# Astrophysical information from the Rayleigh-Jeans Tail of the CMB

Raghunath Ghara  Garrelt Mellema  Saleem Zaroubi

###### Abstract

One of the explanations for the recent EDGES-LOW band 21-cm measurements of a
strong absorption signal around 80 MHz is the presence of an excess radio
background to the Cosmic Microwave Background (CMB). Such excess can be
produced by the decay of unstable particles into small mass dark photons which
have a non-zero mixing angle with electromagnetism. We use the EDGES-LOW band
measurements to derive joint constraints on the properties of the early
galaxies and the parameters of such a particle physics model for the excess
radio background. A Bayesian analysis shows that a high star formation
efficiency and X-ray emission of $4-7\times 10^{48}~{}\rm erg$ per solar mass
in stars are required along with a suppression of star formation in halos with
virial temperatures $\lesssim 2\times 10^{4}$ K. The same analysis also
suggests a 68 percent credible intervals for the mass of the decaying dark
matter particles, it’s lifetime, dark photon mass and the mixing angle of the
dark and ordinary photon oscillation of $[10^{-3.5},10^{-2.4}]$ eV,
$[10^{1.1},10^{2.7}]\times 13.8~{}\rm Gyr$, $[10^{-12.2},10^{-10}]$ eV and
$[10^{-7},10^{-5.6}]$ respectively. This implies an excess radio background
which is $\approx 5.7$ times stronger than the CMB around 80 MHz. This value
is a factor $\sim 3$ higher than the previous predictions which used a
simplified model for the 21-cm signal.

##  1 Introduction

Understanding the properties of the very first sources of light that formed in
the Universe is one of the goals of today’s astronomy. Direct observations of
those early sources such as the first stars or high-mass X-ray binaries
(HMXBs) is currently impossible. However, it has been shown that the
redshifted 21-cm signal from the neutral hydrogen from the epoch when first
sources formed in the Universe (often called the ‘Cosmic Dawn’) contains
substantial information about the early source population [see e.g. 1, 2, for
reviews]. Thus, observations of this signal constitute the most promising
indirect probe of the very first sources of radiation that formed in the
Universe.

Motivated by this, many radio telescope systems have been designed to attempt
to measure the signal. These systems fall into two categories. The first one
consists of large radio interferometers which are capable to measure the
redshift evolution of the spatial fluctuations of the signal in terms of
statistical quantities such as power-spectrum. Radio telescopes systems such
as the Low Frequency Array (LOFAR)111http://www.lofar.org/ [3, 4], the Giant
Metrewave Radio Telescope (GMRT)222http://www.gmrt.tifr.res.in[5, 6], the
Precision Array for Probing the Epoch of Reionization
(PAPER)333http://eor.berkeley.edu/ [7] and the Murchison Widefield Array
(MWA)444http://www.mwatelescope.org/ [8, 9] belong to this category. While
these existing systems are limited by low sensitivity to statistical
characterizations, the future low-frequency component of the Square Kilometre
Array (SKA-Low)555http://www.skatelescope.org/ will have enough sensitivity to
map the spatial fluctuations of the signal in tomographic images [10, 11]. The
second category of systems uses a single antenna to measure the redshift
evolution of the average strength of the signal. Experiments such as EDGES
[12], SARAS [13], BigHorns [14], SciHi [15] and LEDA [16] belong to this type.

Detection of the signal in both observational setups is challenging as the
signal is weaker than the astrophysical foregrounds by several orders of
magnitude [see e.g., 17]. In addition, a long integration time is needed to
reduce instrumental noise [18, 11, 19]. Both of these challenges require
excellent calibration of the radio telescope system in order to permit a
detection. While significant progress towards the detection of the signal has
been made, no independently confirmed detection of the 21 cm signal has been
achieved thus far. However, a number of observational campaigns have provided
upper limits on the strength of the signal or its fluctuations at different
redshifts [see e.g., 6, 20, 21, 3]. Although these upper limits are weak, the
most recent ones are providing unique information about the sources at those
epochs [22, 23] as well as the state of the inter-galactic medium [24, 25].

In 2018 the EDGES team claimed a detection of the redshift evolution of the
global 21-cm signal [26]. The redshift-amplitude profile of the average signal
as measured by the EDGES low-band instrument shows a minimum around redshift
$z\approx 17$ (at a frequency of approximately 80 MHz). While such a minimum
has been predicted by theoretical models of the signal, the measured
absorption strength is several times stronger than the strongest signal that
can be produced in a standard cosmological scenario [see e.g., 27, 28, 29, 30,
31]. While these measurements have not been independently confirmed and remain
debated [e.g. in 32, 33, 34, 35], two processes have been put forward to
explain such a strong signal around redshift 17. The first one postulates that
interaction between baryons and dark matter particles [36, 37, 38, 39]
produced a lower IGM temperature. However, limits from stellar cooling and
fifth force experiments rule out most interaction scenarios that produce
excess cooling of the gas through Rutherford-like scattering with the dark
matter [40, 41]. The parameter space of baryon-dark matter scattering is also
constrained by first star formation [see e.g., 42, 43].

The other proposed process is the existence of an excess radio background
(henceforth RBG) in addition to the Cosmic microwave background (CMB) around
80 MHz, the frequency corresponding to a redshift $z\approx$ 17 [44, 45, 46].
Such a RBG is motivated by measurements with the ARCADE [47] and LWA1 [48]
which have found evidence for it towards the Rayleigh-Jeans part of the CMB.
The LWA1 measurement shows that the RBG at frequencies 40-80 MHz is well
fitted with a power-law with spectral index -2.58$\pm$0.05 and a temperature
of $603^{+102}_{-92}$ mK at 1.42 GHz.

Even though an RBG provides a viable solution for the interpretation of the
EDGES low-band results, its physical origin remains uncertain. Extragalactic
radio point sources are unlikely to produce a RBG strong enough to explain the
measurements by EDGES. Other studies considered galactic origins such as
supermassive black holes [45] or supernovae from the first stars at redshift
$z\gtrsim 17$ [49]. However, the required emissivity of 1-2 GHz photons from
those early sources would need to be $\sim 10^{3}$ times stronger than
observed from local galaxies [49]. An alternative to the astrophysical origins
might be a RBG of cosmological origin. For example, photons produced during
the decay of unstable particles into dark photons with non-zero mixing angle
with electromagnetism [see e.g., 50] remain a viable solution for the EDGES
low-band results.

Studies such as [51] used the EDGES results to put constraints on a uniform
RBG with a synchrotron-like spectrum that existed throughout cosmic history.
However, these authors did not specify the origin of the RBG. In this paper we
instead consider a physically motivated model for the RBG. We will follow the
model of [50] where unstable dark matter particles decay into dark photons
with a small mass and non-vanishing mixing angle with electromagnetism which
increase the photon count at the Rayleigh-Jeans frequencies. [50] constrain
the parameters of their model assuming the required radio background is a
factor $\approx 2$ higher than the one which is produced by the CMB but they
did not include a model for the 21-cm signal. Here, we combine a model of the
21-cm signal with the dark photon model for the RBG to constrain both the
source properties at $z\approx 17$ as well as the parameters of the RBG model.

We have organised the paper in the following way. In Section 2 we describe our
analytical model of the global 21-cm signal, including the model for the RBG
which we use in this study. We present our results in section 3 and conclude
in Section 4. The cosmological parameters which we use in this paper are
$\Omega_{\rm m}=0.32$, $\Omega_{\rm B}=0.049$, $\Omega_{\Lambda}=0.68$,
$h=0.67$, $\sigma_{8}=0.83$ and $n_{\rm s}=0.96$ [52].

##  2 Methodology

###  2.1 Model for the 21-cm signal

The 21-cm signal from the H i gas is measured as the differential brightness
temperature against the background radiation and can be written as

$\delta T_{\rm b}=27~{}x_{\rm HI}(1+\delta_{\rm B})\left(\frac{\Omega_{\rm
B}h^{2}}{0.023}\right)\sqrt[]{\frac{0.15}{\Omega_{\rm
m}h^{2}}\frac{1+z}{10}}\left(1-\frac{T_{\rm rad}}{T_{\rm S}}\right)\,\rm{mK},$
(2.1)

where $x_{\rm HI}$, $\delta_{\rm B}$ and $T_{\rm S}$ denote the neutral
fraction, density contrast, the spin temperature of the hydrogen gas
respectively. $T_{\rm rad}$ is the temperature of the radio background.
$T_{\rm rad}$ becomes the CMB temperature $T_{\gamma}=2.725\times(1+z)$ K in
absence of a RBG. Note that the spin-temperature calculation in our model
depends on collisional coupling, $\rm{Ly{\alpha}}$ coupling and the scattering
with the CMB photons. All these coupling strengths are dependent on $T_{\rm
rad}$ (see e.g. [51]).

We adopt an analytic approach as described below to model the expected 21-cm
signal in the presence of spin temperature fluctuations. The analytical model
[previously used in 53, 41] derives from previous works such as [29, 54]. It
incorporates $\rm{Ly{\alpha}}$, UV, and X-ray photons from sources of
radiation which are presumed to have formed from gas associated with dark
matter halos. The number density of dark matter halos at a given redshift is
determined using the Press-Schechter halo mass function. We assume that star
formation only occurs inside dark matter halos with virial temperatures above
$T_{\rm vir}$. We vary $T_{\rm vir}$ in this study.

The model estimates the volume averaged ionization fractions of the highly
ionized H ii regions ($x_{i}$) and largely neutral gas in the IGM outside
these H ii regions ($x_{e}$). We assume the temperature of the ionized H ii
regions to be $\sim 10^{4}$ K but estimate the gas temperature ($T_{\rm K}$)
of the largely neutral medium outside the H ii regions from the relevant the
heating and cooling processes.

The average number density of UV, X-ray, and $\rm{Ly{\alpha}}$ photons in the
model follows the mass fraction of collapsed objects, $f_{\rm coll}$. The rate
of emission of UV photons per baryon is given by

$\Lambda_{i}=\zeta\frac{{\rm d}f_{\rm coll}}{{\rm d}t}.$ (2.2)

The ionization efficiency parameter $\zeta=N_{\rm ion}\times f_{\rm esc}\times
f_{\star}$ depends on the average number of ionizing photons per baryon
produced in the stars ($N_{\rm ion}$), the star formation efficiency
($f_{\star}$) and the escape fraction of the UV photons ($f_{\rm esc}$). All
these quantities are uncertain during the Cosmic Dawn. Their effects on the
ionization state of the IGM are also degenerate. We therefore fix $N_{\rm
ion}=4000$, which corresponds to population II types of stars, and $f_{\rm
esc}=0.1$, and only vary $f_{\star}$ for modelling reionization. We note that
ionization levels remain very low for the redshifts that we consider here and
thus the ionization state of the IGM does not have a significant impact on the
global 21-cm signal.

Our model also includes X-ray sources which can partially ionize and heat the
neutral IGM at large distances from the H ii regions which form around the
sources. We follow [29] to include their impact. This approach assumes that
the emissivity of X-ray photons from the sources follows the star formation
rate density. The comoving photon emissivity for X-ray sources is modelled as

$\epsilon_{X}(\nu,z)=\epsilon_{X}(\nu)\left(\frac{\rm SFRD}{{\rm
M}_{\odot}~{}\rm yr^{-1}~{}\rm Mpc^{-3}}\right),$ (2.3)

where SFRD is the star formation rate density. We model the SFRD as

${\rm SFRD}(z)=\bar{\rho}_{b}^{0}~{}f_{\star}~{}\frac{d}{dt}f_{\rm coll}(z),$
(2.4)

where $\bar{\rho}_{b}^{0}$ is the cosmic mean baryon density at $z=0$. We use
an X-ray spectral distribution given by

$\epsilon_{X}(\nu)\propto\frac{L_{0}}{h\nu_{0}}\left(\frac{\nu}{\nu_{0}}\right)^{-\alpha_{X}-1}$
(2.5)

with $L_{0}=f_{X}\times 10^{41}~{}\rm erg~{}s^{-1}~{}Mpc^{-3}$,
$h\nu_{0}=1~{}\rm keV$. We choose the proportional constant in equation 2.5
such that the rate of emission of energy in X-ray band (0.1-10 keV in our
case) per unit volume per unit SFRD is $L_{0}$. It is therefore equivalent to
assuming that stellar populations produce a total of $f_{X}\times 3.156\times
10^{48}$ erg ${\rm M}_{\odot}^{-1}$ in the X-ray band. We fix the X-ray
spectral index $\alpha_{X}$ to 0.5 but vary the X-ray parameter $f_{X}$.

The $\rm{Ly{\alpha}}$ background has a crucial role in determining the 21-cm
signal through coupling the spin-temperature of atomic hydrogen with the gas
temperature through Wouthuysen-Field coupling [55]. Here we used the
methodology of [56] to estimate the average $\rm{Ly{\alpha}}$ photon flux. We
assume a power-law spectrum
$\epsilon_{s}(\nu)=f_{\alpha}A_{\alpha}\nu^{-\alpha_{s}-1}$ between
$\rm{Ly{\alpha}}$ and $\rm{Ly{\beta}}$ and between $\rm{Ly{\beta}}$ and the
Lyman limit, where the power-law indices $\alpha_{s}$ can differ between these
two spectral regimes. The spectral index $\alpha_{s}$ between
$\rm{Ly{\alpha}}$ and $\rm{Ly{\beta}}$ is taken to be 0.14 which corresponds
to population II type sources. The normalization factor $A_{\alpha}$ is
estimated such that the number of $\rm{Ly{\alpha}}$ photons per baryon in the
range $\rm{Ly{\alpha}}$–$\rm{Ly{\beta}}$ is 6520 for $f_{\alpha}=1$. The
spectral index in the range $\rm{Ly{\alpha}}$–Lyman limit is adjusted so that
the total number of photons per baryon for this wavelength regime is 9690.

Our model for the 21-cm signal thus has three free parameters, namely the star
formation efficiency $f_{\star}$, the X-ray efficiency $f_{X}$ and minimum
virial temperature for star forming halos, $T_{\rm vir}$. These control the
$\rm{Ly{\alpha}}$ coupling, X-ray heating and the number density of these
sources, respectively.

###  2.2 Model for the excess radio background

We model the RBG following [50]. The first part of this model consists of the
decay of long-lived unstable scalar particles $a$ into light vector particles
$A^{{}^{\prime}}$, which are often called dark photons. If we assume the life
time of the scalar particles $\tau_{a}\gg\tau_{U}$ where $\tau_{U}\approx
13.8\times 10^{9}$ yr is the age of the Universe, then dark matter is a
natural candidate for this scalar field. The energy spectrum of
$A^{{}^{\prime}}$ at redshift $z$ due to the decay $a\rightarrow
2A^{{}^{\prime}}$ can be written as [50],

$\frac{dn_{A^{{}^{\prime}}}}{dw}(w,z)=\frac{2\Omega_{a}\rho_{c}(1+z)^{3}}{m_{a}\tau_{a}wH(\alpha-1)}\Theta(\alpha-1-z)$
(2.6)

where $\Omega_{a}=\Omega_{\rm m}$ in our case, $\rho_{c}$ is the critical
density, $H$ is Hubble rate evaluated at redshift $\alpha-1$ where
$\alpha=m_{a}(1+z)/(2w)$. Here, $m_{a}$ is the mass of the scalar particles
and $w$ is the photon energy.

The second part of this model is the mixing of the dark photons
$A^{{}^{\prime}}$ with ordinary photons $A$. Cosmological
$A^{{}^{\prime}}\leftrightarrow A$ oscillation is the most significant under
resonance conditions, $m_{A^{{}^{\prime}}}=m_{A}(z)\simeq 1.7\times
10^{-14}\times(1+z)^{3/2}X^{1/2}_{e}(z)$ eV where
$m_{A}^{{}^{\prime}}\rightarrow 0$ is the mass of the dark photons and
$m_{A}(z)$ is the plasma mass of the photons at redshift $z$ [57, 58]. The
redshift evolution of the average ionization fraction $X(z)$ is derived from
recfast [59]. The probability of oscillation at the resonance can be written
as [57]

$P_{A\rightarrow A^{{}^{\prime}}}=P_{A^{{}^{\prime}}\rightarrow A}\simeq
1-\exp{\left(-\frac{\pi\epsilon^{2}m^{2}_{A^{{}^{\prime}}}}{w}\times\left|\frac{d{\rm
log}~{}m^{2}_{A}}{dt}\right|^{-1}_{t=t_{\rm res}}\right)}$ (2.7)

where $\epsilon$ is the mixing angle of the dark and ordinary photon
oscillation, $w$ is the energy of the photon, $t_{\rm res}$ is the time of
resonance for $m_{A^{{}^{\prime}}}$. This modifies the number count of CMB
photons as [50],

$\frac{dn_{A}}{dw}\rightarrow\frac{dn_{A}}{dw}\times(1-P_{A\rightarrow
A^{{}^{\prime}}})+\frac{dn_{A}^{{}^{\prime}}}{dw}\times
P_{A^{{}^{\prime}}\rightarrow A}.$ (2.8)

Our physical model for the RBG of cosmological origin thus has four free
parameters, namely $m_{a}$, $\tau_{a}$, $m_{A^{{}^{\prime}}}$ and $\epsilon$.
The decay of dark matter can produce a significant impact on the Rayleigh-
Jeans tail of the CMB if $m_{a}$ remains between $10^{-5}$ to $10^{-1}$ eV.
For $m_{a}\lesssim 10^{-5}$ eV, the produced photons due to the decay of dark
matter are too soft to have an impact on the 21-cm signal at $z\approx 17$.
Assuming that all cold dark matter particles decay into dark photons, the
lifetime $\tau_{a}$ could be infinitely large. We choose $\tau_{a}$ range
between 12 to $10^{7}$ $\tau_{U}$. The larger values will have insignificant
effects on the RJ tail of the CMB while the lower bound is taken from [60]
which only relies on the CMB to obtain the constraint. We choose
$m_{A^{{}^{\prime}}}$ range $10^{-14}-10^{-9}$ eV for which the time when the
resonance occurs falls between recombination and redshift of our interest. The
bounds on $\epsilon$ are sensitive to the bounds on $m_{A^{{}^{\prime}}}$. We
vary $\epsilon$ in the range $10^{-10}-10^{-4}$ which is the most relevant
range for this study.

###  2.3 Bayesian Framework

The main aim of this work is to infer astrophysical and particle physics
information using the global 21-cm signal vs redshift profile as extracted
from EDGES low-band data. As mentioned in the earlier sections, we have a
total of seven free parameters in our combined model, three related to
astrophysical radiation sources and four related to the RBG model. Given a set
of these seven parameters ($\theta$), the above mentioned method generates a
$\delta T_{\rm b}$ vs $z$ profile which we later compare to the measured EDGES
profile in a Bayesian framework. We couple the above mention code to the
cosmomc Markov chain Monte Carlo (MCMC) generic subroutine [61] to explore the
seven-dimensional parameter space. Note that we do not consider all the
frequency channels of EDGES observation, instead, we consider redshifts 21.2,
20., 19.4, 18.9, 17.3, 16.1, 15.2, 14.9, and 14.4 to estimate the
$\chi^{2}(\theta)=\sum_{i}\left(\frac{\delta T_{\rm b,m}(z_{i},\theta)-\delta
T_{\rm b,o}(z_{i})}{\sigma_{\rm o}}\right)^{2}$ where $\delta T_{\rm b,m}$ and
$\delta T_{\rm b,o}$ are the modelled $\delta T_{\rm b}$ and the observed
$\delta T_{\rm b}$ respectively. We assume $\sigma_{\rm o}$ is 25 mK which is
consistent with [32]. We use $-\chi^{2}(\theta)$ as the likelihood in the MCMC
algorithm.

The parameter ranges explored in this study are given in Table 1. We use flat
priors over the ranges of the parameter space. We use one additional flat
prior on $\epsilon$-$m_{A^{{}^{\prime}}}$ which is based on the findings of
COBE-FIRAS [62]. The observed data from COBE-FIRAS has provided strong
constraints on the values of $\epsilon$ as a function of $m_{A^{{}^{\prime}}}$
which exclude a significant part of the parameter space with high values of
$\epsilon$ [63].

Scenarios |  log($f_{\star}$) |  log($f_{X}$) |  log($\frac{T_{\rm vir}}{K}$) |  log($\frac{m_{a}}{\rm eV}$) |  log($\frac{\tau_{a}}{\tau_{U}}$) |  log($\frac{m_{A^{{}^{\prime}}}}{\rm eV}$) |  log($\epsilon$)
---|---|---|---|---|---|---|---
$S_{1}$ | -1. | 0.0 | 4.0 | -3.0 | 2.0 | -11.5 | -6.4
$S_{2}$ | -0.3 | 0.0 | 4.0 | -3.0 | 2.0 | -11.5 | -6.4
$S_{3}$ | -1. | 1.0 | 4.0 | -3.0 | 2.0 | -11.5 | -6.4
$S_{4}$ | -1. | 0.0 | 4.5 | -3.0 | 2.0 | -11.5 | -6.4
$S_{5}$ | -1. | 0.0 | 4.0 | -2.5 | 2.0 | -11.5 | -6.4
$S_{6}$ | -1. | 0.0 | 4.0 | -3.0 | 3.0 | -11.5 | -6.4
$S_{7}$ | -1. | 0.0 | 4.0 | -3.0 | 2.0 | -12.0 | -6.4
$S_{8}$ | -1. | 0.0 | 4.0 | -3.0 | 2.0 | -11.5 | -7.5
MCMC | [-4, -0.3] | [-4, 4] | [3, 5] | [-5, -1] | [1.08, 7] | [-14, -9] | [-10, -4]
Table 1: Sets of seven parameters of the eight different scenarios considered
in Section 3.1. The bottom row shows the range of those seven parameters as
used in the MCMC analysis in section 3.2. Figure 1: Average of the
differential brightness temperature as a function of redshift. These different
models are considered in section 3.1. The thin dot-dashed curve shows the
measured $\delta T_{\rm b}$ vs redshift profile by EDGES-low observation. The
thin double-dot dashed curve shows the $\delta T_{\rm b}$ profile when the
astrophysical parameters are the same as $S_{1}$ and $T_{\rm rad}=T_{\gamma}$.
The black dotted curve shows the $\delta T_{\rm b}$ profile for the best-fit
parameter values from the MCMC analysis in section 3.2. Figure 2: The color
bar shows the logarithm of the ratio of the contribution from the RBG model
and the CMB at a wavelength of 21-cm at redshift 17. Two panels show this as a
function of parameters of the RBG model.

##  3 Results

###  3.1 Exploratory scenarios

We first explore a few scenarios/models to understand the impact of each of
the parameters on the global 21-cm signal from the Cosmic Dawn. In total we
consider eight scenarios with different combinations of the seven parameters.
The parameter values of those scenarios are listed in Table 1. The redshift
evolution of the average differential brightness temperature for each of these
scenarios is shown in Figure 1.

Our fiducial model $S_{1}$ corresponds to $f_{\star}=0.1,f_{X}=1,T_{\rm
vir}=10^{4}~{}{\rm K},m_{a}=10^{-3}~{}{\rm
eV},\tau_{a}=100\times\tau_{U},m_{A^{{}^{\prime}}}=10^{-11.5}~{}{\rm
eV},\epsilon=10^{-6.4}$. For these parameter values, the corresponding $\delta
T_{\rm b}$ vs redshift profile has an absorption trough around redshift
$\approx 17.5$ with an amplitude $\approx 0.5$ K. For reference Figure 1 also
shows the $\delta T_{\rm b}(z)$ profile for the same astrophysical parameters
as in scenario $S_{1}$ but without a RBG (see thin double-dot dashed curve).
In that case the minimum $\delta T_{\rm b}$ remains $\approx-150$ mK at a
redshift $\approx 18$. One can also see that the $\delta T_{\rm b}$ profile of
model $S_{1}$ does not quite match with the measured $\delta T_{\rm b}$
profile from EDGES (see dot-dashed curve). Especially, the width of the
$\delta T_{\rm b}$ profile of the $S_{1}$ model is larger than the width of
the measured $\delta T_{\rm b}$ profile by EDGES-low. One might guess that a
delayed star formation history and more efficient heating of the gas in the
IGM is required to obtain a $\delta T_{\rm b}$ profile that provides a better
fit to the EDGES measurements than the $S_{1}$ model does.

Scenario $S_{2}$ only differs from $S_{1}$ in its value for $f_{\star}$. Its
larger $f_{\star}$ value of $\approx 0.5$ increases the $\rm{Ly{\alpha}}$
coupling as well as the heating due to the X-rays. This results in a shift of
the trough to higher redshift compared to the $S_{1}$ scenario. Also, the
depth of the trough is enhanced.

We choose a higher value of $f_{X}=10$ for scenario $S_{3}$ while all other
parameters have the same value as in $S_{1}$. This causes more efficient X-ray
heating from the beginning of Cosmic Dawn which shifts the trough to higher
redshift while reducing its depth.

Scenario $S_{4}$ has a larger $T_{\rm vir}$ at $10^{4.5}$ K as compared to
$S_{1}$ model. Removing the lower mass halos results in a delay in the global
star formation and an overall reduction of the effects of stars. This shifts
the abosrption trough to lower redshifts and decreases its depth.

After having explored the impact of the astrophysical parameters, we now turn
to the parameters of the RBG model. Scenario $S_{5}$ is identical to $S_{1}$
except for a higher $m_{a}$ value of $10^{-2.5}$ eV. This results in a smaller
number density of the dark photons (see Equ 2.6). This reduces the depth of
the absorption trough and shifts it to a higher redshift.

Increasing the value of $\tau_{a}$ also causes a decrease of number density of
the dark photons and thus has a similar effect as increasing $m_{a}$, as can
be seen in the $\delta T_{\rm b}$ profile for scenario $S_{6}$ which has
$\tau_{a}=10^{3}~{}\tau_{U}$, a value 10 times larger than used in $S_{1}$.

Scenario $S_{7}$ has a smaller $m_{A^{{}^{\prime}}}$ value of $10^{-12}$ eV
compared to $m_{A^{{}^{\prime}}}=10^{-11.5}$ eV in $S_{1}$. This causes a
smaller $P_{A\rightarrow A^{{}^{\prime}}}$ value as $P_{A\rightarrow
A^{{}^{\prime}}}\propto m^{2}_{A^{{}^{\prime}}}$. A similar effect can be
achieved by reducing the value of $\epsilon$ as is done in scenario $S_{8}$
where $\epsilon$ is a bit more than 10 times lower than in $S_{1}$. This leads
to shallower absorption troughs which are shifted to larger redshifts. The
reduction in the RBG for scenario $S_{8}$ is such that its profile resembles
the one in which the RBG is entirely absent.

Figure 2 illustrates the impact of the four RBG parameters on the RBG. The
color bar shows the logarithm of the ratio of the contribution of the RBG and
the CMB at a wavelength of 21-cm at redshift 17. One can see that the ratio
increases for smaller $m_{a}$ and larger $m_{A}^{{}^{\prime}}$. The region in
purple in the right bottom corner of the left panel of Fig 2 represents a part
of the parameter space for which the produced photons are too soft and do not
have an impact on the 21-cm signal at $z\approx 17$. For $m_{a}\lesssim
10^{-13.7}$ eV (purple region on the left hand side of the same panel), the
redshift of resonance is smaller than $z\approx 17$ which makes
$P_{A^{{}^{\prime}}\rightarrow A}=0$. Note that we fix $\epsilon=10^{-6.4}$
and $\tau_{a}=10^{2}~{}\tau_{U}$ for the left panel.

We vary $\epsilon$ and $\tau_{a}$ in the right panel of Fig 2 while we fix
$m_{a}=10^{-3}$ eV and $m_{A^{{}^{\prime}}}=5\times 10^{-10}$ eV. The ratio of
the contribution from the RBG and the CMB increases for smaller $\tau_{a}$ and
$\epsilon$ values. These panels give some idea about which parts of the 4D
parameter space of the RBG cannot satisfy the measured $\delta T_{\rm b}$
profile. However, it is difficult to pinpoint values of the ratio of the
contribution from the RBG and the CMB that are required for the measured
$\delta T_{\rm b}$ profile as that will crucially depend on the three
astrophysical parameters. Thus, we will explore the full seven-dimensional
parameter space in the next section to find the part of the parameter space
that best agrees with the observation.

Parameters | Explored range | Mean | Standard Deviation | Best fit |  68$\%$ limits |  95$\%$ limits
---|---|---|---|---|---|---
$\mathrm{log}(f_{\star})$ | [-4, -0.3] | -0.38 | 0.08 | -0.3 | [-0.4, -0.3] | [-0.55, -0.3]
$\mathrm{log}(f_{X})$ | [-4, 4] | 0.26 | 0.14 | 0.14 | [0.12, 0.34] | [0.01, 0.54]
$\mathrm{log}(\frac{T_{\mathrm{vir}}}{K})$ | [3, 5] | 4.41 | 0.04 | 4.37 | [4.34, 4.4] | [4.28, 4.43]
$\mathrm{log}(\frac{m_{a}}{\mathrm{eV}})$ | [-5, -1] | -3.03 | 0.53 | -3.84 | [-3.5, -2.4] | [-4.1, -2.1]
$\mathrm{log}(\frac{\tau_{a}}{\tau_{U}})$ | [0.5, 7] | 2.27 | 0.9 | 1.48 | [1.1, 2.7] | [1.07, 4.0]
$\mathrm{log}(\frac{m_{A^{{}^{\prime}}}}{\mathrm{eV}})$ | [-14, -9] | -11.3 | 1.1 | -12.76 | [-12.2, -10.0] | [-13.2, -9.4]
$\mathrm{log}(\epsilon)$ | [-10, -4] | -6.3 | 0.64 | -7.38 | [-7., -5.6] | [-7.5, -5.2]
Table 2: 68 and 95 percent credible intervals on the seven parameters used to
model the $\delta T_{\rm b}$ in this study. These limits are obtained from the
measurements of $\delta T_{\rm b}$ vs redshift profile by EDGES-low band
observation.  Figure 3: Constraints on the parameters of this study from the
MCMC analysis using the EDGES observation. The color-bar shows the probability
that models are valid. The solid and dashed curves show the 68 and 95 percent
credible intervals of the consistent models. The diagonal panels show the 1D
marginalized probability distribution of the parameters.

###  3.2 MCMC results

The details of the framework to explore the seven-dimensional parameter space
are given in section 2.3 and the range of the parameters explored in the MCMC
analysis is listed in Table 1. Additional priors to this MCMC analysis are the
allowed parameter space of $\epsilon$ and $m_{A^{{}^{\prime}}}$ derived from
the COBE-FIRAS results and the age of the decaying dark matter particles
$\tau_{a}$, i.e. $\tau_{a}>12\tau_{U}$, as obtained in [60].

Figure 3 shows the outputs of the MCMC analysis and the 68 and 95 percent
credible intervals on the parameters are listed in Table 2. The data is found
to prefer a high star formation efficiency, reaching the maximum value of our
prior range. The 68 percent credible interval for $f_{X}$ is [1.3, 2.2], while
that for $T_{\rm vir}$ is $[2.2,2.5]\times 10^{4}$ K. The limit on $T_{\rm
vir}$ indicates a scenario where the star formation occurs inside the massive
halos only which suggests a delayed star formation history. This is expected
as a smaller value of $T_{\rm vir}$ shifts the $\delta T_{\rm b}$ vs redshift
profile towards a higher redshift (as we have shown by comparing scenario
$S_{1}$ and $S_{4}$ in section 3.1). However, a larger value of $T_{\rm vir}$
corresponds to a smaller value of $f_{\rm coll}$ which tends to produce a
weaker signal due to a weaker $\rm{Ly{\alpha}}$ coupling. In the best-fit star
formation scenario, this is compensated for by a larger value of $f_{\star}$
that enhance the strength of the signal (see the comparison of scenarios
$S_{1}$ and $S_{2}$ in section 3.1). On the other hand, the limits of $f_{X}$
indicate efficient X-ray heating which is required for the quick transition of
the mean signal from absorption to emission at redshift $\lesssim 17$ (see the
comparison of scenarios $S_{1}$ and $S_{3}$ in section 3.1).

It is straightforward to understand that the limits on the astrophysical
parameters as obtained from the MCMC analysis are mainly determined by the
shape of the $\delta T_{\rm b}$ vs redshift profile. As we find that the best-
fit scenario indicates a high value for the $f_{X}$ parameter, one might
expect a non-negligible impact of X-ray heating on the gas at the redshifts of
our interest. On the other hand, the $\rm{Ly{\alpha}}$ coupling is expected to
be not very strong as the MCMC analysis indicates a higher value of $T_{\rm
vir}$ which means a smaller $f_{\rm coll}$ value. These two effects result in
a weaker 21-cm signal compared to the scenario in which the IGM is completely
unheated in the presence of a strong $\rm{Ly{\alpha}}$ background. Thus, one
might expect a stronger RBG will be required as compared to [50] to compensate
for the stronger X-ray heating and weaker $\rm{Ly{\alpha}}$ coupling at these
redshifts.

The 68 percent credible intervals for the four parameters of the RBG, namely
$m_{a}$, $\tau_{a}$, $m_{A^{{}^{\prime}}}$ and $\epsilon$, are
$[10^{-3.5},10^{-2.4}]$ eV, $[10^{1.1},10^{2.7}]\times\tau_{U}$,
$[10^{-12.2},10^{-10.0}]$ eV and $[10^{-7},10^{-5.6}]$. The effective
temperature of the radio background corresponding to the best fit parameter
values at $z\approx 17$ is $T_{\rm rad}\approx 5.7\times T_{\gamma}$. This is
consistent with the $T_{\rm rad}(z\approx 17)\gtrsim 2.9\times T_{\gamma}$ as
found in [51]. This also shows that the data prefer a much stronger RBG than
the one assumed in [50], namely $T_{\rm rad}\approx 2\times T_{\gamma}$.

We show the $\delta T_{\rm b}$ vs redshift profile for the best-fit values of
the parameters in Fig. 1 (thick black dotted curve). It is obvious that it is
still difficult to reproduce the exact shape of the measured curve with the
source model used in this study. To obtain a flat-bottomed $\delta T_{\rm b}$
profile will need exotic source models such as the one discussed in [64]. It
is worth pointing out that even though the source models differ, our results
are consistent with $T_{\rm rad}(z\approx 17)\approx 5.3\times T_{\gamma}$ as
obtained in the best-fit model of [64].

##  4 Discussion & conclusions

In this work we interpret the claimed EDGES detection of a deep 21-cm
absorption signal around $z\approx 17$ in terms of RBG produced by dark matter
particles decaying into dark photons that later oscillate into ordinary
photons due to a nonvanishing mixing angle. This physical model can enhance
the photon number density in the Rayleigh-Jeans tail of the CMB that remains
unconstrained till now. Such an enhanced radio background might be an
explanation for the EDGES-low band results which show an absorption signal a
few factors stronger than theoretical predictions based on standard physics.

The main aim of this work is to find constraints both on the properties of the
astrophysical sources of radiation around redshift $\approx 17$ as well as the
particle physics parameters of the dark photon model for the RBG. For this, we
explored a seven-dimensionial parameter space in which three parameters
describe the astrophysical sources, namely the star formation efficiency
$(f_{\star})$, X-ray heating efficiency $f_{X}$ and the minimum virial
temperature $T_{\rm vir}$ of dark matter halos in which star formation occurs.
The other four parameters determine the RBG, namely the mass of the decaying
dark matter particles $(m_{a})$, the lifetime of the dark matter particles
$(\tau_{a})$, the mass of the dark photons $(m_{A^{{}^{\prime}}})$ and the
mixing angle between dark and ordinary photon oscillation ($\epsilon$).

We explored this seven-dimensional parameter space using a Bayesian framework
to obtain constraints on the parameters using the $\delta T_{\rm b}$ vs
redshift profile as measured by the EDGES-low band experiment. Our main
findings are as follows (numerical values correspond to 68 percent credible
intervals):

  * •

We find that a higher star formation efficiency (0.5 in this case) and an
intermediate value of $f_{X}\approx 1-2$ which means emission of X-ray energy
per stellar mass of $4-7\times 10^{48}~{}\rm erg$ ${\rm M}_{\odot}^{-1}$ is
required to to produce efficient heating at $z\lesssim 17$. At the same time,
one needs a star formation suppression at dark matter halos with virial
temperature $\lesssim 2\times 10^{4}$ K to produce a late start of the $\delta
T_{\rm b}$ profile.

  * •

In addition, one needs a strong radio background which provides tighter
constraints on the parameters of the radio background model. The 68 percent
credible intervals on $m_{a}$, $\tau_{a}$, $m_{A^{{}^{\prime}}}$ and
$\epsilon$ are $[10^{-3.5},10^{-2.4}]$ eV,
$[10^{1.1},10^{2.7}]\times\tau_{U}$, $[10^{-12.2},10^{-10}]$ eV and
$[10^{-7},10^{-5.6}]$ respectively.

  * •

The best fit parameter values suggest a radio background with temperature
$\approx 5.7\times T_{\gamma}$.

The derived constraints on the astrophysical source parameters are in
disagreement with those from studies which assume that the radio background
only consists of the black body radiation of the CMB and instead use an an
excess cooling mechanism to explain the EDGES results. Those scenarios prefer
a lower star formation efficiency combined with a higher cooling rate.

Our results show that the EDGES-low band observation alone excludes a large
part of the parameter space for a RBG due to the decay of dark matter into
dark photons that later oscillate into ordinary photons. However, a
significant part of the parameter space still remains valid. Future probes
such as PIXIE [65] or PRISM [66] will provide stronger constraints on the
parameters of the radio background as well as on the astrophysical parameters.

The findings of this study are based on the assumption that the dark matter
halo mass function (HMF) follows a Press-Schechter (PS) [67] form. However, as
for example shown in [68], the PS-HMF overestimates the number of low mass
halos and underestimates the number of rare high mass halos compared to
results from N-body simulations. At the redshifts we are considering here,
halos with $T_{\mathrm{vir}}\gtrsim 2\times 10^{4}$ K are rare and their
abundance is therefore likely underestimated by the PS-HMF. The HMF from the
ellipsoidal collapse model from Sheth & Tormen (ST) [69] has been shown to
overestimate the number of rare high mass halos compared to numerical results.
We therefore repeated our analysis using the ST-HMF to investigate the impact
of the choice of HMF. For the ST-HMF the 68 percent credible interval limits
for log($f_{\star}$), log($f_{X}$), log($\frac{T_{\rm vir}}{\rm K}$),
log($\frac{m_{a}}{\rm eV}$), log($\frac{\tau_{a}}{\tau_{U}}$),
log($\frac{m_{A^{{}^{\prime}}}}{\rm eV}$) and log($\epsilon$) change to [-0.6,
-0.46], [0.45, 0.78], [4.7, 4.9], [-4.5, -2.1], [1.1, 5.3], [-13.2, -9] and
[-8.1, -5.1] respectively. The best fit values of these parameters are -0.47,
0.73, 4.77, -3.9, 1.59, -12.96 and -7.17 respectively. Clearly the choice of
HMF has a considerable impact on the results, which is not surprising if one
realises that $f_{\rm coll}(T_{\rm vir}=10^{4}\rm K,z=17)$ for the ST HMF is
approximately a factor 4 higher than for the PS HMF. With a similar analogy,
the limits on our model parameters are expected to be in between the limits
from the PS and ST HMF cases if the HMF follows the form as given in [68].

Besides HMF, the obtained constraints on the parameters of this study are also
limited by the uncertainties on the type of sources present at those
redshifts. The source model of this study assumes only population II type
stars. While it is expected that these stars dominate the contribution to the
overall photon emissions at the redshifts of our interest, the contribution
from population III type stars might be also important [51, 70, 71]. The
emission of $\rm{Ly{\alpha}}$, X-ray and radio photons from population III
stars and their remnants can have a significant impact on the shape of the
brightness temperature profile at these redshifts [see, e.g., 72] and thus,
will require a smaller contribution to the overall radio ($\rm{Ly{\alpha}}$
and X-ray) background from the RBG (population II star) model.

One needs to keep in mind that the results of this study are based on the
simple-minded source model which assumes the star formation efficiency
($f_{\star}$) is constant over the range of dark matter halo masses ($M_{\rm
halo}$) and redshifts relevant for the EDGES signal. In general, $f_{\star}$
varies with hosting dark matter halo mass and redshift. Motivated by the
observations of the high-redshift ($z\lesssim 8$) faint galaxy population
[e.g., 73], the halo mass dependencies on $f_{\star}$ are often characterised
by a power-law such as $f_{\star}\propto M^{\alpha_{\star}}_{\rm halo}$ with
positive value of $\alpha_{\star}$. This suppresses (enhances) star-formation
in low-mass (high-mass) halos. The mass-dependent $f_{\star}$ also allows
molecular-cooling low-mass halos to form Population III stars at low
efficiency which is consistent with theoretical predictions [see e.g., 74,
75]. Under such a sophisticated model for star formation as used in
[71]666Note that the constraints in [71] are based on the position of the
21-cm absorption trough at 78 MHz frequency, while we consider the entire
shape of the brightness temperature profile within the redshift range of
EDGES-low. the 68 percent credible limits on $T_{\textup{vir}}$ are lower than
what is obtained in this study. Depending on different feedback mechanisms
such as Lyman-Werner feedback, radiative feedback, etc. the star formation
efficiency can also evolve with redshift, especially when the population III
star contribution is taken into account [see, e.g., 76, 77, 78]. The limits on
our source parameters are expected to change in case $f_{\star}$ changes
rapidly within the redshift window considered in this study (as for example
considered in [64]). For a rapidly evolving $f_{\star}$, a scenario with
smaller values of $T_{\textup{vir}}$ and $f_{\star}$ (similar to the one
inferred in [71]) can also produce a $\delta T_{\rm b}$ profile that matches
the observed one but the shape of the modelled $\delta T_{\rm b}$ profile will
be mainly determined by how $f_{\star}$ evolves with redshift.

Our limits on the RBG are also limited by the uncertainties in the HMF and the
star formation scenarios. For example, the best fit parameter values for the
ST-HMF and constant $f_{\star}$ suggest a radio background with temperature
$\approx 10.6\times T_{\gamma}$ at $z\sim 17$ which is $\approx 2$ factor
larger than what we obtain for the PS-HMF. The required RBG will also differ
with the X-ray heating scenario. For example, in a scenario such as the one
used in [79] where X-ray heating only is active for $z\lesssim 17$ and an
evolving SFR saturates the $\rm{Ly{\alpha}}$ coupling at $z\approx 17$, the
required RBG will be the same as reported in [50].

Lastly, the constraints obtained in this study are strongly dependent on the
model for dark photons. They are of course also dependent on the priors,
especially on the priors for $m_{A^{{}^{\prime}}}$ and $\epsilon$ space [63]
which are based on $A\rightarrow A^{{}^{\prime}}$ transitions and thus in a
sense are the least model-dependent priors. Alternative models for non-
relativistic dark photons such as dark photon dark matter models [80, 81] in
which dark photons comprise dark matter, estimate stronger limits on
$m_{A^{{}^{\prime}}}$ and $\epsilon$ under which our constraints are already
ruled out. Such models therefore are unable to produce a RBG capable of
explaining the EDGES results.

## Acknowledgements

The authors would like to thank Martin Winkler, Tim Linden and an anonymous
referee for insightful comments on this work. RG and SZ acknowledges support
by the Israel Science Foundation (grant no. 255/18). GM is supported by
Swedish Research Council grant 2020-04691.

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$\alpha_{2\ell+1}$ to be $\tau_{\ell}$ oriented towards $a$. Finally, we set
$\alpha_{2d+1}$ to be the diagonal from $v_{b}$ to $v_{j}$. Such a T-path has
zero vertices of $Q_{\ell}$ to the left of $\alpha_{2\ell+1}$ for all
$k<\ell<d$ and has $j-a>0$ vertices of $Q_{d}$ to the left of $\alpha_{2d+1}$.
Together with the inductive hypothesis, we have constructed a T-path with the
appropriate conditions.

Next, given any T-path $w\in\mathbf{P}_{i,j}$ with $w[a:a]\neq Q_{d}$, we
consider $w[i+1:a]$ as a T-path between $v_{i}$ and $v_{b}$ in the polygon
resulting from cutting all subgons not crossed by $(v_{i},v_{b})$. Since the
diagonal $(v_{i},v_{b})$ crosses less than $d$ arcs, we can use the inductive
hypothesis to construct a T-path associated to $w[i+1:a]$ with the conditions
in the claim. By a parallel argument to the previous case, we can extend this
to a T-path from $v_{i}$ to $v_{j}$ with the desired conditions. ∎

Given a matching $w\in\mathbf{P}_{i,j}$, we define $\Phi(w)$ to be the
complete T-path described in the claim. Given two distinct matchings
$w,w^{\prime}$, the images $\Phi(w)$ and $\Phi(w^{\prime})$ will be distinct
T-paths since, for at least one value $1\leq k\leq d$, the steps at index
$2k+1$ in the two T-paths will be distinct. For surjectivity, given a complete
T-path $\alpha$, we can construct a matching $w\in\mathbf{P}_{i,j}$ so that
$\alpha$ satisfies the hypotheses of the claim for $w$. A proof of this would
follow a similar induction as the proof of the Claim.

We check that $wt_{T}(w)=wt(\Phi(w))$. If $w$ uses subgon $Q_{k}$ $m$ times,
the contribution to $wt_{T}(w)$ is $U_{m}(\lambda_{|Q_{k}|})$. In $\Phi(w)$,
step $\alpha_{2k+1}$ will cross $Q_{k}$ and have $m$ vertices to its left.
Thus, $\alpha_{2k+1}$ is a $m$-diagonal in a regular $|Q_{k}|$-gon, so its
contribution to $wt(\Phi(w))$ is also $U_{m}(\lambda_{|Q_{k}|})$. ∎

##  9 Positivity

Since $wt_{T}(w)\geq 0$ for any matching $w$, it follows from Theorem 9 and
Corollary 2 that every frieze pattern from a dissection of a once-punctured
disc or annulus has all positive entries.

######  Corollary 5.

If $\mathcal{F}=\\{m_{i,j}\\}$ is an infinite frieze pattern from a dissected
annulus or once-punctured disc, then for all $i<j$, $m_{i,j}>0$.

###### Proof.

We must show that, for all $i<j$, there is at least one matching
$w\in\mathbf{P}_{i,j}$ with $wt_{T}(w)>0$. From Equation 4, we know that if
any edge from $\mathcal{D}$ is strictly between $v_{i}$ and $v_{j}$, so that
there are ears between the vertices, we can remove these and the sum over
matchings using $wt_{T}$ will be unchanged. From the proof of Equation 7, we
know the same is true for edges weakly between $v_{i}$ and $v_{j}$. Thus, it
suffices to consider the case where there does not exist an edge
$(v_{\ell},v_{m})\in\mathcal{D}$ with $i\leq\ell<m\leq j$.

In this case, for any subgon $\mathcal{P}$ with vertices strictly between
$v_{i}$ and $v_{j}$, a matching can choose $\mathcal{P}$ at most
$|\mathcal{P}|-2$ times. For a concrete example, one could form a matching
$\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$ by setting $\mathcal{P}_{k}$ as
the subgon which has edge $(v_{k},v_{k+1})$ for all $i+1\leq k\leq j-1$. This
is the matching which uses the counterclockwise-most subgon incident to each
vertex.

∎

We consider the positivity of the other frieze patterns of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ \- either those realizable by a quotient
dissection or unrealizable frieze patterns. The following results determine a
larger class of frieze patterns which are guaranteed to be positive.

######  Proposition 5.

Let $\mathcal{F}$ be a frieze pattern where all entries in the quiddity
sequence are at least 2. Then, all entries of $\mathcal{F}$ are positive.

###### Proof.

Baur, Parsons, and Tschabold proved this in the case of positive integers in
[1]. Their proof can be modified for frieze patterns with more general
entries. In particular, we can combine the fact that the 1-periodic frieze
pattern determined by quiddity row $\ldots,2,2,\ldots$ has all positive
entries with Theorem 2.1 of [1] applied to $b\in\mathbb{R}_{\geq 0}$. ∎

######  Corollary 6.

Any sequence $\\{m_{i-1,i+1}\\}$ such that $m_{i-1,i+1}=\sum_{p\in
A_{i}}\lambda_{p}$ and $|A_{i}|>1$ for all $i$ determines a frieze pattern
with all positive entries.

###### Proof.

Recall that $\lambda_{p}=2\cos(\frac{\pi}{p})$, which implies that
$\lambda_{p}\geq 1$, with equality when $p=3$. ∎

Note that this immediately implies there are unrealizable frieze patterns with
all positive entries. One example is the frieze pattern with quiddity cycle
$(2,2\sqrt{2})$. There are also unrealizable frieze patterns with negative
entries; we provide an example below.

$\begin{array}[]{cccccccccccccccccccc}0&&0&&0&&0&&0\\\ &1&&1&&1&&1&\\\
1&&\sqrt{2}&&1&&\sqrt{2}&&1\\\
&\sqrt{2}-1&&\sqrt{2}-1&&\sqrt{2}-1&&\sqrt{2}-1&\\\
2-2\sqrt{2}&&\sqrt{2}-2&&2-2\sqrt{2}&&\sqrt{2}-2&&2-2\sqrt{2}\\\
&&&&\vdots&&&&&&\\\ \end{array}$

Next, we take advantage of the Progression Formulas in [13] to provide another
way to check for positivity of an arbitrary frieze pattern.

######  Proposition 6.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an $n$-periodic, infinite frieze pattern.
Suppose that the first $n$ nontrivial rows are positive; that is $m_{i,j}>0$
whenever $|j-i|\leq n+1$. Moreover, suppose that the first growth coefficient,
$s_{1}=m_{0,n+1}-m_{1,n}$, satisfies $s_{1}\geq 2$. Then, all entries of
$\mathcal{F}$ are positive.

###### Proof.

First, note by Lemma 4.3 of [2] that if $s_{1}\geq 2$, then $s_{k}\geq 2$
where $s_{k}$ is the $k$-th growth coefficient.

Next, we recall a special case of Theorem 5.4 of [13], known as the
Progression Formulas. Let $1\leq i,j\leq n$ and $k\in Z_{\geq 1}$. Moreover,
first assume $i<j$. Then,

$m_{i,j+(k-1)n}=s_{k-1}m_{i,j}+m_{j,i+k(n-1)}$

Similarly, suppose $i\geq j$. Then,

$m_{i,j+kn}=s_{k-1}m_{i,j+n}+m_{j,i+kn}$

We see we can write entries which are below the $n$-th row in terms of entries
which are in higher rows and growth coefficients. Thus, given that the first
$n$ rows are positive, we can conclude by induction that all entries in
$\mathcal{F}$ are positive.

∎

Note that in the example of an unrealizable frieze pattern with negative
entries, the first growth coefficient is $\sqrt{2}$, which is positive but not
larger than 2.

We can use Proposition 6 to check for positivity of frieze patterns. Based on
experiments, we conclude with the following conjecture.

######  Conjecture 1.

The only frieze patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$ with negative
entries are those which fail the realizability test ( Proposition 1). In
particular, all quotient dissections yield frieze patterns with positive
entries.

##  10 Acknowledgements

This research was largely conducted at the 2020 University of Minnesota Twin
Cities REU, supported by NSF RTG grant DMS-1745638. We thank Gregg Musiker for
feedback throughout the process and Libby Farrell for help with the manuscript
and support during the REU.

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