text: ≤ points at x 2.25 mm. The fact that there are bright streaks in this image rather than a smooth cloud suggests that the injection of the polaritons into the cavity occurs unevenly in − momentum space. The asymmetry of the cloud between y and +y may be due to a slight misalignment between the cavity gradient and the injection direction. 1 ) m 2.2 m ( e 2 c n a t s i 1.8 d - x 0 −0.5 0 0.5 y-distance (mm) FIG. 2. (color online) A time-integrated observation of the passing polariton pulse. Coordinates are such that the point of injection is deﬁned at (x,y)=(0,0), and the gradient is approximately − ≈ toward x. Polaritons approach this ﬁeld of view from the left and turn around at x 2.2 mm − before ﬂowing back to x. The sharp cutoﬀ at x =1.7 mm is due to clipping in the spectrometer. To measure the lifetime, the bright jet of polaritons was time-resolved using a Hamamatsu streak camera. To facilitate this, the sample was initially installed such that the gradient was aligned with the horizontal time slit on the streak camera. This enabled us to track a single jet of polaritons while they propagate against the gradient, turn around, and travel backwards, as shown in Figure 3(a). The vertical distance axis in this ﬁgure corresponds to the horizontal x-axis in Figure 2. The trajectory of the polaritons is easily seen in the data, which in this region is well described by a parabolic ﬁt, as expected for the Hamiltonian (3), which is equivalent to that of a ball moving with a constant force due to gravity. Indeed, these data directly demonstrate the in-plane velocity and acceleration of the polaritons during their trajectory. One should note that this region of observation is already more than a millimeter and nearly 200 ps from the injection point, indicating that these polaritons are propagating farther and persisting longer than those in earlier samples, even without conﬁnement in 1D structures, such as used in Ref.12. A simple analysis of this data yields the polariton lifetime after summing in the spatial dimension, as shown in Figure 3(b). The data are well ﬁt by a single exponential with a 6 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: points

text: at

text: x

text: mm.

text: The

text: fact

text: that

text: there

text: are

text: bright

text: streaks

text: in

text: this

text: image

text: rather

text: than

text: a

text: smooth

text: cloud

text: suggests

text: that

text: the

text: injection

text: of

text: the

text: polaritons

text: into

text: the

text: cavity

text: occurs

text: unevenly

text: in

text: momentum

text: space.

text: The

text: asymmetry

text: of

text: the

text: cloud

text: between

text: y

text: and

text: +y

text: may

text: be

text: due

text: to

text: a

text: slight

text: misalignment

text: between

text: the

text: cavity

text: gradient

text: and

text: the

text: injection

text: direction.

text: m

text: m

text: e

text: c

text: n

text: a

text: t

text: s

text: i

text: d

text: x

text: y-distance

text: (mm)

text: FIG.

text: (color

text: online)

text: A

text: time-integrated

text: observation

text: of

text: the

text: passing

text: polariton

text: pulse.

text: Coordinates

text: are

text: such

text: that

text: the

text: point

text: of

text: injection

text: is

text: deﬁned

text: at

text: and

text: the

text: gradient

text: is

text: approximately

text: toward

text: x.

text: Polaritons

text: approach

text: this

text: ﬁeld

text: of

text: view

text: from

text: the

text: left

text: and

text: turn

text: around

text: at

text: x

text: mm

text: before

text: ﬂowing

text: back

text: to

text: x.

text: The

text: sharp

text: cutoﬀ

text: at

text: x

text: mm

text: is

text: due

text: to

text: clipping

text: in

text: the

text: spectrometer.

text: To

text: measure

text: the

text: lifetime,

text: the

text: bright

text: jet

text: of

text: polaritons

text: was

text: time-resolved

text: using

text: a

text: Hamamatsu

text: streak

text: camera.

text: To

text: facilitate

text: this,

text: the

text: sample

text: was

text: initially

text: installed

text: such

text: that

text: the

text: gradient

text: was

text: aligned

text: with

text: the

text: horizontal

text: time

text: slit

text: on

text: the

text: streak

text: camera.

text: This

text: enabled

text: us

text: to

text: track

text: a

text: single

text: jet

text: of

text: polaritons

text: while

text: they

text: propagate

text: against

text: the

text: gradient,

text: turn

text: around,

text: and

text: travel

text: backwards,

text: as

text: shown

text: in

text: Figure

text: The

text: vertical

text: distance

text: axis

text: in

text: this

text: ﬁgure

text: corresponds

text: to

text: the

text: horizontal

text: x-axis

text: in

text: Figure

text: The

text: trajectory

text: of

text: the

text: polaritons

text: is

text: easily

text: seen

text: in

text: the

text: data,

text: which

text: in

text: this

text: region

text: is

text: well

text: described

text: by

text: a

text: parabolic

text: ﬁt,

text: as

text: expected

text: for

text: the

text: Hamiltonian

text: which

text: is

text: equivalent

text: to

text: that

text: of

text: a

text: ball

text: moving

text: with

text: a

text: constant

text: force

text: due

text: to

text: gravity.

text: Indeed,

text: these

text: data

text: directly

text: demonstrate

text: the

text: in-plane

text: velocity

text: and

text: acceleration

text: of

text: the

text: polaritons

text: during

text: their

text: trajectory.

text: One

text: should

text: note

text: that

text: this

text: region

text: of

text: observation

text: is

text: already

text: more

text: than

text: a

text: millimeter

text: and

text: nearly

text: ps

text: from

text: the

text: injection

text: point,

text: indicating

text: that

text: these

text: polaritons

text: are

text: propagating

text: farther

text: and

text: persisting

text: longer

text: than

text: those

text: in

text: earlier

text: samples,

text: even

text: without

text: conﬁnement

text: in

text: structures,

text: such

text: as

text: used

text: in

text: A

text: simple

text: analysis

text: of

text: this

text: data

text: yields

text: the

text: polariton

text: lifetime

text: after

text: summing

text: in

text: the

text: spatial

text: dimension,

text: as

text: shown

text: in

text: Figure

text: The

text: data

text: are

text: well

text: ﬁt

text: by

text: a

text: single

text: exponential

text: with

text: a

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text: Isochrone Comparison using Multi-Band Photometry 5 Table 2 Combined Dataset Photometry RA Dec U σ U B σ B V σ V R σ R I σ I J σ J H σ H KS σ KS 91.88346 24.43455 99.999 9.999 19.878 0.045 18.873 0.037 18.268 0.033 17.688 0.032 16.744 0.018 16.425 0.021 16.134 0.021 92.34281 24.43458 99.999 9.999 19.543 0.045 18.669 0.034 18.051 0.028 17.446 0.026 16.555 0.018 16.100 0.010 15.904 0.012 92.56734 24.43462 99.999 9.999 19.590 0.041 18.605 0.033 18.027 0.027 17.377 0.027 16.517 0.016 16.073 0.010 15.909 0.014 92.43966 24.43469 16.700 0.075 16.422 0.022 15.845 0.019 15.503 0.018 15.099 0.020 14.490 0.017 14.275 0.010 14.084 0.007 92.30543 24.43478 18.854 0.093 18.678 0.030 17.831 0.025 17.273 0.023 16.715 0.024 15.953 0.021 15.585 0.007 15.443 0.009 This table is available in its entirety in machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content. MULTI-BAND PHOTOMETRY OF M35 B. Thompson1, P. Frinchaboy1, K. Kinemuchi2, A. Sarajedini3, and R. Cohen4 Keywords: open clusters and associations: individual (M35) 3. STELLAR STRUCTURE MODELS In this work, four isochrone systems are considered: Dartmouth (Dotter et al. 2007), Y2 (Yi et al. 2001), Padova (Girardi et al. 2002) and PARSEC (Bressan et al. 2012). Each model incorporates diﬀerent physical as- sumptions (i.e. equation of state, radiative and con- ductive opacities), treatment of physical processes (i.e. convective transport, stellar atmospheres) and physical parameters (i.e. solar metallicity, initial He abundance, heavy-element mixture), all of which alter the resulting isochrone shape. Values for the input physics considered in this work are listed in table 3. Before comparing the isochrones to the observed data, some of the values in table 3 can be standardized in order to simplify the ﬁnal comparison. 3.1. Age Figure 4. Left: Interpolation onto a common mass grid for One of the main diﬀerences between isochrone systems the [Fe/H]= +0.07 Dartmouth isochrone. Black dots are origi- used in this work is the range of available ages. M35 nal isochrone grid points, grey are interpolated points in steps of 0.01 M(cid:12). Right: Interpolation in [Fe/H] of dartmouth isochrones has a published age of 178 Myr (Dias et al. 2002), for to the M35 metallicity. Isochrones in grey are the three closest which isochrones are available in the Padova and PAR- grid points used to create the interpolated isochrone. Note that SEC systems. The nearest age in Y2 is 200 Myr, while interpolated isochrones only exist along the main sequence. All the youngest possible isochrone available for Dartmouth isochrones are for an age of 250 Myr. is 250 Myr. This work is interested in how the models and PARSEC, however it is only available for ages > 1 treat main sequence stars, so this age diﬀerence is neg- Gyr, unsuitable for M35. Instead, another method must ligible; most stars on the main sequence will not have be employed for the Dartmouth isochrones. shifted in this 72 Myr span, given that all stars have Starting with the Dartmouth system’s pre-packaged ﬁnished their pre-main sequence evolution. metallicity grid, each isochrone is interpolated in mass, 3.2. Metallicity using a common spacing of 0.01 M(cid:12). For each star in the Many isochrone systems come pre-packaged in rough new isochrone, stellar parameters (log g, L, T) and mag- metallicity grids. For the isochrones to accurately match nitudes are cubically interpolated to the new mass value. observed data, all must be interpolated to the metal- This interpolation only works along the main sequence, licity of M35. Using previous WOCS work in Barrado y where mass increases monotonically. This work is in- Navascu´es et al. (2001) the metallicity of M35 is [Fe/H]= terested in how the isochrone treats the main sequence, − 0.21, which was measured using high-resolution spec- and there is little evolution oﬀ of the main sequence in troscopy of 9 bright stars within the cluster. Interpolated a young cluster like M35, so the loss of giants from the Padova and PARSEC (v1.1) isochrones were pulled from isochrone is acceptable. Once all isochrones are on a the web via the interactive CMD 2.5 interface13. Y2 common mass grid, stellar parameters and magnitudes isochrones include a FORTRAN routine that interpo- are quadratically interpolated in metallicity, using the lates to a speciﬁed metallicity. Dartmouth isochrones nearest three isochr−ones in the grid to the desired [Fe/H]. utilize a similar interpolation web interface14 as Padova For M35, [Fe/H]= 0.50, +0.07−, +0.21 isochrones were used to interpolate the [Fe/H]= 0.21 isochrone. Figure 4 shows the results of each of the interpolation steps. 1 Texas Christian University, Fort Worth, TX 76129 3.3. Atmosphere Models 2 Apache Point Observatory, Sunspot, NM 88349 3 University of Florida, Gainesville, FL 32611 Stars with similar internal parameters (log g, T), but 4 Universidad de Concepcion, Concepcion, Chile diﬀerent color-temperature relations may look highly 13 http://stev.oapd.inaf.it/cgi-bin/cmd discrepant on a cluster CMD. Atmosphere models are 14 http://stellar.dartmouth.edu/%7Emodels/webtools.html standardized across all isochrone systems, allowing for ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Isochrone

label: paragraph

text: Comparison

text: using

text: Multi-Band

text: Photometry

text: Table

text: Combined

label: caption

text: Dataset

text: Photometry

text: RA

label: table

text: Dec

text: U

text: σ

text: U

text: B

text: σ

text: B

text: V

text: σ

text: V

text: R

text: σ

text: R

text: I

text: σ

text: I

text: J

text: σ

text: J

text: H

text: σ

text: H

text: KS

text: σ

text: KS

text: This

text: table

text: is

text: available

text: in

text: its

text: entirety

text: in

text: machine-readable

text: form

text: in

text: the

text: online

text: journal.

text: A

text: portion

text: is

text: shown

text: here

text: for

text: guidance

text: regarding

text: its

text: form

text: and

text: content.

text: MULTI-BAND

text: PHOTOMETRY

text: OF

text: B.

text: P.

text: K.

text: A.

text: and

text: R.

text: Keywords:

text: open

text: clusters

text: and

text: associations:

text: individual

text: STELLAR

text: STRUCTURE

text: MODELS

text: In

text: this

text: work,

text: four

text: isochrone

text: systems

text: are

text: considered:

text: Dartmouth

text: (Dotter

text: et

text: al.

text: (Yi

text: et

text: al.

text: Padova

text: (Girardi

text: et

text: al.

text: and

text: PARSEC

text: (Bressan

text: et

text: al.

text: Each

text: model

text: incorporates

text: diﬀerent

text: physical

text: as-

text: sumptions

text: (i.e.

text: equation

text: of

text: state,

text: radiative

text: and

text: con-

text: ductive

text: opacities),

text: treatment

text: of

text: physical

text: processes

text: (i.e.

text: convective

text: transport,

text: stellar

text: atmospheres)

text: and

text: physical

text: parameters

text: (i.e.

text: solar

text: metallicity,

text: initial

text: He

text: abundance,

text: heavy-element

text: mixture),

text: all

text: of

text: which

text: alter

text: the

text: resulting

text: isochrone

text: shape.

text: Values

text: for

text: the

text: input

text: physics

text: considered

text: in

text: this

text: work

text: are

text: listed

text: in

text: table

text: Before

text: comparing

text: the

text: isochrones

text: to

text: the

text: observed

text: data,

text: some

text: of

text: the

text: values

text: in

text: table

text: can

text: be

text: standardized

text: in

text: order

text: to

text: simplify

text: the

text: ﬁnal

text: comparison.

text: Age

text: Figure

text: Left:

text: Interpolation

text: onto

text: a

text: common

text: mass

text: grid

text: for

text: One

text: of

text: the

text: main

text: diﬀerences

text: between

text: isochrone

text: systems

text: the

text: [Fe/H]=

text: Dartmouth

text: isochrone.

text: Black

text: dots

text: are

text: origi-

text: used

text: in

text: this

text: work

text: is

text: the

text: range

text: of

text: available

text: ages.

text: nal

text: isochrone

text: grid

text: points,

text: grey

text: are

text: interpolated

text: points

text: in

text: steps

text: of

text: Right:

text: Interpolation

text: in

text: [Fe/H]

text: of

text: dartmouth

text: isochrones

text: has

text: a

text: published

text: age

text: of

text: Myr

text: (Dias

text: et

text: al.

text: for

text: to

text: the

text: metallicity.

text: Isochrones

text: in

text: grey

text: are

text: the

text: three

text: closest

text: which

text: isochrones

text: are

text: available

text: in

text: the

text: Padova

text: and

text: PAR-

text: grid

text: points

text: used

text: to

text: create

text: the

text: interpolated

text: isochrone.

text: Note

text: that

text: SEC

text: systems.

text: The

text: nearest

text: age

text: in

text: is

text: Myr,

text: while

text: interpolated

text: isochrones

text: only

text: exist

text: along

text: the

text: main

text: sequence.

text: All

text: the

text: youngest

text: possible

text: isochrone

text: available

text: for

text: Dartmouth

text: isochrones

text: are

text: for

text: an

text: age

text: of

text: Myr.

text: is

text: Myr.

text: This

text: work

text: is

text: interested

text: in

text: how

text: the

text: models

text: and

text: PARSEC,

text: however

text: it

text: is

text: only

text: available

text: for

text: ages

text: treat

text: main

text: sequence

text: stars,

text: so

text: this

text: age

text: diﬀerence

text: is

text: neg-

text: Gyr,

text: unsuitable

text: for

text: Instead,

text: another

text: method

text: must

text: ligible;

text: most

text: stars

text: on

text: the

text: main

text: sequence

text: will

text: not

text: have

text: be

text: employed

text: for

text: the

text: Dartmouth

text: isochrones.

text: shifted

text: in

text: this

text: Myr

text: span,

text: given

text: that

text: all

text: stars

text: have

text: Starting

text: with

text: the

text: Dartmouth

text: system’s

text: pre-packaged

text: ﬁnished

text: their

text: pre-main

text: sequence

text: evolution.

text: metallicity

text: grid,

text: each

text: isochrone

text: is

text: interpolated

text: in

text: mass,

text: Metallicity

text: using

text: a

text: common

text: spacing

text: of

text: For

text: each

text: star

text: in

text: the

text: Many

text: isochrone

text: systems

text: come

text: pre-packaged

text: in

text: rough

text: new

text: isochrone,

text: stellar

text: parameters

text: (log

text: g,

text: L,

text: T)

text: and

text: mag-

text: metallicity

text: grids.

text: For

text: the

text: isochrones

text: to

text: accurately

text: match

text: nitudes

text: are

text: cubically

text: interpolated

text: to

text: the

text: new

text: mass

text: value.

text: observed

text: data,

text: all

text: must

text: be

text: interpolated

text: to

text: the

text: metal-

text: This

text: interpolation

text: only

text: works

text: along

text: the

text: main

text: sequence,

text: licity

text: of

text: Using

text: previous

text: WOCS

text: work

text: in

text: Barrado

text: y

text: where

text: mass

text: increases

text: monotonically.

text: This

text: work

text: is

text: in-

text: Navascu´es

text: et

text: al.

text: the

text: metallicity

text: of

text: is

text: [Fe/H]=

text: terested

text: in

text: how

text: the

text: isochrone

text: treats

text: the

text: main

text: sequence,

text: which

text: was

text: measured

text: using

text: high-resolution

text: spec-

text: and

text: there

text: is

text: little

text: evolution

text: oﬀ

text: of

text: the

text: main

text: sequence

text: in

text: troscopy

text: of

text: bright

text: stars

text: within

text: the

text: cluster.

text: Interpolated

text: a

text: young

text: cluster

text: like

text: so

text: the

text: loss

text: of

text: giants

text: from

text: the

text: Padova

text: and

text: PARSEC

text: isochrones

text: were

text: pulled

text: from

text: isochrone

text: is

text: acceptable.

text: Once

text: all

text: isochrones

text: are

text: on

text: a

text: the

text: web

text: via

text: the

text: interactive

text: CMD

text: common

text: mass

text: grid,

text: stellar

text: parameters

text: and

text: magnitudes

text: isochrones

text: include

text: a

text: FORTRAN

text: routine

text: that

text: interpo-

text: are

text: quadratically

text: interpolated

text: in

text: metallicity,

text: using

text: the

text: lates

text: to

text: a

text: speciﬁed

text: metallicity.

text: Dartmouth

text: isochrones

text: nearest

text: three

text: isochr−ones

text: in

text: the

text: grid

text: to

text: the

text: desired

text: [Fe/H].

text: utilize

text: a

text: similar

text: interpolation

text: web

text: as

text: Padova

text: For

text: [Fe/H]=

text: isochrones

text: were

text: used

text: to

text: interpolate

text: the

text: [Fe/H]=

text: isochrone.

text: Figure

text: shows

text: the

text: results

text: of

text: each

text: of

text: the

text: interpolation

text: steps.

text: Texas

text: Christian

text: University,

text: Fort

text: Worth,

text: TX

text: Atmosphere

text: Models

text: Apache

text: Point

text: Observatory,

text: Sunspot,

text: NM

text: University

text: of

text: Florida,

text: Gainesville,

text: FL

text: Stars

text: with

text: similar

text: internal

text: parameters

text: (log

text: g,

text: T),

text: but

text: Universidad

text: de

text: Concepcion,

text: Concepcion,

text: Chile

text: diﬀerent

text: color-temperature

text: relations

text: may

text: look

text: highly

text: discrepant

text: on

text: a

text: cluster

text: CMD.

text: Atmosphere

text: models

text: are

text: standardized

text: across

text: all

text: isochrone

text: systems,

text: allowing

text: for

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

text: ##LTLine##


text: 3.1 Bounds on Green’s functions 13 (cid:80) (cid:16) where stands for . We write , if there is a constant F (x, U) ∈ ∩Z F (x, y) F 1 C > 0 ∈ y U d such that for all , x, y V L+rL 1 ≤ · · ≤ × on F (x, y) F ( , ) CF (x, y) U(x) U(y). C We shall repeatedly need some properties of , which form part of [1, Lemma 4.4]. Γ L Lemma 3.1 (Properties of Γ ). L (cid:16) (i) . Γ 1 L ≤ ≤ E { ∈ ≤ } (ii) For 1 j 1 s , with = y V : d˜(y) 3jr , 3r L j L+rL L E ≤ sup Γ (x, ) C log(j + 1), ∈ L j x VL+rL ≤ and for , 0 α < 3 ≤ − sup Γ (x, Sh (s , L/(log L)α)) C(log log L)(log L)6 2α. ∈ L L L x VL+rL ∈ (cid:40) (cid:32) (cid:33)(cid:41) (iii) For , x V L+r ≤ d˜(x) d˜(x) ∧ Γ (x, V ) C max (log L)6, log log L . L L L r L We now formulate the key estimate, which shows how both gˆ(q) and Gˆg can be L L dominated from above by the deterministic kernel . See [1, Lemma 4.2] for a proof. Γ L Lemma 3.2. ∈ P (i) There exists a constant such that for all s, C > 0 q κ (cid:22) gˆ(q) CΓ . L L ≤ ≤ (ii) Assume C1(δ, L , L ), and let L L L (log L )2. There exists a constant 0 1 1 1 1 such that for small, C > 0 δ > 0 (cid:22) Gˆg CΓ . L L ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Bounds

text: on

text: Green’s

text: functions

text: where

text: stands

text: for

text: We

text: write

text: if

text: there

text: is

text: a

text: constant

text: F

text: (x,

text: U)

text: ∩Z

text: F

text: (x,

text: y)

text: F

text: C

text: y

text: U

text: d

text: such

text: that

text: for

text: all

text: x,

text: y

text: V

text: L+rL

text: on

text: F

text: (x,

text: y)

text: F

text: CF

text: (x,

text: y)

text: U(x)

text: U(y).

text: C

text: We

text: shall

text: repeatedly

text: need

text: some

text: properties

text: of

text: which

text: form

text: part

text: of

text: Lemma

text: Γ

text: L

text: Lemma

text: (Properties

text: of

text: Γ

text: L

label: list

text: (i)

text: Γ

text: L

text: E

text: (ii)

text: For

text: j

text: s

text: with

text: y

text: V

text: d˜(y)

text: L

text: j

text: L+rL

text: L

text: E

text: sup

text: Γ

text: (x,

text: C

text: log(j

text: L

text: j

text: x

text: VL+rL

text: and

text: for

text: α

text: sup

text: Γ

text: (x,

text: Sh

text: (s

text: L/(log

text: L)α))

text: C(log

text: log

text: L)(log

text: L

text: L

text: L

text: x

text: VL+rL

text: (iii)

text: For

text: x

text: V

text: L+r

text: d˜(x)

text: d˜(x)

text: Γ

text: (x,

text: V

text: C

text: max

text: (log

text: log

text: log

text: L

text: L

text: L

text: L

text: r

text: L

text: We

text: now

text: formulate

text: the

text: key

text: estimate,

text: which

text: shows

text: how

text: both

text: gˆ(q)

text: and

text: Gˆg

text: can

text: be

text: L

text: L

text: dominated

text: from

text: above

text: by

text: the

text: deterministic

text: kernel

text: See

text: Lemma

text: for

text: a

text: proof.

text: Γ

text: L

text: Lemma

text: P

text: (i)

text: There

text: exists

text: a

text: constant

text: such

text: that

text: for

text: all

text: s,

text: C

text: q

text: κ

text: gˆ(q)

text: CΓ

text: L

text: L

text: (ii)

text: Assume

text: L

text: L

text: and

text: let

text: L

text: L

text: L

text: (log

text: L

text: There

text: exists

text: a

text: constant

text: such

text: that

text: for

text: small,

text: C

text: δ

text: Gˆg

text: CΓ

text: L

text: L

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4.4 Proof of the main technical statement 27 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) \ (cid:48) (cid:12) ≤ For the second summand of (22), Lemma 3.1 (ii) gives Gˆv(v, V (v) B ) C, whence t t (cid:48) ≤ (cid:48) Gˆv1 \ (cid:48)∆ π Λ (v) C sup ∆ π Λ (w) . t Vt(v) B Vt(v) L ∈ \ (cid:48) Vt(v) L w V√t(v) B ∈ \ (cid:48) ≤ ∈ Fix w V (v) B . Set η = (cid:8)d(w, ∂V (v)) 2r + d and choose y(cid:9) ∂V (v) such that | − | t t t w t . With w y = η w ∈ | − | ≤ − I(y ) = y ∂V (v) : y y (log L) 5/2s , w t w L we write (cid:88) (cid:48) (cid:88) (cid:48) − ∆ π Λ (w) = ∆ π (w, y) (Λ (y) Λ (y )) Vt(v) L Vt(v) L L w ∈ y ∂Vt(v) (cid:88)(cid:48) − = ∆ π (w, y) (Λ (y) Λ (y )) Vt(v) L L w ∈ y I(yw) (cid:48) − (23) + ∆ π (w, y) (Λ (y) Λ (y )) . Vt(v) L L w ∈ \ y ∂Vt(v) I(yw) (cid:88) (cid:12) (cid:12) ∈ (cid:12) (cid:12)| − | ≤ − For y I(y ), Lemma 4.7 yields Λ (y) Λ (y ) C(log L) 7/3s2 . Therefore, w L L w L (cid:48) | − | ≤ − ∆ π (w, y) Λ (y) Λ (y ) C(log L) 8L2. Vt(v) L L w ∈ y I(yw) { ∈ It remains to handle the second term of (23). To this end, let U(w) = u V (v) : | (cid:48) | } ∈ \ t . Using for the trivial bound ∆ (w, u) > 0 y ∂V (v) I(y ) t w | − | ≤ ≤ − Λ (y) Λ (y ) Λ (y) + Λ (y ) C(log L) 6L2, L L w L L w (cid:88) (cid:12) (cid:12) see (19), we ob(cid:12)tain (cid:12) (cid:48) − ∆ π (w, y) (Λ (y) Λ (y )) Vt(v) L L w ∈ \ y ∂Vt(v) I(yw) ≤ − \ C(log L) 6L2 sup π (u, ∂V (v) I(y )) . ∈ Vt(v) t w u U(w) ∈ ∈ \ If and , then u U(w) y ∂V (v) I(y ) t w | − | ≥ | − | − | − | ≥ − − ≥ − u y y y y u (log L) 5/2s 3r (1/2)(log L) 5/2s . w w L t L (cid:88) For such , we get by standard hitting estimates, see e.g. [1, Lemma 3.2 (ii)], u \ ≤ 1 π (u, ∂V (v) I(y )) Cr | − | Vt(v) t w t u y d ∈ \ y ∂Vt(v) I(yw) ≤ − ≤ − Cr (log L)5/2(s ) 1 C(log L) 9. t L The estimate on the sum can be obtained from [1, Lemma 3.6]. This bounds the second term of (23). We have proven (21) and hence the lemma. (cid:50) Now it is easy to prove the main propagation step. ##LTLine## ##LTLine##

text: Proof

text: of

text: the

text: main

text: technical

text: statement

text: For

text: the

text: second

text: summand

text: of

text: Lemma

text: (ii)

text: gives

text: Gˆv(v,

text: V

text: (v)

text: B

text: C,

text: whence

text: t

text: t

text: π

text: Λ

text: (v)

text: C

text: sup

text: π

text: Λ

text: (w)

text: t

text: Vt(v)

text: B

text: Vt(v)

text: L

text: Vt(v)

text: L

text: w

text: V√t(v)

text: B

text: Fix

text: w

text: V

text: (v)

text: B

text: Set

text: η

text: ∂V

text: (v))

text: d

text: and

text: choose

text: ∂V

text: (v)

text: such

text: that

text: t

text: t

text: t

text: w

text: t

text: With

text: w

text: y

text: η

text: w

text: I(y

text: y

text: ∂V

text: (v)

text: y

text: y

text: (log

text: L)

text: w

text: t

text: w

text: L

text: we

text: write

text: π

text: Λ

text: (w)

text: π

text: (w,

text: y)

text: (Λ

text: (y)

text: Λ

text: (y

text: Vt(v)

text: L

text: Vt(v)

text: L

text: L

text: w

text: y

text: ∂Vt(v)

text: π

text: (w,

text: y)

text: (Λ

text: (y)

text: Λ

text: (y

text: Vt(v)

text: L

text: L

text: w

text: y

text: I(yw)

text: π

text: (w,

text: y)

text: (Λ

text: (y)

text: Λ

text: (y

text: Vt(v)

text: L

text: L

text: w

text: y

text: ∂Vt(v)

text: I(yw)

text: For

text: y

text: I(y

text: Lemma

text: yields

text: Λ

text: (y)

text: Λ

text: (y

text: C(log

text: L)

text: Therefore,

text: w

text: L

text: L

text: w

text: L

text: π

text: (w,

text: y)

text: Λ

text: (y)

text: Λ

text: (y

text: C(log

text: L)

text: Vt(v)

text: L

text: L

text: w

text: y

text: I(yw)

text: It

text: remains

text: to

text: handle

text: the

text: second

text: term

text: of

text: To

text: this

text: end,

text: let

text: U(w)

text: u

text: V

text: (v)

text: t

text: Using

text: for

text: the

text: trivial

text: bound

text: (w,

text: u)

text: y

text: ∂V

text: (v)

text: I(y

text: t

text: w

text: Λ

text: (y)

text: Λ

text: (y

text: Λ

text: (y)

text: Λ

text: (y

text: C(log

text: L)

text: L

text: L

text: w

text: L

text: L

text: w

text: see

text: we

text: π

text: (w,

text: y)

text: (Λ

text: (y)

text: Λ

text: (y

text: Vt(v)

text: L

text: L

text: w

text: y

text: ∂Vt(v)

text: I(yw)

text: C(log

text: L)

text: sup

text: π

text: (u,

text: ∂V

text: (v)

text: I(y

text: Vt(v)

text: t

text: w

text: u

text: U(w)

text: If

text: and

text: then

text: u

text: U(w)

text: y

text: ∂V

text: (v)

text: I(y

text: t

text: w

text: u

text: y

text: y

text: y

text: y

text: u

text: (log

text: L)

text: L)

text: w

text: w

text: L

text: t

text: L

text: For

text: such

text: we

text: get

text: by

text: standard

text: hitting

text: estimates,

text: see

text: e.g.

text: Lemma

text: (ii)],

text: u

text: π

text: (u,

text: ∂V

text: (v)

text: I(y

text: Cr

text: Vt(v)

text: t

text: w

text: t

text: u

text: y

text: d

text: y

text: ∂Vt(v)

text: I(yw)

text: Cr

text: (log

text: C(log

text: L)

text: t

text: L

text: The

text: estimate

text: on

text: the

text: sum

text: can

text: be

text: obtained

text: from

text: Lemma

text: This

text: bounds

text: the

text: second

text: term

text: of

text: We

text: have

text: proven

text: and

text: hence

text: the

text: lemma.

text: Now

text: it

text: is

text: easy

text: to

text: prove

text: the

text: main

text: propagation

text: step.

text: ##LTLine##

text: ##LTLine##


text: 28 4 MEAN SOJOURN TIMES ≥ Lemma 4.9. Assume A0(ε) and B. There exists L = L (η) such that if L ≤ ≤ 0 0 P 1 ∈ L and C1(δ, L , L ) holds, then for L L L (log L )2 and -almost all ω 0 ∩ 0 1 1 1 1 Goodsp OneBadtm, L L ∈ − · E [τ ] [1 f (L), 1 + f (L)] E [τ ] . 0,ω L η η 0 L ∈ P ∩ P ∩ ∩ Proof: We let ω ( )Zd ( s,1)Zd Goodsp OneBadtm. Put p = p . In this proof, ε L L sL/20 we keep the superscript (p) in gˆ(p). By Lemma 4.3 and the perturbation expansion (4), − with ∆ = 1 (Πˆ πˆ(p)), VL E [τ ] = GˆΛ (0) = gˆ(p)Λ (0) + Gˆ∆gˆ(p)Λ (0) = A + A . 0,ω L L L L 1 2 \ B ∪ Set B = V ( tm Sh (L/(log L)2). The term A we split into L L L 1 A = gˆ(p)1 Λ (0) + gˆ(p)1 \ Λ (0). 1 B L VL B L \ ≤ ≤ − Since gˆ(p)(0, V B) C(log L)3 by Lemma 3.1 (ii) and Λ (x) (log L) 6L2, the second L − L summand of A can be bounded by(cid:0)O ((log(cid:0) L) (cid:1)3(cid:1)) E [τ ]. T(cid:0)he main(cid:0) contribu(cid:1)t(cid:1)ion comes 1 0 L from the ﬁrst summand. First notice that − − gˆ(p)1 λ(p)(0) = gˆ(p)1 λ (0) 1 + O s 1 = E [τ ] 1 + O (log L) 6 . B L (cid:2) B (cid:0)L (cid:1) L (cid:0) 0 L (cid:1)(cid:3) ∈ Furthermore, we have for by deﬁnition x B ∈ − − − · Λ (x) 1 f (log L) 3L , 1 + f (log L) 3L λ (x). (cid:2) L(cid:0) (cid:1) η (cid:0) (cid:1) η (cid:0) (cid:1) L (cid:0) (cid:1)(cid:3) Collecting all terms, we conclude that ∈ − − − − − − A 1 O (log L) 3 f (log L) 3L , 1 + O (log L) 3 + f (log L) 3L 1 η η × E [τ ] . 0 L (cid:0) (cid:1) − Lemma 4.8 bounds A by O((log L) 5/3) E [τ ]. Since for L suﬃciently large, 2 0 L − − f (L) > f (log L) 3L + C(log L) 5/3, η η we arrive at ∈ − · E [τ ] = A + A [1 f (L), 1 + f (L)] E [τ ] , 0,ω L 1 2 η η 0 L as claimed. (cid:50) Proposition 4.1 follows now as an immediate consequence of our estimates. Proof of Proposition 4.1: (i) From Lemmata 4.6 and 4.9 we deduce that for large ≥ ≤ ≤ L , if L L and L L L (log L )2, we have under C1(δ, L , L ) and C2(η, L ) 0 1 0 1 1 1 0 1 1 P ∈ − · (E [τ ] / [1 f(L), 1 + f(L)] E [τ ]) 0,ω L 0 L ≤ P ∪ (B(cid:0)adsp ManyBad(cid:1)tm) + P L ∈ −L · ∩ + (E [τ ] / [1 f(L), 1 + f(L)] E [τ ]; Goodsp OneBadtm) 0,ω L 0 L L L ≤ − exp (1/2)(log L)2 . By Proposition 2.1, if is small and is suﬃciently large, holds under δ > 0 L C1(δ, L , L) ≥ ≤ 0 0 A0(ε) for all L L , provided ε ε (δ). This proves the proposition. (cid:50) 0 0

text: MEAN

text: SOJOURN

text: TIMES

text: Lemma

text: Assume

text: and

text: B.

text: There

text: exists

text: L

text: L

text: (η)

text: such

text: that

text: if

text: L

text: P

text: L

text: and

text: L

text: L

text: holds,

text: then

text: for

text: L

text: L

text: L

text: (log

text: L

text: and

text: -almost

text: all

text: ω

text: Goodsp

text: OneBadtm,

text: L

text: L

text: E

text: [τ

text: f

text: (L),

text: f

text: (L)]

text: E

text: [τ

text: L

text: η

text: η

text: L

text: P

text: P

text: Proof:

text: We

text: let

text: ω

text: )Zd

text: Goodsp

text: OneBadtm.

text: Put

text: p

text: p

text: In

text: this

text: proof,

text: ε

text: L

text: L

text: we

text: keep

text: the

text: superscript

text: (p)

text: in

text: gˆ(p).

text: By

text: Lemma

text: and

text: the

text: perturbation

text: expansion

text: with

text: (Πˆ

text: πˆ(p)),

text: VL

text: E

text: [τ

text: GˆΛ

text: gˆ(p)Λ

text: Gˆ∆gˆ(p)Λ

text: A

text: A

text: L

text: L

text: L

text: L

text: B

text: Set

text: B

text: V

text: tm

text: Sh

text: (L/(log

text: The

text: term

text: A

text: we

text: split

text: into

text: L

text: L

text: L

text: A

text: Λ

text: Λ

text: B

text: L

text: VL

text: B

text: L

text: Since

text: V

text: B)

text: C(log

text: by

text: Lemma

text: (ii)

text: and

text: Λ

text: (x)

text: (log

text: L)

text: the

text: second

text: L

text: L

text: summand

text: of

text: A

text: can

text: be

text: bounded

text: L)

text: E

text: [τ

text: comes

text: L

text: from

text: the

text: ﬁrst

text: summand.

text: First

text: notice

text: that

text: λ

text: O

text: s

text: E

text: [τ

text: O

text: (log

text: L)

text: B

text: L

text: B

text: L

text: L

text: Furthermore,

text: we

text: have

text: for

text: by

text: deﬁnition

text: x

text: B

text: Λ

text: (x)

text: f

text: (log

text: L)

text: f

text: (log

text: L)

text: λ

text: (x).

text: η

text: η

text: L

text: Collecting

text: all

text: terms,

text: we

text: conclude

text: that

text: A

text: O

text: (log

text: L)

text: f

text: (log

text: L)

text: O

text: (log

text: L)

text: f

text: (log

text: L)

text: η

text: η

text: E

text: [τ

text: L

text: Lemma

text: bounds

text: A

text: by

text: O((log

text: L)

text: E

text: [τ

text: Since

text: for

text: L

text: suﬃciently

text: large,

text: L

text: f

text: (L)

text: f

text: (log

text: L)

text: C(log

text: L)

text: η

text: η

text: we

text: arrive

text: at

text: E

text: [τ

text: A

text: A

text: f

text: (L),

text: f

text: (L)]

text: E

text: [τ

text: L

text: η

text: η

text: L

text: as

text: claimed.

text: Proposition

text: follows

text: now

text: as

text: an

text: immediate

text: consequence

text: of

text: our

text: estimates.

text: Proof

text: of

text: Proposition

text: (i)

text: From

text: Lemmata

text: and

text: we

text: deduce

text: that

text: for

text: large

text: L

text: if

text: L

text: L

text: and

text: L

text: L

text: L

text: (log

text: L

text: we

text: have

text: under

text: L

text: L

text: and

text: L

text: P

text: (E

text: [τ

text: f(L),

text: f(L)]

text: E

text: [τ

text: L

text: L

text: P

text: P

text: L

text: −L

text: (E

text: [τ

text: f(L),

text: f(L)]

text: E

text: [τ

text: Goodsp

text: OneBadtm)

text: L

text: L

text: L

text: L

text: exp

text: By

text: Proposition

text: if

text: is

text: small

text: and

text: is

text: suﬃciently

text: large,

text: holds

text: under

text: δ

text: L

text: L

text: L)

text: for

text: all

text: L

text: L

text: provided

text: ε

text: ε

text: (δ).

text: This

text: proves

text: the

text: proposition.


text: 33 (cid:88) (cid:0) (cid:1) By the Markov property, using the case and induction in the last step, m = 1 (cid:34) (cid:35) (cid:88)Px,ω τVL(x)(cid:88)> (cid:96)m (cid:0) (cid:1) ≤ ≤···≤ 0 (cid:96)1 (cid:96)m ∞ = (cid:2)E (cid:3) P(cid:88) τ (cid:2) > (cid:96) ; τ (cid:3)> (cid:96) − ≤ ≤···≤ x,ω X(cid:96)m−1,ω VL(x) VL(x) m 1 0 (cid:96)1 (cid:96)m−1 (cid:96)=0 ≤ ≤ sup E τ E τ > (cid:96) − 2mL2m, ∈ z,ω VL(x) x,ω VL(x) m 1 z VL(x) 0≤(cid:96)1≤···≤(cid:96)m−1 if L = L(ω) is suﬃciently large. (cid:50) 5 A quenched invariance principle Here we combine the results on the exit distributions from [1] and our results on the mean sojourn times to prove Theorem 1.2, which provides a functional central limit theorem for the RWRE under the quenched measure. Let us recall the precise statement. P ∈ Assume A0(ε) for small ε > 0, A1 and B√. Then for -a.e. ω Ω, under R R ≥ P , the C( , d)-valued sequence Xn/ n, t 0, converges in law to 0,ω + t − a d-dimensional Brownian motion with diﬀusion matrix D 1Λ, where D is the constant from Theorem 1.1, Λ is given by (2), and Xn is the linear − (cid:98) (cid:99) − t interpolation Xn = X(cid:98) (cid:99) + (tn tn )(X(cid:98) (cid:99) X(cid:98) (cid:99)). t tn tn +1 tn The statement follows if we show that for each real , weak convergence occurs T > 0 R in C([0, T ], d). In order to simplify notation, we will restrict ourselves to T = 1, the general case being a simple generalization of this case. Let us ﬁrst√give a rough (simpliﬁed) idea of our proof. We deﬁne the step size − L = (log n) 1 n. From Theorem 1.1 we infer that the RWRE should have left about n (log n)2/D balls of radius L in the ﬁrst n steps. Proposition 2.1 tells us that for n suﬃciently large , the exit law from each of those balls is close to that of a symmetric n random walk with nearest neighbor kernel . For our limit theorem, this will imply that p Ln we can replace the coarse grained RWRE taking steps of size , i.e. the RWRE observed L n at the successive exit times from balls of radius , by the analogous coarse grained L n random walk with kernel . For the latter, we apply the multidimensional Lindeberg- p Ln Feller limit theorem. Since we know that the kernels p converge to p∞ (see (1) and the Ln comments below Proposition 2.1), we obtain in this way the stated convergence of the one-dimensional distributions. Since our estimates on exit measures and exit times are suﬃciently uniform in in the starting point, multidimensional convergence and tightness then follow from standard arguments. Z 5.1 Construction of coarse grained random walks on d ##LTLine## ##LTLine##

text: By

text: the

text: Markov

text: property,

text: using

text: the

text: case

text: and

text: induction

text: in

text: the

text: last

text: step,

text: m

text: τ

text: τ

text: x,ω

text: VL(x)

text: VL(x)

text: m

text: sup

text: E

text: τ

text: E

text: τ

text: z,ω

text: VL(x)

text: x,ω

text: VL(x)

text: m

text: z

text: VL(x)

text: if

text: L

text: L(ω)

text: is

text: suﬃciently

text: large.

label: section

text: A

text: quenched

text: invariance

text: principle

text: Here

text: we

text: combine

text: the

text: results

text: on

text: the

text: exit

text: distributions

text: from

text: and

text: our

text: results

text: on

text: the

text: mean

text: sojourn

text: times

text: to

text: prove

text: Theorem

text: which

text: provides

text: a

text: functional

text: central

text: limit

text: theorem

text: for

text: the

text: RWRE

text: under

text: the

text: quenched

text: measure.

text: Let

text: us

text: recall

text: the

text: precise

text: statement.

text: P

text: Assume

text: for

text: small

text: ε

text: and

text: B√.

text: Then

text: for

text: -a.e.

text: ω

text: Ω,

text: under

text: R

text: R

text: P

text: the

text: C(

text: d)-valued

text: sequence

text: Xn/

text: n,

text: t

text: converges

text: in

text: law

text: to

text: t

text: a

text: d-dimensional

text: Brownian

text: motion

text: with

text: diﬀusion

text: matrix

text: D

text: where

text: D

text: is

text: the

text: constant

text: from

text: Theorem

text: Λ

text: is

text: given

text: by

text: and

text: Xn

text: is

text: the

text: linear

text: t

text: interpolation

text: Xn

text: (tn

text: tn

text: t

text: tn

text: tn

text: tn

text: The

text: statement

text: follows

text: if

text: we

text: show

text: that

text: for

text: each

text: real

text: weak

text: convergence

text: occurs

text: T

text: R

text: in

text: T

text: d).

text: In

text: order

text: to

text: simplify

text: notation,

text: we

text: will

text: restrict

text: ourselves

text: to

text: T

text: the

text: general

text: case

text: being

text: a

text: simple

text: generalization

text: of

text: this

text: case.

text: Let

text: us

text: ﬁrst√give

text: a

text: rough

text: (simpliﬁed)

text: idea

text: of

text: our

text: proof.

text: We

text: deﬁne

text: the

text: step

text: size

text: L

text: (log

text: n)

text: n.

text: From

text: Theorem

text: we

text: infer

text: that

text: the

text: RWRE

text: should

text: have

text: left

text: about

text: n

text: (log

text: balls

text: of

text: radius

text: L

text: in

text: the

text: ﬁrst

text: n

text: steps.

text: Proposition

text: tells

text: us

text: that

text: for

text: n

text: suﬃciently

text: large

text: the

text: exit

text: law

text: from

text: each

text: of

text: those

text: balls

text: is

text: close

text: to

text: that

text: of

text: a

text: symmetric

text: n

text: random

text: walk

text: with

text: nearest

text: neighbor

text: kernel

text: For

text: our

text: limit

text: theorem,

text: this

text: will

text: imply

text: that

text: p

text: Ln

text: we

text: can

text: replace

text: the

text: coarse

text: grained

text: RWRE

text: taking

text: steps

text: of

text: size

text: i.e.

text: the

text: RWRE

text: observed

text: L

text: n

text: at

text: the

text: successive

text: exit

text: times

text: from

text: balls

text: of

text: radius

text: by

text: the

text: analogous

text: coarse

text: grained

text: L

text: n

text: random

text: walk

text: with

text: kernel

text: For

text: the

text: latter,

text: we

text: apply

text: the

text: multidimensional

text: Lindeberg-

text: p

text: Ln

text: Feller

text: limit

text: theorem.

text: Since

text: we

text: know

text: that

text: the

text: kernels

text: p

text: converge

text: to

text: p∞

text: (see

text: and

text: the

text: Ln

text: comments

text: below

text: Proposition

text: we

text: obtain

text: in

text: this

text: way

text: the

text: stated

text: convergence

text: of

text: the

text: one-dimensional

text: distributions.

text: Since

text: our

text: estimates

text: on

text: exit

text: measures

text: and

text: exit

text: times

text: are

text: suﬃciently

text: uniform

text: in

text: in

text: the

text: starting

text: point,

text: multidimensional

text: convergence

text: and

text: tightness

text: then

text: follow

text: from

text: standard

text: arguments.

text: Z

text: Construction

text: of

text: coarse

text: grained

text: random

text: walks

text: on

text: d

text: ##LTLine##

text: ##LTLine##


text: 5.4 (cid:16)Proof of Theorem 1.2 (cid:17) 39 (cid:12) (cid:12) (cid:101) (cid:12) (cid:12) √ − → (ii) For each (cid:15) > 0, sup| |≤ P Xˆ Xˆ(cid:98) (cid:99) / n > (cid:15) 0. x n2 x,ω kn,t tβn We ﬁrst prove (i). For notational simplicity, we restrict ourselves to the case ; the t = 1 ∈ general case follows exactly the same lines, with replaced everywhere by t [0, 1] β (cid:98) (cid:99) n . For later use, it will be helpful to consider here the supremum over bounded by tβ x n 2n2 instead of n2. We let (Z ) be an i.i.d. sequence of random vect√ors distributed · n,i i=0,...,n | | ≤ according to , independently of the RWRE. Since , it suﬃces q (0, ) (cid:101) Z 2L = o( n) n n,i n to show the statement for ˆ inside the probability replaced by ˆ (tacitly X (cid:101) X + Z βn βn n,0 assuming that ˆ under and are deﬁned on the same probability space, whose X P Z (cid:101) βn x,ω n,0 · · · probability measure we again denote by ). Now let ˆ . Since P Y = Z + + Z (cid:12) (cid:16)(cid:16) (cid:17)· (cid:17)x,ω (cid:16) i n,1 ·(cid:17)(cid:12) n,i Xˆβn + Zn(cid:12)(cid:12),(cid:101)0 has law (Qn)βnqn(x, ) under Px,ω, and x + Yˆi has law (qn)i(x, ),(cid:12)(cid:12)we get √ √ − − (cid:13)(cid:0) − (cid:1) (cid:13) sup P Xˆ + Z x / n > u P (x + Yˆ(cid:13) x)/ n > u (cid:13) | |≤ x,ω βn n,0 βn+1 x 2n2 ≤ − · sup (Q )βn (q )βn q (x, ) . | |≤ n n n 1 x 2n2 ∈(cid:13)(cid:0) (cid:1) (cid:13) | | ≤ For ω (cid:13)A , we obtain by iteration(cid:13), uniformly in x with x 2n2, 1 (cid:13) (cid:13) (cid:13)(cid:0) (cid:1) (cid:13) (cid:13) − · (cid:13) (cid:13) (cid:13) (Q )βn (q )βn q (x, ) (cid:13)(cid:0) (cid:1) (cid:13) ≤ n − n − n 1 · (cid:13) − − − · (cid:13) (Q )βn 1 (Q q ) q (x, ) + (Q )βn 1 (q )βn 1 q2(x, ) ≤ n (cid:107) −n n n· (cid:107) 1 n − −n − n 1· sup (Q q ) q (x, ) + sup (Q )βn 1 (q )βn 1 q (x, ) | |≤ n n n 1 | |≤ n n n 1 x 3n2 x 2n2 ≤ − → → ∞ β (log L ) 9 0 as n . n n √ It remains to show that ˆ converges in distribution to a -dimensional centered Y / n d βn − Gaussian vector with covariance matrix D 1Λ. This will be a consequence of the fol- lowing multidimensional version of the Lindeberg-Feller theorem. (cid:104) (cid:105) ≤ ≤ R Proposition 5.2. Let W , 1 (cid:96) m, be centered and independent d-valued random m,(cid:96) vectors. Put Σ = (σ(ij)) , where σ(ij) = E W (i) W (j) and W (i) is the ith (cid:80) m,(cid:96) m,(cid:96) i,→j=1,∞...,d m,(cid:96) m,(cid:96) m,(cid:96) m,(cid:96) component of . If for , W m m,(cid:96) → (cid:80) (cid:2) (cid:3) (a) m Σ Σ, (cid:96)=1 m,(cid:96) ∈ R | · | | · | → (b) for each v d and each (cid:15) > 0, m E v W 2 ; v W > (cid:15) 0, (cid:96)=1 m,(cid:96) m,(cid:96) · · · → ∞ then converges in distribution as to a -dimensional Gaussian W + +W m d m,1 m,m random vector with mean zero and covariance matrix . Σ Proof of Proposition 5.2: By the Cramér-Wold device, it suﬃces to show that for ∈ R · · · · ﬁxed v d, v (W + + W ) converges in distribution to a Gaussian random m,1 m,m variable with mean zero and variance vT Σv. Under (a) and (b), this follows immediately from the classical one-dimensional Lindeberg-Feller theorem. (cid:50) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: Theorem

text: (ii)

text: For

text: each

text: sup|

text: P

text: Xˆ

text: n

text: x

text: x,ω

text: kn,t

text: tβn

text: We

text: ﬁrst

text: prove

text: (i).

text: For

text: notational

text: simplicity,

text: we

text: restrict

text: ourselves

text: to

text: the

text: case

text: the

text: t

text: general

text: case

text: follows

text: exactly

text: the

text: same

text: lines,

text: with

text: replaced

text: everywhere

text: by

text: t

text: β

text: n

text: For

text: later

text: use,

text: it

text: will

text: be

text: helpful

text: to

text: consider

text: here

text: the

text: supremum

text: over

text: bounded

text: by

text: tβ

text: x

text: n

text: instead

text: of

text: We

text: let

text: (Z

text: be

text: an

text: i.i.d.

text: sequence

text: of

text: random

text: vect√ors

text: distributed

text: n,i

text: according

text: to

text: independently

text: of

text: the

text: RWRE.

text: Since

text: it

text: suﬃces

text: q

text: Z

text: o(

text: n)

text: n

text: n,i

text: n

text: to

text: show

text: the

text: statement

text: for

text: ˆ

text: inside

text: the

text: probability

text: replaced

text: by

text: ˆ

text: (tacitly

text: X

text: X

text: Z

text: βn

text: βn

text: assuming

text: that

text: ˆ

text: under

text: and

text: are

text: deﬁned

text: on

text: the

text: same

text: probability

text: space,

text: whose

text: X

text: P

text: Z

text: βn

text: x,ω

text: probability

text: measure

text: we

text: again

text: denote

text: by

text: Now

text: let

text: ˆ

text: Since

text: P

text: Y

text: Z

text: Z

text: i

text: n,i

text: Xˆβn

text: has

text: law

text: (Qn)βnqn(x,

text: under

text: Px,ω,

text: and

text: x

text: Yˆi

text: has

text: law

text: (qn)i(x,

text: get

text: sup

text: P

text: Xˆ

text: Z

text: x

text: n

text: u

text: P

text: (x

text: x)/

text: n

text: u

text: x,ω

text: βn

text: x

text: sup

text: (Q

text: )βn

text: (q

text: )βn

text: q

text: (x,

text: n

text: n

text: n

text: x

text: For

text: ω

text: we

text: obtain

text: by

text: uniformly

text: in

text: x

text: with

text: x

text: (Q

text: )βn

text: (q

text: )βn

text: q

text: (x,

text: n

text: n

text: n

text: (Q

text: )βn

text: (Q

text: q

text: q

text: (x,

text: (Q

text: )βn

text: (q

text: )βn

text: n

text: −n

text: n

text: n·

text: n

text: −n

text: n

text: 1·

text: sup

text: (Q

text: q

text: q

text: (x,

text: sup

text: (Q

text: )βn

text: (q

text: )βn

text: q

text: (x,

text: n

text: n

text: n

text: n

text: n

text: n

text: x

text: x

text: β

text: (log

text: L

text: as

text: n

text: n

text: n

text: It

text: remains

text: to

text: show

text: that

text: ˆ

text: converges

text: in

text: distribution

text: to

text: a

text: -dimensional

text: centered

text: Y

text: n

text: d

text: βn

text: Gaussian

text: vector

text: with

text: covariance

text: matrix

text: D

text: This

text: will

text: be

text: a

text: consequence

text: of

text: the

text: fol-

text: lowing

text: multidimensional

text: version

text: of

text: the

text: Lindeberg-Feller

text: theorem.

text: R

text: Proposition

text: Let

text: W

text: m,

text: be

text: centered

text: and

text: independent

text: d-valued

text: random

text: vectors.

text: Put

text: Σ

text: (σ(ij))

text: where

text: σ(ij)

text: E

text: W

text: (i)

text: W

text: (j)

text: and

text: W

text: (i)

text: is

text: the

text: ith

text: component

text: of

text: If

text: for

text: W

text: m

text: (a)

text: m

text: Σ

text: Σ,

text: R

text: (b)

text: for

text: each

text: v

text: d

text: and

text: each

text: m

text: E

text: v

text: W

text: v

text: W

text: then

text: converges

text: in

text: distribution

text: as

text: to

text: a

text: -dimensional

text: Gaussian

text: W

text: +W

text: m

text: d

text: m,m

text: random

text: vector

text: with

text: mean

text: zero

text: and

text: covariance

text: matrix

text: Σ

text: Proof

text: of

text: Proposition

text: By

text: the

text: Cramér-Wold

text: device,

text: it

text: suﬃces

text: to

text: show

text: that

text: for

text: R

text: ﬁxed

text: v

text: d,

text: v

text: (W

text: W

text: converges

text: in

text: distribution

text: to

text: a

text: Gaussian

text: random

text: m,m

text: variable

text: with

text: mean

text: zero

text: and

text: variance

text: vT

text: Σv.

text: Under

text: (a)

text: and

text: (b),

text: this

text: follows

text: immediately

text: from

text: the

text: classical

text: one-dimensional

text: Lindeberg-Feller

text: theorem.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 40 5 A QUENCHED INVARIANCE PRINCIPLE · · · We now ﬁnish the proof of (i). Recall that ˆ , where the Y = Z + + Z βn · n,1 n,βn are independent random vectors with law . Since the underlying one-step Z q (0, ) n,(cid:96) n transition kernel p is s(cid:104)ymmetri(cid:105)c, the Z are centered. Moreover, denoting by Z(i) the Ln n,(cid:96) n,(cid:96) th component of , i Z n,(cid:96) (cid:54) E Z(i)Z(j) = 0 for i = j, i, j = 1, . . . , d. n,(cid:96) n,(cid:96) For i = j, we obtai(cid:18)n by(cid:20)(cid:16)deﬁni(cid:17)tio(cid:21)n (cid:20)(cid:16) (cid:17) (cid:21)(cid:19) 1 E (cid:88)Z(i) 2 + · · · + E (cid:90) Z(i) (cid:88)2 n n,1 n,βn β β 2 = n (cid:90) q (0, y)y2 =(cid:88)n ϕ(s) π(pLn)(0, y)y2ds n ∈Z n i n 1 ∈Z VsLn i y d y d β 2 = n (L s)2ϕ(s) π(pLn)(0, y)(y /sL )2ds. (34) n 1 n ∈Z VsLn i n y d We next recall that [1, Lemma 2.1] shows how to recover the kernel out of the p (cid:88) Ln exit measure π(pLn), namely VsLn − 2p = π(pLn)(0, y)(y /sL )2 + O(L 1). Ln ∈Z VsLn i n n y d (cid:18) (cid:20) (cid:21) (cid:20) (cid:21)(cid:19) (cid:16) (cid:17) (cid:16) (cid:17) Replacing by its value, we therefore deduce from (34) that β n 1 E Z(i) 2 + · · · +(cid:0)E (cid:1)Z(i) 2 (cid:0) (cid:1) n n,1 n,βn L2β c − 2p (e ) − = n n ϕ 2p (e ) + O L 1 = Ln i + O (log n) 2 . n Ln i n D √ → → ∞ Since p (e ) p∞(e ) as n , we obtain with W = Z / n, (cid:96) = 1, . . . , β , in Ln i i βn,(cid:96) n,(cid:96) n the notation of(cid:88)Proposition 5.2, βn → − − → ∞ Σ D 1 (2p∞(e )δ (j))d = D 1Λ as n . βn,(cid:96) i i i,j=1 (cid:96)=1 √ ≤ ≤ − Since W 2L / n 2(log n) 1, point (b) √of Proposition 5.2 is trivially fulﬁlled. βn,(cid:96) n · · · Applying this proposition ﬁnally shows that ˆ converges in Y / n = W + + W βn βn,1 βn,βn distribution to a -dimensional centered Gaussian random vector with covariance matrix d − D 1Λ. This ﬁnishes the pr(cid:16)oof of (i). (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) It remains to prove(cid:101)(ii). I(cid:12)n view of Lemm(cid:12) a 5.2, it (cid:12)suﬃces to sh(cid:12)ow that for each (cid:15) > 0, √ − − (cid:98) (cid:99) lim lim sup P Xˆ Xˆ(cid:98) (cid:99) > (cid:15) n; k tβ < θβ = 0. θ↓0 n→∞ |x|≤n2 x,ω kn,t tβn n,t n n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: QUENCHED

text: INVARIANCE

text: PRINCIPLE

text: We

text: now

text: ﬁnish

text: the

text: proof

text: of

text: (i).

text: Recall

text: that

text: ˆ

text: where

text: the

text: Y

text: Z

text: Z

text: βn

text: n,βn

text: are

text: independent

text: random

text: vectors

text: with

text: law

text: Since

text: the

text: underlying

text: one-step

text: Z

text: q

text: n

text: transition

text: kernel

text: p

text: is

text: the

text: Z

text: are

text: centered.

text: Moreover,

text: denoting

text: by

text: Z(i)

text: the

text: Ln

text: th

text: component

text: of

text: i

text: Z

font: default

text: E

text: Z(i)Z(j)

text: for

text: i

text: j,

text: i,

text: j

text: d.

text: For

text: i

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text: 10 2 RESULTS AND CONCEPTS FROM THE STUDY OF EXIT LAWS The main statement for C1 The ﬁrst part of [1, Proposition 1.1] implies the following statement. Proposition 2.1. Assume A1. For δ > 0 small enough, there exist L = L (δ) large ≤ 0 0 and small with the following property: If and is satisﬁed, ε = ε (δ) > 0 ε ε A0(ε) 0 0 ≥ 0 then holds for all . C1 (δ, L , L) L L 0 0 For us, the important implication is that if is satisﬁed, then for any C1(δ, L , L ) ≤ ≤ an(cid:0)d(cid:8) for all (cid:48) ∈ , al(cid:9)l (cid:8)∈ M and(cid:9)a(cid:1)ll0 ∈1P(cid:0)s, (cid:1) 3 L L L [L, 2L] ψ (cid:48) q 1 L ι P ∗ (cid:48) − ∪ ∗ ≤ − (cid:48) D (cid:48) > (log L ) 9 D (cid:48) > δ exp (log L )2 . (11) L ,pL,ψ,q L ,pL In [1, Lemma 2.2] it is shown that the transition kernels form a Cauchy sequence. p L Their limit is given by the kernel p∞ deﬁned in (1), i.e. lim → p (e ) = p∞(e ) for L L i i . From this fact and the last display one can deduce that the diﬀerence in i = 1, . . . , d total variation of the exit laws Π and π(p∞) is small when L is large, in both a smoothed L L and non-smoothed way. See Theorems 0.1 and 0.2 of [1] for precise statements. For us, it will be suﬃcient to keep in mind (11) and the fact that lim → p = p∞. L L We follow the convention of [1] and write “assume ”, if we assume C1(δ, L , L ) ≥ 0 1 for some and some , where can be chosen arbitrarily C1(δ, L , L ) δ > 0 L L δ 0 1 1 0 small and arbitrarily large. L 0 2.4 Good and bad points In [1, Section 2.2], the concept of good and bad points inside is introduced. It turns V L out that for controlling mean sojourn times, we need a stronger notion of “goodness”, see Section . It is however more convenient to ﬁrst recall the original classiﬁcation. ∈ Recall the assignment (10). A point is good (with respect to and ), if x V L δ > 0 • ∈ (cid:13)(cid:16) L (cid:17) (cid:13) For all t [h (x), 2h (x)],(cid:13)with q = p , (cid:13) L L (cid:13) hL(x) (cid:13) − · ≤ Π π(q) (x, ) δ. Vt(x) Vt(x) 1 • (cid:13)(cid:16) (cid:17) (cid:13) If d (x) > 2r, then a(cid:13)dditionally (cid:13) L (cid:13) (cid:13) − · ≤ − Πˆ πˆ(q) πˆ(q)(x, ) (log h (x)) 9 . L L L L 1 ∈ The set of environments where all points are good is denoted . A point x V Good ∈ L L which is not good is called bad, and the set of all bad points inside is denoted x V V B L B L by . = (ω) L L

text: RESULTS

text: AND

text: CONCEPTS

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text: STUDY

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text: EXIT

text: LAWS

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text: Draft version May 26, 2020 Preprint typeset using LATEX style emulateapj v. 12/16/11 GRAVITATIONAL WAVES FROM THE COLLISION OF TIDALLY DISRUPTED STARS WITH MASSIVE BLACK HOLES William E. East 0 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Draft version May 26, 2020 ABSTRACT We use simulations of hydrodynamics coupled with full general relativity to investigate the gravitational waves produced by a star colliding with a massive black hole when the star’s tidal disruption radius lies far outside of the black hole horizon. We consider both main-sequence and white-dwarf compaction stars, and nonspinning black holes, as well as those with near-extremal spin. We study the regime in between where the star can be accurately modeled by a point particle, and where tidal eﬀects completely suppress the gravitational wave signal. We ﬁnd that nonnegligible gravitational waves can be produced even when the star is strongly aﬀected by tidal forces, as well as when it collides with large angular momentum. We discuss the implications that these results have for the potential observation of gravitational waves from these sources with future detectors. Subject headings: black hole physics – gravitation – gravitational waves 1. INTRODUCTION direct captures to those where the star is disrupted, but − − One exciting prospect for upcoming transient surveys not captured, is given by f = (L˜2T /L˜2cap 1) 1, where is the possibility of detecting numerous tidal disruption L˜2 /L˜2 = 8(M /M∗)2/3(M∗/R∗). Thus, for a super- cap T BH events resulting from a star being pulled apart by a massive BH with M = 106 M(cid:12) and a star with solar massive black hole (BH) and then accreted. There have mass and compactioBnH, M∗/R∗ = 2 × 10−6, the ratio of ∼ already been a number of candidate observations of such collisions to disruption events is 0.2. events across the electromagnetic spectrum (Bade et al. A star colliding with a massive BH will most likely 1996; Komossa & Greiner 1999; Gezari et al. 2003, 2006, not produce a bright, fallback accretion-powered electro- 2008; Cappelluti et al. 2009; van Velzen et al. 2011; magnetic transient. However, such an event can produce Bloom et al. 2011; Levan et al. 2011; Zauderer et al. gravitational waves (GWs). Gravitational waves can also 2011; Cenko et al. 2012a,b; Gezari et al. 2012), and with be produced by a star as it passes through periapse and the large number of expected future observations (van is disrupted in a nonmerging encounter (Kobayashi et al. Velzen et al. 2011), there is hope that we will be able use 2004; Haas et al. 2012; Cheng & Evans 2013), or poten- such events to probe the strong-ﬁeld gravity of massive tially by the rebound of a tidally compressed star (Guillo- BHs. These transients could occur not only for main- chon et al. 2009; Stone et al. 2013). Gravitational waves sequence stars falling into supermassive BHs, but also produced by white dwarfs undergoing strong tidal in- possibly for white dwarfs falling into intermediate-mass teractions and mass transfer while orbiting massive BHs BHs (Irwin et al. 2010; Krolik & Piran 2011; Shcherbakov have also been considered (Zalamea et al. 2010; MacLeod et al. 2013). However, complementing the events that et al. 2014). However, such signals will, in general, be produce these electromagnetic transients, there will be weaker than those resulting from collisions. In addi- a population of stars launched on orbits without enough tion, the merger-ringdown signal from a collision event, angular momentum to avoid being swallowed by the BH. if detected, directly reveals the mass and spin of the To get a rough idea of the relative rates, we can assume BH (Berti et al. 2006). The ringdown frequency of su- that the stars ﬁlling the BH loss cone have negligible ∼ permassive BHs with masses M 105–108 M(cid:12) falls orbital energy compared to their kinetic energy when BH in the frequency band of a space-based LISA-like GW orbiting near the BH, and that the diﬀerential rate dΓ (cid:46) instrument, while BHs with masses M 103 M(cid:12) of st∝ars with speciﬁc orbital angular momentum L˜ obeys fall in the band of LIGO (Abramovici etBaHl. 1992), KA- dΓ d(L˜2) (Frank & Rees 1976; Kesden 2012), which GRA (Somiya 2012), and other similar ground-based de- holds in the so-called “pinhole” regime. For a nonspin- tectors. ning BH, parabolic orbits with L˜ < L˜ = 4M (where However, there are several reasons such GW signals cap BH M is the mass of the BH and we use geometric units may be diﬃcult to detect at large distances. To begin BH G = c = 1 throughout, unless otherwise stated) fall into with, for an extreme-mass-ratio system the amplitude of the BH horizon. If we take the Newtonian estimate for the GW is proportional to the mass ratio (for ﬁxed to- the tidal disruption radius r = R∗(M /M∗)1/3 for a tal mass). For extreme-mass-ratio inspirals of compact T BH star of mass M∗ and radius R∗, then the angular momen- objects, considered a promising source for a LISA-like tum of a Newtonian parabolic orbit with a periapse equal GW detector (Amaro-Seoane et al. 2007), the system to r is L˜2 = 2M4/3R∗/M∗1/3. Hence, the proportion of merges slowly through gravitational radiation reaction, T T BH and hence spends many wave cycles orbiting the BH. However, in the collision case, the timescale of the merger is just set by the mass of the BH and is much shorter. weast@stanford.edu

text: Draft

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: GRAVITATIONAL

text: WAVES

text: FROM

text: THE

text: COLLISION

text: OF

text: TIDALLY

text: DISRUPTED

text: STARS

text: WITH

text: MASSIVE

text: BLACK

text: HOLES

text: William

label: author

text: E.

text: East

text: Kavli

text: Institute

text: for

text: Particle

text: Astrophysics

text: and

text: Cosmology,

text: Stanford

text: University,

text: SLAC

text: National

text: Accelerator

text: Laboratory,

text: Menlo

text: Park,

text: CA

text: USA

text: Draft

text: version

text: May

text: ABSTRACT

text: We

label: abstract

text: use

text: simulations

text: of

text: hydrodynamics

text: coupled

text: with

text: full

text: general

text: relativity

text: to

text: investigate

text: the

text: gravitational

text: waves

text: produced

text: by

text: a

text: star

text: colliding

text: with

text: a

text: massive

text: black

text: hole

text: when

text: the

text: star’s

text: tidal

text: disruption

text: radius

text: lies

text: far

text: outside

text: of

text: the

text: black

text: hole

text: horizon.

text: We

text: consider

text: both

text: main-sequence

text: and

text: white-dwarf

text: compaction

text: stars,

text: and

text: nonspinning

text: black

text: holes,

text: as

text: well

text: as

text: those

text: with

text: near-extremal

text: spin.

text: We

text: study

text: the

text: regime

text: in

text: between

text: where

text: the

text: star

text: can

text: be

text: accurately

text: modeled

text: by

text: a

text: point

text: particle,

text: and

text: where

text: tidal

text: eﬀects

text: completely

text: suppress

text: the

text: gravitational

text: wave

text: signal.

text: We

text: ﬁnd

text: that

text: nonnegligible

text: gravitational

text: waves

text: can

text: be

text: produced

text: even

text: when

text: the

text: star

text: is

text: strongly

text: aﬀected

text: by

text: tidal

text: forces,

text: as

text: well

text: as

text: when

text: it

text: collides

text: with

text: large

text: angular

text: momentum.

text: We

text: discuss

text: the

text: implications

text: that

text: these

text: results

text: have

text: for

text: the

text: potential

text: observation

text: of

text: gravitational

text: waves

text: from

text: these

text: sources

text: with

text: future

text: detectors.

text: Subject

text: headings:

text: black

text: hole

text: physics

text: gravitation

text: gravitational

text: waves

text: INTRODUCTION

text: direct

text: captures

text: to

text: those

text: where

text: the

text: star

text: is

text: disrupted,

text: but

text: One

text: exciting

text: prospect

text: for

text: upcoming

text: transient

text: surveys

text: not

text: captured,

text: is

text: given

text: by

text: f

text: where

text: is

text: the

text: possibility

text: of

text: detecting

text: numerous

text: tidal

text: disruption

text: Thus,

text: for

text: a

text: super-

text: cap

text: T

text: BH

text: events

text: resulting

text: from

text: a

text: star

text: being

text: pulled

text: apart

text: by

text: a

text: massive

text: BH

text: with

text: M

text: and

text: a

text: star

text: with

text: solar

text: massive

text: black

text: hole

text: (BH)

text: and

text: then

text: accreted.

text: There

text: have

text: mass

text: and

text: compactioBnH,

text: M∗/R∗

text: the

text: ratio

text: of

text: already

text: been

text: a

text: number

text: of

text: candidate

text: observations

text: of

text: such

text: collisions

text: to

text: disruption

text: events

text: is

text: events

text: across

text: the

text: electromagnetic

text: spectrum

text: (Bade

text: et

text: al.

text: A

text: star

text: colliding

text: with

text: a

text: massive

text: BH

text: will

text: most

text: likely

text: Komossa

text: Greiner

text: Gezari

text: et

text: al.

text: not

text: produce

text: a

text: bright,

text: fallback

text: accretion-powered

text: electro-

text: Cappelluti

text: et

text: al.

text: van

text: Velzen

text: et

text: al.

text: magnetic

text: transient.

text: However,

text: such

text: an

text: event

text: can

text: produce

text: Bloom

text: et

text: al.

text: Levan

text: et

text: al.

text: Zauderer

text: et

text: al.

text: gravitational

text: waves

text: (GWs).

text: Gravitational

text: waves

text: can

text: also

text: Cenko

text: et

text: al.

text: Gezari

text: et

text: al.

text: and

text: with

text: be

text: produced

text: by

text: a

text: star

text: as

text: it

text: passes

text: through

text: periapse

text: and

text: the

text: large

text: number

text: of

text: expected

text: future

text: observations

text: (van

text: is

text: disrupted

text: in

text: a

text: nonmerging

text: encounter

text: (Kobayashi

text: et

text: al.

text: Velzen

text: et

text: al.

text: there

text: is

text: hope

text: that

text: we

text: will

text: be

text: able

text: use

text: Haas

text: et

text: al.

text: Cheng

text: Evans

text: or

text: poten-

text: such

text: events

text: to

text: probe

text: the

text: strong-ﬁeld

text: gravity

text: of

text: massive

text: tially

text: by

text: the

text: rebound

text: of

text: a

text: tidally

text: compressed

text: star

text: (Guillo-

text: BHs.

text: These

text: transients

text: could

text: occur

text: not

text: only

text: for

text: main-

text: chon

text: et

text: al.

text: Stone

text: et

text: al.

text: Gravitational

text: waves

text: sequence

text: stars

text: falling

text: into

text: supermassive

text: BHs,

text: but

text: also

text: produced

text: by

text: white

text: dwarfs

text: undergoing

text: strong

text: tidal

text: in-

text: possibly

text: for

text: white

text: dwarfs

text: falling

text: into

text: intermediate-mass

text: teractions

text: and

text: mass

text: transfer

text: while

text: orbiting

text: massive

text: BHs

text: BHs

text: (Irwin

text: et

text: al.

text: Krolik

text: Piran

text: Shcherbakov

text: have

text: also

text: been

text: considered

text: (Zalamea

text: et

text: al.

text: MacLeod

text: et

text: al.

text: However,

text: complementing

text: the

text: events

text: that

text: et

text: al.

text: However,

text: such

text: signals

text: will,

text: in

text: general,

text: be

text: produce

text: these

text: electromagnetic

text: transients,

text: there

text: will

text: be

text: weaker

text: than

text: those

text: resulting

text: from

text: collisions.

text: In

text: addi-

text: a

text: population

text: of

text: stars

text: launched

text: on

text: orbits

text: without

text: enough

text: tion,

text: the

text: merger-ringdown

text: signal

text: from

text: a

text: collision

text: event,

text: angular

text: momentum

text: to

text: avoid

text: being

text: swallowed

text: by

text: the

text: BH.

text: if

text: detected,

text: directly

text: reveals

text: the

text: mass

text: and

text: spin

text: of

text: the

text: To

text: get

text: a

text: rough

text: idea

text: of

text: the

text: relative

text: rates,

text: we

text: can

text: assume

text: BH

text: (Berti

text: et

text: al.

text: The

text: ringdown

text: frequency

text: of

text: su-

text: that

text: the

text: stars

text: ﬁlling

text: the

text: BH

text: loss

text: cone

text: have

text: negligible

text: permassive

text: BHs

text: with

text: masses

text: M

text: falls

text: orbital

text: energy

text: compared

text: to

text: their

text: kinetic

text: energy

text: when

text: BH

text: in

text: the

text: frequency

text: band

text: of

text: a

text: space-based

text: LISA-like

text: GW

text: orbiting

text: near

text: the

text: BH,

text: and

text: that

text: the

text: diﬀerential

text: rate

text: dΓ

text: instrument,

text: while

text: BHs

text: with

text: masses

text: M

text: of

text: st∝ars

text: with

text: speciﬁc

text: orbital

text: angular

text: momentum

text: L˜

text: obeys

text: fall

text: in

text: the

text: band

text: of

text: LIGO

text: (Abramovici

text: etBaHl.

text: KA-

text: dΓ

text: (Frank

text: Rees

text: Kesden

text: which

text: GRA

text: (Somiya

text: and

text: other

text: similar

text: ground-based

text: de-

text: holds

text: in

text: the

text: so-called

text: “pinhole”

text: regime.

text: For

text: a

text: nonspin-

text: tectors.

text: ning

text: BH,

text: parabolic

text: orbits

text: with

text: L˜

text: L˜

text: (where

text: However,

text: there

text: are

text: several

text: reasons

text: such

text: GW

text: signals

text: cap

text: BH

text: M

text: is

text: the

text: mass

text: of

text: the

text: BH

text: and

text: we

text: use

text: geometric

text: units

text: may

text: be

text: diﬃcult

text: to

text: detect

text: at

text: large

text: distances.

text: To

text: begin

text: BH

text: G

text: c

text: throughout,

text: unless

text: otherwise

text: stated)

text: fall

text: into

text: with,

text: for

text: an

text: extreme-mass-ratio

text: system

text: the

text: amplitude

text: of

text: the

text: BH

text: horizon.

text: If

text: we

text: take

text: the

text: Newtonian

text: estimate

text: for

text: the

text: GW

text: is

text: proportional

text: to

text: the

text: mass

text: ratio

text: (for

text: ﬁxed

text: to-

text: the

text: tidal

text: disruption

text: radius

text: r

text: R∗(M

text: for

text: a

text: tal

text: mass).

text: For

text: extreme-mass-ratio

text: inspirals

text: of

text: compact

text: T

text: BH

text: star

text: of

text: mass

text: M∗

text: and

text: radius

text: R∗,

text: then

text: the

text: angular

text: momen-

text: objects,

text: considered

text: a

text: promising

text: source

text: for

text: a

text: LISA-like

text: tum

text: of

text: a

text: Newtonian

text: parabolic

text: orbit

text: with

text: a

text: periapse

text: equal

text: GW

text: detector

text: (Amaro-Seoane

text: et

text: al.

text: the

text: system

text: to

text: r

text: is

text: Hence,

text: the

text: proportion

text: of

text: merges

text: slowly

text: through

text: gravitational

text: radiation

text: reaction,

text: T

text: T

text: BH

text: and

text: hence

text: spends

text: many

text: wave

text: cycles

text: orbiting

text: the

text: BH.

text: However,

text: in

text: the

text: collision

text: case,

text: the

text: timescale

text: of

text: the

text: merger

text: is

text: just

text: set

text: by

text: the

text: mass

text: of

text: the

text: BH

text: and

text: is

text: much

text: shorter.

text: weast@stanford.edu