Connecting Faint End Slopes of the Lyman-$\alpha$ emitter and Lyman-break Galaxy Luminosity Functions

0.89 MB·English

by  Max Gronke

Mon.Not.R.Astron.Soc.000,1–8(2015) Printed3February2015 (MNLATEXstylefilev2.2)
Connecting Faint End Slopes of the Lyman-α emitter and
Lyman-break Galaxy Luminosity Functions
M. Gronke1(cid:63), M. Dijkstra1, M. Trenti2,3, S. Wyithe3,4
1Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029, 0315 Oslo, Norway
2Kavli Institute for Cosmology and Institute of Astronomy, University of Cambridge, Cambridge, England
3School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
4ARC Centre of Excellence for All-Sky Astrophysics (CAASTRO)
5
1 3February2015
0
2
n ABSTRACT
a We predict Lyman-α (Lyα) luminosity functions (LFs) of Lyα-selected galaxies (Lyα
J emitters,orLAEs)atz =3−6usingthephenomenologicalmodelofDijkstra&Wyithe
0 (2012). This model combines observed UV-LFs of Lyman-break galaxies (LBGs, or
3 drop out galaxies), with constraints on their distribution of Lyα line strengths as a
function of UV-luminosity and redshift. Our analysis shows that while Lyα LFs of

] LAEsaregenerallynotSchechterfunctions,theseprovideagooddescriptionoverthe
O
luminosity range of log (L /ergs−1)=41−44. Motivated by this result, we predict
10 α
C Schechter function parameters at z = 3−6. Our analysis further shows that (i) the
. faint end slope of the Lyα LF is steeper than that of the UV-LF of Lyman-break
h
galaxies, (with a median α <−2.0 at z >4), and (ii) a turn-over in the Lyα LF of
p Lyα ∼
- LAEs at Lyα luminosities 1040 erg s−1 <Lα <∼1041 erg s−1 may signal a flattening of
o
UV-LF of Lyman-break galaxies at −12 > M > −14. We discuss the implications
r UV
t of these results – which can be tested directly with upcoming surveys – for the Epoch
s
of Reionization.
a
[ Key words: galaxies: high-redshift – galaxies: luminosity function, mass function –

cosmology: reionization – ultraviolet: galaxies
1
v
2
2
0
1 INTRODUCTION fraction of their radiation as Lyman-α (Lyα) emission, and
0
this method has been proved to be very efficient in finding
0 Theluminosityfunction(LF)ofgalaxiesprovidesoneofthe
. samples out to z ∼ 7 (e.g. Rhoads et al. 2000; Rhoads &
2 most basic statistical descriptions of a population of galax-
Malhotra 2001; Ouchi et al. 2008; Bond et al. 2009, 2010;
0 ies. It describes the number density of galaxies in a given
Guaitaetal.2010;Kashikawaetal.2011;Hibonetal.2012;
5 luminosityinterval.Generally,theLFiswelldescribedbya
Onoetal.2012;Ota&Iye2012;Rhoadsetal.2012;Shibuya
1 Schechter (1976) function
et al. 2012; Finkelstein et al. 2013; Konno et al. 2014).
:
v (cid:18) L (cid:19)α (cid:18) L (cid:19) (cid:18) L (cid:19)
i φ(L)dL=φ∗ exp − d (1) Galaxiesthathavebeenselected(found)onthebasisof
X L∗ L∗ L∗ their Lyα lines are referred to as ‘Lyα emitters’ (or LAEs).
r with a normalization parameter φ∗, an exponential cutoff LAEsareusefulbecausetheyareselectedonhavingastrong
a at L (cid:38) L∗, and, a power law with faint-end-slope α for LyαlinefluxirrespectiveoftheirassociatedUV-continuum
L(cid:28)L∗.Theparametersdependonwavelengthconsidered, emission. Therefore, LAEs can be fainter in the continuum
galaxy type (e.g., passive versus star forming), and cosmic comparedtoLyman-breakgalaxies,andcomplementgalaxy
time. samples obtained via broadband searches which have been
At high redshift, galaxies are typically identified ei- extensivelycarriedoutwiththeHubbleSpaceTelescopeout
ther through their broadband colors, for example using the toz∼10(e.g.Yan&Windhorst2004;Beckwithetal.2006;
drop-outorLyman-breaktechnique(Steideletal.1996),or Bouwens et al. 2006; Wilkins et al. 2010; Trenti et al. 2011;
through narrow-band searches aimed at detecting emission Grazian et al. 2012; Finkelstein et al. 2012; Bouwens et al.
lines (Partridge & Peebles 1967; Djorgovski et al. 1985) In 2014a; Finkelstein et al. 2014; Oesch et al. 2014; Schmidt
particular, young star forming galaxies emit a significant et al. 2014). Moreover, the sensitivity of the observed Lyα
fluxtointerveningneutralhydrogengasmakesLAEsanex-
cellentprobeoftheEpochofReionization(seee.g.Dijkstra
(cid:63) E-mail:[email protected] 2014, for a review).
(cid:13)c 2015RAS
2 M. Gronke, M. Dijkstra, M. Trenti, S. Wyithe
Since the range of observed Lyα luminosities at high-z 2 METHOD
typically extends only over ∼ 1−1.5 orders of magnitude,
ThenumberdensityofLAEswithluminositiesintheinter-
the shape of the Lyα LF is not strongly constrained and a
val [L ±dL /2] is given by
fitwithaSchechterfunctionleadstosignificantdegeneracy α α
in the parameters. In particular the faint-end slope αLyα is MU(cid:90)V,max
essentially unconstrained: for example, Henry et al. (2012)
φ (L )dL =dL F dM φ(M )P(L |M )
usedasampleofsix(three)LAEstofindα =−1.70+0.73 LAE α α α UV UV α UV
Lyα −0.57
(α = −1.45+0.92) at z = 5.7. Other approaches include MUV,min
Lyα −0.70 (2)
assuming a fixed value for α and resorting to the data
Lyα Here,φ(M )dM denotesthenumberdensityofLBGsas
to constrain the other parameters (van Breukelen et al. UV UV
afunctionofintherangeM ±dM /2.Thisfunctioncan
2005;Dawsonetal.2007;Ouchietal.2008;Huetal.2010; UV UV
be represented by the Schechter function with parameters
Kashikawa et al. 2011; Ciardullo et al. 2012; Zheng et al.
(α ,M∗ ,φ∗ ).
2013).Incontrast,theultraviolet(UV)LFofLyman-break UV UV UV
The term P(L |M )dL is the conditional probabil-
galaxies(LBGs)ismuchbetterconstrainedduetoavailable α UV α
ity that a galaxy has a Lyα luminosity L given an abso-
data stretching over several orders of magnitude in lumi- α
lute UV magnitude M . This conditional probability can
nosity (McLure et al. 2010; Yan et al. 2011; Bouwens et al. UV
be recast in terms of the equivalent width (EW) prob-
2011;Bradleyetal.2012;Oeschetal.2012;Yanetal.2012;
ability density function P(EW|M ) as P(L |M ) =
Lorenzoni et al. 2013; Schenker et al. 2013). For the faint UV α UV
P(EW|M )∂EW ifEW >EW ,whereL andEW are
end slope, the most recent by Bouwens et al. (2014a) finds UV ∂Lα LAE α
related as L = EWL = EW[νL /λ]. Here, the luminos-
α =−1.91±0.09(α =−1.64±0.04)atz∼6(z∼4). α λ ν
UV UV ity/flux densities, frequency and wavelength are evaluated
justlongwardoftheLyαresonanceatλ=(1216+(cid:15))˚A.We
There exists a clear opportunity to connect LAEs and canextrapolatetheseflux/luminositiestotheirvalueswhere
LBGs via the Lyα line emission properties of LBGs. Shap- theUV-continuummeasurementsareusuallymade(seee.g.
ley et al. (2003) provided a probability distribution func- Dijkstra & Westra 2010)1. Furthermore, EWLAE denotes
tion (PDF) of the rest-frame equivalent-width (EW) of the the equivalent width threshold that determines whether a
Lyα line in their sample of ∼ 800 z ∼ 3 LBGs. Dijk- galaxy would make it into an LAE sample. We adopt that
stra & Wyithe (2012) showed that this observed PDF was EWLAE = 0˚A, but note that some surveys adopt colour
well described by an exponential function, and that the criteria for selecting LAEs as large as EWLAE = 64˚A
characteristicscale-lengthofthisfunctionincreasedtowards (see Dijkstra & Wyithe 2012). If EW (cid:54) EWLAE, then
fainter UV-luminosities. While there do not exist equally P(Lα|MUV)=0sinceinthiscasethegalaxydoesnotqual-
well measured PDFs at higher redshifts and/or fainter UV- ify as an LAE. This threshold more closely represents de-
luminosities, recent studies have constrained both the red- tectionthresholdforLyαemittinggalaxiesinspectroscopic
shift and UV-luminosity dependence of the so-called ‘Lyα surveys2 – e.g., with MUSE (Bacon et al. 2010), HETDEX
fraction’, which quantifies the fraction of LBGs for which (Hilletal.2008)and/orVIMOS (Cassataetal.2011,2015).
the Lyα EW exceeds a certain value. The Lyα fractions – Wehaveverifiedthatourmainresultsdonotdependonthis
which represent integrated versions of the full EW PDF – choice3.
increase from z = 2 to z = 6 at fixed M (Stark et al. TheprecedingfactorF inEq.(2)ismerelyanormaliza-
UV
2010; Cassata et al. 2015) and from UV-bright to UV-faint tionconstanttofitthedataand,hence,canbethoughtofas
galaxies(Starketal.2010,2011;Pentericcietal.2011;Ono theratioofpredictedversusthetotalnumberofLAEs.This
et al. 2012; Schenker et al. 2012). factorshouldideallybeF =1.However,Dijkstra&Wyithe
(2012) required that F ∼ 0.5. The origin of this number is
not known (see Dijkstra & Wyithe 2012, for an extensive
There have been several attempts to link the redshift discussion)4,butwestressitonlyaffectsthepredictednor-
evolutionofLBGsandtheirLyαfractionstoLAEluminos- malization linearly and not the predicted faint-end slopes.
ity functions (Dijkstra & Wyithe 2012; Faisst et al. 2014; Hence,thekeyfunctioninouranalysisisP(EW|M ).
UV
Schenker et al. 2014). In this paper, we follow the work
of Dijkstra & Wyithe (2012) and combine the most recent
constraints on UV-LFs & Lyα fractions to make predic- 1 We use the relation Lα = C1EWLUV,ν where LUV,ν ∝
tions for Lyα LFs. Dijkstra & Wyithe (2012) showed that ν−β−2 is the UV luminosity density LUV,ν and C1 ≡
this phenomenological model reproduces observed Lyα LFs να/λα(λUV/λα)−β−2 converts the flux density at λ = (1216+
andtheirredshiftevolutionremarkablywell.Here,wefocus (cid:15))˚A to that at λUV = 1600˚A, which is the wavelength where
specifically on the faint-end slope of the Lyα LF of LAEs, LUV,ν wasmeasured(Dijkstra&Westra2010).
because(i)wecanmakerobustpredictionsforthisfaintend 2 NotethatinpractiseanEWcutislikelystillneededtodistin-
guishbetweenLAEsandlower-z interlopers,suchas[OII]emit-
slope, (ii) as we will show later, this faint end slope can be
highlyrelevantforunderstandingtheEpochofReionization. ters.ThisEWcutcanneverthelessbelowerthanEWLAE∼20˚A
(Leungetal.2015)
3 We have verified that varying EWLAE in the range [0,50]˚A
Thispaperisstructuredasfollows.InSec.2,welayout changesαLyα by∼0.02.
4 The value of F depends weakly on the adopted UV Schechter
our method. We present our results in Sec. 3 and discuss
function parameters.Forexample,Bowleret al. (2014) reported
them in Sec. 4. Finally, we conclude in Sec. 5. The cosmo- slightlydifferentbest-fitvalues,whichdriveF uptoF ∼0.7−0.8.
logical parameters we adopt are Ωm = 0.3,ΩΛ = 0.7,h = TheFinkelsteinetal.(2014)parameters,ontheotherhand,also
0.7,σ8 =0.9. suggestF ∼0.5.
(cid:13)c 2015RAS,MNRAS000,1–8
Faint End Slopes of LAE and LBG LFs 3
Severalfunctionalformshavebeenexploredintheliterature.
��-� �=���
Schenker et al. (2014) compared the maximum likelihood
)
valuesforseveralEWdistributionstotheirKeckMOSFIRE �α
�
(McLeanetal.2012)data,andconcludedthattheexponen- -�� ��-� ● ●
tial distribution introduced by Dijkstra & Wyithe (2012) � ●
�
provides an adequate fit. This functional form is -� ����(��) ● ●
�
(cid:20) EW (cid:21) (�� ��-� ����(���) ●
P(EW|MUV,z)=Nexp −EWc(MUV,z) (3) Ψ ����(�����=����)
with EW = EW +µ (M +M )+µ (z+z )
c c,0 MUV UV UV,0 z 0 ���
wetheersr.eTµhMeUseV,paµrza,mMetUeVr,s0,wze0r,eacnhdosEenWtco,0maartechmtohdeelopbaserravma-- ΔΨ -����
tionsof(Shapleyetal.2003)andStarketal.(2010,2011)as ���� ���� ���� ���� ���� ���� ����
closely as possible. Furthermore, N is a normalization con- ��� (� /����-�)
�� α
stantwhichisforcedtobezerooutsideof[EW , EW ].
min max
Our choice of values for the model parameters is described
in Sec. 3.3 where we present the numerical results. In Ap- Figure 1. Upper panel: The predicted number of LAEs in the
pendix A we show explicitly that the main results in this range log10(cid:0)Lα/ergs−1(cid:1)±dlog10(cid:0)Lα/ergs−1(cid:1) using the UV
LF evolution from Bouwens et al. (2014a) taken at z = 5.7
paperareinsensitivetoboththefunctionalformofP(EW)
(blacksolidline).Thegreydashedlinemarksthefaintendslope
and the parameterization of EW .
c (αLyα = −1.90) and the dashed dotted lines show Schechter
fits to our numerical findings. Once the fit was carried out
over the whole shown luminosity range (blue) and once only
in log (L/ergs−1) = [42,43.5] (green). The red discs are the
10
3 RESULTS z=5.7observationsbyOuchietal.(2008)andtheblackarrows
denotetheMUSE deepfieldandmediumdeepfieldaswellasthe
We first present results in which EW =constant (in § 3.1).
c JWST limitsatthatredshift(seetextfordetails).Lowerpanel:
This allows us to demonstrate that for models in which the
Relativedeviationofthefitstothenumericalresults.
LyαfractiondoesnotevolvewithM ,thefaintendslope
UV
oftheLFofLAEsapproachesthatofLBGs.Wethenpresent
a simplified model in § 3.2 in which the mean Lyα EW- 3.2 Exemplary case where P(EW|MUV) evolves
PDFincreasestowardsfainterUV-luminosityfunction.This with MUV
model demonstrates quantitatively that the faint end slope
If EW depends on M a general analytic solution for
c UV
of the LF of LAEs is steeper than that of LBGs if the Lyα
φ(L )doesnotexist.Forillustrationpurposeswefirstcon-
α
fraction increases towards fainter UV-luminosities. In § 3.3
sider a case in which we replace Eq. (3) with a Dirac-δ dis-
we present the results that we obtained from the EW-PDF
tribution,
given in Eq. (3).
p(EW|L )∝δ(EW −EW ), (7)
UV d
where EW (L ) ≡ EW (L /L∗ )γ. The parameter
d UV d,0 UV UV
3.1 Exemplary case with EWc =const. EWd can be interpreted as the mean of the full PDF. This
δ-function PDF leads5 to
We consider the case P(EW|M )=P(EW), i.e. EW =
UV c
const. ≡ λ/C . Furthermore, we set N = 0 for EW <
1 φ(L )dL ∝dL ×LαUV/(γ+1)
EW . Under these assumptions we find α α α α
LAE (cid:34) (cid:18) L (cid:19)1/(γ+1)(cid:35)
φ(Lα)dLα ∝ (cid:90)∞LαU−V,1νexp(cid:20)−LLU∗V,ν − λLLα (cid:21) dLUV,ν ×exp − C1EWdα,0L∗UV . (8)
UV UV,ν
0 Here, the faint end slope is αLyα = αUV/(γ +1). he Lyα
(4) LF thus has a steeper faint-end slope than the LBG LF,
(cid:32) (cid:115) (cid:33) if γ < 0 (i.e. if EW decreases towards fainter L , as has
L d UV
∝LααUV/2K−αUV 2 λL∗α dLα (5) beenobserved).AlsonotethatweagainobtainαLyα =αUV
UV if EW does not evolve with M .
d UV
where K (x) is the modified Bessel function of the second
n
kind and L∗ is the luminosity corresponding to M∗ .
UV UV 3.3 Realistic case with P(EW|M ) inferred from
UV
Eq.(5)showsthattheLyαLFgenerallydoesnottake-
observations
on a Schechter form. The slope of the LF is given by
√ √ For the model parameters of P(EW|MUV) in Eq. (3) we
αLyα ≡ ddlolgoφg(LLα) =− yKKαUV(−21√(2y)y) (6) adoptthevaluesfromDijkstra&Wyithe(2012)6.Example
α αUV
with y≡Lα/(L∗UVλ). For Lα (cid:28)L∗UVλ we have y(cid:28)1, and 5 For simplicity, we set the minimum and maximum UV lumi-
weobtaintoleadingorderα ≈−Γ(1−α )/Γ(−α )=
Lyα UV UV nositytozeroandinfinity,respectively.
αUV. Thus, having a constant EWc corresponds to an un- 6 Specifically the model parameters related to EWc
changed faint end slope, aLyα =αUV. are given by (EWc,0,µMUV,µz,MUV,0,z0,F) =
(cid:13)c 2015RAS,MNRAS000,1–8
4 M. Gronke, M. Dijkstra, M. Trenti, S. Wyithe
�����=���������-���-� -��� �� �) � ��
� -��� -�
*/��� � ��α���=��������-� α --������ ��α -����� ����� ��α
� � ����
(
-��� *ϕ ����
� � � � � � � � � � � � � � � � � �
� � �
Figure2.EvolutionoftheSchechterparametersoftheUVLFfromBouwensetal.(2014a)(asbluedashedlines)andtheparametersof
thecomputedLyαLF(blacklines).Inparticular,theleftpanelshowsthecharacteristicluminosityL∗ (notethedifferentnormalization
constants),thecentralpanelthefaintendslopeα,andtherightpaneltheoverallnormalizationφ∗.Predictionsatz>6(withintheshaded
grey area) do not account for reionization (see § 4.3). In this region, the black solid lines correspond to models with an uninterrupted
EWevolution,whereasthedashed-dotted linesrepresentamodelinwhichwefreezetheEWevolution,i.e.,EWc(z>6)=EWc(z=6)
(thisassumptionhasbeenadoptedinpreviousworks).
EW-PDFs are shown in Appendix A1. For a more detailed
��� �
motivation of this P(EW) we refer the reader to Dijkstra ��� �� α
& Wyithe (2012). We integrate the UV-LF over the range ��
)
MUV,(min,max) =(−30, −12) when predicting Lyα luminos- �α ���
ity functions, and discuss the impact of varying M in |� ��
Sec. 4. max �� ��� ����
The redshift evolution of the best fit Schechter param- (� ��� (�∼�) ��
eters of the UV LF is taken from Bouwens et al. (2014a)
��
and given as M∗ = −20.89 + 0.12z, φ∗ = 0.48 × ���
UV UV
10−0.19(z−6)10−3cMpc−3, and, α = −1.85−0.09(z−6). -�� -�� -�� -�� -�� -��
UV
Followingtheseanalyses,weuseλ =1600˚Aasrestframe
UV �
��
wavelength in which the UV continuum was measured and
assume a UV spectral slope β = −1.7. This choice for β Figure 3. p(MUV|Lα) ∝ p(Lα|MUV)φ(MUV) versus MUV for
does not affect our results (see Appendix A4 for detailed someexemplaryvaluesofLα.TheblackarrowshowstheJWST
discussion). photometric limit quoted by Windhorst et al. (2006) for z ∼ 6
The upper panel of Fig. 1 shows the resulting number whichcorrespondstoTint.∼106sintegrationtime.
densityofLAEsatz=5.7intheluminosityrangelog L ±
10 α
dlog L /2,i.e.,ψ(L )dlog L ,asafunctionofL .This
10 α α 10 α α
quantity is related to φ(L ) as ψ(L ) = φ(L )L log10
α α α α
(‘log’denotesthenaturallogarithm).Wecomparethesepre-
dictiontothedatafromOuchietal.(2008).Inaddition,we
show theMUSE detectionlimits7 foritsmediumdeep field
(MDF, limiting flux F > 1.1×10−18ergs−1cm−2, integra- ThiscanalsobeseeninthelowerpanelofFig.1,wherewe
display the relative deviation of the fits to the LF.
tion time T = 10h), and, deep field (DF, F > 3.9 ×
int.
10−19ergs−1cm−2,T =80h)surveysaswellasanexem- Fig.2showstheredshiftevolutionoftheSchechterbest
int.
plary JWST8 limit (F (cid:38)10−18ergs−1cm−2, T =104s). fit parameters (as black lines). Predictions for z >6 do not
int.
account for reionization effects and are calculated with an
Figure.1alsoshowstwoSchechterfunctionapproxima-
unaltered EW evolution (solid line) as well as an EW-PDF
tionstoournumericalfindingsfittedoverthefullluminosity-
range shown (in blue) and over log (L /ergs−1) = which does not evolve after z = 6 (dash-dotted line). We
10 α
discuss this result separately in § 4.3. For comparison, we
[40.5,42](ingreen).Althoughwedonotexpecttheresulting
plot the corresponding redshift parameterization of the UV
LyαLFtobeaSchechterfunction(asshowninSec.3.2),it
LF by Bouwens et al. (2014a) (as blue dashed lines). The
providesareasonablefitoverthedisplayedluminosityrange.
left panel shows that L∗ increases by a factor ∼ 2 over the
redshift range z = 3−6, which differs from the redshift
(23˚A,7˚A,6˚A,21.9,−4.0,0.53). The EW-PDF covers the evolution in the characteristic UV-luminosity which drops
range [EWmin,EWmax]. Here, the lower limit EWmin ≡ −a1, by 20%. This difference is driven by the redshift evolu-
where a1(MUV) follows the form a1 = 20˚A for MUV < −21.5, tion in the Lyα EW-PDF, which in turn was inferred from
a1 = (20−6(MUV +21.5)2)˚A for −21.5 (cid:54) MUV (cid:54) −19.0 and the observed redshift-evolution of Lyα ‘fractions’ over this
a1 =−17.5˚A,otherwise(seeDijkstra&Wyithe2012).Weused redshift range. The central panel shows that α < α .
EWmax=1000˚Abutweverifiedthatthischoicedoesnotaffect This is again a consequence of inferred redshiLftyαevolutUioVn
ourresultsquantitatively.
7 MUSE surveylimitstakenfromhttp://muse.univ-lyon1.fr/ of the Lyα-EW PDF (see § 3.2). This figure also illustrates
theclose-to-linearα –z relation.Thisevolutionismostly
IMG/pdf/science_case_gal_formation.pdf. Lyα
8 JWST survey limits obtained from http://www.stsci.edu/ driven by the redshift evolution of αUV. Finally, the right
jwst/science/sensitivity/ panel shows the predicted redshift evolution in φ∗.
(cid:13)c 2015RAS,MNRAS000,1–8
Faint End Slopes of LAE and LBG LFs 5
4 DISCUSSION )�α ��� �=��� �������
4.1 Low-L turnover -� ■ ■ -�
Theintegraloverφ(Lα)dLα divergesforα<−2.Wethere- ��� ��-� ■ -��
�
fore expect that the luminosity function flattens or turns- -� ���� -��
over below some luminosity. The minimum luminosity that �� ��-� (���)
-��
we can account for in our models is (� ����
Ψ (��) -��
L =EW (M ) C L , (9) ��-�
α,min min UV,max 1 UV,min ���� ���� ���� ���� ����
where EWmin = −a1 = 17.5˚A (see footnote 6 in § 3.3) � (����-�)
denotes the minimum equivalent width in our EW-PDF at α
the maximum absolute UV-magnitude (i.e. the lowest UV-
Figure 4. ψ(Lα)dLα versus Lα at z = 3.1 for different values
luminosity). For example, we obtain Lα,min ∼ 1039ergs−1 of MUV,max illustrating the cutoff at low Lyα luminosities. The
for MUV,max = −12. At this luminosity we expect the pre- dashedanddottedlinesshowthe“cutoff”and“deviation”points
dictedLyαluminositytogotozero,asisshowninFigure4. foreachMUV,maxdiscussedinSec.4.DatapointsarefromRauch
An estimate for where we may start to see departures et al. (2008) taken at that redshift (see the text for a discussion
from a power-law slope can be obtained by considering the on the error bars). And the black (grey) arrow denotes planned
conditional probability p(M |L ). Bayes’ theorem states futureMUSE (ultra)deepfieldlimits.
UV α
that p(M |L ) ∝ φ(L |M )φ(M ), of which we show
UV α α UV UV
examples in Fig. 3 for four different values of L . This Fig-
α 4.2 Implications for the Epoch of Reionization
ureillustratesforexamplethatLyαobservationsthatprobe
afluxcorrespondingtoLα =1040 ergs−1 –alevelthatcan Lowluminositygalaxiesareexpectedtoplayamajorrolein
be reached in MUSE ultra deep fields – effectively probe drivingthereionizationoftheUniverse(e.g.Robertsonetal.
galaxies with −14 < MUV < −11, which are fainter than 2010; Trenti et al. 2010; Kuhlen & Faucher-Gigu`ere 2012;
can be probed directly even with the JWST. The JWST Boylan-Kolchinetal.2014).Determiningthefaintendofthe
detection limit shown in Figure 3 is taken from Windhorst LF such as its slope and a turnover luminosity is essential
etal.(2006).Figure3furthershowsthatiftheUV-LFflat- for constraining the volume emissivity of ionizing photons.
tens off at – say – MUV >∼−12 that then the effects should However, even future experiments will have difficulties de-
becomenoticeableinthepredictedLyαluminosityfunction tectingthesegalaxiesdirectlyviatheirUVcontinuumflux.
aroundLα =1040ergs−1,asheregalaxieswithMUV ∼−12 Currentconstraintsrely,therefore,onextrapolationoflocal
dominate the contribution to the Lyα LF. propertiestohigherredshifts(Weiszetal.2014),(relatively
In Figure 4 we make these points more explicit, and few) gravitationally lensed objects (Alavi et al. 2014; Atek
show the predicted faint end of the LAE LF for four val- etal.2014)orinferencesfromgamma-rayburstobservations
ues of MUV,max (calculated with the UV LF parameters at (Trenti et al. 2012). In this work, we have shown that the
z=3.1).ForeachcurvewemarkedLα,minwithdottedlines. Lyα LF can provide an independent probe of the faint end
Forexample,apotentialUVturnoveratMUV ∼−12leads of the UV LF, and that for example the MUSE DF survey
todeviations9oftheLyαLFatLα ∼1040ergs−1andacut- couldalreadydetect(orruleout)aturnoveratMUV (cid:46)−15.
off at Lα ∼1039ergs−1. Figure 4 also contains data points RecentstudieshaveshownthatLyαescapemaybecor-
taken from Rauch et al. (2008). Rauch et al. (2008) per- related with the escape of ionizing photons (Behrens et al.
formed an ultra deep (92-hr) exposure with VLTs FORS2 2014;Verhammeetal.2014),astheescapeofionizingpho-
low resolution spectrograph. The goal of these observations tons requires low HI-column density (N < 1017 cm−2)
HI
was to detect fluorescent Lyα emission from optically thick channels,whichcanalsoprovideescaperoutesforLyαpho-
cloudspoweredbytheionizingbackground.Whiletheirsen- tons.ThefactthatLyαLFsarelikelysteeperthantheUV
sitivity turned out not to be good enough to detect this LFsimpliesthattheLyαvolumeemissivity–andtherefore
fluorescentemission(revisedestimatesoftheionizingback- possiblytheionizingemissivity–areweightedmorestrongly
ground and the conversion efficiency into Lyα), they de- towardslowluminositygalaxies.Thisisconsistentwiththe
tected numerous ultra faint Lyα emitting sources charac- expectation that ionizing photons escape more easily from
terizing their LF down to Lα ∼ 6×1040ergs−1. We com- lowermass–andhencelowerluminosity–galaxies.Asteep
puted the uncertainties with the cosmic variance calcula- faint-end slope of the Lyα LF may therefore provide obser-
tor of Trenti & Stiavelli (2008). These data-points fall on vational support for this scenario.
the predicted LF for M = −16. However we caution
UV,max
thattheturn-overoccursatthelowestluminositydata-point
only, which might suffer from incompleteness (although it 4.3 Predictions for redshifts z=6−8
lies above the detection threshold). In the same figure we
We extrapolated our predictions for the best-fit Schechter
provide the estimated MUSE limits for the deep field (DF)
parameters of the LAE LF to z >6 in two ways (shown in
and the gravitationally lensed ultra deep field (UDF) sur-
Fig.2):(i)inthefirst,weassumethattheEW-PDFcontin-
veys.
uestoevolveasinferredfromtheobservationsatz=3−6.
9 The first deviations in the Lyα LF can be found at Lα,dev. = describes the half width of p(MUV|Lα) at a chosen probability
C1LUV(x)EWc(x) with x ≡ MUV,max−∆MUV. Here, ∆MUV threshold.
(cid:13)c 2015RAS,MNRAS000,1–8
6 M. Gronke, M. Dijkstra, M. Trenti, S. Wyithe
Thismodelisrepresentedbythesolid lines,and,(ii) inthe ACKNOWLEDGEMENTS
second, we ‘freeze’ the EW distribution for z > 6 at the
We thank Masami Ouchi for kindly providing the data
value it had at z=6 (dashed lines). This latter assumption
points shown in Fig. 1. MD and MT thank Alan Dressler
hasbeencommoninpreviousworks(seee.g.Dijkstraetal.
givingapresentationwhichinspiredthisworkattheUCSB
2011; Bolton & Haehnelt 2013; Jensen et al. 2013; Choud-
GLASS meeting in May 2014.
hury et al. 2014; Mesinger et al. 2015). We show results for
thesetwomodelstogetasensefortheuncertaintiesonour
predictions. We stress that we have purposefully not mod-
elled the impact of reionization on the EW-PDF. Reion-
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APPENDIX A: VARYING THE EW
ShapleyA.E.,SteidelC.C.,PettiniM.,AdelbergerK.L.,
DISTRIBUTION
2003, ApJ, 588, 65
Shibuya T., Kashikawa N., Ota K., Iye M., Ouchi M., Fu- In this appendix, we demonstrated that our main results
rusawaH.,ShimasakuK.,HattoriT.,2012,ApJ,752,114 and conclusions do not depend on our assumed EW-PDF.
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Steidel C. C., Giavalisco M., Pettini M., Dickinson M.,
A1 Fiducial P(EW)
Adelberger K. L., 1996, ApJ, 462, L17
Tilvi V., et al., 2014, ApJ, 794, 5 OurdefaultEWdistributionisgivenbyEq.(3).Weplotthe
Trenti M., Stiavelli M., 2008, ApJ, 676, 767 PDFforthreeUVmagnitudesandtworedshiftsinFig.A1.
Trenti M., Stiavelli M., Bouwens R. J., Oesch P., Shull
(cid:13)c 2015RAS,MNRAS000,1–8
8 M. Gronke, M. Dijkstra, M. Trenti, S. Wyithe
A2 Schenker et al. (2014) parameterization ��-�
�=���
As mentioned in Sec. 2, Schenker et al. (2014) suggest an )
α
alternative parameterization of the EW-PDF, namely �
�
A (cid:20) (logEW −µ(β))2(cid:21) -�
P(EW|β)= √2πσemEW exp − 2σ2 . (A1) ��� ��-� ● ●
-
This log-normal PDF possesses the parameters Aem, σ and ��� Schenker+14(1) ●
µ. The latter is given by � Schenker+14(2) ●
µ(β)=µ +µ (β−2.0), (A2) (Ψ MUVfreeze ●
α s βnotconst. ●
where β is the UV continuum slope. Schenker et al. (2014) ��-�
found their EW distribution to depend more strongly on β ���� ���� ���� ���� ���� ���� ����
thanon(MUV,z),andthereforeconstrainedP(EW|β).We �����(�α/����-�)
can include this parameterization into our formalism if we
map P(EW|β) onto P(EW|M ,z).
UV Figure A2. The Lyα LF at z = 5.7 for different EW distribu-
This mapping is based on three results from Bouwens tions. The solid black line and Ouchi et al. (2008) data points
et al. (2014b): arethesameasshowninFig.1.AscomparisonweshowtheLyα
LFcomputedusingtheSchenkeretal.(2014)parameterizationof
(i) We use their empirical linear correlation between β−
theEWPDFoncewiththefullβ−MUVrelationasgivenin§A2
MUV at z∼7. This relation constrains β(MUV =−19.5)= (blue dashed line) and once with a constant µ(β) =−0.2 over
−2.05±0.09±0.13. MUV
(ii) Furthermore, we apply their measured change per all MUV (green dashed-dotted line). Also shown are our results
when freezing the EW PDF at MUV = −19 (purple) and when
unit redshift ∆β/∆z=−0.1±0.05.
usingtheβ−MUV relationinsteadofaconstantβ (orange).
(iii) Finally,weusetheirmeasurementthat∆β/∆M =
UV
−0.2 (−0.08) for M (cid:54)−19 (>−19).
UV
troduces some additional dispersion in the predicted Lyα
Accordingly, our mapping can be written as
fluxatafixedM .However,varyingβ within[−2.0, −1.5]
UV
β =β +µ(β) (M +19)+µ(β)(z−7), (A3) changestheLyαfluxonlyby1−(λ /λ )1.5−2.0 ∼13%.
0 MUV UV z UV Lyα
ThisdispersionissmallerthanthatintroducedbytheEW-
where µ(β) ≡∆β/∆M and µ(β) ≡∆β/∆z.
MUV UV z PDF. If we replace the constant β with the empirical fit
In Eq. (A1) we used the best parameters by
described in § A2, then our predicted Lyα LF (represented
Schenker et al. (2014), i.e., (A , σ, µ , µ ) =
em α s by the orange line in Fig. A2) is barely any different from
(1.0, 1.3,2.875, −1.125). In addition, since A is de-
em our fiducial model that used β = −1.7 (represented by the
generate with F, we set F = 1. The orange dashed line
black solid line).
in Fig. A2 shows the resulting Lyα LF at z = 5.7. The
agreement in the faint-end between the two procedures is
remarkable.Forgreaterluminosities,however,theSchenker
et al. (2014) parameterization leads to a (much) higher
number density of LAEs. While there are significant
uncertainties in the above procedure, the agreement we
get at the faint end slope is especially encouraging. Future
surveys can be extremely useful in further connecting the
LAE and LBG populations by constraining the bright end
of the LAE luminosity function.
A3 ‘Freezing’ the EW PDF for faint galaxies
Since the evolution in the EW PDF for fainter sources in-
volves a (modest) extrapolation of observationally inferred
P(EW), we have also tested and alternative PDF where
we ‘froze’ the evolution at M = −19. That is, we also
UV
conservatively assume that the EW-PDF stops evolving at
M >−19(eventhoughobservationshintthatthisisnot
UV
the case, see fig. 13 of Stark et al. (2010)). Fig. A2 shows
the resulting Lyα LF (purple line). It is clear that our re-
sultsareonlyaffectedslightly,i.e.,thefaint-end-slopeα
Lyα
is reduced by ∼ 0.05. We also tested this over a variety of
redshifts.
A4 Non-constant UV spectral slope
The spectral slope β is not a constant, but depends on
UV magnitude and redshift (as discussed above). This in-
(cid:13)c 2015RAS,MNRAS000,1–8