text: 14 10 10 8 8 6 6 4 4 y 2 y 2 0 0 −2 −2 −4 −4 −6 −6 −12 −10 −8 −6 −4 −2 0 −12 −10 −8 −6 −4 −2 0 x x (cid:48) FIG. 7: Non-eigenvalue solutions to the ﬁrst Painlev´e transcendent. If y(0) = 1 but y (0) is n√ot one of the − − eigenvalues a , the curve either oscillates about and approaches√the stable asymptotic curve x as in the n − left panel or else it lies above the unstable asymptotic curve + x and passes through an inﬁnite number of double poles as in the right panel. B. Conjectural connection with the power-series constant In conclusion, we point out a possible connection between this work and the power-series con- stant P in the theory of complex variables. (The quest to ﬁnd the value of the power series constant F | | was originated by Hayman [9].) Let be the class of functions f(z) that are analytic for z < 1 | | ∈ F but not analytic for z < r for r > 1. If f , i(cid:88)ts power-series expansion ∞ f(z) = a zk k k=0 has a radius of convergence of 1. We denote ρ (f) as the largest number r such that the partial n sum of f(z), (cid:88) n S (z) = a zk, (38) n k k=0 ##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##

text: y

text: y

text: −2

text: −2

text: −4

text: −4

text: −6

text: −6

text: −8

text: −6

text: −4

text: −2

text: −8

text: −6

text: −4

text: −2

text: x

text: x

text: FIG.

text: Non-eigenvalue

text: solutions

text: to

text: the

text: ﬁrst

text: Painlev´e

text: transcendent.

text: If

text: but

text: y

text: is

text: n√ot

text: one

text: of

text: the

text: eigenvalues

text: a

text: the

text: curve

text: either

text: oscillates

text: about

text: and

text: approaches√the

text: stable

text: asymptotic

text: curve

text: x

text: as

text: in

text: the

text: n

text: left

text: panel

text: or

text: else

text: it

text: lies

text: above

text: the

text: unstable

text: asymptotic

text: curve

text: x

text: and

text: passes

text: through

text: an

text: inﬁnite

text: number

text: of

text: double

text: poles

text: as

text: in

text: the

text: right

text: panel.

text: B.

text: Conjectural

text: connection

text: with

text: the

text: power-series

text: constant

text: In

text: conclusion,

text: we

text: point

text: out

text: a

text: possible

text: connection

text: between

text: this

text: work

text: and

text: the

text: power-series

text: con-

text: stant

text: P

text: in

text: the

text: theory

text: of

text: complex

text: variables.

text: (The

text: quest

text: to

text: ﬁnd

text: the

text: value

text: of

text: the

text: power

text: series

text: constant

text: F

text: was

text: originated

text: by

text: Hayman

text: Let

text: be

text: the

text: class

text: of

text: functions

text: f(z)

text: that

text: are

text: analytic

text: for

text: z

text: F

text: but

text: not

text: analytic

text: for

text: z

text: r

text: for

text: r

text: If

text: f

text: power-series

text: expansion

text: f(z)

text: a

text: zk

text: k

text: has

text: a

text: radius

text: of

text: convergence

text: of

text: We

text: denote

text: ρ

text: (f)

text: as

text: the

text: largest

text: number

text: r

text: such

text: that

text: the

text: partial

text: n

text: sum

text: of

text: f(z),

text: n

text: S

text: (z)

text: a

text: zk,

text: n

text: k

text: ##LTFigure##

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text: 15 | | has at least one zero on z = r. We deﬁne ≡ ρ(f) lim inf ρ (f). (39) →∞ n n The power series constant P is then deﬁned as ≡ P sup ρ(f). (40) ∈F f The precise value for P is not known. However, lower and upper bounds on P have been ≤ ≤ √ established. The power series constant was known to lie in the interval 1 P 2 until Clunie ≤ ≤ and Erd¨os [10] sharpened these bounds to 2 P 2. Buckholtz [11] further sharpened these ≤ ≤ bounds to 1.7 P 121/4, which was optimized by Frank [11] to ≤ ≤ 1.7818 P 1.82. (41) The latter values appear to be the best known values to date. It is astonishing that the value of A in (9) agrees with the best known lower bound for the value of P . We do not know whether our value 25/6 coincides exactly with the lower bound. We leave this observation here as coincidence and hope to elaborate on the precise relation in a future paper [12]. As an illustration, let us compute ρ(f) for some speciﬁc functions. In some rare cases we can sum the entire series and can therefore compute the value exactly. For instance, for the class of functions (cid:88) ∞ f (z) = exp[iπτ(k2 + k)] zk (42) τ k=0 we compute − − 1 + iz iz2 z3 f (z) = (43) 1/4 1 + z4 √ − ± − | | ≈ with zeros z = 1 and z = 1/2(1 + i) 2i 4, such that ρ(f ) = z 1.70002. We can 1 2/3 1/4 2 also compute a value closer to P : − − − − 1 + e34iπ z + ei4π z2 + iz3 iz4 ei4π z5 e34iπ z6 z7 f (z) = , (44) (cid:0) 3/(cid:1)8 1 + z8 ≈ leading to ρ f 1.7804. We must terminate the summation and evaluate ρ (f) for a 3/8 n suﬃciently large value of n. In Fig. 8 we display our numerical results for ρ (f ) obtained from 50 τ≈ the partial sum S (z). The maximum values are ρ (f ) = ρ (f ) 1.7818, which 50 50 0.3780 50 0.8780 coincide with the best known lower bound for P up to the precision of the computation. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: has

text: at

text: least

text: one

text: zero

text: on

text: z

text: r.

text: We

text: deﬁne

label: equation

text: ρ(f)

text: lim

text: inf

text: ρ

text: (f).

text: n

text: n

text: The

text: power

text: series

text: constant

text: P

text: is

text: then

text: deﬁned

text: as

text: P

text: sup

text: ρ(f).

text: ∈F

text: f

text: The

text: precise

text: value

text: for

text: P

text: is

text: not

text: known.

text: However,

text: lower

text: and

text: upper

text: bounds

text: on

text: P

text: have

text: been

text: established.

text: The

text: power

text: series

text: constant

text: was

text: known

text: to

text: lie

text: in

text: the

text: interval

text: P

text: until

text: Clunie

text: and

text: Erd¨os

text: sharpened

text: these

text: bounds

text: to

text: P

text: Buckholtz

text: further

text: sharpened

text: these

text: bounds

text: to

text: P

text: which

text: was

text: optimized

text: by

text: Frank

text: to

text: P

text: The

text: latter

text: values

text: appear

text: to

text: be

text: the

text: best

text: known

text: values

text: to

text: date.

text: It

text: is

text: astonishing

text: that

text: the

text: value

text: of

text: A

text: in

text: agrees

text: with

text: the

text: best

text: known

text: lower

text: bound

text: for

text: the

text: value

text: of

text: P

text: We

text: do

text: not

text: know

text: whether

text: our

text: value

text: coincides

text: exactly

text: with

text: the

text: lower

text: bound.

text: We

text: leave

text: this

text: observation

text: here

text: as

text: coincidence

text: and

text: hope

text: to

text: elaborate

text: on

text: the

text: precise

text: relation

text: in

text: a

text: future

text: paper

text: As

text: an

text: illustration,

text: let

text: us

text: compute

text: ρ(f)

text: for

text: some

text: speciﬁc

text: functions.

text: In

text: some

text: rare

text: cases

text: we

text: can

text: sum

text: the

text: entire

text: series

text: and

text: can

text: therefore

text: compute

text: the

text: value

text: exactly.

text: For

text: instance,

text: for

text: the

text: class

text: of

text: functions

text: f

text: (z)

text: k)]

text: zk

text: τ

text: we

text: compute

text: iz

text: f

text: (z)

text: with

text: zeros

text: z

text: and

text: z

text: i)

text: such

text: that

text: ρ(f

text: z

text: We

text: can

text: also

text: compute

text: a

text: value

text: closer

text: to

text: P

text: z

text: f

text: (z)

text: leading

text: to

text: ρ

text: f

text: We

text: must

text: terminate

text: the

text: summation

text: and

text: evaluate

text: ρ

text: (f)

text: for

text: a

text: n

text: suﬃciently

text: large

text: value

text: of

text: n.

text: In

text: Fig.

text: we

text: display

text: our

text: numerical

text: results

text: for

text: ρ

text: (f

text: obtained

text: from

text: τ≈

text: the

text: partial

text: sum

text: S

text: (z).

text: The

text: maximum

text: values

text: are

text: ρ

text: (f

text: ρ

text: (f

text: which

text: coincide

text: with

text: the

text: best

text: known

text: lower

text: bound

text: for

text: P

text: up

text: to

text: the

text: precision

text: of

text: the

text: computation.

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text: ##LTLine##

text: ##LTLine##

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text: (cid:18) (cid:19) (cid:18) (cid:19) cos θ − sin θ − v(1) = − e ikτ and v(2) = e ikτ . (2.20) sin θ cos θ Again we can see that the main eﬀect of a fast turn consists in an orthogonal transforma- tion of the modes. Hence the power spectrum does not change with respect to the single-ﬁeld slow-roll case. This conclusion is not surprising—during the turn the large-k modes are deep inside the Hubble and mass radii, and they behave as massless ﬁelds in Minkowski space. Note that this is diﬀerent from what happens with respect to modes which are super-horizon when the turn takes place.4 2.2 Evolution of the perturbations at the turn: numerical study Gao, Langlois & Mizuno [28] put forth a simple two-ﬁeld potential which allows studying the eﬀects of a sudden turn of the inﬂationary trajectory. The potential between the two (cid:18) (cid:19) (cid:104) (cid:105) canonically normalised ﬁelds, φ and χ, reads 1 1 ∆θ − − 2 V (φ, χ) = m2 φ2 + M2 cos2 χ (φ φ ) tan Ξ(φ) , (2.21) 2 φ 2 2 0 − with Ξ(φ) = ∆θ arctan [s(φ φ )]. Here, ∆θ is a constant parameter corresponding to 0 the net angle variation during the turn, while adjusting the constant parameter s allows to control the sharpness of the turn. In the following, we ﬁx the parameters of the potential as √ √ − × − − π − m = 10 7 M , M = 2 10 4M , φ = 100 6 M , δθ = , s = 5000 3 M 1 , (2.22) φ P P 0 P 10 P in which we reintroduced the Planck mass for clarity. The shape of the potential corre- sponding to this choice is shown in Figure 1. Indeed, it corresponds to a slowly descending valley with steep slopes exhibiting a turn. With the parameters choice (2.22), the mass of the heavy perturbation (or equivalently, the curvature of the potential across the bottom of the valley) is twenty times larger than the Hubble parameter. It is in this sense that we can refer to the turn as sharp. In order to discuss the evolution of the perturbations numerically, it is convenient to adopt yet another time-dependent basis for the perturbations. Following the same prescription as Gao, Langlois & Mizuno, we will track the evolution of the perturbations, schematically denoted by ϕ , in the directions determined by the eigenvectors of the mass matrix [30, 31, 37] I 4 A notable example is that of double quadratic inﬂation (see, for example, Vernizzi & Wands [35]). It is well known that, in this case, there are appreciable changes to the power spectrum, as well as other observables, such as the scalar spectral index and local non-gaussianity as measured by f (see, for example, Dias & Seery NL [36]). – 9 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: cos

text: θ

text: sin

text: θ

text: e

text: ikτ

text: and

text: e

text: ikτ

text: sin

text: θ

text: cos

text: θ

text: Again

text: we

text: can

text: see

text: that

text: the

text: main

text: eﬀect

text: of

text: a

text: fast

text: turn

text: consists

text: in

text: an

text: orthogonal

text: transforma-

text: tion

text: of

text: the

text: modes.

text: Hence

text: the

text: power

text: spectrum

text: does

text: not

text: change

text: with

text: respect

text: to

text: the

text: single-ﬁeld

text: slow-roll

text: case.

text: This

text: conclusion

text: is

text: not

text: surprising—during

text: the

text: turn

text: the

text: large-k

text: modes

text: are

text: deep

text: inside

text: the

text: Hubble

text: and

text: mass

text: radii,

text: and

text: they

text: behave

text: as

text: massless

text: ﬁelds

text: in

text: Minkowski

text: space.

text: Note

text: that

text: this

text: is

text: diﬀerent

text: from

text: what

text: happens

text: with

text: respect

text: to

text: modes

text: which

text: are

text: super-horizon

text: when

text: the

text: turn

text: takes

text: Evolution

text: of

text: the

text: perturbations

text: at

text: the

text: turn:

text: numerical

text: study

text: Gao,

text: Langlois

text: Mizuno

text: put

text: forth

text: a

text: simple

text: two-ﬁeld

text: potential

text: which

text: allows

text: studying

text: the

text: eﬀects

text: of

text: a

text: sudden

text: turn

text: of

text: the

text: inﬂationary

text: trajectory.

text: The

text: potential

text: between

text: the

text: two

text: canonically

text: normalised

text: ﬁelds,

text: φ

text: and

text: χ,

text: reads

text: ∆θ

text: V

text: (φ,

text: χ)

text: φ2

text: χ

text: (φ

text: φ

text: tan

text: Ξ(φ)

text: φ

text: with

text: Ξ(φ)

text: ∆θ

text: arctan

text: [s(φ

text: φ

text: Here,

text: ∆θ

text: is

text: a

text: constant

text: parameter

text: corresponding

text: to

text: the

text: net

text: angle

text: variation

text: during

text: the

text: turn,

text: while

text: adjusting

text: the

text: constant

text: parameter

text: s

text: allows

text: to

text: control

text: the

text: sharpness

text: of

text: the

text: turn.

text: In

text: the

text: following,

text: we

text: ﬁx

text: the

text: parameters

text: of

text: the

text: potential

text: as

text: π

text: m

text: M

text: M

text: φ

text: M

text: δθ

text: s

text: M

text: φ

text: P

text: P

text: P

text: P

text: in

text: which

text: we

text: reintroduced

text: the

text: Planck

text: mass

text: for

text: clarity.

text: The

text: shape

text: of

text: the

text: potential

text: corre-

text: sponding

text: to

text: this

text: choice

text: is

text: shown

text: in

text: Figure

text: Indeed,

text: it

text: corresponds

text: to

text: a

text: slowly

text: descending

text: valley

text: with

text: steep

text: slopes

text: exhibiting

text: a

text: turn.

text: With

text: the

text: parameters

text: choice

text: the

text: mass

text: of

text: the

text: heavy

text: perturbation

text: (or

text: equivalently,

text: the

text: curvature

text: of

text: the

text: potential

text: across

text: the

text: bottom

text: of

text: the

text: valley)

text: is

text: twenty

text: times

text: larger

text: than

text: the

text: Hubble

text: parameter.

text: It

text: is

text: in

text: this

text: sense

text: that

text: we

text: can

text: refer

text: to

text: the

text: turn

text: as

text: sharp.

text: In

text: order

text: to

text: discuss

text: the

text: evolution

text: of

text: the

text: perturbations

text: numerically,

text: it

text: is

text: convenient

text: to

text: adopt

text: yet

text: another

text: time-dependent

text: basis

text: for

text: the

text: perturbations.

text: Following

text: the

text: same

text: prescription

text: as

text: Gao,

text: Langlois

text: Mizuno,

text: we

text: will

text: track

text: the

text: evolution

text: of

text: the

text: perturbations,

text: schematically

text: denoted

text: by

text: ϕ

text: in

text: the

text: directions

text: determined

text: by

text: the

text: eigenvectors

text: of

text: the

text: mass

text: matrix

text: I

text: A

text: notable

text: example

text: is

text: that

text: of

text: double

text: quadratic

text: inﬂation

text: (see,

text: for

text: example,

text: Vernizzi

text: Wands

text: It

text: is

text: well

text: known

text: that,

text: in

text: this

text: case,

text: there

text: are

text: appreciable

text: changes

text: to

text: the

text: power

text: spectrum,

text: as

text: well

text: as

text: other

text: observables,

text: such

text: as

text: the

text: scalar

text: spectral

text: index

text: and

text: local

text: non-gaussianity

text: as

text: measured

text: by

text: f

text: (see,

text: for

text: example,

text: Dias

text: Seery

text: NL

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1: This ﬁgure illustrates the corresponding approximation scheme. The dotted lines represent the degenerate tunnels. According to Assumption (B) on the process, the probability that a path stays in these tunnels for a positive time period is zero, a crucial fact used in the proof of Proposition 5.2. (cid:84) (cid:0) (cid:1) ∈ ↓ (cid:62) Let x Aε,δ, and δ be a sequence such that δ 0. Then for each n δ>0(cid:54) (cid:54)n ⊂ (cid:12) − n (cid:12) | − | (cid:62) 1ε., tBhyerecoemxipstasc0tnesssnw<e tcnan ﬁ1nsducah tshuabtse[sqnu,etnnc]e ((cid:12)(cid:12)sx 1, t T)ε,δonf (cid:12)(cid:12)(san,dt )xtsnuchxtshnat (s , t ) converges to som(cid:84)e (s, t). The condition xnl n−l x (cid:62)n εnthen implies nl nl tnl snl that s < t. Therefore, for ﬁxed u, v with s < u < v < t, there exists some N ∈ N such that [u, v] ⊂(cid:92) (cid:62) (cid:0)[s , t (cid:1)], and hence l N nl nl [u, v] ⊂ x−1 Tε,δnl (cid:26) (cid:27)  (cid:62) (cid:91) (cid:91) l N   − − R − × 2k 1 × R − = x 1 i 1 ε d i . ∈Z (cid:54) (cid:54) 2 (cid:0)(cid:84) (cid:1) k 1 i d In particular, this implies (5.4) and by Assumption (B) we have P Aε,δ = δ>0 0. (cid:26) (cid:27) Now we are going to show tha(cid:12)(cid:12)t for each(cid:12)(cid:12)ε, δ, √ ∈ − (cid:62) ⊂ x W : sup xε,δ x 11 dε Aε,δ. (5.5) (cid:54) (cid:54) u u 0 u 1 To see this, ﬁrst notice that if x belongs to the left hand side of (5.5), then e(cid:12)(cid:12)ither (cid:12)(cid:12) ∈ (cid:54) (cid:54) (1) there exi√sts some u [τε−,δ , τε,δ] for some 1 k Nε,δ, such that − (cid:62) k 1 k xε,δ x 11 dε; or u u 16 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: This

label: caption

text: ﬁgure

text: illustrates

text: the

text: corresponding

text: approximation

text: scheme.

text: The

text: dotted

text: lines

text: represent

text: the

text: degenerate

text: tunnels.

text: According

text: to

text: Assumption

text: (B)

text: on

text: the

text: process,

text: the

text: probability

text: that

text: a

text: path

text: stays

text: in

text: these

text: tunnels

text: for

text: a

text: positive

text: time

text: period

text: is

text: zero,

text: a

text: crucial

text: fact

text: used

text: in

text: the

text: proof

text: of

text: Proposition

text: Let

text: x

text: Aε,δ,

text: and

text: δ

text: be

text: a

text: sequence

text: such

text: that

text: δ

text: Then

text: for

text: each

text: n

text: n

text: tcnan

text: tshuabtse[sqnu,etnnc]e

text: t

text: T)ε,δonf

text: )xtsnuchxtshnat

text: (s

text: t

text: converges

text: to

text: (s,

text: t).

text: The

text: condition

text: xnl

text: n−l

text: x

text: εnthen

text: implies

text: nl

text: nl

text: tnl

text: snl

text: that

text: s

text: t.

text: Therefore,

text: for

text: ﬁxed

text: u,

text: v

text: with

text: s

text: u

text: v

text: t,

text: there

text: exists

text: some

text: N

text: N

text: such

text: that

text: [u,

text: v]

text: t

text: and

text: hence

text: l

text: N

text: nl

text: nl

text: [u,

text: v]

text: x−1

text: Tε,δnl

text: l

text: N

text: R

text: R

text: x

text: i

text: ε

text: d

text: i

text: ∈Z

text: k

text: i

text: d

text: In

text: particular,

text: this

text: implies

text: and

text: by

text: Assumption

text: (B)

text: we

text: have

text: P

text: Aε,δ

text: Now

text: we

text: are

text: going

text: to

text: show

text: for

text: δ,

text: x

text: W

text: sup

text: xε,δ

text: x

text: dε

text: Aε,δ.

text: u

text: u

text: u

text: To

text: see

text: this,

text: ﬁrst

text: notice

text: that

text: if

text: x

text: belongs

text: to

text: the

text: left

text: hand

text: side

text: of

text: then

text: there

text: exi√sts

text: some

text: u

text: [τε−,δ

text: τε,δ]

text: for

text: some

text: k

text: Nε,δ,

text: such

text: that

text: k

text: k

text: xε,δ

text: x

text: dε;

text: or

text: u

text: u

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:0) (cid:1) Deﬁnition(cid:110) 2.3. Given p (cid:62) 1. Let GΩ Rd denote the completion(cid:111)of the set p (cid:98) (cid:99) S(cid:98) (cid:99)(x) := π( p ) (S(x)) : x has bounded total variation p with respect to the p-variation metric d . GΩ (Rd) is called the space of geo- p p metric p-rough paths. In [8], Coutin and Qian showed that under certain conditions on the decor- (cid:0) (cid:1) relation of the increment of a Gaussian process, with probability one the lifting of the dyadic piecewise linear interpolation of the Gaussian sample paths in GΩ Rd is a Cauchy sequence under the p-variation metric. In [12], P. Friz p and N. Victoir extended this result to a larger class of Gaussian processes un- der certain regularity condition on the covariance function. Moreover, they showed that the lifting of any sequence of piecewise linear interpolation of the (cid:0) (cid:1) Gaussian sample paths in GΩ converges to the same limit. From here on- p wards, this limit will be known as the canonical lifting of the Gaussian process in GΩ Rd . A fundamental example of these results is fractional Brownian p motion with Hurst parameter H > 1/4. It follows from Theorem 2.1 that the signature S(x) ∈ T(Rd) of fractional Brownian motion with H > 1/4 is well- 0,1 deﬁned for almost surely through the canonical lifting. A detailed study on the geometric rough path nature of many interesting and important stochastic processes can be found in [12]. 3 Main Results In this section we are going to state main results of the paper and illustrate the idea of proofs. { ∈ } Let X = X : t [0, 1] be a d-dimensional continuous stochastic process t (cid:62) starting at the origin, where d 2. We will always assume that X is realized on the path space (W, B(W), P),where W is the space of Rd-valued continuous B paths over [0, 1] starting at the origin, (W) is the Borel σ-algebra over W, and P is the law of X. In the rest of this paper, we will make the following assumptions on the law P (cid:0) (cid:0)(cid:0) (cid:1)(cid:1)(cid:1) (cid:16) (cid:17) . Assumption (A): There exists a P-null set N and a map S : W\N → 0 0 C (cid:52); T Rd , such that for each x ∈ W\N and (s, t) ∈ ∆, π S (x) = − 0 1 s,t x x and S (x) is the multiplicative extension of some geometric rough path X (stee Tsheorem 2.1). We will call such a map S a P-almost sure lifting. The inte- grals with respect to x will then be deﬁned as integrating against the geometric rough path X. Assumption (B): For any 0 < t < 1, the law of x is absolutely continuous with respect to(cid:80)the Lebesgue measure. t Assumption (C): For any open cube H ⊂ Rd, there exists a diﬀerential one form φ = d φ dxi supported on the closure of H, such that for any (cid:54) (cid:54) i=1 i 0 s < t 1, if we let 5

text: Given

text: p

text: Let

text: GΩ

text: Rd

text: denote

text: the

text: the

text: set

text: p

text: π(

text: p

text: (S(x))

text: x

text: has

text: bounded

text: total

text: variation

text: p

text: with

text: respect

text: to

text: the

text: p-variation

text: metric

text: d

text: GΩ

text: (Rd)

text: is

text: called

text: the

text: space

text: of

text: geo-

text: p

text: p

text: metric

text: p-rough

text: paths.

text: In

text: Coutin

text: and

text: Qian

text: showed

text: that

text: under

text: certain

text: conditions

text: on

text: the

text: decor-

text: relation

text: of

text: the

text: increment

text: of

text: a

text: Gaussian

text: process,

text: with

text: probability

text: one

text: the

text: lifting

text: of

text: the

text: dyadic

text: piecewise

text: linear

text: interpolation

text: of

text: the

text: Gaussian

text: sample

text: paths

text: in

text: GΩ

text: Rd

text: is

text: a

text: Cauchy

text: sequence

text: under

text: the

text: p-variation

text: metric.

text: In

text: P.

text: Friz

text: p

text: and

text: N.

text: Victoir

text: extended

text: this

text: result

text: to

text: a

text: larger

text: class

text: of

text: Gaussian

text: processes

text: un-

text: der

text: certain

text: regularity

text: condition

text: on

text: the

text: covariance

text: function.

text: Moreover,

text: they

text: showed

text: that

text: the

text: lifting

text: of

text: any

text: sequence

text: of

text: piecewise

text: linear

text: interpolation

text: of

text: the

text: Gaussian

text: sample

text: paths

text: in

text: GΩ

text: converges

text: to

text: the

text: same

text: limit.

text: From

text: here

text: on-

text: p

text: wards,

text: this

text: limit

text: will

text: be

text: known

text: as

text: the

text: canonical

text: lifting

text: of

text: the

text: Gaussian

text: process

text: in

text: GΩ

text: Rd

text: A

text: fundamental

text: example

text: of

text: these

text: results

text: is

text: fractional

text: Brownian

text: p

text: motion

text: with

text: Hurst

text: parameter

text: H

text: It

text: follows

text: from

text: Theorem

text: that

text: the

text: signature

text: S(x)

text: T(Rd)

text: of

text: fractional

text: Brownian

text: motion

text: with

text: H

text: is

text: well-

text: deﬁned

text: for

text: almost

text: surely

text: through

text: the

text: canonical

text: lifting.

text: A

text: detailed

text: study

text: on

text: the

text: geometric

text: rough

text: path

text: nature

text: of

text: many

text: interesting

text: and

text: important

text: stochastic

text: processes

text: can

text: be

text: found

text: in

text: Main

text: Results

text: In

text: this

text: section

text: we

text: are

text: going

text: to

text: state

text: main

text: results

text: of

text: the

text: paper

text: and

text: illustrate

text: the

text: idea

text: of

text: proofs.

text: Let

text: X

text: X

text: t

text: be

text: a

text: d-dimensional

text: continuous

text: stochastic

text: process

text: t

text: starting

text: at

text: the

text: origin,

text: where

text: d

text: We

text: will

text: always

text: assume

text: that

text: X

text: is

text: realized

text: on

text: the

text: path

text: space

text: (W,

text: B(W),

text: P),where

text: W

text: is

text: the

text: space

text: of

text: Rd-valued

text: continuous

text: B

text: paths

text: over

text: starting

text: at

text: the

text: origin,

text: (W)

text: is

text: the

text: Borel

text: σ-algebra

text: over

text: W,

text: and

text: P

text: is

text: the

text: law

text: of

text: X.

text: In

text: the

text: rest

text: of

text: this

text: paper,

text: we

text: will

text: make

text: the

text: following

text: assumptions

text: on

text: the

text: law

text: P

text: Assumption

text: (A):

text: There

text: exists

text: a

text: P-null

text: set

text: N

text: and

text: a

text: map

text: S

text: W\N

text: C

text: T

text: Rd

text: such

text: that

text: for

text: each

text: x

text: W\N

text: and

text: (s,

text: t)

text: π

text: S

text: (x)

text: s,t

text: x

text: x

text: and

text: S

text: (x)

text: is

text: the

text: multiplicative

text: extension

text: of

text: some

text: geometric

text: rough

text: path

text: X

text: (stee

text: Tsheorem

text: We

text: will

text: call

text: such

text: a

text: map

text: S

text: a

text: P-almost

text: sure

text: lifting.

text: The

text: inte-

text: grals

text: with

text: respect

text: to

text: x

text: will

text: then

text: be

text: deﬁned

text: as

text: integrating

text: against

text: the

text: geometric

text: rough

text: path

text: X.

text: Assumption

text: (B):

text: For

text: any

text: t

text: the

text: law

text: of

text: x

text: is

text: absolutely

text: continuous

text: with

text: respect

text: Lebesgue

text: measure.

text: t

text: Assumption

text: (C):

text: For

text: any

text: open

text: cube

text: H

text: Rd,

text: there

text: exists

text: a

text: diﬀerential

text: one

text: form

text: φ

text: d

text: φ

text: dxi

text: supported

text: on

text: the

text: closure

text: of

text: H,

text: such

text: that

text: for

text: any

text: i

text: s

text: t

text: if

text: we

text: let


text: Theorem 3.2. Let P be the law of a Gaussian process satisfying conditions speciﬁed in Section 6. Then P satisﬁes Assumption (A), (B), (C). In particular, the result holds for fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge. Before going into the mathematical proofs, we ﬁrst describe the strategy informally. The approximation scheme we are going to use is an adaptation from the work of Le Jan and Qian [17]. However, the main diﬃculties are in the development of each step in the non-Markov setting, which will be clear in the detailed proofs. Step One. Prove that if two paths have the same signature, then the iterated integrals of the paths along any ﬁnite sequence of smooth one forms are the same. Following [17], these iterated integrals along one forms will be called extended signatures. Step Two. Decompose the Euclidean space Rd into disjoint identical open cubes with small tunnels between them. For each such cube, we deﬁne a diﬀer- ential one form supported on the closure of the cube according to Assumption (C). Step Three. Show that, for each path x outside a P-null set, the ordered sequence of cubes visited by x corresponds to the unique maximal sequence of diﬀerential one forms along which the extended signature of x is nonzero. This together with step one allows us to recover the ordered sequence of cubes visited by x from its signature. Step Four. Construct a polygonal approximation of x by joining the centers of cubes visited by x in order. This polygonal path will be parametrized so that it is at the center of the cube at the time when the cube is ﬁrst visited by x. Show that with probability one, as the size of cubes tends to zero, the polygonal path converges to the original path x under the uniform topology. Step Five. Since the signature is invariant under the reparametrization of the path, it is not possible to recover the exact visit times of the cubes. If two paths have the same signature, then the corresponding polygonal paths constructed in (3) are only equal up to a reparametrization. Therefore, we need to introduce a variant of the Fréchet distance on W measuring the distance of two paths modulo parametrization. We should also prove that outside a P-null set this is (cid:48) indeed a metric. It will then follow from step four that if two paths x and x have the same signature, their corresponding approximation paths converge to (cid:48) the same limit under this metric, which will imply that x and x are equal up to a reparametrization. For the Gaussian case, Assumption (A) is veriﬁed from [12] and Assumption (B) is trivial by deﬁnition. By using the Malliavin calculus, for each open cube ´ H we will explicitly construct a diﬀerential one form φ supported on H such → that the functional x t φ(dx ) has a density conditioned on the set AH . s u s,t This certainly veriﬁes Assumption (C). 7 ##LTLine##

text: Theorem

text: Let

text: P

text: be

text: the

text: law

text: of

text: a

text: Gaussian

text: process

text: satisfying

text: conditions

text: speciﬁed

text: in

text: Section

text: Then

text: P

text: satisﬁes

text: Assumption

text: (A),

text: (B),

text: (C).

text: In

text: particular,

text: the

text: result

text: holds

text: for

text: fractional

text: Brownian

text: motion

text: with

text: Hurst

text: parameter

text: H

text: the

text: Ornstein-Uhlenbeck

text: process

text: and

text: the

text: Brownian

text: bridge.

text: Before

text: going

text: into

text: the

text: mathematical

text: proofs,

text: we

text: ﬁrst

text: describe

text: the

text: strategy

text: informally.

text: The

text: approximation

text: scheme

text: we

text: are

text: going

text: to

text: use

text: is

text: an

text: adaptation

text: from

text: the

text: work

text: of

text: Le

text: Jan

text: and

text: Qian

text: However,

text: the

text: main

text: diﬃculties

text: are

text: in

text: the

text: development

text: of

text: each

text: step

text: in

text: the

text: non-Markov

text: setting,

text: which

text: will

text: be

text: clear

text: in

text: the

text: detailed

text: proofs.

text: Step

text: One.

text: Prove

text: that

text: if

text: two

text: paths

text: have

text: the

text: same

text: signature,

text: then

text: the

text: iterated

text: integrals

text: of

text: the

text: paths

text: along

text: any

text: ﬁnite

text: sequence

text: of

text: smooth

text: one

text: forms

text: are

text: the

text: same.

text: Following

text: these

text: iterated

text: integrals

text: along

text: one

text: forms

text: will

text: be

text: called

text: extended

text: signatures.

text: Step

text: Two.

text: Decompose

text: the

text: Euclidean

text: space

text: Rd

text: into

text: disjoint

text: identical

text: open

text: cubes

text: with

text: small

text: tunnels

text: between

text: them.

text: For

text: each

text: such

text: cube,

text: we

text: deﬁne

text: a

text: diﬀer-

text: ential

text: one

text: form

text: supported

text: on

text: the

text: closure

text: of

text: the

text: cube

text: according

text: to

text: Assumption

text: (C).

text: Step

text: Three.

text: Show

text: that,

text: for

text: each

text: path

text: x

text: outside

text: a

text: P-null

text: set,

text: the

text: ordered

text: sequence

text: of

text: cubes

text: visited

text: by

text: x

text: corresponds

text: to

text: the

text: unique

text: maximal

text: sequence

text: of

text: diﬀerential

text: one

text: forms

text: along

text: which

text: the

text: extended

text: signature

text: of

text: x

text: is

text: nonzero.

text: This

text: together

text: with

text: step

text: one

text: allows

text: us

text: to

text: recover

text: the

text: ordered

text: sequence

text: of

text: cubes

text: visited

text: by

text: x

text: from

text: its

text: signature.

text: Step

text: Four.

text: Construct

text: a

text: polygonal

text: approximation

text: of

text: x

text: by

text: joining

text: the

text: centers

text: of

text: cubes

text: visited

text: by

text: x

text: in

text: order.

text: This

text: polygonal

text: path

text: will

text: be

text: parametrized

text: so

text: that

text: it

text: is

text: at

text: the

text: center

text: of

text: the

text: cube

text: at

text: the

text: time

text: when

text: the

text: cube

text: is

text: ﬁrst

text: visited

text: by

text: x.

text: Show

text: that

text: with

text: probability

text: one,

text: as

text: the

text: size

text: of

text: cubes

text: tends

text: to

text: zero,

text: the

text: polygonal

text: path

text: converges

text: to

text: the

text: original

text: path

text: x

text: under

text: the

text: uniform

text: topology.

text: Step

text: Five.

text: Since

text: the

text: signature

text: is

text: invariant

text: under

text: the

text: reparametrization

text: of

text: the

text: path,

text: it

text: is

text: not

text: possible

text: to

text: recover

text: the

text: exact

text: visit

text: times

text: of

text: the

text: cubes.

text: If

text: two

text: paths

text: have

text: the

text: same

text: signature,

text: then

text: the

text: corresponding

text: polygonal

text: paths

text: constructed

text: in

text: are

text: only

text: equal

text: up

text: to

text: a

text: reparametrization.

text: Therefore,

text: we

text: need

text: to

text: introduce

text: a

text: variant

text: of

text: the

text: Fréchet

text: distance

text: on

text: W

text: measuring

text: the

text: distance

text: of

text: two

text: paths

text: modulo

text: parametrization.

text: We

text: should

text: also

text: prove

text: that

text: outside

text: a

text: P-null

text: set

text: this

text: is

text: indeed

text: a

text: metric.

text: It

text: will

text: then

text: follow

text: from

text: step

text: four

text: that

text: if

text: two

text: paths

text: x

text: and

text: x

text: have

text: the

text: same

text: signature,

text: their

text: corresponding

text: approximation

text: paths

text: converge

text: to

text: the

text: same

text: limit

text: under

text: this

text: metric,

text: which

text: will

text: imply

text: that

text: x

text: and

text: x

text: are

text: equal

text: up

text: to

text: a

text: reparametrization.

text: For

text: the

text: Gaussian

text: case,

text: Assumption

text: (A)

text: is

text: veriﬁed

text: from

text: and

text: Assumption

text: (B)

text: is

text: trivial

text: by

text: deﬁnition.

text: By

text: using

text: the

text: Malliavin

text: calculus,

text: for

text: each

text: open

text: cube

text: H

text: we

text: will

text: explicitly

text: construct

text: a

text: diﬀerential

text: one

text: form

text: φ

text: supported

text: on

text: H

text: such

text: that

text: the

text: functional

text: x

text: t

text: φ(dx

text: has

text: a

text: density

text: conditioned

text: on

text: the

text: set

text: AH

text: s

text: u

text: s,t

text: This

text: certainly

text: veriﬁes

text: Assumption

text: (C).

text: ##LTLine##


text: − − Figure 1: NO case. Scatter plot points in the planes δ m (left), m m (center), 1 ee 1 − − θ m (right) satisfying successful strong thermal leptogenesis for Np,−i = 10 1 (red), 13 1 B L − − 10 2 (green) and 10 3 (blu). In all panels the vertical gray band is the Planck m upper 1 bound eq. (3). In the left panel points are plotted for M = 2 and the red solid line is Ω − the analytic lower bound mlb(δ) (cf. eq. (25)) for Np,−i, = 10 1. While the points in the 1 B L left and central panels have been obtained for uniform random values of the three mixing angles generated within the 3σ ranges eq. (6), in the right panel they have been left free (the horizontal band indicates the 3 σ range in eq. (6) for θ ). In the central panel the 13 vertical lines indicate the m values above which 99% of scatter plot points are found (see 1 central panel in Fig. 4). the lower bound, we performed a scatter plot analysis in the space of the 13 parameters (m , 6 in U, 6 in Ω) for M = 1, 2, 5, 10. The results are shown in Fig. 1. for three values 1 Ω − − − of Np,−i = 10 1, 10 2, 10 3 (respectively the red, green and blue points). One can see that B L − for Np,−i = 10 1 the minimum values of m in the left panel at diﬀerent values of δ are B L 1 much higher than the analytic estimation (one has to compare the red points with the red solid line). The reason is due to the fact that the lower bound is saturated for very special | | | | choices of Ω such that max[ Ω2 ], max[ Ω2 ] are as close as possible to the maximum value 11 21 M but at the same time not to suppress too much the CP asymmetry ε needed to have Ω 2τ successful leptogenesis. This is conﬁrmed by Fig. 2 where in the three panels we have ≡ | | | | plotted β ε /ε¯(M ), Ω2 and Ω2 for M = 2. We have made a focused search (by 2τ 2τ 2 11 22 Ω ﬁne-tuning the parameters) managing to ﬁnd a point (the red diamond) where m is very 1 close to the lower bound. For th√is point β gets considerably reduced since it corresponds 2τ ∝ to a situation where the term m in the ﬂavoured decay parameters becomes negligible 1 and the strong thermal co√nditio√n is satisﬁed for a very special condition, basically the ∝ | | | | eq. (22) when the terms m , K are neglected in the right-hand side and Ω , Ω 1 1τ 11 21 become maximal, that leads to a CP asymmetry suppression. We have also performed a scatter plot letting the mixing angles to vary within the 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: NO

text: case.

text: Scatter

text: plot

text: points

text: in

text: the

text: planes

text: δ

text: m

text: (left),

text: m

text: m

text: (center),

text: ee

text: θ

text: m

text: (right)

text: satisfying

text: successful

text: strong

text: thermal

text: leptogenesis

text: for

text: Np,−i

text: (red),

text: B

text: L

text: (green)

text: and

text: (blu).

text: In

text: all

text: panels

text: the

text: vertical

text: gray

text: band

text: is

text: the

text: Planck

text: m

text: upper

text: bound

text: eq.

text: In

text: the

text: left

text: panel

text: points

text: are

text: plotted

text: for

text: M

text: and

text: the

text: red

text: solid

text: line

text: is

text: Ω

text: the

text: analytic

text: lower

text: bound

text: mlb(δ)

text: (cf.

text: eq.

text: for

text: Np,−i,

text: While

text: the

text: points

text: in

text: the

text: B

text: L

text: left

text: and

text: central

text: panels

text: have

text: been

text: obtained

text: for

text: uniform

text: random

text: values

text: of

text: the

text: three

text: mixing

text: angles

text: generated

text: within

text: the

text: 3σ

text: ranges

text: eq.

text: in

text: the

text: right

text: panel

text: they

text: have

text: been

text: left

text: free

text: (the

text: horizontal

text: band

text: indicates

text: the

text: σ

text: range

text: in

text: eq.

text: for

text: θ

text: In

text: the

text: central

text: panel

text: the

text: vertical

text: lines

text: indicate

text: the

text: m

text: values

text: above

text: which

text: of

text: scatter

text: plot

text: points

text: are

text: found

text: (see

text: central

text: panel

text: in

text: Fig.

text: the

text: lower

text: bound,

text: we

text: performed

text: a

text: scatter

text: plot

text: analysis

text: in

text: the

text: space

text: of

text: the

text: parameters

text: (m

text: in

text: U,

text: in

text: Ω)

text: for

text: M

text: The

text: results

text: are

text: shown

text: in

text: Fig.

text: for

text: three

text: values

text: Ω

text: of

text: Np,−i

text: (respectively

text: the

text: red,

text: green

text: and

text: blue

text: points).

text: One

text: can

text: see

text: that

text: B

text: L

text: for

text: Np,−i

text: the

text: minimum

text: values

text: of

text: m

text: in

text: the

text: left

text: panel

text: at

text: diﬀerent

text: values

text: of

text: δ

text: are

text: B

text: L

text: much

text: higher

text: than

text: the

text: analytic

text: estimation

text: (one

text: has

text: to

text: compare

text: the

text: red

text: points

text: with

text: the

text: red

text: solid

text: line).

text: The

text: reason

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: lower

text: bound

text: is

text: saturated

text: for

text: very

text: special

text: choices

text: of

text: Ω

text: such

text: that

text: max[

text: Ω2

text: max[

text: Ω2

text: are

text: as

text: close

text: as

text: possible

text: to

text: the

text: maximum

text: value

text: M

text: but

text: at

text: the

text: same

text: time

text: not

text: to

text: suppress

text: too

text: much

text: the

text: CP

text: asymmetry

text: ε

text: needed

text: to

text: have

text: Ω

text: 2τ

text: successful

text: leptogenesis.

text: This

text: is

text: conﬁrmed

text: by

text: Fig.

text: where

text: in

text: the

text: three

text: panels

text: we

text: have

text: plotted

text: β

text: ε

text: /ε¯(M

text: Ω2

text: and

text: Ω2

text: for

text: M

text: We

text: have

text: made

text: a

text: focused

text: search

text: (by

text: 2τ

text: 2τ

text: Ω

text: ﬁne-tuning

text: the

text: parameters)

text: managing

text: to

text: ﬁnd

text: a

text: point

text: (the

text: red

text: diamond)

text: where

text: m

text: is

text: very

text: close

text: to

text: the

text: lower

text: bound.

text: For

text: th√is

text: point

text: β

text: gets

text: considerably

text: reduced

text: since

text: it

text: corresponds

text: 2τ

text: to

text: a

text: situation

text: where

text: the

text: term

text: m

text: in

text: the

text: ﬂavoured

text: decay

text: parameters

text: becomes

text: negligible

text: and

text: the

text: strong

text: thermal

text: co√nditio√n

text: is

text: satisﬁed

text: for

text: a

text: very

text: special

text: condition,

text: basically

text: the

text: eq.

text: when

text: the

text: terms

text: m

text: K

text: are

text: neglected

text: in

text: the

text: right-hand

text: side

text: and

text: Ω

text: Ω

text: 1τ

text: become

text: maximal,

text: that

text: leads

text: to

text: a

text: CP

text: asymmetry

text: suppression.

text: We

text: have

text: also

text: performed

text: a

text: scatter

text: plot

text: letting

text: the

text: mixing

text: angles

text: to

text: vary

text: within

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≡ | | Figure 2: NO case. Results of the scatter plots for M = 2 for β ε /ε¯(M ), Ω2 | | Ω 2τ 2τ 2 11 and Ω2 versus m (same colour code as in Fig. 1). 21 1 Figure 3: NO case. Results of the scatter plots for M = 2 for the four relevant ﬂavoured Ω decay parameters K , K , K , K versus m (conventions as in Fig. 1). The horizontal 2τ 1τ 1µ 1e 1 (cid:39) dashed line indicates the value K (Np,i = 0.03) 13 (cf. eq. (10)). st ∆α whole range of physical values with no experimental constraints. In the right panel of − Fig. 1 we show the results in the plane m θ . One can see how the smallness of θ 1 13 13 is crucial for the existence of the lower bound. This can be well understood analytically ∝ | | considering that in the expression for K0,max there are two terms U 2 (cf. eq. (26)). 1e e3 In Fig. 3 we also show the results for the values of the three K (α = e, µ, τ) and for 1α K , the four relevant ﬂavoured decay parameters, for M = 2. First of all one can see 2τ Ω how the values of the ﬂavoured decay parameters respect the strong thermal conditions eq. (19). However, the most important plot is that one for K , showing how for values 1e (cid:46) m 10 meV the maximum value of K gets considerably reduced until it falls below 1 1e K , indicated by the horizontal dashed line for Np,−i = 0.1, at the m lower bound st B L 1 value (very closely realised by the red diamond point). It is also clear that already below ∼ 10 meV the possibility to realise strong thermal leptogenesis requires a high ﬁne tuning (cid:46) (cid:39) (cid:46) in the parameters since in this case K K0,max 4 M K for large asymmetries 1e 1e Ω st and not too unreasonably high values of M . This is well illustrated in Fig. 4 where we Ω plotted the distribution of the m values from the scatter plots for M = 1, 2, 5, 10 and 1 Ω 12 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: NO

text: case.

text: Results

text: of

text: the

text: scatter

text: plots

text: for

text: M

text: for

text: β

text: ε

text: /ε¯(M

text: Ω2

text: Ω

text: 2τ

text: 2τ

text: and

text: Ω2

text: versus

text: m

text: (same

text: colour

text: code

text: as

text: in

text: Fig.

text: Figure

text: NO

text: case.

text: Results

text: of

text: the

text: scatter

text: plots

text: for

text: M

text: for

text: the

text: four

text: relevant

text: ﬂavoured

text: Ω

text: decay

text: parameters

text: K

text: K

text: K

text: K

text: versus

text: m

text: (conventions

text: as

text: in

text: Fig.

text: The

text: horizontal

text: 2τ

text: 1τ

text: 1µ

text: dashed

text: line

text: indicates

text: the

text: value

text: K

text: (Np,i

text: (cf.

text: eq.

text: st

text: ∆α

text: whole

text: range

text: of

text: physical

text: values

text: with

text: no

text: experimental

text: constraints.

text: In

text: the

text: right

text: panel

text: of

text: Fig.

text: we

text: show

text: the

text: results

text: in

text: the

text: plane

text: m

text: θ

text: One

text: can

text: see

text: how

text: the

text: smallness

text: of

text: θ

text: is

text: crucial

text: for

text: the

text: existence

text: of

text: the

text: lower

text: bound.

text: This

text: can

text: be

text: well

text: understood

text: analytically

text: considering

text: that

text: in

text: the

text: expression

text: for

text: there

text: are

text: two

text: terms

text: U

text: (cf.

text: eq.

text: In

text: Fig.

text: we

text: also

text: show

text: the

text: results

text: for

text: the

text: values

text: of

text: the

text: three

text: K

text: (α

text: e,

text: µ,

text: τ)

text: and

text: for

text: 1α

text: K

text: the

text: four

text: relevant

text: ﬂavoured

text: decay

text: parameters,

text: for

text: M

text: First

text: of

text: all

text: one

text: can

text: see

text: 2τ

text: Ω

text: how

text: the

text: values

text: of

text: the

text: ﬂavoured

text: decay

text: parameters

text: respect

text: the

text: strong

text: thermal

text: conditions

text: eq.

text: However,

text: the

text: most

text: important

text: plot

text: is

text: that

text: one

text: for

text: K

text: showing

text: how

text: for

text: values

text: m

text: meV

text: the

text: maximum

text: value

text: of

text: K

text: gets

text: considerably

text: reduced

text: until

text: it

text: falls

text: below

text: K

text: indicated

text: by

text: the

text: horizontal

text: dashed

text: line

text: for

text: Np,−i

text: at

text: the

text: m

text: lower

text: bound

text: st

text: B

text: L

text: value

text: (very

text: closely

text: realised

text: by

text: the

text: red

text: diamond

text: point).

text: It

text: is

text: also

text: clear

text: that

text: already

text: below

text: meV

text: the

text: possibility

text: to

text: realise

text: strong

text: thermal

text: leptogenesis

text: requires

text: a

text: high

text: ﬁne

text: tuning

text: in

text: the

text: parameters

text: since

text: in

text: this

text: case

text: K

text: M

text: K

text: for

text: large

text: asymmetries

text: Ω

text: st

text: and

text: not

text: too

text: unreasonably

text: high

text: values

text: of

text: M

text: This

text: is

text: well

text: illustrated

text: in

text: Fig.

text: where

text: we

text: Ω

text: plotted

text: the

text: distribution

text: of

text: the

text: m

text: values

text: from

text: the

text: scatter

text: plots

text: for

text: M

text: and

text: Ω

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: The terms ∆p , ∆p and ∆p , with ∆p + ∆p + ∆p = 0, take into account the pe pµ pτ pe pµ pτ possibility of diﬀerent ﬂavour compositions of the pre-existing leptons and anti-leptons. This would lead to initial values of the pre-existing α asymmetries that are not necessarily just a fraction of Np,−i . The presence of these terms depends on the speciﬁc mechanism B L that produced the pre-existing asymmetry. For example in leptogenesis itself they are in general present, they are the so called phantom terms. However, this indeﬁniteness has just a very small eﬀect on the results. If the ∆p -terms are not present, then in pα − − (cid:28) principle very special ﬂavour conﬁgurations with 1 p0 ⊥, 1 p0 ⊥ 1 could also lead to eτ µτ (cid:29) 2 2 a wash-out of the pre-existing asymmetries without the need to impose K , K 1. We 1e 1µ will comment on this possibility but for the time being we will assume that these terms are present. In this case the condition of successful strong thermal leptogenesis translates into the straightforward set of conditions (cid:38) (cid:38) (cid:46) K , K K (Np,i ), K K (Np,i ), K 1 . (19) 1e 1µ st ∆e,µ 2τ st ∆τ 1τ These conditions guarantee a washout of the electron and muon asymmetries, only possi- (cid:28) ble in the three-ﬂavoured regime at T 109 GeV, and at the same time also a wash-out of the tauon asymmetry in the two-ﬂavoured regime. The latter is still compatible with a generation of a sizeable tauon asymmetry from N decays. This is the only possibility 2 [13]. It should be noticed that in the N -dominated scenario the existence of the heaviest 2 RH neutrino N is necessary in order to have an interference of tree level N decays with 3 2 one-loop N decay graphs containing virtual N yielding suﬃciently large ε . There- 2 3 2α (cid:28) fore, within the N -dominated scenario, where by deﬁnition M 109 GeV, one has a 2 1 phenomenological reason to have at least three RH neutrino species [15]. 6 (cid:46) × 3.2 Case M 5 1011 GeV 3 (cid:46) × If M 5 1011 GeV, then the heaviest RH neutrinos N can contribute to wash-out 3 3 the tauon component together with the next-to-lightest RH neutrinos N . In this way, 2 for the relic value of the pre-existing asymmetry, one obtains (α = e, µ) − N∆p,τf = (p0pτ + ∆ppτ) e 38π (K1τ+K2τ+K3τ) NBp,−i L , (20) → ∞ 6 In the limit M , when N decouples and a two RH neutrino scenario is eﬀectively recovered with 3 3 → m = 0, one has β 0 (cf. eq. (15)). In this limit the only possibility to realise successful leptogenesis 1 2α is to have sizeable CP asymmetries from the interference terms with the lightest RH neutrinos that we ∝ neglected when we wrote eq. (15). These terms are M and successful leptogenesis necessarily requires 1 (cid:38) × in the end a lower bound M 2 1010 GeV [25]. However, then in this case the N -produced asymmetry 1 1 not only cannot be neglected but typically dominates on the N -produced asymmetry and moreover, more 2 importantly for us, strong thermal leptogenesis cannot be realised [13]. This well illustrates that in the N -dominated scenario, the presence of a (coupled) N is necessary for successful leptogenesis. 2 3 8 ##LTLine## ##LTLine##

text: The

text: terms

text: ∆p

text: ∆p

text: and

text: ∆p

text: with

text: ∆p

text: ∆p

text: ∆p

text: take

text: into

text: account

text: the

text: pe

text: pµ

text: pτ

text: pe

text: pµ

text: pτ

text: possibility

text: of

text: diﬀerent

text: ﬂavour

text: compositions

text: of

text: the

text: pre-existing

text: leptons

text: and

text: anti-leptons.

text: This

text: would

text: lead

text: to

text: initial

text: values

text: of

text: the

text: pre-existing

text: α

text: asymmetries

text: that

text: are

text: not

text: necessarily

text: just

text: a

text: fraction

text: of

text: Np,−i

text: The

text: presence

text: of

text: these

text: terms

text: depends

text: on

text: the

text: speciﬁc

text: mechanism

text: B

text: L

text: that

text: produced

text: the

text: pre-existing

text: asymmetry.

text: For

text: example

text: in

text: leptogenesis

text: itself

text: they

text: are

text: in

text: general

text: present,

text: they

text: are

text: the

text: so

text: called

text: phantom

text: terms.

text: However,

text: this

text: indeﬁniteness

text: has

text: just

text: a

text: very

text: small

text: eﬀect

text: on

text: the

text: results.

text: If

text: the

text: ∆p

text: -terms

text: are

text: not

text: present,

text: then

text: in

text: pα

text: principle

text: very

text: special

text: ﬂavour

text: conﬁgurations

text: with

text: could

text: also

text: lead

text: to

text: eτ

text: µτ

text: a

text: wash-out

text: of

text: the

text: pre-existing

text: asymmetries

text: without

text: the

text: need

text: to

text: impose

text: K

text: K

text: We

text: 1µ

text: will

text: comment

text: on

text: this

text: possibility

text: but

text: for

text: the

text: time

text: being

text: we

text: will

text: assume

text: that

text: these

text: terms

text: are

text: present.

text: In

text: this

text: case

text: the

text: condition

text: of

text: successful

text: strong

text: thermal

text: leptogenesis

text: translates

text: into

text: the

text: straightforward

text: set

text: of

text: conditions

text: K

text: K

text: K

text: (Np,i

text: K

text: K

text: (Np,i

text: K

text: 1µ

text: st

text: ∆e,µ

text: 2τ

text: st

text: ∆τ

text: 1τ

text: These

text: conditions

text: guarantee

text: a

text: washout

text: of

text: the

text: electron

text: and

text: muon

text: asymmetries,

text: only

text: possi-

text: ble

text: in

text: the

text: three-ﬂavoured

text: regime

text: at

text: T

text: GeV,

text: and

text: at

text: the

text: same

text: time

text: also

text: a

text: wash-out

text: of

text: the

text: tauon

text: asymmetry

text: in

text: the

text: two-ﬂavoured

text: regime.

text: The

text: latter

text: is

text: still

text: compatible

text: with

text: a

text: generation

text: of

text: a

text: sizeable

text: tauon

text: asymmetry

text: from

text: N

text: decays.

text: This

text: is

text: the

text: only

text: possibility

text: It

text: should

text: be

text: noticed

text: that

text: in

text: the

text: N

text: -dominated

text: scenario

text: the

text: existence

text: of

text: the

text: heaviest

text: RH

text: neutrino

text: N

text: is

text: necessary

text: in

text: order

text: to

text: have

text: an

text: interference

text: of

text: tree

text: level

text: N

text: decays

text: with

text: one-loop

text: N

text: decay

text: graphs

text: containing

text: virtual

text: N

text: yielding

text: suﬃciently

text: large

text: ε

text: There-

text: 2α

text: fore,

text: within

text: the

text: N

text: -dominated

text: scenario,

text: where

text: by

text: deﬁnition

text: M

text: GeV,

text: one

text: has

text: a

text: phenomenological

text: reason

text: to

text: have

text: at

text: least

text: three

text: RH

text: neutrino

text: species

text: Case

text: M

text: GeV

text: If

text: M

text: GeV,

text: then

text: the

text: heaviest

text: RH

text: neutrinos

text: N

text: can

text: contribute

text: to

text: wash-out

text: the

text: tauon

text: component

text: together

text: with

text: the

text: next-to-lightest

text: RH

text: neutrinos

text: N

text: In

text: this

text: way,

text: for

text: the

text: relic

text: value

text: of

text: the

text: pre-existing

text: asymmetry,

text: one

text: obtains

text: (α

text: e,

text: µ)

text: N∆p,τf

text: ∆ppτ)

text: e

text: NBp,−i

text: L

text: In

text: the

text: limit

text: M

text: when

text: N

text: decouples

text: and

text: a

text: two

text: RH

text: neutrino

text: scenario

text: is

text: eﬀectively

text: recovered

text: with

text: m

text: one

text: has

text: β

text: (cf.

text: eq.

text: In

text: this

text: limit

text: the

text: only

text: possibility

text: to

text: realise

text: successful

text: leptogenesis

text: 2α

text: is

text: to

text: have

text: sizeable

text: CP

text: asymmetries

text: from

text: the

text: interference

text: terms

text: with

text: the

text: lightest

text: RH

text: neutrinos

text: that

text: we

text: neglected

text: when

text: we

text: wrote

text: eq.

text: These

text: terms

text: are

text: M

text: and

text: successful

text: leptogenesis

text: necessarily

text: requires

text: in

text: the

text: end

text: a

text: lower

text: bound

text: M

text: GeV

text: However,

text: then

text: in

text: this

text: case

text: the

text: N

text: -produced

text: asymmetry

text: not

text: only

text: cannot

text: be

text: neglected

text: but

text: typically

text: dominates

text: on

text: the

text: N

text: -produced

text: asymmetry

text: and

text: moreover,

text: more

text: importantly

text: for

text: us,

text: strong

text: thermal

text: leptogenesis

text: cannot

text: be

text: realised

text: This

text: well

text: illustrates

text: that

text: in

text: the

text: N

text: -dominated

text: scenario,

text: the

text: presence

text: of

text: a

text: (coupled)

text: N

text: is

text: necessary

text: for

text: successful

text: leptogenesis.

text: ##LTLine##

text: ##LTLine##


text: Aﬃne extractors over large ﬁelds with exponential error ∗ † ‡ Jean Bourgain Zeev Dvir Ethan Leeman Abstract We describe a construction of explicit aﬃne extractors over large ﬁnite ﬁelds with expo- nentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions. 1 Introduction F (cid:55)→ { } F An aﬃne extractor is a mapping E : n 0, 1 m, with the ﬁeld of q elements, such that ⊂ F q q for any subspace V n of some ﬁxed dimension k, the output of E on a uniform sample from q V is distributed close to uniformly over the image. More precisely, if X is a random variable V distributed uniformly on V , then E(X ) is ε-close, in statistical distance1, to the uniform { } V distribution over 0, 1 m (here, and in the following, we will often identify a random variable with its distribution). It is easy to show that a random function E will be an aﬃne extractor. However, constructing explicit families of aﬃne extractors is a challenging problem which is still open for many settings of the parameters. By explicit, we mean that the mapping E can be computed deterministically and eﬃciently, given the parameters n, k and q. The task of constructing explicit aﬃne extractor is an instance of a more general set of prob- lems in which one has a combinatorial or algebraic object possessing certain ‘nice’ properties, one would expect to have in a random (or generic) object, and wishes to come up with an explicit instance of such an object. Other examples include expander graphs [RVW02, LPS88], Ramsey graphs [BRSW06], Error correcting codes, and other variants of algebraic extractors (e.g., ex- tractors for polynomial sources [DGW09, BSG12] or varieties [Dvi12]). Explicit constructions ∗ School of Mathematics, Institute for Advanced Study, Princeton, NJ. Email: bourgain@ias.edu. Research supported by NSF grant DMS 1301619. † Department of Computer Science and Department of Mathematics, Princeton University, Princeton NJ. Email: zeev.dvir@gmail.com. Research partially supported by NSF grants CCF-0832797, CCF-1217416 and by the Sloan fellowship. ‡ Department of Mathematics, University of Texas at Austin. Email: eleeman@math.utexas.edu 1The statistical distance between two distributions P and Q on a ﬁnite domain Ω is deﬁned as | − | max ⊆ P(S) Q(S) . We say that P is ε-close to Q if the statistical distance between P and Q is at most S Ω ε. 1 ##LTLine##

text: Aﬃne

label: title

text: extractors

text: over

text: large

text: ﬁelds

text: with

text: exponential

text: error

label: author

text: Jean

text: Bourgain

text: Zeev

text: Dvir

text: Ethan

text: Leeman

text: Abstract

text: We

label: abstract

text: describe

text: a

text: construction

text: of

text: explicit

text: aﬃne

text: extractors

text: over

text: large

text: ﬁnite

text: ﬁelds

text: with

text: expo-

text: nentially

text: small

text: error

text: and

text: linear

text: output

text: length.

text: Our

text: construction

text: relies

text: on

text: a

text: deep

text: theorem

text: of

text: Deligne

text: giving

text: tight

text: estimates

text: for

text: exponential

text: sums

text: over

text: smooth

text: varieties

text: in

text: high

text: dimensions.

label: section

text: Introduction

text: F

text: F

text: An

text: aﬃne

text: extractor

text: is

text: a

text: mapping

text: E

text: n

text: m,

text: with

text: the

text: ﬁeld

text: of

text: q

text: elements,

text: such

text: that

text: F

text: q

text: q

text: for

text: any

text: subspace

text: V

text: n

text: of

text: some

text: ﬁxed

text: dimension

text: k,

text: the

text: output

text: of

text: E

text: on

text: a

text: uniform

text: sample

text: from

text: q

text: V

text: is

text: distributed

text: close

text: to

text: uniformly

text: over

text: the

text: image.

text: More

text: precisely,

text: if

text: X

text: is

text: a

text: random

text: variable

text: V

text: distributed

text: uniformly

text: on

text: V

text: then

text: E(X

text: is

text: ε-close,

text: in

text: statistical

text: to

text: the

text: uniform

text: V

text: distribution

text: over

text: m

text: (here,

text: and

text: in

text: the

text: following,

text: we

text: will

text: often

text: identify

text: a

text: random

text: variable

text: with

text: its

text: distribution).

text: It

text: is

text: easy

text: to

text: show

text: that

text: a

text: random

text: function

text: E

text: will

text: be

text: an

text: aﬃne

text: extractor.

text: However,

text: constructing

text: explicit

text: families

text: of

text: aﬃne

text: extractors

text: is

text: a

text: challenging

text: problem

text: which

text: is

text: still

text: open

text: for

text: many

text: settings

text: of

text: the

text: parameters.

text: By

text: explicit,

text: we

text: mean

text: that

text: the

text: mapping

text: E

text: can

text: be

text: computed

text: deterministically

text: and

text: eﬃciently,

text: given

text: the

text: parameters

text: n,

text: k

text: and

text: q.

text: The

text: task

text: of

text: constructing

text: explicit

text: aﬃne

text: extractor

text: is

text: an

text: instance

text: of

text: a

text: more

text: general

text: set

text: of

text: prob-

text: lems

text: in

text: which

text: one

text: has

text: a

text: combinatorial

text: or

text: algebraic

text: object

text: possessing

text: certain

text: ‘nice’

text: properties,

text: one

text: would

text: expect

text: to

text: have

text: in

text: a

text: random

text: (or

text: generic)

text: object,

text: and

text: wishes

text: to

text: come

text: up

text: with

text: an

text: explicit

text: instance

text: of

text: such

text: an

text: object.

text: Other

text: examples

text: include

text: expander

text: graphs

text: Ramsey

text: graphs

text: Error

text: correcting

text: codes,

text: and

text: other

text: variants

text: of

text: algebraic

text: extractors

text: (e.g.,

text: ex-

text: tractors

text: for

text: polynomial

text: sources

text: or

text: varieties

text: Explicit

text: constructions

text: School

text: of

text: Mathematics,

text: Institute

text: for

text: Advanced

text: Study,

text: Princeton,

text: NJ.

text: Email:

text: bourgain@ias.edu.

text: Research

text: supported

text: by

text: NSF

text: grant

text: DMS

text: Department

text: of

text: Computer

text: Science

text: and

text: Department

text: of

text: Mathematics,

text: Princeton

text: University,

text: Princeton

text: NJ.

text: Email:

text: zeev.dvir@gmail.com.

text: Research

text: partially

text: supported

text: by

text: NSF

text: grants

text: and

text: by

text: the

text: Sloan

text: fellowship.

text: Department

text: of

text: Mathematics,

text: University

text: of

text: Texas

text: at

text: Austin.

text: Email:

text: eleeman@math.utexas.edu

text: statistical

text: distance

text: between

text: two

text: distributions

text: P

text: and

text: Q

text: on

text: a

text: ﬁnite

text: domain

text: Ω

text: is

text: deﬁned

text: as

text: max

text: P(S)

text: Q(S)

text: We

text: say

text: that

text: P

text: is

text: ε-close

text: to

text: Q

text: if

text: the

text: statistical

text: distance

text: between

text: P

text: and

text: Q

text: is

text: at

text: most

text: S

text: Ω

text: ε.

text: ##LTLine##


text: 2 T. Richardson & M. Fairbairn niﬁcant [as they are in Sculptor (Amorisco & Evans 2012b)]. ≡ − σ2(r) Previous work (Richardson & Fairbairn 2013) demonstrated β(r) 1 t , (1) that in general analogues to the Jeans equations for fourth 2σ2(r) r moments of the velocity distribution (Merriﬁeld & Kent where σ2 and σ2 = 2σ2 = 2σ2 represent the variance 1990) cannot be solved with the anisotropy parameter β r t θ φ of the radial and tangential velocity distributions respec- and gravitational potential alone. For a unique solution we tively, imply diﬀerent gravitational potentials. Since we can- must specify a new parameter such as not measure β, we cannot ultimately know what the gravi- (cid:104) (cid:105) tational potential is (Wolf et al. 2010). (cid:48) − 3 v2v2 β = 1 (cid:104)r (cid:105)t (2) Because of the recent improvements in data sets for 2 v4 r dwarf spheroidals there has recently been a number of new (cid:104) (cid:105) (cid:104) (cid:105) dynamical methods that supersede the standard Jeans anal- that relates fourth order moments v4 and v2v2 = (cid:104) (cid:105) (cid:104) (cid:105) r r t ysis. Metallicity information for stars in dwarf spheroidals 2 v2v2 = 2 v2v2 and therefore allows us to characterize r θ r φ can separate them into distinct stellar sub-components the relative shape (i.e. relative kurtosis as opposed to rela- (Battaglia et al. 2006; Walker et al. 2009; Amorisco & Evans tive dispersion) of the radial and tangential velocity distri- 2012a). By using the Jeans equation to simultaneously ﬁt butions. Eﬀorts to break the classical degeneracy with β are two stellar sub-populations to the same potential, Battaglia therefore compromised by a new degeneracy of solutions for (cid:48) et al. (2008) were able to break the β degeneracy to some diﬀerent choices of β . extent in Sculptor and found that rising anisotropy models In this work we consider a simpler solution that is not and cored density proﬁles were favoured. Subsequent stud- compromised in this way. Higher order analogues of the pro- ies with the projected Virial theorem (Amorisco et al. 2013; jected Virial theorem 2K +W = 0 have been derived (Mer- z z Agnello & Evans 2012) and β-independent mass estimators riﬁeld & Kent 1990; Kent 1990) that give global constraints (Walker & Penarrubia 2011) were able to exclude cuspy den- on velocity moments by eﬀectively integrating the spherical sity proﬁles in Fornax and Sculptor with a high statistical Jeans equations over all radii. While this results in a loss signiﬁcance. Use of the projected Virial theorem on multiple of spatial information it can be used instead of the Jeans tracer populations of globular clusters has also recently been equations to ﬁnd out what density proﬁles are compatible applied to nearby elliptical galaxies (Agnello et al. 2014). with the LOS velocity data. Interestingly, fourth moment The simplicity of these analytic methods coupled with claims analogues to the projected Virial theorem do not depend on (Laporte et al. 2013) that they can be successful on simula- the relative shape of the radial and tangential velocity distri- (cid:48) tions of triaxial haloes (though see Kowalczyk et al. (2013) butions β (a fourth moment quantity) but only the relative for a challenge of this result) gives compelling evidence for widths β (a second moment quantity) and the gravitational cored proﬁles in dwarf spheroidals. potential Φ. As such the potential terms for fourth order pro- In spite of this, a lack of independent evidence from jected Virial theorem estimators can be evaluated with the alternative single-component approaches to support these same parameter set as the second order Jeans equation with- (cid:48) claims has left the cusp/core debate in dwarf spheroidal out introducing new parameters such as β . In essence, the galaxies open. Strigari et al. (2010) claimed that DM haloes fourth order Virial equations give us two extra constraints from the Aquarius simulation can provide an acceptable on the LOS data for free and without loss of generality. It single-component ﬁt to the photometric and kinematic data has been shown that this result extends to nth order (Kent in Milky Way dwarf spheroidals. Other single-component 1990) such that n Virial equations for moments of the LOS (cid:104) (cid:105) analyses with Bayesian hierarchical modelling (Martinez velocity distribution v2n are independent of the nth order z 2013) and Schwarzschild orbit-based modelling (Breddels & anisotropy parameter β (see Richardson & Fairbairn (2013) n Helmi 2013; Jardel & Gebhardt 2013) ﬁnd no strong ev- for deﬁnition) and depend only on parameters β with j < n. j idence for universal cusps or cores when considering the In this work, we will investigate the use of higher or- Milky Way’s dwarf spheroidals as a whole. In contrast to der analogues of the projected Virial theorem in conjunc- these sophisticated techniques, our approach is to try and tion with conventional Jeans analyses to try and break the § ﬁnd the simplest analytic method possible that can break classic degeneracy problem. In 2, we will introduce two new the β degeneracy with a single tracer component and with- shape parameters ζ and ζ which are constructed from the A B out a loss of generality. fourth order projected Virial theorems and which replace the By ﬁtting to the kurtosis of LOS velocity data in addi- kurtosis as a measure of the shape of the LOS velocity dis- tion to the velocity dispersion, there is an extra constraint tribution that can be estimated from the data. We test the that can distinguish between diﬀerent ﬁts to the standard statistical properties of these Virial shape parameters with Jeans equation. Joint ﬁts to the dispersion and kurtosis simulated data and show that these new measures seem to with second and fourth order Jeans equations (Merriﬁeld & be more robust to systematics than the kurtosis in radial Kent 1990) have been conducted on galaxy clusters (L(cid:32) okas bins. § & Mamon 2003), dwarf spheroidals (L(cid:32) okas et al. 2005) and In 3, we will perform an illustrative analysis on Sculp- recently giant elliptical galaxies (Napolitano et al. 2014). tor with constant β to highlight the key properties of the State-of-the-art observations of Milky Way dwarf spheroidal Virial shape parameters and to gain an intuition for how galaxies (Helmi et al. 2006; Walker et al. 2009) now oﬀer they can break the β degeneracy. Finally, we do a gener- § LOS velocities and projected radii for thousands of tracer alized analysis in 4 with non-constant β to see which den- stars. As data sets continue to expand, the inclusion of sity proﬁles are consistent with the Jeans equation and both higher velocity moments is not only possible but also useful, fourth order projected Virial theorems. Implications of key § especially if deviations from Gaussianity are statistically sig- assumptions such as spherical symmetry are discussed in 5. (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine##

text: T.

text: Richardson

text: M.

text: Fairbairn

text: niﬁcant

text: [as

text: they

text: are

text: in

text: Sculptor

text: (Amorisco

text: Evans

text: Previous

text: work

text: (Richardson

text: Fairbairn

text: demonstrated

text: β(r)

text: t

text: that

text: in

text: general

text: analogues

text: to

text: the

text: Jeans

text: equations

text: for

text: fourth

text: r

text: moments

text: of

text: the

text: velocity

text: distribution

text: (Merriﬁeld

text: Kent

text: where

text: σ2

text: and

text: σ2

text: represent

text: the

text: variance

text: cannot

text: be

text: solved

text: with

text: the

text: anisotropy

text: parameter

text: β

text: r

text: t

text: θ

text: φ

text: of

text: the

text: radial

text: and

text: tangential

text: velocity

text: distributions

text: respec-

text: and

text: gravitational

text: potential

text: alone.

text: For

text: a

text: unique

text: solution

text: we

text: tively,

text: imply

text: diﬀerent

text: gravitational

text: potentials.

text: Since

text: we

text: can-

text: must

text: specify

text: a

text: new

text: parameter

text: such

text: as

text: not

text: measure

text: β,

text: we

text: cannot

text: ultimately

text: know

text: what

text: the

text: gravi-

text: tational

text: potential

text: is

text: (Wolf

text: et

text: al.

text: β

text: Because

text: of

text: the

text: recent

text: improvements

text: in

text: data

text: sets

text: for

text: r

text: dwarf

text: spheroidals

text: there

text: has

text: recently

text: been

text: a

text: number

text: of

text: new

text: dynamical

text: methods

text: that

text: supersede

text: the

text: standard

text: Jeans

text: anal-

text: that

text: relates

text: fourth

text: order

text: moments

text: and

text: r

text: r

text: t

text: ysis.

text: Metallicity

text: information

text: for

text: stars

text: in

text: dwarf

text: spheroidals

text: and

text: therefore

text: allows

text: us

text: to

text: characterize

text: r

text: θ

text: r

text: φ

text: can

text: separate

text: them

text: into

text: distinct

text: stellar

text: sub-components

text: the

text: relative

text: shape

text: (i.e.

text: relative

text: kurtosis

text: as

text: opposed

text: to

text: rela-

text: (Battaglia

text: et

text: al.

text: Walker

text: et

text: al.

text: Amorisco

text: Evans

text: tive

text: dispersion)

text: of

text: the

text: radial

text: and

text: tangential

text: velocity

text: distri-

text: By

text: using

text: the

text: Jeans

text: equation

text: to

text: simultaneously

text: ﬁt

text: butions.

text: Eﬀorts

text: to

text: break

text: the

text: classical

text: degeneracy

text: with

text: β

text: are

text: two

text: stellar

text: sub-populations

text: to

text: the

text: same

text: potential,

text: Battaglia

text: therefore

text: compromised

text: by

text: a

text: new

text: degeneracy

text: of

text: solutions

text: for

text: et

text: al.

text: were

text: able

text: to

text: break

text: the

text: β

text: degeneracy

text: to

text: some

text: diﬀerent

text: choices

text: of

text: β

text: extent

text: in

text: Sculptor

text: and

text: found

text: that

text: rising

text: anisotropy

text: models

text: In

text: this

text: work

text: we

text: consider

text: a

text: simpler

text: solution

text: that

text: is

text: not

text: and

text: cored

text: density

text: proﬁles

text: were

text: favoured.

text: Subsequent

text: stud-

text: compromised

text: in

text: this

text: way.

text: Higher

text: order

text: analogues

text: of

text: the

text: pro-

text: ies

text: with

text: the

text: projected

text: Virial

text: theorem

text: (Amorisco

text: et

text: al.

text: jected

text: Virial

text: theorem

text: +W

text: have

text: been

text: derived

text: (Mer-

text: z

text: z

text: Agnello

text: Evans

text: and

text: β-independent

text: mass

text: estimators

text: riﬁeld

text: Kent

text: Kent

text: that

text: give

text: global

text: constraints

text: (Walker

text: Penarrubia

text: were

text: able

text: to

text: exclude

text: cuspy

text: den-

text: on

text: velocity

text: moments

text: by

text: eﬀectively

text: integrating

text: the

text: spherical

text: sity

text: proﬁles

text: in

text: Fornax

text: and

text: Sculptor

text: with

text: a

text: high

text: statistical

text: Jeans

text: equations

text: over

text: all

text: radii.

text: While

text: this

text: results

text: in

text: a

text: loss

text: signiﬁcance.

text: Use

text: of

text: the

text: projected

text: Virial

text: theorem

text: on

text: multiple

text: of

text: spatial

text: information

text: it

text: can

text: be

text: used

text: instead

text: of

text: the

text: Jeans

text: tracer

text: populations

text: of

text: globular

text: clusters

text: has

text: also

text: recently

text: been

text: equations

text: to

text: ﬁnd

text: out

text: what

text: density

text: proﬁles

text: are

text: compatible

text: applied

text: to

text: nearby

text: elliptical

text: galaxies

text: (Agnello

text: et

text: al.

text: with

text: the

text: LOS

text: velocity

text: data.

text: Interestingly,

text: fourth

text: moment

text: The

text: simplicity

text: of

text: these

text: analytic

text: methods

text: coupled

text: with

text: claims

text: analogues

text: to

text: the

text: projected

text: Virial

text: theorem

text: do

text: not

text: depend

text: on

text: (Laporte

text: et

text: al.

text: that

text: they

text: can

text: be

text: successful

text: on

text: simula-

text: the

text: relative

text: shape

text: of

text: the

text: radial

text: and

text: tangential

text: velocity

text: distri-

text: tions

text: of

text: triaxial

text: haloes

text: (though

text: see

text: Kowalczyk

text: et

text: al.

text: butions

text: β

text: (a

text: fourth

text: moment

text: quantity)

text: but

text: only

text: the

text: relative

text: for

text: a

text: challenge

text: of

text: this

text: result)

text: gives

text: compelling

text: evidence

text: for

text: widths

text: β

text: (a

text: second

text: moment

text: quantity)

text: and

text: the

text: gravitational

text: cored

text: proﬁles

text: in

text: dwarf

text: spheroidals.

text: potential

text: Φ.

text: As

text: such

text: the

text: potential

text: terms

text: for

text: fourth

text: order

text: pro-

text: In

text: spite

text: of

text: this,

text: a

text: lack

text: of

text: independent

text: evidence

text: from

text: jected

text: Virial

text: theorem

text: estimators

text: can

text: be

text: evaluated

text: with

text: the

text: alternative

text: single-component

text: approaches

text: to

text: support

text: these

text: same

text: parameter

text: set

text: as

text: the

text: second

text: order

text: Jeans

text: equation

text: with-

text: claims

text: has

text: left

text: the

text: cusp/core

text: debate

text: in

text: dwarf

text: spheroidal

text: out

text: introducing

text: new

text: parameters

text: such

text: as

text: β

text: In

text: essence,

text: the

text: galaxies

text: open.

text: Strigari

text: et

text: al.

text: claimed

text: that

text: DM

text: haloes

text: fourth

text: order

text: Virial

text: equations

text: give

text: us

text: two

text: extra

text: constraints

text: from

text: the

text: Aquarius

text: simulation

text: can

text: provide

text: an

text: acceptable

text: on

text: the

text: LOS

text: data

text: for

text: free

text: and

text: without

text: loss

text: of

text: generality.

text: It

text: single-component

text: ﬁt

text: to

text: the

text: photometric

text: and

text: kinematic

text: data

text: has

text: been

text: shown

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text: result

text: extends

text: to

text: nth

text: order

text: (Kent

text: in

text: Milky

text: Way

text: dwarf

text: spheroidals.

text: Other

text: single-component

text: such

text: that

text: n

text: Virial

text: equations

text: for

text: moments

text: of

text: the

text: LOS

text: analyses

text: with

text: Bayesian

text: hierarchical

text: modelling

text: (Martinez

text: velocity

text: distribution

text: are

text: independent

text: of

text: the

text: nth

text: order

text: z

text: and

text: Schwarzschild

text: orbit-based

text: modelling

text: (Breddels

text: anisotropy

text: parameter

text: β

text: (see

text: Richardson

text: Fairbairn

text: n

text: Helmi

text: Jardel

text: Gebhardt

text: ﬁnd

text: no

text: strong

text: ev-

text: for

text: deﬁnition)

text: and

text: depend

text: only

text: on

text: parameters

text: β

text: with

text: j

text: n.

text: j

text: idence

text: for

text: universal

text: cusps

text: or

text: cores

text: when

text: considering

text: the

text: In

text: this

text: work,

text: we

text: will

text: investigate

text: the

text: use

text: of

text: higher

text: or-

text: Milky

text: Way’s

text: dwarf

text: spheroidals

text: as

text: a

text: whole.

text: In

text: contrast

text: to

text: der

text: analogues

text: of

text: the

text: projected

text: Virial

text: theorem

text: in

text: conjunc-

text: these

text: sophisticated

text: techniques,

text: our

text: approach

text: is

text: to

text: try

text: and

text: tion

text: with

text: conventional

text: Jeans

text: analyses

text: to

text: try

text: and

text: break

text: the

text: ﬁnd

text: the

text: simplest

text: analytic

text: method

text: possible

text: that

text: can

text: break

text: classic

text: degeneracy

text: problem.

text: In

text: we

text: will

text: introduce

text: two

text: new

text: the

text: β

text: degeneracy

text: with

text: a

text: single

text: tracer

text: component

text: and

text: with-

text: shape

text: parameters

text: ζ

text: and

text: ζ

text: which

text: are

text: constructed

text: from

text: the

text: A

text: B

text: out

text: a

text: loss

text: of

text: generality.

text: fourth

text: order

text: projected

text: Virial

text: theorems

text: and

text: which

text: replace

text: the

text: By

text: ﬁtting

text: to

text: the

text: kurtosis

text: of

text: LOS

text: velocity

text: data

text: in

text: addi-

text: kurtosis

text: as

text: a

text: measure

text: of

text: the

text: shape

text: of

text: the

text: LOS

text: velocity

text: dis-

text: tion

text: to

text: the

text: velocity

text: dispersion,

text: there

text: is

text: an

text: extra

text: constraint

text: tribution

text: that

text: can

text: be

text: estimated

text: from

text: the

text: data.

text: We

text: test

text: the

text: that

text: can

text: distinguish

text: between

text: diﬀerent

text: ﬁts

text: to

text: the

text: standard

text: statistical

text: properties

text: of

text: these

text: Virial

text: shape

text: parameters

text: with

text: Jeans

text: equation.

text: Joint

text: ﬁts

text: to

text: the

text: dispersion

text: and

text: kurtosis

text: simulated

text: data

text: and

text: show

text: that

text: these

text: new

text: measures

text: seem

text: to

text: with

text: second

text: and

text: fourth

text: order

text: Jeans

text: equations

text: (Merriﬁeld

text: be

text: more

text: robust

text: to

text: systematics

text: than

text: the

text: kurtosis

text: in

text: radial

text: Kent

text: have

text: been

text: conducted

text: on

text: galaxy

text: clusters

text: okas

text: bins.

text: Mamon

text: dwarf

text: spheroidals

text: okas

text: et

text: al.

text: and

text: In

text: we

text: will

text: perform

text: an

text: illustrative

text: analysis

text: on

text: Sculp-

text: recently

text: giant

text: elliptical

text: galaxies

text: (Napolitano

text: et

text: al.

text: tor

text: with

text: constant

text: β

text: to

text: highlight

text: the

text: key

text: properties

text: of

text: the

text: State-of-the-art

text: observations

text: of

text: Milky

text: Way

text: dwarf

text: spheroidal

text: Virial

text: shape

text: parameters

text: and

text: to

text: gain

text: an

text: intuition

text: for

text: how

text: galaxies

text: (Helmi

text: et

text: al.

text: Walker

text: et

text: al.

text: now

text: oﬀer

text: they

text: can

text: break

text: the

text: β

text: degeneracy.

text: Finally,

text: we

text: do

text: a

text: gener-

text: LOS

text: velocities

text: and

text: projected

text: radii

text: for

text: thousands

text: of

text: tracer

text: alized

text: analysis

text: in

text: with

text: non-constant

text: β

text: to

text: see

text: which

text: den-

text: stars.

text: As

text: data

text: sets

text: continue

text: to

text: expand,

text: the

text: inclusion

text: of

text: sity

text: proﬁles

text: are

text: consistent

text: with

text: the

text: Jeans

text: equation

text: and

text: both

text: higher

text: velocity

text: moments

text: is

text: not

text: only

text: possible

text: but

text: also

text: useful,

text: fourth

text: order

text: projected

text: Virial

text: theorems.

text: Implications

text: of

text: key

text: especially

text: if

text: deviations

text: from

text: Gaussianity

text: are

text: statistically

text: sig-

text: assumptions

text: such

text: as

text: spherical

text: symmetry

text: are

text: discussed

text: in

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##