# A 21-cm power spectrum at 48 MHz, using the Owens Valley Long Wavelength
Array

H. Garsden1,2, L. Greenhill1, G. Bernardi3,4,5, A. Fialkov6,7, D.C. Price1,8,
D. Mitchell9, J. Dowell10, M. Spinelli11,12 and F.K. Schinzel13
1Harvard-Smithsonian Center for Astrophysics, MS42, 60 Garden Street,
Cambridge MA 02138 USA
2Astronomy Unit, Queen Mary University of London, Mile End Road, London E1
4NS, United Kingdom
3INAF-Istituto di Radioastronomia, via Gobetti 101, 40129, Bologna, Italy
4Department of Physics & Electronics, Artillery Road, Rhodes University,
Grahamstown, South Africa
5South African Radio Astronomy Observatory, FIR street, Observatory, Cape
Town, South Africa
6Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
7Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK
8International Centre for Radio Astronomy Research, Curtin University,
Bentley, WA 6102, Australia
9CSIRO Astronomy & Space Science, Australia Telescope National Facility, P.O.
Box 76, Epping, NSW 1710, Australia
10Department of Physics and Astronomy, University of New Mexico, 210 Yale Blvd
NE, Albuquerque, New Mexico
11INAF-Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, 34143
Trieste, Italy
12IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2,
34014 Trieste, Italy
13National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801,
USA;University of New Mexico E-mail<EMAIL_ADDRESS>(HG)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Large-aperture Experiment to detect the Dark Age (LEDA) was designed to
measure the 21-cm signal from neutral hydrogen at Cosmic Dawn,
$z\approx$15-30. Using observations made with the $\approx$ 200 m diameter
core of the Owens Valley Long Wavelength Array (OVRO-LWA), we present a 2-D
cylindrical spatial power spectrum for data at 43.1-53.5 MHz ($z_{\rm
median}\approx 28$) incoherently integrated for 4 hours, and an analysis of
the array sensitivity. Power from foregrounds is localized to a “wedge” within
$k_{\perp},k_{\parallel}$ space. After calibration of visibilities using 5
bright compact sources including Vir A, we measure $\Delta^{2}(k)\approx
2\times 10^{12}\ \mathrm{mK}^{2}$ outside the foreground wedge, where an
uncontaminated cosmological signal would lie, in principle. The measured
$\Delta^{2}(k)$ is an upper limit that reflects a combination of thermal
instrumental and sky noise, and unmodelled systematics that scatter power from
the wedge, as will be discussed. By differencing calibrated visibilities for
close pairs of frequency channels, we suppress foreground sky structure and
systematics, extract thermal noise, and use a mix of coherent and incoherent
integration to simulate a noise-dominated power spectrum for a 3000 h
observation and $z=16-37$. For suitable calibration quality, the resulting
noise level, $\Delta^{2}(k)\approx 100$ mK2 (k = 0.3 Mpc-1), would be
sufficient to detect peaks in the 21-cm spatial power spectrum due to early
Ly-$\alpha$ and X-ray sources, as predicted for a range of theoretical model
parameters.

###### keywords:

cosmology: observations – dark ages, reionization, first stars – techniques:
interferometric; software: simulations

††pubyear: 2021††pagerange: A 21-cm power spectrum at 48 MHz, using the Owens
Valley Long Wavelength Array–C

##  1 Introduction

Observations of Cosmic Dawn and the Epoch of Reionization (EoR) are key to
unveiling properties of the first population of stars and galaxies. The role
of first galaxies in the process of reionization can be probed by the
measurements of Gunn-Peterson troughs in the spectra of high-redshift quasars
(e.g., Fan et al., 2006; Greig et al., 2017; Greig et al., 2019), Ly-$\alpha$
emission from Lyman Break galaxies (Mason et al., 2018) and the measurement of
the total optical depth to Thomson scattering of the Cosmic Microwave
Background (CMB) radiation (e.g., Planck Collaboration et al., 2018). The
emergent picture shows growing evidence of late reionization ending at
$z\approx 6$ (e.g., Weinberger et al., 2019). However, these measurements
cannot reveal much about other processes that occur at early times, such as
the heating of the Intergalactic Medium (IGM) by the first X-ray sources or
the onset of primordial star formation in proto-galaxies.

These processes can be constrained via observations of the redshifted 21-cm
line of neutral hydrogen produced at a rest-frame frequency 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-$\alpha$ radiation from stars (Wouthuysen,
1952; Field, 1958) and the intensity of the radio background at 21 cm (e.g.,
Feng & Holder, 2018). Therefore, this signal can be used to constrain the
process of primordial star and galaxy formation (e.g., probe the effect of
radiative feedback mechanisms at high redshifts, Fialkov et al., 2013), and
probe both thermal and ionization histories of the gas.

The sky-averaged (global) 21-cm signal traces the mean evolution of the
Universe and can act as a cosmic clock, revealing the timing of major cosmic
events such as the onset of star formation. The first tentative detection of
the global 21-cm signal at 78 MHz (corresponding to a redshift of $z\approx
17$) was recently reported by the EDGES collaboration (Bowman et al., 2018).
This detection is not compatible with theoretical predictions based on
standard astrophysical and cosmological assumptions (e.g., Cohen et al., 2017,
explored a large grid of 21-cm global signals, broadly varying properties of
astrophysical sources within acceptable ranges), calling for alternative
theories at Cosmic Dawn. The proposed solutions range from models with over-
cooling of gas via dark matter interactions (see Barkana, 2018, and the
citation history of that work) to the existence of extra radio background
radiation in addition to the CMB (e.g., Bowman et al., 2018; Feng & Holder,
2018), which is usually assumed to be the only source of background radiation.
Moreover, doubts about the signal being of cosmological origin have been
expressed, with possible explanations including instrumental systematic errors
(Hills et al., 2018; Sims & Pober, 2019; Singh & Subrahmanyan, 2019), ground
plane interference (Bradley et al., 2019), and polarized foregrounds (Spinelli
et al., 2019). Other existing global 21-cm experiments are attempting to
verify this detection at high redshifts, including the Large-aperture
Experiment to Detect the Dark Ages (Bernardi et al., 2016; Price et al.,
2018), and efforts in progress with experiments such as SARAS3,
REACH111https://www.astro.phy.cam.ac.uk/research/research-projects/reach,
EDGES Mid-Band, and MIST222http://www.physics.mcgill.ca/mist/. At lower
redshifts only upper limits on the global 21-cm signal have been published so
far (Monsalve et al., 2017; Singh et al., 2017).

Power spectra of 21-cm fluctuations contains information on the initial
fluctuations in density and velocity fields, spatial distribution of sources,
and their spectral properties. Power spectrum measurements can provide an
important validation of the EDGES signal and aid in its theoretical
interpretation (Fialkov et al., 2018; Reis et al., 2020). If the EDGES signal
is imprinted by astrophysical or cosmological processes at Cosmic Dawn, power
spectra from $z\approx 17$ could carry a consistent signature. So far, major
experimental effort has been focused on observing the EoR, with several
instruments yielding upper limits in the range $z\approx 6-9$, including LOFAR
(Mertens et 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, first attempts to measure
the power spectrum have been made by Ewall-Wice et al. (2016) who, using 6 h
of data from the MWA, reported a sensitivity of $10^{8}$ mK2 on comoving
scales $k\lesssim 0.5$ h Mpc-1, at redshifts $11.6\lesssim z\lesssim 17.9$ (a
few orders of magnitude above the expected 21-cm power spectrum). Eastwood et
al. (2019) used 28 h collected by the Owens Valley Radio Observatory Long
Wavelength Array (OVRO-LWA, Hallinan et al., 2015), and found an upper limit
of $10^{8}$ mK2 at $k\lesssim 0.10$ Mpc-1 at $z=18.4$. The AARTFAAC Cosmic
Explorer program (Gehlot et al., 2020), using the LOFAR telescope, reported an
upper limit of $7.3\times 10^{7}$ mK2 at $k=0.144$ h cMpc-1, $z=17.9-18.6$,
from 2 h of observations. Finally, 14 hours collected by LOFAR-LBA were used
to obtain upper limits of $2.1\times 10^{8}$ mK2 and $2.2\times 10^{8}$ mK2 at
$k\approx 0.038$ h Mpc-1 for two observed fields (3C220 and NCP) at
$z=19.8-25.2$ (Gehlot et al., 2019). Next-generation experiments such as HERA
(DeBoer et al., 2017) and the SKA (Koopmans et al., 2015) are designed to
detect the 21-cm power spectrum from a wide range of redshifts and scales,
including both the EoR and Cosmic Dawn.

This paper reports results from the Large-aperture Experiment to detect the
Dark Age (LEDA). LEDA is working towards a detection of 21-cm fluctuations in
the CMB signal at Cosmic Dawn ($z\approx 15-30$), in both global (sky-
averaged) spectra and spatial power spectra, using observations from OVRO-LWA.
We are currently focused on two experiments: 1.Validation of the recent
detection by Bowman et al. (2018), and extraction of any other 21-cm
absorption/emission profiles observed at Cosmic Dawn; 2. Generation and
analysis of power spectra of the 21-cm signal at Cosmic Dawn, and analysis of
the sensitivity of power spectra as a function of observation integration
time.

A paper reporting on progress on the first experiment is under review
(Spinelli et al., 2020); this paper reports results from the second
experiment.

The first power spectrum obtained from OVRO-LWA observations (Eastwood et al.,
2019, mentioned above) was an angular power spectrum generated from 28 h of
integrated data using m-mode analysis (Shaw et al., 2014), and filtering of
foregrounds using the double Karhunen-Loéve transform. In this paper we also
use observations from OVRO-LWA, but generate power spectra using the “delay
spectrum” method (Parsons et al., 2012b), which produces a 2-D cylindrical
power spectrum. Instead of removing foregrounds, the delay spectrum method
isolates foreground power in a “wedge” region of the power spectrum, outside
of which the 21-cm signal may be detected, given enough power spectrum
sensitivity. Foreground isolation is inherent in the delay spectrum method,
and relies on the spectral smoothness of foreground emission. The method also
has the advantage that it involves minimal manipulation of telescope data, and
is easy to implement. It does, however, have limitations, which we will
discuss below. We operate at a higher redshift of $z\approx 28$, compared to
Eastwood et al. (2019), producing a power spectrum at a redshift higher than
any other reported to date. On the other hand, we use a smaller number of
integrated observations than Eastwood et al. (2019), namely 4 h.

We will show that the sensitivity of our power spectrum is low compared to
other reported experiments, but that this can be improved by integrating a
larger number of observations, and varying the method of integration. If some
or all observations are coherently integrated before power spectrum
generation, the noise in the power spectrum is substantially reduced. However
the timing of observations is crucial for coherent integration, since
observations must be nearly identical, apart from noise. We report the results
of simulations and analysis showing that 21-cm Cosmic Dawn fluctuations can be
detected using OVRO-LWA observations that are combined using a mix of coherent
and incoherent integration, and if the observations and power spectrum are
free of any systematics. The number of observations required to detect 21-cm
fluctuations is large, i.e. 3000 h in total, but it can be obtained within a
reasonable time frame using a scheme that we will describe.

The paper is organized as follows: Section 2 describes the Owens Valley Long
Wavelength Array (OVRO-LWA) and data processing; Section 3 describes the
observations used for power spectrum generation, and how they were selected;
Section 4 presents the delay spectrum method; Section 5 shows the power
spectra generated from observations, and discusses their characteristics;
Section 6 describes simulations of noise in power spectra, introduces coherent
integration, and reports analysis of the sensitivity of OVRO-LWA; Section 7
compares that sensitivity against theoretical models of Cosmic Dawn and
demonstrates that OVRO-LWA, and thus compact interferometers, could be used to
study Cosmic Dawn; Section 8 summarizes the results and future work.

For this work we assume a flat universe with the following cosmological
parameter values:
$\Omega_{\mathrm{M}}=0.31,\Omega_{\mathrm{\Lambda}}=0.69,H_{\mathrm{0}}=0.68$.

##  2 The OVRO-LWA Telescope and Processing Pipeline

Figure 1: $uv$-coverage of the OVRO-LWA baselines used for power spectrum
generation, obtained from a frequency channel of width 24 kHz at a frequency
of 57 MHz.

OVRO-LWA is a wide field, low frequency, drift-scan, non-redundant, imaging
interferometer situated in Owens Valley, California, at a latitude of
$37.2398^{\circ}$ and a longitude of $-118.2817^{\circ}$. Its design and
hardware components are based on that of the Long Wavelength Array (LWA) in
New Mexico (Taylor et al., 2012). OVRO-LWA provides data for research in
several areas of observational astronomy and cosmology, including: cosmic rays
(Romero-Wolf et al., 2019; Carvalho et al., 2019; Monroe et al., 2020), radio
transients surveys (Anderson et al., 2019a), solar weather and coronal mass
ejections (Hallinan & Anderson, 2017; Chhabra et al., 2019), exoplanet
searches (Anderson & Hallinan, 2017), radio counterparts to gamma ray bursts
and gravitational waves (Anderson et al., 2018; Anderson et al., 2019b),
emission from compact 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 (Price et al.,
2018), and Cosmic Dawn 21-cm power spectra (Eastwood et al., 2019).

The telescope consists of 256 dual-antenna stands, of the same type used in
the Long Wavelength Array (Taylor et al., 2012); only the OVRO-LWA core,
consisting of 219 stands, was used for experiments reported here. The 219
stands provide 23871 baselines, with lengths 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 $uv$-coverage of the OVRO-LWA core at 57 MHz is shown in Figure 1;
$uv$ lengths range from 0.9 to 40.4 wavelengths. The resolution of the
telescope is $1.4^{\circ}$ at 57 MHz.

Signals from the antennas are processed by an FX correlator on site at the
Owens Valley Radio Observatory (Kocz et al., 2015). The observing band is
split into 22 sub-bands each containing 109 channels of width 24 kHz, 2398
channels in all. Each stand has a folded cross-dipole, for observing X (east-
west) and Y (north-south) polarizations. Cross-correlation of antennas
generates four polarization products: XX, XY, YX, YY. Only XX and YY
polarizations are used for our power spectrum analysis, and they are treated
separately.

Antenna signals are correlated, integrated, and saved as visibilities every 9
s, thus producing a single “observation” every 9 s. Multiple observations are
typically integrated for further processing, e.g. for generating power
spectra. Observations are calibrated using the Real Time System (RTS), which
performs self-calibration against a point source sky model (Mitchell et al.,
2008). The RTS uses a beam model described in LWA Memo
178333http://www.phys.unm.edu/ lwa/memos/memo/lwa0178a.pdf. RTS contains an
algorithm to detect and flag narrow-band RFI in each sub-band, by finding
channels whose amplitude varies significantly from the median amplitude of the
sub-band (Mitchell et al., 2010). No other automated flagging is applied;
further flagging of data is achieved by a visual inspection of antenna
autocorrelations. Post-calibration quality checks (described below), are used
to select observations to be used for science experiments.

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 as the 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 ringed in 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. The two 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-west of 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 is unidentified. The images contain Stokes I observed over a bandwidth of 10.464 MHz centered at 48.324 MHz. They were generated using NRAO CASA with Briggs 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 200 m, is 96 Jy/beam.  Source | Location | Flux Density at | Flux Density at
---|---|---|---
| (RA/DEC) | 74 MHz (Jy) | 48.324 MHz (Jy)
Virgo A | 12:30:49 | 1253.4 | 1689.03
| +12:23:28 |  |
1411+5212 | 14:11:20 | 120.27 | 162.07
| +52:12:04 |  |
1229+0202 | 12:29:06 | 149.96 | 202.08
| +02:02:56 |  |
1504+2600 | 15:04:59 | 129.86 | 174.99
| +26:00:46 |  |
0813+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-LWA
observations used for power spectrum generation. Locations and flux densities
were obtained from the VLA Low-Frequency Sky Survey at 74 MHz, and flux
densities converted by RTS to OVRO-LWA frequencies by assuming a spectral
index of -0.7. The flux densities at 74 MHz and 48.324 MHz are shown.

##  3 Observations used for Power Spectrum Generation

We use 4 hours of non-contiguous (in time) observations, selected from all
observations made during May 2018, between LST 11:55 and 13:15. Of the 219
stands available for power spectrum observations, only 165 were deemed usable
over the entire 4 hours, the rest being flagged due to hardware issues and
outages. We also flagged 509 baselines due to suspected cross-talk between
their antenna cable connections at backend electronic processing units444M.
Eastwood, personal communication.. During calibration, short baselines were
downweighted so that any observed diffuse emission does not interfere with
calibration by RTS, which lacks a diffuse emission model.

During observations, both the Sun and Galactic Center were below the horizon,
and two strong sources, Cygnus A and Cassiopeia A, were on or below the
horizon. The brightest source in the sky was Virgo A, at an elevation of
$\approx 63^{\circ}$. We used Virgo A and four other sources for calibration.
These five sources were selected based on the following criteria: 1\. at an
LST of 12:30, they are above an elevation of $30^{\circ}$, 2. at an LST of
12:30, they are determined by RTS to be the five brightest sources after
primary beam correction. The LST of 12:30 in these criteria was chosen as it
is the approximate middle of the LST range of 11:55 to 13:15. All observations
were separately calibrated using the same five sources. The RTS was configured
to use the VLA Low-Frequency Sky Survey at 74 MHz (Cohen et al., 2007) to
determine source flux densities and locations; the sources are detailed in
Table 1. To obtain flux density at frequencies other than 74 MHz we assume a
spectral index of -0.7 for all sources. It is possible that the spectrum of
these sources turns over at lower frequencies, but only $\approx 4.5\%$ of
sources in the GLEAM catalog do so (Callingham et al., 2017), and Virgo A does
not (de Gasperin et al., 2012), so we do not attempt to correct for this.

A simple method was used to select the 4 h of observations used for power
spectrum generation, based on the quality of the observations and their
suitability for use in power spectra. The method is described in Appendix B.
This produced a set of observations that are not necessarily contiguous in
time, and recorded on different days. Figure 2 shows Stokes I images of the
sky generated from one observation, at LST = 12:30. The sky is the same in
both images, but the right image marks the calibration sources in green, and
other sources of interest in red and yellow (see caption). The images were
obtained using NRAO CASA software555https://casa.nrao.edu.

For power spectra we use a frequency band of 10.464 MHz centered at 48.324
MHz. That band was chosen because it is less subject to RFI and produces the
highest signal-to-noise ratio over the OVRO-LWA band. It does not cover the
EDGES detection frequency, but for our sensitivity analysis, using
simulations, we will do so by expanding the frequency range to cover almost
the entire OVRO-LWA band. Over the 10.464 MHz band, the 21-cm signal is
observed at $z=25-31$. This corresponds to a time range of 35 million years,
when the Universe was 100 Myr old. Power spectra of 21-cm fluctuations should
ideally be obtained over a time range when the Universe is not significantly
evolving, which is unlikely over $z=25-31$, so a narrower frequency band is
more appropriate. However, that will produce a power spectrum with a less
resolution, and we prefer to maintain a higher resolution for analysis of our
first power spectrum (see Pober et al. (2013) for a similar argument).

Figure 3: The power spectra generated from the 4 h of observations described in the text, and plots of $\Delta^{2}(k)$ obtained from the power spectra. The top row shows the power spectra, the second row shows the same power spectra but restricts the $k_{\parallel}$ range to $-0.31$ to $0.31$ Mpc-1, so that the wedge can be seen in more detail. Power values are colored using a log scale. The third row plots $\Delta^{2}(k)$ values at various $k$, obtained from within the Cosmic Dawn window of the power spectra (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. The left column shows the power spectrum generated from polarization XX, the right column from polarization YY. Property | Value
---|---
Power spectrum dimensions (pixels) | 102 horizontal, 108 vertical
$k_{\perp}$ range | 0.001 to 0.018 Mpc-1
$k_{\perp}$ resolution | 0.000175 Mpc-1
$k_{\parallel}$ range | -1.06 to 1.04 Mpc-1
$k_{\parallel}$ resolution | 0.02 Mpc-1
Delay range | -5113.64 to 5018.94 ns
Delay resolution | 95 ns
Horizon delay at max $k_{\perp}$ | 852 ns
$uv$ lengths used | 1 to 33
Table 2: Size, ranges, and resolution of the power spectrum images in the top row of Figure 3. | Pol XX | Pol YY
---|---|---
Observed $P(\mathbf{k})$, foreground wedge | $1.1\times 10^{18}$ | $1.7\times 10^{18}$
Observed $P(\mathbf{k})$, Cosmic Dawn window | $5.2\times 10^{14}$ | $6.7\times 10^{14}$
Ratio of $P(\mathbf{k})$ above | 2039 | 2490
$\Delta^{2}(k=0.3)$, Cosmic Dawn window | $1.7\times 10^{12}$ | $2.4\times 10^{12}$
Table 3: The top two rows show the average power in the power spectra
generated from 4 h of observations (Figure 3), from the foreground wedge
(first row), and the Cosmic Dawn window (second row). The third row shows the
ratio of the power in the foreground wedge to the power in the Cosmic Dawn
window. The bottom row shows values of dimensionless power $\Delta^{2}(k=0.3)$
obtained from the Cosmic Dawn window.

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
$k_{\perp}$ values of 0.005, 0.01, 0.15 Mpc-1. The black lines indicate the
location of the horizon at these $k_{\perp}$ values. The bottom row shows a
horizontal cut made at $k_{\parallel}=0.53\,\mathrm{Mpc}^{-1}$. The left
column is from power spectra generated from polarization XX, the right column
from polarization YY.

##  4 Power Spectrum Generation

OVRO-LWA observes 21-cm fluctuations at redshifted 21-cm frequencies, 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 foregrounds from 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. Examples of the
first are reported in Gehlot et al. (2019), Eastwood et al. (2019) and Barry
et al. (2019); examples of the second in Pober et al. (2013), Thyagarajan et
al. (2015a), Paul et al. (2016) and Ewall-Wice et al. (2016). We implement
foreground isolation by generating a 2-D, cylindrically averaged, power
spectrum using the delay spectrum method Parsons et al. (2012b).

The delay spectrum method applies a Fourier transform to baselines
visibilities, producing a delay spectrum:

$\mathcal{V}_{\mathrm{b}}(\tau)=\int V_{\mathrm{b}}(\nu)e^{-2\pi\nu\tau}d\nu,$
(1)

where $V_{\mathrm{b}}(\nu)$ are visibilities for a baseline (arrayed by
frequency), and $\tau$ is the Fourier conjugate of $\nu$. $\tau$ represents
the time delay of foreground signals between the two antennas of the baseline,
and foregrounds signals are confined to values of $\tau$ less than the horizon
delay, thus producing the “wedge”. The baseline visibilities
$V_{\mathrm{b}}(\nu)$ measure the spatial Fourier transform perpendicular to
the line of sight, and can all be assigned a wavenumber $k_{\perp}$. Equation
1 also approximates the spatial Fourier transform of the 21-cm signal along
the line-of-sight, with a wavenumber $k_{\parallel}$ assigned to $\tau$.

The delay spectra from multiple baselines are converted to power and binned by
$k_{\perp}$, producing a 2-D power spectrum indexed by
($k_{\perp},k_{\parallel})$. Power is calculated using the equation

$P(\mathbf{k})\approx|\mathcal{V}|^{2}\left(\frac{{\lambda}^{2}}{2k_{\mathrm{B}}}\right)^{2}\frac{X^{2}Y}{\Omega_{\mathrm{pp}}B_{\mathrm{pp}}}$
(2)

(Parsons et al., 2012a; Parsons et al., 2014; Parsons, 2017), where
$P(\mathbf{k}$) is the power at $\mathbf{k}$ =
($k_{\perp},k_{\parallel}),\lambda$ is the mean wavelength of the band,
$k_{\mathrm{B}}$ is Boltzmann’s constant, $X^{2}Y$ converts angles and
frequency intervals to comoving distance (Hogg, 1999), $\Omega_{\mathrm{pp}}$
is the integral over the power-squared beam 666Power-squared indicated by the
double p subscript, and $B_{\mathrm{pp}}$ is the effective bandwidth.
$B_{\mathrm{pp}}=\int|w|^{2}d\nu$, where $w$ is the window function applied to
the delay transform. A window function is used to avoid spectral leakage
produced by taking the Fourier transform of a finite length of non-periodic
data; we use a Blackman-Harris window. We use a beam model of the LWA antenna
dipole (Dowell, 2011) to obtain $\Omega_{\mathrm{pp}}$. The units of
$P(\mathbf{k})$ are $\mathrm{mK}^{2}\mathrm{Mpc}^{3}$. $P(\mathbf{k})$ is
often converted to dimensionless power

$\Delta^{2}(k)=\frac{k^{3}}{2\pi^{2}}P(\mathbf{k}),$ (3)

where $k=\sqrt{k_{\perp}^{2}+k_{\parallel}^{2}}$, and $\Delta^{2}(k)$ has
units of $\mathrm{mK}^{2}$. We will use both power values in this paper.

The delay spectrum method has limitations due its approximation of the spatial
Fourier transform along the line-of-sight. The delay spectrum method works
best for narrow-beam, short-baseline, interferometers (Parsons et al., 2012b),
whereas OVRO-LWA is a wide-field, long-baseline interferometer. Liu et al.
(2014, Eqn. 9) have determined that the delay-spectrum method may be used for
observations where

$b\theta_{\mathrm{0}}\ll\frac{c}{B_{\mathrm{band}}},$ (4)

where $b$ is the length of a baseline, $\theta_{\mathrm{0}}$ is the width of
the telescope primary beam (1.8 radians), and $B_{\mathrm{band}}$ is the
bandwidth over which the observations are made. Interpreting “$\ll$” as “one-
half”, Equation 4 is violated by 99.66% of the OVRO-LWA baselines used for our
power spectrum generation, indicating that the baselines are too long and/or
the bandwidth too wide. Reducing the maximum baseline length will reduce the
$k_{\perp}$ range; reducing the bandwidth will reduce the $k_{\parallel}$
range. For the power spectra reported in this work, we ignore these
restrictions, so as to obtain a power spectrum with an extent that allows for
some analysis. We will deal with them in future work (see Section 8).

###  4.1 Additional processing steps

Before generating a delay spectrum from each baseline, flagged channels must
be dealt with. Flagged channels represent missing values in the array
$V_{\mathrm{b}}(\nu)$ (Equation 1), which produce a delay spectrum convolved
with a PSF. This PSF may be deconvolved using a CLEAN algorithm (Högbom,
1974). We use an implementation of CLEAN developed by the HERA
Consortium777pspec_prep.py, https://github.com/HERA-Team for just this
purpose. CLEAN is applied to the delay spectrum within the horizon, including
a buffer of 95 ns outside the horizon; 95 ns being the resolution of $\tau$ in
the delay spectrum of OVRO-LWA baselines (Table 2).

After CLEAN, the visibilities in $V_{\mathrm{b}}(\nu)$ are averaged by groups
of 4, to reduce noise. This reduces the frequency resolution by a factor of 4.
The bandwidth of 10.464 MHz, which initially contains 436 channels of width 24
kHz, is thus transformed to 109 channels of width 96 kHz.

Instead of converting each baseline delay-spectrum to power, we follow Pober
et al. (2013) and multiply observing-time adjacent (i.e. LST adjacent) pairs
of the same baseline, to reduce noise. The LSTs of each multiplied pair must
be within the time smearing limits for OVRO-LWA (see Appendix C), since they
are being coherently combined via the multiplication. Fortunately, after
ordering all observations by LST, this applies to 99.9% of the observations
comprising the 4 h.

##  5 Results

Figure 3 shows the power spectrum generated from the 4 h of observations
described in Section 3, for polarizations XX and YY. Baselines are binned,
i.e.averaged, along the $k_{\perp}$ axis, using a bin size of 0.000175 Mpc-1,
being the $k_{\perp}$ of a baseline with u $=0.5$ at the lowest observing
frequency. There is no binning along the $k_{\parallel}$ axis. The blue lines
are the horizon delay, which increases with increasing $k_{\perp}$, producing
the “wedge” within which foreground emission is isolated. $k_{\parallel}$
wavenumbers are approximated from the delays $\tau$ of the delay transform;
these delays are plotted on the y-axis on the right. $k_{\perp}$ values
approximately correlate with baseline $uv$ lengths, which are shown on the top
x-axis.

The top row of Figure 3 displays the power spectra using the complete range of
$k_{\parallel}$ values derived from the observing frequency band (10.464 MHz);
the second row is the same power spectrum but restricts the $k_{\parallel}$
range to $\pm 0.3$ Mpc-1 so that the wedge can be seen in more detail. A
summary of the important properties of the power spectrum images in the top
row of Figure 3 is in Table 2. The third row of Figure 3 shows the
dimensionless power $\Delta^{2}(k)$ extracted from the Cosmic Dawn window, at
various values of $k$ (the algorithm for extracting $\Delta^{2}(k)$ is
described in Appendix A).

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
in the Cosmic Dawn window. The top section indicates linear features that are
approximately horizontal, with a slight downward slope, and run across almost
the entire $k_{\perp}$ range. The top section has been clipped to
$10^{14}<P(\mathbf{k})<10^{16}$ mK2Mpc3, and rescaled to make the features
clearer.

Figure 4 shows vertical and horizontal cuts through the power spectra. The top
row depicts vertical cuts at three $k_{\perp}$ values of 0.005, 0.01, 0.015
Mpc-1. The location of the horizon at the different $k_{\perp}$ values is
shown by the black vertical lines. The bottom row of Figure 4 depicts a
horizontal cut through the power spectra at a constant $k_{\parallel}=0.53$
Mpc-1. These cuts show that the power drops by a factor of 104 mK2Mpc3 as
$k_{\parallel}$ moves away from 0, and that the power in the Cosmic Dawn
window is fairly flat by $k_{\perp}$. The average power levels $P(\mathbf{k})$
and $\Delta^{2}(k)$ within the foreground wedge and Cosmic Dawn window are
listed in Table 3.

Figure 6: Histograms of noise values ($\Delta V$) obtained from differencing
visibilities in adjacent frequency channels, over all observations used to
generate power spectra. These represent the distribution of thermal noise in
OVRO-LWA (although some systematics are present in the noise distributions, as
described in the text). The top row is obtained from the XX polarization
visibilities, the bottom row is from polarization 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 the right.

Figure 7: Noise-only power spectra generated from 4 h of simulated noise-only
observations, and $\Delta^{2}(k)$ values obtained from them. The visibilities
in the 4 h observations are replaced with noise values drawn from a Gaussian
distribution with zero mean, and standard deviation obtained from the
histograms of OVRO-LWA thermal noise in Figure 6. These simulated noise-only
observations are used to generate the power spectra shown. Different
distributions are used for XX and YY polarization noise visibilities,
producing separate power spectra for XX polarization (left column) and YY
polarization (right column). The top row shows the power spectra, the bottom
row plots $\Delta^{2}(k)$ values obtained from Cosmic Dawn window within each
power spectrum (the regions from which the values are obtained are described
in Appendix A).

Figure 8: Simulated noise-only power spectra, the same as for Figure 7, but some observations have been coherently integrated before power spectrum generation, as described in the text (Section 6.3). | Pol XX | Pol YY
---|---|---
Real/Imag values (fit all data) | 1313.0/1312.8 | 1543.3/1543.1
Real/Imag values (fit $\pm 2000$ Jy) | 1265.8/1265.6 | 1472.2/1472.1
Table 4: The standard deviations ($\sigma$) of Gaussian fits to the
distributions of noise values obtained from the 4 h of observations (Figure
6). The units are Jy. The values in the first row are obtained from fitting
the histograms using the complete x-axis range of $\pm 8000$ Jy; the values in
the bottom row are obtained from fitting in the range $\pm 2000$ Jy. Noise
power in 4 h of observations,
---
incoherently integrated
| Pol XX | Pol YY
$P(\mathbf{k})$ | $2.8\times 10^{13}$ | $3.8\times 10^{13}$
$\Delta^{2}(k=0.3)$ | $4.7\times 10^{10}$ | $6.3\times 10^{10}$
Noise power in 4 h of observations when
partial coherent integration is used
| Pol XX | Pol YY
$P(\mathbf{k})$ | $1.3\times 10^{7}$ | $1.8\times 10^{7}$
$\Delta^{2}(k=0.3)$ | $2.3\times 10^{4}$ | $3.2\times 10^{4}$
Table 5: Values of $P(\mathbf{k})$ and $\Delta^{2}(k)$ extracted from simulated noise-only power spectra. The top two data rows are obtained from the power spectra in Figure 7, which simulates the noise in a power spectrum generated from 4 h of incoherently integrated observations. The bottom two data rows are obtained from Figure 8, which simulates the noise in a power spectrum generated from 4 h of partially coherently integrated observations (the integration method is described in Section 6.3).  System | Total | Redshift |  $k$ for which | Reported power
---|---|---|---|---
| observing time |  | power reported |
OVRO-LWA (Eastwood et al., 2019) | 28 h |  $z=18.4$ (73.152 MHz) |  $k=0.1$ Mpc-1 |  $\Delta^{2}<10^{8}$ mK2
LOFAR LBA (Gehlot et al., 2019) | 14 h |  $z=19.8-25.2$ ($54-68$ MHz) |  $k\approx 0.038$ h Mpc-1 |  $\Delta^{2}<2.2\times 10^{8}$ mK2
MWA (Ewall-Wice et al., 2016) | 3 h |  $z=11.6-17.9$ ($75-113$ MHz) |  $k=0.5$ h Mpc-1 |  $\Delta^{2}<10^{8}$ mK2
AARTFAAC (Gehlot et al., 2020) | 2 h |  $z=17.9-18.6$ ($72-75$ MHz) |  $k=0.144$ h cMpc-1 |  $\Delta^{2}<7\times 10^{7}$ mK2
This work | 4 h |  $z=25-31$ ($43-54$ MHz) |  $k=0.3$ Mpc-1 |  $\Delta^{2}<2\times 10^{12}$ mK2
Table 6: Summary of 21-cm power spectrum limits obtained from observations
using different telescopes.

###  5.1 Discussion

Figure 3 demonstrates that wedge-type power spectra can be generated from
OVRO-LWA observations. The power spectra show good isolation of foregrounds
within the horizon, although there is some spillover. Our power values are
high compared to those reported by Eastwood et al. (2019), who obtained a
value of $\Delta^{2}(k)\approx 10^{8}$ mK2 at $k=0.1\,\mathrm{Mpc}^{-1}$,
compared to our value of $\Delta^{2}(k)\approx 10^{12}$ mK2 at
$k=0.3\,\mathrm{Mpc}^{-1}$. Other telescopes also report lower power; a
summary of these is shown in Table 6. Note that other experiments were using
different observing frequencies, total observing time, and telescopes. One
reason for the higher power that we obtain is due to our lower observing
frequency. At a lower frequency, foreground emission is brighter (Spinelli et
al., 2020), so that systematics and foreground leakage will raise the level of
the power spectrum generally.

A visual inspection of the power spectra shows that horizon spillover can be
separated into two distinct regions, based on the amplitude of the spillover
and its extent beyond the horizon. The regions are depicted in Figure 5. Both
regions exist at all $k_{\perp}$ values. Region 1 is bounded by the horizon
and the dashed line; it extends for approximately $\Delta
k=0.04\,\mathrm{Mpc}^{-1},\Delta\tau=192\,\mathrm{ns}$, beyond the horizon,
and is commonly seen in other power spectra (e.g. Pober et al., 2013;
Thyagarajan et al., 2015a, b). Spillover in Region 1 could be due to the
window function applied to the delay transform, beam chromaticity, intrinsic
spectral structure in foreground emission, RFI, and calibration errors (Kern
et al., 2020b) . The use of a window function on the delay transform is
essential to avoid spectral leakage due to the non-periodic nature of the
baseline visibilities. The kernel width of the Blackman Harris window we apply
to our delay transforms is $288$ ns or $\Delta k_{\parallel}=0.06$, indicating
it is a substantial contributor to Region 1 spillover.

Region 2 extends from Region 1 to the dashed-dot line; it extends for $\Delta
k=0.12\,\mathrm{Mpc}^{-1},\Delta\tau=577\,\mathrm{ns}$ beyond Region 1. The
reason for the spillover in Region 2 is not known but we suggest it is due to
foreground power scattered by systematics that have not been removed by data
quality checks or calibration. Unremoved systematics are also likely
responsible for the faint lines that can be seen in the upper part of the
power spectrum. These are highlighted in Figure 5, upper section ("Linear
features"). The lines begin near $k_{\perp}=0,k_{\parallel}=1$ and drop
linearly and shallowly across the power spectrum to
$k_{\perp}=0.018,k_{\parallel}=0.18$.

Systematics can have many causes, including unmodelled diffuse emission, an
inadequate beam model and/or source model for calibration, RFI, cable
reflections, cross-coupling and mutual coupling, which we discuss as follows.

Models of point sources are incomplete or approximate at OVRO-LWA frequencies.
We generated calibration models using the VLA Low-Frequency Sky Survey at 74
MHz, calculating source flux densities at 48 MHz by assuming a spectral index
of -0.7, and assuming the location of emission at 48 MHz is the same as it is
for 74 MHz. These assumptions likely lead to inaccurate models. Diffuse
emission also exists within our observations, but we have not modelled it for
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.

OVRO-LWA observations are affected by cross-coupling of antenna signals, where
cables are in close proximity as they enter backend hardware processing units.
It is known to impact 509 baselines, and these are ignored when generating
power spectra, but they may not be the only ones affected. A more detailed
investigation is needed.

Electromagnetic interaction between neighboring antennas (mutual coupling) may
be present in OVRO-LWA observations, including coupling between the X and Y
dipoles on the same OVRO-LWA stand, leading to polarization leakage. A study
of mutual coupling in the LWA reported that it exists, but has no consistent
positive or negative effect (Ellingson, 2011), however more investigation of
these effects is needed for OVRO-LWA.

Electrical signals travelling via cable from an antenna to backend components
may reflect from those components, introducing sinusoidal structure in the
antenna frequency response. For the most part, cross-correlations between
antennas (i.e. visibilities) do not suffer this systematic as the noise waves
along a pair of cables are not correlated. However, systematics due to cable
reflections have been reported in other power spectra (e.g Beardsley et al.,
2016) and may be present in ours. The linear features in Figure 5 may be due
to cable reflections, but in that case we expect the features to be
horizontal.

RFI flaggers such as AOFlagger (Offringa et al., 2012) will provide better RFI
detection compared to the simple algorithm implemented within RTS. Flaggers
will be implemented in the future within the OVRO-LWA pipeline. More
sophisticated beam models can be generated through the use of simulation
packages (e.g. FEKO888https://altairhyperworks.com/product/Feko/Applications-
Antenna-Design, NEC999https://www.qsl.net/4nec2/,
CST101010https://uk.mathworks.com/products/connections/product_detail/cst-
microwave-studio.html), direct measurements of the radiation field of the
antenna (for example by using drones (Üstüner et al., 2014) or satellites
(Neben et al., 2015)), or by observing radio sources transiting through the
fringes of the interferometer (Parsons et al., 2016).

Methods for the detection, analysis, and removal of systematics in power
spectra are being developed by other groups and reported in the literature.
For example, Kern et al. (2020b) demonstrates the usefulness of analysing
antenna-based gains in delay and fringe-rate space, using gain-smoothing to
suppress delay modes that appear to be abnormal, and using temporal analysis
and temporal smoothing of gains using autocorrelations. Cross-coupling can be
removed by applying a high-pass filter in fringe rate space (Kern et al.,
2020a). Window functions can be analysed for their contribution to horizon
spillover, and alternative window functions selected (Lanman et al., 2020).
Use of these and other methods will improve the quality of power spectra
generated from OVRO-LWA.

Figure 9: A scheme for making observations with a drift-scan interferometer
that can be partially coherently integrated. Observe for $N$ short time
Intervals ($LST_{\mathrm{i}}\rightarrow LST_{\mathrm{i+t_{A}}}$) each day,
repeat the observations over several days, and coherently integrate all
observations within each Interval across all days. The time range
$t_{\mathrm{A}}$ is the time-averaging limit. The resulting $N$ coherently
integrated observations must be combined incoherently.

##  6 The sensitivity of OVRO-LWA

We now turn to the analysis of the sensitivity of power spectra generated from
OVRO-LWA using the delay spectrum method, and investigate the sensitivity that
can be achieved using a large number of observations and a different
integration strategy.

We determine the level of noise in OVRO-LWA observations – “noise” includes
thermal instrumental and sky noise. We use simulations to generate power
spectra containing only noise; these allow us to find the noise floor in our
power spectra from observations (Figure 3). We then investigate the noise
floor in power spectra generated from observations that are partly coherently
integrated before power spectrum generation. Coherent integration should
significantly reduce the noise level in power spectra.

###  6.1 Measuring the OVRO-LWA thermal noise

The noise in our observations is extracted from observed visibilities using a
simple data-differencing technique (Bernardi et al. (2010); Chatterjee &
Bharadwaj (2019)) that subtracts visibilities in adjacent frequency channels.
We assume that subtracting visibilities at slightly different frequencies will
subtract out the sky signal and systematics, leaving only subtracted noise
values. From the distribution of these noise values we obtain the statistical
properties of the noise, from which we can generate noise-only power spectra
using simulations. The noise level in these power spectra is interpreted as
the level of noise that would be present in power spectra generated from
telescope observations. For visibility differencing, we use all the
visibilities in all the calibrated observations used for the power spectra in
Figure 3, ignoring flagged visibilities.

Figure 6 show histograms of the values obtained from frequency differencing,
for XX and YY polarizations, and for the real and imaginary parts of the
values. The distributions are close to Gaussian, with a zero mean, giving us
confidence that we have in fact extracted mostly thermal noise. However, there
are wings in the data distribution indicating an excess of values around $\pm
3000$ Jy, and the data peak is higher than the fitted peak. To investigate
these, we generated images of the visibility differences from one of the OVRO-
LWA observations. The images mostly contained noise, but showed that Virgo A,
and the RFI sources on the horizon (Figure 2), were not always completely
removed, leading to an excess of values at the location of the wings.

The standard deviations $\sigma$ of the Gaussian fits to the noise histograms
are listed in Table 4, top row. These fits include the wings previously
mentioned; if the wings were not present, we would expect a narrower
distribution and smaller $\sigma$ values. To estimate what these would be, we
fit a Gaussian to the noise histograms in the x-axis range $\pm 2000$ Jy only.
The standard deviations for these fits are in Table 4, second row, and produce
$\sigma$ that are about 5% smaller than those resulting from fitting the
entire x-range.

To be conservative, we use the higher values of $\sigma$ for our noise
simulations.

###  6.2 Measuring the noise floor in the power spectra generated from 4 h of
telescope observations (incoherent integration)

We simulate noise-only observations by replacing all the visibilities in all
the 4 h of telescope observations with noise values drawn from a Gaussian
distribution with zero mean and standard deviation listed in Table 4.
Baselines that were flagged in the observations are flagged in the simulated
noise-only observations, but flagged channels are filled in with a noise
value. A “noise-only” power spectrum is generated from the 4 h of simulated
noise-only observations; these power spectra are shown in Figure 7, and the
power values extracted from them are listed in Table 5. The power spectra
generated from observations have a value of $P(\mathbf{k})\approx 5\times
10^{14}$ in the Cosmic Dawn window (Table 3), but the noise-only power spectra
have $P(\mathbf{k})\approx 3\times 10^{13}$ in the same region, demonstrating
that there are systematics in the observed data.

All the power spectra presented so far were generated using incoherent
integration; we now turn to experiments with coherent integration.

###  6.3 Measuring the noise floor in the power spectra generated from 4 h of
telescope observations (if coherent integration is used)

We follow a method suggested by Paul et al. (2014), where observations are
made within several short time intervals $t_{A}$, repeated over many days.
Each time interval begins at the same LST each day, and all observations made
during time $t_{A}$, at the same LST, may be coherently integrated. The scheme
is depicted in Figure 9. $t_{A}$ is the “time-averaging limit”; the length of
time over which observations may be safely coherently integrated (see Appendix
C). By following this method, $N$ coherently integrated observations are
produced, which must then be incoherently integrated. We refer to this scheme
as “partial coherent integration”.

We note that it may be possible to apply this mix of incoherent/coherent
integration to the 4 h of telescope observations that were used to generate
the power spectra in Figure 3, but it depends on the LSTs of the observations,
as they must fit into appropriate time intervals as described above. An
investigation of this possibility, and publication of power spectra that may
result, is reserved for future work.

To demonstrate the use of the partial coherent integration method, and the
noise reduction that can be achieved, we use the 4 h of simulated noise-only
observations from the previous Section, altering the observation LSTs so we
can apply the method. At 48.342 MHz, $t_{\mathrm{A}}=36$ s (Appendix C), so we
simulate observing at 4 different LSTs ($N=4$, Figure 9) for 36 s each, over
100 days ($D=100$). It is assumed that the thermal noise in the telescope does
not vary over the 100 days, so that all noise values can be obtained from the
distributions in Figure 6.

Figure 8 shows the power spectrum generated as a result, and Table 5 (bottom
two rows) shows the average $\Delta^{2}(k)$ extracted from the power spectra.
These show that we can reduce the noise level in the power spectrum by a
factor of $\approx 10^{6}$, when using partial coherent integration, instead
of fully incoherent integration.

###  6.4 Measuring the noise floor in a power spectra generated from 3000 h of
telescope observations (partial coherent integration)

Following on from the previous experiment, we obtain the noise level in power
spectra generated from 3000 h of partially coherently integrated observations.
These will be compared with the power in 21-cm fluctuations in the next
Section.

We set the number of observing days to $D=750$, or 2 years. An observing
schedule over 2 yrs may be difficult to sustain, but feasible. For OVRO-LWA it
will also generate 169 TiB of data per sub-band, or 3.5 PiB in total, also
feasible using a large computer cluster. We fix the observing time per day to
4 h, so as to avoid transits of the Galactic center and other bright sources.
We assume there are no transits of local bodies. We again assume that the
noise properties of the telescope do not change over the 750 days, and that
there are also no changes in telescope performance, availability, and
reliability. The time-averaging limit $t_{\mathrm{A}}$ varies by frequency, so
the number of Intervals $N$ (Figure 9) must be different for each sub-band, to
accumulate 4 h per day. At $48.342$ MHz with $t_{\mathrm{A}}=36$ s, $N$ = 400;
at 30 MHz with $t_{\mathrm{A}}=54$ s, $N$ = 267.

We use a frequency range of 37.848-84.912 MHz ($z=15.7-36.5$), dividing it
into 18 sub-bands, each of width of 5.232 MHz. Each sub-band overlaps the next
by 2.616 MHz, and are separated in frequency by 2.616 MHz. The center
frequency is used to determine the redshift of the sub-band.

Observations at different frequencies will have different noise properties. We
assume that the noise distribution will be Gaussian as depicted in Figure 6,
but that the $\sigma$ of the noise distribution will vary by frequency. Since
the noise distributions were derived from observed visibilities, we assume
that $\sigma$ varies as a power law with exponent -0.7, as was assumed for the
source flux densities in Section 3. Therefore, $\sigma_{\nu}$ for the noise
distribution at frequency $\nu$ is calculated as:

$\sigma_{\nu}=\sigma_{48.324}*(\nu/48.324\times 10^{6})^{-0.7}.$ (5)

A power spectrum is generated for each sub-band, and the value of
$\Delta^{2}(k)$ at $k=0.3$ extracted from the power spectra. This gives a
$\Delta^{2}(k)$ noise level for a range of redshifts, which can be compared
against expected power based on theoretical models of the cosmological 21-cm
signal, in the next Section.

Generating 3000 hours of noise-only observations is very compute intensive,
and generates 3.5 PiB of data, so we implement it in a GPGPU. Since the
observations all contain the same number of baselines and channels (none are
flagged), a GPGPU thread is created for every channel in every baseline, and
each thread generates and manipulates noise values “on the fly”, avoiding the
need to store the values. A full description of the GPU implementation will be
published elsewhere.

##  7 Investigating Cosmic Dawn using OVRO-LWA Observations

From the simulations described in the previous Section, we obtain the noise
power in OVRO-LWA 21-cm power spectra, generated from 3000 h of partially
coherently integrated observations, over a range of redshifts. If noise is the
only contributor to power in the Cosmic Dawn window, apart from the 21-cm
fluctuations, then the noise power determines whether a 21-cm detection is
possible. In this Section we assume that this is so, and discuss in detail
what may be detected according to theoretical models. However, ensuring that
noise is the only other source of power in the Cosmic Dawn window requires the
elimination of all systematics that we have discussed above. This may
difficult at present, as methods that exist for removal of systematics are
still being proven (e.g. see Kern et al., 2020b), but it is plausible in the
future, at least, we suggest, to a level where systematic effects are
insignificant. We base the following discussion on this “best-case” scenario.

Being a high-redshift instrument, OVRO-LWA is by design observing the 21-cm
signal from the epoch of primordial star and black hole formation. Thus,
observations with this instrument could be used to infer properties of the
first sources as well as to constrain exotic physics at Cosmic Dawn.

The noise power $\Delta^{2}(k=0.3)$ from 3000 h of partially coherently
integrated observations is plotted as crosses in Figure 10, for the redshift
range $17\leq 1+z\leq 38$. To demonstrate that OVRO-LWA is able to detect the
21-cm signal at these redshifts, the Figure also contains $\Delta^{2}(k)$ for
a sample of theoretical models, including the envelope of all possible signals
in the standard astrophysical scenario (as explained below, and in Cohen et
al., 2017), and an exotic physics case which was proposed to explain the deep
EDGES absorption feature (here for illustrative purposes we show a model with
enhanced radio background over the CMB, see Fialkov & Barkana, 2019). The
models are described in the Figure caption. Some of the standard astrophysical
models, along with the exotic scenario, are well above the sensitivity of
OVRO-LWA at these redshifts, indicating that OVRO-LWA is expected to be
sensitive enough to dig deep into the discovery space of the predicted 21-cm
signals.

Let us first examine the standard astrophysical case, i.e., models that assume
the CMB as background radiation, with a conventional cold dark matter
scenario, and hierarchical structure formation. The astrophysical model
contains seven parameters (see Cohen et al., 2020, for a recent summary of the
astrophysical modeling), including: star formation efficiency $f_{\ast}$,
minimum halo mass suitable for star formation (or, equivalently, minimum
circular velocity of such halos, $V_{\mathrm{c}}$); X-ray efficiency of
sources compared to their present-day counterparts $f_{\mathrm{X}}$; the
spectral properties of X-ray sources (namely, the slope of X-ray spectral
energy distribution, $\alpha$, and the minimum frequency,
$\nu_{\mathrm{min}}$). We model the process of reionization using two more
free parameters (the total CMB optical depth, $\tau$, and the mean free path
of UV photons, $R_{\rm mfp}$); however, reionization does not play a
significant role at the high-redshift regime probed by OVRO-LWA.

Astrophysics is in its simplest form at high redshifts accessible using OVRO-
LWA. With first stars forming in small and rare dark matter halos, and owing
to the absence of massive galaxies and AGN, there are relatively few processes
that affect the early 21-cm signal. Thus, constraining the 21-cm signal at
high redshifts with OVRO-LWA would offer one of the purest probes of
primordial star formation. The most prominent feature of the 21-cm signal from
this epoch is the high-redshift peak in the power spectrum imprinted by the
non-homogeneous Ly-$\alpha$ field (Barkana & Loeb, 2005). The amplitude and
central frequency of the peak depend on just two parameters in our case
($f_{\ast}$ and $V_{\mathrm{c}}$), and the peak is above the sensitivity of
OVRO-LWA for models with efficient star formation in small dark matter halos.
Thus, using 3000 h of observations, OVRO-LWA can probe properties of the first
star forming halos, star formation efficiency, and put constraints on stellar
feedback. The second effect that can be constrained by OVRO-LWA is the onset
of the IGM heating by the first X-ray sources. Although in some scenarios
X-ray heating is delayed, in many of the examined models X-ray sources turn on
early and could imprint heating fluctuations in the gas temperature in the
OVRO-LWA band. Examining a set of $\approx 11000$ models with different
combinations of astrophysical parameters, we find that the strongest signal to
noise in the OVRO-LWA band has a model with efficient star formation
($f_{\ast}=50\%$), with a minimum mass of star forming halos of
$M_{\mathrm{h}}\approx 3\times 10^{8}\,M_{\odot}$, and close to present-day
X-ray heating efficiency (magenta curve in Figure 10). For this model both the
Ly-$\alpha$ peak at $z\approx 22$ and the X-ray peak at $z\approx 16.5$ can be
measured by OVRO-LWA.

For the other standard models shown in Figure 10 (shades of blue), either the
X-ray or the Ly-$\alpha$ peaks can be detected. We compare the case of dark
matter halos with $V_{\mathrm{c}}=4.2$, 24.2 and 52.1 km/s, for otherwise
fixed parameters. The Ly-$\alpha$ peak for $V_{\mathrm{c}}=4.2$ km/s occurs at
high redshifts where the sensitivity of OVRO-LWA is weak, and, thus, cannot be
observed. However, the rise of the power imprinted by the temperature
fluctuations can be measured in the highest frequency channels. For the other
two cases ($V_{\mathrm{c}}=24.2$ and 52.1 km/s) the Ly-$\alpha$ 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 is shifted to higher
frequencies out of the OVRO-LWA band. Finally, varying X-ray parameters can
also play a role; compare the case of $V_{\mathrm{c}}=24.2$ km/s with weak
X-ray sources (solid blue, $f_{\mathrm{X}}=0.07$) to a case with the same
$V_{\mathrm{c}}$ but stronger X-ray heating ($f_{\mathrm{X}}=1.7$, dashed
blue). Because the effects of heating and Ly-$\alpha$ coupling on the 21-cm
signal anti-correlate, more efficient X-ray heating results in a lower
Ly-$\alpha$ peak which is difficult to detect, but also in an earlier and
stronger X-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 be boosted by a few
orders of magnitude (e.g., Fialkov et al., 2018; Fialkov & Barkana, 2019; Reis
et al., 2020). The exact amount of the enhancement relative to a standard
scenario with the same astrophysical parameters, depends on the underlying
theory which explains the EDGES observation. Owing to this boost, it would be
much easier to detect such signals. As an illustration, we show one example of
such a signal in Figure 10 (orange line) for a theory in which the 21-cm
signal is enhanced due to an excess radio background over the CMB at high
redshifts (from Fialkov & Barkana, 2019).

Finally, in Figure 11, we compare the sensitivity level obtained from 3000 h
of observations, with the power obtained from the 4 h of observations (Section
5), and the power obtained from other telescopes as reported in the literature
(Table 6). We include a sample of models with excess radio background (Fialkov
& Barkana, 2019).

Figure 10: The simulated noise level of OVRO-LWA from 3000 h of partially
coherently integrated observations at $k=0.3$ Mpc-1 (crosses), plotted on top
of a selection of 21-cm models (note the x-axis is in log scale). We show
standard models with $V_{\mathrm{c}}=4.2$ km/s (light blue), $24.2$ km/s
(blue) and $52.1$ km/s (dark blue) for fixed values of $f_{\ast}=3\%$,
$f_{\mathrm{X}}=0.07$, $\alpha=1.5$, $\nu_{\mathrm{min}}=0.1$ keV (the EoR
parameters, although they do not matter much at these redshifts, are chosen to
be $R_{\mathrm{mfp}}=30$ Mpc and $\tau=0.064$). For comparison we also show
the case of higher $f_{\mathrm{X}}=1.7$ for $V_{\mathrm{c}}=24.2$ km/s (dashed
blue). We also show the model with maximum signal to noise in the LEDA band
(magenta, out of examined 11164 modes). This case has the following values of
the astrophysical parameters: $f_{\ast}=50\%$, $V_{\mathrm{c}}=35.5$ km/s,
$f_{\mathrm{X}}=1$ with $\alpha=1.5$ and $\nu_{\mathrm{min}}=0.1$ keV,
$R_{\mathrm{mfp}}=40$ Mpc and $\tau=0.07$. We also show the envelope of all
possible signals in the standard astrophysical scenario (thick gray lines,
Cohen et al., 2018). Finally, we show an external radio background model
(orange) that has the maximum signal to noise for LEDA: $f_{\ast}=12\%$,
$V_{\mathrm{c}}=4.2$ km/s, $f_{\mathrm{X}}=0.01$ with hard X-ray SED,
$\tau=0.056$ and the excess radio background of 1.2% over the CMB at 1.42 GHz
(from Fialkov & Barkana, 2019).  Figure 11: We show our limit of $2\times
10^{12}$ mK2 at $z=28$ and at $k=0.3$ Mpc-1 obtained from 4 h of observations
(black triangle) in the context of limits from other high redshift 21-cm
probes including LOFAR (NCP, red line) derived from 14 h of data at 54-68 MHz
($z=19.8-25.2$) at $k\approx 0.038$ h Mpc-1 (Gehlot et al., 2019), MWA (green
line) derived from 3 h of data at 75-113 MHz (corresponding to $z=11.6-17.9$)
at $k=0.5$ h Mpc-1 (Ewall-Wice et al., 2016), OVRO-LWA (blue triangle)
obtained using 28 h of data at 73.152 MHz ($z=18.4$) at $k=0.1$ h Mpc-1
(Eastwood et al., 2019), and AARTFAAC (black line) obtained using 2 h of data
at $72-75$ MHz ($z=17.9-18.6$) at $k=0.144$ h cMpc-1 (Gehlot et al., 2020). We
also show the predicted OVRO-LWA sensitivity curve from the 3000 h of noise
data (black crosses) obtained as described in the previous Section, and a
hundred randomly selected 21-cm power spectra modeled with extra radio
background, from Fialkov & Barkana (2019).

##  8 Conclusions

We have demonstrated that the delay spectrum method can be used to generate
power spectra from OVRO-LWA observations, although they contain systematics
that prevent us from reaching the level of 21-cm fluctuations in the Cosmic
Dawn window. Many systematics can influence power spectrum sensitivity,
including: inaccurate sky models for calibration (including unmodelled diffuse
mission), undetected RFI, inaccurate beam model (shape and chromaticity), gain
calibration errors, cross-coupling and mutual coupling, and cable reflections;
all of which are present, or likely to be present, in our power spectra.
Mitigating systematics in the context of 21-cm power spectra is an active area
of research, and we will borrow promising techniques developed for use with
other telescopes, such as the MWA, HERA, SKA, and LOFAR.

We have also demonstrated that a compact, imaging interferometer is able to
generate a 21-cm power spectrum, indicating that such interferometers have a
role to play in detecting the 21-cm signal in power spectra.

We find that the power level in 21-cm fluctuations at 48.324 MHz is
$\Delta^{2}(k)\approx 2\times 10^{12}\,\mathrm{mK}^{2}$ at
$k=0.3\,\mathrm{Mpc}^{-1}$, after incoherently integrating 4 h of
observations. Simulations of OVRO-LWA thermal noise indicate that the noise
floor in the power spectra is $\Delta^{2}(k)\approx 5\times 10^{10}\
\mathrm{mK}^{2}$. This level is too high to detect 21-cm fluctuations at
Cosmic Dawn, but we may take the value of $\Delta^{2}(k)\approx 2\times
10^{12}\,\mathrm{mK}^{2}$ as an upper limit. Simulations show that if we were
able to implement our partial coherent integration method, and apply it to
3000 h of observations, the noise floor would drop to $10^{2}$ mK2, allowing
for a detection according to theoretical models.

However the noise simulations are simplistic. Firstly, properties of the noise
were obtained from visibility differencing. A more rigorous treatment would
involve precise measurements and simulations of the noise in telescope
components (e.g. as for the MWA, Ung et al., 2020) followed by an analysis of
how this converts to noise power $\Delta^{2}(k)$ in a 21-cm power spectrum.
Secondly, observing over a period of many years, to obtain observations that
can be coherently integrated, will require that the telescope remains stable,
along with atmospheric and environmental conditions, and that observations are
not affected by transient events and unwanted objects passing through the
field of view. It is unlikely these can all be controlled, so some
observations may 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 suitable for the
frequency range and baseline lengths that we have used. Long baselines can be
excluded from power spectra, which will result in a shortened $k_{\perp}$
range. Restricting the frequency range is problematic as it will result in few
frequency channels being used for power spectrum generation. For example,
using Equation 4 and again interpreting $\ll$ 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, compared to
the 436 that were used for the power spectra in this paper, resulting in fewer
$k_{\parallel}$ modes available for analysis.

The redshift range over which a power spectrum can made at high redshift, so
that it measures a static Universe, is uncertain, and will not be determined
until we have more knowledge of the actual rate of change of state of the
early Universe, perhaps from global 21-cm experiments. The range will depend
on the redshift of the observation; at $z\approx 9$, Pober et al. (2013)
suggest a redshift range of $\Delta z\approx 0.5$ as being acceptable. At
$z\approx 28$, as used here, a redshift range of 0.5 corresponds to a
frequency range of 0.8 MHz, which allows for only 33 OVRO-LWA channels to be
used for power spectrum generation. This and the discussion in the previous
paragraph indicate that the OVRO-LWA frequency resolution should be increased
for future work.

We may also expand future work to include other methods of power spectrum
generation, such as removing foregrounds from observations using foreground
models (e.g. as in Gehlot et al., 2019), or adopting the m-mode analysis used
by Eastwood et al. (2018). However, many of the issues mentioned above still
remain, and should be addressed before doing so.

## Acknowledgements

The authors thank the referee, whose constructive and helpful comments enabled
us to improve the quality of the paper. AF was supported by a Royal Society
University Research Fellowship. HG was supported in part through a visiting
scientist position at the Smithsonian Astrophysical Observatory (SAO). Phases
of the LEDA project have been supported by NSF grants AST/1616709,
AST/1106059, OIA/1125087, and PHY/0835713. The Owens Valley Radio Observatory
(OVRO) is operated by the California Institute of Technology. The OVRO-LWA
project was enabled by the kind early donation of Deborah Castleman and Harold
Rosen. LG thanks members of the SAO Receiver Lab for their generous
collaboration and OVRO site staff for advice kindly offered, critical and
routine assistance, and collegial spirit.

## Data Availability

The data used to generate the results reported in this paper are not publicly
available, but will be shared, on reasonable request to the corresponding
author.

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##  Appendix A Regions used to obtain power values from power spectra

Regions |  Purpose |  Motivation
---|---|---
|  Applied only to power spectra generated from telescope observations (not simulations). These are for obtaining the mean value of $P(\mathbf{k})$ in the foreground wedge and Cosmic Dawn window. There are 2 Cosmic Dawn window regions, and one foreground wedge region. |  The Cosmic Dawn window regions avoid the leakage from the wedge, and artifacts at large $k_{\parallel}$ (positive and negative). They cover a region where the power is fairly uniform, but still may include artifacts. The foreground wedge region avoids the heightened power at low $k_{\perp}$.
|  Applied only to power spectra generated from telescope observations (not simulations). When calculating $\Delta^{2}(k)$, use only power in these regions. |  Similar to the previous Cosmic Dawn window regions, but they include spillover close to the horizon.
|  Applied only to noise-only power spectra generated from simulated noise-only observations. For obtaining the mean value of $P(\mathbf{k})$ in the Cosmic Dawn window. |  The regions are chosen to correspond to areas of the power spectrum where the noise is uniform and has its lowest value. The noise is not uniform over the power spectrum, there is a slight increase at $k_{\parallel}=0$, and an increase at low and high $k_{\perp}$ because the density of baselines there is low.
|  Applied only to noise-only power spectra generated from simulated noise-only observations. When calculating $\Delta^{2}(k)$, use only values in these regions. They expand the regions shown in the previous row, so that $\Delta^{2}(k)$ can be calculated for a wider range of $k$. |  Similar to the Cosmic Dawn window regions in row 2, but they extend close to the top/bottom edges of the power spectrum.
Table A1: Regions in the power spectrum used for obtaining power
$P(\mathbf{k})$ and $\Delta^{2}(k)$

When extracting values from a power spectrum, for example to report statistics
or generate $\Delta^{2}(k)$, we use only certain regions. The regions avoid
artifacts in the power spectrum and select areas appropriate for the required
statistic; for example, statistics for the Cosmic Dawn window avoid the
foreground wedge and high power at low $k_{\perp}$. Table A1 shows the regions
used for each type of data analysis; they are the area in white plotted on top
of a power spectrum. The first two rows in the table apply only to power
spectra generated from observations, the next two rows apply only to power
spectra generated from simulations of noise.

When a value for $P(\mathbf{k})$ is reported for a power spectrum, this is the
average value of $P(\mathbf{k})$ within the region being discussed. To obtain
$\Delta(k)^{2}$ values from a region, all the pixels within the region are
assigned a $k$ value, converted to dimensionless power via Equation 3, and
binned (averaged) by $k$ using 16 logarithmic bins within the range of the
obtained $k$ values.

##  Appendix B Selection of telescope observations used for power spectrum
generation

The selection criteria are simple and heuristic. Observations for integrated
power spectra are selected first on the accuracy of their calibration, and
then on the quality of the power spectrum that each observation produces.

Each 9 s observation is calibrated, and a calibration score assigned, based on
how closely the calibrated observation reproduces the flux densities of 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 a score
obtained from the power spectrum that the observation generates. A high score
indicates: (a) That the power outside the wedge is low compared to the power
inside the wedge, and (b) that the power outside the horizon is flat, i.e.
does not vary much but maintains a constant value. The scores are obtained
from mean and RMS values of the power in regions in the power spectrum (Table
A1, row 1).

The 4 hours of observations selected for power spectrum generation are the 4
hours of observations that scored the highest in the previous step.

##  Appendix C Time averaging limit for OVRO-LWA

Coherent integration of visibilities recorded at different times may result in
amplitude loss, due to sky sources changing position relative to
interferometer fringes. This is referred to as time-smearing or time-averaging
loss. When observing continuously, only visibilities recorded within a short
time range $t_{A}$ may be coherently integrated.

Following Equations 16.5 and 16.6 in Thompson et al. (2017), the fractional
loss in amplitude, $\xi$, in integrated visibilities observed over some time
range $T$, for an east-west oriented baseline, can be expressed as

$\xi=1-\mathrm{sinc}(\pi\omega_{\mathrm{e}}\mathrm{cos}\delta[D_{\mathrm{EW}}\nu/c]T),$
(6)

where $D_{\mathrm{EW}}$ is the length of the baseline, $\omega_{\mathrm{e}}$
is the sidereal rate, $\delta$ the observing declination of a source, and
$\nu$ the observing frequency. For an interferometer containing many
baselines, the most restrictive value for $T$ is found by setting
$D_{\mathrm{EW}}$ equal to the maximum east-west baseline length, and setting
$\delta$ to 0.

For the OVRO-LWA telescope observations that were used for power spectrum
generation, we allow that the fractional loss in amplitude can be no more than
0.01. Therefore we set $\xi=0.01$, $D_{\mathrm{EW}}=200$ m, $\delta=0$, and
$\nu=48.324$ MHz, and solve for $T$, giving $t_{A}=33$ s. Observations whose
times are separated by no more than 33 s may be coherently combined. We
increase it to 36 s so as to be a multiple of the OVRO-LWA observation
integration time (9 s). The time limit for other frequencies can be similarly
obtained.