# Direction-dependent effects on global 21-cm detection

Yash Agrawal  Raman Research Institute, CV Raman Avenue, Sadashivanagar,
Bangalore, 560080, India  K. Kavitha  Raman Research Institute, CV Raman
Avenue, Sadashivanagar, Bangalore, 560080, India  Saurabh Singh  Raman
Research Institute, CV Raman Avenue, Sadashivanagar, Bangalore, 560080, India

###### Abstract

Cosmic dawn represents critical juncture in cosmic history when the first
population of stars emerged. The astrophysical processes that govern this
transformation need to be better understood. The detection of redshifted 21-cm
radiation emitted from neutral hydrogen during this era offers a direct window
into the thermal and ionization state of the universe. This emission manifests
as differential brightness between spin temperature and the cosmic microwave
background (CMB). SARAS experiment aims to detect the sky-averaged signal in
the frequency range 40-200 MHz. SARAS’s unique design and operation strategy
to float the antenna over a water body minimizes spectral features that may
arise due to stratified ground beneath the antenna. However, the antenna
environment can be prone to configuration changes due to variations in
critical design parameters such as conductivity and antenna tilts. In this
paper, we connect the variations in antenna properties to signal detection
prospects. By using realistic simulations of a direction and frequency-
dependent radiation pattern of the SARAS antenna and its transfer function, we
establish critical parameters and estimate bias in the detectability of
different models of the global 21-cm signal. We find a correlation between the
nature of chromaticity in antenna properties and the bias in the recovered
spectral profiles of 21-cm signals. We also find stringent requirements for
transfer function corrections, which can otherwise make detection prospects
prohibitive. We finally explore a range of critical parameters that allow
robust signal detection.

cosmology: reionization — cosmology: first stars — methods: data analysis —
methods: statistical

##  1 Introduction

One of the poorly understood aspects of cosmic evolution is the formation of
the first stars and galaxies. Cosmic dawn and subsequent epoch of reionization
(EoR) were the periods in cosmic history when the first population of stars
came into existence, marking the end of the dark ages (Barkana & Loeb, 2001;
Miralda-Escudé, 2003). Ionizing radiation from these sources eventually
depleted the neutral hydrogen in the universe (Zaroubi, 2013). The thermal
history of baryons during the dark ages is well-constrained and primarily
governed by the universe’s expansion, proton and electron recombination, and
interaction of the residual free electrons and cosmic microwave background
(CMB). However, when the first sources of radiation formed, they changed the
trajectory of the thermal and ionization state of the intergalactic medium
(IGM) (Ciardi & Ferrara, 2005). The astrophysical processes during cosmic dawn
and EoR need to be better understood (Fan et al., 2006).

One promising technique for studying the thermal and ionization properties of
the universe during this era is the 21-cm spin-flip line from neutral hydrogen
(Shaver et al., 1999; Furlanetto et al., 2006; Pritchard & Loeb, 2012;
Furlanetto et al., 2004). Beyond cosmic dawn, this spin transition firmly
links itself with the thermal temperature of hydrogen gas under the assumption
of strong coupling due to the Wouthuysen Field (WF) effect (Wouthuysen, 1952;
Field, 1959). Detection of sky-averaged 21-cm radiation during this era can
place stringent constraints on several astrophysical parameters (Pritchard &
Loeb, 2008, 2010; Ciardi & Madau, 2003). This radiation, observed as the
global 21-cm signal, presents itself as a mean departure of spin temperature
from CMB temperature at the rest frame frequency of 1420 MHz. However, the sky
averaged 21-cm signal component corresponding to cosmological redshift
$6<z<35$ is redshifted to metre wavelength and lies between $\sim$ 40 MHz to
200 MHz. Several ground-based experiments such as LEDA (Bernardi et al.,
2015), EDGES (Monsalve et al., 2017), SARAS 3 (Nambissan T. et al., 2021),
SCI-HI (Voytek et al., 2014), REACH (de Lera Acedo et al., 2022), $PRI^{Z}M$
(Philip et al., 2019), and MIST (Monsalve et al., 2023a) are attempting to
detect the cosmological signal. Notably, EDGES collaboration claimed the
detection of a flattened Gaussian profile (Bowman et al., 2018), which has
been refuted by SARAS 3 observations (Singh et al., 2022). In view of ground-
based challenges, several space-based experiments are also proposed, including
PRATUSH (Sathyanarayana Rao et al., 2023), DAPPER (Burns, 2021), DSL (Shi et
al., 2022), and LuSEE ‘Night’ (Bale et al., 2023).

The SARAS experiment is custom-designed to observe the 21-cm signal from
cosmic dawn and EoR. In its current third edition, SARAS is optimized to
operate in the frequency range of 40 MHz to 90 MHz. SARAS 3 antenna sits on a
styrofoam raft floating on a water body. The motivation behind the choice of
water as a medium directly below the antenna to a considerable depth is that
water provides a homogeneous medium, avoiding significant spectral features.
Additionally, placing an antenna over water, rather than over the ground,
would achieve a higher radiation efficiency because of the higher permittivity
of water relative to soil (Raghunathan et al., 2021). However, this
configuration makes the antenna susceptible to several perturbations like
variations in raft height, tilts, and water’s conductivity. The systematics
involved will be affected by both the orientation of the antenna and
variations in water properties.

The primary goal of this work is to assess the capability of the SARAS 3
radiometer in constraining different modes of global 21-cm signals under these
perturbations. A direct approach to achieving this involves simulating
realistic sky observations and attempting to recover signals of different
spectral profiles.

Section 2 we detail the electromagnetic simulations carried out for the SARAS
3 antenna. This will be followed by Section 3, where we investigate the
spectral nature of the sky model that we are using to simulate our data using
an achromatic beam. In Section 4, we simulate data under different
perturbations to baseline SARAS 3 configuration and assess the systematics
introduced. We further correlate results with primary beam derivatives .We
further evaluate the systematics caused by a miscalibrated return loss of the
antenna in Section 5. Finally, in Section 6, we inject a range of 21-cm
signals and employ data models and assess the signal extraction capabilities
of the current version of SARAS under different perturbations. Finally
conclusions, derived from the cumulative outcomes of all previous sections,
are presented in Section 7. These interpretations, while primarily centered
around SARAS 3 simulation cases, hold general applicability across experiments
of similar nature.

##  2 Electromagnetic simulations of the antenna

###  2.1 Description of the SARAS 3 antenna

The SARAS 3 antenna (Raghunathan et al., 2021) floats on a freshwater lake
medium below, compared to previous deployments over soil medium. Such media
avoid systematic effects due to multipath propagation between different layers
and make it easier to model the antenna and its environments
electromagnetically. The medium is characterized by its dimensions,
conductivity, and dielectric constant. This is in contrast to stratified
ground layers, where a detailed profile of each layer is required for large
volumes (Spinelli et al., 2022; Monsalve et al., 2023b). The baseline design
of SARAS 3 was decided to maximize its spectral smoothness in coupling sky
power to the receiver, governed by the return loss. Further, the design
minimized variation of its primary beam with frequency as chromatic beams can
introduce significant spectral structures complicating foreground modeling
(Sims et al., 2023; Mahesh et al., 2021).

However, changes in the environment surrounding the antenna and perturbations
in the medium on which the antenna is deployed can introduce systematic
errors. We find that antenna properties are most susceptible to the
conductivity of the water medium, the height of the antenna above the surface
of the water, and the orientation/tilt of the antenna with respect to the
horizon. Figure 1 shows the schematic of the antenna floating over the water
medium.

Figure 1: Schematic of SARAS 3 antenna floating on ideal lake environment

Electromagnetic simulations are performed using CST 111Computer Simulation
Technology— www.cst.com microwave to simulate these varying environments and
capture their effects. CST’s Integral equation solver and Multilayer solver
can be used for these simulations. CST’s multilayer solver was preferred for
analysis as it was found to be more efficient in performing the simulation
compared to the integral equation solver in terms of computation time. It also
allowed for modeling multiple layers, including the lake and the lake bed
layer, into simulations when required.

###  2.2 Baseline antenna configuration and its variations

The antenna was initially designed for deploying over freshwater medium of low
conductivity $<0.05~{}S/m$ using an electromagnetically transparent raft whose
height is 200 mm (Raghunathan et al., 2021). However, the conductivity of the
water medium may vary according to the location, as SARAS 3 can be deployed at
different sites depending on their radio quietness. Also, the water medium
over which the antenna is deployed might not be stable during windy weather
conditions. An unstable water surface might lead to vertical movement of the
raft on the surface of the water, in addition to introducing variations in the
orientation of the antenna axis, tilting it in different directions away from
the zenith. As discussed above, we consider three changing parameters: water
conductivity, antenna reflector height from water, and the tilt angle of the
raft, which consequently affects the antenna’s tilt. Based on realistic field
conditions, we vary conductivity from that of fresh water to saline, in the
range of $0.001$ to $4~{}S/m$. The height of the antenna above water is varied
from 200 mm to 50 mm. Finally, the tilt of the antenna is changed from 0 to 6
degrees. The conductivity of $0.001~{}S/m$ with an antenna 200 mm above water
with no tilt is taken as the baseline or ideal configuration. We simulate
antenna characteristics, namely its primary beam pattern and return loss, over
a range of the above-mentioned degrees of variations. We evaluate the primary
beam over 40-90 MHz, spaced every 1 MHz, assuming open add space boundary
conditions followed by a perfectly matched layer.

##  3 Investigating the Smoothness of the Sky Model

###  3.1 Sky Model: GMOSS

Before investigating the antenna variation impact of the observed sky, we
begin with modeling the intrinsic sky weighted by an achromatic beam as the
best-case scenario. We have used a physically motivated sky model to simulate
the data. Global MOdel for the radio Sky Spectrum (GMOSS) predicts radio
brightness across the frequency range of 22 MHz to 23 GHz by constructing a
physically motivated model of the relevant radiative processes (Rao et al.,
2016). These processes involve galactic and extra-galactic emissions, which
are constrained by all-sky measurements. Previous works have shown that these
foregrounds can be modeled with a maximally smooth function (Sathyanarayana
Rao et al., 2017), unlike the global signal, which is predicted to have
spectral features and inflection points. Here, we generate mock observations
using GMOSS and estimate levels of spectral features intrinsic to the
foregrounds in 40-90 MHz.

###  3.2 Data Simulation and analysis

To investigate the spectral smoothness of the sky model GMOSS, we generate
beam-weighted sky temperature from the sky map for a primary beam of form
$cos^{2}\theta$ for a frequency range of 40 to 90 MHz, where $\theta$ is the
elevation angle. Such achromatic forms of primary beams can be realized using
electrically small antennas. The achromatic nature of the $\cos^{2}$ beam
preserves the spectral integrity of the intrinsic sky spectrum. The beam
weighted sky is given by,

$T_{w}(\nu,t)=\frac{\int_{\phi=0}^{\phi=2\pi}\int_{\theta=0}^{\theta=\pi}T_{F}(\theta,\phi,\nu,t)G(\theta,\phi,\nu)\;\
\sin\theta d\theta d\phi}{\int_{0}^{2\pi}\int_{0}^{\pi}G(\theta,\phi,\nu)\;\
\sin\theta d\theta d\phi}$ (1)

where, $T_{F}(\theta,\phi,\nu,t)$ is brightness of the foregrounds from GMOSS
and $G(\theta,\phi,\nu)=cos^{2}\theta$.

We generate a beam-weighted sky over a 24-hour period at a location in the
Hanle region of India (32°48’46”N 78°52’16” E). Figure 2 shows the GMOSS sky
(panel (a)) and beam-weighted foreground variation over 24 hours (panel (b)).
Note that we do not consider solar or ionospheric effects since our primary
aim is to evaluate foreground mode mixing. Subsequently, we calculate the
average brightness along the temporal axis as shown in 2 panel (c). We further
introduce a subdominant thermal noise of sub-milli kelvin root mean square
(r.m.s). Therefore, such a setup provides robust estimates of chromaticity in
intrinsic foregrounds. In this and the upcoming sections, we will utilize this
approach to produce simulated data for all variations of antenna
configurations.

We use a logarithmic polynomial model for the antenna temperature (Pritchard &
Loeb, 2010; Singh et al., 2022; Bowman et al., 2018) as given by,

$T_{w}(\nu)=10^{\sum_{n=0}^{m}\alpha_{n}(\log_{10}\nu)^{n}},$ (2)

where $T_{w}(\nu)$ is beam-weighted foreground temperature, $\alpha_{n}$ is
the coefficient associated with each term in the polynomial in the log domain
with $(m+1)$ terms. The polynomial in this fit is constrained to be maximally
smooth. A maximally smooth polynomial is a class of constrained functions that
do not have inflection points or zero crossings in higher-order derivatives,
i.e.,

$\frac{d^{m}y}{dx^{m}}\geq 0\quad\text{or}\quad\frac{d^{m}y}{dx^{m}}<0,$ (3)

where $m$ denotes $m^{th}$ derivative of the function $y$ for $m>2$.

The same formalism has been used in the work of Sathyanarayana Rao et al.
(2017) and further refined in Bevins et al. (2021)

Basinhopping (Wales & Doye, 1997) is employed to optimize the parameters,
minimizing the variance as shown in Equation 4. The optimization process
involves fitting polynomials of various orders. The requirement for maximal
smoothness restricts the fit from exhibiting any complex spectral structures.
This ensures that going to a higher order of the polynomial does not fit out
the 21-cm signal, as is often the case with unconstrained polynomials. We
decide the order by recording the reduction in the residual r.m.s with each
additional term and stopping when the model accuracy hits the thermal noise
floor or when the r.m.s saturates due to the constrained nature of the
polynomial. The residuals thus obtained give an estimation of the spectral
smoothness of the sky model. The residuals’ r.m.s value is used as an
indicator to assess the smoothness of fit.

$\sigma^{2}=\frac{\sum_{i=1}^{N}\left(10^{\sum_{n=0}^{m}\alpha_{n}(\log_{10}\nu)^{n}}-T_{w}(\nu)\right)}{N}$
(4)

where N is the number of data points.

###  3.3 Results

Figure 2: (a) GMOSS sky map Mollweide projection at 80 MHz, (b) Variation of
beam-weighted sky with time and frequency, for the primary beam of form
$\cos^{2}\theta$. Color represents brightness temperature in kelvin units for
both panels (a) and (b); (c) Time-averaged foregrounds (d) Residuals (blue)
after fitting maximally smooth function to panel (c), red plus-markers show
the noise r.m.s.

We find that the residuals, after subtracting the fit go down to the thermal
noise level. Figure 2, panel (c) shows the time-averaged foregrounds, while
panel (d) shows the residuals obtained after fitting the simulated data with
the model in Equation 2. Therefore, with a wide achromatic beam, foregrounds
are demonstrated to be spectrally smooth to at least sub-mK level. Therefore,
foregrounds, if observed through achromatic beams, can allow separation of the
global 21-cm signal. This further implies that any excess spectral features
through realistic beams in the following sections can be attributed to the
chromaticity in the primary beam.

##  4 Foregrounds as seen through realistic beams

In Section 3, we demonstrated the spectral smoothness of GMOSS. We also infer,
from the magnitudes of residuals, that the faint global signal can, in
principle, be separated from the bright foregrounds. In this section, we
replicate the procedure outlined in Section 3, but with the convolution of the
sky model using various perturbations applied to the SARAS 3 beam, capturing
realistic variations.

###  4.1 Ideal vs. Perturbed Beam

As mentioned in Section 2, the SARAS 3 antenna sits on the raft floating above
a deep water body under ideal conditions. SARAS 3 is optimized for low-
conductivity water. The beam properties depend on the antenna’s orientation
and the surrounding medium’s dielectric properties. However, several
perturbations could change the nature of the beam and other antenna
characteristics. These variations include changes in the conductivity, height
of the raft and orientation of the antenna. Our objective is to examine the
systematic effects introduced by these perturbations. Table 1 shows different
cases with changing parameters. In Figure 3, we show the spectral and spatial
changes in the beam under different conditions. We note that the beam width
varies with frequency and perturbations, while the azimuthal symmetry of the
primary beam is preserved in all the cases except the tilt variation.

Figure 3: Panels (a), (c), and (e) show the spatial variation of the beams
with changes in conductivity, tilt, and raft height at 70 MHz respectively.
Panels (b), (d), and (f) show the beam full-width-at-half-maximum (FWHM) over
the frequency range 40 MHz to 90 MHz.

Case No | Conductivity (S/m) | Tilt (°) | Raft Height (mm) | Residual r.m.s (mK) | Residuals with miscalibrated S11 (mK)
---|---|---|---|---|---
1 | 0.001 | 0 | 200 | 0.05 | 0.05
2 | 0.002 | 0 | 200 | 0.05 | 0.06
3 | 0.005 | 0 | 200 | 0.04 | 2
4 | 0.01 | 0 | 200 | 0.09 | 11
5 | 0.05 | 0 | 200 | 0.03 | 3
6 | 0.1 | 0 | 200 | 0.04 | 8
7 | 1 | 0 | 200 | 0.07 | 171
8 | 4 | 0 | 200 | 0.03 | 7
9 | 0.001 | 2 | 200 | 0.05 | 154
10 | 0.001 | 4 | 200 | 0.05 | 119
11 | 0.001 | 6 | 200 | 0.09 | 65.7
12 | 0.001 | 0 | 150 | 51.5 | 294
13 | 0.001 | 0 | 100 | 56.14 | 3912
14 | 0.001 | 0 | 50 | 0.06 | 8076

Table 1: Different cases for generating perturbed SARAS primary beams. Columns
2 – 4 are the parameters used in the simulation, namely, water conductivity,
antenna tilt, and raft height. The same case numbers will be used throughout
the paper to refer to specific SARAS beam cases. Column 5 lists the residuals
obtained after modeling foregrounds as described in Section 4. Column 6 shows
the residuals obtained after fitting foregrounds that have miscalibrated S11,
as described in Section 5

.

###  4.2 Data Simulation and Analysis

To simulate the data, following Equation 1, we take $T_{F}(\theta,\phi,\nu)$
to be the brightness of the foregrounds from GMOSS, however
$G(\theta,\phi,\nu)$ is replaced with the realistic SARAS beam cases as listed
in Table 1.

We introduce a free constant parameter, c, in our model for the beam-weighted
sky generated via SARAS beam perturbations. The antenna temperature for the
SARAS beam cases is given by

$T_{w}(\nu)=10^{\sum_{n=0}^{m}\alpha_{n}(\log_{10}\nu)^{n}}+c,$ (5)

where $T_{w}(\nu)$ is antenna temperature for the beam-weighted sky with SARAS
beam, which is being modeled as a logarithmic polynomial and a constant
parameter $c$. The logarithm of Bayesian evidence for the model with and
without the free constant parameter differs by 20, providing compelling
support for the constant parameter as the more suitable choice for the data
(Kass & Raftery, 1995) with realistic beams.

###  4.3 Results

The r.m.s of residuals presented in Table 1 reveals that mostly all beam
scenarios reach the noise floor level. Nevertheless, unlike $cos^{2}\theta$
beam, these instances necessitate one inflection point that indicates excess
spectral complexity. This could potentially arise due to systematics induced
by beam chromaticity. We find raft height to be the most critical parameter,
resulting in relatively high residuals. Notably, in Table 1 corresponding to
cases 12 and 13, the residual r.m.s values following the modeling are four
orders of magnitude higher than the others. However, while r.m.s of residuals
represent the power contained in the systematics, their spectral profile is of
utmost significance. We address this in Section 6, where we jointly model the
simulated data with different variations of the global 21-cm signal.

###  4.4 Beam derivative as a measure of chromaticity

In Section 4.3, we noted the features arising due to variations in the primary
beam. This could be a potential result of beam chromaticity, making a
comprehensive analysis of chromaticity essential for a better understanding.
While the beam varies with elevation, azimuth, and frequency, it’s the latter
that can result in the added spectral complexity. We, therefore, define beam
derivative along the frequency axis for each elevation and azimuth angle. Such
analysis of beam chromaticity has been done in Mozdzen et al. (2016). These
beam derivatives are shown in Figure 4 for the ideal case, along with
variation in the raft height and tilts. In the subsequent panels, enhanced
chromaticity for perturbed cases is obvious from the beam derivative. We
quantify the beam derivative into a chromaticity factor. It is computed by
estimating zero crossings in the beam derivative at each elevation and azimuth
angle. A derivative contrast is computed by finding local maxima and minima
between two successive zero crossings, followed by summing it up for all
angles as shown in Equation 6.

$\small\begin{gathered}\textit{Chromaticty}\\\\[-3.0pt]
\textit{factor}\end{gathered}=\sum_{\theta=0}^{\pi}\sum_{\phi=0}^{2\pi}\sum_{\mathrm{\begin{subarray}{c}\text{number}\\\
\text{of zero}\\\ \text{crossings}\end{subarray}}}\left(\frac{\Delta
G_{\begin{subarray}{c}local\\\ max\end{subarray}}}{\Delta f}-\frac{\Delta
G_{\begin{subarray}{c}local\\\ min\end{subarray}}}{\Delta f}\right)$ (6)

In Figure 5, we compare chromaticity factors as computed above to the r.m.s of
residuals obtained after fitting out the beam-weighted sky with a maximally
smooth function allowing no inflections. Adopting a maximally smooth function
with no inflection brings up spectrally rich structures, enabling a true
estimate of chromaticity introduced in each case. Expectedly, we find
correlations between them, further indicating that excess spectral structure
arises due to the rate of change of beam in the spectral domain. However, we
also notice a non-linear relation between them for raft heights, indicating
the importance of forward modeling foregrounds.

Figure 4: Panel (a) shows the beam derivative for the antenna in an ideal
environment, panel (b) shows the beam derivative for the case where the raft
height reduces to 100 mm, and panel (c) shows the derivatives when the raft
gets tilted to 6 degrees from the zenith due to instabilities in the operating
environment. The derivatives clearly capture the additional chromaticities
introduced due to these perturbations. Figure 5: Residual r.m.s. after fitting
maximally smooth function with no inflection points versus chromaticity factor
for different cases as discussed in Equation 6. Numbers show different
configurations in Table 1. Conductivity, tilt, and raft height variations are
plotted in blue, orange, and green, respectively.

##  5 Analyzing Variation in Return Loss

Return loss governs the coupling of power available at the antenna into the
receiver chain. We quantify this coupling via reflection efficiency defined
as,

$\eta_{r}=1-|\Gamma_{a}^{2}|$ (7)

where $\Gamma_{a}$ is the complex reflection coefficient, which is the linear
equivalent of return loss or S11 in the log scale. The observed brightness,
$T_{a}=\eta_{r}T_{W}$, clearly shows any spectral features in the return loss
are imprinted on the observed data. Although experiments account for this
effect, alterations in the antenna configuration can lead to variations in
$\eta_{r}$. For example, changes in conditions around the antenna can directly
impact $\eta_{r}$. Therefore, any miscalibration due to changed conditions
between measurements of $\eta_{r}$ and observations can introduce additional
spectral features.

This section will assess the systematics introduced by miscalibrated S11
corrections. Simulated plots of S11 for different SARAS cases are shown in
Figure 6, panel (a), (c), and (e).

Figure 6: Panels (a), (c), and (e) show the simulated S11 for conductivity,
raft height, and raft tilt variations, respectively. Panels (b), (d), and (f)
denote the respective S11 miscalibration factors.

###  5.1 Data Simulation and Analysis

We take simulated data for each case just like Section 4, but we correct the
return loss with the baseline SARAS case, which corresponds to case 1 in Table
1. The simulated dataset is described by the Equation 8,

$T_{A}(\nu,t)=\left(\frac{1-|\Gamma(\nu)|^{2}}{1-|\Gamma_{ref}(\nu)|^{2}}\right)T_{w}(\nu,t),$
(8)

where $T_{w}(\nu,t)$ is taken from Equation 1, with $G(\theta,\phi,\nu)$ is
being the respective SARAS beam. $\Gamma$ and $\Gamma_{ref}$ are the
reflection coefficients from the perturbed case and baseline case,
respectively. Equation 8 represents data that takes into account the return
losses but is corrected by a mismatched S11 measurement. Such miscalibration
is representative of scenarios where there is a temporal variation in $\Gamma$
due to changes in medium property or antenna configuration.

The data is subsequently fitted to the same maximally smooth function as
described in Equation 5.

###  5.2 Results

As seen in Table 1, the residuals have clearly increased for each case for a
mismatched S11 correction. It must be noted that the residuals we see here are
a combined effect of beam chromaticity and mismatched S11 correction. In
certain instances, the residuals spike significantly, reaching as high as 8 K,
rendering signal recovery unfeasible. This shows the significance of carrying
out in-situ measurements of S11, factoring in the temporal variations that can
introduce large miscalibration errors.

##  6 extracting global signal

As seen in Section 4, there is an enhancement in spectral complexity of the
beam-weighted foregrounds when convolving the radio sky with the SARAS ideal
beam and its variations. One inflection point is deemed necessary for modeling
the spectrum to reach the noise level. Nevertheless, there are notably
elevated residuals for two cases involving raft heights of 150 mm and 100 mm
above the water surface.

In the present section, we aim to extend the modeling and investigate the
impact of this excess spectral structure on signal detection. We inject a
physically plausible 21-cm signal into the data and attempt to recover it
jointly with the foregrounds.

###  6.1 Data Simulation and Signal Profiles

Our model for observed sky comprises beam-weighted foregrounds, global 21-cm
signal, and thermal noise given by the,

$T_{B}(\nu,t)=T_{w}(\nu,t)+T_{21}(\nu)+T_{n}(\nu),$ (9)

where $T_{n}(\nu)$ is thermal noise. We assume bandpass calibration
systematics are sub-dominant.

It should be noted that the anticipated angular fluctuations in the global
signal amount to 1-2 degrees in the sky (Liu et al., 2013). Conversely, SARAS
employs a wide beam characterized by FWHM values spanning from 40 to 47
degrees. In light of this comparison, $T_{21}$ can be generally considered to
exhibit negligible angular dependence when contrasted with the SARAS beam
width. To simulate the data, we take SARAS beam-weighted sky temperature from
GMOSS and add one of the plausible signals. We take a range of theoretically
plausible signals with varying spectral features. The selection of signals is
governed by spectral profiles, which correspond to different astrophysical
parameters such as star formation efficiency $f_{*}$, virial circular velocity
$V_{c}$, X-ray efficiency of X-ray sources $f_{X}$, spectral energy density
SED, and CMB optical depth $\tau$. These astrophysical parameters govern their
spectral shapes, which were originally discussed in Cohen et al. (2017). We
take a subset of signals listed along with their astrophysical parameters in
Table 2. The profiles represent a range of detectability over a frequency
range of 40-90 MHz, with signals exhibiting absorption features in different
parts of the band with varying amplitudes. We have also included the best-fit
flattened profile from Bowman et al. (2018). Sharp spectral features represent
the strongest detection possibility. Figure 7 shows different signals taken
for this exercise.

Figure 7: (a) Different 21-cm signals with varying spectral features. Signals 1, 2, and 3 exhibit in-band absorption profiles with deep absorption features. Signal 4, even though in-band, has a shallow absorption feature compared to 1, 2, and 3. Signal 5 exhibits partially out-of-band absorption due to late star formation. Their astrophysical parameters are listed in Table 2. (b) Best-fit flattened profile from Bowman et al. (2018) Signal | $f_{*}$ |  $V_{c}$ (Km s-1) | $f_{X}$ | SED | $\tau$
---|---|---|---|---|---
1 | 0.05 | 4.2 | 0.1 | 2 | 0.082
2 | 0.5 | 16.5 | 0.1 | 2 | 0.082
3 | 0.5 | 35.5 | 1 | 1 | 0.066
4 | 0.005 | 4.2 | 50 | 1 | 0.066
5 | 0.05 | 35.5 | 1 | 1 | 0.083
Table 2: Astrophysical parameters for the global 21-cm signals shown in Figure
7: star formation efficiency $f_{*}$, virial circular velocity $V_{c}$, X-ray
efficiency $f_{X}$, spectral energy density SED, and CMB optical depth $\tau$.
The same number identifier will be employed across the paper to denote each
specific signal.

###  6.2 Extraction Methodology and Analysis

We model the simulated data with a log-log polynomial component and a scale
factor multiplied by the injected global signal as shown in Equation 10

$T_{ant}(\nu)=10^{\sum_{n=0}^{m}\alpha_{n}(\log_{10}\nu)^{n}}+c+k~{}T_{21cm}(\nu),$
(10)

where $T_{ant}(\nu)$ is the antenna temperature, and we use the logarithmic
term plus a constant parameter $c$ to model the foregrounds, and $k$ is the
scale factor associated with the global signal.

The convergence of the value of $k$ towards unity indicates the successful
recovery of the injected signal, while its inconsistency with unity points to
either low sensitivity or degeneracy between signal and foreground residuals.
Similar method was adopted in Monsalve et al. (2017), Singh et al. (2018) and
Singh et al. (2022).

We utilize emcee (Foreman-Mackey et al., 2013) to employ a Bayesian approach
to signal detection. Initially, the data is fit to Equation 10 using basin-
hopping, obtaining the best-fit parameters. These parameter values are
subsequently used as initial guesses for MCMC which yield marginalized scale
factors with uncertainties along with 2D probability distribution.

###  6.3 Results

Figure 8: Injected (dashed) and extracted (solid) profiles of different 21-cm signals. Each number represents a unique global signal model as shown in Figure 7. Each panel shows a primary beam specific to a configuration in Table 1. (a) Baseline SARAS configuration as defined in the first row of Table 1, (b) higher water conductivity configuration of 0.01 S/m (row 4), (c) case of 6-degree tilt on low conductivity water (row 11), and (d) raft height reduced from 200 mm to 50 mm (row 14). | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Conductivity Variation Cases
1 | $0.86^{+0.038}_{-0.040}$ | $1.04^{+0.012}_{-0.011}$ | $1.02^{+0.007}_{-0.007}$ | $1.13^{+0.01}_{-0.01}$ | $0.7^{+0.022}_{-0.030}$
2 | $0.79^{+0.042}_{-0.042}$ | $1.07^{+0.007}_{-0.007}$ | $1.05^{+0.006}_{-0.006}$ | $1.52^{+0.018}_{-0.014}$ | $0.69^{+0.018}_{-0.024}$
3 | $0.47^{+0.042}_{-0.050}$ | $0.99^{+0.007}_{-0.008}$ | $1.05^{+0.007}_{-0.007}$ | $1.29^{+0.011}_{-0.010}$ | $0.83^{+0.010}_{-0.012}$
4 | $0.57^{+0.040}_{-0.040}$ | $0.78^{+0.014}_{-0.014}$ | $1.07^{+0.007}_{-0.007}$ | $1.89^{+0.041}_{-0.031}$ | $0.85^{+0.012}_{-0.012}$
5 | $0.75^{+0.03}_{-0.03}$ | $1.01^{+0.007}_{-0.006}$ | $1.02^{+0.006}_{-0.006}$ | $1.14^{+0.009}_{-0.009}$ | $0.93^{+0.008}_{-0.008}$
6 | $0.42^{+0.042}_{-0.050}$ | $1.02^{+0.007}_{-0.008}$ | $1.05^{+0.006}_{-0.006}$ | $1.30^{+0.011}_{-0.011}$ | $0.8^{+0.010}_{-0.011}$
7 | $0.74^{+0.037}_{-0.044}$ | $1.08^{+0.007}_{-0.007}$ | $1.07^{+0.006}_{-0.006}$ | $1.39^{+0.014}_{-0.013}$ | $0.72^{+0.012}_{-0.015}$
8 | $0.89^{+0.025}_{-0.025}$ | $1.01^{+0.006}_{-0.006}$ | $0.98^{+0.006}_{-0.007}$ | $1.04^{+0.009}_{-0.009}$ | $0.98^{+0.007}_{-0.007}$
Tilt Variation Cases
9 | $1.10^{+0.050}_{-0.078}$ | $0.80^{+0.015}_{-0.016}$ | $0.99^{+0.69}_{-0.22}$ | $0.66^{+0.34}_{-0.35}$ | $0.96^{+0.016}_{-0.019}$
10 | $1.03^{+0.045}_{-0.047}$ | $1.13^{+0.028}_{-0.028}$ | $0.99^{+0.012}_{-0.012}$ | $0.80^{+0.034}_{-0.034}$ | $0.99^{+0.017}_{-0.018}$
11 | $0.72^{+0.047}_{-0.046}$ | $0.86^{+0.016}_{-0.017}$ | $1.103^{+0.009}_{-0.009}$ | $1.55^{+0.035}_{-0.032}$ | $1.09^{+0.018}_{-0.025}$
Raft Height Variation Cases
12 | $-84.36^{+0.008}_{-0.008}$ | $21.72^{+0.002}_{-0.002}$ | $-0.067^{+0.003}_{-0.003}$ | $5.50^{+0.002}_{-0.002}$ | $-9.18^{+0.002}_{-0.002}$
13 | $-174^{+0.005}_{-0.005}$ | $49.06^{+0.002}_{-0.002}$ | $-6.69^{+0.001}_{-0.001}$ |  $-0.318^{+0.008}_{-0.007}$ $-2.336^{+0.009}_{-0.029}$ | $8.06^{+0.092}_{-0.092}$
14 | $1.03^{+0.049}_{-0.055}$ | $0.75^{+0.015}_{-0.015}$ | $0.99^{+0.042}_{-0.012}$ | $0.89^{+0.021}_{-0.021}$ | $0.94^{+0.013}_{-0.014}$
Table 3: Best-fit scale factor after joint fitting beam-weighted foregrounds
and 21-cm signals. Each row denotes variation in the primary beam due to
change in antenna configuration, and different columns are different models of
global 21-cm signal as shown in Figure 7.

The scale factor value recovered in the case of an achromatic beam is
consistent with unity for all the injected 21-cm signals, establishing that
global 21-cm signal recovery is feasible in the absence of complex spectral
features in the foregrounds through joint modeling.

Scale factors obtained from the joint fitting of different SARAS beam-weighted
foreground and global 21-cm signal profiles are reported in Table 3. In most
cases, we find reasonable recovery of the injected signal with a scale factor
close to unity. However, some combinations of beams and signals show clear
bias. Such bias strongly indicates some degeneracy between foregrounds and
global signal profiles. Figure 8 shows injected and recovered signal profiles.
Upon closer investigation, distinct trends emerge from Table 3. Various signal
profiles for the same beam have demonstrated varying susceptibilities to
spectral distortion, as seen in Figure 8. It is evident from the scale factor
values from Table 3 that these biases in recovery depend on the spectral
nature of both foregrounds and signal profiles. Signals 1 and 5 consistently
exhibit a biased recovery, yielding values below 1 across all conductivity
variation cases. The bias ranges from 2 % to as high as 64 %. This can be
explained by spectrally smooth profiles of these signals, which are most
likely to be degenerate with beam-weighted sky profiles. Conversely, signal 4
has shown recovery with positive bias in all conductivity variations. The bias
ranges from 4 % to 89 % depending on degeneracy. The distortion of profiles
for all the signals is quite prominent in cases 12 and 13 of Table 3,
corresponding to reduced raft heights of 100 and 150 mm. These raft height
variation cases have proven to generate the most spectral complexities based
on their residual values in Table 1. Similarly, antenna tilt cases varying
between 2 - 6 degrees exhibit varying levels of distortion, largely influenced
by their chromaticity and the spectral profile of the signal itself.
Expectedly, the best-case scenarios emerge for signals 2 and 3, which have
distinct absorption features within the band. These signals exhibit the least
bias of a few percent for various conductivity cases.

Our further investigation involved a joint fitting absorption profile claimed
by the EDGES collaboration (Bowman et al., 2018). It’s important to highlight
that, except for situations where the raft height was set at 100 mm and 150
mm, the injected profile Bowman et al. (2018) remained consistently close to
unity across all cases. In most instances, its distinctive flattened
absorption feature did not exhibit degeneracy with the foreground.

##  7 Discussions and conclusions

The paper addressed diverse aspects of systematics that could emerge due to
direction-dependent primary beam response and their impact on global 21-cm
signal recovery. Our investigation began with understanding the spectral
nature of the intrinsic foregrounds and the effect of realistic beam-
weighting. This revealed the following key takeaways:

  * •

Foregrounds exhibit spectral smoothness to sub-mK level, limited by thermal
noise, with a wide achromatic beam. This was demonstrated via modeling
foregrounds by maximally smooth functions.

  * •

However, with several realistic variations of the SARAS beam, the spectral
complexity increased to the point that one inflection point was deemed
necessary. These spectral features were more prominent in some cases of beam
variations, owing to the varying chromatic nature of different beams.

  * •

The additional systematics due to the frequency dependence of the primary beam
was also correlated with beam derivatives for most variations.

  * •

The magnitude of systematics was further exacerbated due to a few percent
erroneous correction of S11 or transfer function, highlighting the importance
of in-situ measurements.

  * •

Ultimately, examining the influence of these additional features on the
retrieval of the global 21-cm signals with varying spectral profiles showcased
successful recovery of the signals in most cases. In a few cases, biased or
distorted extraction indicated some degeneracy between foregrounds, signal
profiles, and primary beams. The susceptibility to distortion showed trends
based on signal profiles and the spectral complexity of the beam-weighted
foregrounds. We observed that signals with sharp in-band features showed the
most robust recovery. Conversely, signals with broad-band or partially out-of-
band profiles seemed more prone to bias due to degeneracy with foregrounds and
beams.

The findings discussed above underscore the complex interplay between
foregrounds, direction-dependent primary beam, and spectral behavior of 21-cm
signals. The profile of 21-cm signal is crucial to consider since a plethora
of signal models are still unconstrained. Detectability prospects can
significantly vary based on the selection of frequency range and signal model.
For example, signals with absorption features, albeit with a lower amplitude,
might have higher detection prospects compared to deeper signals with wider
widths. Parameter variations with respect to the SARAS radiometer also inform
about the allowed degrees of freedom that can lead to a robust signal
detection.

The results also emphasize the importance of the nature of primary beam
chromaticity towards signal detection. Different experiments across the globe
employ unique antenna designs with different levels of beam chromaticities.
While acknowledging that corrections for primary beams may be non-trivial, the
impact of the choice of primary beams on signal detection prospects cannot be
overemphasized. Given the relation between direction-dependent gain of the
antenna and its surroundings, it becomes increasingly important to
electromagnetically model the response with rigorous characterization of the
terrain chosen for conducting experiments.

## Acknowledgments

The authors sincerely thank the Raman Research Institute and the CMB
Distortion Lab members for providing the resources and conducive research
environment necessary for this study. The authors are grateful to Mayuri S Rao
for her invaluable insights and collaborative efforts throughout this research
endeavor. They also thank Vishakha Pandharpure for the cross-verification of
several EM simulations used in this work. YA extended special appreciation to
Gunjan Tomar, Rajorshi Chandra, Kinjal Roy, and others for their unwavering
support during the project.

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