# Dust Obscuration in Lyman Break Galaxies at ${z\sim 4}$

I-Ting Ho11affiliation: Institute of Astronomy & Astrophysics, Academia
Sinica, P.O. Box 23-141, Taipei 10617, Taiwan , Wei-Hao Wang11affiliation:
Institute of Astronomy & Astrophysics, Academia Sinica, P.O. Box 23-141,
Taipei 10617, Taiwan , Glenn E. Morrison22affiliation: Institute for
Astronomy, University of Hawaii, Honolulu, HI 96822, USA 33affiliation:
Canada-France-Hawaii Telescope, Kamuela, HI 96743, USA , and Neal A.
Miller44affiliation: Department of Astronomy, University of Maryland, College
Park, MD 20742, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Measuring star formation rates (SFRs) in high-$z$ galaxies with their rest-
frame ultraviolet (UV) continuum can be uncertain because of dust obscuration.
Prior studies had used the submillimeter emission at $850~{}\mu\rm m$ to
determine the intrinsic SFRs of rest-frame UV selected galaxies, but the
results suffered from the low sensitivity and poor resolution ($\sim
15\arcsec$). Here, we use ultradeep Very Large Array 1.4 GHz images with $\sim
1\arcsec$–$2\arcsec$ resolutions to measure the intrinsic SFRs. We perform
stacking analyses in the radio images centered on $\sim 3500$ Lyman break
galaxies (LBGs) at $z\sim 4$ in the Great Observatories Origins Deep Survey-
North and South fields selected with Hubble Space Telescope/Advanced Camera
for Surveys data. The stacked radio flux is very low, $0.08\pm 0.15$ $\mu$Jy,
implying a mean SFR of $6\pm 11~{}{\rm M_{\sun}~{}yr^{-1}}$. This is
comparable to the uncorrected mean UV SFRs of $\sim 5~{}{\rm
M_{\sun}~{}yr^{-1}}$, implying that the $z\sim 4$ LBGs have little dust
extinction. The low SFR and dust extinction support the previous results that
$z\sim 4$ LBGs are in general not submillimeter galaxies. We further show that
there is no statistically significant excess of dust-hidden star-forming
components within $\sim 22$ kpc from the LBGs.

###### Subject headings:

dust, extinction — galaxies: evolution — galaxies: high-redshift — radio
continuum: galaxies

††slugcomment: ApJ accepted, August 2010

##  1\. INTRODUCTION

Understanding galaxy evolution in the early Universe requires large samples of
various kinds of high-$z$ galaxies. One important technique enabling
selections of $z>2.5$ star-forming galaxies is the Lyman break technique
(Cowie et al., 1988; Songaila et al., 1990; Lilly et al., 1991; Steidel &
Hamilton, 1993; Steidel et al., 1995; also see a review in Giavalisco, 2002).
The UltraViolet (UV) spectrum of a high-redshift star-forming galaxy usually
exhibits a Lyman continuum discontinuity at 912 Å, which is caused by
absorption of H I in the stellar atmosphere of massive stars, and interstellar
and intergalactic media. By using this feature, large samples of Lyman break
galaxies (LBGs) can be selected in various redshift ranges by searching for
sudden brightness dropouts between two adjacent broad-band images.

An important property of LBGs is their star formation rates (SFRs). However,
determining SFRs for $z>3$ systems is still challenging because most of the
diagnostics at low redshifts are unavailable when light becomes dimmer and
redshifts to wavelengths that are harder to observe. One of the most commonly
used method to determine SFRs at high redshift is to measure the rest-frame UV
continuum which shifts at $z>3$ to the optical and near-infrared (e.g.,
Kennicutt, 1998). Unfortunately, the UV continuum can be easily attenuated by
dust, which results in underestimation of the SFRs. Bouwens et al. (2009)
attempted to determine the correction factor by using the observed correlation
between the ratio of far-infrared (FIR) to UV flux and the UV spectral slope
(Meurer et al., 1999). They found that UV continuum of $z\sim 4$ LBGs in the
Great Observatories Origins Deep Survey-North (GOODS-N) and South (GOODS-S)
fields underestimates SFRs by factors of $\sim 3$–6. However, this result is
debatable since the correlation exhibits significant scatter in different
populations of galaxies (e.g., Cortese et al., 2006; Howell et al., 2010).

A more direct way to estimate the SFRs is to deduce them at longer wavelengths
where light is not affected by dust obscuration. Efforts had been made using
the submillimeter $850~{}\mu\rm m$ emission to determine the intrinsic SFRs in
LBGs (Peacock et al., 2000; Chapman et al., 2000; Webb et al., 2003). However,
these results have low signal-to-noise ratios (S/Ns), presumably because the
submillimeter single-dish maps were not deep enough, a consequence of
instrumental, sky, and confusion noises (i.e., uncertainties contributed by
nearby bright sources or faint undetected sources). In addition, an assumption
of dust temperature is required to estimate the total infrared (IR)
luminosity. These all make the estimate of SFRs based on submillimeter
measurements quite uncertain.

The radio wavelength is an alternative probe of SFRs. At 1.4 GHz, the radio
continuum is dominated by synchrotron radiation from relativistic electrons
produced by supernovae. By converting the 1.4 GHz luminosity to total IR
luminosity with the well-known radio—FIR correlation (Condon, 1992), it is
possible to estimate intrinsic SFRs (Kennicutt, 1998). The radio—FIR
correlation is fairly insensitive to dust temperature. The angular resolution
in the radio can be very high ($\sim 1\arcsec$–$2\arcsec$), meaning very
little confusion noise. Because of these great advantages, this radio-based
method has been used in various high-z studies (e.g., Reddy & Steidel, 2004;
Wang et al., 2006; Carilli et al., 2008; Pannella et al., 2009).

The Very Large Array (VLA) provides high resolution in the radio, but its
sensitivity is insufficient for directly detecting normal star-forming
galaxies at $z\gtrsim 1.5$. Therefore, stacking radio fluxes of large samples
of LBGs to determine their mean SFRs is a necessary approach. Reddy & Steidel
(2004) stacked $z\sim 2$ LBGs and found the dust correction to be $\sim 4.5$.
Using a similar method, Carilli et al. (2008) determined the dust correction
for $z\sim 3$ LBGs to be 1.8.

In order to determine the intrinsic SFRs for $z\sim 4$ LBGs, we performed
radio stacking analyses in the GOODS-N and GOODS-S fields. The GOODS Hubble
Space Telescope/Advance Camera for Surveys (HST/ACS) data (Giavalisco et al.,
2004) were used to locate $\sim 3500$ LBGs in the fields, and deep VLA 1.4 GHz
images (Miller et al., 2008; Morrison et al., 2010) were used to measure and
stack the radio fluxes. We note that although radio stacking of $z\sim 4$ LBGs
had also been attempted by Carilli et al. (2008), we used much deeper optical
images ($\sim 2$ mag deeper at V band ) and radio images ($\sim 15\%-40\%$
deeper). We are therefore probing much closer to the typical members in the
$z\sim 4$ LBG population, as in Bouwens et al. (2009).

In this paper, we first describe the observational data in Section 2. In
Section 3, we describe our methods of selecting LBGs, stacking analysis, and
estimations of SFRs using radio and UV. Finally, we compare the two different
SFRs and discuss the implications in Section 4, and give a summary in Section
5. We assume $\Omega_{0}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70~{}\rm
km~{}s^{-1}~{}Mpc^{-1}$. All magnitudes are in the AB magnitude system.

##  2\. DATA

###  2.1. HST/ACS

The HST/ACS multiband imaging in the GOODS fields (Giavalisco et al., 2004)
consists of four passbands: F435W, F606W, F775W, and F850LP, which are
referred to as $B_{435}$, $V_{606}$, $i_{775}$, and $z_{850}$, respectively.
We adopt the v2.0 source catalogs for selecting our LBG samples. Quantities
measured with SExtractor (Bertin & Arnouts, 1996) “automatic aperture”
(MAG_AUTO, FLUX_AUTO, etc.) are used to approximate total values.

###  2.2. VLA

The VLA 1.4 GHz images of the GOODS-N and GOODS-S fields are described in more
details in Morrison et al. (2010) and Miller et al. (2008), respectively. In
brief, the North field has an rms noise of $\sim 4~{}\mu\rm Jy~{}beam^{-1}$ at
the center, and $<6~{}\mu\rm Jy~{}beam^{-1}$ at the edges of the ACS fields.
The South field has an rms noise of $\sim 6~{}\mu\rm Jy~{}beam^{-1}$ at the
center, and $<7.5~{}\mu\rm Jy~{}beam^{-1}$ at the edges of the fields. These
sensitivities have been corrected for the primary beam response. The beam FWHM
of the North field is $1\farcs 7\times 1\farcs 6$, which corresponds to $\rm
11.8~{}kpc\times 11.1~{}kpc$ at $z=4$. The beam FWHM of the South field is
$2\farcs 8\times 1\farcs 6$, which corresponds to $\rm 19.5~{}kpc\times
11.1~{}kpc$ at $z=4$. Data Release 2 of the South field is used here.

##  3\. METHOD

###  3.1. Sample Selection

“B-dropout” galaxies at $z\sim 4$ are selected utilizing the redshifted Lyman
break located between $B_{435}$ and $V_{606}$. We adopt the well established
criteria (e.g., Beckwith et al., 2006; Bouwens et al., 2007):

$B_{435}-V_{606}>1.1,\\\ $ $B_{435}-V_{606}>(V_{606}-z_{850})+1.1,\\\ $
$V_{606}-z_{850}<1.6,\\\ $ $S/N(V_{606})>5,\ \ \ {\rm and}\ \ S/N(i_{775})>3.$

In addition, compact objects (SExtractor stellarity indices greater than 0.8)
with $i_{775}<$ 26.5 are rejected from our sample to prevent stellar
contamination. A color-color diagram illustrating the sample selection is
shown in Figure 1. In total, we selected 1778 and 1679 B-dropouts in the North
and South fields, respectively. These selection criteria efficiently select
LBGs between $z\sim 3$ and $z\sim 4.5$, with a mean redshift of 3.8 (see more
details in Bouwens et al., 2007). The number of LBGs selected by us is very
similar to that in Bouwens et al. (2007), who used similar GOODS ACS data.

Figure 1.— A color-color diagram showing the $B$-dropouts selection criteria.
Objects in the GOODS fields are plotted with dots. Intersection of the three
color selection criteria described in Section 3.1 are shaded in grey. The
selected LBGs are shown with red dots. Note that some of the objects inside
the grey region are not selected as LBGs because they do not meet the other
criteria (e.g., low S/N or too compact).

###  3.2. Stacking Analyses

We measured the radio fluxes of the $z\sim 4$ LBGs with aperture photometry at
their optical positions. Small offsets ($<0^{\prime\prime}.3$) between the
coordinates of the radio and optical images had been corrected utilizing their
source catalogs. Apertures of two different sizes were used here: small-
apertures with radii equal to the beam FWHMs ($\sim 11$ kpc at $z=4$); large-
apertures with radii equal to twice the beam FWHMs ($\sim 22$ kpc at $z=4$).
Here we first focus on the small aperture results and we will discuss the
large aperture results in Section 4.4. Elliptical apertures were used for the
South field to match its beam shape. Fluxes measured with our aperture
photometry were calibrated with the radio catalogs (Miller et al., 2008;
Morrison et al., 2010). For sources in the radio catalogs, we measured their
fluxes with our aperture photometry and computed the median of the ratios
between our fluxes and the catalog fluxes. The ratios, which were found to be
very close to unity for the aperture sizes we used, were applied back to our
radio fluxes.

Sources with radio fluxes $>100~{}\mu\rm Jy$ were rejected from our
measurements because their radio fluxes do not likely reflect their true SFRs.
There are eight such sources. If they are normal star-forming galaxies at
$z=4$, their IR luminosity would be $\gtrsim 5\times 10^{13}~{}L_{\sun}$ (or
SFR $\gtrsim 7000~{}{\rm M_{\sun}~{}yr^{-1}}$), which is unusually large. We
inspected these sources individually. They either are affected by nearby
bright radio sources that do not appear to be at the same redshifts of the
LBGs, or do not appear to be associated with known submillimeter galaxies
(SMGs) in the samples of Wang et al. (2004), Perera et al. (2008), Devlin et
al. (2009), and Weißet al. (2009). Therefore, they are highly unlikely high-
redshift ultraluminous starbursting galaxies. To avoid the bias from unrelated
nearby bright sources and from active galactic nuclei (AGNs), we did not
include these eight sources in our stacking analyses. For those sources with
radio fluxes $<100~{}\mu\rm Jy$, we then averaged their radio fluxes and
subtracted a background value (see below) from the means to get the final
stacked radio fluxes.

The uncertainties of our stacked fluxes were estimated with Monte Carlo
simulations, with an assumption that the galaxies are distributed randomly
over the map. We measured the mean radio fluxes at random positions, and the
number of random positions is the same as that of the $<100~{}\mu\rm Jy$
B-dropouts. The same apertures were used and the same rejection criterion of
$<100~{}\mu\rm Jy$ was adopted. This measurement is repeated 10,000 times, and
the mean radio fluxes have a fairly Gaussian-like distribution. We then
calculated the mean and the dispersion of these 10,000 measurements. The mean
was subtracted from the stacked radio flux of the LBG sample to form the final
stacked radio flux. This subtraction is to account for the effect of imperfect
clean and chance projection of random radio sources in our flux apertures.
Likewise, the dispersion can then represent the uncertainty of the stacked
radio flux.

There are two caveats in the above procedures. First, if $z\sim 4$ LBGs are
clustered at scales similar to the sizes of our apertures, our assumption that
LBGs are randomly distributed would break down. Our stacking method would then
overestimate the radio fluxes. We investigated clustering with the same method
used by Marsden et al. (2009). We found that the variance-to-mean ratio of
source numbers inside randomly placed $r<10\arcsec$ circles of our sample
exceeds that of random distribution by less than 0.1, and becomes smaller with
smaller radii. Therefore, we conclude that there is no significant small-scale
clustering in our LBG sample and the flux overestimate caused by small-scale
clustering can be neglected.

Second, the faulty estimate of uncertainties can be caused by concentration of
sources in high or low sensitivity regions (i.e., the slightly uneven
sensitivity distribution caused by the primary beam falloff in the GOODS-N or
by the mosaicking in the GOODS-S). Instead of stacking at random positions to
estimate the uncertainty, we also carried out the simulation based on the
source positions of the real LBG sample. We added an offset to the positions
of the LBGs and stacked their fluxes. We repeated this for 10,000 times with
different random offsets of $15\arcsec-60\arcsec$, and calculated the mean and
the dispersion. The results are within 30% to those measured from random
positions, suggesting that LBGs are not concentrated in specific regions.

With the above methods, we measured the final stacked fluxes and flux errors
of the LBGs in the two fields separately. For each aperture size, we combined
the two final stacked fluxes by weighting them with the inverse of the square-
errors to form the combined flux. The results are summarized in Table 1. The
stacked radio signal is very weak, consistent with zero within the noise.

###  3.3. Star Formation Rate

####  3.3.1 Radio SFR

The stacked radio fluxes can be converted to SFRs. Assuming a universal
synchrotron emission spectral index of $\alpha=-0.8$, we convert the stacked
radio fluxes $S_{\rm stack}$ to the rest-frame 1.4 GHz luminosity densities
$L_{\rm 1.4GHz}$, i.e.,

$L_{\rm 1.4GHz}=4\pi d_{l}^{2}\sl S_{\rm stack}\sl(1+z)^{-(1+\alpha)},$ (1)

where $d_{l}$ is the luminosity distance and $z$ is the mean redshift of 3.8.
With the local radio—FIR correlation (Helou et al., 1985; Condon, 1992),
$L_{\rm 1.4GHz}$ can be converted to FIR luminosity $L_{\rm FIR(40-120)}$,
approximately the total luminosity between $\lambda=42.5~{}\mu\rm m$ and
$\lambda=122.5~{}\mu\rm m$, with the following relation:

$q=log{{L_{\rm FIR(40-120)}}\over{3.75\times 10^{12}\rm~{}W}}\sl-log{{{L_{\rm
1.4GHz}}}\over{\rm W~{}Hz^{-1}}},$ (2)

with $q=2.34\pm 0.01$(Yun et al., 2001). Although this value was derived
locally ($z\lesssim 0.15$), recent studies suggest weak evolution out to
$z\sim 2$ (Sargent et al., 2010; Ivison et al., 2010a, b). Extrapolating $q$
to $z=4$ using the relation $q\propto(1+z)^{\gamma}$ with $\gamma=-0.04\pm
0.03$ (Ivison et al., 2010b) implies that $L_{\rm FIR}$ (and therefore radio
SFRs) will be lowered by only $\sim 30\%\pm 20\%$, which is insignificant
compared to the uncertainties in our results.

SFRs can then be estimated with the conversion in Kennicutt (1998):

${{\rm SFR\over({\rm M_{\sun}~{}yr^{-1}})}}=4.5\times 10^{-44}{L_{\rm
IR(8-1000)}\over\rm erg~{}s^{-1}},$ (3)

where $L_{\rm IR(8-1000)}$ is approximately the total luminosity between
$8~{}\mu\rm m$ and $1000~{}\mu\rm m$. This estimation is within $\sim 30\%$ to
other published calibrations. We derived the ratio between $L_{\rm
FIR(40-120)}$ and $L_{\rm IR(8-1000)}$ by computing these two quantities on
model spectral energy distributions of six nearby normal and starburst
galaxies (Silva et al., 1998), using their original definitions over the
Infrared Astronomical Satellite(IRAS) bands (see the summary in Sanders &
Mirabel, 1996). The $L_{\rm IR(8-1000)}$ to $L_{\rm FIR(40-120)}$ ratio has a
range of $1.71-2.32$, with a weak anti-correlation between the ratio and IR
luminosity and a mean of 2.05. This mean ratio is slightly higher than that
used in Yun et al. (2001) of 1.5, which is based on measurements in luminous
IRAS galaxies and starburst galaxies (Sanders et al., 1991; Meurer et al.,
1999; Calzetti et al., 2000), but is closer to the values in nearby normal
spiral galaxies and low luminosity starbursts ($\sim 2$ for M 51 and M100, and
2.3 for M 82, based on the Silva et al. templates). Using this value, we
derived the SFRs listed in Table 2, which are consistent with normal galaxies
or low-luminosity starbursts.

Table 1Mean Radio Fluxes

Field | Small Apertureaafootnotemark:  | Large Aperturebbfootnotemark:
---|---|---
| $F_{\rm LBG}$ | $F_{\rm random}$ | $F_{\rm LBG}$ | $F_{\rm random}$
| ($\mu\rm Jy$) | ($\mu\rm Jy$) | ($\mu\rm Jy$) | ($\mu\rm Jy$)
GOODS-N | $-0.05\pm 0.18$ | $0.36\pm 0.18$ | $0.29\pm 0.28$ | $1.10\pm 0.28$
GOODS-S | $0.36\pm 0.26$ | $0.10\pm 0.26$ | $0.54\pm 0.54$ | $-0.03\pm 0.54$
Combined | $0.08\pm 0.15$ | - | $0.34\pm 0.25$ | -

11footnotetext: Apertures with radii equal to the beam FWHM.

22footnotetext: Apertures with radii equal to twice the beam FWHM.

####  3.3.2 UV SFR

The $i_{775}$ and $V_{606}$ bands correspond to rest-frame UV, and can be used
to calculate SFRs, uncorrected for extinction (Madau et al., 1998), with

${\rm SFR}({\rm M_{\sun}~{}yr^{-1}})={\sl L_{\rm UV}({\rm
ergs~{}s^{-1}~{}Hz^{-1}})\over C},$ (4)

where $L_{\rm UV}$ is the UV luminosity density at the same redshift of 3.8,
and $C$ is $8.0\times 10^{27}$ and $7.9\times 10^{27}$ at 1500Å and 2800Å,
respectively. We list in Table 2 the mean UV SFRs of the B-dropouts with radio
fluxes $<100~{}\mu\rm Jy$. The uncorrected SFRs derived from $i_{775}$ and
$V_{606}$ are remarkably similar to each other as well as to that derived from
the radio stacking analyses, indicating very little extinction in the observed
rest-frame UV emission.

Table 2Star Formation Rates Derived with Various Fluxes | Radio | Optical
---|---|---
| Small aperture | Large aperture | $i_{775}$ | $z_{850}$
SFRs(${\rm M_{\sun}~{}yr^{-1}}$) | $6.0\pm 11.0$ | $25.7\pm 18.8$ | 4.92 | 5.14

##  4\. RESULT AND DISCUSSION

###  4.1. Dust extinction in $z\sim 4$ LBGs

In Figure 2, we plot the $z_{850}$ magnitudes versus the small-aperture radio
fluxes and the corresponding SFRs of all the B-dropouts in the two fields,
with the red points indicating the mean ($<100~{}\mu\rm Jy$ sources). As can
be seen in the figure and Table 2, the average SFRs deduced from the radio and
the rest-frame UV are consistent within measurement uncertainties. The
relatively small SFR derived from the radio flux suggests little extinction in
the $z\sim 4$ LBGs. Unfortunately, although this is by far the deepest radio
stacking on $z\sim 4$ LBGs ($\sim 3\times$ deeper than that in Carilli et al.,
2008), the error in the stacked radio flux is still too large to pin down the
extinction correction. An extinction correction of $>6$ is ruled out at 2.2
$\sigma$ and an extinction correction of $>4$ is ruled out at 1.3 $\sigma$.
Bouwens et al. (2009) derived extinction corrections of $\sim 3-6$ for $z\sim
4$ LBGs based on their UV continuum slopes. Our result slightly favors a lower
value and is marginally ($\sim 2~{}\sigma$) consistent with the result of
Bouwens et al. (2009).

We note that an even stricter upper limit on the radio SFR can be placed if
any of the following is true: (1) the radio—FIR correlation $q$ factor
exhibits a decline with redshift (see Section 4.3); or (2) the conversion from
$L_{\rm FIR(40-120)}$ to $L_{\rm IR(8-1000)}$ is overestimated (see Section
3.3.1); or (3) there is AGN contribution in our radio SFR (see Section 4.2).
The combined effect of all the above may not be negligible and can make the
extinction correction for $z\sim 4$ LBGs even lower. However, this remains to
be tested by future observations.

Carilli et al. (2008) found a low extinction correction of 1.8 on $z\sim 3$
LBGs. On the other hand, on $z\sim 4$ LBGs, they had a 2 $\sigma$ detection of
$0.83\pm 0.42$ $\mu$Jy on 1447 $B$-dropouts, which also suggests a low
extinction correction. This radio flux is much higher than ours ($0.08\pm
0.15$ $\mu$Jy) but their ground-based $V$-band limiting magnitude is also much
shallower than ours (by $\sim 2$ mag). In other words, we are probing
different LBG luminosities. Comparing our $z\sim 4$ result with the $z\sim 4$
result in Carilli et al. (2008) is therefore not straightforward.

Figure 2.— HST $z_{850}$ magnitude vs. 1.4 GHz VLA flux of the $\sim 3500$
B-dropouts in the GOODS fields. Orange symbols indicate objects not included
in our stacking analyses. The radio fluxes are measured with small apertures
with radii equal to the beam FWHMs ($\sim 11~{}\rm kpc$ at $z=4$). The
$z_{850}$ uncertainty of each data point is indicated with vertical bars. The
radio uncertainties, which depend on the locations on the maps, are not shown
to avoid confusion. Instead, we plot the typical radio flux uncertainties for
galaxies close to the center of the 2 fields at the lower-right corner for
reference (upper: North, lower: South). The inferred SFR corresponding to the
measured flux density at 1.4 GHz (see text for details) is shown on the upper
x-axis. The average of all the black data points are shown with the red
symbols, with the radio uncertainties estimated from the MonteCarlo
simulation. The black curves indicate the locus of equal SFR deduced from the
$z_{850}$ and the radio. The color coded curves, from left to right, indicate
1/6, 1/4, 1/2, 2, 4, and 6 times dust corrections from the UV to radio SFR. A
blow up to the stacked data point is shown as an insert.

###  4.2. AGN contamination

To avoid AGN contamination, a flux cut of $<100~{}\mu\rm Jy$ was set and eight
sources were rejected (one in the north and seven in the south). If these
sources comprise all the AGNs in our sample, our sample would have a very low
AGN fraction of $\sim 0.2\%$. For comparison, the AGN fraction in LBGs at
$z\sim 3$ was estimated to be $\sim 3\%$, based on optical spectroscopy
(Shapley et al., 2003). This difference could be due to the fact that our LBGs
are much less luminous than those in the spectroscopic sample.

It is possible to roughly separate AGNs and star forming galaxies based on the
radio power. In the local radio luminosity function (e.g., Mauch & Sadler,
2007), the 1.4 GHz power that divides AGNs and starbursts is $\sim
10^{23}~{}\rm W~{}Hz^{-1}$. Cowie et al. (2004) show that this AGN/starburst
division increases to $\sim 10^{24}~{}\rm W~{}Hz^{-1}$ at $z\sim 1$. This
power corresponds to $\sim 10$ $\mu$Jy for $z=4$. However, the increasing
trend observed by Cowie et al. (2004) can continue to $z\gg 1$, moving the
AGN/starburst division to a radio flux greater than 10 $\mu$Jy. Furthermore,
high-redshift flat-spectrum radio sources (presumably AGNs) have a more
negative $K$-correction than that of steep-spectrum sources (starbursts). This
also increases the apparent flux division between AGNs and starbursts at
observed-frame 1.4 GHz (by roughly a factor of 2). We thus expect sources
below our 100 $\mu$Jy cut to be dominated by star forming galaxies.

It is interesting to test what happens if we assume that $>50$ $\mu$Jy sources
are also AGNs and remove them from the stacking analyses. To do this, we
rejected 13 sources that are brighter than 50 $\mu$Jy and introduced the same
50 $\mu$Jy cut in the Monte Carlo simulation. We found a stacked flux of
$0.05\pm 0.14$ $\mu$Jy, corresponding to a mean SFR of $3.8\pm 10.5$ ${\rm
M_{\sun}~{}yr^{-1}}$. The stacked flux decreases further if we keep lowering
the flux cut but the change is well within the noise. Nevertheless, it is
obvious that we will get a lower mean SFR if we assume more AGN contribution.
Although we are limited by noise here, our conclusions that $z\sim 4$ LBGs
have low SFRs and the mean extinction correction seems to be lower than 5 are
thus still valid.

###  4.3. The Radio—FIR Correlation

Carilli et al. (2008) discussed the use of the local radio—FIR correlation to
derive SFRs of high-redshift galaxies. They mentioned several possibilities
that this correlation may be different at high redshift. Indeed, a weak
evolution has been suggested by recent studies (Sargent et al., 2010; Ivison
et al., 2010a, b). However, as mentioned in Section 3.3.1, even if we
extrapolate the evolution based on a declining $q$ (Ivison et al., 2010b) and
adopt a $L_{\rm IR(8-1000)}$ to $L_{\rm FIR(40-120)}$ ratio that is 25% lower,
our main conclusions would not change and this would make the SFR of the LBGs
even lower. Furthermore, the local radio—FIR correlation is only used to
calibrate the relation between SFR and radio power. Even if there is an
evolution in the radio—FIR correlation at high redshift, we cannot be certain
that the evolution arises from the radio part of the correlation and affects
the radio—SFR relation. In short, we see neither evidence nor strong argument
that the radio SFR is significantly biased by the assumption of the local
radio–FIR correlation.

###  4.4. Implications for SMGs

The relatively low dust correction and SFRs imply that $z\sim 4$ LBGs in
general do not have extreme star formation activities enshrouded in dust,
unlike that in SMGs. If the total SFR of our $z\sim 4$ LBGs is only
contributed by SMGs with SFR of $1000~{}{\rm M_{\sun}~{}yr^{-1}}$ and if other
non-SMG LBGs have no star forming activities, then there can be no more than
$21\pm 39$ SMGs in our LBG sample. In comparison, the source counts at
$850~{}\mu\rm m$ (Coppin et al., 2006) suggest $\sim 350$ SMGs in a field of
the combined GOODS size, with $S_{850}\rm>1.4~{}mJy$ and a mean SFR of
$1000~{}{\rm M_{\sun}~{}yr^{-1}}$. This means that even if the total SFR of
our LBG sample is dominated by $z\sim 4$ SMGs, they can only account for
$<10\%$ of the entire SMG population at $S_{850}\rm>1.4~{}mJy$. This implies
that either LBGs and SMGs have little overlap at $z\sim 4$ or most SMGs are at
$z\ll 4$. We note that previous studies in the submillimeter showed that LBGs
are in general not SMGs (Chapman et al., 2000; Peacock et al., 2000; Webb et
al., 2003), and that most SMGs are at $z<3.5$ (Chapman et al., 2003, 2005).
Our results are consistent with these.

Another scenario related to SMGs worth investigating is dust-hidden companions
sitting close to the LBGs. An example of this is the $z=4.5$ SMG in Capak et
al. (2008). This galaxy is independently selected as an LBG, and has an
ultraluminous dusty component showing up at $>3.6\mu\rm m$ next to its rest-
frame UV emission. To know whether similar systems are common at $z\sim 4$, we
performed the same radio stacking using the large apertures with radii twice
the beam FWHM ($\sim 22$ kpc at $z=4$) to measure the radio fluxes. As shown
in Figure 3 and Table 2, the radio SFR is slightly higher but still consistent
with the UV SFR, implying that there is no statistically significant excess of
dust-hidden star-forming components close to LBGs.

Figure 3.— Same as Figure 2, but the radio fluxes are measured with apertures
of radii twice the beam FWHM ($\sim 22~{}\rm kpc$ at $z=4$).

##  5\. Summary

We employed radio stacking analyses on $\sim 3500$ LBGs at $z\sim 4$ in the
GOODS-N and GOODS-S fields. The mean radio flux was converted to FIR
luminosity via the well-known radio—FIR correlation, and then converted to an
intrinsic SFR, which is $6\pm 11{\rm M_{\sun}~{}yr^{-1}}$. By comparing with
SFRs estimated from rest-frame UV, we found a maximum UV extinction correction
of $\sim 6$. This is roughly consistent with that derived from UV continuum
slopes by Bouwens et al. (2009) with a very similar sample. The low extinction
correction confirms that $z\sim 4$ LBGs are in general not SMGs. We also
investigated the possibility of finding dust-hidden companions close to the
LBGs at $z\sim 4$. The stacked radio flux within $\sim 22\rm~{}kpc$ to the
LBGs implies a mean SFR that is still consistent with the UV inferred SFR,
which suggests no statistically significant excess of dust-hidden star-forming
components close to LBGs.

We thank the referee for providing comments that improve this paper. W.-H.W.
and I.-T.H. acknowledge a grant from the National Science Council of Taiwan
(98-2112-M-001-003-MY2).

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