The Importance of Wide-field Foreground Removal for 21 cm Cosmology: A Demonstration With Early MWA Epoch of Reionization Observations

5.3 MB·

by  J. C. Pober

Draft version January 26, 2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE IMPORTANCE OF WIDE-FIELD FOREGROUND REMOVAL FOR 21 CM COSMOLOGY: A
DEMONSTRATION WITH EARLY MWA EPOCH OF REIONIZATION OBSERVATIONS
J. C. Pober1,2,25, B. J. Hazelton2, A. P. Beardsley3,2, N. A. Barry2, Z. E. Martinot2, I. S. Sullivan2,
M. F. Morales2, M. E. Bell4, G. Bernardi5,6,7, N. D. R. Bhat8,9, J. D. Bowman3, F. Briggs10, R. J. Cappallo11,
P. Carroll2, B. E. Corey11, A. de Oliveira-Costa,12A. A. Deshpande13, Joshua. S. Dillon14,15,16, D. Emrich8,
A. M. Ewall-Wice16, L. Feng16,12, R. Goeke12, L. J. Greenhill7, J. N. Hewitt16,12, L. Hindson17,
N. Hurley-Walker8, D. C. Jacobs3,25, M. Johnston-Hollitt17, D. L. Kaplan18, J. C. Kasper7,19, Han-Seek Kim20,9,
P. Kittiwisit3, E. Kratzenberg11, N. Kudryavtseva8, E. Lenc4,9, J. Line20,9, A. Loeb7, C. J. Lonsdale11,
M. J. Lynch8, B. McKinley10, S. R. McWhirter11, D. A. Mitchell9,21, E. Morgan12, A. R. Neben16, D. Oberoi22,
A. R. Offringa10,9, S. M. Ord8,9, Sourabh Paul13, B. Pindor20,9, T. Prabu13, P. Procopio20, J. Riding20,
6
A. E. E. Rogers11, A. Roshi23, Shiv K. Sethi13, N. Udaya Shankar13, K. S. Srivani13, R. Subrahmanyan13,9,
1 M. Tegmark16, Nithyanandan Thyagarajan3, S. J. Tingay8,9, C. M. Trott8,9, M. Waterson24, R. B. Wayth8,9,
0 R. L. Webster20,9, A. R. Whitney11, A. Williams8, C. L. Williams16, J. S. B. Wyithe20,9
2
Draft version January 26, 2016
n
a ABSTRACT
J
Inthispaperwepresentobservations,simulations,andanalysisdemonstratingthedirectconnection
2
betweenthelocationofforegroundemissionontheskyanditslocationincosmologicalpowerspectra
2
from interferometric redshifted 21 cm experiments. We begin with a heuristic formalism for under-

standing the mapping of sky coordinates into the cylindrically averaged power spectra measurements
]
M usedby21cmexperiments, withafocusontheeffectsoftheinstrumentbeamresponseandtheasso-
ciated sidelobes. We then demonstrate this mapping by analyzing power spectra with both simulated
I and observed data from the Murchison Widefield Array. We find that removing a foreground model
.
h which includes sources in both the main field-of-view and the first sidelobes reduces the contamina-
p tion in high k modes by several percent relative to a model which only includes sources in the main
(cid:107)
- field-of-view, with the completeness of the foreground model setting the principal limitation on the
o
amount of power removed. While small, a percent-level amount of foreground power is in itself more
r
t than enough to prevent recovery of any EoR signal from these modes. This result demonstrates that
s
foregroundsubtractionforredshifted21cmexperimentsistrulyawide-fieldproblem, andalgorithms
a
[ and simulations must extend beyond the main instrument field-of-view to potentially recover the full
21 cm power spectrum.

1
Subject headings: cosmology: observations — dark ages, reionization, first stars — techniques: inter-
v
ferometric
7
7
1
6 1Department of Physics, Brown University, Providence, RI
0 02912,USA
. 2Department of Physics, University of Washington, Seattle,
1 WA98195,USA
0 3School of Earth and Space Exploration, Arizona State
6 University,Tempe,AZ85287,USA
1 4Sydney Institute for Astronomy, School of Physics, The
UniversityofSydney,NSW2006,Australia
v: 5SquareKilometreArraySouthAfrica(SKASA),Pinelands
7405,SouthAfrica
i
X 6Department of Physics and Electronics, Rhodes University,
Grahamstown6140,SouthAfrica
r 7Harvard-Smithsonian Center for Astrophysics, Cambridge,
a MA02138,USA 17SchoolofChemical&PhysicalSciences,VictoriaUniversity
8InternationalCentreforRadioAstronomyResearch,Curtin ofWellington,Wellington6140,NewZealand
University,Bentley,WA6102,Australia 18Department of Physics, University of Wisconsin–
9ARCCentreofExcellenceforAll-skyAstrophysics(CAAS- Milwaukee,Milwaukee,WI53201,USA
TRO) 19Department of Atmospheric, Oceanic and Space Sciences,
10Research School of Astronomy and Astrophysics, Aus- UniversityofMichigan,AnnArbor,MI48109,USA
tralianNationalUniversity,Canberra,ACT2611,Australia 20SchoolofPhysics, TheUniversityofMelbourne, Parkville,
11MITHaystackObservatory,Westford,MA01886,USA VIC3010,Australia
12Kavli Institute for Astrophysics and Space Research, 21CSIROAstronomyandSpaceScience(CASS),POBox76,
Massachusetts Institute of Technology, Cambridge, MA 02139, Epping,NSW1710,Australia
USA 22National Centre for Radio Astrophysics, Tata Institute for
13RamanResearchInstitute,Bangalore560080,India FundamentalResearch,Pune411007,India
14Department of Astronomy, University of California Berke- 23National Radio Astronomy Observatory, Charlottesville
ley,Berkeley,CA94720,USA andGreenbank,USA
15Berkeley Center for Cosmological Physics, University of 24SKA Organisation, Jodrell Bank Observatory, Lower
CaliforniaBerkeley,Berkeley,CA94720,USA Withington,Macclesfield,SK119DL,UK
16Department of Physics, Massachusetts Institute of Tech- 25National Science Foundation Astronomy and Astrophysics
nology,Cambridge,MA02139,USA PostdoctoralFellow
2
1. INTRODUCTION ing maps from the Green Bank Telescope and optical
galaxy surveys (Chang et al. 2010; Masui et al. 2013;
Amajorgoalofmodernexperimentalcosmologyisthe
Switzer et al. 2013). Analysis techniques for recover-
detection of 21 cm emission from neutral hydrogen at
ing the signal focus on the relative spectral smoothness
highredshifts. Dependingontheredshiftsstudied,these
of the foreground emission as an axis for distinguishing
observations can probe a wide range of physical and as-
these contaminants from the 21 cm emission. Over the
trophysicalphenomena. Observationsat∼100–200MHz
past decade, a large body of literature has worked to de-
(z ∼ 6 − 13 in the 21 cm line) probe the Epoch of
veloppipelinesthatcansubtractforegroundsourcesfrom
Reionization(EoR)—thereionizationoftheintergalac-
21 cm data sets (e.g. Morales et al. 2006, Bowman et al.
ticmedium(IGM)byultravioletphotonsemittedbythe
2009, Liu et al. 2009, Liu & Tegmark 2011, Chapman
first stars and galaxies. Observations at higher frequen-
etal.2012,Dillonetal.2013,Chapmanetal.2013,Wang
cies (lower redshifts) trace the neutral hydrogen that re-
et al. 2013). More recently, however, studies of the chro-
mainsingalactichalos,andprovidealowresolution“in-
matic interaction of an interferometer with foreground
tensity map” of large scale structure and, potentially,
emission have demonstrated that smooth spectrum fore-
the Baryon Acoustic Oscillation (BAO) features in the
grounds will occupy an anisotropic wedge-like region of
power spectrum. At lower frequencies (higher redshifts),
cylindrical (k ,k ) Fourier space, leaving an “EoR win-
one begins to trace the birth of the first stars during ⊥ (cid:107)
dow” above the wedge where the 21 cm signal can be
“Cosmic Dawn” and even the preceding Dark Ages. For
cleanly observed (Datta et al. 2010; Vedantham et al.
reviewsofthe21cmcosmologytechniqueandtheassoci-
2012; Morales et al. 2012; Parsons et al. 2012b; Trott
atedsciencedrivers,seeFurlanettoetal.(2006),Morales
et al. 2012; Thyagarajan et al. 2013; Liu et al. 2014a,b).
&Wyithe(2010),Pritchard&Loeb(2012),andZaroubi
Thesepredictionshavesincebeenconfirmedindatasets
(2013).
from PAPER and the MWA (Pober et al. 2013a; Dil-
A large number of experiments seeking to detect the
lon et al. 2014; Parsons et al. 2014; Jacobs et al. 2015;
power spectra of 21 cm fluctuations are already opera-
Ali et al. 2015; Thyagarajan et al. 2015a), although sig-
tional or being commissioned, including the LOw Fre-
nificantly more sensitive observations will be necessary
quency ARray (LOFAR; Yatawatta et al. 2013; van
to see if the window remains uncontaminated down to
Haarlem et al. 2013)26, 21 CentiMeter Array (21CMA;
the level of the 21 cm signal. Pober et al. (2014) demon-
Zheng et al. 2012)27, the Giant Metrewave Radio Tele-
stratethatwhilecurrentEoRobservatories(PAPER,the
scope EoR Experiment (GMRT; Paciga et al. 2013)28,
MWA,andLOFAR)donotpossessthesensitivitytode-
the MIT Epoch of Reionization Experiment (MITEoR;
tect the 21 cm signal with this pure “foreground avoid-
Zheng et al. 2014), the Donald C. Backer Precision Ar-
ance”technique,next-generationexperimentslikeHERA
rayforProbingtheEpochofReionization(PAPER;Par-
and the SKA-low can yield high fidelity power spectrum
sons et al. 2010)29, and the Murchison Widefield Array measurements using this approach, and begin to place
(MWA; Lonsdale et al. 2009; Tingay et al. 2013; Bow- constraints on the physics of reionization.35 However,
man et al. 2013)30, all of which are targeting the sig- the cosmological signal strength peaks on large scales,
nal from the EoR. A number of additional experiments so that k modes within the wedge can have significantly
are also under construction or planned, such as the low- more21cmpowerthanmodeswithinthewindow. Pober
frequency Square Kilometre Array (SKA-low; Mellema et al. (2014) show that if foregrounds can be subtracted
et al. 2013)31 and the Hydrogen Epoch of Reionization from 21 cm data sets, allowing the recovery of k modes
Array (HERA; Pober et al. 2014)32 at EoR and Cosmic fromwithinthewedge,thenthesignificanceofanypower
Dawn redshifts, and BAOs from Integrated Neutral Gas spectrum measurement can be substantially boosted —
Observations (BINGO; Battye et al. 2013), TianLai33, enabling the current generation of 21 cm experiments to
BAORadio(Ansarietal.2012a,b),theCanadianHydro- make a detection.
genIntensityMappingExperiment(CHIME;Shawetal. Continued research into foreground subtraction algo-
2014)34, and the BAO Broadband and Broad-beam ex- rithms is therefore clearly well motivated. As of yet,
periment (BAOBAB; Pober et al. 2013b) at lower red- no technique — whether subtracting a model of the sky
shifts. or using a parametrized fit in frequency — has demon-
At all redshifts, however, 21 cm experiments are lim- strated that foreground emission in actual observations
ited by both the inherent faintness of the cosmological can be removed to the thermal noise level of current in-
signal and the presence of foregrounds which can exceed struments(althoughtheEoRwindowhasto-dateproven
the 21 cm emission by as much as 5 orders of magnitude relatively free of foregrounds when care is taken to limit
inbrightnesstemperature(Santosetal.2005;Yatawatta leakagefromthewedge(Poberetal.2013b;Parsonsetal.
et al. 2013; Bernardi et al. 2013; Pober et al. 2013a). 2014; Jacobs et al. 2015; Ali et al. 2015)). The purpose
As such, the only current detection of HI at cosmolog- of this work is to investigate some of the wide-field ef-
ical distances comes from cross-correlation studies us- fectsthatcomplicatetheremovalofforegroundemission
using data from the MWA. In particular, we focus on
26 http://www.lofar.org the contribution of sources outside the main lobe of the
27 http://21cma.bao.ac.cn/index.html instrument primary beam(inthiswork, weusethe term
28 http://www.ncra.tifr.res.in/ncra/gmrt
primarybeamtorefertotheall-skypowerpatternofthe
29 http://eor.berkeley.edu
antenna or tile element, including sidelobes). Far from
30 http://www.mwatelescope.org
31 http://www.skatelescope.org
32 http://reionization.org 35 Although Pober et al. 2014 focused on results from EoR-
33 http://tianlai.bao.ac.cn frequency experiments, the “wedge” and “EoR window” break-
34 http://chime.phas.ubc.ca downisgenericforall21cmstudies(Poberetal.2013b).
3
thepointingcenter, chromaticeffectsintheinterferome- is dominated by the intrinsic spectra of the sources, so
terresponsebecomestronger;sourcesoutinthesidelobes that a significant amount of foreground emission maps
oftheprimarybeamthereforecreateforegroundcontam- to low k modes, reflective of their inherent (smooth)
(cid:107)
inationinhigherk modesthansourcesnearthepointing frequency spectrum. However, the chromatic response
(cid:107)
center. Here, we explore this effect in more detail. of the interferometer still affects the observed emission,
Thispaperisstructuredasfollows. In§2, welayouta leading to a wedge feature analogous to that of the de-
heuristicderivationofhowtheinstrumentprimarybeam lay spectrum approach, but with more of the emission
entersinmeasurementsofthe21cmpowerspectrumand concentrated at low k (Morales et al. 2012; Dillon et al.
(cid:107)
how foregrounds are distributed throughout the (k⊥,k(cid:107)) 2015). Because of the brightness of foreground emission,
plane. In §3, we briefly describe the MWA and the data thiswedgestilldominatesany21cmsignalinthemodes
analyzed in this study. In §4, we build on the pedagog- it occupies.
ical nature of the previous analysis through simulated Explorations of these wide-field effects in actual data
MWA power spectra using a sky model containing a sin- have been more limited. Thyagarajan et al. (2015b,a)
glepointsourceofemission. Bychangingthelocationof studied both simulated and actual MWA observations
this source, we demonstrate these primary beam effects using the delay spectrum technique and found an excel-
in a realistic but controlled fashion. In §5, we describe lent match between the two, demonstrating a good un-
the calibration, pre-processing, and foreground subtrac- derstandingofbothforegroundemissionandtheprimary
tionappliedtotheobserveddatabeforemakingapower beamoftheMWA.Theyalsofoundthattheforeshorten-
spectrum. The main result is presented in §6, where we ing of baseline lengths when projected towards the hori-
compare power spectra made from our data, where we zon creates sensitivity to diffuse emission normally re-
havebothsubtractedaforegroundmodelwhichincludes solved out on longer baselines. Diffuse foregrounds are
sources in the beam sidelobes and one which does not. brightenoughthattheycanbedetecteddespitethesmall
We discuss the implications of these results for future (butnon-zero)responseoftheMWAelementtowardsthe
foreground subtraction efforts in §7 and conclude in §8. horizon. This led to what they dubbed the “pitch-fork”
effect,aforegroundsignatureindelayspacewherebright
2. WIDE-FIELDEFFECTSINTHEEORPOWER
SPECTRUM emission from within the main field of view appeared at
low delays and emission from the horizon at high delays.
Although many 21 cm experiments have wide fields
This work studies similar effects using an imaging
of view, only recently have studies focused on how wide-
power spectrum approach and will confirm that the sky-
fieldeffectsmightcomplicatemeasurementsofthe21cm
position to k mapping still holds. We will also focus
power spectrum. Theoretical work has identified the (cid:107)
ontheabilitytosubtract foregroundemissionawayfrom
foreground wedge described above and provided a for-
themainfieldofviewtolowerthecontaminationinhigh
malism for mapping the position of foreground emission
k modes. In this section, however, we use the delay-
on the sky to k modes of the 21 cm power spectrum (cid:107)
spectrum formalism (Parsons & Backer 2009; Parsons
(Vedantham et al. 2012; Morales et al. 2012; Parsons
et al. 2012b) to provide a general framework for under-
et al. 2012b; Trott et al. 2012; Thyagarajan et al. 2013;
standingtheseeffects. Westressthatthedelayspectrum
Liu et al. 2014a,b). Broadly speaking, there are two fla-
provides a straightforward, pedagogical way to interpret
vors of 21 cm power spectrum analysis: a “delay spec-
power spectrum results, since the wide-field chromatic
trum” approach, where the line-of-sight Fourier trans-
effects appear at first-order. As argued in Morales et al.
form is done on individual visibilities — and is therefore
(2012); Trott et al. (2012); Liu et al. (2014a,b), and as
not strictly orthogonal to the transverse directions be-
will confirmed with data below, these wide-field effects
cause of the frequency dependence of an individual visi-
are generic to all interferometric 21 cm experiments.
bility — and an “imaging” approach, where the line-of-
Thebasicpremiseofthedelay-spectrumtechniquepre-
sight Fourier transform spans multiple visibilities and is
trulyorthogonaltothetransversedirectionsonthesky.36 sented in Parsons et al. (2012b) is that the square of the
frequency Fourier transform of a single baseline’s visibil-
Afulldiscussionofthedifferencesbetweenthesetwoap-
proaches is outside the scope of this work, but previous ity spectrum (i.e. the delay spectrum, V˜b(τ)) approx-
analyses have shown that the wedge and the mapping imates a measurement of the cosmological power spec-
from foreground sky position to k modes of the power trum (to within a proportionality factor):
(cid:107)
spectrum remains valid for both frameworks. In the de-
lay spectrum approach, the chromatic dependence of an |V˜b(τ)|2 ∝P(k⊥,k(cid:107)), (1)
individual baseline is completely preserved, so that all
where
foreground emission at a given location maps to a given (cid:90)
k(cid:107) mode. An imaging approach, however, removes the V˜b(τ)= dν Vb(ν) e2πiντ (2)
mapping between delay and sky position by projecting
out the frequency sine wave for a known geometric de-
is the delay spectrum, τ is delay, ν is frequency, V is a
lay. In an imaging power spectrum, frequency structure
visibility, and the subscript b indicates that the visibili-
36 The terminology of an “imaging” power spectrum is poten- ties are from a single baseline.
tially misleading, but it has become somewhat standard in the Intuitively, this relation is well-motivated. To a good
community. The key feature is not that an image of the sky is approximation, a single baseline b probes a single trans-
made, but rather that visibilities are gridded into the uv plane verse scale, and thus a single k mode. And, since cos-
and the frequency Fourier transform is taken in a direction truly ⊥
mological redshifting of the 21 cm line maps observed
orthogonal to u and v. The nomenclature of an “imaging” power
spectrum arises because the gridded uv data is only 2D spatial frequenciesintoline-of-sightdistances,theFouriertrans-
Fouriertransformawayfromanimage. form of the frequency spectrum approximates a range of
4
k modes. Put more succinctly, for an interferometer, where the length of baseline b sets k , and τ ∝k . This
(cid:107) ⊥ (cid:107)
baseline length b maps to cosmological k⊥ and delay τ analysisthereforeimpliesthatsourcesatlargedelays(i.e.
maps to k(cid:107). sourcesneartheedgesofthefieldofview,byEquation5)
Thepowerofthissimpleformalismisthatwecannow contaminate the highest k modes of the wedge (k ,k )
(cid:107) ⊥ (cid:107)
map the effects of the primary beam, which enter into space. Althoughnotalwaysstatedasdirectly,thisresult
a visibility measurement in a well-known way, to cosmo- wasalsofoundinMoralesetal.(2012);Vedanthametal.
logical Fourier space and the power spectrum P(k⊥,k(cid:107)). (2012); Thyagarajan et al. (2013) and Liu et al. (2014a)
We begin with the form of a visibility in the flat-sky ap- using entirely independent formalisms.
proximation (Thompson et al. 2001)37: Equation8alsoshowsthemainresultwewishedtode-
(cid:90) rive in this section: the (smooth spectrum) sky emission
V (ν)= dldmA(l,m,ν)I(l,m,ν)e−2πi(ul+vm), (3) I(l,m) which appears in each delay mode is multiplica-
b
tivelyattenuatedbytheprimarybeamoftheinstrument.
Therefore, the foreground emission which contaminates
where A is the primary beam, I is the sky brightness
those k modes measured by a single baseline will itself
distribution, l and m are direction cosines on the sky, (cid:107)
be attenuated by a (distorted) slice through the square
ν is frequency, and u and v are the projected baseline
of the primary beam of the instrument. This result is
lengths on the ground plane measured in wavelengths.
schematically illustrated in Figure 1. Note that because
We can rewrite this expression in terms of the geometric
delay τ (Parsons & Backer 2009):
g
(cid:90)
V (ν)= dl dm A(l,m,ν) I(l,m,ν) e−2πiντg, (4) Horizon Limit
b
EoR Window
where
b·sˆ 1
τg = c = c(bxl+bym), (5) First Sidelobe
b ≡ (b ,b ) is the baseline vector measured in meters First Null
x y
(i.e. u ≡ (u,v) = νb/c), and sˆ ≡ (l,m). Doing the
Primary Field of View
delay-transform given by Equation 2 gives us a delay-
spectrum:
The Wedge
(cid:90)
V˜ (τ)= dl dm dν A(l,m,ν) I(l,m,ν) e−2πiν(τg−τ).
b
(6)
If we make the pedagogical assumption that both A and
I areindependentoffrequency,wecanstraightforwardly Figure 1. Aschematicdiagramoftheeffectsdiscussedhere. The
do the delay transform integral38: primarybeamattenuatesforegroundsinthek(cid:107) direction.
(cid:90)
V˜ (τ)= dl dm A(l,m) I(l,m) δ(τ −τ). (7) delay space is a one-dimensional projection of the sky
b g
coordinates (c.f. Equation 5), the attenuating beam in
a k spectrum will vary depending on the orientation
SinceEquation5relatesthegeometricdelayτ toaspe- (cid:107)
g of the baseline. On an east/west baseline, for example,
cificsetofskydirectioncosines(l,m),thedeltafunction
the delay axis probes the relative east/west position of
selectsasubsetofskypositionswhichcontributetoeach
the source and is insensitive to north/south translations
τ mode in the delay spectrum, albeit with a baseline-
insourcepositions. Suchabaselinewillthereforeclearly
dependentnon-trivialmappingbetweenskypositionand
showtheeffectsoftheeasternandwesternprimarybeam
τ. It is always true, however, that sources which appear
sidelobes in its k spectrum. Similar logic applies to a
at high delays are those which are far from the pointing (cid:107)
centeroftheinstrument(hencethename“horizonlimit” north/southbaselineandthenorthernandsouthernpri-
given to the maximum delay a source can appear at in mary beam sidelobes. Delays on a northeast/southwest
Parsonsetal.2012b). FollowingEquation1, wecansay: baseline, however, probe northeast/southwest sky po-
sition, and thus the east-west translation of a source
(cid:20)(cid:90) (cid:21)2 throughtheeasternandwesternsidelobesdoesnotcause
P(k⊥,k(cid:107))∝ dl dm A(l,m) I(l,m) δ(τg−τ) , as rapid a change in k(cid:107). The net effect is that when all
baselines of the same magnitude are averaged into a k
(8) ⊥
bin,thesedifferentk sidelobepatternsaddupandsmear
(cid:107)
out the location of the sidelobes.
37 Althoughuseoftheflat-skyapproximationtoderiveawide-
Animportantbutsubtlepointisthattheabovederiva-
field interferometric effect may seem ill-motivated, it greatly sim-
tionformappingsky-coordinatesintokspacewasstrictly
plifies the math in this pedagogical treatment. See Parsons et al.
(2012a,b)andThyagarajanetal.(2015b,a)foradiscussionofthe for flat spectrum emission. As shown in Parsons et al.
subtletiesintroducedbythecurvedskyintothedelayformalism. (2012b), any spectral structure — whether intrinsic to
38 Parsons et al. (2012b) showed that both the frequency- the source or the instrumental response — introduces a
dependenceofAandI createaconvolvingkernel,broadeningthe
convolving kernel that broadens the footprint of each k
footprint of each delay mode. The ramifications of this effect are (cid:107)
discussedbelow,butonlycomplicatethepedagogicalnatureofthe mode in cosmological Fourier space. While this kernel is
currentanalysis. narrowforsmooth-spectrumforegrounds,spectralstruc-
5
ture in the 21 cm signal spreads the 21 cm power across outside the primary field of view.
a wide range of k modes. This is equivalent to saying
(cid:107)
thatthe21cmsignalintrinsicallyhaspoweronthesecos- 3. OBSERVATIONSWITHTHEMURCHISONWIDEFIELD
ARRAY
mological scales. The situation for foreground emission
is different, however. Although power spectrum plots The Murchison Widefield Array in Australia consists
are labeled with axes of (k ,k ) with units of hMpc−1, of 128 tiles antenna elements, and each tile is composed
⊥ (cid:107)
these cosmological scalings apply only to the 21 cm sig- of 16 dual-polarization dipole antennas; the array con-
nal. Theanalysisaboveshowshowforegroundsmapinto figuration is shown in Figure 2. The tile element has
this space, and how the primary beam affects this map-
ping. The primary beam of the instrument does still act 1500
asawindowfunctionandcanaffecthighk modesofthe
(cid:107)
cosmologicalsignal;however,thecosmologicalsignalhas
been shown to be relatively featureless on the scale of
1000
this kernel (c.f. Parsons et al. 2012b), rendering this ef-
fectverysmall. Regardless,the21cmsignalisanall-sky
signal with real intrinsic k structure. There is therefore
(cid:107) 500
always 21 cm signal at the peak beam response, so there
will always be power at all k(cid:107) modes truly intrinsic to m)
thecosmologicalsignal. Thispointwillbediscussedfur- (
ther in §7, where we consider the possibility of detecting orth 0
N
21 cm emission at k modes where the foregrounds fall
(cid:107)
in the nulls of the primary beam.
The very wide and relatively smooth primary beam of −500
the PAPER instrument makes the predicted foreground
attenuation difficult to see in the analysis of Pober et al.
(2013a). However, for instruments like the MWA and −1000
LOFAR, which use tiles of dipoles to increase the sys-
temgainandnarrowthesizeoftheprimarybeam,there
should be two clear effects visible in the power spec- −1500
−1500 −1000 −500 0 500 1000 1500
tra. First, there should be significant attenuation of the East(m)
wedgeforegroundemissionbeforethehorizonlimit,since
Figure 2. MWA-128arrayconfiguration; eachsquarerepresents
the instrument field of view is significantly smaller than onetileof16dipoles.
2π steradians, as is seen in Dillon et al. (2014). Sec-
ondly, at higher k values than those corresponding to the effect of significantly narrowing the MWA’s field
(cid:107)
the main beam of the instrument, foreground emission of view over that from a single dipole, but also intro-
should appear coming from the sidelobes of the primary duces significant regular sidelobes in the primary beam.
beam. These two effects can be seen in the delay-space Figure 3 shows three MWA tiles; every tile is aligned
simulations of different antenna elements presented in North/South,sothesidelobesfromeachtileappearwith
Thyagarajan et al. (2015b). For an imaging power spec- nearly the same orientation.
trum technique which averages baselines together, the
second effect will be less clear for an instrument like the
MWA, in which all the dipoles and tiles are oriented in
the same direction. In this case, the sidelobes are al-
waysorientedNorth/SouthandEast/West;asexplained
above,however,thebeamfootprintink ,willdifferfrom
(cid:107)
baseline-to-baseline depending on that baseline’s orien-
tation relative to sidelobe pattern. This will have the
effect of smearing out the sidelobe across a wider range
of k modes than would be seen in an instrument with
(cid:107)
circularly symmetric sidelobes, but as we will show, the
feature is still quite visible in the power spectrum.
The structure of the remainder of this paper is as fol- Figure 3. Three MWA tiles, each consisting of 16 dual-
lows. First, in §3, we describe the MWA instrument and polarizationdipoleelementsina4×4grid.
observations in more detail. With this context provided,
weprovidetheresultsoftwoprincipalanalyses. In§4,we The data used in this work was taken with the MWA
present simulated MWA power spectra made from a sky on 23 Aug. 2013 (Julian Date 2456528) over the course
consisting of a single point source. By moving the posi- of approximately three hours from 16:47:27 to 19:58:24
tion of this source from simulation to simulation, we can UTC.Theobservationsweretakenoverafrequencyband
see the primary beam effects described above in a con- centered on 182.415 MHz, with a total bandwidth of
trolledfashion. In§5,weuseobservationsfromtheMWA 30.72 MHz divided into 24 1.28 MHz coarse channels,
to analyze these primary beam features and present the whichareeachfurtherdividedinto76840kHzfinechan-
power spectra of this data in §6. In particular, we fo- nels.
cus on the effect of subtracting sources from sidelobes The data used in this analysis span a total of 6
30 minute-long pointings, where an analog beamformer
6
steers the main lobe of the primary beam to nearly Before presenting the full analysis of this data set, we
the same sky coordinates for each pointing. The sky will first investigate the effects of the location of celes-
is then allowed to drift overhead for 30 minutes before tial emission on the cosmological power spectrum and
re-pointing. The data within each pointing are saved the wedge in particular. In this section, we will simulate
as individual “snapshot” observations, each lasting 112 visibilities for a single point source and calculate the de-
seconds, with individual integrations of 0.5 s. Figure 4 pendenceofthepowerspectrumonthesource’slocation.
shows the tile primary beam at three different beam- VisibilitiesaresimulatedusingtheFastHolographicDe-
former pointings: the beginning of the observation, a convolution (FHD) software package.39 Visibility simu-
zenith-phased pointing, and the end of the observation. lation is one of several functions in FHD; as described
Since each pointing changes the overall primary beam below, FHD also performs calibration and source sub-
traction on our actual data. As a simulator, FHD con-
structs a uv space model of the sky and integrates small
regionsoftheuvplaneusingtheholographicbeamkernel
(Morales & Matejek 2009) to create model visibilities.
For this analysis, we simulate visibilities for all the
baselines in the MWA in 768 fine frequency channels
spanning the observed 30.72 MHz frequency band. We
onlysimulateone112secondsnapshotwhentheprimary
beam is pointed at zenith (i.e. the snapshot shown in
the middle panel of Figure 4). In addition to reduc-
ing the computational demand of the simulations, using
onlyonesnapshotallowsustoseethesidelobespatterns
most clearly, since integrating over a longer amount of
time means including data when the array had a differ-
ent pointing and primary beam.
We conduct four simulations, each consisting of one
radio point source at a different location on the sky; the
locationssimulatedareshowninFigure5. Foreachsimu-
lation,theinherentfluxdensityofthesourceisincreased
relative to source D (located at zenith) by the inverse
of the primary beam response at its location. In other
words, eachsourcesimulatedhasthesameapparent flux
density. This choice places all the final power spectra on
the same scale, allowing for easier comparison.
InFigure6,weshowthe2D(k ,k )powerspectrafor
⊥ (cid:107)
each of the four simulations described above. We show
only the power spectrum from one of the two linear po-
larized dipoles of the MWA; the power spectrum for the
other polarization are quite similar. Letters correspond
to the source labels in Figure 5. To make the power
spectrum, the simulated visibilities are imaged by FHD
and then analyzed by the (cid:15)ppsilon pipeline described in
Hazeltonetal.,(inprep.).40 Formoreinformationonthe
dataproductstransferredbetweenFHDand(cid:15)ppsilon,see
Jacobs et al., (in prep.).
The effect of source position on the power spectrum
is clear and agrees with the intuition developed in §2.
Source D is located directly at zenith, with the subse-
Figure 4. Primary beam responses of the MWA tiles at several quent sources offset to higher declination (with right as-
pointings. Whitecontoursshowthebeamresponseover-plottedon cension held fixed). In Figure 6, source D exhibits no
the Haslam et al. (1982) all-sky map; contour levels are 0.01, 0.1,
wedge feature. (The power at high k values is due to
0.25,0.5,and0.75ofpeakbeamresponse. Althoughthesidelobes ⊥
move over the course of the observation, the main field of view pooruvcoverageonthesescalesandisdescribedinmore
remains relatively constant. Top: The first (earliest) pointing in detail below.) Sources C and B show a clear wedge fea-
the3hourdataanalyzedhere. Center: Thezenith-phasedpointing ture arise as the source is moved away from zenith, and
nearthecenterofthe3hours. Bottom: Thelast(latest)pointing
the power spectrum of source A — where the source is
ofthedataset.
located in the sidelobe of the primary beam — shows a
concentrationofpoweroutsidethemainfieldofview(in-
response of the instrument, the sidelobe patterns in the
dicated by the dashed black line) but inside the horizon
finalintegratedpowerspectrumwillbesmeared. Aswill
limit (solid black line). This feature is in exact accord
be shown below, however, the effects of the sidelobes
are still quite visible despite the changing primary beam
shape. 39 Source code publicly available at
https://github.com/miguelfmorales/FHD.
40 Source code publicly available at
4. PEDAGOGICALSIMULATIONS https://github.com/miguelfmorales/eppsilon.
7
A
A
B
C
D
B
Figure 5. Positions of the four sources simulated. Source loca-
tionsareinred;blackcontoursshowthe1%primarybeamlevels.
Note that there are four independent simulations, each consisting
ofonepointsourceonly. Letterscorrespondtothepowerspectra
inFigure6.
with our predictions. Simulations using sources offset in
right ascension (instead of declination) show the same
effect, as do sources with offsets in both right ascension
and declination: power moves to higher k as the source C
(cid:107)
moves further from field center.
5. DATAANALYSIS
In this section we present the full analysis of the three
hours of MWA data described in §3. The data is pro-
cessed through the same imaging and power spectrum
analyses (done by FHD and (cid:15)ppsilon, respectively) ap-
plied to the simulations. However, there are initial pre-
processing,calibration,andforegroundsubtractionsteps
applied to the data, which we describe here.
5.1. Pre-processing
Pre-processing of the data uses the custom-built Cot-
D
ter pipeline, which performs time averaging of the inte-
grations to 2 s and frequency averaging of the narrow
band channels to 80 kHz (Offringa et al. 2015). Cot-
ter also uses the aoflagger code to flag and remove
RFI (Offringa et al. 2010, 2012). Cotter also performs a
bandpass correction, removing the spectral shape within
each coarse channel as well as correcting for variations
in digital gain between the coarse channels. Finally, the
dataareconvertedfromanMWA-specificdataformatto
uvfits files.
5.2. Calibration and Imaging
After the pre-processing, data are further calibrated
and imaged using the FHD software package. FHD was Figure 6. (k⊥,k(cid:107))powerspectraofthesimulatedpointsources.
LetterscorrespondtosourcepositionsinFigure5. Thesolidblack
designed for interferometers with wide fields of view and
line shows the horizon limit; the dashed black line indicates the
direction dependent gains like the MWA and uses the main field of view. The wedge feature is absent for source D, lo-
holographic beam pattern to grid visibilities to the uv catedexactlyatzenith,andpowermoveshigherink asthesource
(cid:107)
plane. FHD also keeps track of the gridding statistics moves further from the center of the field of view. Note that the
in the uv plane to allow for full propagation of errors schematicFigure1isplottedwithlinearaxes,whereasthisFigure
uses logarithmic axes, which cause the horizon and field of view
through the image and into the power spectrum.
linestobeparallel.
8
In this analysis, we do not use FHD to perform a de- the primary beam does change with pointing (c.f. Fig-
convolution and construct a source model from the data ure 4). MWACS also avoids the Galactic plane, which
itself as was described in Sullivan et al. (2012); rather, reduces the number of sources in the model at the early
we input a catalog of point sources and use FHD to cal- and late pointings to ∼7000.
culate model visibilities. In all calculations, FHD uses a Itisalsoimportanttonotethatourskymodelassumes
simulated primary beam model including the effects of a fixed spectral index of -0.8 for each source. Although
mutual coupling between dipoles in a tile (Sutinjo et al. the actual sources on the sky will have some spectral
2015). structure, the fact that we include minimal frequency-
FHD also applies a calibration to the data, using the dependence in the model serves to strengthen the ar-
source model provided to solve for frequency-dependent, guments below: subtracting a nearly achromatic fore-
per-tile, per polarization complex gain parameters. Us- ground model removes power from chromatic (i.e. high
ing an iterative approach, we reduce the number of free k )modesofthepowerspectrum. Thisisacleardemon-
(cid:107)
parameters by averaging the calibration solutions into a stration of the inherent chromaticity of the interferome-
bandpass model that is updated on a per-pointing (i.e. ter response pattern.
30 minute) basis. Depending on the position of a tile in
the array, one of five different length cables is used to 6. POWERSPECTRA
return the signal for central processing; we find it nec- We now present the power spectra of this data gen-
essary to calculate a different bandpass model for each erated by the (cid:15)ppsilon code. With observational data,
type of cable in the system. We also fit and remove (cid:15)ppsilon empirically calculates the noise level in the vis-
a per-antenna polynomial (quadratic in amplitude, lin- ibilities and fully propagates errors in the visibilities
ear in phase) that varies on a per-snapshot (112 second) through to the 3D power spectrum. The important re-
timescale, as well as a fit for a known ripple caused by a sults here are the cylindrically averaged 2D power spec-
reflection within a 150 m cable. This particular cable is tra,showninFigure8. Inthisfigure,thelefthandpanel
notpresentinalltiles,sotherippleisonlyremovedfrom shows the power spectrum with only sources in the pri-
those which contain this cable; reflections from cables of mary lobe removed, while the center panel shows the
other lengths on other tiles appear to have much smaller power spectrum where sources are also subtracted from
amplitude, although work is in progress to remove these the sidelobes. In order to enhance the subtle difference
effects as well. betweenthetwopanels,wesubtractthepowerspectrum
For the present analysis, we image each snapshot at includingsidelobesourceremovalfromthepowerspectra
each frequency channel and make 3D image cubes in which removes only main lobe sources (i.e. we subtract
HEALPix (G´orski et al. 2005). Each snapshot cube is thecenterpanelfromtheleft-handpanel). Notethatwe
then summed in image space to make a final integrated perform the subtraction in full 3D (k ,k ,k ) space be-
x y z
cube for power spectrum analysis. forebinninginto2D(k ,k )space. Weplottheresultof
⊥ (cid:107)
thissubtractionintherighthandpanelofFigure8. Most
5.3. Foreground Subtraction
of the difference randomly fluctuates between positive
It is through FHD that model visibilities are also sub- (blue) and negative (red) values, showing no systematic
tracted from the data. We use two sets of model visibili- change of the power spectrum in these regions. How-
tiesgeneratedfromacustom-madepointsourcecatalog. ever, the consistently blue region shows that subtracting
In the main field of view, the catalog contains sources sources from the sidelobes removes a non-trivial amount
generated from FHD deconvolution outputs and an ad- of power (as much as 10% compared to the power spec-
vancedmachine-learningsourceidentifierdesignedtore- trawithnosidelobesourcesubtraction,althoughtypical
ject spurious sources (Carroll, et al., in prep.). Outside values are ∼ 1%) from the region where the sidelobe is
themainfieldofview,ourcatalogcombinessourcesfrom expected: outside the main lobe (dashed black line) but
MWA Commissioning Survey (MWACS; Hurley-Walker withinthehorizon(solidline). Sincethesizeofthemain
et al. 2014), the Culgoora catalog (Slee 1995), and the lobe is frequency and pointing dependent, the dashed
Molongolo Reference Catalog (Large et al. 1981) In one blacklineisonlyanapproximatemarker;thepowerthat
model we only include ∼ 4600 sources that fall within is removed from k modes below this line is consistent
(cid:107)
the primary lobe of the MWA beam; in the other, we with being sidelobe power from a range of frequencies
includeallsourcesoutthroughthefirstsidelobe(∼8500 and pointing centers.
sources). An image of all the sources included during Althoughnottheprimarygoalofthispaper, thereare
the zenith-phased snapshot is shown in Figure 7. There a few additional features in the power spectra that war-
are two effects that serve to limit the number of sources rant explanation.
included in our model. First, we use a primary beam
threshold cut: any sources that fall where the beam re- • The horizontal lines running across the EoR win-
sponse is less than 1% of the peak response are not in- dowaretheeffectofthecoarsechannelizationused
cluded in the model. Second, because it is a composite bytheMWA.Betweeneach1.28MHzcoarsechan-
of several surveys, the completeness of our catalog is not nel are two 80 kHz channels which are flagged due
uniform over the sky. In particular, MWACS does not to low signal response and potential aliasing con-
cover the full declination range of the observations here; cerns. This flagging in frequency has the effect
theeffectisthatfewersourcesareremovedfromthelower of introducing covariance into the line-of-sight k
(cid:107)
declinations of the southern sidelobe, and very few are modes, whichareeffectivelyproducedbyaFourier
in the northern sidelobe. This has the effect of introduc- transform of the frequency axis. This additional
ing a small time-dependence in the number of sources covariance has the effect of coupling power from
included in our model, since the declination coverage of the wedge into higher k modes. Because the flag-
(cid:107)
9
Figure 7. Thesourcesusedforcalibrationandsubtraction. Thisimageshowsthesourcepositionsduringthezenith-phasedpointing. Any
sourceswherethebeamresponseisgreaterthan1%ofthepeakvalueduringthezenithpointingareincludedinourmodel. Thesidelobes
areclearlydistinguishablefromthemainbeam. ThedeclinationrangeoftheMWACSsurveyisthe−10◦ to−55◦,whichaccountsforthe
dropinsourcedensityoutsidethisinterval.
ging is at regular intervals, this additional power noisy measurements of certain modes. Therefore,
also appears at regular intervals in k (the appear- whilethesemodesappeartohaveveryhighpower,
(cid:107)
ance of non-regular spacing comes from the loga- they also have associated very large error bars. A
rithmic scale on the y-axis). Work is underway on plotoftheerrorscalculatedby(cid:15)ppsilonfortheXX
algorithms which can reduce this covariance using polarization is shown in Figure 9.
priors on the fact that the power comes from k
(cid:107)
modes within the wedge. • There are blue/purple regions outside the wedge
which are negative. This is because (cid:15)ppsilon cross-
• Theverticallines,whichareespeciallyprevalentat multiplies the even time samples in the data set
high k⊥ modes come from the uv coverage of the withtheoddtimesamples(withthesamplesinter-
MWA.TheMWAhasexceptionallydensecoverage leavedonatimescaleoftwoseconds); thishasthe
atlowk⊥duetoitslargenumberofshortbaselines. effect of removing the positive-definite noise bias
However,athigherk ((cid:38) 10−1 hMpc−1)thereare that would result from squaring the entire data
⊥
gaps in the coverage, which results in particularly set. Alternating positive and negative values cor-
10
Main Lobe Sources Only (XX Pol) Main and Sidelobe Sources (XX Pol) Difference (XX Pol)
Main Lobe Sources Only (YY Pol) Main and Sidelobe Sources (YY Pol) Difference (YY Pol)
Figure 8. (k ,k )powerspectraofthedata. XXlinearpolarizationisonthetoprow,YYonthebottom. Thesolidblacklineshowsthe
⊥ (cid:107)
horizonlimit;thedashedblacklineindicatesthemainfieldofview. Left: Powerspectrawhereonlysourcesinthemainlobeofthebeam
are used to calibrate and then subtracted from the data. Center: Power spectra where sources in both the main lobe and the sidelobes
are used to calibrate and then subtracted. Right: The difference between the left and center plots. (Note the data are differenced in 3D
(kx,ky,kz)spaceandthenaveragedink⊥ annuli.) Althoughtheleftandcenterpanelsappearindistinguishable,subtractingthemreveals
asignificantdifferenceoutsidethefirstnulloftheprimarybeam. Theconsistentlyblueregionshowsthatremovingsourcesinthesidelobes
hasremovedpowerathighk outsidethemainfieldofview.
(cid:107)
respond to noise dominated regions. 7. DISCUSSION
Through the advances in our understanding of EoR
• Most obviously, a large amount of foreground
foregrounds (i.e. the “wedge” and “EoR window”
power remains in the power spectra. This is
paradigm), we now have a model for the detailed im-
not surprising, as our analysis only subtracted a
pact of sources far from pointing center on the recovery
few thousand point sources, ignoring diffuse emis-
of the 21 cm power spectrum. This work demonstrates
sion both from the Galaxy and unresolved point
that sources outside the main field of view are a signifi-
sources. Subtracting models of this emission will
cant contaminant of the modes of interest in the 21 cm
clearly be necessary for any possibility of recover-
power spectrum, even for an “imaging” power spectrum
ing21cmsignalfrominsidethewedge. Theeffects
analysis. It is therefore worthwhile to heuristically con-
concerning sidelobes presented in this work, how-
siderthedetailedpatternsourcesfarfrompointingcenter
ever, are still quite important: the additional frac-
leave in cylindrically averaged (k ,k ) space. While not
tion of emission removed when including sidelobe ⊥ (cid:107)
all of the conclusions below directly follow from the em-
sources is more than enough to swamp the EoR
pirical power spectra analyzed here, the formalism pre-
signal which might have a peak power spectrum
sentedin§2,thedelayspectrumanalysesinThyagarajan
brightness on order of 106 mK2h−3Mpc3.