text: ∼ ± (2011) estimate that the choice of the luminosity inte- We ﬁnd a z 1 f (Lyα) of 0.7% 0.4%. In Fig- esc gration limit may alter the computed luminosity density ure , we s∼how that the fesc(Lyα) computed from our by at most 60% (for further discussion of this issue see best ﬁt z 1 Sche∼chter function is consistent with the Hayes et al. 2011). Bouwens et al. (2010) derive cosmic f (Lyα) value of 0.9% interpolated from the power esc SFR densities from extinction corrected rest-frame UV law ﬁts of Hayes et al. (2011) and Blanc ±et al. (2011). LFs. We do not attempt to correct for IGM absorption, In Figure , we also show f (Lyα) = 1.0% 0.6% (open ∼ esc which should be negligible for our z 1 sample. circle) computed from our ﬁtted LF limited to regions Both Hayes et al. (2011) and Blanc et al. (2011) have with deep X-ray data. Hayes et al. (2011) argue that the compiled measurements from various Lyα LF studies rising trend in the volumetric Lyα escape fraction with − spanning a redshift range of z = 0.3 7.7 to determine increasing redshift is consistent with expectations due to the evolution of fesc(Lyα). Both groups ﬁt power laws the general decline of dust content in star-forming g∼alax- toDtrhaefstevdeartsiaonanMdayﬁ2n6d, 2e0v2i0dence for a rapidly increas- ies. W∼e have now constrained the fesc(Lyα) from z 0.3 ing f (Lyα), which approaches 100% at high redshifts to z 2 and ﬁnd results consistent with this inferred P∼repersicnt typeset using LATEX style emulateapj v. 08/22/09 (z 9). trend. ∼ Z 1 LYα EMITTERS I. THE LUMINOSITY FUNCTION1, 2, 3, 4 Isak G. B. Wold5, Amy J. Barger5,6,7, and Lennox L. Cowie7 Subject headings: cosmology: observations 7. SUMMARY nosities comparable to LAEs found in the early universe. ∼ With the optical data in hand, Paper II will compare We presented a catalog of 135 candidate z 1 LAEs. our sample’s physical properties to a UV-selected con- We obtained optical spectral data for 90% of our sam- ∼ trol sample. ple. We found that only 13% of our sample are ei- ther spurious, stars, or strong CIVλ1549 emitters. Com- bining these optical data with the UV spectra and X- We would like to thank the referee for their critical ray imaging data, we found that 49% of our Lyα emit- reading of the paper and useful suggestions for improv- ters are AGNs. Eliminating AGNs and LAEs with ing it. We gratefully acknowledge support from a NASA EW (Lyα)<20 ˚A gives a ﬁnal sample of 60 star-forming Keck PI Data Award, administered by the NASA Exo- r ∼ LAEs, which we used to compute the z 1 Lyα LF and planet Science Institute, a Wisconsin Space Grant Con- ∼ the z 1 Lyα volumetric escape fraction (f (Lyα) = sortium Graduate Fellowship, and a Sigma Xi Grant in ± ∼esc 0.7% 0.4%). We note that no improved z 1 Lyα LFs Aid of Research Award (I.G.B.W.). We also appreci- are possible for the foreseeable future due to the lack of ate support from the University of Wisconsin Research UV telescopes. Our best ﬁt LF implies a signiﬁcant in- Committee with funds granted by the Wisconsin Alumni ∼ ∼ crease in L between z 0.3 and z 1 without much Research Foundation and the David and Lucile Packard (cid:63) change in the luminosity density. This requires a rapid Foundation (A.J.B.), as well as from NSF grant AST- ∼ ∼ increase in φ between z 1 and z 2. It is clear that 0709356 (L.L.C.). We thank Michael Cooper for supply- (cid:63) the intrinsic properties of the LAEs must be changing ing spectra from the Arizona CDFS Environment Survey ∼ − rapidly in the z 1 2 interval, resulting in a rapid in- (ACES). This research has made use of the NASA/IPAC crease in the Lyα escape fraction. Our cataloged sources Extragalactic Database (NED), which is operated by the oﬀer the best opportunity to study emitters with lumi- Jet Propulsion Laboratory, California Institute of Tech- nology, under contract with the National Aeronautics 1 Based in part on data obtained from the Mikulski Archive for and Space Administration. Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA REFERENCES contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Oﬃce of Space Science via grant Alexander, D. M., Bauer, F. E., Brandt, W. N., et al. 2003, AJ, NNX09AF08G and by other grants and contracts. 126, 539 2 The W. M. Keck Observatory is operated as a scientiﬁc Atek, H., Kunth, D., Schaerer, D., et al. 2009, A&A, 506, L1 partnership among the the California Institute of Technology, the Atek, H., Siana, B., Scarlata, C., et al. 2011, ApJ, 743, 121 University of California, and NASA, and was made possible by Barger, A. J., Cowie, L. L., Brandt, W. N., et al. 2002, AJ, 124, the generous ﬁnancial support of the W. M. Keck Foundation. 1839 3 Some of the observations reported in this paper were obtained Barger, A. J., Cowie, L. L., & Wold, I. G. B. 2012, ApJ, 749, 106 with the Southern African Large Telescope (SALT). Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393 4 Based in part on zCOSMOS observations carried out using Blanc, G. A., Adams, J. J., Gebhardt, K., et al. 2011, ApJ, 736, the Very Large Telescope at the ESO Paranal Observatory under 31 Programme ID: LP175.A-0839. Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2010, ApJ, 5 Department of Astronomy, University of Wisconsin- 709, L133 Madison, 475 North Charter Street, Madison, WI 53706, Brocklehurst, M. 1971, MNRAS, 153, 471 USA; wold@astro.wisc.edu, barger@astro.wisc.edu 6 Department of Physics and Astronomy, University of Hawaii, Burgh, E. B., Nordsieck, K. H., Kobulnicky, H. A., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) 2505 Correa Road, Honolulu, HI 96822, USA 7 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Conference Series, Vol. 4841, Society of Photo-Optical Drive, Honolulu, HI 96822, USA Instrumentation Engineers (SPIE) Conference Series, ed. M. Iye & A. F. M. Moorwood, 1463–1471

text: estimate

text: that

text: the

text: choice

text: of

text: the

text: luminosity

text: inte-

text: We

text: ﬁnd

text: a

text: z

text: f

text: (Lyα)

text: of

text: In

text: Fig-

text: esc

text: gration

text: limit

text: may

text: alter

text: the

text: computed

text: luminosity

text: density

text: ure

text: we

text: s∼how

text: that

text: the

text: fesc(Lyα)

text: computed

text: from

text: our

text: by

text: at

text: most

text: (for

text: further

text: discussion

text: of

text: this

text: issue

text: see

text: best

text: ﬁt

text: z

text: Sche∼chter

text: function

text: is

text: consistent

text: with

text: the

text: Hayes

text: et

text: al.

text: Bouwens

text: et

text: al.

text: derive

text: cosmic

text: f

text: (Lyα)

text: value

text: of

text: interpolated

text: from

text: the

text: power

text: esc

text: SFR

text: densities

text: from

text: extinction

text: corrected

text: rest-frame

text: UV

text: law

text: ﬁts

text: of

text: Hayes

text: et

text: al.

text: and

text: Blanc

text: ±et

text: al.

text: LFs.

text: We

text: do

text: not

text: attempt

text: to

text: correct

text: for

text: IGM

text: absorption,

text: In

text: Figure

text: we

text: also

text: show

text: f

text: (Lyα)

text: (open

text: esc

text: which

text: should

text: be

text: negligible

text: for

text: our

text: z

text: sample.

text: circle)

text: computed

text: from

text: our

text: ﬁtted

text: LF

text: limited

text: to

text: regions

text: Both

text: Hayes

text: et

text: al.

text: and

text: Blanc

text: et

text: al.

text: have

text: with

text: deep

text: X-ray

text: data.

text: Hayes

text: et

text: al.

text: argue

text: that

text: the

text: compiled

text: measurements

text: from

text: various

text: Lyα

text: LF

text: studies

text: rising

text: trend

text: in

text: the

text: volumetric

text: Lyα

text: escape

text: fraction

text: with

text: spanning

text: a

text: redshift

text: range

text: of

text: z

text: to

text: determine

text: increasing

text: redshift

text: is

text: consistent

text: with

text: expectations

text: due

text: to

text: the

text: evolution

text: of

text: fesc(Lyα).

text: Both

text: groups

text: ﬁt

text: power

text: laws

text: the

text: general

text: decline

text: of

text: dust

text: content

text: in

text: star-forming

text: g∼alax-

text: for

text: a

text: rapidly

text: increas-

text: ies.

text: W∼e

text: have

text: now

text: constrained

text: the

text: fesc(Lyα)

text: from

text: z

text: ing

text: f

text: (Lyα),

text: which

text: approaches

text: at

text: high

text: redshifts

text: to

text: z

text: and

text: ﬁnd

text: results

text: consistent

text: with

text: this

text: inferred

text: P∼repersicnt

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: (z

text: trend.

text: Z

text: LYα

text: EMITTERS

text: I.

text: THE

text: LUMINOSITY

text: Isak

text: G.

text: B.

text: Amy

text: J.

text: and

text: Lennox

text: L.

text: Subject

text: headings:

text: cosmology:

text: observations

text: SUMMARY

text: nosities

text: comparable

text: to

text: LAEs

text: found

text: in

text: the

text: early

text: universe.

text: With

text: the

text: optical

text: data

text: in

text: hand,

text: Paper

text: II

text: will

text: compare

text: We

text: presented

text: a

text: catalog

text: of

text: candidate

text: z

text: LAEs.

text: our

text: sample’s

text: physical

text: properties

text: to

text: a

text: UV-selected

text: con-

text: We

text: obtained

text: optical

text: spectral

text: data

text: for

text: of

text: our

text: sam-

text: trol

text: sample.

text: ple.

text: We

text: found

text: that

text: only

text: of

text: our

text: sample

text: are

text: ei-

text: ther

text: spurious,

text: stars,

text: or

text: strong

text: emitters.

text: Com-

text: bining

text: these

text: optical

text: data

text: with

text: the

text: UV

text: spectra

text: and

text: X-

text: We

text: would

text: like

text: to

text: thank

text: the

text: referee

text: for

text: their

text: critical

text: ray

text: imaging

text: data,

text: we

text: found

text: that

text: of

text: our

text: Lyα

text: emit-

text: reading

text: of

text: the

text: paper

text: and

text: useful

text: suggestions

text: for

text: improv-

text: ters

text: are

text: AGNs.

text: Eliminating

text: AGNs

text: and

text: LAEs

text: with

text: ing

text: it.

text: We

text: gratefully

text: acknowledge

text: support

text: from

text: a

text: NASA

text: EW

text: ˚A

text: gives

text: a

text: ﬁnal

text: sample

text: of

text: star-forming

text: Keck

text: PI

text: Data

text: Award,

text: administered

text: by

text: the

text: NASA

text: Exo-

text: r

text: LAEs,

text: which

text: we

text: used

text: to

text: compute

text: the

text: z

text: Lyα

text: LF

text: and

text: planet

text: Science

text: Institute,

text: a

text: Wisconsin

text: Space

text: Grant

text: Con-

text: the

text: z

text: Lyα

text: volumetric

text: escape

text: fraction

text: (f

text: (Lyα)

text: sortium

text: Graduate

text: Fellowship,

text: and

text: a

text: Sigma

text: Xi

text: Grant

text: in

text: ∼esc

text: We

text: note

text: that

text: no

text: improved

text: z

text: Lyα

text: LFs

text: Aid

text: of

text: Research

text: Award

text: (I.G.B.W.).

text: We

text: also

text: appreci-

text: are

text: possible

text: for

text: the

text: foreseeable

text: future

text: due

text: to

text: the

text: lack

text: of

text: ate

text: support

text: from

text: the

text: University

text: of

text: Wisconsin

text: Research

text: UV

text: telescopes.

text: Our

text: best

text: ﬁt

text: LF

text: implies

text: a

text: signiﬁcant

text: in-

text: Committee

text: with

text: funds

text: granted

text: by

text: the

text: Wisconsin

text: Alumni

text: crease

text: in

text: L

text: between

text: z

text: and

text: z

text: without

text: much

text: Research

text: Foundation

text: and

text: the

text: David

text: and

text: Lucile

text: Packard

text: change

text: in

text: the

text: luminosity

text: density.

text: This

text: requires

text: a

text: rapid

text: Foundation

text: (A.J.B.),

text: as

text: well

text: as

text: from

text: NSF

text: grant

text: AST-

text: increase

text: in

text: φ

text: between

text: z

text: and

text: z

text: It

text: is

text: clear

text: that

text: (L.L.C.).

text: We

text: thank

text: Michael

text: Cooper

text: for

text: supply-

text: the

text: intrinsic

text: properties

text: of

text: the

text: LAEs

text: must

text: be

text: changing

text: ing

text: spectra

text: from

text: the

text: Arizona

text: CDFS

text: Environment

text: Survey

text: rapidly

text: in

text: the

text: z

text: interval,

text: resulting

text: in

text: a

text: rapid

text: in-

text: (ACES).

text: This

text: research

text: has

text: made

text: use

text: of

text: the

text: NASA/IPAC

text: crease

text: in

text: the

text: Lyα

text: escape

text: fraction.

text: Our

text: cataloged

text: sources

text: Extragalactic

text: Database

text: (NED),

text: which

text: is

text: operated

text: by

text: the

text: oﬀer

text: the

text: best

text: opportunity

text: to

text: study

text: emitters

text: with

text: lumi-

text: Jet

text: Propulsion

text: Laboratory,

text: California

text: Institute

text: of

text: Tech-

text: nology,

text: under

text: contract

text: with

text: the

text: National

text: Aeronautics

text: Based

text: in

text: part

text: on

text: data

text: obtained

text: from

text: the

text: Mikulski

text: Archive

text: for

text: and

text: Space

text: Administration.

text: Space

text: Telescopes

text: (MAST).

text: STScI

text: is

text: operated

text: by

text: the

text: Association

text: of

text: Universities

text: for

text: Research

text: in

text: Astronomy,

text: Inc.,

text: under

text: NASA

text: REFERENCES

text: contract

text: Support

text: for

text: MAST

text: for

text: non-HST

text: data

text: is

text: provided

text: by

text: the

text: NASA

text: Oﬃce

text: of

text: Space

text: Science

text: via

text: grant

text: Alexander,

text: D.

text: M.,

text: Bauer,

text: F.

text: E.,

text: Brandt,

text: W.

text: N.,

text: et

text: al.

text: AJ,

text: and

text: by

text: other

text: grants

text: and

text: contracts.

text: The

text: W.

text: M.

text: Keck

text: Observatory

text: is

text: operated

text: as

text: a

text: scientiﬁc

text: Atek,

text: H.,

text: Kunth,

text: D.,

text: Schaerer,

text: D.,

text: et

text: al.

text: A&A,

text: partnership

text: among

text: the

text: the

text: California

text: Institute

text: of

text: Technology,

text: the

text: Atek,

text: H.,

text: Siana,

text: B.,

text: Scarlata,

text: C.,

text: et

text: al.

text: ApJ,

text: University

text: of

text: California,

text: and

text: NASA,

text: and

text: was

text: made

text: possible

text: by

text: Barger,

text: A.

text: J.,

text: Cowie,

text: L.

text: L.,

text: Brandt,

text: W.

text: N.,

text: et

text: al.

text: AJ,

text: the

text: generous

text: ﬁnancial

text: support

text: of

text: the

text: W.

text: M.

text: Keck

text: Foundation.

text: Some

text: of

text: the

text: observations

text: reported

text: in

text: this

text: paper

text: were

text: obtained

text: Barger,

text: A.

text: J.,

text: Cowie,

text: L.

text: L.,

text: Wold,

text: I.

text: G.

text: B.

text: ApJ,

text: with

text: the

text: Southern

text: African

text: Large

text: Telescope

text: (SALT).

text: Bertin,

text: E.,

text: Arnouts,

text: S.

text: A&AS,

text: Based

text: in

text: part

text: on

text: zCOSMOS

text: observations

text: carried

text: out

text: using

text: Blanc,

text: G.

text: A.,

text: Adams,

text: J.

text: J.,

text: Gebhardt,

text: K.,

text: et

text: al.

text: ApJ,

text: the

text: Very

text: Large

text: Telescope

text: at

text: the

text: ESO

text: Paranal

text: Observatory

text: under

text: Programme

text: ID:

text: Bouwens,

text: R.

text: J.,

text: Illingworth,

text: G.

text: D.,

text: Oesch,

text: P.

text: A.,

text: et

text: al.

text: ApJ,

text: Department

text: of

text: Astronomy,

text: University

text: of

text: Wisconsin-

text: Madison,

text: North

text: Charter

text: Street,

text: Madison,

text: WI

text: Brocklehurst,

text: M.

text: MNRAS,

text: USA;

text: wold@astro.wisc.edu,

text: barger@astro.wisc.edu

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Hawaii,

text: Burgh,

text: E.

text: B.,

text: Nordsieck,

text: K.

text: H.,

text: Kobulnicky,

text: H.

text: A.,

text: et

text: al.

text: in

text: Society

text: of

text: Photo-Optical

text: Instrumentation

text: Engineers

text: (SPIE)

text: Correa

text: Road,

text: Honolulu,

text: HI

text: USA

text: Institute

text: for

text: Astronomy,

text: University

text: of

text: Hawaii,

text: Woodlawn

text: Conference

text: Series,

text: Vol.

text: Society

text: of

text: Photo-Optical

text: Drive,

text: Honolulu,

text: HI

text: USA

text: Instrumentation

text: Engineers

text: (SPIE)

text: Conference

text: Series,

text: ed.

text: M.

text: Iye

text: A.

text: F.

text: M.

text: Moorwood,


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text: 9 Figure 6. Rest-frame EWs vs. log Lyα luminosity for the full sample. truncated below 20 ˚A, the maximum likelihood estimate of the scale length is the mean of the rest-frame EW val- ± ues minus 20 ˚A. We ﬁnd a scale length of 80 10 ˚A for the full sample, where we computed the error using the parameterized bootstrap method. Separating the sam- ± ple by luminosity gives a scale length of 91 14 ˚A for sources with logarithmic Lyα luminosities above 42.8 and ± 68 10 ˚A for those with lower luminosities. This is consis- tent with the distribution being invariant as a function of Lyα luminosity, as has been found in the higher redshift samples (Ciardullo et al. 2012). ∼ We compare our sample at z 1 with a sample at z = 2.25 from Nilsson et al. (2009) (Figure 7(b)) and a sample at z = 3.1 from Ciardullo et al. (2012) (Figure 7(c)). The Nilsson et al. sample has a logarithmic luminosity − limit of 42.4 erg s 1, which is very similar to that of the present sample, while the Ciardullo et al. sample covers − − 42 erg s 1 to just above 43 erg s 1, making it slightly fainter, on average. The Nilsson et al. and Ciardullo et al. maximum likelihood estimates of the rest-frame EW ± scale length are both 56 7 ˚A. Nilsson et al. also give ± a least squares ﬁt value of 56 7 ˚A, which is consistent within the errors. However, when Ciardullo et al. (2012) allowed for bi- ases in their sample, such as those introduced by the ﬁlter shape, then their rest-frame EW scale length rose to 64+−10 ˚A. When Ciardullo et al. analyzed an alternate 7 sample at z = 3.1 from Gronwall et al. (2007) in the same way, they found 76+−11 ˚A. From the combined sam- 8 ples, they found 70+−7 ˚A. Ciardullo et al. also reanalyzed 5 the z = 2.1 sample from Guaita et al. (2010) in a self- consistent way and found 50+−96 ˚A. The Ciardullo et al. Figure 7. Histograms o∼f the rest-frame EWs in 10 ˚A bins. (a) z = 2.1 value is consistent with the Nilsson et al. (2009) The present sample at z 1. (b) The Nilsson et al. (2009) sample at z = 2.25. (c) The Ciardullo et al. (2012) sample at z = 3.1. z = 2.25 value. Based on an Anderson-Darling test, they In each case, the blue curve shows the exponential distribution concluded that there is a signiﬁcant diﬀerence between computed using a maximum likelihood procedure. The rest-frame ± ∼ ± ± the z = 3.1 and z = 2.1 samples. scale length is 80 10 ˚A for z 1, 56 7 ˚A for z = 2.25, and 56 7 ˚A The Ciardullo et al. (2012) analysis emphasizes that for z = 3.1. diﬀerences inherent in the methodology and ﬁtting proce- (A color version of this ﬁgure is available in the online journal.) dure are important. In particular, comparisons between the present sample, where the line ﬂuxes are computed from the spectra and the continua from the broadband magnitudes, and samples such as the Ciardullo et al. sample, which are based purely on imaging data, can be ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## 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text: In Figure 8, we show the rest-frame EW scale lengths measured from the maximum likelihood procedures for the present sample (black square), the z = 2.25 Nilsson et al. (2009) sample (green triangle), and the z = 2.1 Guaita et al. (2010), z = 3.1 Ciardullo et al. (2012), and z = 3.1 Gronwall et al. (2007) samples (blue diamonds; values quoted from Ciardullo et al.) While there may ∼ be a drop at z 2, overall the scale length shows little variation with redshift. This is in sharp contrast to the analysis of Zheng et al. (2013), who found a redshift evo- lution of (1 + z)1.7. The Zheng et al. result appears to be due primarily to their misinterpretation of the scale length given by Cowie et al. (2010) for their z = 0.3 ± GALEX LAE sample. Zheng et al. used 23.7 2.2, which was what Cowie et al. gave for their continuum selected sample. However, as Cowie et al. emphasized in their Section 5.4, this needs to be corrected for missing high Figure 8. Maximum likelihood ∼estimates of the rest-frame EW scale length from the present z 1 sample (black square), the EW sources before it can be compared with Lyα lumi- z = 2.25 Nilsson et al. (2009) sample (green triangle), and the nosity selected samples. In Figure 8, we show the Cowie z = 2.1 Guaita et al. (2010), z = 3.1 Gronwall et al. (2007), and et al. corrected scale length of 75 ˚A (red circle). (Because z = 3.1 Ciardullo et al. (2012) samples (blue diamonds; values quoted from Ciardullo et al.) vs. redshift. The red circle shows the of the large systematic errors, we do not show an error scale length from the z = 0.3 GALEX LAE sample of Cowie et al. bar.) This is fully consistent with there being a constant (2010) after correction for their continuum selection, as described value of the scale length with redshift. in their Section 5.4. (A color version of this ﬁgure is available in the online journal.) tricky (e.g., Zheng et al. 2013). Nevertheless, the present samDrpalfetavpepresaiorns tMoaiynd26ic,a2t0e20that, if there is indeed a nar- rowPrienpgrinotftytpheeserteussti-nfgraLAmTEeXEstWyledemisutlraitbeuaptjiovn. 0f8r/o2m2/0z9 = 3 to z = 2, then it has reversed by z = 1. ∼ Z 1 LYα EMITTERS I. THE LUMINOSITY FUNCTION1, 2, 3, 4 Isak G. B. Wold5, Amy J. Barger5,6,7, and Lennox L. Cowie7 Subject headings: cosmology: observations ∼ 6. DISCUSSION redshift of z 1 (see Figure 9, red∼diamonds). We have now increased the sample size of z 1 LAEs to 60 galax- In Figure 9, we show the dramatic increase in the ∼ ies (see Figure 9, black circles). Our relatively nearby observed Lyα luminosities to a redshift of z 1 fol- ∼ ∼ z 1 sample will facilitate the study of LAEs with lu- lowed by relatively no luminosity evolution from z 1 to ∼ minosities analogous to high-redshift LAEs. In Paper II, z 6. This large luminosity boost was previously noted ∼ we will use the available spectroscopic and imaging data by Cowie et al. (2011) with a sample of 5 LAEs at a to constrain the properties of our sample and to serve as 1 Based in part on data obtained from the Mikulski Archive for a baseline for studies of higher redshift LAEs. Space Telescopes (MAST). STScI is operated by the Association In order to quantify the evolution in L(cid:63), we have held of Universities for Research in Astronomy, Inc., under NASA the slope α ﬁxed and found the best Schechter (1976) − contract NAS5-26555. Support for MAST for non-HST data function ﬁt to our data. We assume a slope of α = 1.6, is provided by the NASA Oﬃce of Space Science via grant which allows us to compare directly our results to the NNX09AF08G and by other grants and contracts. ∼ − ± 2 The W. M. Keck Observatory is operated as a scientiﬁc LAE galaxy LF −at z ±0.3 (α = 1.6, log L(cid:63) = 41.81 partnership among the the California Institute of Technology, the 0.09, log φ = 3.77 0.08; Cowie et al. 2010). We (cid:63) ± − ± University of California, and NASA, and was made possible by obtain log L = 43.0 0.2 and log φ = 4.8 0.3 the generous ﬁnancial support of the W. M. Keck Foundation. (cid:63) (cid:63) 3 Some of the observations reported in this paper were obtained (see Figure 5(a)). Limiting our sample to±regions with with the Southern African Large Telescope (SALT). deep X−-ray±data, we obtain log L(cid:63) = 42.8 0.2 and log 4 Based in part on zCOSMOS observations carried out using φ = 4.5 0.3 (see Figure 5(b)). (cid:63) ∼ the Very Large Telescope at the ESO Paranal Observatory under Both our L (z 1) values are roughly consistent with Programme ID: LP175.A-0839. (cid:63) ∼ 5 Department of Astronomy, University of Wisconsin- L(cid:63) values found for LAE galaxy LFs at z 3 (log L(cid:63) = Madison, 475 North Charter Street, Madison, WI 53706, 42.7, Gronwall et al. 2007; log L(cid:63) = 42.8, Ouchi et al. USA; wold@astro.wisc.edu, barger@astro.wisc.edu 2008; log L = 42.8, Ciardullo et al. 2012) and imply an 6 Department of Physics and Astronomy, University of Hawaii, L increase(cid:63)by a factor of ∼ 15 from z ∼ 0.3 to z ∼ 1 2505 Correa Road, Honolulu, HI 96822, USA (cid:63) ∼ ± ∼ 7 Institute for Astronomy, University of Hawaii, 2680 Woodlawn (from±log L(cid:63)(z 0.3) = 41.81 0.09 to log L(cid:63)(z 1) = Drive, Honolulu, HI 96822, USA 43.0 0.2). 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text: ON LINEAR DEPENDENCE OVER COMPLETE DIFFERENTIAL ALGEBRAIC VARIETIES JAMES FREITAG*, OMAR LEO´N SA´NCHEZ, AND WILLIAM SIMMONS** Abstract. We extend Kolchin’s results from [11] on linear dependence over pro- jective varieties in the constants, to linear dependence over arbitrary complete dif- ferential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and suﬃcient conditions given by the vanishing of a certain system of diﬀerential-polynomials equations. We also discuss some conjec- tural questions around completeness and the catenary problem. Keywords: diﬀerential algebraic geometry, model theory AMS 2010 Mathematics Subject Classiﬁcation: 12H05. 03C98 1. Introduction The study of complete diﬀerential algebraic varieties began in the 1960’s during the development of diﬀerential algebraic geometry, and received the attention of various authors over the next several decades [18, 3, 4, 11]. On the other hand, even though model theory had signiﬁcant early interactions with diﬀerential algebra, it was not until recently that the topic has been the subject of various works using the model- theoretic perspective [20, 8, 27, 19]. Up until the last couple of years, relatively few examples of complete diﬀerential algebraic varieties were known. The development of a valuative criterion for com- pleteness led to a variety of new examples, see for instance [20] and [8]. Subsequently, more examples have been discovered [27] using various algebraic techniques in con- junction with the valuative criterion (in Section we present an example of the third author’s, which shows that there are zero-dimensional projective diﬀerential algebraic varieties which are not complete). To verify that a given projective diﬀerential variety V is complete one has to verify that for any quasiprojective diﬀerential variety Y , the second projection: Date: May 26, 2020. *This material is based upon work supported by an American Mathematical Society Mathe- matical Research Communities award and the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship, award number 1204510. **This material is based upon work supported by an American Mathematical Society Mathemat- ical Research Communities award. 1 ##LTLine##

text: ON

text: LINEAR

text: DEPENDENCE

text: OVER

text: COMPLETE

text: DIFFERENTIAL

text: ALGEBRAIC

text: VARIETIES

text: JAMES

text: FREITAG*,

text: OMAR

text: LEO´N

text: SA´NCHEZ,

text: AND

text: WILLIAM

text: SIMMONS**

text: Abstract.

text: We

label: abstract

text: extend

text: Kolchin’s

text: results

text: from

text: on

text: linear

text: dependence

text: over

text: pro-

text: jective

text: varieties

text: in

text: the

text: constants,

text: to

text: linear

text: dependence

text: over

text: arbitrary

text: complete

text: dif-

text: ferential

text: varieties.

text: We

text: show

text: that

text: in

text: this

text: more

text: general

text: setting,

text: the

text: notion

text: of

text: linear

text: dependence

text: still

text: has

text: necessary

text: and

text: suﬃcient

text: conditions

text: given

text: by

text: the

text: vanishing

text: of

text: a

text: certain

text: system

text: of

text: diﬀerential-polynomials

text: equations.

text: We

text: also

text: discuss

text: some

text: conjec-

text: tural

text: questions

text: around

text: completeness

text: and

text: the

text: catenary

text: problem.

text: Keywords:

text: diﬀerential

text: algebraic

text: geometry,

text: model

text: theory

text: AMS

text: Mathematics

text: Subject

text: Classiﬁcation:

label: section

text: Introduction

text: The

text: study

text: of

text: complete

text: diﬀerential

text: algebraic

text: varieties

text: began

text: in

text: the

text: during

text: the

text: development

text: of

text: diﬀerential

text: algebraic

text: geometry,

text: and

text: received

text: the

text: attention

text: of

text: various

text: authors

text: over

text: the

text: next

text: several

text: decades

text: On

text: the

text: other

text: hand,

text: even

text: though

text: model

text: theory

text: had

text: signiﬁcant

text: early

text: interactions

text: with

text: diﬀerential

text: algebra,

text: it

text: was

text: not

text: until

text: recently

text: that

text: the

text: topic

text: has

text: been

text: the

text: subject

text: of

text: various

text: works

text: using

text: the

text: model-

text: theoretic

text: perspective

text: Up

text: until

text: the

text: last

text: couple

text: of

text: years,

text: relatively

text: few

text: examples

text: of

text: complete

text: diﬀerential

text: algebraic

text: varieties

text: were

text: known.

text: The

text: development

text: of

text: a

text: valuative

text: criterion

text: for

text: com-

text: pleteness

text: led

text: to

text: a

text: variety

text: of

text: new

text: examples,

text: see

text: for

text: instance

text: and

text: Subsequently,

text: more

text: examples

text: have

text: been

text: discovered

text: using

text: various

text: algebraic

text: techniques

text: in

text: con-

text: junction

text: with

text: the

text: valuative

text: criterion

text: (in

text: Section

text: we

text: present

text: an

text: example

text: of

text: the

text: third

text: author’s,

text: which

text: shows

text: that

text: there

text: are

text: zero-dimensional

text: projective

text: diﬀerential

text: algebraic

text: varieties

text: which

text: are

text: not

text: complete).

text: To

text: verify

text: that

text: a

text: given

text: projective

text: diﬀerential

text: variety

text: V

text: is

text: complete

text: one

text: has

text: to

text: verify

text: that

text: for

text: any

text: quasiprojective

text: diﬀerential

text: variety

text: Y

text: the

text: second

text: projection:

text: Date:

text: May

text: *This

text: material

text: is

text: based

text: upon

text: work

text: supported

text: by

text: an

text: American

text: Mathematical

text: Society

text: Mathe-

text: matical

text: Research

text: Communities

text: award

text: and

text: the

text: National

text: Science

text: Foundation

text: Mathematical

text: Sciences

text: Postdoctoral

text: Research

text: Fellowship,

text: award

text: number

text: **This

text: material

text: is

text: based

text: upon

text: work

text: supported

text: by

text: an

text: American

text: Mathematical

text: Society

text: Mathemat-

text: ical

text: Research

text: Communities

text: award.

text: ##LTLine##


text: ON LINEAR DEPENDENCE AND COMPLETENESS 11 As p is diﬀerentially bi-homogeneous, it determines a (multi-)projective diﬀerential 1 ⊆ P × P algebraic variety W n n. It is clear that the coordinate projection maps have the form W π1 π2 (cid:127)(cid:127) (cid:34)(cid:34) V ld(V ) Further, ld(V ) = π (W ), and, since V is complete, ld(V ) is closed in the Kolchin P 2 topology of n and deﬁned over K. Next we prove that ld(V ) is irreducible over K. Let a¯ = (a , . . . , a ) be a repre- 0 (cid:54) n sentative of a generic point of V over K and ﬁx j such that a = 0. Pick elements ∈ ≤ ≤ (cid:54) (cid:88) j (cid:104) (cid:105) u U for 0 k n and k = j which are ∆-algebraically independent over K a¯ . k Let − − u = u a 1a , j k j k (cid:54) k=j ⊆ and u¯ = (u , . . . , u ). One can see that (a¯, u¯) is a representative of a point in W P × P 0 n · · · ∈ n n, so that [u : : u ] ld(V ). 0 · · · n We claim that [u : : u ] is a generic point of ld(V ) over K (this will show that 0 n ld(V ) is irreducible over K). To show this it suﬃces to show that (a¯, u¯) is a generic ∈ { } point of p ; i.e., it suﬃces to show that for every p K z¯, y¯ if p(a¯, u¯) = 0 then ∈ 1 p p . 1 ∈ { } Let p K z¯, y¯ be any diﬀerential polyn(cid:34)omial. By(cid:35)the diﬀerential division algo- ∈ { } (cid:88) rithm there exists p K z¯, y¯ not involving y such that 0 j ≡ zep p mod y z (cid:80) j 0 i i ≤ ≤ 0 i n ∈ N for some e . Thus, we can write p as a ﬁnite sum p M where each M 0 ∈ { } M is a diﬀerential monomial in (y ) ≤ ≤ (cid:54) and p K z¯ . Now, as (u ) ≤ ≤ (cid:54) k 0 k n,k=j (cid:104) (cid:105) M k 0 k n,k=j are diﬀerential algebraic independent over K a¯ , it follows that if p(a¯, u¯) = 0 then ∈ p (a¯) = 0 for all M (and hence p p, since a¯ is a generic point of p). This implies M ∈ · { M} ∈ that, if p(a¯, u¯) = 0, p p K z¯, y¯ and so p p , as desired. 0 1 (cid:3) For the following proposition we will make use of the following fact of Kolchin’s about the Kolchin polynomial of the diﬀerential tangent space (we refer the reader to [13] for the deﬁnition and basic properties of diﬀerential tangent spaces). Fact 4.5. Let V be an irreducible diﬀerential algebraic variety deﬁned over K with generic point v¯. Then ω = ω , T∆V V v¯

text: ON

text: LINEAR

text: DEPENDENCE

text: AND

text: COMPLETENESS

text: As

text: p

text: is

text: diﬀerentially

text: bi-homogeneous,

text: it

text: determines

text: a

text: (multi-)projective

text: diﬀerential

text: P

text: P

text: algebraic

text: variety

text: W

text: n

text: n.

text: It

text: is

text: clear

text: that

text: the

text: coordinate

text: projection

text: maps

text: have

text: the

text: form

text: W

text: π1

text: π2

font: default

text: V

text: ld(V

text: Further,

text: ld(V

text: π

text: (W

text: and,

text: since

text: V

text: is

text: complete,

text: ld(V

text: is

text: closed

text: in

text: the

text: Kolchin

text: P

text: topology

text: of

text: n

text: and

text: deﬁned

text: over

text: K.

text: Next

text: we

text: prove

text: that

text: ld(V

text: is

text: irreducible

text: over

text: K.

text: Let

text: a¯

text: (a

text: a

text: be

text: a

text: repre-

text: n

text: sentative

text: of

text: a

text: generic

text: point

text: of

text: V

text: over

text: K

text: and

text: ﬁx

text: j

text: such

text: that

text: a

text: Pick

text: elements

text: j

text: u

text: U

text: for

text: k

text: n

text: and

text: k

text: j

text: which

text: are

text: ∆-algebraically

text: independent

text: over

text: K

text: a¯

text: k

text: Let

text: u

text: u

text: a

text: j

text: k

text: j

text: k

text: k=j

text: and

text: u¯

text: (u

text: u

text: One

text: can

text: see

text: that

text: (a¯,

text: u¯)

text: is

text: a

text: representative

text: of

text: a

text: point

text: in

text: W

text: P

text: P

text: n

text: n

text: n,

text: so

text: that

text: [u

text: u

text: ld(V

text: n

text: We

text: claim

text: that

text: [u

text: u

text: is

text: a

text: generic

text: point

text: of

text: ld(V

text: over

text: K

text: (this

text: will

text: show

text: that

text: n

text: ld(V

text: is

text: irreducible

text: over

text: K).

text: To

text: show

text: this

text: it

text: suﬃces

text: to

text: show

text: that

text: (a¯,

text: u¯)

text: is

text: a

text: generic

text: point

text: of

text: p

text: i.e.,

text: it

text: suﬃces

text: to

text: show

text: that

text: for

text: every

text: p

text: K

text: z¯,

text: y¯

text: if

text: p(a¯,

text: u¯)

text: then

text: p

text: p

text: Let

text: p

text: K

text: z¯,

text: y¯

text: be

text: any

text: diﬀerential

text: diﬀerential

text: division

text: algo-

text: rithm

text: there

text: exists

text: p

text: K

text: z¯,

text: y¯

text: not

text: involving

text: y

text: such

text: that

text: j

text: zep

text: p

text: mod

text: y

text: z

text: j

text: i

text: i

text: i

text: n

text: N

text: for

text: some

text: e

text: Thus,

text: we

text: can

text: write

text: p

text: as

text: a

text: ﬁnite

text: sum

text: p

text: M

text: where

text: each

text: M

text: M

text: is

text: a

text: diﬀerential

text: monomial

text: in

text: (y

text: and

text: p

text: K

text: z¯

text: Now,

text: as

text: (u

text: k

text: k

text: n,k=j

text: M

text: k

text: k

text: n,k=j

text: are

text: diﬀerential

text: algebraic

text: independent

text: over

text: K

text: a¯

text: it

text: follows

text: that

text: if

text: p(a¯,

text: u¯)

text: then

text: p

text: (a¯)

text: for

text: all

text: M

text: (and

text: hence

text: p

text: p,

text: since

text: a¯

text: is

text: a

text: generic

text: point

text: of

text: p).

text: This

text: implies

text: M

text: M}

text: that,

text: if

text: p(a¯,

text: u¯)

text: p

text: p

text: K

text: z¯,

text: y¯

text: and

text: so

text: p

text: p

text: as

text: desired.

text: For

text: the

text: following

text: proposition

text: we

text: will

text: make

text: use

text: of

text: the

text: following

text: fact

text: of

text: Kolchin’s

text: about

text: the

text: Kolchin

text: polynomial

text: of

text: the

text: diﬀerential

text: tangent

text: space

text: (we

text: refer

text: the

text: reader

text: to

text: for

text: the

text: deﬁnition

text: and

text: basic

text: properties

text: of

text: diﬀerential

text: tangent

text: spaces).

text: Fact

text: Let

text: V

text: be

text: an

text: irreducible

text: diﬀerential

text: algebraic

text: variety

text: deﬁned

text: over

text: K

text: with

text: generic

text: point

text: v¯.

text: Then

text: ω

text: ω

text: T∆V

text: V

text: v¯


text: 26 JAMES FREITAG, OMAR LEO´N SA´NCHEZ, AND WILLIAM SIMMONS [25] Abraham Seidenberg. Abstract diﬀerential algebra and the analytic case. Proceedings of the American Mathematical Society, 9(1):159–164, 1958. [26] Abraham Seidenberg. Abstract diﬀerential algebra and the analytic case. ii. Proceedings of the American Mathematical Society, pages 689–691, 1969. [27] William D. Simmons. Completeness of ﬁnite-rank diﬀerential varieties. PhD thesis, University of Illinois at Chicago, 2013. [28] D.V. Trushin. Inheritance of properties of spectra. Mathematical Notes, 88(5-6):868–878, 2010. [29] Ronald A. Walker. Linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized wronskians. Linear Algebra and its Applications, 408(0):151 – 160, 2005. [30] Kenneth Wolsson. A condition equivalent to linear dependence for functions with vanishing wronskian. Linear Algebra and its Applications, 116(0):1 – 8, 1989. freitag@math.berkeley.edu, Department of Mathematics, University of Califor- nia, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840 oleonsan@math.mcmaster.ca, Department of Mathematics and Statistics, McMas- ter University, 1280 Main St W, Hamilton, ON L8S 4L8 simmons@math.ucla.edu, Mathematics Department, University of California, Los Angeles, Math Sciences Building 6363, Los Angeles, CA 90095

text: JAMES

text: FREITAG,

text: OMAR

text: LEO´N

text: SA´NCHEZ,

text: AND

text: WILLIAM

text: SIMMONS

text: Abraham

text: Seidenberg.

text: Abstract

text: diﬀerential

text: algebra

text: and

text: the

text: analytic

text: case.

text: Proceedings

text: of

text: the

text: American

text: Mathematical

text: Society,

text: Abraham

text: Seidenberg.

text: Abstract

text: diﬀerential

text: algebra

text: and

text: the

text: analytic

text: case.

text: ii.

text: Proceedings

text: of

text: the

text: American

text: Mathematical

text: Society,

text: pages

text: William

text: D.

text: Simmons.

text: Completeness

text: of

text: ﬁnite-rank

text: diﬀerential

text: varieties.

text: PhD

text: thesis,

text: University

text: of

text: Illinois

text: at

text: Chicago,

text: D.V.

text: Trushin.

text: Inheritance

text: of

text: properties

text: of

text: spectra.

text: Mathematical

text: Notes,

text: Ronald

text: A.

text: Walker.

text: Linear

text: dependence

text: of

text: quotients

text: of

text: analytic

text: functions

text: of

text: several

text: variables

text: with

text: the

text: least

text: subcollection

text: of

text: generalized

text: wronskians.

text: Linear

text: Algebra

text: and

text: its

text: Applications,

text: Kenneth

text: Wolsson.

text: A

text: condition

text: equivalent

text: to

text: linear

text: dependence

text: for

text: functions

text: with

text: vanishing

text: wronskian.

text: Linear

text: Algebra

text: and

text: its

text: Applications,

text: freitag@math.berkeley.edu,

text: Department

text: of

text: Mathematics,

text: University

text: of

text: Califor-

text: nia,

text: Berkeley,

text: Evans

text: Hall,

text: Berkeley,

text: CA

text: oleonsan@math.mcmaster.ca,

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: McMas-

text: ter

text: University,

text: Main

text: St

text: W,

text: Hamilton,

text: ON

text: simmons@math.ucla.edu,

text: Mathematics

text: Department,

text: University

text: of

text: California,

text: Los

text: Angeles,

text: Math

text: Sciences

text: Building

text: Los

text: Angeles,

text: CA