text: 2 When considering the possibility of a phase transition from Sect. . In Sect. , we discuss our solutions to the Tolman- hadronic matter to quark matter in compact star interiors, the Oppenheimer-Volkoff equation with parameterizations of our phenomenon of high-mass twins presents itself. The term hybrid EoS and how observations constrain our choices of pa- mass twins refers to the existence of two compact star conﬁg- rameterization. A discussion and concluding remarks will be urations that have the same mass but distinctly different radii. given in Sect. and Sect. , respectively. The appearance of It is related to the occurrence of a “third family” of compact a stable branch of compact stars, with quark-matter core at stars, besides white dwarfs and ordinary neutron stars, which high masses, which is disconnected from the stable neutron is separated from the latter by an unstable branch in the mass- star branch and is called the “third family” of stable hybrid radius diagram. As discussed already by Gerlach in 1968 [9], stars, is a main result of the new relativistic density functional the third family is related to the behavior of the high-density based EoS introduced in this work. EoS, which may exhibit a phase transition. The transition to quark matter or to pion condensed matter has been considered in the context of a third family of stars by Ka¨mpfer [10, 11], II. HYBRID STAR MATTER WITH EXCLUDED VOLUME and later this issue was taken up again, e.g., in Refs. [12– EFFECTS 15]. When the mass twins occur in a mass range that cov- ers the presently best known high mass of the pulsar PSR ± J0348+0432 with M = 2.01 0.04 M(cid:12) [16], we will speak The effective relativistic density-functional (RDF) ap- of high-mass twin (HMT) stars. proach has been successfully applied in describing warm, The existence of HMTs hinges on the nature of the phase dense asymmetric nuclear matter with light clusters [25] as transition in these dense systems. In order for this phe- well as other systems where the fundamental interaction be- nomenon to appear, the EoS of the compact star matter must tween constituents is not a priori known. Thus, this RDF ap- have a strong ﬁrst-order phase transition [8, 17, 18] character- proach is adopted and applied here to the hybrid quark-hadron ized by a large jump in energy density of the two phases also matter EoS. The ﬁnal hybrid EoS is comprised of two separate denoted as a large latent heat. In order to support the mass RDF EoS for each phase of matter, combined by utilizing a twin phenomenon, a necessary condition is [17, 19, 20] Maxwell phase transition construction. The correct thermodynamic treatment of quark matter de- pends on the physical constraints of the system which are to be ∆ε > 1 + 3 Ptrans , (1) probed. For large volumes of quark matter that exist for long ε 2 2 ε times, i.e. the thermodynamic limit, the conserved quantities trans trans are quark number (or baryon number), electron charge, color where ∆ε is the latent heat, and εtrans (Ptrans ) is the energy charge, and lepton number. Each of these conserved quanti- density (pressure) at the onset of the deconﬁnement phase ties has an associated chemical potential. These large systems transition. As mentioned, the existence of HMTs would im- must remain charge- and color-neutral, which determines the ply the existence of at least one critical endpoint in the QCD values for µQ and µC. This leaves a three-dimensional phase- phase diagram [18, 21]. The existence of high mass twins can space parametrized by the baryon chemical potential (µB), the potentially be observationally conﬁrmed. If the radii of two lepton chemical potential (µl), and the temperature (T). compact stars with sufﬁciently accurately measured high mass A. Relativistic density-functional approach like, e.g., PSR J0348+0432 [16] and PSR J1614-2230 [22, 23] could be measured (for instance by the NICER mission, which The RDF approach to interacting quark matter can be ob- has recently been launched and mounted on the International tained using the path integral approach to the partition func- Space Station [24]) and turn out signiﬁcantly different from tion, which analogous to the treatment of the Walecka model each other, then this would present evidence for HMTs that of nuclear matter in [26] takes the form imply a strong ﬁrst-order transition in compact star matter. (cid:90) (cid:40)(cid:90) (cid:90) (cid:41) In order to obtain HMTs, it is necessary to fulﬁll the fol- lowing constraints: • Z D D β L a strong stiffening of nuclear matter at supersaturation = q¯ q exp dτ d3x [ + q¯γ µˆq] , (2) eﬀ 0 densities, 0 V • a sufﬁciently soft behavior of quark matter at the where in the two-ﬂavor case hadron-to-quark-matter phase transition, in order to (cid:18) (cid:19) have a sufﬁciently large latent heat ∆ε [8], and • a sufﬁcient stiffening of quark matter above the decon- q q = u , (3) ﬁnement transition. q d These properties characterize the class of hybrid EoS we are investigating in the present work. and µˆ = diag(µ , µ ) is the diagonal matrix of the chem- u d This paper is organized in the following way. We begin ical potentials conjugate to the conserved numbers of up and by outlining our relativistic density-functional approach in down quarks. The effective Lagrangian density is given by ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: When

text: considering

text: the

text: possibility

text: of

text: a

text: phase

text: transition

text: from

text: Sect.

text: In

text: Sect.

text: we

text: discuss

text: our

text: solutions

text: to

text: the

text: Tolman-

text: hadronic

text: matter

text: to

text: quark

text: matter

text: in

text: compact

text: star

text: interiors,

text: the

text: Oppenheimer-Volkoff

text: equation

text: with

text: parameterizations

text: of

text: our

text: phenomenon

text: of

text: high-mass

text: twins

text: presents

text: itself.

text: The

text: term

text: hybrid

text: EoS

text: and

text: how

text: observations

text: constrain

text: our

text: choices

text: of

text: pa-

text: mass

text: twins

text: refers

text: to

text: the

text: existence

text: of

text: two

text: compact

text: star

text: conﬁg-

text: rameterization.

text: A

text: discussion

text: and

text: concluding

text: remarks

text: will

text: be

text: urations

text: that

text: have

text: the

text: same

text: mass

text: but

text: distinctly

text: different

text: radii.

text: given

text: in

text: Sect.

text: and

text: Sect.

text: respectively.

text: The

text: appearance

text: of

text: It

text: is

text: related

text: to

text: the

text: occurrence

text: of

text: a

text: “third

text: family”

text: of

text: compact

text: a

text: stable

text: branch

text: of

text: compact

text: stars,

text: with

text: quark-matter

text: core

text: at

text: stars,

text: besides

text: white

text: dwarfs

text: and

text: ordinary

text: neutron

text: stars,

text: which

text: high

text: masses,

text: which

text: is

text: disconnected

text: from

text: the

text: stable

text: neutron

text: is

text: separated

text: from

text: the

text: latter

text: by

text: an

text: unstable

text: branch

text: in

text: the

text: mass-

text: star

text: branch

text: and

text: is

text: called

text: the

text: “third

text: family”

text: of

text: stable

text: hybrid

text: radius

text: diagram.

text: As

text: discussed

text: already

text: by

text: Gerlach

text: in

text: stars,

text: is

text: a

text: main

text: result

text: of

text: the

text: new

text: relativistic

text: density

text: functional

text: the

text: third

text: family

text: is

text: related

text: to

text: the

text: behavior

text: of

text: the

text: high-density

text: based

text: EoS

text: introduced

text: in

text: this

text: work.

text: EoS,

text: which

text: may

text: exhibit

text: a

text: phase

text: transition.

text: The

text: transition

text: to

text: quark

text: matter

text: or

text: to

text: pion

text: condensed

text: matter

text: has

text: been

text: considered

text: in

text: the

text: context

text: of

text: a

text: third

text: family

text: of

text: stars

text: by

text: Ka¨mpfer

text: II.

text: HYBRID

text: STAR

text: MATTER

text: WITH

text: EXCLUDED

text: VOLUME

text: and

text: later

text: this

text: issue

text: was

text: taken

text: up

text: again,

text: e.g.,

text: in

text: Refs.

text: EFFECTS

text: When

text: the

text: mass

text: twins

text: occur

text: in

text: a

text: mass

text: range

text: that

text: cov-

text: ers

text: the

text: presently

text: best

text: known

text: high

text: mass

text: of

text: the

text: pulsar

text: PSR

text: with

text: M

text: we

text: will

text: speak

text: The

text: effective

text: relativistic

text: density-functional

text: (RDF)

text: ap-

text: of

text: high-mass

text: twin

text: (HMT)

text: stars.

text: proach

text: has

text: been

text: successfully

text: applied

text: in

text: describing

text: warm,

text: The

text: existence

text: of

text: HMTs

text: hinges

text: on

text: the

text: nature

text: of

text: the

text: phase

text: dense

text: asymmetric

text: nuclear

text: matter

text: with

text: light

text: clusters

text: as

text: transition

text: in

text: these

text: dense

text: systems.

text: In

text: order

text: for

text: this

text: phe-

text: well

text: as

text: other

text: systems

text: where

text: the

text: fundamental

text: interaction

text: be-

text: nomenon

text: to

text: appear,

text: the

text: EoS

text: of

text: the

text: compact

text: star

text: matter

text: must

text: tween

text: constituents

text: is

text: not

text: a

text: priori

text: known.

text: Thus,

text: this

text: RDF

text: ap-

text: have

text: a

text: strong

text: ﬁrst-order

text: phase

text: transition

text: character-

text: proach

text: is

text: adopted

text: and

text: applied

text: here

text: to

text: the

text: hybrid

text: quark-hadron

text: ized

text: by

text: a

text: large

text: jump

text: in

text: energy

text: density

text: of

text: the

text: two

text: phases

text: also

text: matter

text: EoS.

text: The

text: ﬁnal

text: hybrid

text: EoS

text: is

text: comprised

text: of

text: two

text: separate

text: denoted

text: as

text: a

text: large

text: latent

text: heat.

text: In

text: order

text: to

text: support

text: the

text: mass

text: RDF

text: EoS

text: for

text: each

text: phase

text: of

text: matter,

text: combined

text: by

text: utilizing

text: a

text: twin

text: phenomenon,

text: a

text: necessary

text: condition

text: is

text: Maxwell

text: phase

text: transition

text: construction.

text: The

text: correct

text: thermodynamic

text: treatment

text: of

text: quark

text: matter

text: de-

text: pends

text: on

text: the

text: physical

text: constraints

text: of

text: the

text: system

text: which

text: are

text: to

text: be

text: ∆ε

text: Ptrans

text: probed.

text: For

text: large

text: volumes

text: of

text: quark

text: matter

text: that

text: exist

text: for

text: long

text: ε

text: ε

text: times,

text: i.e.

text: the

text: thermodynamic

text: limit,

text: the

text: conserved

text: quantities

text: trans

text: trans

text: are

text: quark

text: number

text: (or

text: baryon

text: number),

text: electron

text: charge,

text: color

text: where

text: ∆ε

text: is

text: the

text: latent

text: heat,

text: and

text: εtrans

text: (Ptrans

text: is

text: the

text: energy

text: charge,

text: and

text: lepton

text: number.

text: Each

text: of

text: these

text: conserved

text: quanti-

text: density

text: (pressure)

text: at

text: the

text: onset

text: of

text: the

text: deconﬁnement

text: phase

text: ties

text: has

text: an

text: associated

text: chemical

text: potential.

text: These

text: large

text: systems

text: transition.

text: As

text: mentioned,

text: the

text: existence

text: of

text: HMTs

text: would

text: im-

text: must

text: remain

text: charge-

text: and

text: color-neutral,

text: which

text: determines

text: the

text: ply

text: the

text: existence

text: of

text: at

text: least

text: one

text: critical

text: endpoint

text: in

text: the

text: QCD

text: values

text: for

text: µQ

text: and

text: µC.

text: This

text: leaves

text: a

text: three-dimensional

text: phase-

text: phase

text: diagram

text: The

text: existence

text: of

text: high

text: mass

text: twins

text: can

text: space

text: parametrized

text: by

text: the

text: baryon

text: chemical

text: potential

text: (µB),

text: the

text: potentially

text: be

text: observationally

text: conﬁrmed.

text: If

text: the

text: radii

text: of

text: two

text: lepton

text: chemical

text: potential

text: (µl),

text: and

text: the

text: temperature

text: (T).

text: compact

text: stars

text: with

text: sufﬁciently

text: accurately

text: measured

text: high

text: mass

text: A.

text: Relativistic

text: density-functional

text: approach

text: like,

text: e.g.,

text: PSR

text: and

text: PSR

text: could

text: be

text: measured

text: (for

text: instance

text: by

text: the

text: NICER

text: mission,

text: which

text: The

text: RDF

text: approach

text: to

text: interacting

text: quark

text: matter

text: can

text: be

text: ob-

text: has

text: recently

text: been

text: launched

text: and

text: mounted

text: on

text: the

text: International

text: tained

text: using

text: the

text: path

text: integral

text: approach

text: to

text: the

text: partition

text: func-

text: Space

text: Station

text: and

text: turn

text: out

text: signiﬁcantly

text: different

text: from

text: tion,

text: which

text: analogous

text: to

text: the

text: treatment

text: of

text: the

text: Walecka

text: model

text: each

text: other,

text: then

text: this

text: would

text: present

text: evidence

text: for

text: HMTs

text: that

text: of

text: nuclear

text: matter

text: in

text: takes

text: the

text: form

text: imply

text: a

text: strong

text: ﬁrst-order

text: transition

text: in

text: compact

text: star

text: matter.

text: In

text: order

text: to

text: obtain

text: HMTs,

text: it

text: is

text: necessary

text: to

text: fulﬁll

text: the

text: fol-

text: lowing

text: constraints:

text: Z

text: D

text: D

text: β

text: L

text: a

text: strong

text: stiffening

text: of

text: nuclear

text: matter

text: at

text: supersaturation

text: q¯

text: q

text: exp

text: dτ

text: q¯γ

text: µˆq]

text: eﬀ

text: densities,

text: V

text: a

text: sufﬁciently

text: soft

text: behavior

text: of

text: quark

text: matter

text: at

text: the

text: where

text: in

text: the

text: two-ﬂavor

text: case

text: hadron-to-quark-matter

text: phase

text: transition,

text: in

text: order

text: to

text: have

text: a

text: sufﬁciently

text: large

text: latent

text: heat

text: ∆ε

text: and

text: a

text: sufﬁcient

text: stiffening

text: of

text: quark

text: matter

text: above

text: the

text: decon-

text: q

text: q

text: u

text: ﬁnement

text: transition.

text: q

text: d

text: These

text: properties

text: characterize

text: the

text: class

text: of

text: hybrid

text: EoS

text: we

text: are

text: investigating

text: in

text: the

text: present

text: work.

text: and

text: µˆ

text: diag(µ

text: µ

text: is

text: the

text: diagonal

text: matrix

text: of

text: the

text: chem-

text: u

text: d

text: This

text: paper

text: is

text: organized

text: in

text: the

text: following

text: way.

text: We

text: begin

text: ical

text: potentials

text: conjugate

text: to

text: the

text: conserved

text: numbers

text: of

text: up

text: and

text: by

text: outlining

text: our

text: relativistic

text: density-functional

text: approach

text: in

text: down

text: quarks.

text: The

text: effective

text: Lagrangian

text: density

text: is

text: given

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 via A-Projection is computationally expensive because with N different apertures, there are N(N-1)/2 separate primary beams (one per unique baseline), each of which This process takes place during the regridding of the vis- − has to be precomputed and cached at different parallactic ibilities, so that A 1 replaces the weighting function that † angles. Taking the example of the ﬁnal, full ALMA array evenly samples the u-v plane. However, AA may con- (with 66 antennas) and sky rotations calculated at 5◦ tain ze†ro valued components. As such, formal division increments, over 25,000 separate convolution functions by AA is an extremely dangerous computation to make, would have to calculated over a 4 hour observation. as it has the potential to ﬁll the u-v plane with divide- Computing one baseline beam and rotating it incurs by-zero errors and create an entirely erroneous image. errors, so each one must be computed separately. The uv- Instead, the Fourier transform relationship between the resolution at which this must be calculated also depends aperture and the primary beam is used to rewrite (7) and on the particular setup of an individual imaging run. All (8) as in all, it is clear that it is of immense practical value to understand precisely when this full detail is needed and † ∗ when various approximations will sufﬁce. Vcor = A Vobs (9) Our results provide the following guidelines: for dy- namic ranges of less than 1000, the accuracy of the primary beam model (beyond physical aperture size) (cid:10)F † B2 AA (10) simply does not matter. At dynamic ranges beyond 1000, pointing offsets will begin to affect ﬁnal image quality, followed by changes in beam shape due to illumination offsets beyond dynamic ranges of 5000. Finally, paral- B2 is later factored out of the ﬁnal image to recover lactic angle rotation and the beam differences between I , minimizing the divide-by-zero errors while main- true the DA/DV antennas will become appreciable at dynamic taining the integrity of the A-projection relationship. ranges beyond 10,000. This method allows for a great deal of instrumental image correction prior to actually entering the image V. FUTURE WORK domain, assuming an accurate model of the aperture functions A is used. Future tests could investigate the beneﬁts of A-projection in image correction [3]. A-projection uses a model of the Another subject for further exploration would be the aperture in the initial steps of imaging and deconvolution utilization of the full complex beam in imaging in order in order to remove its effects from the ﬁnal image. The to better understand the propagation of primary beam idea goes that the baseline apertures A have contributed errors through the Stokes polarization planes, such as various effects into the visibilities as they are observed, the effect of beam squint, which would appear in the as described in (1). In order to remove those effects, the Stokes Q plane in ALMA data. While our simulation observed visibilities must be convolved with the inverse used the full complex apertures to construct complex of the baseline aperture function A−1, as seen in (7). beams, only the real component of the primary beam The baseline aperture and its inverse will then form a was used in imaging as we were focusing solely on the unity matrix, returning the true visibilities, which can be Stokes I plane. Furthermore, the complicated complex Fourier transformed into the true image. structure of the aperture functions and their signiﬁcant antenna-to-antenna variations (as seen in the real and imaginary RMS apertures presented in Fig. 1) will lead − ∗ V = A 1 V (7) to residual phase structure in the complex primary beam true obs which will affect Stokes I imaging as well. Since the computational cost of A-Projection increases † signiﬁcantly if antenna-to-antenna variation is included, − A A 1 = † (8) it is of interest to determine tolerance levels at which, AA ##LTLine##

label: paragraph

text: via

text: A-Projection

text: is

text: computationally

text: expensive

text: because

text: with

text: N

text: different

text: apertures,

text: there

text: are

text: separate

text: primary

text: beams

text: (one

text: per

text: unique

text: baseline),

text: each

text: of

text: which

text: This

text: process

text: takes

text: place

text: during

text: the

text: regridding

text: of

text: the

text: vis-

text: has

text: to

text: be

text: precomputed

text: and

text: cached

text: at

text: different

text: parallactic

text: ibilities,

text: so

text: that

text: A

text: replaces

text: the

text: weighting

text: function

text: that

text: angles.

text: Taking

text: the

text: example

text: of

text: the

text: ﬁnal,

text: full

text: ALMA

text: array

text: evenly

text: samples

text: the

text: u-v

text: plane.

text: However,

text: AA

text: may

text: con-

text: (with

text: antennas)

text: and

text: sky

text: rotations

text: calculated

text: at

text: 5◦

text: tain

text: ze†ro

text: valued

text: components.

text: As

text: such,

text: formal

text: division

text: increments,

text: over

text: separate

text: convolution

text: functions

text: by

text: AA

text: is

text: an

text: extremely

text: dangerous

text: computation

text: to

text: make,

text: would

text: have

text: to

text: calculated

text: over

text: a

text: hour

text: observation.

text: as

text: it

text: has

text: the

text: potential

text: to

text: ﬁll

text: the

text: u-v

text: plane

text: with

text: divide-

text: Computing

text: one

text: baseline

text: beam

text: and

text: rotating

text: it

text: incurs

text: by-zero

text: errors

text: and

text: create

text: an

text: entirely

text: erroneous

text: image.

text: errors,

text: so

text: each

text: one

text: must

text: be

text: computed

text: separately.

text: The

text: uv-

text: Instead,

text: the

text: Fourier

text: transform

text: relationship

text: between

text: the

text: resolution

text: at

text: which

text: this

text: must

text: be

text: calculated

text: also

text: depends

text: aperture

text: and

text: the

text: primary

text: beam

text: is

text: used

text: to

text: rewrite

text: and

text: on

text: the

text: particular

text: setup

text: of

text: an

text: individual

text: imaging

text: run.

text: All

text: as

text: in

text: all,

text: it

text: is

text: clear

text: that

text: it

text: is

text: of

text: immense

text: practical

text: value

text: to

text: understand

text: precisely

text: when

text: this

text: full

text: detail

text: is

text: needed

text: and

text: when

text: various

text: approximations

text: will

text: sufﬁce.

text: Vcor

text: A

text: Vobs

text: Our

text: results

text: provide

text: the

text: following

text: guidelines:

text: for

text: dy-

text: namic

text: ranges

text: of

text: less

text: than

text: the

text: accuracy

text: of

text: the

text: primary

text: beam

text: model

text: (beyond

text: physical

text: aperture

text: size)

text: AA

text: simply

text: does

text: not

text: matter.

text: At

text: dynamic

text: ranges

text: beyond

text: pointing

text: offsets

text: will

text: begin

text: to

text: affect

text: ﬁnal

text: image

text: quality,

text: followed

text: by

text: changes

text: in

text: beam

text: shape

text: due

text: to

text: illumination

text: offsets

text: beyond

text: dynamic

text: ranges

text: of

text: Finally,

text: paral-

text: is

text: later

text: factored

text: out

text: of

text: the

text: ﬁnal

text: image

text: to

text: recover

text: lactic

text: angle

text: rotation

text: and

text: the

text: beam

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text: ##LTLine##


text: a) The study of synchronization in populations of coupled biological oscillators is fundamental to many areas of biology to include neuroscience, cardiac dynamics and circadian rhythms. Studying these systems may involve tracking the concentration of hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. However, for many of these systems the behaviors of interest occur on a collective or macroscopic scale. We deﬁne a new macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we ﬁnd in experimental measurements of circadian gene expression and several mathematical models for coupled biological oscillators. We characterize the emergence of this structure through a simple argument and demonstrate its applicability to stochastic and heterogeneous systems of coupled oscillators. Finally, we perform the macroscopic reduction for the heterogeneous stochastic Kuramoto equation and compare the low-dimensional macroscopic model with numerical results from the high-dimensional microscopic model. PACS numbers: Keywords: The study of coupled oscillators is important for many However, many biological systems contain thousands biological and physical systems, including neural net- of oscillators, making even the phase model a very high- works, circadian rhythms and power grids1–3. Mathe- dimensional representation of the dynamical system. A matical models of these coupled oscillator systems can recent breakthrough occurred when Ott and Antonsen be extremely high-dimensional, having at least as many discovered an ansatz that may be used for a family degrees of freedom as the number of oscillators. However, of Kuramoto-like systems to derive a low-dimensional this microscale complexity is belied by the elegant sim- model for the macroscopic behavior of the system9. If plicity which emerges at the macroscopic scale in many the ansatz holds, the long-time behavior of a system → ∞ coupled oscillator systems. Quite generally, these sys- of N oscillators can accurately be described by tems demonstrate a phase transition as the coupling be- two diﬀerential equations, one for the mean phase of tween the oscillators is strengthened leading to the emer- the coupled oscillators, and the other for their collec- gence of a self-organized synchronized state4. tive amplitude10. Despite the hundreds of recent papers This emergence of a synchronized state from the dy- on the Ott-Antonsen (OA) ansatz, the authors are not namics of a very high-dimensional dynamical system, aware of any carefully done experiments to test whether suggests that a low dimensional representation of this this powerful ansatz holds for biological systems. system should be possible. A major step in this di- In this work we test the applicability of the Ott- rection was proposed by Art Winfree in 1967 when he Antonsen ansatz using a recent experimental data set col- intuitively grasped that for systems of weakly coupled lected from circadian oscillator neurons and through sim- oscillators the time evolution of each limit cycle oscilla- ulations of several models of coupled biological oscillators11. tor may be described by a single phase variable5. This We ﬁnd the core assumptions which allow for the deriva- method is now known as phase reduction and has been tion of the macroscopic model using the OA ansatz are applied to study of many coupled oscillator systems1,6,7. not valid in our test systems. However, we ﬁnd a dif- In the following years, Kuramoto formalized the math- ferent, but related, ansatz is capable of describing the ematical procedure for phase reduction and used it to de- data well. Using a simple argument we are able to rive his now famous model for N coupled heterogeneous demonstrate the validity of our ansatz for a wide-class of oscillators, models. Finally, we demonstrate how our ansatz may be (cid:88) used to derive macroscopic models for coupled oscillator systems. K N − φ˙ = ω + sin(φ φ ), i = 1, N (1) i i N j i j=1 RESULTS where φ gives the phase of the ith oscillator, K the i coupling strength and ω gives the natural frequency of The development of the Ott-Antonsen ansatz initiated i the oscillator6. The natural frequencies of the oscillators a revolution in the coupled oscillator literature12. The are typically assumed to be drawn from some distribu- method was ﬁrst presented as an ansatz which could be tion g(ω) which reﬂects the heterogeneity in the oscillator used to reduce Kuramoto (and closely related systems) population. The Kuramoto model captures the essen- to a low-dimensional set of macroscopic equations9. Re- tial features of many coupled oscillator systems and has markably, Ott and Antonsen were also able to show their been used to study the phase transition to synchrony in procedure captures all the long-time attractors of these detail8. systems10. The fact that this procedure allows for the ##LTLine##

text: a)

text: The

label: abstract

text: study

text: of

text: synchronization

text: in

text: populations

text: of

text: coupled

text: biological

text: oscillators

text: is

text: fundamental

text: to

text: many

text: areas

text: of

text: biology

text: to

text: include

text: neuroscience,

text: cardiac

text: dynamics

text: and

text: circadian

text: rhythms.

text: Studying

text: these

text: systems

text: may

text: involve

text: tracking

text: the

text: concentration

text: of

text: hundreds

text: of

text: variables

text: in

text: thousands

text: of

text: individual

text: cells

text: resulting

text: in

text: an

text: extremely

text: high-dimensional

text: description

text: of

text: the

text: system.

text: However,

text: for

text: many

text: of

text: these

text: systems

text: the

text: behaviors

text: of

text: interest

text: occur

text: on

text: a

text: collective

text: or

text: macroscopic

text: scale.

text: We

text: deﬁne

text: a

text: new

text: macroscopic

text: reduction

text: for

text: networks

text: of

text: coupled

text: oscillators

text: motivated

text: by

text: an

text: elegant

text: structure

text: we

text: ﬁnd

text: in

text: experimental

text: measurements

text: of

text: circadian

text: gene

text: expression

text: and

text: several

text: mathematical

text: models

text: for

text: coupled

text: biological

text: oscillators.

text: We

text: characterize

text: the

text: emergence

text: of

text: this

text: structure

text: through

text: a

text: simple

text: argument

text: and

text: demonstrate

text: its

text: applicability

text: to

text: stochastic

text: and

text: heterogeneous

text: systems

text: of

text: coupled

text: oscillators.

text: Finally,

text: we

text: perform

text: the

text: macroscopic

text: reduction

text: for

text: the

text: heterogeneous

text: stochastic

text: Kuramoto

text: equation

text: and

text: compare

text: the

text: low-dimensional

text: macroscopic

text: model

text: with

text: numerical

text: results

text: from

text: the

text: high-dimensional

text: microscopic

text: model.

text: PACS

text: numbers:

text: Keywords:

text: The

text: study

text: of

text: coupled

text: oscillators

text: is

text: important

text: for

text: many

text: However,

text: many

text: biological

text: systems

text: contain

text: thousands

text: biological

text: and

text: physical

text: systems,

text: including

text: neural

text: net-

text: of

text: oscillators,

text: making

text: even

text: the

text: phase

text: model

text: a

text: very

text: high-

text: works,

text: circadian

text: rhythms

text: and

text: power

text: Mathe-

text: dimensional

text: representation

text: of

text: the

text: dynamical

text: system.

text: A

text: matical

text: models

text: of

text: these

text: coupled

text: oscillator

text: systems

text: can

text: recent

text: breakthrough

text: occurred

text: when

text: Ott

text: and

text: Antonsen

text: be

text: extremely

text: high-dimensional,

text: having

text: at

text: least

text: as

text: many

text: discovered

text: an

text: ansatz

text: that

text: may

text: be

text: used

text: for

text: a

text: family

text: degrees

text: of

text: freedom

text: as

text: the

text: number

text: of

text: oscillators.

text: However,

text: of

text: Kuramoto-like

text: systems

text: to

text: derive

text: a

text: low-dimensional

text: this

text: microscale

text: complexity

text: is

text: belied

text: by

text: the

text: elegant

text: sim-

text: model

text: for

text: the

text: macroscopic

text: behavior

text: of

text: the

text: If

text: plicity

text: which

text: emerges

text: at

text: the

text: macroscopic

text: scale

text: in

text: many

text: the

text: ansatz

text: holds,

text: the

text: long-time

text: behavior

text: of

text: a

text: system

text: coupled

text: oscillator

text: systems.

text: Quite

text: generally,

text: these

text: sys-

text: of

text: N

text: oscillators

text: can

text: accurately

text: be

text: described

text: by

text: tems

text: demonstrate

text: a

text: phase

text: transition

text: as

text: the

text: coupling

text: be-

text: two

text: diﬀerential

text: equations,

text: one

text: for

text: the

text: mean

text: phase

text: of

text: tween

text: the

text: oscillators

text: is

text: strengthened

text: leading

text: to

text: the

text: emer-

text: the

text: coupled

text: oscillators,

text: and

text: the

text: other

text: for

text: their

text: collec-

text: gence

text: of

text: a

text: self-organized

text: synchronized

text: tive

text: Despite

text: the

text: hundreds

text: of

text: recent

text: papers

text: This

text: emergence

text: of

text: a

text: synchronized

text: state

text: from

text: the

text: dy-

text: on

text: the

text: Ott-Antonsen

text: (OA)

text: ansatz,

text: the

text: authors

text: are

text: not

text: namics

text: of

text: a

text: very

text: high-dimensional

text: dynamical

text: system,

text: aware

text: of

text: any

text: carefully

text: done

text: experiments

text: to

text: test

text: whether

text: suggests

text: that

text: a

text: low

text: dimensional

text: representation

text: of

text: this

text: this

text: powerful

text: ansatz

text: holds

text: for

text: biological

text: systems.

text: system

text: should

text: be

text: possible.

text: A

text: major

text: step

text: in

text: this

text: di-

text: In

text: this

text: work

text: we

text: test

text: the

text: applicability

text: of

text: the

text: Ott-

text: rection

text: was

text: proposed

text: by

text: Art

text: Winfree

text: in

text: when

text: he

text: Antonsen

text: ansatz

text: using

text: a

text: recent

text: experimental

text: data

text: set

text: col-

text: intuitively

text: grasped

text: that

text: for

text: systems

text: of

text: weakly

text: coupled

text: lected

text: from

text: circadian

text: oscillator

text: neurons

text: and

text: through

text: sim-

text: oscillators

text: the

text: time

text: evolution

text: of

text: each

text: limit

text: cycle

text: oscilla-

text: ulations

text: of

text: several

text: models

text: of

text: coupled

text: biological

text: tor

text: may

text: be

text: described

text: by

text: a

text: single

text: phase

text: This

text: We

text: ﬁnd

text: the

text: core

text: assumptions

text: which

text: allow

text: for

text: the

text: deriva-

text: method

text: is

text: now

text: known

text: as

text: phase

text: reduction

text: and

text: has

text: been

text: tion

text: of

text: the

text: macroscopic

text: model

text: using

text: the

text: OA

text: ansatz

text: are

text: applied

text: to

text: study

text: of

text: many

text: coupled

text: oscillator

text: not

text: valid

text: in

text: our

text: test

text: systems.

text: However,

text: we

text: ﬁnd

text: a

text: dif-

text: In

text: the

text: following

text: years,

text: Kuramoto

text: formalized

text: the

text: math-

text: ferent,

text: but

text: related,

text: ansatz

text: is

text: capable

text: of

text: describing

text: the

text: ematical

text: procedure

text: for

text: phase

text: reduction

text: and

text: used

text: it

text: to

text: de-

text: data

text: well.

text: Using

text: a

text: simple

text: argument

text: we

text: are

text: able

text: to

text: rive

text: his

text: now

text: famous

text: model

text: for

text: N

text: coupled

text: heterogeneous

text: demonstrate

text: the

text: validity

text: of

text: our

text: ansatz

text: for

text: a

text: wide-class

text: of

text: oscillators,

text: models.

text: Finally,

text: we

text: demonstrate

text: how

text: our

text: ansatz

text: may

text: be

text: used

text: to

text: derive

text: macroscopic

text: models

text: for

text: coupled

text: oscillator

text: systems.

text: K

text: N

text: φ˙

text: ω

text: sin(φ

text: φ

text: i

text: N

text: i

text: i

text: N

text: j

text: i

text: RESULTS

text: where

text: φ

text: gives

text: the

text: phase

text: of

text: the

text: ith

text: oscillator,

text: K

text: the

text: i

text: coupling

text: strength

text: and

text: ω

text: gives

text: the

text: natural

text: frequency

text: of

text: The

text: development

text: of

text: the

text: Ott-Antonsen

text: ansatz

text: initiated

text: i

text: the

text: The

text: natural

text: frequencies

text: of

text: the

text: oscillators

text: a

text: revolution

text: in

text: the

text: coupled

text: oscillator

text: The

text: are

text: typically

text: assumed

text: to

text: be

text: drawn

text: from

text: some

text: distribu-

text: method

text: was

text: ﬁrst

text: presented

text: as

text: an

text: ansatz

text: which

text: could

text: be

text: tion

text: g(ω)

text: which

text: reﬂects

text: the

text: heterogeneity

text: in

text: the

text: oscillator

text: used

text: to

text: reduce

text: Kuramoto

text: (and

text: closely

text: related

text: systems)

text: population.

text: The

text: Kuramoto

text: model

text: captures

text: the

text: essen-

text: to

text: a

text: low-dimensional

text: set

text: of

text: macroscopic

text: Re-

text: tial

text: features

text: of

text: many

text: coupled

text: oscillator

text: systems

text: and

text: has

text: markably,

text: Ott

text: and

text: Antonsen

text: were

text: also

text: able

text: to

text: show

text: their

text: been

text: used

text: to

text: study

text: the

text: phase

text: transition

text: to

text: synchrony

text: in

text: procedure

text: captures

text: all

text: the

text: long-time

text: attractors

text: of

text: these

text: The

text: fact

text: that

text: this

text: procedure

text: allows

text: for

text: the

text: ##LTLine##


text: 21 cm models 3 0.14 MH15 Late/Default 1.0 (AGN only) (galaxy-dominated) 0.12 Very Late 50.8 (AGN-dominated) 0.8 0.10 50.6 0.08 50.4 0.6 0.06 50.2 0.4 0.04 50.0 0.2 0.02 49.8 Madau & Haardt 2015 WMAP 2013 Late/Default (total) Planck 2015 Very Late (total) 0.00 Planck 2016 49.6 Very Late (AGN) 0.0 4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14 z z z Figure 1. Evolution of the volume-weighted ionization fraction QV (left panel), the electron cattering optical depth τ (middle panel), and the hydrogen-ionizing emissivity n˙ion (right panel) in the AGN-dominated “Very Late” reionization model considered in this paper and the galaxies-dominated “Late/Default” model of Kulkarni et al. (2016). The AGN-dominated model is shown by red curves in left and middle panels, and red points and open circles in the right panel. The galaxy-dominated model is shown by the grey curves and points. The dashed black curve in the left panel shows the model of Madau & Haardt (2015), which includes emission only from AGN; the red curve in the right panel shows the corresponding emissivity. In the right panel, red ﬁlled circles denote ionizing emissivity from AGN in our model; red open circles refer to the total ionizing emissivity, which also includes contribution from star-forming galaxies. × − transfer simulations.(cid:63) This creates cells with excess neutral simulation is 2.3 108 h 1M(cid:12); the maximum halo mass is × − hydrogen fraction, thereby reducing the mean free path of 3.1 1012 h 1M(cid:12). Lyman continuum photons. The mean free path λmfp enters To model ionizing emission by AGN, we assume that in Equation (3) via n˙ion, which is related to the photoionization high-mass haloes that host luminous AGN, the total number rate by (Kuhlen & Faucher-Gigu`ere 2012; Becker & Bolton of photons Nγ is proportional to the black hole mass Mbh. In 2013) (cid:18) (cid:19) order to estimate the mass of black holes in these haloes, we follow the approach of Kulkarni & Loeb (2012) and employ n˙ion = (1 +ΓzH)2IQσHVλmfp αbα+s 3 , (4) t(ohdfeemﬁMnasebsdh–Maσsratethlaerteicdo(cid:18)isnrhc(uicfltfa.rzMviaseolgo&icviet(cid:19)Kynimbaty2v0i1r6ia)l. Trahdeiuvsi)riafolrvealohcaitlyo αs iswthheersepσeHctirsalthinedheyxdorof gtehne piohnoitzoiniognsiozautricoens cartoλss-<se9c1ti2on˚A, vc = 23.4 km s−1 108hM−1M(cid:20) (cid:12) 1/3 (cid:21) (cid:18) (cid:19) and αb is the spectral index of the ionizing “background” × Ω ∆ 1/6 1 + z 1/2 within ionized regions. m c , (5) Ωz 18π2 10 The Sherwood simulation suite has been run using the m energy- and entropy-conserving TreePM smoothed particle where hydrodynamical (SPH) code p-gadget-3, which is an up- dated version of the gadget-2 code (Springel et al. 2001; Ω (1 + z)3 Ωz = m , (6) Springel 2005). Our base sim−ulation was performed in a pe- m Ωm(1 + z)3 + ΩΛ + Ωk(1 + z)2 riodic cube of length 160 h 1cMpc on a side. The num- ber of gas and dark matter particles were both initially and ∆c is the overdensity of the halo relative to the 20483. This corresponds to a dark matter particle mass critical density, given by × − oMfgMas d=m 6=.383×.44106 1h0−71Mh(cid:12)1.MIn(cid:12)thaendregdasshifptarratincglee mrealesvsanotf ∆ = 18π2 + 82d − 39d2, (7) c to this paper, we use snapshots of the particle positions at z = 10, 8, and 7. Haloes are identiﬁed using the friends-of- − where d = Ωz 1 (Barkana & Loeb 2001). Further, we friends algorithm. At z = 7, the minimum halo mass in our m equate the halo virial velocity with the circular velocity v c of its constituent spheroid and obtain the velocity dispersion (cid:104) (cid:105) of the spheroid using the relation (Ferrarese 2002) (cid:63) This self-shielding is insensitive to the presence of hard ionizing photons, such as those from AGN, due to diminished ionization ≈ σ 0.84 − cross-section (Rahmati et al. 2013). vc 314 208 km s−1 km s 1. 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text: 4 Kulkarni et al. a suitable reionization history to calibrate our simulation. Equation (8) combined with the measured Mbh–σ rela- As discussed above in relation to Equation (3), this calibra- tion at redshift(cid:20)z = 0 (Tremain(cid:21)e et al. 2002) tion will provide us with the photoionization rate and ion- ized hydrogen fraction throughout our simulation box. The σ − ≈ ×Mbh 1/4.02 , (9) AMGadNa-udo&miHnaaaterddtio(n2i0z1in5)g geimveisssirvisietyteovaolruetiioonnizcaotniosindehriesdtobryy 208 km s 1 1.56 108M(cid:12) that is very close to the “Very Late” reionization history as described by Kulkarni et al. (2016). For ease of comparison gives (cid:18) (cid:19) (cid:20) (cid:21) we thus choose this reionization history to calibrate our sim- ulation. The red curve in the left panel of Figure 1 shows Mbh = 0.12 Mhalo 1.6 Ωm ∆c 0.8 (1 + z)2.4. (10) the evolution of the volume-weighted ionized fraction QV 108M(cid:12) 1012M(cid:12) Ωz 18π2 in this Very Late model. For comparison, the grey curve in m this panel shows the evolution of Q in the “Late/Default” V We assume that haloes with mass below a threshold model of Kulkarni et al. (2016). In the Very Late model, ion- ∝ mass Mq have Nγ(M) = Nγgal(M) Mhalo. Ionizing photon†s ized regions overlap and the universe is completely reionized from these low-mass haloes are sourced by star formation. at z = 6, similar to the Late/Default model, but Q evolves V On the other hand, high-mass haloes with mass greater than more rapidly at z > 6 (Kulkarni et al. 2016). This model ∝ the threshold Mq have Nγ(M) = Nγagn(M) Mbh where agrees reasonably well with the background photoionization Mbh is given by Equation (10). These high-mass haloes pro- rate determined from the Lyα forest at z < 6 (Faucher- duce(cid:82)ionizing photons due to AGN. The ratio Gigu`ere et al. 2009; Becker & Bolton 2013) and from quasar ∼ (cid:82) proximity zones at z 6 (Wyithe & Bolton 2011; Calver- r ≡ MMqmax dMNγagn(M) dN/dM , (11) leetyael.t2a0l1.52;0C11h)a,rdailnbeeitt awli.th20n15o)t.abTlehedviﬀaelureenocfesth(ePuelcehcwtreoinn MMmqin dMNγgal(M) dN/dM scattering optical depth to the last scattering surface in this model is τ = 0.055, in good agreement with the most recent ± quantiﬁes the relative photon contribution of AGN and Planck measurement (τ = 0.058 0.012; Planck Collabora- galaxies. Our AGN models are thus described by two pa- tion 2016), as seen in the middle panel of Figure 1. rameters r and Mq. (Appendix gives further details on our 3 RESULTS: 21 CM SIGNAL AGN ionizing emissivity model.) The top panel of Figure 2 shows the evolution of the In our ﬁducial AGN-dominated model, we ﬁx the value 21 cm brightness temperature from redshift z = 10 to 6 of the threshold mass M to that corresponding to a circu- q in our ﬁducial v > 175 km/s AGN-dominated model. The lar velocity of v = 175 km/s. (We will discuss the eﬀect on c c 21 cm brightness temperature is approximated as our results of changing this threshold to v = 150 km/s and c v = 200 km/s below.) At lower circular velocities, cold gas c mass available to grow supermassive black holes can decrease Tb(x) = TbxHI(x)∆(x), (12) rapidly due to an increasing eﬀect of supernova feedback (e.g., Kauﬀmann & Haehnelt 2000; Haehnelt & Kauﬀmann ≈ 2002; Brook et al. 2012). This is reﬂected in a drop in the where the mean temperature Tb 22mK[(1 + z)/7]1/2 black hole mass function for black hole masses smaller than (Choudhury et al. 2009). The above relation neglects the ∼ Mbh 107 M(cid:12), particularly for z > 1 (Merloni & Heinz impact of redshift space distortions due to peculiar veloci- 2008; Kelly & Merloni 2012). With M ﬁxed, a desired total ties, and possible ﬂuctuations in the spin temperature, i.e., q AGN emissivity is achieved in the model by setting the value it implicitly assumes that the spin temperature is much of the parameter r. We calibrate the AGN emissivity evolu- greater than the CMB temperature and that the Lyα cou- tion to values close to the ﬁt by Madau & Haardt (2015) to pling is suﬃciently complete throughout the IGM. This is a the integrated 1 Ry emissivity from AGN down to UV lu- good approximation in the redshift range considered here, minosities of 0.01L∗. This emissivity evolution is shown by when the global ionized fraction is greater than a few per the red curve in the right panel of Figure 1. In this panel, cent (Pritchard & Loeb 2012; Majumdar et al. 2014; Ghara red ﬁlled circles denote ionizing emissivity from AGN in our et al. 2015). For comparison, the middle panel of Figure 2 model; red open circles refer to the total ionizing emissivity, shows the evolution of the 21 cm brightness in the galaxies- which also includes contribution from star-forming galax- dominated Very Late model considered by Kulkarni et al. ies. The ionizing emissivity of AGN in our model closely (2016). The reionization history of this model is identical matches that from the model of Madau & Haardt (2015). to that of our AGN-dominated model, so the diﬀerences in We also have some contribution to n˙ion from star-forming the brightness distribution between the top and middle pan- galaxies in our model, particularly at z = 10, as seen from els of Figure 2 arise solely due to diﬀerences in the source the red open circles in Figure 1. For comparison, the grey model. The AGN-dominated model has fewer, larger, and points in Figure 1 show the photon emissivity in the galaxy- more clustered ionized regions than the galaxies-dominated dominated “Late/Default” model of Kulkarni et al. (2016). model (cf. McQuinn et al. 2007). Figure 2 also shows the Having chosen a source model, we now need to choose galaxies-dominated Late/Default model of Kulkarni et al. † While AGN have harder spectra than star-forming galaxies, the eﬀect of harder photons on the structure of the hydrogen 2015; Kakiichi et al. 2016). The excursion set method therefore ionization fronts is small (Thomas & Zaroubi 2008; Ghara et al. remains applicable. MNRAS 000, 1–9 (2016) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Kulkarni

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text: 10 Kulkarni et al. 6.0 5.0 1.6 z = 7 z = 7 vc = 200 km/s 5.5 z = 8 z = 8 vc = 175 km/s z = 10 4.5 z = 10 1.4 vc = 150 km/s 5.0 1.2 4.0 4.5 1.0 4.0 3.5 0.8 3.5 3.0 0.6 3.0 2.5 0.4 2.5 2.0 2.0 0.2 10.6 11.0 11.4 11.8 160 180 200 220 240 6 7 8 9 10 log10 (Mhalo/M(cid:12)) vc [km/s] z Figure A1. Our model for the photon contribution Nγ for AGN at various redshifts as a function of halo mass (left panel) and circular velocity (middle panel). The right panel shows the evolution of halo mass corresponding to the three circular velocity thresholds considered in this paper. Our chosen reionization model dictates the amplitude of the Nγ–Mhalo relation, while its slope is governed by the black hole mass model of Equation (10). galaxies. Our AGN models are thus described by two param- eters r and M . We ﬁx the value of the threshold mass M q q to that corresponding to a circular velocity of v = 175 km/s c in our ﬁducial model, but also consider the eﬀect of varying this threshold to v = 150 km/s and v = 200 km/s in Fig- c c ure 3. The left panel of Figure A1 shows the N assignment γ for AGN in our ﬁducial model (v = 175 km/s) at redshifts c ∝ z = 7, 8, and∝10. Belo∝w the threshold mass, Nγ Mhalo and above it Nγ Mbh Mh1a.6lo, following Equation (10). The middle panel of Figure A1 shows N as a function of the γ halo circular velocity. We see that N sharply increases at γ v = 175 km/s. This velocity corresponds to a diﬀerent halo c mass at each of the three redshifts considered here, as seen in the right panel of Figure A1. The magnitude of N increases γ with redshift to compensate for the decreasing number den- sity of haloes above the velocity threshold. As described in Section , where we discuss our calibration procedure, the required total ionizing emissivity is dictated by our chosen reionization model. This paper has been typeset from a TEX/LATEX ﬁle prepared by the author. MNRAS 000, 1–9 (2016) N gol γ 01 N gol γ 01 ] M 1101[ M (cid:12) olah ##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##

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