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The TeV Transparency Problem
Sandesh Kalantre
Contents
1 Introduction
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2 Modelling the EBL
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3 Calculation of the Optical Depth
3.1 Cross-Section for Pair Production . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Expression for Optical Depth τ . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 τ for the EBL model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Conclusion
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Introduction
Interaction of light from VHE (E >100 GeV) sources with low-energy photons in the intergalactic space by the pair production process γ + γ → e+ e− leads to a decrease in the
intensity of VHE sources. This intergalactic photon density is generally termed as EBL
(Extra-Galactic Background Light). Even though this photon density is not well known,
lower limits can be set by the resolved emission from galaxy counts. The pair-production
process, γ + γ → e+ e− is a well-understood process in the Standard Model and hence the
cross-sections can be calculated analytically. One would expect that one can correlate the decrease in VHE spectrum intensity to the observed quantity of EBL. However, as it turns out
there is some mechanism which suppresses the pair-production process leading to decreased
absorption in the TeV spectra. This is the TeV Transparency Problem. [1]
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Modelling the EBL
Energetic photons propagating through the intergalactic space undergo pair production with
low energy photons of the extragalactic background light (EBL) in the optical (mostly starlight) and infrared (re-emitted light from warm and cold dust). The EBL is defined as
the emission in 0.1 to 1000 µm range. It comprises the integrated light from resolved and
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unresolved extragalactic sources, and the light from any truly diffuse background, excluding
the cosmic microwave background (CMB).
The EBL is generally modelled by estimating the direct starlight from stars by using
the observed density of stars as well as taking into account the initial mass function (IMF),
star-formation rates and dust-extinction. The stars are modelled as perfect black bodies and
spectra over the range of masses from 0.1Msun to 100Msun are summed over based on the
IMF. [2].
The starlight that is absorbed by dust is reradiated in the infrared. There are generally
thought to be three major dust components in the interstellar medium
1. Large grain component that absorbs optical light and re radiates in the far-IR,
found in and around star-forming regions
2. Small grain component that absorbs the far-UV and re radiates in the near-IR,
located throughout the disk of spiral galaxies and is responsible for most of the observed
dust radiation
3. Polycyclic aromatic hydrocarbons (PAHs) which emit as broad emission lines and
are not generally in thermodynamic equilibrium with their environment
It is assumed the dust emits as a combination of three blackbodies,representing three
types of dust: a 40K blackbody representing warm, large dust grains, a 70 K blackbody
representing hot, small dust grains, and a 450 K blackbody representing PAHs.
Based on such a model, the EBL spectrum looks as follows:
Figure 1: EBL Intensity at z = 0.0
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3.1
Calculation of the Optical Depth
Cross-Section for Pair Production
Figure 2: Pair Production [3]
The interaction between two photons with energies Eγ and , will lead to the creation of a
particle anti-particle pair when the total γ−ray energy in the center of momentum of the
system exceeds the rest frame energy of the two particles. [3]. The threshold energy for e+ e−
pair is given by
th =
2(me c2 )2
Eγ (1 − µ)
where µ = cos θ.
The cross-section for pair-production can be calculated to be
3σT
1+β
2
2
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σ(Eγ , , µ) =
(1 − β ) 2β(β − 2) + (3 − β ) ln
16
1−β
p
where β = 1 − th
.
3.2
Expression for Optical Depth τ
Once we have the EBL intensity as a function of the wavelength and the redshift, it is a
straightforward calculation for the optical depth.
The optical depth of a γ-ray photon at an observed energy Eγ , emitted by a source at
redshift z due to this process is given by:
Z z
Z −1
Z
1−µ ∞
0 dl
τ (Eγ , z) =
dz 0
dn (, z 0 )(1 + z 0 )3 σ(β 0 , z 0 )
0
dz
2
0
1
th
where n (, z) = dnd(,z) is the specific comoving number density of background photons with
energy at redshift z, and the (1 + z)3 term represents its conversion to a proper number
density.
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Explanation of the integral
1. The last integral integrates the cross section over all possible energies from threshold
2 2
ec )
to ∞, where the 1 + z takes care of the fact that the observed
energy 0th = Eγ2(m
(1−µ)(1+z)
Eγ had higher energy at redshift z.
2. The 1−µ
integral takes care of the of all the angles between the γ-ray photon and EBL
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photon somehow.
3. The third integral integrates over the path length.
3.3
τ for the EBL model
Figure 3: τ for various z [3]
We observe that,
1. The behaviour of τ can be modelled to three piecewise linear equations. This is probably because of the the presence of three peaks in the EBL spectrum.
2. For energies above 10 TeV, τ 1, and so the VHE spectrum quickly drops to zero.
3. The cutoff(the energy value above which the VHE spectrum quickly drops to zero) is
higher for lower z.
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Conclusion
The TeV Transparency problem can be solved by assuming the existence of a particle called
axion which the photon converts to in the presence of a magnetic field.
Other probable causes and solutions for TeV transparency problem include
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1. The VHE spectra themselves have a variation in the spectral index over redshift. This
would imply that high redshift spectra should have a lower spectral index (should be
flatter).
2. We are overestimating the EBL. This is unlikely to be true since lower bounds can be
derived from direct observations.
3. Violation of Lorentz invariance: Pair production is suppressed at higher energies because of a different high energy theory.
References
[1] Manuel Meyer Dieter Horns. “Indications for a pair-production anomaly from the propagation of VHE gamma-rays”. In: (). arXiv: arXiv:1201.4711 [astro-ph].
[2] Justin D. Finke, Soebur Razzaque, and Charles D. Dermer. “Modeling the Extragalactic
Background Light from Stars and Dust”. In: The Astrophysical Journal 712.1 (2010),
p. 238. url: http://stacks.iop.org/0004-637X/712/i=1/a=238.
[3] Frank Krennrich Eli Dwek. “The Extragalactic Background Light and the Gamma-ray
Opacity of the Universe”. In: (). arXiv: arXiv:1209.4661 [astro-ph].
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