text: Vek/snotohP X-ray models for tori in AGNs 3 105 104 103 cosθ =0−0.1 in cosθ =0.9−1.0 102 in MY cosθ =0−0.1 in MY cosθ =0.9−1.0 in 101 100 101 102 100 101 102 E (keV) E (keV) − Figure 2. Spectra under MY09’s assumptions with NH = 1024 (left) and 1025 cm 2 (right). Only scattered continua are shown for clarity. The scattered ﬂuorescent photons are treated as the portion of emission lines and not shown here. the edge-on direction, the scatterings should occur at the rim of the torus; otherwise, they will be absorbed by the near side of the torus. For the BN11 geometry, the torus extends to the centre and the column densities are the same for diﬀerent incli- nation angles. As a result, if the low-energy photons are scattered and can escape to the observer, they can only be scattered very near to the centre; otherwise, they will be absorbed before reaching a large radius. Therefore, the re- ﬂection component is only visible when the inclination is slightly larger than the half-opening angle (a few degrees). The possible trajectory of the scattered photons is indicated Figure 3. Geometry of BN11. The red and blue arrows indicate by the arrows in Figure 3. For the edge-on case, the scattered the possible trajectories of low-energy scattered photons that can escape to face-on and edge-on directions, respectively. The yellow region is obscured by the near side of the torus. If the scat- arrows indicate the trajectories of low-energy scattered photons tered photons intend to escape to the edge-on direction, they that will be absorbed by the near side of the torus for edge-on should be scattered at a large radius and follow the way indi- directions. cated by the blue arrows in Figure 3. However, such photons should be rare, since most of them will be absorbed before reaching the large radius. This is also the reason for the sig- clination angles. Under the geometry of MY09, since the niﬁcant decrease found in the Figure 9 in IK09. We plot the central part of the torus is empty, the whole inner surface of positions of the scatterings in Figure 9, which are indeed the torus is directly illuminated by the central source. As a concentrated in the central part of the torus. The region result, a considerable part of the inner wall is visible when is very small compared with the outer radius (2 pc) of the the inclination angle is slightly larger than the half-opening torus. Therefore, this reﬂection component in the edge-on angle of the torus (the blue arrows in Figure 1); even for the direction should be very weak under the geometry of BN11 edge-on case, the rim of the inner wall is still visible but the and IK09, as shown by our simulations in Figure 4. majority of reﬂection in low-energy band is absorbed by the As mentioned in Section 2.2, the EW(Fe Kα) of BN11 is near side of the torus (the blue and yellow arrows in Figure also diﬀerent from our simulations. For optically thin case, 1). This can explain the trend observed in Figure 6. EW(Fe Kα) can be analytically calculated, e.g. by equation To further support the above explanation, in Figure 8 (5) in MY09, and nearly isotropic. Thus, we take this ana- we also plot the positions of the scatterings of the observed lytical result as the benchmark test of BN11’s model and our photons in 1-2 keV, i.e. the photons have experienced scat- simulations. Under the assumption in Figure 3 of BN11, the ◦ terings and ﬁnally escaped to the observer. The distribu- analytical EW(Fe Kα) is 3.7 eV for Γ = 2.0, θ = 60 , and − tor tions of the positions of the scatterings for two directions N = 1022 cm 2. The EW(Fe Kα) of our simulation under H are shown. The scatterings of low-energy photons can only the same assumption is well consistent with 3.7 eV (at better occur at the skin of the torus; otherwise, they will be ab- than 1% level). However, the EW of BN11’s model is about sorbed in the body of the torus. As the inclination angle 4.4 eV for the same case. Since their continua at 6.4 keV are moves to the edge-on direction, the visible part also moves consistent with our results at 1% level even for the edge-on − to the rim. If the scattered photons intend to escape from case of N = 1024 cm 2, there should be some problems in H (cid:13) c 2002 RAS, MNRAS 000, 1–5 ##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## 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text: Vek/snotohP

text: X-ray

text: models

text: for

text: tori

text: in

text: AGNs

text: cosθ

text: in

text: cosθ

text: in

text: MY

text: cosθ

text: in

text: MY

text: cosθ

text: in

text: E

text: (keV)

text: E

text: (keV)

text: Figure

text: Spectra

text: under

text: assumptions

text: with

text: NH

text: (left)

text: and

text: cm

text: (right).

text: Only

text: scattered

text: continua

text: are

text: shown

text: for

text: clarity.

text: The

text: scattered

text: ﬂuorescent

text: photons

text: are

text: treated

text: as

text: the

text: portion

text: of

text: emission

text: lines

text: and

text: not

text: shown

text: here.

text: the

text: edge-on

text: direction,

text: the

text: scatterings

text: should

text: occur

text: at

text: the

text: rim

text: of

text: the

text: torus;

text: otherwise,

text: they

text: will

text: be

text: absorbed

text: by

text: the

text: near

text: side

text: of

text: the

text: torus.

text: For

text: the

text: geometry,

text: the

text: torus

text: extends

text: to

text: the

text: centre

text: and

text: the

text: column

text: densities

text: are

text: the

text: same

text: for

text: diﬀerent

text: incli-

text: nation

text: angles.

text: As

text: a

text: result,

text: if

text: the

text: low-energy

text: photons

text: are

text: scattered

text: and

text: can

text: escape

text: to

text: the

text: observer,

text: they

text: can

text: only

text: be

text: scattered

text: very

text: near

text: to

text: the

text: centre;

text: otherwise,

text: they

text: will

text: be

text: absorbed

text: before

text: reaching

text: a

text: large

text: radius.

text: Therefore,

text: the

text: re-

text: ﬂection

text: component

text: is

text: only

text: visible

text: when

text: the

text: inclination

text: is

text: slightly

text: larger

text: than

text: the

text: half-opening

text: angle

text: (a

text: few

text: degrees).

text: The

text: possible

text: trajectory

text: of

text: the

text: scattered

text: photons

text: is

text: indicated

text: Figure

text: Geometry

text: of

text: The

text: red

text: and

text: blue

text: arrows

text: indicate

text: by

text: the

text: arrows

text: in

text: Figure

text: For

text: the

text: edge-on

text: case,

text: the

text: scattered

text: the

text: possible

text: trajectories

text: of

text: low-energy

text: scattered

text: photons

text: that

text: can

text: escape

text: to

text: face-on

text: and

text: edge-on

text: directions,

text: respectively.

text: The

text: yellow

text: region

text: is

text: obscured

text: by

text: the

text: near

text: side

text: of

text: the

text: torus.

text: If

text: the

text: scat-

text: arrows

text: indicate

text: the

text: trajectories

text: of

text: low-energy

text: scattered

text: photons

text: tered

text: photons

text: intend

text: to

text: escape

text: to

text: the

text: edge-on

text: direction,

text: they

text: that

text: will

text: be

text: absorbed

text: by

text: the

text: near

text: side

text: of

text: the

text: torus

text: for

text: edge-on

text: should

text: be

text: scattered

text: at

text: a

text: large

text: radius

text: and

text: follow

text: the

text: way

text: indi-

text: directions.

text: cated

text: by

text: the

text: blue

text: arrows

text: in

text: Figure

text: However,

text: such

text: photons

text: should

text: be

text: rare,

text: since

text: most

text: of

text: them

text: will

text: be

text: absorbed

text: before

text: reaching

text: the

text: large

text: radius.

text: This

text: is

text: also

text: the

text: reason

text: for

text: the

text: sig-

text: clination

text: angles.

text: Under

text: the

text: geometry

text: of

text: since

text: the

text: niﬁcant

text: decrease

text: found

text: in

text: the

text: Figure

text: in

text: We

text: plot

text: the

text: central

text: part

text: of

text: the

text: torus

text: is

text: empty,

text: the

text: whole

text: inner

text: surface

text: of

text: positions

text: of

text: the

text: scatterings

text: in

text: Figure

text: which

text: are

text: indeed

text: the

text: torus

text: is

text: directly

text: illuminated

text: by

text: the

text: central

text: source.

text: As

text: a

text: concentrated

text: in

text: the

text: central

text: part

text: of

text: the

text: torus.

text: The

text: region

text: result,

text: a

text: considerable

text: part

text: of

text: the

text: inner

text: wall

text: is

text: visible

text: when

text: is

text: very

text: small

text: compared

text: with

text: the

text: outer

text: radius

text: pc)

text: of

text: the

text: the

text: inclination

text: angle

text: is

text: slightly

text: larger

text: than

text: the

text: half-opening

text: torus.

text: Therefore,

text: this

text: reﬂection

text: component

text: in

text: the

text: edge-on

text: angle

text: of

text: the

text: torus

text: (the

text: blue

text: arrows

text: in

text: Figure

text: even

text: for

text: the

text: direction

text: should

text: be

text: very

text: weak

text: under

text: the

text: geometry

text: of

text: edge-on

text: case,

text: the

text: rim

text: of

text: the

text: inner

text: wall

text: is

text: still

text: visible

text: but

text: the

text: and

text: as

text: shown

text: by

text: our

text: simulations

text: in

text: Figure

text: majority

text: of

text: reﬂection

text: in

text: low-energy

text: band

text: is

text: absorbed

text: by

text: the

text: As

text: mentioned

text: in

text: Section

text: the

text: EW(Fe

text: Kα)

text: of

text: is

text: near

text: side

text: of

text: the

text: torus

text: (the

text: blue

text: and

text: yellow

text: arrows

text: in

text: Figure

text: also

text: diﬀerent

text: from

text: our

text: simulations.

text: For

text: optically

text: thin

text: case,

text: This

text: can

text: explain

text: the

text: trend

text: observed

text: in

text: Figure

text: EW(Fe

text: Kα)

text: can

text: be

text: analytically

text: calculated,

text: e.g.

text: by

text: equation

text: To

text: further

text: support

text: the

text: above

text: explanation,

text: in

text: Figure

text: in

text: and

text: nearly

text: isotropic.

text: Thus,

text: we

text: take

text: this

text: ana-

text: we

text: also

text: plot

text: the

text: positions

text: of

text: the

text: scatterings

text: of

text: the

text: observed

text: lytical

text: result

text: as

text: the

text: benchmark

text: test

text: of

text: model

text: and

text: our

text: photons

text: in

text: keV,

text: i.e.

text: the

text: photons

text: have

text: experienced

text: scat-

text: simulations.

text: Under

text: the

text: assumption

text: in

text: Figure

text: of

text: the

text: terings

text: and

text: ﬁnally

text: escaped

text: to

text: the

text: observer.

text: The

text: distribu-

text: analytical

text: EW(Fe

text: Kα)

text: is

text: eV

text: for

text: Γ

text: θ

text: and

text: tor

text: tions

text: of

text: the

text: positions

text: of

text: the

text: scatterings

text: for

text: two

text: directions

text: N

text: cm

text: The

text: EW(Fe

text: Kα)

text: of

text: our

text: simulation

text: under

text: H

text: are

text: shown.

text: The

text: scatterings

text: of

text: low-energy

text: photons

text: can

text: only

text: the

text: same

text: assumption

text: is

text: well

text: consistent

text: with

text: eV

text: (at

text: better

text: occur

text: at

text: the

text: skin

text: of

text: the

text: torus;

text: otherwise,

text: they

text: will

text: be

text: ab-

text: than

text: level).

text: However,

text: the

text: EW

text: of

text: model

text: is

text: about

text: sorbed

text: in

text: the

text: body

text: of

text: the

text: torus.

text: As

text: the

text: inclination

text: angle

text: eV

text: for

text: the

text: same

text: case.

text: Since

text: their

text: continua

text: at

text: keV

text: are

text: moves

text: to

text: the

text: edge-on

text: direction,

text: the

text: visible

text: part

text: also

text: moves

text: consistent

text: with

text: our

text: results

text: at

text: level

text: even

text: for

text: the

text: edge-on

text: to

text: the

text: rim.

text: If

text: the

text: scattered

text: photons

text: intend

text: to

text: escape

text: from

text: case

text: of

text: N

text: cm

text: there

text: should

text: be

text: some

text: problems

text: in

text: H

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Mon. Not. R. Astron. Soc. 000, 1–17 (2002) Printed May 27, 2020 (MN LATEX style ﬁle v2.2) Constraint on the time variation of the ﬁne-structure constant with the SDSS-III BOSS DR12 quasar sample / † Franco D. Albareti,1(cid:63) Johan Comparat,1 Carlos M. Gutie´rrez,2,3 Francisco Prada,1,4,5 Isabelle Paˆris,6 David Schlegel,7 Mart´ın Lo´pez-Corredoira,2,3 Donald P. Schneider,8,9 Arturo Manchado,2,3,10 D. A. Garc´ıa-Herna´ndez,2,3 Patrick Petitjean11 and Jian Ge12 1 Instituto de F´ısica Teo´rica (UAM/CSIC), Universidad Auto´noma de Madrid, Cantoblanco, E-28049 Madrid, Spain 2 Instituto de Astrof´ısica de Canarias (IAC), La Laguna, E-38205 Tenerife, Spain 3 Departamento de Astrof´ısica, Universidad de La Laguna, La Laguna, E-38206 Tenerife, Spain 4 Campus of International Excellence UAM+CSIC, Cantoblanco, E-28049 Madrid, Spain 5 Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Glorieta de la Astronom´ıa, E-18080 Granada, Spain 6 INAF, Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34131 Trieste, Italy 7 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720, USA 8 Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 9 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 10 CSIC, Spain 11 Institut d’Astrophysique de Paris, CNRS-UPMC, UMR7095, 98bis bd Arago, 75014 Paris, France 12 Department of Astronomy, University of Florida, Gainesville, FL 32611-2055, USA Accepted for publication in MNRAS. ABSTRACT From the Sloan Digital Sky Survey (SDSS) Data Release 12, which covers the full Baryonic Oscillation Spectroscopic Survey (BOSS) footprint, we investigate the possi- ble variation of the ﬁne-structure constant over cosmological time-scales. We analyse the largest quasar sample considered so far in the literature, which contains 13 175 spectra (10 363 from SDSS-III/BOSS DR12 + 2812 from SDSS-II DR7) with redshift z < 1. We apply±the em×issi−on-line method on the [O iii] doublet (λλ 4960, 5008 Å) and obtain ∆α/α = (0.9 1.8) 10 5 for the relative variation of the ﬁne-structure constant. We also investi- gate the possible sources of systematics: misidentiﬁcation of the lines, sky OH lines, H β and broad line contamination, Gaussian and Voigt ﬁtting proﬁles, optimal wavelength range for the Gaussian ﬁts, chosen polynomial order for the continuum spectrum, signal-to-noise ratio and good quality of the ﬁts. The uncertainty of the measurement is dominated by the sky subtrac- tion. The results presented in this work, being systematics limited, have suﬃcient statistics to ≈ constrain robustly the variation of the ﬁne-structure constant in redshift bins (∆z 0.06) over the last 7.9 Gyr. In addition, we study the [Ne iii] doublet (λλ 3869, 3968 Å) present in 462 quasar spectra and discuss the systematic eﬀects on using these emission lines to constrain the − ﬁne-structure constant variation. Better constraints on ∆α/α (< 10 6) using the emission-line method would be possible with high-resolution spectroscopy and large galaxy/qso surveys. Key words: line: proﬁles – quasars: emission lines – cosmology: observations – surveys – large-scale structure of Universe. 1 INTRODUCTION mensionless constants of physical theories. Fundamental constants of physics could be thought of as parameters which enter in our Since Dirac’s philosophical argument (Dirac 1937) against the description of Nature but they cannot be predicted with our cur- ﬁxed value of fundamental constants of Nature, several experi- rent theories and should be measured. Dirac’s idea is based on the ments have been performed to constrain possible variation on di- unlikely fact that the most fundamental constants of the Universe have a certain ﬁxed value (at a given energy) with no apparent re- lation with the real world. It is more likely that their present val- (cid:63) ‘la Caixa’-Severo Ochoa Scholar. † ues are the result of a dynamical process, which had yielded the E-mail: franco.albareti@uam.es (cid:13) c 2002 RAS

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text: de

text: la

text: Astronom´ıa,

text: Granada,

text: Spain

text: INAF,

text: Osservatorio

text: Astronomico

text: di

text: Trieste,

text: Via

text: G.

text: B.

text: Tiepolo

text: Trieste,

text: Italy

text: Lawrence

text: Berkeley

text: National

text: Laboratory,

text: Cyclotron

text: Road,

text: Berkeley,

text: CA,

text: USA

text: Department

text: of

text: Astronomy

text: and

text: Astrophysics,

text: The

text: Pennsylvania

text: State

text: University,

text: University

text: Park,

text: PA

text: USA

text: Institute

text: for

text: Gravitation

text: and

text: the

text: Cosmos,

text: The

text: Pennsylvania

text: State

text: University,

text: University

text: Park,

text: PA

text: USA

text: CSIC,

text: Spain

text: Institut

text: d’Astrophysique

text: de

text: Paris,

text: CNRS-UPMC,

text: bd

text: Arago,

text: Paris,

text: France

text: Department

text: of

text: Astronomy,

text: University

text: of

text: Florida,

text: Gainesville,

text: FL

text: USA

text: Accepted

text: for

text: publication

text: in

text: MNRAS.

text: ABSTRACT

text: From

text: the

text: Sloan

text: Digital

text: Sky

text: Survey

text: (SDSS)

text: Data

text: Release

text: which

text: covers

text: the

text: full

text: Baryonic

text: Oscillation

text: Spectroscopic

text: Survey

text: (BOSS)

text: footprint,

text: we

text: investigate

text: the

text: possi-

text: ble

text: variation

text: of

text: the

text: ﬁne-structure

text: constant

text: over

text: cosmological

text: time-scales.

text: We

text: analyse

text: the

text: largest

text: quasar

text: sample

text: considered

text: so

text: far

text: in

text: the

text: literature,

text: which

text: contains

text: spectra

text: from

text: SDSS-III/BOSS

text: from

text: SDSS-II

text: with

text: redshift

text: z

text: We

text: apply±the

text: em×issi−on-line

text: method

text: on

text: the

text: [O

text: iii]

text: doublet

text: (λλ

text: Å)

text: and

text: obtain

text: ∆α/α

text: for

text: the

text: relative

text: variation

text: of

text: the

text: ﬁne-structure

text: constant.

text: We

text: also

text: investi-

text: gate

text: the

text: possible

text: sources

text: of

text: systematics:

text: misidentiﬁcation

text: of

text: the

text: lines,

text: sky

text: OH

text: lines,

text: H

text: β

text: and

text: broad

text: line

text: contamination,

text: Gaussian

text: and

text: Voigt

text: ﬁtting

text: proﬁles,

text: optimal

text: wavelength

text: range

text: for

text: the

text: Gaussian

text: ﬁts,

text: chosen

text: polynomial

text: order

text: for

text: the

text: continuum

text: spectrum,

text: signal-to-noise

text: ratio

text: and

text: good

text: quality

text: of

text: the

text: ﬁts.

text: The

text: uncertainty

text: of

text: the

text: measurement

text: is

text: dominated

text: by

text: the

text: sky

text: subtrac-

text: tion.

text: The

text: results

text: presented

text: in

text: this

text: work,

text: being

text: systematics

text: limited,

text: have

text: suﬃcient

text: statistics

text: to

text: constrain

text: robustly

text: the

text: variation

text: of

text: the

text: ﬁne-structure

text: constant

text: in

text: redshift

text: bins

text: (∆z

text: over

text: the

text: last

text: Gyr.

text: In

text: addition,

text: we

text: study

text: the

text: [Ne

text: iii]

text: doublet

text: (λλ

text: Å)

text: present

text: in

text: quasar

text: spectra

text: and

text: discuss

text: the

text: systematic

text: eﬀects

text: on

text: using

text: these

text: emission

text: lines

text: to

text: constrain

text: the

text: ﬁne-structure

text: constant

text: variation.

text: Better

text: constraints

text: on

text: ∆α/α

text: using

text: the

text: emission-line

text: method

text: would

text: be

text: possible

text: with

text: high-resolution

text: spectroscopy

text: and

text: large

text: galaxy/qso

text: surveys.

text: Key

text: words:

text: line:

text: proﬁles

text: quasars:

text: emission

text: lines

text: cosmology:

text: observations

text: surveys

text: large-scale

text: structure

text: of

text: Universe.

text: INTRODUCTION

text: mensionless

text: constants

text: of

text: physical

text: theories.

text: Fundamental

text: constants

text: of

text: physics

text: could

text: be

text: thought

text: of

text: as

text: parameters

text: which

text: enter

text: in

text: our

text: Since

text: Dirac’s

text: philosophical

text: argument

text: (Dirac

text: against

text: the

text: description

text: of

text: Nature

text: but

text: they

text: cannot

text: be

text: predicted

text: with

text: our

text: cur-

text: ﬁxed

text: value

text: of

text: fundamental

text: constants

text: of

text: Nature,

text: several

text: experi-

text: rent

text: theories

text: and

text: should

text: be

text: measured.

text: Dirac’s

text: idea

text: is

text: based

text: on

text: the

text: ments

text: have

text: been

text: performed

text: to

text: constrain

text: possible

text: variation

text: on

text: di-

text: unlikely

text: fact

text: that

text: the

text: most

text: fundamental

text: constants

text: of

text: the

text: Universe

text: have

text: a

text: certain

text: ﬁxed

text: value

text: (at

text: a

text: given

text: energy)

text: with

text: no

text: apparent

text: re-

text: lation

text: with

text: the

text: real

text: world.

text: It

text: is

text: more

text: likely

text: that

text: their

text: present

text: val-

text: ‘la

text: Caixa’-Severo

text: Ochoa

text: Scholar.

text: ues

text: are

text: the

text: result

text: of

text: a

text: dynamical

text: process,

text: which

text: had

text: yielded

text: the

text: E-mail:

text: franco.albareti@uam.es

text: c

text: RAS


text: 4 F. D. Albareti et al. Outliers_z-eps-conovuetrltieedr-stkoy.-pedpfs-converted-to.pdf | | | − | Figure 2. Left-hand panel: data points (1416) for which (cid:15) = δλz/(1 + z) δλ0 , namely the absolute value of the diﬀerence between the measured line separation at redshift z in rest frame and the local one, is bigger than 1Å plotted as a function of redshift (and the wavelength observed for [O iii] 4960). We | | compare with a typical sky spectrum: the [O iii] positions for these spectra correlate with the sky emission lines. Hence, these high values of (cid:15) are due to bad sky subtractions and/or low S/N. These spectra are removed. Right-hand panel: a spectrum removed from the sample because of the sky emission-line criteria. ± For this quasar, we get (cid:15) = 1.2 0.6Å. The weak [O iii] line is aﬀected by the subtraction of the 7995 Å OH sky emission line, indicated by the vertical red dashed line. range of the BOSS optical spectrograph and the position of the Fig. 2 (left-hand panel) shows that the distribution of these outliers [O iii] lines. This criterion decreases the sample down to 45802 is correlated with a typical sky spectrum. From a visual inspec- quasars. tion, we observed that these spectra have low S/N, and they are in fact contaminated by sky emission line subtraction (see right- (ii) S/N[O iii] 5008 > 10. We impose a mild constraint on the hand panel of Fig. 2). This eﬀect causes us to discard 1416 spec- signal-to-noise ratio (S/N) of the stronger [O iii] line (5008 Å) in tra (12% of the previous 11779 quasars). Finally, we have 10363 order to preserve a large number of spectra. Constraints on the ex- quasar spectra (our ‘ﬁducial sample’). pected width and amplitudes of the lines help in avoiding misidenti- ﬁcations of the [O iii] doublet (see Section ). This selection reduces the sample from 45802 to 13023 objects. The presence of broad Hβ emission line (4861 Å) near the weak [O iii] line 4960 Å could produce a blueshift in the deter- (iii) Non-converging ﬁts. Since we analyse spectra with low mination of the [O iii] line position. This could mimic a positive S/N, there are some cases where the Gaussian ﬁt to the lines does variation on the ﬁne-structure constant. Therefore, a constraint on not converge. 1244 spectra are discarded, leaving us with 11779 the strength and/or width of the Hβ emission line has been im- spectra. posed on previous investigations (Bahcall et al. 2004; Gutie´rrez & (iv) Sky emission lines. Strong atmospheric lines, for instance Lo´pez-Corredoira 2010; Rahmani et al. 2014). However, we do not the O i 5578 Å line, are poorly or not completely removed by the restrict any characteristic of the Hβ line in our ﬁducial sample. SDSS sky subtraction algorithm. This may lead to a wrong iden- We obtain a weighted mean for ∆α/α using as weights the uncer- tiﬁcation of the [O iii] lines and to include low S/N[O iii] spectra tainty in ∆α/α computed with the standard errors for the position of (Gutie´rrez & Lo´pez-Corredoira 2010). Both eﬀects will produce the lines derived from the Gaussian ﬁts. The contamination of Hβ outliers. We use the SDSS sky mask for Lyα forest studies which is automatically taken into account. For instance, a broad Hβ line contains 872 lines (see Delubac et al. 2015, for more details) to near the [O iii] 4960 line means a bad Gaussian ﬁt. Thus, we obtain remove spectra whose [O iii] lines lie within a particular distance larger errors in the position of the line centroids and, consequently, from the strongest sky lines. Even though we vary the distance in ∆α/α. In Section , we analyse several samples where the S/NHβ [O iii] – sky lines, use diﬀerent set of sky lines (according to their is constrained to check that the Hβ contamination has little weight intensity), or evaluate other conditions (S/N, ﬁt errors, etc.) to re- on the ﬁnal constraint value. − move aﬀected spectra; we usually eliminate 3 5 good spectra for An electronic table is published along with the paper which each bad spectra eliminated. Thus, these tests decrease signiﬁcantly contains all the information of each spectrum from our ﬁducial the number of quasars while not being very eﬀective: typically 50% sample of 10363 quasars (see Appendix A). of the outliers are not removed. Thus, we decided to eliminate all The distribution of the selected quasars in redshift accord- spectra for which the separation between both lines diﬀer by more ing to their selected S/N[O iii] 5008 is plotted in Fig. 1 (right-hand than 1 Å from the local value (see the last paragraph in Section ). panel). Fig. 3 (left-hand panel) displays a composite image built (cid:13) c 2002 RAS, MNRAS 000, 1–17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: D.

text: Albareti

text: et

text: al.

font: CJPSAC+txtt

label: caption

text: Figure

text: Left-hand

text: panel:

text: data

text: points

text: for

text: which

font: NBEANG+rtxmi

font: PXCPBB+rtxr

text: z)

text: δλ0

text: namely

text: the

text: absolute

text: value

text: of

text: the

text: diﬀerence

text: between

text: the

text: measured

text: line

text: separation

text: at

text: redshift

text: z

text: in

text: rest

text: frame

text: and

text: the

text: local

text: one,

text: is

text: bigger

text: than

text: 1Å

text: plotted

text: as

text: a

text: function

text: of

text: redshift

text: (and

text: the

text: wavelength

text: observed

text: for

text: [O

text: iii]

font: GYZEWU+rtxsc

text: We

text: compare

text: with

text: a

text: typical

text: sky

text: spectrum:

text: the

text: [O

text: iii]

text: positions

text: for

text: these

text: spectra

text: correlate

text: with

text: the

text: sky

text: emission

text: lines.

text: Hence,

text: these

text: high

text: values

text: of

text: are

text: due

text: to

text: bad

text: sky

text: subtractions

text: and/or

text: low

text: S/N.

text: These

text: spectra

text: are

text: removed.

text: Right-hand

text: panel:

text: a

text: spectrum

text: removed

text: from

text: the

text: sample

text: because

text: of

text: the

text: sky

text: emission-line

text: criteria.

text: For

text: this

text: quasar,

text: we

text: get

text: The

text: weak

text: [O

text: iii]

text: line

text: is

text: aﬀected

text: by

text: the

text: subtraction

text: of

text: the

text: Å

text: OH

text: sky

text: emission

text: line,

text: indicated

text: by

text: the

text: vertical

text: red

text: dashed

text: line.

text: range

text: of

text: the

text: BOSS

text: optical

text: spectrograph

text: and

text: the

text: position

text: of

text: the

text: Fig.

text: (left-hand

text: panel)

text: shows

text: that

text: the

text: distribution

text: of

text: these

text: outliers

text: [O

text: iii]

text: lines.

text: This

text: criterion

text: decreases

text: the

text: sample

text: down

text: to

text: is

text: correlated

text: with

text: a

text: typical

text: sky

text: spectrum.

text: From

text: a

text: visual

text: inspec-

text: quasars.

text: tion,

text: we

text: observed

text: that

text: these

text: spectra

text: have

text: low

text: S/N,

text: and

text: they

text: are

text: in

text: fact

text: contaminated

text: by

text: sky

text: emission

text: line

text: subtraction

text: (see

text: right-

text: (ii)

text: S/N[O

text: iii]

text: We

text: impose

text: a

text: mild

text: constraint

text: on

text: the

text: hand

text: panel

text: of

text: Fig.

text: This

text: eﬀect

text: causes

text: us

text: to

text: discard

text: spec-

text: signal-to-noise

text: ratio

text: (S/N)

text: of

text: the

text: stronger

text: [O

text: iii]

text: line

text: Å)

text: in

text: tra

text: of

text: the

text: previous

text: quasars).

text: Finally,

text: we

text: have

text: order

text: to

text: preserve

text: a

text: large

text: number

text: of

text: spectra.

text: Constraints

text: on

text: the

text: ex-

text: quasar

text: spectra

text: (our

text: ‘ﬁducial

text: sample’).

text: pected

text: width

text: and

text: amplitudes

text: of

text: the

text: lines

text: help

text: in

text: avoiding

text: misidenti-

text: ﬁcations

text: of

text: the

text: [O

text: iii]

text: doublet

text: (see

text: Section

text: This

text: selection

text: reduces

text: the

text: sample

text: from

text: to

text: objects.

text: The

text: presence

text: of

text: broad

text: Hβ

text: emission

text: line

text: Å)

text: near

text: the

text: weak

text: [O

text: iii]

text: line

text: Å

text: could

text: produce

text: a

text: blueshift

text: in

text: the

text: deter-

text: (iii)

text: Non-converging

text: ﬁts.

text: Since

text: we

text: analyse

text: spectra

text: with

text: low

text: mination

text: of

text: the

text: [O

text: iii]

text: line

text: position.

text: This

text: could

text: mimic

text: a

text: positive

text: S/N,

text: there

text: are

text: some

text: cases

text: where

text: the

text: Gaussian

text: ﬁt

text: to

text: the

text: lines

text: does

text: variation

text: on

text: the

text: ﬁne-structure

text: constant.

text: Therefore,

text: a

text: constraint

text: on

text: not

text: converge.

text: spectra

text: are

text: discarded,

text: leaving

text: us

text: with

text: the

text: strength

text: and/or

text: width

text: of

text: the

text: Hβ

text: emission

text: line

text: has

text: been

text: im-

text: spectra.

text: posed

text: on

text: previous

text: investigations

text: (Bahcall

text: et

text: al.

text: Gutie´rrez

text: (iv)

text: Sky

text: emission

text: lines.

text: Strong

text: atmospheric

text: lines,

text: for

text: instance

text: Lo´pez-Corredoira

text: Rahmani

text: et

text: al.

text: However,

text: we

text: do

text: not

text: the

text: O

text: i

text: Å

text: line,

text: are

text: poorly

text: or

text: not

text: completely

text: removed

text: by

text: the

text: restrict

text: any

text: characteristic

text: of

text: the

text: Hβ

text: line

text: in

text: our

text: ﬁducial

text: sample.

text: SDSS

text: sky

text: subtraction

text: algorithm.

text: This

text: may

text: lead

text: to

text: a

text: wrong

text: iden-

text: We

text: obtain

text: a

text: weighted

text: mean

text: for

text: ∆α/α

text: using

text: as

text: weights

text: the

text: uncer-

text: tiﬁcation

text: of

text: the

text: [O

text: iii]

text: lines

text: and

text: to

text: include

text: low

text: S/N[O

text: iii]

text: spectra

text: tainty

text: in

text: ∆α/α

text: computed

text: with

text: the

text: standard

text: errors

text: for

text: the

text: position

text: of

text: (Gutie´rrez

text: Lo´pez-Corredoira

text: Both

text: eﬀects

text: will

text: produce

text: the

text: lines

text: derived

text: from

text: the

text: Gaussian

text: ﬁts.

text: The

text: contamination

text: of

text: Hβ

text: outliers.

text: We

text: use

text: the

text: SDSS

text: sky

text: mask

text: for

text: Lyα

text: forest

text: studies

text: which

text: is

text: automatically

text: taken

text: into

text: account.

text: For

text: instance,

text: a

text: broad

text: Hβ

text: line

text: contains

text: lines

text: (see

text: Delubac

text: et

text: al.

text: for

text: more

text: details)

text: to

text: near

text: the

text: [O

text: iii]

text: line

text: means

text: a

text: bad

text: Gaussian

text: ﬁt.

text: Thus,

text: we

text: obtain

text: remove

text: spectra

text: whose

text: [O

text: iii]

text: lines

text: lie

text: within

text: a

text: particular

text: distance

text: larger

text: errors

text: in

text: the

text: position

text: of

text: the

text: line

text: centroids

text: and,

text: consequently,

text: from

text: the

text: strongest

text: sky

text: lines.

text: Even

text: though

text: we

text: vary

text: the

text: distance

text: in

text: ∆α/α.

text: In

text: Section

text: we

text: analyse

text: several

text: samples

text: where

text: the

text: S/NHβ

text: [O

text: iii]

text: sky

text: lines,

text: use

text: diﬀerent

text: set

text: of

text: sky

text: lines

text: (according

text: to

text: their

text: is

text: constrained

text: to

text: check

text: that

text: the

text: Hβ

text: contamination

text: has

text: little

text: weight

text: intensity),

text: or

text: evaluate

text: other

text: conditions

text: (S/N,

text: ﬁt

text: errors,

text: etc.)

text: to

text: re-

text: on

text: the

text: ﬁnal

text: constraint

text: value.

text: move

text: aﬀected

text: spectra;

text: we

text: usually

text: eliminate

text: good

text: spectra

text: for

text: An

text: electronic

text: table

text: is

text: published

text: along

text: with

text: the

text: paper

text: which

text: each

text: bad

text: spectra

text: eliminated.

text: Thus,

text: these

text: tests

text: decrease

text: signiﬁcantly

text: contains

text: all

text: the

text: information

text: of

text: each

text: spectrum

text: from

text: our

text: ﬁducial

text: the

text: number

text: of

text: quasars

text: while

text: not

text: being

text: very

text: eﬀective:

text: typically

text: sample

text: of

text: quasars

text: (see

text: Appendix

text: A).

text: of

text: the

text: outliers

text: are

text: not

text: removed.

text: Thus,

text: we

text: decided

text: to

text: eliminate

text: all

text: The

text: distribution

text: of

text: the

text: selected

text: quasars

text: in

text: redshift

text: accord-

text: spectra

text: for

text: which

text: the

text: separation

text: between

text: both

text: lines

text: diﬀer

text: by

text: more

text: ing

text: to

text: their

text: selected

text: S/N[O

text: iii]

text: is

text: plotted

text: in

text: Fig.

text: (right-hand

text: than

text: Å

text: from

text: the

text: local

text: value

text: (see

text: the

text: last

text: paragraph

text: in

text: Section

text: panel).

text: Fig.

text: (left-hand

text: panel)

text: displays

text: a

text: composite

text: image

text: built

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Fine-structure constant with BOSS 7 OIIIlines-eps-convNeerItIeIdl-itnoe.sp-defps-converted-to.pdf Fquigausarrei5s.∆[αO/iαiiG[]Oa(ulsiesifi]t-h=an(d2.p3a±ne7l).6a)n×d 1[N0−e4i,ii∆]α(r/iαghV[Oot-ighitiain]d=pa(n3e.3l)±lin1e2s.6f)o×r S1D0−S4Sa-Jn1d2∆14α1/7α.G[8N0au+ess2ii9i]31=43(.349, a±t 8re)d×sh1i0ft−4z,=∆α0/.0α6V[3No.igeTtihii]e m=ea(3su7r±ed9∆) ×α/1α0−fo4.r Tthhies measured ∆α/α for [Ne iii] is not consistent with zero regardless of the proﬁle; see Fig. 14 and last paragraph of Section for discussion. Each panel shows the ﬂux density for each pixel with their respective error bars (solid symbols), together with the Gaussian ﬁt (dotted red curve) and the pseudo-Voigt proﬁle (thick grey curve) to each of the lines. The ﬁtting procedure (described in the text) only takes into account the experimental data (solid symbols) weighted by their error bars. Notice how the deviation of the line centroid position derived from our Gaussian ﬁt (vertical solid line) with respect to the expected position of the line (vertical dashed line) according to the visual redshift provided by the DR12Q catalogue are well correlated for the same pair of [O iii] and [Ne iii] lines, and for the diﬀerent set of lines. The green shaded vertical areas highlight the uncertainty for the expected position of the lines due to the quasar redshift error ≈ × − ( 5 10 4). Also shown is a fourth-order spline interpolation to the spectrum after subtracting the continuum (thin solid line). The [Ne iii] lines are weaker by one order of magnitude than the [O iii] lines, which is usually the case for all the spectra showing both pair of lines. The weak line near the stronger [Ne iii] line is blended with He i (3889.75Å) and Hζ (3890.16Å). ﬂux errors for each pixel, i.e., we use the ivar column found in each interpolation to the stronger [O iii] 5008 line, then ﬁtted this inter- spectrum as weights for the ﬁt. Our ﬁnal centroid measurement for polation to the weaker 4960 line by adjusting the amplitude and each considered line corresponds to the centroid of the Gaussian ﬁt separation of the proﬁle. We have modiﬁed this method by using a done in the last step of the adopted procedure. We also derive an Gaussian ﬁt to the stronger line rather than a third-order spline. error for ∆α/α using the standard errors for the centre position of the Gaussians. This is our main method for measuring α. Although we have described four diﬀerent methods, the main (ii) Voigt proﬁle method. results for ∆α/α presented in this work are based on the Gaussian ﬁtting method, while the other three are used only for comparison Following the same procedure than when using a Gaussian pro- (see Section ). ﬁle, we make the ﬁt with a Voigt proﬁle instead of a Gaussian. More precisely, we use a pseudo-Voigt proﬁle which is a linear combi- Finally, our ﬁnal result for ∆α/α and its error is obtained in the same way as in Chand et al. (2005), namely we compute a weighted nation of a Gaussian and a Lorentzian proﬁle. Then, we have one more parameter, i.e. the amplitude of the Lorentzian function, while mean and a weighted standard deviation, where the errors for ∆α/α of each spectrum are used as weights. its width and its position are the same as those for the Gaussian proﬁle. In Fig. 5, we depict the [O iii] and [Ne iii] lines for the same quasar spectrum to illustrate the Gaussian and Voigt ﬁtting 3.3 Simulated spectra methods. (iii) Integration method. In order to test the robustness and accuracy of our method- Here, the centroids of the lines are obtained by integrating ology, we generate realizations of quasar spectra using as noise a around 1σ from the position of the ﬁtted Gaussian proﬁle. This normal distribution centred at the ﬂux value, and taking the error technique provides indications of whether there is Hβ con≈tamina- in each pixel as the standard deviation. From our ﬁducial sample tion. However, due to the mid-resolution of the spectra R 2000, (10363 quasars), we simulate 100 realizations for each spectrum this method is not very accurate. (> a million in total). This number of realizations provides rea- (iv) Modiﬁed Bahcall method. sonable statistics to derive an error from the standard deviation of In Bahcall et al. (2004) the authors used a diﬀerent approach the measurements on the realizations of each real spectrum, while ∼ to compute the line positions. They performed a third-order spline the computation time remains reasonable ( 2 d) using a standard- (cid:13) c 2002 RAS, MNRAS 000, 1–17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fine-structure

text: constant

text: with

text: BOSS

text: a±t

text: Tthhies

text: measured

text: ∆α/α

text: for

text: [Ne

text: iii]

text: is

text: not

text: consistent

text: with

text: zero

text: regardless

text: of

text: the

text: proﬁle;

text: see

text: Fig.

text: and

text: last

text: paragraph

text: of

text: Section

text: for

text: discussion.

text: Each

text: panel

text: shows

text: the

text: ﬂux

text: density

text: for

text: each

text: pixel

text: with

text: their

text: respective

text: error

text: bars

text: (solid

text: symbols),

text: together

text: with

text: the

text: Gaussian

text: ﬁt

text: (dotted

text: red

text: curve)

text: and

text: the

text: pseudo-Voigt

text: proﬁle

text: (thick

text: grey

text: curve)

text: to

text: each

text: of

text: the

text: lines.

text: The

text: ﬁtting

text: procedure

text: (described

text: in

text: the

text: text)

text: only

text: takes

text: into

text: account

text: the

text: experimental

text: data

text: (solid

text: symbols)

text: weighted

text: by

text: their

text: error

text: bars.

text: Notice

text: how

text: the

text: deviation

text: of

text: the

text: line

text: centroid

text: position

text: derived

text: from

text: our

text: Gaussian

text: ﬁt

text: (vertical

text: solid

text: line)

text: with

text: respect

text: to

text: the

text: expected

text: position

text: of

text: the

text: line

text: (vertical

text: dashed

text: line)

text: according

text: to

text: the

text: visual

text: redshift

text: provided

text: by

text: the

text: catalogue

text: are

text: well

text: correlated

text: for

text: the

text: same

text: pair

text: of

text: [O

text: iii]

text: and

text: [Ne

text: iii]

text: lines,

text: and

text: for

text: the

text: diﬀerent

text: set

text: of

text: lines.

text: The

text: green

text: shaded

text: vertical

text: areas

text: highlight

text: the

text: uncertainty

text: for

text: the

text: expected

text: position

text: of

text: the

text: lines

text: due

text: to

text: the

text: quasar

text: redshift

text: error

text: Also

text: shown

text: is

text: a

text: fourth-order

text: spline

text: interpolation

text: to

text: the

text: spectrum

text: after

text: subtracting

text: the

text: continuum

text: (thin

text: solid

text: line).

text: The

text: [Ne

text: iii]

text: lines

text: are

text: weaker

text: by

text: one

text: order

text: of

text: magnitude

text: than

text: the

text: [O

text: iii]

text: lines,

text: which

text: is

text: usually

text: the

text: case

text: for

text: all

text: the

text: spectra

text: showing

text: both

text: pair

text: of

text: lines.

text: The

text: weak

text: line

text: near

text: the

text: stronger

text: [Ne

text: iii]

text: line

text: is

text: blended

text: with

text: He

text: i

text: and

text: Hζ

text: ﬂux

text: errors

text: for

text: each

text: pixel,

text: i.e.,

text: we

text: use

text: the

text: ivar

text: column

text: found

text: in

text: each

text: interpolation

text: to

text: the

text: stronger

text: [O

text: iii]

text: line,

text: then

text: ﬁtted

text: this

text: inter-

text: spectrum

text: as

text: weights

text: for

text: the

text: ﬁt.

text: Our

text: ﬁnal

text: centroid

text: measurement

text: for

text: polation

text: to

text: the

text: weaker

text: line

text: by

text: adjusting

text: the

text: amplitude

text: and

text: each

text: considered

text: line

text: corresponds

text: to

text: the

text: centroid

text: of

text: the

text: Gaussian

text: ﬁt

text: separation

text: of

text: the

text: proﬁle.

text: We

text: have

text: modiﬁed

text: this

text: method

text: by

text: using

text: a

text: done

text: in

text: the

text: last

text: step

text: of

text: the

text: adopted

text: procedure.

text: We

text: also

text: derive

text: an

text: Gaussian

text: ﬁt

text: to

text: the

text: stronger

text: line

text: rather

text: than

text: a

text: third-order

text: spline.

text: error

text: for

text: ∆α/α

text: using

text: the

text: standard

text: errors

text: for

text: the

text: centre

text: position

text: of

text: the

text: Gaussians.

text: This

text: is

text: our

text: main

text: method

text: for

text: measuring

text: α.

text: Although

text: we

text: have

text: described

text: four

text: diﬀerent

text: methods,

text: the

text: main

text: (ii)

text: Voigt

text: proﬁle

text: method.

text: results

text: for

text: ∆α/α

text: presented

text: in

text: this

text: work

text: are

text: based

text: on

text: the

text: Gaussian

text: ﬁtting

text: method,

text: while

text: the

text: other

text: three

text: are

text: used

text: only

text: for

text: comparison

text: Following

text: the

text: same

text: procedure

text: than

text: when

text: using

text: a

text: Gaussian

text: pro-

text: (see

text: Section

text: ﬁle,

text: we

text: make

text: the

text: ﬁt

text: with

text: a

text: Voigt

text: proﬁle

text: instead

text: of

text: a

text: Gaussian.

text: More

text: precisely,

text: we

text: use

text: a

text: pseudo-Voigt

text: proﬁle

text: which

text: is

text: a

text: linear

text: combi-

text: Finally,

text: our

text: ﬁnal

text: result

text: for

text: ∆α/α

text: and

text: its

text: error

text: is

text: obtained

text: in

text: the

text: same

text: way

text: as

text: in

text: Chand

text: et

text: al.

text: namely

text: we

text: compute

text: a

text: weighted

text: nation

text: of

text: a

text: Gaussian

text: and

text: a

text: Lorentzian

text: proﬁle.

text: Then,

text: we

text: have

text: one

text: more

text: parameter,

text: i.e.

text: the

text: amplitude

text: of

text: the

text: Lorentzian

text: function,

text: while

text: mean

text: and

text: a

text: weighted

text: standard

text: deviation,

text: where

text: the

text: errors

text: for

text: ∆α/α

text: of

text: each

text: spectrum

text: are

text: used

text: as

text: weights.

text: its

text: width

text: and

text: its

text: position

text: are

text: the

text: same

text: as

text: those

text: for

text: the

text: Gaussian

text: proﬁle.

text: In

text: Fig.

text: we

text: depict

text: the

text: [O

text: iii]

text: and

text: [Ne

text: iii]

text: lines

text: for

text: the

text: same

text: quasar

text: spectrum

text: to

text: illustrate

text: the

text: Gaussian

text: and

text: Voigt

text: ﬁtting

text: Simulated

text: spectra

text: methods.

text: (iii)

text: Integration

text: method.

text: In

text: order

text: to

text: test

text: the

text: robustness

text: and

text: accuracy

text: of

text: our

text: method-

text: Here,

text: the

text: centroids

text: of

text: the

text: lines

text: are

text: obtained

text: by

text: integrating

text: ology,

text: we

text: generate

text: realizations

text: of

text: quasar

text: spectra

text: using

text: as

text: noise

text: a

text: around

text: 1σ

text: from

text: the

text: position

text: of

text: the

text: ﬁtted

text: Gaussian

text: proﬁle.

text: This

text: normal

text: distribution

text: centred

text: at

text: the

text: ﬂux

text: value,

text: and

text: taking

text: the

text: error

text: technique

text: provides

text: indications

text: of

text: whether

text: there

text: is

text: Hβ

text: con≈tamina-

text: in

text: each

text: pixel

text: as

text: the

text: standard

text: deviation.

text: From

text: our

text: ﬁducial

text: sample

text: tion.

text: However,

text: due

text: to

text: the

text: mid-resolution

text: of

text: the

text: spectra

text: R

text: quasars),

text: we

text: simulate

text: realizations

text: for

text: each

text: spectrum

text: this

text: method

text: is

text: not

text: very

text: accurate.

text: a

text: million

text: in

text: total).

text: This

text: number

text: of

text: realizations

text: provides

text: rea-

text: (iv)

text: Modiﬁed

text: Bahcall

text: method.

text: sonable

text: statistics

text: to

text: derive

text: an

text: error

text: from

text: the

text: standard

text: deviation

text: of

text: In

text: Bahcall

text: et

text: al.

text: the

text: authors

text: used

text: a

text: diﬀerent

text: approach

text: the

text: measurements

text: on

text: the

text: realizations

text: of

text: each

text: real

text: spectrum,

text: while

text: to

text: compute

text: the

text: line

text: positions.

text: They

text: performed

text: a

text: third-order

text: spline

text: the

text: computation

text: time

text: remains

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text: d)

text: using

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text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Fine-structure constant with BOSS 9 Voigt-eps-converteedr-rtoor.spVdofigt-eps-converted-to.pdf Figure 7. Left-hand panel: measurements of ∆α/α using Gaussian and Voigt ﬁtting proﬁles. Non-compatible measurements at 1σ are shown as red crosses (6.5% of the total). Right-hand panel: errors from the Gaussian and Voigt ﬁtting. Non-compatible measurements at 1σ are shown as red crosses. is the case by considering quasars whose [O iii] line widths are the Gaussian ﬁts by the R2 coeﬃcient. All the considered samples show same within a relative fraction. For more than half of our ﬁducial values for ∆α/α consistent with no variation in α. sample, the [O iii] line widths diﬀer by less than 10% (see Table 2). (vi) Broad lines. We also study samples where the width of both − (b) Amplitude ratio: atomic physics states that the amplitude ratio lines is less than a certain value (in kms 1). These samples are between the [O iii] 5008 and [O iii] 4960 lines is 2.98±(Storey & consistent w−ith no variation of α. Samples built from narrow lines Zeippen 2000) (as quoted in Section , we obtain 2.96 0.02syst). < 300 kms 1 may be more aﬀected by misidentiﬁcation of [O iii] Thus, we consider diﬀerent samples where this ratio diﬀers by less lines as sky lines. than a certain amount from 2.98 (see Table 2). All the samples con- (vii) Diﬀerent methods for measuring the [O iii] line position. sidered in this test yield results for ∆α/α compatible with zero. We compare the results obtained by the methods to measure the Fig. 8 displays the Gaussian widths and ﬂuxes of both [O iii] emis- position of the [O iii] lines described in Section . Since not all the sion lines for our ﬁducial sample. methods provide an error for the measurement, we cannot calcu- (ii) Windows for the Gaussian ﬁts. We use a wavelength range of late a weighted mean, and it is necessary to select a more restricted 2σ around each [O iii] line in order to obtain the ﬁnal Gaussian ﬁt sample. Then, we consider a sample where the diﬀerence between to the line proﬁles. We study how our results depend on this choice. the widths of the lines is less than 25%, the amplitude ratio is con- By considering a larger wavelength interval, the results are more af- strained to diﬀer from the theoretical value 2.98 (Storey & Zeip- fected by the Hβ contamination and possible asymmetries on the pen 2000) by less than 0.5, and the S/NHβ is smaller than half the line wings. The diﬀerences in the number of spectra for these−sam- S/N[O iii] 4960. ples [which are obtained by applying the selection criteria (i) (iv) (viii) Gaussian versus Voigt proﬁles. We compare the results discussed in Section ] arise because of the criteria concerning the for 8485 quasars from our ﬁducial sample after dropping 1878 non-converging ﬁts and the sky emission lines described in Section spectra with non-converging Voigt ﬁts (this reduction increases . the statistical error). We also compute a ‘mixed’ value for ∆α/α where for each spectrum we use the value for the variation of (iii) Hβ contamination. We analyse samples where the ratio be- ttwheevenalSue/NfoHrβ∆aαnd/αS/dNec[Oreaiisi]es49a6s0wisecpolnascteraminoerde.sDtreinspgietnetthcoenfsatcrtatihnatst t(h∆eαﬁ/αne)V-soitgrtu.cture constant with smaller error, either (∆α/α)Gauss or on Hβ, it is always consistent with no variation in α within the er- We have also analysed the standard deviation and errors of rors. This analysis demonstrates that the strength and/or width of the results for ∆α/α as a function of redshift (Fig. 9). Even though the Hβ line do not aﬀect substantially the result for ∆α/α when a we have imposed a constraint on our initial sample based on the weighted mean is adopted. sky emission lines, the standard deviation and errors still correlate (iv) Continuum subtraction. We use a seventh-order polynomial with the sky. In particular, for the correlation with the moving stan- to subtract the continuum spectrum. We examine if the polynomial dard deviation, this means that the precision in our measurement of order has important eﬀects on our measurements. Our values for ∆α/α along the whole redshift interval is limited by the sky sub- ∆α/α and their errors are only slightly aﬀected by the chosen poly- traction algorithm. nomial order. 5 RESULTS (v) Goodness of Gaussian ﬁts. We quantify the quality of the (cid:13) c 2002 RAS, MNRAS 000, 1–17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fine-structure

text: constant

text: with

text: BOSS

text: Figure

text: Left-hand

text: panel:

text: measurements

text: of

text: ∆α/α

text: using

text: Gaussian

text: and

text: Voigt

text: ﬁtting

text: proﬁles.

text: Non-compatible

text: measurements

text: at

text: 1σ

text: are

text: shown

text: as

text: red

text: crosses

text: of

text: the

text: total).

text: Right-hand

text: panel:

text: errors

text: from

text: the

text: Gaussian

text: and

text: Voigt

text: ﬁtting.

text: Non-compatible

text: measurements

text: at

text: 1σ

text: are

text: shown

text: as

text: red

text: crosses.

text: is

label: list

text: the

text: case

text: by

text: considering

text: quasars

text: whose

text: [O

text: iii]

text: line

text: widths

text: are

text: the

text: Gaussian

text: ﬁts

text: by

text: the

text: coeﬃcient.

text: All

text: the

text: considered

text: samples

text: show

text: same

text: within

text: a

text: relative

text: fraction.

text: For

text: more

text: than

text: half

text: of

text: our

text: ﬁducial

text: values

text: for

text: ∆α/α

text: consistent

text: with

text: no

text: variation

text: in

text: α.

text: sample,

text: the

text: [O

text: iii]

text: line

text: widths

text: diﬀer

text: by

text: less

text: than

text: (see

text: Table

text: (vi)

text: Broad

text: lines.

text: We

text: also

text: study

text: samples

text: where

text: the

text: width

text: of

text: both

text: (b)

text: Amplitude

text: ratio:

text: atomic

text: physics

text: states

text: that

text: the

text: amplitude

text: ratio

text: lines

text: is

text: less

text: than

text: a

text: certain

text: value

text: (in

text: kms

text: These

text: samples

text: are

text: between

text: the

text: [O

text: iii]

text: and

text: [O

text: iii]

text: lines

text: is

text: consistent

text: w−ith

text: no

text: variation

text: of

text: α.

text: Samples

text: built

text: from

text: narrow

text: lines

text: Zeippen

text: (as

text: quoted

text: in

text: Section

text: we

text: obtain

text: kms

text: may

text: be

text: more

text: aﬀected

text: by

text: misidentiﬁcation

text: of

text: [O

text: iii]

text: Thus,

text: we

text: consider

text: diﬀerent

text: samples

text: where

text: this

text: ratio

text: diﬀers

text: by

text: less

text: lines

text: as

text: sky

text: lines.

text: than

text: a

text: certain

text: amount

text: from

text: (see

text: Table

text: All

text: the

text: samples

text: con-

text: (vii)

text: Diﬀerent

text: methods

text: for

text: measuring

text: the

text: [O

text: iii]

text: line

text: position.

text: sidered

text: in

text: this

text: test

text: yield

text: results

text: for

text: ∆α/α

text: compatible

text: with

text: zero.

text: We

text: compare

text: the

text: results

text: obtained

text: by

text: the

text: methods

text: to

text: measure

text: the

text: Fig.

text: displays

text: the

text: Gaussian

text: widths

text: and

text: ﬂuxes

text: of

text: both

text: [O

text: iii]

text: emis-

text: position

text: of

text: the

text: [O

text: iii]

text: lines

text: described

text: in

text: Section

text: Since

text: not

text: all

text: the

text: sion

text: lines

text: for

text: our

text: ﬁducial

text: sample.

text: methods

text: provide

text: an

text: error

text: for

text: the

text: measurement,

text: we

text: cannot

text: calcu-

text: (ii)

text: Windows

text: for

text: the

text: Gaussian

text: ﬁts.

text: We

text: use

text: a

text: wavelength

text: range

text: of

text: late

text: a

text: weighted

text: mean,

text: and

text: it

text: is

text: necessary

text: to

text: select

text: a

text: more

text: restricted

text: 2σ

text: around

text: each

text: [O

text: iii]

text: line

text: in

text: order

text: to

text: obtain

text: the

text: ﬁnal

text: Gaussian

text: ﬁt

text: sample.

text: Then,

text: we

text: consider

text: a

text: sample

text: where

text: the

text: diﬀerence

text: between

text: to

text: the

text: line

text: proﬁles.

text: We

text: study

text: how

text: our

text: results

text: depend

text: on

text: this

text: choice.

text: the

text: widths

text: of

text: the

text: lines

text: is

text: less

text: than

text: the

text: amplitude

text: ratio

text: is

text: con-

text: By

text: considering

text: a

text: larger

text: wavelength

text: interval,

text: the

text: results

text: are

text: more

text: af-

text: strained

text: to

text: diﬀer

text: from

text: the

text: theoretical

text: value

text: (Storey

text: Zeip-

text: fected

text: by

text: the

text: Hβ

text: contamination

text: and

text: possible

text: asymmetries

text: on

text: the

text: pen

text: by

text: less

text: than

text: and

text: the

text: S/NHβ

text: is

text: smaller

text: than

text: half

text: the

text: line

text: wings.

text: The

text: diﬀerences

text: in

text: the

text: number

text: of

text: spectra

text: for

text: these−sam-

text: S/N[O

text: iii]

text: ples

text: [which

text: are

text: obtained

text: by

text: applying

text: the

text: selection

text: criteria

text: (i)

text: (iv)

text: (viii)

text: Gaussian

text: versus

text: Voigt

text: proﬁles.

text: We

text: compare

text: the

text: results

text: discussed

text: in

text: Section

text: arise

text: because

text: of

text: the

text: criteria

text: concerning

text: the

text: for

text: quasars

text: from

text: our

text: ﬁducial

text: sample

text: after

text: dropping

text: non-converging

text: ﬁts

text: and

text: the

text: sky

text: emission

text: lines

text: described

text: in

text: Section

text: spectra

text: with

text: non-converging

text: Voigt

text: ﬁts

text: (this

text: reduction

text: increases

text: the

text: statistical

text: error).

text: We

text: also

text: compute

text: a

text: ‘mixed’

text: value

text: for

text: ∆α/α

text: where

text: for

text: each

text: spectrum

text: we

text: use

text: the

text: value

text: for

text: the

text: variation

text: of

text: (iii)

text: Hβ

text: contamination.

text: We

text: analyse

text: samples

text: where

text: the

text: ratio

text: be-

text: t(h∆eαﬁ/αne)V-soitgrtu.cture

text: constant

text: with

text: smaller

text: error,

text: either

text: (∆α/α)Gauss

text: or

text: on

text: Hβ,

text: it

text: is

text: always

text: consistent

text: with

text: no

text: variation

text: in

text: α

text: within

text: the

text: er-

text: We

text: have

text: also

text: analysed

text: the

text: standard

text: deviation

text: and

text: errors

text: of

text: rors.

text: This

text: analysis

text: demonstrates

text: that

text: the

text: strength

text: and/or

text: width

text: of

text: the

text: results

text: for

text: ∆α/α

text: as

text: a

text: function

text: of

text: redshift

text: (Fig.

text: Even

text: though

text: the

text: Hβ

text: line

text: do

text: not

text: aﬀect

text: substantially

text: the

text: result

text: for

text: ∆α/α

text: when

text: a

text: we

text: have

text: imposed

text: a

text: constraint

text: on

text: our

text: initial

text: sample

text: based

text: on

text: the

text: weighted

text: mean

text: is

text: adopted.

text: sky

text: emission

text: lines,

text: the

text: standard

text: deviation

text: and

text: errors

text: still

text: correlate

text: (iv)

text: Continuum

text: subtraction.

text: We

text: use

text: a

text: seventh-order

text: polynomial

text: with

text: the

text: sky.

text: In

text: particular,

text: for

text: the

text: correlation

text: with

text: the

text: moving

text: stan-

text: to

text: subtract

text: the

text: continuum

text: spectrum.

text: We

text: examine

text: if

text: the

text: polynomial

text: dard

text: deviation,

text: this

text: means

text: that

text: the

text: precision

text: in

text: our

text: measurement

text: of

text: order

text: has

text: important

text: eﬀects

text: on

text: our

text: measurements.

text: Our

text: values

text: for

text: ∆α/α

text: along

text: the

text: whole

text: redshift

text: interval

text: is

text: limited

text: by

text: the

text: sky

text: sub-

text: ∆α/α

text: and

text: their

text: errors

text: are

text: only

text: slightly

text: aﬀected

text: by

text: the

text: chosen

text: poly-

text: traction

text: algorithm.

text: nomial

text: order.

text: RESULTS

text: (v)

text: Goodness

text: of

text: Gaussian

text: ﬁts.

text: We

text: quantify

text: the

text: quality

text: of

text: the

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5.2 Inside the funnels of marked vertices Recall that for each marked vertex v ∈ M, we know at least of one vertex on ∂P such that v is its farthest neighbor. For any marked vertex v, let u , . . . , u 1 k−1 be the vertices of P such that v = f(u ) and assume that they appear in this i order when traversing ∂P clockwise. Let u and u be the neighbors of u and 0 k 1 u other than u and u , respectively. Note that both u u and u u k−1 2 k−2 0 1 k−1 k are transition edges of P. Thus, we can assume that their transition hourglasses have been computed. Let C = (u , . . . , u ) and consider the funnel S (C ). We call C the v 0 k v v v main chain of S (C ) while π(u , v) and π(v, u ) are referred to as the walls v v k 0 of the funnel. Because v = f(u ) = f(u ), we know that v is a vertex 1 k−1 of both Hu0u1 and Huk−1uk. By deﬁnition, we have π(v, u0) ⊂ Hu0u1 and π(v, uk) ⊂ Huk−1uk. Thus, we can explicitly compute both paths π(v, u0) and π(v, uk) in O(|Hu0u1| + |Huk−1uk|) time. So, overall, the funnel Sv(Cv) can be constructed in O(k + |Hu0u1| + |Huk−1uk|) time. Recall that, by Lemma 2.6, the total sum of the complexities of the transition hourglasses is O(n). In particular, we can bound the total time needed to construct the funnels of all marked vertices by O(n). Since the complexity of the wal(cid:32)ls of these funn(cid:33)els is bounded by the com- (cid:88) (cid:88) plexity of the transition hourglasses used to compute them, we get that |S (C )| = O n + |H | = O(n). v v ab v∈M ab∈E Lemma 5.3. Let x be a point in P. If v = f(x) is a vertex of M, then x ∈ S (C ). v v Proof. Since f(u ) (cid:54)= f(u ), C is a transition chain. Moreover, C contains 0 k v v R(v) ∩ ∂P by deﬁnition. Therefore, Lemma 2.7 implies that R(v) ⊂ S (C ). v v Since v = f(x), we know that x ∈ R(v) and hence that x ∈ S (C ). v v We now proceed to split a given funnel into O(|S (C )|) apexed triangles that v v encode the distance function from v. To this end, we compute the shortest-path tree T of v in S (C ) in O(|S (C )|) time [11]. We consider the tree T to be v v v v v v rooted at v and assume that for each node u of this tree we have stored the geodesic distance |π(u, v)|. Start an Eulerian tour from v walking in a clockwise order of the edges. Let Let w be the ﬁrst leaf of T found, and let w and w be the next two vertices 1 v 2 3 visited in the traversal. Two cases arise: Case 1: w , w , w makes a right turn. We deﬁne s as the ﬁrst point hit by 1 2 3 the ray apexed at w that shoots in the direction opposite to w . 2 3 We claim that w and s lie on the same edge of the boundary of S (C ). 1 v v Otherwise, there would be a vertex u visible from w inside the wedge with 2 apex w spanned by w and w . Note that the ﬁrst edge of the path π(u, v) 2 1 3 is the edge uw . Therefore, uw belongs to the shortest-path T contradicting 2 2 v the Eulerian order in which the vertices of this tree are visited as u should be 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Inside

text: the

text: funnels

text: of

text: marked

text: vertices

text: Recall

text: that

text: for

text: each

text: marked

text: vertex

text: v

text: M,

text: we

text: know

text: at

text: least

text: of

text: one

text: vertex

text: on

text: ∂P

text: such

text: that

text: v

text: is

text: its

text: farthest

text: neighbor.

text: For

text: any

text: marked

text: vertex

text: v,

text: let

text: u

text: u

text: be

text: the

text: vertices

text: of

text: P

text: such

text: that

text: v

text: f(u

text: and

text: assume

text: that

text: they

text: appear

text: in

text: this

text: i

text: order

text: when

text: traversing

text: ∂P

text: clockwise.

text: Let

text: u

text: and

text: u

text: be

text: the

text: neighbors

text: of

text: u

text: and

text: k

text: u

text: other

text: than

text: u

text: and

text: u

text: respectively.

text: Note

text: that

text: both

text: u

text: u

text: and

text: u

text: u

text: k

text: are

text: transition

text: edges

text: of

text: P.

text: Thus,

text: we

text: can

text: assume

text: that

text: their

text: transition

text: hourglasses

text: have

text: been

text: computed.

text: Let

text: C

text: (u

text: u

text: and

text: consider

text: the

text: funnel

text: S

text: (C

text: We

text: call

text: C

text: the

text: v

text: k

text: v

text: v

text: v

text: main

text: chain

text: of

text: S

text: (C

text: while

text: π(u

text: v)

text: and

text: π(v,

text: u

text: are

text: referred

text: to

text: as

text: the

text: walls

text: v

text: v

text: k

text: of

text: the

text: funnel.

text: Because

text: v

text: f(u

text: f(u

text: we

text: know

text: that

text: v

text: is

text: a

text: vertex

text: of

text: both

text: and

text: By

text: deﬁnition,

text: we

text: have

text: π(v,

text: and

text: π(v,

text: uk)

text: Thus,

text: we

text: can

text: explicitly

text: compute

text: both

text: paths

text: π(v,

text: and

text: π(v,

text: uk)

text: in

text: time.

text: So,

text: overall,

text: the

text: funnel

text: Sv(Cv)

text: can

text: be

text: constructed

text: in

text: O(k

text: time.

text: Recall

text: that,

text: by

text: Lemma

text: the

text: total

text: sum

text: of

text: the

text: complexities

text: of

text: the

text: transition

text: hourglasses

text: is

text: O(n).

text: In

text: particular,

text: we

text: can

text: bound

text: the

text: total

text: time

text: needed

text: to

text: construct

text: the

text: funnels

text: of

text: all

text: marked

text: vertices

text: by

text: O(n).

text: Since

text: the

text: complexity

text: of

text: the

text: of

text: these

text: is

text: bounded

text: by

text: the

text: com-

text: plexity

text: of

text: the

text: transition

text: hourglasses

text: used

text: to

text: compute

text: them,

text: we

text: get

text: that

text: |S

text: (C

text: O

text: n

text: |H

text: O(n).

text: v

text: v

text: ab

text: v∈M

text: ab∈E

text: Lemma

text: Let

text: x

text: be

text: a

text: point

text: in

text: P.

text: If

text: v

text: f(x)

text: is

text: a

text: vertex

text: of

text: M,

text: then

text: x

text: S

text: (C

text: v

text: v

text: Proof.

text: Since

text: f(u

text: f(u

text: C

text: is

text: a

text: transition

text: chain.

text: Moreover,

text: C

text: contains

text: k

text: v

text: v

text: R(v)

text: ∂P

text: by

text: deﬁnition.

text: Therefore,

text: Lemma

text: implies

text: that

text: R(v)

text: S

text: (C

text: v

text: v

text: Since

text: v

text: f(x),

text: we

text: know

text: that

text: x

text: R(v)

text: and

text: hence

text: that

text: x

text: S

text: (C

text: v

text: v

text: We

text: now

text: proceed

text: to

text: split

text: a

text: given

text: funnel

text: into

text: O(|S

text: (C

text: apexed

text: triangles

text: that

text: v

text: v

text: encode

text: the

text: distance

text: function

text: from

text: v.

text: To

text: this

text: end,

text: we

text: compute

text: the

text: shortest-path

text: tree

text: T

text: of

text: v

text: in

text: S

text: (C

text: in

text: O(|S

text: (C

text: time

text: We

text: consider

text: the

text: tree

text: T

text: to

text: be

text: v

text: v

text: v

text: v

text: v

text: v

text: rooted

text: at

text: v

text: and

text: assume

text: that

text: for

text: each

text: node

text: u

text: of

text: this

text: tree

text: we

text: have

text: stored

text: the

text: geodesic

text: distance

text: |π(u,

text: v)|.

text: Start

text: an

text: Eulerian

text: tour

text: from

text: v

text: walking

text: in

text: a

text: clockwise

text: order

text: of

text: the

text: edges.

text: Let

text: Let

text: w

text: be

text: the

text: ﬁrst

text: leaf

text: of

text: T

text: found,

text: and

text: let

text: w

text: and

text: w

text: be

text: the

text: next

text: two

text: vertices

text: v

text: visited

text: in

text: the

text: traversal.

text: Two

text: cases

text: arise:

text: Case

text: w

text: w

text: w

text: makes

text: a

text: right

text: turn.

text: We

text: deﬁne

text: s

text: as

text: the

text: ﬁrst

text: point

text: hit

text: by

text: the

text: ray

text: apexed

text: at

text: w

text: that

text: shoots

text: in

text: the

text: direction

text: opposite

text: to

text: w

text: We

text: claim

text: that

text: w

text: and

text: s

text: lie

text: on

text: the

text: same

text: edge

text: of

text: the

text: boundary

text: of

text: S

text: (C

text: v

text: v

text: Otherwise,

text: there

text: would

text: be

text: a

text: vertex

text: u

text: visible

text: from

text: w

text: inside

text: the

text: wedge

text: with

text: apex

text: w

text: spanned

text: by

text: w

text: and

text: w

text: Note

text: that

text: the

text: ﬁrst

text: edge

text: of

text: the

text: path

text: π(u,

text: v)

text: is

text: the

text: edge

text: uw

text: Therefore,

text: uw

text: belongs

text: to

text: the

text: shortest-path

text: T

text: contradicting

text: v

text: the

text: Eulerian

text: order

text: in

text: which

text: the

text: vertices

text: of

text: this

text: tree

text: are

text: visited

text: as

text: u

text: should

text: be

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##