text: EPJ manuscript No. (will be inserted by the editor) Quark matter nucleation in neutron stars and astrophysical implications Ignazio Bombaci1,2,3, Domenico Logoteta2, Isaac Vidan˜a4 and Constan¸ca Providˆencia4 1 Dipartimento di Fisica “E. Fermi”, Universita` di Pisa, Largo B. Pontecorvo, 3, I-56127 Pisa, Italy 2 INFN, Sezione di Pisa, Largo B. Pontecorvo, 3, I-56127 Pisa, Italy 3 European Gravitational Observatory, Via E. Amaldi, I-56021 S. Stefano a Macerata, Cascina Italy 4 CFisUC, Department of Physics, University of Coimbra, PT-3004-516 Coimbra, Portugal Received: date / Revised version: date Abstract. A phase of strong interacting matter with deconﬁned quarks is expected in the core of massive neutron stars. We investigate the quark deconﬁnement phase transition in cold (T = 0) and hot β-stable hadronic matter. Assuming a ﬁrst order phase transition, we calculate and compare the nucleation rate and the nucleation time due to quantum and thermal nucleation mechanisms. We show that above a threshold value of the central pressure a pure hadronic star (HS) (i.e. a compact star with no fraction of deconﬁned quark matter) is metastable to the conversion to a quark star (QS) (i.e. a hybrid star or a strange star). This process liberates an enormous amount of energy, of the order of 1053 erg, which causes a powerful neutrino burst, likely accompanied by intense gravitational waves emission, and possibly by a second delayed (with respect to the supernova explosion forming the HS) explosion which could be the energy source of a powerful gamma-ray burst (GRB). This stellar conversion process populates the QS branch of compact stars, thus one has in the Universe two coexisting families of compact stars: pure hadronic stars and quark stars. We introduce the concept of critical mass M for cold HSs and proto-hadronic stars cr (PHSs), and the concept of limiting conversion temperature for PHSs. We show that PHSs with a mass M < M could survive the early stages of their evolution without decaying to QSs. Finally, we discuss cr the possible evolutionary paths of proto-hadronic stars. PACS. 97.60.Jd Neutron stars – 25.75.Nq Quark deconﬁnement, quark-gluon plasma production, and phase transitions – 26.60.Kp Equations of state of neutron-star matter – 98.38.Mz Supernova remnants – 98.70.Rz γ-ray bursts – 64.60.Qb Nucleation 1 Introduction Current high precision numerical calculations of QCD on a space-time lattice at zero baryon chemical potential (zero baryon density) have shown that at high tempera- Neutron stars, the compact remnants of core-collapse ture and for physical values of the quark masses, the tran- supernova, are the densest macroscopic objects in the Uni- sition to quark gluon plasma is a crossover [4–6] rather verse. They represent the limit beyond which gravity over- than a real phase transition. whelm all the other forces of nature and lead to the for- Unfortunately, present lattice QCD calculations at ﬁ- mation of a black hole. In fact, neutron star structure nite baryon chemical potential are plagued with the so calculations (see e.g. Refs. [1–3]) based on a large variety called “sign problem”, which makes them unrealizable by of modern equations of state (EOS) of hadronic matter, all presently known lattice methods. Thus, to explore the predict a maximum stellar central density (the one for the QCD phase diagram at low T and high µ, it is necessary to maximum mass star conﬁguration) in the range of 4 – 8 ∼ × invoke some approximations in QCD or to apply a QCD times the saturation density ( 2.8 1014 g/cm3) of nu- eﬀective model [7–16]. In this region of the T-µ plane, sev- clear matter. Thus the core of a neutron star is one of the eral QCD inspired models suggest the deconﬁnement tran- best candidates in the Universe where a phase of strong sition to be a ﬁrst-order phase transition [17,18]. In this interacting matter with deconﬁned quarks could be found, domain of the QCD phase diagram, many possible color and these compact stars can be viewed as natural labo- superconducting phases of quark matter are expected [19, ratories to test the low temperature T and high baryon 20] and matter might be characterized by the formation of chemical potential µ region of the QCD phase diagram. diﬀerent crystalline structures [21,22]. It is worth mention- ing that recent promising attempts to describe the whole Send oﬀprint requests to: ignazio.bombaci@unipi.it QCD phase diagram within a uniﬁed model [23–26] (see ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EPJ

text: manuscript

text: No.

text: (will

text: be

text: inserted

text: by

text: the

text: editor)

text: Quark

label: title

text: matter

text: nucleation

text: in

text: neutron

text: stars

text: and

text: astrophysical

text: implications

text: Ignazio

label: author

text: Domenico

text: Isaac

text: and

text: Constan¸ca

text: Dipartimento

text: di

text: Fisica

text: “E.

text: Fermi”,

text: Universita`

text: di

text: Pisa,

text: Largo

text: B.

text: Pontecorvo,

text: Pisa,

text: Italy

text: INFN,

text: Sezione

text: di

text: Pisa,

text: Largo

text: B.

text: Pontecorvo,

text: Pisa,

text: Italy

text: European

text: Gravitational

text: Observatory,

text: Via

text: E.

text: Amaldi,

text: S.

text: Stefano

text: a

text: Macerata,

text: Cascina

text: Italy

text: CFisUC,

text: Department

text: of

text: Physics,

text: University

text: of

text: Coimbra,

text: Coimbra,

text: Portugal

text: Received:

label: date

text: date

text: Revised

text: version:

text: date

text: Abstract.

text: A

text: phase

text: of

text: strong

text: interacting

text: matter

text: with

text: deconﬁned

text: quarks

text: is

text: expected

text: in

text: the

text: core

text: of

text: massive

text: neutron

text: stars.

text: We

text: investigate

text: the

text: quark

text: deconﬁnement

text: phase

text: transition

text: in

text: cold

text: (T

text: and

text: hot

text: β-stable

text: hadronic

text: matter.

text: Assuming

text: a

text: ﬁrst

text: order

text: phase

text: transition,

text: we

text: calculate

text: and

text: compare

text: the

text: nucleation

text: rate

text: and

text: the

text: nucleation

text: time

text: due

text: to

text: quantum

text: and

text: thermal

text: nucleation

text: mechanisms.

text: We

text: show

text: that

text: above

text: a

text: threshold

text: value

text: of

text: the

text: central

text: pressure

text: a

text: pure

text: hadronic

text: star

text: (HS)

text: (i.e.

text: a

text: compact

text: star

text: with

text: no

text: fraction

text: of

text: deconﬁned

text: quark

text: matter)

text: is

text: metastable

text: to

text: the

text: conversion

text: to

text: a

text: quark

text: star

text: (QS)

text: (i.e.

text: a

text: hybrid

text: star

text: or

text: a

text: strange

text: star).

text: This

text: process

text: liberates

text: an

text: enormous

text: amount

text: of

text: energy,

text: of

text: the

text: order

text: of

text: erg,

text: which

text: causes

text: a

text: powerful

text: neutrino

text: burst,

text: likely

text: accompanied

text: by

text: intense

text: gravitational

text: waves

text: emission,

text: and

text: possibly

text: by

text: a

text: second

text: delayed

text: (with

text: respect

text: to

text: the

text: supernova

text: explosion

text: forming

text: the

text: HS)

text: explosion

text: which

text: could

text: be

text: the

text: energy

text: source

text: of

text: a

text: powerful

text: gamma-ray

text: burst

text: (GRB).

text: This

text: stellar

text: conversion

text: process

text: populates

text: the

text: QS

text: branch

text: of

text: compact

text: stars,

text: thus

text: one

text: has

text: in

text: the

text: Universe

text: two

text: coexisting

text: families

text: of

text: compact

text: stars:

text: pure

text: hadronic

text: stars

text: and

text: quark

text: stars.

text: We

text: introduce

text: the

text: concept

text: of

text: critical

text: mass

text: M

text: for

text: cold

text: HSs

text: and

text: proto-hadronic

text: stars

text: cr

text: (PHSs),

text: and

text: the

text: concept

text: of

text: limiting

text: conversion

text: temperature

text: for

text: PHSs.

text: We

text: show

text: that

text: PHSs

text: with

text: a

text: mass

text: M

text: M

text: could

text: survive

text: the

text: early

text: stages

text: of

text: their

text: evolution

text: without

text: decaying

text: to

text: QSs.

text: Finally,

text: we

text: discuss

text: cr

text: the

text: possible

text: evolutionary

text: paths

text: of

text: proto-hadronic

text: stars.

text: PACS.

text: Neutron

text: stars

text: Quark

text: deconﬁnement,

text: quark-gluon

text: plasma

text: production,

text: and

text: phase

text: transitions

text: Equations

text: of

text: state

text: of

text: neutron-star

text: matter

text: Supernova

text: remnants

text: γ-ray

text: bursts

text: Nucleation

text: Introduction

text: Current

text: high

text: precision

text: numerical

text: calculations

text: of

text: QCD

text: on

text: a

text: space-time

text: lattice

text: at

text: zero

text: baryon

text: chemical

text: potential

text: (zero

text: baryon

text: density)

text: have

text: shown

text: that

text: at

text: high

text: tempera-

text: Neutron

text: stars,

text: the

text: compact

text: remnants

text: of

text: core-collapse

text: ture

text: and

text: for

text: physical

text: values

text: of

text: the

text: quark

text: masses,

text: the

text: tran-

text: supernova,

text: are

text: the

text: densest

text: macroscopic

text: objects

text: in

text: the

text: Uni-

text: sition

text: to

text: quark

text: gluon

text: plasma

text: is

text: a

text: crossover

text: rather

text: verse.

text: They

text: represent

text: the

text: limit

text: beyond

text: which

text: gravity

text: over-

text: than

text: a

text: real

text: phase

text: transition.

text: whelm

text: all

text: the

text: other

text: forces

text: of

text: nature

text: and

text: lead

text: to

text: the

text: for-

text: Unfortunately,

text: present

text: lattice

text: QCD

text: calculations

text: at

text: ﬁ-

text: mation

text: of

text: a

text: black

text: hole.

text: In

text: fact,

text: neutron

text: star

text: structure

text: nite

text: baryon

text: chemical

text: potential

text: are

text: plagued

text: with

text: the

text: so

text: calculations

text: (see

text: e.g.

text: Refs.

text: based

text: on

text: a

text: large

text: variety

text: called

text: “sign

text: problem”,

text: which

text: makes

text: them

text: unrealizable

text: by

text: of

text: modern

text: equations

text: of

text: state

text: (EOS)

text: of

text: hadronic

text: matter,

text: all

text: presently

text: known

text: lattice

text: methods.

text: Thus,

text: to

text: explore

text: the

text: predict

text: a

text: maximum

text: stellar

text: central

text: density

text: (the

text: one

text: for

text: the

text: QCD

text: phase

text: diagram

text: at

text: low

text: T

text: and

text: high

text: µ,

text: it

text: is

text: necessary

text: to

text: maximum

text: mass

text: star

text: conﬁguration)

text: in

text: the

text: range

text: of

text: invoke

text: some

text: approximations

text: in

text: QCD

text: or

text: to

text: apply

text: a

text: QCD

text: times

text: the

text: saturation

text: density

text: of

text: nu-

text: eﬀective

text: model

text: In

text: this

text: region

text: of

text: the

text: T-µ

text: plane,

text: sev-

text: clear

text: matter.

text: Thus

text: the

text: core

text: of

text: a

text: neutron

text: star

text: is

text: one

text: of

text: the

text: eral

text: QCD

text: inspired

text: models

text: suggest

text: the

text: deconﬁnement

text: tran-

text: best

text: candidates

text: in

text: the

text: Universe

text: where

text: a

text: phase

text: of

text: strong

text: sition

text: to

text: be

text: a

text: ﬁrst-order

text: phase

text: transition

text: In

text: this

text: interacting

text: matter

text: with

text: deconﬁned

text: quarks

text: could

text: be

text: found,

text: domain

text: of

text: the

text: QCD

text: phase

text: diagram,

text: many

text: possible

text: color

text: and

text: these

text: compact

text: stars

text: can

text: be

text: viewed

text: as

text: natural

text: labo-

text: superconducting

text: phases

text: of

text: quark

text: matter

text: are

text: expected

text: ratories

text: to

text: test

text: the

text: low

text: temperature

text: T

text: and

text: high

text: baryon

text: and

text: matter

text: might

text: be

text: characterized

text: by

text: the

text: formation

text: of

text: chemical

text: potential

text: µ

text: region

text: of

text: the

text: QCD

text: phase

text: diagram.

text: diﬀerent

text: crystalline

text: structures

text: It

text: is

text: worth

text: mention-

text: ing

text: that

text: recent

text: promising

text: attempts

text: to

text: describe

text: the

text: whole

text: Send

text: oﬀprint

text: requests

text: to:

text: ignazio.bombaci@unipi.it

text: QCD

text: phase

text: diagram

text: within

text: a

text: uniﬁed

text: model

text: (see

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Ignazio Bombaci et al.: Quark matter nucleation in neutron stars and astrophysical implications 13 89. I. Vidan˜a, I. Bombaci, A. Polls. A. Ramos, Astron. and Astrophys. 399, 687 (2003). 90. J. Margueron, I. Vidan˜a, I. Bombaci, Phys. Rev. C 68, 055806 (2003). 91. I. Bombaci, Astron. and Astrophys. 305, 871 (1996). 92. J.E. Horvath, O.G. Benvenuto, H. Vucetich, Phys. Rev. D 45, 3865 (1992). 93. J. E. Horvath, Phys. Rev. D 49, 5590 (1994). 94. M.L. Olesen, J. Madsen, Phys. Rev. D 49, 2698 (1994). 95. T. Harko, K.S. Cheng, P.S. Tang, Astrophys. Jour. 608, 945 (2004). 96. J.S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969). 97. J.S. Langer, L.A. Turski, Phys. Rev. A 8, 3230 (1973). 98. L. Csernai, J.I. Kapusta, Phys. Rev. D 46, 1379 (1992). 99. R. Venugopalan, A.P. Vischer, Phys. Rev. E 49, 5849 (1994). 100. I. Sagert, T. Fischer, M. Hempel, G. Pagliara, J. Schaﬀner-Bielich, A. Mezzacappa, F.-K. Thieleman, M. Liebend¨orfer, Phys. Rev. Lett. 102, 081101 (2009). 101. K. Nakazato, K. Sumiyoshi, S. Yamada, 2008, Phys. Rev. D 77, 103006 (2008). 102. B. Dasgupta, T. Fischer, S. Horiuchi, M. Liebendo¨rfer, A. Mirizzi, Phys. Rev. D, 81, 103005 (2010). 103. S. Bhattacharyya, Adv. Space Res. 45, 949 (2010). 104. A.W. Steiner, J.M. Lattimer, E.F. Brown, Astrophys. Jour. 722, 33 (2010). 105. S. Bhattacharyya, A.V. Thampan, I. Bombaci, Astr. and Astrophys. 372, 925 (2001).

text: Ignazio

label: paragraph

text: Bombaci

text: et

text: al.:

text: Quark

text: matter

text: nucleation

text: in

text: neutron

text: stars

text: and

text: astrophysical

text: implications

label: reference

text: I.

text: Vidan˜a,

text: I.

text: Bombaci,

text: A.

text: Polls.

text: A.

text: Ramos,

text: Astron.

text: and

text: Astrophys.

text: J.

text: Margueron,

text: I.

text: Vidan˜a,

text: I.

text: Bombaci,

text: Phys.

text: Rev.

text: C

text: I.

text: Bombaci,

text: Astron.

text: and

text: Astrophys.

text: J.E.

text: Horvath,

text: O.G.

text: Benvenuto,

text: H.

text: Vucetich,

text: Phys.

text: Rev.

text: D

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text: 4 Ignazio Bombaci et al.: Quark matter nucleation in neutron stars and astrophysical implications Table 2. Same as Table 1, but with trapped neutrinos. 205 60 Q T n ∗ n P NJL CDM Q− H− 0 MeV MeV fm 3 fm 3 MeV/fm3 50 200 0 0.00 0.603 0.516 113.77 Q* Phase Q* Phase 40 5 0.65 0.601 0.514 113.11 195 10 2.87 0.594 0.509 110.69 30 15 6.78 0.580 0.499 106.17 190 Hadronic Phase 20 12.65 0.560 0.483 99.18 Hadronic Phase 20 25 20.21 0.534 0.462 90.12 30 29.88 0.502 0.434 78.65 185 10 180 0 4 Quark matter nucleation in cold hadronic 0 5 10 15 20 25 30 0 5 10 15 20 25 30 T [MeV] T [MeV] stars Fig. 4. Phase equilibrium curve between the β-stable hadronic phase and the Q*-phase, neutrino-free matter. The EOS for the hadronic phase is obtained using the GM1 model with x = Initially, we assume that the compact star survives ∗ σ 0.7, the EOS for the Q -phase using the NJL model (left panel) the early stages of its evolution as a pure hadronic star, or the Chromo Dielectric model (right panel). and we study quark matter nucleation in cold (T = 0) neutrino-free hadronic matter. The case of quark mat- Q ter nucleation at ﬁnite temperature in neutrino-free and Table 1. The speciﬁc (i.e. per baryon) latent heat and neutrino-trapped matter will be discussed in the Section the phase number densities nH and nQ∗ at phase equilibrium. 6. GM1 EOS with x = 0.6 for the hadronic phase, MIT bag σ In our scenario, we consider a purely hadronic star model with B = 85 MeV/fm3 for the quark phase. Results for whose central pressure is increasing due to spin-down or neutrino-free matter. due to mass accretion, e.g., from a companion star. As Q T n ∗ n P the central pressure exceeds the deconﬁnement threshold Q− H− 0 MeV MeV fm 3 fm 3 MeV/fm3 value P , a virtual drop of quark matter in the Q*-phase 0 can be formed in the central region of the star. As soon 0 0.00 0.453 0.366 39.95 as a real drop of Q*-matter is formed, it will grow very 5 0.56 0.451 0.364 39.74 rapidly and the original Hadronic Star will be converted 10 2.40 0.447 0.358 38.58 to an Hybrid Star or to a Strange Star, depending on the 15 5.71 0.439 0.348 36.55 detail of the EOS for quark matter employed to model the 20 10.60 0.428 0.334 33.77 phase transition. 25 17.17 0.414 0.316 30.36 30 25.44 0.398 0.294 26.53 In a cold (T = 0) and neutrino-free pure hadronic star the formation of the ﬁrst drop of QM could take place solely via a quantum nucleation process. The basic quan- tity needed to calculate the nucleation time is the en- where W˜ H (W˜ Q∗) and S˜H (S˜Q∗) denote the enthalpy per ergy barrier separating the Q*-phase from the metastable baryon and entropy per baryon for the hadroQn (quark) hadronic phase. This energy barrier, which represents the phase, respectively. The speciﬁc latent heat and the diﬀerence in the free energy of the system with and with- hadron and quark baryon number densities nH and nQ∗ out a Q*-matter droplet, can be written as [74,75] at phase equilibrium are reported in Tables 1 and 2 in the case of the GM1 –B85 equation of state. Results in R 4 − R R 0.6 U( ) = πn ∗(µ ∗ µ ) 3 + 4πσ 2 (7) Table 1 refer to neutrino-free matter, whereas those in Ta- 3 Q Q H ble 2 refer to matter with trapped neutrinos. As expected R for a ﬁrst-order phase transition, one has a discontinuity where is the radius of the droplet (supposed to jump in the phase number densities: in our particular case be spherical), and σ is the surface tension for the sur- nQ∗(T, P0) > nHQ(T, P0). This result, together with the face separating the hadron from the Q*-phase. ThRe en- positive value of (i.e. the deconﬁnement phase transi- ergy barrier has a maximum at the critical radius = − c tion absorbs heat), tells us (see Eq. (5)) that the phase 2σ/[n ∗(µ µ ∗)]. We neglected the term associated Q H Q transition temperature decreases with pressure (as in the with the curvature energy and also the terms connected melting of ice). with the electrostatic energy, since they are known to only The eﬀect of neutrino trapping on the phase equilib- introduce small corrections [75,33]. The value of the sur- rium properties of the system can be seen comparing the face tension σ for the interface separating the quark and results reported in Tables 1 and 2. As we see, the phase hadron phase is poorly known, and typically values used − − number densities n and n ∗ at phase equilibrium are in the literature range within 10 50 MeV fm 2 [76]). H Q Q − shifted to higher values, and the speciﬁc latent heat is Larger values of σ in the range 50–160 MeV fm 2 have increased with respect to the neutrino-free matter case. also been obtained in the literature [77,78]. Clearly these ]3-mf VeM[ P 0 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ignazio

text: Bombaci

text: et

text: al.:

text: Quark

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text: nucleation

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text: Results

text: for

text: whose

text: central

text: pressure

text: is

text: increasing

text: due

text: to

text: spin-down

text: or

text: neutrino-free

text: matter.

text: due

text: to

text: mass

text: accretion,

text: e.g.,

text: from

text: a

text: companion

text: star.

text: As

text: Q

text: T

text: n

text: n

text: P

text: the

text: central

text: pressure

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text: the

text: deconﬁnement

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text: Q−

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text: MeV

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text: fm

text: value

text: P

text: a

text: virtual

text: drop

text: of

text: quark

text: matter

text: in

text: the

text: Q*-phase

text: can

text: be

text: formed

text: in

text: the

text: central

text: region

text: of

text: the

text: star.

text: As

text: soon

text: as

text: a

text: real

text: drop

text: of

text: Q*-matter

text: is

text: formed,

text: it

text: will

text: grow

text: very

text: rapidly

text: and

text: the

text: original

text: Hadronic

text: Star

text: will

text: be

text: converted

text: to

text: an

text: Hybrid

text: Star

text: or

text: to

text: a

text: Strange

text: Star,

text: depending

text: on

text: the

text: detail

text: of

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text: neutrino-free

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text: hadronic

text: star

text: the

text: formation

text: of

text: the

text: ﬁrst

text: drop

text: of

text: QM

text: could

text: take

text: place

text: solely

text: via

text: a

text: quantum

text: nucleation

text: process.

text: The

text: basic

text: quan-

text: tity

text: needed

text: to

text: calculate

text: the

text: nucleation

text: time

text: is

text: the

text: en-

text: where

text: W˜

text: H

text: (W˜

text: Q∗)

text: and

text: S˜H

text: (S˜Q∗)

text: denote

text: the

text: enthalpy

text: per

text: ergy

text: barrier

text: separating

text: the

text: Q*-phase

text: from

text: the

text: metastable

text: baryon

text: and

text: entropy

text: per

text: baryon

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text: the

text: hadroQn

text: (quark)

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text: hadron

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text: densities

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text: a

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text: can

text: be

text: written

text: as

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text: are

text: reported

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text: those

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text: Q

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text: ble

text: refer

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text: matter

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text: trapped

text: neutrinos.

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text: expected

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text: for

text: a

text: ﬁrst-order

text: phase

text: transition,

text: one

text: has

text: a

text: discontinuity

text: where

text: is

text: the

text: radius

text: of

text: the

text: droplet

text: (supposed

text: to

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text: densities:

text: in

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text: be

text: spherical),

text: and

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text: the

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text: for

text: the

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text: nQ∗(T,

text: nHQ(T,

text: This

text: result,

text: together

text: with

text: the

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text: from

text: the

text: Q*-phase.

text: ThRe

text: en-

text: positive

text: value

text: of

text: (i.e.

text: the

text: deconﬁnement

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text: ergy

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text: absorbs

text: heat),

text: tells

text: us

text: (see

text: Eq.

text: that

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text: ∗(µ

text: µ

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text: neglected

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text: term

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text: Q

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text: decreases

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text: pressure

text: (as

text: in

text: the

text: with

text: the

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text: and

text: also

text: the

text: terms

text: connected

text: melting

text: of

text: ice).

text: with

text: the

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text: energy,

text: since

text: they

text: are

text: known

text: to

text: only

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text: of

text: neutrino

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text: the

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text: introduce

text: small

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text: properties

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text: the

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text: be

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text: σ

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text: the

text: interface

text: separating

text: the

text: quark

text: and

text: results

text: reported

text: in

text: Tables

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text: As

text: we

text: see,

text: the

text: phase

text: hadron

text: phase

text: is

text: poorly

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text: used

text: number

text: densities

text: n

text: and

text: n

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text: are

text: in

text: the

text: literature

text: range

text: within

text: MeV

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text: shifted

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text: and

text: the

text: speciﬁc

text: latent

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text: range

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text: also

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text: the

text: literature

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text: these

text: VeM[

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text: Ignazio Bombaci et al.: Quark matter nucleation in neutron stars and astrophysical implications 5 1.8 10 1.6 QS ● Mcr ● 9 P = P0 = 245 MeV/fm3 250 MeV/fm3 1.4 ❋ 8 1.2 7 1 HS 6 0.8 5 0.6 4 255 MeV/fm3 0.4 3 260 MeV/fm3 0.2 2 7 8 9 10 11 12 13 14 15 1 R [km] 00 5 10 15 20 25 30 35 40 45 50 Fig. 6. (Color on line) Mass-radius relation for pure hadronic R [fm] star (HS) and hybrid star (QS) conﬁgurations. The conﬁgu- ration marked with an asterisk represents the HS for which ∞ Fig. 5. Potential energy of the QM drop as a function of the τq = (i.e. Pc = P0). The conversion process of the HS, with radius of the drop for several pressures above P0 at T = 0. The a gravitational mass equal to Mcr, into the ﬁnal QS is denoted hadronic phase is described with the GM3 model whereas for by the full circles connected by a dashed line. Results are rel- ∗ the Q phases is employed the MIT bag model with ms = 150 ative to the GM1 model with xσ = 0.6 for the hadronic phase MeV, B = 152.45 MeV/fm3. The surface tension σ is taken and the MIT bag model EOS with B = 85 MeV/fm3 for the equal to 30 MeV/fm2. quark phase. The surface tension is σ = 30 MeV/fm2×. Stellar masses are in unit of the mass of the sun, M = 1.989 1033 g. sun large values of the surface tension disfavor or inhibit quark matter nucleation in hadronic stars [50]. The nucleation time, i.e. the time needed to form the the HS, where the pressure and temperature (when we ﬁrst drop of the Q*-phase, can be straightforwardly evalu- will consider the ﬁnite T case) can be considered constant ated within a semi-classical approach [75,31,32]. First one and equal to their central values. The uncertainty in the computes, in the Wentzel -Kramers -Brillouin (WKB) ap- value of Nc is expected to be within one or two orders of proximation, the ground state energy E and the oscilla- magnitude. In any case, all the qualitative features of our 0 R tion frequency ν of the dro(cid:20)p in the p(cid:21)otential well U( ). scenario will not be aﬀected by this uncertainty [31–33]. 0 Then, the probability of tunneling is given by As a consequence of the surface eﬀects it is necessary to − have an overpressure ∆P = P P > 0 with respect to the 0 p = exp −A(E0) (8) bulk transition point P0 to create a drop of deconﬁnement 0 ¯h quark matter in the hadronic environment. The higher the overpressure, the easier to nucleate the ﬁrst drop of ∗ where A is the action under the potential barrier which Q matter. In other words, the higher the mass of the in a relativ(cid:90)istic f(cid:112)ramework reads metastable pure hadronic star, the shorter the time to nucleate a quark matter drop at the center of the star. R As an illustrative example, we plot in Fig. 5 the po- 2 + M R − R R − R R A(E) = [2 ( )c2 + E U( )][U( ) E]d , tential energy U( ) for the formation of a quark mat- c R − ter droplet for diﬀerent values of the stellar central pres- (9) sure P above the deconﬁnement threshold value P . The c 0 R (cid:18) (cid:19) curves in Fig. 5 are relative to a given set of EOS for with ± the classical turning points and the two phases of dense matter and to a ﬁxed value of the surface tension σ (see ﬁgure caption). As expected the M R − n ∗ 2 R ( ) = 4πρ 1 Q 3 (10) potential barrier is lowered as central pressure increases. H nH Thus a pure hadronic star, having a central pressure Pc larger than the transition pressure P for the formation of 0 the droplet eﬀective mass, with ρ and n the hadron H H the Q*-phase, is metastable [31–33] to the “decay” (con- energy density and the hadron baryon number density, version) to a quark star (QS) i.e. to a stellar conﬁguration respectively. The nucleation time is then equal to in which deconﬁned quark matter is present. − These metastable HSs have a mean-life time which is τ = (ν p N ) 1 , (11) q 0 0 c related to the nucleation time to form the ﬁrst critical- size drop of deconﬁned matter in their interior (the ac- ∼ where N 1048 is the number of nucleation centers tual mean-life time of the HS will depend on the mass c ≤ ∼ expected in the innermost part (r R 100 m) of accretion or on the spin-down rate which modiﬁes the nuc 4 ]VeM 01[ )R(U M/ M nus G ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Ignazio

text: Bombaci

text: et

text: al.:

text: Quark

text: matter

text: nucleation

text: in

text: neutron

text: stars

text: and

text: astrophysical

text: implications

text: QS

text: Mcr

text: P

text: HS

text: R

text: [km]

text: Fig.

text: (Color

text: on

text: line)

text: Mass-radius

text: relation

text: for

text: pure

text: hadronic

text: R

text: [fm]

text: star

text: (HS)

text: and

text: hybrid

text: star

text: (QS)

text: conﬁgurations.

text: The

text: conﬁgu-

text: ration

text: marked

text: with

text: an

text: asterisk

text: represents

text: the

text: HS

text: for

text: which

text: Fig.

text: Potential

text: energy

text: of

text: the

text: QM

text: drop

text: as

text: a

text: function

text: of

text: the

text: τq

text: (i.e.

text: Pc

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text: conversion

text: process

text: of

text: the

text: HS,

text: with

text: radius

text: of

text: the

text: drop

text: for

text: several

text: pressures

text: above

text: at

text: T

text: The

text: a

text: gravitational

text: mass

text: equal

text: to

text: Mcr,

text: into

text: the

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text: QS

text: is

text: denoted

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text: phase

text: is

text: described

text: with

text: the

text: model

text: whereas

text: for

text: by

text: the

text: full

text: circles

text: connected

text: by

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text: line.

text: Results

text: are

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text: phases

text: is

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text: the

text: MIT

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text: with

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text: the

text: model

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text: masses

text: are

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text: time,

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text: have

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text: time

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text: is

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text: (ν

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text: size

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text: ]VeM

text: )R(U

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text: M

text: nus

text: G

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text: 8 Ignazio Bombaci et al.: Quark matter nucleation in neutron stars and astrophysical implications Table 3. Critical mass and energy released in the conversion process of an HS into a QS. Results are relative to the GM1 4000 model with x = 0.7 for the hadronic phase and the extended σ 3500 T = 0 bag model EOS of Ref. [56–58] for the quark phase, using dif- T = 10 MeV ferent values for the eﬀective bag constant Beff and the per- 3000 T = 20 MeV turbative QCD correction term a . For the quark masses we 4 2500 use m = 100 MeV, m = m = 0. The value of the critical s u d gravitational (baryonic) mass of the HS sequence is reported on 2000 the column labeled M (M ), whereas those of the mass of cr B,cr 1500 the ﬁnal QS formed in the stellar conversion process of the crit- ical mass HS are shown on the column labeled Mfin. The col- 1000 umn labeled MQS denotes the maximum gravitational mass max 500 of the QS sequence. Finally the energy released in the stel- lUanrictosnovfeBrseioffn apnrodcσessaraereMsehVow/fnmo3natnhdeMcoelVum/fnml2abreeslpedecEticvoenlvy.. 00 1 2 3 4R [fm]5 6 7 8 9 All stellar masses are given in units of the mass of the sun, × Fig. 10. Energy barrier for a virtual drop of the Q*-phase M = 1.989 1033 g, and E is given in units of 1053 erg. sun conv in β-stable neutrino-free hadronic matter as a function of the The surface tension is σ = 10 MeV/fm2. droplet radius and for diﬀerent temperatures for a ﬁxed pres- sure P = 57 MeV/fm3. Results are relative to the GM1 –B85 B a M M M MQS E 0.6 eff 4 cr B,cr fin max conv equation of state and surface tension σ = 30 MeV/fm2. 37.63 0.65 2.003 2.312 1.848 2.318 2.77 50.72 0.65 2.033 2.354 1.954 2.013 1.41 6 Quark matter nucleation in proto-hadronic 41.14 0.70 1.891 2.158 1.731 2.229 2.86 47.20 0.70 1.940 2.225 1.814 2.088 2.25 stars A neutron star at birth (proto-neutron star) is very hot Table 4. Same as Table 3, but for a value of the surface tension (T = 10 – 30 MeV) with neutrinos being still trapped in σ = 30 MeV/fm2. the stellar interior [86,87,71,88–90]. Subsequent neutrino B a M M M MQS E diﬀusion causes deleptonization and heats the stellar mat- eff 4 cr B,cr fin max conv ter to an approximately uniform entropy per baryon S˜ = 1 37.63 0.65 2.021 2.328 1.859 2.318 2.90 – 2 (in units of the Boltzmann’s constant k ). Depending B 50.72 0.65 2.039 2.362 1.960 2.013 1.41 on the stellar composition, during this stage neutrino es- 41.14 0.70 1.995 2.301 1.832 2.229 2.91 cape can lead the more “massive” stellar conﬁgurations to 47.20 0.70 1.973 2.270 1.846 2.088 2.27 the formation of a black hole [91,71]. However, if the mass of the star is suﬃciently small, the star will remain stable and it will cool to temperatures well below 1 MeV within (a) if the nucleation time τ(MmHaSx) associated to the max- a cooling time tcool ∼ a few 102 s, as the neutrinos con- imum mass conﬁguration for the hadronic star sequence tinue to carry energy away from the stellar material [86, is of the same order or much larger than the age of the 71,88]. Thus in a proto-neutron star, the quark deconﬁne- universe T , then univ ment phase transition will be likely triggered by a ther- mal nucleation process [92–95]. In fact, for suﬃciently high M = MHS , (13) lim max temperatures, thermal nucleation is a much more eﬃcient process with respect to the quantum nucleation mecha- in other words, the limiting mass in this case coincides nism. with the Oppenheimer - Volkoﬀ maximum mass for the In Ref. [39,41] we established the physical conditions hadronic star sequence. (b) If the critical mass M is smaller than MHS (i.e. under which a newborn hadronic star (proto-hadronic star, cr max τ(MHS ) < 1 yr), thus the limiting mass for compact stars PHS) could survive the early stages of its evolution with- max out ”decaying” to a quark star. is equal to the largest value between the critical mass for According to the Langer theory [96,97] of homoge- the HS and the maximum m(cid:2) ass for the(cid:3)quark star (HyS neous nucleation the thermal nucleation rate can be writ- or SS) sequence ten [97] as Mlim = max Mcr , MmQaSx . (14) κ − R I = Ω exp( U( , T)/T) (16) 2π 0 c (c) Finally, one must consider an “intermediate” situation for which 1yr < τ(MHS )(cid:2) < T . As(cid:3)the reader can max univ where κ is the so-called dynamical prefactor, which easily realize, now R is related to the growth rate of the drop radius near R Mlim = max MmHaSx , MmQaSx , (15) the critical radius ( c), and Ω0 is the so-called statistical prefactor, which measures the phase-space volume of the R depending on the details of the EOS which could give saddle-point region around . We have used [39,41] for κ c MHS > MQS or vice versa. and Ω the expressions derived in Refs. [98,99], where the max max 0 ]VeM[ )R(U ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ignazio

text: Bombaci

text: et

text: al.:

text: Quark

text: matter

text: nucleation

text: in

text: neutron

text: stars

text: and

text: astrophysical

text: implications

text: Table

text: Critical

text: mass

text: and

text: energy

text: released

text: in

text: the

text: conversion

text: process

text: of

text: an

text: HS

text: into

text: a

text: QS.

text: Results

text: are

text: relative

text: to

text: the

text: model

text: with

text: x

text: for

text: the

text: hadronic

text: phase

text: and

text: the

text: extended

text: σ

text: T

text: bag

text: model

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text: Krings,

text: F.

text: Calabrese,

text: C.

text: Ratti,

text: and

text: V.D.

text: Blondel.

text: Urban

text: gravity:

text: a

text: model

text: for

text: inter-city

text: telecommunication

text: ﬂows.

text: J.

text: Stat.

text: Mech.

text: Theory

text: Exp.,

text: A.P.

text: Masucci,

text: J.

text: Serras,

text: A.

text: Johansson,

text: and

text: M.

text: Batty.

text: Gravity

text: versus

text: radiation

text: models:

text: On

text: the

text: importance

text: of

text: scale

text: and

text: heterogeneity

text: in

text: commuting

text: ﬂows.

text: Phys.

text: Rev.

text: E,

text: X.

text: Liang,

text: J.

text: Zhao,

text: L.

text: Dong,

text: and

text: K.

text: Xu.

text: Unraveling

text: the

text: origin

text: of

text: exponential

text: law

text: in

text: intra-urban

text: human

text: mobility.

text: Sci.

text: Rep.,

text: Y.

text: Yang,

text: C.

text: Herrera,

text: N.

text: Eagle,

text: and

text: M.C.

text: Gonz´alez.

text: Limits

text: of

text: predictability

text: in

text: commuting

text: ﬂows

text: in

text: the

text: absence

text: of

text: data

text: for

text: calibration.

text: Sci.

text: Rep.,

text: N.M.

text: Truscott,

text: J.

text: andAND

text: Ferguson.

text: Evaluating

text: the

text: adequacy

text: of

text: gravity

text: models

text: as

text: a

text: description

text: of

text: human

text: mobility

text: for

text: epidemic

text: modelling.

text: PLoS

text: Comput.

text: Biol.,

text: F.

text: Calabrese,

text: G.

text: Di

text: Lorenzo,

text: L.

text: Liu,

text: and

text: C.

text: Ratti.

text: Estimating

text: origin-destination

text: ﬂows

text: using

text: mobile

text: phone

text: location

text: data.

text: IEEE

text: Perv.

text: Comp.,

text: V.

text: Palchykov,

text: M.

text: Mitrovi´c,

text: H.

text: Jo,

text: J.

text: Saram¨aki,

text: and

text: R.K.

text: Pan.

text: Inferring

text: human

text: mobility

text: using

text: communication

text: patterns.

text: Sci.

text: Rep.,

text: L.

text: Alexander,

text: S.

text: Jiang,

text: M.

text: Murga,

text: and

text: M.C.

text: Gonz´alez.

text: Origin–destination

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text: and

text: time

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text: day

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text: from

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text: phone

text: data.

text: Transportation

text: Research

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text: C:

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text: Technologies,

text: M.

text: Tizzoni,

text: P.

text: Bajardi,

text: A.

text: Decuyper,

text: G.

text: Kon

text: Kam

text: King,

text: C.M.

text: Schneider,

text: V.

text: Blondel,

text: Z.

text: Smoreda,

text: M.C.

text: Gonz´alez,

text: and

text: V.

text: Colizza.

text: On

text: the

text: use

text: of

text: human

text: mobility

text: proxies

text: for

text: modeling

text: epidemics.

text: PLoS

text: Comput

text: Biol,

text: L.

text: Bengtsson,

text: J.

text: Gaudart,

text: X.

text: Lu,

text: S.

text: Moore,

text: E.

text: Wetter,

text: K.

text: Sallah,

text: S.

text: Rebaudet,

text: and

text: R.

text: Piarroux.

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text: data

text: to

text: predict

text: the

text: spatial

text: spread

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text: cholera.

text: Sci.

text: Rep.,

text: S.

text: Riley,

text: K.

text: Eames,

text: V.

text: Isham,

text: D.

text: Mollison,

text: and

text: P.

text: Trapman.

text: Five

text: challenges

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text: Epidemics,

text: B.

text: Gon¸calves

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text: Perra.

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text: Models.

text: Springer,

text: Flickr.

text: Twitter.

text: Foursquare.

text: A.

text: Noulas,

text: S.

text: Scellato,

text: C.

text: Mascolo,

text: and

text: M.

text: Pontil.

text: An

text: empirical

text: study

text: of

text: geographic

text: user

text: activity

text: patterns

text: in

text: foursquare.

text: In

text: Proc.

text: of

text: the

text: Int’l

text: AAAI

text: Conference

text: on

text: Weblogs

text: and

text: Social

text: Media,

text: pages

text: M.

text: Lenormand,

text: M.

text: Picornell,

text: O.G.

text: Cantu´-Ros,

text: A.

text: Tugores,

text: T.

text: Louail,

text: R.

text: Herranz,

text: M.

text: Barthelemy,

text: E.

text: Fr´ıas-

text: Mart´ınez,

text: and

text: J.J.

text: Ramasco.

text: Cross-checking

text: diﬀerent

text: sources

text: of

text: mobility

text: information.

text: PLoS

text: ONE,

text: P.A.

text: Grabowicz,

text: J.J.

text: Ramasco,

text: B.

text: Gon¸calves,

text: and

text: V.M.

text: Egu´ıluz.

text: Entangling

text: mobility

text: and

text: interactions

text: in

text: social

text: media.

text: PLoS

text: ONE,

text: A.

text: Llorente,

text: M.

text: Garcia-Herranz,

text: M.

text: Cebrian,

text: and

text: E.

text: Moro.

text: Social

text: media

text: ﬁngerprints

text: of

text: unemployment.

text: PloS

text: one,

text: D.

text: Barchiesi,

text: T.

text: Preis,

text: S.

text: Bishop,

text: and

text: H.S.

text: Moat.

text: Modelling

text: human

text: mobility

text: patterns

text: using

text: photographic

text: data

text: shared

text: online.

text: Royal

text: Society

text: Open

text: Science,

text: B.

text: Thomee,

text: D.A.

text: Shamma,

text: G.

text: Friedland,

text: B.

text: Elizalde,

text: K.

text: Ni,

text: D.

text: Poland,

text: D.

text: Borth,

text: and

text: L.

text: Li.

text: The

text: new

text: data

text: and

text: new

text: challenges

text: in

text: multimedia

text: research.

text: arXiv

text: preprint


text: total number of users who travelled between i and j. The distance between the locations of two consecutive pictures taken by the same user ranges from a few meters to thousands of kilometers, showing a heavy-tailed behavior with an exponential cutoﬀ (see the Supplementary material for more details). Thus, it is clear that users’ traces can provide information about very diﬀerent types of human displacements, ranging from a short walk within a city to long distance trips or international travel. In each case, and when modelling a particular type of mobility, it is important to consider only the relevant traces for that type of movements, otherwise, the eﬀects of diﬀerent activities will be mixed. We analyzed the eﬀect of distance on both the daily commuting and the air travel, and we set a maximum trip distance of 100 km for daily commuting in the U.S. and a minimum trip distance of 500 km for U.S. air travel. The analysis is described in Methods: Distance thresholds. After aggregating the trips by basin and ﬁltering by distance, we obtain the Flickr ﬂow matrices denoted as F = (fr ) for the air travel network, and F = (fc ) for the commuting network. Here, fc and fr represent the r ij c ij ij ij number of Flickr users travelling from basin i to basin j in each of the networks. Model deﬁnition and ﬁtting. A model of human mobility should be able to predict real mobility ﬂows. The aim of our work was to predict mobility ﬂows of the air travel network and the U.S. commuting network, denoted as Y = (r ) and Y = (c ) respectively. Here, r represents the number of travellers between two airports i r ij c ij ij and j, while c represents the daily amount of commuters between any two counties i and j. Adopting a machine ij learning approach, we represent this prediction problem as a regression task in which the model is ﬁrst trained, and then it is used to estimate the real ﬂows, Y or Y , which are the so called target values. r c The classical gravity model (described in the Methods) estimates the target values as directly proportional to some powers of the population of the origin and destination basins and inversely proportional to some increasing function of the distance between them (typically, a power-law or exponential function). We will indicate this model P P as G(α, β, γ; ), where α, β and γ are the three exponents in the gravity law, and = (p ) is a vector containing i the populations of the basins. Analogously, travel ﬂows estimated from Flickr traces can be described by a model in which we assume that the value y , representing the mobility ﬂow between two basins i and j, is proportional to the number of users’ trips ij · from i to j. We represent this model as C F, where F represents the Flickr ﬂow matrix at the corresponding F resolution level (i.e., F or F ), and C is a constant. r c F Our approach is to combine the ﬂows of human mobility estimated from the two models under a stacked regression procedure [44] in which each model is ﬁtted alone, and then a linear regression determines the weight of each of them. In this way we combine the Flickr traces and the gravity model to improve the prediction of the real mobility patterns. The incorporation of the traces will be deﬁned by a linear function (we omit the subscript indices to generalize to any of the two resolution levels): P · P · H(α, β, γ, A, B; ) = A G(α, β, γ; ) + B F, where H = (h ) represents our hybrid model, G is the gravity model and F is the Flickr ﬂows matrix; α, β, γ, A, B ij are real-valued ﬁtting parameters. We will ﬁt the model by minimizing the following loss function L: (cid:107) − P (cid:107) L = Y H(α, β, γ, A, B; ) , (cid:107) · (cid:107) where denotes the Frobenius norm. Again, we removed the subscript indices, so that this equation represents what is done at each resolution level. The minimization process is made under the stacked regression assumption that each of the two component models can be ﬁtted alone, and then a linear regression determines the weight of each of them. It is important to notice that, as Y and Y are sparse matrices, we only evaluate L for those r c connections for which there is a positive ﬂow of travellers. Model evaluation. The performance of a learning model is measured by its capacity to generalize to ﬂows that were not known during the training step. Then, in order to validate our model we used a cross-validation strategy in which we ﬁtted the model using only part of the target ﬂows Y (training set) and then we tested its performance using the remaining ﬂows (test set). We measured the performance in terms of Pearson correlation coeﬃcient ρ and the determination coeﬃcient r2 between the target ﬂows and the predicted ﬂows. Our procedure follows a 10-fold cross-validation scheme [45]: the real dataset is divided into 10 parts or folds, and for each fold we use the 10% of the target values as test set, and the remaining 90% as training set. Thus,

text: total

text: number

text: of

text: users

text: who

text: travelled

text: between

text: i

text: and

text: j.

text: The

text: distance

text: between

text: the

text: locations

text: of

text: two

text: consecutive

text: pictures

text: taken

text: by

text: the

text: same

text: user

text: ranges

text: from

text: a

text: few

text: meters

text: to

text: thousands

text: of

text: kilometers,

text: showing

text: a

text: heavy-tailed

text: behavior

text: with

text: an

text: exponential

text: cutoﬀ

text: (see

text: the

text: Supplementary

text: material

text: for

text: more

text: details).

text: Thus,

text: it

text: is

text: clear

text: that

text: users’

text: traces

text: can

text: provide

text: information

text: about

text: very

text: diﬀerent

text: types

text: of

text: human

text: displacements,

text: ranging

text: from

text: a

text: short

text: walk

text: within

text: a

text: city

text: to

text: long

text: distance

text: trips

text: or

text: international

text: travel.

text: In

text: each

text: case,

text: and

text: when

text: modelling

text: a

text: particular

text: type

text: of

text: mobility,

text: it

text: is

text: important

text: to

text: consider

text: only

text: the

text: relevant

text: traces

text: for

text: that

text: type

text: of

text: movements,

text: otherwise,

text: the

text: eﬀects

text: of

text: diﬀerent

text: activities

text: will

text: be

text: mixed.

text: We

text: analyzed

text: the

text: eﬀect

text: of

text: distance

text: on

text: both

text: the

text: daily

text: commuting

text: and

text: the

text: air

text: travel,

text: and

text: we

text: set

text: a

text: maximum

text: trip

text: distance

text: of

text: km

text: for

text: daily

text: commuting

text: in

text: the

text: U.S.

text: and

text: a

text: minimum

text: trip

text: distance

text: of

text: km

text: for

text: U.S.

text: air

text: travel.

text: The

text: analysis

text: is

text: described

text: in

text: Methods:

text: Distance

text: thresholds.

text: After

text: aggregating

text: the

text: trips

text: by

text: basin

text: and

text: ﬁltering

text: by

text: distance,

text: we

text: obtain

text: the

text: Flickr

text: ﬂow

text: matrices

text: denoted

text: as

text: F

text: (fr

text: for

text: the

text: air

text: travel

text: network,

text: and

text: F

text: (fc

text: for

text: the

text: commuting

text: network.

text: Here,

text: fc

text: and

text: fr

text: represent

text: the

text: r

text: ij

text: c

text: ij

text: ij

text: ij

text: number

text: of

text: Flickr

text: users

text: travelling

text: from

text: basin

text: i

text: to

text: basin

text: j

text: in

text: each

text: of

text: the

text: networks.

text: Model

text: deﬁnition

text: and

text: ﬁtting.

text: A

text: model

text: of

text: human

text: mobility

text: should

text: be

text: able

text: to

text: predict

text: real

text: mobility

text: ﬂows.

text: The

text: aim

text: of

text: our

text: work

text: was

text: to

text: predict

text: mobility

text: ﬂows

text: of

text: the

text: air

text: travel

text: network

text: and

text: the

text: U.S.

text: commuting

text: network,

text: denoted

text: as

text: Y

text: (r

text: and

text: Y

text: (c

text: respectively.

text: Here,

text: r

text: represents

text: the

text: number

text: of

text: travellers

text: between

text: two

text: airports

text: i

text: r

text: ij

text: c

text: ij

text: ij

text: and

text: j,

text: while

text: c

text: represents

text: the

text: daily

text: amount

text: of

text: commuters

text: between

text: any

text: two

text: counties

text: i

text: and

text: j.

text: Adopting

text: a

text: machine

text: ij

text: learning

text: approach,

text: we

text: represent

text: this

text: prediction

text: problem

text: as

text: a

text: regression

text: task

text: in

text: which

text: the

text: model

text: is

text: ﬁrst

text: trained,

text: and

text: then

text: it

text: is

text: used

text: to

text: estimate

text: the

text: real

text: ﬂows,

text: Y

text: or

text: Y

text: which

text: are

text: the

text: so

text: called

text: target

text: values.

text: r

text: c

text: The

text: classical

text: gravity

text: model

text: (described

text: in

text: the

text: Methods)

text: estimates

text: the

text: target

text: values

text: as

text: directly

text: proportional

text: to

text: some

text: powers

text: of

text: the

text: population

text: of

text: the

text: origin

text: and

text: destination

text: basins

text: and

text: inversely

text: proportional

text: to

text: some

text: increasing

text: function

text: of

text: the

text: distance

text: between

text: them

text: (typically,

text: a

text: power-law

text: or

text: exponential

text: function).

text: We

text: will

text: indicate

text: this

text: model

text: P

text: P

text: as

text: G(α,

text: β,

text: γ;

text: where

text: α,

text: β

text: and

text: γ

text: are

text: the

text: three

text: exponents

text: in

text: the

text: gravity

text: law,

text: and

text: (p

text: is

text: a

text: vector

text: containing

text: i

text: the

text: populations

text: of

text: the

text: basins.

text: Analogously,

text: travel

text: ﬂows

text: estimated

text: from

text: Flickr

text: traces

text: can

text: be

text: described

text: by

text: a

text: model

text: in

text: which

text: we

text: assume

text: that

text: the

text: value

text: y

text: representing

text: the

text: mobility

text: ﬂow

text: between

text: two

text: basins

text: i

text: and

text: j,

text: is

text: proportional

text: to

text: the

text: number

text: of

text: users’

text: trips

text: ij

text: from

text: i

text: to

text: j.

text: We

text: represent

text: this

text: model

text: as

text: C

text: F,

text: where

text: F

text: represents

text: the

text: Flickr

text: ﬂow

text: matrix

text: at

text: the

text: corresponding

text: F

text: resolution

text: level

text: (i.e.,

text: F

text: or

text: F

text: and

text: C

text: is

text: a

text: constant.

text: r

text: c

text: F

text: Our

text: approach

text: is

text: to

text: combine

text: the

text: ﬂows

text: of

text: human

text: mobility

text: estimated

text: from

text: the

text: two

text: models

text: under

text: a

text: stacked

text: regression

text: procedure

text: in

text: which

text: each

text: model

text: is

text: ﬁtted

text: alone,

text: and

text: then

text: a

text: linear

text: regression

text: determines

text: the

text: weight

text: of

text: each

text: of

text: them.

text: In

text: this

text: way

text: we

text: combine

text: the

text: Flickr

text: traces

text: and

text: the

text: gravity

text: model

text: to

text: improve

text: the

text: prediction

text: of

text: the

text: real

text: mobility

text: patterns.

text: The

text: incorporation

text: of

text: the

text: traces

text: will

text: be

text: deﬁned

text: by

text: a

text: linear

text: function

text: (we

text: omit

text: the

text: subscript

text: indices

text: to

text: generalize

text: to

text: any

text: of

text: the

text: two

text: resolution

text: levels):

text: P

text: P

text: H(α,

text: β,

text: γ,

text: A,

text: B;

text: A

text: G(α,

text: β,

text: γ;

text: B

text: F,

text: where

text: H

text: (h

text: represents

text: our

text: hybrid

text: model,

text: G

text: is

text: the

text: gravity

text: model

text: and

text: F

text: is

text: the

text: Flickr

text: ﬂows

text: matrix;

text: α,

text: β,

text: γ,

text: A,

text: B

text: ij

text: are

text: real-valued

text: ﬁtting

text: parameters.

text: We

text: will

text: ﬁt

text: the

text: model

text: by

text: minimizing

text: the

text: following

text: loss

text: function

text: L:

text: P

text: L

text: Y

text: H(α,

text: β,

text: γ,

text: A,

text: B;

text: where

text: denotes

text: the

text: Frobenius

text: norm.

text: Again,

text: we

text: removed

text: the

text: subscript

text: indices,

text: so

text: that

text: this

text: equation

text: represents

text: what

text: is

text: done

text: at

text: each

text: resolution

text: level.

text: The

text: minimization

text: process

text: is

text: made

text: under

text: the

text: stacked

text: regression

text: assumption

text: that

text: each

text: of

text: the

text: two

text: component

text: models

text: can

text: be

text: ﬁtted

text: alone,

text: and

text: then

text: a

text: linear

text: regression

text: determines

text: the

text: weight

text: of

text: each

text: of

text: them.

text: It

text: is

text: important

text: to

text: notice

text: that,

text: as

text: Y

text: and

text: Y

text: are

text: sparse

text: matrices,

text: we

text: only

text: evaluate

text: L

text: for

text: those

text: r

text: c

text: connections

text: for

text: which

text: there

text: is

text: a

text: positive

text: ﬂow

text: of

text: travellers.

text: Model

text: evaluation.

text: The

text: performance

text: of

text: a

text: learning

text: model

text: is

text: measured

text: by

text: its

text: capacity

text: to

text: generalize

text: to

text: ﬂows

text: that

text: were

text: not

text: known

text: during

text: the

text: training

text: step.

text: Then,

text: in

text: order

text: to

text: validate

text: our

text: model

text: we

text: used

text: a

text: cross-validation

text: strategy

text: in

text: which

text: we

text: ﬁtted

text: the

text: model

text: using

text: only

text: part

text: of

text: the

text: target

text: ﬂows

text: Y

text: (training

text: set)

text: and

text: then

text: we

text: tested

text: its

text: performance

text: using

text: the

text: remaining

text: ﬂows

text: (test

text: set).

text: We

text: measured

text: the

text: performance

text: in

text: terms

text: of

text: Pearson

text: correlation

text: coeﬃcient

text: ρ

text: and

text: the

text: determination

text: coeﬃcient

text: between

text: the

text: target

text: ﬂows

text: and

text: the

text: predicted

text: ﬂows.

text: Our

text: procedure

text: follows

text: a

text: cross-validation

text: scheme

text: the

text: real

text: dataset

text: is

text: divided

text: into

text: parts

text: or

text: folds,

text: and

text: for

text: each

text: fold

text: we

text: use

text: the

text: of

text: the

text: target

text: values

text: as

text: test

text: set,

text: and

text: the

text: remaining

text: as

text: training

text: set.

text: Thus,