arXiv subjects

Renyue Cen

Publications and source records attributed to Renyue Cen.

At least 91 records · Page 5Linked to original sources

arXiv2016

Constraint on Matter Power Spectrum on $10^6-10^9M_\odot$ Scales from ${\largeτ_e}$

An analysis of the physics-rich endgame of reionization at $z=5.7$ is performed, utilizing jointly the observations of the Ly$α$ forest, the mean free path of ionizing photons, the luminosity function of galaxies and new physical insight. We find that an upper limit on ${\rm τ_e}$ provides a constraint on the minimum mean free path (of ionizing photons) that is primarily due to dwarf galaxies, which in turn yields a new and yet the strongest constraint on the matter power spectrum on $10^6-10^9M_\odot$ scales. With the latest Planck measurements of ${\rm τ_e = 0.055 \pm 0.009}$, we can place an upper limit of $(8.9\times 10^6, 3.8\times 10^7, 4.2\times 10^8)M_\odot$ on the lower cutoff mass of the halo mass function, or equivalent a lower limit on warm dark matter particle mass ${\rm m_x \ge (15.1, 9.8, 4.6)keV}$ or on sterile neutrino mass ${\rm m_s \ge (161, 90, 33)keV}$, at $(1, 1.4, 2.2)σ$ confidence level, respectively.

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arXiv2016

UV Absorption Line Ratios in Circumgalactic Medium at Low Redshift in Realistic Cosmological Hydrodynamic Simulations

Utilizing high-resolution cosmological hydrodynamic simulations we investigate various ultra-violet absorption lines in the circumgalactic medium of star forming galaxies at low redshift, in hopes of checking and alleviating the claimed observational conundrum of the ratio of NV to OVI absorbers, among others. We find a satisfactory agreement between simulations and extant observational data with respect to the ratios of the following four line pairs examined, NV/OVI, SiIV/OVI, NIII/OVI and NII/OVI. For the pairs involving nitrogen lines, we examine two cases of nitrogen abundance, one with constant N/O ratio and the other with varying N/O ratio, with the latter motivated by theoretical considerations of two different synthetic sources of nitrogen that is empirically verified independently. Along a separate vector, for all line pairs, we examine two cases of radiation field, one with the Haardt-Madau background radiation field and the other with an additional local radiation field sourced by hot gas in the host galaxy. In all cases, two-sample Kolmogorov-Smirnov tests indicate excellent agreements. We find that the apparent agreements between simulations and observations will be strongly tested, if the bulk of current upper limits of various line ratios are turned into actual detections. We show that an increase in observational sensitivity by 0.2 dex will already start to significantly constrain the models.

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arXiv2016

C IV and He II Line Emission of Lyman Alpha Blobs: Powered by Shock Heated Gas

Utilizing {\it ab initio} ultra-high resolution hydrodynamical simulations, we investigate the properties of the interstellar and circum-galactic medium of Ly$α$ Blobs (LABs) at $z=3$, focusing on three important emission lines: Ly$α$ 1216Å, \heii 1640Å and \civ 1449Å. Their relative strengths provide a powerful probe of the thermodynamic properties of the gas when confronted with observations. By adjusting the dust attenuation effect using one parameter and matching the observed size-luminosity relation of LABs using another parameter, we show that our simulations can reproduce the observed \civ/\lya\ and \heii/\lya\ ratios adequately. This analysis provides the first successful physical model to account for simultaneously the LAB luminosity function, luminosity-size relation, and the \civ/Ly$α$ and \heii/Ly$α$ ratios, with only two parameters. The physical underpinning for this model is that, in addition to the stellar component for the \lya\ emission, the \lya\ and \civ\ emission lines due to shock heated gas are primarily collisional excitation driven and the \heii\ emission line collisional ionization driven. We find that the density, temperature and metallicity of the gas responsible for each emission line is significantly distinct, in a multi-phase interstellar and circumgalactic medium that is shock-heated primarily by supernovae and secondarily by gravitational accretion of gas.

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arXiv2016

Reconstructing the redshift evolution of escaped ionizing flux from early galaxies with Planck and HST observations

While galaxies at $6 \lesssim z \lesssim 10$ are believed to dominate the epoch of cosmic reionization, the escape fraction of ionizing flux $f_\mathrm{esc}$ and the photon production rate $\dot n_γ$ from these galaxies must vary with redshift to simultaneously match CMB and low-redshift observations. We constrain $f_\mathrm{esc}(z)$ and $\dot n_γ(z)$ with Planck 2015 measurements of the Thomson optical depth $τ$, recent low multipole E-mode polarization measurements from Planck 2016, SDSS BAO data, and $3 \lesssim z \lesssim 10$ galaxy observations. We compare different galaxy luminosity functions that are calibrated to HST observations, using both parametric and non-parametric statistical methods that marginalize over the effective clumping factor $C_\mathrm{HII}$, the LyC production efficiency $ξ_\mathrm{ion}$, and the time-evolution of the UV limiting magnitude $dM_\mathrm{SF}/dz$. Using a power-law model, we find $f_\mathrm{esc} \lesssim 0.5$ at $z=8$ with slope $β\gtrsim 2.0$ at $68\%$ confidence with little dependence on the galaxy luminosity function or data, although there is non-negligible probability for no redshift evolution $β\sim 0$ or small escape fraction $f_\mathrm{esc} \sim 10^{-2}$. A non-parametric form for $f_\mathrm{esc}(z)$ evolves significantly with redshift, yielding $f_\mathrm{esc} \sim 0.2, 0.3, 0.6$ at $z=6,9,12$, respectively. However, a model-independent reconstruction of $\dot n_γ(z)$ predicts a suppressed escaped photon production rate at $z=9$ for the latest Planck data compared to the other models, implying a quicker period of reionization. We find evidence for redshift evolution in the limiting magnitude of the galaxy luminosity function for empirical models of the galaxy luminosity function.

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arXiv2016

Testing Models of Quasar Hosts With Strong Gravitational Lensing by Quasar Hosts

We perform a statistical analysis of strong gravitational lensing by quasar hosts of background galaxies, in the two competing models of dark matter halos of quasars, HOD and CS models. Utilizing the BolshoiP Simulation we demonstrate that strong gravitational lensing provides a potentially very powerful test of models of quasar hosting halos. For quasars at $z=0.5$, the lensing probability by quasars of background galaxies in the HOD model is higher than that of the CS model by two orders of magnitude or more for lensing image separations in the range of $θ\sim 1.2-12~$arcsec. To observationally test this, we show that, as an example, at the depth of the CANDELS wide field survey and with a quasar sample of $1000$ at $z=0.5$, the two models can be differentiated at $3-4σ$ confidence level.

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arXiv2016

Upper Limit on Star Formation and Metal Enrichment in Minihalos

An analysis of negative radiative feedback from resident stars in minihalos is performed. It is found that the most effective mechanism to suppress star formation is provided by infrared photons from resident stars via photo-detachment of ${\rm H^-}$. It is shown that a stringent upper bound on (total stellar mass, metallicity) of ($\sim 1000{\rm M_\odot}$, $-3.3\pm 0.2$) in any newly minted atomic cooling halo can be placed, with the actual values possibly significantly lower. This has both important physical ramifications on formation of stars and supermassive black seeds in atomic cooling halos at high redshift, pertaining to processes of low temperature metal cooling, dust formation and fragmentation, and direct consequences on the faint end galaxy luminosity function at high redshift and cosmological reionization. The luminosity function of galaxies at the epoch of reionization may be substantially affected due to the combined effect of a diminished role of minihalos and an enhanced contribution from Pop III stars in atomic cooling halos. Upcoming results on reionization optical depth from Planck High-Frequency Instrument data may provide a significant constraint on and a unique probe of this star formation physical process in minihalos. As a numerical example, in the absence of significant contributions from minihalos with virial masses below $1.5\times 10^{8}{\rm M_\odot}$ the reionization optical depth is expected to be no greater than $0.065$, whereas allowing for minihalos of masses as low as ($10^7{\rm M_\odot}$, $10^{6.5}{\rm M_\odot}$) to form stars unconstrained by this self-regulation physical process, the reionization optical depth is expected to exceed $(0.075,0.085)$, respectively.

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arXiv2016

Formation of globular clusters in atomic-cooling halos via rapid gas condensation and fragmentation during the epoch of reionization

We investigate the formation of metal-poor globular clusters (GCs) at the center of two dark matter halos with $M_{\rm halo} \sim4\times10^7 M_\odot$ at $z>10$ using cosmological radiation-hydrodynamics simulations. We find that very compact ($\lesssim$ 1 pc) and massive ($\sim6\times10^5 M_\odot$) clusters form rapidly when pristine gas collapses isothermally with the aid of efficient Ly$α$ emission during the transition from molecular-cooling halos to atomic-cooling halos. Because the local free-fall time of dense star-forming gas is very short ($\ll 1\,{\rm Myr}$), a large fraction of the collapsed gas is turned into stars before stellar feedback processes blow out the gas and shut down star formation. Although the early stage of star formation is limited to a small region of the central star-forming disk, we find that the disk quickly fragments due to metal enrichment from supernovae. Sub-clusters formed in the fragmented clouds eventually merge with the main cluster at the center. The simulated clusters closely resemble the local GCs in mass and size but show a metallicity spread that is much wider than found in the local GCs. We discuss a role of pre-enrichment by Pop III and II stars as a potential solution to the latter issue. Although not without shortcomings, it is encouraging that a naive blind (not tuned) cosmological simulation presents a possible channel for the formation of at least some massive GCs.

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arXiv2016

The Physical Nature of the Most Metal-Poor Damped Lyman Alpha Systems

Utilizing the high-resolution, large-scale LAOZI cosmological simulations we investigate the nature of the metal-poor (${\rm [Z/H]<-2}$) damped Lyman alpha systems (mpDLA) at $z=3$. The following physical picture of mpDLAs emerges. The majority of mpDLAs inhabit regions $\ge 20$~kpc from the host galaxy center on infalling cold gas streams originating from the intergalactic medium, with infall velocity of $\sim 100$ km/s and temperature of $\sim 10^{4}$ K. For each host galaxy, on average, about $1\%$ of the area within a radius $150$~kpc is covered by mpDLAs. The mpDLAs are relatively diffuse ($n_{\rm{gas}} \sim 10^{-2}$ cm$^{-3}$), Jeans quasi-stable, and have very low star formation rate ($\dotΣ \le 10^{-4} M_{\odot} \rm{\ yr}^{-1} \rm{\ kpc}^{-2}$). As mpDLAs migrate inward to the galaxy center, they mix with high metallicity gas and stellar outflows in the process, removing themselves from the metal-poor category and rendering the central ($\le 5$ kpc) regions of galaxies devoid of mpDLAs. Thus, the central regions of the host galaxies are populated by mostly metal-rich DLAs instead of mpDLAs. All observables of the simulated mpDLAs are in excellent agreement with observations, except the gas density, which is about a factor of ten lower than the value inferred observationally. However, the observationally inferred value is based on simplified assumptions that are not borne out in the simulations.

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arXiv2015

Inflow Generated X-ray Corona Around Supermassive Black Holes and Unified Model for X-ray Emission

Three-dimensional hydrodynamic simulations, covering the spatial domain from hundreds of Schwarzschild radii to $2\ \mathrm{pc}$ around the central supermassive black hole of mass $10^8 M_\odot$, with detailed radiative cooling processes, are performed. Generically found is the existence of a significant amount of shock heated, high temperature ($\geq 10^8\ \mathrm{K}$) coronal gas in the inner ($\leq 10^4 r_\mathrm{sch}$) region. It is shown that the composite bremsstrahlung emission spectrum due to coronal gas of various temperatures are in reasonable agreement with the overall ensemble spectrum of AGNs and hard X-ray background. Taking into account inverse Compton processes, in the context of the simulation-produced coronal gas, our model can readily account for the wide variety of AGN spectral shape, which can now be understood physically. The distinguishing feature of our model is that X-ray coronal gas is, for the first time, an integral part of the inflow gas and its observable characteristics are physically coupled to the concomitant inflow gas. One natural prediction of our model is the anti-correlation between accretion disk luminosity and spectral hardness: as the luminosity of SMBH accretion disk decreases, the hard X-ray luminosity increases relative to the UV/optical luminosity.

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arXiv2015

SCORCH I: The Galaxy-Halo Connection in the First Billion Years

SCORCH (Simulations and Constructions of the Reionization of Cosmic Hydrogen) is a new project to study the Epoch of Reionization (EoR). In this first paper, we probe the connection between observed high-redshift galaxies and simulated dark matter halos to better understand the abundance and evolution of the primary source of ionizing radiation. High-resolution N-body simulations are run to quantify the abundance of dark matter halos as a function of mass $M$, accretion rate $\dot{M}$, and redshift $z$. A new fit for the halo mass function $dn/dM$ is $\approx 20\%$ more accurate at the high-mass end where bright galaxies are expected to reside. A novel approach is used to fit the halo accretion rate function $dn/d\dot{M}$ in terms of the halo mass function. Abundance matching against the observed galaxy luminosity function is used to estimate the luminosity-mass relation and the luminosity-accretion-rate relation. The inferred star formation efficiency is not monotonic with $M$ nor $\dot{M}$, but reaches a maximum value at a characteristic mass $\sim 2 \times 10^{11}\ M_\odot$ and a characteristic accretion rate $\sim 6 \times 10^2\ M_\odot/{\rm yr}$ at $z \approx 6$. We find a universal EoR luminosity-accretion-rate relation and construct a fiducial model for the galaxy luminosity function. The Schechter parameters evolve such that $ϕ_\star$ decreases, $M_\star$ is more positive (fainter), and $α$ is more negative (steeper) at higher redshifts. We forecast for the upcoming James Webb Space Telescope and show that with apparent magnitude limit $m_{\rm AB} \approx 31\ (32)$, it can observe $\gtrsim 11\ (24)$ unlensed galaxies per square degree per unit redshift at least down to $M_\star$ at $z \lesssim 13\ (14)$.

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arXiv2015

The Non-Linear Power Spectrum of the Lyman Alpha Forest

The Lyman alpha forest power spectrum has been measured on large scales by the BOSS survey in SDSS-III at $z\sim 2.3$, has been shown to agree well with linear theory predictions, and has provided the first measurement of Baryon Acoustic Oscillations at this redshift. However, the power at small scales, affected by non-linearities, has not been well examined so far. We present results from a variety of hydrodynamic simulations to predict the redshift space non-linear power spectrum of the Lyman Alpha transmission for several models, testing the dependence on resolution and box size. A new fitting formula is introduced to facilitate the comparison of our simulation results with observations and other simulations. The non-linear power spectrum has a generic shape determined by a transition scale from linear to non-linear anisotropy, and a Jeans scale below which the power drops rapidly. In addition, we predict the two linear bias factors of the Lyman Alpha forest and provide a better physical interpretation of their values and redshift evolution. The dependence of these bias factors and the non-linear power on the amplitude and slope of the primordial fluctuations power spectrum, the temperature-density relation of the intergalactic medium, and the mean Lyman Alpha transmission, as well as the redshift evolution, is investigated and discussed in detail. A preliminary comparison to the observations shows that the predicted redshift distortion parameter is in good agreement with the recent determination of Blomqvist et al., but the density bias factor is lower than observed. We make all our results publicly available in the form of tables of the non-linear power spectrum that is directly obtained from all our simulations, and parameters of our fitting formula.

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arXiv2015

Supernova Feedback and the Hot Gas Filling Fraction of the Interstellar Medium

Supernovae (SN), the most energetic stellar feedback mechanism, are crucial for regulating the interstellar medium (ISM) and launching galactic winds. We explore how supernova remnants (SNRs) create a multiphase medium by performing 3D hydrodynamical simulations at various SN rates, $S$, and ISM average densities, $\bar{n}$. The evolution of a SNR in a self-consistently generated three-phase ISM is qualitatively different from that in a uniform or a two-phase warm/cold medium. By travelling faster and further in the low-density hot phase, the domain of a SNR increases by $>10^{2.5}$. Varying $\bar{n}$ and $S$, we find that a steady state can only be achieved when the hot gas volume fraction $f_{\rm{V,hot}}\lesssim 0.6 \pm 0.1 $. Above that level, overlapping SNRs render connecting topology of the hot gas, and the ISM is subjected to thermal runaway. Photoelectric heating (PEH) has a surprisingly strong impact on $f_{\rm{V,hot}}$. For $\bar{n}\gtrsim 3 \cm-3 $, a reasonable PEH rate is able to suppress the thermal runaway. Overall, we determine the critical SN rate for the onset of thermal runaway to be $S_{\rm{crit}} = 200 (\bar{n}/1\cm-3)^k (E_{\rm{SN}}/10^{51}\erg)^{-1} \kpc^{-3} \myr-1$, where $k = (1.2,2.7)$ for $\bar{n} \leq 1$ and $> 1\cm-3 $, respectively. We present a fitting formula of the ISM pressure $P(\bar{n}$, $S$), which can be used as an effective equation of state in cosmological simulations. Despite the 5 orders of magnitude span of $(\bar{n},S)$, the average Mach number varies little: $\mathcal{M} \approx \ 0.5\pm 0.2, \ 1.2\pm 0.3,\ 2.3\pm 0.9$ for the hot, warm and cold phases, respectively.

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arXiv2015

Don't Forget the Forest for the Trees: The Stellar-Mass Halo-Mass Relation in Different Environments

The connection between dark matter halos and galactic baryons is often not well-constrained nor well-resolved in cosmological hydrodynamical simulations. Thus, Halo Occupation Distribution (HOD) models that assign galaxies to halos based on halo mass are frequently used to interpret clustering observations, even though it is well-known that the assembly history of dark matter halos is related to their clustering. In this paper we use high-resolution hydrodynamical cosmological simulations to compare the halo and stellar mass growth of galaxies in a large-scale overdensity to those in a large-scale underdensity (on scales of about 20 Mpc). The simulation reproduces assembly bias, that halos have earlier formation times in overdense environments than in underdense regions. We find that the stellar mass to halo mass ratio is larger in overdense regions in central galaxies residing in halos with masses between 10$^{11}$-10$^{12.9}$ M$_{\odot}$. When we force the local density (within 2 Mpc) at z=0 to be the same for galaxies in the large-scale over- and underdensities, we find the same results. We posit that this difference can be explained by a combination of earlier formation times, more interactions at early times with neighbors, and more filaments feeding galaxies in overdense regions. This result puts the standard practice of assigning stellar mass to halos based only on their mass, rather than considering their larger environment, into question.

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arXiv2015

Testing Dark Matter Halo Models of Quasars With Thermal Sunyaev-Zeldovich Effect

A statistical analysis of stacked Compton$-y$ maps of quasar hosts with a median redshift of $1.5$ using Millennium Simulation is performed to address two issues, one on the feedback energy from quasars and the other on testing dark matter halo models for quasar hosts. On the first, we find that, at the resolution of FWHM=$10$ arcmin obtained by Planck data, the observed thermal Sunyaev-Zeldovich (tSZ) effect can be entirely accounted for and explained by the thermal energy of halos sourced by gravitational collapse of halos, without a need to invoke additional, large energy sources, such as quasar or stellar feedback. Allowing for uncertainties of dust temperature in the calibration of observed Comton$-y$ maps, the maximum additional feedback energy is $\sim 25\%$ of that previously suggested. Second, we show that, with FWHM=$1$ arcmin beam, tSZ measurements will provide a potentially powerful test of quasar-hosting dark matter halo models, limited only by possible observational systematic uncertainties, not by statistical ones, even in the presence of possible quasar feedback.

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arXiv2015

Towards simulating star formation in turbulent high-z galaxies with mechanical supernova feedback

Feedback from supernovae is essential to understanding the self-regulation of star formation in galaxies. However, the efficacy of the process in a cosmological context remains unclear due to excessive radiative losses during the shock propagation. To better understand the impact of SN explosions on the evolution of galaxies, we perform a suite of high-resolution (12 pc), zoom-in cosmological simulations of a Milky Way-like galaxy at z=3 with adaptive mesh refinement. We find that SN explosions can efficiently regulate star formation, leading to the stellar mass and metallicity consistent with the observed mass-metallicity relation and stellar mass-halo mass relation at z~3. This is achieved by making three important changes to the classical feedback scheme: i) the different phases of SN blast waves are modelled directly by injecting radial momentum expected at each stage, ii) the realistic time delay of SNe, commencing at as early as 3 Myr, is required to disperse very dense gas before a runaway collapse sets in at the galaxy centre via mergers of gas clumps, and iii) a non-uniform density distribution of the ISM is taken into account below the computational grid scale for the cell in which SN explodes. The last condition is motivated by the fact that our simulations still do not resolve the detailed structure of a turbulent ISM in which the fast outflows can propagate along low-density channels. The simulated galaxy with the SN feedback model shows strong outflows, which carry approximately ten times larger mass than star formation rate, as well as smoothly rising circular velocity. Other feedback models that do not meet the three conditions form too many stars, producing a peaked rotation curve. Our results suggest that understanding the structure of the turbulent ISM may be crucial to assess the role of SN and other feedback processes in galaxy formation theory.

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arXiv2015

Coevolution Between Supermassive Black Holes and Bulges Is Not Via Internal Feedback Regulation But By Rationed Gas Supply Due To Angular Momentum Distribution

We reason that, without physical fine-tuning, neither the supermassive black holes (SMBHs) nor the stellar bulges can self-regulate or inter-regulate by driving away already fallen cold gas to produce the observed correlation between them. We suggest an alternative scenario where the observed mass ratios of the SMBHs to bulges reflect the angular momentum distribution of infallen gas such that the mass reaching the stable accretion disc is a small fraction of that reaching the bulge region, averaged over the cosmological time scales. We test this scenario using high resolution, large-scale cosmological hydrodynamic simulations (without AGN feedback), assuming the angular momentum distribution of gas landing in the bulge region to yield a Mestel disc that is supported by independent simulations resolving the Bondi radii of SMBHs. A mass ratio of $0.1-0.3\%$ between the very low angular momentum gas that free-falls to the sub-parsec region to accrete to the SMBH and the overall star formation rate is found. This ratio is found to increase with increasing redshift to within a factor of $\sim 2$, suggesting that the SMBH to bulge ratio is nearly redshift independent, with a modest increase with redshift, a testable prediction. Furthermore, the duty cycle of active galactic nuclei (AGN) with high Eddington ratios is expected to increase significantly with redshift. Finally, while SMBHs and bulges are found to coevolve on $\sim 30-150$Myr time scales or longer, there is indication that, on shorer time scales, the SMBH accretion rate and star formation may be less correlated.

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arXiv2015

On the Diffuse Lyman-alpha Halo Around Lyman-alpha Emitting Galaxies

Ly$α$ photons scattered by neutral hydrogen atoms in the circumgalactic media or produced in the halos of star-forming galaxies are expected to lead to extended Ly$α$ emission around galaxies. Such low surface brightness Ly$α$ halos (LAHs) have been detected by stacking Ly$α$ images of high-redshift star-forming galaxies. We study the origin of LAHs by performing radiative transfer modeling of nine $z=3.1$ Lyman-Alpha Emitters (LAEs) in a high resolution hydrodynamic cosmological galaxy formation simulation. We develop a method of computing the mean Ly$α$ surface brightness profile of each LAE by effectively integrating over many different observing directions. Without adjusting any parameters, our model yields an average Ly$α$ surface brightness profile in remarkable agreement with observations. We find that observed LAHs cannot be accounted for solely by photons originating from the central LAE and scattered to large radii by hydrogen atoms in the circumgalactic gas. Instead, Ly$α$ emission from regions in the outer halo is primarily responsible for producing the extended LAHs seen in observations, which potentially includes both star-forming and cooling radiation. With the limit on the star formation contribution set by the ultra-violet (UV) halo measurement, we find that cooling radiation can play an important role in forming the extended LAHs. We discuss the implications and caveats of such a picture.

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arXiv2015

Quantifying Distributions of Lyman Continuum Escape Fraction

Simulations have indicated that most of the escaped Lyman continuum photons escape through a minority of solid angles with near complete transparency, with the remaining majority of the solid angles largely opaque, resulting in a very broad and skewed probability distribution function (PDF) of the escape fraction when viewed at different angles. Thus, the escape fraction of Lyman continuum photons of a galaxy observed along a line of sight merely represents the properties of the interstellar medium along that line of sight, which may be an ill-representation of true escape fraction of the galaxy averaged over its full sky. Here we study how Lyman continuum photons escape from galaxies at $z=4-6$, utilizing high-resolution large-scale cosmological radiation-hydrodynamic simulations. We compute the PDF of the mean escape fraction ($\left $) averaged over mock observational samples, as a function of the sample size, compared to the true mean (had you an infinite sample size). We find that, when the sample size is small, the apparent mean skews to the low end. For example, for a true mean of 6.7%, an observational sample of (2,10,50) galaxies at $z=4$ would have have 2.5% probability of obtaining the sample mean lower than $\left =$(0.007%, 1.8%, 4.1%) and 2.5% probability of obtaining the sample mean being greater than (43%, 18%, 11%). Our simulations suggest that at least $\sim$ 100 galaxies should be stacked in order to constrain the true escape fraction within 20% uncertainty.

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