Small-scale Intensity Mapping: Extended Ly$\alpha$, H$\alpha$ and Continuum emission as a Probe of Halo Star Formation in High-redshift Galaxies

TL;DR

This paper models the extended Lyα, Hα, and continuum emission around high-redshift galaxies as primarily caused by faint satellite sources, providing insights into galaxy formation, star formation rates, and reionization.

Contribution

It introduces an analytic model linking satellite clustering to extended emission halos, offering a new way to interpret observations and constrain satellite populations and photon escape fractions.

Findings

UV and Lyα profiles match observations at 20-40 kpc from LAEs.

Extended halos reveal undetectable satellite sources.

Constraints on photon production and escape fractions relevant for reionization.

Abstract

Lyman alpha halos are observed ubiquitously around star-forming galaxies at high redshift, but their origin is still a matter of debate. We demonstrate that the emission from faint unresolved satellite sources, $M_{\rm UV} \gtrsim -17$, clustered around the central galaxies may play a major role in generating spatially extended Ly$\alpha$, continuum (${\rm UV + VIS}$) and H$\alpha$ halos. We apply the analytic formalism developed in Mas-Ribas & Dijkstra (2016) to model the halos around Lyman Alpha Emitters (LAEs) at $z=3.1$, for several different satellite clustering prescriptions. In general, our UV and Ly$\alpha$ surface brightness profiles match the observations well at $20\lesssim r \lesssim 40$ physical kpc from the centers of LAEs. We discuss how our profiles depend on various model assumptions and how these can be tested and constrained with future H$\alpha$ observations by the James Webb Space Telescope (JWST). Our analysis shows how spatially extended halos constrain (i) the presence of otherwise undetectable satellite sources, (ii) the integrated, volumetric production rates of Ly$\alpha$ and LyC photons, and (iii) their population-averaged escape fractions. These quantities are all directly relevant for understanding galaxy formation and evolution and, for high enough redshifts, cosmic reionization.

Tables

Table 1. Table 1 Ly α 𝛼 \alpha  and UV luminosity function parameters

	Ly$\alpha$	${\mathrm{UV}}_{1500}$	Units
${\varphi }^{\ast }$	$0.92$	$1.56$	$\left({10}^{-3}{\mathrm{Mpc}}^{-3}{\log }_{10}{L}^{-1}/{\mathrm{UVmag}}^{-1}\right)$
$L^{\ast }/M^{\ast }$ (a)	$5.8\times {10}^{42}$	$-20.87$	$\left(\mathrm{erg}{s}^{-1}/\mathrm{UV}\mathrm{mag}\right)$
$\alpha$	$-1.50$	$-1.67$
$L_{\mathrm{Ly}\alpha }^{\min }/M_{\mathrm{UV}}^{\min }$ (a)	${10}^{38}$	$-13$	$\left(\mathrm{erg}{s}^{-1}/\mathrm{UV}\mathrm{mag}\right)$
$L_{\mathrm{Ly}\alpha }^{\max }/M_{\mathrm{UV}}^{\max }$ (a)	${10}^{44}$	$-24$	$\left(\mathrm{erg}{s}^{-1}/\mathrm{UV}\mathrm{mag}\right)$

Table 2. Table 2 NIRCam parameters

	H$\alpha$	VIS	Units
FOV	$9.68$	$9.68$	$\left({\mathrm{arcmin}}^2\right)$
$A_{\mathrm{aper}}$	$25$	$25$	$\left(m^2\right)$
filter	F323N	F335M
${\lambda }_{\mathrm{obs}}\left(z=3.9\right)$	$3.237$	$3.362$	$(\mu$m)
BW	$0.038$	$0.352$	$(\mu$m)
$R$	$\sim 85$	$\sim 10$
$\eta$	$0.285$	$0.458$
$t_{\exp }$	${10}^4$	${10}^4$	(s)

Equations

$$S{B}_x(b)=\frac{2}{(1+z{)}^4}{\int }_b^{R_{max}^{\alpha }}{\overline{ϵ}}_x^{\mathrm{sat}}[1+{\xi }_x(r)]f_{\mathrm{esc}}^x\frac{r\mathrm{d}r}{\sqrt{r^2-b^2}},$$

$$R_{\max }^{\alpha }=\frac{1}{2}\frac{c}{H(z)}\frac{\mathrm{d}{\nu }_{\alpha }}{{\nu }_{\alpha }(z)}.$$

$$S{B}_{\mathrm{UV}}(b)=\frac{2}{(1+z{)}^3}{\int }_b^{R_{max}^{\mathrm{UV}}}{\overline{ϵ}}_{\mathrm{UV}}^{\mathrm{sat}}[1+{\xi }_{\mathrm{UV}}(r)]f_{\mathrm{esc}}^{\mathrm{UV}}\frac{r\mathrm{d}r}{\sqrt{r^2-b^2}},$$

$$S{B}_{\mathrm{VIS}}(b)=\frac{(1+z)}{\mathrm{E}{\mathrm{W}}_{\mathrm{H}\alpha }}\frac{{\lambda }_{\mathrm{H}\alpha }^2}{c}S{B}_{\mathrm{H}\alpha }(b).$$

$${ϵ}_x^{\mathrm{sat}}=C_x{\rho }_{\mathrm{SFR}}^{\mathrm{sat}},$$

$$C_{x}=\left\{\begin{array}[]{ll}1.30\times 10^{42}\hskip 2.84526pt\frac{{\rm erg}\hskip 2.84526pt{\rm yr}}{{\rm s}\hskip 2.84526ptM_{\odot}}&(\mbox{Ly}\alpha)~{};\\ 1.26\times 10^{41}\hskip 2.84526pt\frac{{\rm erg}\hskip 2.84526pt{\rm yr}}{{\rm s}\hskip 2.84526ptM_{\odot}}&(\mbox{H}\alpha)~{};\\ 8.00\times 10^{27}\hskip 2.84526pt\frac{{\rm erg}\hskip 2.84526pt{\rm yr}}{{\rm s}\hskip 2.84526ptM_{\odot}\hskip 2.84526pt{\rm Hz}}&(\mbox{UV})~{}.\end{array}\right.$$

$$\rho (r)=\frac{{\delta }_c{\rho }_c(z)}{r/r_s(1+r/r_s{)}^2},$$

$${\delta }_c=\frac{\Delta }{3}\frac{c_{\mathrm{NFW}}^3}{\log (1+c_{\mathrm{NFW}})-c_{\mathrm{NFW}}/(1+c_{\mathrm{NFW}})},$$

$$r_h={\left(\frac{3M_h}{4\pi \Delta {\rho }_c(z)}\right)}^{1/3}.$$

$$n(r)=(1+z{)}^3{\int }_{r_{\min }}^{r_{\max }}4\pi r^2[1+\xi (r)]\mathrm{d}r{\int }_{L_{\min }}^{L_{\max }}\varphi (L)\mathrm{d}L,$$

$$\varphi (L)\mathrm{d}L={\varphi }^{\ast }(L/L^{\ast }{)}^{\alpha }\exp \left(L/L^{\ast }\right)\mathrm{d}(L/L^{\ast }).$$

$$N_s$$

$$N_{\mathrm{sky}}$$

Full text

1. Introduction

Pioneering studies revealed the presence of diffuse Ly$\alpha$ emission in the halo of several star-forming galaxies (Møller & Warren, 1998; Fynbo et al., 1999, 2001; Rauch et al., 2008). Nowadays, this faint emission is being proved to be nearly ubiquitously in galaxies at high redshift, 3~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}z~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}5, by means of stacking analyses (Steidel et al., 2011; Matsuda et al., 2012; Feldmeier et al., 2013; Momose et al., 2014, 2015), and due to the sensitivity and spatial resolution improvement of instruments such as MUSE (Bacon et al., 2014). A clear understanding on the origin of these extended Ly$\alpha$ halos (LAHs; hereafter) is relevant because it yields information about the physical conditions of the circumgalactic medium (CGM) and, in turn, on the processes governing the formation and evolution of galaxies (Bahcall & Spitzer, 1969).

The main mechanisms contributing to the existence of LAHs are the cooling of gas accreted onto the galaxies, and star formation. Star formation, additionally, can be divided into two processes: (i) The nebular Ly$\alpha$ radiation produced in the interstellar medium (ISM) diffusing outwards to the CGM via scattering, and (ii) the ionizing photons escaping the center of the galaxy which produce Ly$\alpha$ radiation in the neutral CGM via fluorescence.

The Ly$\alpha$ cooling radiation produced by the inflowing gas accreted onto the central galaxy has been investigated by several authors (Haiman et al., 2000; Kereš et al., 2005; Dekel & Birnboim, 2006; Shull et al., 2009), but the significance of cooling is still difficult to predict accurately and remains uncertain (Fardal et al., 2001; Yang et al., 2006; Dijkstra & Loeb, 2009; Faucher-Giguère et al., 2010; Cantalupo et al., 2012; Rosdahl & Blaizot, 2012; Lake et al., 2015). The scattering of nebular Ly$\alpha$ photons produced in the HII regions of the central galaxy likely plays a major role in the observed Ly$\alpha$ surface brightness profiles at small distances from the center (a few tens of kpc; Laursen & Sommer-Larsen, 2007; Laursen et al., 2009; Steidel et al., 2011; Zheng et al., 2011b; Wisotzki et al., 2016; Xue et al., 2017, Leclercq et al., in prep.) but, at large impact parameter, scattering from the central galaxy alone usually cannot account for the totality of the observed emission (e.g., Lake et al., 2015, see also Dijkstra & Kramer 2012). Similarly, we demonstrated in Mas-Ribas & Dijkstra (2016) that the fluorescent effect of the central galaxy cannot explain the observed surface brightness profiles at distances r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}20 physical kpc.

The non-linear clustering of objects derived from the hierarchical Cold Dark Matter model of structure formation predicts that a significant fraction of the faint sources likely reside around more massive, brighter galaxies. Therefore, star-forming regions and galaxies surrounding the central galaxy (satellite sources) may provide additional contributions to the extended halos at large distances from the center, r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}30 pkpc, via the nebular radiation produced ‘in-situ’ in their ISM, and inducing fluorescent emission in the CGM of the central galaxy (e.g., Shimizu & Umemura, 2010; Matsuda et al., 2012; Lake et al., 2015; Momose et al., 2015, see Maiolino et al. 2017 for a recent detection of star formation within outflows). Although most of the satellites are probably too faint to be resolved individually, their overall collective emission may be detectable, similarly to the method of intensity mapping on large scales (e.g., Chang et al., 2010; Visbal & Loeb, 2010; Carilli, 2011; Gong et al., 2011; Silva et al., 2013; Doré et al., 2014; Pullen et al., 2014; Croft et al., 2016; Li et al., 2016). We addressed the relevance of satellite sources in Mas-Ribas & Dijkstra (2016), accounting for the clustering of ionizing radiation which, in turn, yields to enhanced fluorescent Ly$\alpha$ emission. Our results demonstrated that fluorescence alone cannot explain the observed profiles but its contribution can be up to $\sim 50\%$ out to $r\sim 30$ pkpc if conditions of high escape fraction of ionizing photons and cold gas covering factor are accomplished.

In the present work, we focus on the nebular emission (‘in-situ’ production) from the satellite sources. This analysis is important because, as we will demonstrate, we are able to reproduce the observed Ly$\alpha$ and UV surface brightness profiles, which supports the notion that faint satellite sources can explain the extended LAHs. We self-consistently also predict H$\alpha$ and continuum surface brightness profiles for different models and parameters, which will be testable with future JWST observations. We show how the observations of H$\alpha$ surface brightness profiles will serve to clearly distinguish between the mechanisms that give rise to spatially extended emission, and will place constraints on halo star formation, in addition to the current UV measurements.

Obtaining tighter constraints to the presence of radiation sources in the halo of more massive galaxies allows for assessing the important role that faint objects played in the total cosmic photon budget (see, e.g., Nestor et al., 2011, 2013; Alavi et al., 2014; Garel et al., 2016) and, for high enough redshifts, their contribution to the reionization of the Universe (Kuhlen & Faucher-Giguère, 2012; Robertson et al., 2013). Interestingly, Croft et al. (2016) recently reported an excess of Ly$\alpha$ emission resulting from their cross-correlation between Ly$\alpha$ surface brightness and quasars from the Sloan Digital Sky Survey III (SDSS-III; Eisenstein et al., 2011) Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al., 2013). Croft et al. (2016) argue that, if their measured Ly$\alpha$ emission is driven by star formation, this results in a star formation rate density $\sim 30$ times larger than what is obtained from LAE surveys, although consistent with dust-corrected UV continuum analyses. The star formation scenario, however, needs to invoke an escape fraction for Ly$\alpha$ $\sim 100\%$, and strong radiative transfer effects. Our work can be viewed as a complementary experiment at smaller scales, where we ‘cross-correlate’ deeper Ly$\alpha$ intensity images with LAEs.

We perform calculations considering the spatially extended emission observed around Lyman Alpha Emitters (LAEs) at redshift $z=3.1$, which allows for a comparison with the results by Momose et al. (2014) and Matsuda et al. (2012). Our paper is structured as follows: In § 2, we detail the formalism and adopted values for the parameters in the calculation of the surface brightness profiles for the continua, H$\alpha$ and Ly$\alpha$. We present the results for several models in § 3, and provide a discussion in § 4, before concluding in § 5. Appendix A addresses the implications of the luminosity function parameter values, in terms of spatial and luminosity distribution of satellite sources around the central galaxy. In Appendix B, we detail the calculations of the signal-to-noise ratio for our predicted observations with JWST.

We assume a flat $\Lambda$CDM cosmology with values ${\rm\Omega_{\Lambda}=0.7}$, ${\rm\Omega_{m}}=0.3$ and ${\rm H_{0}=68\,km\,s^{-1}\,Mpc^{-1}}$.

2. Formalism

We present a simple analytic formalism that works with integrated properties of the entire emitting population, which allows to circumvent the modelling of individual sources when calculating the surface brightness profiles.

We demonstrated in Mas-Ribas & Dijkstra (2016) that the fluorescent radiation from a central galaxy with ${\rm SFR\sim 10\,M_{\odot}\,yr^{-1}}$ only dominates at distances ~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}20-30 pkpc from the center, and at a level that strongly depends on the characteristics of the circumgalactic gas. In addition, at such small distances, the profile of the central galaxy is significantly driven by the point-spread function (PSF) of the instrument (Momose et al., 2014). Owing to these uncertainties, we here ignore the central galaxy and limit our calculations to distances $>10$ pkpc.

We use a similar formalism to that applied in Mas-Ribas & Dijkstra (2016), to which we refer the reader for details. Briefly, the Ly$\alpha$ and H$\alpha$ surface brightness at impact parameter $b$ equals

[TABLE]

where ‘$x$’ stands for Ly$\alpha$ or H$\alpha$. The factor ${(1+z)^{-4}}$ accounts for the surface brightness dimming. The factor $\bar{\epsilon}^{\rm sat}_{x}$ denotes the integrated volume emissivity in satellite galaxies (see § 2.1), the term $[1+\xi_{x}(r)]$ denotes the boost in emissivity due to clustering of sources around the central galaxy (see § 2.2), and $f_{\rm esc}^{x}$ denotes the escape fraction (see § 2.3). Finally, the value for the upper limit of the integral extends to infinity for the Abel transformation used above but we limit its value accounting for the line-shift due to the expansion of the universe as

[TABLE]

$H(z)$ denotes the Hubble parameter at a given redshift, $c$ is the speed of light and d$\nu_{\alpha}/\nu_{\alpha}=0.02$ accounts for the line-shift for apertures in narrowband surveys of $\sim 100\,{\rm\AA}$, e.g., Matsuda et al. (2012). This approach implies $R_{\rm max}^{\alpha}\sim 3$ pMpc, but we have tested that our results show only differences of a factor $\sim 2$ at large distances, r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}80-100 pkpc, when setting the upper limit within the range $300\,{\rm pkpc}<R_{\rm max}^{\alpha}<5$ pMpc.

We calculate the UV surface brightness at $1500\,\,{\rm\AA}$ rest-frame as

[TABLE]

where we use the parameters for UV radiation, and have multiplied Eq. 1 by $(1+z)$, since the UV surface brightness is measured as a flux density (in units of inverse frequency) per unit solid angle.

We compute the surface brightness for the visible continuum (VIS) as

[TABLE]

We derive the VIS emission using the H$\alpha$ equivalent width because visible radiation is not commonly used as a star formation estimator, therefore not providing a relation between star formation and luminosity at a specific wavelength, unlike Ly$\alpha$, H$\alpha$ and UV in Eq. 6 (see Kennicutt & Evans, 2012, for a complete review). We assume a flat spectrum around H$\alpha$ and a line equivalent width ${\rm EW_{H\alpha}=300\,\,{\rm\AA}}$ (rest-frame; e.g., Mármol-Queraltó et al., 2016, and references therein). In Eq. 4, $\lambda_{\rm H\alpha}$ and $c$ represent the H$\alpha$ wavelength at rest and the speed of light, respectively, applied to obtain the surface brightness in units of inverse frequency. For completeness, we will also explore the ranges $450\geq{\rm EW_{H\alpha}\,[\AA]\geq 150}$ and $700\geq{\rm EW_{H\alpha}\,[\AA]\geq 50}$.

2.1. Volume Emissivity, $\bar{\epsilon}_{x}^{\rm sat}$

The integrated volumetric emissivity (i.e., volume emissivity) in faint satellites is given by

[TABLE]

where $\rho^{\rm sat}_{\rm SFR}$ denotes the star formation rate density in faint satellites. We are interested in the contribution to the star formation rate density from sources fainter than $M_{\rm UV}\equiv M^{\rm sat}_{\rm UV}=-17$, which corresponds roughly to the minimum UV luminosity of unlensed galaxies that can be detected directly (e.g., Bouwens et al., 2015; Finkelstein et al., 2015). For a UV luminosity function with faint-end slope $\alpha=-1.7\,(-1.5)$, this approach translates to extrapolating the LF to $M_{\rm UV}\sim-12\,(M_{\rm UV}\sim-10)$ (Kuhlen & Faucher-Giguère, 2012; Alavi et al., 2016; Lapi et al., 2017; Livermore et al., 2017). The integrated cosmic star formation rate density in the observed population of star forming galaxies is $\rho_{\rm SFR}\sim 0.1\,{\rm M_{\odot}\,yr^{-1}\,cMpc^{-3}}$ at $z\sim 3$ (see, e.g., Hopkins & Beacom, 2006; Bouwens et al., 2015; Khaire & Srianand, 2015; Robertson et al., 2015). We assume that $\rho^{\rm sat}_{\rm SFR}=\rho_{\rm SFR}$, for simplicity. This assumption depends in detail on the faint-end slope of the UV luminosity function (LF) at $M_{\rm UV}>M^{\rm sat}_{\rm UV}$, on $M^{\rm sat}_{\rm UV}$ itself, and the UV magnitude down to which we integrate this LF. The precise value for $\rho^{\rm sat}_{\rm SFR}$ is, therefore, highly uncertain, and our results scale linearly with the value for this parameter. The constant $C_{x}$ represents the standard conversion factor from SFR into UV luminosity density, H$\alpha$ and Ly$\alpha$ luminosities, and is given by

[TABLE]

The conversion factor for the UV continuum comes from Madau et al. (1998), and for H$\alpha$ from Kennicutt (1998). We obtain the conversion factor for Ly$\alpha$ from H$\alpha$, assuming the common $L_{\rm Ly\alpha}=8.7L_{\rm H\alpha}$ ratio (Brocklehurst, 1971; Barnes et al., 2014; Dijkstra, 2014), which assumes case-B recombination. We caution that these conversion factors, especially for Ly$\alpha$, can vary depending on the metallicity, initial mass function (IMF), and ages of the stellar population (Raiter et al., 2010; Mas-Ribas et al., 2016). Additionally, the value of $C_{\rm Ly\alpha}$ strongly depends on the Ly$\alpha$ rest-frame equivalent width of the sources. We will demonstrate in § 4.1 that accounting for this dependence over the faint satellite population has a significant impact on the results.

2.2. Clustering of Emission, $[1+\xi_{x}(r)]$

The cross-correlation function of emission around LAEs is proportional to the matter density field and can be written as $\xi_{x}(r)=b_{x}(r)b_{\rm LAE}(r)\xi(r)$. The term $\xi(r)$ denotes the non-linear dark matter correlation function obtained using CAMB (Lewis et al., 2000). The terms $b_{\rm LAE}(r)$ and $b_{x}(r)$ are the distance-dependent LAE and emission biases, respectively. We discuss these terms below.

2.2.1 The ${\rm LAE}$ bias, $b_{\rm LAE}(r)$

Following Mas-Ribas & Dijkstra 2016 (see their Appendix B for details) our fiducial model (solid black line in Figure 1) adopts $b_{\rm LAE}(r)$ based on observations by Ouchi et al. (2010), who measured $b_{\rm LAE}(r)$ to increase to $b_{\rm LAE}(r)\sim 10$ down to $r\sim 20$ pkpc. We tested in our previous work that using this clustering we obtained an overdensity $\delta_{\rm LAE}\sim 1.5$, averaged over a radial distance of $2\,{\rm Mpc\,h^{-1}}$ from the central galaxy, consistent with the values reported by Matsuda et al. (2012). To quantify how much our results depend on extrapolating $b_{\rm LAE}(r)$ down to smaller scales, we have also repeated our calculations, but limiting the $b_{\rm LAE}(r)$ to a maximum value of 10. This model is represented in Figure 1 as the dotted red line. In addition, the observational uncertainties reported by Ouchi et al. (2010) for the bias at $r\sim 20$ kpc are of the order $\sim 50\%$, consistent at a $2\sigma$ level with the bias obtained assuming a power-law correlation function. Owing to these large uncertainties for the bias at small scales, we also explore other clustering prescriptions in § 2.2.3 below.

2.2.2 Emission bias, $b_{x}(r)$

The term $b_{\alpha}(r)$ expresses the distance-dependent bias of the Ly$\alpha$ emission, which we assume to differ from that of LAEs by a constant, i.e., $b_{\alpha}(r)=k\,b_{\rm LAE}(r)$. The bias $b_{\alpha}(r)$ represents the Ly$\alpha$ luminosity-weighted average of the entire satellite population. Its value thus depends on the faint-end slope of the Ly$\alpha$ luminosity function ($\alpha_{{\rm Ly}\alpha}$, which is likely steeper than the UV-LF; see, e.g., Gronke et al., 2015; Konno et al., 2016), although, as long as $\alpha_{{\rm Ly}\alpha}>-2$, we expect that the bias is set by the most luminous satellites with $M_{\rm UV}\sim M_{\rm UV}^{\rm sat}=-17$. Gronke et al. (2015) have shown that observational constraints on $M_{\rm UV}$-dependent Ly$\alpha$ equivalent width (EW) PDFs imply that the faintest LAEs ($L_{\rm Ly\alpha}\sim 10^{42}$ erg s*-1*) are associated with galaxies with $M_{\rm UV}\sim-17.5$ (see their Figure 3). This result suggests that the UV-brightest satellites may cluster like LAEs, with $k\sim 1$, although fainter sources might present values $k>1$. Croft et al. (2016) argue that $b_{\alpha}(r)$ might be further boosted by radiative transfer effects due to the resonant nature of the Ly$\alpha$ radiation (e.g., Zheng et al., 2011a). To be conservative, we adopt $k=1$ in our fiducial model111It is worth pointing out that the value of $k$ formally cannot be chosen independently of the escape fraction (see § 2.3); a high $k$ value implies that radiative transfer in the CGM/IGM is important. In order to reproduce the observed Ly$\alpha$ LFs of LAEs, one then requires that $f_{\rm esc}^{\rm Ly\alpha}\sim 1$ (see Zheng et al., 2010)..

Our fiducial model assumes that for both, UV and H$\alpha$ emission, we have $b_{\rm UV}(r)=b_{\rm H\alpha}(r)=b_{\alpha}(r)=b_{\rm LAE}(r)$, i.e $k=1$. This choice is motivated by the discussion above, while noticing that radiative transfer cannot further enhance $k$ in these cases. Our predicted surface brightness profiles again scale linearly with $k$.

2.2.3 Alternative clustering prescriptions

We consider two alternative clustering prescriptions:

The distribution of satellites follows that of dark matter in a Navarro-Frenk-White (NFW) profile (Navarro et al., 1997), normalized to be the same as the other clustering estimators at $r\sim 60$ pkpc (similar to the value of the virial radius for the central galaxy). This model is represented by the green line in Figure 1. The density profile in the NFW model equals

[TABLE]

where $\rho_{c}(z)$ is the critical density of the universe at redshift $z$, $\delta_{c}$ is the characteristic overdensity and $r_{s}$ is the scale radius of the dark matter halo. The overdensity $\delta_{c}$ can be expressed as

[TABLE]

where $\Delta=18\pi^{2}$ is the density contrast from Bryan & Norman (1998) and $c_{\rm NFW}=4$ is the concentration parameter at $z=3.03$ from Zhao et al. (2009). We obtain the scale radius from the expression $c_{\rm NFW}\equiv r_{h}/r_{s}$, where $r_{h}$ is the halo virial radius, computed as

[TABLE]

This expression emerges from considering that the mean density of the halos within the virial radius is $\Delta\rho_{c}(z)$ (Sadoun et al., 2016). We have assumed an LAE halo mass $\log M_{h}=11.5\,{\rm M_{\odot}}$, consistent with the observed range of LAE masses in Ouchi et al. (2010). 2. 2.

We extrapolate the common LAE power-law two-point correlation function, with scale length $r_{0}=2.5\,{\rm Mpc\,h^{-1}}$ and power-law index $\alpha_{c}=-1.8$ (e.g., Gawiser et al., 2007; Kovač et al., 2007; Ouchi et al., 2003, 2010; Guaita et al., 2010; Bielby et al., 2015) down to small scales. This clustering profile is denoted by the blue solid line in Figure 1. The power-law function presents differences with our fiducial model at distances above $\sim 60$ pkpc and at tens of pkpc from the center. In this last region is where the non-linear clustering effects, not captured by the power-law, are important, therefore higher values for the fiducial function are expected.

2.3. Escape Fraction, $f_{\rm esc}^{x}$

The Ly$\alpha$ escape fraction, $f_{\rm esc}^{\rm Ly\alpha}$, has been constrained observationally to be $f_{\rm esc}^{\rm Ly\alpha}\sim 20\%$ at $z\sim 3$ from the Ly$\alpha$ and UV luminosity functions (e.g. Blanc et al., 2011; Hayes et al., 2011), and Ly$\alpha$ and star formation analysis (Dijkstra & Jeeson-Daniel, 2013). However, we caution that all the observations have constrained the ‘effective’ escape fraction, which denotes the fraction of Ly$\alpha$ photons that reaches the observer. As mentioned previously, in some models all Ly$\alpha$ photons escape from the ISM, but then scatter in the CGM/IGM to form halos (in these same models Ly$\alpha$ radiative transfer causes $k>1$). These photons would not have been considered in traditional measurements of Ly$\alpha$ luminosity functions (up to a fraction 40\%-~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}90\% of the total Ly$\alpha$ flux, as argued by Wisotzki et al., 2016, see also Drake et al. 2016) and, therefore, not considered for current observational constraints on $f_{\rm esc}^{\rm Ly\alpha}$. Also, there is observational evidence that the Ly$\alpha$ escape fraction increases towards lower UV-luminosities (e.g., Japelj et al., 2016, see also Dijkstra et al. 2016 and references therein). While observations find $f_{\rm esc}^{\rm Ly\alpha}\sim 20\%$, we consider this value likely a lower-limit and adopt the range $0.1\leq f_{\rm esc}^{{\rm Ly}\alpha}\leq 1.0$ throughout, with a fiducial value $f_{\rm esc}^{\rm Ly\alpha}=40\%$.

We adopt the same range and conservative fiducial value for UV and H$\alpha$ escape fractions. We generally expect that $f_{\rm esc}^{\rm UV}\geq f_{\rm esc}^{\rm Ly\alpha}$ because UV photons are not affected by radiative transfer effects, i.e., resonant scattering that increases the chance to be destroyed by dust (see, e.g., Figure 7 in Garel et al., 2015). The escape fraction of H$\alpha$ can be even larger than that of UV, due to the wavelength dependence of the dust extinction curve (see, e.g., Pei, 1992; Calzetti et al., 1994, 2000; Gordon et al., 2003).

3. Surface brightness profiles

We present the resulting surface brightness profiles below. It is important to keep in mind that these results are degenerate in the product of emissivity, escape fraction and bias, $\bar{\epsilon}^{\rm sat}_{x}\,f_{\rm esc}^{x}\,b_{x}$, where $x$ refers to UV, H$\alpha$ and Ly$\alpha$.

3.1. Ly$\alpha$

The left panel in Figure 2 shows the predicted Ly$\alpha$ surface brightness profile at $r>10$ pkpc. The black solid line denotes the fiducial model, and the shaded areas indicate the range of surface brightness profiles we get by varying $0.2\leq f_{\rm esc}^{\rm Ly\alpha}\leq 0.7$ (dark) and $0.1\leq f_{\rm esc}^{\rm Ly\alpha}\leq 1.0$ (light). These ranges give an idea of the effect of a possible radial variation of the escape fraction due to the decrease of neutral gas with distance. The blue, green and dotted red lines represent the power-law, NFW and ‘bias-limited’ models, respectively (for our fiducial choice $f_{\rm esc}^{\rm Ly\alpha}=0.4$). The light blue dots represent the data and uncertainties from the observations by Momose et al. (2014) at $z=3.1$, which are not reliable at $r>40$ pkpc due to systematics (and therefore represented with open circles; Momose et al., 2014, see also Feldmeier et al. 2013). Magenta dots represent the data in the LAE overdensity bin $2.5<\delta_{\rm LAE}<5.5$ by Matsuda et al. (2012), which we also used in Mas-Ribas & Dijkstra (2016) given the value of our LAE overdensity.

Our fiducial model reproduces the observations well within the range 20~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc. At shorter distances, the Ly$\alpha$ surface brightness may be enhanced by resonantly scattered Ly$\alpha$ that escapes from the central LAE and/or by fluorescence (Mas-Ribas & Dijkstra, 2016). Systematics may in turn affect the data at r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc, although the fiducial model reproduces the data from Matsuda et al. (2012) at these scales remarkably well. The other clustering prescriptions reproduce the observed surface brightness levels to within a factor of $\sim 2$ at 20~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc. In general, they give rise to flatter surface brightness profiles, which reflects that in these models $\xi_{\alpha}$ is flatter at r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}100 pkpc. The impact of the different clustering prescriptions becomes more severe at r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}20 pkpc. However, as we mentioned previously, here we expect the surface brightness profile to be enhanced by Ly$\alpha$ and LyC photons that escaped from the central LAE.

3.2. UV

The right panel in Figure 2 shows the predicted UV surface brightness profiles. We use the same symbols and colors as in the left panel. The horizontal dashed line shows the UV surface brightness level below which the data by Momose et al. (2014) is affected by systematics. This figure shows that at 20~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc our fiducial model predicts a UV profile above the observations by a factor of $\sim 3$, while the other clustering prescriptions lie within a factor of $\sim 1.5-2$. Our fiducial model thus results in an excess - by a factor of $\sim 3$

• of UV emission in the halos of LAEs. This result may reflect an overestimated star formation rate density in faint galaxies (i.e., $\rho^{\rm sat}_{\rm SFR}$). Based on analysis and modeling of Hubble Space Telescope observations aimed at detecting long-duration gamma-ray bursts host galaxies at high redshift, Trenti et al. (2012) inferred that $\sim 30\%$ ($\sim 40\%$) of the total star formation at $z\sim 3$ (z~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}5) occurs in galaxies too faint to be directly detected. This result is broadly consistent with the difference observed here, although the reduced emissivity value would also result in Ly$\alpha$ profiles below the observations by the same factor if no other parameters are tuned. An overestimated escape fraction $f_{\rm esc}^{\rm UV}$ would produce the same effect, though we consider this possibility unlikely. Alternatively, we may have overestimated the abundance of sources in the halo of LAEs due to clustering. We investigate the predicted luminosity and spatial distributions of satellites for various models, and the dependence on luminosity function parameter values in Appendix A. In § 4.1, we further discuss the significant effect of a likely evolution of the Ly$\alpha$ rest-frame equivalent with the UV magnitude of the sources.

3.3. H$\alpha$

Our predictions can be tested with future observations of H$\alpha$ surface brightness profiles since H$\alpha$ falls into the wavelength range covered by the James Webb Space Telescope (JWST; Gardner et al., 2006). In addition, H$\alpha$ does not resonantly scatter, which simplifies interpreting its surface brightness profile compared to Ly$\alpha$, and enables distinguishing between the different possible origins of LAHs.

The left panel in Figure 3 displays the predicted H$\alpha$ surface brightness profiles. Our fiducial model is represented by the solid black line and (conservatively) assumes $f_{\rm esc}^{\rm H\alpha}=0.4$. The impact of varying $f_{\rm esc}^{\rm H\alpha}$ and other models are shown in the same way as in Figure 2. The fiducial profile rises above $10^{-19}\,{\rm erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$ at distances r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}20 pkpc. The red and blue error bars represent the predicted uncertainties on the surface brightness profile, as if it was observed by the near infrared camera (NIRCam) onboard JWST considering the two observational strategies described below. The surface brightness uncertainties are derived from the signal-to-noise ratio (SNR), which decreases from ${\rm SNR}\sim 14$ ($\sim 48$) at $r=20$ pkpc to ${\rm SNR}\sim 0.4$ ($\sim 1.4$) at $r=80$ pkpc for halos around observed LAEs (star-forming galaxies, SFGs). We detail the calculations of the SNR in Appendix B. The left panel in Figure 3 indicates that the H$\alpha$ emission predicted by the various models can be detected up to distances r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}80 pkpc when stacking the SFGs expected in the field of view (FOV)222Zhang et al. (2016) have already demonstrated in a recent work the strength of the stacking technic applied to H$\alpha$ radiation around low redshift galaxies.. Considering uniquely the emission around observed LAEs and our adopted observational strategy, NIRCam can prove the halos up to $r\sim 40$ pkpc, yielding upper limits at larger distances (see below). However, the presence of star formation at large distances from the central LAEs can be assessed up to $r\sim 80$ pkpc with observations of H$\alpha$ and visible continuum radiation around star-forming galaxies (red error bars in both panels of Figure 3.

3.3.1 NIRCam H$\alpha$ observations of LAEs

The Multi-object Spectroscopy333https://jwst-docs.stsci.edu/display/JTI/NIRSpec+Multi+Object+Spectroscopy (MOS) observing mode of the near infrared spectrograph (NIRSpec) would be desirable for our observations, given the large FOV, high spectral resolution (up to $R\sim 2700$), and the obtention of the spectra over a broad wavelength range. However, observations of nearby areas of the sky with contiguous (in the direction of dispersion) elements of the Micro-shutter Assembly (MSA) result in spectra overlapping. The Integral-field Unit444https://jwst-docs.stsci.edu/display/JTI/NIRSpec+IFU+Spectroscopy (IFU) spectroscopy mode circumvents this problem with the use of 3-dimensional spectral imaging data cubes but, in this case, the FOV is smaller than the expected halo of a single galaxy (FOV$\sim 3"\times 3"$). Owing to the impracticability of the above modes, we consider the imaging capabilities of NIRCam for our calculations.

We adopt the narrow-band (NB) filter F323N, with a bandpass of $0.038\,\mu{\rm m}$, resulting in a resolution $R\sim 85$. We use this filter because it is the one closer to the H$\alpha$ wavelength of interest, but we note that it is centered at a wavelength $3.2\,\mu{\rm m}$, corresponding precisely to an H$\alpha$ redshift $z=3.9$. For this calculations, we assume the previous H$\alpha$ flux and surface brightness at $z=3.1$, but we recalculate the sky background555For the calculation of the background, we adopt Eq. 22 in the NIRSpec technical note http://www.stsci.edu/~tumlinso/nrs_sens_2852.pdf. at $z=3.9$, obtaining ${\rm SB_{sky}\,(3.20\,\mu{\rm m})=4\times 10^{-20}\,erg\,s^{-1}\,cm^{-2}\,\,{\rm\AA}^{-1}\,arcsec^{-2}}$, consistent with the estimates by Giavalisco et al. (2002) for HST and the Spitzer/IRAC measurements by Krick et al. (2012). We set the observing time to $10^{4}$ s.

We calculate the number of LAEs observable simultaneously in the FOV of NIRCam (FOV$=2\times 2^{\prime}.2\times 2^{\prime}.2$) as follows: We integrate the LAE luminosity function at $z=3.1$ by Ouchi et al. (2008), with the parameters in Table 1, for the luminosity range $10^{42}\leq L_{\rm Ly\alpha}\,({\rm erg\,s^{-1}})\leq 10^{44}$. This calculation yields a space density of LAEs $n_{\rm LAE}\sim 2\times 10^{-3}\,{\rm cMpc^{-3}}$, in agreement with the findings by Ciardullo et al. (2006). The selected filter results in a redshift depth $\Delta z=0.058$ centered at $z=3.9$, giving rise to the simultaneous observation of $\sim 7$ LAEs in the FOV.

Considering LAEs with luminosities $L_{\rm LAE}>10^{42}\,{\rm erg\,s^{-1}}$, we can prove LAHs up to distances $\sim 40$ pkpc, covering entirely the range of radii out to which the extended emission has been detected around LAEs. However, LAEs with these luminosities account for a small fraction of the total star-forming galaxy population. JWST surveys, as those already proposed by the NIRSpec and NIRCam GTO teams in the GOODS and CANDELS fields666https://confluence.stsci.edu/display/STUCP/JWST+Guaranteed+Time+Observers+Cycle+1+Plans, will detect a larger number of star-forming galaxies by means of the continuum and H$\alpha$ radiation. We show below that stacking a larger sample of galaxies will enable proving extended H$\alpha$ emission at larger distances from the center of galaxies, and reaching low surface brightness levels, useful for assessing the role of cooling radiation.

3.3.2 NIRCam H$\alpha$ observations of SFGs

We predict the extended H$\alpha$ emission around star-forming galaxies using the same observing configuration as above, but we estimate the number of SFGs as follows: We integrate the UV luminosity function with the parameters by Kuhlen & Faucher-Giguère (2012) listed in Table 1 within the range -24\leq{M_{\rm UV}}~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}-17, resulting in a space density $n_{\rm SFG}\sim 2\times 10^{-2}\,{\rm cMpc^{-3}}$. The upper limit, $M_{\rm UV}\sim-17$, rises from considering $L_{\rm UV_{1500}}^{\rm min}\sim 0.025\,L_{\rm UV}^{*}$, and is consistent with the current limit of (unlensed) galaxy surveys (Finkelstein et al., 2015). The obtained space density results in the simultaneous observation of $\sim 86$ SFGs in the FOV.

We refer the reader to Appendix B for a detailed description of the signal-to-noise ratio calculations for the two above strategies.

3.4. VIS

The right panel in Figure 3 shows our predicted surface brightness profiles for the visible (VIS) continuum, with colors and labels as in the left panel. In this case, the shaded areas display the regions $450\geq{\rm EW_{H\alpha}\,[\AA]\geq 150}$ and $700\geq{\rm EW_{H\alpha}\,[\AA]\geq 50}$. JWST observations of the continuum radiation, in the visible wavelength range around $\sim 6800\,\,{\rm\AA}$ rest-frame, will enable proving star formation at large distances in the halos observing SFGs. Additionally, the VIS profiles will complement the UV profiles at large distances, allowing a better comparison of the different continua and line profiles which, in turn, unveils the contribution of the different processes yielding LAHs (see § 4.2).

3.4.1 NIRCam VIS observations of LAEs

We follow the previous observational strategies, using the SNR calculations presented in Appendix B, and the instrumental parameters listed in Table 2.

We consider the same sample of LAEs as in § 3.3.1, and the medium-band filter F335M, centered at $3.362\,\mu$m and with a bandpass $0.352\,\mu$m, resulting in a resolution $R\sim 10$. Since this filter is broader than that used to obtain the sample of LAEs, the observational depth will be larger, i.e., the number of galaxies falling into the filter band is larger than that of LAEs. This (undesired) additional number of galaxies may require the modelling of the sources and the removal of extra flux.

3.4.2 NIRCam VIS observations of SFGs

For the observations of star-forming galaxies, we follow the same procedure and sample of galaxies as in § 3.3.2. We use the same filter as above, F335M, noticing that the same modelling of sources just discussed will also be necessary in this case.

4. Discussion

We discuss below the differences between the parameters for faint satellites and brighter galaxies, and the dependence of our results on these values (§ 4.1). In § 4.2 we show how the comparison between the H$\alpha$, Ly$\alpha$ and continuum profiles breaks the degeneracies between the different mechanisms that give rise to the extended halos.

4.1. The EW-PDF($M_{\rm UV}$), Ly$\alpha$ duty cycle, and

$C_{{\rm Ly}\alpha}$

For any fixed choice of satellite clustering, tuning the model to reproduce the observed Ly$\alpha$ surface brightness profile will cause it to overshoot the UV surface brightness profile (by a factor of up to $\sim 1.5-3$, depending on the clustering model, see Figure 2). This effect can be easily remedied by requiring that $f_{\rm esc}^{{\rm Ly}\alpha}>f_{\rm esc}^{\rm UV}$. However, resonant scattering typically enlarges the total path that Ly$\alpha$ photons travel through dusty, multiphase media, which increases the probability that these photons are destroyed by dust grains, relative to that of the continuum (see, e.g., Laursen et al., 2013; Gronke & Dijkstra, 2014). We therefore consider that it is not reasonable to require that $f_{\rm esc}^{{\rm Ly}\alpha}>f_{\rm esc}^{\rm UV}$ for the entire population.

It is more likely that our adopted conversion factors from star formation rate density to integrated volume emissivity ($C_{x}$ in Eq. 6) differ somewhat. Our current choices for $C_{{\rm Ly}\alpha}$ and $C_{\rm UV}$ imply that all star-forming galaxies produce a Ly$\alpha$ line with a rest-frame equivalent width of EW$\sim 80$  Å  (see, e.g., Dijkstra & Westra, 2010), but the EW of the Ly$\alpha$ line can be larger by a factor of a few for very young stellar populations (e.g., Schaerer, 2003). $C_{x}$, especially $C_{{\rm Ly}\alpha}$, can be increased for lower metallicity, low SFR galaxies and/or for more top-heavy IMFs (Raiter et al., 2010; Forero-Romero & Dijkstra, 2013; Mas-Ribas et al., 2016, see also the review by Kennicutt & Evans 2012 for H$\alpha$ and UV). This interpretation is supported by the short duty-cycle of Ly$\alpha$ selected LAEs reported by Ouchi et al. (2010), which illustrates that the larger EW objects are dominated by young stellar populations.

Our results suggest that, in order to simultaneously reproduce the observed Ly$\alpha$ and UV surface brightness profiles, we need the population averaged rest-frame EW to be $\sim 1.5-3$ larger, i.e., we need $\langle{\rm EW}\rangle\sim 120-240$ Å  for Ly$\alpha$ . ‘Population averaged’ here refers to an average over all satellite galaxies with M_{\rm UV}~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}-17. In the left panel of Figure 4, we have reduced the UV surface brightness profiles of the fiducial and power-law models by a factor of 3 and 2, respectively, to obtain a good fit to the data. In the right panel of the same figure, we present the corresponding Ly$\alpha$ profiles, for different values of $\langle{\rm EW}\rangle$. The dashed and solid black lines denote the power-law and fiducial profiles, respectively, when considering the same escape fraction for Ly$\alpha$ and UV, as in our previous calculations, i.e., $\langle{\rm EW}\rangle\sim 80$ Å. In this case, the profiles fall below the observations as expected. Considering $\langle{\rm EW}\rangle\sim 160$ Å, the fiducial model (solid blue line) reproduces the data by Matsuda et al. (2012) well for r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}30 pkpc, but is slightly lower at larger distances. Accounting for the contribution of the central galaxy, the power-law model (dashed blue line) may match the data at r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc although is above the observations by Matsuda et al. (2012) at larger distances. If we consider $\langle{\rm EW}\rangle\sim 240$ Å  (dashed and solid yellow lines), the models matches the data well at any distance, but if the central galaxy is added, they may overpredict the profiles. Therefore, we conclude from this calculation that an average equivalent width around $\langle{\rm EW}\rangle\sim 160$ Å  may provide a reasonable fit to the data, although the exact value depends on the specific model and contribution of the central galaxy.

Is this $\langle{\rm EW}\rangle$ requirement reasonable? There is strong observational support that the Ly$\alpha$ EW-PDF evolves towards fainter UV-luminosities. Dijkstra & Wyithe (2012) presented a fitting formula for $P({\rm EW}|M_{\rm UV},z)$ constrained by observations. The solid black line in Figure 5 displays $\langle{\rm EW}\rangle$ (rest-frame) as a function of $M_{\rm UV}$ as given by this fitting formula (see Schenker et al., 2014, for an alternative parametrization). Figure 5 shows that for bright LBGs (M_{\rm UV}~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}-22, see Figure A1-A3 of Dijkstra & Wyithe 2012, which is based in data by Shapley et al. 2003), $\langle{\rm EW}\rangle\sim 0$. However, $\langle{\rm EW}\rangle$ rapidly rises towards lower UV luminosities (based on data by Stark et al. 2010) and reaches $\langle{\rm EW}\rangle\sim 55$ Å  at $M_{\rm UV}\sim-19$. Due to the lack of observational data, Dijkstra & Wyithe 2012 adopted (conservatively) the same slope as in the range $M_{\rm UV}\leq-21.5$ for the region $M_{\rm UV}\geq-19$ for the evolution of equivalent width. This modest extrapolation can partially account for simultaneously matching the UV and Ly$\alpha$ surface brightness profiles. However, the evolution may be steeper than assumed by these authors (see, e.g., the recent work at $z=7$ by Ota et al., 2017). The dashed red line in Figure 5 extrapolates the evolution observed by Stark et al. (2010) in the range -19~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}M_{\rm UV}~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}-22, reaching the required value to match both profiles, $\langle{\rm EW}\rangle=240$, quickly after $M_{\rm UV}\sim-16$. The shaded red region shows the area between these two evolutions.

4.2. Comparing H$\alpha$, Ly$\alpha$, and Continuum profiles

Joint analyses of H$\alpha$, Ly$\alpha$ and continuum surface brightness profiles are very useful because they enable disentangling the possible origins of the extended emission. The differences will rise from the physical mechanisms that can yield photons of these three wavelength bands. (i) Continuum radiation is a direct tracer of star formation because it is only produced in the ISM, and it is not a resonant transition. (ii) H$\alpha$ is also produced in the ISM via recombinations following hydrogen ionization but, in addition, can be produced far from the star-forming regions if ionizing photons reach those distances and ionize the more distant gas (fluorescence). (iii) Ly$\alpha$ can be produced via the two previous mechanisms, but also by collisional excitation of neutral hydrogen accreted into the central galaxy (gravitational cooling). Additionally, Ly$\alpha$ is a resonant transition, which allows the Ly$\alpha$ photons to scatter away from the sites where they are produced. The flow chart and plots of Figure 6 represent a simple method to identify the mechanisms playing a role in the extended emission. The idealized diffuse halo in the left part of the figure shows extended Ly$\alpha$ (in blue) but compact H$\alpha$ and continuum emission (in red and green, respectively). This scenario is a clear indication of scattering and/or cooling, as we describe below. The middle plot illustrates a more extended H$\alpha$ halo compared to that of the continuum, indicating that fluorescence is important. When star formation occurs far from the center, the continuum will also appear more extended, as schematically illustrated in the right plot. H$\alpha$ and Ly$\alpha$ halos will also be extended in this case accounting for the nebular radiation of the satellite sources, and can be subject to the extra contribution of fluorescence, scattering and/or cooling. Additional information can be obtained from the radial profiles as follows:

A strong suppression of the continuum and H$\alpha$ surface brightness compared to our predictions at a fixed Ly$\alpha$ surface brightness favors the scattering and cooling models. Models that purely invoke scattering to explain spatially extended Ly$\alpha$ halos cannot produce extended continuum and H$\alpha$ halos. Cooling gives rise to H$\alpha$ and UV halos that are suppressed by a factor of $\sim 10$ compared to our predictions here (see Dijkstra 2014 for a review discussing the H$\alpha$ and UV continuum signatures of cooling radiation). This scenario corresponds to the 2D plot on the left part of Figure 6, where Ly$\alpha$ emission appears more extended than the H$\alpha$ and continuum. 2. 2.

Comparing H$\alpha$ and Ly$\alpha$ surface brightness profiles constrains to what extent scattering affects the Ly$\alpha$ surface brightness profile. This is because the volume emissivity of Ly$\alpha$ and H$\alpha$ closely track each other, while only Ly$\alpha$ photons undergo resonant scattering. Scattering systematically flattens the Ly$\alpha$ surface brightness profile, as the Ly$\alpha$ photons diffuse outwards prior to escape.

Comparing H$\alpha$ and continuum surface brightness profiles can determine the importance of nebular against fluorescent emission: if no ionizing photons escape from either the central LAE or the satellites, then the H$\alpha$ is produced in dense nebulae surrouding O and B stars, and we expect the continuum and H$\alpha$ volume emissivity to closely track each other (co-spatial green and red regions in the left plot of Figure 6). Instead, if ionizing photons escape efficiently from low luminosity galaxies (as expected theoretically and observationally; e.g., Wise et al., 2014; Japelj et al., 2016, see also Dijkstra et al. 2016 and references therein), then we expect gas in the CGM to fluoresce in response to the enhanced local ionizing radiation field (see Mas-Ribas & Dijkstra 2016). In this case, each satellite galaxy is more extended in H$\alpha$ than in the continuum. The resulting overall H$\alpha$ surface brightness profile should also be more extended (middle plot in Figure 6 where the red region presents a larger area than the green one).

5. Conclusions

We have quantified the contribution of faint (satellite) galaxies ($M_{\rm UV}>-17$) to spatially extended Ly$\alpha$ and UV halos around star forming galaxies. We have applied the analytic formalism developed in Mas-Ribas & Dijkstra (2016) to model the halos around Lyman Alpha Emitters (LAEs) at $z=3.1$, for several different satellite clustering prescriptions. The predicted surface brightness depends linearly on the product $\bar{\epsilon}^{\rm sat}_{x}\,f_{\rm esc}^{x}\,b_{x}$, where $x$ refers to UV, H$\alpha$ and Ly$\alpha$. Here, $\bar{\epsilon}^{\rm sat}_{x}$ denotes the integrated emissivity of faint galaxies, which is directly linked to the star formation rate density in these galaxies (see § 2.1), $b_{x}$ denotes the emission bias (see § 2.2.2), and $f^{x}_{\rm esc}$ denotes the escape fraction (see § 2.3). Our main results are as follows:

• •

All our models give rise to spatially extended Ly$\alpha$ and UV halos at a level that is broadly consistent with observations at 20~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc from the centers of LAEs, for a reasonable choice of the product $\bar{\epsilon}^{\rm sat}_{x}\,f_{\rm esc}^{x}\,b_{x}$. The flatness of the surface brightness profiles depends on the clustering prescription at small scales (r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}100 pkpc). This result supports the notion that faint satellite sources can explain the extended emission, and constrains their presence in the halo of more massive galaxies.

• •

For any fixed choice of satellite clustering, the ratio between predicted and observed surface brightness at 20~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}r~{}\hbox to0.0pt{<\hss}{\lower 4.30554pt\hbox{\sim}}40 pkpc is higher for UV than for Ly$\alpha$. In other words, any given model which we tune to perfectly reproduce the observed Ly$\alpha$ surface brightness profile will overshoot the predicted UV surface brightness profile (by a factor of up to $\sim 3$, depending on the clustering model). We discussed in § 4.1 that this implies that we need the average Ly$\alpha$ EW (rest-frame) of satellite galaxies to lie around $\langle{\rm EW}\rangle\sim 120-240$ Å, the lower end of which is consistent with the observed evolution of the Ly$\alpha$ EW-PDF as a function of $M_{\rm UV}$ (see Figure 5 and Dijkstra & Wyithe 2012). We found that extrapolating the observed evolution of EW with $M_{\rm UV}$ can, at least partially, accommodate these differences.

• •

Because there exist multiple alternative explanations for the presence of extended Ly$\alpha$ halos around star forming galaxies (incl. scattering, cooling, fluorescence; see § 1), it is important to investigate whether there are observables that distinguish between different mechanisms. We have therefore also predicted H$\alpha$ surface brightness profiles. Our calculations demonstrate that JWST will be able to probe H$\alpha$ surface brightness profiles out to distances r~{}\hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}80 pkpc and at levels down to ${\rm SB_{H\alpha}\sim 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$. These H$\alpha$ observations will enable breaking the degeneracies between the different mechanisms that give rise to extended halos.

We generally expect a progressive steepening of the surface brightness profiles from Ly$\alpha$ to H$\alpha$ to continuum. The exact quantitative steepening depends on how efficiently ionizing photons escape from the central galaxy, and its surrounding satellites, and also the distribution of self-shielding gas in the CGM of the central galaxy and in the central parts of the satellite sources. These more detailed calculations are beyond the scope of our current work. Observations of extended halos complement other recently proposed ways to constraint escape fractions, such as using H$\beta$ EWs (Zackrisson et al., 2013, 2016), Ly$\alpha$ line profies (Verhamme et al., 2015; Dijkstra et al., 2016; Verhamme et al., 2016), and covering factor values (Jones et al., 2012, 2013; Leethochawalit et al., 2016; Reddy et al., 2016). We will apply our method to investigate the average ionizing escape fraction of galaxies during the epoch of cosmic reionization in an upcoming work. We also plan to further constrain our modeling by including predictions for the spatial distribution (radial offset) of long-duration GRBs from the center of the dark-matter host halo, which will depend on the star formation rate and metallicity of the faint satellites (Trenti et al., 2015). Current success rates for the detection of GRB host galaxies at $z\sim 3-5$ are $\sim 60\%$ (Greiner et al., 2015), thus it might be possible that a fraction of the ‘host-less’ GRBs inhabits and probes faint (undetected) satellite sources.

Our results have focussed on using a LAE as the central galaxy. The reason for using LAEs is that there exists good observational data for Ly$\alpha$ halos. However, as Ly$\alpha$ halos appear ubiquitously around star forming galaxies (Steidel et al., 2011; Wisotzki et al., 2016), our analysis can be applied to different populations, allowing for a better understanding on the physical processes governing galaxy formation and evolution, and the role played by faint, undetected sources to the cosmic photon budget at different epochs.

acknowledgements

LMR is grateful to the ENIGMA group at the MPIA in Heidelberg for kind hospitality and inspiring discussions. We thank the anonymous referee for a prompt report and careful reading that improved the clarity of our work. Thanks to Joop Schaye and Zoltán Haiman for comments on the likely distance dependence of the escape fraction, and Nobunari Kashikawa for pointing clustering aspects. We thank Joel Primack, David Sobral, kyoung-Soo Lee, Lutz Wisotzki, Hakon Dahle, Ainar Drews, Marcia and George Rieke, Benjamin Racine, Dan Stark, Johan Fynbo, Eros Vanzella, Jorryt Matthee and Michal Michalowski for their comments and suggestions. MD is grateful to the Astronomy Department in UCSB for kind hospitality.

Appendix A Appendix A: Distribution of satellite sources

We calculate the number of sources in concentric shells at a distance $r$ around the central galaxy for our fiducial clustering model using the expression

[TABLE]

where $r_{\rm min}$, $r_{\rm max}$ and $L_{\rm min}$, $L_{\rm max}$ denote the radial limits of the shell and luminosity limits, respectively, and $\phi(L){\rm d}L$ is the Schechter luminosity function (Schechter, 1976)

[TABLE]

We run the Poisson distribution on the obtained $n(r)$ values to randomly draw a distribution of $10\,000$ integer numbers of galaxies at every concentric shell. The parameters of the luminosity functions are quoted in Table 1, and are taken from the fitting formula by Kuhlen & Faucher-Giguère (2012) for UV, and from Ouchi et al. (2008) for Ly$\alpha$777For comparison, we use the two luminosity functions because, although the parameters of the UV and Ly$\alpha$ luminosity functions are related, this relation depends on several assumptions and is not entirely understood (see, e.g., Figure 4 in Garel et al., 2015)..

Figure 7 displays the distribution of sources in the range $10\leq r\leq 160$ pkpc, covering the entire region of interest, using the parameters in Table 1. Left panel represents the distribution of Ly$\alpha$ luminosities and the right one of UV magnitudes. Every panel quotes the average number of galaxies from Eq. A1, $n(r)$. Our fiducial model predicts $\sim 1-2$ sources in the range $L_{Ly\alpha}\sim 10^{40}-10^{41}\,{\rm erg\,s^{-1}}$, the number increasing for fainter luminosities. For UV, we obtain $\sim 1$ ($\sim 2$) sources with magnitudes $M_{\rm UV}\sim-17\,(-16)$. We have also computed the distribution of galaxies at different radial distances (not shown); the number of galaxies decreases considerably outwards from the center, as expected given the profile of the correlation function, and luminosity and magnitude distributions display similar profiles than those in Figure 7.

We obtain different emissivity values when integrating the luminosity functions compared to those computed previously from the star-formation rate density (a factor $\sim 4.6$ reduction for Ly$\alpha$, and $\sim 1.78$ increase for UV). These differences may be interpreted as corrections to the adopted values for any of the parameters $\bar{\epsilon}_{x}$, $f_{\rm esc}^{x}$ and/or $b_{x}$, but changes in the expressions relating emissivity and star formation may (partially or totally) also accommodate such differences, since they depend on the adopted IMF and other parameters (Hopkins & Beacom, 2006). In addition, we assess below the effect introduced by varying parameters of the luminosity functions: Setting a minimum Ly$\alpha$ luminosity of ${L_{\rm Ly\alpha}}=10^{34}\,{\rm erg\,s^{-1}}$ and an extreme minimum UV magnitude $M_{\rm UV}\sim-3$  results in emissivity changes by less than $1\%$ ($10\%$) for Ly$\alpha$ (UV) functions compared to the previous case. We obtain, however, thousands of sources in the halo. With lower limits set to ${L_{\rm Ly\alpha}}=10^{41}\,{\rm erg\,s^{-1}}$ and $M_{\rm UV}\sim-17$, the increase of UV emissivity is now lower, a factor $\sim 1.35$, and Ly$\alpha$ is lower by a factor, $\sim 5.45$. The total average number of sources is $\sim 3$ ($\sim 1$) for UV (Ly$\alpha$), all with luminosities corresponding to the lower limits. As mentioned above, several works indicate a steeper Ly$\alpha$ faint-end slope than the ones in Table 1 (Gronke et al., 2015; Sobral et al., 2016). Considering $\alpha=-1.8$ for both functions, the UV emissivity is now above by a factor $\sim 2.5$ and Ly$\alpha$ below by a factor $\sim 2$ when compared to the emissivity from star formation. This result demonstrates that the values for the emissivity are more sensible to changes of the faint-end slope than in the lower limits of the luminosity functions.

Appendix B Appendix B: NIRCam signal-to-noise calculation

We calculate the signal-to-noise ratio in our observations as ${\rm SNR}=N_{s}/\sqrt{N_{s}+N_{\rm sky}}$. We ignore the instrumental noise and systematics since these depend on the observational methodology, i.e., number of exposures, number of pixels for source and background calculations, rms fluctuations in the detector response after flat-fielding, or the use or not of auxiliary calibration data for the dark-current subtraction. Given the large FOV, we expect our noise to be dominated by photons instead of systematics. Some of the systematics are accounted for in the system throughput parameter $\eta$ and, in any case, we check our results with the on-line calculator tool (see below). $N_{s}$ and $N_{\rm sky}$ are the azimuthally averaged photon counts for the sources and sky, respectively, and are computed as

[TABLE]

We have used the on-line JWST Exposure Time Calculator888https://jwst.etc.stsci.edu/ (ETC) and have found that the results are consistent with our calculations. We find that a line flux of $f_{\rm H\alpha}\sim 5\times 10^{-19}\,{\rm erg\,s^{-1}\,cm^{-2}}$ and $t_{\rm exp}\sim 10^{4}$ s correspond to 1$\sigma$ (${\rm SNR=1}$), although this flux can vary by a factor of a few when accounting for different readout modes (see, e.g., http://www.stsci.edu/jwst/instruments/nircam/docarchive/JWST-STScI-001721.pdf).