text: streamer

text: channels.

text: A

text: zoom-in

text: of

text: the

text: mesh

text: around

text: a

text: streamer

text: head

text: at

text: t

text: is

text: shown

text: in

text: ﬁgure

text: for

text: dereﬁnement

text: when

text: ∆x

text: µm

text: and

text: α∆x

text: The

text: quadtree

text: mesh

text: for

text: this

text: example

text: was

text: constructed

text: from

text: boxes

text: containing

text: cells.

text: The

text: model

text: described

text: above

text: is

text: used

text: to

text: simulate

text: discharges

text: in

text: a

text: domain

text: of

text: see

text: ﬁgure

text: Initially,

text: a

text: density

text: n

text: of

text: approximately

text: cm

text: electrons

text: and

text: ions

text: is

text: present

text: between

text: y

text: mm

text: and

text: y

text: mm,

text: elsewhere

text: the

text: density

text: is

text: zero.

text: The

text: precise

text: density

text: in

text: each

text: cell

text: is

text: drawn

text: using

text: random

text: numbers,

text: by

text: taking

text: samples

text: from

text: a

text: normal

text: distribution

text: with

text: mean

text: and

text: variance

text: n

text: with

text: ∆x

text: in

text: the

text: region

text: of

text: the

text: initial

text: condition.

text: For

text: n

text: as

text: we

text: have

text: here,

text: this

text: approximates

text: the

text: Poisson

text: distribution

text: of

text: physical

text: particle

text: noise

text: (when

text: the

text: simulation

text: would

text: be

text: truly

text: At

text: y

text: cm

text: the

text: domain

text: is

text: grounded,

text: and

text: at

text: y

text: a

text: background

text: ﬁeld

text: of

text: MV/m

text: is

text: applied

text: through

text: a

text: Neumann

text: condition

text: for

text: the

text: electric

text: potential;

text: therefore

text: the

text: electrons

text: drift

text: downward

text: in

text: the

text: ﬁeld.

text: The

text: electron

text: and

text: ion

text: density

text: at

text: the

text: y-boundaries

text: are

text: set

text: to

text: zero,

text: and

text: the

text: domain

text: has

text: periodic

text: boundary

text: conditions

text: in

text: the

text: x-direction.

text: Figure

text: shows

text: how

text: the

text: electron

text: density

text: and

text: the

text: electric

text: ﬁeld

text: evolve

text: in

text: time.

text: At

text: ﬁrst,

text: the

text: pre-ionized

text: layer

text: grows

text: rather

text: homogeneously

text: downwards

text: due

text: to

text: electron

text: drift

text: and

text: creation

text: of

text: more

text: charges

text: through

text: impact

text: ionization,

text: but

text: because

text: small

text: inhomogeneities

text: locally

text: enhance

text: the

text: electric

text: ﬁeld

text: the

text: layer

text: quickly

text: destabilizes

text: into

text: streamer

text: channels.

text: The

text: faster

text: channels

text: electrically

text: screen

text: the

text: slower

text: ones,

text: reducing

text: the

text: number

text: of

text: active

text: channels

text: over

text: time.

text: Figure

text: shows

text: a

text: zoom

text: of

text: the

text: adaptively

text: reﬁned

text: mesh

text: at

text: t

text: ns.

text: Conclusion

text: outlook

text: This

text: paper

text: describes

text: Aﬁvo,

text: a

text: framework

text: for

text: parallel

text: simulations

text: on

text: adap-

text: tively

text: reﬁned

text: quadtree/octree

text: grids

text: with

text: a

text: geometric

text: multigrid

text: solver.

text: We

text: have

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: for sprites and lightning, Journal of Geophysical Research 115, a00E43. doi:10.1029/2009ja014867. URL http://dx.doi.org/10.1029/2009JA014867 [5] S. Pancheshnyi, P. S´egur, J. Capeill`ere, A. Bourdon, Numerical sim- ulation of ﬁlamentary discharges with parallel adaptive mesh reﬁne- ment, Journal of Computational Physics 227 (13) (2008) 6574–6590. doi:10.1016/j.jcp.2008.03.020. URL http://dx.doi.org/10.1016/j.jcp.2008.03.020 [6] C. Montijn, W. Hundsdorfer, U. Ebert, An adaptive grid reﬁnement strat- egy for the simulation of negative streamers, Journal of Computational Physics 219 (2) (2006) 801–835. doi:10.1016/j.jcp.2006.04.017. URL http://dx.doi.org/10.1016/j.jcp.2006.04.017 [7] C. Li, U. Ebert, W. Hundsdorfer, Spatially hybrid computations for streamer discharges: II. fully 3D simulations, Journal of Computational Physics 231 (3) (2012) 1020–1050. doi:10.1016/j.jcp.2011.07.023. URL http://dx.doi.org/10.1016/j.jcp.2011.07.023 [8] A. Luque, U. Ebert, Density models for streamer discharges: Beyond cylindrical symmetry and homogeneous media, Journal of Computational Physics 231 (3) (2012) 904–918. doi:10.1016/j.jcp.2011.04.019. URL http://dx.doi.org/10.1016/j.jcp.2011.04.019 [9] S. Nijdam, J. Teunissen, E. Takahashi, U. Ebert, The role of free electrons in the guiding of positive streamers, Plasma Sources Science and Technol- ogy 25 (4) (2016) 044001. doi:10.1088/0963-0252/25/4/044001. URL http://dx.doi.org/10.1088/0963-0252/25/4/044001 [10] A. S. A. et al., Boxlib, [Online; accessed 22-July-2015]. URL https://ccse.lbl.gov/BoxLib/index.html [11] P. Colella, D. T. Graves, J. N. Johnson, N. D. Keen, T. J. Ligocki, D. F. Martin, P. W. McCorquodale, D. Modiano, P. O. Schwartz, T. D. Stern- berg, B. V. Straalen, Chombo software package for AMR applications - design document (2011). URL https://apdec.org/designdocuments/ChomboDoc/ChomboDesign/ chomboDesign.pdf [12] R. D. Hornung, A. M. Wissink, S. R. Kohn, Managing complex data and geometry in parallel structured amr applications, Engineering with Com- puters 22 (3-4) (2006) 181–195. doi:10.1007/s00366-006-0038-6. URL http://dx.doi.org/10.1007/s00366-006-0038-6 [13] P. MacNeice, K. M. Olson, C. Mobarry, R. de Fainchtein, C. Packer, Paramesh: A parallel adaptive mesh reﬁnement community toolkit, Com- puter Physics Communications 126 (3) (2000) 330–354. doi:10.1016/s0010- 4655(99)00501-9. URL http://dx.doi.org/10.1016/S0010-4655(99)00501-9 22

text: for

text: sprites

text: and

text: lightning,

text: Journal

text: of

text: Geophysical

text: Research

text: URL

text: S.

text: Pancheshnyi,

text: P.

text: S´egur,

text: J.

text: Capeill`ere,

text: A.

text: Bourdon,

text: Numerical

text: sim-

text: ulation

text: of

text: ﬁlamentary

text: discharges

text: with

text: parallel

text: adaptive

text: mesh

text: reﬁne-

text: ment,

text: Journal

text: of

text: Computational

text: Physics

text: URL

text: C.

text: Montijn,

text: W.

text: Hundsdorfer,

text: U.

text: Ebert,

text: An

text: adaptive

text: grid

text: reﬁnement

text: strat-

text: egy

text: for

text: the

text: simulation

text: of

text: negative

text: streamers,

text: Journal

text: of

text: Computational

text: Physics

text: URL

text: C.

text: Li,

text: U.

text: Ebert,

text: W.

text: Hundsdorfer,

text: Spatially

text: hybrid

text: computations

text: for

text: streamer

text: discharges:

text: II.

text: fully

text: simulations,

text: Journal

text: of

text: Computational

text: Physics

text: URL

text: A.

text: Luque,

text: U.

text: Ebert,

text: Density

text: models

text: for

text: streamer

text: discharges:

text: Beyond

text: cylindrical

text: symmetry

text: and

text: homogeneous

text: media,

text: Journal

text: of

text: Computational

text: Physics

text: URL

text: S.

text: Nijdam,

text: J.

text: Teunissen,

text: E.

text: Takahashi,

text: U.

text: Ebert,

text: The

text: role

text: of

text: free

text: electrons

text: in

text: the

text: guiding

text: of

text: positive

text: streamers,

text: Plasma

text: Sources

text: Science

text: and

text: Technol-

text: ogy

text: URL

text: A.

text: S.

text: A.

text: et

text: al.,

text: Boxlib,

text: [Online;

text: accessed

text: URL

text: P.

text: Colella,

text: D.

text: T.

text: Graves,

text: J.

text: N.

text: Johnson,

text: N.

text: D.

text: Keen,

text: T.

text: J.

text: Ligocki,

text: D.

text: F.

text: Martin,

text: P.

text: W.

text: McCorquodale,

text: D.

text: Modiano,

text: P.

text: O.

text: Schwartz,

text: T.

text: D.

text: Stern-

text: berg,

text: B.

text: V.

text: Straalen,

text: Chombo

text: software

text: package

text: for

text: AMR

text: applications

text: design

text: document

text: URL

text: chomboDesign.pdf

text: R.

text: D.

text: Hornung,

text: A.

text: M.

text: Wissink,

text: S.

text: R.

text: Kohn,

text: Managing

text: complex

text: data

text: and

text: geometry

text: in

text: parallel

text: structured

text: amr

text: applications,

text: Engineering

text: with

text: Com-

text: puters

text: URL

text: P.

text: MacNeice,

text: K.

text: M.

text: Olson,

text: C.

text: Mobarry,

text: R.

text: de

text: Fainchtein,

text: C.

text: Packer,

text: Paramesh:

text: A

text: parallel

text: adaptive

text: mesh

text: reﬁnement

text: community

text: toolkit,

text: Com-

text: puter

text: Physics

text: Communications

text: URL


text: – If all the 2D children of a box are ﬂagged for removal, and the box itself not for reﬁnement, then the children are removed. – Only leaves can be removed (because the grid changes by at most one level at a time). – Boxes cannot be removed if that would violate 2:1 balance. When boxes are added or removed in the reﬁnement procedure, the mesh connectivity is automatically updated, and the array containing all the boxes is automatically resized when necessary. The removal of boxes can create holes in this array, which are automatically ﬁlled when their number exceeds a threshold. The boxes are then also sorted (per level) according to their Morton index [19]. After updating the reﬁnement, information on the added and removed boxes per level is returned, so that a user can set values on the new boxes. Removed boxes are no longer present, so it is recommended that users perform a restriction operation (transfer from ﬁne to coarse) before adapting the mesh, as this is not done automatically. 2.4. Ghost cells The usage of ghost cells has two main advantages: algorithms can operate without special care for the boundaries, and they can do so in parallel. In Aﬁvo each box always has one layer of ghost cells around its cell-centered variables. For numerical operations that depend on the nearest neighbors, such as comput- ing a second order Laplacian or centered diﬀerences, one ghost cell is enough. When additional ghost values are required, these can directly be accessed due to the shared-memory parallelism. Aﬁvo supports a single layer of ghost cells around boxes, including corners (and edges in 3D). A user has to provide two routines for ﬁlling ghost cells on the sides of boxes, one for physical boundaries and one for reﬁnement boundaries. A couple of such routines are also included with the framework. For each box, ghost cells are ﬁrst ﬁlled on the sides, then on the edges (only present in 3D) and ﬁnally on the corners. For each side of a box, there are three options: • If there is a neighbor at the same reﬁnement level, copy from it. • If there is a physical boundary, call the user’s routine for boundary con- ditions. • If there is a reﬁnement boundary, call the user’s routine for reﬁnement boundaries. For the edge and corner ghost cells values are copied if a neighbor at the same reﬁnement level is present. If there is no such neighbor, for example due to a physical or reﬁnement boundary, these ghost cells are ﬁlled using linear extrapolation. The extrapolation procedure is illustrated in ﬁgure 3, for a corner ghost cell in 2D. A nice property of this approach is that if one later uses bilinear interpolation using the points (a, b, c, d) the result is equivalent to a linear interpolation based on points (a, b, c). 8

label: list

text: If

text: all

text: the

text: children

text: of

text: a

text: box

text: are

text: ﬂagged

text: for

text: removal,

text: and

text: the

text: box

text: itself

text: not

text: for

text: reﬁnement,

text: then

text: the

text: children

text: are

text: removed.

text: Only

text: leaves

text: can

text: be

text: removed

text: (because

text: the

text: grid

text: changes

text: by

text: at

text: most

text: one

text: level

text: at

text: a

text: time).

text: Boxes

text: cannot

text: be

text: removed

text: if

text: that

text: would

text: violate

text: balance.

text: When

text: boxes

text: are

text: added

text: or

text: removed

text: in

text: the

text: reﬁnement

text: procedure,

text: the

text: mesh

text: connectivity

text: is

text: automatically

text: updated,

text: and

text: the

text: array

text: containing

text: all

text: the

text: boxes

text: is

text: automatically

text: resized

text: when

text: necessary.

text: The

text: removal

text: of

text: boxes

text: can

text: create

text: holes

text: in

text: this

text: array,

text: which

text: are

text: automatically

text: ﬁlled

text: when

text: their

text: number

text: exceeds

text: a

text: threshold.

text: The

text: boxes

text: are

text: then

text: also

text: sorted

text: (per

text: level)

text: according

text: to

text: their

text: Morton

text: index

text: After

text: updating

text: the

text: reﬁnement,

text: information

text: on

text: the

text: added

text: and

text: removed

text: boxes

text: per

text: level

text: is

text: returned,

text: so

text: that

text: a

text: user

text: can

text: set

text: values

text: on

text: the

text: new

text: boxes.

text: Removed

text: boxes

text: are

text: no

text: longer

text: present,

text: so

text: it

text: is

text: recommended

text: that

text: users

text: perform

text: a

text: restriction

text: operation

text: (transfer

text: from

text: ﬁne

text: to

text: coarse)

text: before

text: adapting

text: the

text: mesh,

text: as

text: this

text: is

text: not

text: done

text: automatically.

text: Ghost

text: cells

text: The

text: usage

text: of

text: ghost

text: cells

text: has

text: two

text: main

text: advantages:

text: algorithms

text: can

text: operate

text: without

text: special

text: care

text: for

text: the

text: boundaries,

text: and

text: they

text: can

text: do

text: so

text: in

text: parallel.

text: In

text: Aﬁvo

text: each

text: box

text: always

text: has

text: one

text: layer

text: of

text: ghost

text: cells

text: around

text: its

text: cell-centered

text: variables.

text: For

text: numerical

text: operations

text: that

text: depend

text: on

text: the

text: nearest

text: neighbors,

text: such

text: as

text: comput-

text: ing

text: a

text: second

text: order

text: Laplacian

text: or

text: centered

text: diﬀerences,

text: one

text: ghost

text: cell

text: is

text: enough.

text: When

text: additional

text: ghost

text: values

text: are

text: required,

text: these

text: can

text: directly

text: be

text: accessed

text: due

text: to

text: the

text: shared-memory

text: parallelism.

text: Aﬁvo

text: supports

text: a

text: single

text: layer

text: of

text: ghost

text: cells

text: around

text: boxes,

text: including

text: corners

text: (and

text: edges

text: in

text: A

text: user

text: has

text: to

text: provide

text: two

text: routines

text: for

text: ﬁlling

text: ghost

text: cells

text: on

text: the

text: sides

text: of

text: boxes,

text: one

text: for

text: physical

text: boundaries

text: and

text: one

text: for

text: reﬁnement

text: boundaries.

text: A

text: couple

text: of

text: such

text: routines

text: are

text: also

text: included

text: with

text: the

text: framework.

text: For

text: each

text: box,

text: ghost

text: cells

text: are

text: ﬁrst

text: ﬁlled

text: on

text: the

text: sides,

text: then

text: on

text: the

text: edges

text: (only

text: present

text: in

text: and

text: ﬁnally

text: on

text: the

text: corners.

text: For

text: each

text: side

text: of

text: a

text: box,

text: there

text: are

text: three

text: options:

text: If

text: there

text: is

text: a

text: neighbor

text: at

text: the

text: same

text: reﬁnement

text: level,

text: copy

text: from

text: it.

text: If

text: there

text: is

text: a

text: physical

text: boundary,

text: call

text: the

text: user’s

text: routine

text: for

text: boundary

text: con-

text: ditions.

text: If

text: there

text: is

text: a

text: reﬁnement

text: boundary,

text: call

text: the

text: user’s

text: routine

text: for

text: reﬁnement

text: boundaries.

text: For

text: the

text: edge

text: and

text: corner

text: ghost

text: cells

text: values

text: are

text: copied

text: if

text: a

text: neighbor

text: at

text: the

text: same

text: reﬁnement

text: level

text: is

text: present.

text: If

text: there

text: is

text: no

text: such

text: neighbor,

text: for

text: example

text: due

text: to

text: a

text: physical

text: or

text: reﬁnement

text: boundary,

text: these

text: ghost

text: cells

text: are

text: ﬁlled

text: using

text: linear

text: extrapolation.

text: The

text: extrapolation

text: procedure

text: is

text: illustrated

text: in

text: ﬁgure

text: for

text: a

text: corner

text: ghost

text: cell

text: in

text: A

text: nice

text: property

text: of

text: this

text: approach

text: is

text: that

text: if

text: one

text: later

text: uses

text: bilinear

text: interpolation

text: using

text: the

text: points

text: (a,

text: b,

text: c,

text: d)

text: the

text: result

text: is

text: equivalent

text: to

text: a

text: linear

text: interpolation

text: based

text: on

text: points

text: (a,

text: b,

text: c).


text: (cid:112) (cid:28) In the inﬂation phase, c = (cid:15)M2, thus z = 2(cid:15)a2M2. We set (cid:15) 1 as a constant during 1 p p (cid:48)(cid:48) ≈ inﬂation. Then z /z (2 + 3(cid:15))/τ2(cid:2). The solution of Eq. (42) is (cid:3) √ − πτ · − · − u = C H(1)( kτ) + C H(2)( kτ) (48) 2 2 21 ν2 22 ν2 ≈ with ν 3/2 + (cid:15). 2 (cid:48) (cid:48) We require that u (τ ) = u (τ ) and u (τ ) = u (τ ), with τ approximately corre- 1 (cid:114)m 2 m 1 m 2 m m (cid:104) (cid:105) sponding to the beginning time of inﬂation phase, and we obtain − (cid:114) − i − π(cid:104) − − − − (cid:105) C = e ikτm 2kτ H(2−) ( kτ ) + (2ν 1 2ikτ )H(2)( kτ ) , (49) 21 4 2kτ m ν2 1 m 2 m ν2 m m − i − π − − − − C = e ikτm 2kτ H(1−) ( kτ ) + (2ν 1 2ikτ )H(1)( kτ ) . (50) 22 4 2kτ m ν2 1 m 2 m ν2 m m The power spectrum of ζ is given by · | − | (cid:28) (cid:0) P(cid:1)R = PRinf C C 2 , k aH , (51) 21 22 · − where PRinf = Hi2nf k 3 2ν2 is the power spectrum of scalar perturbation modes that 8π2M2(cid:15) aH p exit horizon during inﬂation. We see that for the perturbation modes exiting horizon in − (cid:28) | − | (cid:39) − ∼ the Genesis phase, kτ 1, C C 2 ( kτ )2, thus PR k2 is strong blue- m 21 22 m − (cid:29) tilted, while for the perturbation modes exiting horizon in the inﬂation phase, kτ 1, m | − | (cid:39) ∼ − − C C 2 1, thus PR k3 2ν2 = k 2(cid:15) is ﬂat with a slightly red tilt. 21 22 Tensor perturbation is unaﬀected by the R(3)δg00 operator. Its quadratic action is given in Eq. (4) with Q = 1 and c2 = 1. The spectrum of primordial GWs can be calculated T T (cid:48)(cid:48) (cid:48)(cid:48) similarly, see also Ref. [6]. Since z /z = a /a, we have T T · | − | (cid:28) (cid:0) (cid:1) P = P inf C C 2 , k aH , (52) T T 21 22 · − where P inf = 2Hi2nf k 3 2ν2 is the power spectrum of tensor perturbation modes that exit T π2M2 aH p horizon during inﬂation. Thus the spectrum of primordial GWs has a shape similar to that of the scalar perturbation. C. Numerical simulation 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: inﬂation

text: phase,

text: c

text: thus

text: z

text: We

text: set

text: as

text: a

text: constant

text: during

text: p

text: p

text: inﬂation.

text: Then

text: z

text: /z

text: The

text: solution

text: of

text: Eq.

text: is

text: πτ

text: u

text: C

text: kτ)

text: C

text: kτ)

text: ν2

text: ν2

text: with

text: ν

text: We

text: require

text: that

text: u

text: (τ

text: u

text: (τ

text: and

text: u

text: (τ

text: u

text: (τ

text: with

text: τ

text: approximately

text: corre-

text: m

text: m

text: m

text: m

text: sponding

text: to

text: the

text: beginning

text: time

text: of

text: inﬂation

text: phase,

text: and

text: we

text: obtain

text: i

text: C

text: e

text: ikτm

text: kτ

text: kτ

text: m

text: ν2

text: m

text: m

text: ν2

text: m

text: m

text: i

text: π

text: C

text: e

text: ikτm

text: kτ

text: kτ

text: m

text: ν2

text: m

text: m

text: ν2

text: m

text: m

text: The

text: power

text: spectrum

text: of

text: ζ

text: is

text: given

text: by

text: PRinf

text: C

text: C

text: k

text: aH

text: where

text: PRinf

text: k

text: is

text: the

text: power

text: spectrum

text: of

text: scalar

text: perturbation

text: modes

text: that

text: aH

text: p

text: exit

text: horizon

text: during

text: inﬂation.

text: We

text: see

text: that

text: for

text: the

text: perturbation

text: modes

text: exiting

text: horizon

text: in

text: the

text: Genesis

text: phase,

text: kτ

text: C

text: C

text: kτ

text: thus

text: PR

text: is

text: strong

text: blue-

text: m

text: m

text: tilted,

text: while

text: for

text: the

text: perturbation

text: modes

text: exiting

text: horizon

text: in

text: the

text: inﬂation

text: phase,

text: kτ

text: m

text: C

text: C

text: thus

text: PR

text: k

text: is

text: ﬂat

text: with

text: a

text: slightly

text: red

text: tilt.

text: Tensor

text: perturbation

text: is

text: unaﬀected

text: by

text: the

text: operator.

text: Its

text: quadratic

text: action

text: is

text: given

text: in

text: Eq.

text: with

text: Q

text: and

text: The

text: spectrum

text: of

text: primordial

text: GWs

text: can

text: be

text: calculated

text: T

text: T

text: similarly,

text: see

text: also

text: Ref.

text: Since

text: z

text: /z

text: a

text: /a,

text: we

text: have

text: T

text: T

text: P

text: P

text: inf

text: C

text: C

text: k

text: aH

text: T

text: T

text: where

text: P

text: inf

text: k

text: is

text: the

text: power

text: spectrum

text: of

text: tensor

text: perturbation

text: modes

text: that

text: exit

text: T

text: aH

text: p

text: horizon

text: during

text: inﬂation.

text: Thus

text: the

text: spectrum

text: of

text: primordial

text: GWs

text: has

text: a

text: shape

text: similar

text: to

text: that

text: of

text: the

text: scalar

text: perturbation.

text: C.

text: Numerical

text: simulation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: Y.

text: F.

text: Cai,

text: D.

text: A.

text: Easson

text: and

text: R.

text: Brandenberger,

text: JCAP

text: [hep-th]].

text: Y.

text: Wan,

text: T.

text: Qiu,

text: F.

text: P.

text: Huang,

text: Y.

text: F.

text: Cai,

text: H.

text: Li

text: and

text: X.

text: Zhang,

text: JCAP

text: no.

text: [gr-qc]].

text: A.

text: Nicolis,

text: R.

text: Rattazzi

text: and

text: E.

text: Trincherini,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: G.

text: W.

text: Horndeski,

text: Int.

text: J.

text: Theor.

text: Phys.

text: C.

text: Deﬀayet,

text: X.

text: Gao,

text: D.

text: A.

text: Steer

text: and

text: G.

text: Zahariade,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: T.

text: Kobayashi,

text: M.

text: Yamaguchi

text: and

text: J.

text: Yokoyama,

text: Prog.

text: Theor.

text: Phys.

text: [hep-th]].

text: M.

text: Libanov,

text: S.

text: Mironov

text: and

text: V.

text: Rubakov,

text: JCAP

text: no.

text: [hep-th]].

text: T.

text: Kobayashi,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-th]].

text: A.

text: Ijjas

text: and

text: P.

text: J.

text: Steinhardt,

text: Phys.

text: Lett.

text: B

text: [gr-qc]].

text: A.

text: Ijjas

text: and

text: P.

text: J.

text: Steinhardt,

text: Phys.

text: Rev.

text: Lett.

text: no.

text: [gr-qc]].

text: R.

text: Kolevatov

text: and

text: S.

text: Mironov,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-th]].

text: S.

text: Akama

text: and

text: T.

text: Kobayashi,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-

text: th]].

text: Y.

text: Cai,

text: Y.

text: Wan,

text: H.

text: G.

text: Li,

text: T.

text: Qiu

text: and

text: Y.

text: S.

text: Piao,

text: JHEP

text: [gr-qc]].

text: P.

text: Creminelli,

text: D.

text: Pirtskhalava,

text: L.

text: Santoni

text: and

text: E.

text: Trincherini,

text: JCAP

text: no.

text: [hep-th]].

text: C.

text: Cheung,

text: P.

text: Creminelli,

text: A.

text: L.

text: Fitzpatrick,

text: J.

text: Kaplan

text: and

text: L.

text: Senatore,

text: JHEP

text: [hep-th]].

text: G.

text: Gubitosi,

text: F.

text: Piazza

text: and

text: F.

text: Vernizzi,

text: JCAP

text: [JCAP

text: [hep-th]].

text: F.

text: Piazza

text: and

text: F.

text: Vernizzi,

text: Class.

text: Quant.

text: Grav.

text: [hep-th]].

text: J.

text: Gleyzes,

text: D.

text: Langlois,

text: F.

text: Piazza

text: and

text: F.

text: Vernizzi,

text: JCAP


text: In the cubic Galileon case, condition (9) is − − Hγ γ2 γ˙ > 0 . (15) ∼ In the bounce model, H < 0 in the contracting phase, and in the Genesis model, H 0 − − in the Genesis phase, both suggest Hγ γ2 γ˙ < 0 2. Thus c2 < 0 is inevitable in the s corresponding phases, so the nonsingular models based on the cubic Galileon is pathological. We see the Genesis model in the cubic Galileon version again in detail. During the slow ∼ − expansion (Genesis phase), H 1/( t)n with the constant n > 1. Thus γ˙ (cid:39) H˙ ∼ − − (cid:29) ( t)n 1 1, (16) Hγ H2 (cid:28) which implies Hγ γ˙ . Thus with (15), we see that c2 < 0 is inevitable in the slow expansion s (cid:28) phase. It seems that if n = 1, Hγ γ˙ might be avoided. However, when n = 1, we have − ∼ − → H = p/( t) and a 1/( t)p with constant p, thus a 0 in the inﬁnite past. From (16), we see that c2 > 0 requires p = H2/H˙ > 1. Therefore, the universe is singular, or from (cid:82) s another point of view, it is geodesically incomplete since the aﬃne parameter of the graviton → −∞ t geodesics f adt is ﬁnite for p > 1 when t . ti (cid:32) i (cid:33) In the EFT case, condition (9) requires Q˙ Q˙ − c2 γ2 − T γ + m˜ 4 γ T + Hγ γ˙ Q > 0 . (17) Q Q Q m˜ 4 T m˜ 4 m˜ 4 We might have c2 > 0, only if (considering only the case where either Q or Q is modiﬁed) s T m˜ 4 Q˙ − γ˙ T > c2 γ H + , (18) Q T γ T Q˙ c2 γ − γ˙ or m˜ 4 < T H + (initially Q < 0) . (19) Q Q γ m˜ 4 m˜ 4 m˜ 4 − Generally, Hγ + γ˙ > 0, as in Genesis model and bounce model. Thus the solution (18) ˙ suggests Q > 0, so that we will have Q = 0 in inﬁnite past. Thus based on (12) and (18), T T it seems that though the pathology can be cured by applying Q , Q = 0 is inevitable. A T T ∼ model with (18) has been proposed by Kobayashi [34] (Q −1 , p > n > 1). During the (cid:16) (cid:17) T ( t)p ∼ ∼ − Genesis γ H 1/( t)n, n > 1, (17) is − 1 γ˙ ˙ Q /Q = n/p < 1. (20) T T γ 2 In the case where γ grows from 0 initially, (15) is also obeyed no more. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: cubic

text: Galileon

text: case,

text: condition

text: is

text: Hγ

text: γ2

text: γ˙

text: In

text: the

text: bounce

text: model,

text: H

text: in

text: the

text: contracting

text: phase,

text: and

text: in

text: the

text: Genesis

text: model,

text: H

text: in

text: the

text: Genesis

text: phase,

text: both

text: suggest

text: Hγ

text: γ2

text: γ˙

text: Thus

text: is

text: inevitable

text: in

text: the

text: s

text: corresponding

text: phases,

text: so

text: the

text: nonsingular

text: models

text: based

text: on

text: the

text: cubic

text: Galileon

text: is

text: pathological.

text: We

text: see

text: the

text: Genesis

text: model

text: in

text: the

text: cubic

text: Galileon

text: version

text: again

text: in

text: detail.

text: During

text: the

text: slow

text: expansion

text: (Genesis

text: phase),

text: H

text: t)n

text: with

text: the

text: constant

text: n

text: Thus

text: γ˙

text: H˙

text: t)n

text: Hγ

text: which

text: implies

text: Hγ

text: γ˙

text: Thus

text: with

text: we

text: see

text: that

text: is

text: inevitable

text: in

text: the

text: slow

text: expansion

text: s

text: phase.

text: It

text: seems

text: that

text: if

text: n

text: Hγ

text: γ˙

text: might

text: be

text: avoided.

text: However,

text: when

text: n

text: we

text: have

text: H

text: p/(

text: t)

text: and

text: a

text: t)p

text: with

text: constant

text: p,

text: thus

text: a

text: in

text: the

text: inﬁnite

text: past.

text: From

text: we

text: see

text: that

text: requires

text: p

text: Therefore,

text: the

text: universe

text: is

text: singular,

text: or

text: from

text: s

text: another

text: point

text: of

text: view,

text: it

text: is

text: geodesically

text: incomplete

text: since

text: the

text: aﬃne

text: parameter

text: of

text: the

text: graviton

text: t

text: geodesics

text: f

text: adt

text: is

text: ﬁnite

text: for

text: p

text: when

text: t

text: ti

text: i

text: In

text: the

text: EFT

text: case,

text: condition

text: requires

text: Q˙

text: Q˙

text: γ2

text: T

text: γ

text: m˜

text: γ

text: T

text: Hγ

text: γ˙

text: Q

text: Q

text: Q

text: Q

text: m˜

text: T

text: m˜

text: m˜

text: We

text: might

text: have

text: only

text: if

text: (considering

text: only

text: the

text: case

text: where

text: either

text: Q

text: or

text: Q

text: is

text: modiﬁed)

text: s

text: T

text: m˜

text: Q˙

text: γ˙

text: T

text: γ

text: H

text: Q

text: T

text: γ

text: T

text: Q˙

text: γ

text: γ˙

text: or

text: m˜

text: T

text: H

text: (initially

text: Q

text: Q

text: Q

text: γ

text: m˜

text: m˜

text: m˜

text: Generally,

text: Hγ

text: γ˙

text: as

text: in

text: Genesis

text: model

text: and

text: bounce

text: model.

text: Thus

text: the

text: solution

text: suggests

text: Q

text: so

text: that

text: we

text: will

text: have

text: Q

text: in

text: inﬁnite

text: past.

text: Thus

text: based

text: on

text: and

text: T

text: T

text: it

text: seems

text: that

text: though

text: the

text: pathology

text: can

text: be

text: cured

text: by

text: applying

text: Q

text: Q

text: is

text: inevitable.

text: A

text: T

text: T

text: model

text: with

text: has

text: been

text: proposed

text: by

text: Kobayashi

text: (Q

text: −1

text: p

text: n

text: During

text: the

text: T

text: t)p

text: Genesis

text: γ

text: H

text: t)n,

text: n

text: is

text: γ˙

text: Q

text: /Q

text: n/p

text: T

text: T

text: γ

text: In

text: the

text: case

text: where

text: γ

text: grows

text: from

text: initially,

text: is

text: also

text: obeyed

text: no

text: more.

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text: ∼ Initially, Q −1 = 0. T ( t)p In solution (19), Q must cross 0 at t to Q > 0, as pointed out by Cai et.al [39] m˜ 4 m˜ 4 m˜ 4 and Creminelli et.al [40]. Around t , Q˙ > c2 γ must be satisﬁed. m˜ 4 m˜ 4 T In (17), if Q > 0 throughout, m˜ 4 Q˙ c2 γ − γ˙ m˜ 4 > T H + (21) Q Q γ m˜ 4 m˜ 4 is obtained. Thus, similar to (18), we have Q = 0 (which deﬁnitely requires γ = 0) in m˜ 4 ∼ − ∼ − the inﬁnite past. In the Genesis model, Q 1/( t)p and γ 1/( t)n with p > n, since m˜ 4 ˙ ˙ Q /Q > γ˙ /γ. However, p > n indicates Q < γ in the inﬁnite past (Q = 0), which m˜ 4 m˜ 4 m˜ 4 m˜ 4 violates the inequality (21). Thus Q > 0 throughout seems unworkable. m˜ 4 Nonsingular cubic Galileon models Initially γ < 0 γ > 0 throughout √ × Crossing 0 for γ? √ √ c2 < 0 is inevitable (“no-go”)? s (cid:39) − Phase with c2 < 0 (Pathological phase) γ˙ > 0 around γ 0 Hγ γ˙ < γ2 s Curing pathology in EFT Conditions of c2 > 0 (11) (17) s Applying Q (12) (18) T Applying Q (13) (19) m˜ 4 TABLE I: Pathology in nonsingular cubic Galileon cosmological models and its cure in EFT by either Q or Q . T m˜ 4 III. APPLICATION TO GENESIS-INFLATION In this section, we will build a nonsingular model with the solution (19), in which the slow-roll inﬂation is preceded by a Genesis phase. A Genesis phase is a slowly expanding (cid:28) − phase originating form the Minkowski vacuum with a drastic violation of NEC, i.e., (cid:15) 1, thus the energy density is increasing with the expansion of the universe and hence is free from the initial singularity [16][17] (see also [20]). As will be shown below, our model cannot only 7 ##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: Initially,

text: Q

text: −1

text: T

text: t)p

text: In

text: solution

text: Q

text: must

text: cross

text: at

text: t

text: to

text: Q

text: as

text: pointed

text: out

text: by

text: Cai

text: et.al

text: m˜

text: m˜

text: m˜

text: and

text: Creminelli

text: et.al

text: Around

text: t

text: Q˙

text: γ

text: must

text: be

text: satisﬁed.

text: m˜

text: m˜

text: T

text: In

text: if

text: Q

text: throughout,

text: m˜

text: Q˙

text: γ

text: γ˙

text: m˜

text: T

text: H

text: Q

text: Q

text: γ

text: m˜

text: m˜

text: is

text: obtained.

text: Thus,

text: similar

text: to

text: we

text: have

text: Q

text: (which

text: deﬁnitely

text: requires

text: γ

text: in

text: m˜

text: the

text: inﬁnite

text: past.

text: In

text: the

text: Genesis

text: model,

text: Q

text: t)p

text: and

text: γ

text: t)n

text: with

text: p

text: n,

text: since

text: m˜

text: Q

text: /Q

text: γ˙

text: /γ.

text: However,

text: p

text: n

text: indicates

text: Q

text: γ

text: in

text: the

text: inﬁnite

text: past

text: (Q

text: which

text: m˜

text: m˜

text: m˜

text: m˜

text: violates

text: the

text: inequality

text: Thus

text: Q

text: throughout

text: seems

text: unworkable.

text: m˜

text: Nonsingular

text: cubic

text: Galileon

text: models

text: Initially

text: γ

text: γ

text: throughout

text: Crossing

text: for

text: γ?

text: is

text: inevitable

text: (“no-go”)?

text: s

text: Phase

text: with

text: (Pathological

text: phase)

text: γ˙

text: around

text: γ

text: Hγ

text: γ˙

text: γ2

text: s

text: Curing

text: pathology

text: in

text: EFT

text: Conditions

text: of

text: s

text: Applying

text: Q

text: T

text: Applying

text: Q

text: m˜

text: TABLE

text: I:

text: Pathology

text: in

text: nonsingular

text: cubic

text: Galileon

text: cosmological

text: models

text: and

text: its

text: cure

text: in

text: EFT

text: by

text: either

text: Q

text: or

text: Q

text: T

text: m˜

text: III.

text: APPLICATION

text: TO

text: GENESIS-INFLATION

text: In

text: this

text: section,

text: we

text: will

text: build

text: a

text: nonsingular

text: model

text: with

text: the

text: solution

text: in

text: which

text: the

text: slow-roll

text: inﬂation

text: is

text: preceded

text: by

text: a

text: Genesis

text: phase.

text: A

text: Genesis

text: phase

text: is

text: a

text: slowly

text: expanding

text: phase

text: originating

text: form

text: the

text: Minkowski

text: vacuum

text: with

text: a

text: drastic

text: violation

text: of

text: NEC,

text: i.e.,

text: thus

text: the

text: energy

text: density

text: is

text: increasing

text: with

text: the

text: expansion

text: of

text: the

text: universe

text: and

text: hence

text: is

text: free

text: from

text: the

text: initial

text: singularity

text: (see

text: also

text: As

text: will

text: be

text: shown

text: below,

text: our

text: model

text: cannot

text: only

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