text: yrtemmysA epyT ytniatrecnU yrtemmysA epyT ytniatrecnU 7 1.0 1.0 0.12 0.7 0.10 0.5 0.6 0.5 0.5 0.08 0.0 0.4 0.0 0.06 0.3 0.04 -0.5 0.2 -0.5 0.02 0.1 -1.0 0.0 -1.0 0.00 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Acceptance Asymmetry Flux Asymmetry FIG. 3: The suppression fraction (SF) for DC-II is illustrated coloured coded in the z-axis. Left shows the evolution of SF − against both the reactor acceptance asymmetry (x-axis), deﬁned as (Ω Ω )/(Ω + Ω ), and the reactor uncertainty − R2 R1 R2 R1 type asymmetry (y-axis), deﬁned as (δc δu)/(δc + δu). The iso-ﬂux condition is met whenever the acceptance asymmetry ∼ → is 0, leading to a SF 0 (violet), regardless of the nature of reactor uncertainties, thus demonstrating that the near is a geometrical perfect monitor of the far. The DC geometry is such that the iso-ﬂux-ness is partially met along the projection − ≈ − ∼ (Ω Ω )/(Ω + Ω ) 0.28, hence SF is, at most, 0.12. Right shows the evolution of SF against both the reactor R2 R1 R2 R1 − power ﬂux asymmetry (x-axis), deﬁned as (Φ Φ )/(Φ + Φ ) (i.e. the ﬂux diﬀerence between reactor R1 and R2), and R2 R1 R2 R1 the reactor uncertainty type asymmetry (y-axis), as deﬁned before. Note that the maximal value possible of SF is the same ∼ 0.12, as explained. Total cancellation (i.e. lowest SF) is found if the uncertainties are reactor correlated maximally or either reactor is oﬀ. This is because both cases are mathematically identical as to having one eﬀective reactor as source, perfectly monitored by the near, regardless of the experimental setup geometry. (cid:88) √ − the 1/ N scaling of the remaining totally uncorrelated NR 1 error, wheRre NR = 2. Snf = (cid:88)Ω(niR) Ω(fiR) ; Multi-Detector: the RENO Conﬁguration i>1 − (cid:48) NR 1 S = Ω(i) Ω(j); Generalising Eq. 10 to account for a larger number of nf (cid:54) nR fR i,j>1;i=j reactors (N = 6), RENO is becomes our next case with R two detector sites and N = 6, as illustrated in Fig. 1. R and Ω(i) is a generalization from the Eq. 5, to obtain The new general expression is dR (cid:18) (cid:19) (cid:88) Ω Φ ∼ L2 Φ − SF2 = 1 + Sd2 + 2k(Sd + Sd(cid:48) ) Ω(diR) = ΩddRR1iΦRRi1 = Ld2dRRi1ΦRR1i ; i = 2 . . . NR 1 (13) (1 + S )2 d d=n,f (cid:48) − 1 + S + k(S + S + S ) where reactor R1 has been arbitrary chosen as reference 2 nf n f nf (12) reactor. Nevertheless, choosing a diﬀerent reactor leave (1 + S )(1 + S ) n f Eq. 12 unchanged. The iso-ﬂux condition, originally in Eq. 11, is now more where we have introduced the following notation generally expressed as (cid:88) NR−1 Ω(niR) = Ω(fiR) (14) Sd = (cid:88) Ω(diR) ; i>1 − which, like in the case of DC-II, would cause SF to go NR 1 Sd2 = (cid:88) (Ω(diR) )2; to zero, regardless of any other error dependence, upon imposition, if the geometry of the experimental site con- i>1 − sidered allows. (cid:48) NR 1 S = Ω(i) Ω(j); Upon introducing the geometry of RENO, we can sum- d dR dR (cid:54) marise our ﬁnding as follows, i,j>1;i=j ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: yrtemmysA

text: epyT

text: ytniatrecnU

text: yrtemmysA

text: epyT

text: ytniatrecnU

text: Acceptance

text: Asymmetry

text: Flux

text: Asymmetry

text: FIG.

text: The

text: suppression

text: fraction

text: (SF)

text: for

text: DC-II

text: is

text: illustrated

text: coloured

text: coded

text: in

text: the

text: z-axis.

text: Left

text: shows

text: the

text: evolution

text: of

text: SF

text: against

text: both

text: the

text: reactor

text: acceptance

text: asymmetry

text: (x-axis),

text: deﬁned

text: as

text: (Ω

text: Ω

text: )/(Ω

text: Ω

text: and

text: the

text: reactor

text: uncertainty

text: type

text: asymmetry

text: (y-axis),

text: deﬁned

text: as

text: (δc

text: δu)/(δc

text: δu).

text: The

text: iso-ﬂux

text: condition

text: is

text: met

text: whenever

text: the

text: acceptance

text: asymmetry

text: is

text: leading

text: to

text: a

text: SF

text: (violet),

text: regardless

text: of

text: the

text: nature

text: of

text: reactor

text: uncertainties,

text: thus

text: demonstrating

text: that

text: the

text: near

text: is

text: a

text: geometrical

text: perfect

text: monitor

text: of

text: the

text: far.

text: The

text: DC

text: geometry

text: is

text: such

text: that

text: the

text: iso-ﬂux-ness

text: is

text: partially

text: met

text: along

text: the

text: projection

text: (Ω

text: Ω

text: )/(Ω

text: Ω

text: hence

text: SF

text: is,

text: at

text: most,

text: Right

text: shows

text: the

text: evolution

text: of

text: SF

text: against

text: both

text: the

text: reactor

text: power

text: ﬂux

text: asymmetry

text: (x-axis),

text: deﬁned

text: as

text: (Φ

text: Φ

text: )/(Φ

text: Φ

text: (i.e.

text: the

text: ﬂux

text: diﬀerence

text: between

text: reactor

text: and

text: and

text: the

text: reactor

text: uncertainty

text: type

text: asymmetry

text: (y-axis),

text: as

text: deﬁned

text: before.

text: Note

text: that

text: the

text: maximal

text: value

text: possible

text: of

text: SF

text: is

text: the

text: same

text: as

text: explained.

text: Total

text: cancellation

text: (i.e.

text: lowest

text: SF)

text: is

text: found

text: if

text: the

text: uncertainties

text: are

text: reactor

text: correlated

text: maximally

text: or

text: either

text: reactor

text: is

text: oﬀ.

text: This

text: is

text: because

text: both

text: cases

text: are

text: mathematically

text: identical

text: as

text: to

text: having

text: one

text: eﬀective

text: reactor

text: as

text: source,

text: perfectly

text: monitored

text: by

text: the

text: near,

text: regardless

text: of

text: the

text: experimental

text: setup

text: geometry.

text: the

text: N

text: scaling

text: of

text: the

text: remaining

text: totally

text: uncorrelated

text: NR

text: error,

text: wheRre

text: NR

text: Snf

text: Ω(fiR)

text: Multi-Detector:

text: the

text: RENO

text: Conﬁguration

text: NR

text: S

text: Ω(i)

text: Ω(j);

text: Generalising

text: Eq.

text: to

text: account

text: for

text: a

text: larger

text: number

text: of

text: nf

text: nR

text: fR

text: reactors

text: (N

text: RENO

text: is

text: becomes

text: our

text: next

text: case

text: with

text: R

text: two

text: detector

text: sites

text: and

text: N

text: as

text: illustrated

text: in

text: Fig.

text: R

text: and

text: Ω(i)

text: is

text: a

text: generalization

text: from

text: the

text: Eq.

text: to

text: obtain

text: The

text: new

text: general

text: expression

text: is

text: dR

text: Ω

text: Φ

text: Φ

text: Ω(diR)

text: i

text: NR

text: S

text: d

text: d=n,f

text: S

text: k(S

text: S

text: S

text: where

text: reactor

text: has

text: been

text: arbitrary

text: chosen

text: as

text: reference

text: nf

text: n

text: f

text: nf

text: reactor.

text: Nevertheless,

text: choosing

text: a

text: diﬀerent

text: reactor

text: leave

text: S

text: S

text: n

text: f

text: Eq.

text: unchanged.

text: The

text: iso-ﬂux

text: condition,

text: originally

text: in

text: Eq.

text: is

text: now

text: more

text: where

text: we

text: have

text: introduced

text: the

text: following

text: notation

text: generally

text: expressed

text: as

text: Ω(niR)

text: Ω(fiR)

text: Sd

text: Ω(diR)

text: which,

text: like

text: in

text: the

text: case

text: of

text: DC-II,

text: would

text: cause

text: SF

text: to

text: go

text: NR

text: (Ω(diR)

text: to

text: zero,

text: regardless

text: of

text: any

text: other

text: error

text: dependence,

text: upon

text: imposition,

text: if

text: the

text: geometry

text: of

text: the

text: experimental

text: site

text: con-

text: sidered

text: allows.

text: NR

text: S

text: Ω(i)

text: Ω(j);

text: Upon

text: introducing

text: the

text: geometry

text: of

text: RENO,

text: we

text: can

text: sum-

text: d

text: dR

text: dR

text: marise

text: our

text: ﬁnding

text: as

text: follows,

text: ##LTFigure##

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text: Goldberg showed that Skip-Gram with Negative Sampling model(SGNS) is im- plicitly factorizing a word-context matrix [1] by assuming that dim→−ens→−ionality d · is suﬃciently large. In other words, we can assign each product v c a value independently of the others. In SGNS model, we have →− →− ∈ · 1 P((v, c) D) = σ( v c ) = −→− ·→− 1 + e v c Suppose we choose k negative samples for each node-context pair (v, c) ac- cordin(cid:88)g to(cid:88)the distribution PD(cN) = #|(DcN| ). Then the objective function for SGNS can be written as →− →− →− →− · · E − · l = (cid:88) (cid:88) #(v, c)(log σ( v c ) + k (cid:88)∼ [log(cid:88)σ( v c )]) ∈ ∈ cN PD v V c VC →− →− →− →− · · #(c ) − · = (cid:88) (cid:88) #(v, c)(log σ( v c ) + k #(v) | N| log σ( v c ) ∈ ∈ ∈ ∈ D v V c VC v V cN VC →− →− →− →− · · · #(c) − · = (#(v, c)(log σ( v c ) + k #(v) | | log σ( v c ) ∈ ∈ D v V c VC →− →− · Denote x = v c . By solving ∂l = 0, we have ∂x →− →− · | | · #(v, c) D − v c = x = log · log k #(v) #(c) #(v|i,|cj) − Thus we have M = log D log k. M can be interpreted as Point- ij #|(v|i)·#|(cj|) ij D D wise Mutual Information(PMI) of node-context pair (v , c ) shifted by log k. i j 2.2 Softmax Since both Negative Sampling and Hierarchical Softmax are variants of soft- max, we pay more attention to softmax model a→−nd g→−ive a further discussion in · next section. We also assume that the values of v c are independent. In softmax model, (cid:80) →− →− · ∈ e v c P((v, c) D) = →− →− · (cid:48) (cid:48)∈ e v c c VC (cid:88) (cid:88) And the objective function is (cid:80) →− →− · · e v c l = #(v, c) log →− →− · (cid:48) v∈V c∈VC c(cid:48)∈VC e v c →− →− · After extracting all terms associated to v c (cid:88)as l(v, c), we have (cid:80) →− →− (cid:80) →− →− · · e v c e v c˜ l(v, c) = #(v, c) log →− →− →− →− + #(v, c˜) log →− →− →− →− · (cid:48) · · (cid:48) · c(cid:48)∈VC,c(cid:48)(cid:54)=c e v c + e v c c˜∈VC,c˜(cid:54)=c c(cid:48)∈VC,c(cid:48)(cid:54)=c e v c + e v c 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Goldberg

label: paragraph

text: showed

text: that

text: Skip-Gram

text: with

text: Negative

text: Sampling

text: model(SGNS)

text: is

text: im-

text: plicitly

text: factorizing

text: a

text: word-context

text: matrix

text: by

text: assuming

text: that

text: dim→−ens→−ionality

text: d

text: is

text: suﬃciently

text: large.

text: In

text: other

text: words,

text: we

text: can

text: assign

text: each

text: product

text: v

text: c

text: a

text: value

text: independently

text: of

text: the

text: others.

text: In

text: SGNS

text: model,

text: we

text: have

text: P((v,

text: c)

text: D)

text: σ(

text: v

text: c

text: e

text: v

text: c

text: Suppose

text: we

text: choose

text: k

text: negative

text: samples

text: for

text: each

text: node-context

text: pair

text: (v,

text: c)

text: ac-

text: distribution

text: PD(cN)

text: #|(DcN|

text: Then

text: the

text: objective

text: function

text: for

text: SGNS

text: can

text: be

text: written

text: as

text: E

text: l

text: #(v,

text: c)(log

text: σ(

text: v

text: c

text: k

text: v

text: c

text: cN

text: PD

text: v

text: V

text: c

text: VC

text: #(c

text: #(v,

text: c)(log

text: σ(

text: v

text: c

text: k

text: #(v)

text: N|

text: log

text: σ(

text: v

text: c

text: D

text: v

text: V

text: c

text: VC

text: v

text: V

text: cN

text: VC

text: #(c)

text: (#(v,

text: c)(log

text: σ(

text: v

text: c

text: k

text: #(v)

text: log

text: σ(

text: v

text: c

text: D

text: v

text: V

text: c

text: VC

text: Denote

text: x

text: v

text: c

text: By

text: solving

text: ∂l

text: we

text: have

text: ∂x

text: #(v,

text: c)

text: D

text: v

text: c

text: x

text: log

text: log

text: k

text: #(v)

text: #(c)

text: #(v|i,|cj)

text: Thus

text: we

text: have

text: M

text: log

text: D

text: log

text: k.

text: M

text: can

text: be

text: interpreted

text: as

text: Point-

text: ij

text: #|(v|i)·#|(cj|)

text: ij

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text: D

text: wise

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text: k.

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text: Softmax

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text: Negative

text: Sampling

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text: softmax

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text: next

text: section.

text: We

text: also

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text: that

text: the

text: values

text: of

text: v

text: c

text: are

text: independent.

text: In

text: softmax

text: model,

text: e

text: v

text: c

text: P((v,

text: c)

text: D)

text: e

text: v

text: c

text: c

text: VC

text: And

text: the

text: objective

text: function

text: is

text: e

text: v

text: c

text: l

text: #(v,

text: c)

text: log

text: v∈V

text: c∈VC

text: e

text: v

text: c

text: After

text: extracting

text: all

text: terms

text: associated

text: to

text: v

text: c

text: l(v,

text: c),

text: we

text: have

text: e

text: v

text: c

text: e

text: v

text: c˜

text: l(v,

text: c)

text: #(v,

text: c)

text: log

text: #(v,

text: c˜)

text: log

text: e

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text: ##LTLine##

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text: − where η = diag( 1, 1, ...1) is the Minkowski metric. The metric is assumed to be ab − compatible with the connection ωa , namely η | = 0, and thus ω = ω , where | bc ab c abc bac denotes covariant diﬀerentiation with respect to ωa . The teleparallelism condition is bc − imposed by assuming that the curvature of ωa is zero, i.e. Ra = ωa ωa + − bc bµν bν,µ bµ,ν ωa ωc ωa ωc = 0 (note however that the curvature R¯a constructed from the cµ bν cν bµ bµν Christoﬀel connection is not zero). One way to realize this condition is by considering the Weitzenb¨ock connection, which is deﬁned in terms of the vielbein in all coordinate frames as ωλ = e λea , and therefore ωa = 0 in the preferred frame e that deﬁnes the µν a µ,ν bc a connection. Then, the following torsion scalar can be deﬁned as 1 1 − T = T µνλT + T µνλT T νµT λ (2.8) 4 µνλ 2 λνµ ν λµ = SµνλT , (2.9) µνλ where the tensor Sµνλ is deﬁned by 1K 1 − − Sµνλ = νλµ + (gµλT ρν gµνT ρλ) = Sµλν (2.10) 2 2 ρ ρ and K 1 − − −K − = (T T T ) = = ω Γ (2.11) µνλ λµν νλµ µνλ νµλ µνλ µνλ 2 is the contorsion tensor. In the teleparallel formulation, torsion is used to express the (cid:90) (cid:90) (cid:16) (cid:17) action (2.1) in an equivalent form (up to boundary terms) as − 1 − 1 S = dDx e T dDx e gµν∂ φ ∂ φ + V . (2.12) min 2κ2 2 µ ν D − Since T diﬀers from R¯ by boundary terms, namely T = R¯ + 2T νµ with ; denoting ν ;µ covariant diﬀerentiation with respect to the Christoﬀel connection, the above action (2.12) indeed gives exactly the same equations with the action (2.1). For example, the hairy black hole solution (2.2), (2.4), (2.5) can be obtained from the action (2.12) in four dimensions. In this work we are interested in ﬁnding local spherically symmetric solutions in which the presence of the torsion could leave an explicit signature on the solution, other than from the trivial one found from the action (2.12). To achieve this, the action (2.12) has to be modiﬁed in such a way as to include a coupling of torsion to the scalar ﬁeld. The easiest way to modify the action (2.12) is to add a direct coupling of the scalar ﬁeld to torsion (cid:90) (cid:90) (cid:16) (cid:17) − 1 − 1 S − = dDx e T dDx e gµν∂ φ ∂ φ + V + ξf(φ)T . (2.13) non min 2κ2 2 µ ν D – 5 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: η

text: diag(

text: is

text: the

text: Minkowski

text: metric.

text: The

text: metric

text: is

text: assumed

text: to

text: be

text: ab

text: compatible

text: with

text: the

text: connection

text: ωa

text: namely

text: η

text: and

text: thus

text: ω

text: ω

text: where

text: bc

text: ab

text: c

text: abc

text: bac

text: denotes

text: covariant

text: diﬀerentiation

text: with

text: respect

text: to

text: ωa

text: The

text: teleparallelism

text: condition

text: is

text: bc

text: imposed

text: by

text: assuming

text: that

text: the

text: curvature

text: of

text: ωa

text: is

text: zero,

text: i.e.

text: Ra

text: ωa

text: ωa

text: bc

text: bµν

text: bν,µ

text: bµ,ν

text: ωa

text: ωc

text: ωa

text: ωc

text: (note

text: however

text: that

text: the

text: curvature

text: R¯a

text: constructed

text: from

text: the

text: cµ

text: bν

text: cν

text: bµ

text: bµν

text: Christoﬀel

text: connection

text: is

text: not

text: zero).

text: One

text: way

text: to

text: realize

text: this

text: condition

text: is

text: by

text: considering

text: the

text: Weitzenb¨ock

text: connection,

text: which

text: is

text: deﬁned

text: in

text: terms

text: of

text: the

text: vielbein

text: in

text: all

text: coordinate

text: frames

text: as

text: ωλ

text: e

text: λea

text: and

text: therefore

text: ωa

text: in

text: the

text: preferred

text: frame

text: e

text: that

text: deﬁnes

text: the

text: µν

text: a

text: µ,ν

text: bc

text: a

text: connection.

text: Then,

text: the

text: following

text: torsion

text: scalar

text: can

text: be

text: deﬁned

text: as

text: T

text: T

text: µνλT

text: T

text: µνλT

text: T

text: νµT

text: λ

text: µνλ

text: λνµ

text: ν

text: λµ

text: SµνλT

text: µνλ

text: where

text: the

text: tensor

text: Sµνλ

text: is

text: deﬁned

text: by

text: Sµνλ

text: νλµ

text: (gµλT

text: ρν

text: gµνT

text: ρλ)

text: Sµλν

text: ρ

text: ρ

text: and

text: K

text: −K

text: (T

text: T

text: T

text: ω

text: Γ

text: µνλ

text: λµν

text: νλµ

text: µνλ

text: νµλ

text: µνλ

text: µνλ

text: is

text: the

text: contorsion

text: tensor.

text: In

text: the

text: teleparallel

text: formulation,

text: torsion

text: is

text: used

text: to

text: express

text: the

text: action

text: in

text: an

text: equivalent

text: form

text: (up

text: to

text: boundary

text: terms)

text: as

text: S

text: dDx

text: e

text: T

text: dDx

text: e

text: gµν∂

text: φ

text: φ

text: V

text: min

text: µ

text: ν

text: D

text: Since

text: T

text: diﬀers

text: from

text: R¯

text: by

text: boundary

text: terms,

text: namely

text: T

text: R¯

text: νµ

text: with

text: denoting

text: ν

text: ;µ

text: covariant

text: diﬀerentiation

text: with

text: respect

text: to

text: the

text: Christoﬀel

text: connection,

text: the

text: above

text: action

text: indeed

text: gives

text: exactly

text: the

text: same

text: equations

text: with

text: the

text: action

text: For

text: example,

text: the

text: hairy

text: black

text: hole

text: solution

text: can

text: be

text: obtained

text: from

text: the

text: action

text: in

text: four

text: dimensions.

text: In

text: this

text: work

text: we

text: are

text: interested

text: in

text: ﬁnding

text: local

text: spherically

text: symmetric

text: solutions

text: in

text: which

text: the

text: presence

text: of

text: the

text: torsion

text: could

text: leave

text: an

text: explicit

text: signature

text: on

text: the

text: solution,

text: other

text: than

text: from

text: the

text: trivial

text: one

text: found

text: from

text: the

text: action

text: To

text: achieve

text: this,

text: the

text: action

text: has

text: to

text: be

text: modiﬁed

text: in

text: such

text: a

text: way

text: as

text: to

text: include

text: a

text: coupling

text: of

text: torsion

text: to

text: the

text: scalar

text: ﬁeld.

text: The

text: easiest

text: way

text: to

text: modify

text: the

text: action

text: is

text: to

text: add

text: a

text: direct

text: coupling

text: of

text: the

text: scalar

text: ﬁeld

text: to

text: torsion

text: S

text: dDx

text: e

text: T

text: dDx

text: e

text: gµν∂

text: φ

text: φ

text: V

text: ξf(φ)T

text: non

text: min

text: µ

text: ν

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:17)(cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) 2(cid:16) − (cid:17) (cid:16) 1 − (cid:17) 1 4ξφ φ,ρ (eS λνe κ) eb + e Tδλ SνκλT + 4ξ eTφ φ,λ+eS νλ(φ φ,κ) κ2 ,ρ κ b ,ν µ 4 µ νκµ 2 ,µ µ ,κ ,ν D 1 − − 2 − +e φ φ,ρδλ φ φ,λ + V δλ 4ξφ φ,ρ eSdcaω e λeb = 0 . (2.17) 2 ,ρ µ ,µ µ κ2 ,ρ bdc a µ D We mention that the above expressions contain also the connection ω . Assuming for abc simplicity the Weitzenbo¨ck connection, that is setting ω = 0, which is a solution of the abc equation of motion R (ωa ) = 0, we acquire the ﬁeld equations containing only the abcd bc vielbein, namely (cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) 2 (cid:20)− (cid:21) 1(cid:16) − (cid:17) 4ξφ φ,ρ (eS λνe κ) eb + e T δλ SνκλT κ2 ,ρ κ b ,ν µ 4 µ νκµ D 1 1 − +4ξ eT φ φ,λ + eS νλ(φ φ,κ) + e φ φ,ρδλ φ φ,λ + V δλ = 0 . (2.18) 2 ,µ µ ,κ ,ν 2 ,ρ µ ,µ µ Addition(cid:90)ally, v(cid:26)ariation of the action (2.1(cid:27)4) with(cid:90)respect to the scalar ﬁeld φ gives (cid:2) (cid:3) (cid:2) (cid:3) − − dV − − δ S = dDx e(1 2ξT )φ,µ e δφ dDx e(1 2ξT )φ,µδφ , (2.19) φ ,µ dφ ,µ (cid:2) (cid:3) from where the scalar-ﬁeld equation of motion is obtained as − − dV e(1 2ξT )φ,µ e = 0 . (2.20) ,µ dφ Summarizing, equations (2.18) are the gravity ﬁeld equations, while equation (2.20) is the Klein-Gordon one for the scalar ﬁeld. Observe that T being a scalar can be identiﬁed with the Hubble parameter in a cosmological Friedmann-Robertson-Walker (FRW) back- ground as T = G = 6H2. This implies that the derivative coupling T gµν∂ φ ∂ φ on a 00 µ ν FRW background will give the same cosmological evolution as the derivative coupling of the scalar ﬁeld to Einstein tensor Gµν∂ φ ∂ φ. However, general cosmological perturba- µ ν tions will diﬀerentiate the two models because they will excite also the spatial components of the Einstein tensor. In the next section we will study local spherically symmetric solutions of the ﬁeld equations (2.18) and (2.20) and we expect to ﬁnd solutions with features diﬀerent than the local black hole solutions of theories with a scalar ﬁeld coupled to Einstein tensor with a derivative coupling [50–54]. 3 Spherically symmetric solutions In this section we look for spherically symmetric solutions of the theory with a deriva- tive coupling of the torsion to the scalar ﬁeld in four dimensions. We consider the general spherically symmetric metric ansatz of the form – 7 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: 4ξφ

text: φ,ρ

text: (eS

text: λνe

text: κ)

text: eb

text: e

text: Tδλ

text: SνκλT

text: 4ξ

text: eTφ

text: φ,λ+eS

text: νλ(φ

text: φ,κ)

text: κ2

text: ,ρ

text: κ

text: b

text: ,ν

text: µ

text: µ

text: νκµ

text: ,µ

text: µ

text: ,κ

text: ,ν

text: D

text: +e

text: φ

text: φ,ρδλ

text: φ

text: φ,λ

text: V

text: δλ

text: 4ξφ

text: φ,ρ

text: eSdcaω

text: e

text: λeb

text: ,ρ

text: µ

text: ,µ

text: µ

text: κ2

text: ,ρ

text: bdc

text: a

text: µ

text: D

text: We

text: mention

text: that

text: the

text: above

text: expressions

text: contain

text: also

text: the

text: connection

text: ω

text: Assuming

text: for

text: abc

text: simplicity

text: the

text: Weitzenbo¨ck

text: connection,

text: that

text: is

text: setting

text: ω

text: which

text: is

text: a

text: solution

text: of

text: the

text: abc

text: equation

text: of

text: motion

text: R

text: (ωa

text: we

text: acquire

text: the

text: ﬁeld

text: equations

text: containing

text: only

text: the

text: abcd

text: bc

text: vielbein,

text: namely

text: 4ξφ

text: φ,ρ

text: (eS

text: λνe

text: κ)

text: eb

text: e

text: T

text: δλ

text: SνκλT

text: κ2

text: ,ρ

text: κ

text: b

text: ,ν

text: µ

text: µ

text: νκµ

text: D

text: eT

text: φ

text: φ,λ

text: eS

text: νλ(φ

text: φ,κ)

text: e

text: φ

text: φ,ρδλ

text: φ

text: φ,λ

text: V

text: δλ

text: ,µ

text: µ

text: ,κ

text: ,ν

text: ,ρ

text: µ

text: ,µ

text: µ

text: of

text: the

text: action

text: to

text: the

text: scalar

text: ﬁeld

text: φ

text: gives

text: dV

text: δ

text: S

text: dDx

text: )φ,µ

text: e

text: δφ

text: dDx

text: )φ,µδφ

text: φ

text: ,µ

text: dφ

text: ,µ

text: from

text: where

text: the

text: scalar-ﬁeld

text: equation

text: of

text: motion

text: is

text: obtained

text: as

text: dV

text: )φ,µ

text: e

text: ,µ

text: dφ

text: Summarizing,

text: equations

text: are

text: the

text: gravity

text: ﬁeld

text: equations,

text: while

text: equation

text: is

text: the

text: Klein-Gordon

text: one

text: for

text: the

text: scalar

text: ﬁeld.

text: Observe

text: that

text: T

text: being

text: a

text: scalar

text: can

text: be

text: identiﬁed

text: with

text: the

text: Hubble

text: parameter

text: in

text: a

text: cosmological

text: Friedmann-Robertson-Walker

text: (FRW)

text: back-

text: ground

text: as

text: T

text: G

text: This

text: implies

text: that

text: the

text: derivative

text: coupling

text: T

text: gµν∂

text: φ

text: φ

text: on

text: a

text: µ

text: ν

text: FRW

text: background

text: will

text: give

text: the

text: same

text: cosmological

text: evolution

text: as

text: the

text: derivative

text: coupling

text: of

text: the

text: scalar

text: ﬁeld

text: to

text: Einstein

text: tensor

text: Gµν∂

text: φ

text: φ.

text: However,

text: general

text: cosmological

text: perturba-

text: µ

text: ν

text: tions

text: will

text: diﬀerentiate

text: the

text: two

text: models

text: because

text: they

text: will

text: excite

text: also

text: the

text: spatial

text: components

text: of

text: the

text: Einstein

text: tensor.

text: In

text: the

text: next

text: section

text: we

text: will

text: study

text: local

text: spherically

text: symmetric

text: solutions

text: of

text: the

text: ﬁeld

text: equations

text: and

text: and

text: we

text: expect

text: to

text: ﬁnd

text: solutions

text: with

text: features

text: diﬀerent

text: than

text: the

text: local

text: black

text: hole

text: solutions

text: of

text: theories

text: with

text: a

text: scalar

text: ﬁeld

text: coupled

text: to

text: Einstein

text: tensor

text: with

text: a

text: derivative

text: coupling

text: Spherically

text: symmetric

text: solutions

text: In

text: this

text: section

text: we

text: look

text: for

text: spherically

text: symmetric

text: solutions

text: of

text: the

text: theory

text: with

text: a

text: deriva-

text: tive

text: coupling

text: of

text: the

text: torsion

text: to

text: the

text: scalar

text: ﬁeld

text: in

text: four

text: dimensions.

text: We

text: consider

text: the

text: general

text: spherically

text: symmetric

text: metric

text: ansatz

text: of

text: the

text: form

text: ##LTLine##

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text: ##LTLine##


text: 5. INTERPRETATION AND DISCUSSION the planetary detection and verify whether the planet could be found without previous transit knowledge. A With our measured mass of 1.87+−0.27 M⊕ and a radius range of Doppler amplitudes of the planet from 1.31 to ± 0.26 − of 1.20 0.09 R⊕ (H13), Kepler-78b has a bulk density 1.96 m s 1 are reported, which is broadly consistent with − of 6.0+−1.9 g cm 3, suggesting a rocky composition similar the result of this work as well as that of H13 and P13. 1.4 − Hatzes (2014) reports the planet can be identiﬁed with- to Earth (ρ⊕ = 5.52 g cm 3). Using the two-component, rock-iron models from Fortney et al. (2007), we estimate ± an iron fraction of 32 26%, consistent with an Earth-like composition (iron mass fraction of 0.319, McDonough & Sun 1995). These simpliﬁed models consider Kepler-78b as an iron core surrounded by a rocky mantel, and ac- count for compression that is important for higher planet masses. We ignore the eﬀect of an atmosphere on the ra- dius of Kepler-78b due to its equilibrium temperature of 1500-3000 K (Sanchis-Ojeda et al. 2013). We now speculate about the implications of these mea- surements to the composition of Kepler-78b. In Figure 5, we illustrate that the best-ﬁt composition of Kepler-78b is indistinguishable from the composition of Earth. How- ever, Kepler-78b’s range in iron mass fraction stretches from almost purely rock (a Moon-like composition) to 60% iron (a Mercury-like composition). This is consis- tent with a range of rocky solar system planets but dis- tinguishable from other solid celestial bodies (such as M-type asteroids, which are almost 100% iron by mass, or comets, made up of rock and ice with no iron at all). Since its best-ﬁt composition and mass are more simi- lar to Earth than any other solar system body, Kepler- Fig. 5.— A graph of mass versus radius for rocky planets. The constraints on Kepler-78b are shown by the red point. The es- 78b might have had a formation process that was similar timation of Kepler-78b by H13 is shown as the light gray error to Earth’s. The proximity of Kepler-78 to its host star bars. Composition curves ranging from a pure water (blue) to pure (0.009 AU) makes it unlikely that it formed in its current iron composition (black) have been plotted. Earth-like (67% rock, location, and migration to its current orbit is likely. 33% iron) and Mercury-like (67% iron, 33% rock) compositions are denoted by green and brown curves, respectively. Solar sys- The improvement in mass determination of this model tem planets are shown as green squares. Other well-characterized exoplanets are plotted as black points. Exoplanet masses, radii, relative to H13 is shown in Figure 5. Despite the fact and their associated errors are from the Exoplanet Orbit Database that the mass measured by this method is a 6.5-σ mea- (http://exoplanets.org; downloaded on 23 August 2014). Planets surement compared to H13’s 4-σ and P13’s 6-σ detection, with fractional mass uncertainties of over 50% are not shown. the new measurement of Kepler-78b’s density is as pre- cise as those obtained by either of the two competing out prior knowledge of the system using a modiﬁed ver- teams in 2013, who estimated its density to be 5.3+−2.0 sion of the parametric method originally tested by P13, (H13) and 5.57+−31..03 g cm−3 (P13). Relative to P13, t1h.6e tahnedﬂroepatoirntgs cahmunakssooﬀfs1e.t3m1 o±d0el.,24toMa⊕na,liynzceonbsoitshtednattaatsetthse, errors on the density in this study are 15% smaller, al- 1-σ level with the mass calculated in both this study and though only marginally smaller than that of H13. This is the previous studies. The spread in the Doppler ampli- because even though the new mass measurement is some- tudes reported by Hatzes (2014) illustrates that slight what more precise, the error on the density is driven three diﬀerences in the choice of noise model has a signiﬁ- times more strongly by the error on the radius than the cant eﬀect on the planetary signal extracted, and that mass. Higher cadence photometry of the planetary tran- the error on the planetary mass measurement has likely sit would allow for ingress and egress of the system to be been underestimated. Thus, by starting with the sim- observed, breaking the degeneracy between the impact plest possible nonparametric descriptions of the data, we parameter and transit depth. Estimates of the planet’s ﬁnd a description of the data that is minimally complex density could also be improved with more accurate mea- yet strongly supported by Bayesian information criteria. surements of the stellar mass, as converting RV measure- Furthermore, the structure of the best ﬁt GP kernel as ments into a planetary mass is directly dependent on the well as the success of sharing kernel parameters to de- stellar mass. scribe both datasets supports our belief that we have robustly measured the astrophysical variability of the Hatzes (2014) tested several traditional models on the Draft version May 27, 2020 Kepler-78 system, and can obtain a mass measurement same RV data analyzed here to check the robustness of for Kepler-78b that is consistent with previous work. Preprint typeset using LATEX style emulateapj v. 12/16/11 DETERMINING THE MASS OF KEPLER-78b WITH NONPARAMETRIC GAUSSIAN PROCESS ESTIMATION ##LTFigure## ##LTFigure##

label: section

text: INTERPRETATION

text: AND

text: DISCUSSION

text: the

text: planetary

text: detection

text: and

text: verify

text: whether

text: the

text: planet

text: could

text: be

text: found

text: without

text: previous

text: transit

text: knowledge.

text: A

text: With

text: our

text: measured

text: mass

text: of

text: M⊕

text: and

text: a

text: radius

text: range

text: of

text: Doppler

text: amplitudes

text: of

text: the

text: planet

text: from

text: to

text: of

text: R⊕

text: has

text: a

text: bulk

text: density

text: m

text: s

text: are

text: reported,

text: which

text: is

text: broadly

text: consistent

text: with

text: of

text: g

text: cm

text: suggesting

text: a

text: rocky

text: composition

text: similar

text: the

text: result

text: of

text: this

text: work

text: as

text: well

text: as

text: that

text: of

text: and

text: Hatzes

text: reports

text: the

text: planet

text: can

text: be

text: identiﬁed

text: with-

text: to

text: Earth

text: (ρ⊕

text: g

text: cm

text: Using

text: the

text: two-component,

text: rock-iron

text: models

text: from

text: Fortney

text: et

text: al.

text: we

text: estimate

text: an

text: iron

text: fraction

text: of

text: consistent

text: with

text: an

text: Earth-like

text: composition

text: (iron

text: mass

text: fraction

text: of

text: McDonough

text: Sun

text: These

text: simpliﬁed

text: models

text: consider

text: as

text: an

text: iron

text: core

text: surrounded

text: by

text: a

text: rocky

text: mantel,

text: and

text: ac-

text: count

text: for

text: compression

text: that

text: is

text: important

text: for

text: higher

text: planet

text: masses.

text: We

text: ignore

text: the

text: eﬀect

text: of

text: an

text: atmosphere

text: on

text: the

text: ra-

text: dius

text: of

text: due

text: to

text: its

text: equilibrium

text: temperature

text: of

text: K

text: (Sanchis-Ojeda

text: et

text: al.

text: We

text: now

text: speculate

text: about

text: the

text: implications

text: of

text: these

text: mea-

text: surements

text: to

text: the

text: composition

text: of

text: In

text: Figure

text: we

text: illustrate

text: that

text: the

text: best-ﬁt

text: composition

text: of

text: is

text: indistinguishable

text: from

text: the

text: composition

text: of

text: Earth.

text: How-

text: ever,

text: range

text: in

text: iron

text: mass

text: fraction

text: stretches

text: from

text: almost

text: purely

text: rock

text: (a

text: Moon-like

text: composition)

text: to

text: iron

text: (a

text: Mercury-like

text: composition).

text: This

text: is

text: consis-

text: tent

text: with

text: a

text: range

text: of

text: rocky

text: solar

text: system

text: planets

text: but

text: dis-

text: tinguishable

text: from

text: other

text: solid

text: celestial

text: bodies

text: (such

text: as

text: M-type

text: asteroids,

text: which

text: are

text: almost

text: iron

text: by

text: mass,

text: or

text: comets,

text: made

text: up

text: of

text: rock

text: and

text: ice

text: with

text: no

text: iron

text: at

text: all).

text: Since

text: its

text: best-ﬁt

text: composition

text: and

text: mass

text: are

text: more

text: simi-

text: lar

text: to

text: Earth

text: than

text: any

text: other

text: solar

text: system

text: body,

text: Kepler-

text: Fig.

text: A

text: graph

text: of

text: mass

text: versus

text: radius

text: for

text: rocky

text: planets.

text: The

text: constraints

text: on

text: are

text: shown

text: by

text: the

text: red

text: point.

text: The

text: es-

text: might

text: have

text: had

text: a

text: formation

text: process

text: that

text: was

text: similar

text: timation

text: of

text: by

text: is

text: shown

text: as

text: the

text: light

text: gray

text: error

text: to

text: Earth’s.

text: The

text: proximity

text: of

text: to

text: its

text: host

text: star

text: bars.

text: Composition

text: curves

text: ranging

text: from

text: a

text: pure

text: water

text: (blue)

text: to

text: pure

text: AU)

text: makes

text: it

text: unlikely

text: that

text: it

text: formed

text: in

text: its

text: current

text: iron

text: composition

text: (black)

text: have

text: been

text: plotted.

text: Earth-like

text: rock,

text: location,

text: and

text: migration

text: to

text: its

text: current

text: orbit

text: is

text: likely.

text: iron)

text: and

text: Mercury-like

text: iron,

text: rock)

text: compositions

text: are

text: denoted

text: by

text: green

text: and

text: brown

text: curves,

text: respectively.

text: Solar

text: sys-

text: The

text: improvement

text: in

text: mass

text: determination

text: of

text: this

text: model

text: tem

text: planets

text: are

text: shown

text: as

text: green

text: squares.

text: Other

text: well-characterized

text: exoplanets

text: are

text: plotted

text: as

text: black

text: points.

text: Exoplanet

text: masses,

text: radii,

text: relative

text: to

text: is

text: shown

text: in

text: Figure

text: Despite

text: the

text: fact

text: and

text: their

text: associated

text: errors

text: are

text: from

text: the

text: Exoplanet

text: Orbit

text: Database

text: that

text: the

text: mass

text: measured

text: by

text: this

text: method

text: is

text: a

text: mea-

text: downloaded

text: on

text: August

text: Planets

text: surement

text: compared

text: to

text: and

text: detection,

text: with

text: fractional

text: mass

text: uncertainties

text: of

text: over

text: are

text: not

text: shown.

text: the

text: new

text: measurement

text: of

text: density

text: is

text: as

text: pre-

text: cise

text: as

text: those

text: obtained

text: by

text: either

text: of

text: the

text: two

text: competing

text: out

text: prior

text: knowledge

text: of

text: the

text: system

text: using

text: a

text: modiﬁed

text: ver-

text: teams

text: in

text: who

text: estimated

text: its

text: density

text: to

text: be

text: sion

text: of

text: the

text: parametric

text: method

text: originally

text: tested

text: by

text: and

text: g

text: Relative

text: to

text: tahnedﬂroepatoirntgs

text: errors

text: on

text: the

text: density

text: in

text: this

text: study

text: are

text: smaller,

text: al-

text: level

text: with

text: the

text: mass

text: calculated

text: in

text: both

text: this

text: study

text: and

text: though

text: only

text: marginally

text: smaller

text: than

text: that

text: of

text: This

text: is

text: the

text: previous

text: studies.

text: The

text: spread

text: in

text: the

text: Doppler

text: ampli-

text: because

text: even

text: though

text: the

text: new

text: mass

text: measurement

text: is

text: some-

text: tudes

text: reported

text: by

text: Hatzes

text: illustrates

text: that

text: slight

text: what

text: more

text: precise,

text: the

text: error

text: on

text: the

text: density

text: is

text: driven

text: three

text: diﬀerences

text: in

text: the

text: choice

text: of

text: noise

text: model

text: has

text: a

text: signiﬁ-

text: times

text: more

text: strongly

text: by

text: the

text: error

text: on

text: the

text: radius

text: than

text: the

text: cant

text: eﬀect

text: on

text: the

text: planetary

text: signal

text: extracted,

text: and

text: that

text: mass.

text: Higher

text: cadence

text: photometry

text: of

text: the

text: planetary

text: tran-

text: the

text: error

text: on

text: the

text: planetary

text: mass

text: measurement

text: has

text: likely

text: sit

text: would

text: allow

text: for

text: ingress

text: and

text: egress

text: of

text: the

text: system

text: to

text: be

text: been

text: underestimated.

text: Thus,

text: by

text: starting

text: with

text: the

text: sim-

text: observed,

text: breaking

text: the

text: degeneracy

text: between

text: the

text: impact

text: plest

text: possible

text: nonparametric

text: descriptions

text: of

text: the

text: data,

text: we

text: parameter

text: and

text: transit

text: depth.

text: Estimates

text: of

text: the

text: planet’s

text: ﬁnd

text: a

text: description

text: of

text: the

text: data

text: that

text: is

text: minimally

text: complex

text: density

text: could

text: also

text: be

text: improved

text: with

text: more

text: accurate

text: mea-

text: yet

text: strongly

text: supported

text: by

text: Bayesian

text: information

text: criteria.

text: surements

text: of

text: the

text: stellar

text: mass,

text: as

text: converting

text: RV

text: measure-

text: Furthermore,

text: the

text: structure

text: of

text: the

text: best

text: ﬁt

text: GP

text: kernel

text: as

text: ments

text: into

text: a

text: planetary

text: mass

text: is

text: directly

text: dependent

text: on

text: the

text: well

text: as

text: the

text: success

text: of

text: sharing

text: kernel

text: parameters

text: to

text: de-

text: stellar

text: mass.

text: scribe

text: both

text: datasets

text: supports

text: our

text: belief

text: that

text: we

text: have

text: robustly

text: measured

text: the

text: astrophysical

text: variability

text: of

text: the

text: Hatzes

text: tested

text: several

text: traditional

text: models

text: on

text: the

text: Draft

text: version

text: May

text: system,

text: and

text: can

text: obtain

text: a

text: mass

text: measurement

text: same

text: RV

text: data

text: analyzed

text: here

text: to

text: check

text: the

text: robustness

text: of

text: for

text: that

text: is

text: consistent

text: with

text: previous

text: work.

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: DETERMINING

text: THE

text: MASS

text: OF

text: WITH

text: NONPARAMETRIC

text: GAUSSIAN

text: PROCESS

text: ESTIMATION

text: ##LTFigure##

text: ##LTFigure##


text: Kepler-78b Mass Measurement with Gaussian Process 5 3. METHODS hood of those hyperparameter values, and then iterating through parameter space until the most likely values are We model the photometric variations observed by Ke- found. The squared exponential kernel, for example, de- pler with a GP regression with a quasi-periodic kernel. ﬁnes a covariance matrix throug(cid:20)h a(cid:16)n operat(cid:17)or(cid:21), We then model the radial velocity measurements with a quasi-periodic kernel GP regression, using the kernel pe- − − t t 2 rhiyopderhpyaprearmpaertaermθeter., θTphhoits, atossutrmaiens tthheatRVtheGpPrepdeormioid- Σij = k(ti, tj) = h2exp i λ j , (1) RV nant signal observed in both datasets comes from the star, and thus will modulate at the stellar rotation pe- where h is the covariance amplitude, and λ the covari- riod. When we model the photometric data, we recover a ance length scale. The amplitude observed is described period hyperparameter that agrees with other autocorre- by h, while λ is a characteristic timescale over which the lation analyses of the Kepler photometry (Sanchis-Ojeda data is going to be correlated. et al. 2013), supporting this assumption. We create a GP regression of each RV dataset with a quasi-periodic ker- We discuss other GP kernels and the inferred physical nel, where the period hyperparameter has been trained meaning of their hyperparameters in Table 2. on the period hyperparameter of the photometric regres- The logarithm of the posterior likelihood of the GP sion. We then ﬁt all the RV GP hyperparameters, h, regression is calculated as θ, w, and λ in the quasi-periodic case, σ, a white noise parameter, as well as the Doppler amplitude K of a Ke- L −1 − − 1 | | − n plerian signal at the known orbital period and phase of log[ (r)] = rTΣ 1r log Σ log(2π), (2) the planet, to the RV data simultaneously. We ﬁnd that 2 2 2 all of our GP + Keplerian models are able to recover where r is the vector of residuals after removal of the the planetary Doppler amplitude K at a value consistent (optional) mean function, Σ is the covariance matrix, L with previous work. and n the number of data points. A prior term, , prior can be added to the likelihood to account for any priors 3.1. Gaussian Process Regression: Concept placed on the hyperparameters. For example, we apply (cid:104) (cid:105) the Gaussian prior Gaussian process is a nonparametric method to de- (cid:16) (cid:17) scribe a dataset by evaluating correlations between n data points through a covariance kernel. This kernel L −1 θtrue−θ 2 describes the relationship of each point in the data×set prior = e 2 σθ , (3) to each other point, and can be expressed as an n n matrix (subsequently referred to as the covariance ma- trix). The kernel is a function of hyperparameters. More to restrict the hyperparameter θ for the RV data, as complicated kernels can have more hyperparameters that the period of the correlated noise signal is equivalent to characterize diﬀerent qualities of the correlations in the the stellar rotation period P in both the photometric data, such as various periods, characteristic amplitudes rot and RV datasets. We restrict the period hyperparame- and length scales, etc. ter θ in the RV regressions within a Gaussian of width σ θ Gaussian process regression is widely used in the ﬁeld determined by the posterior distribution of the period hy- of machine learning (Neal 1997; Herbrich et al. 2003; perparameter found for our photometric GP regression. Quin˜onero-Candela & Rasmussen 2005; Wang et al. Prior knowledge of this hyperparameter helps to ensure 2008). Gibson et al. (2012) introduced the technique to convergence of the other hyperparameters of the RV re- the ﬁeld of exoplanets through analysis of transmission gression, as the RV data is not as well sampled as the spectroscopy to model correlated noise in the instru- photometry. mental systematics of HST/NICMOS, as described in § This likelihood calculation can be used to identify the 1. Concurrently with this work, Haywood et al. (2014) best ﬁt GP hyperparameters. For more details on how have demonstrated the technique of GP modeling of RV the covariance kernel and parameter boundary conditions and photometric signals for the CoRoT-7 planetary sys- § were chosen, refer to 4.1. For a more complete descrip- tem, ﬁrst modeling the photometry with a GP and then tion of Gaussian process regression and posterior likeli- using the photometric GP hyperparameters to train the hood evaluation, see Rasmussen & Williams (2006). initial RV GP hyperparameters. Haywood et al. (2014) demonstrated that in the case of CoRoT-7b, parametric 3.3. Gaussian Process and Stellar Activity spot models such as those used by H13 and P13 gave in- correct masses and uncertainties. Thus, it is important The presence of magnetic features on the stellar sur- to test many time series techniques, and further explore face induces variation in a star’s RV that can mimic a the novel application of GPs in Doppler analysis. We ex- planet’s orbital RV signal. Starspots are regions of high pand upon the GP training technique used in Haywood magnetic ﬁelds on the surface of a star that appear rel- et al. (2014) here. atively cooler and darker than the surrounding photo- sphere. As the starspots move across the disk of a star 3.2. Gaussian Process Covariance Kernel Choice (due to stellar rotation), the ﬂux balance between the redshifted and blueshifted halves of the star is broken, Finding the best GP regression requires choosing a producing a shift in the centroid of the spectral lines. kernel and initial hyperparameters, evaluating the likeli- This shift corresponds to an apparent RV signal and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Mass

text: Measurement

text: with

text: Gaussian

text: Process

text: METHODS

text: hood

text: of

text: those

text: hyperparameter

text: values,

text: and

text: then

text: iterating

text: through

text: parameter

text: space

text: until

text: the

text: most

text: likely

text: values

text: are

text: We

text: model

text: the

text: photometric

text: variations

text: observed

text: by

text: Ke-

text: found.

text: The

text: squared

text: exponential

text: kernel,

text: for

text: example,

text: de-

text: pler

text: with

text: a

text: GP

text: regression

text: with

text: a

text: quasi-periodic

text: kernel.

text: ﬁnes

text: a

text: covariance

text: matrix

text: We

text: then

text: model

text: the

text: radial

text: velocity

text: measurements

text: with

text: a

text: quasi-periodic

text: kernel

text: GP

text: regression,

text: using

text: the

text: kernel

text: pe-

text: t

text: t

text: rhiyopderhpyaprearmpaertaermθeter.,

text: θTphhoits,

text: atossutrmaiens

text: tthheatRVtheGpPrepdeormioid-

text: Σij

text: k(ti,

text: tj)

text: i

text: λ

text: j

text: RV

text: nant

text: signal

text: observed

text: in

text: both

text: datasets

text: comes

text: from

text: the

text: star,

text: and

text: thus

text: will

text: modulate

text: at

text: the

text: stellar

text: rotation

text: pe-

text: where

text: h

text: is

text: the

text: covariance

text: amplitude,

text: and

text: λ

text: the

text: covari-

text: riod.

text: When

text: we

text: model

text: the

text: photometric

text: data,

text: we

text: recover

text: a

text: ance

text: length

text: scale.

text: The

text: amplitude

text: observed

text: is

text: described

text: period

text: hyperparameter

text: that

text: agrees

text: with

text: other

text: autocorre-

text: by

text: h,

text: while

text: λ

text: is

text: a

text: characteristic

text: timescale

text: over

text: which

text: the

text: lation

text: analyses

text: of

text: the

text: Kepler

text: photometry

text: (Sanchis-Ojeda

text: data

text: is

text: going

text: to

text: be

text: correlated.

text: et

text: al.

text: supporting

text: this

text: assumption.

text: We

text: create

text: a

text: GP

text: regression

text: of

text: each

text: RV

text: dataset

text: with

text: a

text: quasi-periodic

text: ker-

text: We

text: discuss

text: other

text: GP

text: kernels

text: and

text: the

text: inferred

text: physical

text: nel,

text: where

text: the

text: period

text: hyperparameter

text: has

text: been

text: trained

text: meaning

text: of

text: their

text: hyperparameters

text: in

text: Table

text: on

text: the

text: period

text: hyperparameter

text: of

text: the

text: photometric

text: regres-

text: The

text: logarithm

text: of

text: the

text: posterior

text: likelihood

text: of

text: the

text: GP

text: sion.

text: We

text: then

text: ﬁt

text: all

text: the

text: RV

text: GP

text: hyperparameters,

text: h,

text: regression

text: is

text: calculated

text: as

text: θ,

text: w,

text: and

text: λ

text: in

text: the

text: quasi-periodic

text: case,

text: σ,

text: a

text: white

text: noise

text: parameter,

text: as

text: well

text: as

text: the

text: Doppler

text: amplitude

text: K

text: of

text: a

text: Ke-

text: L

text: −1

text: n

text: plerian

text: signal

text: at

text: the

text: known

text: orbital

text: period

text: and

text: phase

text: of

text: log[

text: (r)]

text: rTΣ

text: log

text: Σ

text: the

text: planet,

text: to

text: the

text: RV

text: data

text: simultaneously.

text: We

text: ﬁnd

text: that

text: all

text: of

text: our

text: GP

text: Keplerian

text: models

text: are

text: able

text: to

text: recover

text: where

text: r

text: is

text: the

text: vector

text: of

text: residuals

text: after

text: removal

text: of

text: the

text: the

text: planetary

text: Doppler

text: amplitude

text: K

text: at

text: a

text: value

text: consistent

text: (optional)

text: mean

text: function,

text: Σ

text: is

text: the

text: covariance

text: matrix,

text: L

text: with

text: previous

text: work.

text: and

text: n

text: the

text: number

text: of

text: data

text: points.

text: A

text: prior

text: term,

text: prior

text: can

text: be

text: added

text: to

text: the

text: likelihood

text: to

text: account

text: for

text: any

text: priors

text: Gaussian

text: Process

text: Regression:

text: Concept

text: placed

text: on

text: the

text: hyperparameters.

text: For

text: example,

text: we

text: apply

text: the

text: Gaussian

text: prior

text: Gaussian

text: process

text: is

text: a

text: nonparametric

text: method

text: to

text: de-

text: scribe

text: a

text: dataset

text: by

text: evaluating

text: correlations

text: between

text: n

text: data

text: points

text: through

text: a

text: covariance

text: kernel.

text: This

text: kernel

text: L

text: −1

text: θtrue−θ

text: describes

text: the

text: relationship

text: of

text: each

text: point

text: in

text: the

text: data×set

text: prior

text: e

text: σθ

text: to

text: each

text: other

text: point,

text: and

text: can

text: be

text: expressed

text: as

text: an

text: n

text: n

text: matrix

text: (subsequently

text: referred

text: to

text: as

text: the

text: covariance

text: ma-

text: trix).

text: The

text: kernel

text: is

text: a

text: function

text: of

text: hyperparameters.

text: More

text: to

text: restrict

text: the

text: hyperparameter

text: θ

text: for

text: the

text: RV

text: data,

text: as

text: complicated

text: kernels

text: can

text: have

text: more

text: hyperparameters

text: that

text: the

text: period

text: of

text: the

text: correlated

text: noise

text: signal

text: is

text: equivalent

text: to

text: characterize

text: diﬀerent

text: qualities

text: of

text: the

text: correlations

text: in

text: the

text: the

text: stellar

text: rotation

text: period

text: P

text: in

text: both

text: the

text: photometric

text: data,

text: such

text: as

text: various

text: periods,

text: characteristic

text: amplitudes

text: rot

text: and

text: RV

text: datasets.

text: We

text: restrict

text: the

text: period

text: hyperparame-

text: and

text: length

text: scales,

text: etc.

text: ter

text: θ

text: in

text: the

text: RV

text: regressions

text: within

text: a

text: Gaussian

text: of

text: width

text: σ

text: θ

text: Gaussian

text: process

text: regression

text: is

text: widely

text: used

text: in

text: the

text: ﬁeld

text: determined

text: by

text: the

text: posterior

text: distribution

text: of

text: the

text: period

text: hy-

text: of

text: machine

text: learning

text: (Neal

text: Herbrich

text: et

text: al.

text: perparameter

text: found

text: for

text: our

text: photometric

text: GP

text: regression.

text: Quin˜onero-Candela

text: Rasmussen

text: Wang

text: et

text: al.

text: Prior

text: knowledge

text: of

text: this

text: hyperparameter

text: helps

text: to

text: ensure

text: Gibson

text: et

text: al.

text: introduced

text: the

text: technique

text: to

text: convergence

text: of

text: the

text: other

text: hyperparameters

text: of

text: the

text: RV

text: re-

text: the

text: ﬁeld

text: of

text: exoplanets

text: through

text: analysis

text: of

text: transmission

text: gression,

text: as

text: the

text: RV

text: data

text: is

text: not

text: as

text: well

text: sampled

text: as

text: the

text: spectroscopy

text: to

text: model

text: correlated

text: noise

text: in

text: the

text: instru-

text: photometry.

text: mental

text: systematics

text: of

text: HST/NICMOS,

text: as

text: described

text: in

text: This

text: likelihood

text: calculation

text: can

text: be

text: used

text: to

text: identify

text: the

text: Concurrently

text: with

text: this

text: work,

text: Haywood

text: et

text: al.

text: best

text: ﬁt

text: GP

text: hyperparameters.

text: For

text: more

text: details

text: on

text: how

text: have

text: demonstrated

text: the

text: technique

text: of

text: GP

text: modeling

text: of

text: RV

text: the

text: covariance

text: kernel

text: and

text: parameter

text: boundary

text: conditions

text: and

text: photometric

text: signals

text: for

text: the

text: planetary

text: sys-

text: were

text: chosen,

text: refer

text: to

text: For

text: a

text: more

text: complete

text: descrip-

text: tem,

text: ﬁrst

text: modeling

text: the

text: photometry

text: with

text: a

text: GP

text: and

text: then

text: tion

text: of

text: Gaussian

text: process

text: regression

text: and

text: posterior

text: likeli-

text: using

text: the

text: photometric

text: GP

text: hyperparameters

text: to

text: train

text: the

text: hood

text: evaluation,

text: see

text: Rasmussen

text: Williams

text: initial

text: RV

text: GP

text: hyperparameters.

text: Haywood

text: et

text: al.

text: demonstrated

text: that

text: in

text: the

text: case

text: of

text: parametric

text: Gaussian

text: Process

text: and

text: Stellar

text: Activity

text: spot

text: models

text: such

text: as

text: those

text: used

text: by

text: and

text: gave

text: in-

text: correct

text: masses

text: and

text: uncertainties.

text: Thus,

text: it

text: is

text: important

text: The

text: presence

text: of

text: magnetic

text: features

text: on

text: the

text: stellar

text: sur-

text: to

text: test

text: many

text: time

text: series

text: techniques,

text: and

text: further

text: explore

text: face

text: induces

text: variation

text: in

text: a

text: star’s

text: RV

text: that

text: can

text: mimic

text: a

text: the

text: novel

text: application

text: of

text: GPs

text: in

text: Doppler

text: analysis.

text: We

text: ex-

text: planet’s

text: orbital

text: RV

text: signal.

text: Starspots

text: are

text: regions

text: of

text: high

text: pand

text: upon

text: the

text: GP

text: training

text: technique

text: used

text: in

text: Haywood

text: magnetic

text: ﬁelds

text: on

text: the

text: surface

text: of

text: a

text: star

text: that

text: appear

text: rel-

text: et

text: al.

text: here.

text: atively

text: cooler

text: and

text: darker

text: than

text: the

text: surrounding

text: photo-

text: sphere.

text: As

text: the

text: starspots

text: move

text: across

text: the

text: disk

text: of

text: a

text: star

text: Gaussian

text: Process

text: Covariance

text: Kernel

text: Choice

text: (due

text: to

text: stellar

text: rotation),

text: the

text: ﬂux

text: balance

text: between

text: the

text: redshifted

text: and

text: blueshifted

text: halves

text: of

text: the

text: star

text: is

text: broken,

text: Finding

text: the

text: best

text: GP

text: regression

text: requires

text: choosing

text: a

text: producing

text: a

text: shift

text: in

text: the

text: centroid

text: of

text: the

text: spectral

text: lines.

text: kernel

text: and

text: initial

text: hyperparameters,

text: evaluating

text: the

text: likeli-

text: This

text: shift

text: corresponds

text: to

text: an

text: apparent

text: RV

text: signal

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 Grunblatt et al. 4. RESULTS TABLE 3 ± Planetary Properties We ﬁnd a Doppler amplitude of Kepler-78b of 1.86 − 0.25 m s 1, corresponding to a mass of 1.87+−0.27 M⊕ 0.26 Property Value using the quasi-periodic kernel with common temporal hyperparameters, excepting amplitudes and white noise Name Kep±ler-78b terms, for the two RV datasets. We ﬁnd that the quasi- Radius, Rpl 1.20± 0.09R⊕ Orbital period P 0.35500744 0.00000006 days orb ± − periodic kernel provides the best ﬁt to the Kepler pho- Doppler amplitude, K 1.86 0.25 m s 1 tometry based on visual signal inspection, and the valid- Mass, M 1.87+−0.27 M⊕ ity of the ﬁt is conﬁrmed through χ2 calculation. Based Density, ρpl 6.0+−1.90.g26cm−3 red pl 1.±4 on a Bayesian analysis, we ﬁnd the strongest evidence for Iron fraction 32 26% a quasi-periodic kernel regression of the radial velocity data over simpler kernels as well, and that stronger evi- Note. — The name, radius, and orbital period values in Table 3 are dence exists for one set of temporal hyperparameters to taken from Howard et al. (2013). All other values have been calculated for this work. describe both radial velocity datasets as opposed to two, suggesting that the dominant signal in both RV datasets is indeed astrophysical. This supports the ﬁndings of Haywood et al. (2014). We provide the adopted model ther of the other models, with a ∆AIC=8 between it and parameters for the best-ﬁt quasi-periodic RV GP regres- the next lowest AIC value, that for the quasi-periodic, sion + Keplerian orbit models chosen in Table 4. uncoupled model, and ∆AIC=14 for the coupled model with a squared exponential kernel. The relative likelihood 4.1. GP kernel selection − of one model to another is given by e ∆AIC/2, indicating We ﬁnd a planetary mass within 1-σ of our adopted that the uncoupled quasi-periodic model is 1% as likely mass result for all models tested. We make the assump- as the coupled model, and the coupled squared exponen- tion that the predominant signal observed in the photo- tial kernel model is 0.09% as likely. The independent metric and both RV datasets comes from the star, and GP, squared exponential kernel had a ∆AIC=19 when thus will modulate at the stellar rotation period. We compared to the chosen model. We also tested removing ﬁnd that when we model the photometric data with a the white noise parameter from the squared exponential periodic or quasi-periodic GP kernel, we recover a pe- model, ﬁnding that doing so raised the BIC by over 20 riod hyperparameter consistent with previous estimates points and the AIC by over 15. Thus, we determined of the stellar rotation period, supporting our assumption that the white noise parameter was justiﬁed for all ker- about the predominant photometric signal. Similarly, nels. Strictly speaking, these information criteria require when we model the RV datasets with a quasi-periodic the assumption that the noise in the data is independent GP with no priors, we recover a period hyperparameter and identically distributed, which we know is not the consistent with the stellar rotation period or its alias. case. However, since we use the same data for all BIC Thus, we conclude that the quasi-periodic GP regression and AIC comparisons, they provide a valuable compar- is both well motivated and eﬀective for describing the ison between the diﬀerent GP kernels. We report all predominant signal in our datasets. relevant AIC and BIC values in Table 5. We calculated BIC and AIC values for all of our RV If we are modeling the stellar activity in both datasets, models. We test the three kernels of interest both with we would expect the hyperparameters for our simulta- shared hyperparameters as well as with fully indepen- neous models to be the same, with the exception of the dent hyperparameters in order to ensure that the varia- non-temporal amplitude hyperparameter h and the white tion we observe is consistent between datasets and thus noise term σ, which would be diﬀerent but related be- astrophysical, rather than local, in origin. Furthermore, tween the models due to the diﬀerent appearance of the we test our squared exponential kernel models with and disk of the star in diﬀerent wavelength regimes. We ex- without a white noise term for each dataset. We report plored the relation of the GP kernel amplitude hyperpa- these BIC and AIC values in Table 5. rameters, whose relationship we evaluated by calculating a, the ratio of the HARPS-N amplitude to the HIRES We ﬁnd that the quasi-periodic model we choose has ≥ − ≥ amplitude. The convergence of a to a best-ﬁt value sug- AIC and BIC values with ∆BIC 0.5 and ∆AIC 8 gests that there is a steady relation in the ranges of the between it and all other models tested. A ∆BIC value RV signals observed by HIRES and HARPS-N. Addi- of less than 2 suggests two models are indistinguishably tionally, the value of a is broadly consistent with the likely, whereas a ∆BIC greater than 10 suggests strong results of Desort et al. (2007), who illustrate that the evidence for the model with a lower BIC value. Interest- expected radial velocity contribution from starspots on ingly, the BIC for the quasiperiodic and squared expo- a K dwarf is larger at shorter wavelengths, resulting in nential kernels are almost identical, while there is signif- a larger signal in HIRES than in HARPS-N, as HARPS- icantly less evidence for the periodic model (BIC of peri- N has signiﬁcant wavelength coverage in the 600-700nm odic model 18 points higher than SE and quasi-periodic regime that HIRES does not. We show the distribution models). Furthermore, the lowest BIC of all was found of a along with the other parameters of our MCMC cal- for the squared exponential kernel with two independent − culation in Figure 4. Any variation due explicitly to the GP regressions, with a ∆BIC = 0.5 as compared to the correlated stellar activity should thus cause our models chosen model. However, the AIC of the quasi-periodic, to have a shape (period and phase) that is consistent be- coupled GP model is signiﬁcantly lower than that of ei- tween the two datasets. We allow the reader to conﬁrm ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Grunblatt

text: et

text: al.

text: RESULTS

text: TABLE

text: Planetary

text: Properties

text: We

text: ﬁnd

text: a

text: Doppler

text: amplitude

text: of

text: of

text: m

text: s

text: corresponding

text: to

text: a

text: mass

text: of

text: M⊕

text: Property

text: Value

text: using

text: the

text: quasi-periodic

text: kernel

text: with

text: common

text: temporal

text: hyperparameters,

text: excepting

text: amplitudes

text: and

text: white

text: noise

text: Name

text: terms,

text: for

text: the

text: two

text: RV

text: datasets.

text: We

text: ﬁnd

text: that

text: the

text: quasi-

text: Radius,

text: Rpl

text: Orbital

text: period

text: P

text: days

text: orb

text: periodic

text: kernel

text: provides

text: the

text: best

text: ﬁt

text: to

text: the

text: Kepler

text: pho-

text: Doppler

text: amplitude,

text: K

text: m

text: s

text: tometry

text: based

text: on

text: visual

text: signal

text: inspection,

text: and

text: the

text: valid-

text: Mass,

text: M

text: M⊕

text: ity

text: of

text: the

text: ﬁt

text: is

text: conﬁrmed

text: through

text: χ2

text: calculation.

text: Based

text: Density,

text: ρpl

text: red

text: pl

text: on

text: a

text: Bayesian

text: analysis,

text: we

text: ﬁnd

text: the

text: strongest

text: evidence

text: for

text: Iron

text: fraction

text: a

text: quasi-periodic

text: kernel

text: regression

text: of

text: the

text: radial

text: velocity

text: data

text: over

text: simpler

text: kernels

text: as

text: well,

text: and

text: that

text: stronger

text: evi-

text: Note.

text: The

text: name,

text: radius,

text: and

text: orbital

text: period

text: values

text: in

text: Table

text: are

text: dence

text: exists

text: for

text: one

text: set

text: of

text: temporal

text: hyperparameters

text: to

text: taken

text: from

text: Howard

text: et

text: al.

text: All

text: other

text: values

text: have

text: been

text: calculated

text: for

text: this

text: work.

text: describe

text: both

text: radial

text: velocity

text: datasets

text: as

text: opposed

text: to

text: two,

text: suggesting

text: that

text: the

text: dominant

text: signal

text: in

text: both

text: RV

text: datasets

text: is

text: indeed

text: astrophysical.

text: This

text: supports

text: the

text: ﬁndings

text: of

text: Haywood

text: et

text: al.

text: We

text: provide

text: the

text: adopted

text: model

text: ther

text: of

text: the

text: other

text: models,

text: with

text: a

text: between

text: it

text: and

text: parameters

text: for

text: the

text: best-ﬁt

text: quasi-periodic

text: RV

text: GP

text: regres-

text: the

text: next

text: lowest

text: AIC

text: value,

text: that

text: for

text: the

text: quasi-periodic,

text: sion

text: Keplerian

text: orbit

text: models

text: chosen

text: in

text: Table

text: uncoupled

text: model,

text: and

text: for

text: the

text: coupled

text: model

text: with

text: a

text: squared

text: exponential

text: kernel.

text: The

text: relative

text: likelihood

text: GP

text: kernel

text: selection

text: of

text: one

text: model

text: to

text: another

text: is

text: given

text: by

text: e

text: indicating

text: We

text: ﬁnd

text: a

text: planetary

text: mass

text: within

text: of

text: our

text: adopted

text: that

text: the

text: uncoupled

text: quasi-periodic

text: model

text: is

text: as

text: likely

text: mass

text: result

text: for

text: all

text: models

text: tested.

text: We

text: make

text: the

text: assump-

text: as

text: the

text: coupled

text: model,

text: and

text: the

text: coupled

text: squared

text: exponen-

text: tion

text: that

text: the

text: predominant

text: signal

text: observed

text: in

text: the

text: photo-

text: tial

text: kernel

text: model

text: is

text: as

text: likely.

text: The

text: independent

text: metric

text: and

text: both

text: RV

text: datasets

text: comes

text: from

text: the

text: star,

text: and

text: GP,

text: squared

text: exponential

text: kernel

text: had

text: a

text: when

text: thus

text: will

text: modulate

text: at

text: the

text: stellar

text: rotation

text: period.

text: We

text: compared

text: to

text: the

text: chosen

text: model.

text: We

text: also

text: tested

text: removing

text: ﬁnd

text: that

text: when

text: we

text: model

text: the

text: photometric

text: data

text: with

text: a

text: the

text: white

text: noise

text: parameter

text: from

text: the

text: squared

text: exponential

text: periodic

text: or

text: quasi-periodic

text: GP

text: kernel,

text: we

text: recover

text: a

text: pe-

text: model,

text: ﬁnding

text: that

text: doing

text: so

text: raised

text: the

text: BIC

text: by

text: over

text: riod

text: hyperparameter

text: consistent

text: with

text: previous

text: estimates

text: points

text: and

text: the

text: AIC

text: by

text: over

text: Thus,

text: we

text: determined

text: of

text: the

text: stellar

text: rotation

text: period,

text: supporting

text: our

text: assumption

text: that

text: the

text: white

text: noise

text: parameter

text: was

text: justiﬁed

text: for

text: all

text: ker-

text: about

text: the

text: predominant

text: photometric

text: signal.

text: Similarly,

text: nels.

text: Strictly

text: speaking,

text: these

text: information

text: criteria

text: require

text: when

text: we

text: model

text: the

text: RV

text: datasets

text: with

text: a

text: quasi-periodic

text: the

text: assumption

text: that

text: the

text: noise

text: in

text: the

text: data

text: is

text: independent

text: GP

text: with

text: no

text: priors,

text: we

text: recover

text: a

text: period

text: hyperparameter

text: and

text: identically

text: distributed,

text: which

text: we

text: know

text: is

text: not

text: the

text: consistent

text: with

text: the

text: stellar

text: rotation

text: period

text: or

text: its

text: alias.

text: case.

text: However,

text: since

text: we

text: use

text: the

text: same

text: data

text: for

text: all

text: BIC

text: Thus,

text: we

text: conclude

text: that

text: the

text: quasi-periodic

text: GP

text: regression

text: and

text: AIC

text: comparisons,

text: they

text: provide

text: a

text: valuable

text: compar-

text: is

text: both

text: well

text: motivated

text: and

text: eﬀective

text: for

text: describing

text: the

text: ison

text: between

text: the

text: diﬀerent

text: GP

text: kernels.

text: We

text: report

text: all

text: predominant

text: signal

text: in

text: our

text: datasets.

text: relevant

text: AIC

text: and

text: BIC

text: values

text: in

text: Table

text: We

text: calculated

text: BIC

text: and

text: AIC

text: values

text: for

text: all

text: of

text: our

text: RV

text: If

text: we

text: are

text: modeling

text: the

text: stellar

text: activity

text: in

text: both

text: datasets,

text: models.

text: We

text: test

text: the

text: three

text: kernels

text: of

text: interest

text: both

text: with

text: we

text: would

text: expect

text: the

text: hyperparameters

text: for

text: our

text: simulta-

text: shared

text: hyperparameters

text: as

text: well

text: as

text: with

text: fully

text: indepen-

text: neous

text: models

text: to

text: be

text: the

text: same,

text: with

text: the

text: exception

text: of

text: the

text: dent

text: hyperparameters

text: in

text: order

text: to

text: ensure

text: that

text: the

text: varia-

text: non-temporal

text: amplitude

text: hyperparameter

text: h

text: and

text: the

text: white

text: tion

text: we

text: observe

text: is

text: consistent

text: between

text: datasets

text: and

text: thus

text: noise

text: term

text: σ,

text: which

text: would

text: be

text: diﬀerent

text: but

text: related

text: be-

text: astrophysical,

text: rather

text: than

text: local,

text: in

text: origin.

text: Furthermore,

text: tween

text: the

text: models

text: due

text: to

text: the

text: diﬀerent

text: appearance

text: of

text: the

text: we

text: test

text: our

text: squared

text: exponential

text: kernel

text: models

text: with

text: and

text: disk

text: of

text: the

text: star

text: in

text: diﬀerent

text: wavelength

text: regimes.

text: We

text: ex-

text: without

text: a

text: white

text: noise

text: term

text: for

text: each

text: dataset.

text: We

text: report

text: plored

text: the

text: relation

text: of

text: the

text: GP

text: kernel

text: amplitude

text: hyperpa-

text: these

text: BIC

text: and

text: AIC

text: values

text: in

text: Table

text: rameters,

text: whose

text: relationship

text: we

text: evaluated

text: by

text: calculating

text: a,

text: the

text: ratio

text: of

text: the

text: HARPS-N

text: amplitude

text: to

text: the

text: HIRES

text: We

text: ﬁnd

text: that

text: the

text: quasi-periodic

text: model

text: we

text: choose

text: has

text: amplitude.

text: The

text: convergence

text: of

text: a

text: to

text: a

text: best-ﬁt

text: value

text: sug-

text: AIC

text: and

text: BIC

text: values

text: with

text: ∆BIC

text: and

text: ∆AIC

text: gests

text: that

text: there

text: is

text: a

text: steady

text: relation

text: in

text: the

text: ranges

text: of

text: the

text: between

text: it

text: and

text: all

text: other

text: models

text: tested.

text: A

text: ∆BIC

text: value

text: RV

text: signals

text: observed

text: by

text: HIRES

text: and

text: HARPS-N.

text: Addi-

text: of

text: less

text: than

text: suggests

text: two

text: models

text: are

text: indistinguishably

text: tionally,

text: the

text: value

text: of

text: a

text: is

text: broadly

text: consistent

text: with

text: the

text: likely,

text: whereas

text: a

text: ∆BIC

text: greater

text: than

text: suggests

text: strong

text: results

text: of

text: Desort

text: et

text: al.

text: who

text: illustrate

text: that

text: the

text: evidence

text: for

text: the

text: model

text: with

text: a

text: lower

text: BIC

text: value.

text: Interest-

text: expected

text: radial

text: velocity

text: contribution

text: from

text: starspots

text: on

text: ingly,

text: the

text: BIC

text: for

text: the

text: quasiperiodic

text: and

text: squared

text: expo-

text: a

text: K

text: dwarf

text: is

text: larger

text: at

text: shorter

text: wavelengths,

text: resulting

text: in

text: nential

text: kernels

text: are

text: almost

text: identical,

text: while

text: there

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text: signif-

text: a

text: larger

text: signal

text: in

text: HIRES

text: than

text: in

text: HARPS-N,

text: as

text: HARPS-

text: icantly

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text: evidence

text: for

text: the

text: periodic

text: model

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text: N

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text: signiﬁcant

text: wavelength

text: coverage

text: in

text: the

text: odic

text: model

text: points

text: higher

text: than

text: SE

text: and

text: quasi-periodic

text: regime

text: that

text: HIRES

text: does

text: not.

text: We

text: show

text: the

text: distribution

text: models).

text: Furthermore,

text: the

text: lowest

text: BIC

text: of

text: all

text: was

text: found

text: of

text: a

text: along

text: with

text: the

text: other

text: parameters

text: of

text: our

text: MCMC

text: cal-

text: for

text: the

text: squared

text: exponential

text: kernel

text: with

text: two

text: independent

text: culation

text: in

text: Figure

text: Any

text: variation

text: due

text: explicitly

text: to

text: the

text: GP

text: regressions,

text: with

text: a

text: ∆BIC

text: as

text: compared

text: to

text: the

text: correlated

text: stellar

text: activity

text: should

text: thus

text: cause

text: our

text: models

text: chosen

text: model.

text: However,

text: the

text: AIC

text: of

text: the

text: quasi-periodic,

text: to

text: have

text: a

text: shape

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text: phase)

text: that

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text: coupled

text: GP

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text: signiﬁcantly

text: lower

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text: datasets.

text: We

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text: the

text: reader

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text: conﬁrm

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text: 1 Introduction Entanglement entropy is an important notion in many-body quantum system [1, 2]. For a bipartite system, the entanglement entropy of the subsystem A is deﬁned to be the Von Neumann entropy of the reduced density matrix − S = Tr ρ log ρ , (1.1) EE A A where the reduced density matrix ρ = Tr ρ is obtained by smearing the degrees of freedom in A B | (cid:105)(cid:104) | B complementary to A. When the total system is in the vacuum ρ = 0 0 , the entanglement entropy of A equals to the one of its complement S (A) = S (B). (1.2) EE EE At a ﬁnite temperature, however, such equality does not hold anymore. The entanglement entropy has been studied in various condensed matter systems [3], and also in the context of AdS/CFT correspondence [4,5]. The computation of the entanglement entropy from its deﬁnition becomes a formidable task when the number of degrees of freedom in the system is huge. Especially in quantum ﬁeld theory with inﬁnite degrees of freedom, it is more convenient to compute the R´enyi entropy via the Replica trick [6]. The R´enyi entropy is deﬁned to be − 1 S = − Trρn . (1.3) n n 1 A It is related to the entanglement entropy by the relation S = lim S (1.4) EE → n n 1 if the analytic continuation of the limit is available. In ﬁeld theory, the entropy is deﬁned with respect to a spacial submanifold at a ﬁxed time. In two dimensional spacetime, the submanifold could be interval(s). However, after Wick rotation, the Euclideanized ﬁeld theory is deﬁned in a complex plane and the R´enyi entropy becomes − 1 Z S = − n , (1.5) n n 1 Zn 1 in which Z is the partition function on a n-sheeted Riemann surface resulted from the pasting n of n complex plane along the branch cuts (intervals). There has been a long history to study the entanglement entropy in quantum ﬁeld theory. In the dimension higher than two, the entanglement entropy has been found to obey the area law (for a nice review, see [7]). But we are only allowed to compute the entanglement entropy 2 ##LTLine## ##LTLine## ##LTLine##

text: Introduction

text: Entanglement

text: entropy

text: is

text: an

text: important

text: notion

text: in

text: many-body

text: quantum

text: system

text: For

text: a

text: bipartite

text: system,

text: the

text: entanglement

text: entropy

text: of

text: the

text: subsystem

text: A

text: is

text: deﬁned

text: to

text: be

text: the

text: Von

text: Neumann

text: entropy

text: of

text: the

text: reduced

text: density

text: matrix

text: S

text: Tr

text: ρ

text: log

text: ρ

text: EE

text: A

text: A

text: where

text: the

text: reduced

text: density

text: matrix

text: ρ

text: Tr

text: ρ

text: is

text: obtained

text: by

text: smearing

text: the

text: degrees

text: of

text: freedom

text: in

text: A

text: B

text: B

text: complementary

text: to

text: A.

text: When

text: the

text: total

text: system

text: is

text: in

text: the

text: vacuum

text: ρ

text: the

text: entanglement

text: entropy

text: of

text: A

text: equals

text: to

text: the

text: one

text: of

text: its

text: complement

text: S

text: (A)

text: S

text: (B).

text: EE

text: EE

text: At

text: a

text: ﬁnite

text: temperature,

text: however,

text: such

text: equality

text: does

text: not

text: hold

text: anymore.

text: The

text: entanglement

text: entropy

text: has

text: been

text: studied

text: in

text: various

text: condensed

text: matter

text: systems

text: and

text: also

text: in

text: the

text: context

text: of

text: AdS/CFT

text: correspondence

text: The

text: computation

text: of

text: the

text: entanglement

text: entropy

text: from

text: its

text: deﬁnition

text: becomes

text: a

text: formidable

text: task

text: when

text: the

text: number

text: of

text: degrees

text: of

text: freedom

text: in

text: the

text: system

text: is

text: huge.

text: Especially

text: in

text: quantum

text: ﬁeld

text: theory

text: with

text: inﬁnite

text: degrees

text: of

text: freedom,

text: it

text: is

text: more

text: convenient

text: to

text: compute

text: the

text: R´enyi

text: entropy

text: via

text: the

text: Replica

text: trick

text: The

text: R´enyi

text: entropy

text: is

text: deﬁned

text: to

text: be

text: S

text: Trρn

text: n

text: n

text: A

text: It

text: is

text: related

text: to

text: the

text: entanglement

text: entropy

text: by

text: the

text: relation

text: S

text: lim

text: S

text: EE

text: n

text: n

text: if

text: the

text: analytic

text: continuation

text: of

text: the

text: limit

text: is

text: available.

text: In

text: ﬁeld

text: theory,

text: the

text: entropy

text: is

text: deﬁned

text: with

text: respect

text: to

text: a

text: spacial

text: submanifold

text: at

text: a

text: ﬁxed

text: time.

text: In

text: two

text: dimensional

text: spacetime,

text: the

text: submanifold

text: could

text: be

text: interval(s).

text: However,

text: after

text: Wick

text: rotation,

text: the

text: Euclideanized

text: ﬁeld

text: theory

text: is

text: deﬁned

text: in

text: a

text: complex

text: plane

text: and

text: the

text: R´enyi

text: entropy

text: becomes

text: Z

text: S

text: n

text: n

text: n

text: Zn

text: in

text: which

text: Z

text: is

text: the

text: partition

text: function

text: on

text: a

text: n-sheeted

text: Riemann

text: surface

text: resulted

text: from

text: the

text: pasting

text: n

text: of

text: n

text: complex

text: plane

text: along

text: the

text: branch

text: cuts

text: (intervals).

text: There

text: has

text: been

text: a

text: long

text: history

text: to

text: study

text: the

text: entanglement

text: entropy

text: in

text: quantum

text: ﬁeld

text: theory.

text: In

text: the

text: dimension

text: higher

text: than

text: two,

text: the

text: entanglement

text: entropy

text: has

text: been

text: found

text: to

text: obey

text: the

text: area

text: law

text: (for

text: a

text: nice

text: review,

text: see

text: But

text: we

text: are

text: only

text: allowed

text: to

text: compute

text: the

text: entanglement

text: entropy

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) − (cid:28) (cid:28) where (z z ) M β. Since 2 1 iM − − − − − − ∼ − − − − − − dz(z z1) nk (z (z2 1)) (1 nk ) (1 e 2πink ) log (z1 (z2 1)), (A.27) iM 2(k) the leading singular term in W is 2 ∼ sin πk − − − k | − − W¯ 2(k) i n η(τ) 3ϑ ( (1 ) τ) log (z (z 1)), (A.28) 2 (cid:12) π (cid:12) 1 n 1 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and (cid:12) (cid:12) W 2(k) ∼ 2 sin πk − − − 2 n ( log(z (z 1))). (A.29) 1 2 1(k) π W 1 References [1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cam- bridge university press, 2010. [2] D. Petz, Quantum information theory and quantum statistics. Springer, 2008. [3] L. Amico, R. Fazio, A. Osterloh and V. Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80, 517 (2008). [4] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys.Rev.Lett. 96 (2006) 181602, arXiv:hep-th/0603001 [hep-th]. [5] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP 0608 (2006) 045, arXiv:hep-th/0605073 [hep-th]. [6] J. Callan, Curtis G. and F. Wilczek, “On geometric entropy,” Phys.Lett. B333 (1994) 55–61, arXiv:hep-th/9401072 [hep-th]. [7] J. Eisert, M. Cramer and M. B. Plenio, “Area laws for the entanglement entropy - a review,” Rev. Mod. Phys. 82, 277 (2010). [8] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum ﬁeld theory,” J.Stat.Mech. 0406 (2004) P06002, arXiv:hep-th/0405152 [hep-th]. [9] P. Calabrese, J. Cardy and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal ﬁeld theory,” J. Stat. Mech. 0911, P11001 (2009) [arXiv:0905.2069 [hep-th]]. [10] P. Calabrese, J. Cardy and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal ﬁeld theory II,” J. Stat. Mech. 1101, P01021 (2011) [arXiv:1011.5482 [hep-th]]. 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: (z

text: z

text: M

text: β.

text: Since

text: iM

text: dz(z

text: nk

text: (z

text: nk

text: e

text: log

text: iM

text: the

text: leading

text: singular

text: term

text: in

text: W

text: is

text: sin

text: πk

text: k

text: W¯

text: i

text: n

text: η(τ)

text: 3ϑ

text: τ)

text: log

text: (z

text: (z

text: π

text: n

text: and

text: W

text: sin

text: πk

text: n

text: log(z

text: (z

text: π

text: W

text: References

text: M.

text: A.

text: Nielsen

text: and

text: I.

text: L.

text: Chuang,

text: Quantum

text: computation

text: and

text: quantum

text: information.

text: Cam-

text: bridge

text: university

text: press,

text: D.

text: Petz,

text: Quantum

text: information

text: theory

text: and

text: quantum

text: statistics.

text: Springer,

text: L.

text: Amico,

text: R.

text: Fazio,

text: A.

text: Osterloh

text: and

text: V.

text: Vedral,

text: “Entanglement

text: in

text: many-body

text: systems,”

text: Rev.

text: Mod.

text: Phys.

text: S.

text: Ryu

text: and

text: T.

text: Takayanagi,

text: “Holographic

text: derivation

text: of

text: entanglement

text: entropy

text: from

text: AdS/CFT,”

text: Phys.Rev.Lett.

text: [hep-th].

text: S.

text: Ryu

text: and

text: T.

text: Takayanagi,

text: “Aspects

text: of

text: Holographic

text: Entanglement

text: Entropy,”

text: JHEP

text: [hep-th].

text: J.

text: Callan,

text: Curtis

text: G.

text: and

text: F.

text: Wilczek,

text: “On

text: geometric

text: entropy,”

text: Phys.Lett.

text: [hep-th].

text: J.

text: Eisert,

text: M.

text: Cramer

text: and

text: M.

text: B.

text: Plenio,

text: “Area

text: laws

text: for

text: the

text: entanglement

text: entropy

text: a

text: review,”

text: Rev.

text: Mod.

text: Phys.

text: P.

text: Calabrese

text: and

text: J.

text: L.

text: Cardy,

text: “Entanglement

text: entropy

text: and

text: quantum

text: ﬁeld

text: theory,”

text: J.Stat.Mech.

text: [hep-th].

text: P.

text: Calabrese,

text: J.

text: Cardy

text: and

text: E.

text: Tonni,

text: “Entanglement

text: entropy

text: of

text: two

text: disjoint

text: intervals

text: in

text: conformal

text: ﬁeld

text: theory,”

text: J.

text: Stat.

text: Mech.

text: [hep-th]].

text: P.

text: Calabrese,

text: J.

text: Cardy

text: and

text: E.

text: Tonni,

text: “Entanglement

text: entropy

text: of

text: two

text: disjoint

text: intervals

text: in

text: conformal

text: ﬁeld

text: theory

text: II,”

text: J.

text: Stat.

text: Mech.

text: [hep-th]].

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##