A 21-cm power spectrum at 48 MHz, using the Owens Valley Long Wavelength Array
Hugh Garsden, Lincoln Greenhill, Gianni Bernardi, Anastasia Fialkov, Daniel C. Price, Daniel Mitchell, Jayce Dowell, Marta Spinelli, Frank K. Schinzel
MMNRAS , 1–18 (2021) Preprint 3 March 2021 Compiled using MNRAS L A TEX style file v3.0
A 21-cm power spectrum at 48 MHz, using the Owens Valley LongWavelength Array
H. Garsden , ★ , L. Greenhill , G. Bernardi , , , A. Fialkov , , D.C. Price , , D. Mitchell , J. Dowell ,M. Spinelli , and F.K. Schinzel Harvard-Smithsonian Center for Astrophysics, MS42, 60 Garden Street, Cambridge MA 02138 USA Astronomy Unit, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom INAF-Istituto di Radioastronomia, via Gobetti 101, 40129, Bologna, Italy Department of Physics & Electronics, Artillery Road, Rhodes University, Grahamstown, South Africa South African Radio Astronomy Observatory, FIR street, Observatory, Cape Town, South Africa Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia CSIRO Astronomy & Space Science, Australia Telescope National Facility, P.O. Box 76, Epping, NSW 1710, Australia Department of Physics and Astronomy, University of New Mexico, 210 Yale Blvd NE, Albuquerque, New Mexico INAF-Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, 34143 Trieste, Italy IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA;University of New Mexico
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The Large-aperture Experiment to Detect the Dark Age (LEDA) was designed to measure brightness temperature fluctuations inthe Cosmic Microwave Background due to 21-cm absorption/emission by neutral hydrogen in the early Universe. Measurementsof 21-cm fluctuations will provide constraints on astrophysical processes during Cosmic Dawn ( 𝑧 ≈ − 𝑧 ≈ Δ ( 𝑘 ) ≈ × mK in the Cosmic Dawn window. Given thata value of Δ ( 𝑘 ) ≈
100 mK is predicted from theory, we conclude that our power spectrum is dominated by telescope thermalnoise and systematic effects. By modelling the OVRO-LWA thermal noise, we show that OVRO-LWA has sufficient sensitivityfor a 21-cm detection if 3000 hrs of observations are integrated using a mix of incoherent and coherent averaging. We show thatOVRO-LWA will then be able to detect theoretically predicted Ly- 𝛼 and X-ray peaks in 21-cm power spectra at Cosmic Dawn,according to theoretical models. Key words: cosmology: observations – dark ages, reionization, first stars – techniques: interferometric; computing: GPU, HPC
Observations of Cosmic Dawn and the Epoch of Reionization (EoR)are key to unveiling properties of the first population of stars andgalaxies. The role of first galaxies in the process of reionizationcan be probed by the measurements of Gunn-Peterson troughs inthe spectra of high-redshift quasars (e.g., Fan et al. 2006; Greig et al.2017, 2019), Ly- 𝛼 emission from Lyman Break galaxies (Mason et al.2018) and the measurement of the total optical depth to Thomsonscattering of the Cosmic Microwave Background (CMB) radiation(e.g., Planck Collaboration et al. 2018). The emergent picture showsgrowing evidence of late reionization ending at 𝑧 ∼ ★ E-mail: [email protected] (HG) berger et al. 2019). However, these measurements cannot reveal muchabout other processes that occur at early times, such as the heatingof the Intergalactic Medium (IGM) by the first X-ray sources or theonset of primordial star formation in proto-galaxies.These processes can be constrained via observations of the red-shifted 21-cm line of neutral hydrogen produced at a rest-framefrequency of 1.42 GHz (e.g., Furlanetto et al. 2006; Barkana 2016),which can be observed using low frequency radio telescopes today.In addition to being sensitive to the process of reionization, the 21-cm signal depends on the temperature of the IGM (e.g., Madau et al.1997), Ly- 𝛼 radiation from stars (Wouthuysen 1952; Field 1958) andthe intensity of the radio background at 21 cm (e.g., Feng & Holder2018). Therefore, this signal can be used to constrain the process ofprimordial star and galaxy formation (e.g., probe the effect of radia- © a r X i v : . [ a s t r o - ph . C O ] M a r H. Garsden et al. tive feedback mechanisms at high redshifts, Fialkov et al. 2013), andprobe both thermal and ionization histories of the gas.The sky-averaged (global) 21-cm signal traces the mean evolutionof the Universe and can act as a cosmic clock, revealing the timing ofmajor cosmic events such as the onset of star formation. The first ten-tative detection of the global 21-cm signal at 78 MHz (correspondingto a redshift of 𝑧 ∼
17) was recently reported by the EDGES collab-oration (Bowman et al. 2018). This detection is not compatible withtheoretical predictions based on standard astrophysical and cosmo-logical assumptions (e.g., Cohen et al. 2017, explored a large gridof 21-cm global signals, broadly varying properties of astrophysicalsources within acceptable ranges), calling for alternative theories atCosmic Dawn. The proposed solutions range from models with over-cooling of gas via dark matter interactions (see Barkana 2018, andthe citation history of that work) to the existence of extra radio back-ground radiation in addition to the CMB (e.g., Bowman et al. 2018;Feng & Holder 2018), which is usually assumed to be the only sourceof background radiation. Moreover, doubts about the signal being ofcosmological origin have been expressed, with possible explanationsincluding instrumental systematic errors (Hills et al. 2018; Sims &Pober 2019; Singh & Subrahmanyan 2019), ground plane interfer-ence (Bradley et al. 2019), and polarized foregrounds (Spinelli et al.2019). Other existing global 21-cm experiments are attempting toverify this detection at high redshifts, including the Large-apertureExperiment to Detect the Dark Ages (LEDA, Bernardi et al. 2016;Price et al. 2018), and efforts in progress with experiments suchas SARAS3, REACH , EDGES Mid-Band, and MIST . At lowerredshifts only upper limits on the global 21-cm signal have beenpublished so far (Monsalve et al. 2017; Singh et al. 2017).Power spectra of 21-cm fluctuations contains information on theinitial fluctuations in density and velocity fields, spatial distributionof sources, and their spectral properties. Power spectrum measure-ments can provide an important validation of the EDGES signal andaid in its theoretical interpretation (Fialkov et al. 2018; Reis et al.2020). If the EDGES signal is imprinted by astrophysical or cosmo-logical processes at Cosmic Dawn, power spectra from 𝑧 ∼
17 couldcarry a consistent signature. So far, major experimental effort hasbeen focused on observing the EoR, with several instruments yield-ing upper limits in the range 𝑧 ∼ −
9, including LOFAR (Mertenset al. 2020), MWA (Trott et al. 2020; Barry et al. 2019; Li et al.2019), PAPER (Kolopanis et al. 2019), and GMRT (Paciga et al.2013). At the higher redshifts corresponding to Cosmic Dawn, firstattempts to measure the power spectrum have been made by Ewall-Wice et al. (2016) who, using 6 hr of data from the MWA, reporteda sensitivity of 10 mK on comoving scales 𝑘 (cid:46) . − , atredshifts 11 . (cid:46) 𝑧 (cid:46) . mK at 𝑘 (cid:46) .
10 Mpc − at 𝑧 = .
4. The AARTFAAC CosmicExplorer program (Gehlot et al. 2020), using the LOFAR telescope,reported an upper limit of (8549 mK) at 𝑘 = .
144 h cMpc − , 𝑧 ≈
18, from 2 hr of observations. Finally, 14 hours collected byLOFAR-LBA were used to obtain upper limits of 14561 mK and14886 mK at 𝑘 ∼ .
038 h Mpc − for two observed fields (3C220and NCP) at 𝑧 = . − . from a wide range of redshifts and scales, including both the EoRand Cosmic Dawn.This paper reports results from the Large-aperture Experimentto Detect the Dark Age (LEDA). LEDA is working towards a de-tection of 21-cm fluctuations in the CMB signal at Cosmic Dawn( 𝑧 (cid:46) −
40) , in both sky-averaged spectra and spatial power spec-tra, using observations from OVRO-LWA. We are currently focusedon two experiments: 1.Validation of the recent detection by Bowmanet al. (2018), and extraction of any other 21-cm absorption/emissionprofiles observed at Cosmic Dawn; 2. Generation and analysis ofpower spectra of the 21-cm signal at Cosmic Dawn, and analysis ofthe sensitivity of power spectra as a function of observation integra-tion time.A paper reporting on progress on the first experiment is underreview (Spinelli et al. 2020); this paper reports results from thesecond experiment.The first power spectrum obtained from OVRO-LWA observa-tions (Eastwood et al. 2019) was an angular power spectrum gener-ated from 28hr of integrated data using m-mode analysis (Shaw et al.2014), and removal of foregrounds using the double Karhunen-Loévetransform. In this paper we also use observations from OVRO-LWA,but generate power spectra using the “delay spectrum” method (Par-sons et al. 2012b), which produces a 2-D cylindrical power spectrumof a 3-D cosmological volume. Instead of removing foregrounds, thedelay spectrum method isolates foreground power in a “wedge” re-gion of the power spectrum, outside of which the 21-cm signal maybe detected, given enough power spectrum sensitivity. Foregroundisolation is inherent in the delay spectrum method, and relies on thespectral smoothness of foreground emission. The method also has theadvantage that it involves minimal manipulation of telescope data,and is easy to implement. It does, however, have limitations, whichwe will discuss below. We operate at a higher redshift of 𝑧 ≈ MNRAS , 1–18 (2021) Figure 1. 𝑢𝑣 -coverage of the OVRO-LWA baselines used for power spectrumgeneration, obtained from a frequency channel of width 24 kHz at a frequencyof 57 MHz. used to study Cosmic Dawn; Section 8 summarizes the results andfuture work.For this work we assume a flat universe with the following cosmo-logical parameter values: Ω 𝑀 = . , Ω Λ = . , 𝐻 = . OVRO-LWA is a wide field, low frequency, drift-scan, non-redundantimaging interferometer situated in Owens Valley, California, at a lat-itude of 37 . ◦ and longitude of − . ◦ . Its design and hard-ware components are based on that of the Long Wavelength Array(LWA) in New Mexico (Taylor et al. 2012). OVRO-LWA providesdata for research in several areas of observational astronomy and cos-mology, including: cosmic rays (Romero-Wolf et al. 2019; Carvalhoet al. 2019; Monroe et al. 2020), radio transients surveys (Andersonet al. 2019a), solar weather and coronal mass ejections (Hallinan &Anderson 2017; Chhabra et al. 2019), exoplanet searches (Ander-son & Hallinan 2017), radio counterparts to gamma ray bursts andgravitational waves (Anderson et al. 2018, 2019b), emission fromcompact binaries (Callister et al. 2019), properties of the ionosphere(Shume et al. 2017), sky surveys at low frequencies (Eastwood &Hallinan 2018), Cosmic Dawn 21-cm global signal detection (Priceet al. 2018), and Cosmic Dawn 21-cm power spectra (Eastwood et al.2019).The telescope consists of 256 dual-antenna stands, of the sametype used in the Long Wavelength Array (Taylor et al. 2012). Ofthese stands, 219 are used for this study, the others being used byother research groups. The 219 stands provide 23871 baselines, withlengths from 4.8 to 212.4 meters. The observing bandwidth is 27.384-84.912 MHz (z=15.7 to 50.9 for the 21-cm HI line). The 𝑢𝑣 -coverageof the telescope at 57 MHz is shown in Figure 2; 𝑢𝑣 lengths rangefrom 0.9 to 40.4 wavelengths. The resolution of the telescope is 1 . ◦ at 57 MHz.Signals from the antennas are processed by an FX correlator on Source Location Flux at Flux at(RA/DEC) 74 MHz (Jy) 48.324 MHz (Jy)Virgo A 12:30:49 1253.4 1689.03+12:23:281411+5212 14:11:20 120.27 162.07+52:12:041229+0202 12:29:06 149.96 202.08+02:02:561504+2600 15:04:59 129.86 174.99+26:00:460813+4813 08:13:36.34 136.40 183.80+48:13:01
Table 1.
Details of the five sources used for calibration of the OVRO-LWAobservations used for power spectrum generation. Locations and fluxes wereextracted from the VLA Low-Frequency Sky Survey at 74 MHz, and fluxesconverted to OVRO-LWA frequencies by assuming a spectral index of -0.7.The fluxes at 74 MHz and 48.324 MHz are shown. site at the Owens Valley Radio Observatory (Kocz et al. 2015). Theobserving band is split into 22 sub-bands each containing 109 chan-nels of width 24 kHz, 2398 channels in all. Each stand has a foldedcross-dipole, for observing X (east-west) and Y (north-south) polar-izations. Cross-correlation of antennas generates four polarizationproducts: XX, XY, YX, YY. Only XX and YY polarizations are usedfor our power spectrum analysis, and they are treated separately. Inpractice, some stands, baselines and channels are not used, as theymay contain bad data caused by hardware faults, known systematics,or RFI.Correlated data, i.e. visibilities, are integrated over 9s time in-tervals, each interval producing what we will refer to a single “ob-servation”. Multiple observations are typically integrated for furtherprocessing, e.g. for generating power spectra. Observations are cal-ibrated using the Real Time System (RTS), which performs self-calibration against a point source sky model (for details see Mitchellet al. 2008). The RTS uses a beam model described in LWA Memo178 . RTS contains an algorithm to detect and flag narrow-band RFIin each sub-band, by finding channels whose amplitude varies sig-nificantly from the median amplitude of the sub-band (Mitchell et al.2010). No other automated flagging is applied; further flagging ofdata is achieved by a visual inspection of antenna autocorrelations.Post-calibration quality checks (described below), are used to selectobservations to be used for science experiments. We use 4 hours of non-contiguous (in time) calibrated observations,selected from observations made during May 2018, between LST11:55 and 13:15. Of the 219 stands available in OVRO-LWA, only165 were used, the rest being flagged due to hardware issues and out-ages. We also flagged 509 baselines due to suspected cross-talk be-tween their antenna cable connections at backend electronic process-ing units . During calibration, short baselines were downweighted so M. Eastwood, personal communication. MNRAS , 1–18 (2021)
H. Garsden et al.
Figure 2.
An image of the sky generated from an OVRO-LWA observation used for power spectrum generation, at LST = 12:30. The right image is the same asthe left, but is marked with sources of interest. The 5 brightest sources (after primary beam correction) at this LST were used for calibration. They are ringedin green, and include Virgo A, the brightest source, which is at zenith. The 20 next brightest sources, after the 5 used for calibration, are ringed in yellow. Thetwo sources on the horizon, ringed in red, are intermittent RFI sources; they are possibly located in the towns of Bishop or Mammoth Lakes, both north-westof OVRO-LWA at distances of 18 and 76 km respectively. The source on the horizon at the top is Cassiopeia A, the source near the horizon at the bottom isunidentified. The images contain Stokes I observed over a bandwidth of 10.464 MHz centered at 48.324 MHz. They were generated using NRAO CASA withBriggs weighting applied (robust=0.5). To enhance the visibility of the dimmer sources, an Arcsinh color scale is used, and the image is clipped to a range of-90 to 300 Jy/beam. The image has not been CLEANed. The confusion limit for OVRO-LWA at this frequency, based on the longest baseline length of 200m, is96 Jy/beam. that any observed diffuse emission does not interfere with calibrationby RTS, which lacks a diffuse emission model.During observations, both the Sun and Galactic Center were belowthe horizon, and two strong sources, Cygnus A and Cassiopeia A,were on or below the horizon. The brightest source in the sky wasVirgo A, at an elevation of ≈ ◦ . We used five sources for calibration,one of which was Virgo A. The five sources were selected based onthe following criteria: 1. at an LST of 12:30, they are above anelevation of 30 ◦ , 2. at an LST of 12:30, they are determined by RTSto be the five brightest sources after primary beam correction. TheLST of 12:30 in these criteria was chosen as it is the approximatemiddle of the LST range of 11:55 to 13:15. All observations wereseparately calibrated using the same five sources. Source locationsand fluxes were obtained from the VLA Low-Frequency Sky Surveyat 74 MHz (Cohen et al. 2007); these are detailed in Table 1. Toobtain fluxes at frequencies other than 74 MHz we assume a spectralindex of -0.7 for all sources. It is possible that the spectrum of thesesources turns over at lower frequencies, but only ≈ .
5% of sourcesin the GLEAM catalog do so (Callingham et al. 2017), and Virgo Adoes not (de Gasperin et al. 2012).A simple method was used to select the 4 hrs of observations usedfor power spectrum generation, based on the quality of the observa-tions and their suitability for use in power spectra. The method isdescribed in Appendix B. This produced a set of observations thatare not necessarily contiguous in time, and recorded on differentdays. Figure 2 shows Stokes I images of the sky generated from oneobservation, at LST = 12:30. The sky is the same in both images,but the right image marks the calibration sources in green, and othersources of interest in red and yellow (see caption). The images wereobtained using NRAO CASA software.For power spectra we use a frequency band of 10.464 MHz cen-tered at 48.324 MHz. That band was chosen because it is less subjectto RFI and produces the highest signal-to-noise ratio over the OVRO-LWA band. It does not cover the EDGES detection frequency, but for
Property ValuePower spectrum dimensions (pixels) 102 horizontal, 108 vertical 𝑘 ⊥ range 0.001 to 0.018 Mpc − 𝑘 ⊥ resolution 0.000175 Mpc − 𝑘 (cid:107) range -1.06 to 1.04 Mpc − 𝑘 (cid:107) resolution 0.02 Mpc − Delay range -5113.64 to 5018.94 nsDelay resolution 95 nsHorizon delay at max 𝑘 ⊥
852 ns 𝑢𝑣 lengths used 1 to 33 Table 2.
Size, ranges, and resolution of the power spectrum images in the toprow of Figure 3. our noise analysis, using simulations, we will expand the frequencyrange to cover the whole OVRO-LWA band. Over the 10.464 MHzband, the 21-cm signal is observed at 𝑧 = −
31. This correspondsto a time range of 35 million years, when the Universe was 100 Myrold. Power spectra of 21-cm fluctuations should ideally be obtainedover a time range when the Universe is not significantly evolving,which is unlikely over 𝑧 = −
31, so a narrower frequency bandis more appropriate. However, that will produce a power spectrumwith a less resolution, and we prefer to maintain a higher resolutionfor analysis of our first power spectra (see Pober et al. (2013) for asimilar argument).
MNRAS000
MNRAS000 , 1–18 (2021) Figure 3.
The power spectra generated from the 4 hrs of observations described in the text, and plots of Δ ( 𝑘 ) obtained from the power spectra. The top rowshows the power spectra, the second row show the same power spectra but restricts the 𝑘 (cid:107) range to − .
31 to 0 .
31 Mpc − , so that the wedge can be seen in moredetail. Power values are colored using a log scale. The third row plots Δ ( 𝑘 ) values at various 𝑘 , obtained from within the Cosmic Dawn window of the powerspectra (the exact region used is described in Appendix A). Some important properties of the power spectrum images in the top row are listed in Table 2. Theleft column shows the power spectrum generated from polarization XX, the right column from polarization YY. OVRO-LWA observes 21-cm fluctuations at redshifted 21-cm fre-quencies, as well as foregrounds emitting at the same frequencies;foregrounds signals are dominated by Galactic synchrotron and free-free emission, and extragalactic sources. To separate the foregroundsfrom the 21-cm signal in power spectra, several methods have been used (Morales et al. 2019), which fall broadly into two categories:foreground removal, and foreground isolation. The first uses mod-els of the foregrounds and/or statistical techniques to remove fore-grounds from observational data, leaving the 21-cm signal in theresidual, from which a power spectrum can be generated. The sec-ond generates a power spectrum without foreground removal, butrelies on the spectral properties of foregrounds to isolate them in a
MNRAS , 1–18 (2021)
H. Garsden et al.
Figure 4.
Vertical and horizontal cuts through the power spectra in the top row of Figure 3. The top row panels contain 3 vertical cuts made at 𝑘 ⊥ values of 0.005,0.01, 0.15 Mpc − . The black lines indicate the location of the horizon at these 𝑘 ⊥ values. The bottom row shows a horizontal cut made at 𝑘 (cid:107) = .
53 Mpc − . Theleft column is from power spectra generated from polarization XX, the right column polarization YY.Pol XX Pol YYObserved 𝑃 ( k ) , foreground wedge 1 . × . × Observed 𝑃 ( k ) , Cosmic Dawn window 5 . × . × Ratio of 𝑃 ( k ) above 2039 2490 Δ ( 𝑘 = . ) , Cosmic Dawn window 1 . × . × Table 3.
The top two rows show the average power in the power spectragenerated from 4 hrs of observations (Figure 3), from the foreground wedge(first row), and the Cosmic Dawn window (second row). The third row showsthe ratio of the power in the foreground wedge to the power in the CosmicDawn window. The bottom row shows values of dimensionless power Δ ( 𝑘 = . ) obtained from the Cosmic Dawn window. region of the power spectrum, leaving the rest of the power spec-trum foreground free. Examples of the first are reported in Gehlotet al. (2019), Eastwood et al. (2019) and Barry et al. (2019); exam-ples of the second in Pober et al. (2013), Thyagarajan et al. (2015a),and Ewall-Wice et al. (2016). We implement foreground isolation bygenerating 2-D, cylindrically averaged, power spectra using the delayspectrum method, which isolates foregrounds in a region known asthe “wedge”. Interferometer visibilities 𝑉 ( 𝑢, 𝑣, 𝑤, 𝜈 ) are amplitudes of transverseFourier modes of the sky, at observing frequency 𝜈 , and 𝑢𝑣 plane co-ordinates ( 𝑢, 𝑣, 𝑤 ) . For the purpose of this discussion we ignore the 𝑤 term, implying observation of a “flat” sky perpendicular to the lineof sight. Because the 21-cm signal is isotropic, baseline orientationis not important, and ( 𝑢, 𝑣 ) is reduced to u = |( 𝑢, 𝑣 )| by averagingpower in baselines with the same u. Visibilities 𝑉 ( 𝑢, 𝜈 ) then form a1-D power spectrum dependent on frequency 𝜈 . If they are Fouriertransformed by frequency via: V ( u , 𝜂 ) = ∫ 𝑉 ( u , 𝜈 ) 𝑒 − 𝜋𝜈𝜂 𝑑𝜈, (1)the resulting u , 𝜂 are the wavenumbers of Fourier modes perpendic-ular to, and parallel to, the line of sight. By converting them to unitsMpc − , they become the wavenumbers 𝑘 ⊥ , 𝑘 (cid:107) of Fourier modes ofa 2-D Fourier transform of a cosmological volume. Spectral power 𝑃 is generated by squaring V and converting to units of mK Mpc (more detail below). The result is a 2-D power spectrum 𝑃 ( 𝑘 ⊥ , 𝑘 (cid:107) ) or 𝑃 ( k ) . In practice, the transform in Equation 1 is difficult to implement,because 𝑉 ( u , 𝜈 ) do not lie over a regular grid. To make the Fouriertransform, one may, for example, re-grid the visibilities, use non-equispaced Fourier transforms, least square spectral analysis (Gehlotet al. 2019), or periodgrams (Barry et al. 2019). The delay spectrummethod takes a different approach, and approximates Equation 1 bygenerating a delay spectrum (Morales & Hewitt 2004; Parsons &Backer 2009; Parsons et al. 2012b) from every baseline. The delay MNRAS000
The top two rows show the average power in the power spectragenerated from 4 hrs of observations (Figure 3), from the foreground wedge(first row), and the Cosmic Dawn window (second row). The third row showsthe ratio of the power in the foreground wedge to the power in the CosmicDawn window. The bottom row shows values of dimensionless power Δ ( 𝑘 = . ) obtained from the Cosmic Dawn window. region of the power spectrum, leaving the rest of the power spec-trum foreground free. Examples of the first are reported in Gehlotet al. (2019), Eastwood et al. (2019) and Barry et al. (2019); exam-ples of the second in Pober et al. (2013), Thyagarajan et al. (2015a),and Ewall-Wice et al. (2016). We implement foreground isolation bygenerating 2-D, cylindrically averaged, power spectra using the delayspectrum method, which isolates foregrounds in a region known asthe “wedge”. Interferometer visibilities 𝑉 ( 𝑢, 𝑣, 𝑤, 𝜈 ) are amplitudes of transverseFourier modes of the sky, at observing frequency 𝜈 , and 𝑢𝑣 plane co-ordinates ( 𝑢, 𝑣, 𝑤 ) . For the purpose of this discussion we ignore the 𝑤 term, implying observation of a “flat” sky perpendicular to the lineof sight. Because the 21-cm signal is isotropic, baseline orientationis not important, and ( 𝑢, 𝑣 ) is reduced to u = |( 𝑢, 𝑣 )| by averagingpower in baselines with the same u. Visibilities 𝑉 ( 𝑢, 𝜈 ) then form a1-D power spectrum dependent on frequency 𝜈 . If they are Fouriertransformed by frequency via: V ( u , 𝜂 ) = ∫ 𝑉 ( u , 𝜈 ) 𝑒 − 𝜋𝜈𝜂 𝑑𝜈, (1)the resulting u , 𝜂 are the wavenumbers of Fourier modes perpendic-ular to, and parallel to, the line of sight. By converting them to unitsMpc − , they become the wavenumbers 𝑘 ⊥ , 𝑘 (cid:107) of Fourier modes ofa 2-D Fourier transform of a cosmological volume. Spectral power 𝑃 is generated by squaring V and converting to units of mK Mpc (more detail below). The result is a 2-D power spectrum 𝑃 ( 𝑘 ⊥ , 𝑘 (cid:107) ) or 𝑃 ( k ) . In practice, the transform in Equation 1 is difficult to implement,because 𝑉 ( u , 𝜈 ) do not lie over a regular grid. To make the Fouriertransform, one may, for example, re-grid the visibilities, use non-equispaced Fourier transforms, least square spectral analysis (Gehlotet al. 2019), or periodgrams (Barry et al. 2019). The delay spectrummethod takes a different approach, and approximates Equation 1 bygenerating a delay spectrum (Morales & Hewitt 2004; Parsons &Backer 2009; Parsons et al. 2012b) from every baseline. The delay MNRAS000 , 1–18 (2021) spectrum is defined as: V 𝑏 ( 𝜏 ) = ∫ 𝑉 𝑏 ( 𝜈 ) 𝑒 − 𝜋𝜈𝜏 𝑑𝜈, (2)where 𝑏 indicates that 𝑉 𝑏 ( 𝜈 ) are visibilities for a baseline, and 𝜏 , theFourier conjugate of 𝜈 in this case, is the delay between antennas ofemission from foreground sources. A source in direction s relative toa baseline with vector b , has delay 𝜏 = b · s 𝑐 . (3)Sources at zenith have zero delay, and sources on the horizon havedelay ± 𝜏 𝐻 , which varies by baseline length, and is the maximumpossible delay for a baseline. For “ wedge” power spectra, the rangeof frequencies 𝜈 is chosen so that 𝜏 in Equation 2 extends beyond ± 𝜏 𝐻 . Any foreground emission present in 𝑉 𝑏 ( 𝜈 ) must appear atdelays | 𝜏 | < | 𝜏 𝐻 | , with the Cosmic Dawn window lying outside thisrange.The key difference between Equations 1 and 2 is that for Equa-tion 2, the Fourier transform is made on visibilities recorded at thesame baseline length, and for Equation 1, the transform is made onvisibilities recorded at the same 𝑘 ⊥ . Equation 1 is strictly correctfor a spatial Fourier transform, but Equation 2 is easy to implement,and results in foreground emission being confined to a region of theFourier transform centered around the DC component ( 𝜏 = 𝑘 ⊥ wavenumber can be assigned to each baseline, derived from thevalue of u for the centre frequency of the observing band, and the 𝑘 (cid:107) wavenumbers can be approximated from 𝜏 .A 2-D Fourier transform is produced by binning the delay spectrafor all baselines in order of increasing 𝑘 ⊥ , producing a 2-D powerspectrum that approximates 𝑃 ( k ) . The horizon delay of baselinesincreases with increasing 𝑘 ⊥ , producing foreground isolation withinan expanding horizon, or “wedge” within the power spectrum. Theoutside of the wedge is used for detection and measurement of the21-cm signal.Delay spectrum visibilities V are converted to power using 𝑃 ( k ) ≈ |V| (cid:18) 𝜆 𝑘 𝐵 (cid:19) 𝑋 𝑌 Ω 𝑝 𝑝 𝐵 𝑝 𝑝 (4)(Parsons et al. 2012a, 2014; Parsons 2017), where 𝜆 is the mean wave-length of the band, 𝑘 𝐵 is Boltzmann’s constant, 𝑋 𝑌 converts anglesand frequency intervals to comoving distance, Ω 𝑝 𝑝 is the integralover the power-squared beam, and 𝐵 𝑝 𝑝 is the effective bandwidth. 𝐵 𝑝 𝑝 = ∫ | 𝑤 | 𝑑𝜈 , where 𝑤 is the window function applied to thedelay transform. A window function is used to avoid spectral leak-age produced by taking the Fourier transform of a finite length ofnon-periodic data. We use a Blackman-Harris window, giving 𝐵 𝑝 𝑝 = 2 .
72 MHz for an observed bandwith of 10.464 MHz. We use abeam model contained within the LWA Software Library to ob-tain Ω 𝑝 𝑝 = . 𝑋 𝑌 is derived from cosmological distances usingHogg (1999), and has value of 4 × Mpc /[ str ∗ GHz ] . The unitsof 𝑃 ( k ) are mK Mpc . 𝑃 ( k ) is often converted to dimensionless power Δ ( 𝑘 ) = 𝑘 𝜋 𝑃 ( k ) , (5)where 𝑘 = √︃ 𝑘 ⊥ + 𝑘 (cid:107) , and Δ ( 𝑘 ) has units of mK . We will use bothpower values in this paper. https://fornax.phys.unm.edu/lwa/trac/wiki The use of the delay spectrum method has limitations. Ignoring the 𝑤 co-ordinate in ( 𝑢, 𝑣, 𝑤 ) and assuming a flat sky, is not strictly validfor a wide-field instrument such as OVRO-LWA. The delay spectrummethod works best for narrow-beam, short- baseline, interferometers(Parsons et al. 2012b), and Liu et al. (2014, Eqn. 9) have quantifiedthese limitations as satisfying 𝑏𝜃 (cid:28) 𝑐𝐵 𝑏𝑎𝑛𝑑 , (6)where 𝑏 is the length of the baseline, 𝜃 is the width of the telescopeprimary beam (1.8 radians), and 𝐵 𝑏𝑎𝑛𝑑 is the bandwidth over whichthe power spectrum is made. Interpreting “ (cid:28) ” as “one-half”, Equa-tion 6 is violated by 99.66% of the OVRO-LWA baselines used forour power spectrum generation, indicating that the baselines are toolong and/or the bandwidth too wide. Reducing the maximum base-line length will reduce the 𝑘 ⊥ range, reducing the bandwidth willreduce the 𝑘 (cid:107) range. For the power spectra reported in this work, weignore these restrictions, so as to obtain a power spectrum with anextent that allows for some analysis. We will deal with them in futurework (see Section 8). Before generating a delay spectrum from each baseline, flagged chan-nels must be dealt with. Flagged channels represent missing valuesin the array 𝑉 𝑏 ( 𝜈 ) (Equation 2), which produce a delay spectrumconvolved with a PSF, in the same way that missing visibilities inan interferometric observation produce an image convolved with aPSF. The PSF in the delay transform can be deconvolved using thesame CLEAN algorithm (Högbom 1974) that is applied to images.We use an implementation of CLEAN developed by the HERA Con-sortium . CLEAN is applied to the delay spectrum in between thehorizon delays ( − 𝜏 𝐻 , 𝜏 𝐻 ) , including a buffer of 95 ns delay outsidethe horizon at − 𝜏 𝐻 and 𝜏 𝐻 . A buffer width of 𝜏 =
95 ns was chosenbecause that is the resolution of 𝜏 in the delay spectrum of OVRO-LWA baselines. CLEAN is an iterative algorithm, and we direct itto stop when the CLEAN residual varies from the previous CLEANresidual by a factor of less than 10 − . When CLEAN has finished, thearray 𝑉 𝑏 ( 𝜈 ) contains no missing values. A Blackman-Harris windowis applied to the delay spectrum as it is being cleaned.After CLEAN, the visibilities in 𝑉 𝑏 ( 𝜈 ) are averaged by groups of4, to reduce noise. This reduces the frequency resolution by a factorof 4. The bandwidth of 10.464 MHz, which initially contains 436channels of width 24 kHz, is thus transformed to 109 channels ofwidth 96 kHz.After applying the delay transform to baselines within an obser-vation, the baselines may be converted to power using Equation 4.However, we follow Pober et al. (2013) and multiply observing-timeadjacent (i.e. LST adjacent) pairs of observations to reduce noise.Given observations A, B, C, D, . . . ordered by LST, we produceA × B ∗ , C × D ∗ , . . . . Although our observations are not real-timecontiguous, they are closely separated by LST. The LSTs of each mul-tiplied pair must be within the time smearing limits for OVRO-LWA(see Appendix C), since they are being coherently combined via themultiplication. Fortunately, this applies to 99.9% of the observationscomprising the 4 hrs. The multiplication achieves noise reductionbecause visibilities recorded within the time-smearing limit shouldbe similar apart from noise, and their (complex) product will includetheir average power plus cross-power/noise terms and cross-noise pspec_prep.py , https://github.com/HERA-TeamMNRAS , 1–18 (2021) H. Garsden et al.
Figure 5.
The power spectrum for XX polarization, (Figure 3, top left image)overlaid with features of interest. Regions 1 and 2 are regions of spillover inthe Cosmic Dawn window, and the effect of bandpass ripples is also indicated. terms. When baseline delay spectra are binned, i.e. averaged, to forma power spectrum, these noise terms will integrate down.
Figure 3 shows the power spectrum generated from the 4 hrs ofobservations described in section 3, for polarizations XX and YY. Thepositive and negative 𝑘 (cid:107) values on the y-axis correspond to positiveand negative modes of the delay (Fourier) transform. Baselines arebinned, i.e.averaged, along the 𝑘 ⊥ axis, using a bin size of 0.000175Mpc − , being the 𝑘 ⊥ of a baseline with u = . 𝑘 (cid:107) axis. The blue lines arethe horizon delay, which increases with increasing 𝑘 ⊥ , producing the“wedge” within which foreground emission is isolated. As explainedin section 4.1, 𝑘 (cid:107) wavenumbers are approximated from the delays 𝜏 of the delay transform; these delays are plotted on the y-axis on theright. 𝑘 ⊥ values correlate with baseline 𝑢𝑣 lengths, which are shownon the top x-axis.The top row of Figure 3 displays the power spectra using thecomplete range of 𝑘 (cid:107) values derived from the observing frequencyband (10.464 MHz); the second row is the same power spectrum butrestricts the 𝑘 (cid:107) range to ± . − so that the wedge can be seenin more detail. A summary of the important properties of the powerspectrum images in the top row of Figure 3 is in Table 2. The thirdrow of Figure 3 shows the dimensionless power Δ ( 𝑘 ) extracted fromthe Cosmic Dawn window, at various values of 𝑘 (the algorithm forextracting Δ ( 𝑘 ) is described in Appendix A).Figure 4 shows vertical and horizontal cuts through the powerspectra. The top row depicts vertical cuts at three 𝑘 ⊥ values of 0.005,0.01, 0.015, with 𝑃 ( k ) on the y-axis and 𝑘 (cid:107) on the x-axis. Thelocation of the horizon at the different 𝑘 ⊥ values is shown by thegray vertical lines. The bottom row of Figure 4 depicts a horizontalcut through the power spectra, this time with 𝑘 ⊥ on the x-axis, at aconstant 𝑘 (cid:107) = .
53. These cuts show that the power drops by a factorof 10 as 𝑘 (cid:107) moves away from 0, and that the power in the CosmicDawn window is fairly flat by 𝑘 ⊥ . The average power levels 𝑃 ( k ) Pol XX Pol YYReal values 1313.0 1543.3Imag values 1312.8 1543.1
Table 4.
The standard deviations of the distributions of noise values (Figure5) in the 4hr observation, measured by visibility differencing. The units areJy. Noise power in 4 hrs of observationsPol XX Pol YY 𝑃 ( k ) . × . × Δ ( 𝑘 = . ) . × . × Noise power in 4 hrs of observations whenpartial coherent integration is usedPol XX Pol YY 𝑃 ( k ) . × . × Δ ( 𝑘 = . ) . × . × Table 5.
Values of 𝑃 ( k ) and Δ ( 𝑘 ) extracted from simulated noise-onlypower spectra. The top two data rows are obtained from the power spectrain Figure 5, which simulates the noise in a power spectrum generated from4 hrs of incoherently integrated observations. The bottom two data rowsare obtained from Figure 5, which simulates the noise in a power spectrumgenerated from 4 hrs of partially coherently integrated observations (theintegration method is described in section 6.3). and Δ ( 𝑘 ) within the foreground wedge and Cosmic Dawn windoware listed in Table 3. Figure 3 demonstrates that wedge-type power spectra can be gener-ated from OVRO-LWA observations. The power spectra show goodisolation of foregrounds within the horizon, although there is somespillover. Our power values are high compared to those reported fromother telescopes, and compared to those from OVRO-LWA reportedby Eastwood et al. (2019), who obtained a value of Δ ( 𝑘 ) ≈ mK at 𝑘 = . − , compared to our value of 10 mK at thesame 𝑘 . Other experiments also report lower power; a summary ofthese is shown in Table 7. Note that other experiments were using dif-ferent observing frequencies and total observing time, and differenttelescopes with different properties. One reason for the higher powerthat we obtain is due to our lower observing frequency. At a lowerfrequency, foreground emission is brighter (Spinelli et al. 2020), sothat systematics and foreground leakage will raise the level of thepower spectrum generally.A visual inspection of the power spectra shows that horizonspillover can be separated into two distinct regions, based on theamplitude of the spillover and its extent beyond the horizon. Theregions are depicted in Figure 5. Both regions exist at all 𝑘 ⊥ values.Region 1 is bounded by the horizon and the dashed line; it extendsfor approximately Δ 𝑘 = .
04 Mpc − , 𝜏 =
192 ns, beyond the hori-zon, and is commonly seen in other power spectra (Pober et al. 2013;Thyagarajan et al. 2015a,b). Spillover in Region 1 could be due to thewindow function applied to the delay transform, beam chromaticity,
MNRAS000
MNRAS000 , 1–18 (2021) Figure 6.
Histograms of noise values ( Δ 𝑉 ) obtained from differencing visibilities in adjacent frequency channels, over all observations used to generate powerspectra. These represent the distribution of thermal noise in OVRO-LWA. The top row is obtained from the XX polarization visibilities, the bottom row is frompolarization YY. The real and imaginary values of the complex noise values are shown separately; the real values in the left column, and imaginary values in theright. intrinsic spectral structure in foreground emission, RFI, and calibra-tion errors (Kern et al. 2020b) . The use of a window function onthe delay transform is essential to avoid spectral leakage due to thenon-periodic nature of the baseline visibilities. The kernel width ofthe Blackman Harris window we apply to our delay transforms is 288ns or Δ 𝑘 = .
06, indicating it is a substantial contributor to Region1 spillover.Region 2 extends from Region 1 to the dashed-dot line; it extendsfor Δ 𝑘 = .
12 Mpc − , 𝜏 =
577 ns beyond Region 1. The reason forthe spillover in Region 2 is not known but we suggest it is due to fore-ground power scattered by calibration errors. Calibration errors canbe caused by many factors including unmodelled diffuse emission,an inadequate beam model, RFI, an inadequate source model, cablereflections, cross-coupling and mutual coupling, which we discussnext.Models of point sources are incomplete or approximate at OVRO-LWA frequencies. We generated calibration models using the VLALow-Frequency Sky Survey at 74 MHz, calculating source fluxesat 48 MHz by assuming a spectral index of -0.7, and assuming thelocation of emission at 48 MHz is the same as it is for 74 MHz.These assumptions likely lead to inaccurate models. Diffuse emissionalso exists within our observations, but we have not modelled itfor calibration. Using more accurate source models, and including diffuse emission using a Global Sky Model (de Oliveira-Costa et al.2008; Dowell et al. 2017), will improve calibration.Cross-coupling, or cross-talk, occurs when electronic componentsare close enough for them to transfer signal from one to the other.Cross-coupling exists in OVRO-LWA where antenna cable connec-tions are in close proximity as cables enter backend hardware pro-cessing units. It is known to impact 509 baselines, and these areignored when generating power spectra, but the 509 baselines maynot constitute the complete set, and more detailed investigation isneeded.Mutual coupling is the electromagnetic interaction between neigh-boring antennas, which affects antenna efficiency and radiation pat-terns (when compared to an antenna in isolation). A study of mutualcoupling in the LWA reported that it exists, but has no consistentpositive or negative effect (Ellingson 2011). Coupling between theX and Y dipoles on the same OVRO-LWA stand is also possible, re-sulting in polarization leakage. Again, more investigation is neededfor OVRO-LWA.Electrical signals travelling via cable from an antenna to back-end components may reflect from those components, introducingsinusoidal structure in the antenna frequency response. These areclearly evident for some OVRO-LWA antennas. For the most part,cross-correlations between antennas (i.e. visibilities) do not suf-
MNRAS , 1–18 (2021) H. Garsden et al.
Figure 7.
Noise-only power spectra generated from 4 hrs of simulated noise-only observations, and Δ ( 𝑘 ) values obtained from them. The visibilities in the4hr observations are replaced with noise values drawn from a Gaussian distribution with 0 mean, and standard deviation obtained from the histograms ofOVRO-LWA thermal noise in Figure 5. These simulated noise-only observations are used to generate the power spectra shown. Different distributions are usedfor XX and YY polarization noise visibilities, producing separate power spectra for XX polarization (left column) and YY polarization (right column). The toprow shows the power spectra, the bottom row plots Δ ( 𝑘 ) values obtained from Cosmic Dawn window within each power spectrum (the regions from which thevalues are obtained are described in Appendix A). fer this systematic as the noise waves along a pair of cables arenot correlated. However, we have found that cable reflections dohave a slight impact on our power spectrum. In Figure 3, there isa slight change in power noticeable as a faint line that begins at 𝑘 ⊥ = , 𝑘 (cid:107) = 𝑘 ⊥ = . , 𝑘 (cid:107) = .
18 ("Bandpass ripple" in Figure 5).The location of this feature correlates with ripples in the frequencyresponse of some antennas.RFI flaggers such as AOFlagger (Offringa et al. 2012) will providebetter RFI detection compared to the simple algorithm implementedwithin RTS. These will be implemented in the future within theOVRO-LWA pipeline. More sophisticated beam models can be gen-erated through the use of simulation packages (e.g. FEKO , NEC ,CST ), direct measurements of the radiation field of the antenna (forexample by using drones (Üstüner et al. 2014) or satellites (Nebenet al. 2015)), or by observing radio sources transiting through thefringes of the interferometer (Parsons et al. 2016). Beam measure-ments will be undertaken in the future.The issues discussed above can be tackled directly, but they canalso be mitigated by more sophisticated data processing. Methods for https://altairhyperworks.com/product/Feko/Applications-Antenna-Design https://uk.mathworks.com/products/connections/product_detail/cst-microwave-studio.html the detection, analysis, and removal of systematics in power spectraare being developed by other groups and reported in the literature.For example, Kern et al. (2020b) demonstrates the usefulness ofanalysing antenna-based gains in delay and fringe-rate space, usinggain-smoothing to suppress delay modes that appear to be abnor-mal, and using temporal analysis and temporal smoothing of gainsusing autocorrelations. Cross-coupling can be removed by applyinga high-pass filter in fringe rate space (Kern et al. 2020a). Windowfunctions can be analysed for their contribution to horizon spillover,and alternative window functions selected (Lanman et al. 2020). Useof these and other methods will improve the quality of power spectragenerated from OVRO-LWA. We now turn to the analysis of the sensitivity of power spectra gen-erated from OVRO-LWA using the delay spectrum method, and in-vestigate the sensitivity that can be achieved using a large number ofobservations and a different integration strategy.We determine the level of thermal noise in OVRO-LWA, and usesimulations to generate power spectra containing only noise; theseallow us to find the noise floor in our power spectra from observations(Figure 3). We then investigate the noise floor in power spectra gen-erated from observations that are partly coherently integrated beforepower spectrum generation. All the power spectra discussed so far
MNRAS000
MNRAS000 , 1–18 (2021) Figure 8.
Simulated noise-only power spectra, the same as for Figure 5, but some observations have been coherently integrated before power spectrum generation,as described in the text (Section 6.3).
LST LST + t A LST LST + t A LST N LST N + t A ... Day 1
LST LST + t A LST LST + t A LST N LST N + t A ... Day 2
LST LST + t A LST LST + t A LST N LST N + t A ... Day D ...
Symbols
Observe continuouslyfor time t A Coherently integrate allobservations within the box
Interval 1Interval 2Interval N ...
Figure 9.
A scheme for making observations, with a drift-scan interferometer, that can be coherently integrated. Observe for several short time intervals eachday (
𝐿𝑆𝑇 𝑖 → 𝐿𝑆𝑇 𝑖 + 𝑡 𝐴 ), repeat the observations over several days, and coherently integrate each interval across all the days. The time interval 𝑡 𝐴 is thetime-averaging limit. MNRAS , 1–18 (2021) H. Garsden et al. have been generated using incoherent integration. For our purposes,incoherent and coherent integration are defined as follows: • Incoherent integration:
Convert all observations to power,then average them. • Coherent integration:
Average all the observations, then con-vert the result to power.Coherent integration will treat each observation as a 2-D matrix ofvisibilities indexed by baseline and frequency, and calculate a matrixaverage of many observations. Coherent integration of observationsshould produce substantial noise reduction, because the noise invisibilities are directly summed, rather than summing the mixedpower and noise terms as is done for incoherent integration (Section4.3). Hence the noise floor in power spectra will be much lower,increasing the possibility of a 21-cm signal detection.
The noise in our observations is extracted from observed visibilitiesusing a simple data-differencing technique (Bernardi et al. (2010);Chatterjee & Bharadwaj (2019)) that subtracts visibilities in adjacentfrequency channels. We assume that subtracting visibilities at slightlydifferent frequencies will subtract out the sky signal and systematics,leaving only subtracted noise values. Even if these values do containresidual sky signal and systematics, we assume they are an upperlimit on the thermal noise level, and so we lose nothing by usingthem as “noise” values. From the distribution of these noise valueswe obtain the statistical properties of the noise, from which we cangenerate noise-only power spectra using simulations. The noise levelin these power spectra is interpreted as the level of noise that wouldbe present in power spectra generated from telescope observations.For visibility differencing, we use all the visibilities in all the cali-brated observations used for the power spectra in Figure 3, ignoringflagged visibilities. Figure 5 show histograms of the values obtainedfrom visibility differencing, for XX and YY polarizations, and forthe real and imaginary parts of the values. The distributions are veryclose to Gaussian, with a zero mean, giving us confidence that wehave in fact extracted mostly thermal noise. The standard deviations 𝜎 of the histograms are listed in Table 4, from which we can obtainthe standard deviation of the noise. If subtracted noise values havestandard deviation 𝜎 , then the noise values have standard deviation 𝜎 /√
2, because variances add when two distributions are added orsubtracted.
We simulate noise-only observations by replacing all the visibilitiesin all the 4 hrs of telescope observations with noise values drawnfrom a Gaussian distribution with zero mean and standard deviationlisted in Table 4. Baselines that were flagged in the 4 hrs of telescopeobservations are flagged in the 4 hrs of simulated noise-only obser-vations, but flagged channels are filled in with a noise value. A powerspectrum is generated from the 4 hrs of simulated noise-only obser-vations; these are shown in Figure 5, and the power values extractedfrom them are listed in Table 5. The power spectra generated from 4hrs of telescope observations have a value of 𝑃 ( k ) ≈ × in theCosmic Dawn window (Table 3), but the noise simulation indicatesthat the noise level is 𝑃 ( k ) ≈ × , demonstrating that there aresystematics in the observed data. All the power spectra presented so far were generated using inco-herent integration of observations. We now turn to experiments withcoherent integration. For a drift-scan instrument, a given baseline can only be integratedfor a limited period of time before the rotation of the sky causessmearing effects that decrease sensitivity. We refer to this limit as thetime-averaging limit 𝑡 𝐴 ; it is longer than the 9s interval that produceseach OVRO-LWA observation (and hence each simulated noise-onlyobservation), so multiple time-contiguous observations may be inte-grated, if their time range is less than 𝑡 𝐴 . The time-averaging limitfor OVRO-LWA is defined and calculated in Appendix C.Observations made at the same LST on different days may alsobe coherently integrated, since they observe the same sky (assuminglocal bodies such as the Sun are always absent). This, along with thetime-averaging limit 𝑡 𝐴 , means that observations recorded duringthe interval 𝐿𝑆𝑇 → 𝐿𝑆𝑇 + 𝑡 𝐴 , even on different days, may all becoherently integrated, producing a single integrated observation. If 𝑁 different intervals are used within a day, i.e. 𝐿𝑆𝑇 𝑖 → 𝐿𝑆𝑇 𝑖 + 𝑡 𝐴 ( 𝑖 = . . . 𝑁 ), repeated over many days, then 𝑁 integrated observationscan produced. A depiction of the scheme is shown in Figure 9. The 𝑁 observations that result from this procedure cannot be coherentlycombined, and are therefore incoherently integrated to form a powerspectrum, using the delay spectrum method. We refer to this methodas “partial coherent integration”.We note that it may be possible to apply this mix of incoher-ent/coherent integration to the 4 hrs of telescope observations thatwere used to generate the power spectra in Figure 3, but it depends onthe LSTs of the observations, as they must fit into appropriate timeintervals as described above. An investigation of this possibility, andpublication of power spectra that may result, is reserved for futurework.The number 𝑁 , the time-averaging limit 𝑡 𝐴 , and the number ofdays 𝐷 over which the observations are repeated, determine the totalobserving time 𝑇 = 𝑁 × 𝐷 × 𝑡 𝐴 . 𝑡 𝐴 is fixed by observing frequency,but 𝑁 and 𝐷 may be varied, for the same 𝑇 . For the purpose ofreducing noise, it is desirable to have 𝑁 as small possible, whichresults in 𝐷 being large, and thus an observing schedule that couldstretch over many years. A trade-off between 𝑁 and 𝐷 is necessary,depending on what value of 𝐷 is practical.To demonstrate the use of the partial coherent integration method,and the noise reduction that can be achieved, we use the 4 hrs ofsimulated noise-only observations from the previous section, alteringthe observation LSTs so we can apply the method with 𝑁 = 𝐷 = 𝑡 𝐴 = 𝑇 = × × = Δ ( 𝑘 ) extracted from the power spectra. Theseshow that we can reduce the noise level in the power spectrum by afactor of ≈ , when partial coherent integration is used. MNRAS000
We simulate noise-only observations by replacing all the visibilitiesin all the 4 hrs of telescope observations with noise values drawnfrom a Gaussian distribution with zero mean and standard deviationlisted in Table 4. Baselines that were flagged in the 4 hrs of telescopeobservations are flagged in the 4 hrs of simulated noise-only obser-vations, but flagged channels are filled in with a noise value. A powerspectrum is generated from the 4 hrs of simulated noise-only obser-vations; these are shown in Figure 5, and the power values extractedfrom them are listed in Table 5. The power spectra generated from 4hrs of telescope observations have a value of 𝑃 ( k ) ≈ × in theCosmic Dawn window (Table 3), but the noise simulation indicatesthat the noise level is 𝑃 ( k ) ≈ × , demonstrating that there aresystematics in the observed data. All the power spectra presented so far were generated using inco-herent integration of observations. We now turn to experiments withcoherent integration. For a drift-scan instrument, a given baseline can only be integratedfor a limited period of time before the rotation of the sky causessmearing effects that decrease sensitivity. We refer to this limit as thetime-averaging limit 𝑡 𝐴 ; it is longer than the 9s interval that produceseach OVRO-LWA observation (and hence each simulated noise-onlyobservation), so multiple time-contiguous observations may be inte-grated, if their time range is less than 𝑡 𝐴 . The time-averaging limitfor OVRO-LWA is defined and calculated in Appendix C.Observations made at the same LST on different days may alsobe coherently integrated, since they observe the same sky (assuminglocal bodies such as the Sun are always absent). This, along with thetime-averaging limit 𝑡 𝐴 , means that observations recorded duringthe interval 𝐿𝑆𝑇 → 𝐿𝑆𝑇 + 𝑡 𝐴 , even on different days, may all becoherently integrated, producing a single integrated observation. If 𝑁 different intervals are used within a day, i.e. 𝐿𝑆𝑇 𝑖 → 𝐿𝑆𝑇 𝑖 + 𝑡 𝐴 ( 𝑖 = . . . 𝑁 ), repeated over many days, then 𝑁 integrated observationscan produced. A depiction of the scheme is shown in Figure 9. The 𝑁 observations that result from this procedure cannot be coherentlycombined, and are therefore incoherently integrated to form a powerspectrum, using the delay spectrum method. We refer to this methodas “partial coherent integration”.We note that it may be possible to apply this mix of incoher-ent/coherent integration to the 4 hrs of telescope observations thatwere used to generate the power spectra in Figure 3, but it depends onthe LSTs of the observations, as they must fit into appropriate timeintervals as described above. An investigation of this possibility, andpublication of power spectra that may result, is reserved for futurework.The number 𝑁 , the time-averaging limit 𝑡 𝐴 , and the number ofdays 𝐷 over which the observations are repeated, determine the totalobserving time 𝑇 = 𝑁 × 𝐷 × 𝑡 𝐴 . 𝑡 𝐴 is fixed by observing frequency,but 𝑁 and 𝐷 may be varied, for the same 𝑇 . For the purpose ofreducing noise, it is desirable to have 𝑁 as small possible, whichresults in 𝐷 being large, and thus an observing schedule that couldstretch over many years. A trade-off between 𝑁 and 𝐷 is necessary,depending on what value of 𝐷 is practical.To demonstrate the use of the partial coherent integration method,and the noise reduction that can be achieved, we use the 4 hrs ofsimulated noise-only observations from the previous section, alteringthe observation LSTs so we can apply the method with 𝑁 = 𝐷 = 𝑡 𝐴 = 𝑇 = × × = Δ ( 𝑘 ) extracted from the power spectra. Theseshow that we can reduce the noise level in the power spectrum by afactor of ≈ , when partial coherent integration is used. MNRAS000 , 1–18 (2021) Following on from the previous experiment, we investigate the noiselevel that may be achieved using a much larger total observing time 𝑇 , and whether a detection of the 21-cm signal could be made inthat case. We use the full OVRO-LWA band, dividing it into 21 sub-bands, each of width of 5.232 MHz. Each sub-band overlaps the nextby 2.616 MHz.A different value for the standard deviation 𝜎 of the noise distri-bution is required for each sub-band, because they are at differentfrequencies. The sub-band 𝜎 are estimated from the 𝜎 obtained at48.324 Hz (Section 6.1), using the following method. Since noisevalues were derived from observed visibilities, we assume they willscale by a power law with spectral index -0.7, as was assumed for thesource fluxes in Section 3. Therefore, 𝜎 𝜈 for the noise distribution atfrequency 𝜈 is calculated as: 𝜎 𝜈 = 𝜎 . ∗ ( 𝜈 / . × ) − . . (7)We fix the total observing time to 3000 hrs, and apply the partialincoherent integration method with number of days 𝐷 = 𝑡 𝐴 varies by frequency, 𝑁 must nowbe different for each sub-band, so that each sub-band accumulates 4hrs per day. At 48 .
342 Mhz, with 𝑡 𝐴 = 𝑡 𝐴 = ×
750 (days) × = .For each sub-band, after coherent integration is applied, 𝑁 𝑏 inte-grated noise-only observations are produced, where 𝑏 indicates that 𝑁 varies by sub-band. A power spectrum is generated from the 𝑁 𝑏 observations for each sub-band, and the value of Δ ( 𝑘 ) at 𝑘 = . Δ ( 𝑘 ) noise level for a range of redshifts,which can be compared against expected power based on theoreticalmodels of the cosmological 21-cm signal, in the next section.Generating 3000 hours of noise-only observations is very computeintensive, and generates 3.5 PiB of data, so we implement it in aGPGPU. Since the observations all contain the same number ofbaselines and channels (none are flagged), a GPGPU thread is createdfor every channel in every baseline, and each thread generates andmanipulates noise values “on the fly”, avoiding the need to storethe values. A full description of the GPU implementation will bepublished elsewhere. Being a high-redshift instrument, OVRO-LWA is by design observingthe 21-cm signal from the epoch of primordial star and black holeformation. Thus, observations with this instrument could be usedto infer properties of the first sources as well as to constrain exoticphysics at Cosmic Dawn. Actually slightly over 3000 hrs, due to the fact that 𝑡 𝐴 must be a multipleof 9s (see Appendix C), giving slightly over 4 hrs per day. The noise power Δ ( 𝑘 = . ) from 3000 hrs of partially coher-ently integrated observations is plotted as crosses in Figure 7, for theredshift range 18 ≤ + 𝑧 ≤
38. To demonstrate that OVRO-LWAis able to detect the 21-cm signal at these redshifts, the Figure alsocontains Δ ( 𝑘 ) for a sample of theoretical models, including the en-velope of all possible signals in the standard astrophysical scenario(as explained below, and in Cohen et al. 2017), and an exotic physicscase which was proposed to explain the deep EDGES absorption fea-ture (here for illustrative purposes we show a model with enhancedradio background over the CMB, see Fialkov & Barkana 2019). Themodels are described in the Figure caption. Some of the standard as-trophysical models, along with the exotic scenario, are well above thesensitivity of OVRO-LWA at these redshifts, indicating that OVRO-LWA is expected to be sensitive enough to dig deep into the discoveryspace of the predicted 21-cm signals.Let us first examine the standard astrophysical case, i.e., modelsthat assume the CMB as background radiation, with a conventionalcold dark matter scenario, and hierarchical structure formation. Theastrophysical model contains seven parameters (see Cohen et al.2020, for a recent summary of the astrophysical modeling), includ-ing: star formation efficiency 𝑓 ∗ , minimum halo mass suitable forstar formation (or, equivalently, minimum circular velocity of suchhalos, 𝑉 𝑐 ); X-ray efficiency of sources compared to their present-daycounterparts 𝑓 𝑋 ; the spectral properties of X-ray sources (namely,the slope of X-ray spectral energy distribution, 𝛼 , and the minimumfrequency, 𝜈 𝑚𝑖𝑛 ). We model the process of reionization using twomore free parameters (the total CMB optical depth, 𝜏 , and the meanfree path of UV photons, 𝑅 mfp ); however, reionization does not playa significant role at the high-redshift regime probed by OVRO-LWA.Astrophysics is in its simplest form at high redshifts accessibleusing OVRO-LWA. With first stars forming in small and rare darkmatter halos, and owing to the absence of massive galaxies andAGN, there are relatively few processes that affect the early 21-cmsignal. Thus, constraining the 21-cm signal at high redshifts withOVRO-LWA would offer one of the purest probes of primordial starformation. The most prominent feature of the 21-cm signal fromthis epoch is the high-redshift peak in the power spectrum imprintedby the non-homogeneous Ly- 𝛼 field (Barkana & Loeb 2005). Theamplitude and central frequency of the peak depend on just two pa-rameters in our case ( 𝑓 ∗ and 𝑉 𝑐 ), and the peak is above the sensitivityof OVRO-LWA for models with efficient star formation in small darkmatter halos. Thus, using 3000 hours of observations, OVRO-LWAcan probe properties of the first star forming halos, star formation effi-ciency, and put constraints on stellar feedback. The second effect thatcan be constrained by OVRO-LWA is the onset of the IGM heatingby the first X-ray sources. Although in some scenarios X-ray heatingis delayed, in many of the examined models X-ray sources turn onearly and could imprint heating fluctuations in the gas temperaturein the OVRO-LWA band. Examining a set of ∼ 𝑓 ∗ = 𝑀 ℎ ∼ × 𝑀 (cid:12) , and close to present-day X-rayheating efficiency (magenta curve in Figure 7). For this model boththe Ly- 𝛼 peak at 𝑧 ∼
22 and the X-ray peak at 𝑧 ∼ . 𝛼 peaks can be detected. We compare thecase of dark matter halos with 𝑉 𝑐 = .
2, 24.2 and 52.1 km/s, forotherwise fixed parameters. The Ly- 𝛼 peak for 𝑉 𝑐 = . MNRAS , 1–18 (2021) H. Garsden et al.
System Total Redshift 𝑘 for which Reported powerobserving time power reportedOVRO-LWA (Eastwood et al. 2019) 28 hr 𝑧 = . 𝑘 = . − Δ < mK LOFAR LBA (Gehlot et al. 2019) 14hr 𝑧 = . − . −
68 MHz) 𝑘 ∼ .
038 h Mpc − Δ < . × mK MWA (Ewall-Wice et al. 2016) 3 hr 𝑧 = . − . −
113 MHz) 𝑘 = . − Δ < mK AARTFAAC (Gehlot et al. 2020) 2 hr 𝑧 = . − . −
75 MHz) 𝑘 = .
144 h cMpc − Δ < × mK This work 4hr 𝑧 = −
31 (43 −
54 MHz) 𝑘 = . − Δ < × mK Table 6.
Summary of limits obtained from observations using different telescopes. the temperature fluctuations can be measured in the highest frequencychannels. For the other two cases ( 𝑉 𝑐 = . 𝛼 peak is at lower redshifts and happens to be above the noise curve,and, thus, is detectable. However, for these models the X-ray peak isshifted to higher frequencies out of the OVRO-LWA band. Finally,varying X-ray parameters can also play a role; compare the case of 𝑉 𝑐 = . 𝑓 𝑋 = .
07) to a casewith the same 𝑉 𝑐 but stronger X-ray heating ( 𝑓 𝑋 = .
7, dashed blue).Because the effects of heating and Ly- 𝛼 coupling on the 21-cm signalanti-correlate, more efficient X-ray heating results in a lower Ly- 𝛼 peak which is difficult to detect, but also in an earlier and strongerX-ray peak that shifts into the OVRO-LWA band.If the anomalously deep global signal detected by EDGES Low-Band is astrophysical, the 21-cm power spectrum is expected to beboosted by a few orders of magnitude (e.g., Fialkov et al. 2018; Fi-alkov & Barkana 2019; Reis et al. 2020). The exact amount of theenhancement relative to a standard scenario with the same astrophys-ical parameters, depends on the underlying theory which explains theEDGES observation. Owing to this boost, it would be much easier todetect such signals. As an illustration, we show one example of sucha signal in Figure 7 (orange line) for a theory in which the 21-cmsignal is enhanced due to an excess radio background radiation overthe CMB at high redshifts (from Fialkov & Barkana 2019).Finally, in Figure 7, we compare the sensitivity level obtainedfrom 3000 hrs of observations, with the power obtained from the 4hrs of observations (section 5), and the power obtained from othertelescopes as reported in the literature. We include a sample of modelswith excess radio background (Fialkov & Barkana 2019). We have demonstrated that the delay spectrum method can be usedto generate power spectra from OVRO-LWA observations, althoughthey contain systematics that prevent us from reaching the level of21-cm fluctuations in the Cosmic Dawn window. Many systemat-ics can influence power spectrum sensitivity, including: inaccuratesky models for calibration (including unmodelled diffuse mission),undetected RFI, inaccurate beam model (shape and chromaticity),gain calibration errors, cross-coupling and mutual coupling, and ca-ble reflections; all which are present, or likely to be present, in ourpower spectra. Mitigating systematics in the context of 21-cm powerspectra is an active area of research, and we will borrow promis-ing techniques developed for use with other telescopes, such as theMWA, HERA, SKA, and LOFAR.We find that the power level in 21-cm fluctuations at 48.324MHz is Δ ( 𝑘 ) ≈ × mK at 𝑘 = . − , after inco-herently integrating 4 hrs of observations. Simulations of OVRO-
16 20 25 30 35 40 -2 P ( k ) k / [ m K ] Figure 10.
The simulated noise level of OVRO-LWA from 3000 hrs of par-tially coherently integrated observations at 𝑘 = . − (crosses), plot-ted on top of a selection of 21-cm models. We show standard models with 𝑉 𝑐 = . . . 𝑓 ∗ = 𝑓 𝑋 = . 𝛼 = . 𝜈 𝑚𝑖𝑛 = . 𝑅 𝑚 𝑓 𝑝 =
30 Mpc and 𝜏 = . 𝑓 𝑋 = . 𝑉 𝑐 = . 𝑓 ∗ = 𝑉 𝑐 = . 𝑓 𝑋 = 𝛼 = . 𝜈 𝑚𝑖𝑛 = . 𝑅 𝑚 𝑓 𝑝 =
40 Mpc and 𝜏 = .
07. We also show the envelopeof all possible signals in standard astrophysical scenario (thick grey lines,Cohen et al. 2018). Finally, we show an external radio background model(orange) that has the maximum signal to noise for LEDA 𝑓 ∗ = 𝑉 𝑐 = . 𝑓 𝑋 = .
01 with hard X-ray SED, 𝜏 = .
056 and the excess radiobackground of 1.2% over the CMB at 1.42 GHz (from Fialkov & Barkana2019).MNRAS000
056 and the excess radiobackground of 1.2% over the CMB at 1.42 GHz (from Fialkov & Barkana2019).MNRAS000 , 1–18 (2021)
10 15 20 25 30 35 40 P ( k ) k / [ m K ] Figure 11.
We show our limit of 2 × mK at 𝑧 =
28 and at 𝑘 = . − obtained from 4hr of observations (black triangle) in the context oflimits from other high redshift 21-cm probes including LOFAR (NCP, redline) derived using 14 h of data at 54-68 MHz ( 𝑧 = . − .
2) at 𝑘 ∼ . 𝑧 = . − .
9) at 𝑘 = . − (Ewall-Wiceet al. 2016), and LWA (blue triangle) showing the datapoint from Eastwoodet al. (2019) obtained using 28 h of data at 73.152 MHz ( 𝑧 = .
4) at 𝑘 = .
15h Mpc − . We also show the predicted OVRO-LWA sensitivity curve from the3000 hr of noise data (black crosses) obtained as described in the previoussection, and a hundred randomly selected 21-cm power spectra modeled withextra radio background, from Fialkov & Barkana (2019). LWA thermal noise indicate that the noise floor in the power spectrais Δ ( 𝑘 ) (cid:46) × mK . This level is too high to detect 21-cm fluctuations at Cosmic Dawn, but we may take the value of Δ ( 𝑘 ) ≈ × mK as an upper limit. Simulations show that ifwe were able to implement our partial coherent integration method,and apply it to 3000 hrs of observations, the noise floor would dropto 10 mk , allowing for a detection according to theoretical models.However the noise simulations are simplistic. Firstly, properties ofthe noise were obtained from visibility differencing. A more rigoroustreatment would involve precise measurements and simulations of thenoise in telescope components (e.g. as for the MWA, Ung et al. 2020)followed by an analysis of how this converts to noise power Δ ( 𝑘 ) ina 21cm power spectrum. Secondly, observing over a period of manyyears, to obtain observations that can be coherently integrated, willrequire that the telescope remains stable, along with atmospheric andenvironmental conditions, and that observations are not affected bytransient events and unwanted objects passing through the field ofview. It is unlikely these can all be controlled, so some observationsmay need to be discarded, increasing the total observing time to compensate, or their effects will have to be mitigated in the data.These issues will have to be planned for.We also mentioned that the delay spectrum technique is not suit-able for the frequency range and baseline lengths that we have used.Long baselines can be excluded from power spectra, which will resultin a shortened 𝑘 ⊥ range. Restricting the frequency range is problem-atic as it will result in few frequency channels being used for powerspectrum generation. For example, using Equation 6 and again inter-preting (cid:28) as “one-half”, and limiting baseline lengths to 50 meters,the allowable bandwidth is 1.6 MHz. This means only 70 OVRO-LWA frequency channels (of width 24 kHz) can be used, comparedto the 436 that were used for the power spectra in this paper, resultingin fewer 𝑘 (cid:107) modes available for analysis.The redshift range over which a power spectrum can made at highredshift, so that it measures a static Universe, is uncertain, and willnot be determined until we have more knowledge of the actual rateof change of state of the early Universe, perhaps from global 21-cmexperiments. The range will depend on the redshift of the observation;at 𝑧 ≈
9, Pober et al. (2013) suggest a redshift range of Δ 𝑧 ≈ . 𝑧 ≈
28, as used here, a redshift range of 0.5corresponds to a frequency range of 0.8 MHz, which allows for only33 OVR0-LWA channels to be used for power spectrum generation.This and the discussion in the previous paragraph indicate that theOVRO-LWA frequency resolution should be increased for futurework.We may also expand future work to include other methods ofpower spectrum generation, such as removing foregrounds from ob-servations using foreground models (e.g. as in Gehlot et al. 2019), oradopting the m-mode analysis used by Eastwood et al. (2018). How-ever, many of the issues mentioned above still remain, and should beaddressed before doing so.
ACKNOWLEDGEMENTS
We acknowledge the helpful advice and software algorithms obtainedfrom the HERA and MWA consortia. AF was supported by the RoyalSociety University Research Fellowship.Correlated telescope data was produced by the Owens Valley Ra-dio Observatory operated by the California Institute of Technology.Further processing and simulations were made using the LEDA Com-puting Cluster at the Harvard-Smithsonian Center for Astrophysics.
DATA AVAILABILITY
The data used to generate the results reported in this paper are notpublicly available, but will be shared, on reasonable request to thecorresponding author.
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APPENDIX A: REGIONS USED TO OBTAIN POWERVALUES FROM POWER SPECTRA
When extracting values from a power spectrum, for example to reportstatistics or generate Δ ( 𝑘 ) , we use only certain regions. The regionsavoid artifacts in the power spectrum and select areas appropriatefor the required statistic; for example, statistics for the Cosmic Dawnwindow avoid the foreground wedge and high power at low 𝑘 ⊥ . TableA1 shows the regions used for each type of data analysis; they are thearea in white plotted on top of a power spectrum. The first two rows in MNRAS000
When extracting values from a power spectrum, for example to reportstatistics or generate Δ ( 𝑘 ) , we use only certain regions. The regionsavoid artifacts in the power spectrum and select areas appropriatefor the required statistic; for example, statistics for the Cosmic Dawnwindow avoid the foreground wedge and high power at low 𝑘 ⊥ . TableA1 shows the regions used for each type of data analysis; they are thearea in white plotted on top of a power spectrum. The first two rows in MNRAS000 , 1–18 (2021) Regions Purpose MotivationApplied only to power spectra generated fromtelescope observations (not simulations). Theseare for obtaining the mean value of 𝑃 ( k ) in theforeground wedge and Cosmic Dawn window.There are 2 Cosmic Dawn window regions, andone foreground wedge region. The Cosmic Dawn window regions avoid theleakage from the wedge, and artifacts at large 𝑘 (cid:107) (positive and negative). They cover a regionwhere the power is fairly uniform, but still mayinclude artifacts. The foreground wedge regionavoids the heightened power at low 𝑘 ⊥ .Applied only to power spectra generated fromtelescope observations (not simulations). Whencalculating Δ ( 𝑘 ) , use only values in these re-gions. Similar to the previous Cosmic Dawn windowregions, but they include spillover close to thehorizon.Applied only to noise-only power spectra gen-erated from simulated noise-only observations.For obtaining the mean value of 𝑃 ( k ) in theCosmic Dawn window. The regions are chosen to correspond to areas ofthe power spectrum where the noise is uniformand has its lowest value. The noise should be sta-tistically the same in the vertical ( 𝑘 (cid:107) ) direction,but there may be a slight DC offset at 𝑘 (cid:107) =
0, sothe inner region is avoided.Applied only to noise-only power spectra gen-erated from simulated noise-only observations.When calculating Δ ( 𝑘 ) , use only values inthese regions. They expand the regions shownin the previous row, so that Δ ( 𝑘 ) can be calcu-lated for a wider range of 𝑘 . Similar to the Cosmic Dawn window regions inrow 2, but they extend close to the top/bottomedges of the power spectrum. Table A1.
Regions in the power spectrum used for obtaining power 𝑃 ( k ) and Δ ( 𝑘 ) the table apply only to power spectra generated from observations, thenext two rows apply only to power spectra generated from simulationsof noise.When a value for 𝑃 ( k ) is reported for a power spectrum, thisis the average value of 𝑃 ( k ) within the region being discussed. Toobtain Δ ( 𝑘 ) values from a region, all the pixels within the region areassigned a 𝑘 value, converted to dimensionless power via Equation5, and binned (averaged) by 𝑘 using 16 logarithmic bins within therange of the obtained 𝑘 values. APPENDIX B: SELECTION OF TELESCOPEOBSERVATIONS USED FOR POWER SPECTRUMGENERATION
The selection criteria are simple and heuristic. Observations for in-tegrated power spectra are selected first on the accuracy of theircalibration, and then on the quality of the power spectrum that eachobservation produces.Each observation is calibrated, and a calibration score assigned,
MNRAS , 1–18 (2021) H. Garsden et al. based on how closely the calibrated observation reproduces the fluxesof the calibration sources. A score of 1 indicates perfect calibration,and we select all observations with a score of 0.8 to 1.2.Each observation selected by the previous step is then assigned ascore obtained from the power spectrum that the observation gener-ates. A high score indicates: (a) That the power outside the horizon islow compared to the power inside the horizon, and (b) that the poweroutside the horizon is flat, i.e. does not vary much but maintains aconstant value. The scores are obtained from mean and RMS valuesof the power in regions in the power spectrum (Table A1, row 1).The 4 hours of observations selected for power spectrum gener-ation are the 4 hours of observations that scored the highest in theprevious step.
APPENDIX C: TIME AVERAGING LIMIT FOR OVRO-LWA
Coherent integration of visibilities recorded at different times mayresult in amplitude loss, due to sky sources changing position relativeto interferometer fringes. This is referred to as time-smearing ortime-averaging loss. When observing continuously, only visibilitiesrecorded within a time limit may be coherently integrated.Following Equations 16.5 and 161.6 in Thompson et al. (2017),the fractional loss in amplitude, 𝜉 , in integrated visibilities observedover some time range T, for an east-west oriented baseline, can beexpressed as 𝜉 = − 𝑠𝑖𝑛𝑐 ( 𝜋𝜔 𝑒 𝑐𝑜𝑠𝛿 [ 𝐷 𝐸𝑊 𝜈 / 𝑐 ] 𝑇 ) , (C1)where 𝐷 𝐸𝑊 is the length of the baseline, 𝜔 𝑒 is the sidereal rate, 𝛿 the observing declination of a source, and 𝜈 the observing frequency.For an interferometer containing many baselines, the most restrictivevalue for T is found by setting 𝐷 𝐸𝑊 equal to the maximum east-westbaseline length, and setting 𝛿 to 0.For the OVRO-LWA telescope observations that were used forpower spectrum generation, we allow that the fractional loss inamplitude can be no more than 0.01. Therefore we set 𝜉 = . 𝐷 𝐸𝑊 = 𝛿 =
0, and 𝜈 = .
324 MHz, and solve for 𝑇 , giving 𝑇 =
33s (referred to in the text as the time-averaging limit 𝑡 𝐴 ). Ob-servations whose times are separated by no more than 33s may becoherently combined. We increase it to 36s so as to be multiple ofthe OVRO-LWA observation integration time (9s). The time limit forother frequencies can be similarly obtained. This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000