Draft version October 10, 2018Preprint typeset using LATEX style emulateapj v. 01/23/15

THE HYDROGEN EPOCH OF REIONIZATION ARRAY DISH II: CHARACTERIZATION OF SPECTRALSTRUCTURE WITH ELECTROMAGNETIC SIMULATIONS AND ITS SCIENCE IMPLICATIONS.

Aaron Ewall-Wice 1,2, Richard Bradley3,4, David Deboer5, Jacqueline Hewitt1,2, Aaron Parsons5, JamesAguirre6, Zaki S. Ali5, Judd Bowman7, Carina Cheng5, Abraham R. Neben1,2, Nipanjana Patra5, Nithyanandan

Thyagarajan 7, Mariet Venter8,9, Eloy de Lera Acedo10, Joshua S. Dillon5,11, Roger Dickenson3, PhillipDoolittle3, Dennis Egan3, Mike Hedrick3, Patricia Klima3, Saul Kohn6, Patrick Schaffner3, John Shelton3,

Benjamin Saliwanchik12, H.A. Taylor3, Rusty Taylor3, Max Tegmark1,2, Butch Wirt3,1MIT Kavli Institute for Cosmological Physics, Cambridge, MA, 02139 USA

2MIT Dept. of Physics, Cambridge, MA, 02139 USA3National Radio Astronomy Obs., Charlottesville, VA

4Dept. of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA5Dept. of Astronomy, University of California, Berkeley, CA, USA

6Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA7Arizona State University, School of Earth and Space Exploration, Tempe, AZ 85287, USA8Dept. of Electrical and Electronic Engineering, Stellenbosch University, Stellenbosch, SA

9SKA South Africa, Cape Town, SA10Cavendish Laboratory, University of Cambridge, Cambridge, UK

11Berkeley Center for Cosmological Physics, Berkeley, CA, USA and12Astrophysics and Cosmology Research Unit, University of KwaZulu-Natal, Durban, SA

Draft version October 10, 2018

ABSTRACT

We use time-domain electromagnetic simulations to determine the spectral characteristics of theHydrogen Epoch of Reionization Arrays (HERA) antenna. These simulations are part of a multi-faceted campaign to determine the effectiveness of the dish’s design for obtaining a detection ofredshifted 21 cm emission from the epoch of reionization. Our simulations show the existence ofreflections between HERA’s suspended feed and its parabolic dish reflector that fall below -40 dBat 150 ns and, for reasonable impedance matches, have a negligible impact on HERA’s ability toconstrain EoR parameters. It follows that despite the reflections they introduce, dishes are effectivefor increasing the sensitivity of EoR experiments at relatively low cost. We find that electromagneticresonances in the HERA feed’s cylindrical skirt, which is intended to reduce cross coupling and beamellipticity, introduces significant power at large delays (−40 dB at 200 ns) which can lead to someloss of measurable Fourier modes and a modest reduction in sensitivity. Even in the presence of thisstructure, we find that the spectral response of the antenna is sufficiently smooth for delay filtering tocontain foreground emission at line-of-sight wave numbers below k‖ . 0.2hMpc−1, in the region wherethe current PAPER experiment operates. Incorporating these results into a Fisher Matrix analysis,we find that the spectral structure observed in our simulations has only a small effect on the tightconstraints HERA can achieve on parameters associated with the astrophysics of reionization.

1. INTRODUCTION

Observations of the redshifted 21 cm radiation fromneutral hydrogen in the intergalactic medium (IGM)have the potential to illuminate the hitherto unobserveddark ages and cosmic dawn, revolutionizing our under-standing of the first UV and X-ray sources in the uni-verse and how they influenced galactic evolution (seeFurlanetto et al. (2006), Morales & Wyithe (2010), andPritchard & Loeb (2012) for reviews). Major experi-mental endeavors are underway to detect the 21 cm sig-nal, with most focusing on the epoch of reionization(EoR) during which UV photons from early galaxies con-verted the hydrogen in the universe from neutral to ion-ized. One approach involves measuring the sky-averagedglobal signal and is being pursued by experiments suchas EDGES (Bowman & Rogers 2010), LEDA (Greenhill& Bernardi 2012; Bernardi et al. 2015), DARE (Burnset al. 2012), SciHi (Voytek et al. 2014), ZEBRA (Ekers& Subrahmanyan 2012), SARAS (Patra et al. 2015), andBIGHORNS (Sokolowski et al. 2015) either in their plan-ning stages or already taking data. The global signal isalso potentially observable with a zero-spacing interfer-

ometer (Presley et al. 2015; Singh et al. 2015; Venumad-hav et al. 2015). Another strategy is to observe spatialfluctuations in the 21 cm emission using radio interfer-ometers. A first generation of such experiments are at-tempting to obtain upper limits or a first statistical de-tection of the power spectrum of 21 cm brightness tem-perature fluctuations. These include the Giant Metre-wave Telescope (GMRT) (Paciga et al. 2013), the LowFrequency ARray (LOFAR), (van Haarlem et al. 2013),the Murchison Widefield Array (MWA) (Tingay et al.2013), the MIT Epoch of Reionization Experiment (MI-TEoR) (Zheng et al. 2013), and the Precision Array forProbing the Epoch of Reionization (PAPER) (Parsonset al. 2010). Already, many of these experiments arebeginning to yield upper limits on the 21 cm signal (Dil-lon et al. 2013; Parsons et al. 2014; Jacobs et al. 2015;Dillon et al. 2015a; Trott et al. 2016; Ewall-Wice et al.2016b) and significant scientific results. The most strin-gent power-spectrum upper limit of ≈ 500 mK2 by PA-PER (Ali et al. 2015) is able to rule out a number ofscenarios in which the intergalactic medium received lit-tle or no heating from X-rays (Pober et al. 2015; Greig

arX

iv:1

602.

0627

7v2

[as

tro-

ph.C

O]

14

Sep

2016

2 Ewall-Wice et al.

et al. 2015b). As these current observatories reach theirsensitivity limits, work is beginning on the next gener-ation of interferometers, which will have the sensitivityrequired for a robust detection of the 21 cm power spec-trum. These include the Square Kilometer Array (SKA-1Low) (Mellema et al. 2013) and the Hydrogen Epoch ofReionization Array (HERA)(Pober et al. 2014; DeBoeret al. 2016; Dillon & Parsons 2016) .

The primary challenge to obtaining a high redshift de-tection of the cosmological signal through both of thesemethods is the existence of foregrounds that are ≈ 105

times brighter (Bernardi et al. 2009; Pober et al. 2013a;Dillon et al. 2014). Fortunately, it is expected that theforegrounds are spectrally smooth. In power-spectrumexperiments, smooth foregrounds are naturally containedto a finite region of Fourier space, corresponding to largeline of sight scales, known as the wedge (Datta et al. 2010;Vedantham et al. 2012; Parsons et al. 2012b; Thyagara-jan et al. 2013; Liu et al. 2014a,b).

Since the location of each foreground on the sky de-termines its position in the wedge, with sources near thehorizon being introduced at line of sight scales closest tothe EoR window, the angular response of an instrumenthas significant implications on the amount of side-lobepower that is leaked into the EoR window. In Thyagara-jan et al. (2015b,a) and Pober et al. (2016), it is foundthat the response of the antenna, near the horizon, hasa significant impact on foreground contamination andthat centralized beams with highly suppressed sidelobesare preferable.

Beyond leakage from wedge sidelobes, any structure inthe frequency response of the instrument is imprinted onthe foregrounds and has the potential to leak power intothe EoR window at small line of sight scales, maskingthe signal. Indeed, sub-percent spectral features in theanalogue and digital signal chains on the initial build-outof the MWA are proving to be a significant calibrationchallenge (Dillon et al. 2015a; Ewall-Wice et al. 2016b;Beardsley et al. 2016).

In principle, spectral structure in the bandpass of aninstrument may be removed in calibration. However,many approaches rely on detailed models of the fore-grounds themselves and imperfections in these modelsmay introduce spurious spectral structure (e.g. Barryet al. (2016); Patil et al. (2016); Ewall-Wice et al. inpreparation). Redundant calibration (Wieringa 1992;Liu et al. 2010; Zheng et al. 2014) avoids relying on adetailed sky models but in its current state would beunable to correct direction dependent chromaticity thatvaries from antenna to antenna. Furthermore, directiondependent calibration requires imaging which is not per-formed in the delay power spectrum technique (Parsonset al. 2012b) that HERA is designed for. Because ofour limited knowledge of low frequency foregrounds andthe fidelity of calibration algorithms, it is important todesign experiments whose bandpasses are as devoid ofspectral structure as possible.

Drawing from the lessons of the PAPER, MWA, andMITEoR experiments, HERA (Pober et al. 2014; DeBoeret al. 2016; Dillon & Parsons 2016) is a next genera-tion 21 cm experiment designed to achieve a two-ordersof magnitude improvement in sensitivity over current ef-forts, allowing it to make a robust detection of the 21 cmpower spectrum during the EoR. Much of this sensitivity

Fig. 1.— The HERA antenna element (bottom) uses a parabolicdish to achieve an order of magnitude increase in collecting areaover the PAPER antenna (top). The sleeved dipole in the centerof the PAPER backplane is identical to the sleeved dipole beingsuspended under the cylindrical skirt over the vertex of the HERAdish. The suspended feed arrangement has the potential to intro-duce intra-antenna reflections which we explore in this work.

increase is enabled by moving the collecting area of theinstrument into short baselines, with more modes out-side of the wedge, and a switch from PAPER’s skirteddipoles and the MWA’s phased dipole arrays to an an-tenna element that consists of a feed suspended over alarge reflecting parabolic dish. In Fig. 1 we show a PA-PER antenna and one of the initial 19 HERA dishes cur-rently being deployed in South Africa at the same site. Acentral requirement for HERA’s dish design is that theantenna have a response that is sufficiently smooth infrequency and a narrow beam, leaving the EoR windowfree of supra-horizon emission.

This paper and its companions (Neben et al. 2016; Pa-tra et al. in preparation; Thyagarajan et al. 2016) de-scribe a multi-pronged campaign to assess the require-ments and performance of the HERA dish for isolat-ing foregrounds within the wedge. We accomplish thisby establishing antenna specifications with simulationsof foregrounds using the Precision Radio InterferometrySimulator (PRISim1)(Thyagarajan et al. 2016) and ver-ifying that the HERA primary antenna element meetsthese specifications with reflectometry (Patra et al. inpreparation) and ORBCOMM beam mapping (Nebenet al. 2016). In this work, we present the results of time-

1 https://github.com/nithyanandan/PRISim

The HERA Dish II: Electromagnetic Simulations and Science Implications 3

domain electromagnetic simulations that are intended topredict the degree of spectral structure due to reflectionsin the HERA dish, assess the impact of this spectralstructure on the leakage of foregrounds into the EoRwindow, and to verify consistency with the reflectome-try measurements taken in the field.

This paper is organized as follows. In § 2 we lay outour analytic framework, describing the impact of signal-path delays (from reflections or otherwise) on foregroundleakage in delay-transform power spectra. In § 3 we de-scribe our time-domain electromagnetic simulations ofthe HERA dish element and how we extract the voltageresponse function. In § 4 we describe the results andtheir origin in the antenna geometry. We also verify oursimulation framework by comparing its prediction for theS11 reflection coefficient of the HERA dish to direct fieldmeasurements described in Patra et al. (in preparation).In § 5 we apply our electromagnetic simulation resultsto a foreground model to determine the extent that theHERA dish’s chromatic structure compromises the EoRwindow and the impact on HERA’s ability to constrainreionization parameters. We summarize our conclusionsin § 6.

2. THE IMPACT OF REFLECTIONS ONDELAY-TRANSFORM POWER SPECTRA

In this section, we show how delayed signals in the ana-log signal path of an antenna lead to foreground contam-ination of the EoR window. Intuitively, any reflectionsin the signal path introduce frequency ripples in the gainof the instrument. Since time delay is the Fourier dual tofrequency (Parsons et al. 2012b), reflections with largerdelays introduce ripples with shorter periods. Any high-delay frequency structure imprinted on the much brighterforegrounds has the potential to mimic and swamp thesignal unless it is brought below the ratio between theforegrounds and the weak cosmological signal. Equa-tions describing the effect of direction independent re-flections in an interferometers signal chain downstreamof the feed are derived in Ewall-Wice et al. (2016b). Wenow extend this analysis by considering the direction de-pendent reflections that can occur within the antennaelement. We start by denoting the electric field of radia-tion arriving from direction s at the ith antenna elementat position xi with angular frequency ω as s(s,xi, ω) andthe time-domain field as s(s,xi, t). Using the fact thats(s,xi, t) = s(s,0, t−τi) ≡ s(s, t−τi), where τi = s·xi/c,we omit xi from our notation, referring to s(s,xi, t) ass(s, t− τi).

Reflections within the signal chain of each antenna aregenerally described by a complex direction-dependent re-flection coefficient, ri(s, ω) which we refer to as ri(s, τ)in the delay domain. The effect of each reflection withdelay τ is to add the signal to itself multiplied by ri anddelayed by τ . The voltage signal measured at the ith an-tenna element, vi, is the integral over solid angle of theelectric fields arriving from all directions. The presenceof reflections introduces a convolution of the electric fieldentering the antenna (delayed by τi) with ri(s, τ):

vi(t) =

∫dΩ

∫dτ ri(s, τ)s(s, t− τi − τ). (1)

Applying the Fourier convolution theorem, the Fourier

transform of this equation gives vi(ω) as the simple an-gular integral of the product of si(s, ω) and ri(s, ω).

vi(ω) =

∫dΩ ri(s, ω)s(s, ω)e−iωτi . (2)

The correlator of a radio interferometer records thetime-averaged product of the Fourier transfored voltagestreams of the ith and jth antennas. The time averagedcorrelation between the two antennas is

V ′ij(ω) =⟨vi(ω)v∗j (ω)

⟩t

=

∫dΩ ri(s, ω)r∗j (s, ω)I(s, ω)e−iω∆τij

=

∫dΩ ri(s, ω)r∗j (s, ω)I(s, ω)e−iωbij ·s/c, (3)

where ∆τij = τi − τj = (xi − xj) · s/c, I(s, ω) =⟨|s(s, ω)|2

⟩t

is the intensity, bij = (xi − xj), and the’prime’ on Vij indicates that this is the measured visibil-ity with antenna gains applied rather than the intrinsicvisibility of the sky. Here, we have invoked the fact thatelectromagnetic waves arriving from different directionsare incoherent (Thompson et al. 1986). This equationis the familiar interferometry equation (Thompson et al.1986), allowing us to see that ri(s, ω) is mathematicallyequivalent to the voltage beam of the antenna. This canbe explained more intuitively by the fact that any compo-nent of the instrument that multiplies the signal in thefrequency domain will convolve the signal in the delaydomain. Convolutions of the signal in the time-domainare mathematically identical to the effect of reflections.Setting a specification on reflections is hence equivalentto setting a specification on the spectral smoothness ofthe antenna’s voltage beam.

In order to filter spectrally smooth foregrounds fromthe signal, many experiments are employing the de-lay transform over frequency, defined as (Parsons et al.2012b)

Vij(τ) =1

2π

∫dωVij(ω)eiωτ . (4)

Applying this to equation 3, we obtain

Vij(τ) =1

2π

∫dΩ

∫dωri(s, ω)r∗j (s, ω)I(s, ω)e−iω(bij ·s/c−τ).

(5)Let us examine the quantity within the angular integral.Assuming that the voltage beam is perfectly flat in itsspectrum and ignoring the smooth spectral structure inI(s, ω) we see that for fixed s, the ω integral will result ina dirac-delta function in delay, centered at τ = bij · s/c.hence each source located at s on the sky is mapped to aline τ = bij ·s/c, resulting in the much discussed “wedge”(Datta et al. 2010; Vedantham et al. 2012; Parsons et al.2012b; Morales et al. 2012; Thyagarajan et al. 2013; Liuet al. 2014a,b).

The presence of a realistic frequency dependent beamcauses each source line to be convolved in delay withthe direction dependent kernel. To see this, we carryout the ω integral in equation 5 and apply the FourierConvolution theorem.

Vij(τ) =

∫dΩ

∫dτ ′I(s,bij · s/c− τ − τ ′)R(s, τ ′), (6)

4 Ewall-Wice et al.

where

Rij(s, τ) =

∫dτ ′ri(s, τ

′ − τ)r∗j (s, τ ′). (7)

Thus, each source line in the wedge is convolved with

an antenna kernel, Rij , which is itself the convolution ofthe time-reversed delay response of the voltage beam ofantenna i with the complex conjugate of voltage beam ofantenna j. In the remainder of this paper, we will often

refer to Rij as the power kernel applied to a visibility. Wedemonstrate the effect of foreground smearing in Fig. 2for a simple model with only three sources. Without re-flections, the sources would form lines intersecting zeroin the two dimensional space with the baseline length,b on the x-axis and the delay, τ on the y-axis. Withthe reflections, the sources are smeared out, leading tosupra-horizon contamination. For the sake of simplic-ity, we now consider the case where the beam can beseparated into independent angular and frequency com-ponents, ri(s, ω) = gi(ω)ai(s). For such a case, everyline in Fig. 2 would be convolved with the same delaydependent shape, normalized to the gain of ai(s). In thissituation, we have

Vij(τ) =

∫dτ ′∫dτ ′′gi(τ

′ − τ ′′)g∗j (τ ′′)V aij(τ − τ ′), (8)

where V aij is the visibility for the achromatic voltage pat-tern, ai(s). We can gain further insight into the behaviorof the delay kernel arising from chromaticity by assumingthat gi(τ = 0) gi(τ > 0), which should be the case atlarge delays for the smooth bandpasses our instrumentsare designed to have.

Vij(τ) ≈ gi(0)

∫dτ ′g∗j (τ ′)V aij(τ − τ ′)

+ g∗j (0)

∫dτ ′gi(−τ ′)V aij(τ − τ ′). (9)

Hence, to first order, the impact of reflections is toconvolve the delay-transformed visibility with the volt-age beam of the instrument. This may be a somewhatcounter-intuitive result since naive dimensional analysismight predict that the power-kernel is proportional tothe square of the delay-response. This linear relation re-quires that the voltage beam fall roughly five orders ofmagnitude (the same as the dynamic range between fore-grounds and signal) in the regions of delay space that wewant to measure the signal.

3. ELECTROMAGNETIC SIMULATIONS OF THE HERADISH ELEMENT

Having formally derived the impact of reflections onforeground visibilities, we are in a position to investigatetheir existence in the HERA dish. In this paper, we as-sume a separable beam kernel whose response is givenat zenith and whose high-delay components are sourcedby intra-dish reflections (equation 9). The impact of thedirection dependent kernel which includes side-lobe chro-maticity is investigated in (Thyagarajan et al. 2016). Inthis section, we describe the setup and parameters of oursimulations (§ 3.1), and how we extract the voltage re-sponse function of the dish (§ 3.2).

3.1. The Simulations

The time-dependent electromagnetic behavior of theantenna was modeled using Microwave Studio, a com-mercial numerical simulation package produced by Com-puter Simulation Technology (CST). The model consistsof an idealized 14 m diameter paraboloid reflector with afeed structure placed at the focus (4.5 m above the sur-face). The feed is a dual linear polarized sleeved dipoleidentical to that used as an element of PAPER but witha cylindrical skirt used in place of the angled planar re-flectors. The dipole arms are modeled as copper tubingwith aluminum disks and reflector surfaces. For sim-ulation, the entire model was encapsulated in vacuumdielectric with radiative boundaries and discrete ports of125 Ω impedance were defined at the terminals of the or-thogonal dipoles. The simulation is conducted in a boxencapsulating the dish geometry plus three wavelengths(at 100 MHz) on all sides with open boundary conditions.

Microwave Studio divides the model into approxi-mately 94 million discrete hexahedral cells where the fieldis calculated over time using Finite Integration Tech-nique incorporating the Perfect Boundary Approxima-tion2 (Workflow and Solver Overflow Document, CSTMicrowave Studio, 2015, Chapter 3.) in response to a100-200 MHz broadband pulse of about 40 ns in dura-tion. The model was excited in two ways: 1) via a E-Wpolarized plane wave entering the model along the boresight, and 2) from one of the discrete terminals (whoseanalysis we focus on in § 4.5). The voltage responses atthe terminals are monitored for these two cases over a du-ration of 500 ns as the excitation pulse travels throughoutthe model.

In Fig. 3 we show the geometry of the electromagneticsimulation. A 150 MHz plane wave with a Gaussian en-velope is initialized above the dish vertex traveling in the−z direction. It is reflected by the dish before enteringthe dipole feed, hidden below the cylindrical skirt in thisfigure. We record the electric field voltage of the planewave at the feed output terminals as a function of time,plotted as a red line in Fig. 3 along with the output volt-age at the terminals for the feed polarized parallel to theplan wave (black line). The delay between the centralenvelope of the plane wave and the voltage output ofthe feed is ≈ 30 ns which corresponds to the round triptravel time from the feed to the dish vertex and back.However, while the input plane wave, modulated by agaussian, falls off rapidly after the first ≈ 20 ns after itspeak, we see that the voltage output decays far moreslowly due to reflections and resonances within the an-tenna structure. We are able to get a qualitative feel forthe amplitude of the reflections by inspecting the falloffof the time domain voltage response and see that after60 ns it reaches ≈ −25 dB.

3.2. Deconvolving the Response Function

We can do much better than this. From equation 1,we know that the voltage output results from the convo-lution of the plane wave input with the voltage gain ofthe antenna. Since we know the input wave, a straight-forward application of the Fourier convolution theoremallows us to determine the voltage response.

2 This proprietary technique is used by CST to reduce the num-ber of simulation cells.

The HERA Dish II: Electromagnetic Simulations and Science Implications 5

Fig. 2.— A cartoon demonstration of the impact on foregrounds of the frequency dependent beam. Left: The location of three, spectrallyflat, sources in delay space assuming a frequency independent beam (no reflections in the antenna element). Right: the presence ofchromaticity due to delayed signal within the antenna smears the source in delay with the kernel given by equation 7. Since the frequencyresponse of the dish is a function of direction on the sky, the shape of the delay kernel is different for each source line. We see that thissmearing can lead to substantial supra-horizon emission. In this paper, we consider a direction independent delay-kernel that is sourceprimarily by reflections while a more general direction-dependent kernel is explored in Thyagarajan et al. (2016)

Fig. 3.— An illustration of our simulation products and their origin in the HERA antenna geometry. A plane wave is injected from abovethe feed. The electric field of the plane wave at the feed terminals (red line) along with the voltage output is recorded (black line). Thefeed in our simulation is situated 5 m above the bottom of the dish, hence there is a ≈ 30 ns delay between when the plane wave passesthe terminal for the first time (A) and when it is first absorbed in the dipole (B), leading to the voltage response. Of concern to 21 cmexperiments are the subsequent reflections between the feed and the dish (C) which can lead to larger delays that contaminate the EoRwindow.

Since our simulation is sampled in finite time steps,we will adopt discretized notation for this section. Inparticular, our simulation consists of N samples, evenlyspaced by dτ at times τn = n×dτ . We denote the outputvoltage at the feed terminals at time τn as vn. Rewritingthe convolution in equation 1 in discrete notation, wehave

vn(s) =∑m

rm(s)sn−m(s). (10)

We may undo this convolution by taking a discreteFourier transform (DFT) of both v and s in time, divid-ing them in Fourier space, and taking an inverse DFT

back. Symbolically,

r(s) = F−1

[F v(s)

F s(s)

], (11)

where F is the Fourier transform matrix for a 1d vectorof length N .

Fmn = e−2πimn/N . (12)

In Fig. 4 we show the amplitude of the Fourier transformof our Gaussian input, centered at 150 MHz along withthe voltage response. Since our input only has supportbetween ≈ 20 and 280 MHz, the direct ratio of our volt-age response and input wave is dominated by numericalnoise outside of this range. We eliminate these artifacts

6 Ewall-Wice et al.

by multiplying our ratio by a Blackman-Harris windowbetween 100 MHz and 200 MHz and set our estimate tozero elsewhere. From a physical standpoint, this is sensi-ble since 21 cm experiments only observe a limited band-width. PAPER’s correlator, which will initially serve asthe HERA backend samples over a 100 MHz instanta-neous frequency interval and analog filtering is appliedto prevent aliasing.

50 100 150 200 250 300f (MHz)

0.0

0.2

0.4

0.6

0.8

1.0

Frequency

Resp

onse

(arb

itra

ry u

nit

s)

InputWave

VoltageResponse

Bandpass

FilteredBandpass

Fig. 4.— The absolute value of the Fourier transform of thevoltage output from our dish simulations (green line) and the inputwave (blue line), normalized to the amplitude of the input waveat 150 MHz. The ratio between input and output is plotted as ared line. Since our input is limited to frequencies between ≈ 20and 280 MHz, there are significant numerical artifacts in the ratiothat causes divergence towards the plot edges. To eliminate thisnoise, we multiply by a Blackman-Harris window between 100 and200 MHz and set our estimate to zero elsewhere (cyan line).

4. SIMULATION RESULTS

We now discuss the results of our simulations. We fo-cus on the time-domain die off of the voltage responseand the resulting power kernel (§ 4.1), comparing it toan identical time-domain simulation of the skirted dipoleantenna used by PAPER. In § 4.2 we investigate the de-pendence of the power kernel on frequency to determinewhether specific parts of the HERA bandpass are moreaffected by reflections than others. In § 4.3 we determinethe origin of excess response at long delays in our simu-lations. In § 4.4, we discuss potential trade-offs in elim-inating this structure from the antenna by determiningthe impact of removing components of the HERA feedthat are responsible for the contamination. Finally, in§ 4.5 we verify our simulation framework by comparinga separate time domain simulation of S11 of the HERAdish to a direct field measurement with a vector networkanalyzer (VNA).

4.1. The Time Domain Response of the HERA Dish

Applying equation 11 to our simulation, we obtain esti-mates of the time-domain voltage response of the HERAdish towards zenith which we plot in Fig. 5. We also con-duct a time-domain simulation of the voltage output inresponse to an identical input plane wave for the skirteddipole PAPER antenna (pictured above the HERA dish

in Fig. 1) in order to determine whether the presenceof the parabolic dish introduces reflections and spectralstructure in excess of previous successful antenna de-signs.3 We inspect the absolute value of v for both thePAPER and HERA antennas in Fig. 5. Since non-zerovalues of v at negative delays violates causality, we as-sume such features are sourced by artifacts such as side-lobes of the zero-delay peak and/or numerical precisionnoise which sets a limit of ≈ −60 dB on the dynamicrange of our method. We first note peaks in the HERAcurve that are spaced by ≈ 35 ns and are absent in thePAPER simulation. Another significant difference be-tween the two curves is a knee in the HERA gain at≈ 120 ns that is not present in PAPER, leading to an in-crease in gain by approximately 20 dB at 200 ns. Becausethis knee does not exhibit nodes at 35 ns intervals it isprobably not sourced by reflections in the antenna-feedgeometry but by reflections involving a geometric lengththat is not resolved by the 100 MHz bandwidth of thesimulation.

Our next step is to compute the power kernel, R, givenby equation 7 by performing a convolution of the time-reversed voltage response with the complex conjugatevoltage response. In Fig. 6 we show the resulting powerkernel for PAPER and HERA. Since both voltage gainsdrop rapidly with increasing delay, the approximation inequation 9 holds quite well and we see that the kernelsfall off at a rate similar to the voltage response.

200 100 0 100 200 300τ (ns)

60

50

40

30

20

10

0

|v(τ

)| (

dB

)

HERA Dish

PAPER Antenna

HERA DishNo Cylinder

HERA DishDipole Only

-0.11 -0.05 0.00 0.05 0.11 0.16k (hMpc−1)

Fig. 5.— The Fourier transform of the simulated voltage re-sponse of the HERA Dish (solid black line) and the PAPER an-tenna element (dashed black line). Reflections in the HERA dishelement lead to significantly enhanced power above ∼ 50 ns. Sincenegative delays should be devoid of signal, they allow us to deter-mine the dynamic range of our simulations which have a numericalnoise/sidelobes floor of −60 dB.

4.2. The Delay Response of Subbands

3 To date, PAPER has produced the most stringent limits on the21 cm power spectrum leading us to use its design as a standard.

The HERA Dish II: Electromagnetic Simulations and Science Implications 7

200 100 0 100 200 300τ (ns)

60

50

40

30

20

10

0

Pow

er

Kern

el (d

B)

HERA Dish

HERA DishNo Cylinder.

HERA DishDipole Only.

PAPER Antenna

-0.11 -0.05 0.00 0.05 0.11 0.16k (hMpc−1)

Fig. 6.— The absolute value of the power kernel for the HERAdish (solid black line) and for the PAPER antenna element (dashedblack line) calculated using equation 7. While an antenna can onlyphysically have a voltage response at positive delays, the delaykernel is formed from the convolution of one antenna with thetime reversed conjugate response function of the other. Hence, the

power kernel for two identical antennas will have R(τ) = R∗(−τ).

Because the 21 cm brightness temperature fluctuationsevolve over redshift intervals of ∆z & 0.5 (Zaldarriagaet al. 2004), experiments will sub-divide their bands into≈ 10 MHz intervals for power spectrum estimates at mul-tiple redshifts. Thus the localization of reflections withinthe HERA band will determine whether all or only asubset of redshifts are affected. To determine whetherthe reflections are localized in frequency, we computethe voltage delay response and power kernel for threedifferent subbands: 100− 130 MHz, 130− 160 MHz, and160−190 MHz. In order to maintain decent resolution ofthe kernel itself, we use frequency ranges that are severaltimes larger than the actual subbands that will be usedfor EoR power spectrum estimation (≈ 10 MHz). Theresulting power kernels are plotted in Fig. 7. The centrallobe of the subband kernels is significantly wider due tothe the wider window functions incurred by the reducedbandwidth. We find that the shallow long-term falloffis only visible within the central 130 − 160 MHz band,indicating that long term reflections are isolated around150 MHz.

To further illustrate the isolation of fine frequencystructure in the center of the bandpass and to verifythat our observations are not an artifact of our reduc-tion of the simulation outputs, we fit 10 MHz intervals ofthe absolute value of the simulated gains to a sixth orderpolynomial and inspect the residuals in Fig. 8. We findthat the gain residuals on the sixth order fit are an orderof magnitude greater on the 145−155 MHz subband thanany other frequency interval.

4.3. The Origin of the Knee.

In Fig. 5 we see that the long-term delay response ofthe HERA antenna differs from the PAPER design intwo regards, the existence of node-like structures spaced

0 50 100 150 200 250 300 350 400τ (ns)

60

50

40

30

20

10

0

Pow

er

Kern

el (d

B)

100-130 MHz

130-160 MHz

160-190 MHz

100-200 MHz

0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.19 0.21

k (hMpc−1 )

Fig. 7.— The power kernel for the three subbands discussed in§ 4.2 along with the kernel for the full bandwidth response function.While the long term falloff from reflections is prominent between130 − 160 MHz, it appears at a much lower level in the other twosubbands which fall below the central subband by ∼ 20 dB at ∼300 ns. k‖ values for each delay are computed at 150 MHz. Thewider central lobe below 150 ns for the subband gains is due to thelower delay resolution from the smaller bandwidth. We also showthe delay kernel for 100 MHz bandwidth (black thick line).

4 2 0 2 4

f-f0 (MHz)

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

Fract

ional R

esi

dual

(×

10

-4)

f0 =130 MHz

f0 =140 MHz

f0 =150 MHz

f0 =160 MHz

f0 =169 MHz

f0 =180 MHz

Fig. 8.— Residuals on the absolute value of the gain over sev-eral subbands after fitting to a sixth order polynomial. Consistentwith our findings in Fig. 7, the fine frequency residuals in the 145-155 MHz subband are over an order of magnitude greater thanthose in the other subbands.

every ∼ 35 ns, corresponding to the round trip delay be-tween the dish and the feed, and a long-term knee thatdominates the response function after 100 ns but doesnot exibit any periodicity that might be associated withthe feed-dish geometry. Here we establish the origins ofthe long time-scale structure using simulations. Reflec-tometry measurements conducted on the isolated feedin several different configurations (Patra et al. in prepa-ration) independently discover and verify these origins.To determine whether this knee-like structure originatesfrom feed-dish reflections, we perform a simulation of aplane wave incident along the bore-sight the HERA feedwithout the dish and compare it with the time-domainresponse for the feed-and-dish system in Fig. 10. Thefeed only simulation lacks the reflections associated withthe dish and feed configuration (present at small delays)

8 Ewall-Wice et al.

CylinderBackplane

DipoleFig. 9.— A closeup rendering of the HERA feed which is sus-

pended over the reflector, illustrating the cylindrical skirt, thebackplane, and the dipole. Long time-scale spectral structure arisesfrom electrical oscillations within the cylindrical cavity.

but retains the large-delay knee. Thus, the dominantlarge time-scale contamination of the HERA band-passis not intrinsic to the arrangement of a feed suspendedover the dish.

We isolate the source of the knee within the feed struc-ture (shown in Fig. 9) by running simulations of thedipole feed with only a back-plane along with a sleeveddipole in isolation. We show both of these voltage re-sponses in Fig. 10. When the cylindrical skirt is removedfrom the feed, the knee vanishes. Because of the narrowband nature of the knee (§ 4.2), its presence only whenthe cylinder is attached, and its exponential falloff (char-acteristic of a damped harmonic oscillator), we concludethat the cylindrical skirt is behaving as a resonant cavityand the resulting stored oscillating fields are responsiblefor the majority of the fine-scale spectral structure inthe HERA antenna’s gain. We estimate the quality fac-tor of the resonance to be ≈ 6.5 from the time constantof the exponential falloff which yields a resonance widthof ≈ 20 MHz.

Since the termination of the dipole could be signifi-cantly different from the termination used in the simu-lation (100Ω), we also investigate the impact of the ter-mination impedance of the dipole on the delay-response.we vary the termination impedance between 50 to 500 Ωfor several plane-wave simulations and show the result-ing voltage responses in Fig. 11. We find that chang-ing the termination impedance has a significant impacton the time-domain response below ≈ 150 ns in the re-gion that is dominated by dish-feed reflections. Sincethe termination impedance does not affect resonance ofthe feed cavity, its effect is small beyond ≈ 200 ns. Onlyin the extreme, 500 Ω case do the reflections extend toappreciable delays. Since delays over ≈ 250 ns are re-sponsible for contaminating the EoR window, termina-tion impedance will only impact HERA’s sensitivity foran extremely poor match.

Simulations combining the dish with the analog signal-chain show that matching networks can reduce the levelsfeed-dish reflections below those observed in this paper,even under realistic termination conditions (Fagnoni &de Lera Acedo 2016)4.

4 See Fig. 4. This paper investigates a different variant of theHERA feed in which the resonance is absent but whose primary

50 0 50 100 150 200 250 300 350τ (ns)

50

40

30

20

10

0

|v(τ

)| (

dB

)

Feed and Dish

Feed Only

Feed Only No Cylinder

Dipole Only

-0.03 0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.19k (hMpc−1)

Fig. 10.— The absolute value of the time-domain voltage re-sponse of the cylindrical dipole feed compared to the absolute valueof the voltage response of the feed suspended over the dish. As wemight expect, the ∼ 35 nm lobed structures associated with feed-dish reflections are absent from the simulation of the feed only.However, the knee like feature after ∼ 100 ns is. This indicatesthat the most severe spectral contamination in the current HERAdesign does not originate in reflections between the feed and thedish but rather within the feed itself.

50 0 50 100 150 200 250 300 350 400τ (ns)

50

40

30

20

10

0

Volt

age R

esp

onse

(dB

)

50 Ω

100 Ω

125 Ω

200 Ω

500 Ω

-0.03 0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.19 0.21k (hMpc−1)

Fig. 11.— The time-domain response of the HERA antenna to-wards a plane wave incident from zenith for a variety of terminationimpedances. As we vary the termination impedance, the structurewhich is dominated by feed-dish reflections, below ≈ 150 ns, variessignificantly but leaves delays greater than ≈ 200 ns unchanged.Only in the extreme, 500 Ω case do the reflections extend to largedelays. Since structure above 250 ns is primarily reponsible for con-taminating the EoR window (14), the termination impedance hasa relatively small impact on HERA’s overall sensitivity.

beam properties are still being investigated.

The HERA Dish II: Electromagnetic Simulations and Science Implications 9

Since the feed properties dominate the antenna’slong-term response, we investigate the voltage response(Fig. 5) and the power kernel (Fig. 6) of the HERA an-tenna without the cylindrical skirt and a bare dipole.Removing the cylinder leads to a ≈ 10 dB improvementin the antenna response after 100 ns and eliminating thebackplane as well yields a 20 to 30 dB improvement.Hence, the delay response of the HERA antenna can po-tentially be improved by modifying the feed. There areseveral trade-offs in polarization and sensitivity that aremade when we remove feed components which we nowdiscuss.

4.4. Tradeoffs in Beam Properties.

We determined above that the long-delay knee in thegain of the HERA dish is primarily caused by resonancein the cylindrical skirt and backplane that surround thePAPER dipole. These structures are added to enhancethe symmetry of its beam, to improve the effective areaof the antenna, and reduce cross coupling(DeBoer et al.2016). It is therefore worth examining the impact thatremoving these components has on the properties of theHERA beam.

We first examine the trade-off that is made in polar-ization when we remove elements of the original design.Faraday rotation is capable of imparting fine-scale fre-quency structure into the Q/U polarized Stokes visibil-ities. Antennas in which the X and Y polarized beamsare not identical can leak stokes Q/U visibilities and thespectral structure they contain into the Stokes-I visibili-ties from which delay power spectra are formed, poten-tially masking the 21 cm signal (Jelic et al. 2010; Mooreet al. 2013, 2015). It is shown in Moore et al. (2013) andMoore et al. (2015) that this leakage is given approxi-mately by

Pleak ≈A−A+

PP ≡ ξPP , (13)

where PP is the linearly polarized power spectrum, and

A± =

∫dΩ|Axx(s)±Ayy(s)|2, (14)

where Axx is the x polarized beam and Ayy is the they polarized beam. In Table 1 we give ξ at 150 MHz,computed for the three configurations of the HERA feeddiscussed in § 4.3. We see that the feed with the cylin-der has the smallest ellipticity with ξ ≈ 3 × 10−3 overmuch of the band. Polarization measurements fromAsad et al. (2015) indicate a polarized power spectra ofPP ≈ 103 mk2 within the EoR window. Thus, an an-tenna with ξ ≈ 3×10−3 would produce leakage of severalmk2 which is roughly an order of magnitude below typi-cal reionization peak amplitudes of several tens of mK2.However, the feed variant in which the cylinder has beenremoved has an ellipticity that is roughly ≈ 4× largerand a bare dipole, ≈ 10× larger. These enhanced ellip-ticities would bring the level of the polarization leakageto or above the level predicted in Asad et al. (2015). Onthe other hand, it is possible the LST binning and aver-aging over many nights with different rotation measurescan suppress polarization leakage by a factor of ≈ 103 inthe power spectrum (Moore et al. 2015).

The second impact that we consider, is the effective

TABLE 1Ellipticity and effective area of the HERA antenna for

different feed configurations.

Feed Configuration Ellipticity, ξ Effective Area, Aeff, (m2)

Cylindrical and Backplane 3.5× 10−3 56.7No Cylinder 1.5× 10−2 97.2No Cylinder or Backplane 2.8× 10−2 41.9

area of the antenna, given by (Wilson et al. 2009),

Aeff =λ2∫

dΩA(s). (15)

In Table 1, we see that the effective area at 150 MHz is ac-tually enhanced by the removal of the cylinder but is neg-atively impacted by the removal of the backplane whichintroduces a significant back-lobe, reducing the beam’sdirectivity. As detailed in Neben et al. (2016), this re-duction in the effective area of the dish lowers HERA’ssensitivity by a factor of order unity.

The development of the HERA feed is still underwayand it is possible that improvements in the feed will al-low for the 150 MHz resonance to be eliminated whilemaintaining the illumination and beam-symmetry of theoriginal feeds.

4.5. Verifying Our Framework with S11 Measurementsand Simulations

We now assess the accuracy of our time-domain simu-lation framework by comparing simulations to measure-ments of the S11 parameter of the HERA dish. Up untilnow, our simulations have derived the voltage responseof the dish using simulations of an incoming plane waveas is the case for radio signals arriving from objects atcosmological distances. It is possible to probe the gain ofthe dish using objects in the far field such as known ra-dio sources (Thyagarajan et al. 2011; Pober et al. 2012;Colegate et al. 2015) or constellations of ORBCOMMsatellites (Neben et al. 2015, 2016). However naturalradio sources are too weak to probe the dish responseat the . 10−4 level necessary to verify our simulationsand the ORBCOMM technique can only be used to mapthe gain at the 137 MHz ORBCOMM transmission fre-quency. Work is currently underway to use broad-bandtransmitters flown into the far field of the dish by drones(Jacobs et al. in preparation) but this system is still un-der development. Reflectometry of the dish with a VNAis a straightforward alternative used in Patra et al. (inpreparation) to estimate the gains directly. Rather thancomparing their estimate of the gain with our predictedgains (which is done in their paper), we set up a time do-main simulation to compute the S11 parameter of HERAantenna and compare it to direct S11 measurements.S11 refers to the complex ratio between a voltage sig-

nal transmitted into the feed terminals and the volt-age reflected back as a function of frequency, S11(f) ≡vtrans(f)/vrecv(f). Measurements, described in furtherdetail in Patra et al. (in preparation), of S11 were takenon a prototype HERA dish at the National Radio Astron-omy Observatory Green Bank facility using an Agilent8753D VNA. The VNA was connected to the antenna’sbalanced feed terminal via a 30.5 m length of RG-8X-LLcoaxial cable and M/A Com HH-128 180 degree hybridjunction. The VNA was calibrated at the balanced end of

10 Ewall-Wice et al.

0 100 200 300 400 500τ (ns)

60

50

40

30

20

10

0

S11

(dB

)

HERA S11 SimulationHERA S11 Measurement

Fig. 12.— A comparison between time-domain simulations (blackline) and measurements (grey line) of S11 for the HERA dish. Wealso show an S11 measurement with the cables leading from theVNA to the feed terminated by an open circuit which allows us toprobe the dynamic range of the measurement. We use the standarddeviation of the open measurement (grey dashed line) between 200and 400 ns as our systematic floor (grey shaded region). We findvery good agreement between our S11 measurement and the sim-ulation, validating the predictions of our simulations. Both thesimulations and measurements in this figure were derived from de-lay transforms over 100 MHz.

the hybrid junction using a set of termination standardshaving SMA connectors. A small adapter from SMA tothe post terminals on the feed was not included in thecalibration. The VNA excites the terminals with a band-limited (100-200 MHz) pulse, and the complex reflectedvoltage at the calibration plane is measured as a functionof frequency. The reflection for the case of open termi-nals was measured to confirm the dynamic range andresolution of the measurement. While our simulationsare terminated with an impedance of 125 Ω, the VNAmeasurements are terminated at 100 Ω which only has asmall effect on the large-delay response of the antenna(Fig. 11).

Because S11 is defined as a ratio in the frequency do-main in a form identical to our voltage gains, we mayrun time-domain simulations similar to those describedin § 3.1 except rather than simulating an incoming planewave, we simulate the excitation of the feed terminals bya delta-gap impedance port and record output voltage asa function of time. We calculate S11 from the simulationin the same manner that we obtained r using equation 11with vrecv taking the place of v in the numerator insidethe Fourier transform and vtrans taking the place of s inthe denominator.

In Fig. 12 we show the simulated amplitude of S11 asa function of frequency for the HERA dish, observing adistinctive two peaked structure before a steep die off indelay that transitions to a shallower falloff at ≈ 150 ns.The first peak is due to the reflection of the input waveoff of the back of the feed while the second, roughly 35 nslater arises from the transmitted component of the inputwave reflecting off of the dish and arriving back at thefeed. The ensuing long term die off arises from reflec-tions within the feed and dish structure and for reasonsthat will be elaborated on in Patra et al. (in prepara-

tion) corresponds very closely to our simulations of thedish gain itself (compare with Fig. 5). We get a sense ofthe dynamic range of the measurement by unhooking theSMA adapter that attaches the sleeved dipole feed to thecable from our VNA, forming an open circuit that shouldideally give a reflection coefficient of ≈ 1 at zero delaywith no reflections at any other times. In this measure-ment, we find noise-like structure at ≈ −50 dB. Below≈ 500 ns, we found that this structure does not integratedown with time, leading us to conclude that it is causedby systematics, likely uncalibrated low level reflectionsin the VNA-feed cables. The level of this noise sets thesystematic floor in our measurements which we show asa grey shaded region in Fig. 12. We find that in theregion where the S11 measurement is above the system-atics floor, there is good agreement with our simulations(within several dB).

5. THE EFFECT OF THE SPECTRAL STRUCTURE DUETO REFLECTIONS AND RESONANCES IN THE HERA

DISH ON FOREGROUND LEAKAGE AND SENSITIVITY

We can now explore the impact of the dish’s perfor-mance on the leakage of foregrounds out of the wedge,and into the EoR window. Beyond the delay kernels con-sidered in this paper and Patra et al. (in preparation), theextent of leakage will depend both on the angular struc-ture of the primary beam, which is established throughmeasurements and simulations in Neben et al. (2016) andthe brightness of the foregrounds themselves Thyagara-jan et al. (2016). In this section, we investigate the am-plitude of foreground leakage as a function of delay giventhe angular primary beam model and our simulation ofthe dish’s spectral structure. We start by performinga physically motivated extrapolation of the delay struc-ture observed in our simulations to cover the range of co-moving scales relevant to power spectrum measurements(§ 5.1). In § 5.2, we combine this extrapolated delay re-sponse with simulations of foregrounds to determine theoverall level of foreground power in HERA’s visibilities.This leakage will cause large-scale line-of-sight Fouriermodes to be contaminated by foregrounds and hence in-accessible to the foreground filtering approach. Since thesignal-to-noise ratio is maximized at the smallest k val-ues, the loss of these modes will reduce the significance ofthe power spectrum detection and negatively impact theoverall bottom line of the science that HERA can accom-plish. We explore the impact of the chromaticity due toreflections on science using the Fisher matrix formalismin § 5.3.

5.1. Extrapolating the Bandpass and Power Kernel

Our simulations of the dish response only extend to≈ 400 ns. However, interferometers are expected to bemost sensitive to the 21 cm power spectrum at comovingscales between k ≈ 0.1 − 0.5 hMpc−1, corresponding tothe delays between 180 and 900 ns at z = 8. To extrap-olate out to the higher delays in this range, we assumethat the response function is dominated by a sum of fieldoscillations within the feed’s cylindrical skirt and reflec-tions between the feed and the dish.

r = ro + rr (16)

The long term falloff after the knee, which is dominatedby oscillations within the cylinder, appears as a line on

The HERA Dish II: Electromagnetic Simulations and Science Implications 11

a linear-log plot (Fig. 5). This is the response we wouldexpect for a damped harmonic oscillator. We thus modelthe voltage response from the cylindrical skirt as an ex-ponential

ro = AoX− τ150MHz

2Qo (17)

where Q is the quality factor of the cavity which we de-termine to be ≈ 6.5.

We see in Fig. 5 that when the cylinder and backplaneare removed, the response of the antenna falls off muchfaster, in line with the response below ≈ 130 ns. Like theresponse from the cylinder, this falloff also goes exponen-tially which we now show is also indicative of reflections.

We let Γd represent the reflection coefficient of the dishvertex and Γf represent the reflection coefficient of thefeed in the presence of each other. An electromagneticwave incident on the feed, at t = 0, is accepted with am-plitude (1+Γf ). The reflected component travels back tothe dish and acquires an amplitude of (ΓfΓd) before re-turning at time, τd later where (1 + Γf ) will be acceptedand Γf will be reflected back towards the dish. Sum-ming the infinite series of reflections, the time dependentvoltage at the feed is

vr(t) = (1 + Γf )∑m

(ΓfΓd)ms(t−mτd), (18)

which implies that

rr(τ) = (1 + Γf )∑m

(ΓfΓd)mδD(τ −mτd). (19)

In practice, each delta-function in equation 21 is con-volved with the window function arising from the finitebandwidth of the instrument. For our simulations, whichhave a bandwidth of 100 MHz, the width of the win-dow function is ≈ 10 ns, partially filling in the gaps be-tween each reflection. Since the number of reflections ism = t/τd, than we can write the long-term delay responsein discrete form as

r(ndτ) ≈ (ΓfΓd)ndτ/τd (20)

which is an exponential in time. We thus model theresponse due to reflections between the feed and the dishas an exponential

rr = ArX(τ/30ns)r . (21)

To extrapolate beyond 400 ns, we fit the voltage re-sponse to the sum of two exponentials for ro and rr andset the feed response beyond the maximum time our sim-ulation to this power law sum. In our pedagogical treat-ment, we have assumed that the reflection coefficientsare frequency independent which is not the case in reallife. The impact of frequency dependent reflection coeffi-cients is to replace the summands in equation 19 with theFourier transform of (ΓfΓd)

m evaluated at t = (τ−mτd).As long as (ΓfΓd)

m is compact in delay space, whichis the case if the reflection coefficients evolve smoothlywith frequency, than our pedagogical approximation stillyields the power law in equation 21.

5.2. The Impact of delay response of the HERAAntenna on Foreground Contamination

Given the chromaticity due to reflections and reso-nances in the HERA antenna, what Fourier modes willstill be accessible with the delay filtering technique? Toanswer this question, we combine our extrapolated sim-ulations of the HERA dish’s spectral structure with sim-ulations of foregrounds.

The foreground model is discussed in detail in Thya-garajan et al. (2015b) but we describe them briefly herefor the reader’s convenience. It consists of two ma-jor components: diffuse synchrotron emission from ourGalaxy whose structure is described by the Global SkyModel (GSM) of de Oliveira-Costa et al. (2008) and apopulation of point sources including objects from theNRAO Sky Survey (NVSS) (Condon et al. 1998) at1.4 GHz and the Sydney University Mologolo Sky Survey(SUMMS) (Bock et al. 1999) at 843 MHz. Their fluxesare extrapolated to the observed 100-200 MHz band us-ing a spectral index of 〈α〉 = −0.83 determined in Mauchet al. (2003). Visibilities are computed from the diffuseand point source models assuming an achromatic angu-lar response for the HERA beam at 137 MHz describedin Neben et al. (2016). We compute two sets of visibil-ities: one in which the spectral structure of the dish isassumed to be completely flat, and another in which thebeam is multiplied by the frequency dependent gain atzenith determined by our simulations.

The foreground filtering procedure employed by PA-PER and HERA involves delay transforming the visi-bilities and performing a 1D CLEAN (Parsons & Backer2009; Parsons et al. 2012b) which discovers and subtractsforegrounds within the horizon plus a small buffer, allow-ing for the suppression of foreground side-lobes in delayspace. The level of foreground subtraction possible bythis procedure is limited by the thermal noise level onthe visibility, which in turn depends on the number oftime steps and redundant baselines that are averagedbefore performing the CLEANing step. In this work, weassume that each visibility is CLEANed independentlywith a twenty minute cadence. Each short baseline iscoherent for ≈ 10 s per night (Ali et al. 2015) so this isequivalent to each local sidereal time (LST) being inte-grated for ≈ 120 nights before CLEANing. If we wereto average over redundant baselines this number woulddrop enormously. The standard deviation on the real andimaginary part of a single delay transformed visibility is(Morales & Hewitt 2004)

σV =

√2BkBTsysAe√τ

, (22)

where Ae is the effective area of the dish, B is the band-width, Tsys is the system temperature, τ is the inte-gration time, and kB is the Boltzmann constant. ForTsys we use the equation Tsys = 100K + Tsky where100 K is the temperature of the PAPER receiver andTsky = 60(λ/1 meter)2.55 is the sky temperature (Rogers& Bowman 2008; Fixsen et al. 2011). For Ae we use thevalue of 98 m2 determined in Neben et al. (2016). WeCLEAN down to five times the thermal noise level. Sub-tracting a model of the sky is an alternative to CLEAN-ing that could reach below the noise level on a singlevisibility but also assumes a priori knowledge of the fore-grounds and the instrumental response. In Fig 13 wecompare the delay transform of visibilities before and af-

12 Ewall-Wice et al.

ter CLEANing at the LST of 4 hours. While CLEANingis able to remove structure within the horizon, it does notreduce any of the power leaked outside of the horizon bythe chromaticity of the dish. If we knew the spectralresponse of the dish perfectly, then we might be able toCLEAN with this kernel and remove the supra-horizonstructure though the true direction-dependent nature ofthe chromaticity would complicate this task.

To form estimates of the 21 cm power spectrum, wesplit each visibility into Blackman-Harris windowed sub-bands centered at redshift intervals of ∆z = 0.5 and eachwith a noise equivalent bandwidth of 10 MHz5. The flatsky approximation allows us to Fourier transform eachinterval in frequency, square, and multiply by a set ofprefactors to obtain a power spectrum estimate (Parsonset al. 2012a, 2014),

P (k) =

(2kBλ2

)2X2Y

BppΩpp|V (u)|2. (23)

Here, λ is the central wavelength of the observation, kBis the Boltzmann constant, Bpp is the integral of thesquare of the bandpass, and Ωpp is the integral of theprimary beam squared over solid angle. X and Y arelinear factors converting between native interferometryand cosmological coordinates, defined through the rela-tion 2πu = 2π(u, v, η) = (Xkx, Xky, Y kz).

As a drift scan instrument, HERA will observe thesky at many LSTs within the declination stripe thatpasses through its primary beam, averaging over thepower spectrum estimate at each LST. It is well doc-umented that foreground power varies significantly overLST (Thyagarajan et al. 2015b), hence such an estimatewill either filter or weight LSTs in a way that minimizesthe impact of the most contaminated observations. Forour analysis, we focus on a single LST of 4 hours which isa patch of sky with low Galactic contamination. Such apatch is representative of the kind that HERA will focusmost of its observing time on.

Computing the power spectra, we inspect the ampli-tude of foregrounds given the delay-response and angu-lar pattern of the HERA dish for baselines of two differ-ent lengths in Fig. 14. In both baselines, we find thatthe residuals after CLEANing tend to be at similar lev-els except at the subband centered at z = 8.5 (150 MHz)where the cylinder resonance is present. In bands outsideof the resonance, foreground power outside of the horizonis dominated by side-lobes of the finite window functionused in the Fourier transform. Hence, if we remove powerinside of the horizon through cleaning, the level of thisleakage is reduced. Power that is not reduced by the 1dclean is caused by intrinsic super-horizon structure in theantenna gain. Over the resonance, foreground residualsremain above the signal level out to k‖ = 0.23hMpc−1.Several other baselines especially those oriented entirelyin the E-W direction, which we do not show, containup to two orders of magnitude greater galactic contam-ination as is noted in Thyagarajan et al. (2015a). Indata analysis, these baselines would be down-weightedor discarded so that they do not bias our final estimate.

5 The end to end width of each Fourier transformed interval is20 MHz, however the noise equivalent width of this interval (givenby the integral of the bandpass squared) is only ≈ 0.5 the full widthsince the Blackman-Harris window suppresses the edge channels.

400 200 0 200 400

delay (ns)

103

104

105

106

107

108

109

1010

Abs(V)

(Jy-H

z)

HI signal

Foregrounds

Foregrounds with Reflections

Foreground Residuals

Foreground Residuals with Reflections

-0.21 -0.11 0.00 0.11 0.21k|| (hMpc−1)

7.3 m East12.6 m NorthLST=-20 hr

Fig. 13.— The absolute magnitude of a 100-200 MHz delay trans-formed visibility from a 14-meter baseline (blue line) comparedto the same visibility (green line) contaminated by the delay-response observed in our simulations of the HERA dish. We seethat the extended delay kernel smooths out structure, originatingfrom foregrounds, within the horizon. For HERA, we expect touse the delay-CLEAN to remove foregrounds. However, the depthof CLEANing is limited by the noise level on a single baseline.We show the foreground residuals from a CLEAN down to the 5σnoise level after 20 minutes of integration. Since CLEANing can-not distinguish between foregrounds and signal, it should only beperformed within a narrow region of delay-space, close to the hori-zon and cannot remove the broad wings leaked by the resonanceunless it is accurately modeled.

Outside of the resonance, beam chromaticity is domi-nated by beam-antenna reflections and since the levelof foreground contamination is nearly identical to theachromatic beam, we conclude that feed-dish reflectionsdo not pose a significant limitation HERA’s ability tomeasure the EoR power spectrum. Dishes are thereforea viable strategy for scaling the collecting area of 21 cminterferometry experiments at low cost.

The level of the foregrounds in Fig. 14 is conserva-tive since no attempt has been made to apply inversecovariance weighting techniques (Tegmark 1997; Liu &Tegmark 2011; Dillon et al. 2013; Parsons et al. 2014; Liuet al. 2014a,b; Dillon et al. 2015b,a; Trott et al. 2016) orfringe rate filtering (Parsons et al. 2015). Applications ofinverse covariance filters in recent PAPER observationshave yielded reductions in foreground power by greaterthan an order of magnitude (Ali et al. 2015), hence oursimulations show that even with the presence of the en-hanced spectral structure from reflections, HERA will beable to isolate foregrounds well below the level of thermalnoise in most of the EoR window.

5.3. The Implications of Reflections and Resonances onEoR Science

A primary near-term goal of 21 cm EoR observationsis to obtain information about the nature of the sourcesthat drove reionization. Since the amplitude of the 21 cm

The HERA Dish II: Electromagnetic Simulations and Science Implications 13

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

P(k

) (h

−1M

pc)

3 m

K2

z=7.5

7.3 m East, 12.6 m North

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

z=7.5

14.6 m East, 25.3 m North

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

P(k

) (h

−1M

pc)

3 m

K2

z=8.0

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

z=8.0

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

P(k

) (h

−1M

pc)

3 m

K2

z=8.5

-0.4 -0.2 0.0 0.2 0.4101102103104105106107108109

101010111012101310141015

z=8.5

-0.4 -0.2 0.0 0.2 0.4

k (hMpc−1)

101102103104105106107108109

101010111012101310141015

P(k

) (h

−1M

pc)

3 m

K2

z=9.0

-0.4 -0.2 0.0 0.2 0.4

k (hMpc−1)

101102103104105106107108109

101010111012101310141015

z=9.0

-708 -354 0 354 708delay (ns)

-708 -354 0 354 708delay (ns)

-729 -364 0 364 729 -729 -364 0 364 729

-749 -374 0 374 749 -749 -374 0 374 749

-768 -384 0 384 768 -768 -384 0 384 768

Achromatic Beam Chromatic Beam Achromatic Residuals Chromatic Residuals HI Signal

Fig. 14.— The delay transform power spectrum for different baselines over several redshifts with (green solid line) and without (bluesolid line) the presence of the simulated chromaticity in the HERA dish. Each estimate is computed using the square of a Blackman-Harriswindowed delay transform (equation 23) over a noise equivalent bandwidth of 10 MHz. We also show the power spectra of foregroundresiduals after CLEANing the entire 100 MHz band with (dashed cyan line) and without (red dashed line) the simulated spectral structure.For all subbands, except z = 8.5, we find that the delay response of the HERA antenna has a negligible effect on the k‖ where theforegrounds drop below the signal level. This is the same subband where enhanced spectral structure due to resonance in the feed cylinderis observed in our simulations (Figs. 7 and 8). Outside of the resonance, the chromaticity of the antenna, which is primarily sourcedby feed-dish reflections, is practically identical to the achromatic model. It follows that feed-dish reflections have a negligible impact onHERA’s 21 cm power spectrum sensitivity. For the achromatic beam, power outside of the horizon is leaked by the window-function of thefinite-band Fourier transform. Thus, reducing power inside of the horizon does reduce the level of contamination outside of the windowexcept when the structure is intrinsic to the frequency gain of the antenna as is the case at z=8.5 where structure is present at super-horizondelays due to the feed resonance.

14 Ewall-Wice et al.

signal is largest at smaller k values, a loss of large scalesignal due to foreground contamination eliminates themodes on which HERA would otherwise have the great-est signal to noise detections, reducing its overall sen-sitivity. In this section, we use the Fisher Matrix for-malism to estimate the impact of the HERA’s intra-dishreflections on its sensitivity and ability to constrain theastrophysics of reionization. The Fisher Matrix allowsus to forecast the covariances and errors on reionizationparameters given errors on power spectrum observationsdue to the uncertainties caused by cosmic variance andthermal noise which is in turn determined by the uv cov-erage and observing time of the interferometer. The co-variance between the parameters, θ, of an astrophysicalmodel is given by the inverse of the Fisher matrix, Fwhich for Gaussian and independently determined powerspectrum bins may be written approximately as (Poberet al. 2014),

Fij ≈∑k,z

1

σ2(k, z)

∂∆2(k, z)

∂θi

∂∆2(k, z)

∂θj, (24)

where ∆2(k, z) is the power spectrum amplitude for somek − z bin and σ2(k, z) is the variance of the power spec-trum estimate. We write this equation as approximatesince it ignores additive terms arising from the depen-dence of σ2(k, z) on model parameters due to cosmicvariance which only contribute at the ≈ 1% level (Ewall-Wice et al. 2016a).

To simulate ∆2, we use the publicly available21cmFAST6 code (Mesinger et al. 2011) which generatesrealizations of the 21 cm brightness temperature fieldusing the excursion set formalism of Furlanetto et al.(2004). We employ a popular three parameter modelof reionization (Mesinger et al. 2012) with the followingvariables

• ζ: The “ionization efficiency” is defined in Furlan-etto et al. (2004) to be the inverse of the mass col-lapse fraction necessary to ionize a region and iscomputed from a number of other physical parame-ters including the fraction of collapsed baryons thatform stars and the UV photon escape fraction. Be-cause ζ acts as an efficiency parameter, its primaryeffect is to change the timing of reionization. Wechoose a fiducial value of ζ = 20, though expectedvalues range anywhere between 5 and 50 (Songaila& Cowie 2010).

• Rmfp: The presence of Lyman limit systems andother potential absorbers within HII regions causesUV photons to have a finite mean free path denotedby Rmfp. In the 21cmFAST framework, HII regionscease to grow after reaching the radius of Rmfp,primarily impacting the morphology of the signal.We choose a fiducial value of Rmfp = 15 Mpc whichis in line with recent simulations accounting for thesubgrid physics of absorption (Sobacchi & Mesinger2014).

• Tminvir : The minimal mass of dark matter halos that

hosted ionizing sources. While in principle, ha-los with virial temperatures as small as 102 K are

6 http://homepage.sns.it/mesinger/DexM___21cmFAST.html

thought to be able to form stars (Haiman et al.1996; Tegmark et al. 1997), thermal and mechani-cal feedback have been seen to raise this limit to ashigh as 105 K (Springel & Hernquist 2003; Mesinger& Dijkstra 2008; Okamoto et al. 2008). We choosea fiducial value of Tmin

vir = 1.5 × 104 K which is setby the atomic line cooling threshold.

In order to account for the degeneracies in the powerspectrum between heating from X-rays and reionizationfrom UV photons, we also marginalize over three ad-ditional parameters that describe the impact of heat-ing from early X-ray sources as explored in (Ewall-Wiceet al. 2016a). These are the X-ray heating efficiency,fX ; the maximal energy of X-ray photons that are selfabsorbed by the ISM of early galaxies, νmin; and thespectral slope, α which are taken to have fiducial val-ues of 1, 0.3 keV, and −1.2 respectively. We choose toparameterize our model in terms of the fractional differ-ence of each variable from its fiducial value so that, forexample, θζ = (ζ−ζfid)/ζfid and compute the derivativesin equation 24 by performing a linear fit to realizationsof the 21 cm power spectrum calculated by 21cmFAST atθi = ±10−2,±5× 10−2,±10−1, and ±2× 10−2.

For each measurement in the uv plane, the standarddeviation of a power spectrum measurement (σ2(k, z))is given by the direct sum of sample variance and ther-mal noise (McQuinn et al. 2006) which in turn dependson the primary beam of the instrument and the timespent sampling each uv cell. For our analysis, we as-sume that the uv plane is sampled by circular apertureswith effective areas of 98 m2 and that τ(k) is determinedby a drift scan in which non-instantaneously redundantbaselines are combined within each uv cell. We computethe standard deviation of each (k, z) bin for the proposed350-element deployment of HERA (HERA-350) using thepublic 21cmSense7 code (Pober et al. 2013b, 2014).

We have seen in Fig. 14 that the simulated chromatic-ity of the dish leaks foregrounds beyond the horizon tovarying degrees depending on the subband with the res-onance occuring at z = 8.5(150 MHz). In this subband,foregrounds exceed the signal out to ∼ 380 ns which fora 14.6 m baseline is ≈ 330 ns beyond the horizon. At allother redshifts, the leakage due to reflections only ex-tends to ≈ 250 ns beyond the horizon. To determine theimpact of the observed reflections on HERA’s ability toconstrain the astrophysics of reionization, we vary thecutoff in delay below which cosmological Fourier modeswill be unobservable. To determine this cutoff, we con-sider three different scenarios for beam chromaticity thatcapture a range of possibilities informed by our simula-tions.

• Optimistic: The cylinder resonance is miti-gated. In our most optimistic scenario, we assumethat a more advanced feed design is able to mitigatethe 150 MHz resonance in the cylinder while pre-serving polarization symmetry and effective area.In this case the foregrounds pass below the levelof the signal at 250 ns beyond the horizon at allredshifts between z = 7 and 12, consistent withwhat is observed in the bands where the reflections

7 https://github.com/jpober/21cmSense

The HERA Dish II: Electromagnetic Simulations and Science Implications 15

10-1 100

k (hMpc−1)

100

101

102

∆2 (

mK

2)

z=7.51 0 - 1 1 0 0

100

101

102

∆2 (

mK

2)

z=8.51 0 - 1 1 0 0

100

101

102

∆2 (

mK

2)

z=9.5

Optimistic Pessimistic Signal

Fig. 15.— The 1σ thermal noise levels achieved by HERA-350 atthree different redshifts with (grey line) and without (black line)the presence of beam chromaticity due to the chromaticity observedin this work. We compare these noise levels to the fiducial powerspectrum signal (red line). We saw in Fig. 14 that with reflections,foregrounds exceed the signal level out to k = 0.23hMpc−1 at z =8.5 which we assume are unusable, forcing us to ignore modes outto 350 ns beyond the horizon, leading to the sensitivity projectedin the red curve. The absence of these reflections allows us to workwithin 250 ns of the horizon (green curve), leading to an increasein sensitivity by a factor of ≈ 1.5.

are less severe or when they are not present at all.Since the achromatic beam stays above the fore-grounds below ≈ 250 ns as well, this number isset by the width of the Blackman-Harris windowfunction and more optimal estimators of the powerspectrum may reduce it substantially (see Ali et al.(2015)). Hence, even our optimistic scenario is onthe conservative side, making it more pessimisticthan previous Fisher analyses (Pober et al. 2014;Liu et al. 2016; Liu & Parsons 2016; Ewall-Wiceet al. 2016a). In this scenario, we assume that onlymodes within 250 ns of the horizon are unobserv-

-0.2

0.0

0.2

∆Rmfp

Rmfp

Pessimistic

Moderate

Optimistic

-0.2 0.0 0.2∆ζ

ζ

-0.2

0.0

0.2

∆Tmin

vir

Tmin

vir

-0.2 0.0 0.2∆Rmfp

Rmfp

Fig. 16.— 95% confidence regions for reionization parametersassuming 1000 hours of observation on HERA-350 between theredshifts between 7.5 and 12. The presence of the reflections leadsto an increase in the major axes of these confidence regions by afactor of one to two.

able.

• Moderate: The Simulations accurately cap-ture the chromaticity of the dish. In this case,we assume that the impedance match is sufficientfor the contamination from reflections to pass be-low the level of the signal at 250 ns beyond the hori-zon except at redshift 8.5 where the feed resonancecauses foregrounds to extend above the signal to350 ns beyond the horizon. In this scenario, we as-sume that modes within 250 ns of the horizon areunobservable except for redshift 8.5 where modesbelow 350 ns beyond the horizon are unobservable.This is the scenario that would follow directly fromour simulation results.

• Pessimistic: Structure is present in all sub-bands. In this scenario, we assume that theresonance is still present in the neighborhood of150 MHz and that a very poor impedance match(similar to the 500 Ω case in Fig. 11) is obtained,allowing for significant contamination from reflec-tions over the entire band, beyond the resonance.For this case, structure would be present out to≈ 350 ns over the entire 100-200 MHz frequencyrange covered by HERA. In this scenario, we as-sume that modes within 350 ns of the horizon areunobservable across all redshift intervals.

In Fig. 15, we compare the level of 1σ thermal noise forour optimistic and pessimistic scenarios to the amplitudeof the 21 cm signal at several different redshifts. Thesmallest k modes lost to foreground contamination arethe highest signal to noise measurements that HERA isexpected to obtain. Their absence impacts sensitivityin two ways. By leading to a reduction in the maximal

16 Ewall-Wice et al.

7 8 9 10 11 12

zmax

10-2

10-1

100

Fract

ional Err

or

ζ

7 8 9 10 11 12

zmax

Rmfp

7 8 9 10 11 12

zmax

Tminvir

Optimistic Moderate Pessimistic

Fig. 17.— Fractional Errors on reionization and heating parameters as a function of maximal observed redshift out to the low end ofHERA-350’s initial observing band at z = 12. The presence of strong reflections contained within a small subband at z = 8.5 has a minimalimpact on our overall constraints on reionization parameters. If these reflections are not localized they can worsen our sensitivity to someparameters such as Tmin

vir by a factor of two.

signal to noise ratio of ≈ 1.5 and a reduction in the totalnumber of modes that the instrument is able to measure.

Folding our calculations of thermal noise and thederivatives of ∆2 into equation 24 and inverting, we ob-tain the covariance matrix for model parameters. Weshow the 95% confidence ellipses for the reionization pa-rameters in Fig. 16. The presence of the resonance withina limited sub-band about z = 8.5 leads to an almostneglible increase in the extent of our confidence inter-vals while its presence across the entire band causes thelengths and widths of our confidence ellipses to increaseby a factor of ≈ 2. The diagonal elements of our covari-ance give us error bars on each parameter which we plotin Fig. 17. We see that similar to our confidence regions,the error bars on reionization parameters for our opti-mistic and moderate scenarios are nearly indestinguish-able. In our worst case where the resonance is presentacross the entire band, we see an increase in our errorbars by a factor of ≈ 2.

The error bars on reionization parameters, even for ourmost optimistic model, are a factor of a few larger thanthe “moderate” errors predicted in previous works usinglikelihood analyses such as Pober et al. (2014), Greig& Mesinger (2015); Greig et al. (2015b,a) Ewall-Wiceet al. (2016a), Liu et al. (2016); Liu & Parsons (2016).There are several reasons for this. Firstly, in our mostoptimistic scenario, we assumed that foregrounds causethe signal to be inaccessible below 250 ns beyond the edgeof the wedge which corresponds to kmin ≈ 0.15hMpc−1

at z = 8.5 while in previous works a minimal comovingkmin ≈ 0.1hMpc−1 was used. Secondly, previous studiesassumed a fully illuminated HERA aperture, which for a14 m diameter dish predicts an effective area of ≈ 155 m2.Electromagnetic simulations and ORBCOMM mappingof the angular beam pattern of the HERA dish showthat the effective area of the antenna element is actually≈ 98 m2 at 137 MHz which leads to an increase in theoverall thermal noise levels by a factor of ≈ 1.5 (Nebenet al. 2016).

Although conservative, our analysis shows that thelevel of the HERA antenna’s delay response is not aninsurmountable obstacle for the 21 cm power spectrummeasurement. Even if the additional systematics intro-duced by the feed resonance are present over the entireband, HERA will obtain a & 10σ detection of the powerspectrum and be capable of establishing precision con-straints on the properties of the sources that reionizedthe IGM.

6. CONCLUSIONS

In this paper, we have derived the impact of spectralstructure from reflections in the analog signal chain of aninterferometer on foreground contamination of the 21 cmsignal. We have used simulations of electromagneticwaves incident from zenith on the primary antenna ele-ment on HERA to determine the extent to which spectralstructure due to reflections in the dish leak foregroundsinto the EoR window. The results of our simulations ofthe dish’s voltage response are broadly consistent withreflectometry measurements (Patra et al. in preparation)and can be summarized in the following points.

• Additional spectral structure, in excess of that ob-served in the antenna design of HERA’s predeces-sor, PAPER, is observed in our simulations. Be-cause this structure is present in simulations with-out the dish and is dramatically reduced when thefeed’s cylindrical enclosure is removed, we concludethat this structure arises primarily from a reso-nance within the feed cylinder. On the other hand,contamination from reflections between the feedand the parabolic dish, which are present outside ofthe 150 MHz resonance, are relatively minor. Suchreflections were perceived as a significant risk tothe use of dish antennas in EoR studies and oursimulations show that their impact on array sensi-tivity is small and 21 cm experiments can be eco-nomically scaled up through the use of dishes. Fur-ther adjustments to the feed design aim to remove

The HERA Dish II: Electromagnetic Simulations and Science Implications 17

the cylindrical resonance while preserving collect-ing area and polarization match.

• The resonance which is responsible for the majorityof contamination has a finite width of ≈ 10 MHzof the HERA band. Because estimates of thepower spectrum are obtained from sub-intervals of≈ 10 MHz, the structure that we have simulatedwill only impact a single ∆z = 0.5 redshift inter-val.

• Combining simulations of the HERA antenna’sdelay-response with the foreground model of Thya-garajan et al. (2016), we find that a resonance inthe feed extends foregrounds above the level of thecosmological signal to ≈ 350 ns beyond the horizonwhile without the resonance (but with reflections),foregrounds extend above the signal to ≈ 250 nsbeyond the horizon (Fig. 14). These forecasts areconservative in that we do not attempt to inverselyweight the foregrounds by their covariances or ap-ply additional mitigation algorithms such as delayrate filtering.

• If beam chromaticity is contained within a 10 MHzsubband around 150 MHz, then the overall con-straints that HERA will be able to place on the as-trophysics of reionization are minimally impacted.If the resonance were to extend across the entireband, our constraints on reionization parameterswill suffer a two-fold increase in uncertainty but

still remain on the order of 10%.

Our results are very encouraging since they show thatthe HERA design is capable of isolating the foregroundsto sufficiently low delays to make a high significance de-tection of the 21 cm power spectrum during the EoR.Since reflections are not a major source of spectral struc-ture, feeds suspended over dishes are a viable strategyfor increasing the collecting area of HI interferometers ata modest cost. Our results also emphasize the impor-tance of mitigating any spectral structure in the analogsignal chain. The contamination from the feed resonancemasking our signal over some redshifts is mathematicallyidentical to Fourier modes introduced by any instrumen-tal spectral structure. Hence it is important that reflec-tions be stringently suppressed at all steps in the signalchain beyond the dish to ensure that bright foregroundsfall below the EoR signal. A systematic definition ofspecifications on the spectral structure necessary to ob-serve the EoR power spectrum at different k values ispresented in Thyagarajan et al. (2016).

ACKNOLEDGMENTS

We would like to thank Danny Jacobs, JonathanPober, Gianni Bernardi, Peter Williams, and Jeff Zhengfor their helpful comments. This work was supported byNSF grants AST-1440343 and AST-1410484, the Mar-ble Astrophysics Fund, and the MIT School of Science.ARP acknowledges support from NSF CAREER award1352519. JSD acknowledges support from a BerkeleyCenter for Cosmological Physics Fellowship. AEW ac-knowledges support from an NSF Graduate Research Fel-lowship under Grant No. 1122374.

REFERENCES

Ali, Z. S., Parsons, A. R., Zheng, H., et al. 2015, ApJ, 809, 61Asad, K. M. B., Koopmans, L. V. E., Jelic, V., et al. 2015,

MNRAS, 451, 3709Barry, N., Hazelton, B., Sullivan, I., Morales, M. F., & Pober,

J. C. 2016, MNRAS, 461, 3135Beardsley, A. P., Hazelton, B. J., Sullivan, I. S., et al. 2016,

ArXiv e-prints, arXiv:1608.06281Bernardi, G., McQuinn, M., & Greenhill, L. J. 2015, ApJ, 799, 90Bernardi, G., de Bruyn, A. G., Brentjens, M. A., et al. 2009,

A&A, 500, 965Bock, D. C.-J., Large, M. I., & Sadler, E. M. 1999, AJ, 117, 1578Bowman, J. D., & Rogers, A. E. E. 2010, Nature, 468, 796Burns, J. O., Lazio, J., Bale, S., et al. 2012, Advances in Space

Research, 49, 433Colegate, T. M., Sutinjo, A. T., Hall, P. J., et al. 2015, ArXiv

e-prints, arXiv:1501.05017Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ,

115, 1693Datta, A., Bowman, J. D., & Carilli, C. L. 2010, ApJ, 724, 526de Oliveira-Costa, A., Tegmark, M., Gaensler, B. M., et al. 2008,

MNRAS, 388, 247DeBoer, D. R., Parsons, A. R., Aguirre, J. E., et al. 2016, ArXiv

e-prints, arXiv:1606.07473Dillon, J. S., Liu, A., & Tegmark, M. 2013, Phys. Rev. D, 87,

043005Dillon, J. S., & Parsons, A. R. 2016, ArXiv e-prints,

arXiv:1602.06259Dillon, J. S., Liu, A., Williams, C. L., et al. 2014, Phys. Rev. D,

89, 023002Dillon, J. S., Neben, A. R., Hewitt, J. N., et al. 2015a,

Phys. Rev. D, 91, 123011Dillon, J. S., Tegmark, M., Liu, A., et al. 2015b, Phys. Rev. D,

91, 023002

Ekers, R., & Subrahmanyan, R. 2012, CORE, ZEBRA, SARAS;The Zero Spacing Interferometers, http://www.caastro.org/files/30/1496724333/ekers_eor.pdf

Ewall-Wice, A., Hewitt, J., Mesinger, A., et al. 2016a, MNRAS,458, 2710

Ewall-Wice, A., Dillon, J. S., Hewitt, J. N., et al. 2016b,MNRAS, 460, 4320

Fagnoni, N., & de Lera Acedo, E. 2016, ArXiv e-prints,arXiv:1606.08701

Fixsen, D. J., Kogut, A., Levin, S., et al. 2011, ApJ, 734, 5Furlanetto, S. R., Oh, S. P., & Briggs, F. H. 2006, Phys. Rep.,

433, 181Furlanetto, S. R., Zaldarriaga, M., & Hernquist, L. 2004, ApJ,

613, 1Greenhill, L. J., & Bernardi, G. 2012, ArXiv e-prints,

arXiv:1201.1700Greig, B., & Mesinger, A. 2015, MNRAS, 449, 4246Greig, B., Mesinger, A., & Koopmans, L. V. E. 2015a, ArXiv

e-prints, arXiv:1509.03312Greig, B., Mesinger, A., & Pober, J. C. 2015b, ArXiv e-prints,

arXiv:1509.02158Haiman, Z., Thoul, A. A., & Loeb, A. 1996, ApJ, 464, 523Jacobs, D., et al. in preparationJacobs, D. C., Pober, J. C., Parsons, A. R., et al. 2015, ApJ, 801,

51Jelic, V., Zaroubi, S., Labropoulos, P., et al. 2010, MNRAS, 409,

1647Liu, A., & Parsons, A. R. 2016, MNRAS, 457, 1864Liu, A., Parsons, A. R., & Trott, C. M. 2014a, Phys. Rev. D, 90,

023018—. 2014b, Phys. Rev. D, 90, 023019Liu, A., Pritchard, J. R., Allison, R., et al. 2016, Phys. Rev. D,

93, 043013Liu, A., & Tegmark, M. 2011, Phys. Rev. D, 83, 103006

18 Ewall-Wice et al.

Liu, A., Tegmark, M., Morrison, S., Lutomirski, A., &Zaldarriaga, M. 2010, MNRAS, 408, 1029

Mauch, T., Murphy, T., Buttery, H. J., et al. 2003, MNRAS, 342,1117

McQuinn, M., Zahn, O., Zaldarriaga, M., Hernquist, L., &Furlanetto, S. R. 2006, ApJ, 653, 815

Mellema, G., Koopmans, L. V. E., Abdalla, F. A., et al. 2013,Experimental Astronomy, 36, 235

Mesinger, A., & Dijkstra, M. 2008, MNRAS, 390, 1071Mesinger, A., Furlanetto, S., & Cen, R. 2011, MNRAS, 411, 955Mesinger, A., McQuinn, M., & Spergel, D. N. 2012, MNRAS, 422,

1403Moore, D., Aguirre, J. E., Parsons, A., et al. 2015, ArXiv

e-prints, arXiv:1502.05072Moore, D. F., Aguirre, J. E., Parsons, A. R., Jacobs, D. C., &

Pober, J. C. 2013, ApJ, 769, 154Morales, M. F., Hazelton, B., Sullivan, I., & Beardsley, A. 2012,

ApJ, 752, 137Morales, M. F., & Hewitt, J. 2004, ApJ, 615, 7Morales, M. F., & Wyithe, J. S. B. 2010, ARA&A, 48, 127Neben, A. R., Bradley, R. F., Hewitt, J. N., et al. 2015, Radio

Science, 50, 614—. 2016, ArXiv e-prints, arXiv:1602.03887Okamoto, T., Gao, L., & Theuns, T. 2008, MNRAS, 390, 920Paciga, G., Albert, J. G., Bandura, K., et al. 2013, MNRAS, 433,

639Parsons, A., Pober, J., McQuinn, M., Jacobs, D., & Aguirre, J.

2012a, ApJ, 753, 81Parsons, A. R., & Backer, D. C. 2009, AJ, 138, 219Parsons, A. R., Liu, A., Ali, Z. S., & Cheng, C. 2015, ArXiv

e-prints, arXiv:1503.05564Parsons, A. R., Pober, J. C., Aguirre, J. E., et al. 2012b, ApJ,

756, 165Parsons, A. R., Backer, D. C., Foster, G. S., et al. 2010, AJ, 139,

1468Parsons, A. R., Liu, A., Aguirre, J. E., et al. 2014, ApJ, 788, 106Patil, A. H., Yatawatta, S., Zaroubi, S., et al. 2016, ArXiv

e-prints, arXiv:1605.07619Patra, N., Subrahmanyan, R., Sethi, S., Udaya Shankar, N., &

Raghunathan, A. 2015, ApJ, 801, 138Patra et al. in preparation, ApJPober, J. C., Parsons, A. R., Jacobs, D. C., et al. 2012, AJ, 143,

53Pober, J. C., Parsons, A. R., Aguirre, J. E., et al. 2013a, ApJ,

768, L36Pober, J. C., Parsons, A. R., DeBoer, D. R., et al. 2013b, AJ,

145, 65

Pober, J. C., Liu, A., Dillon, J. S., et al. 2014, ApJ, 782, 66Pober, J. C., Ali, Z. S., Parsons, A. R., et al. 2015, ApJ, 809, 62Pober, J. C., Hazelton, B. J., Beardsley, A. P., et al. 2016, ArXiv

e-prints, arXiv:1601.06177Presley, M. E., Liu, A., & Parsons, A. R. 2015, ApJ, 809, 18Pritchard, J. R., & Loeb, A. 2012, Reports on Progress in

Physics, 75, 086901Rogers, A. E. E., & Bowman, J. D. 2008, AJ, 136, 641Singh, S., Subrahmanyan, R., Udaya Shankar, N., &

Raghunathan, A. 2015, ApJ, 815, 88Sobacchi, E., & Mesinger, A. 2014, MNRAS, 440, 1662Sokolowski, M., Tremblay, S. E., Wayth, R. B., et al. 2015,

PASA, 32, 4Songaila, A., & Cowie, L. L. 2010, ApJ, 721, 1448Springel, V., & Hernquist, L. 2003, MNRAS, 339, 312Tegmark, M. 1997, Phys. Rev. D, 55, 5895Tegmark, M., Silk, J., Rees, M. J., et al. 1997, ApJ, 474, 1Thompson, A. R., Moran, J. M., & Swenson, G. W. 1986,

Interferometry and synthesis in radio astronomyThyagarajan, N., Helfand, D. J., White, R. L., & Becker, R. H.

2011, ApJ, 742, 49Thyagarajan, N., Parsons, A. R., DeBoer, D. R., et al. 2016, ApJ,

825, 9Thyagarajan, N., Udaya Shankar, N., Subrahmanyan, R., et al.

2013, ApJ, 776, 6Thyagarajan, N., Jacobs, D. C., Bowman, J. D., et al. 2015a,

ApJ, 807, L28—. 2015b, ApJ, 804, 14Tingay, S. J., Goeke, R., Bowman, J. D., et al. 2013, PASA, 30, 7Trott, C. M., Pindor, B., Procopio, P., et al. 2016, ApJ, 818, 139

van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013,A&A, 556, A2

Vedantham, H., Udaya Shankar, N., & Subrahmanyan, R. 2012,ApJ, 745, 176

Venumadhav, T., Chang, T.-C., Dore, O., & Hirata, C. M. 2015,ArXiv e-prints, arXiv:1512.05248

Voytek, T. C., Natarajan, A., Jauregui Garcıa, J. M., Peterson,J. B., & Lopez-Cruz, O. 2014, ApJ, 782, L9

Wieringa, M. H. 1992, Experimental Astronomy, 2, 203Wilson, T. L., Rohlfs, K., & Huttemeister, S. 2009, Tools of Radio

Astronomy (Springer-Verlag), doi:10.1007/978-3-540-85122-6Zaldarriaga, M., Furlanetto, S. R., & Hernquist, L. 2004, ApJ,

608, 622Zheng, H., Tegmark, M., Buza, V., et al. 2013, ArXiv e-prints,

arXiv:1309.2639—. 2014, MNRAS, 445, 1084