text: Extreme Ultraviolet Emission Lines at High Redshift 7 ± (0.8 0.1) in the z = 1.863 galaxy 863 348, the most extreme A1689 A1689 A1689 M0451 A1689 CIII] emitter in our sample. Two other CIII] emitters in our sample show CIV detections with CIV/CIII] ﬂux ratios of 0.4 (860 359) f /f 876 330 863 348 860 359 1.1 881 329 line CIII] and 0.5 (881 329). The rest-frame CIV equivalent widths of these three systems span 3.3-8.1 A˚ . NVλ1240 <∼1.1 <∼0.2 <∼0.6 ... <∼2.9 ± Ultimately we would like to characterise the distribution of NIV]λ1487 0.5 0.2 <∼0.1 <∼0.1 ... <∼0.7 ± ± ± nebular CIV/CIII] ﬂux ratios (an indicator of the hardness of the CIVλ1549 <∼0.6 0.8 0.1 0.4 0.1 ... 0.5 0.3 ± ionising spectrum) in dwarf star forming galaxies. While CIV is He IIλ1640 <∼0.5 <∼0.1 <∼0.2 0.5 0.1 <∼0.5 ± ± only signiﬁcantly detected in 3 of 17 systems, it should be em- OIII]λ1661 <∼0.4 0.2 0.1 <∼0.2 0.2 0.1 <∼0.5 ± ± ± ± phasized that non-detection of CIV in our sample does not neces- OIII]λ1666 0.5 0.2 0.6 0.1 0.3 0.1 0.3 0.1 <∼0.5 sarily indicate a low (<∼0.5) CIV/CIII] ratio. Given that CIV ap- NIII]λ1750 <∼0.3 <∼±0.1 <∼±0.1 <∼±0.1 <∼0.3 pears to typically be at least a factor of two fainter than CIII], we [SiIII]λ1883 <∼0.3 0.2 0.1 0.1 0.1 0.2 0.1 <∼0.5 ± ± can only detect CIV emission in those galaxies with very large [SiIII]λ1892 <∼0.3 <∼0.1 0.1 0.1 0.2 0.1 <∼0.5 equivalent width CIII] emission. In the eight galaxies with rest- frame CIII] equivalent widths in excess of 7.0 A˚ , three systems Table 4. Rest-UV emission ﬂux ratios (relative to the blended CIII]λ1908 have CIV detections with CIV/CIII] ﬂux ratios greater than 0.4 doublet). If line is not detected signiﬁcantly, we list the 2σ upper ﬂux ra- (881 329,860 359,863 348). One of the undetected galaxies has a tio limit. No CIVλ1549 ﬂux measurement is possible for MACS0451-1.1 owing to prominent interstellar absorption feature. strong CIV absorption component (854 362) which precludes iden- tiﬁcation of nebular emission in our low resolution spectra. In the remaining 4, we place 2σ upper limits of 0.6-2.3 on the CIV/CIII] powerful emission line target at z = 3.6 (Fosbury et al. 2003). ± ﬂux ratios and thus cannot rule out the presence of CIV at 40-50% While we do detect NIV] in 876 330 (with a ﬂux 0.5 0.2 that of the ﬂux of CIII]. Thus among the large equivalent width CIII] emit- CIII]), we do not detect it in any of the other extreme line emitters. ters, powerful CIV emission appears to be present with a CIV/CIII] Finally, many of the emission features seen in our low lu- ﬂux ratio greater than 0.4 at least 50% of the time. minosity sample are also often observed in narrow-lined AGNs. Given that the energy required to doubly ionise helium (54.4 The AGN composite spectrum constructed in Hainline et al. (2012) eV) is only slightly greater than the energy required to triply-ionise suggests an average CIV equivalent width of 16.3 A˚ , more than a carbon (47.9 eV), we might expect to detect nebular He IIλ1640 factor of two larger than the most extreme sources in our sample. emission in several of the extreme CIV emitting galaxies in our The narrow-line AGN composite also shows NV (5.6 A˚ ) and He sample. But as is clear from Figure 2, many of the extreme CIII] II (8.1 A˚ ), both of which are not seen in most of the low lumi- and CIV emitting galaxies in our sample do not have detectable nosity galaxies in our sample. An additional point of comparison He II emission. The best limit comes from the lack of strong He II is the CIV/CIII] ratio which probes the hardness of the radiation emission in 863 348, the galaxy with the largest equivalent width ﬁeld. The average CIV/CIII] ﬂux ratio found for narrow-line AGN CIII] and CIV emission. Adopting a 2σ upper limit on the He II in the ‘class A’ composite spectrum presented in Alexandroff et al. ﬂux, we ﬁnd that 863 348 has a He II/CIII] ratio of <∼ 0.1 and a (2013) is 7.5, more than an order of magnitude larger than most of rest-frame equivalent width of <∼ 1.4 A˚ . The only conﬁdent He II the galaxies in our sample. Thus while the low luminosity systems detection is in the CIII] emitting galaxy MACS 0451-1.1. The He studied in this paper may have a harder spectrum than more lumi- ± II / CIII] ﬂux ratio in this system (0.5 0.1) is very similar to that nous galaxies at high redshift, they are considerably less extreme found for the metal poor galaxy BX 418 in Erb et al. (2010). As than typical narrow-lined AGNs. However we do note that there are with BX 418, stellar winds might contribute signiﬁcantly to the He several narrow line AGN in the sample considered in Alexandroff et II line in MACS 0451-1.1. A higher resolution (and larger S/N) al. (2013) with CIV/CIII] ratios close to those spanned by our sam- spectra would be required to disentangle the nebular and stellar ple (see their Figure 8), so AGN contribution cannot be discounted. § components of the line. In 5, we will explore whether the high ionisation emission fea- After CIII], the most commonly detected emission line (other tures seen in Figure 2 can be powered entirely by low metallicity than Lyα) is the OIII]λλ1661, 1666 doublet. We commonly de- stars without requiring radiation from additional energetic sources tect OIII]λ1666 emission with ﬂuxes as large as 50-60% those (i.e. fast radiative shocks, X-ray binaries, AGN). Ultimately more of CIII]. The OIII]λ1661 emission line is generally weaker than data will be required to constrain the possible contribution of AGN. OIII]λ1666, with ﬂuxes typically not greater than 20% those of CIII]. Among the ﬁve systems with the best quality spectra (and 3.4 Lyα emission of CIII] Emitters those listed in Table 4), rest-frame equivalent widths range be- tween <∼ 0.5 A˚ and 1.5 A˚ for OIII]λ1661 and 1.2 A˚ and 7.4 A˚ The CIII] emitters in our sample are almost always accompanied for OIII]λ1666. The OIII]λ1666 equivalent widths in our sample by powerful Lyα emission (Figure 4a) with rest-frame equivalent are considerably greater than the equivalent width (0.23 A˚ ) mea- widths ranging between 50 and 150 A˚ (see Table 2). The link be- (cid:39) sured in luminous star forming galaxies at z 3 (Shapley et al. tween the equivalent width of CIII] and that of Lyα was ﬁrst re- (cid:39) 2003). The most extreme UV line emitters (863 348, 876 330), alised through examination of z 3 LBG spectra in Shapley et show OIII]λ1666 emission with equivalent widths between 3 and al. (2003). By grouping galaxies by their Lyα equivalent width, × 6 greater than that measured in the metal poor star forming sys- Shapley et al. (2003) demonstrated that the CIII] equivalent width tem BX 418 (Erb et al. 2010). increases from 0.41 A˚ to 5.37 A˚ as the Lyα equivalent width in- − We also note the presence of weak emission from NIV] λ1487 creases from 14.92 A˚ (absorption) to 52.63 A˚ (emission). and Si III] λλ1883,1892 in several of our sources. Where detected, The 17 low luminosity galaxies considered in this paper ex- the Si III] doublet typically is seen with a ﬂux of 10-20% that of tend the results of Shapley et al. (2003) to larger Lyα and CIII] CIII]. Recent work has revealed strong NIV] emission in a strong equivalent widths (Figure 4b), conﬁrming the close relationship be- Lyα emitting galaxy at z = 5.6 (Vanzella et al. 2010) and in a tween the two quantities. Our spectra generally reveal Lyα equiv- (cid:13) c 2013 RAS, MNRAS 000, 1–19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Extreme

text: Ultraviolet

text: Emission

text: Lines

text: at

text: High

text: Redshift

text: in

text: the

text: z

text: galaxy

text: the

text: most

text: extreme

label: table

text: CIII]

text: emitter

text: in

text: our

text: sample.

text: Two

text: other

text: CIII]

text: emitters

text: in

text: our

text: sample

text: show

text: CIV

text: detections

text: with

text: CIV/CIII]

text: ﬂux

text: ratios

text: of

text: f

text: /f

text: line

text: CIII]

text: and

text: The

text: rest-frame

text: CIV

text: equivalent

text: widths

text: of

text: these

text: three

text: systems

text: span

text: A˚

text: Ultimately

text: we

text: would

text: like

text: to

text: characterise

text: the

text: distribution

text: of

text: nebular

text: CIV/CIII]

text: ﬂux

text: ratios

text: (an

text: indicator

text: of

text: the

text: hardness

text: of

text: the

text: ionising

text: spectrum)

text: in

text: dwarf

text: star

text: forming

text: galaxies.

text: While

text: CIV

text: is

text: He

text: only

text: signiﬁcantly

text: detected

text: in

text: of

text: systems,

text: it

text: should

text: be

text: em-

text: phasized

text: that

text: non-detection

text: of

text: CIV

text: in

text: our

text: sample

text: does

text: not

text: neces-

text: sarily

text: indicate

text: a

text: low

text: CIV/CIII]

text: ratio.

text: Given

text: that

text: CIV

text: ap-

text: pears

text: to

text: typically

text: be

text: at

text: least

text: a

text: factor

text: of

text: two

text: fainter

text: than

text: CIII],

text: we

text: can

text: only

text: detect

text: CIV

text: emission

text: in

text: those

text: galaxies

text: with

text: very

text: large

text: equivalent

text: width

text: CIII]

text: emission.

text: In

text: the

text: eight

text: galaxies

text: with

text: rest-

text: frame

text: CIII]

text: equivalent

text: widths

text: in

text: excess

text: of

text: A˚

text: three

text: systems

text: Table

text: Rest-UV

text: emission

text: ﬂux

text: ratios

text: (relative

text: to

text: the

text: blended

text: have

text: CIV

text: detections

text: with

text: CIV/CIII]

text: ﬂux

text: ratios

text: greater

text: than

text: doublet).

text: If

text: line

text: is

text: not

text: detected

text: signiﬁcantly,

text: we

text: list

text: the

text: 2σ

text: upper

text: ﬂux

text: ra-

text: One

text: of

text: the

text: undetected

text: galaxies

text: has

text: a

text: tio

text: limit.

text: No

text: ﬂux

text: measurement

text: is

text: possible

text: for

text: owing

text: to

text: prominent

text: interstellar

text: absorption

text: feature.

text: strong

text: CIV

text: absorption

text: component

text: which

text: precludes

text: iden-

text: tiﬁcation

text: of

text: nebular

text: emission

text: in

text: our

text: low

text: resolution

text: spectra.

text: In

text: the

text: remaining

text: we

text: place

text: 2σ

text: upper

text: limits

text: of

text: on

text: the

text: CIV/CIII]

text: powerful

text: emission

text: line

text: target

text: at

text: z

text: (Fosbury

text: et

text: al.

text: ﬂux

text: ratios

text: and

text: thus

text: cannot

text: rule

text: out

text: the

text: presence

text: of

text: CIV

text: at

text: While

text: we

text: do

text: detect

text: NIV]

text: in

text: (with

text: a

text: ﬂux

text: that

text: of

text: the

text: ﬂux

text: of

text: CIII].

text: Thus

text: among

text: the

text: large

text: equivalent

text: width

text: CIII]

text: emit-

text: CIII]),

text: we

text: do

text: not

text: detect

text: it

text: in

text: any

text: of

text: the

text: other

text: extreme

text: line

text: emitters.

text: ters,

text: powerful

text: CIV

text: emission

text: appears

text: to

text: be

text: present

text: with

text: a

text: CIV/CIII]

text: Finally,

text: many

text: of

text: the

text: emission

text: features

text: seen

text: in

text: our

text: low

text: lu-

text: ﬂux

text: ratio

text: greater

text: than

text: at

text: least

text: of

text: the

text: time.

text: minosity

text: sample

text: are

text: also

text: often

text: observed

text: in

text: narrow-lined

text: AGNs.

text: Given

text: that

text: the

text: energy

text: required

text: to

text: doubly

text: ionise

text: helium

text: The

text: AGN

text: composite

text: spectrum

text: constructed

text: in

text: Hainline

text: et

text: al.

text: eV)

text: is

text: only

text: slightly

text: greater

text: than

text: the

text: energy

text: required

text: to

text: triply-ionise

text: suggests

text: an

text: average

text: CIV

text: equivalent

text: width

text: of

text: A˚

text: more

text: than

text: a

text: carbon

text: eV),

text: we

text: might

text: expect

text: to

text: detect

text: nebular

text: He

text: factor

text: of

text: two

text: larger

text: than

text: the

text: most

text: extreme

text: sources

text: in

text: our

text: sample.

text: emission

text: in

text: several

text: of

text: the

text: extreme

text: CIV

text: emitting

text: galaxies

text: in

text: our

text: The

text: narrow-line

text: AGN

text: composite

text: also

text: shows

text: NV

text: A˚

text: and

text: He

text: sample.

text: But

text: as

text: is

text: clear

text: from

text: Figure

text: many

text: of

text: the

text: extreme

text: CIII]

text: II

text: A˚

text: both

text: of

text: which

text: are

text: not

text: seen

text: in

text: most

text: of

text: the

text: low

text: lumi-

text: and

text: CIV

text: emitting

text: galaxies

text: in

text: our

text: sample

text: do

text: not

text: have

text: detectable

text: nosity

text: galaxies

text: in

text: our

text: sample.

text: An

text: additional

text: point

text: of

text: comparison

text: He

text: II

text: emission.

text: The

text: best

text: limit

text: comes

text: from

text: the

text: lack

text: of

text: strong

text: He

text: II

text: is

text: the

text: CIV/CIII]

text: ratio

text: which

text: probes

text: the

text: hardness

text: of

text: the

text: radiation

text: emission

text: in

text: the

text: galaxy

text: with

text: the

text: largest

text: equivalent

text: width

text: ﬁeld.

text: The

text: average

text: CIV/CIII]

text: ﬂux

text: ratio

text: found

text: for

text: narrow-line

text: AGN

text: CIII]

text: and

text: CIV

text: emission.

text: Adopting

text: a

text: 2σ

text: upper

text: limit

text: on

text: the

text: He

text: II

text: in

text: the

text: ‘class

text: A’

text: composite

text: spectrum

text: presented

text: in

text: Alexandroff

text: et

text: al.

text: ﬂux,

text: we

text: ﬁnd

text: that

text: has

text: a

text: He

text: II/CIII]

text: ratio

text: of

text: and

text: a

text: is

text: more

text: than

text: an

text: order

text: of

text: magnitude

text: larger

text: than

text: most

text: of

text: rest-frame

text: equivalent

text: width

text: of

text: A˚

text: The

text: only

text: conﬁdent

text: He

text: II

text: the

text: galaxies

text: in

text: our

text: sample.

text: Thus

text: while

text: the

text: low

text: luminosity

text: systems

text: detection

text: is

text: in

text: the

text: CIII]

text: emitting

text: galaxy

text: MACS

text: The

text: He

text: studied

text: in

text: this

text: paper

text: may

text: have

text: a

text: harder

text: spectrum

text: than

text: more

text: lumi-

text: II

text: CIII]

text: ﬂux

text: ratio

text: in

text: this

text: system

text: is

text: very

text: similar

text: to

text: that

text: nous

text: galaxies

text: at

text: high

text: redshift,

text: they

text: are

text: considerably

text: less

text: extreme

text: found

text: for

text: the

text: metal

text: poor

text: galaxy

text: BX

text: in

text: Erb

text: et

text: al.

text: As

text: than

text: typical

text: narrow-lined

text: AGNs.

text: However

text: we

text: do

text: note

text: that

text: there

text: are

text: with

text: BX

text: stellar

text: winds

text: might

text: contribute

text: signiﬁcantly

text: to

text: the

text: He

text: several

text: narrow

text: line

text: AGN

text: in

text: the

text: sample

text: considered

text: in

text: Alexandroff

text: et

text: II

text: line

text: in

text: MACS

text: A

text: higher

text: resolution

text: (and

text: larger

text: S/N)

text: al.

text: with

text: CIV/CIII]

text: ratios

text: close

text: to

text: those

text: spanned

text: by

text: our

text: sam-

text: spectra

text: would

text: be

text: required

text: to

text: disentangle

text: the

text: nebular

text: and

text: stellar

text: ple

text: (see

text: their

text: Figure

text: so

text: AGN

text: contribution

text: cannot

text: be

text: discounted.

text: components

text: of

text: the

text: line.

text: In

text: we

text: will

text: explore

text: whether

text: the

text: high

text: ionisation

text: emission

text: fea-

text: After

text: CIII],

text: the

text: most

text: commonly

text: detected

text: emission

text: line

text: (other

text: tures

text: seen

text: in

text: Figure

text: can

text: be

text: powered

text: entirely

text: by

text: low

text: metallicity

text: than

text: Lyα)

text: is

text: the

text: doublet.

text: We

text: commonly

text: de-

text: stars

text: without

text: requiring

text: radiation

text: from

text: additional

text: energetic

text: sources

text: tect

text: emission

text: with

text: ﬂuxes

text: as

text: large

text: as

text: those

text: (i.e.

text: fast

text: radiative

text: shocks,

text: X-ray

text: binaries,

text: AGN).

text: Ultimately

text: more

text: of

text: CIII].

text: The

text: emission

text: line

text: is

text: generally

text: weaker

text: than

text: data

text: will

text: be

text: required

text: to

text: constrain

text: the

text: possible

text: contribution

text: of

text: AGN.

text: with

text: ﬂuxes

text: typically

text: not

text: greater

text: than

text: those

text: of

text: CIII].

text: Among

text: the

text: ﬁve

text: systems

text: with

text: the

text: best

text: quality

text: spectra

text: (and

text: Lyα

text: emission

text: of

text: CIII]

text: Emitters

text: those

text: listed

text: in

text: Table

text: rest-frame

text: equivalent

text: widths

text: range

text: be-

text: tween

text: A˚

text: and

text: A˚

text: for

text: and

text: A˚

text: and

text: A˚

text: The

text: CIII]

text: emitters

text: in

text: our

text: sample

text: are

text: almost

text: always

text: accompanied

text: for

text: The

text: equivalent

text: widths

text: in

text: our

text: sample

text: by

text: powerful

text: Lyα

text: emission

text: (Figure

text: with

text: rest-frame

text: equivalent

text: are

text: considerably

text: greater

text: than

text: the

text: equivalent

text: width

text: A˚

text: mea-

text: widths

text: ranging

text: between

text: and

text: A˚

text: (see

text: Table

text: The

text: link

text: be-

text: sured

text: in

text: luminous

text: star

text: forming

text: galaxies

text: at

text: z

text: (Shapley

text: et

text: al.

text: tween

text: the

text: equivalent

text: width

text: of

text: CIII]

text: and

text: that

text: of

text: Lyα

text: was

text: ﬁrst

text: re-

text: The

text: most

text: extreme

text: UV

text: line

text: emitters

text: alised

text: through

text: examination

text: of

text: z

text: LBG

text: spectra

text: in

text: Shapley

text: et

text: show

text: emission

text: with

text: equivalent

text: widths

text: between

text: and

text: al.

text: By

text: grouping

text: galaxies

text: by

text: their

text: Lyα

text: equivalent

text: width,

text: greater

text: than

text: that

text: measured

text: in

text: the

text: metal

text: poor

text: star

text: forming

text: sys-

text: Shapley

text: et

text: al.

text: demonstrated

text: that

text: the

text: CIII]

text: equivalent

text: width

text: tem

text: BX

text: (Erb

text: et

text: al.

text: increases

text: from

text: A˚

text: to

text: A˚

text: as

text: the

text: Lyα

text: equivalent

text: width

text: in-

text: We

text: also

text: note

text: the

text: presence

text: of

text: weak

text: emission

text: from

text: NIV]

text: creases

text: from

text: A˚

text: (absorption)

text: to

text: A˚

text: (emission).

text: and

text: Si

text: III]

text: in

text: several

text: of

text: our

text: sources.

text: Where

text: detected,

text: The

text: low

text: luminosity

text: galaxies

text: considered

text: in

text: this

text: paper

text: ex-

text: the

text: Si

text: III]

text: doublet

text: typically

text: is

text: seen

text: with

text: a

text: ﬂux

text: of

text: that

text: of

text: tend

text: the

text: results

text: of

text: Shapley

text: et

text: al.

text: to

text: larger

text: Lyα

text: and

text: CIII]

text: CIII].

text: Recent

text: work

text: has

text: revealed

text: strong

text: NIV]

text: emission

text: in

text: a

text: strong

text: equivalent

text: widths

text: (Figure

text: conﬁrming

text: the

text: close

text: relationship

text: be-

text: Lyα

text: emitting

text: galaxy

text: at

text: z

text: (Vanzella

text: et

text: al.

text: and

text: in

text: a

text: tween

text: the

text: two

text: quantities.

text: Our

text: spectra

text: generally

text: reveal

text: Lyα

text: equiv-

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: − p(c) = 365580800000000c18 2065812736000000c17 + 5257074184960000c16 − − 7950536566252800c15 + 7939897360159392c14 5501379135910008c13 − +2703932242407045c12 947252378063088c11 + 234371204926092c10 − − 40028929618536c9 + 4535144373717c8 317453745456c7 − +11493047016c6 83177280c5 + 167184c4. The polynomial p factors into c4 and an irreducible factor of degree 14. The c4 factor corresponds to degenerate drawings with c = 0 and with u and u drawn at the 2 3 same point of the plane; since the expected drawing is of a different type, we can ignore this factor. Let f(c) be the factor of degree 14. We can convert f to a monic polynomial using the same techniques as before, producing the following polynomial: g(x) = x14f(258/x)/167184 − = x14 128360x13 +4575935386x12 − 32609554186008x11 +120191907907039173x10 − 273701889217560990672x9 +413454551042624579937072x8 − 431130685015107552530542464x7 +317510974076480215971285088080x6 − 166668765204034179394613907054336x5 +62060780922813932272692806330099712x4 − 16033136614269762618278694793639526400x3 +2735179704826314422602131722817699840000x2 − 277301626082465808611849917345431552000000x +12660899181603462048518168020372684800000000. The polynomial g can be algorithmically veriﬁed to be irreducible via a computer algebra system. Its discriminant is · · · · · · · 2188 390 526 725 1312 43156 9870049872 1426547617972 · · 200409943514532 19082702490530411267805114306612 · 2068784364712376186850628387585613, 20

label: equation

text: p(c)

text: The

label: paragraph

text: polynomial

text: p

text: factors

text: into

text: and

text: an

text: irreducible

text: factor

text: of

text: degree

text: The

text: factor

text: corresponds

text: to

text: degenerate

text: drawings

text: with

text: c

text: and

text: with

text: u

text: and

text: u

text: drawn

text: at

text: the

text: same

text: point

text: of

text: the

text: plane;

text: since

text: the

text: expected

text: drawing

text: is

text: of

text: a

text: different

text: type,

text: we

text: can

text: ignore

text: this

text: factor.

text: Let

text: f(c)

text: be

text: the

text: factor

text: of

text: degree

text: We

text: can

text: convert

text: f

text: to

text: a

text: monic

text: polynomial

text: using

text: the

text: same

text: techniques

text: as

text: before,

text: producing

text: the

text: following

text: polynomial:

text: g(x)

text: The

text: polynomial

text: g

text: can

text: be

text: algorithmically

text: veriﬁed

text: to

text: be

text: irreducible

text: via

text: a

text: computer

text: algebra

text: system.

text: Its

text: discriminant

text: is


text: – A root computation tree is an algebraic computation tree in which the allowable functions include the ability to ﬁnd complex roots of polynomials whose coefﬁ- cients are integers or previously computed values, and to compute complex con- jugates of previously computed values. For instance, this model can compute any algebraic number. As a measure of complexity in this model, we deﬁne the degree of a root computation tree as the maximum degree of any of its polynomials. A bounded-degree root computation tree has its degree bounded by some constant unrelated to the size of its input. Thus, a quadratic computation tree is exactly a bounded-degree root computation tree (of degree two). Our impossibility results and degree lower bounds for these models imply the same results for algorithms in more realistic models of computation that use as a black box the corresponding primitives for constructing and representing algebraic numbers in symbolic computation systems. Because our results are lower bounds, they also apply a fortiori to weaker primitives, such as systems limited to real algebraic numbers, which don’t include complex conjugation. It is important to note that each of the above models can generate algebraic numbers of unbounded degree. For instance, even the quadratic computation tree (compass and straightedge model) can construct regular 2k-gons, whose coordinates are algebraic numbers with degrees that are high powers of two. Thus, to prove lower bounds and impossibility results in these models, it is not sufﬁcient to prove that a problem is described by a high-degree polynomial; additional structure is needed. Algebraic graph theory. In algebraic graph theory, the properties of a graph are examined via the spectra of several matrices associated with the graph. The adjacency × matrix A = adj(G) of a graph G is the n n matrix with A equal to 1 if there is i,j an edge between i and j and 0 otherwise. The degree matrix D = deg(G) of G is the × n n matrix with D = deg(v ). From these two matrices we deﬁne the Laplacian i,i − i − matrix, L = lap(G) = D A, and the transition matrix, T = tran(G) = D 1A. Lemma 1. For a regular graph G, adj(G), lap(G), and tran(G) have the same set of eigenvectors. Lemma 2. For the cycle on n vertices, the eigenvalues of adj(G) are 2 cos(2πk/n), ≤ for 0 k < n. Mo¨bius transformations. We may represent each point p in the plane by a complex number, z, whose real part represents p’s x coordinate and whose imaginary part rep- resents p’s y coordinate. A Mo¨bius transformation is a fractional linear transformation, (cid:55)→ z (az + b)/(cz + d), deﬁned by a 4-tuple (a, b, c, d) of complex numbers, or the complex conjugate of such a transformation. We prove the following in Appendix 1. Lemma 3. Given any two disjoint circles, a Mo¨bius transformation mapping them to two concentric circles can be constructed using a quadratic computation tree. (cid:81) Number theory. The Euler totient function, φ(n), counts the number of integers in − the interval [1, n 1] that are relatively(cid:89)prime to n. It can be calculated from the prime factorization n = pri by the formula i − − φ(n) = pri 1(p 1). i i 3

label: list

text: A

text: root

text: computation

text: tree

text: is

text: an

text: algebraic

text: computation

text: tree

text: in

text: which

text: the

text: allowable

text: functions

text: include

text: the

text: ability

text: to

text: ﬁnd

text: complex

text: roots

text: of

text: polynomials

text: whose

text: coefﬁ-

text: cients

text: are

text: integers

text: or

text: previously

text: computed

text: values,

text: and

text: to

text: compute

text: complex

text: con-

text: jugates

text: of

text: previously

text: computed

text: values.

text: For

text: instance,

text: this

text: model

text: can

text: compute

text: any

text: algebraic

text: number.

text: As

text: a

text: measure

text: of

text: complexity

text: in

text: this

text: model,

text: we

text: deﬁne

text: the

text: degree

text: of

text: a

text: root

text: computation

text: tree

text: as

text: the

text: maximum

text: degree

text: of

text: any

text: of

text: its

text: polynomials.

text: A

text: bounded-degree

text: root

text: computation

text: tree

text: has

text: its

text: degree

text: bounded

text: by

text: some

text: constant

text: unrelated

text: to

text: the

text: size

text: of

text: its

text: input.

text: Thus,

text: a

text: quadratic

text: computation

text: tree

text: is

text: exactly

text: a

text: bounded-degree

text: root

text: computation

text: tree

text: (of

text: degree

text: two).

text: Our

text: impossibility

text: results

text: and

text: degree

text: lower

text: bounds

text: for

text: these

text: models

text: imply

text: the

text: same

text: results

text: for

text: algorithms

text: in

text: more

text: realistic

text: models

text: of

text: computation

text: that

text: use

text: as

text: a

text: black

text: box

text: the

text: corresponding

text: primitives

text: for

text: constructing

text: and

text: representing

text: algebraic

text: numbers

text: in

text: symbolic

text: computation

text: systems.

text: Because

text: our

text: results

text: are

text: lower

text: bounds,

text: they

text: also

text: apply

text: a

text: fortiori

text: to

text: weaker

text: primitives,

text: such

text: as

text: systems

text: limited

text: to

text: real

text: algebraic

text: numbers,

text: which

text: don’t

text: include

text: complex

text: conjugation.

text: It

text: is

text: important

text: to

text: note

text: that

text: each

text: of

text: the

text: above

text: models

text: can

text: generate

text: algebraic

text: numbers

text: of

text: unbounded

text: degree.

text: For

text: instance,

text: even

text: the

text: quadratic

text: computation

text: tree

text: (compass

text: and

text: straightedge

text: model)

text: can

text: construct

text: regular

text: whose

text: coordinates

text: are

text: algebraic

text: numbers

text: with

text: degrees

text: that

text: are

text: high

text: powers

text: of

text: two.

text: Thus,

text: to

text: prove

text: lower

text: bounds

text: and

text: impossibility

text: results

text: in

text: these

text: models,

text: it

text: is

text: not

text: sufﬁcient

text: to

text: prove

text: that

text: a

text: problem

text: is

text: described

text: by

text: a

text: high-degree

text: polynomial;

text: additional

text: structure

text: is

text: needed.

text: Algebraic

text: graph

text: theory.

text: In

text: algebraic

text: graph

text: theory,

text: the

text: properties

text: of

text: a

text: graph

text: are

text: examined

text: via

text: the

text: spectra

text: of

text: several

text: matrices

text: associated

text: with

text: the

text: graph.

text: The

text: adjacency

text: matrix

text: A

text: adj(G)

text: of

text: a

text: graph

text: G

text: is

text: the

text: n

text: n

text: matrix

text: with

text: A

text: equal

text: to

text: if

text: there

text: is

text: i,j

text: an

text: edge

text: between

text: i

text: and

text: j

text: and

text: otherwise.

text: The

text: degree

text: matrix

text: D

text: deg(G)

text: of

text: G

text: is

text: the

text: n

text: n

text: matrix

text: with

text: D

text: deg(v

text: From

text: these

text: two

text: matrices

text: we

text: deﬁne

text: the

text: Laplacian

text: i,i

text: i

text: matrix,

text: L

text: lap(G)

text: D

text: A,

text: and

text: the

text: transition

text: matrix,

text: T

text: tran(G)

text: D

text: Lemma

text: For

text: a

text: regular

text: graph

text: G,

text: adj(G),

text: lap(G),

text: and

text: tran(G)

text: have

text: the

text: same

text: set

text: of

text: eigenvectors.

text: Lemma

text: For

text: the

text: cycle

text: on

text: n

text: vertices,

text: the

text: eigenvalues

text: of

text: adj(G)

text: are

text: for

text: k

text: n.

text: Mo¨bius

text: transformations.

text: We

text: may

text: represent

text: each

text: point

text: p

text: in

text: the

text: plane

text: by

text: a

text: complex

text: number,

text: z,

text: whose

text: real

text: part

text: represents

text: p’s

text: x

text: coordinate

text: and

text: whose

text: imaginary

text: part

text: rep-

text: resents

text: p’s

text: y

text: coordinate.

text: A

text: Mo¨bius

text: transformation

text: is

text: a

text: fractional

text: linear

text: transformation,

text: z

text: (az

text: b)/(cz

text: d),

text: deﬁned

text: by

text: a

text: (a,

text: b,

text: c,

text: d)

text: of

text: complex

text: numbers,

text: or

text: the

text: complex

text: conjugate

text: of

text: such

text: a

text: transformation.

text: We

text: prove

text: the

text: following

text: in

text: Appendix

text: Lemma

text: Given

text: any

text: two

text: disjoint

text: circles,

text: a

text: Mo¨bius

text: transformation

text: mapping

text: them

text: to

text: two

text: concentric

text: circles

text: can

text: be

text: constructed

text: using

text: a

text: quadratic

text: computation

text: tree.

text: Number

text: theory.

text: The

text: Euler

text: totient

text: function,

text: φ(n),

text: counts

text: the

text: number

text: of

text: integers

text: in

text: the

text: interval

text: n

text: that

text: are

text: to

text: n.

text: It

text: can

text: be

text: calculated

text: from

text: the

text: prime

text: factorization

text: n

text: pri

text: by

text: the

text: formula

text: i

text: φ(n)

text: pri

text: i

text: i


text: C Additional Impossibility Results for Spectral Graph Drawing In this appendix, we provide additional impossibility results for spectral graph draw- ing, based on the 12-vertex graph, H, shown in Figure 7. Fig.7. A graph, H, whose adjacency and transition eigenvectors are uncomputable in a radical tree.   The adjacency matrix of H is given by      1    1 1     1 1 1     1 1     1 1     1 1  adj(H) =    1 1     1 1 1    1 1  1 1 1 and its characteristic polynomial can be computed to be q(x) = char(adj(H)) − − − − = (x6 x5 5x4 + 4x3 + 5x2 2x 1) − − − (x6 + x5 5x4 4x3 + 5x2 + 2x 1). Let q and q be the factors of q in the order given above. First, observe that q (x) = −0 1 0 q ( x), which implies that we need only compute the Galois group of q . The poly- 1 0 nomial q is irreducible by Lemma 10, as it produces a prime for 13 integer inputs in 0 the range from 0 to 50. The discriminant of q is 592661 (a prime), and we have the 0 following factorizations of q into irreducible polynomials modulo the primes 13 and 0 7, which do not divide the discriminant: ≡ q (x) (x + 9)(x5 + 3x4 + 7x3 + 6x2 + 3x + 10) (mod 13) 0 ≡ q (x) (x + 2)(x2 + 5x + 5)(x3 + 6x2 + x + 2) (mod 7). 0 By Dedekind’s theorem, the factorization modulo 13 implies the existence of a 5-cycle in Gal(split(q )/Q) and factorization modulo 7 implies the existence of a permutation 0 22 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: C

label: section

text: Additional

text: Impossibility

text: Results

text: for

text: Spectral

text: Graph

text: Drawing

text: In

text: this

text: appendix,

text: we

text: provide

text: additional

text: impossibility

text: results

text: for

text: spectral

text: graph

text: draw-

text: ing,

text: based

text: on

text: the

text: graph,

text: H,

text: shown

text: in

text: Figure

text: A

label: caption

text: graph,

text: H,

text: whose

text: adjacency

text: and

text: transition

text: eigenvectors

text: are

text: uncomputable

text: in

text: a

text: radical

text: tree.

text: The

text: adjacency

text: matrix

text: of

text: H

text: is

text: given

text: by

text: 1

text: adj(H)

text: 1

text: and

text: its

text: characteristic

text: polynomial

text: can

text: be

text: computed

text: to

text: be

text: q(x)

text: char(adj(H))

text: Let

text: q

text: and

text: q

text: be

text: the

text: factors

text: of

text: q

text: in

text: the

text: order

text: given

text: above.

text: First,

text: observe

text: that

text: q

text: (x)

text: −0

text: q

text: x),

text: which

text: implies

text: that

text: we

text: need

text: only

text: compute

text: the

text: Galois

text: group

text: of

text: q

text: The

text: poly-

text: nomial

text: q

text: is

text: irreducible

text: by

text: Lemma

text: as

text: it

text: produces

text: a

text: prime

text: for

text: integer

text: inputs

text: in

text: the

text: range

text: from

text: to

text: The

text: discriminant

text: of

text: q

text: is

text: (a

text: prime),

text: and

text: we

text: have

text: the

text: following

text: factorizations

text: of

text: q

text: into

text: irreducible

text: polynomials

text: modulo

text: the

text: primes

text: and

text: which

text: do

text: not

text: divide

text: the

text: discriminant:

text: q

text: (x)

text: (x

text: (mod

text: q

text: (x)

text: (x

text: x

text: (mod

text: By

text: Dedekind’s

text: theorem,

text: the

text: factorization

text: modulo

text: implies

text: the

text: existence

text: of

text: a

text: in

text: Gal(split(q

text: )/Q)

text: and

text: factorization

text: modulo

text: implies

text: the

text: existence

text: of

text: a

text: permutation

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Fig.10. Concentric circle packings Pack(2, n) for n = 5, 7, 9, 11. Gal(split(g)/Q) = S . So by Lemma 7 the value of b cannot be computed by a radical 8 computation tree. Thus, by Lemma 11 a radical computation tree cannot construct any circle packing of the graph in Figure 9, proving the following theorem. Theorem 12. There exists a graph on sixteen vertices such that constructing a circle packing for the graph on a radical computation tree is not possible. Additional circle packings and their groups. The Galois groups in this section were calculated using Sage. We identiﬁed the low-degree groups by using the PARI library, and the high-degree symmetric groups by using a brute force search for primes satis- fying the conditions of Dedekind’s theorem and Lemma 9. To factor the polynomials arising in these computations we used the FLINT library. The circle packings we consider in this section are variants of the bipyramid. The graph Pack(k, n) is constructed from a cycle of length kn+k by replacing every n+1th vertex with a pair of adjacent vertices. Then we add two additional vertices u and v. As in the bipyramid, each of the vertices in the initial cycle is adjacent to both u and v. For each pair that replaced one of the vertices in the cycle we connect one vertex in the pair to u and one to v. This creates a maximal planar graph with kn + 2k + 2 vertices. The graph in Figure 9 is Pack(2, 5), and Figure 10 depicts several additional graphs of the form Pack(2, n). The next conjecture concerns the graphs of the form Pack(2, n). We tested its correctness for all the graphs of this form up to n = 120. Conjecture 1. For the concentric packing of Pack(2, n), the value of b satisﬁes a poly- nomial such that when 30 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Concentric

text: circle

text: packings

text: n)

text: for

text: n

text: Gal(split(g)/Q)

text: S

text: So

text: by

text: Lemma

text: the

text: value

text: of

text: b

text: cannot

text: be

text: computed

text: by

text: a

text: radical

text: computation

text: tree.

text: Thus,

text: by

text: Lemma

text: a

text: radical

text: computation

text: tree

text: cannot

text: construct

text: any

text: circle

text: packing

text: of

text: the

text: graph

text: in

text: Figure

text: proving

text: the

text: following

text: theorem.

text: Theorem

text: There

text: exists

text: a

text: graph

text: on

text: sixteen

text: vertices

text: such

text: that

text: constructing

text: a

text: circle

text: packing

text: for

text: the

text: graph

text: on

text: a

text: radical

text: computation

text: tree

text: is

text: not

text: possible.

text: Additional

text: circle

text: packings

text: and

text: their

text: groups.

text: The

text: Galois

text: groups

text: in

text: this

text: section

text: were

text: calculated

text: using

text: Sage.

text: We

text: identiﬁed

text: the

text: low-degree

text: groups

text: by

text: using

text: the

text: PARI

text: library,

text: and

text: the

text: high-degree

text: symmetric

text: groups

text: by

text: using

text: a

text: brute

text: force

text: search

text: for

text: primes

text: satis-

text: fying

text: the

text: conditions

text: of

text: Dedekind’s

text: theorem

text: and

text: Lemma

text: To

text: factor

text: the

text: polynomials

text: arising

text: in

text: these

text: computations

text: we

text: used

text: the

text: FLINT

text: library.

text: The

text: circle

text: packings

text: we

text: consider

text: in

text: this

text: section

text: are

text: variants

text: of

text: the

text: bipyramid.

text: The

text: graph

text: Pack(k,

text: n)

text: is

text: constructed

text: from

text: a

text: cycle

text: of

text: length

text: kn+k

text: by

text: replacing

text: every

text: vertex

text: with

text: a

text: pair

text: of

text: adjacent

text: vertices.

text: Then

text: we

text: add

text: two

text: additional

text: vertices

text: u

text: and

text: v.

text: As

text: in

text: the

text: bipyramid,

text: each

text: of

text: the

text: vertices

text: in

text: the

text: initial

text: cycle

text: is

text: adjacent

text: to

text: both

text: u

text: and

text: v.

text: For

text: each

text: pair

text: that

text: replaced

text: one

text: of

text: the

text: vertices

text: in

text: the

text: cycle

text: we

text: connect

text: one

text: vertex

text: in

text: the

text: pair

text: to

text: u

text: and

text: one

text: to

text: v.

text: This

text: creates

text: a

text: maximal

text: planar

text: graph

text: with

text: kn

text: vertices.

text: The

text: graph

text: in

text: Figure

text: is

text: and

text: Figure

text: depicts

text: several

text: additional

text: graphs

text: of

text: the

text: form

text: n).

text: The

text: next

text: conjecture

text: concerns

text: the

text: graphs

text: of

text: the

text: form

text: n).

text: We

text: tested

text: its

text: correctness

text: for

text: all

text: the

text: graphs

text: of

text: this

text: form

text: up

text: to

text: n

text: Conjecture

text: For

text: the

text: concentric

text: packing

text: of

text: n),

text: the

text: value

text: of

text: b

text: satisﬁes

text: a

text: poly-

text: nomial

text: such

text: that

text: when

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 2 C.E. Rusu et al. a system, and is also likely to be a three-image lensed quasar system. This result is a product of an adaptive optics (AO) imaging campaign of gravitationally lensed quasars (Rusu et al. 2011, Rusu et al., in prep.) from the SDSS Quasar Lens Search (SQLS; Oguri et al. 2006; Inada et al. 2012). In Section we summarize the previous observations of this object. In Section we present our adaptive op- tics observations, data reduction and morphological model- ing technique. Section addresses the new multi-band color information, and Section presents our gravitational lens mass models. We summarize our conclusions in Section . Throughout this work, the concordance cosmology with − − − (cid:48)(cid:48) × (cid:48)(cid:48) (cid:48) H = 70 km 1 s 1 Mpc 1, Ω = 0.27 and Ω = 0.73 is Figure 1. Left : 5 5 JHK color combined close- 0 m Λ assumed. up of SDSS J1405+0959. The labels correspond to the text. − 2 PREVIOUSLY AVAILABLE OBSERVATIONS Right : I band image of the s(cid:48)y(cid:48) stem from the discovery paper (Inada et al. 2014, UH88/8k, 0. 235 pixel scale, 450 s). North is SDSS J1405+0959 was identiﬁed in the SDSS Data Re- up and East is to the left. lease 7 (Abazajian et al. 2009) as a two-component system, one of which is a quasar at z = 1.810. The system was reported by Jackson et al. (2012) as a new gravitationally dent point-spread function (PSF) estimator. This star was lensed quasar discovered in the Major UKIDSS–SDSS Cos- selected to have a similar tip-tilt star to that of the system, mic Lens Survey (MUSCLES). This was based on the spec- in terms of brightness and separation (Table 2). (cid:48)(cid:48) tral similarity of the two components, as well as the de- The observations were performed with 8 dithering, in crease in separation with increasing wavelength, suggesting order to remove bad pixels and cosmic rays, and to allow the presence of a lensing galaxy. Based on a tentative detec- for the creation of ﬂat frames and sky frames from the data. tion of the 4000 ˚A break, as well as possib∼le emission lines, Data redu†ction was performed with IRAF(cid:63), using the IRCS the galaxy redshift was estimated to be z 0.66. IMGRED (Minowa et al. 2005) package designed to reduce The system was identiﬁed independently as a lensed data obtained with IRCS, and supplemented with a geomet- quasar candidate from the SQLS (see Inada et al. 2012), rical distortion correction. with follow-up spectroscopy and V , R, I band imaging con- Photometric zero-point calibration was performed using ﬁrming the gravitational lens nature (Inada et al. 2014). the observed standard stars in Table 1. All objects were An additional close-by component, also visible in Jackson corrected for Galactic extinction (Schlegel et al. 1998), and et al. (2012), in the infrared, was identiﬁed as a potential atmospheric extinction relative to the standard star. The secondary lens (see Figure 1). standard star catalogue magnitud(cid:48) es at K band were used 3 ADAPTIVE OPTICS OBSERVATIONS to calibrate the photometry at K ‡band, since the ex−pected diﬀerences based on interpolation are small (0.01 0.03 3.1 Description of the observations and data mag). reduction 3.2 PSF and morphological modeling Adaptive optics observations were performed with the As seen from Figure 1, the adaptive optics observations Infrared Camera and Spectrograph (IRCS; Kobayashi et al. reveal dramatically increased detail compared to Inada et al. 2000) at the Subaru Telescope, in conjunction with the Laser (2014). Two new features were identiﬁed. First, there is a red Guide Star Adaptive Optics system (LGS+AO188; Hayano object marked GX just to the East of galaxy G2. Second, et al. 2008, 2010; Minowa et al. 2012). IRCS was used in the there is a fainter highly elongated object, GY, in between (cid:48)(cid:48) (cid:48)(cid:48) × 0. 0528 pixel scale mode, providing a ﬁeld of view of 54 G1 and G2, which seems to be fragmented into three compo- (cid:48)(cid:48) 54 . AO188 employs a curvature wavefront sensor and a 188- nents, the central one being brightest. As the same structure (cid:48) element bimorph deformable mirror. The LGS system uses was observed in both H and K band, this fragmentation is an artiﬁcial sodium laser guide star for high-order wavefront real, and not due to low signal-to-noise (S/N). sensing (Watanabe et al. 2004). In order to characterize the morphology of the individ- (cid:48) Observations were initially performed in the K band, ual components of this system, as well as extract relative followed a year later by J and H band, in order to obtain astrometric and photometric quantities, it is necessary to photometric redshifts for the newly detected objects. A sum- have an accurate knowledge of the PSF of the system. The mary of the exposure times, observation dates, airmass and PSF is diﬃcult to characterize for most adaptive optics ob- (cid:46) (cid:48) photometric standard stars is provided in Table 1. servations, as it varies signiﬁcantly on scales of 1 , or the Even with the use of a laser guide star, a natural bright size of the isoplanatic angle. On these scales, the probabil- star is still required in the proximity of the science target, in order to perform low-order, tip-tilt mode wavefront cor- rections. Such a tip-tilt star was identiﬁed well within the (cid:63) IRAF is distributed by the National Optical Astronomy Ob- servatory, which is operated by the Association of Universities recommended brightness limit of R = 18 mag, and inside the (cid:48)(cid:48) for Research in Astronomy (AURA) under cooperative agreement recommended separation range of 80 (Table 2). In addition, with the National Science Foundation. as a bright star does not exist in the SDSS J1405+0959 ﬁeld † ∼ (cid:48) The package is available at http://www.naoj.org/Observing/ of view, a separate star located 30 away was observed D‡ataReduction/index.html just after the system (only in the H band), as an indepen- http://www2.keck.hawaii.edu/inst/nirc/UKIRTstds.html (cid:13) c 2014 RAS, MNRAS 000, 1–11 ##LTFigure## ##LTFigure##

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text: Adaptive optics observations of the gravitationally lensed quasar SDSS J1405+0959 9 µﬂuxratio ±±0.060.033.20[0.37]0.050.04±±0.100.043.85[0.37]0.100.05 ±±0.010.043.66[0.37]0.050.01 ±±0.210.035.36[0.37]0.460.04±0.020.280.06±±0.110.035.02[0.37]0.290.03±0.080.440.17±±0.140.025.35[0.37]0.140.05±0.020.280.04±±0.100.035.27[0.37]0.100.02±0.040.320.02±±0.00.026.73[0.37]1.690.05±0.020.210.01±±0.000.006.99[0.37]1.700.06±0.160.280.06 referstotheparameters onangleispositivefrom wisespeciﬁed,allmodels ∆tisthetimedelayand e,respectively.Theerror ncebasedonthemethod ∆t[days] ±±2.52.567.8129.43.62.1±±6.61.954.1103.67.71.8 ±±5.40.881.2106.411.83.1 ±±3.15.735.694.13.93.0±11.176.810.0−±±11.97.277.9105.49.59.9±12.381.17.7±±7.32.235.692.76.03.7±5.977.83.6−±±366.057.288.8240.0±0.783.66.8±±4.69.231.088.40.81.2±16.267.50.0±±19.616.437.690.92.30.0±14.263.82.4 thirdandfourthcolumns ),respectively.ThepositiAuncertainties.Unlessother radius,eistheellipticity, helogarithmicdensityslop werecheckedforconverge e,θ[deg]γ,θ[deg]eγ ±0.007/0.141,0.006±0.004/0.064,0.002 ±±±0.038.30.006[0.18,43.4]0.053,0.035.00.027−±2.6[[0.40],3.8]2.6±0.016/0.030,0.009 ±±±0.046.80.005[0.18,43.4]0.017,0.014.50.013±−±0.043.2[0.54,3.8]0.091.9±0.002/0.032,0.014 ±±±0.016.80.000[0.18,43.4]0.018,0.045.00.015−±2.8[[0.20],3.8]3.0±0.004/0.063,0.014 ±±±0.093.10.0090,[43.4]0.063,0.0010.90.024±−±0.344.00.24,[3.8]0.190.6 hmodel,theﬁrst(second)lineonthe eferstoAB(AC)andF/F(F/FBACgaussianpriorgivenbytheobserved hermalmassmodel,θistheEinsteinEerperturbationtothepotential,andt mentedinglaﬁc,wheremultiplechains entisnotconsideredinthemodels. Table7.Lensingmodels ModelImagesσ,θE ±±2.30.021.SIS+γ(G1)2252.3,0.952.00.01±±1.50.012.2SIS+γ2231.9,0.802.30.01±±1.70.0[169.3,0.43]1.11.0±±3.40.013.2SIE+γ2233.6,0.810.80.01±±2.50.01[170.5,0.43]0.60.00±±5.00.034.2SIS+γ+δ3226.0,0.762.60.02±±±±0.0093.110.30.07...(δ,θ)=(0.035,[35.6])181.2,0.49δ0.0243.97.70.04±±5.00.035.2SIE+γ+δ3234.0,0.824.40.05±−±±±0.00914.09.70.06...(δ,θ)=(0.030,[77.9])186.0,0.52δ0.0078.77.00.04±±22.00.016.2SIS+γ+δ;(χ/d.o.f.=0.25/1)3+2225.0,0.763.80.03±±±±0.0051.74.70.02...(δ,θ)=(0.036,36.8)182.1,0.50δ0.0041.51.70.02±±24.60.037.2SIE+γ+δ;(χ/d.o.f.=0.9/1)3+2222.8,0.740.50.00±±±±0.0021.00.90.01...(δ,θ)=(0.045,43.7)183.0,0.50δ0.0072.14.90.03±0.008.pow.law+γ(G1)+SIS(G2)30.830.06±±±(cid:48)0.14215.20.09...γ=1.83;(χ/d.o.f.=0.3/0)174.9,0.460.000.00.00±0.009.pow.law+γ(G1)+SIE(G2)30.810.08±−±±(cid:48)0.15215.90.02...γ=1.82;(χ/d.o.f.=0/2)169.6,0.480.000.10.06 AllexternalshearsγareconsideredatthepositionofgalaxyG1.Foreac ofG1(G2);also,theﬁrst(second)lineonthesixthandeighthcolumnsr NorthtowardsEast.Valuesinsidesquarebracketsareconstrainedwitha2have(χ/ν=0/0).σrepresentsthevelocitydispersionofthesingularisot(cid:48)µrepresentsthetotalmagniﬁcationfactor;δandγrepresentthethirdord barsweredeterminedusingtheMarkovChainMonteCarlomethodimple describedinGelmanetal.(1995).Externalconvergencefromtheenvironm (cid:13) c 2014 RAS, MNRAS 000, 1–11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Adaptive

text: optics

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text: gravitationally

text: lensed

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text: 11 seen in Fig. 5(a). This implies that the resistivity of error-prone comparison between diﬀerent devices. the contact covered graphene must be very large near Subsequent oxidization of the MgO spin injection bar- the contact-induced CNP. Therefore, the combination of rier furthermore leads to the appearance of a contact- oxide barrier resistance and graphene resistance can now induced charge neutrality point in charge transport. We also favor currents over the thicker parts (see large arrows observe that this Dirac peak is strongly linked to gate in Fig. 10(c)) of the oxide than over the thinner part in dependent spin transport as seen by the increase of the the middle of the electrode (see small arrow in Fig. 10(c)). spin lifetime. We gave a possible explanation for this This notion is further supported as the graphene resis- behavior considering both the Co-to-graphene interac- tance under the thicker oxide is now smaller due to the tion and the spatially inhomogeneous spin and charge larger gate-induced charge carrier density as illustrated current paths through the MgO barrier. This expla- in the doping proﬁle in Fig. 10(c). On the other hand, nation highlights one important aspect: Both, the spin it is very unlikely that spins that are injected over these and charge transport must simultaneously be understood thicker parts of the oxide diﬀuse into a graphene region on a microscopic level in order to unveil relevant spin of the thinner oxide barrier mainly because of its small scattering mechanisms from macroscopic Hanle measure- density of states at the CNP. But it will be exactly these ments of the whole device. This was already partially leftover inner pinhole regions in which spin dephasing and addressed in a previous study that theoretically inves- scattering remains strongest. The reduced interaction of tigated the implications of diﬀerent spin transport pa- the injected spins with these spin hot spots may explain rameters of contact-covered and contact-free graphene the strongly suppressed contact-induced spin scattering regions on the extracted value of the spin lifetime by at this gate voltage which allows for nanosecond spin life- Hanle measurements.6 But the real situation is by far − times as shown in Fig. 6(b). more complicated. For example, there will be p n junctions formed between the diﬀerently doped contact- VI. CONCLUSIONS covered and contact-free graphene regions with large lo- cal electric ﬁelds which might also play an important role In summary, we successfully manipulated the graphene- for spin transport. We especially demonstrated that not to-MgO/Co interface in single and bilayer graphene only diﬀerences between the contact covered and bare spin-valve devices by successive oxygen treatments and graphene regions are important to understand the over- achieved a gradual transition from small towards high all charge and spin transport properties of a device but contact resistance area products. With this the spin also subtle variations within the contact regions which transport properties are enhanced signiﬁcantly. We ob- result from inhomogeneous oxide barriers and pinholes. serve an increase in spin lifetimes and spin resistances The research leading to these results has received by a factor of seven and achieve values of up to 1 ns funding from the DFG through FOR-912 and the Euro- at room temperature. Therefore the limiting factor of pean Union Seventh Framework Programme under grant ◦ contact-induced spin dephasing is demonstrated without agreement n 604391 Graphene Flagship. ∗ Applied Physics Express 6, 073001 (2013). † E-mail address: bernd.beschoten@physik.rwth-aachen.de 13 I. Neumann, J. Van de Vondel, G. Bridoux, M. V. Costache, F. Alzina, C. M. S. Torres, and S. O. Valenzuela, Small 9, 156 ‡ (2013). refercolor 14 M. Dro¨geler, F. Volmer, M. Wolter, B. Terr´es, K. Watanabe, T. Taniguchi, G. Gu¨ntherodt, C. Stampfer, and B. Beschoten, 1 N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. vAarnXWiveee-sp,rNinattsur(e2041448),, 5a7rX1 i(v2:010470)6..2439. 2 T.-Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswa1l5, JM. .SHam. mD., GS.uRim. aAra˜lie,sA, P. .PJa.chZooumde,rM, J..ZIennggla,-MAy.nP´eos,piJn.cCiu.c,Brant, N. Tombros, and B. J. van Wees, ArXiv e-prints (2014), G. Gu¨ntherodt, B. Beschoten, and B. O¨zyilmaz, Phys. Rev. Letta.r1X0i7v:,10440762.40665(62.011). 3 W. Han and R. K. Kawakami, Phys. Rev. Lett. 107, 047207 (21601D1)..Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. Lett. 103, 146801 (2009). 4 S. Jo, D.-K. Ki, D. Jeong, H.-J. Lee, and S. Kettemann, Phys1.7RCev. .EBrtl8e4r,, S0.75K4o5n3sc(2h0u1h1,)M. . Gmitra, and J. Fabian, Phys. Rev. B 80, 041405 (2009). 5 W. Han, J.-R. Chen, D. Wang, K. M. McCreary, H. Wen, A.1G8 .PS. wZahratnzg, Ja.nSdhMi,. Wan.dWRu.,KN.eKwaJwoaukranmali,oNf aPnhoysLicestt1e4rs, 01323,015 (2012). 3443 (2012). 19 W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 6 M. H. D. Guimaraes, A. Veligura, P. J. Zomer, T. Maassen, I. J(2.0V1e0r)a.-Marun, N. Tombros, and B. J. van Wees, Nano Letters 12, 3512 (2012). 20 T. Maassen, J. J. van den Berg, N. IJbema, F. Fromm, T. Seyller, R. Yakimova, and B. J. van Wees, Nano Letters 12, 7 P. J. Zomer, M. H. D. Guimara˜es, N. Tombros, and B. J. van W1ee4s9,8P(h2y0s1.2R).ev. B 86, 161416 (2012). 8 B. Dlubak, M.-B. Martin, C. Deranlot, B. Servet, S. Xavier, R.2M1 aFt.taVnoalm, Mer., SMp.riDnkro¨leg,eCle.r,BEer.gMera,yWni.cAke.,DNe. Hvoener,dFen. PDertireosﬀch, , M. L. Boschen, G. Gu¨ntherodt, and B. Beschoten, Phys. Rev. A. Anane, P. Seneor, and A. Fert, Nat Phys 8, 557 (2012). B 88, 161405 (2013). 9 J. Abel, A. Matsubayashi, J. J. Garramone, and V. P. LaBella,2J2ouTr.nMalaoafsVseanc,uIu.mJ.SVcieernac-eM&arTuenc,hMno.lHog.yDB.:GMuiicmroaerla˜eecst,roannicdsB. J. van Wees, Phys. Rev. B 86, 235408 (2012). and Nanometer Structures 30, 03D115 (2012). 23 T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, Phys. Rev. B 79, 245430 (2009). 10 B. Birkner, D. Pachniowski, A. Sandner, M. Ostler, T. Seyller2,4J.RF.aNboiaunc,hiMa.nCdiKor.gTa,anDi.gaWkei,isAs,ppaln. dPhJy.sE. rLoemtts., 9P6h,y2s5. 3R5e0v3. (2010). B 87, 081405 (2013). 25 G. Giovannetti, P. A. Khomyakov, G. Brocks, V. M. Karpan, J. van den Brink, and P. J. Kelly, Phys. Rev. Lett. 101, 11 M. Wojtaszek, I. J. Vera-Marun, T. Maassen, and B. J. van Wee0s2, 6P8h0y3s.(2R0e0v8.).B 87, 081402 (2013). 12 T. Yamaguchi, Y. Inoue, S. Masubuchi, S. Morikawa, M. Onuk2i6, KP.. GW.aMtaantahbeer,, TJ..CT.aRniegaudc,hi,anRd. MR.oAriy. aB,uharnmdaTn., MPhaychs.idRae,v. B 73, 205412 (2006). ##LTLine## ##LTLine## ##LTLine##

text: seen

text: in

text: Fig.

text: This

text: implies

text: that

text: the

text: resistivity

text: of

text: error-prone

text: comparison

text: between

text: diﬀerent

text: devices.

text: the

text: contact

text: covered

text: graphene

text: must

text: be

text: very

text: large

text: near

text: Subsequent

text: oxidization

text: of

text: the

text: MgO

text: spin

text: injection

text: bar-

text: the

text: contact-induced

text: CNP.

text: Therefore,

text: the

text: combination

text: of

text: rier

text: furthermore

text: leads

text: to

text: the

text: appearance

text: of

text: a

text: contact-

text: oxide

text: barrier

text: resistance

text: and

text: graphene

text: resistance

text: can

text: now

text: induced

text: charge

text: neutrality

text: point

text: in

text: charge

text: transport.

text: We

text: also

text: favor

text: currents

text: over

text: the

text: thicker

text: parts

text: (see

text: large

text: arrows

text: observe

text: that

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text: Dirac

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text: strongly

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text: to

text: gate

text: in

text: Fig.

text: of

text: the

text: oxide

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text: over

text: the

text: thinner

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text: spin

text: transport

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text: by

text: the

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text: the

text: the

text: middle

text: of

text: the

text: electrode

text: (see

text: small

text: arrow

text: in

text: Fig.

text: spin

text: lifetime.

text: We

text: gave

text: a

text: possible

text: explanation

text: for

text: this

text: This

text: notion

text: is

text: further

text: supported

text: as

text: the

text: graphene

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text: behavior

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text: both

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text: the

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text: the

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text: the

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text: and

text: charge

text: larger

text: gate-induced

text: charge

text: carrier

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text: as

text: illustrated

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text: through

text: the

text: MgO

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text: This

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text: Fig.

text: On

text: the

text: other

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text: nation

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text: one

text: important

text: aspect:

text: Both,

text: the

text: spin

text: it

text: is

text: very

text: unlikely

text: that

text: spins

text: that

text: are

text: injected

text: over

text: these

text: and

text: charge

text: transport

text: must

text: simultaneously

text: be

text: understood

text: thicker

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text: the

text: oxide

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text: a

text: microscopic

text: level

text: in

text: order

text: to

text: unveil

text: relevant

text: spin

text: of

text: the

text: thinner

text: oxide

text: barrier

text: mainly

text: because

text: of

text: its

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text: scattering

text: mechanisms

text: from

text: macroscopic

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text: density

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text: the

text: CNP.

text: But

text: it

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text: these

text: ments

text: of

text: the

text: whole

text: device.

text: This

text: was

text: already

text: partially

text: leftover

text: inner

text: pinhole

text: regions

text: in

text: which

text: spin

text: dephasing

text: and

text: addressed

text: in

text: a

text: previous

text: study

text: that

text: theoretically

text: inves-

text: scattering

text: remains

text: strongest.

text: The

text: reduced

text: interaction

text: of

text: tigated

text: the

text: implications

text: of

text: diﬀerent

text: spin

text: transport

text: pa-

text: the

text: injected

text: spins

text: with

text: these

text: spin

text: hot

text: spots

text: may

text: explain

text: rameters

text: of

text: contact-covered

text: and

text: contact-free

text: graphene

text: the

text: strongly

text: suppressed

text: contact-induced

text: spin

text: scattering

text: regions

text: on

text: the

text: extracted

text: value

text: of

text: the

text: spin

text: lifetime

text: by

text: at

text: this

text: gate

text: voltage

text: which

text: allows

text: for

text: nanosecond

text: spin

text: life-

text: Hanle

text: But

text: the

text: real

text: situation

text: is

text: by

text: far

text: times

text: as

text: shown

text: in

text: Fig.

text: more

text: complicated.

text: For

text: example,

text: there

text: will

text: be

text: p

text: n

text: junctions

text: formed

text: between

text: the

text: diﬀerently

text: doped

text: contact-

text: VI.

text: CONCLUSIONS

text: covered

text: and

text: contact-free

text: graphene

text: regions

text: with

text: large

text: lo-

text: cal

text: electric

text: ﬁelds

text: which

text: might

text: also

text: play

text: an

text: important

text: role

text: In

text: summary,

text: we

text: successfully

text: manipulated

text: the

text: graphene-

text: for

text: spin

text: transport.

text: We

text: especially

text: demonstrated

text: that

text: not

text: to-MgO/Co

text: interface

text: in

text: single

text: and

text: bilayer

text: graphene

text: only

text: diﬀerences

text: between

text: the

text: contact

text: covered

text: and

text: bare

text: spin-valve

text: devices

text: by

text: successive

text: oxygen

text: treatments

text: and

text: graphene

text: regions

text: are

text: important

text: to

text: understand

text: the

text: over-

text: achieved

text: a

text: gradual

text: transition

text: from

text: small

text: towards

text: high

text: all

text: charge

text: and

text: spin

text: transport

text: properties

text: of

text: a

text: device

text: but

text: contact

text: resistance

text: area

text: products.

text: With

text: this

text: the

text: spin

text: also

text: subtle

text: variations

text: within

text: the

text: contact

text: regions

text: which

text: transport

text: properties

text: are

text: enhanced

text: signiﬁcantly.

text: We

text: ob-

text: result

text: from

text: inhomogeneous

text: oxide

text: barriers

text: and

text: pinholes.

text: serve

text: an

text: increase

text: in

text: spin

text: lifetimes

text: and

text: spin

text: resistances

text: The

text: research

text: leading

text: to

text: these

text: results

text: has

text: received

text: by

text: a

text: factor

text: of

text: seven

text: and

text: achieve

text: values

text: of

text: up

text: to

text: ns

text: funding

text: from

text: the

text: DFG

text: through

text: and

text: the

text: Euro-

text: at

text: room

text: temperature.

text: Therefore

text: the

text: limiting

text: factor

text: of

text: pean

text: Union

text: Seventh

text: Framework

text: Programme

text: under

text: grant

text: contact-induced

text: spin

text: dephasing

text: is

text: demonstrated

text: without

text: agreement

text: n

text: Graphene

text: Flagship.

text: Applied

text: Physics

text: Express

text: E-mail

text: address:

text: bernd.beschoten@physik.rwth-aachen.de

text: I.

text: Neumann,

text: J.

text: Van

text: de

text: Vondel,

text: G.

text: Bridoux,

text: M.

text: V.

text: Costache,

text: F.

text: Alzina,

text: C.

text: M.

text: S.

text: Torres,

text: and

text: S.

text: O.

text: Valenzuela,

text: Small

text: refercolor

text: M.

text: Dro¨geler,

text: F.

text: Volmer,

text: M.

text: Wolter,

text: B.

text: Terr´es,

text: K.

text: Watanabe,

text: T.

text: Taniguchi,

text: G.

text: Gu¨ntherodt,

text: C.

text: Stampfer,

text: and

text: B.

text: Beschoten,

text: N.

text: Tombros,

text: C.

text: Jozsa,

text: M.

text: Popinciuc,

text: H.

text: T.

text: Jonkman,

text: and

text: B.

text: J.

text: T.-Y.

text: Yang,

text: J.

text: Balakrishnan,

text: F.

text: Volmer,

text: A.

text: Avsar,

text: M.

text: JM.

text: .SHam.

text: mD.,

text: GS.uRim.

text: aAra˜lie,sA,

text: P.

text: .PJa.chZooumde,rM,

text: N.

text: Tombros,

text: and

text: B.

text: J.

text: van

text: Wees,

text: ArXiv

text: e-prints

text: G.

text: Gu¨ntherodt,

text: B.

text: Beschoten,

text: and

text: B.

text: O¨zyilmaz,

text: Phys.

text: Rev.

text: W.

text: Han

text: and

text: R.

text: K.

text: Kawakami,

text: Phys.

text: Rev.

text: Lett.

text: F.

text: Guinea,

text: and

text: A.

text: Brataas,

text: Phys.

text: Rev.

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text: Jo,

text: D.-K.

text: Ki,

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text: Jeong,

text: H.-J.

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text: and

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text: Kettemann,

text: Gmitra,

text: and

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text: Fabian,

text: Phys.

text: Rev.

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text: Han,

text: J.-R.

text: Chen,

text: D.

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text: K.

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text: McCreary,

text: H.

text: Wen,

text: .PS.

text: wZahratnzg,

text: Ja.nSdhMi,.

text: Wan.dWRu.,KN.eKwaJwoaukranmali,oNf

text: W.

text: Han,

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text: Maassen,

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text: Yakimova,

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text: TJ..CT.aRniegaudc,hi,anRd.

text: MR.oAriy.

text: aB,uharnmdaTn.,

text: MPhaychs.idRae,v.

text: B

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