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text: Extraction

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text: Knowledge.

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text: Conference

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text: Xin

text: Jiang,

text: Yunhua

text: Hu,

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text: Hang

text: Li.

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text: Ranking

text: Network

text: Diversity

text: and

text: Economic

text: Development.

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text: Extraction.

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text: Science,

text: May.

text: of

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text: international

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text: SIGIR

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text: and

text: development

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text: information

text: retrieval,

text: Gonenc

text: Ercan

text: and

text: Ilyas

text: Cicekli.

text: Using

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text: age.,

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text: Status

text: Index

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text: Felice

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text: Paper

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text: Automatic

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text: Articles.

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text: International

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text: Paynter,

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text: 6 JEAN GILLIBERT, CHRISTIAN RETORE´ (cid:96) axiom z1:Z1,...,zp:Zp t:B exchange (cid:96) (cid:96) x:A x:A zσ(1):Zσ(1),...,zσ(p):Zσ(p) t:B (σ:permutation) (cid:96) z1:Z1,...,zp:Zp,x:A,x:A t:B contraction (cid:96) z1:Z1,...,zp:Zp,x:A t:B (cid:96) z1:Z1,...,zp:Zp, t:B weakening (cid:96) z1:Z1,...,zp:Zp,x:A t:B (cid:96) z1:Z1,...,zp:Zp,x:A t:B (cid:96) → abstraction / hypothetical reasoning Γ,x:A (λxA.t):A B (cid:96) → (cid:96) z1:Z1,...,zp:Zp, f:A B y1:Y1,...,yk:Yk u:A (cid:96) application / modus ponens z1:Z1,...,zp:Zp,y1:Y1,...,yk:Yk f(u):B FIGURE 1. The Curry-Howard isomorphism [22]. Upper case letters (cid:96) stand for propositions. On the left handside of “ ” one only has type assignments for variables, like x : X. On the right hand side, before the “:” there is a lambda term which encodes the proofs. Leaving out terms, that are on the left hand side of “:” one get a standard natural deduction system for proposition logic as introduced by Gentzen. [15, 16] These rules infer a sequent from one or two sequents a sequent being (cid:96) a sequence of hypo·t·h·eses→and a formula on the right hand side, Z1,..,Zp C meaning: (Z1& Zp) C. inside a topos, and there exists a classifying topos whose category of T models is universal. This is a point-free manifestation of the “space of models of T ”. 4. INTUITIONISTIC PROOFS, TYPED LAMBDA CALCULUS AND CARTESIAN CLOSED CATEGORIES A second link between intutionistic logic and category theory is the following. Proofs in (propositional) intuitionistic logic can be viewed as typed lambda terms, that are functional → programs. The main idea is that a proof of A B maps proofs of A to proofs of B, hence ∧ → it is rather natural to interpret a formula (with , ) by the sets of its proofs. Functions can be applied to arguments of the right type, and composed. To compute such a term, one substitutes the occurrences of the bound variable used to deﬁne the function with the argument of the function: (λxA. tB)(uA) (cid:32)β t[xA: = uA] Values are normal terms i.e. terms that cannot be reduced any further. In ﬁgure we give the proof rules and the corresponding proof terms, that are simply typed lambda terms, a good reference being [20] and [25] on the categorical aspects. Functions in this setting are algorithms (that always converge) that represent total recur- sive functions, the ones whose totality can be proved within the underlying logical system. They are deﬁned intentionally, how could one view them as usual functions? Functions can be viewed as morphisms in a cartesian closed category, cf. deﬁnition 1 above — in this → logical setting, BA is rather written A B. Observe that the logical rules match categorical ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: JEAN

text: GILLIBERT,

text: CHRISTIAN

text: RETORE´

text: axiom

text: t:B

text: exchange

text: x:A

text: x:A

text: t:B

text: (σ:permutation)

text: t:B

text: contraction

text: t:B

text: t:B

text: weakening

text: t:B

text: t:B

text: abstraction

text: hypothetical

text: reasoning

text: Γ,x:A

text: (λxA.t):A

text: B

text: f:A

text: B

text: u:A

text: application

text: modus

text: ponens

text: f(u):B

text: FIGURE

text: The

label: caption

text: Curry-Howard

text: isomorphism

text: Upper

text: case

text: letters

text: stand

text: for

text: propositions.

text: On

text: the

text: left

text: handside

text: of

text: one

text: only

text: has

text: type

text: assignments

text: for

text: variables,

text: like

text: x

text: X.

text: On

text: the

text: right

text: hand

text: side,

text: before

text: the

text: there

text: is

text: a

text: lambda

text: term

text: which

text: encodes

text: the

text: proofs.

text: Leaving

text: out

text: terms,

text: that

text: are

text: on

text: the

text: left

text: hand

text: side

text: of

text: one

text: get

text: a

text: standard

text: natural

text: deduction

text: system

text: for

text: proposition

text: logic

text: as

text: introduced

text: by

text: Gentzen.

text: These

text: rules

text: infer

text: a

text: sequent

text: from

text: one

text: or

text: two

text: sequents

text: a

text: sequent

text: being

text: a

text: sequence

text: of

text: hypo·t·h·eses→and

text: a

text: formula

text: on

text: the

text: right

text: hand

text: side,

text: C

text: meaning:

text: Zp)

text: C.

text: inside

text: a

text: topos,

text: and

text: there

text: exists

text: a

text: classifying

text: topos

text: whose

text: category

text: of

text: T

text: models

text: is

text: universal.

text: This

text: is

text: a

text: point-free

text: manifestation

text: of

text: the

text: “space

text: of

text: models

text: of

text: T

text: INTUITIONISTIC

text: PROOFS,

text: TYPED

text: LAMBDA

text: CALCULUS

text: AND

text: CARTESIAN

text: CLOSED

text: CATEGORIES

text: A

text: second

text: link

text: between

text: intutionistic

text: logic

text: and

text: category

text: theory

text: is

text: the

text: following.

text: Proofs

text: in

text: (propositional)

text: intuitionistic

text: logic

text: can

text: be

text: viewed

text: as

text: typed

text: lambda

text: terms,

text: that

text: are

text: functional

text: programs.

text: The

text: main

text: idea

text: is

text: that

text: a

text: proof

text: of

text: A

text: B

text: maps

text: proofs

text: of

text: A

text: to

text: proofs

text: of

text: B,

text: hence

text: it

text: is

text: rather

text: natural

text: to

text: interpret

text: a

text: formula

text: (with

text: by

text: the

text: sets

text: of

text: its

text: proofs.

text: Functions

text: can

text: be

text: applied

text: to

text: arguments

text: of

text: the

text: right

text: type,

text: and

text: composed.

text: To

text: compute

text: such

text: a

text: term,

text: one

text: substitutes

text: the

text: occurrences

text: of

text: the

text: bound

text: variable

text: used

text: to

text: deﬁne

text: the

text: function

text: with

text: the

text: argument

text: of

text: the

text: function:

text: (λxA.

text: tB)(uA)

text: t[xA:

text: uA]

text: Values

text: are

text: normal

text: terms

text: i.e.

text: terms

text: that

text: cannot

text: be

text: reduced

text: any

text: further.

text: In

text: ﬁgure

text: we

text: give

text: the

text: proof

text: rules

text: and

text: the

text: corresponding

text: proof

text: terms,

text: that

text: are

text: simply

text: typed

text: lambda

text: terms,

text: a

text: good

text: reference

text: being

text: and

text: on

text: the

text: categorical

text: aspects.

text: Functions

text: in

text: this

text: setting

text: are

text: algorithms

text: (that

text: always

text: converge)

text: that

text: represent

text: total

text: recur-

text: sive

text: functions,

text: the

text: ones

text: whose

text: totality

text: can

text: be

text: proved

text: within

text: the

text: underlying

text: logical

text: system.

text: They

text: are

text: deﬁned

text: intentionally,

text: how

text: could

text: one

text: view

text: them

text: as

text: usual

text: functions?

text: Functions

text: can

text: be

text: viewed

text: as

text: morphisms

text: in

text: a

text: cartesian

text: closed

text: category,

text: cf.

text: deﬁnition

text: above

text: in

text: this

text: logical

text: setting,

text: BA

text: is

text: rather

text: written

text: A

text: B.

text: Observe

text: that

text: the

text: logical

text: rules

text: match

text: categorical

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Prepared for submission to JCAP Cusps and Cores in the presence of galactic bulges A. Del Popoloa,b N. Hiotelisc aDipartimento di Fisica e Astronomia, University Of Catania, Viale Andrea Doria 6, 95125 Catania, Italy bInternational Institute of Physics, Universidade Federal do Rio Grande do Norte, 59012-970 Natal, Brazil c1st Experimental Lyceum of Athens, Ipitou 15, Plaka, GR-10557 Athens, Greece E-mail: adelpopolo@oact.inaf.it Abstract. In this paper, we study how the presence of bulge formation in galaxies inﬂuence their inner density proﬁle, by means of an extended version of the Del Popolo (2009) semi- analytical model. As in Del Popolo (2009), the model takes into account the eﬀect of baryons adiabatic contraction, ordered and random angular momentum, dynamical friction, and adds to the previous the eﬀect of gas cooling, star formation, supernova feedback, and reionization. Our model shows that dwarf galaxies are bulgeless, in agreement with observations showing that the large majority of them has no stellar bulges, and are characterized by a ﬂat proﬁle well described by a Burkert proﬁle. We then studied the eﬀect of a bulge, added to the cored DM halo, on the density proﬁle. In the case of a galaxy having a mass 1011M(cid:12) the inner (cid:39) × density proﬁle has a slope α 0.65, for a bulge of 4.5 109M(cid:12) , while if bulge formation is (cid:39) × not considered, the slope would be α 0.55. If the bulge is larger, 6.5 109M(cid:12) the slope is (cid:39) (cid:39) α 0.7. In the case of a larger galaxy with 1012M(cid:12) the slope is α 0.85, while in absence (cid:39) of bulge it is α 0.75. We ﬁnally study how the inner slope α changes with the bulge mass, and we ﬁnd a correlation among the two quantities. The result shows that bulge formation has an important role in shaping the inner DM density proﬁle in agreement with Inoue & Saitoh (2011) result. The result implies that going from Sc to SO Hubble type the slope is slightly steepening due to the bulge formation, and due to the fact that early type galaxies have larger bulges. Keywords: galaxies, formation

text: Prepared

text: for

text: submission

text: to

text: JCAP

text: Cusps

label: title

text: and

text: Cores

text: in

text: the

text: presence

text: of

text: galactic

text: bulges

text: A.

label: author

text: Del

text: Popoloa,b

text: N.

text: Hiotelisc

text: aDipartimento

text: di

text: Fisica

text: e

text: Astronomia,

text: University

text: Of

text: Catania,

text: Viale

text: Andrea

text: Doria

text: Catania,

text: Italy

text: bInternational

text: Institute

text: of

text: Physics,

text: Universidade

text: Federal

text: do

text: Rio

text: Grande

text: do

text: Norte,

text: Natal,

text: Brazil

text: Experimental

text: Lyceum

text: of

text: Athens,

text: Ipitou

text: Plaka,

text: Athens,

text: Greece

text: E-mail:

text: adelpopolo@oact.inaf.it

text: Abstract.

text: In

label: abstract

text: this

text: paper,

text: we

text: study

text: how

text: the

text: presence

text: of

text: bulge

text: formation

text: in

text: galaxies

text: inﬂuence

text: their

text: inner

text: density

text: proﬁle,

text: by

text: means

text: of

text: an

text: extended

text: version

text: of

text: the

text: Del

text: Popolo

text: semi-

text: analytical

text: model.

text: As

text: in

text: Del

text: Popolo

text: the

text: model

text: takes

text: into

text: account

text: the

text: eﬀect

text: of

text: baryons

text: adiabatic

text: contraction,

text: ordered

text: and

text: random

text: angular

text: momentum,

text: dynamical

text: friction,

text: and

text: adds

text: to

text: the

text: previous

text: the

text: eﬀect

text: of

text: gas

text: cooling,

text: star

text: formation,

text: supernova

text: feedback,

text: and

text: reionization.

text: Our

text: model

text: shows

text: that

text: dwarf

text: galaxies

text: are

text: bulgeless,

text: in

text: agreement

text: with

text: observations

text: showing

text: that

text: the

text: large

text: majority

text: of

text: them

text: has

text: no

text: stellar

text: bulges,

text: and

text: are

text: characterized

text: by

text: a

text: ﬂat

text: proﬁle

text: well

text: described

text: by

text: a

text: Burkert

text: proﬁle.

text: We

text: then

text: studied

text: the

text: eﬀect

text: of

text: a

text: bulge,

text: added

text: to

text: the

text: cored

text: DM

text: halo,

text: on

text: the

text: density

text: proﬁle.

text: In

text: the

text: case

text: of

text: a

text: galaxy

text: having

text: a

text: mass

text: the

text: inner

text: density

text: proﬁle

text: has

text: a

text: slope

text: α

text: for

text: a

text: bulge

text: of

text: while

text: if

text: bulge

text: formation

text: is

text: not

text: considered,

text: the

text: slope

text: would

text: be

text: α

text: If

text: the

text: bulge

text: is

text: larger,

text: the

text: slope

text: is

text: α

text: In

text: the

text: case

text: of

text: a

text: larger

text: galaxy

text: with

text: the

text: slope

text: is

text: α

text: while

text: in

text: absence

text: of

text: bulge

text: it

text: is

text: α

text: We

text: ﬁnally

text: study

text: how

text: the

text: inner

text: slope

text: α

text: changes

text: with

text: the

text: bulge

text: mass,

text: and

text: we

text: ﬁnd

text: a

text: correlation

text: among

text: the

text: two

text: quantities.

text: The

text: result

text: shows

text: that

text: bulge

text: formation

text: has

text: an

text: important

text: role

text: in

text: shaping

text: the

text: inner

text: DM

text: density

text: proﬁle

text: in

text: agreement

text: with

text: Inoue

text: Saitoh

text: result.

text: The

text: result

text: implies

text: that

text: going

text: from

text: Sc

text: to

text: SO

text: Hubble

text: type

text: the

text: slope

text: is

text: slightly

text: steepening

text: due

text: to

text: the

text: bulge

text: formation,

text: and

text: due

text: to

text: the

text: fact

text: that

text: early

text: type

text: galaxies

text: have

text: larger

text: bulges.

text: Keywords:

text: galaxies,

text: formation


text: (“shell-crossing”), and as a consequence energy will no longer be an integral of motion. In order to go on with the calculation, studying the shells dynamics, and to ﬁnally obtain the ﬁnal density proﬁle, we assumed that the central potential varies adiabatically, as is usually (cid:20) (cid:21) done (Gunn 1977; Filmore & Goldreich 1984), such that we obtained the proﬁle − ρ (x ) d ln f(x ) 1 ta m i ρ(x) = 1 + (2.1) f(x )3 d ln g(x ) i i being f(x ) = x/x (x ) the collapse factor, x the initial radius, and x (x ) the turn- i m i i m i around radius, given by 1 + δ i x = g(x ) = x − − − (2.2) m i i δ (Ω 1 1) i i where δ indicates the average overdensity in the considered shell, and Ω the density i i parameter. The baryonic fraction was obtained as in McGaugh et al. (2010), f = (M /M )/f = F /f (2.3) d b 500 b b b ± being f = 0.17 0.01 (Komatsu et al. 2009) the universal baryon fraction, f the b d detected baryon fraction, namely the ratio among the known baryons to the amount of baryons that one expects from the cosmic baryon fraction, and F = M /M 5 the baryonic (cid:16) (cid:17) b b 500 fraction. The random angular momentum, j, is expressed in terms of the ratio of the pericentric, r , to apocentric radii, r , e = rmin (Avila-Reese et al. 1998), taking also into account min max rmax the fact that eccentricity is a function of the dynamical state of the system (cid:39) e(r ) 0.8(r /r )0.1 (2.4) max max ta for r < 0.1r , as shown by Ascasibar, Yepes & G¨ottleber (2004). max ta The “ordered angular momentum”, h, arising from tidal torques of large scale structure on the proto-structure, is obtained through the tidal torque theory (TTT) (Hoyle 1953; Peebles 1969; White 1984; Ryden 1988; Eisenstein & Loeb 1995). The steepening eﬀect produced by the DM compression caused by baryons collapse, the so-called adiabatic contraction (AC), was calculated according to the Gnedin et al. (2004) prescriptions. As usual, it is assumed that the density proﬁle of DM and baryons is the same (Mo et al. 1998; Keeton 2001; Treu & Koopmans 2002; Cardone & Sereno 2005; IS11), and it is given by a NFW proﬁle. The ﬁnal distribution of baryons is assumed to be a disk (Blumenthal et al. 1986; Flores et al. 1993; Mo et al. 1998; Klypin et al. 2002; Cardone & Sereno 2005). For precision’s sake, in our calculations, we shall assume Klypin et al. (2002) model for the baryons distribution (see their Section 2.1), and their parameters: disc radial − scale-length r = 3 3.5 kpc, and vertical scale-length z = 400 pc. The bulge, having a d 0 5M denoted the mass in a radius R , within which the mean overdensity is 500 times larger than the 500 500 critical density ρ . The virial mass M (see the following), can be converted to M using White (2001), c vir 500 Hu & Kravtsov (2003) and Lukic et al. (2009) method. – 3 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (“shell-crossing”),

text: and

text: as

text: a

text: consequence

text: energy

text: will

text: no

text: longer

text: be

text: an

text: integral

text: of

text: motion.

text: In

text: order

text: to

text: go

text: on

text: with

text: the

text: calculation,

text: studying

text: the

text: shells

text: dynamics,

text: and

text: to

text: ﬁnally

text: obtain

text: the

text: ﬁnal

text: density

text: proﬁle,

text: we

text: assumed

text: that

text: the

text: central

text: potential

text: varies

text: adiabatically,

text: as

text: is

text: usually

label: equation

text: done

text: (Gunn

text: Filmore

text: Goldreich

text: such

text: that

text: we

text: obtained

text: the

text: proﬁle

text: ρ

text: (x

text: d

text: ln

text: f(x

text: ta

text: m

text: i

text: ρ(x)

text: f(x

text: d

text: ln

text: g(x

text: i

text: i

text: being

text: f(x

text: x/x

text: (x

text: the

text: collapse

text: factor,

text: x

text: the

text: initial

text: radius,

text: and

text: x

text: (x

text: the

text: turn-

text: i

text: m

text: i

text: i

text: m

text: i

text: around

text: radius,

text: given

text: by

text: δ

text: i

text: x

text: g(x

text: x

text: m

text: i

text: i

text: δ

text: (Ω

text: i

text: i

text: where

text: δ

text: indicates

text: the

text: average

text: overdensity

text: in

text: the

text: considered

text: shell,

text: and

text: Ω

text: the

text: density

text: i

text: i

text: parameter.

text: The

text: baryonic

text: fraction

text: was

text: obtained

text: as

text: in

text: McGaugh

text: et

text: al.

text: f

text: (M

text: /M

text: )/f

text: F

text: /f

text: d

text: b

text: b

text: b

text: b

text: being

text: f

text: (Komatsu

text: et

text: al.

text: the

text: universal

text: baryon

text: fraction,

text: f

text: the

text: b

text: d

text: detected

text: baryon

text: fraction,

text: namely

text: the

text: ratio

text: among

text: the

text: known

text: baryons

text: to

text: the

text: amount

text: of

text: baryons

text: that

text: one

text: expects

text: from

text: the

text: cosmic

text: baryon

text: fraction,

text: and

text: F

text: M

text: /M

text: the

text: baryonic

text: b

text: b

text: fraction.

text: The

text: random

text: angular

text: momentum,

text: j,

text: is

text: expressed

text: in

text: terms

text: of

text: the

text: ratio

text: of

text: the

text: pericentric,

text: r

text: to

text: apocentric

text: radii,

text: r

text: e

text: rmin

text: (Avila-Reese

text: et

text: al.

text: taking

text: also

text: into

text: account

text: min

text: max

text: rmax

text: the

text: fact

text: that

text: eccentricity

text: is

text: a

text: function

text: of

text: the

text: dynamical

text: state

text: of

text: the

text: system

text: e(r

text: /r

text: max

text: max

text: ta

text: for

text: r

text: as

text: shown

text: by

text: Ascasibar,

text: Yepes

text: G¨ottleber

text: max

text: ta

text: The

text: “ordered

text: angular

text: momentum”,

text: h,

text: arising

text: from

text: tidal

text: torques

text: of

text: large

text: scale

text: structure

text: on

text: the

text: proto-structure,

text: is

text: obtained

text: through

text: the

text: tidal

text: torque

text: theory

text: (TTT)

text: (Hoyle

text: Peebles

text: White

text: Ryden

text: Eisenstein

text: Loeb

text: The

text: steepening

text: eﬀect

text: produced

text: by

text: the

text: DM

text: compression

text: caused

text: by

text: baryons

text: collapse,

text: the

text: so-called

text: adiabatic

text: contraction

text: (AC),

text: was

text: calculated

text: according

text: to

text: the

text: Gnedin

text: et

text: al.

text: prescriptions.

text: As

text: usual,

text: it

text: is

text: assumed

text: that

text: the

text: density

text: proﬁle

text: of

text: DM

text: and

text: baryons

text: is

text: the

text: same

text: (Mo

text: et

text: al.

text: Keeton

text: Treu

text: Koopmans

text: Cardone

text: Sereno

text: and

text: it

text: is

text: given

text: by

text: a

text: NFW

text: proﬁle.

text: The

text: ﬁnal

text: distribution

text: of

text: baryons

text: is

text: assumed

text: to

text: be

text: a

text: disk

text: (Blumenthal

text: et

text: al.

text: Flores

text: et

text: al.

text: Mo

text: et

text: al.

text: Klypin

text: et

text: al.

text: Cardone

text: Sereno

text: For

text: precision’s

text: sake,

text: in

text: our

text: calculations,

text: we

text: shall

text: assume

text: Klypin

text: et

text: al.

text: model

text: for

text: the

text: baryons

text: distribution

text: (see

text: their

text: Section

text: and

text: their

text: parameters:

text: disc

text: radial

text: scale-length

text: r

text: kpc,

text: and

text: vertical

text: scale-length

text: z

text: pc.

text: The

text: bulge,

text: having

text: a

text: d

text: denoted

text: the

text: mass

text: in

text: a

text: radius

text: R

text: within

text: which

text: the

text: mean

text: overdensity

text: is

text: times

text: larger

text: than

text: the

text: critical

text: density

text: ρ

text: The

text: virial

text: mass

text: M

text: (see

text: the

text: following),

text: can

text: be

text: converted

text: to

text: M

text: using

text: White

text: c

text: vir

text: Hu

text: Kravtsov

text: and

text: Lukic

text: et

text: al.

text: method.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [ [ z=3 _____ z=2 z=1 z=0 [ ] Figure 1. Plot of the density proﬁle of a DM halo having a virial mass M = 109M(cid:12). The evolution vir at z = 3, z = 2, z = 1, z = 0, is represented by the solid line, dotted line, short-dashed line, and long dashed-line, respectively. where (cid:15) = 3.5 (Li et al. 2010). If this gas is re-added to the hot phase, this will disc produce a change in the thermal energy by ∆E = 0.5∆M V 2 (2.10) hot reheat vir In the case ∆E > ∆E , the hot gas will be ejected from the halo, and can be SN hot assumed equal to − ∆E ∆E SN hot ∆M = . (2.11) eject 0.5V 2 vir The halo can accrete the ejected material that becomes part of the hot component related to the central galaxy (De Lucia et al. 2004; Croton et al. 2006). In the present model, the perturbation is made of DM and baryons. The perturbation evolves till the maximum radius and then collapses, ﬁrstly in the DM component, and then in the baryon component. After a given shell reaches turn-around, it starts to collapse. Baryonic clumps exchange angular momentum with DM, cool forming stars, and accrete to the center, giving rise to the bulge (Immeli, Samland, Gerhard, & Westera 2004; Lackner & Ostriker 2010). 3 Results and discussion The results of the calculation are shown in Fig. 1-3. In Fig. 1, we plot the density proﬁle evolution of a bulgeless dwarf galaxy with virial mass M = 109M(cid:12), at diﬀerent z. vir – 5 – ##LTFigure## ##LTLine##

font: default

text: Figure

text: Plot

text: of

text: the

text: density

text: proﬁle

text: of

text: a

text: DM

text: halo

text: having

text: a

text: virial

text: mass

text: M

text: The

text: evolution

text: vir

text: at

text: z

text: z

text: z

text: z

text: is

text: represented

text: by

text: the

text: solid

text: line,

text: dotted

text: line,

text: short-dashed

text: line,

text: and

text: long

text: dashed-line,

text: respectively.

text: where

text: (Li

text: et

text: al.

text: If

text: this

text: gas

text: is

text: re-added

text: to

text: the

text: hot

text: phase,

text: this

text: will

text: disc

text: produce

text: a

text: change

text: in

text: the

text: thermal

text: energy

text: by

text: ∆E

text: V

text: hot

text: reheat

text: vir

text: In

text: the

text: case

text: ∆E

text: ∆E

text: the

text: hot

text: gas

text: will

text: be

text: ejected

text: from

text: the

text: halo,

text: and

text: can

text: be

text: SN

text: hot

text: assumed

text: equal

text: to

text: ∆E

text: ∆E

text: SN

text: hot

text: ∆M

text: eject

text: vir

text: The

text: halo

text: can

text: accrete

text: the

text: ejected

text: material

text: that

text: becomes

text: part

text: of

text: the

text: hot

text: component

text: related

text: to

text: the

text: central

text: galaxy

text: (De

text: Lucia

text: et

text: al.

text: Croton

text: et

text: al.

text: In

text: the

text: present

text: model,

text: the

text: perturbation

text: is

text: made

text: of

text: DM

text: and

text: baryons.

text: The

text: perturbation

text: evolves

text: till

text: the

text: maximum

text: radius

text: and

text: then

text: collapses,

text: ﬁrstly

text: in

text: the

text: DM

text: component,

text: and

text: then

text: in

text: the

text: baryon

text: component.

text: After

text: a

text: given

text: shell

text: reaches

text: turn-around,

text: it

text: starts

text: to

text: collapse.

text: Baryonic

text: clumps

text: exchange

text: angular

text: momentum

text: with

text: DM,

text: cool

text: forming

text: stars,

text: and

text: accrete

text: to

text: the

text: center,

text: giving

text: rise

text: to

text: the

text: bulge

text: (Immeli,

text: Samland,

text: Gerhard,

text: Westera

text: Lackner

text: Ostriker

text: Results

text: and

text: discussion

text: The

text: results

text: of

text: the

text: calculation

text: are

text: shown

text: in

text: Fig.

text: In

text: Fig.

text: we

text: plot

text: the

text: density

text: proﬁle

text: evolution

text: of

text: a

text: bulgeless

text: dwarf

text: galaxy

text: with

text: virial

text: mass

text: M

text: at

text: diﬀerent

text: z.

text: vir

text: ##LTFigure##

text: ##LTLine##


text: Bulge (a) / 1 09 o. 0 -1 -2 -3 8 9 10 11 12 (b) Figure 3. Top panel: plot of the logarithm of the slope of α in terms of the bulge mass for two diﬀerent virial masses: a) virial mass equal to 1011M(cid:12) (solid line), and b) virial mass equal to 1012M(cid:12) (dashed line). Bottom panel: plot of the logarithm of the slope α in terms of the virial mass. The galaxy evolution was started at z = 50, and Fig. 1 plots the dwarf density proﬁle at z = 3 (solid line), z = 2 (dotted line), z = 1 (short-dashed line), and z = 0 (long-dashed line). After the system virializes around z = 10, in the initial evolution stage adiabatic contraction – 7 – ##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##

text: Bulge

text: (a)

text: o.

text: (b)

text: Figure

text: Top

text: panel:

text: plot

text: of

text: the

text: logarithm

text: of

text: the

text: slope

text: of

text: α

text: in

text: terms

text: of

text: the

text: bulge

text: mass

text: for

text: two

text: diﬀerent

text: virial

text: masses:

text: a)

text: virial

text: mass

text: equal

text: to

text: (solid

text: line),

text: and

text: b)

text: virial

text: mass

text: equal

text: to

text: (dashed

text: line).

text: Bottom

text: panel:

text: plot

text: of

text: the

text: logarithm

text: of

text: the

text: slope

text: α

text: in

text: terms

text: of

text: the

text: virial

text: mass.

text: The

text: galaxy

text: evolution

text: was

text: started

text: at

text: z

text: and

text: Fig.

text: plots

text: the

text: dwarf

text: density

text: proﬁle

text: at

text: z

text: (solid

text: line),

text: z

text: (dotted

text: line),

text: z

text: (short-dashed

text: line),

text: and

text: z

text: (long-dashed

text: line).

text: After

text: the

text: system

text: virializes

text: around

text: z

text: in

text: the

text: initial

text: evolution

text: stage

text: adiabatic

text: contraction

text: ##LTFigure##

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text: ##LTLine##

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text: (cid:39) (AC) at z 5 produces a steepening of the proﬁle (not shown in the ﬁgure). The dwarf is bulgeless in agreement with simulations (e.g., Governato et al. 2010) and observations, showing that usually dwarfs have no bulge. The ﬂattening of the proﬁle observed is due to the interplay between DM and baryons, secondary infall, and two-body relaxation. In the infall, baryons clumps interact with DM particles, heating them with the result that they move on external orbits decreasing the (cid:39) inner density. A core forms at the center of the dwarf starting from z 3 and enlarges till (cid:39) z = 0, when the core radius is r 1 kpc. No cusp is observed to reform in the late stage of evolution, like in IS11, but this is expected since in this case there is no bulge, diﬀerently from their simulation. The previous result, as already discussed in DP09, DP12a,b, is in agreement with the Romano-Diaz et al. (2008) simulations. Similarly to other studies (e.g., El-Zant et al. 2001, 2004; Cole et al. 2011) the proﬁle ﬂattening was interpreted as being due to clumps falling to the galaxy center, and DM heating by dynamical friction. In DP12a, we carried a comparison with the Governato et al. (2012) results concerning the formation of a bulgeless dwarf, ﬁnding a good agreement. In Fig. 2a, 2b, we study the eﬀect of a bulge on galaxies of diﬀerent mass. In order to compare the result with IS11 simulations, a bulge is added to the cored DM halo. In the quoted ﬁgures, we plot the evolution of a density proﬁle of a galaxy with M = × vir 1011M(cid:12) (panel a), and of M = 1012M(cid:12) (panel b), both having a bulge mass of 4.5 109M(cid:12). vir In this case the baryon fraction is f = 0.17 (F = 0.03) (see McGaugh et al. 2010; Dai et d b al. 2012) and λ = 0.05 (note that in IS11, λ = 0.1). The density proﬁle is plotted at z = 3 (solid line), z = 2 (dotted line), z = 1 (long- short-dashed line), z = 0.6 (dashed line) (in the case of M = 1011M(cid:12) halo), z = 0.5 (in vir the case of M = 1012M(cid:12) halo), and z = 0 (long-dashed line). vir In the case of the M = 1011M(cid:12) halo (panel a), the core enlarges till a redshift z = 0.6, vir when the slope α reaches its minimum value, while in the case of the M = 1012M(cid:12) halo vir (panel b) α is reached at z = 0.5. After the quoted redshifts, the proﬁle starts to steepen min (cid:39) and the inner slope reaches a value of α 0.65 in the case of the M = 1011M(cid:12) DM halo, (cid:39) vir and α 0.85 in the case of the M = 1012M(cid:12) DM halo. In the case in which no bulge is vir (cid:39) (cid:39) present the values of the inner slope are α 0.6, and α 0.8, respectively. The ﬂattening process giving rise to the formation of a core, and the following steepening happens in a similar way to the IS11 simulation. The result shows that the bulge formation has an important role in the density proﬁle settling. This is due to the deepening of the potential well as bulge forms, dragging matter to the center. Fig. 2a,b, shows two important issues: a) in the presence of a bulge, the ﬁnal density proﬁle is steeper than in the case it is absent. As we show in the following of the paper, there is a correlation among the bulge mass and the inner slope of the density proﬁle; b) galaxies having the same bulge mass but larger virial mass have steeper inner slopes (as already discussed in DP09). The reason of the steepening of the inner density proﬁle with mass, was already ex- plained in DP09. In a few words, less massive proto-structures are born from smaller peaks of the density ﬁeld (Del Popolo & Gambera 1996). Since angular momentum is anti-correlated with the peak height (Hoﬀman 1986), the smaller an object is the larger the angular momen- tum it acquires. As a consequence, particles in the proto-structure are endowed with larger angular momentum and will stay for longer times far away from the proto-structure center, producing a shallower proﬁle (see also William et al. 2004). In DP12a,b, it was shown, that – 8 –

text: (AC)

text: at

text: z

text: produces

text: a

text: steepening

text: of

text: the

text: proﬁle

text: (not

text: shown

text: in

text: the

text: ﬁgure).

text: The

text: dwarf

text: is

text: bulgeless

text: in

text: agreement

text: with

text: simulations

text: (e.g.,

text: Governato

text: et

text: al.

text: and

text: observations,

text: showing

text: that

text: usually

text: dwarfs

text: have

text: no

text: bulge.

text: The

text: ﬂattening

text: of

text: the

text: proﬁle

text: observed

text: is

text: due

text: to

text: the

text: interplay

text: between

text: DM

text: and

text: baryons,

text: secondary

text: infall,

text: and

text: two-body

text: relaxation.

text: In

text: the

text: infall,

text: baryons

text: clumps

text: interact

text: with

text: DM

text: particles,

text: heating

text: them

text: with

text: the

text: result

text: that

text: they

text: move

text: on

text: external

text: orbits

text: decreasing

text: the

text: inner

text: density.

text: A

text: core

text: forms

text: at

text: the

text: center

text: of

text: the

text: dwarf

text: starting

text: from

text: z

text: and

text: enlarges

text: till

text: z

text: when

text: the

text: core

text: radius

text: is

text: r

text: kpc.

text: No

text: cusp

text: is

text: observed

text: to

text: reform

text: in

text: the

text: late

text: stage

text: of

text: evolution,

text: like

text: in

text: but

text: this

text: is

text: expected

text: since

text: in

text: this

text: case

text: there

text: is

text: no

text: bulge,

text: diﬀerently

text: from

text: their

text: simulation.

text: The

text: previous

text: result,

text: as

text: already

text: discussed

text: in

text: is

text: in

text: agreement

text: with

text: the

text: Romano-Diaz

text: et

text: al.

text: simulations.

text: Similarly

text: to

text: other

text: studies

text: (e.g.,

text: El-Zant

text: et

text: al.

text: Cole

text: et

text: al.

text: the

text: proﬁle

text: ﬂattening

text: was

text: interpreted

text: as

text: being

text: due

text: to

text: clumps

text: falling

text: to

text: the

text: galaxy

text: center,

text: and

text: DM

text: heating

text: by

text: dynamical

text: friction.

text: In

text: we

text: carried

text: a

text: comparison

text: with

text: the

text: Governato

text: et

text: al.

text: results

text: concerning

text: the

text: formation

text: of

text: a

text: bulgeless

text: dwarf,

text: ﬁnding

text: a

text: good

text: agreement.

text: In

text: Fig.

text: we

text: study

text: the

text: eﬀect

text: of

text: a

text: bulge

text: on

text: galaxies

text: of

text: diﬀerent

text: mass.

text: In

text: order

text: to

text: compare

text: the

text: result

text: with

text: simulations,

text: a

text: bulge

text: is

text: added

text: to

text: the

text: cored

text: DM

text: halo.

text: In

text: the

text: quoted

text: ﬁgures,

text: we

text: plot

text: the

text: evolution

text: of

text: a

text: density

text: proﬁle

text: of

text: a

text: galaxy

text: with

text: M

text: vir

text: (panel

text: a),

text: and

text: of

text: M

text: (panel

text: b),

text: both

text: having

text: a

text: bulge

text: mass

text: of

text: vir

text: In

text: this

text: case

text: the

text: baryon

text: fraction

text: is

text: f

text: (F

text: (see

text: McGaugh

text: et

text: al.

text: Dai

text: et

text: d

text: b

text: al.

text: and

text: λ

text: (note

text: that

text: in

text: λ

text: The

text: density

text: proﬁle

text: is

text: plotted

text: at

text: z

text: (solid

text: line),

text: z

text: (dotted

text: line),

text: z

text: (long-

text: short-dashed

text: line),

text: z

text: (dashed

text: line)

text: (in

text: the

text: case

text: of

text: M

text: halo),

text: z

text: (in

text: vir

text: the

text: case

text: of

text: M

text: halo),

text: and

text: z

text: (long-dashed

text: line).

text: vir

text: In

text: the

text: case

text: of

text: the

text: M

text: halo

text: (panel

text: a),

text: the

text: core

text: enlarges

text: till

text: a

text: redshift

text: z

text: vir

text: when

text: the

text: slope

text: α

text: reaches

text: its

text: minimum

text: value,

text: while

text: in

text: the

text: case

text: of

text: the

text: M

text: halo

text: vir

text: (panel

text: b)

text: α

text: is

text: reached

text: at

text: z

text: After

text: the

text: quoted

text: redshifts,

text: the

text: proﬁle

text: starts

text: to

text: steepen

text: min

text: and

text: the

text: inner

text: slope

text: reaches

text: a

text: value

text: of

text: α

text: in

text: the

text: case

text: of

text: the

text: M

text: DM

text: halo,

text: vir

text: and

text: α

text: in

text: the

text: case

text: of

text: the

text: M

text: DM

text: halo.

text: In

text: the

text: case

text: in

text: which

text: no

text: bulge

text: is

text: vir

text: present

text: the

text: values

text: of

text: the

text: inner

text: slope

text: are

text: α

text: and

text: α

text: respectively.

text: The

text: ﬂattening

text: process

text: giving

text: rise

text: to

text: the

text: formation

text: of

text: a

text: core,

text: and

text: the

text: following

text: steepening

text: happens

text: in

text: a

text: similar

text: way

text: to

text: the

text: simulation.

text: The

text: result

text: shows

text: that

text: the

text: bulge

text: formation

text: has

text: an

text: important

text: role

text: in

text: the

text: density

text: proﬁle

text: settling.

text: This

text: is

text: due

text: to

text: the

text: deepening

text: of

text: the

text: potential

text: well

text: as

text: bulge

text: forms,

text: dragging

text: matter

text: to

text: the

text: center.

text: Fig.

text: shows

text: two

text: important

text: issues:

text: a)

text: in

text: the

text: presence

text: of

text: a

text: bulge,

text: the

text: ﬁnal

text: density

text: proﬁle

text: is

text: steeper

text: than

text: in

text: the

text: case

text: it

text: is

text: absent.

text: As

text: we

text: show

text: in

text: the

text: following

text: of

text: the

text: paper,

text: there

text: is

text: a

text: correlation

text: among

text: the

text: bulge

text: mass

text: and

text: the

text: inner

text: slope

text: of

text: the

text: density

text: proﬁle;

text: b)

text: galaxies

text: having

text: the

text: same

text: bulge

text: mass

text: but

text: larger

text: virial

text: mass

text: have

text: steeper

text: inner

text: slopes

text: (as

text: already

text: discussed

text: in

text: The

text: reason

text: of

text: the

text: steepening

text: of

text: the

text: inner

text: density

text: proﬁle

text: with

text: mass,

text: was

text: already

text: ex-

text: plained

text: in

text: In

text: a

text: few

text: words,

text: less

text: massive

text: proto-structures

text: are

text: born

text: from

text: smaller

text: peaks

text: of

text: the

text: density

text: ﬁeld

text: (Del

text: Popolo

text: Gambera

text: Since

text: angular

text: momentum

text: is

text: anti-correlated

text: with

text: the

text: peak

text: height

text: (Hoﬀman

text: the

text: smaller

text: an

text: object

text: is

text: the

text: larger

text: the

text: angular

text: momen-

text: tum

text: it

text: acquires.

text: As

text: a

text: consequence,

text: particles

text: in

text: the

text: proto-structure

text: are

text: endowed

text: with

text: larger

text: angular

text: momentum

text: and

text: will

text: stay

text: for

text: longer

text: times

text: far

text: away

text: from

text: the

text: proto-structure

text: center,

text: producing

text: a

text: shallower

text: proﬁle

text: (see

text: also

text: William

text: et

text: al.

text: In

text: it

text: was

text: shown,

text: that


text: Strong particle-hole asymmetry of charge instabilities in doped Mott insulators Mat´ıas Bejas1, Andr´es Greco1 and Hiroyuki Yamase2,3 1Facultad de Ciencias Exactas, Ingenier´ıa y Agrimensura and Instituto de F´ısica Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000 Rosario, Argentina 2National Institute for Materials Science, Tsukuba 305-0047, Japan 3Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany E-mail: bejas@ifir-conicet.gov.ar, agreco@fceia.unr.edu.ar and yamase.hiroyuki@nims.go.jp Abstract. We study possible charge instabilities in doped Mott insulators by employing the two-dimensional t-J model with a positive value of the next nearest- (cid:48) neighbor hopping integral t on a square lattice, which is applicable to electron- doped cuprates. Although the d-wave charge density wave (ﬂux phase) and d-wave Pomeranchuk instability (nematic order) are dominant instabilities for a negative (cid:48) t that corresponds to hole-doped cuprates, we ﬁnd that those instabilities are strongly suppressed and become relevant only rather close to half ﬁlling. Instead, various types of bond orders with modulation vectors close to (π, π) are dominant in a moderate doping region. Phase separation is also enhanced, but it can be suppressed substantially by the nearest-neighbor Coulomb repulsion without aﬀecting the aforementioned charge instabilities. PACS numbers: 71.10.Hf, 75.25.Dk, 74.72.Ek, 74.72.Kf Keywords: electron- and hole-doped cuprates, pseudogap, charge orders Submitted to: New J. Phys. 1. Introduction High-temperature cuprate superconductors are realized by carrier doping into antiferromagnetic Mott insulators, and superconductivity is characterized by d-wave symmetry. The cuprate superconductors are layered materials; and the electronic properties in the CuO plane, where Cu sites form a square lattice, hold the key to 2 high-temperature superconductivity. Its essential physics is believed to be contained in the two-dimensional t-J and Hubbard models on a square lattice [1, 2]. Despite these common views, the underlying physics of cuprate superconductivity remains highly elusive.

text: Strong

text: particle-hole

text: asymmetry

text: of

text: charge

text: instabilities

text: in

text: doped

text: Mott

text: insulators

text: Mat´ıas

text: Andr´es

text: and

text: Hiroyuki

text: de

text: Ciencias

text: Exactas,

text: Ingenier´ıa

text: y

text: Agrimensura

text: and

text: Instituto

text: de

text: F´ısica

text: Rosario

text: (UNR-CONICET),

text: Avenida

text: Pellegrini

text: Rosario,

text: Argentina

text: Institute

text: for

text: Materials

text: Science,

text: Tsukuba

text: Japan

text: for

text: Solid

text: State

text: Research,

text: Stuttgart,

text: Germany

text: E-mail:

text: bejas@ifir-conicet.gov.ar,

text: agreco@fceia.unr.edu.ar

text: and

text: yamase.hiroyuki@nims.go.jp

text: Abstract.

text: We

text: study

text: possible

text: charge

text: instabilities

text: in

text: doped

text: Mott

text: insulators

text: by

text: employing

text: the

text: two-dimensional

text: t-J

text: model

text: with

text: a

text: positive

text: value

text: of

text: the

text: next

text: nearest-

text: neighbor

text: hopping

text: integral

text: t

text: on

text: a

text: square

text: lattice,

text: which

text: is

text: applicable

text: to

text: electron-

text: doped

text: cuprates.

text: Although

text: the

text: d-wave

text: charge

text: density

text: wave

text: (ﬂux

text: phase)

text: and

text: d-wave

text: Pomeranchuk

text: instability

text: (nematic

text: order)

text: are

text: dominant

text: instabilities

text: for

text: a

text: negative

text: t

text: that

text: corresponds

text: to

text: hole-doped

text: cuprates,

text: we

text: ﬁnd

text: that

text: those

text: instabilities

text: are

text: strongly

text: suppressed

text: and

text: become

text: relevant

text: only

text: rather

text: close

text: to

text: half

text: ﬁlling.

text: Instead,

text: various

text: types

text: of

text: bond

text: orders

text: with

text: modulation

text: vectors

text: close

text: to

text: (π,

text: π)

text: are

text: dominant

text: in

text: a

text: moderate

text: doping

text: region.

text: Phase

text: separation

text: is

text: also

text: enhanced,

text: but

text: it

text: can

text: be

text: suppressed

text: substantially

text: by

text: the

text: nearest-neighbor

text: Coulomb

text: repulsion

text: without

text: aﬀecting

text: the

text: aforementioned

text: charge

text: instabilities.

text: PACS

text: numbers:

text: Keywords:

text: electron-

text: and

text: hole-doped

text: cuprates,

text: pseudogap,

text: charge

text: orders

text: Submitted

text: to:

text: New

text: J.

text: Phys.

text: Introduction

text: High-temperature

text: cuprate

text: superconductors

text: are

text: realized

text: by

text: carrier

text: doping

text: into

text: antiferromagnetic

text: Mott

text: insulators,

text: and

text: superconductivity

text: is

text: characterized

text: by

text: d-wave

text: symmetry.

text: The

text: cuprate

text: superconductors

text: are

text: layered

text: materials;

text: and

text: the

text: electronic

text: properties

text: in

text: the

text: CuO

text: plane,

text: where

text: Cu

text: sites

text: form

text: a

text: square

text: lattice,

text: hold

text: the

text: key

text: to

text: high-temperature

text: superconductivity.

text: Its

text: essential

text: physics

text: is

text: believed

text: to

text: be

text: contained

text: in

text: the

text: two-dimensional

text: t-J

text: and

text: Hubbard

text: models

text: on

text: a

text: square

text: lattice

text: Despite

text: these

text: common

text: views,

text: the

text: underlying

text: physics

text: of

text: cuprate

text: superconductivity

text: remains

text: highly

text: elusive.


text: Particle-hole asymmetry in doped Mott insulators 15 [13] Anderson P W, Lee P A, Randeria M, Rice T M, Trivedi N and Zhang F C 2004 J. Phys. Cond. Matter 16 R755 [14] Lee P A, Nagaosa N and Wen X G 2006 Rev. Mod. Phys. 78 17 [15] Chakravarty S, Laughlin R B, Morr D K and Nayak C 2001 Phys. Rev. B 63 094503 [16] Cappelluti E and Zeyher R 1999 Phys. Rev. B 59 6475 [17] Greco A 2009 Phys. Rev. Lett. 103 217001 [18] Bejas M, Buzon G, Greco A and Foussats A 2011 Phys. Rev. B 83 014514 [19] Greco A and Bejas M 2011 Phys. Rev. B 83 212503 [20] Varma C M 1999 Phys. Rev. Lett. 83 3538 [21] Varma C M 2006 Phys. Rev. B 73 155113 [22] Yamase H 2009 Phys. Rev. B 79 052501 [23] Hackl A and Vojta M 2009 Phys. Rev. B 80 220514(R) [24] Yamase H and Metzner W 2012 Phys. Rev. Lett. 108 186405 [25] Castellani C, Castro C D and Grilli M 1995 Phys. Rev. Lett. 75 4650 [26] Becca F, Tarquini M, Grilli M and Castro C D 1996 Phys. Rev. B 54 12443 [27] Hashimoto M, He R H, Tanaka K, Testaud J P, Meevasana W, Moore R G, Lu D, Yao H, Yoshida Y, Eisaki H, Devereaux T P, Hussain Z and Shen Z X 2010 Nat. Phys. 6 414 [28] He R H, Hashimoto M, Karapetyan H, Koralek J D, Hinton J P, Testaud J P, Nathan V, Yoshida Y, Yao H, Tanaka K, Meevasana W, Moore R G, Lu D H, Mo S K, Ishikado M, Eisaki H, Hussain Z, Devereaux T P, Kivelson S A, Orenstein J, Kapitulnik A and Shen Z X 2011 Science 331 1579–1583 [29] Kivelson S A, Bindloss I P, Fradkin E, Oganesyan V, Tranquada J M, Kapitulnik A and Howald C 2003 Rev. Mod. Phys. 75 1201 [30] Vojta M 2009 Adv. Phys. 58 699 [31] Emery V J and Kivelson S A 1993 Physica C 209 597 [32] Wu T, Mayaﬀre H, Kr¨amer S, Horvati´c M, Berthier C, Hardy W N, Liang R, Bon D A and Julien M H 2011 Nature 477 191 [33] Ghiringhelli G, Tacon M L, Minola M, Blanco-Canosa S, Mazzoli C, Brookes N B, Luca G M D, Frano A, Hawthorn D G, He F, Loew T, Sala M M, Peets D C, Salluzzo M, Schierle E, Sutarto R, Sawatzky G A, Weschke E, Keimer B and Braicovich L 2012 Science 337 821 [34] Chang J, Blackburn E, Holmes A T, Christensen N B, Larsen J, Mesot J, Liang R, Bonn D A, Hardy W N, Watenphul A, v Zimmermann M, Forgan E M and Hayden S M 2012 Nat. Phys. 8 871 [35] Comin R, Frano A, Yee M M, Yoshida Y, Eisaki H, Schierle E, Weschke E, Sutarto R, He F, Soumyanarayanan A, He Y, Tacon M L, Elﬁmov I S, Hoﬀman J E, Sawatzky G A, Keimer B and Damascelli A 2014 Science 343 390 [36] da Silva Neto E H, Aynajian P, Frano A, Comin R, Schierle E, Weschke E, Gyenis A, Wen J, Schneeloch J, Xu Z, Ono S, Gu G, Tacon M L and Yazdani A 2014 Science 343 393 [37] Tranquada J M, Sternlieb B J, Axe J D, Nakamura Y and Uchida S 1995 Nature (London) 375 561 [38] Bejas M, Greco A and Yamase H 2012 Phys. Rev. B 86 224509 [39] Metlitski M A and Sachdev S 2010 Phys. Rev. B 82 075127 [40] Metlitski M A and Sachdev S 2010 New J. Phys. 12 105007 [41] Holder T and Metzner W 2012 Phys. Rev. B 85 165130 [42] Husemann C and Metzner W 2012 Phys. Rev. B 86 085113 [43] Efetov K B, Meier H and P´epin C 2013 Nat. Phys. 9 442 [44] Bulut S, Atkinson W A and Kampf A P 2013 Phys. Rev. B 88 155132 [45] Sachdev S and Placa R L 2013 Phys. Rev. Lett. 111 027202 [46] Armitage N P, Fournier P and Greene R L 2010 Rev. Mod. Phys. 82 2421 [47] Lee W S, Lee J J, Nowadnick E A, Gerber S, Tabis W, Huang S W, Strocov V N, Motoyama E M, Yu G, Moritz B, Huang H Y, Wang R P, Huang Y B, Wu W B, Chen C T, Huang D J, Greven

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text: Model reduction for slow–fast stochastic systems with metastable behavioura) Maria Bruna,1,1, b) S. Jonathan Chapman,1 and Matthew J. Smith1 (Dated: 26 May 2020) The quasi-steady-state approximation (or stochastic averaging principle) is a useful tool in the study of mul- tiscale stochastic systems, giving a practical method by which to reduce the number of degrees of freedom in a model. The method is extended here to slow–fast systems in which the fast variables exhibit metastable behaviour. The key parameter that determines the form of the reduced model is the ratio of the timescale for the switching of the fast variables between metastable states to the timescale for the evolution of the slow variables. The method is illustrated with two examples: one from biochemistry (a fast-species-mediated chem- ical switch coupled to a slower-varying species), and one from ecology (a predator–prey system). Numerical simulations of each model reduction are compared with those of the full system. PACS numbers: Keywords: I. INTRODUCTION sudden switch in the dominant vegetation type, but the carbon content of the soil may respond relatively slowly. This raises questions about how to eﬃciently and accu- Understanding the impact of noise on nonlinear dy- rately simulate the dynamics of such systems: can the namical systems has been an active ﬁeld of research for evolution of the slowly evolving components be predicted many years, with a wide range of applications in physics, without simulating the full evolution of all the fast com- chemistry, biology, ecology and earth science. Although ponents? the addition of noise sometimes does not change the In this paper we bring together these two features, qualitative dynamics and can be modelled by adding a multiple timescales and metastability, and consider noisy stochastic perturbation to the deterministic solution tra- systems with slow and fast metastable components. We jectory (as in the linear noise approximation? ), there is develop a method to remove the fast degrees of freedom a growing number of applications in which noise has been while retaining the metastable behaviour in the resulting shown to be crucial to explain features which cannot be mathematical description of the slow processes. captured by deterministic models. For example, in de- Most of the work done to remove fast degrees of free- terministic systems with multiple stable steady states dom from stochastic multiscale systems is based on ex- the addition of even a small amount of noise causes tending the principle of slow or centre manifold theory these states to become metastable, with the stochas- of deterministic dynamical systems. This assumes that tic system undergoing random transitions between the the phase space can be decomposed into fast variables deterministic steady states. Examples include genetic x which relax very quickly and slow variables y, which regulatory networks,? lactose utilisation networks and change more slowly: the bet-hedging in bacteria.? ? This behaviour cannot be captured by the deterministic model,? but can be dx explained successfully with a stochastic model.? (cid:15) = F(x, y), x(0) = x0, (1a) dt A common feature of many complex dynamical sys- dy = G(x, y), y(0) = y , (1b) tems is the presence of processes evolving on widely sep- dt 0 arated timescales. This can present a challenge for nu- merical simulation. Often we are only interested in the where (cid:15) (cid:28) 1 quantiﬁes the separation of timescales. behaviour of solutions on a long timescale,? but in prin- The standard singular perturbation theory, based on ciple to determine this we need to resolve processes occur- Fenichel’s theory,? consists of taking the limit (cid:15) → 0 ring on the fastest timescale. Metastability complicates and assuming that the fast variables x have equilibrated this further, since the switching rate between states in- onto the centre manifold, F(x, y) = 0. The ﬂow on troduces a new implicit timescale to the process. For this slow manifold (with reduced dimensionality) is then example, several ecological systems are believed to be given by able to switch between alternative states, such as be- tween tree and grass-dominated vegetation.? ? A few dy key mechanisms may be suﬃcient to explain the observed = G(h(y), y), y(0) = y , (2) 0 dt bi-stability of these systems, but a wide range of other ecosystem components (such as the resident bird com- munity) respond to these dynamics, and may do so at where F(h(y), y) = 0.? Note that the projection onto diﬀerent rates; for example the population size of a resi- the slow manifold is valid even if F(x, y) = 0 has more dent bird community may respond relatively quickly to a than one solution for y ﬁxed; given the initial condition ##LTLine## ##LTLine## ##LTLine##

text: Model

text: reduction

text: for

text: slow–fast

text: stochastic

text: systems

text: with

text: metastable

text: behavioura)

text: Maria

text: b)

text: S.

text: Jonathan

text: and

text: Matthew

text: J.

text: (Dated:

label: date

text: May

text: The

text: quasi-steady-state

text: approximation

text: (or

text: stochastic

text: averaging

text: principle)

text: is

text: a

text: useful

text: tool

text: in

text: the

text: study

text: of

text: mul-

text: tiscale

text: stochastic

text: systems,

text: giving

text: a

text: practical

text: method

text: by

text: which

text: to

text: reduce

text: the

text: number

text: of

text: degrees

text: of

text: freedom

text: in

text: a

text: model.

text: The

text: method

text: is

text: extended

text: here

text: to

text: slow–fast

text: systems

text: in

text: which

text: the

text: fast

text: variables

text: exhibit

text: metastable

text: behaviour.

text: The

text: key

text: parameter

text: that

text: determines

text: the

text: form

text: of

text: the

text: reduced

text: model

text: is

text: the

text: ratio

text: of

text: the

text: timescale

text: for

text: the

text: switching

text: of

text: the

text: fast

text: variables

text: between

text: metastable

text: states

text: to

text: the

text: timescale

text: for

text: the

text: evolution

text: of

text: the

text: slow

text: variables.

text: The

text: method

text: is

text: illustrated

text: with

text: two

text: examples:

text: one

text: from

text: biochemistry

text: (a

text: fast-species-mediated

text: chem-

text: ical

text: switch

text: coupled

text: to

text: a

text: slower-varying

text: species),

text: and

text: one

text: from

text: ecology

text: (a

text: predator–prey

text: system).

text: Numerical

text: simulations

text: of

text: each

text: model

text: reduction

text: are

text: compared

text: with

text: those

text: of

text: the

text: full

text: system.

text: PACS

text: numbers:

text: Keywords:

text: I.

text: INTRODUCTION

text: sudden

text: switch

text: in

text: the

text: dominant

text: vegetation

text: type,

text: but

text: the

text: carbon

text: content

text: of

text: the

text: soil

text: may

text: respond

text: relatively

text: slowly.

text: This

text: raises

text: questions

text: about

text: how

text: to

text: eﬃciently

text: and

text: accu-

text: Understanding

text: the

text: impact

text: of

text: noise

text: on

text: nonlinear

text: dy-

text: rately

text: simulate

text: the

text: dynamics

text: of

text: such

text: systems:

text: can

text: the

text: namical

text: systems

text: has

text: been

text: an

text: active

text: ﬁeld

text: of

text: research

text: for

text: evolution

text: of

text: the

text: slowly

text: evolving

text: components

text: be

text: predicted

text: many

text: years,

text: with

text: a

text: wide

text: range

text: of

text: applications

text: in

text: physics,

text: without

text: simulating

text: the

text: full

text: evolution

text: of

text: all

text: the

text: fast

text: com-

text: chemistry,

text: biology,

text: ecology

text: and

text: earth

text: science.

text: Although

text: ponents?

text: the

text: addition

text: of

text: noise

text: sometimes

text: does

text: not

text: change

text: the

text: In

text: this

text: paper

text: we

text: bring

text: together

text: these

text: two

text: features,

text: qualitative

text: dynamics

text: and

text: can

text: be

text: modelled

text: by

text: adding

text: a

text: multiple

text: timescales

text: and

text: metastability,

text: and

text: consider

text: noisy

text: stochastic

text: perturbation

text: to

text: the

text: deterministic

text: solution

text: tra-

text: systems

text: with

text: slow

text: and

text: fast

text: metastable

text: components.

text: We

text: jectory

text: (as

text: in

text: the

text: linear

text: noise

text: approximation?

text: there

text: is

text: develop

text: a

text: method

text: to

text: remove

text: the

text: fast

text: degrees

text: of

text: freedom

text: a

text: growing

text: number

text: of

text: applications

text: in

text: which

text: noise

text: has

text: been

text: while

text: retaining

text: the

text: metastable

text: behaviour

text: in

text: the

text: resulting

text: shown

text: to

text: be

text: crucial

text: to

text: explain

text: features

text: which

text: cannot

text: be

text: mathematical

text: description

text: of

text: the

text: slow

text: processes.

text: captured

text: by

text: deterministic

text: models.

text: For

text: example,

text: in

text: de-

text: Most

text: of

text: the

text: work

text: done

text: to

text: remove

text: fast

text: degrees

text: of

text: free-

text: terministic

text: systems

text: with

text: multiple

text: stable

text: steady

text: states

text: dom

text: from

text: stochastic

text: multiscale

text: systems

text: is

text: based

text: on

text: ex-

text: the

text: addition

text: of

text: even

text: a

text: small

text: amount

text: of

text: noise

text: causes

text: tending

text: the

text: principle

text: of

text: slow

text: or

text: centre

text: manifold

text: theory

text: these

text: states

text: to

text: become

text: metastable,

text: with

text: the

text: stochas-

text: of

text: deterministic

text: dynamical

text: systems.

text: This

text: assumes

text: that

text: tic

text: system

text: undergoing

text: random

text: transitions

text: between

text: the

text: the

text: phase

text: space

text: can

text: be

text: decomposed

text: into

text: fast

text: variables

text: deterministic

text: steady

text: states.

text: Examples

text: include

text: genetic

text: x

text: which

text: relax

text: very

text: quickly

text: and

text: slow

text: variables

text: y,

text: which

text: regulatory

text: networks,?

text: lactose

text: utilisation

text: networks

text: and

text: change

text: more

text: slowly:

text: the

text: bet-hedging

text: in

text: bacteria.?

text: This

text: behaviour

text: cannot

text: be

text: captured

text: by

text: the

text: deterministic

text: model,?

text: but

text: can

text: be

text: dx

text: explained

text: successfully

text: with

text: a

text: stochastic

text: model.?

text: F(x,

text: y),

text: dt

text: A

text: common

text: feature

text: of

text: many

text: complex

text: dynamical

text: sys-

text: dy

text: G(x,

text: y),

text: y

text: tems

text: is

text: the

text: presence

text: of

text: processes

text: evolving

text: on

text: widely

text: sep-

text: dt

text: arated

text: timescales.

text: This

text: can

text: present

text: a

text: challenge

text: for

text: nu-

text: merical

text: simulation.

text: Often

text: we

text: are

text: only

text: interested

text: in

text: the

text: where

text: quantiﬁes

text: the

text: separation

text: of

text: timescales.

text: behaviour

text: of

text: solutions

text: on

text: a

text: long

text: timescale,?

text: but

text: in

text: prin-

text: The

text: standard

text: singular

text: perturbation

text: theory,

text: based

text: on

text: ciple

text: to

text: determine

text: this

text: we

text: need

text: to

text: resolve

text: processes

text: occur-

text: Fenichel’s

text: theory,?

text: consists

text: of

text: taking

text: the

text: limit

text: ring

text: on

text: the

text: fastest

text: timescale.

text: Metastability

text: complicates

text: and

text: assuming

text: that

text: the

text: fast

text: variables

text: x

text: have

text: equilibrated

text: this

text: further,

text: since

text: the

text: switching

text: rate

text: between

text: states

text: in-

text: onto

text: the

text: centre

text: manifold,

text: F(x,

text: y)

text: The

text: ﬂow

text: on

text: troduces

text: a

text: new

text: implicit

text: timescale

text: to

text: the

text: process.

text: For

text: this

text: slow

text: manifold

text: (with

text: reduced

text: dimensionality)

text: is

text: then

text: example,

text: several

text: ecological

text: systems

text: are

text: believed

text: to

text: be

text: given

text: by

text: able

text: to

text: switch

text: between

text: alternative

text: states,

text: such

text: as

text: be-

text: tween

text: tree

text: and

text: grass-dominated

text: vegetation.?

text: A

text: few

text: dy

text: key

text: mechanisms

text: may

text: be

text: suﬃcient

text: to

text: explain

text: the

text: observed

text: G(h(y),

text: y),

text: y

text: dt

text: bi-stability

text: of

text: these

text: systems,

text: but

text: a

text: wide

text: range

text: of

text: other

text: ecosystem

text: components

text: (such

text: as

text: the

text: resident

text: bird

text: com-

text: munity)

text: respond

text: to

text: these

text: dynamics,

text: and

text: may

text: do

text: so

text: at

text: where

text: F(h(y),

text: y)

text: Note

text: that

text: the

text: projection

text: onto

text: diﬀerent

text: rates;

text: for

text: example

text: the

text: population

text: size

text: of

text: a

text: resi-

text: the

text: slow

text: manifold

text: is

text: valid

text: even

text: if

text: F(x,

text: y)

text: has

text: more

text: dent

text: bird

text: community

text: may

text: respond

text: relatively

text: quickly

text: to

text: a

text: than

text: one

text: solution

text: for

text: y

text: ﬁxed;

text: given

text: the

text: initial

text: condition

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