Preparation for population studies with VHE data 

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Blazar sequence: BL Lac vs. FSRQ

From a statistical study based on a large selection of blazars (126 in total), Fossati et al. [40] found out the observed radio luminosity and the synchrotron (low energy bump) peak within blazars SED are anti-correlated. This study brought to existence the « blazar sequence » shown in Figure 2.2 (right) which unified the two classes of blazars: from bright and low frequency flat spectrum radio quasar (FSRQ) to dim and high frequency BL Lacertae (BL Lac). A more detailed classification of the blazar sequence is defined with respect to the synchrotron peak frequency peak,s:
• FSRQ with peak,s 1012−1013 Hz.
• LBL (low frequency BL Lac) with peak,s 1013−1014 Hz.
• IBL (intermediate frequency BL Lac) with peak,s 1015−1016 Hz.
• HBL (high frequency BL Lac) with peak,s 1017−1018 Hz.
Later on, Ghisellini et al. [41] performed a revision of this sequence with a total of 747 blazars, also shown in Figure 2.2 (left). In this re-edition, the gamma-ray luminosity is preferred to the radio one for the classification. This choice is motivated by current understanding of physical processes at the origin of the high energy emission. Although there is no general consensus, it is commonly believed the especially large flux observed in FSRQs at high energy is caused by the presence of important external photons fields. They thus introduce a significant inverse Compton emission which boosts the SED high energy bump. Alternatively, SSC processes are usually enough to model BL Lacs emission and make up for a standard picture.
Either way, although HBLs emit the most energetic photons and are therefore the most interesting types of blazar for LIV studies, they tend to be observed at the low redshifts due to their lower luminosity. Other types of blazars are therefore still considered as strong candidates as they offer an acceptable compromise between energy and distance.

Physical processes in relativistic jets

This section will be dedicated to the blazars emission mechanisms, focusing on one of the most commonly accepted scenario. As discussed above blazars SED are dominated by two non- Figure 2.2: Sequences of blazar SEDs. The original sequence (right) provides a classification based on the radio luminosity (low energy synchrotron bump) while the revised version (left) is based on the gamma-ray luminosity (high energy inverse Compton bump). Credit: [41]. thermal components, the low energy one being attributed to synchrotron processes while the high energy one is believed to arise from inverse Compton processes involving leptons found in the knot plasma. Either way, the two components can only be explained when particles are found to be highly energetic, which indicates the presence of efficient and powerful acceleration mechanisms. Only the leptonic processes will be discussed in this work, providing with a brief overview and essential concepts and formulae that will be used later on. We invite the reader to refer to [42, 43] for a broader and more detailed review of current knowledge on blazar emission mechanisms (including hadronic and lepto-hadronic models).

First order Fermi acceleration: shock mechanism

A second and more efficient scenario enables particle acceleration through shock waves. In this picture, the magnetic mirror is played by the inhomogeneities in the shocked (upstream) and inert (downstream) media as illustrated in Figure 2.3 (right). Once again, this process is better grasped with the help of changes of reference frame. Whether we are placing ourselves in the upstream or downstream reference frame (i.e. where the corresponding medium is at rest), the moving medium will have a head-on motion. A particle coming from the downstream (resp upstream) medium at rest and passing through the shock front will then see the upstream (resp. downstream) medium as a magnetic cloud coming towards it. From the second order Fermi mechanism, this head-on motion induces an energy gain for the particle. The diffusion effects provoked by the magnetic turbulences (inhomogeneities) in the medium decreases the probability of particle escape. Changing the reference to the rest frame of the upstream (resp downstream) medium, the particle is likely to be once again reflected on the other medium, which is also seen with a head-on motion. As a consequence, as long as the particle cannot escape the shock system, it will undergo multiple reflections. The critical difference with the second order mechanism is that all the collisions are now head-on. For each round-trip, the particle will gain an average energy < E E >/ r−1 r vshockc / v c shock, (2.10).
with r the compression factor between the upstream and downstream media, v =vup−vdown > 0 the relative velocity between the upstream and downstream media, and vshock = shockc the shock front velocity. In the knotted relativistic jet picture, the shock front is for instance assimilated to the transition between a knot and the large scale jet. Although this process also requires « pre-accelerated » particles, it provides faster acceleration than the second oder Fermi mechanism.

External inverse Compton

Other than the synchrotron field, inverse Compton processes can occur in blazars from the interaction between leptons and a photon fields external to the jet or the AGN itself. The additional number of photons brought by external fields will contribute to generate very high energy photons and boost the flux in the SED inverse Compton bump. External photon fields can originate from:
• the accretion disc radiating in the optical to UV spectrum.
• the BLR as it scatters the accretion disc radiation in the UV spectrum with a possible extension to X-rays.
• the dust torus radiating in the IR and radio spectrum.
• the large scale jet radiating in the radio spectrum from synchrotron processes possibly up to X-rays.
• the host galaxy mostly in the optical spectrum.
• the EBL mostly in the microwave to optical spectrum.
Although the large scale jet emission would take the form of a power law, external photon fields can generally be modeled with a blackbody emission centered on a specific temperature. A fraction of the external photons – those that are directed or reflected head-on towards the knot – should be blue-shifted in the knot’s rest frame. As a consequence, the external photons can reach higher energies. In particular, photons emitted by the accretion disc or scattered by the BLR can reach keV energies (X-rays) and therefore be boosted up to TeV energies through inverse Compton interactions with the knot’s leptons.

VHE gamma-ray astronomy

Typically ranging from a few hundred keV to hundreds of TeV, gamma-rays are the most energetic form of light. Most photons are produced through thermal processes. The hotter the source, the higher the frequency of the radiated light. However, objects can hardly get hot enough to produce highly energetic gamma-rays1 such that they are mostly generated through non-thermal mechanisms including:
• radioactive decay typically procuding E MeV gamma-rays.
• electron-positron annihilation (e++e− −! + ) creating two photons of minimum energy E = 511 keV in the annihilation rest frame.
• pion decay (0 −! + ) with an energy distribution peaking at E 70 MeV.
• synchrotron radiation producing gamma-rays with energies that depend on the energy of the charged particle moving in the magnetic field, and the magnetic field strength (c.f. Section • inverse Compton scattering producing gamma-rays which energies depend on the initial photon and lepton energies (c.f. Section
• bremsstrahlung radiation by decelerating charged particles which produce gamma-rays with energies that depend on the degree of deceleration.
Gamma-ray astronomy focuses on the most energetic photons with the goal to uncover the nature of objects, and phenomena at their origin. As they propagate through the cosmos, these gamma-rays may interact with non relativistic matter via three interaction channels:
• photoelectric effect (ionisation of an atom) – dominant up to E 50 keV.
• Compton scattering ( +e− −!e−+ ) – dominant in the range E 0.1−5 MeV.
• pair production ( + −!e−+e+) – dominant above E 5 MeV.
The fraction of emitted gamma-rays that survived their trip en route to Earth are later massively screened by the Earth atmosphere. As a consequence, direct gamma-ray detection with groundbased instruments is impossible. There are two ways to work around this issue: either bypass the atmosphere with space-based direct detection, or ground-based indirect detection with byproducts of gamma-ray interactions with the atmosphere.

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Direct detection: satellite-embarked instruments

Direct detection while bypassing the atmosphere requires to send a detector in space, embarked on board a satellite. Due to financial and technical reasons, the weight and size of such detectors is greatly limited by rocket standards, with typical scales of 1 m3 and a few hundred kilograms. The detectors’ collection area and internal structure are therefore limited in size and weight, greatly restricting their potential. However, the field of view (FoV) of such instruments is generally very large (typically a few steradians2) such that a large fraction of the sky can be monitored at each orbit.
The gamma–ray flux emerging from the ensemble of astrophysical gamma-ray sources is characterised by a decreasing power law energy spectrum. As a consequence, this flux gets dimmer as photon energy increases. Therefore, a collection area close to 1 m2 in size has a very small probability to catch photons with energy exceeding a few hundreds GeV.
The detection principle is a destructive one where gamma-rays go through a limited number of heavy material layers in order to provoke interactions, and die out by depositing all or part of their energy in the detector. The penetration depth – distance at which the photon starts to interact with the medium – increases with photon energy such that very energetic gamma-rays can cross the detector through and through, and go undetected. This major limitation inherent to gamma-ray space-telescopes defines a differenciation between high energy (HE) and very high energy (VHE) photons. Direct detections are limited to the detection of HE photons only.

Table of contents :

I Scientific Framework 
1 Quantum gravity and departures from Lorentz invariance tested with photons from astrophysical sources 
1.1 Quantum gravity built on effective field theories
1.1.1 String theories
1.1.2 Loop quantum gravity
1.2 Departures from Lorentz invariance
1.2.1 Breaking Lorentz symmetry
1.2.2 Deforming Lorentz symmetry
1.3 Phenomenology
1.3.1 Time delays
1.3.2 Kinematic interactions and threshold effects
1.4 Astrophysical probes
1.4.1 Gamma-ray bursts
1.4.2 Pulsars
1.4.3 Flaring active galactic nuclei
1.5 LIV vs source intrinsic effects
2 Blazars 
2.1 Characteristics
2.1.1 Superluminal motion and relativistic beaming
2.1.2 Blazar emission
2.1.3 Blazar sequence: BL Lac vs. FSRQ
2.2 Physical processes in relativistic jets
2.2.1 Acceleration
2.2.2 Leptonic radiation processes Synchrotron Inverse Compton External inverse Compton
3 Gamma-ray astronomy with imaging atmospheric Cherenkov telescopes 
3.1 VHE gamma-ray astronomy
3.1.1 Direct detection: satellite-embarked instruments
3.1.2 Indirect detection: ground-based imaging atmospheric Cherenkov telescopes Extensive air showers Cherenkov radiation Imaging atmospheric Cherenkov telescopes
3.2 H.E.S.S.: the high energy stereoscopic system
3.2.1 Overview of the H.E.S.S. array
3.2.2 Structure and optical system
3.2.3 Data acquisition Trigger system Calibration
3.2.4 Analysis Event reconstruction Signal extraction Spectral and temporal analysis
3.3 CTA: the Cherenkov telescope array
II Modelisation of blazar emission and interpretation of intrinsic delays 
4 Intrinsic time delays in blazars 
4.1 Time-dependent modeling of blazar
4.1.1 Generating a flare Homogeneous one-zone SSC model Extended scenario Domains of parameters
4.1.2 Generating astrophysical observables: the AGNES simulator Lepton spectrum Energy spectrum Light curves and intrinsic time delays
4.2 Properties of intrinsic time delays
4.2.1 Regimes in the SSC scenario
4.2.2 Impact of model parameters on intrinsic delays SSC scenario Extended scenario
4.2.3 Observability of non-zero intrinsic delays
5 Discrimination between intrinsic and LIV-induced time delays 
5.1 Multi-frequency study: gamma-rays vs. X-rays
5.1.1 Euclidian distance study Building a powerful tool Dependency on model parameters
5.1.2 Hysteresis study: a sensitive tool
5.2 LIV injection
5.2.1 Impact on delays and euclidian distances
5.2.2 Impact on hysteresis
5.3 LIV-modified EBL absorption: extreme scenarii
5.4 Observational perspectives
III Preparation for population studies with VHE data 
6 Searches for Lorentz invariance violation signatures with time of flight studies 
6.1 Analysis methods
6.1.1 Single data set transformation
6.1.2 Comparison between data subsets
6.1.3 Strengths and limitations
6.2 State of the art
6.2.1 Up-to-date limits
6.2.2 Future prospects
7 Method development and validation for future population studies 
7.1 The maximum likelihood method
7.1.1 Building a probability density function
7.1.2 The special case of pulsars
7.1.3 Background treatment
7.1.4 IRF treatment Acceptance Energy resolution Multi-era treatment Optimising the computational time
7.1.5 Combination
7.1.6 Confidence intervals
7.2 Lag distance models
7.3 Selected sources and simulation parameters
7.4 Tests and calibration
7.4.1 At n = 0: tabulation settings
7.4.2 For n 6= 0: calibration
7.5 Statistical and systematic uncertainties
7.6 Results and discussion on the QG energy scale
7.6.1 Individual sources and combinations
7.6.2 Subluminal vs. superluminal
7.6.3 Systematic uncertainties
7.6.4 Lag-distance models
7.6.5 Comparison with older published limits (subluminal)
7.7 Summary and perspective
A Solution of the time dependent SSC differential equation 
B Convergence and calibration plots produced with the LIVelihood software 


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