text: – 7 – Table 1. Detected Spectral Lines ± ± ± Ion Rest wavelength Observed wavelength 1σ Doppler velocity 1σ FWHM 1σ − − (˚A) (˚A) (km s 1) (km s 1) ± ± ± H I 3889.06 3889.03 0.035 -2.2 2.70 29.76 2.783 ± ± ± H I 3970.08 3970.16 0.122 5.8 9.19 60.18 7.853 ± ± ± He I 4026.19 4026.43 0.017 17.4 1.29 29.93 3.291 ‡ N III 4097.36 4097.59 16.7 31.97 ‡∗ N III 4097.36 4096.95 -29.9 78.2 ± ± ± H I 4101.73 4101.82 0.013 6.2 0.91 43.93 1.031 ‡ N III 4103.39 4103.63 17.7 39.31 ‡∗ N III 4103.39 4102.94 -32.8 79.4 ± ± ± Si IV 4116.10 4115.69 0.130 -30.2 9.50 102.70 15.827 ± ± ± H I 4340.47 4340.55 0.004 5.2 0.29 52.63 7.542 † ± ± ± H I 4340.47 4338.73 0.371 -120.4 25.62 3533.02 36.013 ± ± ± [O III] 4363.21 4363.28 0.044 4.9 3.03 33.60 4.356 ‡ He I 4388.15 4387.88 -18.4 72.49 ± ± ± He I 4471.48 4471.72 0.006 16.4 0.43 29.90 0.898 ± ± ± S IV 4485.64 4485.61 0.055 -2.2 3.65 49.86 7.024 ± ± ± S IV 4504.11 4503.86 0.084 -16.6 5.57 52.98 11.734 ± ± ± He II 4541.49 4540.86 0.044 -41.3 2.88 75.32 10.595 ± ± ± N III 4634.14 4633.86 0.045 -18.1 2.92 76.98 9.911 ± ± ± N III 4640.64 4640.41 0.107 -14.8 6.93 90.31 8.120 ± ± ± C III 4647.42 4647.52 0.015 6.7 0.98 40.45 2.084 ± ± ± C III 4650.25 4650.35 0.057 6.4 3.69 38.23 5.538 ± ± ± He II 4685.80 4685.35 0.015 -28.7 0.98 122.71 3.173 ± ± ± He I 4713.15 4713.30 0.037 9.7 2.38 34.73 3.931 ‡ He II 4859.32 4858.72 -36.9 50.16 ± ± ± H I 4861.35 4861.32 0.002 -1.9 0.10 43.72 3.626 † ± ± ± H I 4861.35 4860.44 0.026 -56.0 1.58 2241.28 62.489 ± ± ± He I 4921.93 4922.00 0.016 4.5 0.98 36.30 1.535 ± ± ± [O III] 4958.91 4958.85 0.031 -3.7 1.86 29.62 4.329 ± ± ± [O III] 5006.84 5006.33 0.052 -30.9 3.10 32.51 4.197 ‡ He I 5015.68 5015.90 13.2 20.32 ± ± ± He II 5411.52 5410.65 0.061 -48.0 3.40 116.45 9.634 ± ± ± C III 5695.92 5695.69 0.021 -11.9 1.13 73.26 1.353 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Detected

text: Spectral

text: Lines

text: Ion

text: Rest

text: wavelength

text: Observed

text: wavelength

text: 1σ

text: Doppler

text: velocity

text: 1σ

text: FWHM

text: 1σ

text: (˚A)

text: (˚A)

text: (km

text: s

text: (km

text: s

text: H

text: I

text: H

text: I

text: He

text: I

text: N

text: III

text: N

text: III

text: H

text: I

text: N

text: III

text: N

text: III

text: Si

text: IV

text: H

text: I

text: H

text: I

text: [O

text: III]

text: He

text: I

text: He

text: I

text: S

text: IV

text: S

text: IV

text: He

text: II

text: N

text: III

text: N

text: III

text: C

text: III

text: C

text: III

text: He

text: II

text: He

text: I

text: He

text: II

text: H

text: I

text: H

text: I

text: He

text: I

text: [O

text: III]

text: [O

text: III]

text: He

text: I

text: He

text: II

text: C

text: III

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RLO systems powered by BHs in young SCs 11 4. DISCUSSION accretion and/or the presence of beaming would aﬀect the optical counterpart signiﬁcantly and cannot be ac- 4.1. Ultraluminous X-ray sources and simulated RLO counted for by our current models. The tolerance ranges binaries − in V and B V account for the fact that some observed Figures 6 and 7 show that the simulated RLO systems ULX counterparts are known to vary signiﬁcantly (e.g. describe a well-deﬁned pattern in the CMD. In partic- NGC 1313 X-2, Mucciarelli et al. 2007; Gris´e et al. 2008; ular, RLO-EBs and RLO-PBs behave in two diﬀerent Liu et al. 2009; Ho IX X-1, Gris´e et al. 2011). ways, since the former are generally redder than the lat- We do not ﬁnd any simulated systems that satisfy the ter. In the CMD, RLO-MSBHs follow the same trend as aforementioned requirements in the case of both M83 RLO-EBs, since all RLO-MSBHs but one formed from XMM1 and NGC 3034 ULX5. M83 XMM1 is one of dynamical exchange. On the other hand, RLO-MSBHs the three sources for which the V magnitude was derived represent the high-luminosity envelope of RLO-EBs in from the F606W ﬁlter, likely introducing a larger uncer- the CMD. tainty. In Table 7, we report the simulated system which These results provide important insights to understand is closer to the observed counterpart of M83 XMM1: it − − the observations of bright X-ray binaries in young SCs. has ∆ − = 0.28 and ∆ = 0.48. B V V Comparing the simulations with a complete set of data In the case of NGC 3034 ULX5, all the simulated is beyond the aims of this paper. In this section, we will systems that match the criteria on optical magnitude (cid:28) focus on a speciﬁc class of X-ray binaries, the ULXs. We and color have L 0.1 L . In Table 7, we Edd X,max chose the ULXs because they are mostly associated with report the simulated system which is closer to the ob- star forming regions and/or young SCs (e.g. Zezas et served counterpart according to ∆ − and ∆ , but has ∼ B V V al. 2002; Kaaret et al. 2004; Berghea 2009; Mapelli et al. L 0.02 L . NGC 3034 ULX5 (also known Edd X,max 2010; Poutanen et al. 2013; Berghea et al. 2013), because as M82 X-1) is an outlier of our sample, under many they seem to anti-correlate with the local metallicity (e.g. respects. First, the intrinsic extinction is expected to Pakull & Mirioni 2002; Zampieri et al. 2004; Soria et be very high, but cannot be reliably quantiﬁed (G13). al. 2005; Swartz, Tennant & Soria 2009; Mapelli et al. Second, the best-matching simulated system has much 2009; Mapelli et al. 2010; Mapelli et al. 2011a; Kaaret & longer orbital period (5000 d) than the observed one (62 Feng 2013; Prestwich et al. 2013) and because they have d, Kaaret et al. 2006). Third, the maximum observed X- an estimated luminosity (assuming isotropy) higher than ray luminosity is by far the highest in the sample (Feng & the Eddington luminosity of a 10 M(cid:12) BH. For the high Kaaret 2010). This source likely belongs to a completely luminosity and for the anti-correlation with metallicity, diﬀerent class (an intermediate-mass BH has been re- Mapelli et al. (2009) proposed that (a fraction of) ULXs quired to explain some of its properties, e.g. Kaaret et are associated with MSBHs (see also Zampieri & Roberts al. 2001; Patruno et al. 2006; Casella et al. 2008; Feng & 2009; Mapelli et al. 2010; Mapelli et al. 2011a). Kaaret 2010). Unfortunately, identiﬁed counterparts are available In the other cases, we come to some remarkable re- in the literature only for a limited number of ULXs. sults. The counterparts of M81 X-6 and NGC 1313 X-1 The most recent and homogeneous sample of optical are matched only by simulated RLO-PBs, the counter- ULX counterparts has been reported by Gladstone et parts of NGC 1313 X-2, NGC 2403 X-1 and NGC 5204 al. (2013, hereafter G13). Among the 22 ULXs studied X-1 are matched by both simulated RLO-PBs and sim- in G13 that have Chandra observations, and that have ulated RLO-EBs, while the counterparts of Ho IX X-1 at least one detected possible counterpart from Hubble and IC 342 X-1 are matched only by simulated RLO- Space Telescope (HST) data, we selected nine sources (see EBs. Considering the three observed counterparts that Table 6), based on the criteria described in Appendix . are matched by both RLO-PBs and RLO-EBs, the RLO- Five of the nine considered sources (Ho IX X-1, NGC 1313 EBs always have the highest L (because they host Edd X-2, IC 342 X-1, NGC 5204 X-1 and NGC 3034 ULX5) more massive BHs). Thus, the RLO-EBs have the ad- have multiple possible counterparts in the Chandra error vantage that they do not require an exceedingly high box. In Table 6, and in Figures 6 and 7, we report only super-Eddington factor in these three cases. the counterparts that were selected based on the criteria If we require not only a good matching between ob- described in Appendix . In Figures 6 and 7, we show served and simulated optical counterparts, but also that ≥ apparent magnitudes obtained positioning both the ob- L L /3 (i.e. only a mild super-Eddington Edd X,max served counterparts and the simulated ones at a distance factor5), then Ho IX X-1, NGC 1313 X-2, IC 342 X-1 of 5 Mpc. and (marginally) NGC 5204 X-1 can be matched only by From Figures 6 and 7, it is apparent that the ob- simulated RLO-MSBHs. The mass of the best-matching − served counterparts populate the same regions in the MSBHs is in the 25 77 M(cid:12) range. In the following, we CMD as the simulated ones. In Table 7, we list all focus on these four sources. the simulated systems that have Eddington luminosity ≥ HO IX X-1 L 0.1 L (where L is the maximum ob- Edd X,max X,max served X-ray luminosity) and that diﬀer from the ob- In the case of Ho IX X-1, a MSBH accretor is par- | | sfeerrevnecdec|bouetnwteere|pnarstims buylat∆edVan<d0o.5bs(ewrhveedreV∆Vmiasgtnhiteuddief)- t(iLcXu,lamralxy =fav2o.8re×d 1b0y40theerghsig−h1)o.bTsehrevebdesXt--mraaytclhuimnginRosLitOy and by ∆ − < 0.2 (where ∆ − is the diﬀerence be- B V B V− tween simulated and observed B V color). The re- 5 We chose to focus on mildly super-Eddington systems not be- ≥ quirement that L 0.1 L is necessary because cause we think that the considered ULXs cannot be strongly super- Edd X,max Eddington, but just because our code assumes Eddington-limited our models have been calculated under the assumption accretion. In a forthcoming study, we will generalize our results to of sub-Eddington accretion: a strongly super-Eddington super-Eddington systems.

text: RLO

label: paragraph

text: systems

text: powered

text: by

text: BHs

text: in

text: young

text: SCs

label: section

text: DISCUSSION

text: accretion

text: and/or

text: the

text: presence

text: of

text: beaming

text: would

text: aﬀect

text: the

text: optical

text: counterpart

text: signiﬁcantly

text: and

text: cannot

text: be

text: ac-

text: Ultraluminous

text: X-ray

text: sources

text: and

text: simulated

text: RLO

text: counted

text: for

text: by

text: our

text: current

text: models.

text: The

text: tolerance

text: ranges

text: binaries

text: in

text: V

text: and

text: B

text: V

text: account

text: for

text: the

text: fact

text: that

text: some

text: observed

text: Figures

text: and

text: show

text: that

text: the

text: simulated

text: RLO

text: systems

text: ULX

text: counterparts

text: are

text: known

text: to

text: vary

text: signiﬁcantly

text: (e.g.

text: describe

text: a

text: well-deﬁned

text: pattern

text: in

text: the

text: CMD.

text: In

text: partic-

text: NGC

text: Mucciarelli

text: et

text: al.

text: Gris´e

text: et

text: al.

text: ular,

text: RLO-EBs

text: and

text: RLO-PBs

text: behave

text: in

text: two

text: diﬀerent

text: Liu

text: et

text: al.

text: Ho

text: IX

text: Gris´e

text: et

text: al.

text: ways,

text: since

text: the

text: former

text: are

text: generally

text: redder

text: than

text: the

text: lat-

text: We

text: do

text: not

text: ﬁnd

text: any

text: simulated

text: systems

text: that

text: satisfy

text: the

text: ter.

text: In

text: the

text: CMD,

text: RLO-MSBHs

text: follow

text: the

text: same

text: trend

text: as

text: aforementioned

text: requirements

text: in

text: the

text: case

text: of

text: both

text: RLO-EBs,

text: since

text: all

text: RLO-MSBHs

text: but

text: one

text: formed

text: from

text: and

text: NGC

text: is

text: one

text: of

text: dynamical

text: exchange.

text: On

text: the

text: other

text: hand,

text: RLO-MSBHs

text: the

text: three

text: sources

text: for

text: which

text: the

text: V

text: magnitude

text: was

text: derived

text: represent

text: the

text: high-luminosity

text: envelope

text: of

text: RLO-EBs

text: in

text: from

text: the

text: ﬁlter,

text: likely

text: introducing

text: a

text: larger

text: uncer-

text: the

text: CMD.

text: tainty.

text: In

text: Table

text: we

text: report

text: the

text: simulated

text: system

text: which

text: These

text: results

text: provide

text: important

text: insights

text: to

text: understand

text: is

text: closer

text: to

text: the

text: observed

text: counterpart

text: of

text: it

text: the

text: observations

text: of

text: bright

text: X-ray

text: binaries

text: in

text: young

text: SCs.

text: has

text: and

text: B

text: V

text: V

text: Comparing

text: the

text: simulations

text: with

text: a

text: complete

text: set

text: of

text: data

text: In

text: the

text: case

text: of

text: NGC

text: all

text: the

text: simulated

text: is

text: beyond

text: the

text: aims

text: of

text: this

text: paper.

text: In

text: this

text: section,

text: we

text: will

text: systems

text: that

text: match

text: the

text: criteria

text: on

text: optical

text: magnitude

text: focus

text: on

text: a

text: speciﬁc

text: class

text: of

text: X-ray

text: binaries,

text: the

text: ULXs.

text: We

text: and

text: color

text: have

text: L

text: L

text: In

text: Table

text: we

text: Edd

text: X,max

text: chose

text: the

text: ULXs

text: because

text: they

text: are

text: mostly

text: associated

text: with

text: report

text: the

text: simulated

text: system

text: which

text: is

text: closer

text: to

text: the

text: ob-

text: star

text: forming

text: regions

text: and/or

text: young

text: SCs

text: (e.g.

text: Zezas

text: et

text: served

text: counterpart

text: according

text: to

text: and

text: but

text: has

text: B

text: V

text: V

text: al.

text: Kaaret

text: et

text: al.

text: Berghea

text: Mapelli

text: et

text: al.

text: L

text: L

text: NGC

text: (also

text: known

text: Edd

text: X,max

text: Poutanen

text: et

text: al.

text: Berghea

text: et

text: al.

text: because

text: as

text: is

text: an

text: outlier

text: of

text: our

text: sample,

text: under

text: many

text: they

text: seem

text: to

text: anti-correlate

text: with

text: the

text: local

text: metallicity

text: (e.g.

text: respects.

text: First,

text: the

text: intrinsic

text: extinction

text: is

text: expected

text: to

text: Pakull

text: Mirioni

text: Zampieri

text: et

text: al.

text: Soria

text: et

text: be

text: very

text: high,

text: but

text: cannot

text: be

text: reliably

text: quantiﬁed

text: al.

text: Swartz,

text: Tennant

text: Soria

text: Mapelli

text: et

text: al.

text: Second,

text: the

text: best-matching

text: simulated

text: system

text: has

text: much

text: Mapelli

text: et

text: al.

text: Mapelli

text: et

text: al.

text: Kaaret

text: longer

text: orbital

text: period

text: d)

text: than

text: the

text: observed

text: one

text: Feng

text: Prestwich

text: et

text: al.

text: and

text: because

text: they

text: have

text: d,

text: Kaaret

text: et

text: al.

text: Third,

text: the

text: maximum

text: observed

text: X-

text: an

text: estimated

text: luminosity

text: (assuming

text: isotropy)

text: higher

text: than

text: ray

text: luminosity

text: is

text: by

text: far

text: the

text: highest

text: in

text: the

text: sample

text: (Feng

text: the

text: Eddington

text: luminosity

text: of

text: a

text: BH.

text: For

text: the

text: high

text: Kaaret

text: This

text: source

text: likely

text: belongs

text: to

text: a

text: completely

text: luminosity

text: and

text: for

text: the

text: anti-correlation

text: with

text: metallicity,

text: diﬀerent

text: class

text: (an

text: intermediate-mass

text: BH

text: has

text: been

text: re-

text: Mapelli

text: et

text: al.

text: proposed

text: that

text: (a

text: fraction

text: of)

text: ULXs

text: quired

text: to

text: explain

text: some

text: of

text: its

text: properties,

text: e.g.

text: Kaaret

text: et

text: are

text: associated

text: with

text: MSBHs

text: (see

text: also

text: Zampieri

text: Roberts

text: al.

text: Patruno

text: et

text: al.

text: Casella

text: et

text: al.

text: Feng

text: Mapelli

text: et

text: al.

text: Mapelli

text: et

text: al.

text: Kaaret

text: Unfortunately,

text: identiﬁed

text: counterparts

text: are

text: available

text: In

text: the

text: other

text: cases,

text: we

text: come

text: to

text: some

text: remarkable

text: re-

text: in

text: the

text: literature

text: only

text: for

text: a

text: limited

text: number

text: of

text: ULXs.

text: sults.

text: The

text: counterparts

text: of

text: and

text: NGC

text: The

text: most

text: recent

text: and

text: homogeneous

text: sample

text: of

text: optical

text: are

text: matched

text: only

text: by

text: simulated

text: RLO-PBs,

text: the

text: counter-

text: ULX

text: counterparts

text: has

text: been

text: reported

text: by

text: Gladstone

text: et

text: parts

text: of

text: NGC

text: NGC

text: and

text: NGC

text: al.

text: hereafter

text: Among

text: the

text: ULXs

text: studied

text: are

text: matched

text: by

text: both

text: simulated

text: RLO-PBs

text: and

text: sim-

text: in

text: that

text: have

text: Chandra

text: observations,

text: and

text: that

text: have

text: ulated

text: RLO-EBs,

text: while

text: the

text: counterparts

text: of

text: Ho

text: IX

text: at

text: least

text: one

text: detected

text: possible

text: counterpart

text: from

text: Hubble

text: and

text: IC

text: are

text: matched

text: only

text: by

text: simulated

text: RLO-

text: Space

text: Telescope

text: (HST)

text: data,

text: we

text: selected

text: nine

text: sources

text: (see

text: EBs.

text: Considering

text: the

text: three

text: observed

text: counterparts

text: that

text: Table

text: based

text: on

text: the

text: criteria

text: described

text: in

text: Appendix

text: are

text: matched

text: by

text: both

text: RLO-PBs

text: and

text: RLO-EBs,

text: the

text: RLO-

text: Five

text: of

text: the

text: nine

text: considered

text: sources

text: (Ho

text: IX

text: NGC

text: EBs

text: always

text: have

text: the

text: highest

text: L

text: (because

text: they

text: host

text: Edd

text: IC

text: NGC

text: and

text: NGC

text: more

text: massive

text: BHs).

text: Thus,

text: the

text: RLO-EBs

text: have

text: the

text: ad-

text: have

text: multiple

text: possible

text: counterparts

text: in

text: the

text: Chandra

text: error

text: vantage

text: that

text: they

text: do

text: not

text: require

text: an

text: exceedingly

text: high

text: box.

text: In

text: Table

text: and

text: in

text: Figures

text: and

text: we

text: report

text: only

text: super-Eddington

text: factor

text: in

text: these

text: three

text: cases.

text: the

text: counterparts

text: that

text: were

text: selected

text: based

text: on

text: the

text: criteria

text: If

text: we

text: require

text: not

text: only

text: a

text: good

text: matching

text: between

text: ob-

text: described

text: in

text: Appendix

text: In

text: Figures

text: and

text: we

text: show

text: served

text: and

text: simulated

text: optical

text: counterparts,

text: but

text: also

text: that

text: apparent

text: magnitudes

text: obtained

text: positioning

text: both

text: the

text: ob-

text: L

text: L

text: (i.e.

text: only

text: a

text: mild

text: super-Eddington

text: Edd

text: X,max

text: served

text: counterparts

text: and

text: the

text: simulated

text: ones

text: at

text: a

text: distance

text: then

text: Ho

text: IX

text: NGC

text: IC

text: of

text: Mpc.

text: and

text: (marginally)

text: NGC

text: can

text: be

text: matched

text: only

text: by

text: From

text: Figures

text: and

text: it

text: is

text: apparent

text: that

text: the

text: ob-

text: simulated

text: RLO-MSBHs.

text: The

text: mass

text: of

text: the

text: best-matching

text: served

text: counterparts

text: populate

text: the

text: same

text: regions

text: in

text: the

text: MSBHs

text: is

text: in

text: the

text: range.

text: In

text: the

text: following,

text: we

text: CMD

text: as

text: the

text: simulated

text: ones.

text: In

text: Table

text: we

text: list

text: all

text: focus

text: on

text: these

text: four

text: sources.

text: the

text: simulated

text: systems

text: that

text: have

text: Eddington

text: luminosity

text: HO

text: IX

text: L

text: L

text: (where

text: L

text: is

text: the

text: maximum

text: ob-

text: Edd

text: X,max

text: X,max

text: served

text: X-ray

text: luminosity)

text: and

text: that

text: diﬀer

text: from

text: the

text: ob-

text: In

text: the

text: case

text: of

text: Ho

text: IX

text: a

text: MSBH

text: accretor

text: is

text: par-

text: sfeerrevnecdec|bouetnwteere|pnarstims

text: t(iLcXu,lamralxy

text: and

text: by

text: (where

text: is

text: the

text: diﬀerence

text: be-

text: B

text: V

text: B

text: V−

text: tween

text: simulated

text: and

text: observed

text: B

text: V

text: color).

text: The

text: re-

text: We

text: chose

text: to

text: focus

text: on

text: mildly

text: super-Eddington

text: systems

text: not

text: be-

text: quirement

text: that

text: L

text: L

text: is

text: necessary

text: because

text: cause

text: we

text: think

text: that

text: the

text: considered

text: ULXs

text: cannot

text: be

text: strongly

text: super-

text: Edd

text: X,max

text: Eddington,

text: but

text: just

text: because

text: our

text: code

text: assumes

text: Eddington-limited

text: our

text: models

text: have

text: been

text: calculated

text: under

text: the

text: assumption

text: accretion.

text: In

text: a

text: forthcoming

text: study,

text: we

text: will

text: generalize

text: our

text: results

text: to

text: of

text: sub-Eddington

text: accretion:

text: a

text: strongly

text: super-Eddington

text: super-Eddington

text: systems.


text: RLO systems powered by BHs in young SCs 19 Marlowe H., Kaaret P., Lang C., Feng H., Gris´e F., Miller N., Rappaport S. A., Podsiadlowski Ph., Pfahl E. 2005, MNRAS, Cseh D., Corbel S., Mushotzky R. F., 2014, MNRAS, accepted, 356, 401 arXiv:1407.6254 Ripamonti E., Mapelli M., Zampieri L., Colpi M. 2011, AN, 332, Middleton, M. J., Walton, D. J., Roberts, T. P., Heil, L. 2014, 418 MNRAS, 438, L51 Roberts T. P., Gladstone J. C., Goulding A. D., Swinbank A. M., Mucciarelli P., Zampieri L., Treves A., Turolla R., Falomo R. Ward M. J., Goad M. R., Levan A. J. 2011, AN, 332, 398 2007, ApJ, 658, 999 Sepinsky J., Kalogera V., Belczynski K. 2005, ApJ, 621L, 37 Nelemans G., Yungelson L. R., Portegies Zwart S. F., Verbunt F. Sigurdsson S., Phinney E. S. 1993, ApJ, 415, 631 2001, A&A, 365, 491 Sigurdsson S., Phinney E. S. 1995, ApJS, 99, 609 Pakull M. W., Mirioni L. 2002, astro-ph/0202488 Silva-Villa E., Larsen S. S. 2010, A&A, 516, 10 Parker R. J., Goodwin S. P. 2007, MNRAS, 380, 1271 Sollima A., Carballo-Bello J. A., Beccari G., Ferraro F. R., Fusi Patruno A., Portegies Zwart S., Dewi J., Hopman C. 2006, Pecci F., Lanzoni B. 2010, MNRAS, 401, 577 MNRAS, 370, L6 Soria R., Cropper M., Pakull M., Mushotzky R., Wu K. 2005, Patruno A., Zampieri L. 2008, MNRAS, 386, 543 MNRAS, 356, 12 Patruno A., Zampieri L. 2010, MNRAS, 403, L69 Sutton, A. D., Roberts, T. P., Middleton, M. J. 2013, MNRAS, Pfalzner S. 2009, A&A, 498, L37 435, 1758 Pintore F., Zampieri L. 2012, MNRAS, 420, 1107 Swartz D. A., Tennant A. F., Soria R. 2009, ApJ, 703, 159 Pintore F., Zampieri L., Wolter A., Belloni T. 2014, MNRAS, Swartz D. A., Soria R., Tennant A. F., Yukita M. 2011, ApJ, 741, 439, 3461 49 Podsiadlowski Ph., Rappaport S., Pfahl E. D. 2002, ApJ, 565, Tao L., Feng H., Gris´e F., Kaaret Ph. 2011, ApJ, 737, 81 1107 Terashima Y., Inoue H., Wilson A. S. 2006, ApJ, 645, 264 Podsiadlowski Ph., Rappaport S., Han Z. 2003, MNRAS, 341, 385 van der Hucht K. A. 1991, in ‘Wolf-Rayet Stars and Interrelations Porras A., Christopher M., Allen L., Di Francesco J., Megeath S. with Other Massive Stars in Galaxies’: Proceedings of the Th., Myers Ph. C. 2003, AJ, 126, 1916 143rd Symposium of the International Astronomical Union. Portegies Zwart S. F., Verbunt F. 1996, A&A, 309, 179 Edited by Karel A. van der Hucht and Bambang Hidayat. Portegies Zwart S. F., Verbunt F., Ergma E. 1997, A&A, 321, 207 International Astronomical Union. Symposium no. 143, Kluwer Portegies Zwart S. F., McMillan S. L. W., Hut P., Makino J. Academic Publishers, Dordrecht, p.19 2001, MNRAS, 321, 199 Vink J. S., de Koter A., Lamers H. J. G. L. M. 2001, A&A, 369, Portegies Zwart S. F., McMillan S. L. W. 2002, ApJ, 576, 899 574 Portegies Zwart S. F., McMillan S. L. W., Makino J. 2007, Vink J. S., de Koter A. 2005, A&A, 442, 587 MNRAS, 374, 95 Voss R., Nielsen M. T. B., Nelemans G., Fraser M., Smartt S. J. Portegies Zwart S. F., McMillan S. L. W., Gieles M. 2010, 2011, MNRAS, 418, L124 ARA&A, 48, 431 Winter L. M., Mushotzky R. F., Reynolds C. S. 2006, ApJ, 649, Poutanen J., Fabrika S., Valeev A. F., Sholukhova O., Greiner J. 730 2013, MNRAS, 432, 506 Zampieri L., Mucciarelli P., Falomo R., Kaaret P., Di Stefano R., Prestwich A. H., Tsantaki M., Zezas A., Jackson F., Roberts T. Turolla R., Chieregato M., Treves A. 2004, ApJ, 603, 523 P., Foltz R., Linden T., Kalogera V. 2013, ApJ, 769, 92 Zampieri L., Roberts T. 2009, MNRAS, 400, 677 Ramsey C. J., Williams R. M. Gruendl R. A., Chen C.-H. R., Zezas A., Fabbiano G., Rots A. H., Murray S. S. 2002, ApJ, 577, Chu Y.-H., Wang Q. D. 2006, ApJ, 641, 241 710 Rana V. et al. 2014, ApJ, submitted, arXiv:1401.4637 Zuo Z.-Y., Li X.-D. 2010, MNRAS, 405, 2768 Rangelov B., Chandar R., Prestwich A., Whitmore B. C. 2012, ApJ, 758, 99

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text: systems

text: powered

text: by

text: BHs

text: in

text: young

text: SCs

text: Marlowe

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text: Kaaret

text: P.,

text: Lang

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text: Feng

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text: Gris´e

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text: Colpi

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text: Belloni

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text: MNRAS,

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text: Rappaport

text: S.,

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text: MNRAS,

text: van

text: der

text: Hucht

text: K.

text: A.

text: in

text: ‘Wolf-Rayet

text: Stars

text: and

text: Interrelations

text: Porras

text: A.,

text: Christopher

text: M.,

text: Allen

text: L.,

text: Di

text: Francesco

text: J.,

text: Megeath

text: S.

text: with

text: Other

text: Massive

text: Stars

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text: Proceedings

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text: Th.,

text: Myers

text: Ph.

text: C.

text: AJ,

text: Symposium

text: of

text: the

text: International

text: Astronomical

text: Union.

text: Portegies

text: Zwart

text: S.

text: F.,

text: Verbunt

text: F.

text: A&A,

text: Edited

text: by

text: Karel

text: A.

text: van

text: der

text: Hucht

text: and

text: Bambang

text: Hidayat.

text: Portegies

text: Zwart

text: S.

text: F.,

text: Verbunt

text: F.,

text: Ergma

text: E.

text: A&A,

text: International

text: Astronomical

text: Union.

text: Symposium

text: no.

text: Kluwer

text: Portegies

text: Zwart

text: S.

text: F.,

text: McMillan

text: S.

text: L.

text: W.,

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text: P.,

text: Makino

text: J.

text: Academic

text: Publishers,

text: Dordrecht,

text: MNRAS,

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text: S.,

text: de

text: Koter

text: A.,

text: Lamers

text: H.

text: J.

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text: M.

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text: Koter

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text: J.,

text: Fabrika

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text: F.,

text: Sholukhova

text: O.,

text: Greiner

text: J.

text: MNRAS,

text: Zampieri

text: L.,

text: Mucciarelli

text: P.,

text: Falomo

text: R.,

text: Kaaret

text: P.,

text: Di

text: Stefano

text: R.,

text: Prestwich

text: A.

text: H.,

text: Tsantaki

text: M.,

text: Zezas

text: A.,

text: Jackson

text: F.,

text: Roberts

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text: Turolla

text: R.,

text: Chieregato

text: M.,

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text: A.

text: ApJ,

text: P.,

text: Foltz

text: R.,

text: Linden

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text: Williams

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text: We want the solution to be regular at the origin and thus require F = 0. The leading − − anharmonic correction to this potential is (1/2)m2 (φ φ )3/φ , so the harmonic approx- − (cid:28) A A A imation is valid as long as φ φ φ . Thus S is a radius such that φ(S) = φ (1 + (cid:15)), A A A where (cid:15) < 1 is a suitably chosen constant. In the region S < r < R , we can approximate A the eﬀective potential by the density term alone and therefore the ﬁeld will have the form given by equation (A.11). (cid:48) Now we sew all these pieces together, matching φ and φ at r = S to ﬁnd the constants (cid:48) C and D, then matching φ and φ again at r = RA determines the constant α and the radius S. The result is    φ , (cid:18) (cid:16) (cid:17) (cid:19) r < S ,  A  −  φ + 1 MA r3 3S2r+2S3 , S < r < R , φ = A 8πRA M rRA2 A (A.15) − − 3 − φbg 4π1RA MMA 1 RSA RrA e mbgr , RA < r , (cid:114) and − 8π M S = R 1 R φ . (A.16) A A bg 3 M A (cid:28) We have made two approximations here: The ﬁrst is m R 1, the same approxi- bg A (cid:29) mation that we made in the case of the weakly perturbing ball. The second is φ φ , bg A which is well justiﬁed here because we are considering a ball of solid material surrounded by ≤ ≤ a vacuum. The scalar ﬁeld has the strongly perturbed proﬁle provided 0 S R , which A is equivalent to 3 M ≥ A φ . (A.17) bg 8πR M A Khoury and Weltman[5] called this the thin-shell regime because the value of the scalar ﬁeld drops from φ to φ over a thin region near the surface of the ball. bg A We ﬁnd it convenient to write the scalar ﬁeld outside the ball in a universal form for both weakly and strongly perturbing objects: − 1 M R − φ = φ λ A A e mbgr (A.18) bg A 4πR M r A (cid:40) where 1 , ρ R2 < 3Mφ , λA = 1 − S3 ≈ 3Mφbg , ρARA2 > 3Mφbg . (A.19) RA3 ρARA2 A A bg The parameter λ determines how responsive the chameleon ﬁeld is to the object. A The chameleon ﬁeld pulls a point test particle towards the spherical test mass with acceleration 1 a = ∂ φ . (A.20) φ r M – 12 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: want

text: the

text: solution

text: to

text: be

text: regular

text: at

text: the

text: origin

text: and

text: thus

text: require

text: F

text: The

text: leading

text: anharmonic

text: correction

text: to

text: this

text: potential

text: is

text: (φ

text: φ

text: so

text: the

text: harmonic

text: approx-

text: A

text: A

text: A

text: imation

text: is

text: valid

text: as

text: long

text: as

text: φ

text: φ

text: φ

text: Thus

text: S

text: is

text: a

text: radius

text: such

text: that

text: φ(S)

text: φ

text: A

text: A

text: A

text: where

text: is

text: a

text: suitably

text: chosen

text: constant.

text: In

text: the

text: region

text: S

text: r

text: R

text: we

text: can

text: approximate

text: A

text: the

text: eﬀective

text: potential

text: by

text: the

text: density

text: term

text: alone

text: and

text: therefore

text: the

text: ﬁeld

text: will

text: have

text: the

text: form

text: given

text: by

text: equation

text: Now

text: we

text: sew

text: all

text: these

text: pieces

text: together,

text: matching

text: φ

text: and

text: φ

text: at

text: r

text: S

text: to

text: ﬁnd

text: the

text: constants

text: C

text: and

text: D,

text: then

text: matching

text: φ

text: and

text: φ

text: again

text: at

text: r

text: RA

text: determines

text: the

text: constant

text: α

text: and

text: the

text: radius

text: S.

text: The

text: result

text: is

text: φ

text: r

text: S

text: A

text: φ

text: MA

text: S

text: r

text: R

text: φ

text: A

text: M

text: A

text: φbg

text: MMA

text: RSA

text: RrA

text: e

text: mbgr

text: RA

text: r

text: and

text: 8π

text: M

text: S

text: R

text: R

text: φ

text: A

text: A

text: bg

text: M

text: A

text: We

text: have

text: made

text: two

text: approximations

text: here:

text: The

text: ﬁrst

text: is

text: m

text: R

text: the

text: same

text: approxi-

text: bg

text: A

text: mation

text: that

text: we

text: made

text: in

text: the

text: case

text: of

text: the

text: weakly

text: perturbing

text: ball.

text: The

text: second

text: is

text: φ

text: φ

text: bg

text: A

text: which

text: is

text: well

text: justiﬁed

text: here

text: because

text: we

text: are

text: considering

text: a

text: ball

text: of

text: solid

text: material

text: surrounded

text: by

text: a

text: vacuum.

text: The

text: scalar

text: ﬁeld

text: has

text: the

text: strongly

text: perturbed

text: proﬁle

text: provided

text: S

text: R

text: which

text: A

text: is

text: equivalent

text: to

text: M

text: A

text: φ

text: bg

text: M

text: A

text: Khoury

text: and

text: called

text: this

text: the

text: thin-shell

text: regime

text: because

text: the

text: value

text: of

text: the

text: scalar

text: ﬁeld

text: drops

text: from

text: φ

text: to

text: φ

text: over

text: a

text: thin

text: region

text: near

text: the

text: surface

text: of

text: the

text: ball.

text: bg

text: A

text: We

text: ﬁnd

text: it

text: convenient

text: to

text: write

text: the

text: scalar

text: ﬁeld

text: outside

text: the

text: ball

text: in

text: a

text: universal

text: form

text: for

text: both

text: weakly

text: and

text: strongly

text: perturbing

text: objects:

text: M

text: R

text: φ

text: φ

text: λ

text: A

text: A

text: e

text: mbgr

text: bg

text: A

text: M

text: r

text: A

text: where

text: ρ

text: λA

text: A

text: A

text: bg

text: The

text: parameter

text: λ

text: determines

text: how

text: responsive

text: the

text: chameleon

text: ﬁeld

text: is

text: to

text: the

text: object.

text: A

text: The

text: chameleon

text: ﬁeld

text: pulls

text: a

text: point

text: test

text: particle

text: towards

text: the

text: spherical

text: test

text: mass

text: with

text: acceleration

text: a

text: φ

text: φ

text: r

text: M

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≡ For the strongly perturbing solution the ﬁeld reaches φ φ (ρ ) within a radius cyl min cyl S < R . In this region we can approximate the right hand side of equation (A.22) as −cyl ≡ m2 (φ φ ), where m m (ρ ). Then the solution within r < S is cyl cyl cyl min cyl φ(r) = φ + EK (m r) + F I (m r) , (A.27) cyl 0 cyl 0 cyl where E and F are integration constants and we set E = 0 to insure regularity at the origin. Just as in the spherical case this harmonic approximation is valid as long as − (cid:28) φ φ φ . Thus S is a radius such that φ(S) = φ (1 + (cid:15)) where (cid:15) < 1 is a suitably cyl cyl cyl chosen constant. In the region S < r < R , we can approximate the eﬀective potential by cyl the density term alone and therefore the ﬁeld will have the form given by equation (A.24). (cid:48) We sew all of these parts of the solution together by ensuring that φ and φ are continuous at r = S and r = Rcyl. (cid:16) (cid:0) (cid:1)(cid:17) φ , (cid:18) (cid:19) r < S ,  cyl  φ + ρcylr2 1 − S2 + 2S2 ln S , S < r < R , φ = cyl 4M r2 r2 r cyl (A.28) φ − ρcylRc2yl 1 − S2 K (m r) , R < r , bg 2M R2 0 bg cyl cyl and the surfa(cid:32)ce S is det(cid:33)er(cid:18)mined by: (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) 4Mφ − S2 − − m R 2S2 S bg bg cyl = 1 1 2γ 2 ln + ln (A.29) ρ R2 R2 E 2 R2 R cyl cyl cyl cy; cyl Whether the cylinder is weakly or strongly perturbing, as we have just seen, the ﬁeld − ρ R2 outside it takes the form of φ(r) = φ cyl cyl K (m r) and the acceleration of a test bg 2M 0 bg object is −∂rφ(r) −ρcylRc2ylmbg (cid:39) −ρcylRc2yl a = = K (m r) . (A.30) φ M 2M2 1 bg 2M2r (cid:28) Here, the last step makes use of our usual approximation m r 1, and shows that bg the cylinder produces a 1/r force. In this limit we can also ﬁnd simpliﬁed expressions for S and a using Eq. (A.29) φ ρcylRc2yl φbg = , (A.31) 2M ln(m R /2) bg cyl − S2 − 2Mφ bg 1 = . (A.32) R2 ρ R2 ln(m R /2) cy cyl cyl bg cyl B The Chameleon Force Between two Extended Sources In this section we compute the force exerted by ball A on a test object, ball B, of mass M , radius R and density ρ , this discussion was ﬁrst presented in Reference[21]. We B B B – 14 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: the

text: strongly

text: perturbing

text: solution

text: the

text: ﬁeld

text: reaches

text: φ

text: φ

text: (ρ

text: within

text: a

text: radius

text: cyl

text: min

text: cyl

text: S

text: R

text: In

text: this

text: region

text: we

text: can

text: approximate

text: the

text: right

text: hand

text: side

text: of

text: equation

text: as

text: −cyl

text: (φ

text: φ

text: where

text: m

text: m

text: (ρ

text: Then

text: the

text: solution

text: within

text: r

text: S

text: is

text: cyl

text: cyl

text: cyl

text: min

text: cyl

text: φ(r)

label: equation

text: φ

text: EK

text: (m

text: r)

text: F

text: I

text: (m

text: r)

text: cyl

text: cyl

text: cyl

text: where

text: E

text: and

text: F

text: are

text: integration

text: constants

text: and

text: we

text: set

text: E

text: to

text: insure

text: regularity

text: at

text: the

text: origin.

text: Just

text: as

text: in

text: the

text: spherical

text: case

text: this

text: harmonic

text: approximation

text: is

text: valid

text: as

text: long

text: as

text: φ

text: φ

text: φ

text: Thus

text: S

text: is

text: a

text: radius

text: such

text: that

text: φ(S)

text: φ

text: where

text: is

text: a

text: suitably

text: cyl

text: cyl

text: cyl

text: chosen

text: constant.

text: In

text: the

text: region

text: S

text: r

text: R

text: we

text: can

text: approximate

text: the

text: eﬀective

text: potential

text: by

text: cyl

text: the

text: density

text: term

text: alone

text: and

text: therefore

text: the

text: ﬁeld

text: will

text: have

text: the

text: form

text: given

text: by

text: equation

text: We

text: sew

text: all

text: of

text: these

text: parts

text: of

text: the

text: solution

text: together

text: by

text: ensuring

text: that

text: φ

text: and

text: φ

text: are

text: continuous

text: at

text: r

text: S

text: and

text: r

text: Rcyl.

text: φ

text: r

text: S

text: cyl

text: φ

text: ln

text: S

text: S

text: r

text: R

text: φ

text: cyl

text: r

text: cyl

text: φ

text: K

text: (m

text: r)

text: R

text: r

text: bg

text: bg

text: cyl

text: cyl

text: and

text: the

text: S

text: is

text: by:

text: m

text: R

text: S

text: bg

text: bg

text: cyl

text: 2γ

text: ln

text: ln

text: ρ

text: E

text: R

text: cyl

text: cyl

text: cyl

text: cy;

text: cyl

text: Whether

text: the

text: cylinder

text: is

text: weakly

text: or

text: strongly

text: perturbing,

text: as

text: we

text: have

text: just

text: seen,

text: the

text: ﬁeld

text: ρ

text: outside

text: it

text: takes

text: the

text: form

text: of

text: φ(r)

text: φ

text: cyl

text: cyl

text: K

text: (m

text: r)

text: and

text: the

text: acceleration

text: of

text: a

text: test

text: bg

text: bg

text: object

text: is

text: −∂rφ(r)

text: a

text: K

text: (m

text: r)

text: φ

text: M

text: bg

text: Here,

text: the

text: last

text: step

text: makes

text: use

text: of

text: our

text: usual

text: approximation

text: m

text: r

text: and

text: shows

text: that

text: bg

text: the

text: cylinder

text: produces

text: a

text: force.

text: In

text: this

text: limit

text: we

text: can

text: also

text: ﬁnd

text: simpliﬁed

text: expressions

text: for

text: S

text: and

text: a

text: using

text: Eq.

text: φ

text: φbg

text: ln(m

text: R

text: bg

text: cyl

text: bg

text: ρ

text: ln(m

text: R

text: cy

text: cyl

text: cyl

text: bg

text: cyl

text: B

text: The

text: Chameleon

text: Force

text: Between

text: two

text: Extended

text: Sources

text: In

text: this

text: section

text: we

text: compute

text: the

text: force

text: exerted

text: by

text: ball

text: A

text: on

text: a

text: test

text: object,

text: ball

text: B,

text: of

text: mass

text: M

text: radius

text: R

text: and

text: density

text: ρ

text: this

text: discussion

text: was

text: ﬁrst

text: presented

text: in

text: We

text: B

text: B

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: (cid:18) (cid:19) − ∂ φ i A F = M ∂ Φ + λ . (B.26) i B i A B M To evaluate the gradients of Φ and φ it is convenient to switch to coordinates centred A A on spherical ball A, with ball B located at radius r. Then GM A ∂ Φ = , (B.27) r A r2 M A ∂ φ = λ , (B.28) r A A 4πMr2 (cid:32) (cid:33) (cid:18) (cid:19) and the total attractive force between the balls is GM M M 2 A B P F = 1 + 2λ λ , (B.29) r r2 A B M where we have used the deﬁnition of the reduced Planck mass 8πM2 G = 1. When the ∼ P balls are weakly perturbing, λ , λ 1, the chameleon force is larger than the gravitational A B attraction by the potentially very large factor 2(M /M)2. However nearly all tests of gravity P (cid:28) to date involve macroscopic objects for which λ , λ 1. A B C Inside a Vacuum Chamber Experiments to measure the force on a beam or cloud of atoms require an ultra-high vacuum chamber to protect the atoms from collisions with the gas in the atmosphere. The chameleon force depends on λ λ and hence on the background value of φ (see equation A B (A.19)). So we need to determine φ inside the vacuum chamber. Within the wall of the bg chamber, the density ρ is high, and the scalar ﬁeld has a correspondingly low equilibrium wall ∼ value φ = φ (ρ ) given by equation (A.6). In the vacuum, where the density is ρ − wall min wall vac 10 16ρ , the scalar ﬁeld rises towards a much higher equilibrium value φ = φ (ρ ), wall eq min vac but we need to determine whether it can reach this equilibrium in a chamber of limited size L. The ﬁeld adapts to the change of density between the walls and the vacuum over a char- acteristic distance of 1/m (ρ ). Thus we expect the ﬁeld at the centre of the chamber to min vac (cid:38) − adapt to the vacuum value only if the chamber is large enough to satisfy L Λ5/4M3/4ρ 3/4. × − − vac With a chamber of 0.1 m radius (L = 5 1014 GeV 1) and a vacuum of 10 10 Torr of hy- − (cid:18)− × (cid:19)− (cid:18) (cid:19) drogen (ρ = 10 14 kg m 3 = 5 10 35 GeV4) this requires vac Λ 5/4 M 3/4 (cid:46) × − − 5 10 11 . (C.1) 10 3 eV M P Equation (C.1) is only satisﬁed for particularly low values of Λ and M. Over most of the parameter space, the opposite is true and therefore the scalar ﬁeld is smaller than the equilibrium value φ throughout the vacuum region. In this case we can neglect the vac last term in equation (A.5) and approximate the equation of motion over the region of the ∇ vacuum chamber as 2φ = (∂/∂φ)(Λ5/φ). We expect that the value of φ at the centre of – 18 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: φ

text: i

text: A

text: F

text: M

text: Φ

text: λ

text: i

text: B

text: i

text: A

text: B

text: M

text: To

text: evaluate

text: the

text: gradients

text: of

text: Φ

text: and

text: φ

text: it

text: is

text: convenient

text: to

text: switch

text: to

text: coordinates

text: centred

text: A

text: A

text: on

text: spherical

text: ball

text: A,

text: with

text: ball

text: B

text: located

text: at

text: radius

text: r.

text: Then

text: GM

text: A

text: Φ

text: r

text: A

text: M

text: A

text: φ

text: λ

text: r

text: A

text: A

text: and

text: the

text: total

text: attractive

text: force

text: between

text: the

text: balls

text: is

text: GM

text: M

text: M

text: A

text: B

text: P

text: F

text: 2λ

text: λ

text: r

text: A

text: B

text: M

text: where

text: we

text: have

text: used

text: the

text: deﬁnition

text: of

text: the

text: reduced

text: Planck

text: mass

text: G

text: When

text: the

text: P

text: balls

text: are

text: weakly

text: perturbing,

text: λ

text: λ

text: the

text: chameleon

text: force

text: is

text: larger

text: than

text: the

text: gravitational

text: A

text: B

text: attraction

text: by

text: the

text: potentially

text: very

text: large

text: factor

text: However

text: nearly

text: all

text: tests

text: of

text: gravity

text: P

text: to

text: date

text: involve

text: macroscopic

text: objects

text: for

text: which

text: λ

text: λ

text: A

text: B

text: C

text: Inside

text: a

text: Vacuum

text: Chamber

text: Experiments

text: to

text: measure

text: the

text: force

text: on

text: a

text: beam

text: or

text: cloud

text: of

text: atoms

text: require

text: an

text: ultra-high

text: vacuum

text: chamber

text: to

text: protect

text: the

text: atoms

text: from

text: collisions

text: with

text: the

text: gas

text: in

text: the

text: atmosphere.

text: The

text: chameleon

text: force

text: depends

text: on

text: λ

text: λ

text: and

text: hence

text: on

text: the

text: background

text: value

text: of

text: φ

text: (see

text: equation

text: A

text: B

text: So

text: we

text: need

text: to

text: determine

text: φ

text: inside

text: the

text: vacuum

text: chamber.

text: Within

text: the

text: wall

text: of

text: the

text: bg

text: chamber,

text: the

text: density

text: ρ

text: is

text: high,

text: and

text: the

text: scalar

text: ﬁeld

text: has

text: a

text: correspondingly

text: low

text: equilibrium

text: wall

text: value

text: φ

text: φ

text: (ρ

text: given

text: by

text: equation

text: In

text: the

text: vacuum,

text: where

text: the

text: density

text: is

text: ρ

text: wall

text: min

text: wall

text: vac

text: the

text: scalar

text: ﬁeld

text: rises

text: towards

text: a

text: much

text: higher

text: equilibrium

text: value

text: φ

text: φ

text: (ρ

text: wall

text: eq

text: min

text: vac

text: but

text: we

text: need

text: to

text: determine

text: whether

text: it

text: can

text: reach

text: this

text: equilibrium

text: in

text: a

text: chamber

text: of

text: limited

text: size

text: L.

text: The

text: ﬁeld

text: adapts

text: to

text: the

text: change

text: of

text: density

text: between

text: the

text: walls

text: and

text: the

text: vacuum

text: over

text: a

text: char-

text: acteristic

text: distance

text: of

text: (ρ

text: Thus

text: we

text: expect

text: the

text: ﬁeld

text: at

text: the

text: centre

text: of

text: the

text: chamber

text: to

text: min

text: vac

text: adapt

text: to

text: the

text: vacuum

text: value

text: only

text: if

text: the

text: chamber

text: is

text: large

text: enough

text: to

text: satisfy

text: L

text: vac

text: With

text: a

text: chamber

text: of

text: m

text: radius

text: (L

text: GeV

text: and

text: a

text: vacuum

text: of

text: Torr

text: of

text: hy-

text: drogen

text: (ρ

text: kg

text: m

text: this

text: requires

text: vac

text: Λ

text: M

text: eV

text: M

text: P

text: Equation

text: is

text: only

text: satisﬁed

text: for

text: particularly

text: low

text: values

text: of

text: Λ

text: and

text: M.

text: Over

text: most

text: of

text: the

text: parameter

text: space,

text: the

text: opposite

text: is

text: true

text: and

text: therefore

text: the

text: scalar

text: ﬁeld

text: is

text: smaller

text: than

text: the

text: equilibrium

text: value

text: φ

text: throughout

text: the

text: vacuum

text: region.

text: In

text: this

text: case

text: we

text: can

text: neglect

text: the

text: vac

text: last

text: term

text: in

text: equation

text: and

text: approximate

text: the

text: equation

text: of

text: motion

text: over

text: the

text: region

text: of

text: the

text: vacuum

text: chamber

text: as

text: 2φ

text: We

text: expect

text: that

text: the

text: value

text: of

text: φ

text: at

text: the

text: centre

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 a b Log acceleration g c 10 1 (cid:72) (cid:144) (cid:76) d e V (cid:76) e 0 3 (cid:45) 0 1 (cid:144)0 (cid:76) (cid:72) 10 (cid:45)5 g o L (cid:45)1 (cid:45)10 (cid:45)(cid:45)(cid:45)6402246 (cid:45)2 (cid:45)14 (cid:45)12 (cid:45)10 (cid:45)8 (cid:45)6 (cid:45)4 (cid:45)2 0 Log M Mp 10 Figure 4: Contour plot showing accel(cid:72)era(cid:144) tio(cid:76)n of rubidium atoms (normalised to the acceler- ation g of free fall on earth) due to the chameleon force outside a sphere of radius R = 1 cm A and screening factor λ = 3MAφbg . The atom and sphere are placed centrally in a 10 cm- A ρARA2 − − radius vacuum chamber containing 10 10 Torr of hydrogen (as for Fig. 1). The Λ M area above the heavy solid black line will be excluded by a ﬁrst atom interferometer experiment − measuring 10 6g. With modest attention to systematic errors this can move down to the ≥ heavy dashed white line. For Λ 10 meV, atom interferometry could sense chameleon physics up to the Planck mass M . We calculate that measurements on atoms and neutrons P near surfaces already exclude the top-left corner, as indicated by the light-weight lines. (a) deﬂection of Rb atoms by a cylinder [24, 25]. (b) Deﬂection of Na atoms between parallel plates [27]. (c) Energy gradient for Rb atoms near a surface [29]. (d) Frequency shift of har- monically trapped Rb atoms near a plane surface [30]. (e) Energy shift of neutrons bouncing between plane surfaces [9]. All of these contours have a sloping region at high values of M/M , where the atom/neutron is unshielded, and a ﬂat, M-insensitive region where the p shielding factor λ falls below unity. Note the accelerations in the contour plot relate to the B proposed interferometry experiment, not to the accelerations of the atoms and neutrons in experiments (a - e). We have shown that external sources are shielded by the vacuum envelope, but that a force can be produced using a cm-sized source mass inside. We ﬁnd that individual atoms can sense the chameleon ﬁeld without screening it and are consequently very sensitive detectors of the ﬁeld. We use our results to impose new limits on the chameleon parameters, derived from existing force measurements on atoms and neutrons, and we show that most of the open chameleon parameter space is within experimental reach using current methods of atom interferometry. – 7 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: a

text: b

text: Log

text: acceleration

text: g

text: c

text: d

text: e

text: V

text: e

text: g

text: o

text: L

text: Log

text: M

text: Mp

text: Figure

text: Contour

label: caption

text: plot

text: showing

text: of

text: rubidium

text: atoms

text: (normalised

text: to

text: the

text: acceler-

text: ation

text: g

text: of

text: free

text: fall

text: on

text: earth)

text: due

text: to

text: the

text: chameleon

text: force

text: outside

text: a

text: sphere

text: of

text: radius

text: R

text: cm

text: A

text: and

text: screening

text: factor

text: λ

text: The

text: atom

text: and

text: sphere

text: are

text: placed

text: centrally

text: in

text: a

text: cm-

text: A

text: radius

text: vacuum

text: chamber

text: containing

text: Torr

text: of

text: hydrogen

text: (as

text: for

text: Fig.

text: The

text: Λ

text: M

text: area

text: above

text: the

text: heavy

text: solid

text: black

text: line

text: will

text: be

text: excluded

text: by

text: a

text: ﬁrst

text: atom

text: interferometer

text: experiment

text: measuring

text: With

text: modest

text: attention

text: to

text: systematic

text: errors

text: this

text: can

text: move

text: down

text: to

text: the

text: heavy

text: dashed

text: white

text: line.

text: For

text: Λ

text: meV,

text: atom

text: interferometry

text: could

text: sense

text: chameleon

text: physics

text: up

text: to

text: the

text: Planck

text: mass

text: M

text: We

text: calculate

text: that

text: measurements

text: on

text: atoms

text: and

text: neutrons

text: P

text: near

text: surfaces

text: already

text: exclude

text: the

text: top-left

text: corner,

text: as

text: indicated

text: by

text: the

text: light-weight

text: lines.

text: (a)

text: deﬂection

text: of

text: Rb

text: atoms

text: by

text: a

text: cylinder

text: (b)

text: Deﬂection

text: of

text: atoms

text: between

text: parallel

text: plates

text: (c)

text: Energy

text: gradient

text: for

text: Rb

text: atoms

text: near

text: a

text: surface

text: (d)

text: Frequency

text: shift

text: of

text: har-

text: monically

text: trapped

text: Rb

text: atoms

text: near

text: a

text: plane

text: surface

text: (e)

text: Energy

text: shift

text: of

text: neutrons

text: bouncing

text: between

text: plane

text: surfaces

text: All

text: of

text: these

text: contours

text: have

text: a

text: sloping

text: region

text: at

text: high

text: values

text: of

text: M/M

text: where

text: the

text: atom/neutron

text: is

text: unshielded,

text: and

text: a

text: ﬂat,

text: M-insensitive

text: region

text: where

text: the

text: p

text: shielding

text: factor

text: λ

text: falls

text: below

text: unity.

text: Note

text: the

text: accelerations

text: in

text: the

text: contour

text: plot

text: relate

text: to

text: the

text: B

text: proposed

text: interferometry

text: experiment,

text: not

text: to

text: the

text: accelerations

text: of

text: the

text: atoms

text: and

text: neutrons

text: in

text: experiments

text: (a

text: e).

text: We

text: have

text: shown

text: that

text: external

text: sources

text: are

text: shielded

text: by

text: the

text: vacuum

text: envelope,

text: but

text: that

text: a

text: force

text: can

text: be

text: produced

text: using

text: a

text: cm-sized

text: source

text: mass

text: inside.

text: We

text: ﬁnd

text: that

text: individual

text: atoms

text: can

text: sense

text: the

text: chameleon

text: ﬁeld

text: without

text: screening

text: it

text: and

text: are

text: consequently

text: very

text: sensitive

text: detectors

text: of

text: the

text: ﬁeld.

text: We

text: use

text: our

text: results

text: to

text: impose

text: new

text: limits

text: on

text: the

text: chameleon

text: parameters,

text: derived

text: from

text: existing

text: force

text: measurements

text: on

text: atoms

text: and

text: neutrons,

text: and

text: we

text: show

text: that

text: most

text: of

text: the

text: open

text: chameleon

text: parameter

text: space

text: is

text: within

text: experimental

text: reach

text: using

text: current

text: methods

text: of

text: atom

text: interferometry.

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text: Acknowledgements We would like to thank Justin Khoury for valuable discussions during the preparation of this work. CB is supported by a Royal Society University Research Fellowship. EAH is supported by a Royal Society Research Professorship. EJC is supported in part by the UK STFC. References [1] Copeland, E. J., Sami, M., & Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753 (2006). [2] Joyce, A., Jain, B., Khoury, J., & Trodden, M. Beyond the Cosmological Standard Model. arXiv:1407.0059 [astro-ph.CO]. [3] Adelberger,E. G., Gundlach, J. H., Heckel, B. R., Hoedl, S. & Schlamminger, S. Torsion balance experiments: A low-energy frontier of particle physics. Prog. Part. Nucl. Phys. 62, 102 (2009). [4] Khoury, J. & Weltman, A. Chameleon ﬁelds: Awaiting surprises for tests of gravity in space. Phys. Rev. Lett. 93, 171104 (2004). [5] Khoury, J. & Weltman, A. Chameleon cosmology. Phys. Rev. D 69, 044026 (2004). [6] Brax, P., van de Bruck, C., Davis, A. -C., Khoury, J. & Weltman, A. Detecting dark energy in orbit - The Cosmological chameleon. Phys. Rev. D 70, 123518 (2004). [7] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513 (2012) 1 [arXiv:1106.2476 [astro-ph.CO]]. [8] Mota, D. F. & Shaw, D. J. Evading equivalence principle violations, cosmological and other experimental constraints in scalar ﬁeld theories with a strong coupling to matter. Phys. Rev. D 75, 063501 (2007). [9] Jenke, T., et al. Gravity Resonance Spectroscopy Constrains Dark Energy and Dark Matter Scenarios Phys. Rev. Lett. 112, 151105 (2014). [10] P. Brax and G. Pignol, Phys. Rev. Lett. 107 (2011) 111301 [arXiv:1105.3420 [hep-ph]]. [11] P. Brax, G. Pignol and D. Roulier, Phys. Rev. D 88 (2013) 083004 [arXiv:1306.6536 [quant-ph]]. [12] Abbott, T. et al. The Dark Energy Survey http://arxiv.org/pdf/astro-ph/0510346.pdf (2005). [13] Amendola, L. et al. Cosmology and fundamental physics with the Euclid satellite. Living Rev. Rel 16, 6 (2013). [14] Planck Collaboration. Planck 2013 results. XVI. Cosmological parameters. http://arxiv.org/abs/1303.5076. [15] Gannouji, R., Moraes, B., Mota, D. F., Polarski, D., Tsujikawa, S. & Winther, H. A. Chameleon dark energy models with characteristic signatures. Phys. Rev. D 82, 124006 (2010). [16] Upadhye, A. Dark energy ﬁfth forces in torsion pendulum experiments. Phys. Rev. D 86, 102003 (2012). [17] Brax, P. & Burrage, C. Atomic precision tests and light scalar couplings. Phys. Rev. D 83, 035020 (2011). [18] Jain, B., Vikram, V. & Sakstein, J. Astrophysical tests of modiﬁed gravity: Constraints from distance indicators in the nearby universe. Astrophys. J. 779, 39 (2013). [19] Steﬀen, J. H. et al. [GammeV Collaboration], Laboratory constraints on chameleon dark energy and power-law ﬁelds. Phys. Rev. Lett. 105, 261803 (2010). – 8 –

text: Acknowledgements

text: We

text: would

text: like

text: to

text: thank

text: Justin

text: Khoury

text: for

text: valuable

text: discussions

text: during

text: the

text: preparation

text: of

text: this

text: work.

text: CB

text: is

text: supported

text: by

text: a

text: Royal

text: Society

text: University

text: Research

text: Fellowship.

text: EAH

text: is

text: supported

text: by

text: a

text: Royal

text: Society

text: Research

text: Professorship.

text: EJC

text: is

text: supported

text: in

text: part

text: by

text: the

text: UK

text: STFC.

text: References

text: Copeland,

text: E.

text: J.,

text: Sami,

text: M.,

text: Tsujikawa,

text: S.

text: Dynamics

text: of

text: dark

text: energy.

text: Int.

text: J.

text: Mod.

text: Phys.

text: D

text: Joyce,

text: A.,

text: Jain,

text: B.,

text: Khoury,

text: J.,

text: Trodden,

text: M.

text: Beyond

text: the

text: Cosmological

text: Standard

text: Model.

text: [astro-ph.CO].

text: Adelberger,E.

text: G.,

text: Gundlach,

text: J.

text: H.,

text: Heckel,

text: B.

text: R.,

text: Hoedl,

text: S.

text: Schlamminger,

text: S.

text: Torsion

text: balance

text: experiments:

text: A

text: low-energy

text: frontier

text: of

text: particle

text: physics.

text: Prog.

text: Part.

text: Nucl.

text: Phys.

text: Khoury,

text: J.

text: Weltman,

text: A.

text: Chameleon

text: ﬁelds:

text: Awaiting

text: surprises

text: for

text: tests

text: of

text: gravity

text: in

text: space.

text: Phys.

text: Rev.

text: Lett.

text: Khoury,

text: J.

text: Weltman,

text: A.

text: Chameleon

text: cosmology.

text: Phys.

text: Rev.

text: D

text: Brax,

text: P.,

text: van

text: de

text: Bruck,

text: C.,

text: Davis,

text: A.

text: -C.,

text: Khoury,

text: J.

text: Weltman,

text: A.

text: Detecting

text: dark

text: energy

text: in

text: orbit

text: The

text: Cosmological

text: chameleon.

text: Phys.

text: Rev.

text: D

text: T.

text: Clifton,

text: P.

text: G.

text: Ferreira,

text: A.

text: Padilla

text: and

text: C.

text: Skordis,

text: Phys.

text: Rept.

text: [astro-ph.CO]].

text: Mota,

text: D.

text: F.

text: Shaw,

text: D.

text: J.

text: Evading

text: equivalence

text: principle

text: violations,

text: cosmological

text: and

text: other

text: experimental

text: constraints

text: in

text: scalar

text: ﬁeld

text: theories

text: with

text: a

text: strong

text: coupling

text: to

text: matter.

text: Phys.

text: Rev.

text: D

text: Jenke,

text: T.,

text: et

text: al.

text: Gravity

text: Resonance

text: Spectroscopy

text: Constrains

text: Dark

text: Energy

text: and

text: Dark

text: Matter

text: Scenarios

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Brax

text: and

text: G.

text: Pignol,

text: Phys.

text: Rev.

text: Lett.

text: [hep-ph]].

text: P.

text: Brax,

text: G.

text: Pignol

text: and

text: D.

text: Roulier,

text: Phys.

text: Rev.

text: D

text: [quant-ph]].

text: Abbott,

text: T.

text: et

text: al.

text: The

text: Dark

text: Energy

text: Survey

text: Amendola,

text: L.

text: et

text: al.

text: Cosmology

text: and

text: fundamental

text: physics

text: with

text: the

text: Euclid

text: satellite.

text: Living

text: Rev.

text: Rel

text: Planck

text: Collaboration.

text: Planck

text: results.

text: XVI.

text: Cosmological

text: parameters.

text: Gannouji,

text: R.,

text: Moraes,

text: B.,

text: Mota,

text: D.

text: F.,

text: Polarski,

text: D.,

text: Tsujikawa,

text: S.

text: Winther,

text: H.

text: A.

text: Chameleon

text: dark

text: energy

text: models

text: with

text: characteristic

text: signatures.

text: Phys.

text: Rev.

text: D

text: Upadhye,

text: A.

text: Dark

text: energy

text: ﬁfth

text: forces

text: in

text: torsion

text: pendulum

text: experiments.

text: Phys.

text: Rev.

text: D

text: Brax,

text: P.

text: Burrage,

text: C.

text: Atomic

text: precision

text: tests

text: and

text: light

text: scalar

text: couplings.

text: Phys.

text: Rev.

text: D

text: Jain,

text: B.,

text: Vikram,

text: V.

text: Sakstein,

text: J.

text: Astrophysical

text: tests

text: of

text: modiﬁed

text: gravity:

text: Constraints

text: from

text: distance

text: indicators

text: in

text: the

text: nearby

text: universe.

text: Astrophys.

text: J.

text: Steﬀen,

text: J.

text: H.

text: et

text: al.

text: [GammeV

text: Collaboration],

text: Laboratory

text: constraints

text: on

text: chameleon

text: dark

text: energy

text: and

text: power-law

text: ﬁelds.

text: Phys.

text: Rev.

text: Lett.


text: 13 a) b) -0.3 0.8 − A(ω, k) = (1/π) Tr Im GRPA(ω, k). (54) -0.35 0.7 -0.4 0.6 -0.45 Since GRPA(ω, k) in Eq. (53) is a rational function, -0.5 0.5 the spectral function A(ω, k) is a sum of Lorentzians. -0.55 Deep in the Mott phase, the strong coupling spectral 0.4 -0.6 function agrees with that obtained from DMFT, plotted 0.3 -0.65 in Figure 10 a),b) for a ﬁnite cylinder geometry. Note (cid:82) -0.7 0.2 that the resolution of the edge states is dependent on the -π 0 π -π 0 π inverse lifetime δ. In Figure 10 c) we plot the density of ≡ c) states A(ω) d2k A(ω, k) for two values of δ. The gap between quasiparticle (hole) bands disappears when the ∼ inverse lifetime approaches the bandwidth, δ t . In the (cid:28)1 opposite regime, a clear gap is present for δ t . We 1 note ﬁnally that using larger clusters [117, 118] yields a GRPA whose qualitative features are similar to those of Eq. (53). In particular, this approach will not yield an estimate for the quasiparticle lifetime 1/δ. 2. Chern number of particle or hole excitations (cid:28) We assume that δ t , such that the GRPA(ω, k) 1 has well deﬁned quasiparticle and quasihole peaks. We − use Greek indices α = +, to denote the upper and FIG. 10. (Color online) Typical spectral functions deep in lower subbands. We denote the quasiparticle dispersion the Mott domain, for t = 3U/100, t = U/100, µ = U/2: 1 2 relation as ω (k) and the quasihole dispersion relation +,α a) and b) quasihole and quasiparticle branches of the spec- ω− (k). Quasiparticle and quasihole poles arise from tral function obtained from DMFT for a cylinder geometry ,α the equation exhibiting edge modes for δ = U/1000. c) density of states A(ω) (arbitrary units) in a torus geometry for two values of δ: U/1000 (solid black line) and U/100 (thick dashed red line). − λ [ω± (k), k] = 0, (55) Quasiparticle and quasihole bands are centered at U µ and ,α − µ, respectively. where λ denotes any one of the two eigenvalues of − [GRPA(ω, k)] 1 obtained from Eq. (53). We are inter- for each cluster j. In what follows, we assume that ested in band Chern numbers, which arise from the clusters are identical, and therefore we will drop the clus- Ishikawa-Matsuyama formula [119] of the many-body ter index denoting the local Green’s function simply by Hall conductiv(cid:90)ity (cid:2) (cid:3) [GRPA(iω )] . n ab Note that Eq. (52) reduces to Eq. (27) of Subsec. if − d2kdω − − we consider singl−e-site clusters. The spectral function σ = (cid:15)ijTr ∂ G∂ G 1G∂ G 1 . (56) has a pole at nU µ with residue (n + 1) and a pole at xy 8π2 0 i j − − − (n 1)U µ with residue n. If the cluster comprises the unit cell, hybridization from the intracluster kinetic The summation over indices i, j = 0, 1, 2 is implicit term results in pairs of quasiparticle and quasihole poles. and (cid:15)ij is the antisymmetric tensor. Integrations are per- We commit to clusters consisting of a single unit cell. formed over the Brillouin zone and over real frequencies In this case, the RPA approximation to the single particle ω. We ha≡ve denoted partial derivatives as ∂j = ∂/∂kj, Gre(cid:2)en’s function i(cid:3)s [117, 118] where Uk0 ω, and k1,2 denote momentum. Let (ω, k) be the unitary transformation that diago- − nalizes [GRPA(ω, k)] 1, th(cid:88)at is GRPA(iω , k) −1 = [GRPA(iω )]−1 − H(cid:48) (k). (53) n n H − U U† [GRPA] 1 = λ δ . (57) ab aα α αβ βb This is the equivalent of Eq. (30) in the cluster per- αβ turbation theory language. The diﬀerence is that now all H(cid:48) We introduce t(cid:88)he matrix of Berry gauge ﬁelds three of [GRPA(iω , k)] , [GRPA(iω )] and [ (k)] × n ab n ab H ab are 2 2 matrices acting on the sublattice basis. Tracing A U U† over sublattice indices in Eq. (53), we obtain the spectral j = i ∂ , for j = 0, 1, 2. (58) αβ aα j βa function a ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a)

text: b)

text: A(ω,

text: k)

text: Tr

text: Im

text: GRPA(ω,

text: k).

text: Since

text: GRPA(ω,

text: k)

text: in

text: Eq.

text: is

text: a

text: rational

text: function,

text: the

text: spectral

text: function

text: A(ω,

text: k)

text: is

text: a

text: sum

text: of

text: Lorentzians.

text: Deep

text: in

text: the

text: Mott

text: phase,

text: the

text: strong

text: coupling

text: spectral

text: function

text: agrees

text: with

text: that

text: obtained

text: from

text: DMFT,

text: plotted

text: in

text: Figure

text: a),b)

text: for

text: a

text: ﬁnite

text: cylinder

text: geometry.

text: Note

text: that

text: the

text: resolution

text: of

text: the

text: edge

text: states

text: is

text: dependent

text: on

text: the

text: -π

text: π

text: -π

text: π

text: inverse

text: lifetime

text: δ.

text: In

text: Figure

text: c)

text: we

text: plot

text: the

text: density

text: of

text: c)

text: states

text: A(ω)

text: A(ω,

text: k)

text: for

text: two

text: values

text: of

text: δ.

text: The

text: gap

text: between

text: quasiparticle

text: (hole)

text: bands

text: disappears

text: when

text: the

text: inverse

text: lifetime

text: approaches

text: the

text: bandwidth,

text: δ

text: t

text: In

text: the

text: opposite

text: regime,

text: a

text: clear

text: gap

text: is

text: present

text: for

text: δ

text: t

text: We

text: note

text: ﬁnally

text: that

text: using

text: larger

text: clusters

text: yields

text: a

text: GRPA

text: whose

text: qualitative

text: features

text: are

text: similar

text: to

text: those

text: of

text: Eq.

text: In

text: particular,

text: this

text: approach

text: will

text: not

text: yield

text: an

text: estimate

text: for

text: the

text: quasiparticle

text: lifetime

text: Chern

text: number

text: of

text: particle

text: or

text: hole

text: excitations

text: We

text: assume

text: that

text: δ

text: t

text: such

text: that

text: the

text: GRPA(ω,

text: k)

text: has

text: well

text: deﬁned

text: quasiparticle

text: and

text: quasihole

text: peaks.

text: We

text: use

text: Greek

text: indices

text: α

text: to

text: denote

text: the

text: upper

text: and

text: FIG.

text: (Color

text: online)

text: Typical

text: spectral

text: functions

text: deep

text: in

text: lower

text: subbands.

text: We

text: denote

text: the

text: quasiparticle

text: dispersion

text: the

text: Mott

text: domain,

text: for

text: t

text: t

text: µ

text: relation

text: as

text: ω

text: (k)

text: and

text: the

text: quasihole

text: dispersion

text: relation

text: +,α

text: a)

text: and

text: b)

text: quasihole

text: and

text: quasiparticle

text: branches

text: of

text: the

text: spec-

text: ω−

text: (k).

text: Quasiparticle

text: and

text: quasihole

text: poles

text: arise

text: from

text: tral

text: function

text: obtained

text: from

text: DMFT

text: for

text: a

text: cylinder

text: geometry

text: ,α

text: the

text: equation

text: exhibiting

text: edge

text: modes

text: for

text: δ

text: c)

text: density

text: of

text: states

text: A(ω)

text: (arbitrary

text: units)

text: in

text: a

text: torus

text: geometry

text: for

text: two

text: values

text: of

text: δ:

text: (solid

text: black

text: line)

text: and

text: (thick

text: dashed

text: red

text: line).

text: λ

text: [ω±

text: (k),

text: k]

text: Quasiparticle

text: and

text: quasihole

text: bands

text: are

text: centered

text: at

text: U

text: µ

text: and

text: ,α

text: µ,

text: respectively.

text: where

text: λ

text: denotes

text: any

text: one

text: of

text: the

text: two

text: eigenvalues

text: of

text: [GRPA(ω,

text: k)]

text: obtained

text: from

text: Eq.

text: We

text: are

text: inter-

text: for

text: each

text: cluster

text: j.

text: In

text: what

text: follows,

text: we

text: assume

text: that

text: ested

text: in

text: band

text: Chern

text: numbers,

text: which

text: arise

text: from

text: the

text: clusters

text: are

text: identical,

text: and

text: therefore

text: we

text: will

text: drop

text: the

text: clus-

text: Ishikawa-Matsuyama

text: formula

text: of

text: the

text: many-body

text: ter

text: index

text: denoting

text: the

text: local

text: Green’s

text: function

text: simply

text: by

text: Hall

text: [GRPA(iω

text: n

text: ab

text: Note

text: that

text: Eq.

text: reduces

text: to

text: Eq.

text: of

text: Subsec.

text: if

text: we

text: consider

text: singl−e-site

text: clusters.

text: The

text: spectral

text: function

text: σ

text: G∂

text: G

text: G

text: has

text: a

text: pole

text: at

text: nU

text: µ

text: with

text: residue

text: (n

text: and

text: a

text: pole

text: at

text: xy

text: i

text: j

text: (n

text: µ

text: with

text: residue

text: n.

text: If

text: the

text: cluster

text: comprises

text: the

text: unit

text: cell,

text: hybridization

text: from

text: the

text: intracluster

text: kinetic

text: The

text: summation

text: over

text: indices

text: i,

text: j

text: is

text: implicit

text: term

text: results

text: in

text: pairs

text: of

text: quasiparticle

text: and

text: quasihole

text: poles.

text: and

text: is

text: the

text: antisymmetric

text: tensor.

text: Integrations

text: are

text: per-

text: We

text: commit

text: to

text: clusters

text: consisting

text: of

text: a

text: single

text: unit

text: cell.

text: formed

text: over

text: the

text: Brillouin

text: zone

text: and

text: over

text: real

text: frequencies

text: In

text: this

text: case,

text: the

text: RPA

text: approximation

text: to

text: the

text: single

text: particle

text: ω.

text: We

text: ha≡ve

text: denoted

text: partial

text: derivatives

text: as

text: ∂j

text: ∂/∂kj,

text: function

text: where

text: ω,

text: and

text: denote

text: momentum.

text: Let

text: (ω,

text: k)

text: be

text: the

text: unitary

text: transformation

text: that

text: diago-

text: nalizes

text: [GRPA(ω,

text: k)]

text: is

text: GRPA(iω

text: k)

text: −1

text: [GRPA(iω

text: (k).

text: n

text: n

text: H

text: U

text: U†

text: [GRPA]

text: λ

text: δ

text: ab

text: aα

text: α

text: αβ

text: βb

text: This

text: is

text: the

text: equivalent

text: of

text: Eq.

text: in

text: the

text: cluster

text: per-

text: αβ

text: turbation

text: theory

text: language.

text: The

text: diﬀerence

text: is

text: that

text: now

text: all

text: We

text: introduce

text: matrix

text: of

text: Berry

text: gauge

text: ﬁelds

text: three

text: of

text: [GRPA(iω

text: k)]

text: [GRPA(iω

text: and

text: (k)]

text: n

text: ab

text: n

text: ab

text: H

text: ab

text: are

text: matrices

text: acting

text: on

text: the

text: sublattice

text: basis.

text: Tracing

text: A

text: U

text: U†

text: over

text: sublattice

text: indices

text: in

text: Eq.

text: we

text: obtain

text: the

text: spectral

text: j

text: i

text: for

text: j

text: αβ

text: aα

text: j

text: βa

text: function

text: a

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##