H<EMAIL_ADDRESS>XXXXXXX/PAR.20XX.00000 1 A New Approach to
Calculate Coronal Electron Density: Simplified van de Hulst’s Method–9

# A New Approach to Calculate Coronal Electron Density: Simplified van de
Hulst’s Method

H. Çakmak${}^{1\cc}$0000-0002-1959-6049
1İstanbul University  Faculty of Science  Department of Astronomy and Space
Sciences  34116  Beyazıt  İstanbul  Türkiye

(2023; 00.00.2023 00.00.2023 00.00.2023 00.00.2023 00.00.0000 )

###### Abstract

Determining the electron density is a challenging task in solar corona
studies, as it requires certain assumptions to be made, such as symmetric,
homogeneous and radial distribution, thermal equilibrium, etc. In such
studies, the observed $K$ corona brightness is based on the coronal electron
density. An important paper on the calculation of electron density was
published in 1950 by van de Hulst in an article titled “The Electron Density
of the Solar Corona”. The author developed a method with some assumptions to
calculate the electron density from the observed $K$ corona brightness. We
presented here, a new simplified calculation method for the coronal electron
density is presented. The integral equation solution given by van de Hulst is
interpreted from a different perspective and the $K$ coronal electron density
is calculated using only observational data without making any additional
adjustments such as successive approximations and multiple attempts.

###### keywords:

Sun: corona – scattering – polarization, Astrometry and celestial mechanics:
eclipses

††volume: 0

##  1 Introduction

Theoretical studies on the solar corona began with a published article by
Schuster (1879). This work investigated the brightness and polarisation of the
solar corona with regards to various particle distributions within the corona.
The majority of the fundamental mathematical issues were resolved with the
explanations provided here. According to this, the corona light is the
composite of all the light scattered by the free electrons in the line of
sight direction. The polarisation of the corona light results from this
phenomenon. Minnaert (1930) further developed Schuster’s theory by taking into
account the limb darkening effect of the observed solar disc. Additionally,
the equations for the relation between electron density and brightness were
provided. Baumbach (1937, 1938) introduced the first general formula for the
electron density of the solar corona from photometric observations as a
function of the solar radius;

$N(r)=10^{8}\Bigg{(}\frac{0.036}{r^{1.5}}+\frac{1.55}{r^{6}}+\frac{2.99}{r^{16}}\Bigg{)}$
(1)

where $N$ is the electron density in cm3, and $r$ is the distance from the
solar disc expressed in solar radius. Subsequently, corona light intensity was
analysed by Allen (1946) and van de Hulst (1950) according to the minimum and
maximum phases of the solar cycle. They provided two distinct corona models.
The type of corona during cycle maximum exhibits nearly spherical brightness
distribution, and most coronal structures show a symmetric arrangement across
the solar disc (see Figure 1, right panel). In contrast, the type of corona
during cycle minimum exhibits a concentration of coronal structures in the
equatorial and polar regions (Figure 1, left panel) and features with
asymmetric brightness distribution. Furthermore, Saito et al. (1970) developed
an empirical function of the electron density which also depends on the
heliographic latitude as follows;

$\begin{split}N_{\text{e}}(r,\phiup)&=\frac{3.09\times
10^{8}}{r^{16}}\big{(}1-0.5\ \sin\phiup\big{)}\\\ &+\frac{1.58\times
10^{8}}{\mathit{r}^{6}}\big{(}1-0.95\ \sin\phiup\big{)}\\\
&+\frac{0.0251\times 10^{8}}{r^{2.5}}\big{(}1-1.0\
\sin^{0.5}\phiup\big{)}\end{split}$ (2)

where $N_{\rm e}$ is the electron density in cm3 and $\phi$ is the
heliographic latitude. The equation allows us to compute the electron density
asymmetrically across the solar disc. It is thus possible to use this equation
to represent changes in coronal brightness based on corona type during solar
minimum or maximum by adjusting the $\rm sin\phi$ coefficients (see Appendix B
for details).

Figure 1: Appearance of the solar corona according to solar cycle phase. Left:
minimum phase (on 4 October 1995), Right: maximum phase (on 21 June 2001).

A dataset of corona brightness from eclipse observations is typically
necessary to develop the formulas presented above. In order to achieve the
most accurate results, intricate computations and various approaches are
necessary. For instance, the outcomes of the method, which are outlined in the
following section, were obtained through successive approximations and
extensive trials, each time improving the computations slightly. Similarly,
another approach, which incorporates van de Hulst’s model, was devised by von
Klüber (1958). This article derives the $K$ corona luminance and the corona
electron density through the assumption that the polarization arises from $K$
corona light and that $F$ corona light is unpolarized. Consequently, the
subsequent equation was formulated as

$K\,P_{\text{K}}=P_{\text{K+F}}(K+F)=K_{\text{t}}-K_{\text{r}}$ (3)

where $K+F$ represents the total corona brightness. $P_{\text{K}}$ and
$P_{\text{K+F}}$ refer to the degree of polarisation of the $K$ corona and the
total corona, respectively. Furthermore, $K_{\text{t}}$ and $K_{\text{r}}$
represent the tangential and radial components of the $K$ corona brightness,
respectively. Similar complex computations, as in Van de Hulst’s method, were
carried out during this study. First, the electron density of the corona was
determined for the $K_{\text{t}}-K_{\text{r}}$ component. Subsequently, the
$K_{\text{t}}$ component was calculated through a reverse calculation of Van
de Hulst’s equation. Using these values of the $K_{\text{t}}-K_{\text{r}}$ and
$K_{\text{t}}$ components, the observational corona brightness $K$ was
obtained (for detailed information, refer to article von Klüber, 1958).

The new approach presented here has simpler steps compared to the methods
mentioned above. The electron density of the corona is computed without time-
consuming calculations, using only the luminosity $K$ and the degree of
polarization of the corona. As an approximation for the calculations, two new
electron densities $N_{\text{t-r}}$ and $N_{\text{t}}$ are defined for the
components $K_{\text{t}}-K_{\text{r}}$ and $K_{\text{t}}$, respectively.

Nowadays, as a result of the developing technological possibilities, different
methods have been developed to calculate coronal electron densities (Bemporad,
2020; Del Zanna et al., 2023). However, these methods are quite different from
the method described here, in terms of both observation type and electron
density calculation technique. In addition, due to the lack of numerical
results on the electron density for the equatorial and polar regions in these
studies, it was not possible to make a comparison with the results given here.

The van de Hulst approach to determining electron density is concisely
outlined in Section 2. A full explanation of about new approximation is given
in Section 3. Subsequently, Section 4 presents the validation of the new
method utilizing model values from Table 5A in van de Hulst’s article. In
Section 5, an instance of the new method’s application is showcased, featuring
the numerical values acquired during the full solar eclipse on the 29th of
March, 2006. The Discussion section concludes the article by detailing the
benefits and advantages of the novel methodology.

##  2 van de Hulst’s Method for Calculating the Electron Density

The content of this section is a brief overview of the author’s original
article, providing only a basic outline of the method. For more comprehensive
information, it is advisable to refer to van de Hulst (1950). Most of the
explanations given here, such as formulae and figures, are also necessary for
a better understanding of the new approach presented in the following section.

The observed brightness of the corona is assumed to be the light scattered by
the free electrons (Schuster, 1879; Baumbach, 1937). Therefore, this
brightness should be directly proportional to the density of electrons in the
corona. In order to visualise this scenario, consider a single ray of light
hitting a vibrating electron (at point P) and reflecting in the direction of
the observer (Figure 2). Here, $r$ and $x$ represent the actual distance and
projected distance of the light from the centre of the disc, respectively,
whilst $\theta$ denotes the angle separating the incoming light direction and
the line of sight. Applying the formula presented by van de Hulst (1950), the
total intensity of the light scattered per second per unit solid angle by a
column with a cross-section of 1 cm2 is determined using

Figure 2: Geometrical representation of both the scattered light incident on a
vibrating electron and its intensity components (reconstructed from van de
Hulst, 1950).

$K(x)=C\int\limits_{x}^{\infty}N(r)\Biggr{\\{}\bigg{(}2-\frac{x^{2}}{r^{2}}\bigg{)}A(r)+\frac{x^{2}}{r^{2}}B(r)\Biggr{\\}}\frac{rdr}{\sqrt{r^{2}-x^{2}}}$
(4)

the equations below are expressed in terms of the tangential and radial
components of this reflected light,

$\begin{split}K_{\text{t}}(x)&=C\int\limits_{x}^{\infty}N(r)A(r)\frac{rdr}{\sqrt{r^{2}-x^{2}}}\\\\[6.45831pt]
K_{\text{t}}(x)-K_{\rm
r}(x)&=C\int\limits_{x}^{\infty}N(r)\biggr{\\{}A(r)-B(r)\biggr{\\}}\frac{x^{2}dr}{r\sqrt{r^{2}-x^{2}}}\end{split}$
(5)

where $A$ and $B$ represent the lengths of semi-major and semi-minor axis of
the vibration ellipsoid, respectively. The constant C is equal to 3/4
$R_{\odot}\sigma=3.44\times 10^{-14}$cm3, where $R_{\odot}\,(=6.96\times
10^{10}\text{cm})$ represents the solar radius, and $\sigma\,(=0.66\times
10^{-24}\text{cm}^{2}$) is the electron scattering cross-section. The primary
issue here is to divide the known $K(x)$ intensity into two parts
$K_{\text{t}}(x)$ and $K_{\text{r}}(x)$, and solve the integrals in such a way
that both equations produce the same electron density $N(r)$. van de Hulst
originally defined the coronal brightness components $K_{\text{t}}(r)$ and
$K_{\text{t}}(r)-K_{\text{r}}(r)$ through the use of coronal intensity $K(x)$
and its model polarization degree $p(x)$ as

$\displaystyle K_{\text{t}}(x)$ $\displaystyle=1/2[1+p(x)]K(x)$ (6)
$\displaystyle K_{\text{t}}(x)-K_{\text{r}}(x)$ $\displaystyle=p(x)K(x)$ (7)

This can also be expressed as a power series in the form of

$\displaystyle K_{\text{t}}(x)$ $\displaystyle=\sum_{s}h_{s}\,x^{-s}$ (8)
$\displaystyle K_{\text{t}}(x)-K_{\text{r}}(x)$
$\displaystyle=\sum_{s}k_{s}\,x^{-s}$ (9)

where $\sum_{s}h_{s}\,x^{-s}$ represents with three elements, namely A x-a \+
B x-b \+ C x-c. van de Hulst then made another approach and assumed that the
solution of integrals given in Equation 5 was of the following form;

$\displaystyle r\,C\,N(r)\,A(r)$
$\displaystyle=\sum_{s}\frac{h_{s}}{a_{\text{-1}}}\,r^{-s}$
$\displaystyle=K_{\text{t}}(r)$ (10) $\displaystyle
r\,C\,N(r)\big{\\{}A(r)-B(r)\big{\\}}$
$\displaystyle=\sum_{s}\frac{k_{s}}{a_{\text{s+1}}}\,r^{-s}$
$\displaystyle=K_{\text{t}}(r)-K_{\text{r}}(r)$ (11)

where

$a_{\rm
s}=\int\limits_{0}^{\pi/2}\sin^{n}\theta\,d\theta=\frac{\pi}{2^{n+1}}\frac{n!}{\\{(n/2)!\\}^{2}}$
(12)

The electron densities can now be calculated from Equations 10 and 11.
Firstly, the coefficients $h_{\text{s}}$, $k_{\text{s}}$ and $s$ of Equations
8 and 9 are obtained by making polynomial fit to the calculated values of
Equations 6 and 7 separately. Then, the right-hand sides of Equations 10 and
11 are calculated respectively by using a new polynomial function formed with
these coefficients. $r$, C, $A(r)$ and $B(r)$ are precomputable values in this
approach (refer to van de Hulst (1950)’s article for calculation of these
values). At this point, the calculated electron densities $N(r)$ in both
Equations 10 and 11 must show the same value. If not, a method of successive
approximations is used, replacing both $K_{\text{t}}(r)$ and
$K_{\text{t}}(r)-K_{\text{r}}(r)$ by a reduction as small as

$\displaystyle K_{\text{t}}(r)$
$\displaystyle=\big{\\{}1+\epsilon\,p\big{\\}}K_{\text{t}}^{{}^{\prime}}$ (13)
$\displaystyle K_{\text{t}}(r)-K_{\text{r}}(r)$
$\displaystyle=\big{\\{}1+\epsilon\,(1+p)\big{\\}}(K_{\text{t}}^{{}^{\prime}}-K_{\text{r}}^{{}^{\prime}})$
(14)

where $\epsilon$ is a value not exceeding $\pm 0.05$.
$K_{\text{t}}^{{}^{\prime}}$ and
$K_{\text{t}}^{{}^{\prime}}-K_{\text{r}}^{{}^{\prime}}$ are pre-computed
values of $K_{\text{t}}(r)$ and $K_{\text{t}}(r)-K_{\text{r}}(r)$. These
computations are repeated, altering $\epsilon$ in each iteration, until both
$K_{\text{t}}(r)$ and $K_{\text{t}}(r)-K_{\text{r}}(r)$ show the same electron
density in Equation 10 and 11. Although van de Hulst has achieved satisfactory
results for the electron densities of the model corona using this methodology,
its practical application is rather challenging and requires multiple attempts
of unspecified numbers.

##  3 New Approach for Coronal Electron Density

The values of $K_{\text{t}}(x)$ and $K_{\text{t}}(x)-K_{\text{r}}(x)$ computed
in Equations 6 and 7 are numerically different from each other. Thus, this
difference should also be valid for Equations 10 and 11. Therefore, the
electron density $N(r)$ in each equation must be different as well. At this
point, as a novel approach, the value of $N(r)$ in each equation is given a
different nomenclature, defined as

$\displaystyle N_{\text{t}}(r)$
$\displaystyle=\frac{1}{r\,\text{C}}\frac{K_{\text{t}}(r)}{A(r)}$ (15)
$\displaystyle N_{\text{t-r}}(r)$
$\displaystyle=\frac{1}{r\,\text{C}}\frac{K_{\text{t}}(r)-K_{\text{r}}(r)}{A(r)-B(r)}$
(16)

where $N_{\text{t}}(r)$ represents the electron density for $K_{\text{t}}(r)$
and $N_{\text{t-r}}(r)$ represents $K_{\text{t}}(r)-K_{\text{r}}(r)$. On the
other hand, by eliminating the $p(x)K(x)$ values in Equations 6 and 7, the
corona intensity $K(x)$ can be expressed in terms of $K_{\text{t}}(x)$ and
$K_{\text{t}}(x)-K_{\text{r}}(x)$ as

$K_{\text{t}}+K_{\text{r}}=K=2\,K_{\text{t}}-(K_{\text{t}}-K_{\text{r}})$ (17)

Considering Equation 4 or Equations 10 and 11, the corona intensity $K(r)$ is
linearly proportional to the electron density $N(r)$. Therefore, any valid
conclusion drawn between the components of the $K(r)$ corona (Equation 17) can
also be drawn between the components of the electron density (Equations 15 and
16). Accordingly, the electron density $N$ can be expressed in a similar
manner as

$N=2\,N_{\text{t}}-N_{\text{t - r}}$ (18)

With this newly derived equation, it is now possible to calculate the electron
density for a known $K(x)$ intensity using only the values of the
$K_{\text{t}}(r)$ and $K_{\text{t}}(r)-K_{\text{r}}(r)$ components.

##  4 Validating the New Approach

The newly developed method was tested using the $K$ corona brightness and the
polarization degree values of the van de Hulst model as an observational
corona values. The model corona values such as $K_{\text{t}}+K_{\text{r}}$,
$K_{\text{t}}$ and $K_{\text{t}}-K_{\text{r}}$ are taken from Table 5A in van
de Hulst (1950)’s article. First, the polynomial coefficients $h_{\text{s}}$,
$k_{\text{s}}$ and $s$ in Equations 8 and 9 were obtained by fitting a
separate curve to the brightness values $K_{\text{t}}$ and
$K_{\text{t}}-K_{\text{r}}$. Then, new polynomial functions are created using
these new coefficients produced by $h_{\text{s}}/a_{\text{s-1}}$ and
$k_{\text{s}}/a_{\text{s+1}}$ (shown in Equation 10 and 11). These new
functions are referred to as “generated functions” (GFs), and are represented
as

$f(K_{\text{t}}-K_{\text{r}})=\sum_{s}\frac{k_{\text{s}}}{a_{\text{s+1}}}\,r^{\text{-s}}\quad\text{and}\quad
f(K_{\text{t}})=\sum_{s}\frac{h_{\text{s}}}{a_{\text{s-1}}}\,r^{\text{-s}}$
(19)

After calculating the electron densities $N_{\text{t}}$ and $N_{\text{t-r}}$
for the brightnesses $K_{\text{t}}$ and $K_{\text{t}}-K_{\text{r}}$ using
Equations 15 and 16, the total electron density is determined by combining
these values with Equation 18. The results obtained for the equatorial region
of van de Hulst model are shown in Table 1. The $K$ corona brightness values
of the minimum type model are shown on the left side of the table. Also, the
coefficients (A, B, C, a, b and c) of the fitted function for each component
are listed under its column at the left-bottom side. For the fitting process,
a three-element polynomial is employed, given by (A $r^{\text{-a}}$ \+ B
$r^{\text{-b}}$ \+ C $r^{\text{-c}}$). Furthermore, values for
$a_{\text{s+1}}$ and $a_{\text{s-1}}$, calculated using Equation 12 with $a$,
$b$ and $c$ coefficients, are listed at bottom of the left side. The computed
values of GFs $f(K_{\text{t}}$) and $f(K_{\text{t}}-K_{\text{r}}$) are
presented in the first two columns of the right side of Table 1 and the
coefficients utilized to construct these functions are listed underneath these
values. The calculated electron densities $N_{\text{t}}$ and $N_{\text{t-r}}$,
and the total electron density $N$ are exhibited in their corresponding
columns on the right side of Table 1.

The electron density calculations were repeated for the polar region of the
van de Hulst model, resulting in the same level of agreement. Table 4 presents
the electron density values attained by the new method for both the equatorial
and polar regions of the minimum type corona, alongside the values obtained by
van de Hulst (1950). From the table, it can be seen that the new method’s
values match closely with those of the van de Hulst model for both the
equatorial and polar regions. A similar comparison was made using the $K$
corona values and polarization degree values of the Allen (1973). The same
agreement was also achieved for these values. The computed values of the new
method for Allen (1973) values are shown in Table 2 and Table 4, respectively.

Table 1: Calculation results for the equatorial region of the minimum-type
corona of van de Hulst (1950) computed using the new approach. Brightness is
in units of 10${}^{-8}I_{\odot}$, and electron density is in units of 106
cm-3.

$r$ | $K$ | $P_{\text{K}}$ | $K_{\text{t}}-K_{\text{r}}$ | $K_{\text{t}}$ | $f(K_{\text{t}}-K_{\text{r}})$ | $f(K_{\text{t}})$ | $N_{\text{t-r}}$ | $N_{\text{t}}$ | $N$
---|---|---|---|---|---|---|---|---|---
1.0 | 300.40 | 0.18 | 54.40 | 177.40 | 104.03 | 470.10 | 239.66 | 226.68 | 213.7
1.03 | 202.80 | 0.24 | 48.00 | 125.40 | 111.69 | 328.27 | 171.88 | 176.05 | 180.2
1.06 | 141.30 | 0.28 | 39.10 | 90.20 | 94.66 | 230.57 | 128.04 | 130.48 | 132.9
1.1 | 91.10 | 0.32 | 29.30 | 60.20 | 70.99 | 148.07 | 87.24 | 88.68 | 90.1
1.2 | 37.10 | 0.41 | 15.14 | 26.12 | 35.53 | 58.50 | 39.40 | 39.18 | 38.9
1.3 | 18.50 | 0.46 | 8.56 | 13.53 | 19.72 | 28.63 | 21.28 | 21.02 | 20.8
1.5 | 6.20 | 0.54 | 3.34 | 4.77 | 7.46 | 9.69 | 8.24 | 8.29 | 8.3
1.7 | 2.57 | 0.59 | 1.51 | 2.04 | 3.33 | 4.06 | 3.90 | 3.96 | 4.0
2.0 | 0.85 | 0.62 | 0.53 | 0.69 | 1.19 | 1.35 | 1.55 | 1.55 | 1.5
2.6 | 0.16 | 0.66 | 0.11 | 0.13 | 0.23 | 0.23 | 0.37 | 0.34 | 0.3
3.0 | 0.07 | 0.65 | 0.05 | 0.06 | 0.09 | 0.09 | 0.17 | 0.15 | 0.1
4.0 | 0.02 | 0.61 | 0.01 | 0.02 | 0.02 | 0.01 | 0.04 | 0.03 | 0.02
|  |  | $k_{\text{s}}$ | $h_{\text{s}}$ |  $k_{\text{s}}$/$a_{\text{s+1}}$ |  $h_{\text{s}}$/$a_{\text{s-1}}$ |  |  |
|  | A | 40.50 | 71.98 | 90.03 | 143.68 |  |  |
|  | B | 26.66 | 121.28 | 83.22 | 398.10 |  |  |
|  | C | -12.77 | -15.86 | -69.22 | -71.68 |  |  |
|  | a | 6.25 | 6.74 | 6.25 | 6.74 |  |  |
|  | b | 13.80 | 17.42 | 13.80 | 17.42 |  |  |
|  | c | 44.63 | 32.58 | 44.63 | 32.58 |  |  |
|  |  | $a_{\text{s+1}}$ | $a_{\text{s-1}}$ |  |  |  |  |
|  | for a | 0.4499 | 0.5010 |  |  |  |  |
|  | for b | 0.3203 | 0.3046 |  |  |  |  |
|  | for c | 0.1845 | 0.2213 |  |  |  |  |

Table 2: Calculation results for the equatorial region of the minimum-type
corona of Allen (1973) computed using the new approach. Brightness is in units
of 10${}^{-8}I_{\odot}$, and electron density is in units of 106 cm-3.

$r$ | $K$ | $P_{\text{K}}$ | $K_{\text{t}}-K_{\text{r}}$ | $K_{\text{t}}$ | $f(K_{\text{t}}-K_{\text{r}})$ | $f(K_{\text{t}})$ | $N_{\text{t-r}}$ | $N_{\text{t}}$ | $N$
---|---|---|---|---|---|---|---|---|---
1.01 | 269.15 | 0.22 | 58.41 | 163.78 | 175.31 | 532.27 | 317.89 | 271.40 | 224.9
1.03 | 199.53 | 0.23 | 46.09 | 122.81 | 115.52 | 333.43 | 177.77 | 178.82 | 179.9
1.06 | 144.54 | 0.25 | 36.28 | 90.41 | 86.75 | 226.47 | 117.34 | 128.16 | 139.0
1.1 | 102.33 | 0.28 | 28.24 | 65.29 | 65.84 | 157.67 | 80.91 | 94.43 | 107.9
1.2 | 44.67 | 0.33 | 14.74 | 29.70 | 35.16 | 71.74 | 39.00 | 48.05 | 57.1
1.4 | 12.02 | 0.40 | 4.85 | 8.43 | 11.59 | 18.79 | 12.55 | 14.95 | 17.3
1.6 | 4.68 | 0.42 | 1.97 | 3.33 | 4.43 | 6.43 | 5.03 | 5.89 | 6.7
1.8 | 2.00 | 0.39 | 0.77 | 1.38 | 1.90 | 2.77 | 2.30 | 2.86 | 3.4
2.0 | 1.00 | 0.34 | 0.34 | 0.67 | 0.89 | 1.43 | 1.16 | 1.64 | 2.1
2.2 | 0.60 | 0.30 | 0.18 | 0.39 | 0.45 | 0.85 | 0.62 | 1.07 | 1.5
2.5 | 0.27 | 0.26 | 0.07 | 0.17 | 0.18 | 0.46 | 0.28 | 0.66 | 1.0
3.0 | 0.10 | 0.20 | 0.02 | 0.06 | 0.05 | 0.21 | 0.09 | 0.37 | 0.7
4.0 | 0.03 | 0.13 | 0.004 | 0.02 | 0.01 | 0.07 | 0.01 | 0.17 | 0.3

Table 3: Comparison of electron densities obtained using the new method with those from the van de Hulst (1950) values. Electron density is in units of 106 cm-3. (a) | van de Hulst $N$ | This study $N$
---|---|---
$r$ | Equator | Polar | Equator | Polar
1.0 | 227.0 | 174.0 | 213.7 | 170.3
1.03 | 178.0 | 127.0 | 180.2 | 127.0
1.06 | 132.0 | 87.2 | 132.9 | 86.9
1.1 | 90.0 | 53.2 | 90.1 | 52.9
1.2 | 39.8 | 16.3 | 39.0 | 15.8
1.3 | 21.2 | 5.98 | 20.8 | 5.8
1.5 | 8.3 | 1.4 | 8.3 | 1.5
1.7 | 4.0 | 0.542 | 4.0 | 0.620
2.0 | 1.580 | 0.196 | 1.544 | 0.199
2.6 | 0.374 | 0.040 | 0.317 | 0.030
3.0 | 0.176 | 0.017 | 0.131 | 0.010
4.0 | 0.050 | 0.004 | 0.021 | 0.001

Table 4: Comparison of electron densities obtained using the new method with
those from the Allen (1973) values. Electron density is in units of 106 cm-3.

(b) | Allen $N$ | This study $N$
---|---|---
$r$ | Equator | Polar | Equator | Polar
1.01 | 251.2 | 199.5 | 224.9 | 215.4
1.03 | 177.8 | 131.8 | 179.9 | 150.1
1.06 | 125.9 | 95.5 | 139.0 | 87.7
1.1 | 91.2 | 64.6 | 107.9 | 57.6
1.2 | 46.8 | 19.9 | 57.1 | 26.2
1.4 | 15.1 | 4.4 | 17.3 | 3.4
1.6 | 6.8 | 1.3 | 6.7 | 0.5
1.8 | 3.6 | 0.6 | 3.4 | 0.1
2.0 | 2.0 | 0.3 | 2.1 | 0.06
2.2 | 1.3 | 0.2 | 1.5 | 0.04
2.5 | 0.6 | 0.1 | 1.0 | 0.02
3.0 | 0.3 | 0.05 | 0.7 | 0.01
4.0 | 0.1 | 0.02 | 0.3 | 0.001

##  5 Calculated Electron Densities of the 2006 Solar Eclipse

The newly developed method was utilized to compute the electron density of the
solar corona as observed during the total eclipse on March 29, 2006, in
Türkiye. This eclipse observation was carried out with the 8-inch Meade
telescope by the staff of the Astronomy and Space Sciences Department of
Istanbul University in the Manavgat district of Antalya. During the eclipse
event, observations of white light polarization were conducted, and eclipse
photographs were taken at three different polarization angles, 0∘, 60∘, and
120∘. A total of 15 photos were taken during totality with an interval of 3m
30s between 11h 55s 10m and 11h 58s 40m UT. Five different exposure times were
used in these shoots; 1/2, 1/4, 1/30, 1/60, and 1/125 second. In addition,
images of the solar disc were taken at different diaphragm openings before the
eclipse for brightness calibration and exposure times used here were the same
as those used during the eclipse. After performing brightness calibration and
computation of Stokes parameters using polarization images, the total corona
brightness ($K+F$) and polarization degree ($P_{\text{K+F}}$) of the 2006
eclipse obtained by considering the sky with instrumental contribution and
active chromospheric regions (see Appendix A for details).

Figure 3: (a) Isophotes of total corona brightness (values are in units of
10${}^{-9}I_{\odot}$) and (b) isolines of polarization degree (values are in
percent) of the 29 March 2006 solar eclipse.

The isophote plots of total corona brightness and its polarization degree are
shown separately in Figure 3. The numbers on the isophote lines are in units
of 10${}^{-9}I_{\odot}$ for the total brightness, and in percent for the
polarization degree. The values obtained for both parameters at specific
distances from the solar disc are given in Table 5.

The $K$ corona brightness values are obtained by subtracting $F$ corona model
values of van de Hulst (1950) from the observed $K+F$ total brightness values.
This process has been carried out with the assumption that $F$ corona does not
change much from one solar cycle to another (Kulijanishvili & Kapanadze, 2005;
Morgan & Habbal, 2007). Also, the polarization degree $P_{\text{K}}$ of the
$K$ corona is determined using the equation of von Klüber (1958), which is
given by

$P_{\rm K}=P_{\rm K+F}\Bigg{(}\frac{K+F}{K}\Bigg{)}.$ (20)

For this eclipse, the electron densities in the equatorial and polar regions
were calculated using the $K$ corona brightness and its polarization degree.
The obtained results are shown in Table 6A for the equatorial region and Table
6B for the polar region. The calculated electron densities of the equatorial
(black circle) and polar (black triangle) regions are shown in Figure 4 in
comparison with the observational values of Newkirk (1967) and Allen (1973)
and model electron density values of van de Hulst (1950) and Saito et al.
(1970). It is clear from the figure that the electron density values observed
in the 2006 eclipse are in good agreement with comparison values given. Any
discrepancies between the observed and model values may be due to the
asymmetric brightness distribution caused by the asymmetric distribution of
the solar material. This is clearly illustrated in Figure 3a, where the
equatorial and polar intensity distributions are compared.

Figure 4: Comparison of electron densities in the equatorial and polar regions
during the solar eclipse on 29 March 2006 with selected observational data and
model values of van de Hulst (1950) and Saito et al. (1970)

Table 5: Observed total corona brightness $K+F$ and polarization degree
$P_{\text{K+F}}$ values of 29 March 2006 eclipse.

|  $K+F$ corona ($\times 10^{-9}I_{\odot}$) |  |  $P_{\text{K+F}}$ (%)
---|---|---|---
$r$ | Equa. | Polar |  | Equa. | Polar
1.10 | 1311 | 648 |  | 19.0 | 32.4
1.15 | 1046 | 364 |  | 26.7 | 36.3
1.20 | 782 | 201 |  | 36.4 | 33.2
1.25 | 513 | 128 |  | 44.7 | 25.9
1.30 | 293 | 90 |  | 46.1 | 19.6
1.35 | 180 | 70 |  | 39.8 | 15.0
1.40 | 125 | 57 |  | 32.4 | 13.9
1.45 | 94 | 48 |  | 26.8 | 12.3
1.50 | 74 | 42 |  | 22.5 | 11.6
1.55 | 61 | 38 |  | 19.4 | 10.7
1.60 | 52 | – |  | 17.1 | –
1.65 | 45 | – |  | 15.8 | –
1.70 | 40 | – |  | 14.6 | –

Table 6: Values used for electron density calculation (left side), and results
from Eclipse 2006 (right side). $K$ values are in units of $10^{-9}I_{\odot}$,
and $P$ is in percent.

$A-$ equatorial region
---
$r$ | $K$ | $P_{\text{K}}$ | $K_{\text{t}}-K_{\text{r}}$ | $K_{\text{t}}$ | $f(K_{\text{t}}-K_{\text{r}})$ | $f(K_{\text{t}})$ | $N_{\text{t-r}}$ | $N_{\text{t}}$ | $N_{\text{equ}}$
1.10 | 158.6 | 0.196 | 31.1 | 94.8 | 160.5 | 269.3 | 197.2 | 161.3 | 125.4
1.15 | 96.6 | 0.285 | 27.5 | 62.0 | 103.4 | 171.5 | 118.9 | 109.0 | 99.0
1.20 | 72.6 | 0.403 | 28.9 | 50.2 | 68.0 | 111.4 | 75.4 | 74.6 | 73.9
1.25 | 46.0 | 0.517 | 23.8 | 34.9 | 45.5 | 73.7 | 49.5 | 51.8 | 54.1
1.30 | 24.9 | 0.558 | 13.9 | 19.4 | 30.9 | 49.6 | 33.4 | 36.4 | 39.5
1.35 | 14.4 | 0.506 | 7.3 | 10.8 | 21.3 | 33.9 | 23.0 | 26.0 | 28.9
1.40 | 9.4 | 0.433 | 4.1 | 6.7 | 14.9 | 23.6 | 16.2 | 18.7 | 21.3
1.45 | 6.6 | 0.376 | 2.5 | 4.6 | 10.6 | 16.6 | 11.6 | 13.7 | 15.8
1.50 | 5.0 | 0.330 | 1.6 | 3.3 | 7.6 | 11.8 | 8.4 | 10.1 | 11.8
1.55 | 3.9 | 0.298 | 1.2 | 2.6 | 5.5 | 8.5 | 6.2 | 7.5 | 8.9
1.60 | 3.2 | 0.274 | 0.9 | 2.1 | 4.0 | 6.2 | 4.6 | 5.7 | 6.8
1.65 | 2.4 | 0.263 | 0.6 | 1.5 | 3.0 | 4.6 | 3.5 | 4.3 | 5.2
1.70 | 1.9 | 0.249 | 0.5 | 1.2 | 2.2 | 3.4 | 2.6 | 3.3 | 4.0
$B-$ polar region
$r$ | $K$ | $P_{\text{K}}$ | $K_{\text{t}}-K_{\text{r}}$ | $K_{\text{t}}$ | $f(K_{\text{t}}-K_{\text{r}})$ | $f(K_{\text{t}})$ | $N_{\text{t-r}}$ | $N_{\text{t}}$ | $N_{\text{pol}}$
1.10 | 54.8 | 0.395 | 21.6 | 38.2 | 76.5 | 115.4 | 94.0 | 69.1 | 44.2
1.15 | 28.5 | 0.425 | 12.1 | 20.3 | 36.0 | 58.0 | 41.4 | 36.9 | 32.4
1.20 | 13.7 | 0.382 | 5.2 | 9.4 | 18.3 | 30.6 | 20.3 | 20.5 | 20.7
1.25 | 7.6 | 0.316 | 2.4 | 5.0 | 10.0 | 16.9 | 10.9 | 11.9 | 12.8
1.30 | 4.7 | 0.275 | 1.3 | 3.0 | 5.9 | 9.8 | 6.4 | 7.2 | 8.0
1.35 | 3.3 | 0.252 | 0.8 | 2.1 | 3.7 | 5.9 | 4.0 | 4.5 | 5.1
1.40 | 2.5 | 0.240 | 0.6 | 1.6 | 2.4 | 3.7 | 2.6 | 3.0 | 3.3
1.45 | 2.1 | 0.258 | 0.5 | 1.3 | 1.7 | 2.5 | 1.8 | 2.0 | 2.2
1.50 | 1.0 | 0.287 | 0.3 | 0.7 | 1.2 | 1.7 | 1.3 | 1.4 | 1.6
1.55 | 0.5 | 0.266 | 0.1 | 0.3 | 0.9 | 1.2 | 1.0 | 1.0 | 1.1
1.60 | 0.3 | 0.264 | 0.1 | 0.2 | 0.6 | 0.9 | 0.7 | 0.8 | 0.8

##  6 Discussion

The main challenge of the newly developed method is to determine the optimal
coefficients ($h_{\text{s}},k_{\text{s}},s$) of the power function in
transition from Equations 8–9 to Equations 10–11. This involves a demanding
phase of performing numerous fitting curve tests to determine the appropriate
coefficients of the three- or two-element power function. Using these
coefficients, the observational values of $K_{\text{t}}(x)$ and
$K_{\text{t}}(x)-K_{\text{r}}(x)$, which depend on the projection distance,
are converted into the values of $K_{\text{t}}(r)$ and
$K_{\text{t}}(r)-K_{\text{r}}(r)$, which depend on the true distance.

Figure 5: (a) Power function curve fitted without any constraints, (b) power
function curve fitted with delimited coefficients, (c) screenshot of a program
showing parameters utilized to fit the power function curve under different
configurations.

During the fitting process, it is crucial to ensure that the fitted curve
passes through the overall distribution of the observation points. Attempting
to fit the curve close to every observation point is generally ineffective
(see Figure 5a) due to inevitable observational errors that cause scattering
in the values. Thus, a solution should be devised for the general trend of
these observation points (see Figure 5b). There are two possible techniques to
accomplish this task. The first option involves fitting a two-element power
function curve to the values of a curve obtained by fitting a polynomial
equation with one element or six or more elements. The second option involves
fitting a power function curve with restricted parameters where each
coefficient has a boundary between specified values (refer to Figure 5c). The
second method, which is preferred in this study, provides a straightforward
and fulfilling solution without requiring additional experimentation. However,
it can be challenging to adjust the limits for limiting coefficients in a
consistent manner across different eclipse data. Nevertheless, this is a
common occurrence, as each eclipse typically has a unique distribution with
its own distinct characteristics. Once the coefficients for the power function
are determined for the observational values, generating GFs and obtaining the
electron density become straightforward steps in this method.

When examining the test results of the electron density for van de Hulst
(1950) obtained by the newly introduced method shown in Table 4, it is evident
that the compared values are in very good agreement. This fact is more
apparent in Figure 6a, which confirms the accuracy level of the new method.
The similar agreement is also seen for the values of Allen (1973) given in
Tables 2 and 4, respectively. As highlighted in Figure 6, the electron density
values of both the model and the novel method are mainly distributed along the
line. When checked for compatibility in individual regions, the coefficient of
determination $R^{2}$ is 0.9974 for the equatorial region and 0.9996 for the
polar region of van de Hulst (1950), while 0.9845 for the equatorial region
and 0.9853 for the polar region of Allen (1973).

Figure 6: Comparison of electron density values from (a) van de Hulst’s model
and (b) Allen’s values with those derived from this study. The plots below
illustrate the numerical disparity between these two approaches.

Although the results from Eclipse 2006 are quite satisfactory, It would be
better to retest this new method with the data from other eclipse
observations, especially with the other eclipse results from other researchers
who have their own observational electron density data. Thus, the accuracy of
the new method will be confirmed by finding similar or conclusive results for
these data. For example, when the new method was tested with van de Hulst
model values in Section 4, satisfactory electron density results were
obtained. Thus, the probability of finding similar consistent results with
other observational data is quite high. This should be examined, particularly
by other researchers who have measured electron densities using their methods.
In order to check the new approach with different observational values, it is
intended to contact more than one researcher investigating this topic in the
future.

Peer Review:

Externally peer-reviewed.

Author Contribution:

Conception/Design of study - H.Ç.; Data Analysis/Interpretation - H.Ç.;
Drafting Manuscript - H.Ç.; Critical Revision of Manuscript - H.Ç.; Final
Approval and Accountability - H.Ç.

Conflict of Interest:

Authors declared no conflict of interest.

Financial Disclosure:

Authors declared no financial support.

## Acknowledgements

Thanks to especially every staff who took part in the 2006 solar eclipse
observation. Thanks also to the anonymous referees for their valuable
suggestions and comments that improved the manuscript. This work was supported
by the Istanbul University Scientific Research Projects Commission with
project numbers 24242 and 470/27122005.

## References

  * Allen (1946) Allen C. W., 1946, MNRAS, 106, 137
  * Allen (1973) Allen C., 1973, Astrophysical Quantities 3rd ed.. Athlone Press, London
  * Baumbach (1937) Baumbach S., 1937, Astronomische Nachrichten, 263, 121
  * Baumbach (1938) Baumbach S., 1938, Astronomische Nachrichten, 267, 273
  * Bemporad (2020) Bemporad A., 2020, ApJ, 904, 178
  * Çakmak (2017) Çakmak H., 2017, Sol. Phys., 292, 186
  * Del Zanna et al. (2023) Del Zanna G., et al., 2023, ApJS, 265, 11
  * Kulijanishvili & Kapanadze (2005) Kulijanishvili V. I., Kapanadze N. G., 2005, Sol. Phys., 229, 45
  * Minnaert (1930) Minnaert M., 1930, ZAp, 1, 209
  * Morgan & Habbal (2007) Morgan H., Habbal S. R., 2007, A&A, 471, L47
  * Newkirk (1967) Newkirk Gordon J., 1967, ARA&A, 5, 213
  * Saito et al. (1970) Saito K., Makita M., Nishi K., Hata S., 1970, Annals of the Tokyo Astronomical Observatory, 12, 51
  * Schuster (1879) Schuster A., 1879, MNRAS, 40, 35
  * von Klüber (1958) von Klüber H., 1958, MNRAS, 118, 201
  * van de Hulst (1950) van de Hulst H. C., 1950, Bull. Astron. Inst. Netherlands, 11, 135

##  Appendix A Calculation procedures of the K corona brightness

First of all, as a reminder, all procedures relevant to this section are given
comprehensively in Çakmak (2017)’s article. Please review this article for
more information. A very brief summary of the general steps of this procedure
is given verbally here. In order to determine the brightness of the $K$
corona, a brightness calibration must first be performed. This requires taking
images at different diaphragm openings with the same exposure times as used
for polarized images. Once the intensity calibration function (defined in
Çakmak, 2017) has been obtained by using these solar disc images, the
brightness of the corona in all polarized images is calculated by normalizing
their intensity. Then, the average corona brightness values are determined for
the polar region between latitudes 0∘–30∘ and for the equatorial region
between latitudes 40∘–90∘, separately (Figure 7). At this stage, regions with
chromospheric structure were excluded from the calculation, taking into
account their distance range from the Sun’s surface, to avoid erroneous
increases in brightness.

Figure 7: The latitude ranges for the polar and equatorial regions used in a
single quadrant. Distances to the solar disc are given in units of solar
radius.

##  Appendix B Explanations about latitude-dependent coronal brightness
calculation

A very brief summary of the explanations on this subject from the article of
Saito et al. (1970) is given here. Please refer to that article for more
information. To describe the brightness of an arbitrary point in the solar
corona, the graphical situation shown in Figure 8 is considered. In this
figure, $P$ is the point in question, $P^{\prime}$ is the projection of $P$ on
the celestial plane, $\phi$ is the heliographic latitude of $P$, $\phi_{0}$ is
the projected angle of on the celestial plane, $\theta$ is angle
$OPP^{\prime}$ and $z$ is the line-of-sight length.

Figure 8: Latitude-dependent diagram of an arbitrary point in the solar corona
(reconstructed from Saito et al., 1970).

In that paper, the author obtained the following equations from Equation 5

$\begin{matrix}I_{\text{t}}-I_{\text{r}}=C\
2y\int\limits_{0}^{\pi/2}N(r,\phiup)(A-B)\text{d}\thetaup\\\\[2.15277pt]
I_{\text{t}}=C\
2y\int\limits_{0}^{\pi/2}N(r,\phiup)(A)\frac{\text{d}\thetaup}{\rm\sin^{2}\thetaup}\end{matrix}$
(21)

under certain assumptions with

$\begin{matrix}\sin\phiup=\sin\phiup_{0}\,\sin\thetaup\,,\quad
r\,\sin\thetaup=y\\\\[6.45831pt]
N=\frac{N_{0}}{R^{n}}=N_{0}\,k^{n}\,\sin^{n}\,\thetaup\,,\quad N(r,\phi)=\sum
N_{0,i}\,\frac{1-f_{i}\,\sin^{si}\,\phiup}{r^{ni}}\end{matrix}$

where $k$ is the modulus of the integrals and a number between 0 and 1, $n$ is
a real number, $f_{i}$ and $s_{i}$ are positive real number. As a result of
arranging the Equation 21, the following formula is obtained for latitude-
dependent $K$ corona brightness.

$\begin{split}(I_{\text{t}}\pm I_{\text{r}})_{\phiup_{0}}=(I_{\text{t}}\pm
I_{\text{r}})_{\text{equ}}&-\frac{\sin\phiup_{0}}{k}\,0.5\times 5.365\times
10^{-6}(I_{\text{t}}\pm I_{\text{r}})_{17}\\\
&+\frac{\sin\phiup_{0}}{k}\,0.95\times 2.752\times 10^{-6}(I_{\text{t}}\pm
I_{\text{r}})_{7}\\\ &+\frac{\sin^{0.5}\phiup_{0}}{k^{0.5}}\,1.0\times
0.0436\times 10^{-6}(I_{\text{t}}\pm I_{\text{r}})_{3}\end{split}$ (22)

This equation can be expressed as

$\displaystyle N_{\text{e}}(r,\phiup)$ $\displaystyle=\frac{3.09\times
10^{8}}{r^{16}}\big{(}1-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.5}\
\rm sin\phiup\big{)}$ $\displaystyle+\frac{1.58\times
10^{8}}{\mathit{r}^{6}}\big{(}1-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.95}\
sin\phiup\big{)}$ (23) $\displaystyle+\frac{0.0251\times
10^{8}}{\mathit{r}^{2.5}}\big{(}1-\rm{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1.0}\
sin^{0.5}\phiup\big{)}$

Using this last equation, the coronal electron density can be visualized as
shown Figure 9. As can be seen from Equation B, different profiles for the
coronal electron density can be produced by changing the coefficients of $\rm
sin\phiup$’s shown in red.

Figure 9: Iso-density curves of electrons in the $K$ corona computed with the
Equation B (reconstructed from Saito et al., 1970).


#  Minimal warm inflation and TCC ††thanks: Citation: Authors. Title. Pages….
DOI:000000/11111.

Vahid Kamali,
Department of Physics, Bu-Ali Sina (Avicenna) University, Hamedan 65178,
016016, Iran.
<EMAIL_ADDRESS>
Hossein Moshafi,
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.
O. Box 19395-5531, Tehran, Iran
<EMAIL_ADDRESS>
Saeid Ebrahimi,
Department of Physics, Bu-Ali Sina (Avicenna) University, Hamedan 65178,
016016, Iran.
<EMAIL_ADDRESS>

###### Abstract

The minimal warm inflation model was constructed as a warm inflation setup
with direct interaction between inflaton and (non-Abelian) gauge fields. The
model was shown to be compatible with observation for some forms of potential.
As a result of direct analysis of CMB data, the model presents a reasonable
phase-space of its parameter compatible with observation and the Trans-
planckian censorship conjecture (TCC).

_K_ eywords Warm inflation $\cdot$ TCC

##  1 Introduction

A cosmological theory called the Hot Big Bang (HBB) has been widely accepted
as one that can explain observational data in a systematic way [1, 2]. There
are some inquiries in this context that are best explained by adding a short
period of rapid expansion during the early evolution of the Universe.
Accelerated expansion during early time is generally based on quantum field
theory (QFT), mainly scalar fields [3, 4, 5, 6, 7, 8]. Perturbation of
inflaton as quanta of the scalar field that is responsible for accelerated
expansion is the seed of large-scale structures (LSS) [9, 10, 11, 12, 13]. The
CMB temperature map can also be explained by linear perturbation theory in a
curved manifold (FLRW). The standard inflation model employs a scalar field
that rolls a nearly flat potential during the accelerated phase of expansion
known as slow-roll epoch. It is followed by the reheating epoch, during which
the inflaton has transferred its energy, mainly to light particles [14, 15,
16, 17, 18]. There is an alternative explanation for this two-part sector of
early time evolution, in which the inflaton interacts with light fields in a
thermal bath during slow-roll expansion [19]. Within this so-called warm
inflation scenario, the inflaton-dominated era connects smoothly to the
radiation-dominated era without any separate reheating epoch. Building a QFT
model with a Direct coupling between inflaton and a limited number of light
fields presents some challenges [20]. To resolve the inquiries, the warm
little inflation model was presented, which fits with observations in a weak
dissipative regime [21]. The model has been extended by a scenario compatible
with observations in a high dissipative regime [22]. In Ref. [23], the direct
coupling of inflatons and $SU(2)$ gauge fields, with the radiation equation of
state $P=\frac{\rho}{3}$, is investigated within a warm inflation context for
the first time. In this case, the dissipation parameter is proportional to the
Hubble parameter, keeping it out of swampland and in line with observations
[24, 25, 26]. The idea of using Chern-Simons interaction to find a thermal
bath during early time acceleration epoch was extended by introducing a
minimal warm inflation model with a temperature-dependent dissipation term
[27] and extended in some follow up papers [28, 29, 30]. A recent quantum
gravity conjecture suggests that trans-Planckian modes are not allowed to
cross the horizon during inflation (TCC) which closely constrains the energy
scale of inflation [31, 32] Through a direct observational analysis [33, 34],
we demonstrate that the minimal warm inflation is compatible with TCC and
observations.

##  2 Minimal warm inflation

In order to present minimal warm inflation, a direct coupling term is
introduced between the axion (inflaton) and the gauge field in the Lagrangian
[27].

$\displaystyle\mathcal{L}_{int}=\frac{g_{YM}^{2}}{64\pi^{2}}\frac{\phi}{f}\tilde{G}_{a}^{\mu\nu}G_{\mu\nu}^{a}$
(1)

$g_{YM}$ denotes the gauge coupling and $G_{\mu\nu}$ represents the field
strength of any Yang-Mills group. Inflatons directly coupled to light gauge
fields are able to reach thermal equilibrium suddenly. The process may begin
with a small perturbation or even quantum fluctuations [27]. The effects of
this interaction (1) have a significant impact on the inflaton (axion) field
in the thermal bath, background, and perturbation. The warm inflation scenario
can be described by two coupled field equations.

$\displaystyle\ddot{\phi}+(3H+\Upsilon(\phi,T))\dot{\phi}+V_{,\phi}=0$ (2)
$\displaystyle\dot{\rho}_{\gamma}+4H\rho_{\gamma}=\Upsilon\dot{\phi}^{2}$
$\displaystyle
H^{2}=\frac{1}{3M_{p}^{2}}(\rho_{\phi}+\rho_{\gamma})~{}~{}~{}~{}~{}~{}\rho_{\gamma}=C_{R}T^{4}$

at the background level. A direct interaction (1) yields the coupling term. As
fields evolve at the slow-roll limit,

$\displaystyle\ddot{\phi}$ $\displaystyle\ll$ $\displaystyle 3H\dot{\phi},$
(3) $\displaystyle\dot{\rho}_{\gamma}$ $\displaystyle\ll$ $\displaystyle
4H\rho_{\gamma},$

coupled equations (2) are simplified to

$\displaystyle 3H(1+Q)\dot{\phi}\simeq-V_{,\phi}$ (4) $\displaystyle
4\rho_{\gamma}\simeq 3Q\dot{\phi}^{2}$ $\displaystyle
H^{2}=\frac{V}{3M^{2}_{p}\textbullet}~{}~{}~{}~{}~{}~{}Q=\frac{\Upsilon}{3H}$

According to the minimal warm inflation model, dissipation coefficient
$\Upsilon$, which is originally derived via interaction term (1) in the
Lagrangian, is proportional to the $T^{3}$. We will study the hybrid form of
the potential in detail [27]

$\displaystyle V(\phi)=V_{0}+\frac{1}{2}m^{2}\phi^{2}$ (5)

Using field evolution (4) in the slow-roll limit we can determine the
evolution of dissipation parameter $Q$:

$\displaystyle\frac{d\ln Q}{dN}=\frac{10\epsilon_{V}-\eta_{V}}{7Q}$ (6)
$\displaystyle\eta_{V}=M_{p}^{2}\frac{V_{,\phi\phi}}{V}~{}~{}~{}~{}~{}~{}~{}\epsilon_{V}=\frac{M_{p}^{2}}{2}(\frac{V^{\prime}}{V})^{2}$

which is a crucial relation in our study. For potential (5), using Eq(6), we
can calculate the dissipation parameter $Q$ based on the number of e-foldings
$N$:

$\displaystyle Q=\frac{m^{2}M_{p}^{2}}{V_{0}}(\frac{6}{7}N+1)$ (7)

The power-spectrum of the high dissipative minimal warm inflation model in
terms of dissipation parameter $Q$ [27]

$\displaystyle\Delta_{R}=\frac{\sqrt{3}}{4\pi^{\frac{3}{2}}}\frac{H^{3}T}{\dot{\phi}^{2}}(\frac{Q}{Q_{3}})^{9}Q^{\frac{1}{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{T}{\dot{\phi}^{2}}=\frac{C_{T}}{4C_{R}H}$
(8)

can be combined with relation (7) and a relation for the number of e-folding
$N$ based on wavenumber $\kappa$ [33]:

$\displaystyle
N(k)=56.02-\ln\left(\frac{k}{k_{0}}\right)+\ln\left(\frac{V_{k}^{\frac{1}{2}}}{V_{\rm
end}^{\frac{1}{2}}}\right)+\ln\left(\frac{V_{\rm
end}^{\frac{1}{4}}}{10^{16}\rm{GeV}}\right)$ (9)

Therefore, we can write $\Delta_{R}(\frac{k}{k_{0}})$ of the minimal warm
inflation model in the following way:

$\displaystyle\Delta_{R}\left(\frac{k}{k_{0}}\right)=\tilde{V}_{0}^{-8.5}\tilde{C}_{T}\left[75.5-0.86\ln\left(\frac{k}{k_{0}}\right)-0.21\ln\tilde{V_{0}}\right]^{9.5}$
(10)

where $\tilde{V}_{0}=(\frac{V_{0}}{m^{2}M_{p}^{2}})^{-1}$,
$\tilde{C}_{T}=2.2\times 10^{-5}C_{T}m^{2}$ are dimensionless parameters which
are supposed to be constrained by observational data (see section (4)). We
will discuss what is known as trans-Planckian censorship conjecture (TCC) and
how it constrains the field theories used to explain the inflation epoch. This
would be interesting to show whether MWI is compatible with observational data
and TCC criteria.

##  3 TCC and warm inflation

If we accept string theory as a consistent theory of quantum gravity, we can
introduce a huge landscape of vacua which can be used in low energy IR limit.
Based on swampland program some of the vacua that seem self-consistent and
useful in IR limit are belong to the swampland. In this program there are some
reasonable top-down constraints, in the limit of conjecture not theorem, on
the filed excursion and the energy scale of the field theory that used in
particle physics and cosmology [31, 32]. In the present work we focus on TCC
which has a tight constraints on the energy scale of inflation. We will show
the minimal warm inflation is compatible with criteria of TCC as well as
observation in a wide range of parameter phase space. Sub-Planckian modes have
to be hidden by Hubble horizon during inflation 1:

$\displaystyle\frac{a_{f}}{a_{i}}.l_{p}<\frac{1}{H(t)}$ (11)

Figure 1: During an accelerated epoch all scales stretch. They can cross the
horizon. Sub-plankean scale need more time for horizon crossing. To meet the
TCC criteria (11), the period of accelerated epoch need to be short enough.

which restricts the duration of inflation. If we assume constant Hubble
horizon or quasi-de Siter inflation

$\displaystyle H(t)\simeq H_{f}$ (12)

we can find an upper limit for the energy density of inflation [32]

$\displaystyle V^{\frac{1}{4}}<6\times 10^{8}\rm{GeV}$ (13)

and tensor to scale ratio parameter

$\displaystyle r<10^{-31}$ (14)

which are nearly forbid the slow-roll inflation we need to resolve the
shortcoming of HBB. But it would be relaxed if we consider warm inflation
where the evolution of Hubble parameter during slow-roll epoch has an
important role (1) on the theoretical perturbation parameters. In this case
the tensor-to scalar ratio upper bound is more accessible [35]

$\displaystyle r<10^{-10}$ (15)

But swampland conditions would not be relaxed in the context of warm inflation
[36]. By a direct data analysis we try to find a parameter phase space of
minimal warm inflation compatible with observation as well as TCC constrains.

##  4 Methodology and Observational Data

In this section, we describe the data and the analysis methods used here. We
used most recent publicly available measurement of the CMB, released by
_Planck_ team as _Planck_ 2018 data. For this purpose we used the Cosmic
Microwave Background (CMB) temperature power spectrum (TT), the polarization
power spectrum (EE), the cross-correlation power spectrum of temperature and
polarization (TE), and the _Planck_ low-$\ell$ polarization likelihood (lowE)
which we show by _Planck_ TT+Pol in figures and tables. [1, 2].
The primordial power spectrum of scalar perturbations (Eq. 10) can be
expressed as

$\Delta_{R}\left(\frac{k}{k_{0}}\right)=\tilde{V}_{1}\left[75.5-0.86\ln\left(\frac{k}{k_{0}}\right)-0.21\ln\tilde{V_{0}}\right]^{9.5}$
(16)

which we redefined two parameters for convenience of calculation as follows

$\displaystyle\tilde{V}_{0}=10^{7}\tilde{V}_{0}$ $\displaystyle
10^{27}\frac{\tilde{C}_{T}}{\tilde{V}_{0}^{8.5}}=\tilde{V}_{1}$ (17)

We consider a parameter space with six independent free parameters in our
analysis:

$\mathbf{P}=\left\\{\Omega_{b}h^{2},\Omega_{c}h^{2},100\theta_{\rm
MC},\tau,\tilde{V}_{0},\tilde{V}_{1}\right\\}$

with flat uniform priors on the parameters reported in Table 1. In our
analysis, we assume curvature $\Omega_{K}$ is zero, and the Dark Energy
equation of state is $w=-1$. We set the number of neutrino species
$N_{\nu}=3.046$, and neutrino mass $m_{\nu}=0.06\rm{eV}$.
In our cosmological analysis, the allowed regions are calculated numerically
with modified verision of the CosmoMC Markov Chain Monte Carlo (MCMC) sampler
[34] and the CAMB Boltzmann code. We apply Metropolis-Hasting sampling to 4
chains running in parallel, and use a convergence criterion that obeys
$R-1<0.01$, where the Gelman-Rubin $R$-statistics [37] is the variance of
chain means divided by the mean of chain variances.

Parameter | Symbol | Prior
---|---|---
Cold dark matter density | $\Omega_{\mathrm{c}}h^{2}$ | $[0.001,0.99]$
Baryon density | $\Omega_{\mathrm{b}}h^{2}$ | $[0.005,0.1]$
Warm inflation parameter | $10^{7}\tilde{V}_{0}$ | $[0.1,10.0]$
Warm inflation parameter | $\tilde{V}_{1}$ | $[0.1,10.0]$
Angular scale at decoupling | $100\Theta_{\rm{MC}}$ | $[0.5,10.0]$
Optical depth | $\tau$ | $[0.01,0.8]$
Pivot scale $[{\rm{Mpc}}^{-1}]$ | $k_{\rm pivot}$ | 0.05
Table 1: Flat priors on the cosmological parameters varied in this paper. Parameters |  _Planck_ TT+Pol
---|---
$\Omega_{b}h^{2}$ | $0.02124\pm 0.00012$
$\Omega_{c}h^{2}$ | $0.1387\pm 0.0010$
$10^{7}\tilde{V}_{0}$ | $<0.626$
$\tilde{V}_{1}$ | $4.481^{+0.095}_{-0.20}$
$100\theta_{MC}$ | $1.03888\pm 0.00029$
$\chi^{2}_{\rm CMB}$ | $1359.4\,({\nu\rm{:}\,11.8})$
Table 2: $68\%$ CL constraints for the Warm inflation model, for _Planck_ 2018
temperature and polarization data. Figure 2: Triangular plot for minimal warm
inflation, representing contour plot and likelihoods of parameters of model
$\tilde{V}_{0}$ and $\tilde{V}_{1}$.  Figure 3: Two-dimensional contour plots
for minimal warm inflation, investigating the two parameters of model
$\tilde{V}_{0}$ and $\tilde{V}_{1}$ vs. $\Omega_{m}$ are plotted.  Figure 4:
The power spectrum of temperature anisotropies for minimal warm inflation
model in comparison with standard $\Lambda$CDM model and _Planck_ 2018 power
spectrum is plotted. Below panel shows the difference of MWI power spectrum
and _Planck_ data.

##  5 Discussion and Conclusions

In this paper we have tried to constrain MWI model by using a direct method of
data analysis. The Scalar power-spectrum of warm inflation as a function of
wave number is presented by a non-standard model dependent format. It would be
interesting to constrain the free parameters of the model directly. Best-fit
values and confidence intervals for parameters of MWI model summarized in
Table. 2. As you can see by implementation of just _Planck_ data we can put an
upper bound on parameter $\tilde{V}_{0}<6.26\times 10^{-8}$ which is
consistent with TCC. Same result has shown in Fig. 2. In Fig. 3 we see that
matter density $\Omega_{m}$ shows nearly zero correlation with model
parameters $\tilde{V}_{0}$ and $\tilde{V}_{1}$. Also we find that our model
gives higher matter density than standard inflation.
The result we have found for energy scales

$\displaystyle 4\times 10^{7}<V_{0}^{\frac{1}{4}}<5\times 10^{8}\rm{GeV}$ (18)

of the model is surprisingly compatible with TCC (13) It is the main result of
our discussion.

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