Miguel A. Sánchez Conde (Instituto de Astrofísica de Canarias)

In collaboration with: D. Paneque, E. Bloom, F. Prada and A. Domínguez

[ PHYSICAL REVIEW D 79, 123511 (2009) ]

””Multiicube workshop“ – Padova, March 1 – 5, 2010

2

QED pair creation processes is the dominant process for the cosmological absorption of gamma-rays:

γ + γbackground e+ + e-

Around the TeV region:

Infrared/optical background photons: Extragalactic Background Light (EBL)

For a source at redshift 0.5 and 0.5 TeV, attenuation ~2 orders of magnitude!!

€

ε ≈ 0.5 1TeVE

eV

3C 279   Flat spectrum radio quasar   z=0.54   The most distant AGN in

gamma-rays (>100 GeV)   Push EBL models already to

the limit!

  Modeling of AGN emission mechanisms typically assume spectral index >1.5

[MAGIC Collaboration, Albert et al. 2008]

  Recent gamma observations might already pose substantial challenges to the conventional models to explain the observed source spectra and/or EBL density.

  The VERITAS Collaboration recently claimed a detection above 0.1 TeV coming from 3C66A (z=0.444). EBL-corrected spectrum harder than 1.5 (Acciari+09).

  TeV photons coming from 3C 66A? (Neshpor+98; Stepanyan+02). Difficult to explain with conventional EBL models and physics.

  The lower limit on the EBL at 3.6 µm was recently revised upwards by a factor ∼2, suggesting a more opaque universe (Levenson+08).

  Some sources at z = 0.1 － 0.2 seem to have harder intrinsic energy spectra than previously anticipated (Krennrich+08).

  While it is still possible to explain the above points with conventional physics (EBL, very hard spectrum), the axion/photon oscillation would naturally explain these puzzles:   More high energy photons than expected.

  Softer intrinsic spectrum when including axions.

AGNs located at cosmological distances will be affected by both mixing in the source (e.g. Hooper & Serpico 07) and in the IGMF (De Angelis+07):

A.  Source mixing: flux attenuation B.  IGM mixing: flux attenuation and/or enhancement

In order to observe both effects in the gamma-ray band, we need ultralight axions.

4

where wpl =!

4!"ne/me = 0.37 ! 10!4µeV!

ne/cm!3

the plasma frequency, me the electron mass and ne theelectron density.

Finally, !a is the ALP mass term:

!a =m2

a

2E!" 2.5 ! 10!20m2

a,µeV

"

TeV

E!

#

cm!1. (7)

Note that in Eqs.(4-7) we have introduced the dimen-sionless quantities BmG = B/10!3 G, M11 = M/1011

GeV and mµeV = m/10!6 eV.Since we expect to have not only one coherence do-

main but several domains with magnetic fields di"er-ent from zero and subsequently with a potential pho-ton/axion mixing in each of them, we can derive a totalconversion probability [21] as follows:

P!"a "1

3[1 # exp(#3NP0/2)] (8)

where P0 is given by Eq.(2) and N represents the numberof domains. Note that in the limit where N P0 $ %, thetotal probability saturates to 1/3, i.e. one third of thephotons will convert into ALPs.

It is useful here to rewrite Eq. (2) following Ref. [11],i.e.

P0 =1

1 + (Ecrit/E!)2sin2

$

%

B s

2 M

&

1 +

"

Ecrit

E!

#2

'

( (9)

so that we can define a characteristic energy, Ecrit, givenby:

Ecrit &m2 M

2 B(10)

or in more convenient units:

Ecrit(GeV ) &m2

µeV M11

0.4 BG(11)

where the subindices refer again to dimensionless quan-tities: mµeV & m/µeV , M11 & M/1011 GeV and BG &B/Gauss; m is the e"ective ALP mass m2 & |m2

a # #2pl|.

Recent results from the CAST experiment [5] give a valueof M11 ' 0.114 for axion mass ma ( 0.02 eV. Althoughthere are other limits derived with other methods or ex-periments, the CAST bound is the most general andstringent limit in the range 10!11 eV ) ma ) 10!2

eV.At energies below Ecrit the conversion probability is

small, which means that the mixing will be small. There-fore we must focus our detection e"orts at energies abovethis Ecrit, where the mixing is expected to be large(strong mixing regime). As pointed out in Ref. [11], in thecase of using typical parameters for an AGN in Eq. (11),Ecrit will lie in the GeV range given an ALP mass of theorder of * µeV.

To illustrate how the photon/axion mixing inside thesource works, we show in Figure 2 an example for anAGN modeled by the parameters listed in Table II (ourfiducial model, see Section III). The only di"erence isthe use of an ALP mass of 1 µeV instead of the valuethat appear in that Table, so that we obtain a criticalenergy that lie in the GeV energy range. E"ectively, us-ing Eq. (11) we get Ecrit = 0.19 GeV. Note that themain e"ect is an attenuation in the total expected in-tensity of the source. One can see in Figure 2 a sinu-soidal behavior just below the critical energy. However,it must be noted that a) the oscillation e"ects are small;b) these oscillations only occur when using photons po-larized in one direction while, in reality, the photon fluxesare expected to be rather non-polarized; and c) the abovegiven expressions are approximations and actually onlytheir asymptotic behavior should be taken as exact andwell described by the formulae. Therefore, the chancesof observing sinusoidally-varying energy spectra in as-trophysical source, due to photon/axion oscillations, areessentially zero.

0.01 0.10 1.00 10.00 100.00E (GeV)

0.7

0.8

0.9

1.0

1.1

Inte

nsity

FIG. 2: Example of photon/axion oscillations inside thesource or vicinity, and its e!ect on the source intensity (solidline), which was normalized to 1 in the Figure. We used theparameters given in Table II to model the AGN source, butwe adopted an ALP mass of 1 µeV. This gives Ecrit = 0.19GeV. The dot-dashed line represents the maximum (theoret-ical) attenuation given by Eq. (8), and equal to 1/3.

B. Mixing in the IGMFs

The strength of the Intergalactic Magnetic Fields(IGMFs) is expected to be many orders of magnitudeweaker (*nG) than that of the source and its surround-ings (*G). Consequently, as described by Eq. (11), theenergy at which photon/axion oscillation occurs in theIGM is many orders of magnitude larger than that atwhich oscillation can occur in the source and its vicinity.

•  Axions (pseudoscalar boson) were postulated to solve the strong-CP problem in the 70s. •  Good Dark Matter candidates •  They are expected to oscillate into photons (and viceversa) in the presence of magnetic fields:

(Sánchez-Conde+, PRD 09)

0.1 1.0 10.0 100.0 1000.0E (GeV)

1

10

100

1000

Axio

n b

oost fa

cto

r

  Larger axion boosts for distant sources.   The more attenuating the EBL, the larger the axion boosts.

Attenuation due to source mixing

Intergalactic mixing

Enhancement due to intergalactic mixing

Attenuation due to intergalactic mixing

0.1 1.0 10.0 100.0 1000.0 10000.0E (GeV)

1

Axio

n b

oost fa

cto

r

z=0.536 z=0.117

Axion boost =(Flux w axions) / (Flux w/o axions)

(Sánchez-Conde+, PRD 09)

(Sánchez-Conde+, PRD 09)

Softer intrinsic spectrum with axions

We can observe the spectrum up to higher energies with axions

[3C279 data points from the MAGIC Collaboration, Albert et al. 2008] (Sánchez-Conde & Dominguez, in prep.)

0.1 1.0 10.0 100.0 1000.0E (GeV)

1

10

100

1000

Axio

n b

oost fa

cto

r

Fermi/LAT and/or IACTs Look for intensity drops in the residuals (“best-model”-data). Source model dependent. Powerful, relatively near AGNs.

IACTs observations Look for systematic intensity enhancements at energies where the EBL is important.

Distant (z > 0.2) sources at the highest possible energies (>1 TeV), to push EBL models to the extreme.

Source and EBL model dependent, but very important enhancement expected in some cases.

Fermi/LAT and/or IACTs Look for intensity drops in the residuals. Only depends on the IGMF and axion properties (mass and coupling constant). Independent of the sources -> CLEAR signature!

  If axions exist, they could distort the spectra of astrophysical sources importantly.

  If photon/axion mixing in the IGMFs, then also mixing in the source. For maxion ≈ 10-10 eV -> gamma ray energy range.

  Photon/axion mixing in both the source and the IGM are expected to be at work over several decades in energy -> joint effort of Fermi and current IACTs needed.   Fermi/LAT instrument expected to play a key role, since it will detect thousands of

AGNs (up to z∼5), at energies where the EBL is not important.   IACTs specially important at higher energies (>300 GeV), where the EBL is present.

  Main caveats: the effect of photon/axion oscillations could be attributed to conventional physics in the source and/or propagation of the gamma-rays towards the Earth.

  However, detailed observations of AGNs at different redshifts and different flaring states could be used to identify the signature of an effective photon/axion mixing.

  Axions were postulated to solve the strong CP problem in the 70s.

  Good Dark Matter candidates (axions with masses ≈ meV-µeV could account for the total Dark Matter content).

  They are expected to oscillate into photons (and viceversa) in the presence of magnetic fields:

Photon/axion oscillations are the main vehicle used at present in axion searches (ADMX, CAST…).

Some astrophysical environments fulfill the mixing requirements

AGNs, IGMFs

M11: coupling constant inverse (gαγ/1011 GeV)

BG: magnetic field (G) spc: size region (pc)

with

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

€

15 ⋅ BG ⋅ spcM11

≥1

M11 ≥ 0.114 GeV (CAST limit)

€

15 ⋅ BG ⋅ spcM11

≥1

M11: coupling constant inverse (gαγ/1011 GeV)

BG: magnetic field (G) spc: size region (pc)

with

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

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axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

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axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

HINTS OF THE EXISTENCE OF AXIONLIKE PARTICLES . . . PHYSICAL REVIEW D 79, 123511 (2009)

123511-3

axion coupling strength, F is the electromagnetic field-strength tensor, !F its dual, E the electric field, and B themagnetic field. The axion has the important feature that itsmass ma and coupling constant are inversely related toeach other. There are, however, other predicted stateswhere this relation does not hold; such states are knownas axionlike particles (ALPs). An important and intriguingconsequence of Eq. (1) is that ALPs oscillate into photonsand vice versa in the presence of an electric or magneticfield. In fact this effect represents the keystone in ongoingALP searches carried out by current experiments.

In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will account forthe mixing that takes place inside or near the gamma-raysources together with that one expected to occur in theIGMFs. We will do it under the same consistent frame-work. Furthermore, it is important to remark that it will benecessary to include the EBL in our formalism, in particu-lar in the equations that describe the intergalactic mixing.Its main effect we should remember is an attenuation of thephoton flux, especially at energies above 100 GeV. Weshow in Fig. 1 a diagram that outlines our formalism.Very schematically, the diagram shows the travel of aphoton from the source to the Earth in a scenario withaxions. In the same figure, we show the main physicalcases that one could identify inside our formalism: mixingin both the source and the IGMF, mixing in only one ofthese environments, the effect of the EBL, axion to photonreconversions in the IGMF, etc. A quantitative descriptionof the photon/axion mixing phenomenon in both the sourceand the IGMFs can be found in the next two subsections.

A. Mixing inside and near the source

It has been recently proposed that an efficient conversionfrom photons to ALPs (and vice versa) could take place inor near some astrophysical objects that should host a strongmagnetic field [12].

Given a domain of length s, where there is a roughlyconstant magnetic field and plasma frequency, the proba-bility of a photon of energy E! to be converted into an ALPafter traveling through it can be written as [14,16]

P0 ! ""Bs#2sin2""oscs=2#""oscs=2#2

: (2)

Here "osc is the oscillation wave number,

"2osc ’ ""CM $"pl % "a#2 $ 4"2

B; (3)

"B that gives us an idea of how effective is the mixing, i.e.

"B ! Bt

2M’ 1:7& 10%21M11BmG cm%1; (4)

where Bt the magnetic field component along the polariza-tion vector of the photon and M11 the inverse of thecoupling constant."CM is the vacuum Cotton-Mouton term, i.e.

"CM ! % "

45#

!Bt

Bcr

"2E!

’ %1:3& 10%21B2mG

!E!

TeV

"cm%1; (5)

where Bcr ! m2e=e ’ 4:41& 1013 G is the critical mag-

netic field strength (e is the electron charge)."pl is the plasma term

"pl !w2

pl

2E’ 3:5& 10%20

!ne

103 cm%3

"!TeV

E!

"cm%1; (6)

where wpl !#######################4#"ne=me

p! 0:37&

10%4 $eV###################ne=cm

%3p

is the plasma frequency, me is theelectron mass, and ne is the electron density.Finally, "a is the ALP mass term,

"a !m2

a

2E!’ 2:5& 10%20m2

a;$eV

!TeV

E!

"cm%1: (7)

FIG. 1 (color online). Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF areconsidered under the same consistent framework. Photon to axion oscillations (or vice versa) are represented by a crooked line, whilethe symbols ! and a mean gamma-ray photons and axions, respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are schematically represented by a line that goes from the source to the Earth.

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  Some astrophysical environments fulfill the mixing requirements:

In IGMFs, BG≈10-9 -> Mixing also possible for cosmological distances (spc ≥ 108)

  Important implications for astronomical observations (AGNs, pulsars, GRBs…).

€

15 ⋅ BG ⋅ spcM11

≥1Astrophysical sources with BG·spc ≥ 0.01 will be valid.

M11 ≥ 0.114 GeV (CAST limit)

BG·spc also determines the Emax to which sources can accelerate cosmic rays: Emax= 9.3·1020 ·BG·spc eV (Hillas criterion)

We observe cosmic rays up to 3·1020 eV -> BG·spc up to 0.3 must exist!

0.01 0.10 1.00 10.00 100.00 1000.00E (GeV)

0.7

0.8

0.9

1.0

1.1

Inte

nsity

The main effect is an ATTENUATION of the photon flux above the critical energy:

For typical AGN numbers, the effect is present in gamma-rays below axion masses ≈ 10-6 eV

4

where wpl =!

4!"ne/me = 0.37 ! 10!4µeV!

ne/cm!3

the plasma frequency, me the electron mass and ne theelectron density.

Finally, !a is the ALP mass term:

!a =m2

a

2E!" 2.5 ! 10!20m2

a,µeV

"

TeV

E!

#

cm!1. (7)

Note that in Eqs.(4-7) we have introduced the dimen-sionless quantities BmG = B/10!3 G, M11 = M/1011

GeV and mµeV = m/10!6 eV.Since we expect to have not only one coherence do-

main but several domains with magnetic fields di"er-ent from zero and subsequently with a potential pho-ton/axion mixing in each of them, we can derive a totalconversion probability [21] as follows:

P!"a "1

3[1 # exp(#3NP0/2)] (8)

where P0 is given by Eq.(2) and N represents the numberof domains. Note that in the limit where N P0 $ %, thetotal probability saturates to 1/3, i.e. one third of thephotons will convert into ALPs.

It is useful here to rewrite Eq. (2) following Ref. [11],i.e.

P0 =1

1 + (Ecrit/E!)2sin2

$

%

B s

2 M

&

1 +

"

Ecrit

E!

#2

'

( (9)

so that we can define a characteristic energy, Ecrit, givenby:

Ecrit &m2 M

2 B(10)

or in more convenient units:

Ecrit(GeV ) &m2

µeV M11

0.4 BG(11)

where the subindices refer again to dimensionless quan-tities: mµeV & m/µeV , M11 & M/1011 GeV and BG &B/Gauss; m is the e"ective ALP mass m2 & |m2

a # #2pl|.

Recent results from the CAST experiment [5] give a valueof M11 ' 0.114 for axion mass ma ( 0.02 eV. Althoughthere are other limits derived with other methods or ex-periments, the CAST bound is the most general andstringent limit in the range 10!11 eV ) ma ) 10!2

eV.At energies below Ecrit the conversion probability is

small, which means that the mixing will be small. There-fore we must focus our detection e"orts at energies abovethis Ecrit, where the mixing is expected to be large(strong mixing regime). As pointed out in Ref. [11], in thecase of using typical parameters for an AGN in Eq. (11),Ecrit will lie in the GeV range given an ALP mass of theorder of * µeV.

To illustrate how the photon/axion mixing inside thesource works, we show in Figure 2 an example for anAGN modeled by the parameters listed in Table II (ourfiducial model, see Section III). The only di"erence isthe use of an ALP mass of 1 µeV instead of the valuethat appear in that Table, so that we obtain a criticalenergy that lie in the GeV energy range. E"ectively, us-ing Eq. (11) we get Ecrit = 0.19 GeV. Note that themain e"ect is an attenuation in the total expected in-tensity of the source. One can see in Figure 2 a sinu-soidal behavior just below the critical energy. However,it must be noted that a) the oscillation e"ects are small;b) these oscillations only occur when using photons po-larized in one direction while, in reality, the photon fluxesare expected to be rather non-polarized; and c) the abovegiven expressions are approximations and actually onlytheir asymptotic behavior should be taken as exact andwell described by the formulae. Therefore, the chancesof observing sinusoidally-varying energy spectra in as-trophysical source, due to photon/axion oscillations, areessentially zero.

0.01 0.10 1.00 10.00 100.00E (GeV)

0.7

0.8

0.9

1.0

1.1

Inte

nsity

FIG. 2: Example of photon/axion oscillations inside thesource or vicinity, and its e!ect on the source intensity (solidline), which was normalized to 1 in the Figure. We used theparameters given in Table II to model the AGN source, butwe adopted an ALP mass of 1 µeV. This gives Ecrit = 0.19GeV. The dot-dashed line represents the maximum (theoret-ical) attenuation given by Eq. (8), and equal to 1/3.

B. Mixing in the IGMFs

The strength of the Intergalactic Magnetic Fields(IGMFs) is expected to be many orders of magnitudeweaker (*nG) than that of the source and its surround-ings (*G). Consequently, as described by Eq. (11), theenergy at which photon/axion oscillation occurs in theIGM is many orders of magnitude larger than that atwhich oscillation can occur in the source and its vicinity.

Ecrit= 0.19 GeV (B=1.5 G; maxion=1µeV)

Maximum theoretical attenuation = 1/3

Effect of the Cotton-Mouton term

8

that, as we increase the number of domains, the attenua-tion increases until the size of the domain is “too small”.At this point, the probability of photon/axion conversionis almost zero for the single domains, which reduces theoverall photon/axion conversion.

Finally, we must note here that we neglected theCotton-Mouton term in our calculations (Eq. 5), despiteof the fact that it appears in the formalism presented inSection II A. This e!ect could be indeed really impor-tant to correctly estimate the mixing in the source, veryspecially at higher energies (essentially what happens isthat the e"ciency of the mixing decreases as the energyincreases). However, once we include photon/axion os-cillation in the IGMF, which is the dominant mixing byfar at higher energies and for which the Cotton-Moutonterm is completely negligible, we found that our resultsdo not vary significantly.

TABLE I: Maximum attenuations due to photon/axion oscil-lations in the source obtained for di!erent sizes of the regionwhere the magnetic field is confined (“B region”) and di!er-ent lengths for the coherent domains. Only length domainssmaller than the size of the B region are possible. The Bfield strength used is 1.5 G (see Table II). The photon fluxintensity without ALPs was normalized to 1. In bold face, isthe attenuation given by our fiducial model.

B region (pc) Length domains (pc)3!10!4 3!10!3 0.03 0.3

0.3 0.84 0.67 0.67 0.750.03 0.98 0.84 0.77 -

3!10!3 0.99 0.98 - -

We summarize in Table II the parameters we have con-sidered in order to calculate the total photon/axion con-version in both the source (for the two benchmark AGNs)and in the IGM. As we already mentioned, these valuesrepresent our fiducial model.

The e!ect of existence of ALPs on the total photonflux coming from 3C 279 and from PKS 2155-304 (usingthe fiducial model presented in Table II) can be seen inFigure 4. We carried out the calculations for the twoEBL models cited above: Kneiske best-fit and Primack.The inferred critical energies for the mixing in the sourceare Ecrit = 4.6 eV for 3C 279 and Ecrit = 69 eV forPKS 2155-304, while for the mixing in the IGM we ob-tain Ecrit = 28.5 GeV. The photon attenuation due tophoton/axion mixing inside the source is 16% for 3C 279and 1% for PKS 2155-304, as can be seen above their re-spective critical energies in Figure 4. On the other hand,the photon attenuation due to photon/axion oscillationin the IGM is 30% for the distance of both sources, andit occurs at the same critical energy. The role of the EBLis negligible at this low energy (i.e. below !100 GeV),which means that the intensity curves for the two EBLmodels agree to this energy.

The situation changes above !100 GeV, where thephoton attenuation due to the EBL is noticeable. At

TABLE II: Parameters used to calculate the total pho-ton/axion conversion in both the source (for the two AGNsconsidered, 3c279 and PKS 2155-304) and in the IGM. Thevalues related to 3C 279 were obtained from Ref. [54], whilethose ones for PKS 2155-304 were obtained from Ref. [55].As for the IGM, ed,int was obtained from [56], and Bint waschosen to be well below the upper limit typically given in theliterature (see discussion in the text). This Table representsour fiducial model.

Parameter 3C 279 PKS 2155-304B (G) 1.5 0.1

Source ed (cm!3) 25 160parameters L domains (pc) 0.003 3 ! 10!4

B region (pc) 0.03 0.003z 0.536 0.117

Intergalactic ed,int (cm!3) 10!7 10!7

parameters Bint (nG) 0.1 0.1L domains (Mpc) 1 1

ALP M (GeV) 1.14 ! 1010 1.14 ! 1010

parameters ALP mass (eV) 10!10 10!10

this point, the results depend substantially on the sourcedistance and the EBL model used. A stronger photonattenuation is obtained for the Kneiske best-fit modelagainst the Primack EBL model, as expected. Becausethe strong photon attenuation due to the EBL, the ALPsthat later convert to photons imply a further enhance-ment of the expected photon flux. Therefore, as one cannotice from Figure 4, the existence of ALPs translatesinto a relatively small (!30%) intensity attenuation atlow energies and a large intensity enhancement (severalorders of magnitude, depending on the energy range, dis-tance of the source and chosen EBL model) at high en-ergies.

In order to quantitatively study the e!ect of ALPs onthe total photon intensity, we plot in Figure 5 the dif-ference between the predicted arriving photon intensitywithout including ALPs and that one obtained when in-cluding the photon/axion oscillations (called here the ax-ion boost factor). Again, this was done for our fiducialmodel (Table II) and for the two EBL models describedabove. The plots show di!erences in the axion boost fac-tors obtained for 3C 279 and PKS 2155-304 due mostlyto the redshift di!erence.

In the case of 3C 279, the axion boost is an atten-uation of about 16% below the critical energy (due tomixing inside the source). Above this critical energy andbelow 200-300 GeV, where the EBL attenuation is stillsmall, there is an extra attenuation of about 30% (mix-ing in the IGMF). Above 200-300 GeV the axion boostreaches very high values: at 1 TeV, a factor of !16 forthe Primack EBL model and !140 for the Kneiske best-fit model. As already discussed, the more attenuatingthe EBL model considered, the more relevant the e!ectof photon/axion oscillations in the IGMF, since any ALPto photon reconversion might substantially enhance the

0 500 1000 1500 2000D (Mpc)

1

Inte

nsity

0 500 1000 1500 2000D (Mpc)

0.1

1.0

Inte

nsity

The effect can be an ATTENUATION or an ENHANCEMENT of the photon flux, depending on distance, B field and EBL model considered.

The effect will be present in the gamma-ray band for axion masses ≈ 10-10 eV

B=1 nG; M11=0.7 GeV; D=2 Gpc, Ldom=1 Mpc; Primack EBL model 0 500 1000 1500 2000

D (Mpc)

0.001

0.010

0.100

1.000

Inte

nsity

50 GeV 500 GeV 2 TeV

EBL

photon intensity

axion intensity

•  We compute the photon/axion mixing in N coherent domains with equal size and random B orientation. •  The EBL introduces an additional absorption. The more attenuating the EBL, the more important the mixing in the final intensity.

5

Despite the low magnetic field B, the photon/axion os-cillation can take place due to the large distances, sincethe important quantity defining the probability for thisconversion is the product B ! s, as described by Eq (9).Assuming B "0.1 nG (see below), and M11 = 0.114 (co-incident with the upper limit reported by CAST [5]),then the e!ect can be observationally detectable (Ecrit <1 TeV) only if the ALP mass is ma < 6 ! 10!10 eV. Ifthe axion mass ma was larger than this value, then theconsequences of this oscillation could not be probed withthe current generation of IACTs, that observe up to fewtens of TeV [73]. In our fiducial model (see Table II) weused ma = 10!10 eV, which implies Ecrit = 28.5 GeV.

It is important to stress that at energies larger than 10GeV, and especially larger than 100 GeV, besides the os-cillation to ALPs, the photons should also be a!ected bythe di!use radiation from the Extragalactic BackgroundLight (EBL). The EBL introduces an attenuation in thephoton flux due to e!e+ pair production that comes fromthe interaction of the gamma-ray source photons with in-frared and optical-UV background photons for the ener-gies under consideration [33]. Therefore, it will be neces-sary to modify the above equations to properly accountfor the EBL in our calculations. These equations canbe found in Ref. [10], where the photon/axion mixing inthe IGMF was also studied, although for other purposesand a di!erent energy range. The same equations werealso used in Ref. [12] to study for the first time the pho-ton/axion mixing in the presence of IGMFs for the sameenergy range that we are considering in this work.

There is little information on the strength and mor-phology of the IGMFs. As for the morphology, severalauthors reported that space should be divided into sev-eral domains, each of them with a size for which the mag-netic field is coherent. Di!erent domains will have ran-domly changing directions of B field of about the samestrength [43, 44]. The IGMF strength is constrained tobe smaller than 1 nG [45], which is somewhat supported

by the estimates of "0.3-0.9 nG that can be inferred [46]from recent observations of the Pierre Auger Observa-tory [47]. On the other hand, there is controversy on thepossibility of generating such a strong magnetic field. De-tailed simulations yield IGMFs of the order of 0.01 nGso that they can later reproduce the measured B fields innearby galaxy clusters [48, 49]. Given this controversy,we decided to use a mid-value of 0.1nG in our fiducialmodel (Table II).

In our model, we assume that the photon beam prop-agates over N domains of a given length, the modulus ofthe magnetic field B roughly constant in each of them.We will take, however, randomly chosen orientations,which in practice will be also equivalent to a variation inthe strength of the component of the magnetic field in-volved in the photon/axion mixing. If the photon beamis propagating along the y axis, the oscillation will occurwith magnetic fields in the x and z directions since thepolarization of the photon can only be along those axis.Therefore, we can describe the beam state by the vector(!x, !z, a). The transfer equation will be [10]:

!

"

!x

!z

a

#

$ = eiEy%

T0 e!0y + T1 e!1y + T2 e!2y&

!

"

!x

!z

a

#

$

0

(12)

where:

"0 # $1

2 "",

"1 # $1

4""

'

1 +(

1 $ 4 #2

)

"2 # $1

4 ""

'

1 $(

1 $ 4 #2

)

(13)

T0 #

!

"

sin2$ $ cos$ sin$ 0$ cos$ sin$ cos2$ 0

0 0 0

#

$ T1 #

!

*

*

"

1+"

1!4 #2

2"

1!4 #2cos2$ 1+

"1!4 #2

2"

1!4 #2cos$ sin$ $ #

"1!4 #2

cos$1+

"1!4 #2

2"

1!4 #2cos$ sin$ 1+

"1!4 #2

2"

1!4 #2sin2$ $ #"

1!4 #2sin$

#"1!4 #2

cos$ #"1!4 #2

sin$ $ 1!"

1!4 #2

2"

1!4 #2

#

+

+

$

T2 #

!

*

*

"

$ 1!"

1!4 #2

2"

1!4 #2cos2$ $ 1!

"1!4 #2

2"

1!4 #2cos$ sin$ #"

1!4 #2cos$

$ 1!"

1!4 #2

2"

1!4 #2cos$ sin$ $ 1!

"1!4 #2

2"

1!4 #2sin2$ #"

1!4 #2sin$

$ #"

1!4 #2cos$ $ #

"1!4 #2

sin$ 1+"

1!4 #2

2"

1!4 #2

#

+

+

$

(14)

$ being the angle between the x-axis and B in each singledomain. # a dimensionless parameter equal to:

# #B ""

M% 0.11

,

B

10!9 G

- ,

1011 GeV

M

- ,

""

Mpc

-

(15)

that represents the number of photon/axion oscillationswithin the mean free path of the photon "" . Notice thatif there was no EBL, the quanta beam would be equipar-titioned between the ALP component and the two photonpolarizations after crossing a large number of domains.

5

Despite the low magnetic field B, the photon/axion os-cillation can take place due to the large distances, sincethe important quantity defining the probability for thisconversion is the product B ! s, as described by Eq (9).Assuming B "0.1 nG (see below), and M11 = 0.114 (co-incident with the upper limit reported by CAST [5]),then the e!ect can be observationally detectable (Ecrit <1 TeV) only if the ALP mass is ma < 6 ! 10!10 eV. Ifthe axion mass ma was larger than this value, then theconsequences of this oscillation could not be probed withthe current generation of IACTs, that observe up to fewtens of TeV [73]. In our fiducial model (see Table II) weused ma = 10!10 eV, which implies Ecrit = 28.5 GeV.

It is important to stress that at energies larger than 10GeV, and especially larger than 100 GeV, besides the os-cillation to ALPs, the photons should also be a!ected bythe di!use radiation from the Extragalactic BackgroundLight (EBL). The EBL introduces an attenuation in thephoton flux due to e!e+ pair production that comes fromthe interaction of the gamma-ray source photons with in-frared and optical-UV background photons for the ener-gies under consideration [33]. Therefore, it will be neces-sary to modify the above equations to properly accountfor the EBL in our calculations. These equations canbe found in Ref. [10], where the photon/axion mixing inthe IGMF was also studied, although for other purposesand a di!erent energy range. The same equations werealso used in Ref. [12] to study for the first time the pho-ton/axion mixing in the presence of IGMFs for the sameenergy range that we are considering in this work.

There is little information on the strength and mor-phology of the IGMFs. As for the morphology, severalauthors reported that space should be divided into sev-eral domains, each of them with a size for which the mag-netic field is coherent. Di!erent domains will have ran-domly changing directions of B field of about the samestrength [43, 44]. The IGMF strength is constrained tobe smaller than 1 nG [45], which is somewhat supported

by the estimates of "0.3-0.9 nG that can be inferred [46]from recent observations of the Pierre Auger Observa-tory [47]. On the other hand, there is controversy on thepossibility of generating such a strong magnetic field. De-tailed simulations yield IGMFs of the order of 0.01 nGso that they can later reproduce the measured B fields innearby galaxy clusters [48, 49]. Given this controversy,we decided to use a mid-value of 0.1nG in our fiducialmodel (Table II).

In our model, we assume that the photon beam prop-agates over N domains of a given length, the modulus ofthe magnetic field B roughly constant in each of them.We will take, however, randomly chosen orientations,which in practice will be also equivalent to a variation inthe strength of the component of the magnetic field in-volved in the photon/axion mixing. If the photon beamis propagating along the y axis, the oscillation will occurwith magnetic fields in the x and z directions since thepolarization of the photon can only be along those axis.Therefore, we can describe the beam state by the vector(!x, !z, a). The transfer equation will be [10]:

!

"

!x

!z

a

#

$ = eiEy%

T0 e!0y + T1 e!1y + T2 e!2y&

!

"

!x

!z

a

#

$

0

(12)

where:

"0 # $1

2 "",

"1 # $1

4""

'

1 +(

1 $ 4 #2

)

"2 # $1

4 ""

'

1 $(

1 $ 4 #2

)

(13)

T0 #

!

"

sin2$ $ cos$ sin$ 0$ cos$ sin$ cos2$ 0

0 0 0

#

$ T1 #

!

*

*

"

1+"

1!4 #2

2"

1!4 #2cos2$ 1+

"1!4 #2

2"

1!4 #2cos$ sin$ $ #

"1!4 #2

cos$1+

"1!4 #2

2"

1!4 #2cos$ sin$ 1+

"1!4 #2

2"

1!4 #2sin2$ $ #"

1!4 #2sin$

#"1!4 #2

cos$ #"1!4 #2

sin$ $ 1!"

1!4 #2

2"

1!4 #2

#

+

+

$

T2 #

!

*

*

"

$ 1!"

1!4 #2

2"

1!4 #2cos2$ $ 1!

"1!4 #2

2"

1!4 #2cos$ sin$ #"

1!4 #2cos$

$ 1!"

1!4 #2

2"

1!4 #2cos$ sin$ $ 1!

"1!4 #2

2"

1!4 #2sin2$ #"

1!4 #2sin$

$ #"

1!4 #2cos$ $ #

"1!4 #2

sin$ 1+"

1!4 #2

2"

1!4 #2

#

+

+

$

(14)

$ being the angle between the x-axis and B in each singledomain. # a dimensionless parameter equal to:

# #B ""

M% 0.11

,

B

10!9 G

- ,

1011 GeV

M

- ,

""

Mpc

-

(15)

that represents the number of photon/axion oscillationswithin the mean free path of the photon "" . Notice thatif there was no EBL, the quanta beam would be equipar-titioned between the ALP component and the two photonpolarizations after crossing a large number of domains.

5

Despite the low magnetic field B, the photon/axion os-cillation can take place due to the large distances, sincethe important quantity defining the probability for thisconversion is the product B ! s, as described by Eq (9).Assuming B "0.1 nG (see below), and M11 = 0.114 (co-incident with the upper limit reported by CAST [5]),then the e!ect can be observationally detectable (Ecrit <1 TeV) only if the ALP mass is ma < 6 ! 10!10 eV. Ifthe axion mass ma was larger than this value, then theconsequences of this oscillation could not be probed withthe current generation of IACTs, that observe up to fewtens of TeV [73]. In our fiducial model (see Table II) weused ma = 10!10 eV, which implies Ecrit = 28.5 GeV.

It is important to stress that at energies larger than 10GeV, and especially larger than 100 GeV, besides the os-cillation to ALPs, the photons should also be a!ected bythe di!use radiation from the Extragalactic BackgroundLight (EBL). The EBL introduces an attenuation in thephoton flux due to e!e+ pair production that comes fromthe interaction of the gamma-ray source photons with in-frared and optical-UV background photons for the ener-gies under consideration [33]. Therefore, it will be neces-sary to modify the above equations to properly accountfor the EBL in our calculations. These equations canbe found in Ref. [10], where the photon/axion mixing inthe IGMF was also studied, although for other purposesand a di!erent energy range. The same equations werealso used in Ref. [12] to study for the first time the pho-ton/axion mixing in the presence of IGMFs for the sameenergy range that we are considering in this work.

There is little information on the strength and mor-phology of the IGMFs. As for the morphology, severalauthors reported that space should be divided into sev-eral domains, each of them with a size for which the mag-netic field is coherent. Di!erent domains will have ran-domly changing directions of B field of about the samestrength [43, 44]. The IGMF strength is constrained tobe smaller than 1 nG [45], which is somewhat supported

by the estimates of "0.3-0.9 nG that can be inferred [46]from recent observations of the Pierre Auger Observa-tory [47]. On the other hand, there is controversy on thepossibility of generating such a strong magnetic field. De-tailed simulations yield IGMFs of the order of 0.01 nGso that they can later reproduce the measured B fields innearby galaxy clusters [48, 49]. Given this controversy,we decided to use a mid-value of 0.1nG in our fiducialmodel (Table II).

In our model, we assume that the photon beam prop-agates over N domains of a given length, the modulus ofthe magnetic field B roughly constant in each of them.We will take, however, randomly chosen orientations,which in practice will be also equivalent to a variation inthe strength of the component of the magnetic field in-volved in the photon/axion mixing. If the photon beamis propagating along the y axis, the oscillation will occurwith magnetic fields in the x and z directions since thepolarization of the photon can only be along those axis.Therefore, we can describe the beam state by the vector(!x, !z, a). The transfer equation will be [10]:

!

"

!x

!z

a

#

$ = eiEy%

T0 e!0y + T1 e!1y + T2 e!2y&

!

"

!x

!z

a

#

$

0

(12)

where:

"0 # $1

2 "",

"1 # $1

4""

'

1 +(

1 $ 4 #2

)

"2 # $1

4 ""

'

1 $(

1 $ 4 #2

)

(13)

T0 #

!

"

sin2$ $ cos$ sin$ 0$ cos$ sin$ cos2$ 0

0 0 0

#

$ T1 #

!

*

*

"

1+"

1!4 #2

2"

1!4 #2cos2$ 1+

"1!4 #2

2"

1!4 #2cos$ sin$ $ #

"1!4 #2

cos$1+

"1!4 #2

2"

1!4 #2cos$ sin$ 1+

"1!4 #2

2"

1!4 #2sin2$ $ #"

1!4 #2sin$

#"1!4 #2

cos$ #"1!4 #2

sin$ $ 1!"

1!4 #2

2"

1!4 #2

#

+

+

$

T2 #

!

*

*

"

$ 1!"

1!4 #2

2"

1!4 #2cos2$ $ 1!

"1!4 #2

2"

1!4 #2cos$ sin$ #"

1!4 #2cos$

$ 1!"

1!4 #2

2"

1!4 #2cos$ sin$ $ 1!

"1!4 #2

2"

1!4 #2sin2$ #"

1!4 #2sin$

$ #"

1!4 #2cos$ $ #

"1!4 #2

sin$ 1+"

1!4 #2

2"

1!4 #2

#

+

+

$

(14)

$ being the angle between the x-axis and B in each singledomain. # a dimensionless parameter equal to:

# #B ""

M% 0.11

,

B

10!9 G

- ,

1011 GeV

M

- ,

""

Mpc

-

(15)

that represents the number of photon/axion oscillationswithin the mean free path of the photon "" . Notice thatif there was no EBL, the quanta beam would be equipar-titioned between the ALP component and the two photonpolarizations after crossing a large number of domains.

5

Despite the low magnetic field B, the photon/axion os-cillation can take place due to the large distances, sincethe important quantity defining the probability for thisconversion is the product B ! s, as described by Eq (9).Assuming B "0.1 nG (see below), and M11 = 0.114 (co-incident with the upper limit reported by CAST [5]),then the e!ect can be observationally detectable (Ecrit <1 TeV) only if the ALP mass is ma < 6 ! 10!10 eV. Ifthe axion mass ma was larger than this value, then theconsequences of this oscillation could not be probed withthe current generation of IACTs, that observe up to fewtens of TeV [73]. In our fiducial model (see Table II) weused ma = 10!10 eV, which implies Ecrit = 28.5 GeV.

It is important to stress that at energies larger than 10GeV, and especially larger than 100 GeV, besides the os-cillation to ALPs, the photons should also be a!ected bythe di!use radiation from the Extragalactic BackgroundLight (EBL). The EBL introduces an attenuation in thephoton flux due to e!e+ pair production that comes fromthe interaction of the gamma-ray source photons with in-frared and optical-UV background photons for the ener-gies under consideration [33]. Therefore, it will be neces-sary to modify the above equations to properly accountfor the EBL in our calculations. These equations canbe found in Ref. [10], where the photon/axion mixing inthe IGMF was also studied, although for other purposesand a di!erent energy range. The same equations werealso used in Ref. [12] to study for the first time the pho-ton/axion mixing in the presence of IGMFs for the sameenergy range that we are considering in this work.

There is little information on the strength and mor-phology of the IGMFs. As for the morphology, severalauthors reported that space should be divided into sev-eral domains, each of them with a size for which the mag-netic field is coherent. Di!erent domains will have ran-domly changing directions of B field of about the samestrength [43, 44]. The IGMF strength is constrained tobe smaller than 1 nG [45], which is somewhat supported

by the estimates of "0.3-0.9 nG that can be inferred [46]from recent observations of the Pierre Auger Observa-tory [47]. On the other hand, there is controversy on thepossibility of generating such a strong magnetic field. De-tailed simulations yield IGMFs of the order of 0.01 nGso that they can later reproduce the measured B fields innearby galaxy clusters [48, 49]. Given this controversy,we decided to use a mid-value of 0.1nG in our fiducialmodel (Table II).

In our model, we assume that the photon beam prop-agates over N domains of a given length, the modulus ofthe magnetic field B roughly constant in each of them.We will take, however, randomly chosen orientations,which in practice will be also equivalent to a variation inthe strength of the component of the magnetic field in-volved in the photon/axion mixing. If the photon beamis propagating along the y axis, the oscillation will occurwith magnetic fields in the x and z directions since thepolarization of the photon can only be along those axis.Therefore, we can describe the beam state by the vector(!x, !z, a). The transfer equation will be [10]:

!

"

!x

!z

a

#

$ = eiEy%

T0 e!0y + T1 e!1y + T2 e!2y&

!

"

!x

!z

a

#

$

0

(12)

where:

"0 # $1

2 "",

"1 # $1

4""

'

1 +(

1 $ 4 #2

)

"2 # $1

4 ""

'

1 $(

1 $ 4 #2

)

(13)

T0 #

!

"

sin2$ $ cos$ sin$ 0$ cos$ sin$ cos2$ 0

0 0 0

#

$ T1 #

!

*

*

"

1+"

1!4 #2

2"

1!4 #2cos2$ 1+

"1!4 #2

2"

1!4 #2cos$ sin$ $ #

"1!4 #2

cos$1+

"1!4 #2

2"

1!4 #2cos$ sin$ 1+

"1!4 #2

2"

1!4 #2sin2$ $ #"

1!4 #2sin$

#"1!4 #2

cos$ #"1!4 #2

sin$ $ 1!"

1!4 #2

2"

1!4 #2

#

+

+

$

T2 #

!

*

*

"

$ 1!"

1!4 #2

2"

1!4 #2cos2$ $ 1!

"1!4 #2

2"

1!4 #2cos$ sin$ #"

1!4 #2cos$

$ 1!"

1!4 #2

2"

1!4 #2cos$ sin$ $ 1!

"1!4 #2

2"

1!4 #2sin2$ #"

1!4 #2sin$

$ #"

1!4 #2cos$ $ #

"1!4 #2

sin$ 1+"

1!4 #2

2"

1!4 #2

#

+

+

$

(14)

$ being the angle between the x-axis and B in each singledomain. # a dimensionless parameter equal to:

# #B ""

M% 0.11

,

B

10!9 G

- ,

1011 GeV

M

- ,

""

Mpc

-

(15)

that represents the number of photon/axion oscillationswithin the mean free path of the photon "" . Notice thatif there was no EBL, the quanta beam would be equipar-titioned between the ALP component and the two photonpolarizations after crossing a large number of domains.

No plasma term No CM term Only ΔB

8

that, as we increase the number of domains, the attenua-tion increases until the size of the domain is “too small”.At this point, the probability of photon/axion conversionis almost zero for the single domains, which reduces theoverall photon/axion conversion.

TABLE I: Maximum attenuations due to photon/axion oscil-lations in the source obtained for di!erent sizes of the regionwhere the magnetic field is confined (“B region”) and di!er-ent lengths for the coherent domains. Only length domainssmaller than the size of the B region are possible. The Bfield strength used is 1.5 G (see Table II). The photon fluxintensity without ALPs was normalized to 1. In bold face, isthe attenuation given by our fiducial model.

B region (pc) Length domains (pc)3!10!4 3!10!3 0.03 0.3

0.3 0.84 0.67 0.67 0.750.03 0.98 0.84 0.77 -

3!10!3 0.99 0.98 - -

We summarize in Table II the parameters we have con-sidered in order to calculate the total photon/axion con-version in both the source (for the two benchmark AGNs)and in the IGM. As we already mentioned, these valuesrepresent our fiducial model.

TABLE II: Parameters used to calculate the total pho-ton/axion conversion in both the source (for the two AGNsconsidered, 3c279 and PKS 2155-304) and in the IGM. Thevalues related to 3C 279 were obtained from Ref. [54], whilethose ones for PKS 2155-304 were obtained from Ref. [55].As for the IGM, ed,int was obtained from [56], and Bint waschosen to be well below the upper limit typically given in theliterature (see discussion in the text). This Table representsour fiducial model.

Parameter 3C 279 PKS 2155-304B (G) 1.5 0.1

Source ed (cm!3) 25 160parameters L domains (pc) 0.003 3 ! 10!4

B region (pc) 0.03 0.003z 0.536 0.117

Intergalactic ed,int (cm!3) 10!7 10!7

parameters Bint (nG) 0.1 0.1L domains (Mpc) 1 1

ALP M (GeV) 1.14 ! 1010 1.14 ! 1010

parameters ALP mass (eV) 10!10 10!10

The e!ect of existence of ALPs on the total photonflux coming from 3C 279 and from PKS 2155-304 (usingthe fiducial model presented in Table II) can be seen inFigure 4. We carried out the calculations for the twoEBL models cited above: Kneiske best-fit and Primack.The inferred critical energies for the mixing in the sourceare Ecrit = 4.6 eV for 3C 279 and Ecrit = 69 eV forPKS 2155-304, while for the mixing in the IGM we ob-tain Ecrit = 28.5 GeV. The photon attenuation due to

0.1 1.0 10.0 100.0 1000.0E (GeV)

10-5

10-4

10-3

10-2

10-1

100

Inte

nsity

0.1 1.0 10.0 100.0 1000.0 10000.0E (GeV)

0.01

0.10

1.00

Inte

nsity

FIG. 4: E!ect of photon/axion conversions both inside thesource and in the IGM on the total photon flux coming from3C 279 (z=0.536) and PKS 2155-304 (z=0.117) for two EBLmodels: Kneiske best-fit (dashed line) and Primack (solidline). The expected photon flux without including ALPs isalso shown for comparison (dotted line for Kneiske best-fitand dot-dashed line for Primack).

photon/axion mixing inside the source is 16% for 3C 279and 1% for PKS 2155-304, as can be seen above their re-spective critical energies in Figure 4. On the other hand,the photon attenuation due to photon/axion oscillationin the IGM is 30% for the distance of both sources, andit occurs at the same critical energy. The role of the EBLis negligible at this low energy (i.e. below !100 GeV),which means that the intensity curves for the two EBLmodels agree to this energy.

The situation changes above !100 GeV, where thephoton attenuation due to the EBL is noticeable. Atthis point, the results depend substantially on the sourcedistance and the EBL model used. A stronger photonattenuation is obtained for the Kneiske best-fit modelagainst the Primack EBL model, as expected. Becausethe strong photon attenuation due to the EBL, the ALPsthat later convert to photons imply a further enhance-

CAST limit

ultralight axions

Ecrit,interg = 28.5 GeV (both)

Ecrit,source(3C) = 4.6 eV

Ecrit,source(PKS) = 69 eV

0.1 1.0 10.0 100.0 1000.0E (GeV)

10-6

10-5

10-4

10-3

10-2

10-1

100

Inte

nsity

0.1 1.0 10.0 100.0 1000.0 10000.0E (GeV)

0.01

0.10

1.00

Inte

nsity

•  The critical energy varies accordingly. •  For distant sources, weaker intergalactic B fields could lead to higher axion boosts.

0.1 1.0 10.0 100.0 1000.0E (GeV)

1

10

100

1000

Axio

n b

oost fa

cto

r

0.1 1.0 10.0 100.0 1000.0 10000.0E (GeV)

1

Axio

n b

oost fa

cto

r

0.1 1.0 10.0 100.0 1000.0E (GeV)

1

10

100

1000

Axio

n b

oost fa

cto

r

0.1 1.0 10.0 100.0 1000.0 10000.0E (GeV)

1

Axio

n b

oost fa

cto

r

  If we accurately knew the intrinsic spectrum of the sources and/or the density of the EBL, we should be able to observationally detect axion signatures or to exclude some portions of the parameter space.

  We lack this knowledge... Detection challenging but still possible!

  Before going to axions:   Observe several AGNs located at different redshifts, as well as the same AGN

undergoing different flaring states, from radio to TeV.   Try to describe the observational data with “conventional” theoretical models

for the AGN emission and for the EBL.

  If these “conventional” models for the source emission and EBL fail (important residuals for the best-fit model), then the axion scenario should be explored.

0.1 1.0 10.0 100.0 1000.0E (GeV)

1

10

100

1000

Axio

n b

oost fa

cto

r

1.  Search of e.g. the systematic drop at the SAME energy

2.  Stacking analysis using different sources and periods

3.  In parallel, simulations 4.  Detection? If not, constrains on

the parameter space

The impact of changing B, M, ma

f (B, M, ma)

Critical energy and flux attenuation or enhancement varies accordingly

f(B, M)

€

Ecrit ∝ma2 ⋅ M11

BG

€

mixing∝BG ⋅ M11

B

M

exclusion line

(axion mass fixed)