text: 3 leaving us with (cid:90) (cid:90) (cid:20) (cid:0) (cid:1) D D 1 − 1 (cid:21) − Z = Aa (cid:0)Fa ∆ (A) exp i d(cid:1)d(cid:0)x Fa Faµ(cid:1)ν ∂ Aa ∂ Aa 2 µ µν FP 4 µν 4 µ ν ν µ −1 − L Fa + ∂ Aa ∂ Aa g(cid:15)abcAb Ac + . (2.7) 2 µν µ ν ν µ µ ν gf (cid:104) (cid:105) When the generating functional Z is written in this form we see that there are now two propagators FF and (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) AA as well as two three point functions FAA and AAA (but no mixed propagators AF or four point vertex (cid:104) (cid:105) AAAA .) This possibility of altering the Feynman rules in YM theory will now be exploited when examining the ﬁrst order (Palatini) form of the Einstein-Hilbert action. III. THE FIRST ORDER EINSTEIN-HILBERT ACTION Rather than using g and Γλ as independent ﬁelds in the 1EH Lagrangian of Eq. (1.1), it proves convenient to µν µν use [9] √ − hµν = ggµν (3.1a) (cid:0) (cid:1) and − 1 Gλ = Γλ δλΓσ + δλΓσ (3.1b) µν µν 2 µ νσ ν µσ so that now we have (cid:18) (cid:19) L 1 − = hµν Gλ + − Gλ Gσ Gλ Gσ . (3.2) 1EH µν ,λ d 1 µλ νσ µσ νλ The canonical structure of this action(cid:90) has been examined in refs(cid:90). [9, 10] and the resulting path integral in ref. [11]. Here, we will consider using the Faddeev-Popov path integral [12] D D L L Z = hµν Gλ ∆ (h) exp i ddx [ + ] . (3.3) 1EH µν FP 1EH gf Directly using the form of Eq. (3.2) makes it impossible to deﬁne a propagator for hµν and Gλ . (This is easily seen if L µν one were attempt to ﬁnd a propagator for ﬁelds φ and V λ with the Lagrangian = φV λ.) In ref. [4], hµν is expanded − ,λ about a ﬂat metric ηµν = diag(+, +, +, . . . , ) so that hµν(x) = ηµν + φµν(x); (3.4) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) the propagators φφ , GG , φG and the vertex φGG are given in ref. [3]. However, it is not immediately evident L how this form of Z yields results consistent with those that follow from the 2EH Lagrangian . 1EH (cid:16) (cid:17) 2EH To show this equivalence, we start by writing Eq. (3.2) as L − 1 = Gλ hµν + Mµνπτ(h)Gλ Gσ , (3.5) 1EH µν ,λ 2 λ σ µν πτ where (cid:20) 1 1 (cid:105) Mµνπτ(h) = − (δνδτhµπ + δµδτhνπ + δνδπhµτ + δµδπhντ) λ σ 2 d 1 λ σ λ σ λ σ λ σ − (δτδνhµπ + δτδµhνπ + δπδνhµτ + δπδµhντ) (3.6) λ σ λ σ λ σ λ σ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: leaving

text: us

text: with

text: D

text: D

text: Z

text: Aa

text: (A)

text: exp

text: i

text: Fa

text: Aa

text: Aa

text: µ

text: µν

text: FP

text: µν

text: µ

text: ν

text: ν

text: µ

text: −1

text: L

text: Fa

text: Aa

text: Aa

text: Ac

text: µν

text: µ

text: ν

text: ν

text: µ

text: µ

text: ν

text: gf

text: When

text: the

text: generating

text: functional

text: Z

text: is

text: written

text: in

text: this

text: form

text: we

text: see

text: that

text: there

text: are

text: now

text: two

text: propagators

text: FF

text: and

text: AA

text: as

text: well

text: as

text: two

text: three

text: point

text: functions

text: FAA

text: and

text: AAA

text: (but

text: no

text: mixed

text: propagators

text: AF

text: or

text: four

text: point

text: vertex

text: AAAA

text: This

text: possibility

text: of

text: altering

text: the

text: Feynman

text: rules

text: in

text: YM

text: theory

text: will

text: now

text: be

text: exploited

text: when

text: examining

text: the

text: ﬁrst

text: order

text: (Palatini)

text: form

text: of

text: the

text: Einstein-Hilbert

text: action.

text: III.

text: THE

text: FIRST

text: ORDER

text: EINSTEIN-HILBERT

text: ACTION

text: Rather

text: than

text: using

text: g

text: and

text: Γλ

text: as

text: independent

text: ﬁelds

text: in

text: the

text: Lagrangian

text: of

text: Eq.

text: it

text: proves

text: convenient

text: to

text: µν

text: µν

text: use

text: hµν

text: ggµν

text: and

text: Gλ

text: Γλ

text: δλΓσ

text: δλΓσ

text: µν

text: µν

text: µ

text: νσ

text: ν

text: µσ

text: so

text: that

text: now

text: we

text: have

text: L

text: hµν

text: Gλ

text: Gλ

text: Gσ

text: Gλ

text: Gσ

text: µν

text: ,λ

text: d

text: µλ

text: νσ

text: µσ

text: νλ

text: The

text: canonical

text: structure

text: of

text: this

text: has

text: been

text: examined

text: in

text: and

text: the

text: resulting

text: path

text: integral

text: in

text: ref.

text: Here,

text: we

text: will

text: consider

text: using

text: the

text: Faddeev-Popov

text: path

text: integral

text: D

text: D

text: L

text: L

text: Z

text: hµν

text: Gλ

text: (h)

text: exp

text: i

text: ddx

text: µν

text: FP

text: gf

text: Directly

text: using

text: the

text: form

text: of

text: Eq.

text: makes

text: it

text: impossible

text: to

text: deﬁne

text: a

text: propagator

text: for

text: hµν

text: and

text: Gλ

text: (This

text: is

text: easily

text: seen

text: if

text: L

text: µν

text: one

text: were

text: attempt

text: to

text: ﬁnd

text: a

text: propagator

text: for

text: ﬁelds

text: φ

text: and

text: V

text: λ

text: with

text: the

text: Lagrangian

text: φV

text: λ.)

text: In

text: ref.

text: hµν

text: is

text: expanded

text: ,λ

text: about

text: a

text: ﬂat

text: metric

text: ηµν

text: diag(+,

text: so

text: that

text: hµν(x)

text: ηµν

text: φµν(x);

text: the

text: propagators

text: φφ

text: GG

text: φG

text: and

text: the

text: vertex

text: φGG

text: are

text: given

text: in

text: ref.

text: However,

text: it

text: is

text: not

text: immediately

text: evident

text: L

text: how

text: this

text: form

text: of

text: Z

text: yields

text: results

text: consistent

text: with

text: those

text: that

text: follow

text: from

text: the

text: Lagrangian

text: To

text: show

text: this

text: equivalence,

text: we

text: start

text: by

text: writing

text: Eq.

text: as

text: L

text: Gλ

text: hµν

text: Mµνπτ(h)Gλ

text: Gσ

text: µν

text: ,λ

text: λ

text: σ

text: µν

text: πτ

text: where

text: Mµνπτ(h)

text: (δνδτhµπ

text: δµδτhνπ

text: δνδπhµτ

text: δµδπhντ)

text: λ

text: σ

text: d

text: λ

text: σ

text: λ

text: σ

text: λ

text: σ

text: λ

text: σ

text: (δτδνhµπ

text: δτδµhνπ

text: δπδνhµτ

text: δπδµhντ)

text: λ

text: σ

text: λ

text: σ

text: λ

text: σ

text: λ

text: σ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 [1] M. Ferraris, M. Francaviglia, and C. Reina, Gen. Rel. Grav. 14, 243 (1982). [2] D. G. C. McKeon, Can. J. Phys. 72, 601 (1994); S. Okubo and Y. Tosa, Phys. Rev. D20, 462 (1979); I. L. Buchbinder and I.L. Shapiro, Sov. J. Nucl. Phys. 37, 248 (1983); Acta. Phys. Pol. B16, 103 (1985). [3] F. T. Brandt and D. G. C. McKeon, Phys. Rev. D91, 105006 (2015). [4] D. M. Capper, G. Leibbrandt, and M. Ramon Medrano, Phys. Rev. D8, 4320 (1973). [5] M. Y. Kalmykov, P. I. Pronin and K. V. Stepanyantz, Class. Quant. Grav. 11, 2645 (1994) [6] F. T. Brandt, J. Frenkel, and D. G. C. McKeon, Phys. Rev. D76, 105029 (2007). [7] F. T. Brandt and D. G. C. McKeon, Phys. Rev. D79, 087702 (2009). [8] D. G. C. McKeon, Can. J. Phys. 93, 1164 (2015). [9] D. G. C. McKeon, Int. J. Mod. Phys. A25, 3453 (2010). [10] N. Kiriushcheva and S. V. Kuzmin, Eur. Phys. J. C70, 389 (2010). [11] F. Chishtie and D. G. C. McKeon, Class. Quant. Grav. 29, 235016 (2012). [12] L. D. Faddeev and V. N. Popov, Phys. Lett. B25, 29 (1967). [13] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B44, 189 (1972). [14] G. Leibbrandt, Rev. Mod. Phys. 47, 849 (1975). [15] G. ’t Hooft and M. J. G. Veltman, Annales Poincare Phys. Theor. A20, 69 (1974). [16] M. H. Goroﬀ and A. Sagnotti, Phys.Lett. B160, 81 (1985); Nucl. Phys. B266, 709 (1986). [17] K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Nucl. Phys. B 174 (1980) 345. [18] A. E. M. van de Ven, Nucl. Phys. B378, 309 (1992). [19] R. B. Mann, D. G. C. McKeon, T. G. Steele, and L. Tarasov, Nucl. Phys. B311, 630 (1989). ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: reference

text: M.

text: Ferraris,

text: M.

text: Francaviglia,

text: and

text: C.

text: Reina,

text: Gen.

text: Rel.

text: Grav.

text: D.

text: G.

text: C.

text: McKeon,

text: Can.

text: J.

text: Phys.

text: S.

text: Okubo

text: and

text: Y.

text: Tosa,

text: Phys.

text: Rev.

text: I.

text: L.

text: Buchbinder

text: and

text: I.L.

text: Shapiro,

text: Sov.

text: J.

text: Nucl.

text: Phys.

text: Acta.

text: Phys.

text: Pol.

text: F.

text: T.

text: Brandt

text: and

text: D.

text: G.

text: C.

text: McKeon,

text: Phys.

text: Rev.

text: D.

text: M.

text: Capper,

text: G.

text: Leibbrandt,

text: and

text: M.

text: Ramon

text: Medrano,

text: Phys.

text: Rev.

text: M.

text: Y.

text: Kalmykov,

text: P.

text: I.

text: Pronin

text: and

text: K.

text: V.

text: Stepanyantz,

text: Class.

text: Quant.

text: Grav.

text: F.

text: T.

text: Brandt,

text: J.

text: Frenkel,

text: and

text: D.

text: G.

text: C.

text: McKeon,

text: Phys.

text: Rev.

text: F.

text: T.

text: Brandt

text: and

text: D.

text: G.

text: C.

text: McKeon,

text: Phys.

text: Rev.

text: D.

text: G.

text: C.

text: McKeon,

text: Can.

text: J.

text: Phys.

text: D.

text: G.

text: C.

text: McKeon,

text: Int.

text: J.

text: Mod.

text: Phys.

text: N.

text: Kiriushcheva

text: and

text: S.

text: V.

text: Kuzmin,

text: Eur.

text: Phys.

text: J.

text: F.

text: Chishtie

text: and

text: D.

text: G.

text: C.

text: McKeon,

text: Class.

text: Quant.

text: Grav.

text: L.

text: D.

text: Faddeev

text: and

text: V.

text: N.

text: Popov,

text: Phys.

text: Lett.

text: G.

text: ’t

text: Hooft

text: and

text: M.

text: J.

text: G.

text: Veltman,

text: Nucl.

text: Phys.

text: G.

text: Leibbrandt,

text: Rev.

text: Mod.

text: Phys.

text: G.

text: ’t

text: Hooft

text: and

text: M.

text: J.

text: G.

text: Veltman,

text: Annales

text: Poincare

text: Phys.

text: Theor.

text: M.

text: H.

text: Goroﬀ

text: and

text: A.

text: Sagnotti,

text: Phys.Lett.

text: Nucl.

text: Phys.

text: K.

text: G.

text: Chetyrkin,

text: A.

text: L.

text: Kataev

text: and

text: F.

text: V.

text: Tkachov,

text: Nucl.

text: Phys.

text: B

text: A.

text: E.

text: M.

text: van

text: de

text: Ven,

text: Nucl.

text: Phys.

text: R.

text: B.

text: Mann,

text: D.

text: G.

text: C.

text: McKeon,

text: T.

text: G.

text: Steele,

text: and

text: L.

text: Tarasov,

text: Nucl.

text: Phys.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof of Lemma 4.2. Let F, G, ε be as in the statement of Lemma 4.2. First suppose d = 2. Using Corollary 4.1 of [14] as a starting-point, we can adapt the proof of Lemma 10.5 of [18] to random radius balls, thereby showing that the probability that Z(Φ − ) fails × (1 ε)F to cross the rectangle [0, 3a] [0, a] decays exponentially in a. Given a > 0, specify a sequence D (a), D (a), . . . of rectangles of aspect ratio 3, 1 2 × alternating between horizontal and vertical rectangles, with D (a) = [0, 3a] [0, a], and 1 with D (a) crossing D (a) the short way for each n. By the union bound and the n n+1 exponential decay just mentioned, the probability that for some n there is no long-way crossing of D (a) in Z(Φ − ) decays exponentially in a. Hence the probability that n (1 ε)F × Z∞(Φ − ) fails to include a long-way crossing of [0, 3a] [0, a] is exponentially decaying (1 ε)F × in a. Likewise for the vertical rectangle [0, a] [0, 3a]. Let E denote the event that Z∞(Φ − ) includes long-way crossings of each of the a − − × − (1 ε)F × − − × rectangles [ 3a/2, a/2] [ 3a/2, 3a/2], [a/2, 3a/2] [ 3a/2, 3a/2], [ 3a/2, 3a/2] − − − × − \ [ 3a/2, a/2] and [ 3a/2, 3a/2] [a/2, 3a/2] (whose union is the annulus [ 3a/2, 3a/2]2 − − P ( a/2, a/2)2). By the preceding discussion, 1 [E ] decays exponentially in a. a⊂ − ≥ If E occurs for some a large enough so that L [ a/2, a/2]2, then for any h 0 a (cid:48) | and u > 6a + b the set Z (Φ − , Φ ) is necessarily contained in the square − L (1 ε)F εF+hG [Bu] [ 3a, 3a]2. The case d = 2 of (4.7) follows. ≥ ≤ ≤ ∈ ∞ ∈ Now consider d (cid:90) 3. Suppose 0 h ε2/2. Choose b (0, ) with F Mb and ∈ (cid:48)(cid:48) (cid:48) G Mb±. Set Φ := Φ − + Φ . Recall the deﬁnition (4.3) of R . For u > 0 F+hG (1 ε)F εF+hG K,b set { (cid:48) } (cid:48)(cid:48) N = 1 R{ } (Φ − , Φ ) > u Φ (d(x, r)). u (cid:9) x ,b (1 ε)F εF+hG F+hG [L B ] (cid:48) b ≥ If R (Φ − , Φ ) > u, then N 1 so by Markov’s inequality L,b (1 ε)F εF+hG u P{ (cid:48) } ≤ E R (Φ − , Φ ) > u N . L,b (1 ε)F εF+hG u { (cid:90) (cid:90) (cid:48) } ∈ X Deﬁne the event A := R{ } (Φ − , Φ + δ ) > u , for (x, r) and u > 0. x,r,u x ,b (1 ε)F εF+hG (x,r) We assert that for u > 1 we have E ≤ P N (A )(F + hG)(dr) dx. u x,r,u (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) To see this let us write simply Φ for Φ − , Φ for Φ and Φ for Φ (so Φ = (cid:48) (cid:48)(cid:48)(1 ε)F εF+hG F+∩hG ∅ Φ + Φ ). Suppose (x, r) is a point of Φ that contributes to N . Then B (x) Z∞(Φ) = (cid:48) u r (otherwise R{ } (Φ, Φ ) would be zero). Moreover, B (x) lies in a component of Z(Φﬁn + (cid:48) x ,b r (cid:48) Φ ) that avoids Z∞(Φ) and is not contained in B (x). Let Ψ (resp. Ψ ) be the point (cid:48) ub process Φ (resp. Φ ) with the point (x, r) removed (if it is a part of the point process in ∪ (cid:48) the ﬁrst place). Then Z∞(Ψ) = Z∞(Φ), and there exists a component of Z(Ψﬁn Ψ ) that avoids Z∞(Ψ), intersects B (x), and is not contained in B (x) (at least if u > 1). But this r ub (cid:48) (cid:48) conclusion just says that event A occurs if we identify Ψ, Ψ with Φ, Φ respectively. x,r,u Therefore using the Mecke formula gives us the asserted inequality. P Thus to prove (4.7) it suﬃces to prove that (A ) decays exponentially in u, uni- ∈ (cid:9) ∈ ∈ x,r,u formly over x L B , h [0, ε2/2] and r (0, b]. We now ﬁx such x, h, r. Let K be as b ∈ N (cid:9) ⊂ − in Lemma 4.1 and choose n with L B [ n K, n K]d. ∈ Z 0 − 2b ×R 0− 0 For n let S denote the slab ((n 1)K, nK] d 1 and let H denote the half-space ∪ n ⊂ ≥ (cid:48) ∈ n −∞ ≤ S . Given u with L B − and n n , for ϕ, ϕ Nb set <m n m u b 0 (cid:48) | (cid:48)| W (ϕ, ϕ ) = Z{ }(ϕ , ϕ ∩ ) u,n x [Hn] [Hn Bu] 11

text: Proof

text: of

text: Lemma

text: Let

text: F,

text: G,

text: ε

text: be

text: as

text: in

text: the

text: statement

text: of

text: Lemma

text: First

text: suppose

text: d

text: Using

text: Corollary

text: of

text: as

text: a

text: starting-point,

text: we

text: can

text: adapt

text: the

text: proof

text: of

text: Lemma

text: of

text: to

text: random

text: radius

text: balls,

text: thereby

text: showing

text: that

text: the

text: probability

text: that

text: Z(Φ

text: fails

text: ε)F

text: to

text: cross

text: the

text: rectangle

text: a]

text: decays

text: exponentially

text: in

text: a.

text: Given

text: a

text: specify

text: a

text: sequence

text: D

text: (a),

text: D

text: (a),

text: of

text: rectangles

text: of

text: aspect

text: ratio

text: alternating

text: between

text: horizontal

text: and

text: vertical

text: rectangles,

text: with

text: D

text: (a)

text: a],

text: and

text: with

text: D

text: (a)

text: crossing

text: D

text: (a)

text: the

text: short

text: way

text: for

text: each

text: n.

text: By

text: the

text: union

text: bound

text: and

text: the

text: n

text: exponential

text: decay

text: just

text: mentioned,

text: the

text: probability

text: that

text: for

text: some

text: n

text: there

text: is

text: no

text: long-way

text: crossing

text: of

text: D

text: (a)

text: in

text: Z(Φ

text: decays

text: exponentially

text: in

text: a.

text: Hence

text: the

text: probability

text: that

text: n

text: ε)F

text: Z∞(Φ

text: fails

text: to

text: include

text: a

text: long-way

text: crossing

text: of

text: a]

text: is

text: exponentially

text: decaying

text: ε)F

text: in

text: a.

text: Likewise

text: for

text: the

text: vertical

text: rectangle

text: a]

text: Let

text: E

text: denote

text: the

text: event

text: that

text: Z∞(Φ

text: includes

text: long-way

text: crossings

text: of

text: each

text: of

text: the

text: a

text: ε)F

text: rectangles

text: and

text: (whose

text: union

text: is

text: the

text: annulus

text: P

text: By

text: the

text: preceding

text: discussion,

text: [E

text: decays

text: exponentially

text: in

text: a.

text: a⊂

text: If

text: E

text: occurs

text: for

text: some

text: a

text: large

text: enough

text: so

text: that

text: L

text: then

text: for

text: any

text: h

text: a

text: and

text: u

text: b

text: the

text: set

text: Z

text: (Φ

text: Φ

text: is

text: necessarily

text: contained

text: in

text: the

text: square

text: L

text: ε)F

text: εF+hG

text: [Bu]

text: The

text: case

text: d

text: of

text: follows.

text: Now

text: consider

text: d

text: Suppose

text: h

text: Choose

text: b

text: with

text: F

text: Mb

text: and

text: G

text: Mb±.

text: Set

text: Φ

text: Φ

text: Φ

text: Recall

text: the

text: deﬁnition

text: of

text: R

text: For

text: u

text: F+hG

text: ε)F

text: εF+hG

text: K,b

text: set

text: N

text: R{

text: (Φ

text: Φ

text: u

text: Φ

text: (d(x,

text: r)).

text: u

text: x

text: ,b

text: ε)F

text: εF+hG

text: F+hG

text: [L

text: B

text: b

text: If

text: R

text: (Φ

text: Φ

text: u,

text: then

text: N

text: so

text: by

text: Markov’s

text: inequality

text: L,b

text: ε)F

text: εF+hG

text: u

text: P{

text: E

text: R

text: (Φ

text: Φ

text: u

text: N

text: L,b

text: ε)F

text: εF+hG

text: u

text: X

text: Deﬁne

text: the

text: event

text: A

text: R{

text: (Φ

text: Φ

text: δ

text: u

text: for

text: (x,

text: r)

text: and

text: u

text: x,r,u

text: x

text: ,b

text: ε)F

text: εF+hG

text: (x,r)

text: We

text: assert

text: that

text: for

text: u

text: we

text: have

text: E

text: P

text: N

text: (A

text: )(F

text: hG)(dr)

text: dx.

text: u

text: x,r,u

text: To

text: see

text: this

text: let

text: us

text: write

text: simply

text: Φ

text: for

text: Φ

text: Φ

text: for

text: Φ

text: and

text: Φ

text: for

text: Φ

text: (so

text: Φ

text: ε)F

text: εF+hG

text: F+∩hG

text: Φ

text: Φ

text: Suppose

text: (x,

text: r)

text: is

text: a

text: point

text: of

text: Φ

text: that

text: contributes

text: to

text: N

text: Then

text: B

text: (x)

text: Z∞(Φ)

text: u

text: r

text: (otherwise

text: R{

text: (Φ,

text: Φ

text: would

text: be

text: zero).

text: Moreover,

text: B

text: (x)

text: lies

text: in

text: a

text: component

text: of

text: Z(Φﬁn

text: x

text: ,b

text: r

text: Φ

text: that

text: avoids

text: Z∞(Φ)

text: and

text: is

text: not

text: contained

text: in

text: B

text: (x).

text: Let

text: Ψ

text: (resp.

text: Ψ

text: be

text: the

text: point

text: ub

text: process

text: Φ

text: (resp.

text: Φ

text: with

text: the

text: point

text: (x,

text: r)

text: removed

text: (if

text: it

text: is

text: a

text: part

text: of

text: the

text: point

text: process

text: in

text: the

text: ﬁrst

text: place).

text: Then

text: Z∞(Ψ)

text: Z∞(Φ),

text: and

text: there

text: exists

text: a

text: component

text: of

text: Z(Ψﬁn

text: Ψ

text: that

text: avoids

text: Z∞(Ψ),

text: intersects

text: B

text: (x),

text: and

text: is

text: not

text: contained

text: in

text: B

text: (x)

text: (at

text: least

text: if

text: u

text: But

text: this

text: r

text: ub

text: conclusion

text: just

text: says

text: that

text: event

text: A

text: occurs

text: if

text: we

text: identify

text: Ψ,

text: Ψ

text: with

text: Φ,

text: Φ

text: respectively.

text: x,r,u

text: Therefore

text: using

text: the

text: Mecke

text: formula

text: gives

text: us

text: the

text: asserted

text: inequality.

text: P

text: Thus

text: to

text: prove

text: it

text: suﬃces

text: to

text: prove

text: that

text: (A

text: decays

text: exponentially

text: in

text: u,

text: uni-

text: x,r,u

text: formly

text: over

text: x

text: L

text: B

text: h

text: and

text: r

text: b].

text: We

text: now

text: ﬁx

text: such

text: x,

text: h,

text: r.

text: Let

text: K

text: be

text: as

text: b

text: N

text: in

text: Lemma

text: and

text: choose

text: n

text: with

text: L

text: B

text: n

text: K,

text: n

text: K]d.

text: Z

text: ×R

text: 0−

text: For

text: n

text: let

text: S

text: denote

text: the

text: slab

text: ((n

text: nK]

text: d

text: and

text: let

text: H

text: denote

text: the

text: half-space

text: n

text: n

text: S

text: Given

text: u

text: with

text: L

text: B

text: and

text: n

text: n

text: for

text: ϕ,

text: ϕ

text: Nb

text: set

text: <m

text: n

text: m

text: u

text: b

text: W

text: (ϕ,

text: ϕ

text: Z{

text: }(ϕ

text: ϕ

text: u,n

text: x

text: [Hn]

text: [Hn

text: Bu]


text: and deﬁne the indicator functions (cid:48) { (cid:48) | ∩ (cid:48) (cid:54) ∅} f (ϕ, ϕ ) := 1 Z((ϕ + ϕ ) W (ϕ, ϕ )) = ; u,n (cid:48) { | [S∩n+1] u,n (cid:48) ∅} g (ϕ, ϕ ) := 1 Z∞(ϕ ) W (ϕ, ϕ ) = ; u,n (cid:48) { [Sn+(cid:48)|1] \ u,n (cid:54) ∅} h (ϕ, ϕ ) := 1 Z{ }(ϕ, ϕ ) H = . u,n x [Bu] n (cid:48) ≤ (cid:48) (cid:48) Then we claim that f (ϕ, ϕ ) f (ϕ, ϕ )g (ϕ, ϕ ). Indeed, u,n+1 u,n u,n (cid:48) if f (ϕ, ϕ ) = 0 (4.9) u,n (cid:48) (cid:48) ⊂ −∞ × R − then W (ϕ, ϕ ) = W (ϕ, ϕ ) ( , nK + b] d 1, u,n+1 u,n (cid:48) (cid:48) ∅ while if g (ϕ, ϕ ) = 0 then W (ϕ, ϕ ) = , and in both cases it is not possible for (cid:48) | u,n u,n+1 Z((ϕ + ϕ ) ) to intersect with W . [Sn+2] (cid:48) ≤ (cid:48) u,n+1 Also hu,n+1(ϕ, ϕ ) fu,n(ϕ(cid:89), ϕ ) by (4.9). The(cid:89)refore for n = n0 + 2, n0 + 3, . . . we have − − (cid:48) ≤ n 1 (cid:48) ≤ n 2 (cid:48) (cid:48) h (ϕ, ϕ ) f (ϕ, ϕ ) f (ϕ, ϕ )g (ϕ, ϕ ). (4.10) u,n u,m u,m u,m m=n0 m=n0 F | (cid:48) | Denote by the σ-ﬁeld generated by (Φ − , Φ ). If the conditional n (cid:48) (1 ε)F [Hn] εF+FhG [Hn] expectation of f (Φ − , Φ + δ ) with respect to is at least 1/2, then by u,n (1 ε)F εF+hG (x,r) (cid:48) n Lemma 4.1 with A taken to be W (Φ − , Φ + δ ), the conditional expectation − (cid:48) u,n (1 ε)F εF+hG (x,r) of 1 g (Φ − , Φ + δ ) is at least γ. Hence, setting u,n (1 ε)F εF+hG (x,r) (cid:48) (cid:48) V := f (Φ − , Φ + δ )g (Φ − , Φ + δ ), u,n u,n (1 ε)F εF+hG (x,r) u,n (1 ε)F εF+hG (x,r) we have E | F ≤ − − [V ] max(1 γ, 1/2) = 1(cid:34) γ. (cid:35) u,n n F (cid:89) Also, for each n, V is -measurable and by (4.10) we have u,n+1 n E (cid:34) (cid:48) (cid:35) ≤ E n [hu,n+2(Φ(1−ε)F , ΦεF(cid:89)+hG + δ(x,r))] (cid:89)Vu,i i=n0 − − E E |F × n 1 ≤ − E n 1 = [V ] V (1 γ) V u,n n u,i u,i ≤ · · · ≤ −i=n0 − ≤ − i−=n0 (1 γ)n n0 exp( γ(n n )). 0 Arguing similarly in each of the 2d positive or negative coordinate directions shows that for u a multiple of b we have P P{ (cid:48) | \ (cid:54) ∅} (A ) = Z{ }(Φ − , Φ + δ ) B − = x,r,u ≤ P{ x (1 ε)F ε(cid:48)F+hG|[Bu] \ −(x,r) − u b − (cid:54) ∅} Z{ }(Φ − , Φ ) [ (u b)/d, (u b)/d]d) = ≤ x − (1(cid:98)ε)F− εF+hG [B(cid:99)u]− − 2d exp( γ( (u b)/(dK) n 2)), 0 ≥ which gives us the result (4.7) for d 3. To deduce (4.8), set G = 0 in (4.7), and use the fact that (cid:48) ⊂ (cid:48) Z (Φ − + Φ , 0) Z (Φ − , Φ ). L (1 ε)F εF L (1 ε)F εF 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: deﬁne

text: the

text: indicator

text: functions

font: default

text: f

text: (ϕ,

text: ϕ

text: Z((ϕ

text: ϕ

text: W

text: (ϕ,

text: ϕ

text: u,n

text: u,n

text: g

text: (ϕ,

text: ϕ

text: Z∞(ϕ

text: W

text: (ϕ,

text: ϕ

text: u,n

text: u,n

text: h

text: (ϕ,

text: ϕ

text: Z{

text: }(ϕ,

text: ϕ

text: H

text: u,n

text: x

text: [Bu]

text: n

text: Then

text: we

text: claim

text: that

text: f

text: (ϕ,

text: ϕ

text: f

text: (ϕ,

text: ϕ

text: )g

text: (ϕ,

text: ϕ

text: Indeed,

text: u,n

text: u,n

text: if

text: f

text: (ϕ,

text: ϕ

text: u,n

text: R

text: then

text: W

text: (ϕ,

text: ϕ

text: W

text: (ϕ,

text: ϕ

text: nK

text: b]

text: d

text: u,n

text: while

text: if

text: g

text: (ϕ,

text: ϕ

text: then

text: W

text: (ϕ,

text: ϕ

text: and

text: in

text: both

text: cases

text: it

text: is

text: not

text: possible

text: for

text: u,n

text: Z((ϕ

text: ϕ

text: to

text: intersect

text: with

text: W

text: Also

text: ϕ

text: ϕ

text: by

text: for

text: n

text: we

text: have

text: n

text: n

text: h

text: (ϕ,

text: ϕ

text: f

text: (ϕ,

text: ϕ

text: f

text: (ϕ,

text: ϕ

text: )g

text: (ϕ,

text: ϕ

text: u,n

text: u,m

text: u,m

text: u,m

text: F

text: Denote

text: by

text: the

text: σ-ﬁeld

text: generated

text: by

text: (Φ

text: Φ

text: If

text: the

text: conditional

text: n

text: ε)F

text: [Hn]

text: εF+FhG

text: [Hn]

text: expectation

text: of

text: f

text: (Φ

text: Φ

text: δ

text: with

text: respect

text: to

text: is

text: at

text: least

text: then

text: by

text: u,n

text: ε)F

text: εF+hG

text: (x,r)

text: n

text: Lemma

text: with

text: A

text: taken

text: to

text: be

text: W

text: (Φ

text: Φ

text: δ

text: the

text: conditional

text: expectation

text: u,n

text: ε)F

text: εF+hG

text: (x,r)

text: of

text: g

text: (Φ

text: Φ

text: δ

text: is

text: at

text: least

text: γ.

text: Hence,

text: setting

text: u,n

text: ε)F

text: εF+hG

text: (x,r)

text: V

text: f

text: (Φ

text: Φ

text: δ

text: )g

text: (Φ

text: Φ

text: δ

text: u,n

text: u,n

text: ε)F

text: εF+hG

text: (x,r)

text: u,n

text: ε)F

text: εF+hG

text: (x,r)

text: we

text: have

text: E

text: F

text: [V

text: γ,

text: γ.

text: u,n

text: n

text: F

text: Also,

text: for

text: each

text: n,

text: V

text: is

text: -measurable

text: and

text: by

text: we

text: have

text: n

text: E

text: E

text: n

text: δ(x,r))]

text: E

text: E

text: |F

text: n

text: E

text: n

text: [V

text: V

text: γ)

text: V

text: u,n

text: n

text: u,i

text: u,i

text: γ)n

text: exp(

text: γ(n

text: n

text: Arguing

text: similarly

text: in

text: each

text: of

text: the

text: positive

text: or

text: negative

text: coordinate

text: directions

text: shows

text: that

text: for

text: u

text: a

text: multiple

text: of

text: b

text: we

text: have

text: P

text: P{

text: (A

text: Z{

text: }(Φ

text: Φ

text: δ

text: B

text: x,r,u

text: P{

text: x

text: ε)F

text: −(x,r)

text: u

text: b

text: Z{

text: }(Φ

text: Φ

text: (u

text: b)/d,

text: (u

text: b)/d]d)

text: x

text: εF+hG

text: exp(

text: γ(

text: (u

text: b)/(dK)

text: n

text: which

text: gives

text: us

text: the

text: result

text: for

text: d

text: To

text: deduce

text: set

text: G

text: in

text: and

text: use

text: the

text: fact

text: that

text: Z

text: (Φ

text: Φ

text: Z

text: (Φ

text: Φ

text: L

text: ε)F

text: εF

text: L

text: ε)F

text: εF

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:32) (cid:33) (cid:88) (cid:88) − −| | Dn f(ϕ) = ( 1)n I f ϕ + δ (2.4) z1,...,zn ⊂{ } ∈ zi I 1,...,n i I | | where I denotes the number of elements of I. B X Proposition 2.2. Let µ be a locally ﬁnite measure and ν a ﬁnite signed measure on ( ). → R (cid:12) (cid:90) Let f : N be measurable and boun(cid:12)ded. If µ + aν is a measure for some a > 0, then (cid:12) (cid:12) d+ E E f(Φ) = D f(Φ) ν(dz). (2.5) µ+sν µ z ds s=0 − E If also µ aν is a measure, then f(Φ) is diﬀerentiable in s at s = 0. µ+sν The proof of this perturbation formula can be found in [23, Theorem 2.1] (for the case ν = µ), for ﬁnite measures in [17, Theorem 2.1], and for locally ﬁnite measures and square-integrable functions in [12]. It may also be found in [13]. 3 Main results 3.1 Smoothness of the capacity functional Our ﬁrst result concerns diﬀerentiating the capacity functional θ with respect to L the measure F . This can be useful to compare the percolation properties of diﬀerent radius distributions. For example, in [5] and in [15] the percolation threshold for F a Dirac measure (i.e. balls of ﬁxed radius) has been compared with the percolation threshold for F the sum of two Dirac measures (i.e. for balls of random radius with just two possible values), or with more general F . [5] show that in suﬃciently high dimensions the Dirac measure does not minimise the critical volume fraction (as had been previously conjectured) but do not quantify the phrase ‘suﬃciently high’ and do not rule out the possibility that the Dirac measure minimises the critical volume fraction in low dimensions. With suﬃcient analytic tools, it might be possible to compare diﬀerent radius distributions (perhaps with the same volume fraction) by calculus. For example, we could compare two measures F and F by passing continuously from one to the other. 1 2 Our result gives the directional derivative for θ (F ) as we vary F . If we wish to keep L the total measure (i.e., the density) constant then we need to add to F a signed measure with total measure zero. More generally we may consider adding an arbitrary signed measure G to F . We use notation for the classes of measures from Section and Dn from (2.3). ∈ ∈ Theorem 3.1. Suppose that F M(cid:93) with t (F ) < 1, and G M(cid:93)± is such that F + aG is a measure for some(cid:12)(cid:12)a > 0.(cid:90)L(cid:90)et L ⊂ Rd be ccompact. Then (cid:12) (cid:12) d+ P { ∩ (cid:54) ∅ ∩ ∅} θ (F + hG) = L Z∞(Φ + δ ) = , L Z∞(Φ) = L F (x,r) dh h=0 × G(dr)dx, (3.1) 5 ##LTLine## ##LTLine##

label: equation

text: Dn

text: f(ϕ)

text: I

text: f

text: ϕ

text: δ

text: zi

text: I

text: i

text: I

text: where

text: I

text: denotes

text: the

text: number

text: of

text: elements

text: of

text: I.

text: B

text: X

text: Proposition

text: Let

text: µ

text: be

text: a

text: locally

text: ﬁnite

text: measure

text: and

text: ν

text: a

text: ﬁnite

text: signed

text: measure

text: on

text: R

text: Let

text: f

text: N

text: be

text: measurable

text: and

text: If

text: µ

text: aν

text: is

text: a

text: measure

text: for

text: some

text: a

text: then

text: d+

text: E

text: E

text: f(Φ)

text: D

text: f(Φ)

text: ν(dz).

text: µ+sν

text: µ

text: z

text: ds

text: E

text: If

text: also

text: µ

text: aν

text: is

text: a

text: measure,

text: then

text: f(Φ)

text: is

text: diﬀerentiable

text: in

text: s

text: at

text: s

text: µ+sν

text: The

text: proof

text: of

text: this

text: perturbation

text: formula

text: can

text: be

text: found

text: in

text: Theorem

text: (for

text: the

text: case

text: ν

text: µ),

text: for

text: ﬁnite

text: measures

text: in

text: Theorem

text: and

text: for

text: locally

text: ﬁnite

text: measures

text: and

text: square-integrable

text: functions

text: in

text: It

text: may

text: also

text: be

text: found

text: in

text: Main

text: results

text: Smoothness

text: of

text: the

text: capacity

text: functional

text: Our

text: ﬁrst

text: result

text: concerns

text: diﬀerentiating

text: the

text: capacity

text: functional

text: θ

text: with

text: respect

text: to

text: L

text: the

text: measure

text: F

text: This

text: can

text: be

text: useful

text: to

text: compare

text: the

text: percolation

text: properties

text: of

text: diﬀerent

text: radius

text: distributions.

text: For

text: example,

text: in

text: and

text: in

text: the

text: percolation

text: threshold

text: for

text: F

text: a

text: Dirac

text: measure

text: (i.e.

text: balls

text: of

text: ﬁxed

text: radius)

text: has

text: been

text: compared

text: with

text: the

text: percolation

text: threshold

text: for

text: F

text: the

text: sum

text: of

text: two

text: Dirac

text: measures

text: (i.e.

text: for

text: balls

text: of

text: random

text: radius

text: with

text: just

text: two

text: possible

text: values),

text: or

text: with

text: more

text: general

text: F

text: show

text: that

text: in

text: suﬃciently

text: high

text: dimensions

text: the

text: Dirac

text: measure

text: does

text: not

text: minimise

text: the

text: critical

text: volume

text: fraction

text: (as

text: had

text: been

text: previously

text: conjectured)

text: but

text: do

text: not

text: quantify

text: the

text: phrase

text: ‘suﬃciently

text: high’

text: and

text: do

text: not

text: rule

text: out

text: the

text: possibility

text: that

text: the

text: Dirac

text: measure

text: minimises

text: the

text: critical

text: volume

text: fraction

text: in

text: low

text: dimensions.

text: With

text: suﬃcient

text: analytic

text: tools,

text: it

text: might

text: be

text: possible

text: to

text: compare

text: diﬀerent

text: radius

text: distributions

text: (perhaps

text: with

text: the

text: same

text: volume

text: fraction)

text: by

text: calculus.

text: For

text: example,

text: we

text: could

text: compare

text: two

text: measures

text: F

text: and

text: F

text: by

text: passing

text: continuously

text: from

text: one

text: to

text: the

text: other.

text: Our

text: result

text: gives

text: the

text: directional

text: derivative

text: for

text: θ

text: (F

text: as

text: we

text: vary

text: F

text: If

text: we

text: wish

text: to

text: keep

text: L

text: the

text: total

text: measure

text: (i.e.,

text: the

text: density)

text: constant

text: then

text: we

text: need

text: to

text: add

text: to

text: F

text: a

text: signed

text: measure

text: with

text: total

text: measure

text: zero.

text: More

text: generally

text: we

text: may

text: consider

text: adding

text: an

text: arbitrary

text: signed

text: measure

text: G

text: to

text: F

text: We

text: use

text: notation

text: for

text: the

text: classes

text: of

text: measures

text: from

text: Section

text: and

text: Dn

text: from

text: Theorem

text: Suppose

text: that

text: F

text: with

text: t

text: (F

text: and

text: G

text: is

text: such

text: that

text: F

text: aG

text: is

text: a

text: measure

text: for

text: L

text: Rd

text: be

text: ccompact.

text: Then

text: d+

text: P

text: θ

text: (F

text: hG)

text: L

text: Z∞(Φ

text: δ

text: L

text: Z∞(Φ)

text: L

text: F

text: (x,r)

text: dh

text: G(dr)dx,

text: ##LTLine##

text: ##LTLine##


text: − and the right hand side of (3.1) is ﬁnite. If also F aG is a measure, then θ (F +hG) is { L∩ (cid:54) inﬁnitely diﬀerentiable in a neighbourhood of h = 0, then setting f˜ (ϕ) = 1 L Z∞(ϕ) = ∅} ∈ ∈ N (cid:12) (cid:90) (cid:90) L for ϕ N, for all n we h(cid:12)ave (cid:12) (cid:12) dn · · · E θ (F + hG) = Dn f˜ (Φ) dhn L F (x1,r1),...,(xn,rn) L h=0 × · · · dx G(dr ) dx G(dr ). (3.2) 1 1 n n We shall prove Theorem 3.1 in Section . The identity (3.1) tells us that the perturba- { ∩ (cid:54) ∅} tion formula (2.5) remains valid for f(ϕ) = 1 L Z∞(ϕ) = with µ(dxdr) = dx F (dr) and ν(dxdr) = dx G(dr), though in this case both µ and ν are inﬁnite (but σ-ﬁnite). Our next theorem is a corollary of Theorem 3.1, and signiﬁcantly adds to the known results mentioned in Section concerning continuity of θ (tF ) for ﬁxed F , in the case of L (cid:9) deterministically bounded radii. Recall that the Minkowski diﬀerence A B of two sets ⊂ R { − ∈ ∈ } { } ∈ R A, B d is deﬁned by x y : x A, y B . When B = x for some x d, we − (cid:9) { } write simply A x for A x . ∈ ⊂ R (cid:55)→ Theorem 3.2. Let F M(cid:93), and let L d be compact. Then t θ (tF ) is inﬁnitely ∞ 1 { ∩ (cid:54) ∅} L ∈ diﬀerentiable on (t (F ), ) and setting f˜ (ϕ) = 1 L Z∞(ϕ) = for all ϕ N, we c ∈ N (cid:90) (cid:90) L have for t > t (F ) and n that c dnθ (tF ) · · · E L = Dn f˜ (Φ) dtn tF (x1,r1),...,(xn,rn) L × · · · dx F (dr ) dx F (dr ). (3.3) 1 1 n n (cid:90)(cid:90) In particular, for t > t (F ) we have c d P { ∩ (cid:54) ∅ ∩ ∅} θ (tF ) = L Z∞(Φ + δ ) = , L Z∞(Φ) = (3.4) L (cid:90) tF (x,r) dt × dx F (dr) E | (cid:9) \ (cid:9) | = (Z∞(Φ + δ ) L) (Z∞(Φ) L) F (dr). (3.5) tF (0,r) ⊂ R Proof. Let L d be compact. The inﬁnite diﬀerentiability of θ (tF ), and the formula L ∗ ∗ (3.3) for dn θ (tF ), follow from applying Theorem 3.1 to the measures F and G given ∗ dtn L ∗ by F = tF and G = F ; also (3.4) follows as a special case of (3.3). It remains to prove P (3.5). By sta(cid:90)ti(cid:90)onarity, T Φ has the same distribution as Φ under , so the right side of x tF (3.4) equals (cid:90)(cid:90) E { − ∩ (cid:54) ∅ − ∩ ∅} 1 (L x) Z∞(Φ + δ ) = , (L x) Z∞(Φ) = dx F (dr) tF (0,r) E { ∈ (cid:9) \ (cid:9) } = 1 x (L Z∞(Φ + δ )) (L Z∞(Φ)) dx F (dr), tF (0,r) 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: the

text: right

text: hand

text: side

text: of

text: is

text: ﬁnite.

text: If

text: also

text: F

text: aG

text: is

text: a

text: measure,

text: then

text: θ

text: (F

text: +hG)

text: is

text: L∩

text: inﬁnitely

text: diﬀerentiable

text: in

text: a

text: neighbourhood

text: of

text: h

text: then

text: setting

text: f˜

text: (ϕ)

text: L

text: Z∞(ϕ)

text: N

text: L

text: for

text: ϕ

text: N,

text: for

text: all

text: n

text: we

text: dn

text: E

text: θ

text: (F

text: hG)

text: Dn

text: f˜

text: (Φ)

text: dhn

text: L

text: F

text: L

text: dx

text: G(dr

text: dx

text: G(dr

text: n

text: n

text: We

text: shall

text: prove

text: Theorem

text: in

text: Section

text: The

text: identity

text: tells

text: us

text: that

text: the

text: perturba-

text: tion

text: formula

text: remains

text: valid

text: for

text: f(ϕ)

text: L

text: Z∞(ϕ)

text: with

text: µ(dxdr)

text: dx

text: F

text: (dr)

text: and

text: ν(dxdr)

text: dx

text: G(dr),

text: though

text: in

text: this

text: case

text: both

text: µ

text: and

text: ν

text: are

text: inﬁnite

text: (but

text: σ-ﬁnite).

text: Our

text: next

text: theorem

text: is

text: a

text: corollary

text: of

text: Theorem

text: and

text: signiﬁcantly

text: adds

text: to

text: the

text: known

text: results

text: mentioned

text: in

text: Section

text: concerning

text: continuity

text: of

text: θ

text: (tF

text: for

text: ﬁxed

text: F

text: in

text: the

text: case

text: of

text: L

text: deterministically

text: bounded

text: radii.

text: Recall

text: that

text: the

text: Minkowski

text: diﬀerence

text: A

text: B

text: of

text: two

text: sets

text: R

text: R

text: A,

text: B

text: d

text: is

text: deﬁned

text: by

text: x

text: y

text: x

text: A,

text: y

text: B

text: When

text: B

text: x

text: for

text: some

text: x

text: d,

text: we

text: write

text: simply

text: A

text: x

text: for

text: A

text: x

text: R

text: Theorem

text: Let

text: F

text: and

text: let

text: L

text: d

text: be

text: compact.

text: Then

text: t

text: θ

text: (tF

text: is

text: inﬁnitely

text: L

text: diﬀerentiable

text: on

text: (t

text: (F

text: and

text: setting

text: f˜

text: (ϕ)

text: L

text: Z∞(ϕ)

text: for

text: all

text: ϕ

text: N,

text: we

text: c

text: N

text: L

text: have

text: for

text: t

text: t

text: (F

text: and

text: n

text: that

text: c

text: dnθ

text: (tF

text: E

text: L

text: Dn

text: f˜

text: (Φ)

text: dtn

text: tF

text: L

text: dx

text: F

text: (dr

text: dx

text: F

text: (dr

text: n

text: n

text: In

text: particular,

text: for

text: t

text: t

text: (F

text: we

text: have

text: c

text: d

text: P

text: θ

text: (tF

text: L

text: Z∞(Φ

text: δ

text: L

text: Z∞(Φ)

text: L

text: tF

text: (x,r)

text: dt

text: dx

text: F

text: (dr)

text: E

text: (Z∞(Φ

text: δ

text: L)

text: (Z∞(Φ)

text: L)

text: F

text: (dr).

text: tF

text: R

text: Proof.

text: Let

text: L

text: d

text: be

text: compact.

text: The

text: inﬁnite

text: diﬀerentiability

text: of

text: θ

text: (tF

text: and

text: the

text: formula

text: L

text: for

text: dn

text: θ

text: (tF

text: follow

text: from

text: applying

text: Theorem

text: to

text: the

text: measures

text: F

text: and

text: G

text: given

text: dtn

text: L

text: by

text: F

text: tF

text: and

text: G

text: F

text: also

text: follows

text: as

text: a

text: special

text: case

text: of

text: It

text: remains

text: to

text: prove

text: P

text: By

text: T

text: Φ

text: has

text: the

text: same

text: distribution

text: as

text: Φ

text: under

text: so

text: the

text: right

text: side

text: of

text: x

text: tF

text: equals

text: E

text: (L

text: x)

text: Z∞(Φ

text: δ

text: (L

text: x)

text: Z∞(Φ)

text: dx

text: F

text: (dr)

text: tF

text: E

text: x

text: (L

text: Z∞(Φ

text: δ

text: (L

text: Z∞(Φ))

text: dx

text: F

text: (dr),

text: tF

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 Donahue et al. values. However, the outliers may also be a result of sys- In this section we compare the X-ray morphologies of tematics related to the SZE analysis. Although we have CLASH clusters to the predicted X-ray morphology of developed a rigorous procedure to correct for the large clusters from the MUSIC simulation with similar masses PSF and the large angular scale noise in the SZE images, and redshifts, but no morphological selection. For conve- either or both of these eﬀects could potentially bias our nience, we choose to show the morphological parameters derived PA values. measured for the simulated clusters at z = 0.333. We Two eﬀects could cause the dynamic range of the X-ray veriﬁed that the results are similar for the other red- measurements of concentration and axis ratios to exceed shifts. Each simulated cluster is represented three times those of the same properties measured from lensing and in this sample, for 3 diﬀerent viewing angles. Remem- SZE maps. ber that in this work we are considering the complete 1. The Chandra X-ray maps have an instrumental set of simulated clusters selected only on the basis of the ∼ (cid:48)(cid:48) PSF with a width 1 . The eﬀective PSF is gravitational mass, and therefore the simulated clusters photon limited to be larger, however, the eﬀective plotted in Figure 8 can represent any state of relaxation Chandra PSF for emissivity ﬂuctuations is consid- and virialization. For this reason, the expectation for the erably more compact than the eﬀective resolution CLASH clusters is that their morphological parameters of large scale structures in the lensing and SZE will be in the range of simulated clusters but skewed. maps. In particular, we expect the CLASH clusters should be rounder (i.e. axial ratios near unity), and to have smaller 2. X-ray surface brightness of the X-rays scales like w and power ratios than mean values of the simulated density squared, as opposed to linearly in gas den- clusters. sity (or pressure) for the SZE signal and linearly Scatter plots for the parameters C, w, axial ratio, and in projected total mass density for the lensing sig- power ratios P30 and P40 are presented in Figure 8, nal. As a result, structures of higher density, such in which one can see that the distribution of simulated as the central region of the cluster, will produce clusters is similar to that of the observed CLASH clus- an enhanced X-ray signal compared to the SZE or ters in the power ratios P30 and P40. The two power ra- lensing signal. tios are correlated and the 5 less-relaxed, lensing-selected We ﬁnd the typical axis ratio for CLASH clusters in CLASH clusters have larger power ratios than the 20- ± the X-rays to be 0.88 0.06, which is similar to the SZE object X-ray selected CLASH subsample. On average, ± maps at 0.90 0.06, and somewhat more elongated (at the CLASH clusters are rounder than the simulated clus- ± 500 kpc) in lensing maps 0.80 0.08 (although the LTM ters, in the sense that their axial ratios are closer to unity. ± lensing models are more circular, at 0.92 0.04.) In a This resultis not surprising, since the CLASH X-ray clus- one-to-one comparison, the lensing maps are more elon- ters were chosen in part for their round X-ray isophotes. gated than the projected X-ray emission, but are gener- However, even the irregular, lensing-selected clusters are ally aligned in the same direction. That these clusters rounder than most of the simulated clusters, according to are typically circular is not surprising, since they were the axial-ratio measurements: All of the lensing-selected selected to be nearly circular in the X-ray. That they CLASH clusters have an axial ratio about 0.8, while only are similarly circular in SZ and lensing images then is 15% of the simulated sample has an axial ratio above 0.8. also expected. That the gas is about 10% or so rounder The centroid shifts w of the CLASH clusters are than the projected mass at 500 kpc was predicted by Flo- smaller than most simulated clusters of similar mass, res et al. (e.g. 2007) when the total mass is dominated but they are not≤out of range: only 25% of the MUSIC by collisionless dark matter, and thus the relative ax- clusters have w 0.01 while 60% of the CLASH sample ial ratios (and therefore eccentricities) is consistent with have such small w. gravitational potentials dominated by collisionless dark The range of concentration C of the CLASH clusters is matter. similar to that of the simulated sample. Note that a sig- The SZE estimates of the axis ratios are not partic- niﬁcant subset of simulated clusters have very high con- ularly correlated with the X-ray estimates at the same centrations compared to those observed for CLASH clus- radius, but both estimates have a very similar mean and ters, even for the relaxed cool core clusters in CLASH. ± standard deviation, 0.9 0.06, where the scatter is dom- None of the simulated clusters have concentrations as low inated by the measurement uncertainties. Abell 1423’s as a few of the CLASH strong-lensing selected clusters. SZE axis ratio is an outlier for the sample’s range of To summarize, compared to simulated clusters in MU- SZE measurements, possibly due to the dim SZE signal SIC with a similar mass, CLASH cluster morphologies towards this cluster. The two lowest X-ray axis ratios (i.e are on average rounder and have smaller centroid shifts. the highest elongations) were found for MACS0416 and Their surface brightness concentrations and power ratios MACS0647. These are two of the ﬁve lensing-selected are similar to that of the mass-selected sample of simu- clusters from the CLASH survey (Postman et al. 2012b). lated clusters. Both of these CLASH clusters have evidence for inter- Meneghetti et al. (2014a) deﬁned regularity M for a actions in their optical appearance (at least two bright given simulated cluster in terms of the oﬀsets of a set galaxies in their core, with extended elongated intraclus- of morphological properties, in units of standard devia- ter light in between them.) tions. We refer the reader to Meneghetti et al. (2014a) We have plotted the histograms of centroid and posi- for details and speciﬁc relations, but we brieﬂy review the tion angle diﬀerences in Figure 6 and 7. results relevant to this work here. For each of ﬁve mor- phological parameters, w, eccentricity, P30, and P40, 5.3. Comparison of X-ray Morphological Properties of and 1/C, they ﬁnd the diﬀerence between the log of the CLASH Clusters with Simulations parameter and the log of the mean and divide by the

text: Donahue

text: et

text: al.

text: values.

text: However,

text: the

text: outliers

text: may

text: also

text: be

text: a

text: result

text: of

text: sys-

text: In

text: this

text: section

text: we

text: compare

text: the

text: X-ray

text: morphologies

text: of

text: tematics

text: related

text: to

text: the

text: SZE

text: analysis.

text: Although

text: we

text: have

text: CLASH

text: clusters

text: to

text: the

text: predicted

text: X-ray

text: morphology

text: of

text: developed

text: a

text: rigorous

text: procedure

text: to

text: correct

text: for

text: the

text: large

text: clusters

text: from

text: the

text: MUSIC

text: simulation

text: with

text: similar

text: masses

text: PSF

text: and

text: the

text: large

text: angular

text: scale

text: noise

text: in

text: the

text: SZE

text: images,

text: and

text: redshifts,

text: but

text: no

text: morphological

text: selection.

text: For

text: conve-

text: either

text: or

text: both

text: of

text: these

text: eﬀects

text: could

text: potentially

text: bias

text: our

text: nience,

text: we

text: choose

text: to

text: show

text: the

text: morphological

text: parameters

text: derived

text: PA

text: values.

text: measured

text: for

text: the

text: simulated

text: clusters

text: at

text: z

text: We

text: Two

text: eﬀects

text: could

text: cause

text: the

text: dynamic

text: range

text: of

text: the

text: X-ray

text: veriﬁed

text: that

text: the

text: results

text: are

text: similar

text: for

text: the

text: other

text: red-

text: measurements

text: of

text: concentration

text: and

text: axis

text: ratios

text: to

text: exceed

text: shifts.

text: Each

text: simulated

text: cluster

text: is

text: represented

text: three

text: times

text: those

text: of

text: the

text: same

text: properties

text: measured

text: from

text: lensing

text: and

text: in

text: this

text: sample,

text: for

text: diﬀerent

text: viewing

text: angles.

text: Remem-

text: SZE

text: maps.

text: ber

text: that

text: in

text: this

text: work

text: we

text: are

text: considering

text: the

text: complete

text: The

text: Chandra

text: X-ray

text: maps

text: have

text: an

text: instrumental

text: set

text: of

text: simulated

text: clusters

text: selected

text: only

text: on

text: the

text: basis

text: of

text: the

text: PSF

text: with

text: a

text: width

text: The

text: eﬀective

text: PSF

text: is

text: gravitational

text: mass,

text: and

text: therefore

text: the

text: simulated

text: clusters

text: photon

text: limited

text: to

text: be

text: larger,

text: however,

text: the

text: eﬀective

text: plotted

text: in

text: Figure

text: can

text: represent

text: any

text: state

text: of

text: relaxation

text: Chandra

text: PSF

text: for

text: emissivity

text: ﬂuctuations

text: is

text: consid-

text: and

text: virialization.

text: For

text: this

text: reason,

text: the

text: expectation

text: for

text: the

text: erably

text: more

text: compact

text: than

text: the

text: eﬀective

text: resolution

text: CLASH

text: clusters

text: is

text: that

text: their

text: morphological

text: parameters

text: of

text: large

text: scale

text: structures

text: in

text: the

text: lensing

text: and

text: SZE

text: will

text: be

text: in

text: the

text: range

text: of

text: simulated

text: clusters

text: but

text: skewed.

text: maps.

text: In

text: particular,

text: we

text: expect

text: the

text: CLASH

text: clusters

text: should

text: be

text: rounder

text: (i.e.

text: axial

text: ratios

text: near

text: unity),

text: and

text: to

text: have

text: smaller

text: X-ray

text: surface

text: brightness

text: of

text: the

text: X-rays

text: scales

text: like

text: w

text: and

text: power

text: ratios

text: than

text: mean

text: values

text: of

text: the

text: simulated

text: density

text: squared,

text: as

text: opposed

text: to

text: linearly

text: in

text: gas

text: den-

text: clusters.

text: sity

text: (or

text: pressure)

text: for

text: the

text: SZE

text: signal

text: and

text: linearly

text: Scatter

text: plots

text: for

text: the

text: parameters

text: C,

text: w,

text: axial

text: ratio,

text: and

text: in

text: projected

text: total

text: mass

text: density

text: for

text: the

text: lensing

text: sig-

text: power

text: ratios

text: and

text: are

text: presented

text: in

text: Figure

text: nal.

text: As

text: a

text: result,

text: structures

text: of

text: higher

text: density,

text: such

text: in

text: which

text: one

text: can

text: see

text: that

text: the

text: distribution

text: of

text: simulated

text: as

text: the

text: central

text: region

text: of

text: the

text: cluster,

text: will

text: produce

text: clusters

text: is

text: similar

text: to

text: that

text: of

text: the

text: observed

text: CLASH

text: clus-

text: an

text: enhanced

text: X-ray

text: signal

text: compared

text: to

text: the

text: SZE

text: or

text: ters

text: in

text: the

text: power

text: ratios

text: and

text: The

text: two

text: power

text: ra-

text: lensing

text: signal.

text: tios

text: are

text: correlated

text: and

text: the

text: less-relaxed,

text: lensing-selected

text: We

text: ﬁnd

text: the

text: typical

text: axis

text: ratio

text: for

text: CLASH

text: clusters

text: in

text: CLASH

text: clusters

text: have

text: larger

text: power

text: ratios

text: than

text: the

text: the

text: X-rays

text: to

text: be

text: which

text: is

text: similar

text: to

text: the

text: SZE

text: object

text: X-ray

text: selected

text: CLASH

text: subsample.

text: On

text: average,

text: maps

text: at

text: and

text: somewhat

text: more

text: elongated

text: (at

text: the

text: CLASH

text: clusters

text: are

text: rounder

text: than

text: the

text: simulated

text: clus-

text: kpc)

text: in

text: lensing

text: maps

text: (although

text: the

text: LTM

text: ters,

text: in

text: the

text: sense

text: that

text: their

text: axial

text: ratios

text: are

text: closer

text: to

text: unity.

text: lensing

text: models

text: are

text: more

text: circular,

text: at

text: In

text: a

text: This

text: resultis

text: not

text: surprising,

text: since

text: the

text: CLASH

text: X-ray

text: clus-

text: one-to-one

text: comparison,

text: the

text: lensing

text: maps

text: are

text: more

text: elon-

text: ters

text: were

text: chosen

text: in

text: part

text: for

text: their

text: round

text: X-ray

text: isophotes.

text: gated

text: than

text: the

text: projected

text: X-ray

text: emission,

text: but

text: are

text: gener-

text: However,

text: even

text: the

text: irregular,

text: lensing-selected

text: clusters

text: are

text: ally

text: aligned

text: in

text: the

text: same

text: direction.

text: That

text: these

text: clusters

text: rounder

text: than

text: most

text: of

text: the

text: simulated

text: clusters,

text: according

text: to

text: are

text: typically

text: circular

text: is

text: not

text: surprising,

text: since

text: they

text: were

text: the

text: axial-ratio

text: measurements:

text: All

text: of

text: the

text: lensing-selected

text: selected

text: to

text: be

text: nearly

text: circular

text: in

text: the

text: X-ray.

text: That

text: they

text: CLASH

text: clusters

text: have

text: an

text: axial

text: ratio

text: about

text: while

text: only

text: are

text: similarly

text: circular

text: in

text: SZ

text: and

text: lensing

text: images

text: then

text: is

text: of

text: the

text: simulated

text: sample

text: has

text: an

text: axial

text: ratio

text: above

text: also

text: expected.

text: That

text: the

text: gas

text: is

text: about

text: or

text: so

text: rounder

text: The

text: centroid

text: shifts

text: w

text: of

text: the

text: CLASH

text: clusters

text: are

text: than

text: the

text: projected

text: mass

text: at

text: kpc

text: was

text: predicted

text: by

text: Flo-

text: smaller

text: than

text: most

text: simulated

text: clusters

text: of

text: similar

text: mass,

text: res

text: et

text: al.

text: (e.g.

text: when

text: the

text: total

text: mass

text: is

text: dominated

text: but

text: they

text: are

text: not≤out

text: of

text: range:

text: only

text: of

text: the

text: MUSIC

text: by

text: collisionless

text: dark

text: matter,

text: and

text: thus

text: the

text: relative

text: ax-

text: clusters

text: have

text: w

text: while

text: of

text: the

text: CLASH

text: sample

text: ial

text: ratios

text: (and

text: therefore

text: eccentricities)

text: is

text: consistent

text: with

text: have

text: such

text: small

text: w.

text: gravitational

text: potentials

text: dominated

text: by

text: collisionless

text: dark

text: The

text: range

text: of

text: concentration

text: C

text: of

text: the

text: CLASH

text: clusters

text: is

text: matter.

text: similar

text: to

text: that

text: of

text: the

text: simulated

text: sample.

text: Note

text: that

text: a

text: sig-

text: The

text: SZE

text: estimates

text: of

text: the

text: axis

text: ratios

text: are

text: not

text: partic-

text: niﬁcant

text: subset

text: of

text: simulated

text: clusters

text: have

text: very

text: high

text: con-

text: ularly

text: correlated

text: with

text: the

text: X-ray

text: estimates

text: at

text: the

text: same

text: centrations

text: compared

text: to

text: those

text: observed

text: for

text: CLASH

text: clus-

text: radius,

text: but

text: both

text: estimates

text: have

text: a

text: very

text: similar

text: mean

text: and

text: ters,

text: even

text: for

text: the

text: relaxed

text: cool

text: core

text: clusters

text: in

text: CLASH.

text: standard

text: deviation,

text: where

text: the

text: scatter

text: is

text: dom-

text: of

text: the

text: simulated

text: clusters

text: have

text: concentrations

text: as

text: low

text: inated

text: by

text: the

text: measurement

text: uncertainties.

text: Abell

text: as

text: a

text: few

text: of

text: the

text: CLASH

text: strong-lensing

text: selected

text: clusters.

text: SZE

text: axis

text: ratio

text: is

text: an

text: outlier

text: for

text: the

text: sample’s

text: range

text: of

text: To

text: summarize,

text: compared

text: to

text: simulated

text: clusters

text: in

text: MU-

text: SZE

text: measurements,

text: possibly

text: due

text: to

text: the

text: dim

text: SZE

text: signal

text: SIC

text: with

text: a

text: similar

text: mass,

text: CLASH

text: cluster

text: morphologies

text: towards

text: this

text: cluster.

text: The

text: two

text: lowest

text: X-ray

text: axis

text: ratios

text: (i.e

text: are

text: on

text: average

text: rounder

text: and

text: have

text: smaller

text: centroid

text: shifts.

text: the

text: highest

text: elongations)

text: were

text: found

text: for

text: and

text: Their

text: surface

text: brightness

text: concentrations

text: and

text: power

text: ratios

text: These

text: are

text: two

text: of

text: the

text: ﬁve

text: lensing-selected

text: are

text: similar

text: to

text: that

text: of

text: the

text: mass-selected

text: sample

text: of

text: simu-

text: clusters

text: from

text: the

text: CLASH

text: survey

text: (Postman

text: et

text: al.

text: lated

text: clusters.

text: Both

text: of

text: these

text: CLASH

text: clusters

text: have

text: evidence

text: for

text: inter-

text: Meneghetti

text: et

text: al.

text: deﬁned

text: regularity

text: M

text: for

text: a

text: actions

text: in

text: their

text: optical

text: appearance

text: (at

text: least

text: two

text: bright

text: given

text: simulated

text: cluster

text: in

text: terms

text: of

text: the

text: oﬀsets

text: of

text: a

text: set

text: galaxies

text: in

text: their

text: core,

text: with

text: extended

text: elongated

text: intraclus-

text: of

text: morphological

text: properties,

text: in

text: units

text: of

text: standard

text: devia-

text: ter

text: light

text: in

text: between

text: them.)

text: tions.

text: We

text: refer

text: the

text: reader

text: to

text: Meneghetti

text: et

text: al.

text: We

text: have

text: plotted

text: the

text: histograms

text: of

text: centroid

text: and

text: posi-

text: for

text: details

text: and

text: speciﬁc

text: relations,

text: but

text: we

text: brieﬂy

text: review

text: the

text: tion

text: angle

text: diﬀerences

text: in

text: Figure

text: and

text: results

text: relevant

text: to

text: this

text: work

text: here.

text: For

text: each

text: of

text: ﬁve

text: mor-

text: phological

text: parameters,

text: w,

text: eccentricity,

text: and

text: Comparison

text: of

text: X-ray

text: Morphological

text: Properties

text: of

text: and

text: they

text: ﬁnd

text: the

text: diﬀerence

text: between

text: the

text: log

text: of

text: the

text: CLASH

text: Clusters

text: with

text: Simulations

text: parameter

text: and

text: the

text: log

text: of

text: the

text: mean

text: and

text: divide

text: by

text: the


text: 20 Donahue et al. 7. CONCLUSIONS quickly, potentially leading to changes in axis ratios. Fu- ture work, expanding beyond what the CLASH team The CLASH project (Postman et al. 2012a) has col- reported in Meneghetti et al. (2014b), will be able to lected a signiﬁcant amount of data for a sample of 25 compare simulations of high-mass clusters of galaxies to massive clusters of galaxies with redshifts from 0.2-0.9, these well-characterized CLASH clusters can be made including strong and weak lensing constraints from HST by selecting clusters based on estimated morphological (Merten et al. 2015), weak lensing constraints from Sub- properties and mass or gas temperatures similar to those aru (Umetsu et al. 2014), X-ray observations from Chan- of CLASH clusters. dra and XMM (Donahue et al. 2014), and SZE observa- We also show that the stellar mass of the BCG at small tions from Bolocam (Czakon et al. 2015). To compare the scales (10 kpc), is strongly aligned with the position an- results of the CLASH cluster sample to predictions from gle of the matter distribution on much larger scales (500 simulations, Meneghetti et al. (2014a) selected simulated kpc), probed by lensing, X-ray, and SZE observations. clusters replicating the morphological and temperature To our knowledge, this the ﬁrst time BCG position angle selection of the original CLASH sample. We present here has been demonstrated to align with the position angle of the full uniformly measured X-ray morphological param- the mass distribution in a galaxy-cluster sample with de- eters and uncertainties for the 25 CLASH clusters used in tailed lensing maps. However, the result has its roots in that work. In addition we measure the same parameters mid-20th century astronomy. Alignments of BCGs with in lensing and SZE maps. All parameters are measured the galaxy distribution of their host clusters have been inside radii of 500 kpc, and for a sub-sample of the clus- noted by Abell and others since the 1950s. ters inside 0.5R . The full set of morphological prop- 500 This correlation shows that there is a strong relation- erties are centroids, concentration ratios (C), centroid ship between the assembly of the mass distribution of shifts (w), axial ratios, power ratios (P /P , P /P ), and 3 0 4 0 the stars in the very center of the matter halo, inside position angles. We present means and standard devia- the brightest cluster galaxy, and the distribution of dark tions of these properties for the CLASH sample and for matter and the hot X-ray at large scales. The underly- the X-ray selected CLASH subsample. For the ﬁrst time ing cause of this correlation is likely to be the shape of we demonstrate a strong correlation between these mor- the underlying gravitational potential. Cluster potential phological quantities as measured from lensing, X-ray, − shapes are not expected to be perfectly spherical because and SZE maps at a consistent radial scale of 500 h 1 70 in models of large scale structure matter does not ﬂow kpc (which is about half of R for these clusters of 500 in evenly over 4π steradians but preferentially along ﬁl- galaxies.) aments. Systematic alignment of BCGs with their clus- In order to visualize how typical CLASH clusters are ters suggests that even the very inner regions of a galaxy relative to a complete set of simulated clusters of sim- cluster reﬂect the anisotropy of mass accretion on much ilar mass, we compared high-mass clusters from the larger scales. MUSIC simulations (Meneghetti et al. 2014b) with our CLASH observations. The simulated clusters were mass- selected but are not matched in morphology as they MD was partially supported by an STScI/NASA were in Meneghetti et al. (2014b). The full set of sim- award HST-GO-12065.07-A and NASA ADAP award ulated clusters were expected to be more heterogeneous NNX13AI41G. AB was partially supported by the same in structure and dynamic state in terms of relaxation NASA ADAP award NNX13AI41G and a Chandra than the CLASH sample. The CLASH clusters have archive grant SAO AR3-14013X. ER was supported by similar power ratios, but C, w, and AR indicate that FP7-PEOPLE-2013-IIF (Grant Agreement PIIF-GA- the CLASH clusters indeed appear more regular than 2013-627474) and NSF AST-1210973. SE acknowledges a typical simulated cluster of similar mass. Very early the ﬁnancial contribution from contracts ASI-INAF attempts to simulate X-ray cluster morphologies typi- I/009/10/0 and PRIN-INAF 2012. This work was sup- cally failed to create clusters that were as relaxed as ported in part by National Science Foundation Grant those selected from X-ray observations. The ﬁrst sim- No. PHYS-1066293 and the hospitality of the Aspen ulations had limited cosmological context (simulations Center for Physics. JS was supported by NSF/AST- of individual clusters) or incorrect cosmology, such as 0838261, NASA/NNX11AB07G, and the Norris Foun- a standard CDM universe with Ω = 1 which predicts M dation CCAT Postdoctoral Fellowship. NGC was par- more recent assembly activity, as discussed in (Tsai & tially supported by a NASA Graduate Student Research Buote 1996; Buote & Xu 1997). Some early simulations Fellowship. The Bolocam observations were partially assumed lower matter densities (Evrard et al. 1993; supported by the Gordon and Betty Moore Founda- Mohr et al. 1995) and late time acceleration of the ex- tion. This research made use of the Caltech Submil- pansion (Thomas et al. 1998), but accurate predictions limeter Observatory, which was operated at the time of the distribution of hot gas inside of clusters require by the California Institute of Technology under cooper- baryon prescriptions including the eﬀects of AGN and ative agreement with the National Science Foundation star formation feedback (?). These processes regulate (NSF/AST-0838261). GY acknowledges ﬁnancial sup- the cooling and the heating of the cluster core and have port from MINECO’s grant AYA2012-31101. The MU- signiﬁcant impact on the central region. The morpho- SIC simulations have been performed in the Marenos- logical parameter most aﬀected by baryon processes is trum Supercomputer at the Barcelona Supercomputer the X-ray surface brightness concentration. However, Center, thanks to time granted by the Red Espan˜ola de AGN feedback may also change the shape and distri- Supercomputaci´on. bution of substructures because gas stripping becomes Facilities: Chandra X-ray Observatory, Hubble Space more eﬃcient and merging systems are thermalized more Telescope/ACS, Hubble SpaceTelescope/WFC3, Caltech

text: Donahue

text: et

text: al.

label: section

text: CONCLUSIONS

text: quickly,

text: potentially

text: leading

text: to

text: changes

text: in

text: axis

text: ratios.

text: Fu-

text: ture

text: work,

text: expanding

text: beyond

text: what

text: the

text: CLASH

text: team

text: The

text: CLASH

text: project

text: (Postman

text: et

text: al.

text: has

text: col-

text: reported

text: in

text: Meneghetti

text: et

text: al.

text: will

text: be

text: able

text: to

text: lected

text: a

text: signiﬁcant

text: amount

text: of

text: data

text: for

text: a

text: sample

text: of

text: compare

text: simulations

text: of

text: high-mass

text: clusters

text: of

text: galaxies

text: to

text: massive

text: clusters

text: of

text: galaxies

text: with

text: redshifts

text: from

text: these

text: well-characterized

text: CLASH

text: clusters

text: can

text: be

text: made

text: including

text: strong

text: and

text: weak

text: lensing

text: constraints

text: from

text: HST

text: by

text: selecting

text: clusters

text: based

text: on

text: estimated

text: morphological

text: (Merten

text: et

text: al.

text: weak

text: lensing

text: constraints

text: from

text: Sub-

text: properties

text: and

text: mass

text: or

text: gas

text: temperatures

text: similar

text: to

text: those

text: aru

text: (Umetsu

text: et

text: al.

text: X-ray

text: observations

text: from

text: Chan-

text: of

text: CLASH

text: clusters.

text: dra

text: and

text: XMM

text: (Donahue

text: et

text: al.

text: and

text: SZE

text: observa-

text: We

text: also

text: show

text: that

text: the

text: stellar

text: mass

text: of

text: the

text: BCG

text: at

text: small

text: tions

text: from

text: Bolocam

text: (Czakon

text: et

text: al.

text: To

text: compare

text: the

text: scales

text: kpc),

text: is

text: strongly

text: aligned

text: with

text: the

text: position

text: an-

text: results

text: of

text: the

text: CLASH

text: cluster

text: sample

text: to

text: predictions

text: from

text: gle

text: of

text: the

text: matter

text: distribution

text: on

text: much

text: larger

text: scales

text: simulations,

text: Meneghetti

text: et

text: al.

text: selected

text: simulated

text: kpc),

text: probed

text: by

text: lensing,

text: X-ray,

text: and

text: SZE

text: observations.

text: clusters

text: replicating

text: the

text: morphological

text: and

text: temperature

text: To

text: our

text: knowledge,

text: this

text: the

text: ﬁrst

text: time

text: BCG

text: position

text: angle

text: selection

text: of

text: the

text: original

text: CLASH

text: sample.

text: We

text: present

text: here

text: has

text: been

text: demonstrated

text: to

text: align

text: with

text: the

text: position

text: angle

text: of

text: the

text: full

text: uniformly

text: measured

text: X-ray

text: morphological

text: param-

text: the

text: mass

text: distribution

text: in

text: a

text: galaxy-cluster

text: sample

text: with

text: de-

text: eters

text: and

text: uncertainties

text: for

text: the

text: CLASH

text: clusters

text: used

text: in

text: tailed

text: lensing

text: maps.

text: However,

text: the

text: result

text: has

text: its

text: roots

text: in

text: that

text: work.

text: In

text: addition

text: we

text: measure

text: the

text: same

text: parameters

text: century

text: astronomy.

text: Alignments

text: of

text: BCGs

text: with

text: in

text: lensing

text: and

text: SZE

text: maps.

text: All

text: parameters

text: are

text: measured

text: the

text: galaxy

text: distribution

text: of

text: their

text: host

text: clusters

text: have

text: been

text: inside

text: radii

text: of

text: kpc,

text: and

text: for

text: a

text: sub-sample

text: of

text: the

text: clus-

text: noted

text: by

text: Abell

text: and

text: others

text: since

text: the

text: ters

text: inside

text: The

text: full

text: set

text: of

text: morphological

text: prop-

text: This

text: correlation

text: shows

text: that

text: there

text: is

text: a

text: strong

text: relation-

text: erties

text: are

text: centroids,

text: concentration

text: ratios

text: (C),

text: centroid

text: ship

text: between

text: the

text: assembly

text: of

text: the

text: mass

text: distribution

text: of

text: shifts

text: (w),

text: axial

text: ratios,

text: power

text: ratios

text: (P

text: /P

text: P

text: /P

text: and

text: the

text: stars

text: in

text: the

text: very

text: center

text: of

text: the

text: matter

text: halo,

text: inside

text: position

text: angles.

text: We

text: present

text: means

text: and

text: standard

text: devia-

text: the

text: brightest

text: cluster

text: galaxy,

text: and

text: the

text: distribution

text: of

text: dark

text: tions

text: of

text: these

text: properties

text: for

text: the

text: CLASH

text: sample

text: and

text: for

text: matter

text: and

text: the

text: hot

text: X-ray

text: at

text: large

text: scales.

text: The

text: underly-

text: the

text: X-ray

text: selected

text: CLASH

text: subsample.

text: For

text: the

text: ﬁrst

text: time

text: ing

text: cause

text: of

text: this

text: correlation

text: is

text: likely

text: to

text: be

text: the

text: shape

text: of

text: we

text: demonstrate

text: a

text: strong

text: correlation

text: between

text: these

text: mor-

text: the

text: underlying

text: gravitational

text: potential.

text: Cluster

text: potential

text: phological

text: quantities

text: as

text: measured

text: from

text: lensing,

text: X-ray,

text: shapes

text: are

text: not

text: expected

text: to

text: be

text: perfectly

text: spherical

text: because

text: and

text: SZE

text: maps

text: at

text: a

text: consistent

text: radial

text: scale

text: of

text: h

text: in

text: models

text: of

text: large

text: scale

text: structure

text: matter

text: does

text: not

text: ﬂow

text: kpc

text: (which

text: is

text: about

text: half

text: of

text: R

text: for

text: these

text: clusters

text: of

text: in

text: evenly

text: over

text: 4π

text: steradians

text: but

text: preferentially

text: along

text: ﬁl-

text: galaxies.)

text: aments.

text: Systematic

text: alignment

text: of

text: BCGs

text: with

text: their

text: clus-

text: In

text: order

text: to

text: visualize

text: how

text: typical

text: CLASH

text: clusters

text: are

text: ters

text: suggests

text: that

text: even

text: the

text: very

text: inner

text: regions

text: of

text: a

text: galaxy

text: relative

text: to

text: a

text: complete

text: set

text: of

text: simulated

text: clusters

text: of

text: sim-

text: cluster

text: reﬂect

text: the

text: anisotropy

text: of

text: mass

text: accretion

text: on

text: much

text: ilar

text: mass,

text: we

text: compared

text: high-mass

text: clusters

text: from

text: the

text: larger

text: scales.

text: MUSIC

text: simulations

text: (Meneghetti

text: et

text: al.

text: with

text: our

text: CLASH

text: observations.

text: The

text: simulated

text: clusters

text: were

text: mass-

text: selected

text: but

text: are

text: not

text: matched

text: in

text: morphology

text: as

text: they

text: MD

text: was

text: partially

text: supported

text: by

text: an

text: STScI/NASA

text: were

text: in

text: Meneghetti

text: et

text: al.

text: The

text: full

text: set

text: of

text: sim-

text: award

text: and

text: NASA

text: ADAP

text: award

text: ulated

text: clusters

text: were

text: expected

text: to

text: be

text: more

text: heterogeneous

text: AB

text: was

text: partially

text: supported

text: by

text: the

text: same

text: in

text: structure

text: and

text: dynamic

text: state

text: in

text: terms

text: of

text: relaxation

text: NASA

text: ADAP

text: award

text: and

text: a

text: Chandra

text: than

text: the

text: CLASH

text: sample.

text: The

text: CLASH

text: clusters

text: have

text: archive

text: grant

text: SAO

text: ER

text: was

text: supported

text: by

text: similar

text: power

text: ratios,

text: but

text: C,

text: w,

text: and

text: AR

text: indicate

text: that

text: (Grant

text: Agreement

text: PIIF-GA-

text: the

text: CLASH

text: clusters

text: indeed

text: appear

text: more

text: regular

text: than

text: and

text: NSF

text: SE

text: acknowledges

text: a

text: typical

text: simulated

text: cluster

text: of

text: similar

text: mass.

text: Very

text: early

text: the

text: ﬁnancial

text: contribution

text: from

text: contracts

text: ASI-INAF

text: attempts

text: to

text: simulate

text: X-ray

text: cluster

text: morphologies

text: typi-

text: and

text: PRIN-INAF

text: This

text: work

text: was

text: sup-

text: cally

text: failed

text: to

text: create

text: clusters

text: that

text: were

text: as

text: relaxed

text: as

text: ported

text: in

text: part

text: by

text: National

text: Science

text: Foundation

text: Grant

text: those

text: selected

text: from

text: X-ray

text: observations.

text: The

text: ﬁrst

text: sim-

text: No.

text: and

text: the

text: hospitality

text: of

text: the

text: Aspen

text: ulations

text: had

text: limited

text: cosmological

text: context

text: (simulations

text: Center

text: for

text: Physics.

text: JS

text: was

text: supported

text: by

text: NSF/AST-

text: of

text: individual

text: clusters)

text: or

text: incorrect

text: cosmology,

text: such

text: as

text: and

text: the

text: Norris

text: Foun-

text: a

text: standard

text: CDM

text: universe

text: with

text: Ω

text: which

text: predicts

text: M

text: dation

text: CCAT

text: Postdoctoral

text: Fellowship.

text: NGC

text: was

text: par-

text: more

text: recent

text: assembly

text: activity,

text: as

text: discussed

text: in

text: (Tsai

text: tially

text: supported

text: by

text: a

text: NASA

text: Graduate

text: Student

text: Research

text: Buote

text: Buote

text: Xu

text: Some

text: early

text: simulations

text: Fellowship.

text: The

text: Bolocam

text: observations

text: were

text: partially

text: assumed

text: lower

text: matter

text: densities

text: (Evrard

text: et

text: al.

text: supported

text: by

text: the

text: Gordon

text: and

text: Betty

text: Moore

text: Founda-

text: Mohr

text: et

text: al.

text: and

text: late

text: time

text: acceleration

text: of

text: the

text: ex-

text: tion.

text: This

text: research

text: made

text: use

text: of

text: the

text: Caltech

text: Submil-

text: pansion

text: (Thomas

text: et

text: al.

text: but

text: accurate

text: predictions

text: limeter

text: Observatory,

text: which

text: was

text: operated

text: at

text: the

text: time

text: of

text: the

text: distribution

text: of

text: hot

text: gas

text: inside

text: of

text: clusters

text: require

text: by

text: the

text: California

text: Institute

text: of

text: Technology

text: under

text: cooper-

text: baryon

text: prescriptions

text: including

text: the

text: eﬀects

text: of

text: AGN

text: and

text: ative

text: agreement

text: with

text: the

text: National

text: Science

text: Foundation

text: star

text: formation

text: feedback

text: These

text: processes

text: regulate

text: GY

text: acknowledges

text: ﬁnancial

text: sup-

text: the

text: cooling

text: and

text: the

text: heating

text: of

text: the

text: cluster

text: core

text: and

text: have

text: port

text: from

text: MINECO’s

text: grant

text: The

text: MU-

text: signiﬁcant

text: impact

text: on

text: the

text: central

text: region.

text: The

text: morpho-

text: SIC

text: simulations

text: have

text: been

text: performed

text: in

text: the

text: Marenos-

text: logical

text: parameter

text: most

text: aﬀected

text: by

text: baryon

text: processes

text: is

text: trum

text: Supercomputer

text: at

text: the

text: Barcelona

text: Supercomputer

text: the

text: X-ray

text: surface

text: brightness

text: concentration.

text: However,

text: Center,

text: thanks

text: to

text: time

text: granted

text: by

text: the

text: Red

text: Espan˜ola

text: de

text: AGN

text: feedback

text: may

text: also

text: change

text: the

text: shape

text: and

text: distri-

text: Supercomputaci´on.

text: bution

text: of

text: substructures

text: because

text: gas

text: stripping

text: becomes

text: Facilities:

text: Chandra

text: X-ray

text: Observatory,

text: Hubble

text: Space

text: more

text: eﬃcient

text: and

text: merging

text: systems

text: are

text: thermalized

text: more

text: Telescope/ACS,

text: Hubble

text: Caltech


text: 2. THE CLASH PROJECT AND SAMPLE ple. Quantiﬁed X-ray morphologies were measured from maps for the simulated clusters with procedures identical The CLASH cluster program and strategy are de- to those used on the actual X-ray data (Meneghetti et al. scribed in Postman et al. (2012b). Relevant cluster 2014a). properties are provided in Table 1. CLASH was a Hub- In this paper, we provide quantitative X-ray surface- ble Multi-Cycle Treasury program with multiple sci- brightness morphological parameters for the 25 clusters ence goals. The most relevant CLASH science goal for in the CLASH survey (Postman et al. 2012a), as well as this work was to obtain well-constrained gravitational- similarly-deﬁned morphological parameters derived from lensing mass proﬁles for a sample of 25 massive clusters − gravitational lensing (shear) projected mass maps and of galaxies between redshifts of 0.2 0.9. To avoid SZE gas (Compton “y”-parameter) maps. Two sets of any biases that would be introduced by selecting clus- morphological measurements were made, one inside a ters on the basis of their lensing signal, twenty of the − CLASH clusters were instead selected on the basis of X- ﬁxed metric aperture of 500 h701 kpc and the other in- ray morphology, to have relatively round X-ray isophotes side half the R500 overdensity radius (i.e. R500 is the ra- centered on a prominent BCG. The remaining ﬁve were dius inside which the average mass density is 500 times selected to be systems capable of providing extraordi- the critical density at the redshift of the cluster, so it nary, gravitationally-boosted views of the high redshift is a cosmology- and mass-dependent quantity.) We re- universe. All of the clusters have relatively hot intraclus- constructed the angular scale−corresponding to 0.5R500 ter gas (ICM), with global gas temperatures of kT > 5 in arcseconds from the M500h 1 mass quantity reported keV. This program was allocated 524 orbits over a 3 year in Merten et al. (2015). To avoid ambiguity, we report (cycle) period between May, 2010 and May, 2013. Dur- that speciﬁc angular scale for each CLASH cluster in ing this time, HST observed all 25 clusters with up to 16 Table 1. The ﬁxed metric aperture of 500 kpc has the passbands, utilizing the Wide Field Camera 3 (WFC3) advantage of not changing signiﬁcantly from analysis to Infrared (IR) and UV/Visible (UVIS) channels and the analysis, as well as having been used by previous ob- Advanced Camera for Surveys (ACS). All CLASH clus- servers for similar purposes (e.g., Jeltema et al. 2005; ters also have good to excellent Chandra X-ray data, Cassano et al. 2010). For the CLASH sample, a ﬁxed with at least 6,000-10,000 X-ray events between 0.5-7.0 metric aperture is approximately the same fraction of keV in publicly available datasets. the virial radius for most of the clusters, since these clus- Observations of the CLASH cluster sample have al- ters are similar in mass. Regardless of the aperture we ready shown that their concentration-mass relation is use, the morphologies of the Brightest Cluster Galaxies consistent with Λ-CDM based predictions, once the (BCGs) at kpc scales in the relaxed sample are strikingly CLASH X-ray morphological selection is taken into aligned with gravitational potential elongations within account (Merten et al. 2015; Meneghetti et al. 2014a; these much larger apertures, suggesting a strong relation- Umetsu et al. 2014, 2015). This major result conﬁrmed ship between the assembly of the BCG and the cluster suspicions that previous studies, having selected clusters as a whole. with prominent lensing features, gave mass concentration We also compare CLASH clusters to a broader sample measurements biased higher than those predicted from of simulated clusters from MUSIC-221 (Sembolini et al. simulations based on idealized mass-selection of clusters. 2013), selected to have similar masses by making a cut This eﬀect was suggested in Comerford & Natarajan at a minimum global temperature of 5 keV, without re- (e.g., 2007); Meneghetti et al. (e.g., 2010); Groener gard to morphology. The purpose of this comparison et al. (e.g., 2016). For the CLASH analysis, biases in- is to examine how typical the morphological properties duced by X-ray morphological selection were quantiﬁed of CLASH clusters are of massive, simulated clusters in bySuseblmeictttiendg tcoluAstpeJr0s8frOocmt 2s0im15u; lAacticoenpsteidn1t1hJeasna2m01e6way any dynamical state. Throughout the paper, we assume thPartepthrientytwypeerseetsuesliencgtLeAdTEfXorstiynlecleumsuiolanteainpjtvh.e12C/1L6A/1S1H sam- cosmological para−meters−of ΩM = 0.3, ΩΛ = 0.7, and H = 70h km s 1 Mpc 1 (i.e. h = 1 is the default). 0 70 70 THE MORPHOLOGIES AND ALIGNMENTS OF GAS, MASS, AND THE CENTRAL GALAXIES OF CLASH CLUSTERS OF GALAXIES Megan Donahue1,2, Stefano Ettori3, Elena Rasia4,5, Jack Sayers6, Adi Zitrin6,10, Massimo Meneghetti3,6, G. Mark Voit1, Sunil Golwala6, Nicole Czakon7, Gustavo Yepes8, Alessandro Baldi1, Anton Koekemoer9, Marc 2 donahue@pa.msu.edu 3 Osservatori Astronomico di Bologna, Via Ranzani 1, 40127 Bologna, Italy; INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy 4 Department of Physics, University of Michigan, Ann Arbor, MI, 48109, USA 5 INAF-Osservatorio Astronomico of Trieste, via Tiepolo 11, 34121 Trieste, Italy 6 Division of Physics, Math, and Astronomy, California Insti- tute of Technology, Pasadena, CA 91125 7 Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R. O. C. 8 Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, Cantoblanco, 28049, Madrid, Spain 9 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA 10 Hubble Fellow 21 MUSIC website: http://music.ft.uam.es

text: THE

text: CLASH

text: PROJECT

text: AND

text: SAMPLE

text: ple.

text: Quantiﬁed

text: X-ray

text: morphologies

text: were

text: measured

text: from

text: maps

text: for

text: the

text: simulated

text: clusters

text: with

text: procedures

text: identical

text: The

text: CLASH

text: cluster

text: program

text: and

text: strategy

text: are

text: de-

text: to

text: those

text: used

text: on

text: the

text: actual

text: X-ray

text: data

text: (Meneghetti

text: et

text: al.

text: scribed

text: in

text: Postman

text: et

text: al.

text: Relevant

text: cluster

text: properties

text: are

text: provided

text: in

text: Table

text: CLASH

text: was

text: a

text: Hub-

text: In

text: this

text: paper,

text: we

text: provide

text: quantitative

text: X-ray

text: surface-

text: ble

text: Multi-Cycle

text: Treasury

text: program

text: with

text: multiple

text: sci-

text: brightness

text: morphological

text: parameters

text: for

text: the

text: clusters

text: ence

text: goals.

text: The

text: most

text: relevant

text: CLASH

text: science

text: goal

text: for

text: in

text: the

text: CLASH

text: survey

text: (Postman

text: et

text: al.

text: as

text: well

text: as

text: this

text: work

text: was

text: to

text: obtain

text: well-constrained

text: gravitational-

text: similarly-deﬁned

text: morphological

text: parameters

text: derived

text: from

text: lensing

text: mass

text: proﬁles

text: for

text: a

text: sample

text: of

text: massive

text: clusters

text: gravitational

text: lensing

text: (shear)

text: projected

text: mass

text: maps

text: and

text: of

text: galaxies

text: between

text: redshifts

text: of

text: To

text: avoid

text: SZE

text: gas

text: (Compton

text: “y”-parameter)

text: maps.

text: Two

text: sets

text: of

text: any

text: biases

text: that

text: would

text: be

text: introduced

text: by

text: selecting

text: clus-

text: morphological

text: measurements

text: were

text: made,

text: one

text: inside

text: a

text: ters

text: on

text: the

text: basis

text: of

text: their

text: lensing

text: signal,

text: twenty

text: of

text: the

text: CLASH

text: clusters

text: were

text: instead

text: selected

text: on

text: the

text: basis

text: of

text: X-

text: ﬁxed

text: metric

text: aperture

text: of

text: kpc

text: and

text: the

text: other

text: in-

text: ray

text: morphology,

text: to

text: have

text: relatively

text: round

text: X-ray

text: isophotes

text: side

text: half

text: the

text: overdensity

text: radius

text: (i.e.

text: is

text: the

text: ra-

text: centered

text: on

text: a

text: prominent

text: BCG.

text: The

text: remaining

text: ﬁve

text: were

text: dius

text: inside

text: which

text: the

text: average

text: mass

text: density

text: is

text: times

text: selected

text: to

text: be

text: systems

text: capable

text: of

text: providing

text: extraordi-

text: the

text: critical

text: density

text: at

text: the

text: redshift

text: of

text: the

text: cluster,

text: so

text: it

text: nary,

text: gravitationally-boosted

text: views

text: of

text: the

text: high

text: redshift

text: is

text: a

text: cosmology-

text: and

text: mass-dependent

text: quantity.)

text: We

text: re-

text: universe.

text: All

text: of

text: the

text: clusters

text: have

text: relatively

text: hot

text: intraclus-

text: constructed

text: the

text: angular

text: scale−corresponding

text: to

text: ter

text: gas

text: (ICM),

text: with

text: global

text: gas

text: temperatures

text: of

text: kT

text: in

text: arcseconds

text: from

text: the

text: mass

text: quantity

text: reported

text: keV.

text: This

text: program

text: was

text: allocated

text: orbits

text: over

text: a

text: year

text: in

text: Merten

text: et

text: al.

text: To

text: avoid

text: ambiguity,

text: we

text: report

text: (cycle)

text: period

text: between

text: May,

text: and

text: May,

text: Dur-

text: that

text: speciﬁc

text: angular

text: scale

text: for

text: each

text: CLASH

text: cluster

text: in

text: ing

text: this

text: time,

text: HST

text: observed

text: all

text: clusters

text: with

text: up

text: to

text: Table

text: The

text: ﬁxed

text: metric

text: aperture

text: of

text: kpc

text: has

text: the

text: passbands,

text: utilizing

text: the

text: Wide

text: Field

text: Camera

text: advantage

text: of

text: not

text: changing

text: signiﬁcantly

text: from

text: analysis

text: to

text: Infrared

text: (IR)

text: and

text: UV/Visible

text: (UVIS)

text: channels

text: and

text: the

text: analysis,

text: as

text: well

text: as

text: having

text: been

text: used

text: by

text: previous

text: ob-

text: Advanced

text: Camera

text: for

text: Surveys

text: (ACS).

text: All

text: CLASH

text: clus-

text: servers

text: for

text: similar

text: purposes

text: (e.g.,

text: Jeltema

text: et

text: al.

text: ters

text: also

text: have

text: good

text: to

text: excellent

text: Chandra

text: X-ray

text: data,

text: Cassano

text: et

text: al.

text: For

text: the

text: CLASH

text: sample,

text: a

text: ﬁxed

text: with

text: at

text: least

text: X-ray

text: events

text: between

text: metric

text: aperture

text: is

text: approximately

text: the

text: same

text: fraction

text: of

text: keV

text: in

text: publicly

text: available

text: datasets.

text: the

text: virial

text: radius

text: for

text: most

text: of

text: the

text: clusters,

text: since

text: these

text: clus-

text: Observations

text: of

text: the

text: CLASH

text: cluster

text: sample

text: have

text: al-

text: ters

text: are

text: similar

text: in

text: mass.

text: Regardless

text: of

text: the

text: aperture

text: we

text: ready

text: shown

text: that

text: their

text: concentration-mass

text: relation

text: is

text: use,

text: the

text: morphologies

text: of

text: the

text: Brightest

text: Cluster

text: Galaxies

text: consistent

text: with

text: Λ-CDM

text: based

text: predictions,

text: once

text: the

text: (BCGs)

text: at

text: kpc

text: scales

text: in

text: the

text: relaxed

text: sample

text: are

text: strikingly

text: CLASH

text: X-ray

text: morphological

text: selection

text: is

text: taken

text: into

text: aligned

text: with

text: gravitational

text: potential

text: elongations

text: within

text: account

text: (Merten

text: et

text: al.

text: Meneghetti

text: et

text: al.

text: these

text: much

text: larger

text: apertures,

text: suggesting

text: a

text: strong

text: relation-

text: Umetsu

text: et

text: al.

text: This

text: major

text: result

text: conﬁrmed

text: ship

text: between

text: the

text: assembly

text: of

text: the

text: BCG

text: and

text: the

text: cluster

text: suspicions

text: that

text: previous

text: studies,

text: having

text: selected

text: clusters

text: as

text: a

text: whole.

text: with

text: prominent

text: lensing

text: features,

text: gave

text: mass

text: concentration

text: We

text: also

text: compare

text: CLASH

text: clusters

text: to

text: a

text: broader

text: sample

text: measurements

text: biased

text: higher

text: than

text: those

text: predicted

text: from

text: of

text: simulated

text: clusters

text: from

text: (Sembolini

text: et

text: al.

text: simulations

text: based

text: on

text: idealized

text: mass-selection

text: of

text: clusters.

text: selected

text: to

text: have

text: similar

text: masses

text: by

text: making

text: a

text: cut

text: This

text: eﬀect

text: was

text: suggested

text: in

text: Comerford

text: Natarajan

text: at

text: a

text: minimum

text: global

text: temperature

text: of

text: keV,

text: without

text: re-

text: (e.g.,

text: Meneghetti

text: et

text: al.

text: (e.g.,

text: Groener

text: gard

text: to

text: morphology.

text: The

text: purpose

text: of

text: this

text: comparison

text: et

text: al.

text: (e.g.,

text: For

text: the

text: CLASH

text: analysis,

text: biases

text: in-

text: is

text: to

text: examine

text: how

text: typical

text: the

text: morphological

text: properties

text: duced

text: by

text: X-ray

text: morphological

text: selection

text: were

text: quantiﬁed

text: of

text: CLASH

text: clusters

text: are

text: of

text: massive,

text: simulated

text: clusters

text: in

text: bySuseblmeictttiendg

text: any

text: dynamical

text: state.

text: Throughout

text: the

text: paper,

text: we

text: assume

text: sam-

text: cosmological

text: para−meters−of

text: ΩM

text: ΩΛ

text: and

text: H

text: km

text: s

text: Mpc

text: (i.e.

text: h

text: is

text: the

text: default).

text: THE

text: MORPHOLOGIES

text: AND

text: ALIGNMENTS

text: OF

text: GAS,

text: MASS,

text: AND

text: THE

text: CENTRAL

text: GALAXIES

text: OF

text: CLASH

text: CLUSTERS

text: OF

text: GALAXIES

text: Megan

text: Stefano

text: Elena

text: Jack

text: Adi

text: Massimo

text: G.

text: Mark

text: Sunil

text: Nicole

text: Gustavo

text: Alessandro

text: Anton

text: Marc

text: donahue@pa.msu.edu

text: Osservatori

text: Astronomico

text: di

text: Bologna,

text: Via

text: Ranzani

text: Bologna,

text: Italy;

text: INFN,

text: Sezione

text: di

text: Bologna,

text: viale

text: Berti

text: Pichat

text: Bologna,

text: Italy

text: Department

text: of

text: Physics,

text: University

text: of

text: Michigan,

text: Ann

text: Arbor,

text: MI,

text: USA

text: INAF-Osservatorio

text: Astronomico

text: of

text: Trieste,

text: via

text: Tiepolo

text: Trieste,

text: Italy

text: Division

text: of

text: Physics,

text: Math,

text: and

text: Astronomy,

text: California

text: Insti-

text: tute

text: of

text: Technology,

text: Pasadena,

text: CA

text: Institute

text: of

text: Astronomy

text: and

text: Astrophysics,

text: Academia

text: Sinica,

text: No.

text: Sec.

text: Roosevelt

text: Rd,

text: Taipei

text: Taiwan,

text: R.

text: O.

text: C.

text: Departamento

text: de

text: F´ısica

text: Te´orica,

text: Universidad

text: Aut´onoma

text: de

text: Madrid,

text: Cantoblanco,

text: Madrid,

text: Spain

text: Space

text: Telescope

text: Science

text: Institute,

text: San

text: Martin

text: Drive,

text: Baltimore,

text: MD,

text: USA

text: Hubble

text: Fellow

text: MUSIC

text: website: