THE OPTICALLY UNBIASED GRB HOST (TOUGH) SURVEY. VII. THE HOST GALAXY LUMINOSITY FUNCTION: PROBING THE RELATIONSHIP BETWEEN GRBs AND STAR FORMATION TO REDSHIFT ∼6

The TOUGH survey targeted 69 GRBs in the R and K_s bands with FORS2 and ISAAC reaching limiting magnitudes of R(AB ) ∼ 27.3 mag and ${K}_{s}(\mathrm{AB})\sim 23.4$ mag at a 3σ confidence level. About 80% have a detected host galaxy (Hjorth et al. 2012; see also D. Malesani et al. 2015, in preparation) and thanks to ongoing spectroscopic follow-up observations $\gt 87\%$ now have a measured redshift (Jakobsson et al. 2012; Krühler et al. 2012, 2015). In addition, the hosts between z = 2 and z = 4.5 were targets of a moderately deep spectroscopy campaign to study Lyα in emission, and hosts at $z\lt 1$ were part of a radio survey. These campaigns allowed the investigation of several properties such as the $(R-{K}_{s})$ color and the offset distribution (Hjorth et al. 2012; D. Malesani et al. 2015, in preparation), the redshift distribution (P. Jakobsson et al. 2012), the Lyα recovery rate (Milvang-Jensen et al. 2012), and the unobscured SFR inferred from radio observations for the hosts at $z\lt 1$ (Michałowsk et al. 2012).

The VLT R-band data build the foundation of this paper. We complement this data set with HST observations via a dedicated GRB host program (PI: A. J. Levan), which targeted nearly all $3\lt z\lt 4$ hosts. At z ≲ 0.9 the TOUGH R-band data probe the rest-frame optical, and for these hosts we obtained data in bluer filters with TNG/LRS, Keck/LRIS, GTC/OSIRIS, and Gemini-S/GMOS, or used archival data. A log of the data is shown in Table 1. In addition, for the fields of GRBs 050803 and 050922B (two GRBs with uncertain redshifts), we succeeded in obtaining multi-band data (Table 1) probing the spectral energy distribution (SED) from the rest-frame UV to the near-IR (NIR). Their Spitzer observations are described in detail in Perley et al. (2015a, 2015b).

To secure the field calibration of GRB 060805A we used the 60 inch Palomar telescope, and for GRB 060729 we used the Gamma-ray Optical/Near-infrared Detector (GROND; Greiner et al. 2008) mounted at the MPG/ESO 2.2 m telescope.

Furthermore, we incorporated measurements reported in the literature; specifically we used Perley et al. (2009) for GRB 050416A, Chen et al. (2009) for 050820A, Hjorth et al. (2012) for the K_s band photometry for GRBs 050803 and 050922B, Mangano et al. (2007) for 060614, Krühler et al. (2011) for 070306, and Tanvir et al. (2012) for GRBs 050904, 060522, and 060927.

The ground-based data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding) with dedicated software packages (Keck: customized IDL routine, Gemini: Gemini IRAF package GROND: customized pipeline (for details see Krühler et al. 2008; Yoldaş et al. 2008), and all other data with IRAF; Tody 1986). HST observations, after standard "on-the-fly" processing, were subsequently cleaned for bias striping, introduced due to the replacement electronics after Servicing Mission 4 (2009 May 11–24), and then drizzled with the multidrizzle software (Koekemoer et al. 2003) into final science images. The reduction of the Spitzer data is described in Perley et al. (2015b).

The photometry for the ground-based images was performed as described in D. Malesani et al. (2015, in preparation): if a host was detected at $\gt 5\sigma$ confidence level, we chose a sufficiently large aperture to measure the total flux. For hosts detected at a lower confidence level, we reduced the radius to the stellar FWHM and applied an aperture correction derived from the brighter hosts in the sample.

Once an instrumental magnitude was established it was photometrically calibrated against zeropoints (GRB 051117B), against the brightness of a photometric standard star (GRB 050525A), a number of field stars measured in a similar way (GRBs 060729), or tied to the SDSS DR8 (the rest; Aihara et al. 2011). In some cases, we converted the SDSS photometry into the Bessel system using Lupton,¹⁵ if needed. These measurements were finally corrected for Galactic extinction using the extinction maps by Schlegel et al. (1998) and transformed into the AB system using Blanton & Roweis (2007) and Breeveld et al. (2011). These magnitudes are listed in Table 2.

Table 2.  UV Properties of the TOUGH Sample

			m		${M}_{1600\;\mathring{\rm{A}}}$ b				m		${M}_{1600\;\mathring{\rm{A}}}^{{\rm{b}}}$
GRB	Redshifta	Filter	(mag)	${\beta }_{\mathrm{UV}}$	(mag)	GRB	Redshift	Filter	(mag)	${\beta }_{\mathrm{UV}}$	(mag)
Hosts with Known Redshifts
050315	1.95	R	24.51 ± 0.15	−1.54 ∓ 0.03	−20.08 ± 0.14	060707	3.42	R	25.01 ± 0.06	−1.62 ∓ 0.01	−20.82 ± 0.06
050318	1.444	R	$\gt 26.95$	$\lt -1.92$	$\gt -17.16$	060708	1.92 ± 0.12	R	26.94 ± 0.28	−1.89 ∓ 0.03	−17.75 ± 0.27
050401	2.898	R	26.19 ± 0.31	−1.79 ∓ 0.04	−19.31 ± 0.31	060714	2.71	R	26.46 ± 0.28	$-{1.82}_{+0.04}^{-0.03}$	−18.91 ± 0.28
050406	${2.7}_{-0.41}^{+0.29}$	R	26.76 ± 0.34	−1.86 ∓ 0.04	−18.60 ± 0.34	060719	1.53	R	24.81 ± 0.12	−1.64 ∓ 0.02	−19.27 ± 0.11
050416A	0.653	g'	24.00 ± 0.03	−1.73 ∓ 0.01	−18.26 ± 0.03	060729	0.54	g'	25.30 ± 0.16	−1.92 ∓ 0.02	−16.66 ± 0.15
050502Bc	5.2 ± 0.3	R	$\gt 25.98$	$\lt -1.85$	$\gt -20.61$	060805Ae	0.60	g'	24.61 ± 0.07	−1.83 ∓ 0.01	−17.53 ± 0.06
050525A	0.606	g'	$\gt 25.83$	$\lt -1.95$	$\gt -16.40$	060805Ae	2.36	R	25.26 ± 0.14	−1.65 ∓ 0.02	−19.79 ± 0.13
050714B	2.438	R	25.51 ± 0.20	−1.69 ∓ 0.03	−19.61 ± 0.19	060814	1.92	R	22.96 ± 0.11	−1.19 ∓ 0.03	−21.46 ± 0.10
050730	3.969	F775W	$\gt 27.50$	$\lt -1.98$	$\gt -18.58$	060908	1.88	R	25.66 ± 0.18	−1.74 ∓ 0.03	−18.93 ± 0.17
050801	1.38 ± 0.07	R	$\gt 26.74$	$\lt -1.91$	$\gt -17.26$	060912A	0.94	B	22.59 ± 0.06	$-{1.33}_{+0.02}^{-0.01}$	−20.39 ± 0.06
050819	2.504	R	23.99 ± 0.09	−1.39 ∓ 0.02	−21.13 ± 0.09	060923Af	${2.47}_{-0.52}^{+0.33}$	R	26.12 ± 0.24	−1.78 ∓ 0.03	−19.05 ± 0.23
050820A	2.615	F775W	26.30 ± 0.06	−1.80 ∓ 0.01	−18.96 ± 0.06	060923B	1.51	R	24.15 ± 0.16	−1.53 ∓ 0.03	−19.83 ± 0.14
050822	1.434	R	24.36 ± 0.08	−1.59 ∓ 0.01	−19.54 ± 0.07	060927	5.46	F110W	$\gt 28.23$	$\lt -2.10$	$\gt -18.39$
050824	0.828	B	24.11 ± 0.16	−1.67 ∓ 0.03	−18.71 ± 0.15	061007	1.26	R	24.56 ± 0.17	−1.65 ∓ 0.03	−19.08 ± 0.15
050904	6.295	F850LP	$\gt 26.27$	$\lt -1.95$	$\gt -20.58$	061021	0.35	g'	26.40 ± 0.16	−2.07 ∓ 0.01	−14.65 ± 0.15
050908	3.347	F775W	${27.55}_{-0.24}^{+0.30}$	−1.96 ∓ 0.03	$-{18.22}_{-0.24}^{+0.30}$	061110A	0.76	V	25.25 ± 0.10	−1.85 ∓ 0.01	−17.41 ± 0.09
050915A	2.527	R	24.70 ± 0.16	−1.54 ∓ 0.03	−20.47 ± 0.15	061110B	3.43	F775W	${27.02}_{-0.21}^{+0.26}$	$-{1.91}_{+0.02}^{-0.03}$	$-{18.79}_{-0.21}^{+0.26}$
050922C	2.199	R	$\gt 26.29$	$\lt -1.81$	$\gt -18.64$	061121	1.32	R	22.84 ± 0.03	−1.31 ∓ 0.01	−20.67 ± 0.03
051001	2.43	R	24.53 ± 0.13	−1.51 ∓ 0.03	−20.55 ± 0.12	070103	2.62	R	24.21 ± 0.14	−1.43 ∓ 0.03	$-{21.03}_{-0.14}^{+0.13}$
051006	1.059	R	23.03 ± 0.07	−1.43 ∓ 0.01	−20.03 ± 0.06	070110	2.35	R	25.19 ± 0.11	−1.64 ∓ 0.02	−19.84 ± 0.11
051016B	0.936	g'	23.13 ± 0.03	−1.46 ∓ 0.01	−19.87 ± 0.03	070129	2.34	R	24.23 ± 0.12	−1.45 ∓ 0.03	−20.75 ± 0.11
051117B	0.481	u'	22.91 ± 0.19	−1.65 ∓ 0.03	−18.67 ± 0.18	070224	1.99	R	26.02 ± 0.31	−1.78 ∓ 0.04	$-{18.71}_{-0.29}^{+0.30}$
060115	3.533	F814W	${27.21}_{-0.21}^{+0.27}$	−1.94 ∓ 0.03	−18.65 ± 0.27	070306	1.50	g'	22.90 ± 0.09	−1.18 ∓ 0.03	−21.21 ± 0.09
060218	0.034	u'	20.61 ± 0.12	−2.02 ∓ 0.01	−15.20 ± 0.11	070318	0.84	R	24.60 ± 0.11	−1.76 ∓ 0.01	−18.18 ± 0.10
060306d	1.559	R	24.21 ± 0.08	−1.54 ∓ 0.02	−19.86 ± 0.07	070328	2.06	R	24.55 ± 0.13	−1.54 ∓ 0.03	−20.17 ± 0.12
060522	5.11	F110W	$\gt 27.82$	$\lt -2.05$	$\gt -18.69$	070419B	1.96	R	25.20 ± 0.20	−1.66 ∓ 0.03	−19.44 ± 0.19
060526	3.221	F775W	$\gt 27.52$	$\lt -1.95$	$\gt -18.18$	070506	2.31	R	26.21 ± 0.22	−1.79 ∓ 0.03	−18.83 ± 0.21
060604	2.136	R	25.62 ± 0.18	−1.71 ∓ 0.03	−19.23 ± 0.17	070611	2.04	R	$\gt 27.27$	$\lt -1.92$	$\gt -17.55$
060605	3.773	F775W	${27.48}_{-0.29}^{+0.40}$	−1.97 ∓ 0.03	$-{18.50}_{-0.29}^{+0.40}$	070721B	3.63	$F775W$	${27.69}_{-0.33}^{+0.47}$	$-{1.99}_{+0.03}^{-0.04}$	$-{18.22}_{-0.33}^{+0.47}$
060607A	3.075	R	$\gt 28.05$	$\lt -1.99$	$\gt -17.57$	070802	2.45	R	25.25 ± 0.21	−1.64 ∓ 0.04	−19.88 ± 0.20
060614	0.125	u	24.71 ± 0.30	−2.10 ∓ 0.01	−14.06 ± 0.29
Hosts with Unknown Redshiftsg
050726	3.5	R	$\gt 26.19$	$\lt -1.81$	$\gt -19.68$	061004	3.5	R	$\gt 25.84$	$\lt -1.76$	$\gt -20.03$
050803	3.5	i'	26.29 ± 0.50	$-{1.83}_{+0.07}^{-0.06}$	−19.55 ± 0.50	070330	3.5	R	$\gt 26.19$	$\lt -1.81$	$\gt -19.67$
050922B	3.5	i'	25.20 ± 0.15	−1.67 ∓ 0.03	−20.63 ± 0.15	070621	3.5	R	25.85 ± 0.23	$-{1.77}_{+0.04}^{-0.03}$	−20.02 ± 0.23
060919	3.5	R	25.80 ± 0.26	−1.76 ∓ 0.04	−20.07 ± 0.26	070808	3.5	R	26.85 ± 0.33	−1.90 ± 0.04	−19.01 ± 0.33
060923C	3.5	R	25.49 ± 0.18	−1.71 ∓ 0.03	−20.38 ± 0.18
New Photometric Redshiftsh
050803	3.5 ± 0.5	i'	26.29 ± 0.5	$-{1.83}_{+0.07}^{-0.06}$	−19.55 ± 0.50	050922B	4.5 ± 0.5	i'	25.20 ± 0.15	−1.69 ∓ 0.03	−21.13 ± 0.15

Notes. All magnitudes are corrected for Galactic extinction and were converted into the AB system. For the non-detections we report the 3σ limiting magnitudes. Redshifts were taken from Hjorth et al. (2012), Perley et al. (2013), and Krühler et al. (2015).

^aFor hosts with photometric redshifts, we report the values at the nominal redshifts. ^bThe absolute magnitude was computed through ${M}_{1600\;\mathring{\rm{A}}}$ = $m-\mathrm{DM}(z)-2.5\left({\beta }_{\mathrm{UV}}+2\right)$ $\mathrm{log}\left((1+z)1600\;\mathring{\rm{A}}/{\lambda }_{\mathrm{obs}}\right)$ + $2.5\;\mathrm{log}(1+z)$ where DM is the distance modulus for the assumed cosmology. ^cLuminosity includes a correction for IGM and Lyα absorption. ^dThe Hα emission line is double peaked with the centers of the peaks at z = 1.5585 and z = 1.5597. Without loss of generality we assume z = 1.559. ^eThe host identification is not unique. ^fThe redshift range of 060923A was constructed by combining limits from Jakobsson et al. (2012) and Perley et al. (2013). ^gFor hosts with redshift limits, we report the UV properties at z = 3.5. For the hosts of GRBs 060919, 060923C, 070621, and 070808, the luminosities are strictly speaking upper limits. Note, the redshift of 060923C is between z = 0.86 and z = 3.5. For GRBs 050726 ($z\lt 5.5$), 050803 ($z\lt 6.1$), 050922B ($z\lt 6.1$), 061004 ($z\lt 10$), and 070330 ($z\lt 5.5$) we report their luminosities at z = 3.5 to avoid corrections for IGM and Lyα absorption. ^hListed are the two hosts for which new redshift constraints were obtained through SED fitting. As discussed in Section 3.3 a broad range of physical galaxy parameters and redshifts fits the data within their uncertainties. The likely redshift of GRB 050803 is z ∼ 3.5 and of GRB 050922B z ∼ 4.5.

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The R-band observation of the host galaxy of GRB 050502B is affected by Lyα absorption in the host galaxy and in the intergalactic medium (IGM). To quantify this attenuation, we compared the observed to the expected $(R-I)$ color of the afterglow. Afonso et al. (2011) reported that the X-ray-to-optical SED of the afterglow can be described by a simple power law, ${F}_{\nu }\propto {\nu }^{-\beta }$, with a spectral index of β ∼ 0.9. For this model, the expected ${(R-I)}_{\mathrm{AB}}$ color is 0.22 mag, i.e., the emission received in the R band is dimmed by 0.95 mag, assuming no reddening at the GRB site. Throughout the paper we assume that the host galaxy has the same attenuation. The values reported in Table 2 include this correction.

For the HST images we measured the background-subtracted flux within an aperture with a radius of 025. To quantify the measurement error, we randomly distributed 40 apertures of the same radius within 3'' from the optical afterglow. These apertures had a minimum distance of 06 from any object to avoid source contamination. After that, we computed the standard deviation as a proxy for the photometric error. To account for flux losses we applied aperture corrections that were calculated from the encircled energy (Sirianni et al. 2005). If no host was detected, we measured the 3σ limiting magnitude via ${m}_{\mathrm{limit}}=23.9-{\mathrm{AP}}_{\mathrm{cor}}-2.5\;\mathrm{log}\left({F}_{\nu }+3\sigma \left({F}_{\nu }\right)\right)$ where ${\mathrm{AP}}_{\mathrm{cor}}$ is the aperture correction and ${F}_{\nu }$ is the formal flux density in μJy measured at the position of the optical afterglow with its 1σ error $\sigma \left({F}_{\nu }\right)$.

At z ∼ 3, an angular size of 05 translates to a physical scale of ∼4 kpc. This diameter is at the lower end of the observed size distribution of LBGs (Hathi et al. 2008). Increasing the aperture radius to 03, the average size of an LBG at z ∼ 3 leads to no significant increase in flux. On the contrary, the measurement would have been affected by neighboring objects if the radius exceeded 04.

The analysis of the Spitzer data is described in detail in Perley et al. (2015b); in brief, after downloading the processed PBCD images from the Spitzer Legacy Archive,¹⁶ we modeled and subtracted nearby contaminating sources, and then measured the flux of the host via aperture photometry and used the IRAC handbook zeropoints to convert the instrumental to apparent magnitudes.¹⁷

D. Malesani et al. (2015, in preparation) describe in detail how the hosts were identified in the deep VLT images. The additional data obtained with ground-based telescopes have a similar spatial resolution and do not exceed the limiting magnitudes of the VLT images. In contrast the HST images exceed the VLT images in spatial resolution and depth (see Figure 1). This necessitates repeating the host identification. In the following we can limit the discussion to the z = 3–4 hosts, whereas the host identification of GRB 050820A is discussed in Chen et al. (2009) and Chen (2012), and of GRBs 050904, 060522, and 060927 in Tanvir et al. (2012).

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To identify the most likely host galaxy candidate, we chose the probabilistic approach by Bloom et al. (2002) (For a detailed discussion, see also Perley et al. 2009). This method is based on the observed galaxy density from Hogg et al. (1997) and quantifies the chance probability ${p}_{\mathrm{ch}}$ of finding a galaxy with a certain magnitude and a certain distance from the GRBs. This chance probability is given by

$$\begin{eqnarray*}&&{p}_{\mathrm{ch}}=1-\mathrm{exp}\left(-\pi \times {r}_{\mathrm{eff}}\times \sigma (m)\right)\end{eqnarray*}$$

where ${r}_{\mathrm{eff}}$ is the effective radius and $\sigma (m)$ is the galaxy density for a given observed magnitude m. The effective radius depends on the half-light radius ${r}_{1/2}$, the distance from the GRB r₀, and the localization accuracy. The localization accuracy is defined by the error of aligning the Hubble Space Telescope (HST) images to images of the optical afterglows, which is between 0033 and 0061. The effective radius in Bloom et al. (2002) can hence be re-written as ${r}_{\mathrm{eff}}=2\;{r}_{1/2}$ if ${r}_{0}\lt {r}_{1/2}$ and ${r}_{\mathrm{eff}}={({r}_{0}^{2}+4\;{r}_{1/2}^{2})}^{1/2}$ if ${r}_{0}\gt {r}_{1/2}$. The half-light radii were measured with SExtractor v2.19.5 (Bertin & Arnouts 1996). To limit the number of candidates we set an upper limit of 3'' on the GRB offset and required a chance probability of ${p}_{\mathrm{ch}}\lt 5\%$.

The final candidates are listed in Table 3 (not corrected for Galactic reddening, whereas the unreddended magnitudes are reported in Table 2). The host offsets are consistent with the observed distribution for the TOUGH sample (D. Malesani et al. 2015, in preparation). If no host candidate was detected, we report the nominal flux and the 3σ limiting magnitude. The host identifications of GRBs 060115 and 060605 are not unique, where alternative candidates for each of these GRBs have the same magnitudes within 2σ. For simplicity, we used their weighted means in the further analysis.

Table 3.  Properties of the z = 3–4 Host Galaxies Observed With HST

GRB	α, δ	${r}_{1/2}$	r0	Band	${F}_{\nu }$	Brightness	${p}_{\mathrm{ch}}$
	(J2000)	('')	('')		$(\mathrm{nJy})$	(mag)
050730	⋯	⋯	⋯	F775W	7 ± 7	$\gt 27.6$	⋯
050908	01:21:50.727,	0.05	0.02	$F775W$	29 ± 7	${27.60}_{-0.24}^{+0.30}$	0.002
	−12:57:17.31
060115	03:36:08.314,	0.09	0.28	$F814W$	38 ± 7	${27.27}_{-0.18}^{+0.22}$	0.020
	+17:20:42.80
	03:36:08.351,	0.08	0.44	F814W	26 ± 7	${27.68}_{-0.26}^{+0.34}$	0.052
	+17:20:42.86
060526	⋯	⋯	⋯	F775W	5 ± 7	$\gt 27.66$	⋯
060605	21:28:37.312,	0.09	0.06	F775W	30 ± 9	${27.55}_{-0.28}^{+0.38}$	0.007
	−06:03:30.96
	21:28:37.321,	0.06	0.47	F775W	28 ± 9	${27.63}_{-0.30}^{+0.42}$	0.054
	−06:03:30.56
060607a	⋯	⋯	⋯	F775W	13 ± 8	$\gt 27.48$	⋯
061110B	21:35:40.396,	0.12	0.05	F775W	45 ± 10	${27.10}_{-0.21}^{+0.26}$	0.008
	+06:52:34.30
070721B	02:12:32.935,	0.08	0.20	F775W	25 ± 9	${27.76}_{-0.33}^{+0.47}$	0.018
	−02:11:40.63

Note. For each galaxy we list its half-light radius ${r}_{1/2}$, its projected distance to the GRB r₀, its flux density ${F}_{\nu }$, and the apparent magnitude. If no host candidate was detected, we report the nominal flux density at the afterglow position and the corresponding 3σ limiting magnitude. All magnitudes (but not the flux densities) include an aperture correction but no correction for Galactic reddening. The uncertainty in the reported coordinates is $\sim 0\buildrel{\prime\prime}\over{.} 4$ (comprising the astrometry error of the optical afterglow images and the alignment error of the VLT and HST images). See Section 3.2 for details.

^aThere is in fact a host candidate with a chance probability of ${p}_{\mathrm{ch}}=0.04$ 029 from the afterglow. However, it is only detected in a very small aperture with a radius of 02. The coordinates of the object are R.A., decl.(J2000) = 21:58:50.388, −22:29:46.68 ± 04. Its magnitude is $m(F775W)={28.28}_{-0.35}^{+0.53}$ mag.

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Although the TOUGH sample has a current redshift completeness of 87%, nine hosts remain without precise redshift information (Table 2). We succeeded in obtaining multi-band data for GRBs 050803 and 050922B from 4000 to 42000 Å (Table 1) to model their SEDs and obtain photometric redshifts.

The field of GRB 050803 was observed in the same filters but with a different telescope (Table 2). We built super-stacks for each band by resampling these data to the grid of the Keck/LRIS images (which has the highest spatial resolution) while conserving the flux and weighting the images by their limiting magnitudes. The final extinction-corrected magnitudes of the two hosts are reported in Table 4.

Table 4.  Broad-band Photometry of 050803 and 050922B

GRB 050803		GRB 050922B
	Brightness		Brightness
Band	(mag)	Band	(mag)
g'	$\gt 27.45$	g'	27.50 ± 0.50
R	26.29 ± 0.22	R	26.52 ± 0.22
i'	26.45 ± 0.50	i'	25.18 ± 0.14
F160W	25.74 ± 0.18	z'	25.01 ± 0.34
Ks	$\gt 23.30$	Ks	$\gt 24.00$
3.6 μm	$\gt 24.67$	3.6 μm	24.77 ± 0.36
4.5 μm	$\gt 25.00$	4.5 μm	24.60 ± 0.46

Note. All magnitudes are corrected for Galactic reddening. Non-detections are reported at 3σ confidence level. The K-band photometry was taken from Hjorth et al. (2012).

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We modeled the SED with Le Phare (Arnouts et al. 1999; Ilbert et al. 2006),¹⁸ using a grid of galaxy templates based on Bruzual & Charlot (2003) stellar population-synthesis models with the Chabrier IMF and a Calzetti dust attenuation curve (Calzetti et al. 2000). For a description of the galaxy templates, physical parameters of the galaxy fitting, and their error estimation, we refer to Krühler et al. (2011). To account for zeropoint offsets in the cross calibration and absolute flux scale, a systematic error contribution of 0.05 mag was added in quadrature to the uncertainty introduced by photon noise. Figure 2 displays the observed host SEDs and their best fits.

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Considering the brightness of the two galaxies ($R\gt 26$ mag) and their measurement errors, this fitting does not yield a unique redshift solution for either of the two galaxies. A broad range of physical galaxy parameters and redshifts fit the data within their uncertainties. Of particular interest in our case is the possibility that the galaxies are at $z\gt 3$. A high redshift does in fact provide a good description of the photometry in both cases. The SED of the host of GRB 050922B has significant $g\prime -r\prime$ and $r\prime -i\prime$ colors which are reasonably well explained by the Lyα and Lyman-limit breaks at z ∼ 4.5. Similarly, the red $g\prime -r\prime$ and blue $r\prime -i\prime$ colors of the galaxy hosting GRB 050803 is indicative of a redshift of z ∼ 3.5. Lower redshift solutions ($z\lt 1$) exist as well in both cases, but based on the available photometry, it is at least plausible that both GRBs originated at $z\gt 3$.

Although we have at least one measurement of the rest-frame UV continuum between 1216 and 4000 Å for each host galaxy in our sample, these data probe different parts of the UV continuum; on average they probe the rest-frame at 2140 Å. UV LFs of LBG samples are typically reported at 1500–1700 Å (e.g., Arnouts et al. 2005; Reddy & Steidel 2009; Oesch et al. 2010; Bouwens et al. 2014). To shift the UV luminosities of the TOUGH sample to the common rest-frame at 1600 Å, a K-correction was applied assuming the UV continuum to be power-law shaped (${F}_{\lambda }\propto {\lambda }^{-{\beta }_{\mathrm{UV}}}$). However, LBG surveys have also shown that the spectral slope is luminosity and redshift dependent (e.g., Bouwens et al. 2009, 2010b). To account for that we make use of the parameterization by Trenti et al. (2014) that is based on the findings for LBGs by (Bouwens et al. 2012, and references therein). Since the slope depends on the unknown observed UV luminosity, we solve the inverse problem: we compute the expected apparent magnitudes for a range of UV luminosities ($-30\;\mathrm{mag}\lt M\lt -8\;\mathrm{mag}$) at the redshift of each GRB and select the luminosity that minimizes the difference between the observed and the expected apparent magnitudes. In Table 2 we summarize the best-fit slopes and luminosities.

We note that the UV luminosities in the FUV are highly sensitive to any reddening correction. In LBG surveys it is a common practice to build the LF from the obscured UV luminosities, and therefore without any loss of generality or limitation in the comparison with LBG surveys, we omit any reddening correction.

The exact shape of the unobscured UV continuum is determined by the age of the young stellar population and metallicity. Savaglio et al. (2009, and the update in the GHostS database) and Perley et al. (2013) showed that the observed age distribution extends from a few tens of megayears to two billion years. Though this result is based on heterogeneous samples, the maximum age is consistent with limits we extracted from the TOUGH sample (D. Malesani et al. 2015, in preparation). To check whether the slopes used in our analysis are consistent with these ages, we fit the UV continuum of single-age stellar population models from Bruzual & Charlot (2003) between 1350 and 3600 Å with power-law models. We find that the slopes vary between ${\beta }_{\mathrm{UV}}\sim -2.8$ and ∼3.5 for ages between 0.005 and 2.5 Gyr, in agreement with the values derived in Section 3.4.

We next assess how the assumption about the luminosity and redshift dependence affects our results. We first assume the slope to be luminosity independent. We constrain the redshift evolution using results from Schiminovich et al. (2005), Finkelstein et al. (2012), and Hathi et al. (2013). Between z = 0 and z = 8, the UV slope could be parameterized as

$$\begin{eqnarray}&&{\beta }_{\mathrm{UV}}(z)=(-1.62\pm 0.04)+(-0.07\pm 0.01)\times z\end{eqnarray} \tag{ 1 }$$

for luminosities between $\sim 0.1{L}_{\star }$ and $\sim 1.5{L}_{\star }$. Figure 3 (top panel) displays the difference in the observed luminosity for the different parameterizations of the UV slope. The median difference is 0.01 mag and in most cases negligible. The largest differences of 0.26 mag are observed at z ∼ 1 where the R-band data have the largest distance from the common rest frame at 1600 Å.

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Next we drop the redshift dependency and assume a very young stellar population with a characteristic spectral slope of ${\beta }_{\mathrm{UV}}=-2.5$ for all hosts. The luminosity will increase for all hosts (Figure 3, bottom panel). The largest shift of 0.82 mag is observed at z ∼ 1. However, it is very unlikely that the majority of GRB hosts have such blue UV continua. Several low-redshift hosts are known to have evolved young stellar populations, e.g., GRB 970228 (z = 0.695): age = 1.7 Gyr; GRB 990712 (z = 0.433): ${\rm{age}}=1.1$ Gyr; and GRB 011121 (z = 0.362): age = 2.3 Gyr (Savaglio et al. 2009). In conclusion, though individual hosts may be more or less luminous than inferred from our ansatz, we do not consider our UV slope assumptions to have any systematic effect on the ensemble above z ∼ 1.

Figure 4 (left panel) plots the absolute magnitudes at 1600 Å (detections and upper limits) of the sample GRB hosts versus redshift. The absolute magnitudes of the detected hosts span a wide range in magnitude from −14 to −21.4 mag. The majority of the upper limits on non-detected hosts are as deep as, and deeper than, any of the host detections, particularly at high redshift ($z\gt 3$). All upper limits, except for one host at z = 6.295, are well below the median magnitude of the detected sample.

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It is interesting to note that the brightest hosts, i.e., those above the median absolute magnitude, span a quite limited redshift range between z ∼ 1 and $z\lesssim 3$. Conversely the dimmest 50% of hosts (detections and upper limits) span the entire redshift range of the sample. This evolution is similar to that of the UV-inferred global star formation rate density (e.g., Daddi et al. 2007; Noeske et al. 2007; Rodighiero et al. 2010; Elbaz et al. 2011; Bouwens et al. 2014), which also peaks at z = 2–3.

Above $z\gt 3$, there is a conspicuous absence of any host detection above the median magnitude, except for the single bright ${M}_{1600\;\mathring{\rm{A}}}=-20.8$ mag host (GRB 060707) at z = 3.424, though the host of GRB050922B could have similar luminosity (for details see Section 3.3). At the other end of the redshift scale, below z ∼ 1, there are once again no hosts above the median magnitude. Our unbiased host sample seems to suggest that GRBs favor lower luminosity hosts throughout the entire redshift range in which they are detected and are only found in UV brighter hosts in the range between $1\lt z\lt 3$, though we note (as discussed below) that some of the hosts with unknown redshifts may in reality exist in the higher redshift range.

Figure 5 plots the rate density (number per unit comoving volume per year) of TOUGH GRBs occurring in hosts above and below the overall survey median luminosity (dotted line left panel Figure 4) versus redshift. Although the overall numbers are small, the volume density of the brighter host fraction is lower than that of the fainter fraction at all redshifts except $1\lt z\lt 3$ where bright hosts reverse this trend to become 2.5–3 times more common than those below the global median. We caution that the relatively low numbers in each large and somewhat arbitrary redshift bin prevent any firmer statistical conclusion from being drawn from the binned data. Further analysis of the cumulative distribution function (CDF) of the TOUGH hosts compared to model LFs derived from LBG populations in Section 4.3 investigates these preliminary observations in greater detail.

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In the following sections, we investigate how GRB hosts trace the SFR history, in particular whether they follow the model proposed by Fynbo et al. (2002, see also Jakobsson et al. 2005) of GRBs selecting galaxies from a general population according to their SFR or whether an additional dependency must be invoked, such as metallicity (Stanek et al. 2006; Modjaz et al. 2008; Levesque et al. 2010).

We note that the redshifts of nine hosts are uncertain (Table 2) within defined limits (for details see Jakobsson et al. 2012), one host identification (GRB 060805A) is non-unique, and the IGM correction for one host (GRB 050502B) is uncertain. In the right panel of Figure 4 we plot their possible positions in the ${M}_{1600\;\mathring{\rm{A}}}$–z plane (gray lines for hosts of uncertain z, green data points for the host with uncertain IGM correction, and blue data points for the non-unique host identification), where it can be seen that the plausible distribution of these uncertain hosts shows no discrepancy from the firm detections/limits. We discuss further the effects of these unknown redshift hosts on our results in Section 5.1.

As a first diagnostic to investigate whether GRBs are biased or unbiased tracers of star formation, we investigate the evolution of the median obscured UV luminosity of the hosts. Recently Trenti et al. (2014) presented tracks for the evolution of the median observed UV luminosity of GRB host galaxies for various levels of GRB production bias with respect to host metallicity, characterized by a parameter p which represents a minimum, metal-independent plateau for the efficiency of forming a GRB. Thus a low value of p characterizes a high level of bias toward low metallicity hosts (p = 0 representing a stringent metallicity cutoff), whereas $p\to \infty$ characterizes no metallicity bias.

Figure 6 shows the evolution of the median UV luminosity in unit redshift bins of our sample in comparison to these models. The luminosity depth of our observations is a function of redshift (Figure 4), and we hence recalculated the median values given in Trenti et al. (2014) for the observed luminosity limits in each redshift bin. The faint limit was set to $M-3\sigma (M)$ in each magnitude bin; specifically we compute the median UV luminosity between $-22.5\leqslant M\leqslant -13.2$ at $z\lt 1$, $-22.5\leqslant M\leqslant -16.7$ at $1\leqslant z\lt 3$, $-22.5\leqslant M\leqslant -17.2$ at $3\leqslant z\lt 4.5$. Observed medians and their errors were estimated via a bootstrap method (30,000 samples), where each detected host was represented as a Gaussian centered on the measured luminosity with a width given by the measurement uncertainty. Non-detected hosts were drawn from a uniform distribution between a fiducial magnitude cut and the 3σ limiting detection.

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Though we have only four data points with significant uncertainty, the data appear to favor models of GRB production incorporating significant levels of bias toward low metallicity hosts (as suggested by Vergani et al. 2014; Cucchiara et al. 2015). In particular, the inclusion of any of the hosts of unknown redshift in individual redshift bins always lowers the median magnitude in that bin. Note that these models of metallicity bias for GRB production, which include metallicity-dependent (single star collapsars) and independent channels (binary progenitor)¹⁹ do not imply a fixed fraction of GRB production via each channel, but rather an evolving fraction with redshift, with the metallicity-dependent (collapsar type) channel always being dominant at high redshift (see Trenti et al. 2014 for full details).

In order to go further than just analyzing the median properties of the TOUGH hosts, we construct LFs for GRB hosts from z = 0 to z = 4.5 in appropriate redshift and luminosity bins to compare both with those from LBG surveys and those predicted by the models of Trenti et al. (2014). We select those hosts from our sample that fall in the redshift ranges of the three LBGs surveys in Table 5, and in the range $0\lt z\lt 1$.

Table 5.  Parameters of LBG Luminosity Functions

Redshift	Luminosity	${M}_{1600\;\mathring{\rm{A}},\star }$	Faint-end
Interval	Interval (mag)	(mag)	Slope α	References
$1.0\leqslant z\leqslant 1.5$	−21.50 to −17.83	−20.08 ± 0.36	−1.84 ± 0.15	(1)
$1.9\leqslant z\leqslant 2.7$	−22.83 to −17.83	−20.70 ± 0.11	−1.73 ± 0.07	(2)
$3.0\leqslant z\leqslant 4.5$	−22.69 to −15.94	−21.07 ± 0.08	−1.64 ± 0.04	(3)

Note. Values of LBG LFs in the Schechter parameterization for different redshift intervals. The redshift column shows the interval for which the LFs were applied to.

References. (1) Oesch et al. (2010), (2) Reddy & Steidel (2009) (3) Bouwens et al. (2014).

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Figure 7 shows which part of the ${M}_{1600\;{\rm{\mathring{\rm{A}}}}}$–z parameter space is probed by LBG surveys. The evolution of ${M}_{\star }$ implies that GRB hosts probe the full luminosity range of LBGs between z = 1 and 3, whereas at lower and higher redshifts, GRBs rather probe the faint-end of the observed LBG LF. We note that the luminosity range of GRB host galaxies extends to much fainter galaxies between z = 1 and z = 3 than probed by the LBG surveys in Table 5. However, recent observations by Alavi et al. (2014) found no evidence for a change in the LBG LF parameters at z ∼ 2 down to ${M}_{1500}=-15$ mag, which reassures us in extrapolating the LFs in Table 5 to lower luminosities.

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Figure 8 displays the GRB host galaxy cumulative distributions for the four redshift intervals. Since GRBs are produced by the collapse of very massive stars, it has been suggested that GRBs should select galaxies according to their SFR. In the simplest model, Fynbo et al. (2002) proposed that the UV GRB-host LF should be similar to that of LBG samples weighted by their SFR, where the SFR is proportional to the unobscured UV luminosity, which can be expressed as $\mathrm{SFR}\propto {10}^{0.4\times {A}_{V}({M}_{\mathrm{obs}})}\;{L}_{\mathrm{obs}}$. Following Trenti et al. (2014), we use ${A}_{V}=4.43+0.79\;\mathrm{log}(10)\;{\sigma }_{{\beta }_{\mathrm{UV}}}^{2}+1.99\;{\beta }_{\mathrm{UV}}$ where ${\sigma }_{{\beta }_{\mathrm{UV}}}=0.34$. The cumulative distributions of the SFR-weighted LBG LFs from Table 5 are overlaid in the same plot.

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At low and high redshift, the observed distribution differs significantly from the no metallicity bias (or equivalently, LBG derived) model LFs. In the $3\lt z\lt 4.5$ region, for example, we can reject the null hypothesis of the data being drawn from the SFR weighted LBG LF at a level of 0.01% via a Kolmogorov--Smirnov (KS) test. In the medium redshift ranges ($1\lt z\lt 1.5$ and $1.9\lt z\lt 2.7$) the data are insufficient to distinguish reliably between any of the models, particularly due to some non-detections and the possibility of some of the hosts of unknown redshift being within these bins as discussed in the next section.

There are nine hosts in our sample with redshifts unknown within certain limits. As discussed in Jakobsson et al. (2012), a conservative upper limit of $z\lt 3.5$ can be placed on four of the bursts associated with these hosts via their measured excess (above Galactic) X-ray absorption following the method of Grupe et al. (2007). The remaining five bursts can be at higher redshifts (see Table 2 for details).

In this paper we set a formal redshift limit of $z\lt 3.5$ to avoid corrections for IGM and Lyα absorption. These unknown redshift hosts can therefore exist anywhere along the gray tracks plotted in the right-hand panel of Figure 4. The maximum plausible redshift of z = 3.5 is approximately coincident with the midpoint of our highest redshift LBG comparison sample, and in Figure 8 we also show the effect of placing all the unknown hosts at the midpoint of each LBG comparison range in turn (hatched regions). As can be seen from the bottom panel in this plot, if all unknown redshift hosts are actually at their highest plausible redshift, which may be a likely scenario (see Figure 11 in Hjorth et al. 2012), and all upper limits are treated as detections, then the likelihood (KS probability) of the TOUGH cumulative distribution being consistent with the SFR weighted LBG model rises to 1.0%. Note that this value represents our most conservative limit compared to the more flexible simulation discussed below.

To quantify how many of the TOUGH hosts with unknown redshift would be required to be within the range $3\leqslant z\leqslant 3.5$ in order to make the TOUGH host LF consistent with the SFR-weighted LBG LF, we performed a Monte-Carlo simulation as follows. One of the nine hosts with unknown redshift was chosen at random and assigned a random redshift between $3\leqslant z\leqslant 3.5$. The appropriate host luminosity was then drawn from a normal distribution centered around the observation value with a width (1σ) of the detection error for detected hosts, and a uniform distribution between the upper limit and the luminosity of the faintest host in the TOUGH $3\leqslant z\leqslant 4.5$ sample for those hosts with an upper limit only. This host is added to the TOUGH CDF, and a KS value computed between the new CDF and the SFR weighted LBG-LF. The process is repeated 30,000 times and a mean and median KS value obtained. We then successively add hosts of unknown redshift and repeat the procedure until all nine hosts are included in the CDF. Figure 9 shows how the measured KS value varies with the number of added hosts of unknown redshift.

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As can be seen from Figure 9, at least 3 unknown hosts are required for the TOUGH distribution to fall below the 3σ equivalent rejection probability level of $\lt 0.3\%$ chance of consistency with being drawn from the SFR-weighted LBG (no metallicity bias) LF model. Even when all unknown redshift hosts are included in the simulation, the median KS probability of the distributions being consistent is only 0.8%. The gap may be closed further if some of the unknown hosts are at higher redshifts still ($3.5\lt z\lt 4.5$) since their R-band observations then would have been significantly affected by IGM and Lyα absorption. Nevertheless, the metallicity dependent models of Trenti et al. (2014) appear a better fit to the data, with critical KS rejection levels much lower.

We emphasize again that these models do not represent a fixed fraction of GRB production via metal-dependent and independent channels, but are derived from stellar population syntheses combined with a GRB production efficiency model with respect to metallicity controlled by the value of p. Lower (but non-zero) p values represent a stronger level of bias toward low metallicity hosts, as expected from the collapsar model. We note that the p = 0 model (blue lines) which represents a stringent metallicity plateau for GRB production is effectively ruled out by observations of some high metallicity GRB hosts (e.g., Levesque et al. 2010; Elliott et al. 2013; Schulze et al. 2014), and are plotted as convenient limits for comparison only.

The TOUGH sample is independent of the brightness of the optical afterglow, but the properties of the prompt emission could still bias the sample toward a certain GRB population and hence a certain host galaxy population. We find no correlation between the host properties and energy released at γ-ray energies, ${E}_{\mathrm{iso}}$ (values taken from Butler et al. 2007), or the peak flux photon flux (taken from Sakamoto et al. 2008) at any redshift. At low redshift (where we are sampling the greatest proportion of the GRB γ-ray LF itself) there is also no tendency for brighter bursts to favor dimmer hosts, and we therefore do not believe any flux-limited observational bias of the bursts themselves significantly affects any of our conclusions regarding the hosts.

Previous GRB host surveys have tended to be biased toward bursts with optically brighter afterglows and/or hosts, whereas the TOUGH sample (as stressed throughout this work) is independent of the optical properties of bursts or hosts. To examine whether these previous observational biases are significant or not, we investigate whether the same conclusions could be drawn from a previous heterogeneous sample. To address this question we retrieved all host measurements that probe the rest-frame UV continuum from the GHostS database (Savaglio 2006). Until 2012 January (before the first results from the TOUGH survey were published), the database comprised observations for over 120 hosts. As an example, we build the UV luminosity distributions for this sample and divide them into the same redshift and luminosity bins as the TOUGH sample.

The observed cumulative distributions are shown in the lower panels of Figure 10, along with those of the TOUGH sample in the upper panels. At all redshifts, except perhaps for $1\lt z\lt 1.5$, we see an overabundance of luminous galaxies in the GHostS sample compared to TOUGH. This clearly shows that selection effects from GRB afterglow observations must be carefully taken into account before attempting to extract any ensemble properties. The limiting magnitudes in the GHostS sample are in general also significantly shallower than ours. To be able to extract any meaningful results, GRB-selected galaxy samples need to be carried out to the magnitude limits of LBG surveys, i.e., reaching at least R ∼ 28 mag.

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Our findings let us draw the following picture. GRBs probe the full luminosity range (and below) of LBG surveys at all comparable redshifts. Furthermore, throughout the entire redshift range of our sample, GRB hosts appear consistent with models of GRBs being produced by a combination of a metal-dependent (single star collapsar) channel and an independent (binary progenitor) channel with a relatively high level of bias (low, but non-zero value of p) toward low metallicity hosts (Trenti et al. 2014), though we cannot completely rule out the unbiased models in the range $1\lt z\lt 3$. This behavior may also lend support to the model of the starvation of infalling pristine gas at z ∼ 1–1.5 proposed by Perley et al. (2013), moving the GRB host population away from LBG galaxies as a whole to low-mass UV faint galaxies as redshift decreases.

At $z\gt 3$, we observe a lack of UV luminous GRB hosts. All detected hosts (except one) are at least 3 mag fainter than ${L}_{\star }$ galaxies. If GRB hosts were truly similar to LBGs, we may expect this connection to improve at higher redshifts, where any potential metallicity bias becomes less important in comparison to the overall galaxy population. Indeed the presence of a reasonable fraction of the nine hosts of unknown redshift in this range would reduce this lack of bright hosts, though not entirely to the level of consistency with an SFR weighted LBG LF.

Thus it would seem that at both high and low redshift, TOUGH supports the idea that GRBs are preferentially found in low-metallicity, relatively UV faint hosts compared to the overall galaxy population, and therefore their relationship to the cosmic SFR density history as measured by UV flux limited surveys is not simple. Several previous studies have pointed to the numbers of GRBs observed at high redshift indicating a large amount of hidden (massive) star formation in galaxies that are too faint to be detected in current LBG surveys (Fynbo et al. 1999; Haehnelt et al. 2000; Fynbo et al. 2002; Le Floc'h et al. 2003; Jakobsson et al. 2005, 2012; Kistler et al. 2008; Schmidt 2009).

Alternatively others have suggested that GRB production efficiency with respect to star formation may increase with redshift (Kistler et al. 2009; Butler et al. 2010; Robertson & Ellis 2012; Salvaterra et al. 2012). If this latter alternative of increased GRB efficiency were the sole reason for the change, however, one might expect to still see some bright hosts at high redshift. However, highly star-forming galaxies may well also contain a large amount of dust, both obscuring their intrinsic SFR and making it more difficult to obtain spectroscopic redshifts. Despite targeted and deep searches, no GRB host has yet been detected beyond z ∼ 5, which lends support to the lack of bright hosts at high redshift not being an artifact of our survey. Obviously since high-z GRBs are rare, the influence of small number statistics cannot be avoided. We suggest however, that though an increase in GRB production efficiency with respect to the UV measured cosmic SFR density at high redshift may be a useful calculational tool in simulation studies, it does not represent the detailed picture regarding individual GRB host galaxies, and more unbiased GRB host data are required.

During the review process of this paper, other workers have submitted investigations of the properties of GRB host galaxies. Greiner et al. (2015) presented discussions of the properties of a heterogeneous sample of GRB hosts at z ∼ 3, and Perley et al. (2015b) presented findings on the NIR luminosity distribution z = 0 to z = 5 from the optically unbiased Swift Gamma-Ray Burst Host Galaxy Legacy Survey (SHOALS; Perley et al. 2015a). While the former study concludes that GRBs are unbiased tracers of star formation (compare our Section 5.2 and Figure 10), the latter study comes to a similar conclusion to that presented in this paper: GRB hosts in general appear fainter than a model that solely depends on SFR would predict, with a particularly pronounced lack of NIR luminous galaxies at $z\lt 1$. In all three samples, the number of events at $z\gt 4$ is relatively small, illustrating the need for a more concerted observational effort targeting this regime.