text: 16 Warren et al. could be adjusted to bring the CR-only case’s emission in line with the other two. We reiterate that thermal particles—which are certain to be present in the shocked plasma—participating in the photon production process yields large gains in emission with no tweaking of param- eters needed. Despite the large factor by which the TP & NL mod- els are brighter than the CR-only case, both are fainter in the Fermi-LAT band than most of the afterglows with observed GeV emission. There are reasons to suspect that the model presented here underpredicts the amount of high-energy emission, though. As discussed elsewhere, we assume a weaker magnetic ﬁeld near the subshock than is observed in PIC simulations (where (cid:15) can locally B exceed 0.1, as in Spitkovsky 2008; Sironi et al. 2013). The higher ﬁelds would allow for more, and higher-energy, synchrotron photons for a given electron. This increase in synchrotron photons would, of course, increase the Fig. 8.— Total photon ﬂux above 100 MeV early in the afterglow. observed photon ﬂux; it would also increase the ratio of The high-energy light curve of the TP model is marked with a red line, that of the CR-only case with a blue line, and the light curve synchrotron to SSC photons, allowing the NL and TP of the NL model with a black line. The thin black line at lower models to decay with α = 1.2 for a longer period before right shows a decay index of -1.2. SSC emission is signiﬁcant at 100 MeV energies. On the other hand, an increase in synchrotron and SSC emis- thermal particles in the CR-only case strongly depresses sion would both be associated with stronger cooling, as SSC emission relative to the TP and NL models. Early in discussed in Section , which would reduce the emission the afterglow there are many observations against which at the highest energies. The degree to which these com- our model may be compared; see Panaitescu (2016) for a peting eﬀects balance each other out will be explored in discussion of the two dozen bright bursts that have been future extensions of this model. observed over a signiﬁcant length of time. In that pa- The strongest limits we are aware of on later-time per, a distinction was drawn between bright-soft bursts gamma-ray ﬂux from GRB afterglows are presented (whose initial ﬂux above 100 MeV is high, but which de- in Aliu et al. (2014), who observed GRB 130427A cay rapidly) and faint-hard bursts. The temporal decay ≈ chosen to demarcate the two populations was t−1.2, cor- o≈n three conse≈cutive nights after the GRB ( 20 hr, 44 hr, and 68 hr after the trigger). Their non- responding to a trough in the distribution of observed detections placed upper limits (respectively) of 9, 7, and decay indices. × − − − 3 10 12 erg cm 2 s 1 on ﬂux in the VERITAS range, We plot the expected light curves (in terms of photon E > 100 GeV. The ﬁrst observation, at 20 hours after ﬂux rather than energy ﬂux) from the CR-only case and γ the GRB, would occur between time steps 5 and 6 (green the TP & NL models at photon energies higher than 100 and yellow curves in Figure 4) of our model afterglows. MeV in Figure 8. We note that the CR-only case follows By this point, even the TP and NL models are below the the α = 1.2 guide line very closely. This is entirely sensi- limits established by Aliu et al. (2014), so the VERITAS ble, given its origin as high-energy synchrotron emission, non-detections are consistent with our results. ν > max(ν , ν ), from a power-law distribution of par- m c In spite of the fact that the TP and NL models ticles accelerated by a relativistic shock. In the TP and predict non-observations by VERITAS and minimally- NL shocks, the thermal peak of the SSC spectrum ex- signiﬁcant observations by Fermi, they are well above ceeds the synchrotron spectrum at 100 MeV in energy the sensitivity curves for the Cherenkov Telescope Ar- until well into the afterglow (see Figure 4). The pas- ray (CTA). In Figure 9 we repeat the second, third, and sage of the SSC thermal peak through this energy range fourth time steps from the bottom two panels of Fig- causes a plateau in emission, just as occurred in both ure 4, overlaid to provide a more direct comparison of X-ray and optical in Figure 6. As thermal SSC emission the changes due to acceleration eﬃciency. In addition, gives way to SSC emission from photons originating with we plot the expected sensitivity curve for a half-hour ob- the non-thermal population, the GeV light curve steep- servation by the CTA North site.29 It is clear from the ens. A well-covered GeV light curve that showed a break ﬁgure that our models with thermal particles are well should be a strong indicator of the presence of thermal above the detection threshold for the array, and they re- particles producing SSC emission at these energies. main above the sensitivity curve for approximately 10 ks As with the optical in Figure 6, the presence or ab- after the initial trigger. If, as discussed above, our mod- sence of thermal particles makes a substantial diﬀerence el underpredicts the GeV ﬂux, GRB afterglows would in the ﬂux above 100 MeV. Though the diﬀerence is ini- be detectable even longer and/or to greater distances, tially just a factor of 30, the plateau in the TP & NL models’ emission caused by the the thermal peak rapid- ly increases the disparity. Afterglow parameters ((cid:15)e, (cid:15)B, 29 The CTA curve uses data taken from https://portal.cta- E , fraction of particles in the nonthermal tail, etc.) observatory.org/Pages/CTA-Performance.aspx . iso ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Warren

text: et

text: al.

text: could

text: be

text: adjusted

text: to

text: bring

text: the

text: CR-only

text: case’s

text: emission

text: in

text: line

text: with

text: the

text: other

text: two.

text: We

text: reiterate

text: that

text: thermal

text: particles—which

text: are

text: certain

text: to

text: be

text: present

text: in

text: the

text: shocked

text: plasma—participating

text: in

text: the

text: photon

text: production

text: process

text: yields

text: large

text: gains

text: in

text: emission

text: with

text: no

text: tweaking

text: of

text: param-

text: eters

text: needed.

text: Despite

text: the

text: large

text: factor

text: by

text: which

text: the

text: TP

text: NL

text: mod-

text: els

text: are

text: brighter

text: than

text: the

text: CR-only

text: case,

text: both

text: are

text: fainter

text: in

text: the

text: Fermi-LAT

text: band

text: than

text: most

text: of

text: the

text: afterglows

text: with

text: observed

text: GeV

text: emission.

text: There

text: are

text: reasons

text: to

text: suspect

text: that

text: the

text: model

text: presented

text: here

text: underpredicts

text: the

text: amount

text: of

text: high-energy

text: emission,

text: though.

text: As

text: discussed

text: elsewhere,

text: we

text: assume

text: a

text: weaker

text: magnetic

text: ﬁeld

text: near

text: the

text: subshock

text: than

text: is

text: observed

text: in

text: PIC

text: simulations

text: (where

text: can

text: locally

text: B

text: exceed

text: as

text: in

text: Spitkovsky

text: Sironi

text: et

text: al.

text: The

text: higher

text: ﬁelds

text: would

text: allow

text: for

text: more,

text: and

text: higher-energy,

text: synchrotron

text: photons

text: for

text: a

text: given

text: electron.

text: This

text: increase

text: in

text: synchrotron

text: photons

text: would,

text: of

text: course,

text: increase

text: the

text: Fig.

text: Total

text: photon

text: ﬂux

text: above

text: MeV

text: early

text: in

text: the

text: afterglow.

text: observed

text: photon

text: ﬂux;

text: it

text: would

text: also

text: increase

text: the

text: ratio

text: of

text: The

text: high-energy

text: light

text: curve

text: of

text: the

text: TP

text: model

text: is

text: marked

text: with

text: a

text: red

text: line,

text: that

text: of

text: the

text: CR-only

text: case

text: with

text: a

text: blue

text: line,

text: and

text: the

text: light

text: curve

text: synchrotron

text: to

text: SSC

text: photons,

text: allowing

text: the

text: NL

text: and

text: TP

text: of

text: the

text: NL

text: model

text: with

text: a

text: black

text: line.

text: The

text: thin

text: black

text: line

text: at

text: lower

text: models

text: to

text: decay

text: with

text: α

text: for

text: a

text: longer

text: period

text: before

text: right

text: shows

text: a

text: decay

text: index

text: of

text: SSC

text: emission

text: is

text: signiﬁcant

text: at

text: MeV

text: energies.

text: On

text: the

text: other

text: hand,

text: an

text: increase

text: in

text: synchrotron

text: and

text: SSC

text: emis-

text: thermal

text: particles

text: in

text: the

text: CR-only

text: case

text: strongly

text: depresses

text: sion

text: would

text: both

text: be

text: associated

text: with

text: stronger

text: cooling,

text: as

text: SSC

text: emission

text: relative

text: to

text: the

text: TP

text: and

text: NL

text: models.

text: Early

text: in

text: discussed

text: in

text: Section

text: which

text: would

text: reduce

text: the

text: emission

text: the

text: afterglow

text: there

text: are

text: many

text: observations

text: against

text: which

text: at

text: the

text: highest

text: energies.

text: The

text: degree

text: to

text: which

text: these

text: com-

text: our

text: model

text: may

text: be

text: compared;

text: see

text: Panaitescu

text: for

text: a

text: peting

text: eﬀects

text: balance

text: each

text: other

text: out

text: will

text: be

text: explored

text: in

text: discussion

text: of

text: the

text: two

text: dozen

text: bright

text: bursts

text: that

text: have

text: been

text: future

text: extensions

text: of

text: this

text: model.

text: observed

text: over

text: a

text: signiﬁcant

text: length

text: of

text: time.

text: In

text: that

text: pa-

text: The

text: strongest

text: limits

text: we

text: are

text: aware

text: of

text: on

text: later-time

text: per,

text: a

text: distinction

text: was

text: drawn

text: between

text: bright-soft

text: bursts

text: gamma-ray

text: ﬂux

text: from

text: GRB

text: afterglows

text: are

text: presented

text: (whose

text: initial

text: ﬂux

text: above

text: MeV

text: is

text: high,

text: but

text: which

text: de-

text: in

text: Aliu

text: et

text: al.

text: who

text: observed

text: GRB

text: cay

text: rapidly)

text: and

text: faint-hard

text: bursts.

text: The

text: temporal

text: decay

text: chosen

text: to

text: demarcate

text: the

text: two

text: populations

text: was

text: cor-

text: o≈n

text: three

text: conse≈cutive

text: nights

text: after

text: the

text: GRB

text: hr,

text: hr,

text: and

text: hr

text: after

text: the

text: trigger).

text: Their

text: non-

text: responding

text: to

text: a

text: trough

text: in

text: the

text: distribution

text: of

text: observed

text: detections

text: placed

text: upper

text: limits

text: (respectively)

text: of

text: and

text: decay

text: indices.

text: erg

text: cm

text: s

text: on

text: ﬂux

text: in

text: the

text: VERITAS

text: range,

text: We

text: plot

text: the

text: expected

text: light

text: curves

text: (in

text: terms

text: of

text: photon

text: E

text: GeV.

text: The

text: ﬁrst

text: observation,

text: at

text: hours

text: after

text: ﬂux

text: rather

text: than

text: energy

text: ﬂux)

text: from

text: the

text: CR-only

text: case

text: and

text: γ

text: the

text: GRB,

text: would

text: occur

text: between

text: time

text: steps

text: and

text: (green

text: the

text: TP

text: NL

text: models

text: at

text: photon

text: energies

text: higher

text: than

text: and

text: yellow

text: curves

text: in

text: Figure

text: of

text: our

text: model

text: afterglows.

text: MeV

text: in

text: Figure

text: We

text: note

text: that

text: the

text: CR-only

text: case

text: follows

text: By

text: this

text: point,

text: even

text: the

text: TP

text: and

text: NL

text: models

text: are

text: below

text: the

text: the

text: α

text: guide

text: line

text: very

text: closely.

text: This

text: is

text: entirely

text: sensi-

text: limits

text: established

text: by

text: Aliu

text: et

text: al.

text: so

text: the

text: VERITAS

text: ble,

text: given

text: its

text: origin

text: as

text: high-energy

text: synchrotron

text: emission,

text: non-detections

text: are

text: consistent

text: with

text: our

text: results.

text: ν

text: max(ν

text: ν

text: from

text: a

text: power-law

text: distribution

text: of

text: par-

text: m

text: c

text: In

text: spite

text: of

text: the

text: fact

text: that

text: the

text: TP

text: and

text: NL

text: models

text: ticles

text: accelerated

text: by

text: a

text: relativistic

text: shock.

text: In

text: the

text: TP

text: and

text: predict

text: non-observations

text: by

text: VERITAS

text: and

text: minimally-

text: NL

text: shocks,

text: the

text: thermal

text: peak

text: of

text: the

text: SSC

text: spectrum

text: ex-

text: signiﬁcant

text: observations

text: by

text: Fermi,

text: they

text: are

text: well

text: above

text: ceeds

text: the

text: synchrotron

text: spectrum

text: at

text: MeV

text: in

text: energy

text: the

text: sensitivity

text: curves

text: for

text: the

text: Cherenkov

text: Telescope

text: Ar-

text: until

text: well

text: into

text: the

text: afterglow

text: (see

text: Figure

text: The

text: pas-

text: ray

text: (CTA).

text: In

text: Figure

text: we

text: repeat

text: the

text: second,

text: third,

text: and

text: sage

text: of

text: the

text: SSC

text: thermal

text: peak

text: through

text: this

text: energy

text: range

text: fourth

text: time

text: steps

text: from

text: the

text: bottom

text: two

text: panels

text: of

text: Fig-

text: causes

text: a

text: plateau

text: in

text: emission,

text: just

text: as

text: occurred

text: in

text: both

text: ure

text: overlaid

text: to

text: provide

text: a

text: more

text: direct

text: comparison

text: of

text: X-ray

text: and

text: optical

text: in

text: Figure

text: As

text: thermal

text: SSC

text: emission

text: the

text: changes

text: due

text: to

text: acceleration

text: eﬃciency.

text: In

text: addition,

text: gives

text: way

text: to

text: SSC

text: emission

text: from

text: photons

text: originating

text: with

text: we

text: plot

text: the

text: expected

text: sensitivity

text: curve

text: for

text: a

text: half-hour

text: ob-

text: the

text: non-thermal

text: population,

text: the

text: GeV

text: light

text: curve

text: steep-

text: servation

text: by

text: the

text: CTA

text: North

text: It

text: is

text: clear

text: from

text: the

text: ens.

text: A

text: well-covered

text: GeV

text: light

text: curve

text: that

text: showed

text: a

text: break

text: ﬁgure

text: that

text: our

text: models

text: with

text: thermal

text: particles

text: are

text: well

text: should

text: be

text: a

text: strong

text: indicator

text: of

text: the

text: presence

text: of

text: thermal

text: above

text: the

text: detection

text: threshold

text: for

text: the

text: array,

text: and

text: they

text: re-

text: particles

text: producing

text: SSC

text: emission

text: at

text: these

text: energies.

text: main

text: above

text: the

text: sensitivity

text: curve

text: for

text: approximately

text: ks

text: As

text: with

text: the

text: optical

text: in

text: Figure

text: the

text: presence

text: or

text: ab-

text: after

text: the

text: initial

text: trigger.

text: If,

text: as

text: discussed

text: above,

text: our

text: mod-

text: sence

text: of

text: thermal

text: particles

text: makes

text: a

text: substantial

text: diﬀerence

text: el

text: underpredicts

text: the

text: GeV

text: ﬂux,

text: GRB

text: afterglows

text: would

text: in

text: the

text: ﬂux

text: above

text: MeV.

text: Though

text: the

text: diﬀerence

text: is

text: ini-

text: be

text: detectable

text: even

text: longer

text: and/or

text: to

text: greater

text: distances,

text: tially

text: just

text: a

text: factor

text: of

text: the

text: plateau

text: in

text: the

text: TP

text: NL

text: models’

text: emission

text: caused

text: by

text: the

text: the

text: thermal

text: peak

text: rapid-

text: ly

text: increases

text: the

text: disparity.

text: Afterglow

text: parameters

text: The

text: CTA

text: curve

text: uses

text: data

text: taken

text: from

text: E

text: fraction

text: of

text: particles

text: in

text: the

text: nonthermal

text: tail,

text: etc.)

text: iso

text: ##LTFigure##

text: ##LTLine##

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text: or (cid:15) from ﬁrst principles over a large portion of an The aim of this paper is to build a physically- B afterglow.11 In Lemoine (2013), though, a functional motivated model of a GRB afterglow, from early in form for decaying microturbulence was assumed based the afterglow until very nearly the transrelativistic stage on PIC results. The synchrotron spectrum of particles of deceleration. We pay particular attention to the downstream from the shock was calculated using that nonlinear eﬀects, from the simultaneous acceleration of functional form in a variety of conditions (e.g., gradu- electrons and protons, linking the shock structure and al/rapid decay of turbulence, weak/strong inverse Comp- the cosmic rays being accelerated. Where needed, we ton losses, etc.). The longitudinal extent of the enhanced incorporate results from PIC simulations to guide our − turbulence was some 102 104 ion skin depths. While parametrization of key microphysics. We calculate pho- this is a large distance for PIC simulations, it is ignor- ton spectra for various times, which we compare against ably small compared to the volumes being simulated in observations in the optical, X-ray, and gamm§a-ray bands. this work. The length scale of our Monte Carlo code is The paper is organized as follows. In we discuss − r = γ u m c/(eB ), which is σ 1/2 times larger than the Monte Carlo code used to generate the particle and g0 0 0 p 0 the plasma skin depth; the emission shells are individu- photon spectra to be presented later. The Monte Carlo ally thousands of r in size. Here, u is the shock speed, code is applied post-process to a hydroynamical base to g0 0 m iDsrtahfetpvreortsoionnmMaasys,2c9,, a20n2d0e is the electronic charge. create a seque§nce of afterglow snapshots, in a manner p Preprint typeset using LATEX style emulateapj v. 5/25/10 we describe in . We present our e§xample afterglows§and compare against observations in , and conclude in . NONLINEAR PARTICLE ACCELERATION AND THERMAL PARTICLES IN GRB AFTERGLOWS Donald C. Warren,1 Donald C. Ellison,2 Maxim V. Barkov,1,3,4 Shigehiro Nagataki1,5 Subject headings: 2. THE MONTE CARLO CODE where r = pc/(ZeB) is the gyroradius of a par- The work presented here relies on a Monte Carlo g ticle with charge number Z in the local magnetic ﬁeld code treating ﬁrst-order Fermi acceleration in the vicini- B. That is, we assume that particle scattering occurs in ty of an inﬁnite, steady-state, parallel, collisionless shock. the Bohm limit. This assumption fails in the immediate The code follows the scattering and propagation of parti- vicinity of a relativistic shock, as the magnetic ﬁeld is cles in a turbulent magnetic ﬁeld around a velocity gra- turbulent down to length scales of the ion skin depth, dient. Momentum and energy ﬂuxes are tracked dur- which may be many orders of magnitude smaller than ing propagation, which may then be used to modify the the particle gyroradius (e.g., Sironi et al. 2013; Plotnikov shock structure in an attempt to satisfy the Rankine– et al. 2013). In these conditions the mean free path ap- Hugoniot conditions at all locations. The process has ∝ ∝ been described in detail in, e.g., Ellison et al. (2013) and proaches λmfp p2 rather than p1 (e.g., Plotnikov Warren et al. (2015), and the reader is referred to those et al. 2011). If, on the other hand, the power spectrum papers for a more complete description of the code. In of the magnetic turbulence extends to length scales com- this section we brieﬂy discuss key features, and depar- parable to rg, then scattering proceeds as p1 at all mo- tures from previous work. menta. We do not attempt to address these complexities All particles—regardless of energy or species—are at this time; previous work on this topic has shown that assumed to scatter elastically oﬀ turbulence in the lo- this simple model for particle scattering can adequate- cal ﬂuid frame with a mean free path given by ly reproduce some important PIC results for injection and early acceleration (Ellison et al. 2013). Ellison et al. (2016) discussed the eﬀect of a modiﬁed diﬀusion coeﬃ- λ = r , (1) mfp g cient on both CR spectra and nonlinear smoothing, and we refer the reader to that paper for more information. 11 PIC simulations such as Sironi et al. (2013) and Ardaneh et al. (2015), of course, allow for direct calculation of both (cid:15)e and For now we simply note that the region of intense (cid:15)B. However, these simulations are limited by computational pow- turbulence is contained to a small region around the er to studying time periods vastly shorter than the timescales of shock, and so we assume that the eﬀects of this region on afterglows. the overall model are minimal. In the upstream direction 1 Astrophysical Big Bang Laboratory, RIKEN, Saitama 351- the small-scale Weibel instability is contained to a pre- 0198, Japan; donald.warren@riken.jp cursor extending a short distance ahead of the shock (see 2 Physics Department, North Carolina State University, Box Equation 6 in Sironi et al. 2013). Downstream from the 8202, Raleigh, NC 27695, U.S.A. 3 Institute for Physics and Astronomy, University of Potsdam, shock the turbulence decays rapidly, inverse-cascading − 11476 Potsdam, Germany to larger wavelengths and decreasing in strength as t 1/2 4 DESY, Platanenallee 6, 15378 Zeuthen, Germany 5 Interdisicplinary Theoretical Science Research Group (Lemoine et al. 2013, and references therein). Both scales (iTHES), RIKEN, Saitama 351-0198, Japan are dramatically smaller than the distances considered in

text: or

label: paragraph

text: from

text: ﬁrst

text: principles

text: over

text: a

text: large

text: portion

text: of

text: an

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: build

text: a

text: physically-

text: B

text: In

text: Lemoine

text: though,

text: a

text: functional

text: motivated

text: model

text: of

text: a

text: GRB

text: afterglow,

text: from

text: early

text: in

text: form

text: for

text: decaying

text: microturbulence

text: was

text: assumed

text: based

text: the

text: afterglow

text: until

text: very

text: nearly

text: the

text: transrelativistic

text: stage

text: on

text: PIC

text: results.

text: The

text: synchrotron

text: spectrum

text: of

text: particles

text: of

text: deceleration.

text: We

text: pay

text: particular

text: attention

text: to

text: the

text: downstream

text: from

text: the

text: shock

text: was

text: calculated

text: using

text: that

text: nonlinear

text: eﬀects,

text: from

text: the

text: simultaneous

text: acceleration

text: of

text: functional

text: form

text: in

text: a

text: variety

text: of

text: conditions

text: (e.g.,

text: gradu-

text: electrons

text: and

text: protons,

text: linking

text: the

text: shock

text: structure

text: and

text: al/rapid

text: decay

text: of

text: turbulence,

text: weak/strong

text: inverse

text: Comp-

text: the

text: cosmic

text: rays

text: being

text: accelerated.

text: Where

text: needed,

text: we

text: ton

text: losses,

text: etc.).

text: The

text: longitudinal

text: extent

text: of

text: the

text: enhanced

text: incorporate

text: results

text: from

text: PIC

text: simulations

text: to

text: guide

text: our

text: turbulence

text: was

text: some

text: ion

text: skin

text: depths.

text: While

text: parametrization

text: of

text: key

text: microphysics.

text: We

text: calculate

text: pho-

text: this

text: is

text: a

text: large

text: distance

text: for

text: PIC

text: simulations,

text: it

text: is

text: ignor-

text: ton

text: spectra

text: for

text: various

text: times,

text: which

text: we

text: compare

text: against

text: ably

text: small

text: compared

text: to

text: the

text: volumes

text: being

text: simulated

text: in

text: observations

text: in

text: the

text: optical,

text: X-ray,

text: and

text: gamm§a-ray

text: bands.

text: this

text: work.

text: The

text: length

text: scale

text: of

text: our

text: Monte

text: Carlo

text: code

text: is

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: we

text: discuss

text: r

text: γ

text: u

text: m

text: c/(eB

text: which

text: is

text: σ

text: times

text: larger

text: than

text: the

text: Monte

text: Carlo

text: code

text: used

text: to

text: generate

text: the

text: particle

text: and

text: p

text: the

text: plasma

text: skin

text: depth;

text: the

text: emission

text: shells

text: are

text: individu-

text: photon

text: spectra

text: to

text: be

text: presented

text: later.

text: The

text: Monte

text: Carlo

text: ally

text: thousands

text: of

text: r

text: in

text: size.

text: Here,

text: u

text: is

text: the

text: shock

text: speed,

text: code

text: is

text: applied

text: post-process

text: to

text: a

text: hydroynamical

text: base

text: to

text: m

text: is

text: the

text: electronic

text: charge.

text: create

text: a

text: seque§nce

text: of

text: afterglow

text: snapshots,

text: in

text: a

text: manner

text: p

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: we

text: describe

text: in

text: We

text: present

text: our

text: e§xample

text: afterglows§and

text: compare

text: against

text: observations

text: in

text: and

text: conclude

text: in

text: NONLINEAR

text: PARTICLE

text: ACCELERATION

text: AND

text: THERMAL

text: PARTICLES

text: IN

text: GRB

text: AFTERGLOWS

text: Donald

text: C.

text: Donald

text: C.

text: Maxim

text: V.

text: Shigehiro

text: Subject

text: headings:

text: THE

text: MONTE

text: CARLO

text: CODE

text: where

text: r

text: pc/(ZeB)

text: is

text: the

text: gyroradius

text: of

text: a

text: par-

text: The

text: work

text: presented

text: here

text: relies

text: on

text: a

text: Monte

text: Carlo

text: g

text: ticle

text: with

text: charge

text: number

text: Z

text: in

text: the

text: local

text: magnetic

text: ﬁeld

text: code

text: treating

text: ﬁrst-order

text: Fermi

text: acceleration

text: in

text: the

text: vicini-

text: B.

text: That

text: is,

text: we

text: assume

text: that

text: particle

text: scattering

text: occurs

text: in

text: ty

text: of

text: an

text: inﬁnite,

text: steady-state,

text: parallel,

text: collisionless

text: shock.

text: the

text: Bohm

text: limit.

text: This

text: assumption

text: fails

text: in

text: the

text: immediate

text: The

text: code

text: follows

text: the

text: scattering

text: and

text: propagation

text: of

text: parti-

text: vicinity

text: of

text: a

text: relativistic

text: shock,

text: as

text: the

text: magnetic

text: ﬁeld

text: is

text: cles

text: in

text: a

text: turbulent

text: magnetic

text: ﬁeld

text: around

text: a

text: velocity

text: gra-

text: turbulent

text: down

text: to

text: length

text: scales

text: of

text: the

text: ion

text: skin

text: depth,

text: dient.

text: Momentum

text: and

text: energy

text: ﬂuxes

text: are

text: tracked

text: dur-

text: which

text: may

text: be

text: many

text: orders

text: of

text: magnitude

text: smaller

text: than

text: ing

text: propagation,

text: which

text: may

text: then

text: be

text: used

text: to

text: modify

text: the

text: the

text: particle

text: gyroradius

text: (e.g.,

text: Sironi

text: et

text: al.

text: Plotnikov

text: shock

text: structure

text: in

text: an

text: attempt

text: to

text: satisfy

text: the

text: Rankine–

text: et

text: al.

text: In

text: these

text: conditions

text: the

text: mean

text: free

text: path

text: ap-

text: Hugoniot

text: conditions

text: at

text: all

text: locations.

text: The

text: process

text: has

text: been

text: described

text: in

text: detail

text: in,

text: e.g.,

text: Ellison

text: et

text: al.

text: and

text: proaches

text: λmfp

text: rather

text: than

text: (e.g.,

text: Plotnikov

text: Warren

text: et

text: al.

text: and

text: the

text: reader

text: is

text: referred

text: to

text: those

text: et

text: al.

text: If,

text: on

text: the

text: other

text: hand,

text: the

text: power

text: spectrum

text: papers

text: for

text: a

text: more

text: complete

text: description

text: of

text: the

text: code.

text: In

text: of

text: the

text: magnetic

text: turbulence

text: extends

text: to

text: length

text: scales

text: com-

text: this

text: section

text: we

text: brieﬂy

text: discuss

text: key

text: features,

text: and

text: depar-

text: parable

text: to

text: rg,

text: then

text: scattering

text: proceeds

text: as

text: at

text: all

text: mo-

text: tures

text: from

text: previous

text: work.

text: menta.

text: We

text: do

text: not

text: attempt

text: to

text: address

text: these

text: complexities

text: All

text: particles—regardless

text: of

text: energy

text: or

text: species—are

text: at

text: this

text: time;

text: previous

text: work

text: on

text: this

text: topic

text: has

text: shown

text: that

text: assumed

text: to

text: scatter

text: elastically

text: oﬀ

text: turbulence

text: in

text: the

text: lo-

text: this

text: simple

text: model

text: for

text: particle

text: scattering

text: can

text: adequate-

text: cal

text: ﬂuid

text: frame

text: with

text: a

text: mean

text: free

text: path

text: given

text: by

text: ly

text: reproduce

text: some

text: important

text: PIC

text: results

text: for

text: injection

text: and

text: early

text: acceleration

text: (Ellison

text: et

text: al.

text: Ellison

text: et

text: al.

text: discussed

text: the

text: eﬀect

text: of

text: a

text: modiﬁed

text: diﬀusion

text: coeﬃ-

text: λ

text: r

text: mfp

text: g

text: cient

text: on

text: both

text: CR

text: spectra

text: and

text: nonlinear

text: smoothing,

text: and

text: we

text: refer

text: the

text: reader

text: to

text: that

text: paper

text: for

text: more

text: information.

text: PIC

text: simulations

text: such

text: as

text: Sironi

text: et

text: al.

text: and

text: Ardaneh

text: et

text: al.

text: of

text: course,

text: allow

text: for

text: direct

text: calculation

text: of

text: both

text: and

text: For

text: now

text: we

text: simply

text: note

text: that

text: the

text: region

text: of

text: intense

text: However,

text: these

text: simulations

text: are

text: limited

text: by

text: computational

text: pow-

text: turbulence

text: is

text: contained

text: to

text: a

text: small

text: region

text: around

text: the

text: er

text: to

text: studying

text: time

text: periods

text: vastly

text: shorter

text: than

text: the

text: timescales

text: of

text: shock,

text: and

text: so

text: we

text: assume

text: that

text: the

text: eﬀects

text: of

text: this

text: region

text: on

text: afterglows.

text: the

text: overall

text: model

text: are

text: minimal.

text: In

text: the

text: upstream

text: direction

text: Astrophysical

text: Big

text: Bang

text: Laboratory,

text: RIKEN,

text: Saitama

text: the

text: small-scale

text: Weibel

text: instability

text: is

text: contained

text: to

text: a

text: pre-

text: Japan;

text: donald.warren@riken.jp

text: cursor

text: extending

text: a

text: short

text: distance

text: ahead

text: of

text: the

text: shock

text: (see

text: Physics

text: Department,

text: North

text: Carolina

text: State

text: University,

text: Box

text: Equation

text: in

text: Sironi

text: et

text: al.

text: Downstream

text: from

text: the

text: Raleigh,

text: NC

text: U.S.A.

text: Institute

text: for

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Potsdam,

text: shock

text: the

text: turbulence

text: decays

text: rapidly,

text: inverse-cascading

text: Potsdam,

text: Germany

text: to

text: larger

text: wavelengths

text: and

text: decreasing

text: in

text: strength

text: as

text: t

text: DESY,

text: Platanenallee

text: Zeuthen,

text: Germany

text: Interdisicplinary

text: Theoretical

text: Science

text: Research

text: Group

text: (Lemoine

text: et

text: al.

text: and

text: references

text: therein).

text: Both

text: scales

text: (iTHES),

text: RIKEN,

text: Saitama

text: Japan

text: are

text: dramatically

text: smaller

text: than

text: the

text: distances

text: considered

text: in


text: NLDSA model for GRB afterglows 9 4. EXAMPLE AFTERGLOWS 4.1. Particle spectra and energy densities Having explained the Monte Carlo code (Section ) In Figure 2 we show the spectra of cosmic rays for all and the manner in which it is coupled to a hydrody- three models, in the local plasma frame. For the sake of namical simulation to create an evolving afterglow mod- clarity, the cooled populations that have been swept fur- el (Section ), we will now present the results from three ther downstream (the interior shells of Figure 1) are not test cases. The ﬁrst two models assume that acceleration given. These spectra were taken just downstream from takes place in the test-particle limit, with few enough the subshock at x = 0, to most closely match the spectral particles accelerated that the structure of the shock is indices predicted by analytical work. Though we assume not signiﬁcantly modiﬁed. The particular values of P an isotropic distribution of particles for all photon pro- inj are listed in Table 1. In one model, thermal (uninject- duction processes, this is obviously not the case at the ed) particles are explicitly excluded from the radiating subshock. All particles require several diﬀusion lengths population, though they contribute to the pressure and of transport downstream before their angular distribu- energy density downstream. We refer to this as the “CR- tion relaxes into isotropy. Since the diﬀusion length is an only” test case. In the second model, the “TP” model, increasing function of momentum (Equation 1) higher- all particles are included in the photon production cal- energy particles remain anisotropic for longer distances culations. In the third model, “NL,” injection is eﬃcient downstream. The eﬀects of anisotropic particle distribu- enough that the nonlinear backreaction of accelerated tions will be addressed in future work; for now we merely CRs on the shock structure cannot be ignored. note the assumption being made. We must comment here on the physical reasonable- The spectra presented in Figure 2 do not use arbi- ness of all three models. The CR-only case is not intend- trary units. Since upstream escape is negligible (no par- ed to be physically plausible. There is no way to con- ticles escaped during the ﬁrst six time steps, and even in − serve number, momentum, and energy ﬂuxes in a shock the last two steps the escaping ﬂux was less than 10 6 interacting with the ISM unless a thermal population of the incoming energy ﬂux), and since the diﬃculty of is present. The CR-only case is included only because scattering upstream against a relativistic shock means it mimics the standard synchrotron model for afterglow most particles are downstream, the total number of par- emission, which ignores the thermal population. It serves ticles in each spectrum is the number of particles swept as a control against which our other two models may be up at a given time step. As such, the spectra are not compared; we are not presenting it as a valid alternative only absolutely normalized within each model: they are to either the TP or the NL model. Both the TP and NL normalized between models as well. models are physically possible, and operate at diﬀerent Both the CR-only case and TP model result in ends of the injection eﬃciency spectrum. very simple spectra. This is unsurprising, as they were All three model afterglows use an upstream ambient designed to reproduce the simple power laws used in magnetic ﬁeld B = 3 mG. This is orders of magnitude the standard synchrotron model for afterglows. In the 0 higher than the typical value of 3 µG assumed for the TP model’s spectra, there is a thermal peak25 of parti- Milky Way. There are locations where such high ﬁelds cles that crossed the shock once (but did not enter the may exist in the circumburst environment,24 but this val- acceleration process), and a high energy tail of shock- ue was chosen for a diﬀerent reason: once it is compressed accelerated particles out to a maximum energy. The according to Equation 3, the downstream ﬁeld approach- thermal peak has been excised from the CR-only case, ≈ − es an energy fraction of (cid:15) 10 3, when both rest mass while keeping the normalization of accelerated particles B energy and pressure are considered. This is lower than the same. In all cases the spectral index of the proton ∼ − the traditional value, (cid:15) 10 2, but in line with current distributions decreases with time, as the shock Lorentz B expectations (it is potentially even on the high side; see factor drops (see Keshet & Waxman 2005; Ellison 2005). Santana et al. 2014; Beniamini et al. 2015, and see also Since protons experience no losses while they in- the discussion of (cid:15) − in (Lemoine et al. 2013)). Because teract with the shock, the high-energy turnover of the B we assume only magnetic ﬁeld compression and no addi- CR tail is due to the limit on acceleration time. The tional ampliﬁcation, the value of (cid:15) just downstream of electrons, by contrast, experience signiﬁcant synchrotron B the shock (that is, prior to the expansion described pre- losses in the intense magnetic ﬁelds downstream of the viously) decays with time as the shock decelerates—see shock. The slow increase in the maximum energy of the Equation 3—rather than maintaining a constant value at electron spectrum is due to decreased compression also all times. This temporal decay at the shock front occurs lowering the magnetic ﬁeld and cooling rate. in addition to the expected dilution of magnetic ener- Our Monte Carlo model predicts that protons are gy density as the plasma expands downstream from the easily accelerated beyond the knee in the cosmic ray shock. spectrum; the high magnetic ﬁelds, combined with pitch- angle scattering, conﬁne the protons to a small volume around the shock, and allow for many shock crossing ters as deﬁned in Planck Collaboration et al. (2014). 24 Speciﬁcally, Robishaw et al. (2008) report magnetic ﬁeld cycles in the time-limited scenario presented here. The strengths upwards of 10 mG in starburst galaxies from measure- ments of Zeeman splitting of the 1667 MHz OH megamaser line. 25 The term “thermal” is used only to specify those shocked Additionally, ambient ﬁeld strengths within 100-200 pc of the cen- particles that don’t enter the Fermi acceleration process. The re- ter of the Milky Way may locally reach or exceed mG levels (Fer- sults we obtain in no way require or presume that these particles ri`ere 2009). equilibrate in a thermodynamic sense.

text: NLDSA

text: model

text: for

text: GRB

text: afterglows

label: section

text: EXAMPLE

text: AFTERGLOWS

text: Particle

text: spectra

text: and

text: energy

text: densities

text: Having

text: explained

text: the

text: Monte

text: Carlo

text: code

text: (Section

text: In

text: Figure

text: we

text: show

text: the

text: spectra

text: of

text: cosmic

text: rays

text: for

text: all

text: and

text: the

text: manner

text: in

text: which

text: it

text: is

text: coupled

text: to

text: a

text: hydrody-

text: three

text: models,

text: in

text: the

text: local

text: plasma

text: frame.

text: For

text: the

text: sake

text: of

text: namical

text: simulation

text: to

text: create

text: an

text: evolving

text: afterglow

text: mod-

text: clarity,

text: the

text: cooled

text: populations

text: that

text: have

text: been

text: swept

text: fur-

text: el

text: (Section

text: we

text: will

text: now

text: present

text: the

text: results

text: from

text: three

text: ther

text: downstream

text: (the

text: interior

text: shells

text: of

text: Figure

text: are

text: not

text: test

text: cases.

text: The

text: ﬁrst

text: two

text: models

text: assume

text: that

text: acceleration

text: given.

text: These

text: spectra

text: were

text: taken

text: just

text: downstream

text: from

text: takes

text: place

text: in

text: the

text: test-particle

text: limit,

text: with

text: few

text: enough

text: the

text: subshock

text: at

text: x

text: to

text: most

text: closely

text: match

text: the

text: spectral

text: particles

text: accelerated

text: that

text: the

text: structure

text: of

text: the

text: shock

text: is

text: indices

text: predicted

text: by

text: analytical

text: work.

text: Though

text: we

text: assume

text: not

text: signiﬁcantly

text: modiﬁed.

text: The

text: particular

text: values

text: of

text: P

text: an

text: isotropic

text: distribution

text: of

text: particles

text: for

text: all

text: photon

text: pro-

text: inj

text: are

text: listed

text: in

text: Table

text: In

text: one

text: model,

text: thermal

text: (uninject-

text: duction

text: processes,

text: this

text: is

text: obviously

text: not

text: the

text: case

text: at

text: the

text: ed)

text: particles

text: are

text: explicitly

text: excluded

text: from

text: the

text: radiating

text: subshock.

text: All

text: particles

text: require

text: several

text: diﬀusion

text: lengths

text: population,

text: though

text: they

text: contribute

text: to

text: the

text: pressure

text: and

text: of

text: transport

text: downstream

text: before

text: their

text: angular

text: distribu-

text: energy

text: density

text: downstream.

text: We

text: refer

text: to

text: this

text: as

text: the

text: “CR-

text: tion

text: relaxes

text: into

text: isotropy.

text: Since

text: the

text: diﬀusion

text: length

text: is

text: an

text: only”

text: test

text: case.

text: In

text: the

text: second

text: model,

text: the

text: “TP”

text: model,

text: increasing

text: function

text: of

text: momentum

text: (Equation

text: higher-

text: all

text: particles

text: are

text: included

text: in

text: the

text: photon

text: production

text: cal-

text: energy

text: particles

text: remain

text: anisotropic

text: for

text: longer

text: distances

text: culations.

text: In

text: the

text: third

text: model,

text: “NL,”

text: injection

text: is

text: eﬃcient

text: downstream.

text: The

text: eﬀects

text: of

text: anisotropic

text: particle

text: distribu-

text: enough

text: that

text: the

text: nonlinear

text: backreaction

text: of

text: accelerated

text: tions

text: will

text: be

text: addressed

text: in

text: future

text: work;

text: for

text: now

text: we

text: merely

text: CRs

text: on

text: the

text: shock

text: structure

text: cannot

text: be

text: ignored.

text: note

text: the

text: assumption

text: being

text: made.

text: We

text: must

text: comment

text: here

text: on

text: the

text: physical

text: reasonable-

text: The

text: spectra

text: presented

text: in

text: Figure

text: do

text: not

text: use

text: arbi-

text: ness

text: of

text: all

text: three

text: models.

text: The

text: CR-only

text: case

text: is

text: not

text: intend-

text: trary

text: units.

text: Since

text: upstream

text: escape

text: is

text: negligible

text: (no

text: par-

text: ed

text: to

text: be

text: physically

text: plausible.

text: There

text: is

text: no

text: way

text: to

text: con-

text: ticles

text: escaped

text: during

text: the

text: ﬁrst

text: six

text: time

text: steps,

text: and

text: even

text: in

text: serve

text: number,

text: momentum,

text: and

text: energy

text: ﬂuxes

text: in

text: a

text: shock

text: the

text: last

text: two

text: steps

text: the

text: escaping

text: ﬂux

text: was

text: less

text: than

text: interacting

text: with

text: the

text: ISM

text: unless

text: a

text: thermal

text: population

text: of

text: the

text: incoming

text: energy

text: ﬂux),

text: and

text: since

text: the

text: diﬃculty

text: of

text: is

text: present.

text: The

text: CR-only

text: case

text: is

text: included

text: only

text: because

text: scattering

text: upstream

text: against

text: a

text: relativistic

text: shock

text: means

text: it

text: mimics

text: the

text: standard

text: synchrotron

text: model

text: for

text: afterglow

text: most

text: particles

text: are

text: downstream,

text: the

text: total

text: number

text: of

text: par-

text: emission,

text: which

text: ignores

text: the

text: thermal

text: population.

text: It

text: serves

text: ticles

text: in

text: each

text: spectrum

text: is

text: the

text: number

text: of

text: particles

text: swept

text: as

text: a

text: control

text: against

text: which

text: our

text: other

text: two

text: models

text: may

text: be

text: up

text: at

text: a

text: given

text: time

text: step.

text: As

text: such,

text: the

text: spectra

text: are

text: not

text: compared;

text: we

text: are

text: not

text: presenting

text: it

text: as

text: a

text: valid

text: alternative

text: only

text: absolutely

text: normalized

text: within

text: each

text: model:

text: they

text: are

text: to

text: either

text: the

text: TP

text: or

text: the

text: NL

text: model.

text: Both

text: the

text: TP

text: and

text: NL

text: normalized

text: between

text: models

text: as

text: well.

text: models

text: are

text: physically

text: possible,

text: and

text: operate

text: at

text: diﬀerent

text: Both

text: the

text: CR-only

text: case

text: and

text: TP

text: model

text: result

text: in

text: ends

text: of

text: the

text: injection

text: eﬃciency

text: spectrum.

text: very

text: simple

text: spectra.

text: This

text: is

text: unsurprising,

text: as

text: they

text: were

text: All

text: three

text: model

text: afterglows

text: use

text: an

text: upstream

text: ambient

text: designed

text: to

text: reproduce

text: the

text: simple

text: power

text: laws

text: used

text: in

text: magnetic

text: ﬁeld

text: B

text: mG.

text: This

text: is

text: orders

text: of

text: magnitude

text: the

text: standard

text: synchrotron

text: model

text: for

text: afterglows.

text: In

text: the

text: higher

text: than

text: the

text: typical

text: value

text: of

text: µG

text: assumed

text: for

text: the

text: TP

text: model’s

text: spectra,

text: there

text: is

text: a

text: thermal

text: of

text: parti-

text: Milky

text: Way.

text: There

text: are

text: locations

text: where

text: such

text: high

text: ﬁelds

text: cles

text: that

text: crossed

text: the

text: shock

text: once

text: (but

text: did

text: not

text: enter

text: the

text: may

text: exist

text: in

text: the

text: circumburst

text: but

text: this

text: val-

text: acceleration

text: process),

text: and

text: a

text: high

text: energy

text: tail

text: of

text: shock-

text: ue

text: was

text: chosen

text: for

text: a

text: diﬀerent

text: reason:

text: once

text: it

text: is

text: compressed

text: accelerated

text: particles

text: out

text: to

text: a

text: maximum

text: energy.

text: The

text: according

text: to

text: Equation

text: the

text: downstream

text: ﬁeld

text: approach-

text: thermal

text: peak

text: has

text: been

text: excised

text: from

text: the

text: CR-only

text: case,

text: es

text: an

text: energy

text: fraction

text: of

text: when

text: both

text: rest

text: mass

text: while

text: keeping

text: the

text: normalization

text: of

text: accelerated

text: particles

text: B

text: energy

text: and

text: pressure

text: are

text: considered.

text: This

text: is

text: lower

text: than

text: the

text: same.

text: In

text: all

text: cases

text: the

text: spectral

text: index

text: of

text: the

text: proton

text: the

text: traditional

text: value,

text: but

text: in

text: line

text: with

text: current

text: distributions

text: decreases

text: with

text: time,

text: as

text: the

text: shock

text: Lorentz

text: B

text: expectations

text: (it

text: is

text: potentially

text: even

text: on

text: the

text: high

text: side;

text: see

text: factor

text: drops

text: (see

text: Keshet

text: Waxman

text: Ellison

text: Santana

text: et

text: al.

text: Beniamini

text: et

text: al.

text: and

text: see

text: also

text: Since

text: protons

text: experience

text: no

text: losses

text: while

text: they

text: in-

text: the

text: discussion

text: of

text: in

text: (Lemoine

text: et

text: al.

text: Because

text: teract

text: with

text: the

text: shock,

text: the

text: high-energy

text: turnover

text: of

text: the

text: B

text: we

text: assume

text: only

text: magnetic

text: ﬁeld

text: compression

text: and

text: no

text: addi-

text: CR

text: tail

text: is

text: due

text: to

text: the

text: limit

text: on

text: acceleration

text: time.

text: The

text: tional

text: ampliﬁcation,

text: the

text: value

text: of

text: just

text: downstream

text: of

text: electrons,

text: by

text: contrast,

text: experience

text: signiﬁcant

text: synchrotron

text: B

text: the

text: shock

text: (that

text: is,

text: prior

text: to

text: the

text: expansion

text: described

text: pre-

text: losses

text: in

text: the

text: intense

text: magnetic

text: ﬁelds

text: downstream

text: of

text: the

text: viously)

text: decays

text: with

text: time

text: as

text: the

text: shock

text: decelerates—see

text: shock.

text: The

text: slow

text: increase

text: in

text: the

text: maximum

text: energy

text: of

text: the

text: Equation

text: than

text: maintaining

text: a

text: constant

text: value

text: at

text: electron

text: spectrum

text: is

text: due

text: to

text: decreased

text: compression

text: also

text: all

text: times.

text: This

text: temporal

text: decay

text: at

text: the

text: shock

text: front

text: occurs

text: lowering

text: the

text: magnetic

text: ﬁeld

text: and

text: cooling

text: rate.

text: in

text: addition

text: to

text: the

text: expected

text: dilution

text: of

text: magnetic

text: ener-

text: Our

text: Monte

text: Carlo

text: model

text: predicts

text: that

text: protons

text: are

text: gy

text: density

text: as

text: the

text: plasma

text: expands

text: downstream

text: from

text: the

text: easily

text: accelerated

text: beyond

text: the

text: knee

text: in

text: the

text: cosmic

text: ray

text: shock.

text: spectrum;

text: the

text: high

text: magnetic

text: ﬁelds,

text: combined

text: with

text: pitch-

text: angle

text: scattering,

text: conﬁne

text: the

text: protons

text: to

text: a

text: small

text: volume

text: around

text: the

text: shock,

text: and

text: allow

text: for

text: many

text: shock

text: crossing

text: ters

text: as

text: deﬁned

text: in

text: Planck

text: Collaboration

text: et

text: al.

text: Speciﬁcally,

text: Robishaw

text: et

text: al.

text: report

text: magnetic

text: ﬁeld

text: cycles

text: in

text: the

text: time-limited

text: scenario

text: presented

text: here.

text: The

text: strengths

text: upwards

text: of

text: mG

text: in

text: starburst

text: galaxies

text: from

text: measure-

text: ments

text: of

text: Zeeman

text: splitting

text: of

text: the

text: MHz

text: OH

text: megamaser

text: line.

text: The

text: term

text: “thermal”

text: is

text: used

text: only

text: to

text: specify

text: those

text: shocked

text: Additionally,

text: ambient

text: ﬁeld

text: strengths

text: within

text: pc

text: of

text: the

text: cen-

text: particles

text: that

text: don’t

text: enter

text: the

text: Fermi

text: acceleration

text: process.

text: The

text: re-

text: ter

text: of

text: the

text: Milky

text: Way

text: may

text: locally

text: reach

text: or

text: exceed

text: mG

text: levels

text: (Fer-

text: sults

text: we

text: obtain

text: in

text: no

text: way

text: require

text: or

text: presume

text: that

text: these

text: particles

text: ri`ere

text: equilibrate

text: in

text: a

text: thermodynamic

text: sense.


text: 11 where t¯i = (τi, ξi) is the mirror reﬂection(cid:90)of ti = (ξi, τi) with respect to the (cid:10) (cid:11) diagonal line. Then the inner product between two PDs is given by kernel P X P X P X P X(cid:32) (cid:33)P X (cid:32) P X (cid:33) k ( ( 1), ( 2)) = ( (cid:88)1), ( 2) = Φ ( ( 1))Φ ( ( 2))dt σ ≥ σ σ ξ τ (cid:107) − (cid:107) (cid:107) − (cid:107) 1 − t1 t2 2 − − t1 t¯2 2 = exp i j 2 exp i j 2 . 8πσ 8σ 8σ i,j k is called the persistence state-space (PSS) kernel. Kernel distance D is then σ K deﬁned as a distance between PDs in a feature space [53]: (cid:10) (cid:11) P X P X (cid:107) P X − P X (cid:107) DK( ( 1), ( 2)) = (cid:2) Φσ( ( 1)) Φσ( ( 2)) P X − P X P X − P X = Φ ( ( 1)) Φ ( ( 2), Φ ( ( 1)) Φ ( ( 2) 1/2 σ σ (cid:3) σ σ P X P X P X P X = k ( ( 1), ( 1)) + k ( ( 2), ( 2)) σ σ − P X P X 2k ( ( 1), ( 2)) 1/2. σ The performance of kernel distance depends on the parameter σ. Thus, σ should be chosen carefully. Example 8. The same topological spaces given in Example 6 are used. The PDs of holes are shown in Fig. 6-(b). After kernel smoothing of PDs (c), the kernel distances D are estimated (d, e). As in the bottleneck distances in Fig. 5-(c), K X X X X the kernel distance in (e) can distinguish 1, 2 (no hole) from 3, 4 (three holes). The bottleneck distance does not take into account the diﬀerence in the number of connected components or holes since it compares two PDs through a one-to-one correspondence. However, the kernel distance estimates the diﬀerence of all pairs of points in two PDs. Moreover, the bottleneck distance tends to ignore points that are close to the diagonal line in PD by mapping to auxiliary points although such points may sometimes indicate the underlying topological structure that discriminates topological spaces. 7 Gromov-Hausdorﬀ distance The change of connected components during the ﬁltration has an agglom- erative hierarchical clustering structure based on single linkage [13,14,41]. Each node starts as a single connected component, and pairs of connected components then are merged if the distance between the closest nodes in two disjoint con- nected components is smaller than the ﬁltration value. The distance s between ij the closest nodes in the two disjoint connected components R and R is called 1 2 the single linkage distance (SLD), which is deﬁned as s = min ρ . ij ∈ ∈ lk l R1,k R2 ##LTLine## ##LTLine## ##LTLine##

text: where

text: t¯i

text: (τi,

text: ξi)

text: is

text: the

text: mirror

text: ti

text: (ξi,

text: τi)

text: with

text: respect

text: to

text: the

text: diagonal

text: line.

text: Then

text: the

text: inner

text: product

text: between

text: two

text: PDs

text: is

text: given

text: by

text: kernel

text: P

text: X

text: P

text: X

text: P

text: X

text: P

text: X

text: P

text: X

text: k

text: Φ

text: σ

text: σ

text: σ

text: ξ

text: τ

text: exp

text: i

text: j

text: exp

text: i

text: j

text: 8πσ

text: 8σ

text: 8σ

text: i,j

text: k

text: is

text: called

text: the

text: persistence

text: state-space

text: (PSS)

text: kernel.

text: Kernel

text: distance

text: D

text: is

text: then

text: σ

text: K

text: deﬁned

text: as

text: a

text: distance

text: between

text: PDs

text: in

text: a

text: feature

text: space

text: P

text: X

text: P

text: X

text: P

text: X

text: P

text: X

text: DK(

text: Φσ(

text: Φσ(

text: P

text: X

text: P

text: X

text: P

text: X

text: P

text: X

text: Φ

text: Φ

text: Φ

text: Φ

text: σ

text: σ

text: σ

text: σ

text: P

text: X

text: P

text: X

text: P

text: X

text: P

text: X

text: k

text: k

text: σ

text: σ

text: P

text: X

text: P

text: X

text: σ

text: The

text: performance

text: of

text: kernel

text: distance

text: depends

text: on

text: the

text: parameter

text: σ.

text: Thus,

text: σ

text: should

text: be

text: chosen

text: carefully.

text: Example

text: The

text: same

text: topological

text: spaces

text: given

text: in

text: Example

text: are

text: used.

text: The

text: PDs

text: of

text: holes

text: are

text: shown

text: in

text: Fig.

text: After

text: kernel

text: smoothing

text: of

text: PDs

text: (c),

text: the

text: kernel

text: distances

text: D

text: are

text: estimated

text: (d,

text: e).

text: As

text: in

text: the

text: bottleneck

text: distances

text: in

text: Fig.

text: K

text: X

text: X

text: X

text: X

text: the

text: kernel

text: distance

text: in

text: (e)

text: can

text: distinguish

text: (no

text: hole)

text: from

text: (three

text: holes).

text: The

text: bottleneck

text: distance

text: does

text: not

text: take

text: into

text: account

text: the

text: diﬀerence

text: in

text: the

text: number

text: of

text: connected

text: components

text: or

text: holes

text: since

text: it

text: compares

text: two

text: PDs

text: through

text: a

text: one-to-one

text: correspondence.

text: However,

text: the

text: kernel

text: distance

text: estimates

text: the

text: diﬀerence

text: of

text: all

text: pairs

text: of

text: points

text: in

text: two

text: PDs.

text: Moreover,

text: the

text: bottleneck

text: distance

text: tends

text: to

text: ignore

text: points

text: that

text: are

text: close

text: to

text: the

text: diagonal

text: line

text: in

text: PD

text: by

text: mapping

text: to

text: auxiliary

text: points

text: although

text: such

text: points

text: may

text: sometimes

text: indicate

text: the

text: underlying

text: topological

text: structure

text: that

text: discriminates

text: topological

text: spaces.

text: Gromov-Hausdorﬀ

text: distance

text: The

text: change

text: of

text: connected

text: components

text: during

text: the

text: ﬁltration

text: has

text: an

text: agglom-

text: erative

text: hierarchical

text: clustering

text: structure

text: based

text: on

text: single

text: linkage

text: Each

text: node

text: starts

text: as

text: a

text: single

text: connected

text: component,

text: and

text: pairs

text: of

text: connected

text: components

text: then

text: are

text: merged

text: if

text: the

text: distance

text: between

text: the

text: closest

text: nodes

text: in

text: two

text: disjoint

text: con-

text: nected

text: components

text: is

text: smaller

text: than

text: the

text: ﬁltration

text: value.

text: The

text: distance

text: s

text: between

text: ij

text: the

text: closest

text: nodes

text: in

text: the

text: two

text: disjoint

text: connected

text: components

text: R

text: and

text: R

text: is

text: called

text: the

text: single

text: linkage

text: distance

text: (SLD),

text: which

text: is

text: deﬁned

text: as

text: s

text: min

text: ρ

text: ij

text: lk

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 Fig. 7: (a) An example of a network, (b) its dendrogram, (c) the distance matrix based on Euclidean distance, (d) the single linkage matrix (SLM). Although it is possible to deﬁne GH distance directly from the metric spaces X X 1 = (V, ρ1) and 2 = (V, ρ2), it turns out that it is identical to L∞ distance between networks in (3) thus ignoring the topological structures of the network. This is the main reason the GH distance is deﬁned through the dendrograms [38,41]. Example 9. When GH distance is applied to 100 simulated networks in Example 4, the networks are clustered according to two baseline topological structures with 100 % accuracy. The superior almost perfect performance is due to the fact that GH distance use the topological information of connectedness represented in the corresponding dendrograms. Example 10. GH distance (7) is used in discriminating brain networks in [38,41]. FDG-PET data consists of 24 ADHD, 26 ASD and 11 pediatric controls (Ped- Con). Each group gives a single SLM (Fig. 8). The jackknife resampling is applied in replicating 24, 26, and 11 SLMs from ADHD, ASD, and PedCon respectively. GH distance is used to compute the distance between SLMs in a pairwise fash- ion. Based on the GH distance matrix, SLMs were clustered into 3 groups using Ward hierarchical clustering. The clustering accuracy was 100 %. While the bottleneck distance measures the diﬀerence between networks by encoding their topological information into the PDs, GH distance measures the diﬀerence between networks by embedding the network into the ultrametric space that represents hierarchical clustering structure of network [14]. GH dis- tance can be viewed as measuring distance between corresponding dendrograms of the networks. GH distance has been statistically quantiﬁed using resampling techniques such as jackknife or permutation tests [38,41,42]. 8 KS-test like distance The graph ﬁltration can be quantiﬁed using monotonic function f satisfying ◦ B X ≥ ◦ B X ≥ · · · ≥ ◦ B X f ( ) f ( ) f ( ). (8) 0 ρ(1) ρ(q) ##LTFigure## ##LTFigure##

text: Fig.

text: (a)

label: caption

text: An

text: example

text: of

text: a

text: network,

text: (b)

text: its

text: dendrogram,

text: (c)

text: the

text: distance

text: matrix

text: based

text: on

text: Euclidean

text: distance,

text: (d)

text: the

text: single

text: linkage

text: matrix

text: (SLM).

text: Although

text: it

text: is

text: possible

text: to

text: deﬁne

text: GH

text: distance

text: directly

text: from

text: the

text: metric

text: spaces

text: X

text: X

text: (V,

text: and

text: (V,

text: it

text: turns

text: out

text: that

text: it

text: is

text: identical

text: to

text: L∞

text: distance

text: between

text: networks

text: in

text: thus

text: ignoring

text: the

text: topological

text: structures

text: of

text: the

text: network.

text: This

text: is

text: the

text: main

text: reason

text: the

text: GH

text: distance

text: is

text: deﬁned

text: through

text: the

text: dendrograms

text: Example

text: When

text: GH

text: distance

text: is

text: applied

text: to

text: simulated

text: networks

text: in

text: Example

text: the

text: networks

text: are

text: clustered

text: according

text: to

text: two

text: baseline

text: topological

text: structures

text: with

text: accuracy.

text: The

text: superior

text: almost

text: perfect

text: performance

text: is

text: due

text: to

text: the

text: fact

text: that

text: GH

text: distance

text: use

text: the

text: topological

text: information

text: of

text: connectedness

text: represented

text: in

text: the

text: corresponding

text: dendrograms.

text: Example

text: GH

text: distance

text: is

text: used

text: in

text: discriminating

text: brain

text: networks

text: in

text: FDG-PET

text: data

text: consists

text: of

text: ADHD,

text: ASD

text: and

text: pediatric

text: controls

text: (Ped-

text: Con).

text: Each

text: group

text: gives

text: a

text: single

text: SLM

text: (Fig.

text: The

text: jackknife

text: resampling

text: is

text: applied

text: in

text: replicating

text: and

text: SLMs

text: from

text: ADHD,

text: ASD,

text: and

text: PedCon

text: respectively.

text: GH

text: distance

text: is

text: used

text: to

text: compute

text: the

text: distance

text: between

text: SLMs

text: in

text: a

text: pairwise

text: fash-

text: ion.

text: Based

text: on

text: the

text: GH

text: distance

text: matrix,

text: SLMs

text: were

text: clustered

text: into

text: groups

text: using

text: Ward

text: hierarchical

text: clustering.

text: The

text: clustering

text: accuracy

text: was

text: While

text: the

text: bottleneck

text: distance

text: measures

text: the

text: diﬀerence

text: between

text: networks

text: by

text: encoding

text: their

text: topological

text: information

text: into

text: the

text: PDs,

text: GH

text: distance

text: measures

text: the

text: diﬀerence

text: between

text: networks

text: by

text: embedding

text: the

text: network

text: into

text: the

text: ultrametric

text: space

text: that

text: represents

text: hierarchical

text: clustering

text: structure

text: of

text: network

text: GH

text: dis-

text: tance

text: can

text: be

text: viewed

text: as

text: measuring

text: distance

text: between

text: corresponding

text: dendrograms

text: of

text: the

text: networks.

text: GH

text: distance

text: has

text: been

text: statistically

text: quantiﬁed

text: using

text: resampling

text: techniques

text: such

text: as

text: jackknife

text: or

text: permutation

text: tests

text: KS-test

text: like

text: distance

text: The

text: graph

text: ﬁltration

text: can

text: be

text: quantiﬁed

text: using

text: monotonic

text: function

text: f

text: satisfying

text: B

text: X

text: B

text: X

text: B

text: X

text: f

text: f

text: f

text: ρ(q)

text: ##LTFigure##

text: ##LTFigure##


text: 14 Fig. 8: SLMs of ADHD, ASD and PedCon groups of FDG-PET brain networks (top). Since each group gives a single SLM, the jackknife resampling was used to replicate SLMs. GH distance matrix shows the pairwise distance between net- works, which is half the diﬀerence between SLMs. Three diagonal block matrices shows distinct clusters between the three groups. The number of connected components, the zeroth Betti number β , satisﬁes 0 the monotonicity property (8). The size of the largest cluster satisﬁes a simi- lar but opposite relation of monotonic increase. Many graph theoretic features such as node degree will be also monotonic. However, any other higher order Betti numbers may not be monotonic. The monotonic feature vectors (8) are (cid:82) used in quantifying the brain networks [20,21,39]. In [39], the slope of the linear ◦ B X regression line is ﬁtted in scatter points (ρ , f ( )) ≤ ≤ and used to (j) ρ(j) 1∞j q◦ B X diﬀerentiate diﬀerent clinical populations. In [21], integral f ( ) d(cid:15) is 0 (cid:15) used as the summary statistic instead of the vector (8). X X Given two networks 1 = (V, ρ1)(cid:12) and 2 = (V, ρ2), Kolm(cid:12)ogorove-Smirnov X (cid:12) X (cid:12) (KS) test like distance between 1 and 2 is given by X X ◦ B X − ◦ B X D ( 1, 2) = sup f ( 1) f ( 2) , KS ≥ (cid:15) (cid:15) (cid:15) 0 where f is a monotonic topological feature [20,21,42]. D is motivated by the KS two-sample Kolmogorove-Smirnov (KS) test [10,20,31], which is a nonparametric ##LTFigure## ##LTFigure##

text: Fig.

text: SLMs

text: of

text: ADHD,

text: ASD

text: and

text: PedCon

text: groups

text: of

text: FDG-PET

text: brain

text: networks

text: (top).

text: Since

text: each

text: group

text: gives

text: a

text: single

text: SLM,

text: the

text: jackknife

text: resampling

text: was

text: used

text: to

text: replicate

text: SLMs.

text: GH

text: distance

text: matrix

text: shows

text: the

text: pairwise

text: distance

text: between

text: net-

text: works,

text: which

text: is

text: half

text: the

text: diﬀerence

text: between

text: SLMs.

text: Three

text: diagonal

text: block

text: matrices

text: shows

text: distinct

text: clusters

text: between

text: the

text: three

text: groups.

text: The

text: number

text: of

text: connected

text: components,

text: the

text: zeroth

text: Betti

text: number

text: β

text: satisﬁes

text: the

text: monotonicity

text: property

text: The

text: size

text: of

text: the

text: largest

text: cluster

text: satisﬁes

text: a

text: simi-

text: lar

text: but

text: opposite

text: relation

text: of

text: monotonic

text: increase.

text: Many

text: graph

text: theoretic

text: features

text: such

text: as

text: node

text: degree

text: will

text: be

text: also

text: monotonic.

text: However,

text: any

text: other

text: higher

text: order

text: Betti

text: numbers

text: may

text: not

text: be

text: monotonic.

text: The

text: monotonic

text: feature

text: vectors

text: are

text: used

text: in

text: quantifying

text: the

text: brain

text: networks

text: In

text: the

text: slope

text: of

text: the

text: linear

text: B

text: X

text: regression

text: line

text: is

text: ﬁtted

text: in

text: scatter

text: points

text: (ρ

text: f

text: and

text: used

text: to

text: (j)

text: ρ(j)

text: q◦

text: B

text: X

text: diﬀerentiate

text: diﬀerent

text: clinical

text: populations.

text: In

text: integral

text: f

text: is

text: used

text: as

text: the

text: summary

text: statistic

text: instead

text: of

text: the

text: vector

text: X

text: X

text: Given

text: two

text: networks

text: (V,

text: and

text: (V,

text: X

text: X

text: (KS)

text: test

text: like

text: distance

text: between

text: and

text: is

text: given

text: by

text: X

text: X

text: B

text: X

text: B

text: X

text: D

text: sup

text: f

text: f

text: KS

text: where

text: f

text: is

text: a

text: monotonic

text: topological

text: feature

text: D

text: is

text: motivated

text: by

text: the

text: KS

text: two-sample

text: Kolmogorove-Smirnov

text: (KS)

text: test

text: which

text: is

text: a

text: nonparametric

text: ##LTFigure##

text: ##LTFigure##


text: A New Model Variance Estimator for an Area Level Small Area Model to Solve Multiple Problems Simultaneously Masayo Yoshimori Hirose The Institute of Statistical Mathematics and Partha Lahiri Joint Program in Survey Methodology, University of Maryland, College Park, U.S.A. Abstract The two-level normal hierarchical model (NHM) has played a criti- cal role in the theory of small area estimation (SAE), one of the growing areas in statistics with numerous applications in diﬀerent disciplines. In this paper, we address major well-known shortcomings associated with the empirical best linear unbiased prediction (EBLUP) of a small area mean and its mean squared error (MSE) estimation by consid- ering an appropriate model variance estimator that satisﬁes multiple properties. The proposed model variance estimator simultaneously (i) improves on the estimation of the related shrinkage factors, (ii) protects EBLUP from the common overshrinkage problem, (iii) avoids complex bias correction in generating strictly positive second-order un- biased mean square error (MSE) estimator either by the Taylor series or single parametric bootstrap method. The idea of achieving multiple desirable properties in an EBLUP method through a suitably devised model variance estimator is the ﬁrst of its kind and holds promise in providing good inferences for small area means under the classical lin- ear mixed model prediction framework. The proposed methodology is also evaluated using a Monte Carlo simulation study and real data analysis. Keywords: Adjusted maximum likelihood method; Empirical Bayes; Em- pirical best linear unbiased prediction; Linear mixed model; Second-order unbiasedness. 1

text: A

label: title

text: New

text: Model

text: Variance

text: Estimator

text: for

text: an

text: Area

text: Level

text: Small

text: Area

text: Model

text: to

text: Solve

text: Multiple

text: Problems

text: Simultaneously

text: Masayo

label: author

text: Yoshimori

text: Hirose

text: The

text: Institute

text: of

text: Statistical

text: Mathematics

text: and

text: Partha

text: Lahiri

text: Joint

text: Program

text: in

text: Survey

text: Methodology,

text: University

text: of

text: Maryland,

text: College

text: Park,

text: U.S.A.

text: Abstract

text: The

label: abstract

text: two-level

text: normal

text: hierarchical

text: model

text: (NHM)

text: has

text: played

text: a

text: criti-

text: cal

text: role

text: in

text: the

text: theory

text: of

text: small

text: area

text: estimation

text: (SAE),

text: one

text: of

text: the

text: growing

text: areas

text: in

text: statistics

text: with

text: numerous

text: applications

text: in

text: diﬀerent

text: disciplines.

text: In

text: this

text: paper,

text: we

text: address

text: major

text: well-known

text: shortcomings

text: associated

text: with

text: the

text: empirical

text: best

text: linear

text: unbiased

text: prediction

text: (EBLUP)

text: of

text: a

text: small

text: area

text: mean

text: and

text: its

text: mean

text: squared

text: error

text: (MSE)

text: estimation

text: by

text: consid-

text: ering

text: an

text: appropriate

text: model

text: variance

text: estimator

text: that

text: satisﬁes

text: multiple

text: properties.

text: The

text: proposed

text: model

text: variance

text: estimator

text: simultaneously

text: (i)

text: improves

text: on

text: the

text: estimation

text: of

text: the

text: related

text: shrinkage

text: factors,

text: (ii)

text: protects

text: EBLUP

text: from

text: the

text: common

text: overshrinkage

text: problem,

text: (iii)

text: avoids

text: complex

text: bias

text: correction

text: in

text: generating

text: strictly

text: positive

text: second-order

text: un-

text: biased

text: mean

text: square

text: error

text: (MSE)

text: estimator

text: either

text: by

text: the

text: Taylor

text: series

text: or

text: single

text: parametric

text: bootstrap

text: method.

text: The

text: idea

text: of

text: achieving

text: multiple

text: desirable

text: properties

text: in

text: an

text: EBLUP

text: method

text: through

text: a

text: suitably

text: devised

text: model

text: variance

text: estimator

text: is

text: the

text: ﬁrst

text: of

text: its

text: kind

text: and

text: holds

text: promise

text: in

text: providing

text: good

text: inferences

text: for

text: small

text: area

text: means

text: under

text: the

text: classical

text: lin-

text: ear

text: mixed

text: model

text: prediction

text: framework.

text: The

text: proposed

text: methodology

text: is

text: also

text: evaluated

text: using

text: a

text: Monte

text: Carlo

text: simulation

text: study

text: and

text: real

text: data

text: analysis.

text: Keywords:

text: Adjusted

text: maximum

text: likelihood

text: method;

text: Empirical

text: Bayes;

text: Em-

text: pirical

text: best

text: linear

text: unbiased

text: prediction;

text: Linear

text: mixed

text: model;

text: Second-order

text: unbiasedness.


text: Table 1: Estimates of shrinkage factors B in 3 areas (minimum, median i and max B values) in 1992 and 1993 SAIPE data i 1992 year 1993 year States D RE HL States D RE HL i i DC 31.6940 1.0000 0.9968 DC 38.2260 0.9574 0.9546 HI 11.3470 1.0000 0.9887 OR 12.1880 0.8775 0.8563 CA 1.8830 1.0000 0.7227 CA 2.1560 0.5588 0.4284 Table 2: Estimates of MSEs in 3 areas (minimum, median and max B i values) in 1992 and 1993 SAIPE data 1992 data States D naive.RE DL.RE PB.RE BL.RE Taylor.HL PB.HL i DC 31.69 1.81 1.91 1.80 1.19 2.08 2.07 HI 11.35 1.19 1.45 1.30 0.88 1.48 1.57 CA 1.88 1.26 2.82 1.34 1.20 1.72 1.37 1993 data States D naive.RE DL.RE PB.RE BL.RE Taylor.HL PB.HL i DC 38.23 4.07 4.23 4.14 4.97 4.41 4.33 OR 12.19 3.02 3.39 2.91 3.13 3.52 3.21 CA 2.16 1.64 2.19 1.74 1.72 1.87 1.60 { · · · } problems, we set m = 15 and generate (y , θ ), i = 1, , m using the i i Fay-Herriot model (1). We use the 1992 SAIPE data described in the previous section to de- sign our simulation study. The 15 areas correspond to states with largest sampling variances D . In the simulation, we use x and D for these states i i i from the 1992 SAIPE data and use A = 15.94, which is the median of D i for the 15 states. The weighted least squared estimates of β from the real data including all 50 states and DC are treated as true β for the simulation. We deﬁne the relative bias (RB) and relative root mean squared error (RRMSE) of an estimator Bˆ of B as: i i − (cid:113)E(Bˆ B ) × RB of Bˆ : i i 100; i B i MSE(Bˆi) × RRMSE of Bˆ : 100, i B i − where MSE(Bˆ ) = E(Bˆ B )2. The expectations in the deﬁnitions of RB i i i 16 ##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: Table

text: Estimates

text: of

text: shrinkage

text: factors

text: B

text: in

text: areas

text: (minimum,

text: median

text: i

text: and

text: max

text: B

text: values)

text: in

text: and

text: SAIPE

text: data

text: i

label: table

text: year

text: year

text: States

text: D

text: RE

text: HL

text: States

text: D

text: RE

text: HL

text: i

text: i

text: DC

text: DC

text: HI

text: OR

text: CA

text: CA

text: Table

text: Estimates

text: of

text: MSEs

text: in

text: areas

text: (minimum,

text: median

text: and

text: max

text: B

text: i

text: values)

text: in

text: and

text: SAIPE

text: data

text: data

text: States

text: D

text: naive.RE

text: DL.RE

text: PB.RE

text: BL.RE

text: Taylor.HL

text: PB.HL

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text: estimator of M (θˆEB ). They showed that i i;MOM − E[M (Aˆ )] = M [θˆEB(Aˆ )] + O(m 1), i;approx MOM i i MOM for large m, under regularity conditions. In fact, M (Aˆ) is not second- i;approx order unbiased estimator of M (θˆEB) for any variance component estimators i i proposed in the literature. Bias correction is usually applied to achieve second-order unbiasedness. However, some bias-correction can even yield negative estimates of MSE. See Jiang (2007) and Molina and Rao (2015) for further discussions. Mimicking a Bayesian hyperprior calculation, Laird and Louis (1987) introduced a parametric bootstrap method for measuring uncertainty of an empirical Bayes estimator. While their point estimator is identical to EBLUP, their measure of uncertainty has more of a Bayesian ﬂavor rather than MSE. Butar (1997) [see also Butar and Lahiri 2003] was the ﬁrst to introduce parametric bootstrap method to produce a second-order unbiased MSE estimator in the small area estimation context. Since Butar’s work, a number of papers on parametric bootstrap MSE estimation methods ap- peared in the SAE literature; see Pfeﬀermann and Glickman (2004), Chat- terjee and Lahiri (2007); Hall and Maiti (2006); Pfeﬀerman and Correra (2012). Some of them are the second-order unbiased but not strictly posi- tive. Some adjustments were proposed to make the second-order unbiased double parametric bootstrap MSE estimators strictly positive, but adjusted MSE estimators were not claimed to have the dual property of second-order unbiasedness and strict positivity. As pointed out in Jiang et al. (2016), a proof is not at all trivial and it is not even clear if the adjustments for positivity retain the second-order unbiasedness of the MSE estimators. In this paper, we focus on the estimation of two important area-speciﬁc functions of A — the shrinkage factor B and the MSE of the EBLUP i M (θˆEB). We propose a single area speciﬁc estimator of A, say Aˆ , that i i i simultaneously satisﬁes the following multiple desirable properties under certain mild regularity conditions: Property 1: Obtain a second-order unbiased estimator of B , that is, − i E(Bˆ ) = B + o(m 1), among the class of estimators of B with iden- i i − i tical variance, up to the order O(m 1), where Bˆ = D /(Aˆ + D ). i i i i ≤ Property 2: 0 < inf ≥ Bˆ sup ≥ Bˆ < 1. That is, it protects EBLUP m 1 i m 1 i from overshrinking to the regression estimator, a common problem encountered in the EB method; 5

text: estimator

text: of

text: M

text: (θˆEB

text: They

text: showed

text: that

text: i

text: i;MOM

text: E[M

text: (Aˆ

text: M

text: [θˆEB(Aˆ

text: O(m

text: i;approx

text: MOM

text: i

text: i

text: MOM

text: for

text: large

text: m,

text: under

text: regularity

text: conditions.

text: In

text: fact,

text: M

text: (Aˆ)

text: is

text: not

text: second-

text: i;approx

text: order

text: unbiased

text: estimator

text: of

text: M

text: (θˆEB)

text: for

text: any

text: variance

text: component

text: estimators

text: i

text: i

text: proposed

text: in

text: the

text: literature.

text: Bias

text: correction

text: is

text: usually

text: applied

text: to

text: achieve

text: second-order

text: unbiasedness.

text: However,

text: some

text: bias-correction

text: can

text: even

text: yield

text: negative

text: estimates

text: of

text: MSE.

text: See

text: Jiang

text: and

text: Molina

text: and

text: Rao

text: for

text: further

text: discussions.

text: Mimicking

text: a

text: Bayesian

text: hyperprior

text: calculation,

text: Laird

text: and

text: Louis

text: introduced

text: a

text: parametric

text: bootstrap

text: method

text: for

text: measuring

text: uncertainty

text: of

text: an

text: empirical

text: Bayes

text: estimator.

text: While

text: their

text: point

text: estimator

text: is

text: identical

text: to

text: EBLUP,

text: their

text: measure

text: of

text: uncertainty

text: has

text: more

text: of

text: a

text: Bayesian

text: ﬂavor

text: rather

text: than

text: MSE.

text: Butar

text: [see

text: also

text: Butar

text: and

text: Lahiri

text: was

text: the

text: ﬁrst

text: to

text: introduce

text: parametric

text: bootstrap

text: method

text: to

text: produce

text: a

text: second-order

text: unbiased

text: MSE

text: estimator

text: in

text: the

text: small

text: area

text: estimation

text: context.

text: Since

text: Butar’s

text: work,

text: a

text: number

text: of

text: papers

text: on

text: parametric

text: bootstrap

text: MSE

text: estimation

text: methods

text: ap-

text: peared

text: in

text: the

text: SAE

text: literature;

text: see

text: Pfeﬀermann

text: and

text: Glickman

text: Chat-

text: terjee

text: and

text: Lahiri

text: Hall

text: and

text: Maiti

text: Pfeﬀerman

text: and

text: Correra

text: Some

text: of

text: them

text: are

text: the

text: second-order

text: unbiased

text: but

text: not

text: strictly

text: posi-

text: tive.

text: Some

text: adjustments

text: were

text: proposed

text: to

text: make

text: the

text: second-order

text: unbiased

text: double

text: parametric

text: bootstrap

text: MSE

text: estimators

text: strictly

text: positive,

text: but

text: adjusted

text: MSE

text: estimators

text: were

text: not

text: claimed

text: to

text: have

text: the

text: dual

text: property

text: of

text: second-order

text: unbiasedness

text: and

text: strict

text: positivity.

text: As

text: pointed

text: out

text: in

text: Jiang

text: et

text: al.

text: a

text: proof

text: is

text: not

text: at

text: all

text: trivial

text: and

text: it

text: is

text: not

text: even

text: clear

text: if

text: the

text: adjustments

text: for

text: positivity

text: retain

text: the

text: second-order

text: unbiasedness

text: of

text: the

text: MSE

text: estimators.

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: the

text: estimation

text: of

text: two

text: important

text: area-speciﬁc

text: functions

text: of

text: A

text: the

text: shrinkage

text: factor

text: B

text: and

text: the

text: MSE

text: of

text: the

text: EBLUP

text: i

text: M

text: (θˆEB).

text: We

text: propose

text: a

text: single

text: area

text: speciﬁc

text: estimator

text: of

text: A,

text: say

text: Aˆ

text: that

text: i

text: i

text: i

text: simultaneously

text: satisﬁes

text: the

text: following

text: multiple

text: desirable

text: properties

text: under

text: certain

text: mild

text: regularity

text: conditions:

text: Property

text: Obtain

text: a

text: second-order

text: unbiased

text: estimator

text: of

text: B

text: that

text: is,

text: i

text: E(Bˆ

text: B

text: o(m

text: among

text: the

text: class

text: of

text: estimators

text: of

text: B

text: with

text: iden-

text: i

text: i

text: i

text: tical

text: variance,

text: up

text: to

text: the

text: order

text: O(m

text: where

text: Bˆ

text: D

text: /(Aˆ

text: D

text: i

text: i

text: i

text: i

text: Property

text: inf

text: Bˆ

text: sup

text: Bˆ

text: That

text: is,

text: it

text: protects

text: EBLUP

text: m

text: i

text: m

text: i

text: from

text: overshrinking

text: to

text: the

text: regression

text: estimator,

text: a

text: common

text: problem

text: encountered

text: in

text: the

text: EB

text: method;