matter at the same time. Here is why.

It is usually assumed that dark matter in standard cosmology models is
produced during the radiation-dominated era. If the axion mass $m_{a}\geqslant
10^{-2}~{}eV$, axions were produced thermally, with cosmological abundance

$\Omega_{a}h_{0}^{2}=\frac{m_{a}}{130~{}eV}\left(\frac{10}{g_{*S,F}}\right),$
(88)

where $g_{*S,F}$ is the effective number of relativistic degrees of freedom
when axions freeze out of equilibrium. Turner showed [ref159, ref159a] that
the vast majority of these would have arisen in the early Universe via thermal
mechanisms such as Primakoff scattering and photo-production. The Boltzmann
equation can be solved to give their present comoving number density as
$n_{a}=(830/g_{*S,F})~{}cm^{-3}$ [ref157], where $g_{*S,F}\approx 15$ counts
the number of relativistic degrees of freedom left in the plasma at the time
when axions ”froze out” of equilibrium. The present density parameter
$\Omega_{a}=n_{a}m_{a}/\rho_{crit,0}$ of thermal axions thus leads to the
expression (82) whence it follows

$0.11\leqslant\Omega_{a}\leqslant 0.25.$ (89)

Here we have taken $0.6\leqslant h_{0}\leqslant 0.9$ as usual. This is
comparable to the density of dark matter.

At the same time, there are axion constraints in the so-called nonstandard
thermal histories. As it was shown in [ref188], the most intriguing one among
the well-known non-standard thermal histories (low-temperature reheating and
kination cosmologies) is the LTR cosmology which allows for the fact that
there is currently no direct evidence for radiation domination prior to big-
bang nucleosynthesis [ref189]. According to this scenario, radiation
domination begins as late as 1 MeV, and is preceded by significant entropy
generation. Thermal axion relic abundances are then suppressed, and
cosmological limits to axions are loosened. However, for reheating
temperatures $T_{rh}\leqslant 35~{}MeV$, the large-scale structure limit to
the axion mass is lifted (see Fig. 21). The remaining constraint from the
total density of matter is significantly relaxed in this case. Constraints are
also relaxed for higher reheating temperatures. It turns out that axions will
be produced thermally, with nonstandard cosmological abundance

$\Omega_{a}^{nth}h_{0}^{2}=\frac{m_{a}}{130~{}eV}\left(\frac{10}{g_{*S,F}}\right)\cdot\gamma(T_{rh}/T_{F}),$
(90)

where $T_{F}$ is the decoupling temperature of axions.

Figure 21: (a) Upper limits to the hadronic axion mass from cosmology,
allowing the possibility of a low-temperature-reheating scenario. The green
region shows the region excluded by the constraint $\Omega_{a}h^{2}<0.135$ as
a function of the reheating temperature $T_{rh}$. The red region shows the
additional part of axion parameter space excluded by WMAP1/SDSS data. At low
reheating temperatures, upper limits to the axion mass are loosened. For
$T_{rh}\geqslant 170~{}MeV$, the usual constraints are recovered. Adapted from
[ref188].
(b) Estimated improvement in the accessible axion parameter space from
including more precise measurements of the matter power spectrum (the region
bounded by the yellow line), corresponding to LSST [ref190, ref191], or from
measurements of clustering on smaller length scales, corresponding to
Lyman-$\alpha$ forest measurements (the region bounded by the blue line)
[ref192]. The red region indicates the parameter space excluded by WMAP1/SDSS
measurements. Adapted from [ref188]. Our result (a red star) has the
coordinates $\left\\{T_{rh}=30~{}MeV;m_{a}=17~{}eV\right\\}$.

Using the abundance $\Omega_{a}^{nth}$ normalized by its standard value
$\Omega_{a}$ as a function of the reheating temperature, obtained by Grin et
al. (Fig. 4 in [ref188]) it is easy to derive for the mass $m_{a}=17~{}eV$ at
$T_{rh}=30-35~{}MeV$ (see Fig. 21)

$\frac{\Omega_{a}^{nth}}{\Omega_{a}}=\gamma(T_{rh}/T_{F})\cong
0.2~{}~{}for~{}~{}T_{rh}=30~{}~{}MeV.$ (91)

Hence,

$0.022\leqslant\Omega_{a}^{nth}\leqslant 0.05.$ (92)

which is rather small and comparable to the baryonic density value
$0.027\leqslant\Omega_{b}\leqslant 0.040$ [ref156] as opposed to the standard
case of (89).

Which one of them is preferable when it comes to description of real physics –
the standard expression (89) for the axion dark matter density or the non-
standard one (92)? Although there is no answer to this question nowadays, one
may still hope that, according to [ref188], future limits to axions in the
standard radiation-dominated and LTR thermal histories may follow from
constraints to their contribution to the energy density of relativistic
particles at $T\sim$1 MeV. This is due to the fact that a comparison between
the abundance of 4He and the predicted abundance from standard big-bang
nucleosynthesis (SBBN) places constraints to the radiative content of the
Universe at $T\sim$1 MeV [ref193]. As the paper [ref188] notes, this can be
stated as a constraint to the effective neutrino number ($N_{\nu}^{eff}$),
because at early times, axions contribute to the total relativistic energy
density (through $N_{\nu}^{eff}$), and thus constraints to 4He abundances can
be turned into constraints on $m_{a}$ and $T_{rh}$.

It is known that in terms of the baryon-number density $n_{b}$, the primordial
4He abundance is characterized by the expression $Y_{P}=4n_{He}/n_{b}$. In
order to translate measurements of primordial 4He abundance $Y_{P}$ to
constraints on $m_{a}$ and $T_{rh}$ we use the scaling relation by Grin et al.
[ref188]

$\Delta
N_{\nu}^{eff}(m_{a},T_{rh})=N_{\nu}^{eff}-3=\frac{43}{7}\left\\{(6.25\cdot\Delta
Y_{p}+1)-1\right\\},$ (93)

which describes the relation between the deviation $\Delta N_{\nu}^{eff}$ for
the given values of $m_{a}$ and $T_{rh}$ and the deviation $\Delta Y_{P}$ of
the primordial 4He abundance. Here the deviation $\Delta Y_{P}$ is thought of
as a deviation from the value $Y_{P}=0.2487\pm 0.0006$ accepted for SBBN-
predicted primordial abundances [ref194].

According to the calculations in [ref188], for the values $m_{a}=17~{}eV$,
$T_{rh}=30~{}MeV$ and the effective neutrino number $N_{\nu}^{eff}\cong 3.44$
(Fig. 7 in [ref188]), the deviation $\Delta N_{\nu}^{eff}$ is 0.44.
Substitution of this value into Eq. (93) gives the following value for the
$\Delta Y_{P}$ of the primordial 4He abundance:

$\Delta Y_{p}=0.0044,$ (94)

Let us present some data for the sake of comparison. One careful study gives
the value $Y_{P}=0.2565\pm 0.0010~{}(stat)\pm 0.0050~{}(syst)$ from 93
H${}_{\text{II}}$ regions [ref195], leading to the sensitivity limit
$N_{\nu}^{eff}\cong 3.61$. Interestingly enough, the deviation $\Delta
N_{\nu}^{eff}=0.61$ may be a sign of the higher mass axions existence, since
according to [ref186], for sufficiently high masses, the axionic contribution
saturates to $\Delta N_{\nu}^{eff}=4/7$ at high reheating temperatures. At the
same time the Planck satellite is expected to reach $\Delta Y_{P}=0.013$
[ref196], yielding sensitivity of $N_{\nu}^{eff}\cong 4.04$, while CMBPol (a
proposed future CMB polarization experiment) is expected to approach $\Delta
Y_{P}=0.0039$, leading to the sensitivity limit $N_{\nu}^{eff}\cong 3.30$
[ref188].

It makes it clear that the answer to a question about the preference of one
expression (standard (89) or non-standard (92)) for the axion dark matter
density over another may be searched for only in direction of a sharp increase
in observations precision. According to [ref197], measuring CMB temperature
and polarization with cosmic variance accuracy would allow to constrain
$Y_{P}$ to within 1.5%, or $\Delta Y_{P}\sim 0.0036$ (assuming flatness). Such
an ideal measurement would be able to discriminate between the BBN-guided,
deuterium based helium value and the current lowest direct helium
observations. In other words, ”…if the CMB-determined helium mass fraction
turns out to be as high as suggested by SBBN calculations combined with the
observed deuterium abundance, this could indicate a systematic error in the
present direct astrophysical helium observations. Alternatively, if the CMB
could independently determine the helium value with sufficient precision to
confirm the present low helium value coming from direct observations, then
this would be a smoking gun for new physics” [ref197]. For example, one could
imagine sterile neutrinos appearing within the nonstandard BBN scenarios,
which would agree with present observations of
$\eta_{10}=10^{10}(n_{b}/n_{\gamma})$, while having a low helium mass
fraction. To put it differently, in our opinion, in spite of the future
possible constraints to axions and low-temperature reheating from the helium
abundance and next-generation large-scale-structure surveys, it is the
appearance of the sterile neutrinos that may effectively solve the problem of
missing dark matter in the framework of LTR cosmology.

And finally, returning to our 17 eV axion ”caught” in the extragalactic
background, we may say that regardless of the scenario which provides a
significant portion of the dark matter, experimentally observed invisible
axions (Fig. 20) must have rest masses in the ”semi-visible” range (1500Å \-
20000Å) where they do contribute significantly to the light of the night sky.
In this sense one may think of our findings as a result of the axion mechanism
of Sun luminosity and the ”…nature’s most versatile dark-matter detector: the
light of the night sky” [ref166].

##  6 Relic axion-like archion and cosmic infrared background

It is well known that a direct measurement of the EBL and, particularly, the
cosmic infrared background (CIB) consists in observing the cumulative emission
from various pregalactic objects, protogalaxies, galaxies, cosmic explosions
and decaying elementary particles (including dark matter particles) throughout
the evolution of the Universe and therefore one can provide important
constraints on the integrated cosmological history of star formation [ref198,
ref199] and control the second most important contribution to the cosmic
electromagnetic background after the Cosmic Microwave Background generated at
the time of recombination at a redshift around 1000 [ref199]. This background
is expected to be composed of three main components [ref199]:

  * •

the stellar radiation in galaxies concentrated in the ultraviolet and visible
with a redshifted component in the near InfraRed (IR);

  * •

a fraction of the stellar radiation absorbed by dust either in the galaxies or
in the intergalactic medium;

  * •

the radiation from active galactic nuclei (a fraction of which is also
absorbed by dust and reradiated in the far-IR).

Its detection is a subject of great scientific interest and the main purpose
of the Diffuse Infrared Background Experiment (DIRBE) on the Cosmic Background
Explorer (COBE) space-craft [ref176, ref177, ref178]. These studies resulted
in upper limits on the EBL in the 1.25-100 $\mu m$ region, and in the
detection of a positive isotropic signal at 140 and 240 $\mu m$.

However, there are serious problems with interpretation of some DIRBE data at
far-IR wavelengths. For example [ref198], the energy sources could either be
yet undetected dust-enshrouded galaxies, or extremely dusty star-forming
regions in observed galaxies, and they may be responsible for the observed
iron enrichment in the intracluster medium. Although there is currently no
compelling need to invoke non-nuclear energy sources to explain the COBE data,
their potential contribution to the observed EBL cannot be ruled out. It leads
to a conclusion that the exact star formation history or scenarios required to
produce the EBL at far-IR wavelengths cannot be unambiguously resolved by the
COBE observations and must await future observations [ref198].

The other type of problems is revealed by the interpretation of the DIRBE data
at near-IR wavelengths. For example, in their studies of EBL at near-IR
wavelengths Cambrésy et al. [ref178] obtain a significantly higher cosmic
background than integrated galaxy counts ($3.6\pm 0.8~{}kJy\cdot sr^{-1}$ and
$5.3\pm 1.2~{}kJy\cdot sr^{-1}$ for 1.25 $\mu m$ and 2.2 $\mu m$,
respectively), suggesting either an increase of the galaxy luminosity function
for magnitudes fainter than 30 or the existence of another contribution to the
cosmic background from primeval stars, black holes, or relic particle decay.
However, models predict other possible contributions to the background at
these wavelengths [ref200] such as a burst of star formation either in
primeval galaxies or in Population III stars ($z\approx 10$), very massive
black holes (accreting from a uniform pre-galactic medium at $z\approx 40$),
massive decaying big bang relic particles ($z\approx 300$). In this regard,
Cambresy et al. [ref178] make a natural conclusion that the new constraints in
the near-infrared should encourage revisiting the importance of those
contributions to the CIB in cosmological models.

Figure 22: The spectral intensity $I_{\lambda}(\lambda_{0})$ of the background
radiation from decaying axions as a function of the observed wavelength
$\lambda_{0}$. The curves for values $m_{a}=17.0~{}eV$,
$\zeta=c_{a\gamma\gamma}/0.72=0.03$ and $m_{a}=2.0~{}eV$, $\zeta=9.0$
correspond to upper, median and lower limits on $h_{0}$. Also observational
upper limits (solid symbols) over these wavebands are shown.

In this connection we consider the effect of dark matter in the form of 2 eV
relic particle on the extragalactic background light at near-IR region (Fig.
22). It was obtained on the basis of the intensity (79) of axionic
contributions to the EBL calculation as a function of the wavelength
$\lambda_{0}$. The theoretical fit (green curves in Fig. 22) of the spectral
intensity of the background radiation (79) produced by axion decays,
apparently, describes the known experimental data for the near-infrared band
very well. They include data from ground-based telescope observations (BK86
[ref171]) and the DIRBE instrument aboard the COBE satellite (H98 [ref176],
WR00 [ref177], C01 [ref178]).

Let us consider some properties of such relic particle with the 2 eV mass. It
may be immediately stated that such particle cannot play the role of a sterile
neutrino with the mass of the order of a few eV [ref201]. And the reason is
the following.

It is known that if neutrinos are massive and if the mass eigenstates are not
degenerate, then it is possible to have a radiative decay of the form
$\nu_{s}\to\nu+\gamma$. According to [ref202], this gives for the decay rate
of Majorana neutrinos

$\Gamma_{\nu_{S}\rightarrow\nu\gamma}^{*}=5.52\cdot
10^{-32}\left(\frac{\sin^{2}\theta}{10^{-10}}\right)\left(\frac{m_{S}}{keV}\right)^{5}~{}~{}s^{-1},$
(95)

where $m_{S}$ is the mass eigenstate most closely associated with the sterile
neutrino, and $\theta$ is the mixing angle between the sterile and active
neutrino. The decay of a nonrelativistic sterile neutrino into two (nearly)
massless particles produces a line at the energy $E{\gamma}=m_{S}/2$.

Obviously, in case of the decay reaction $\nu_{s}\to\nu+\gamma$, Eq. (79) can
be generalized to another relic $\nu_{s}$ by multiplying it by

$I_{\lambda}(\lambda_{0})\cdot\frac{1}{2}\cdot\frac{\Omega_{\nu}}{\Omega_{a}}\cdot\frac{\tau_{a\rightarrow\gamma\gamma}^{*}}{\tau_{\nu_{S}\rightarrow\nu\gamma}^{*}},$
(96)

where the 1/2 is a number of photons produced in each $\nu_{s}$ decay,
$\Omega_{\nu}\leqslant 0.014$ is the present total density parameter of the
neutrinos,

$\tau_{\nu_{\chi}\rightarrow\nu\gamma}^{*}=(\Gamma_{\nu_{S}\rightarrow\nu\gamma}^{*})^{-1}=(1.7\cdot
10^{15}t_{Univ})\zeta^{-2}\left(\frac{m_{S}}{keV}\right)^{-5},$ (97)

where

$\zeta^{-2}=\left(\frac{10^{-10}}{\sin^{2}2\theta}\right).$ (98)

Using the generalization (96), it is rather easy to show that a theoretical
fit of spectral intensity $I_{\lambda}$ of the background radiation is several
orders of magnitude lower than the experimental data in near-infrared band
under any reasonable values of the mixing angle $\sin^{2}2\theta$.

This relic particle with the 2 eV mass may be supposed to belong to a class of
axion-like particles with rather exotic properties. One of them may originate
from the fact that in the distant past their birth in the electromagnetic
field, or in other words interaction with a photon, was not suppressed for
some reason, while at the present time there are simply no conditions suitable
for their birth. To put it differently, the mentioned conditions of such
particles’ birth must be completely suppressed nowadays in order our axion
mechanism of Sun luminosity to be possible.

In our opinion, such axion-like particle, provided that it exists, has the
properties most similar to those of the so-called archion [ref203, ref204,
ref205, ref206, ref207, ref208, ref209]. An archion may be very similar to a
hadronic axion with highly suppressed interaction with leptons under certain
conditions. Let us discuss briefly some of its properties below.

As is generally known, in all the models of an invisible axion this particle
appears as a Goldstone boson connected with the phase of a complex
SU(2)$\times$U(1) singlet Higgs field. The axion coupling to the gauge bosons
appears in these models after the U(1)PQ symmetry violation by means of a
mechanism, specified by the non-vanishing color anomaly U(1)PQ \- SU(3)c \-
SU(3)c [ref210].

In its most generic form the Lagrangian of axion interaction with fermions
(quarks and leptons) and photons is

$L=c_{\alpha\beta}\alpha\cdot\bar{F}_{\alpha}(\sin\theta_{\alpha\beta}+i\gamma_{5}\cos\theta_{\alpha\chi})F_{\beta}+g_{a\gamma}aF_{\mu\nu}\bar{F}^{\mu\nu},$
(99)

where $\alpha,\beta=1,2,3$ indices denote the generation of fermions $F$, and
the constants $\theta_{\alpha\beta}$,

$c_{\alpha\beta}\propto f_{a}^{-1}$ (100)

and

$g_{a\gamma}\propto f_{a}^{-1}$ (101)

depend on the axion model chosen.

A model of an archion [ref203, ref204, ref205, ref206, ref207, ref208, ref209]
arose from a model of horizontal unification, the basis of which is expounded
in a monograph by Khlopov [ref210]. This theory includes the global U(1)H
symmetry, the spontaneous breaking of which leads to prediction of a Goldstone
boson of the invisible axion type. Such boson called ”archion” by authors of
[ref205, ref206, ref207, ref208] has the flavor non-diagonal as well as the
flavor diagonal coupling to fermions.

The global U(1)H symmetry in horizontal unification may be identified with the
Peccei–Quinn symmetry U(1)PQ [ref028, ref029, ref030, ref031, ref032, ref032a,
ref033, ref033a], which is due to the fact of triangle anomaly existence in
axial currents U(1)H interaction with gluons [ref210].

In the simplest variant of the horizontal unification (the gauge symmetry

$S(U)_{c}\otimes SU(2)\otimes U(1)\otimes SU(3)_{H}\otimes U(1)_{H}$ (102)

with a minimal set of heavy fermions) the anomaly is compensated, and the
archion remains almost massless. The interaction of the archion with photons
is absent because of the parallel compensation which corresponds to the
current-photon-photon anomaly.

On the other hand, according to [ref210], within any realistic extension of
the horizontal unification model up to the grand unification symmetry, for
example, within the extension to the SU(5)${}_{H}\otimes$SU(3)H symmetry,
there is no compensation because of the extra heavy fermions, so that the
archion appears to be similar to a hadronic axion with strongly suppressed
interaction with leptons.

In the framework of the archion model, the archion-photon coupling constant
$g_{a\gamma}$ which appears in Eqs. (99) and (101) has the following form
[ref210]:

$g_{a\gamma}=\frac{\alpha}{2\pi f_{a}}c_{a\gamma\gamma}=\frac{\alpha}{4\pi
f_{a}}\frac{A_{c}z^{3/2}}{(1+z)^{2}}\left[\frac{A_{em}}{A_{c}}-\frac{2(4+z)}{3(1+z)}\right]^{-1}.$
(103)

Here as usual $z=m_{u}/m_{d}$ is the $u$\- and $d$-quarks mass ratio, $f_{a}$
is the energy scale associated with the breakdown of the U(1)H (or
equivalently, the U(1)PQ symmetry), $A_{c}$ and $A_{em}$ are the color and
electromagnetic anomalies respectively.

Taking into account the value of the $c_{a\gamma}=0.72\zeta$ constant (see
(83) and Fig. 22), which is $\sim$6.5 for the archion mass of 2 eV, it is easy
to estimate the archion-photon coupling constant from (103) and (10):

$g_{a\gamma}^{*}\simeq 2.5\cdot 10^{-9}~{}~{}GeV^{-1}.$ (104)

Obviously, if the archion indeed exists and has such a high archion-photon
coupling constant, it must be of the relic origin only in a sense that the
conditions of such particles birth must be completely suppressed nowadays.
Otherwise, our axion mechanism of Sun luminosity would not be possible, like
we pointed out earlier191919It is necessary to note that regardless of whether
one assumes the existence of the axion mechanism of Sun luminosity or not,
such high value of the archion-photon coupling constant (104) is forbidden by
the DAMA experiment observations [ref146] presented in Fig. 17a and 17b.. The
value of this constant is also very important because according to [ref186],
it is a strong indicator of (a) the effects of axion emission on the evolution
of helium burning low-mass stars [ref187], (b) the effect of decaying relic
axions on the diffuse extragalactic background radiation [ref157, ref166]. In
other words, an archion with the 2 eV mass, characterized by the archion-
photon coupling constant (104), must be relic in order not to violate the
known limitation (87) which is a consequence of combined action of effects (a)
and (b).

For all invisible axion models, including the archion model, Lagrangian of its
interaction with nucleons has the same form [ref209]. Given (104), it is not
hard to estimate the archion-nucleon coupling constant for the 2 eV mass
archion (Fig. 22) using (29)-(31) and (10):

$g_{an}^{*}\leqslant 5.6\cdot 10^{-8}.$ (105)

If the archion exists with such low value of the archion-nucleon coupling
constant, it obviously must have only relic origin in order not to violate the
SN1987A limit ($3\cdot 10^{-7}\leqslant g_{an}\leqslant 10^{-6}$ [ref111,
ref112]).

This rises the question of why would Nature need a relic axion-like archion in
addition to an ordinary hadronic axion. Curiously enough, the answer suggests
itself and is related to the evolutionary formation of the visible large-scale
cosmological structure against the background of the invisible ”dark”
structure.

Apart from the details of this scenario, our axion-like particles may be among
the primary participants of this creative action, if they do exist. Let us
note however that in spite of the fact that the cosmological structure
formation is provided by the single kind of hidden-mass particles – axions,
the ”axion liquid” turns out to be two-component. The visible structure in the
form of galaxies and superclusters is formed by the shortwave component of the
17 eV axions density perturbations spectrum, while the relic thermal 2 eV
archions background plays an important role in sub-shortwave density
perturbations spectrum evolution. The latter includes, in particular, the
massive halos formation beyond the visible parts of the galaxies. Following
[ref210], it is worth mentioning that the so-called phase-space argument by
Tramaine and Gunn [ref211], which gives rise to a known limit on particles
mass in the halo, may be substantially weakened or even omitted for the Bose
gas [ref212, ref213, ref214, ref215].

If all said above is true, it is more than sufficient for a ”sensible”
existence of axions, and particularly, a relic axion-like archion. On the
other hand, except for the indirect indication of axions existence in the form
of Fig. 22, we have no other – direct – arguments. In order to fill this gap a
little, let us try to substantiate the quantitative relations between the
experimental spectral intensities of the background radiation from decaying
axions and archions shown in Fig. 22 theoretically on the basis of a
simplified cosmological model which takes into account the processes of axion
dark matter radiative decay.

###  6.1 Decaying axion and relic archion as two components of luminous dark
matter

Let us consider the flat (for simplicity) Universe after the recombination
consisting of the following components: relic radiation, usual (light,
visible) matter, dark matter (presented by axions and archions), non-relic
radiation (resulting from their decay) and dark energy (described by the
cosmological constant $\Lambda$), then for the corresponding metrics

$ds^{2}=c^{2}dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (106)

where $a(t)$ is the scale factor, the first Friedmann equation reads

$\frac{3\dot{a}^{2}}{c^{2}a^{2}}=\kappa T_{00}+\Lambda,$ (107)

where a dot denotes the derivative with respect to $t$, $\kappa=8\pi
G_{N}/c^{4}$ ($G_{N}$ is the Newtonian gravitational constant) and the $00$
covariant component of the total energy-momentum tensor reads

$T_{00}=\varepsilon_{rr}+\varepsilon_{\nu
m}+\varepsilon_{a2}+\varepsilon_{a17}+\varepsilon_{r2}+\varepsilon_{r17},$
(108)

where, in their turn, $\varepsilon_{rr,\nu m,a1,a17,r2,r17}$ denote energy
densities of relic radiation (assumed to be independent of all other
components), visible matter (also assumed to be independent), archions (with
the mass $m_{a2}=2~{}eV$ and the disintegration constant
$\lambda_{a2}=3.88\cdot 10^{-22}~{}s$, assumed to be completely
nonrelativistic), axions (with the mass $m_{a17}=17~{}eV$ and the
disintegration constant $\lambda_{a17}=1.91\cdot 10^{-22}s^{-1}$, also assumed
to be completely nonrelativistic), radiation resulting from archion decay and
radiation resulting from axion decay respectively.

Here some additional comments should be made. First, the disintegration
constants are estimated simply as $\lambda=1/\tau^{*}$ on basis of the formula
(83). Second, we assume both dark matter components (archions and axions)
completely nonrelativistic, in other words, we completely neglect their
velocities (or temperatures). This assumption is simultaneously in agreement
and disagreement with [ref166], where the formula (185) (which is equivalent
to (79) in our text) contains simultaneously the scent of the standard
cosmological $\Lambda$CDM-model without matter velocities in the denominator
of the integrand and the scent of the velocity dispersion in its numerator. It
seems that this fact does not mean that there is a self-contradiction in the
formula (185), because the numerator may be much more sensitive to the local
velocity of dark matter than the denominator to the global one. However, in
the subsequent analysis we shall be interested in the evolution of the average
non-relic radiation energy density without taking into account its frequency
distribution and the nonzero dark matter temperature (for simplicity).

Substituting (108) into (107) and introducing the standard portions

$\Omega=\frac{\kappa
c^{2}}{3H_{0}^{2}}\varepsilon_{(0)},~{}~{}\Omega_{\Lambda}=\frac{c^{2}}{3H_{0}^{2}}\Lambda,$
(109)

where the subscript $(0)$ corresponds to the current moment of time $t=0$
(this value may be chosen without loss of generality) and $H=\dot{a}/a$ is the
Hubble parameter, we obtain

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}$
$\displaystyle\left(\Omega_{rr}\frac{\varepsilon_{rr}}{\varepsilon_{rr(0)}}+\Omega_{\nu
m}\frac{\varepsilon_{\nu m}}{\varepsilon_{\nu
m(0)}}+\Omega_{a2}\frac{\varepsilon_{a2}}{\varepsilon_{a2(0)}}+\right.$
$\displaystyle\left.+\Omega_{a17}\frac{\varepsilon_{a17}}{\varepsilon_{a17(0)}}+\Omega_{r2}\frac{\varepsilon_{r2}}{\varepsilon_{r2(0)}}+\Omega_{r17}\frac{\varepsilon_{r17}}{\varepsilon_{r17(0)}}+\Omega_{\Lambda}\right).$
(110)

Here $\varepsilon_{rr}\sim 1/a^{4}$, $\varepsilon_{\nu m}\sim 1/a^{3}$,
consequently, as usual,

$\Omega_{rr}\frac{\varepsilon_{rr}}{\varepsilon_{rr(0)}}+\Omega_{\nu
m}\frac{\varepsilon_{\nu m}}{\varepsilon_{\nu
m(0)}}=\Omega_{rr}\left(\frac{a_{0}}{a}\right)^{4}+\Omega_{\nu
m}\left(\frac{a_{0}}{a}\right)^{3},$ (111)

where $a_{0}$ is the current value of the scale factor: $a(0)=a_{0}$. Further,

$\varepsilon_{a2}:\frac{1}{a^{3}}\exp(-\lambda_{a2}t),~{}~{}\varepsilon_{a17}:\frac{1}{a^{3}}\exp(-\lambda_{a17}t)\Longrightarrow\varepsilon_{a2(0)}:\frac{1}{a_{0}^{3}},~{}~{}\varepsilon_{a17(0)}:\frac{1}{a_{0}^{3}},$
(112)

whence

$\Omega_{a2}\frac{\varepsilon_{a2}}{\varepsilon_{a2(0)}}+\Omega_{a17}\frac{\varepsilon_{a17}}{\varepsilon_{a17(0)}}=\Omega_{a2}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)+\Omega_{a17}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a17}t).$
(113)

Finally, combining first and second Friedmann equations, we come to the
following equations:

$d\left[(\varepsilon_{a2}+\varepsilon_{r2})a^{3}\right]+\frac{1}{3}\varepsilon_{r2}d(a^{3})=0,~{}~{}d\left[(\varepsilon_{a17}+\varepsilon_{r17})a^{3}\right]+\frac{1}{3}\varepsilon_{r17}d(a^{3})=0.$
(114)

From the first one we immediately get

$\frac{d(\varepsilon_{a2}a^{3})}{da}+a^{3}\frac{d\varepsilon_{r2}}{da}+4a^{2}\varepsilon_{r2}=0,~{}~{}\frac{d(\varepsilon_{a2}a^{3})}{da}=\varepsilon_{a2(0)}a_{0}^{3}\exp(-\lambda_{a2}t)\frac{(-\lambda_{a2})}{\dot{a}},$
(115)

$\frac{d\varepsilon_{r2}}{da}+\frac{4\varepsilon_{r2}}{a}=\varepsilon_{a2(0)}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)\frac{\lambda_{a2}}{\dot{a}},$
(116)

whence finally

$\displaystyle\varepsilon_{r2}$
$\displaystyle=\frac{1}{a^{4}}\left(\varepsilon_{a2(0)}a_{0}^{3}\lambda_{a2}\int\limits_{a_{0}}^{a}\exp(-\lambda_{a2}t)\frac{a}{\dot{a}}da+\varepsilon_{r2(0)}a_{0}^{4}\right)=$
$\displaystyle=\frac{1}{a^{4}}\left(\varepsilon_{a2(0)}a_{0}^{3}\lambda_{a2}\int\limits_{0}^{t}a(t)\exp(-\lambda_{a2}t)dt+\varepsilon_{r2(0)}a_{0}^{4}\right).$
(117)

Similarly,

$\varepsilon_{r17}=\frac{1}{a^{4}}\left(\varepsilon_{a17(0)}a_{0}^{3}\lambda_{a17}\int\limits_{0}^{t}a(t)\exp(-\lambda_{a17}t)dt+\varepsilon_{r17(0)}a_{0}^{4}\right).$
(118)

The substitution of (111), (113), (117) and (118) into (110) gives the
equation defining the function $a(t)$ satisfying the condition $a(t)=a_{0}$.
Taking into account that the values of both disintegration constants
$\lambda_{a2}$ and $\lambda_{a17}$ are extremely small and the radiation
energy density decreases faster than the nonrelativistic matter energy density
when $a$ increases, one can neglect all (i.e. both relic and non-relic)
radiation contributions and get

$\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}\left[\Omega_{\nu
m}\left(\frac{a_{0}}{a}\right)^{3}+\Omega_{a2}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)+\Omega_{a17}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a17}t)+\Omega_{\Lambda}\right].$
(119)

Introducing the dimensionless quantities

$\tilde{t}=H_{0}t,~{}~{}\tilde{a}=\frac{a}{a_{0}},~{}~{}\tilde{\lambda}_{a2}=\frac{\lambda_{a2}}{H_{0}},~{}~{}\tilde{\lambda}_{a17}=\frac{\lambda_{a17}}{H_{0}},$
(120)

from (112) we obtain

$\left(\frac{1}{\tilde{a}}\frac{d\tilde{a}}{d\tilde{t}}\right)^{2}=\frac{\Omega_{\nu
m}}{\tilde{a}^{3}}+\frac{\Omega_{a2}}{\tilde{a}^{3}}\exp(-\tilde{\lambda}_{a2}\tilde{t})+\frac{\Omega_{a17}}{\tilde{a}^{3}}\exp(-\tilde{\lambda}_{a17}\tilde{t})+\Omega_{\Lambda},~{}~{}\tilde{a}(0)=1.$
(121)

According to the recent observations [ref216], we consider the following
values: $\Omega_{\nu m}=0.046$, $\Omega_{a2}+\Omega_{a17}=0.236$,
$\Omega_{\Lambda}=1-\Omega_{\nu m}-(\Omega_{a2}+\Omega_{a17})=0.718$. For
$\Omega_{a2}$ and $\Omega_{a17}$ we use the values $0.016$ and $0.220$
respectively. Again, due to extreme smallness of $\lambda_{a2}$ and
$\lambda_{a17}$ (in the presence of nonzero $\Lambda$) they do not affect
noticeably the dependence $a(t)$. The following graphs confirm this statement:

Figure 23: The numerical solution of Eq. (121) (the blue firm line) is
indistinguishable from its numerical solution for $\lambda_{a2}=0$ and
$\lambda_{a17}=0$ (the yellow dashed line).

From (117) and (118) we obtain respectively

$\tilde{t}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}},$
(122)

$\tilde{t}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}.$
(123)

Let us introduce the convenient functions

$f_{a2}(\tilde{t})=\frac{\varepsilon_{r2}}{\varepsilon_{a2(0)}}\tilde{a}^{4}-\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}}=\tilde{\lambda}_{a2}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t},$
(124)

$f_{a17}(\tilde{t})=\frac{\varepsilon_{r17}}{\varepsilon_{a17(0)}}\tilde{a}^{4}-\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}=\tilde{\lambda}_{a17}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t},$
(125)

For $a=0$ we get $f_{a2}=-9.26\cdot 10^{-5}$ and $f_{a17}=-4.56\cdot 10^{-5}$.
These negative values may be completely compensated by the proper choice of
$\varepsilon_{r2(0)}$ and $\varepsilon_{r17(0)}$ in order to give
$\varepsilon_{r2}=0$ and $\varepsilon_{r17}=0$ at the same moment of time
(when $a=0$). This has clear physical sense: we assume that at the
recombination moment the non-relic radiation is absent (all radiation, present
at that moment of time, may be considered as the relic one). From this
requirement we immediately obtain $\varepsilon_{r2(0)}=9.26\cdot
10^{-5}\varepsilon_{a2(0)}$ and $\varepsilon_{r17(0)}=4.56\cdot
10^{-5}\varepsilon_{a17(0)}$.

Now let us also depict the helpful graphs of the ratios
$\varepsilon_{r2}/\varepsilon_{a2}$ and $\varepsilon_{r17}/\varepsilon_{a17}$
as functions of $\tilde{t}$ (Fig. 24a, red and green lines respectively):

$\frac{\varepsilon_{r2}}{\varepsilon_{a2}}=\frac{\exp(\tilde{\lambda}_{a2}\tilde{t})}{\tilde{a}}\left(\tilde{\lambda}_{a2}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}}\right),$
(126)

$\frac{\varepsilon_{r17}}{\varepsilon_{a17}}=\frac{\exp(\tilde{\lambda}_{a17}\tilde{t})}{\tilde{a}}\left(\tilde{\lambda}_{a17}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}\right),$
(127)

In particular, for $t=0$ we get $\varepsilon_{r2}/\varepsilon_{a2}=9.26\cdot
10^{-5}$ and $\varepsilon_{r17}/\varepsilon_{a17}=4.56\cdot 10^{-5}$, as it
should be (Fig. 24a).

Figure 24: Graphs of the ratios $\varepsilon_{r2}/\varepsilon_{a2}$ (red),
$\varepsilon_{r17}/\varepsilon_{a17}$ (green) (a) and
$\varepsilon_{r2}/\varepsilon_{r17}$ (b) as functions of time $\tilde{t}$.

Let us also depict the helpful graph of the ratio
$\varepsilon_{r2}/\varepsilon_{r17}$ as a function of $\tilde{t}$ (Fig. 24b).

The theoretical estimate of the ratio $\varepsilon_{r2}/\varepsilon_{r17}$ is,
apparently, equal to

$\left(\frac{\varepsilon_{r2}}{\varepsilon_{r17}}\right)_{theory}\cong 0.15.$
(128)

On the other hand, it is rather easy to estimate the experimental
$\varepsilon_{r2}/\varepsilon_{r17}$ ratio using the data shown in Fig. 22 and
the expression for the spectral intensity $I_{\lambda}(\lambda_{0})$ of the
background radiation from decaying axions:

$\left(\frac{\varepsilon_{r2}}{\varepsilon_{r17}}\right)_{exper}=\frac{\int\limits_{12400}^{50000}I_{\lambda
2}(\lambda_{0})d\lambda_{0}}{\int\limits_{1500}^{15000}I_{\lambda
17}(\lambda_{0})d\lambda_{0}}\cong 0.23.$ (129)

Curiously enough, the theoretical estimate (128) and the experimental one
(129) agree fairly well, which increases one’s optimism as to the essence of
the results obtained in the current section. At the same time, our estimates
require additional thorough verification, especially when it comes to the
experimental justification of the validity and reliability of the data on the
extragalactic background light in near-ultraviolet, optical and near-infrared
bands (1500-10000Å). Indeed, it is a very important step to perform, because
otherwise our unusual results may seem more like just the luckily guessed
rules of calculation, which do not reflect the actual nature of things.

##  7 Summary and Conclusion

In the present paper we present a self-consistent model of the axion mechanism
of Sun luminosity and solar dynamo – geodynamo connection, in the framework of
which we estimate the values of the axion mass ($m_{a}\sim 17~{}eV$) and the
axion coupling constants to photons ($g_{a\gamma}\sim 7.07\cdot
10^{-11}~{}GeV^{-1}$), nucleons ($g_{an}\sim 3.20\cdot 10^{-7}$) and electrons
($g_{ae}\sim 5.28\cdot 10^{-11}$). Their verification on the basis of the
model results comparison with the known experiments is also provided,
including the CAST-, CUORE- and XMASS-experiments.

In order to explain the solar-terrestrial magnetic connection we propose the
axion mechanism explaining the Sun luminosity physics and ”solar dynamo –
geodynamo” connection, where the total energy of axions, which appear in the
Sun core, is initially modulated by the magnetic field of the solar tachocline
zone due to the inverse coherent Primakoff effect and after that it makes its
way towards the Earth where its ”iron” component containing the 14.4 keV solar
axions is resonantly absorbed inside the nickel-iron core of the Earth. It
results in the fact that the variations of the axion intensity play a role of
an energy source and a modulator of the Earth magnetic field. In other words,
the solar axion mechanism is not only responsible for formation of a thermal
energy source in the liquid core of the Earth necessary for generation and
maintenance of the Earth magnetic field, but unlike other alternative
mechanisms [ref015], naturally explains the cause of the experimentally
observed strong negative correlation of the magnetic field in the tachocline
zone of the Sun and the magnetic field of the Earth.

It is necessary to note that obtained estimations can’t be excluded by the
existing experimental data (see Fig. 17a and Fig. 18a) because the effect of
solar axion intensity modulation by temporal variations of the toroidal
magnetic field of the solar tachocline zone discussed above was not taken into
account in these observations (see Fig. 17b and Fig. 18b). On the other hand,
the obtained estimates for the axion-photon coupling and the axion-nucleon
coupling cannot be ruled out by the existing theoretical limitations known as
the globular cluster star limit ($g_{a\gamma}<6\cdot 10^{-11}~{}GeV^{-1}$) and
the red giant star limit ($g_{ae}<3\cdot 10^{-13}$) [ref100], since these
values are highly model-dependent. It actually means that the axion parameters
obtained in the present paper do not contradict any of the known experimental
and theoretical model-independent limitations.

Let us give some major ideas of the paper which formed a basis for the
statement of the problem justification, and the corresponding experimental
data which comport with these ideas either explicitly or implicitly.

###  7.1 Axion mechanism of Sun luminosity

One of the key ideas behind this mechanism is the effect of $\gamma$-quanta
channeling along the magnetic flux tubes (waveguides) in the Sun convective
zone (Fig. 8), which may be represented by the $\gamma$-quanta channeling in
the periodical structure (Fig. 6) in the particular case. A low refraction
(i.e. a high transparency) of the thin magnetic flux tubes is achieved due to
the ultrahigh magnetic pressure (see (16)), induced by the magnetic field of
about 200-400 T (Fig. 9a, adapted from [ref077]). It is noteworthy that
although such strong magnetic fields have never been used for the explanation
or interpretation of the simulation results such as the stability analysis of
tachocline latitudinal differential rotation and coexisting toroidal band
using an MHD analog of the shallow-water model [ref077], they have always been
present implicitly in the very same simulation results, as our analysis shows.
Fig. 9a provides an illustrative example where a hidden part of the ”latitude
– magnetic field of the overshoot tachocline zone” dependence (which is absent
on the original plot in Fig.11 of [ref077]) is added. A right to exist and
physical validity of this part is confirmed by the bare fact of its direct
correspondence (Fig. 9a,b) with the real X-ray image of the Sun in its active
phase (Fig. 9b) obtained in the experiments performed with the Japanese X-ray
telescope Yohkoh (1991-2001) (adapted from [ref014]).

The direct experiments on monitoring the space-time evolution of the magnetic
flux tubes, rising from the deep layers of the convective zone of the Sun
(Fig. 8b, adapted from [ref064]) proved the existence of the ”hollow” ideal
magnetic waveguides for $\gamma$-quanta. These tubes cross the photosphere and
form the solar active regions such as sunspots which are the sources of X-rays
(Fig. 8d, adapted from [ref068]). There are also some theoretical results
substantiating the effect of anchoring magnetic flux tubes in the tachocline
(Fig. 8c, adapted from [ref067]).

Finally, the most impressive evidence of the axion mechanism of Sun luminosity
are the solar images (Fig. 13, adapted from [ref014]) taken at photon energies
from 250 eV up to a few keV from the Japanese X-ray telescope Yohkoh
(1991-2001), which depict the solar X-ray activity during the last maximum of
the 11-year solar cycle (Fig. 13b). It is hard to imagine another model or
considerations which would explain such anomalous X-ray radiation distribution
over the active Sun surface just as well. One should also keep in mind that
the alternative mechanism of axion transformation into $\gamma$-quanta in the
magnetic field of sunspots or other solar active regions is completely ruled
out, since there are no signs of an X-ray bright spot at the disk center (see
Fig. 13a and b), which should otherwise be observed according to [ref217].

###  7.2 Invisible axions and Solar Equator effect

According to the axion mechanism of Sun luminosity, a part of axions that pass
through the tachocline zone near the equator and poles (Fig. 6 and Fig. 13) is
not converted into $\gamma$-quanta by the Primakoff effect because of the
magnetic field vector collinearity to the axion momentum. It is the equatorial
part (Fig. 13) of invisible axions that reaches the Earth, where its ”iron”
component (i.e. the 14.4 keV solar axions) is resonantly absorbed in the Earth
core. The energy of the solar axions supplied to the Earth core in this way
plays a role of a trigger for the generation and maintenance of the
geomagnetic field, thus giving birth to the effect of the steady
anticorrelation between the variations of the Solar magnetic field and
geomagnetic field (Fig. 3).

In this sense, it is appropriate to make a short remark related to the
anticorrelation between the solar and terrestrial magnetic fields variations.
The small variations of TSI ($\Delta$TSI) are apparently produced by the solar
magnetic field variations. At the same time, these variations of the solar
magnetic field drive the equatorial sector width variations (Fig. 13), and
consequently, the ”equatorial” axion flux (see Fig. 12)

$\Delta_{equ}^{a}=0.05\mp\Delta TSI/\langle TSI\rangle.$ (130)

In other words, the ”equator” effect is not only the source of ”invisible”
axions, it also modulates their intensity inversely proportional to the solar
magnetic field change (see (130)), thus maintaining the inverse correlation
between the solar magnetic field and the ”invisible” axions flux (Fig. 12).
The latter is also the cause of the inverse correlation between the solar and
terrestrial magnetic fields variations (Fig. 3).

###  7.3 Axion mechanism of the solar dynamo – geodynamo connection

This mechanism is responsible for the physical (axionic) nature of the steady
anticorrelation between the variations of the solar magnetic field and the
geomagnetic field (the Y-component) which is directly proportional to the
westward drift of magnetic features (Fig. 4, Fig. 5) at the measurement points
(Western Europe and Australia). Figuratively speaking, in this case the
Y-component of the Earth magnetic field acts as a ”measuring instrument” which
tracks the influence of the ”equatorial” 14.4 keV solar axions on thermal and
magnetic processes in the liquid core of the Earth. The physical scenario for
this may be following.

The ”iron” solar axions that are resonantly absorbed in the Earth core,
activate the vertical background motion along the gravity force. This motion,
in its turn, ”pushes” the temperature gradient to the bottom of the liquid
core more or less heavily (depending on the total energy of axions), thereby
changing the temperature profile in the convective medium of the Earth core.
The change in the temperature profile also changes the thermal conditions near
the nuclear georeactor situated at the boundary of the liquid and solid core
of the Earth. Since the power output of such reactor is proportional to
temperature in the range of 3000-5000C∘ typical for the Earth liquid core, it
means that the variations of the Earth core temperature generated by the
mechanism of solar dynamo-geodynamo connection induce the corresponding
variations of the nuclear georeactor thermal power. If the georeactor
hypothesis is true, the fluctuations of georeactor thermal power, induced by
the variations of the absorbed axions energy in the Earth’s core, can
influence the Earth global climate in the form of anomalous temperature jumps
in the following way. Strong fluctuations of the georeactor thermal power can
lead to partial blocking of the convection in the liquid core [ref113, ref114,
ref115] and the change of the angular velocity of liquid geosphere rotation,
thereby changing the angular velocities of the Earth mantle and crust by
virtue of the total angular momentum conservation law. It means that the heat,
or more precisely the dissipation energy caused by friction of the Earth
surface and lower atmosphere, can make a considerable contribution to the
total energy balance of the atmosphere and thereby influence on the Earth
global climate evolution significantly [ref113, ref114, ref115].

On the other hand, it is clear that understanding of the mechanism of the
solar-terrestrial magnetic correlation can become the clue of the so-called
problem of solar power pacemaker related to possible existence of some hidden
but crucial mechanism of the Sun’s energy influence on the fundamental
geophysical processes. It is interesting that the ”tracks” of this mechanism
have been observed for a long time and manifest themselves in different
problems of solar-terrestrial physics, in particular, in climatology where the
mechanisms, by which small changes in the Sun’s energy output during the solar
cycle can cause change in the weather and climate, have been a puzzle and the
subject of intense research in recent decades.

If we also add the fact that the dissipation energy caused by friction of the
Earth surface and the boundary layer has a considerable value, which is enough
to cover the known deficiency of the total energy balance of the atmosphere,
it becomes clear that in the framework of the axion mechanism of the solar
dynamo – geodynamo connection it is the solar magnetic field that is a host
power pacemaker of the Earth global climate [ref113].

###  7.4 Axion-like particle and extragalactic background light

A complete physical ”portrait” of the axion, obtained from the axion mechanism
of Sun luminosity and the solar dynamo – geodynamo connection, actually
predefined the necessary conditions of its detection in astrophysics. For
example, for the search of the axion with the mass $m_{a}=17~{}eV$ in
radiative decays of the $a\to\gamma\gamma$ type, when the decay photons are
emitted at or near a peak wavelength

$\lambda_{a}=\frac{2hc}{m_{a}c^{2}}\cong 1459\text{\AA},$ (131)

we used the known observational data obtained during the study of the diffuse
extragalactic background which, according to [ref116], are more suitable than
the flux from any particular region of the sky.

Surprisingly, our theoretical fit of the spectral intensity of the background
radiation (Fig. 20) produced by axion decays describes the experimental data
in the near ultraviolet and optical bands including the data from several
ground- and satellite-based telescope observations [ref169, ref170, ref171,
ref172, ref173, ref174, ref175, ref176, ref177, ref178, ref179] with very good
accuracy.

In this sense we may say that our result, obtained within the self-consistent
model of Sun luminosity and solar dynamo – geodynamo connection, has a strong
and evident ”experimental” support represented by the astrophysical solar-like
axions, provided that this is not just a fortuitous coincidence of the data
(Fig. 20), and the axions with the mass $m_{a}=17~{}eV$ really exist.

###  7.5 Plausible dark matter candidate: hadronic axion or axion-like arhion?

Along with the identification of the solar-like axion with the 17 eV mass, we
also found the possible relic axion-like archion with the 2 eV mass in near-
infrared band of the extragalactic background (Fig. 22), which behaves
similarly to the hardronic axion with highly suppressed interaction with
leptons under certain conditions.

Discovering both the hadronic axion and the axion-like archion (with similar
properties, but different physical nature) made us think about what we
actually see and identify. Aren’t these two particles just axion-like archions
in reality? All that we know about the World, tells us that Mother Nature,
although is notable for its Darwinian diversity in the elementary particles
zoo, would not ”multiply entities beyond necessity”. Keeping in mind this
principle of Ockham’s razor and our intuitive notion about the laws of
astroparticle physics, let us give some arguments in favour of the statement
that we are really dealing with the axion-like archions and not with hadronic
axions.

The main reason of our doubts is the following. It is known [ref209] that the
model of an archion automatically avoids the serious problems related to the
necessity of a stable relic supermassive quark existence, predicted by the
hadronic axion model. As a result, the so-called ”wild isotopes” should exist
as a consequence of a corresponding nucleosynthesis process, and they are not
observed today, as is known. On the other hand, including the hadronic axion
model into the Grand Unified model should lead to an inevitable existence of a
supermassive lepton associated with the axion. This is due to the fact that
within such unification, a composite of a supermassive quark with an ordinary
light quark would cause a supermassive quark instability as well as the axion
interaction with leptons, which is incompatible with the hadronic axion
properties [ref210]. In other words, the axion just would not be hadronic in
this case.

Everything is completely different within the archion model. In contrast to
the hadronic axion model, the archion model (i.e. the Berezhiani-Saharov-
Khlopov (BSK) model of horizontal unification [ref209]) naturally gives rise
to a supermassive quark instability and a strong suppression of the axion-like
archion interaction with leptons [ref203, ref204, ref205, ref206, ref207,
ref208, ref209, ref210]. At the same time, because of the suppression of the
archion interaction with the lightest generation of leptons
($u$,$d$,$e$,$\nu_{e}$), the existing limitations on the corresponding scale
(the energy scale $f_{PQ}$ associated with the break-down of the U(1) PQ-
symmetry) of the invisible axion are reduced to

$f_{a}\equiv f_{PQ}\sim 10^{6}~{}~{}GeV,$ (132)

which makes the BSK-model of an archion rather close to the hadronic axion
model [ref032, ref032a, ref033, ref033a, ref102, ref103].

It is interesting that for such values of the energy $f_{a}$ in the BSK-model
of horizontal unification the stable massive neutrinos dominance becomes
possible, or more strictly speaking, the stable sterile neutrinos with the
mass about several keV and small mixing angles with the active neutrinos. If
we also take into account that one or more of these gauge-singlet fermions can
have Majorana masses below the electroweak scale, in which case they appear as
sterile neutrinos in the low-energy theory [ref218], it becomes clear that
these particles have fundamental importance extending the Standard Model by
gauge singlet fermions, which can accommodate the neutrino masses.
Furthermore, if one of the sterile neutrinos has mass in the 1–20 keV range
and has small mixing angles with the active neutrinos, such particle is a
plausible candidate for dark matter [ref219]. The same particle could be
produced in the supernova explosion, and its emission from a cooling neutron
star could explain the pulsar kicks, facilitate core collapse supernova
explosions, and affect the formation of the first stars and black holes (e.g.,
[ref218] and Refs. therein). Therefore, there is a strong motivation to search
for signatures of sterile neutrinos in this mass range, especially as the
first results were obtained recently indicating the detection of the sterile
neutrino mass of 5.0$\pm$0.2 keV and a mixing angle in a narrow range for
which neutrino oscillations can produce all of the dark matter [ref220].

We tried to find a trace of the radiative decay of the sterile neutrino in the
form $\nu_{s}\to\nu+\gamma$ as well. As we noted before, one should not be
embarrassed by the fact that the corresponding life-time of the sterile
neutrino is many orders of magnitude larger than the age of the universe (see
(95) and (97)), because sterile neutrinos are produced in the early universe
by neutrino oscillations [ref219] and, possibly, by other mechanisms as well,
and therefore every dark matter halo should contain some fraction of these
particles. In other words, given a large number of particles in these
astrophysical systems, even a small decay width can make them observable via
the photons produced in the radiative decay.

Figure 25: (a) The cosmic XRB spectrum and the predicted contribution from the
active galactic nuclei (AGNs), that gives origin to the X-ray background. Gray
points: HEAO-1 A2 HED data (Gruber et al. [ref221]). Dark green points: HEAO-1
A4 LED (Gruber et al. [ref221]). Cyan points: Rossi-XTE (Revnivtsev et al.
[ref222]). Red bowtie: HEAO-1 A4 MED (Kinzer et al. [ref223]). Blue bowtie:
ROSAT PSPC data (Georgantopoulos et al. [ref224]). Light green bowtie:
BeppoSAX (Vecchi et al. [ref225]). Purple and yellow bowties: Newton-XMM (Lumb
et al. [ref226]; De Luca & Molendi [ref227]). Solid line: synthesis model
spectrum by Haardt-Madau [ref228], produced by a mixture of absorbed (the
short-dashed black line) and unabsorbed (the long-dashed black line) AGNs.
Adopted from [ref228].
(b) The cosmic XRB spectrum as a function of the wavelength $\lambda_{0}$.
Solid purple line: synthesis model spectrum by Haardt-Madau [ref228], produced
by a mixture of absorbed and unabsorbed AGNs. Black points with bars
(crosses): HEAO A-2 results (Wu et al. [ref229]). Solid green line: our fit of
cosmic XRB spectrum. The orange region is a sum of our fit of spectrum (the
green line) and the spectrum of $\gamma$-quanta (the blue shaded region) born
in the supposed radiative decay of the sterile neutrinos.

We limited our search for possible $\gamma$-spectra of the sterile neutrino
decay to the analysis of the experimental results obtained by different
authors during their study of the diffuse X-ray background (Fig. 25a). The
cosmic XRB spectrum in the 0.1 – 1 keV range was particularly interesting in
this sense, because the X-ray spectrum here is not ”distorted” by the
absorption processes, since unabsorbed AGNs dominate in this energy region
(see predictions by the Haardt-Madau model on Fig. 25a). Allowing for the fact
that in this energy region the problem of Lyman-$\alpha$ forest may be
neglected, it becomes possible to use Eq. (79) for theoretical calculation of
the intensity of sterile neutrino contributions to the diffuse X-ray
background as a function of the wavelength $\lambda_{0}$.

It is necessary to mention some specifics of cosmic XRB spectrum calculation
for this region202020The observations performed using the X-ray spectroscopy
of locations in the Universe are not used here, because we calculate the
spectrum of $\gamma$-quanta born in radiative decays $\nu_{S}\to\nu+\gamma$
for different $z$ (integral method (86) [ref166]) in contrast to the X-ray
spectroscopy of emission lines from the cooling stars (so-called differential
method (e.g. [ref157]). First, the XRB spectrum is highly ”damaged” (see Fig.
25a) by multiplicity of observations with different measuring bases, and
possibly, different methodologies. For this reason we decided to use the
observational data obtained in the single HEAO experiment. It covers the whole
spectrum in 1 – 400 keV range, and also provides the measurements of the
cosmic XRB spectrum in 0.1 – 1 keV region [ref229]. Second, when calculating
the $\gamma$-spectrum formed by the radiative decay of sterile neutrinos, we
used Eqs. (79)-(81) adjusted for expressions (95)-(97), which reflect the
radiative features of sterile neutrinos, and assumed their velocity
distribution (see (81)) to be Gaussian with a variance of $\upsilon_{c}\sim
30~{}km/s$ [ref230, ref230a] and their density to be equal [ref231, ref232]

$\Omega_{S}h_{0}^{2}\approx
0.3\left(\frac{\sin^{2}2\theta}{10^{-10}}\right)\left(\frac{m_{S}}{100~{}keV}\right)^{2}.$
(133)

We also introduced a factor of 3 for the cosmic XRB spectrum calculation (79),
which manually takes into account that the bulk of the EBL contributions in
the decaying-neutrino scenario comes from neutrinos which are distributed on
larger scales. We will refer to these collectively as free-streaming
neutrinos, though some of them may actually be associated with more massive
systems such as clusters of galaxies, and a possible impact of the
astrophysical data non-gaussianity. In support of this assumption let us quote
a remark made in paper [ref236], where a factor of 10 was used: ”…the fact
that a significant part of the dark matter at redshifts $z\leqslant 10$ is
concentrated in galaxies and clusters of galaxies just means that the
strongest signal from the dark matter decay should come from the sum of the
signals from the compact sources at $z\leqslant 10$. Taking into account that
the DM decay signal from $z\leqslant 10$ is some two orders of magnitude
stronger than that from $z\geqslant 10$, while the subtraction of resolved
sources reduces the residual X-ray background maximum by a factor of 10, we
argue that it would be wrong to subtract the contribution from the resolved
sources from the XRB observations when looking for the DM decay signal”.

Our theoretical fit (the yellow curve in Fig. 25b) of spectral intensity of
the background radiation (79) produced by sterile neutrino decays describes
the experimental data from HEAO A-2 in near X-ray band [ref221] adequately.
However, this result requires a serious and thorough verification, since the
data by Wu et al. [ref229] are, in fact, the only cosmic XRB spectrum
measurements of the Large Magellanic Clouds in the 0.16 – 3.5 keV region. From
this point of view, this result is more likely to serve as a demonstration of
the possibilities and peculiarities of the sterile neutrino search in the
diffuse extragalactic212121Wu’s analysis [ref229] of the background data
reveals a limit on the mean absolute intensity for the extragalactic emission
$I_{\lambda}(0.16-3.5~{}keV)\sim 5.6\cdot 10^{-8}~{}ergs\cdot
cm^{-2}s^{-1}sr^{-1}$. X-ray background which sometimes may be more suitable
than the flux (in the form of a narrow band) from any particular region of the
sky.

Turning back to the archion properties, let us point out that such sterile
neutrino ($m_{S}=2.75~{}keV$, $\sin^{2}2\theta=5.1\cdot 10^{-8}$) representing
the dark matter with density $\Omega_{S}\approx 0.20$ (see (133) and Fig. 26)
fits into the archion model very well. This is due to the fact that it is the
sterile neutrino appearance that makes it possible to solve the problem of
missing dark matter (see (92)) which is associated with the archions model (in
a case of $m_{S}>m_{a}$) and emerges in LTR cosmology:

Figure 26: Full parameter space constraints for the sterile neutrino
production model based on the diffuse extragalactic X-ray background, assuming
sterile neutrinos constitute the dark matter. To facilitate comparisons, we
adopt many of the conventions used by Abazajian [ref233]. Favored regions are
in red/magenta colors, disfavored and excluded regions are in blue/turquoise
colors. The favored parameters consistent with pulsar kick generation are in
horizontal hatching (Kusenko et al. [ref234, ref235]). Constraints from X-ray
observations include the diffuse X-ray background (turquoise) (Boyarsky et al.
[ref236]). Also shown is the best current constraint from Chandra, from
observations of contributions of dark matter X-ray decay in the cosmic X-ray
background through the CDFN and CDFS (brown contour, ”Unresolved CXB Milky
Way”). Also shown is an estimate of the sensitivity of a 1 Ms observation of
M31 with IXO (yellow). The region at $m_{S}<1.7~{}keV$ is disfavored by
conservative application of constraints from the Lyman-$\alpha$ forest.
Inset: Supernova bound on sterile neutrino masses $m_{S}$ and mixing angles
$\sin^{2}(2\theta)$, where the purple region is excluded by the energy-loss
argument while the green one by the energy-transfer argument [ref231]. The
excluded region will be extended to the dashed (red) line if the build-up of
degeneracy parameter is ignored, i.e., $\eta(t)=0$. The dot-dashed (green)
line represents the sterile neutrinos as dark matter with the correct relic
abundance $\Omega_{S}h_{0}^{2}=0.1$. The red star ($m_{S}=2.75~{}keV$,
$\sin^{2}2\theta=5.1\cdot 10^{-8}$) marks our result of sterile neutrino
parameters identification.

$\Omega_{DM}=\Omega_{S}+\Omega_{a}^{nth}\sim 0.25,$ (134)

where $\Omega_{a}^{nth}\approx 0.04$ is a non-standard density of the axion
dark matter (92).

Briefly summarizing all said above about the problem of hadronic axion and
axion-like archion, in our opinion, the archion model is preferable as
compared to the hadronic axion model, but as the saying is, time will tell.

And finally let us emphasize two the most painful points of the present paper,
at least from its authors’ point of view.

Regardless of the model type, the simultaneous existence of two axion-like
particles (the hadronic axion with the mass 17 eV and the axion-like arhion
with the mass 2 eV) rises a natural question whether there may be two
different energy scales $f_{a}$ associated with the breakdown of the U(1) PQ-
symmetry, one of them coming from the hadronic axion ($f_{a}=3.5\cdot
10^{5}~{}GeV^{-1}$), and another relic one – from the axion-like archion
($f_{a}=3.0\cdot 10^{6}~{}GeV^{-1}$). Or otherwise stated, can the energy have
two scales, or in general, be hierarchical? If it can, then what kind of
consequences of such fundamental symmetry violations hierarchy may be observed
today? However, if one just winks at the possible existence of the axion-like
particle with the 2 eV mass, the problem just vanishes without any
consequences for the results of the present paper.

The second painful point is related to the key problem of the axion mechanism
of Sun luminosity and is stated rather simply: ”Is the process of axion
conversion into $\gamma$-quanta by the Primakoff effect really possible in the
Solar tachocline magnetic field?” This question is directly connected to the
problem of the hollow magnetic flux tubes existence in the convective zone of
the Sun, which are supposed to connect the tachocline with the photosphere.
So, both the theory and experiment have to answer the question of whether
there are the waveguides in the form of the hollow magnetic flux tubes in the
convective zone of the Sun, which are perfectly transparent for
$\gamma$-quanta, or our model of the axion mechanism of Sun luminosity and
solar dynamo – geodynamo connection is built around simply guessed rules of
calculation which do not reflect any real nature of things.

## References

##  Appendix A Appendix I. Effect of $\gamma$-quanta channeling in periodic
structures

It is known [ref-a1.01] that the real part of dielectric susceptibility
$\chi(\omega)$ for the photons with energies exceeding the $K$-electrons
binding energy has the form

$Re\chi(\omega)=-\omega_{e}^{2}/\omega^{2},$ (A.1)

where $\omega_{e}=(4\pi Ne^{2}/m_{e})^{1/2}$ is the electron plasma frequency,
$N$ is the electrons density, $m_{e}$ is the electron mass. Since
$\omega_{e}\sim 10~{}eV$ for the majority of materials, the susceptibility is
small and negative in the X-ray band. It means that X-ray photons may
experience the total reflection on the border of two materials from the one
with the larger value of $|Re~{}\chi(\omega)|$, and the angle of photons
arrival to the border should not exceed $\theta_{c}=|Re~{}\chi|^{1/2}$, where
$\Delta(Re~{}\chi)$ is a dielectric susceptibility step. It is appropriate to
mention that the phenomenon of a small-angle X-ray reflection has been used in
X-ray optic elements for a long time [ref-a1.02], and also for transporting
(e.g. [ref-a1.03]) and turning (e.g. [ref-a1.04]) the X-ray bundles by
cylindrical tubes.

On the other hand, it is also known that according to the optical-mechanical
analogy (Hamilton and Fermat principles identity), in certain cases the
radiation propagation may be described in terms of ray trajectories obeying
the principle of least action. The refractive index, defining a ray
trajectory, plays a role of an external potential in this case. Based on this
analogy, one may conjecture that there exists an effect of high-energy
chargeless particles channeling that leads to the substantial anisotropy after
passing the periodic structures. This assumption is validated using the
example Vinecky and Finegold model problem [ref-a1.51, ref-a1.52] related to
calculation of the ray trajectories and absorption coefficients for the hard
photons in geometrical optics approximation.

###  A.1 Statement of a problem

Let us follow [ref-a1.51, ref-a1.52] and consider $\gamma$-radiation with
frequency $\omega=2\pi c/\lambda$ transmission through a medium under the
following condition:

$\lambda_{e}<\lambda\ll a_{0},$ (A.2)

where $\lambda_{e}=\hbar/mc$ is the electron Compton wavelength, $a_{0}$ is a
typical interatomic distance in the medium.

The left-hand side of the inequality (A.2) lets one consider the quanta as
propagating in the continuous medium with a refraction index $n$. As long as
the right-hand side of (A.2) holds, the incident wave frequency is
substantially higher than the lattice vibration eigenfrequencies (at least for
the higher electron shells of the atoms that form the lattice). Therefore the
scattering electrons may be considered free, and a crystal in the large may be
treated as a frozen spatially-inhomogeneous electron plasma (the impact of
nuclei on the scattering is negligible).

The electron plasma dielectric constant222222For a justification of the term
”dielectric constant” applicability for the X-ray and $\gamma$-bands in the
absence of the Lorentz field averaging over the physically small volume
elements see [ref-a1.05]. may be written down in the form

$\varepsilon=\varepsilon_{1}+i\varepsilon_{2}=1-\frac{\omega_{e}^{2}(\vec{r})}{\omega^{2}+i\omega\omega_{\gamma}},$
(A.3)

where $\omega_{\gamma}$ is the damping parameter which defines the wave
absorption in the system.

For the refraction index

$n=\sqrt{\varepsilon}=n_{1}+in_{2}$

from (A.3) taking into account that $\omega\gg\omega_{e}\gg\omega_{\gamma}$ in
the $\gamma$-band, we derive

$n_{1}\approx
1-\frac{1}{2}\left(\frac{\omega_{e}}{\omega}\right)^{2},~{}~{}n_{2}\approx\frac{1}{2}\frac{\omega_{e}^{2}\omega_{\gamma}}{\omega^{3}}.$
(A.4)

In order to perform the quantitative calculations, let us consider a
simplified model of the one-dimensional lattice (a stack of alternating layers
with different, but constant, densities and uniform $N(\vec{r})$ distribution
along the layer (Fig. A.1)).

Figure A.1: Ray trajectory dependence on the angle $\alpha$ in a long-period
structure: the channeling happens when $\alpha<\alpha_{m}$; when
$\alpha>\alpha_{m}$, the ray crosses the surfaces.

Let us choose the distribution in the $y$ direction (perpendicular to the
layers) in the form:

$N(y)=N_{0}\left(1+\beta^{2}\sin^{2}\pi\frac{y}{a}\right).$ (A.5)

Corresponding expressions for $n_{1}$, $n_{2}$ are

$n_{1}=1-\frac{1}{2}\xi^{2}\left(1+\beta^{2}\sin^{2}\frac{1}{2}k_{a}y,~{}~{}n_{2}=\frac{1}{2}\zeta\xi^{2}\left(1+\beta^{2}\sin^{2}\frac{1}{2}k_{a}y\right)\right),$
(A.6)

where

$\xi^{2}\equiv\frac{\omega_{0}^{2}}{\omega^{2}}=\frac{4\pi
N_{0}e^{2}}{m\omega^{2}}\ll
1,~{}~{}\zeta\equiv\frac{\omega_{\gamma}}{\omega}\ll\xi,~{}~{}k_{a}\equiv\frac{2\pi}{a}.$
(A.7)

So the problem reduces to determining the intensity of the $\gamma$-quanta
beam hitting a sample of the thickness $x$ with the angle $\alpha$ to the
$O_{x}$ axis (Fig. A.1).

We shall use the equation for the intensity of a ray that passed a path $l$ in
the absorbing medium:

$J_{1}=J_{0}\exp\left\\{-2k\int n_{2}(l)dl\right\\},~{}~{}k=\frac{2\pi
n_{1}}{\lambda}\approx\frac{2\pi}{\lambda},$ (A.8)

The integral in (A.8) is taken along the ray trajectory, while the trajectory
is determined by means of the Fermat principle [ref-a1.06]

$\delta\int n_{1}dl=0,$

variation of which yields a system of three nonlinear second-order
differential equations describing the ray equation in a parametric form
$x=x(l)$, $y=y(l)$, $z=z(l)$:

$n_{1}\frac{d^{2}x_{i}}{dl^{2}}+\left(\frac{\partial n_{1}}{\partial
x_{j}}\frac{dx_{j}}{dl}\right)\frac{dx_{i}}{dl}=\frac{\partial n_{1}}{\partial
x_{i}},~{}~{}i,j=1,2,3$ (A.9)

($x_{1}$, $x_{2}$, $x_{3}$ correspond to $x$, $y$, $z$). It follows from the
identical relation $dx_{j}dx_{j}=dl^{2}$ that only two equations of (A.9) are
independent.

###  A.2 Solution of the model problem

For the model (A.5) the ray trajectory lies in one plane $xy$, and the system
(A.9) reduces to the equation

$n_{1}\frac{d^{2}y}{dl^{2}}+\frac{\partial n_{1}}{\partial
y}\left(\frac{dy}{dl}\right)^{2}=\frac{\partial n_{1}}{\partial y}.$ (A.10)

Let us point out some features of the channeling process that directly follow
from Eq. (A.10). Assuming $dy/dl=0$ everywhere along the ray, we derive the
particular solution (A.10) which describes the propagation in the planes
parallel to the planes of constant density. According to (A.10), such motion
is only possible when $dn_{1}/dy=0$, i.e. within the planes $A_{j}$ and
$B_{j}$, corresponding to the minimum and maximum values of $n_{1}(y)$.
Otherwise, the straight motion of a ray along the planes $n_{1}(y)=const$ is
impossible. Particularly, if the ray is initially parallel to the planes, but
lies neither in $A_{j}$, nor in $B_{j}$, it is bent towards the nearest
$B_{j}$, since $n_{1}$ grows in this direction. Therefore, the motion within
the plane $B_{j}$ is stable, while within $A_{j}$ it is not. In other words,
the rays entering the layers along the $O_{x}$ axis are pushed out into the
”channels” of the low electron density (regions with minimum $N(y)$).

The first integral of Eq. (A.10) is as follows:

$\frac{dy}{dl}=\left[1-\left(\frac{n_{1}(y_{0})}{n_{1}(y)}\cos\alpha\right)^{2}\right]^{1/2},~{}~{}\frac{dy}{dx}=\left[\left(\frac{n_{1}(y)}{n_{1}(y_{0})\cos\alpha}\right)^{2}-1\right]^{1/2}.$
(A.11)

Let us use (A.11) for determining the area of the ray motion depending on
initial values of $\alpha$ and $y_{0}$. By taking $dy/dx=0$ in the ray’s
turning point $y=y_{m}$ in (A.11), we write down

$n_{1}^{2}(y_{m})=n_{1}^{2}(y_{0})\cos^{2}\alpha.$ (A.12)

By substituting (A.6) we derive

$\sin^{2}\left(\frac{1}{2}k_{a}y_{m}\right)=\frac{1-\xi^{2}}{(\beta\xi)^{2}}\sin^{2}\alpha+\cos^{2}\alpha\cdot\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right).$
(A.13)

Eq. (A.13), obviously, has a real root $y_{m}$ only in the case

$\frac{1-\xi^{2}}{(\beta\xi)^{2}}\sin^{2}\alpha+\cos^{2}\alpha\cdot\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right)\leqslant
1,$ (A.14)

whence taking into account (A.7) we obtain

$\alpha\leqslant\alpha_{m}(y_{0}),~{}~{}\alpha_{m}(y_{0})=\arctan\left[\beta\xi\cos\left(\frac{1}{2}k_{a}y_{0}\right)\right]\ll
1.$ (A.15)

The $dy/dx$ function zeros have the form

$y_{m}=\pm k_{a}^{-1}\arccos\left(\cos
k_{a}y_{0}-2(\beta\xi)^{-2}\sin^{2}\alpha\right).$ (A.16)

For each $y_{0}$ a ray with the angle of arrival $\alpha<\alpha_{m}(y_{0})$
”oscillates” between the end points $[y_{m},-y_{m}]$ when passing through a
sample, i.e. it ”channels” between two neighbouring planes $A_{j}$ and
$A_{j+1}$. Hence, $\alpha_{m}(y_{0})$ is the maximum angle of channeling for
the given $y_{0}$. According to (A.15), the angle $\alpha_{m}=0$ corresponds
to $y_{0}-y_{m}=(1/2)a$, which means that the values of $y_{0}$ that make the
channeling possible lie in the region

$\left(-\frac{1}{2}a,\frac{1}{2}a\right),$

thus embracing the whole sample facet exposed to the beam.

If $\alpha>\alpha_{m}(y_{0})$, the $dy/dx$ function has no zeros – a ray that
entered under the angle $\alpha>\alpha_{m}(y_{0})$ does not channel – instead
it crosses the layers one by one and is heavily absorbed by them. Fig. A.1
shows the trajectories of both the ”captured” ray that propagates between two
planes throughout the sample and the non-channeling one.

From this analysis the following picture of the channeling emerges. When a
beam hits the sample in the $\alpha=0$ direction, virtually all the rays pass
through the sample without any substantial absorption – including the ones
that propagate within the strongly absorbing layers as they are pushed out
back into the channels. Under small $\alpha\neq 0$ the thin bands of $\delta
y_{0}$ values appear on both sides of the low density planes, where the
incident rays quit channeling by switching to the crossing trajectories. Since
each crossing is accompanied by a loss of some amount of energy, we obtain two
sets of rays – experiencing the weak and strong absorption respectively. With
$\alpha$ growth (and, consequently, the $\delta y_{0}$ bands growth) first set
of rays shrinks, while the second set grows. When
$\alpha=\alpha_{m}(y_{0}=0)$, the channeling region collapses and all the rays
are strongly absorbed. Therefore the value

$\alpha_{M}\equiv\alpha_{m}(y_{0}=0)=\arctan\beta\xi\approx\beta\xi-\beta\frac{\omega_{e}}{\omega}$
(A.17)

characterizes the maximum possible channeling angle, thus determining the
divergence of a beam for a given sample. As seen from (A.17), the value of
$\alpha_{M}$ decreases with the beam rigidity growth!

Let us estimate the value of $\alpha_{M}$ for the plasma medium in the Solar
tachocline zone. Let $\beta\sim 1$ for simplicity. Taking into account that
the plasma frequency in the tachocline is equal to $\omega_{e}\sim 4.6\cdot
10^{16}~{}s^{-2}$ [see (8) in the main part], and the frequency of
$\gamma$-quanta with the energy $\langle E\rangle=4.2~{}keV$ (or $\lambda\sim
4.6\cdot 10^{-9}~{}cm$) is $\omega=w\pi c/\lambda=2\pi\langle
E\rangle/\hbar\sim 4\cdot 10^{19}~{}s^{-1}$, from (A.17) we obtain
$\alpha_{m}\sim 10^{-3}$. For the sake of channeling effect illustration, Fig.
A.1 shows the model results of the photon ($\lambda\sim 10^{-9}~{}cm$) beam
propagation through the layered media with $\alpha_{M}\sim 10^{-4}$ (see
Section A.4).

Let us now integrate the Eqs. (A.11). For the case of (A.4) these equations
are reducible to the following form (up to the first vanishing terms of the
order $\xi^{2}$ under the radical sign):

$\frac{dy}{dl}=p\sin\alpha\sqrt{1-q^{2}\sin^{2}\left(\frac{1}{2}k_{a}y\right)},~{}~{}\frac{dy}{dx}=p\tan\alpha\sqrt{1-q^{2}\sin^{2}\left(\frac{1}{2}k_{a}y\right)},$
(A.18)

where

$p\equiv\sqrt{1+(\beta\xi)^{2}\cot^{2}\alpha\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right)},~{}~{}q\equiv\frac{\beta\xi\cot\alpha}{p}=\left(\sin^{2}\frac{\pi
y_{0}}{a}+\frac{\tan^{2}\alpha}{(\beta\xi)^{2}}\right)^{-1/2}.$ (A.19)

By integrating (A.18) we obtain the following trajectory equation:

$x=x_{j}+\frac{2}{k_{a}\beta\xi}\begin{cases}F\left(\arcsin
q\sin\left(\frac{1}{2}k_{a}y\right),q^{-1}\right)&at~{}~{}q\geqslant 1,\\\
qF\left(\frac{1}{2}k_{a}y,q\right)&at~{}~{}q<1\end{cases}$ (A.20)

Here $F(\varphi,q)$ is an elliptic integral of the first kind, and the inverse
function amF (the Jacobian elliptic function) period is equal to $4K(q)$,
where $K(q)\equiv F(\pi/2,q)$ [ref-a1.07]. Correspondingly, the function
$y(x)$ inverse to (A.20a) is periodical with the period

$x_{\tau}=\frac{8}{k_{a}\beta\xi}K(q^{-1})$ (A.21)

and is associated with the channeling trajectories. For the crossing
trajectories (A.29b) which are translation-invariant under the simultaneous
transformations

$x\rightarrow x+jx_{\tau},~{}~{}y\rightarrow y+ja,~{}~{}j=0,1,2,...,$ (A.22)

we have

$x_{\tau}=\frac{8q}{k_{a}\beta\xi}K(q).$ (A.23)

The regions of channeling and crossing trajectories are delimited by the value
of the angle $\alpha=\alpha_{m}(y_{0})$ corresponding to the value $q=1$. When
$\alpha=\alpha_{m}(y_{0})$, the rays asymptotically approach the ”repelling”
planes $A_{j}$ and $x_{\tau}\to\infty$.

A number of trajectory oscillations along the distance $x$ for the channeling
and crossing rays is, respectively

$N_{\tau}=\frac{x}{x_{\tau}}=\frac{1}{8}k_{a}\beta\xi\begin{cases}K^{-1}(q^{-1})&at~{}~{}q>1,\\\
q^{-1}K^{-1}(q)&at~{}~{}q<1.\end{cases}$ (A.24)

###  A.3 Determination of the absorption coefficient angular dependence

Let us move on to the absorption coefficient calculation in the exponent
(A.8). By means of (A.18), integration over trajectory reduces to integration
over a variable $u=k_{a}y/2$ within the region

$\begin{cases}0\leqslant u\leqslant u_{m}&at~{}~{}q>1,\\\ 0\leqslant
u\leqslant\pi/2&at~{}~{}q<1.\end{cases}$

Let us remind that $u=k_{a}y_{m}/2=\arcsin(q^{-1})$ by virtue of (A.15) and
(A.19) corresponds to the turning points $y_{m}$ of the channeling
trajectories. Further, substituting (A.8) into the expression for $n_{2}$
(A.6), performing integration and multiplying the obtained integrals, which
correspond to the above mentioned regions, by $N_{\tau}$ (A.24), we obtain

$\sigma=k\zeta\xi^{2}\frac{x}{\cos\alpha}\begin{cases}1+\beta^{2}\left(1-\frac{E(q^{-1})}{K(q^{-1})}\right)&at~{}~{}q>1,\\\
1+\beta^{2}\left(1-\frac{E(q)}{K(q)}\right)&at~{}~{}q<1,\\\ \end{cases}$
(A.25)

where $E(q)$ is the complete elliptic integral of the second kind [ref-a1.08].

According to (A.25), the intensity $J(x)$ (see (A.8)) of a ray that passed
through a sample of the thickness $x$ may be written down as

$J(x)=J_{0}\exp(-\sigma
x)=J_{0}\exp\left(-\frac{\chi_{0}}{\cos\alpha}x\right)\cdot
Q(\alpha,y_{0},x),$ (A.26)

where

$Q(\alpha,y_{0},x)=\begin{cases}\exp\left[-\frac{\chi_{0}}{\cos\alpha}\beta^{2}x\left(1-\frac{E(q^{-1})}{K(q^{-1})}\right)\right]&at~{}~{}q>1,\\\
\exp\left[-\frac{\chi_{0}}{q^{2}\cos\alpha}\beta^{2}x\left(1-\frac{E(q)}{K(q)}\right)\right]&at~{}~{}q<1.\end{cases}$
(A.27)

Here the multiplier $J_{0}\exp(-\chi_{0}x/\cos\alpha)$ corresponds to
$\gamma$-rays propagation through a homogeneous medium with the electron
density $N_{e}$ and the absorption coefficient $\chi_{0}=k\zeta\xi^{2}$. An
additional multiplier $Q(\alpha,y_{0},x)$ characterizes the influence of the
medium layering. Fig. A.2 shows the $Q(\alpha,y_{0},x)$ curves for

$\frac{y_{0}}{a}=0,~{}1/8,~{}1/4,~{}3/8,~{}1/2$ (A.28)

with $\lambda=10^{-9}~{}cm$, $\beta=1$ and $\chi_{0}x=2$, obtained by a
numerical calculation using the elliptic integral value tables [ref-a1.08].

Figure A.2: Transmitted beam relative intensity dependence on the arrival
angle $\alpha$ for different values of $y_{0}$ (blue curves; the symmetric
parts of the curves for $\alpha<0$ are not shown). Red curve represents
$Q(\alpha,y_{0},x)$ averaged over all values of $y_{0}$.

As seen in the figure, the ”total transmission” $(Q=1)$ is observed when
$\alpha=0$, $y_{0}=0$ (the propagation in the $B_{j}$ plane), and the maximum
absorption depending on $\beta$ and $(Q\to\exp(-\chi_{0}\beta x\cos\alpha))$
is observed when $\alpha=0$, $y_{0}=(1/2)a$ (the propagation in the $A_{j}$
plane).

The dips on the decaying parts of the transmission curves when
$y_{0}\neq(1/2)a$ correspond to the critical values of the angle
$\alpha=\alpha_{m}(y_{0})$ (A.15), for which the $\gamma$-quanta beam, that
entered a sample, asymptotically approaches the nearest $A_{j}$ plane and is
absorbed in it. With somewhat bigger values of $\alpha$ the rays pass on to
the neighboring inter-plane space and moving within it leave a sample before
they enter the next $A_{j+1}$ plane. The additional narrow maxima on blue
curves (Fig. A.2) correspond to this type of rays.

Finally, when $\alpha>\alpha_{M}$ and, particularly, when $\alpha\to\pi/2$
(propagation across the layers) from (A.18) we obtain $q\to 0$, which
corresponds to $Q\to\exp(-\chi_{0}\beta^{2}x/2\cos\alpha)$, i.e. there is an
additional absorption due to the averaged additional density of the layers
$N_{1}=N_{0}\beta^{2}$.

Fig. A.2 shows a curve $Q(\alpha,y_{0},x)$ averaged over all values of
$y_{0}$. The side dips and maxima are smoothed out, and there is only a
central maximum. It corresponds to a clearly defined primary $\gamma$-quanta
propagation with their angle of arrival within $\alpha_{M}\sim 2\cdot
10^{-4}$. Therefore, the dense layers ”modulating” a material play a role of
an anisotropic filter which passes the $\gamma$-radiation for a narrow range
of angles $\alpha<\alpha_{M}$ only. The physical mechanism of the arising
transparency anisotropy consists in $\gamma$-quanta channeling between the
layers because of the radiation ”refraction” in a heterogeneous electron
plasma.

###  A.4 Channeling effect onset conditions

By definition, channeling occurs for the particles the motion of which in the
transversal phase plane is limited to a region

$\Delta y\Delta p\sim a\alpha\hbar k,$ (A.29)

where $\alpha$ is a channeling angle. According to the uncertainty principle,
the size of this region cannot be less than $\hbar$, i.e.

$ak_{0}\geqslant\alpha^{-1}.$ (A.30)

By letting $\alpha\sim\omega_{e}/\omega$ for the estimation, from (A.30) we
derive

$ak_{0}\geqslant 1,~{}~{}k_{0}\equiv\omega_{0}/c.$ (A.31)

For example, in a monocrystal with $a=a_{0}\sim 3\cdot 10^{-8}~{}cm$,
$\omega_{0}\sim 5\cdot 10^{16}~{}s^{-1}$ we have $a_{0}k_{0}\sim 5\cdot
10^{-2}\ll 1$, i.e. the condition (A.31) does not hold and, consequently, the
channeling is impossible. This prohibition is true for any quanta, since in
the region $\omega\gg\omega_{0}$, where the approximation (A.4) and (A.5) is
applicable, the frequency $\omega$ falls out from the condition (A.31).

However, for a layered system with $a\gg a_{0}$ the condition (A.31) may prove
to be true, and the channeling may be possible. A more strict condition
$ak_{0}\gg 1$ makes the classical description of this effect possible. This is
the case for a long-period structure with $a/a_{0}\gg 10^{2}$ considered
above. It is appropriate to emphasize that we consider here the structures
based on the amorphous matrices which are not monocrystals. Otherwise the
long-period structure would have been overlapped by the short-period
oscillations with the amplitude equal or greater than $\beta N_{0}$, which
would have lead to a noticeable tunnel effect. The question about channeling
possibilities in this case requires a special research.

Thus, the papers [ref-a1.51, ref-a1.52] show that the phenomenon of X-ray and
$\gamma$-radiation channeling exists in layered structures under conditions
when it is possible to use geometric optics. The essence of this phenomenon
lies in the fact that the rays are reflected form the layers with the higher
electron density if they propagate under the small enough angle
($\alpha<\alpha_{M}$) to the layers plane. The initially uniform intensity
distribution in the transversal plane becomes non-uniform, since the rays
concentrate in the ”channels” – the layers with the lower electron density. It
leads to the substantial absorption decrease and deeper radiation penetration
into the sample along the layers than in the case of an arbitrary arrival
angle.

###  A.5 On the account of an absorption impact on X-ray intensity when
channeling through the solar layered structures

As it was shown above, the process of $\gamma$-rays channeling through a
homogeneous medium with the electron density $N_{e}$ and the absorption
coefficient $\chi_{0}=k\zeta\xi^{2}$ may be described by the multiplier
$J_{0}\exp(-\chi_{0}x/\cos\alpha)$ in (A.26).

Calculation of this multiplier for an arbitrary point in the solar convective
zone, obviously, requires the estimation of the average Rosseland free path or
so-called Rosseland opacity (the photon absorption coefficient averaged
according to Rosseland) in these points. On the other hand, it is necessary to
know the radial profiles of the temperature and density in the solar
convective zone in order to calculate the Rosseland free path or Rosseland
opacity.

The transmission of photons with the intensity $J_{0}$ normally incident on a
uniform plasma (A.26) is given by

$T(\nu)=J(\nu)/J_{0}(\nu)=\exp(-\chi_{0}x)=\exp\left[-k(\nu)\rho x\right],$
(A.32)

where $h\nu$ is the photon energy and $J(\nu)$ is the attenuated photon
intensity emerging from the plasma, $k(\nu)$ is the opacity per unit mass
typically measured in units of $cm^{2}/g$, $\rho$ is the density, and $x$ is
the optical path length. For plasmas such as the Sun that are much larger than
the photon mean free path, radiation transport is usually described by the
diffusion approximation [ref-a1.09, ref-a1.10, ref-a1.11] using the Rosseland
mean opacity $k_{R}$,

$\frac{1}{k_{R}}=\int d\nu\frac{1}{k(\nu)}\frac{dB}{dT}\Bigg{/}\int
d\nu\frac{dB}{dT},$ (A.33)

where $B$ is the Planck function, $T$ is the plasma temperature, and the
weighting function $dB/dT$ peaks at roughly 3.8 kT. Near the convective zone
(CZ) boundary $T\sim 190~{}eV$ and $dB/dT$ peaks at $h\nu\sim 750~{}eV$ (Fig.
A.3). The frequency dependent opacity near the CZ boundary calculated using
the opacity project model [ref-a1.12, ref-a1.12a, ref-a1.13] is displayed in
Fig. A.3. Comparison with the weighting function for the Rosseland mean shows
that the most important photon energies are approximately
$300<h\nu<1300~{}eV$.

Figure A.3: Frequency dependent opacity [ref-a1.12, ref-a1.12a, ref-a1.13] for
the 17 element solar composition [ref-a1.14] near the base of the solar
convection zone compared to $dB/dT$. The electron temperature and density were
193 eV and $1\cdot 10^{23}~{}cm^{-3}$, respectively. Adopted from [ref-a1.15].

However, it is necessary to point out some essential peculiarities of the
absorption impact on the X-ray intensity when it channels inside the solar
layered structures based on, e.g. magnetic flux tubes superlattices (see
Appendix B).

First, as it was noted in the body text, the total energy balance of the Sun
is not violated in the framework of the axion mechanism of Sun luminosity, but
the radiation transport changes substantially relative to the standard model
of the Sun – the part of radiation transport not related to axions is very
small ($\sim 0.015\Lambda_{Sun}$). It means that the thermodynamic parameters
(temperature, pressure, plasma density, electron density etc.) are
considerably smaller in the axion model of the Sun as compared to the standard
model. On the other hand, the Rosseland free path or Rosseland opacity
calculation inside the thin magnetic flux tubes, which play the role of
$\gamma$-quanta waveguides formed in the tachocline, must take into account
the new values for the mentioned parameters in the framework of the axion
mechanism of Sun luminosity.

Second, we suppose that the decrease of the pressure in the convective zone,
for example, by an order of magnitude may lead to such decrease of the
pressure inside the magnetic flux tubes that it virtually does not influence
the radiation transport in these tubes. In other words, the Rosseland free
paths of $\gamma$-quanta in the thin magnetic flux tubes is so big that the
corresponding Rosseland opacity tends to zero, and so do the absorption
coefficients in (A.32). The low refractivity, or equally the high transparency
of the thin magnetic flux tubes is achieved due to the high magnetic pressure
(see (16) and Fig. 9) able to compensate the outer pressure of the convective
zone completely (see Appendix B for details).

## References

##  Appendix B Appendix II. On a possibility of the layered structures
formation in the solar convective zone on the basis of the magnetic flux tubes
superlattices

It is natural to ask a question, whether there is a physical possibility for
the above mentioned long-period structures formation in the
magnetohydrodynamical plasma media typical for the solar dynamo evolution.
Curiously enough, such mechanisms do exist. Let us examine some of them
briefly.

###  B.1 Zonal jet streams

The long-period structures in magnetohydrodynamical plasma media may show as
the so-called zonal jet streams, spontaneously generated in turbulent systems.
In fact, zonal jets are very common in nature. Well-known examples are those
in the atmospheres of giant planets and the alternating jet streams found in
the Earth’s world ocean [ref-a2.01]. Zonal flow formation in nuclear fusion
devices are also well studied [ref-a2.02].

As we have already pointed out above, a common feature of these zonal flows is
that they are spontaneously generated in turbulent systems. Because the
Earth’s outer core is believed to be in a turbulent state, it is possible that
there is a zonal flow in the liquid iron of the outer core. It is interesting
that a previously unknown convective regime of the outer core that has a dual
structure comprising inner, sheet-like radial plumes and an outer, westward
cylindrical zonal flow232323Computer simulations have been playing an
important role in the development of our understanding of the geodynamo, but
the direct numerical simulation of the geodynamo with a realistic parameter
regime is still beyond the power of today’s supercomputers. Difficulties in
simulating the geodynamo arise from the extreme conditions of the core, which
are characterized by very large and very small values of the non-dimensional
parameters of the system. Along them, the Ekman number, $E$, has been adopted
as a barometer of the distance of simulations from real core conditions, in
which $E$ is of the order of 10-15. Following the initial computer simulations
of the geodynamo, the Ekman number achieved has been steadily decreasing, with
recent geodynamo simulations performed with $E$ of the order of 10-6
[ref-a2.04]. In work by Miyagoshi et al. [ref-a2.03, ref-a2.04] they present a
geodynamo simulation with the Ekman number of the order of 10-7 – the highest
resolution yet achieved, making use of 4096 processors of the Earth Simulator.
And what is ahead when the magnitude of $E$ becomes closer to its real
value?!, was recently found [ref-a2.03] (Fig. 6 in the body text).

Fig. 6 in the main body of the present paper shows snapshots of the same data
of zonal flow formation in the Earth’s core. The sheet convection structure is
visualized as isosurfaces of the axial vorticity, which are almost straight in
the $z$ direction. The blue curves surrounding the $\omega_{z}$ isosurfaces
are streamlines of the velocity in the outer part of the dual-convection
structure. The ring-like shape of each streamline indicates that the azimuthal
component is dominant.

However, according to (A.26) in A, for the ideal (without absorption) photon
channeling the long-period structures, in which one of the interlaced media
has almost zero density, are necessary. Surprisingly, such long-period (in
terms of density) structures may appear in the plasma media in general, and in
the solar convective zone in particular. We mean here the so-called magnetic
flux tubes, the properties of which are discussed below.

###  B.2 Some properties of the magnetic flux tubes in Sun convective zone

Fig. B.1 shows the results of the three-dimensional solar hexagonal
magnetoconvection simulation [ref-a2.05]. In addition to a mere fact of the
long-period layers formation based on the magnetic flux tubes superlattices,
it is necessary to make sure that the matter density inside the tubes is much
smaller than the density of the outer plasma.

Figure B.1: Vertical field component for $R_{m}=400$ when the imposed field is
vertical, displayed by perspective plots. Four times are shown, for the three
levels $z$ = 0, 1/2, and 1. In this and subsequent plots the time unit is
(cell height)/(max. vertical velocity), i.e. $\leqslant$ 1.4 turnover time.
Adopted from [ref-a2.05].

At the same time, as the analysis in Appendix A shows, for the purposes of
photon channeling it is not necessary to guarantee the layers periodicity. In
other words, the channeling requires a large number of necessarily interlaced,
although not necessarily periodic, layers of different density. The total
crosscut area of such interlaced layers must also be comparable to the photon
beam cross-section.

In this connection below we shall examine some properties of the magnetic flux
tubes in the Sun convective zone, which may serve as a basis for estimating
the temperature, pressure and matter density inside the tubes depending on the
similar plasma parameters outside the tube. We shall also derive the zero
refractivity (or absolute transparency) condition for the effective photon
channeling along the magnetic tubes.

###  B.3 The self-confinement of force-free magnetic fields and energy
conservation law.

Magnetic field $\vec{B}$ alternating along the vertical axis $z$ induces the
vortex electric field in the magnetic flux tube containing a dense plasma. The
charged particles rotation in plasma with the angular velocity $\omega$ leads
to a centrifugal force per unit volume

$|\vec{F}|\sim\rho|\vec{\omega}|^{2}r.$ (B.1)

If we consider this problem in the rotating noninertial reference frame, such
noninertiality, according to the equivalence principle, is equivalent to
”introducing” a radial non-uniform gravitational field with the ”free fall
acceleration”

$g(r)=|\vec{\omega}|^{2}r.$ (B.2)

In such case the pressure difference inside ($p_{int}$) and outside
($p_{ext}$) the rotating ”liquid” of the tube may be treated by analogy with
the regular hydrostatic pressure (Fig. B.1).

Figure B.2: Representation of the ”hydrostatic equilibrium” in the rotating
”liquid” of a magnetic flux tube.

Let us pick a radial ”liquid column” inside the tube as it is shown in
Fig.B.1. Since the ”gravity” is non-uniform in this column, it is equivalent
to a uniform field with the ”free fall acceleration”

$\left\langle g(r)\right\rangle=\frac{1}{2}|\vec{\omega}|^{2}R.$ (B.3)

where $R$ is the tube radius which plays a role of the ”liquid column height”
in our analogy.

By equating the forces acting on the chosen column similar to the hydrostatic
pressure (Fig. B.1), we derive:

$p_{ext}=p_{int}+\frac{1}{2}\rho|\vec{\omega}|^{2}R^{2}.$ (B.4)

This rises the natural question as to what physics is hidden behind the
”centrifugal” pressure. In this relation, let us consider the magnetic field
energy density

$w_{B}=\frac{|\vec{B}|^{2}}{2\mu_{0}},$ (B.5)

where $\mu_{0}$ is the magnetic permeability of vacuum.

Suppose that the total magnetic field energy of the ”growing” tube grows
linearly between the tachocline and the photosphere. In this case if the
average total energy of the magnetic field in the tube transforms into the
kinetic energy of tube matter rotation completely, it is easy to show that

$\left\langle
E_{B}\right\rangle=\frac{1}{2}w_{B}V=\frac{1}{2}\frac{|\vec{B}|^{2}}{2\mu_{0}}V=\frac{I|\vec{\omega}|^{2}}{2},$
(B.6)

where $I=mR^{2}/2$ is the tube’s moment of inertia about the rotation axis,
$m$ and $V$ are the mass and the volume of the tube medium respectively.

Then from (B.6) it follows that

$\frac{\rho|\vec{\omega}|R^{2}}{2}=\frac{|\vec{B}|^{2}}{2\mu_{0}}.$ (B.7)

Finally, substituting (B.7) into (B.4) we obtain the desired relation

$p_{ext}=p_{int}+\frac{|\vec{B}|^{2}}{2\mu_{0}},$ (B.8)

which is exactly equal to a well-known expression by Parker [ref-a2.06],
describing the so-called self-confinement of force-free magnetic fields.

###  B.4 Hydrostatic equilibrium and a sharp tube medium cooling effect

Assume that the tube has the length $l(t)$ by the time $t$. Then its volume is
equal to $S\cdot l(t)$ and the heat capacity is

$c\cdot\rho(t)\cdot Sl(t),$ (B.9)

where $S$ is the tube cross-section, $\rho(t)$ is the density inside the tube,
$c$ is the specific heat capacity.

If the tube becomes longer by $\upsilon(t)dt$ for the time $dt$, then the
magnetic field energy increases by

$\frac{1}{2}\frac{|\vec{B}|}{2\mu_{0}}S\upsilon(t)dt.$ (B.10)

where $\upsilon(t)$ is the tube propagation speed.

The matter inside the tube, obviously, has to cool by the temperature $dT$ so
that the internal energy release maintained the magnetic energy growth.
Therefore, the following equality must hold:

$c\rho(t)l(t)\frac{dT}{dt}S=-\frac{1}{2}\frac{|\vec{B}|^{2}}{2\mu_{0}}\upsilon(t)S.$
(B.11)

Taking into account the fact that the tube grows practically linearly
[ref-a2.07], i.e. $\upsilon t=l$, the Parker relation (B.8) and the tube’s
equation of state

$p_{int}(t)=\frac{\rho}{\mu_{*}}R_{*}T(t)~{}~{}\Longleftrightarrow~{}~{}\rho=\frac{\mu_{*}}{R_{*}}\frac{p_{int}}{T(t)},$

the equality (B.11) may be rewritten (by separation of variables) as follows:

$\frac{dT}{T}=-\frac{R_{*}}{2c\mu_{*}}\left[\frac{p_{ext}}{p_{int}(t)}-1\right]\frac{dt}{t},$
(B.12)

where $\mu_{*}$ is the tube matter molar mass, $R_{*}$ is the universal gas
constant.

After integration of (B.12) we obtain

$\ln\left[\frac{T(t)}{T(0)}\right]=-\frac{R_{*}}{2c\mu_{*}}\int\limits_{0}^{t}\left[\frac{p_{ext}}{p_{int}(\tau)}-1\right]\frac{d\tau}{\tau}.$
(B.13)

It is easy to see that the multiplier ($1/\tau$) in (B.13) assigns the region
near $\tau=0$ in the integral. The integral converges since

$\lim\limits_{\tau\rightarrow
0}\left[\frac{p_{ext}}{p_{int}(\tau)}-1\right]=0.$

Expanding $p_{int}$ into Taylor series and taking into account that
$p_{int}(\tau=0)=p_{ext}$, we obtain

$p_{int}(\tau)=p_{ext}+\frac{dp_{int}(\tau=0)}{d\tau}=p_{ext}(1-\gamma\tau),$
(B.14)

where

$\gamma=-\frac{1}{p_{ext}}\frac{dp_{int}(\tau=0)}{d\tau}=\frac{1}{p_{int}(\tau=0)}\frac{dp_{int}(\tau=0)}{d\tau}.$
(B.15)

From (B.14) it follows that

$\frac{p_{ext}}{p_{int}(\tau)}-1=\frac{1}{1-\gamma\tau}-1\approx\gamma\tau,$
(B.16)

therefore, substituting (B.16) into (B.13), we find its solution in the form

$T(t)=T(0)\exp\left(-\frac{R_{*}}{2c\mu_{*}}\gamma t\right).$ (B.17)

It is extremely important to note here that the solution (B.17) points at the
remarkable fact that at least at the initial stages of the tube formation its
temperature decreases exponentially, i.e. very sharply. The same conclusion
can be made for the pressure and the matter density in the tube. Let us show
it.

Assuming that the relation (B.15) holds not only for $\tau=0$, but also for
small $\tau$ close to zero, we obtain

$p_{int}(t)=p_{int}(0)\exp(-\gamma t),$ (B.18)

i.e. the inner pressure also decreases exponentially, but with the different
exponential factor.

Further, assuming that the heat capacity of a molecule in the tube is

$c=\frac{i}{2}\frac{R_{*}}{\mu_{*}}$ (B.19)

where $i$ is a number of molecule’s degrees of freedom, and substituting the
expressions (B.17) and (B.18) into the tube’s equation of state, we derive the
expression for the matter density in the tube

$\rho(t)=\frac{\mu_{*}}{R_{*}}\frac{p_{int}(0)}{T(0)}\exp\left[-\left(1-\frac{1}{i}\right)\gamma
t\right],$ (B.20)

Taking into account that $i\geqslant 3$, we can see that the density decreases
exponentially just like the temperature (B.17) and pressure (B.18).

###  B.5 Ideal photon channeling (without absorption) conditions inside the
magnetic flux tubes

Let us calculate the inner pressure $p_{int}$ of the magnetic flux tube in the
solar tachocline. In the framework of the standard model of the Sun the
pressure in the tachocline zone is about $\sim 6\cdot 10^{12}~{}Pa$, while the
magnetic field strength reaches $400~{}T$, according to our estimates (Fig. 9
in the body text). Then from the hydrostatic condition by Parker (B.8) it
follows that the inner pressure of the magnetic flux tube in the solar
tachocline is equal to

$p_{int}\sim 6\cdot 10^{12}-4\cdot 10^{10}\simeq 6\cdot 10^{12}~{}~{}[Pa],$
(B.21)

which is comparable to the external pressure.

However, the situation changes drastically within the framework of the ”axion”
model of the Sun. We pointed out earlier (A) that the values of thermodynamic
parameters (temperature, pressure and plasma density) in the ”axion” model are
substantially smaller than the corresponding parameters in the standard model
of the Sun. Let us suppose that the pressure in the tachocline zone in the
”axion” model falls by an order of magnitude and is about $\sim 10^{11}~{}Pa$.
In such case it is easy to see that the magnetic field as strong as $\sim
500~{}T$ compensates the outer pressure almost completely, and the inner
pressure

$p_{int}\sim 10^{11}-O(10^{11})\rightarrow 0~{}~{}[Pa]$ (B.22)

becomes ultralow.

The result (B.22) means that the temperature and the matter density decrease
together with the inner pressure, and the decrease is sharply exponential,
because the exponential factor in (B.16) becomes very large

$\gamma\tau=\frac{p_{ext}}{p_{int}(\tau)}-1\rightarrow\infty.$ (B.23)

Because of the fact that the density, pressure and temperature in the tube are
ultralow, they virtually do not influence the radiation transport in these
tubes at all. In other words, the Rosseland free paths for $\gamma$-quanta
inside the tubes are so big that the Rosseland opacity and the absorption
coefficients in (A.32) tend to zero. The low refractivity (or high
transparency) is achieved for the following limiting condition:

$p_{ext}\simeq\frac{|\vec{B}|^{2}}{2\mu_{0}}.$ (B.24)

The obtained results should not be considered as a proof, but rather as some
trial estimates that validate the substantiation of an almost ideal (without
absorption) photon channeling mechanism along the magnetic flux tubes.

A major disadvantage of our reasoning is that, first of all, our estimates
apply to the initial magnetic tube formation stages, and second, that it lacks
the model calculations of the temperature, pressure and density in the
framework of the ”axion” model of the Sun, and consequently, there is no
comparison with the corresponding parameters of the standard model. On the
other hand, we believe that the idea of $\gamma$-quanta channeling along the
magnetic flux tubes is so physically natural and promising, that these
disadvantages will be overcome in the near future in spite of the obvious
serious difficulties.

## References