1. Introduction
The Standard Model (SM) of strong, weak and electromagnetic interactions based on the gauge group with three sequential families of quarks and leptons has had wonderful success in explaining all known processes involving elementary particles, and the detection of the Higgs boson with the right properties in 2012 represents its crownig.
However nobody would seriously regard the SM as the ultimate theory of fundamental interactions. Apart from more or less aesthetic reasons, electroweak and strong interactions are not unified, and gravity is not even included. The natural expectation from a final theory would rather be a full unification of all four fundamental interactions at the quantum level. Moreover, the need for an extension of the SM is made compelling by the fact that it cannot account for the observational evidence for non-baryonic dark matter—ultimately responsible for the formation of structures in the Universe—as well as for dark energy, presumably triggering the present accelerated cosmic expansion.
Thus, the SM is presently viewed as the low-energy manifestation of some more fundamental and complete theory of all elementary-particle interactions including gravity. Any specific attempt to accomplish this task is characterized by a set of new particles, along with a specific mass spectrum and their interactions with the standard world. This point will be outlined in detail in
Section 2.
Although it is presently impossible to tell which new proposal—out of so many ones—has any chance to successfully describe Nature, it seems remarkable that several attempts along very different directions such as four-dimensional supersymmetric models [
1,
2,
3,
4,
5], multidimensional Kaluza-Klein theories [
6,
7] and especially M theory—which encompasses superstring and superbrane theories—predict the existence of
axion-like particles (ALPs) [
8,
9] (for a very incomplete list of references, see [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]).
Basically, ALPs are very light, pseudo-scalar bosons—denoted by
a—which mainly couple to two photons with a strength
according to the Feynman diagram in
Figure 1.
Owing to their very low mass , they are effectively stable particles (even if they were not, their lifetime would be much longer than the age of the Universe). Additional couplings to fermions and gauge bosons may be present, but they are irrelevant for our forthcoming considerations and will be discarded.
Consider now the Feynman diagram in
Figure 1.
A possibility is that one photon is propagating but the other represents an
external magnetic field
. In such a situation we have
and
conversions, represented by the Feynman diagram in
Figure 2. Note that
could be replaced by an external electric field
, but we will not be interested in this possibility.
Because we can combine two diagrams of the latter kind together—as shown in
Figure 3—this means that
oscillations take place in the presence of an external magnetic field. They are quite similar to flavour oscillations of massive neutrinos, apart from the need of the external magnetic field
in order to compensate for the spin mismatch [
30,
31,
32,
33].
Over the last fifteen years or so, ALPs have attracted an ever growing interest, basically for three different reasons.
In a suitable region of the parameter plane
ALPs turn out to be very good candidates for cold dark matter [
34,
35,
36,
37,
38].
In another region of the parameter plane
—which can overlap with the previous one—ALPs give rise to very interesting astrophysical effects (for a very incomplete list of references, see [
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62,
63,
64,
65,
66,
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79,
80,
81,
82,
83,
84,
85,
86,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116,
117,
118,
119,
120,
121,
122,
123,
124,
125,
126,
127,
128,
129]. In particular, we shall see that ALPs can be
indirectly detected by the new generation of gamma-ray observatories, such as CTA (Cherenkov Telescope Array) [
130], HAWC (High-Altitude Water Cherenkov Observatory) [
131], GAMMA-400 (High-Altitude Water Cherenkov Observatory) [
132], LHAASO (High-Altitude Water Cherenkov Observatory) [
133], TAIGA-HiSCORE (Hundred Square km Cosmic Origin Explorer) [
134] and HERD (High Energy cosmic-Radiation Detection) [
135].
The last reason is that the region of the parameter plane
relevant for astrophysical effects can be probed—and ALPs can be
directly detected—in the laboratory experiment called
shining through the wall within the next few years thanks to the upgrade of ALPS (Any Light Particle Search) II at DESY [
136] and by the STAX experiment [
137]. Alternatively, these ALPs can be observed by the planned IAXO (International Axion Observatory) observatory [
138], as well as with other strategies developed by Avignone and collaborators [
139,
140,
141]. Moreover, if the bulk of the dark matter is made of ALPs they can also be detected by the planned experiment ABRACADABRA (A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus) [
142].
Our aim is to offer a pedagogical and self-contained account of the most important implications of ALPs for high-energy astrophysics.
The paper is structured as follows.
Section 2 contains a brief outline of what can be considered as the standard view about the relation of the SM with the ultimate theory.
Section 3 describes the most important properties of ALPs, which will be used in the subsequent discussions, along with most of the bounds on
and
.
The aim of
Section 4 is to provide all the necessary astrophysical background needed to understand the rest of the paper, which is therefore fully self-contained.
Section 5 describes the propagation of a photon beam in the
-ray band—emitted by a far-away source with redshift
z—in extragalactic space and observed on Earth at energy
. Extragalactic space is supposed to be magnetized, and so
oscillations should take place in the beam as it propagates. Moreover, the extragalactic magnetic field
is modeled as a domain-like structure—which mimics the real physical situation—with the strength of
nearly equal in all domains, the size
of all domains being taken equal and set by the
coherence length while the direction of
jumps
randomly and
abruptly from one domain to the next. Because of the latter fact, this model is called domain-like sharp edge model (DLSHE), and is adequate for beam energies currently detected (up to a few TeV) since the
oscillation length
is very much larger than
. It is just during propagation that an important effect of the presence of ALPs comes about. What happens is that the very-high-energy (VHE,
) beam photons scatter off background infrared/optical/ultraviolet photons, which are nothing but the light emitted by stars during the whole evolution of the Universe, called
extragalactic background light (EBL). Owing to the Breit-Wheeler process
[
143], the beam undergoes a frequency-dependent attenuation
1. However in the presence of
oscillations photons acquire a ‘split personality’: for some time they behave as a true photon—thereby undergoing EBL absorption—but for some time they behave as ALPs, which are totally unaffected by the EBL and propagate
freely. Therefore, now the optical depth
is smaller than in conventional physics. However since the corresponding photon survival probability is
, even a small decrease in
gives rise to a
much larger photon survival probability as compared to the case of conventional physics. This is the crux of the argument, first realized in 2007 by De Angelis, Roncadelli and Mansutti [
52].
Section 6 addresses the so-called
VHE BL LAC spectral anomaly. Basically, even if the EBL absorption is considerably reduced in the presence of ALPs, it nevertheless produces a frequency-dependent dimming of the source. Owing to the Breit-Wheeler process
, the beam undergoes a frequency-dependent attenuation. Thus, if we want to know the emitted spectrum
we have to EBL-deabsorb the observed one
. Correspondingly, we obtain the emitted spectrum which slightly departs from a single power law, hence for simplicity we perform the best-fit
. When we apply this procedure to a sufficiently rich homogeneous sample of VHE sources and perform a statistical analysis of the set of the emitted spectral slopes
for all considered sources, in the absence of ALPs we find that the best-fit regression line is a concave parabola in the
plane. As a consequence, there is a statistical correlation between
and
z. However how can the sources get to know their z so as to tune their
in such a way to reproduce the above statistical correlation? At first sight, one could imagine that this arises from selection effects, but this possibility has been excluded, whence the anomaly in question. So far, only conventional physics has been used. Nevertheless, it has been shown that by putting ALPs into the game with
and
such an anomaly disappears altogether!
Section 7 discusses a vexing question. A particular kind of active galactic nucleus (AGN)—named Flat Spectrum Radio Quasars (FSRQs)—should not emit in the
-ray band above 30 GeV according to conventional physics, but several FSRQs have been detected up to 400 GeV! It is shown that in the presence of
oscillation not only FSRQs emit up to 400 GeV, but in addition their spectral energy distribution comes out to be in perfect agreement with observations! Basically, what happens is that the above mechanism that increases the photon survival probability in extragalactic space works equally well inside FSRQs.
Section 8 is superficially similar to
Section 5, but with a big difference. In 2015 Dobrynina, Kartavtsev and Raffelt [
145] realized that at energies
photon dispersion on the CMB (Cosmic Microwave Background) becomes the leading effect, which causes the
oscillation length to get smaller and smaller as E further increases. Therefore, things change drastically whenever
, because in this case a whole oscillation—or even several oscillations—probe a whole domain, and if it is described unphysically like in the DLSHE model then the results also come out as unphysical. The simplest way out of this problem is to smooth out the edges in such a way that the change of the
direction becomes continuous across the domain edges. After a short description of the new model, the propagation of a VHE photon beam from a far-away source is described in detail.
Section 9 presents a full scenario, which also includes the
conversions also in the source. Actually, the VHE photon/ALP beam emitted by the considered sources crosses a variety of magnetic field structures in very different astrophysical environments where
oscillations occur: inside the BL Lac jet, within the host galaxy, in extragalactic space, and finally inside the Milky Way. For three specific sources a better agreement with observations is achieved as compared to conventional physics.
Section 10 addresses another characteristic effect brought about by photon-ALP interaction, namely the change in the polarization state of photons. Less attention has been paid so far to this effect in the literature. Very recent and interesting results on this topic are discussed.
5. Propagation of ALPs in Extragalactic Space—1
Let us consider a far away VHE blazar at redshift
z which is presently detected by an IACT. We stress that H.E.S.S., MAGIC and VERITAS are sensitive to photons with energy from about
up to a few TeV. As a consequence, we have
and so we can apply the formalism developed in
Section 3, but a small extension is needed in order to take EBL absorption into account.
Our ultimate task is the computation of the photon survival probability
in the presence of ALPs. We have seen that in conventional physics photons undergo EBL absorption, which severely depletes the photon beam when
z is sufficiently large. Clearly, now—owing to the presence of the extragalactic magnetic field—
oscillations will take place in the beam. This means that during its propagation a photon acquires a `split personality’: for some time it behaves as a true photon—thereby undergoing EBL absorption—but for some time it behaves as an ALP, and so it is unaffected by the EBL and propagates freely. Therefore, the optical depth in the presence of ALPs
is now
smaller than in the conventional case. But since the corresponding photon survival probability is
recalling (
52) we conclude that
is
much larger than
evaluated in conventional physics: this is the crux of the argument. As a consequence, far-away sources that are too faint to be detected according to conventional physics would become observable.
In order to be definite—and in view of the discussion to be presented in the next section—we choose the values of some parameters in agreement with the subsequent needs.
As far as the extragalactic magnetic field is concerned, we assume a domain-like structure described in
Section 4.4 with a DLSHE model, since we shall see that
.
5.1. Strategy
Thanks to the fact that is homogeneous in every domain, the beam propagation equation can be solved exactly in every single domain. But due to the nature of the extragalactic magnetic field, the angle of B in each domain with a fixed fiducial direction equal for all domains (which we identify with the z-axis) is a random variable, and so the propagation of the photon/ALP beam becomes a -dimensional stochastic process, where denotes the total number of magnetic domains crossed by the beam. Moreover, we shall see that the whole photon/ALP beam propagation can be recovered by iterating times the propagation over a single magnetic domain, changing each time the value of the random angle. Therefore, we identify the photon survival probability with its value averaged over the angles.
Our discussion is framed within the standard
CDM cosmological model with
and
, and so the redshift is the natural parameter to express distances. In particular, the proper length
extending over the redshift interval
is
Accordingly, the overall structure of the cellular configuration of the extragalactic magnetic field is naturally described by a
uniform mesh in redshift space with elementary step
, which is therefore the same for all domains. This mesh can be constructed as follows. We denote by
the proper length along the
y-direction of the generic
n-th domain, with
. Note that
is the maximal integer contained in the number
, hence
. In order to fix
we consider the domain closest to us, labelled by 1 and—with the help of Equation (
66)—we write its proper length as
, from which we get
. So, once
is chosen in agreement with such a prescription, the size of
all magnetic domains in redshift space is fixed. At this point, two further quantities can be determined. First,
. Second, the proper length of the
n-th domain along the
y-direction follows from Equation (
66) with
. Whence
Manifestly, in order to maximize
we choose
in order to be in the
strong-mixing regime for
. Incidentally, when the external magnetic field is homogeneous—as it is in fact in each single domain—a look at Lagrangian (
8) shows that all results depend on the combination
and not on
and
B separately. It is therefore quite convenient to employ the parameter
in terms of which Equation (
37) can be rewritten as
Because we would like to be in the strong-mixing regime almost everywhere within the VHE band, we take
. What about
? We should keep in mind that
is unknown, but the upper bound on the mean diffuse extragalactic electron density
is provided by the WMAP measurement of the baryon density [
232], which—thanks to Equation (
20)—translates into the upper bound
. Moreover, in order to fix
we use the fact that the result to be derived in the next section requires
. As a consequence, we get
.
5.2. Propagation over a Single Domain
We have to determine and the magnetic field strength in the generic n-th domain.
The first goal can be achieved as follows. Because the domain size is so small as compared to the cosmological standards, we can safely drop cosmological evolutionary effects when considering a single domain. Then as far as absorption is concerned what matters is the mean free path
for the reaction
, and the term
should be inserted into the 11 and 22 entries of the
matrix. In order to evaluate
, we imagine that two hypothetical sources located at both edges of the
n-th domain are observed. Therefore, we apply Equation (
53) to both sources. With the notational simplifications
and
, we have
which upon combination imply that the flux change across the domain in question is
But owing to Equation (
62)
mutatis mutandis implies that Equation (
72) should have the form
and the comparison with Equation (
72) ultimately yields
where the optical depth is evaluated by means of Equation (
58) or more simply taken from [
200].
As for the determination of
, we note that because of the high conductivity of the IGM medium the magnetic flux lines can be thought as frozen inside it [
204,
205]. Therefore, the flux conservation during the cosmic expansion entails that
B scales like
, so that the magnetic field strength in a domain at redshift
z is
[
204,
205]. Hence in the
n-th magnetic domain we have
.
Thus, at this stage the mixing matrix
as explicitly written in the
n-th domain reads
where
is the random angle between
and the fixed fiducial direction along the
z-axis (note that indeed
). Observe that since we are in the strong-mixing regime,
and
can be neglected with respect to the other terms in the mixing matrix. So—apart from
—all other matrix elements entering
are known. Finding the transfer matrix corresponding to
is straightforward even if tedious by using the results reported in
Appendix A. The result is
with
where we have set
with
where
is just
as defined by Equation (
68) and evaluated in the
n-th domain.
5.3. Calculation of the Photon Survival Probability in the Presence of Photon-ALP Oscillations
As we said, our aim is to derive the photon survival probability
from the source at redshift
z to us in the present context. So far, we have dealt with a single magnetic domain but now we enlarge our view so as to encompass the whole propagation process of the beam from the source to us. This goal is trivially achieved thanks to the analogy with non-relativistic quantum mechanics, according to which—for a fixed arbitrary choice of the angles
—the whole transfer matrix describing the propagation of the photon/ALP beam is
Moreover, the probability that a photon/ALP beam emitted by a blazar at
z in the state
will be detected in the state
for the above choice of
is given by
with
.
Since the actual values of the angles
are unknown, the best that we can do is to evaluate the probability entering Equation (
84) as averaged over all possible values of the considered angles, namely
indeed in accordance with the strategy outlined above. In practice, this is accomplished by evaluating the r.h.s. of Equation (
84) over a very large number of realizations of the propagation process (we take 5000 realizations) randomly choosing the values of all angles
for every realization, adding the results and dividing by the number of realizations.
Because the photon polarization cannot be measured at the considered energies, we have to sum the result over the two final polarization states
Moreover, we suppose here that the emitted beam consists
of unpolarized photons, so that the initial beam state is described by the density matrix
We find in this way the photon survival probability
A final remark is in order. It is obvious that the beam follows a single realization of the considered stochastic process at once, but since we do not know which one is actually selected the best we can do is to evaluate the average photon survival probability.
6. VHE BL Lac Spectral Anomaly
After all these preliminary considerations, we are now in a position to discuss an important effect. Actually, we show that VHE astrophysics leads to a first strong hint at ALPs.
Basically, we are going to demonstrate that conventional physics leads to a paradoxical situation concerning the EBL-deabsorbed BL Lac spectra as a function of the source redshift z. But such a situation disappears altogether once the ALPs consistent with the observational bounds enter the game.
As a first step, we have to select a sample of BL Lacs which is suitable for our analysis. They have to meet the following conditions.
We focus our attention on
flaring blazars, which show episodic time variability with their luminosity increasing by more than a factor of two, on the time span from a few hours to a few days: the reason is both their enhanced luminosity—which entails in turn their detectability [
233,
234]—and our desire to consider a homogeneous sample of BL Lacs.
As we shall see, our analysis requires the knowledge of the redshift, the observed spectrum and the energy range wherein every blazar is observed. This information is available only for some of the observed flaring sources.
In order to get rid of evolutionary effects inside blazars we restrict our attention to those with .
It seems a good thing to deal with sources that are as similar as possible. Therefore, we consider only intermediate-frequency peaked (IBL) and high-frequency peaked (HBL) flaring BL Lacs with observed energy .
We are consequently left with a sample
of 39 flaring VHE BL Lacs, which are listed in
Appendix C.
In first approximation, all observed spectra of the VHE blazars in
are fitted by a single power-law—neglecting a possible small curvature of some spectra in their lowest energy part—and so they have the form
where
is the observed energy,
is a
common reference energy while
and
denote the normalization constant and the observed slope, respectively, for a source at redshift
z. Actually,
is generally defined at different energies for different sources. So, for the sake of comparison among all observed spectra normalization constants we need to perform a rescaling
in the observed spectrum of the considered blazars in such a way that
coincides with
at the fiducial energy
for every source in
. Accordingly, Equation (
89) becomes
As already emphasized, these spectra are strongly affected by the EBL, hence if we want to know the
shape of the
emitted spectra they have to be EBL-deabsorbed. We know that the emitted and observed spectra are related by Equation (
53), which we presently rewrite as
where CP stands throughout this section for conventional physics.
6.1. Conventional Physics
Let us start by deriving the emitted spectrum of every source in
, starting from each observed one. As a preliminary step, thanks to Equation (
53) we rewrite Equation (
91) as
Because of the presence of the exponential in the r.h.s. of Equation (
92),
cannot behave as an exact power law. Yet, it turns out to be close to it. Therefore, we best-fit (BF)
to a single power-law expression
over the energy range
where a source is observed, and so
varies inside
(which changes from source to source). Correspondingly, the resulting values of
are plotted in
Figure 8.
We proceed by performing a statistical analysis of all values of
as a function of
z, by employing the least square method and try to fit the data with one parameter (horizontal straight line), two parameters (first-order polynomial), and three parameters (second-order polynomial). In order to test the statistical significance of the fits we evaluate the corresponding
. The values of the
obtained for the three fits are
(one parameter),
(two parameters) and
(three parameters). Thus, data appear to be best-fitted by the second-order polynomial
The best-fit regression line given by Equation (
94) turns out to be a concave parabola shown in
Figure 9.
This is the key-point. In order to appreciate the physical consequences of Equation (
94) we should keep in mind that
is the
exponent of the emitted energy entering
. Hence, in the two extreme cases
and
we have
thereby implying that the hardening of the emitted flux progressively
increases with the redshift. More generally, we have found a
statistical correlation between the
and
z.
However, this result looks physically absurd. How can the sources get to know their z so as to tune their in such a way to reproduce the above statistical correlation? We call the existence of such a correlation the VHE BL Lac spectral anomaly, which of course concerns flaring BL Lac alone. According to physical intuition, we would have expected a straight horizontal best-fit regression line in the plane.
The most natural explanation would be that such an anomaly arises from selection effects, but it has been demonstrated that this is not the case [
118].
6.2. ALPs Enter the Game
As an attempt to get rid of the VHE BL Lac spectral anomaly, we put ALPs into play, with parameters consistent with the previously mentioned bounds. Because the presently operating IACTs reach at most e few TeV, the oscillation length is much larger than the magnetic domain size
and so the propagation model in extragalactic space considered in
Section 5 is fully adequate.
Basically, we go through exactly the same steps described above. That is to say, we rewrite Equation (
92) with
, keeping in mind that now
. Whence
Next, we still best-fit
to a single power law expression
over the energy range
where a source is observed, hence
varies within
. Such a best-fitting procedure is performed for every benchmark value of
and
, namely
,
, and
,
.
Moreover, we carry out again the above statistical analysis of the values of for all blazars in , for any benchmark value of and .
Finally, the statistical significance of each fit can be quantified by computing the corresponding
, whose values are reported in
Table 1 for
,
, and in
Table 2 for
,
. In both tables the values of
are reported for comparison.
The relevance of such a statistical analysis is to single out two preferred situations (corresponding to the minimum of ): one for and the other for . In either case, the results are and a straight best-fit regression line which is exactly horizontal. More in detail, for we get and , while for we find and .
Manifestly, both cases turn out to be very similar. We plot the values of
in
Figure 10 only for the two considered situations.
Because
is our preferred value, we are now in a position to make a sharp prediction of the ALP parameters. Correspondingly—owing to Equation (
69) with
—the ALP mass must be
, since in
Section 5.1 we have seen that
. Moreover, by recalling Equation (
68) with
and the upper bounds on
and
B quoted in
Section 3.6 we get
. Remarkably, these parameters are consistent with the bounds reported in
Section 3.6.
In conclusion, we have indeed succeeded in getting rid of the VHE BL Lac spectral anomaly, since the are on average independent of z. We stress that it is an automatic consequence of the ALP scenario, and not an ad hoc requirement.
A final remark is in order. It is obvious that by effectively changing the EBL level—this is what the ALP actually does— the best-fit regression line also changes. But that it transforms from a concave parabola into a perfectly straight horizontal line looks almost a miracle!
6.3. A New Scenario for Flaring BL Lacs
Besides getting rid of the VHE BL Lac spectral anomaly, the ALP scenario naturally leads to a new view of flaring BL Lacs.
In order to best appreciate this point, it is enlightening to fit the values of
by a horizontal straight regression line, at the cost of relaxing the best-fitting requirement. Accordingly, the scatter of the values of
for
of blazars belonging to
is less than
of the mean value set by horizontal straight regression line in
Figure 11, namely equal to 0.47. Superficially, the VHE BL Lac spectral anomaly problem would be solved—but in reality it is not—since we correspondingly have
which is by far too large.
The result obtained in the presence of
oscillations and
leads to a similar but much more satisfactory picture. In the first place, we are dealing with a horizontal straight
best-fit regression line, and in addition the corresponding
turns out to be considerably smaller. Specifically, the scatter of the values of
for
of the considered blazars is now less than
about the mean value set by
for
and less than
about the mean value set by
for
, namely equal to 0.33 for
and equal to 0.32 for
. This situation is shown in
Figure 12.
We argue that the small scatter in the values of
implies that the physical emission mechanism is the same for all flaring blazars, with the small fluctuations in
arising from the difference of their
internal quantities: after all, no two identical galaxies have ever been found! On the other hand, the larger scatter in the values of
as derived in [
118]—presumably unaffected by photon-ALP oscillations when error bars are taken into account—is naturally traced back to the different environmental state of each flaring source, such as for instance the accretion rate.
A natural question finally arises. How is it possible that the large spread in the
distribution (see
Appendix C) arises from the small scatter in the
distribution shown in
Figure 10? The answer is very simple: most of the scatter in the
distribution arises from the large scatter in the source redshifts.
8. Propagation of ALPs in Extragalactic Space—2
So far, we have been dealing with the DLSHE model for the extragalactic magnetic field since at the VHE currently probed by the IACTs the photon-ALP oscillation length is much larger than the size of the magnetic domains.
However, in 2015 Dobrynina, Kartavtsev and Raffelt [
145] realized that at even larger energies photon dispersion on the CMB (Cosmic Microwave Background) becomes the leading effect, which implies
to decrease as
E further increases. Therefore, things completely change whenever
, since in this case a full oscillation—or even several oscillations—probe a whole domain, and if it is described unphysically like in the DLSHE model then the results come out unphysical as well. Manifestly, this would be a disaster for the VHE observatories of the next generations, which will reach energies up to 100 TeV or even larger.
This problem can be solved by smoothing out the edges in order to make the change of the magnetic field
direction continuous across the domain edges, even if it is still
random, as already stressed in
Section 4.4. Hence, in both cases only a random
single realization of the beam propagation process is
observable at once. We still suppose that photon-ALP oscillations are present in the beam from a blazar at redshift
z, and so the photon survival probability is denoted by
, where
and
are the two angles that fix the direction of
in space at a generic point
y along the beam and perpendicularly to it. In order to achieve our goal we have to resort to a
domain-like smooth-edges (DLSME) model—mentioned in
Section 4.4—wherein the beam propagation equation within a single domain becomes three-dimensional and very difficult to solve analytically. But as shown in [
114] such an equation becomes effectively
two-dimensional. Moreover, according the above two models [
213,
215] the
strength of
should vary rather little in different domains, hence we average it over many domains and attribute in first approximation the resulting value to
each domain, denoting it for simplicity again by
B. Finally, we consistently we take the transverse magnetic field component
.
The two-dimensional beam propagation equation has been solved exactly and analytically [
114]. It turns out that such a solution is
indistinguishable from the numerical solution of the above three-dimensional exact equation (more about this in [
114]). Physically, this amounts to the the whole physics of the problem being confined inside the planes
perpendicular to the beam rather than being spread out throughout the full three-dimensional space. As a consequence,
, where
is the angle between
and a fixed fiducial
z-direction
equal in all domains (namely in all planes
).
8.1. Preliminary Remarks
Broadly speaking, what we said in
Section 3 remains unchanged, apart from two facts.
One is that the mixing matrix depends on
y also in a single domain, and its explicit form is
The meaning of the terms of
is as follows. The contribution from photon dispersion on the CMB is
[
145], the contribution from the EBL absorption is
where
denotes the corresponding photon mean free path inside a single domain (more about this, later), the contribution from the plasma frequency of the ionized intergalactic medium is
while the remaining terms are
and
, just like in
Section 3.2.
The other fact is that we now have three regimes, separated by the
low-energy threshold
which is equal to Equation (
37), and the
high-energy thresholdSpecifically we have:
—This is the
low-energy weak-mixing regime, wherein the terms
dominate. Correspondingly, we have
and
However, since we will not consider this case throughout the paper.
—This is the intermediate-energy or
strong-mixing regime in which the
term dominates. Accordingly, we obtain
Clearly, and are independent of all the energy-dependent terms, and becomes maximal: observe that enters and nowhere else.
—This is the
high-energy weak-mixing regime, which is in a sense a sort of reversed low-energy weak-mixing regime, where however the term
dominates over
. Correspondingly, we get
Manifestly, decreases with increasing E and exhibits oscillations in E: this means that the individual realizations of the beam propagation are also oscillating functions of E. Moreover—since —as E increases the photon-ALP oscillations become unobservable at some point.
8.2. Domain-like Smooth-Edges (DLSME) Model
As we said, we are going to apply the DLSME model to a monochromatic photon/ALP beam of energy
E emitted by a far-away blazar, propagating through extragalactic space and reaching us [
115].
Therefore we briefly summarize this model (see [
114] for a full account). We suppose that there are
domains between the blazar and us, and we number them so that domain 1 is the one closest to the blazar while domain
is the one closest to us. Momentarily, we take all domains with the same length. We denote by
the set of coordinates which defines the beginning (
) and the end (
) of the
n-th domain (
) towards the blazar.
Because of our ignorance about the strength of in every domain and since it is supposed to vary rather little in different domains, we average B over many domains, and next we attribute the resulting value to each of them, so that B—but not —will henceforth be regarded as constant in first approximation.
As we said the problem is actually a two-dimensional one, since what matters is only . Therefore, we denote by the set of angles that forms with the fixed fiducial z-direction in the middle of every domain. Thanks to the previous assumptions also —but not —can be taken as constant in all domains.
Given the fact that changes randomly from one domain to the next, in order for to be continuous all along the beam it is compelling that it has equal values on both sides of every edge, e.g., the one between the n-th and the -th domain. Thus, the emerging picture is that is homogeneous in the central part, but as the distance from the edge with the -th domain decreases we assume that linearly changes thereby becoming equal to on the same edge but in the -th domain. Accordingly, the continuity of the components of along the whole beam is ensured.
A schematic view of this construction is shown in
Figure 15.
In practice, it is useful to define the two quantities
and
as
where
is the
smoothing parameter. The interval
is the region where the angle
changes smoothly from the value
in the
n-th domain to the value
in the (
)-th domain. Manifestly, for
we have
, and we recover the DLSHE model. On the other hand, for
then
becomes the midpoint of the
n-th domain, and likewise
becomes the midpoint of the
-th domain: now the smoothing is maximal, because we never have a constant value of
in any domain. The general case is intermediate—represented by a value of
—so that in the central part of a domain the angle is constant (
) and then it linearly joins the value of the constant angle in the next domain (
). Hence, in a generic interval
we have
According to our conventions, the blazar redshift is , the points and defining the n-th domain have redshift and (), respectively, and we set for the average redshift of the n-th domain. Similarly, the emitted beam has energy , whereas the beam at points and has energy and (), respectively. Finally, we define the average energy in the n-th domain as , and the observer has energy . As usual, (. We stress that at variance with the previous conventions, the observed beam energy is .
8.3. Propagation over a Single Domain
We proceed along the same lines of
Section 5.2, namely we have to account for the EBL absorption and to determine the magnetic field strength
in the generic
n-th domain of size
.
The only novelty is that instead of taking the length of all domains strictly equal, we allow for a small spread. Thus, we assume a probability distribution for the
. Owing to the properties of
at redshift
, we take for the probability density in question the power law
inside the range
, which means that
, indeed allowed by present bounds [
209]. Needless to say, such a choice is largely arbitrary and the corresponding histogram is shown in
Figure 16.
In order to accomplish this task, we just employ the discussion reported in
Section 5.2. Accordingly, we find
with
again given by the FR model, and
where
is the strength of
in the local Universe, namely in the domain closest to the observer (
).
So, the above mixing matrix
in a single
n-th domain is fully determined, and now has
and all terms replaced with those evaluated within the
n-th domain. It can next be inserted into the reduced Schrödinger-like Equation (
19).
8.4. Solution of the Beam Propagation Equation
Our present job is two-fold. In the first place, we have to solve such a reduced Schrödinger equation, which amounts to find its transfer matrix in a single domain. This is the hard part of the game, which is actually the main achievement of [
114]. Next, the overall transfer matrix emerges by properly multiplying the transfer matrices pertaining to all domains, which is a trivial implication of quantum mechanics.
The first one amounts to solving the beam propagation equation inside a
single n-th domain. The solution reads
for an arbitrary choice of the angle
. Unfortunately, the explicit forms of the two transfer matrices in Equation (
134) is much too cumbersome to be reported here, and the reader can found them in [
114] (see its Equations (54) and (91) with
and the appropriate conversions in order to go over from physical space to redshift space).
The second point consists in the evaluation of the
whole transfer matrix from the blazar to us, namely along a single arbitrary realization of the whole beam propagation process. Starting from Equation (
134) the desired equation presently has the form
Note that this product must be ordered in such a way that the transfer matrices with smaller n must be closer to the source.
8.5. Results
We are finally in a position to derive the desired result, namely the photon survival probability from the blazar to us along an arbitrary realization of the whole beam propagation process. This goal is again trivially achieved thanks to the analogy with non-relativistic quantum mechanics, namely by employ again Equation (
84).
As before, the photon polarization cannot be measured at the considered VHE, hence we have to start with the unpolarized beam state and sum the result over the two final polarization states. So, for the reader’s convenience we revert to the same, common notation used in
Section 5.2, namely
,
,
. Accordingly, we have
with
Below, the photon survival probability
is plotted versus the observed energy
for 7 simulated blazars at
, assuming for each one our benchmark values
. In order to simplify notations, we will denote
as
. Thousand random realizations of the propagation process have been considered for each choice of
z,
,
. In all figures a random distribution of the domain length
has been taken: a power law distribution function has been chosen with exponent
and domain length in the interval between the minimal value
and the maximal value
. The resulting average domain length is
. Our results are shown in
Figure 17,
Figure 18,
Figure 19,
Figure 20,
Figure 21,
Figure 22 and
Figure 23. We take the smoothing parameter
for the transition between two adjacent magnetic domains. The dotted-dashed black line corresponds to conventional physics, the solid light-gray line to the median of all realizations of the propagation process and the solid yellow line to a single realization with a random distribution of both the domain lengths and of the orientation angles of the magnetic field inside the domains. The filled area represents the envelope of the results on the percentile of all the possible realizations of the propagation process at 68% (dark blue), 90% (blue) and 99% (light blue), respectively. In the upper-left panel we have taken
, in the upper-right panel
, in the lower-left panel
and in the lower-right panel
[
115]. A rather similar approach has been developed in 2017 by Kartavtsev, Raffelt and Vogel, where however only the average photon survival probability is considered [
102].
9. A Full Scenario
Up until this point we have especially addressed two specific topics which provide two strong hints at the existence of an ALP with
and
. In addition, we have considered the propagation of a photon/ALP beam in extragalactic space both for VHE energies currently observed by the IACTs (
Section 5) and for energies to be measured by the next generation of VHE observatories (
Section 8).
A partial scenario—complementary to the one discussed in
Section 5—has been put forward in 2008 and consists in the conversion
inside a blazar and the reconversion
in the Milky Way, neglecting any possible
oscillation in extragalactic space [
56].
Our present aim is to discuss a full scenario wherein a VHE photon/ALP beam is described from its origin inside a BL Lac to its detection on Earth. An early attempt towards this goal was done in 2009 [
62], but since then much progress has been done. So, our analysis will be performed in the light of the most up-to-date astrophysical information and for energies up to above 50 TeV [
116].
We are going to consider three sources.
Markarian 501 at .
The extreme BL Lac 1ES 0229+200 at .
A simulated source like BL Lac 1ES 0229+200 but at
Recall that BL Lacs have been observed up to .
Manifestly, the emitted VHE photon/ALP beam from the BL Lacs in question crosses a variety of magnetic field structures in very different astrophysical environments: inside the BL Lac jet, within the host galaxy, in extragalactic space, and finally inside the Milky Way. Accordingly, we shall have to evaluate the transfer matrix in each of these structures.
Our strategy is to assume a realistic emitted spectrum for the considered three BL Lacs, and derive their observed spectrum up to above 50 TeV.
9.1. Propagation in the BL Lac Jet
We denote by the region where the VHE photons originate inside the BL Lac jet, with denoting its distance from the central supermassive black hole (SMBH). So, our first step is to evaluate the transfer matrix in the jet region between and the end of the jet , which we denote as .
The region is rather far from the central SMBH, and the jet axis is supposed to coincide with the direction y (as usual). In order to evaluate the photon/ALP beam propagation inside the jet we must know three quantities: (1) the distance from the central SMBH, (2) the transverse magnetic field profile from to , (3) the electron density profile from to .
The Synchrotron Self Compton (SSC) diagnostics as applied to the SED of BL Lacs [
249] allows us to derive realistic values for these quantities. Inside
we find
and for definiteness we choose
. Moreover, we get
, leading in turn to a plasma frequency of
, thanks to Equation (
20). Although there is no direct way to infer a precise value of
, this quantity can be estimated from the size of
—which is assumed as measure of the jet cross-section— thus finding
. For definiteness, we shall take
. Once produced, VHE photons propagate unimpeded out to
where they leave the jet, entering the host galaxy. Within
, what is relevant is the toroidal part of the magnetic field which is transverse to the jet axis [
245,
250,
251]. Its profile is
Concerning the electron density profile, due to the conical shape of the jet our expectation is
The knowledge of the above quantities allows us to compute the entire propagation process of the photon/ALP beam within the jet, namely .
It should be kept in mind that in we consider the photon/ALP beam in a frame co-moving with the jet, so that we must apply the transformation to the beam in order to go to a fixed frame—as it will be performed in the next regions—with being the Lorentz factor. We take .
9.2. Propagation in the Host Galaxy
All the three considered blazars are hosted by elliptical galaxies, which we denote by
. We have already addressed the propagation of a VHE beam in these galaxies in
Section 7.6, finding that even if the beam is in the strong-mixing regime the effect of
oscillations is totally negligible. Therefore, denoting by
and by
the points on the
y axis where the beam enters and exits from the host galaxy, respectively, we have
.
9.3. Propagation in the Extragalactic Space
We let
be the region where the photon/ALP beam propagates in the extragalactic space, i.e., from
up to the border of the Milky Way
. Now the beam behaviour in
is affected by the morphology and strength of the extragalactic magnetic field
. We have already repeatedly considered this issue in great detail, and for the present purposes in
Section 8.
9.4. Propagation in the Milky Way
We denote by the region where the photon/ALP beam propagates inside the Milky Way, i.e., from up to the Earth, whose position is denoted by .
By closely following the strategy described in [
74], we compute
. In order to take into account the structured behaviour of the Galactic magnetic field
we adopt the recent Jansson and Farrar model [
252,
253], which includes a disk and a halo component, both parallel to the Galactic plane, and a poloidal `X-shaped’ component at the galactic center. Its most updated version is described in [
254], where newer polarized synchrotron data and use of different models of the cosmic ray and thermal electron distribution are employed.
The alternative model of the Galactic magnetic field existing in the literature is the one in [
255]. However this model is based mainly on data along the Galactic plane so that the Galactic halo component of
is not accurately determined. For this reason we prefer to use the Jansson and Farrar model. In any case, we have tested the robustness of our results by employing also this model and—even if with some little modifications—our results are qualitatively unchanged.
While the Jansson and Farrar model allows also for a random and a striated component of the field, it turns out that only the regular component is relevant in the present context, since the oscillation length is much larger than the coherence length of the turbulent field.
Inside the Milky Way disk the electron number density is
, resulting in a plasma frequency
owing to Equation (
20): this emerges from a new model for the distribution of the free electrons in the Galaxy [
256]. Moreover, the Galaxy is modeled by an extended thick disk accounting for the so-called warm interstellar medium, a thin disk standing for the Galactic molecular ring, spiral arms (inferred from a new fit to Galactic HII regions), a Galactic Center disk and seven local features counting the Gum Nebula, the Galactic Loop I and the Local Bubble. The model includes also an offset of the Sun from the Galactic plane and a warp of the outer Galactic disk. The Galactic model parameters are obtained from the fit to 189 pulsars with independently determined distances and DMs.
Accordingly, we compute for an arbitrary direction of the line of sight to a given blazar.
9.5. Overall Photon Survival Probability
Because all the transfer matrices in each region are now known, the total transfer matrix
describing the propagation of the photon/ALP beam from the VHE photon production region in the BL Lac jet up to the Earth reads
where of course we have
and
z. Since photon polarization cannot be measured in the VHE gamma-ray band, we have to treat the beam as unpolarized. Therefore, we must use the generalized polarization density matrix
. As a consequence, the overall photon survival probability becomes
where
and
Below—merely for notational convenience—we shall replace simply by .
In order to give the reader a feeling of what happens in the various regions crossed by the photon/ALP beam, in
Figure 24 we plot how the oscillation length
varies as a function of the energy
E in the jet, in the extragalactic space and in the Milky Way. As the upper panel of
Figure 24 shows, the behaviour of
versus
E is strongly affected by the value of
: as expected (see also [
114]), as
decreases—when the distance from the emission region increases—the maximal value of
increases and the energy where the QED vacuum polarization effect becomes important increases as well. Instead, in the central panel of
Figure 24 what happens in extragalactic space is that
starts to decrease because of the effect of the photon dispersion on the CMB, which becomes more and more important as
E increases (for more details see [
114]). Finally, in the lower panel of
Figure 24 we see that in the Milky Way
is almost constant with
E since the QED vacuum polarization effect and photon dispersion on the CMB become subdominant as compared to the photon-ALP mixing in almost all the considered energy range.
9.6. Blazar Spectra
Starting from the intrinsic spectra, we are now in a position to employ the overall photon survival probability in order to derive the observed spectra of the considered three blazars, and from them to infer the corresponding SED in the presence of oscillations all the way from inside the blazar to us. We can thus compare our findings with the results from conventional physics.
The observable physical quantity is the blazar spectrum pertaining to a single random realization of the photon/ALP propagation process. Nevertheless, it is enlightening to contemplate several realizations at once and to compute some of their statistical properties—the median and the area containing the , and of the total number of realizations—in order to check the stability of the result against the distribution of the angles of the extragalactic magnetic field inside each domain with respect to a fixed fiducial direction z equal for all domain. We recall that these angles are independent random variables.
For the three blazars in question, we model their intrinsic spectrum with a power law exponentially truncated at a fixed cut-off energy
as
where
is a normalization constant accounting for the blazar luminosity,
is a reference energy and
is the spectral index.
Markarian 501—This source is a high-frequency peaked blazar (HBL) at redshift
. We use the observational data points from HEGRA [
257] in a condition where Markarian 501 was observed in a high emission state, which allows us to have a very good quality spectrum up to ∼30 TeV. This fact is important for testing our model, since at such high energies it starts to make predictions which depart from conventional physics. In
Figure 25 we report its observed SED when conventional physics alone is considered, and when
oscillations are at work. In order to obtain the SED we take
,
and
in Equation (
146).
1ES 0229+200—This is a BL Lac at redshift
. This is the prototype of the so-called `extreme HBL’ (EHBL) [
258,
259], which exhibit a rather hard VHE observed spectrum up to at least 10 TeV. This fact is particularly interesting since the observed data points at such high energies allow to distinguish between the models based on conventional physics and those containing
oscillations. Future observations with the CTA that can eventually reach energies up to 100 TeV can provide a definitive answer. In
Figure 26 we plot its observed SED both when only conventional physics is taken into account and in the case in which also
oscillations are present. The SED is obtained by taking in Equation (
146)
in the case of conventional physics, and
when also
oscillations are considered, while in either case we choose
and
. Note that
is in agreement with the one derived for the Fermi/LAT spectrum in the recent analysis of [
259].
Extreme BL Lac at —BL Lacs have been observed also at redshift
, and so we assume the existence of an EHBL at redshift
. For this blazar we take a SED similar to the one of 1ES 0229+200, namely
,
and
in Equation (
146) for both cases (conventional physics alone, presence of
oscillations). We consider two possibilities: (1) such BL Lac is observed in the sky along the direction of the galactic pole: in
Figure 27 we plot its observed SED for both cases of presence/absence of photon-ALP interaction; (2) in
Figure 28 we exhibit the corresponding observed SED for the same BL Lac instead observed in the sky along the direction of the galactic plane for both cases of presence/absence of photon-ALP interaction.
9.7. Results
Our results about the SED of the above-considered BL Lacs are exhibited in
Figure 25 and
Figure 28. Generally speaking, the
oscillations give rise to a harder observed spectrum for all three sources as compared to the outcome of conventional physics. We stress that this fact becomes increasingly evident as
E or
z (or both) get larger.
Our findings strongly suggest that
oscillations inside the magnetic field of the BL Lac jet play a key role in starting the propagation in extragalactic space with a sizable amount of already produced ALPs, whose relevance depends both on
E and on
z. This is a rather subtle point and deserves a clear explanation. Superficially, one might expect
to increase with
, according to the physical intuition. This is certainly true as long as the EBL does not play an important role, namely for
E and
z low enough. Needless to say,
conversions in the BL Lac and
back-conversions in the Milky Way help increasing
, but not that much. Suppose instead that both
and
z are fairly large but that
E is not, so that photon dispersion on the CMB can be discarded. Accordingly, the conversion probability increases so that inside each single magnetic domain many
and
conversions take place. But since
z is supposed to be fairly large the EBL level is high, so that most of the photons get absorbed. Such a behaviour is very clearly shown in
Figure 27 and
Figure 28 around
. As the energy increases, photon dispersion on the CMB becomes dominant, which causes a much smaller number of
and
conversions to occur in extragalactic space. By and large, most of the ALPs produced in the BL Lac survive until they enter the Galaxy, whose strong magnetic field allows them to convert to photons. Whence the peak in
Figure 27 and
Figure 28 around
. All figures show that—as
E progressively increases beyond
—the area covered by the realizations of the photon/ALP propagation process gradually reduces. The reason is that the EBL absorption becomes so high at those energies that almost all the photons in each extragalactic magnetic field domain are absorbed and only the ones reconverted from ALPs inside the Galaxy are observed (as previously mentioned). Therefore, the parameter space of the model—
orientation angles, domain lengths
—gets reduced, and this fact reduces the available area that can be covered by the realizations of the propagation process.
Observe that in all figures we draw the CTA sensitivity curve for the South site and 50 h of observation. Because the sensitivity curves are relay on conservative criteria [
262,
263] we expect that the theoretical spectral features—look at the peak in
Figure 27 and
Figure 28 around
—which are close to the sensitivity curve should anyhow be detectable by the CTA.
For the reader’s convenience, we would like to briefly summarize our results. We have investigated the propagation of a photon/ALP beam originating well inside a BL Lac jet and traveling in the jet magnetic field, in the host galaxy magnetic field, in the extragalactic magnetic field, and in the Milky Way magnetic field up to us. We see from Markarian 501 (see
Figure 25) that conventional physics does not fit the highest energy point of the SED while the model including
oscillations naturally matches the data. For 1ES 0229+200 (see
Figure 26) the model including
oscillations fits the data remarkably well, especially the highest energy data points of the SED. As far as the simulated 1ES 0229+200 at
is concerned, the situation is striking: only
oscillations predict the peak around E = (10–30) TeV of the SED, while conventional physics prediction is many order of magnitude below.
As it is evident from
Figure 27 and
Figure 28—as the redshift increases—at high energies the difference between the results from conventional physics alone, and the model including
oscillations becomes more and more dramatic. This is especially true when sizable
conversions take place inside a blazar, since then most of the emitted ALPs can become photons only inside the Milky Way magnetic field. In particular, for very distant BL Lacs we predict a peak in the energy spectra at
as it is evident from
Figure 27 and
Figure 28 for a BL Lac at
. In addition, the energy oscillations in the observed spectrum—clearly recognizable in the figures—are a clear-cut feature of our scenario, which can be observed provided that the detector has enough energy resolution: they arise from the photon dispersion on the CMB.
A competitive scenario capable to reduce the optical depth is the Lorentz invariance violation (LIV) which could predict a somewhat similar peak in the BL Lac spectra above ∼20
[
264,
265]. But the two scenarios can be distinguished since the LIV does
not predict any spectral energy oscillatory behavior [
117].
At this point some remarks look compelling.
The jet parameters (, ) are affected by uncertainties, and the amount of produced ALPs in this region clearly reflects this fact. Nevertheless, we have checked that the final spectra qualitatively possess the above-mentioned features regardless of the choice of the jet parameters, provided of course that they are realistic.
Even if we consider very low values of the extragalactic magnetic field—namely —the considered model predicts the above-mentioned features even if partially reduced, in particular concerning the amplitude of the energy oscillations. However, the peak in the spectra at remains unaffected at high redshift.
The electromagnetic cascade proposed to mimic
oscillation effects in blazar spectra [
266] can work only for
, which is indeed quite close to the
lower limits [
208,
209,
210]. Still, for
the charged particles produced in the cascade are deflected by
and the resulting additional photon flux turns out to be very likely irrelevant (for more details, see e.g., [
267]). This argument also applies to the possible additional
pairs produced in the process
.
For
the infrared radiation from dust present inside the Milky Way could play a moderate role in absorbing photons [
268]. But this effect is irrelevant for us and can be safely discarded. Basically, the resulting absorption is substantial only inside the Galactic plane and a few degrees above and below it, hence only ALPs converted to photons in the Galactic plane close to the outer border of the Milky Way disk fully undergo such an effect. Actually, two points should be be stressed. (1) It goes without saying that when the line of sight to the blazar lies outside the galactic plane the considered effect is totally irrelevant. (2) Even for the photon/ALP beam entering the Milky Way along the Galactic plane the
oscillations reduce photon absorption, thereby considerably weakening dust absorption.