Method development and validation for future population studies 

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Kinematic interactions and threshold effects

A MDR can lead to a modification of kinematic interactions between particles. In particular, it can change the energy threshold at which a given physical process is allowed to happen. As mentioned before, the extent of these modifications depends on whether the Lorentz invariance is properly broken or simply deformed. LIV and DSR formalisms will therefore steer different phenomenological predictions. Processes that are forbidden by special relativity – i.e. for which the energy threshold is not defined – are also forbidden in DSR approaches. In contrast, LIV approaches can steer a new energy threshold thanks to the existence of a preferred frame of reference. Newly allowed processes notably include vacuum Cherenkov radiation (e± ! e± ), photon decay in vacuum ( !e+e−) or photon splitting ( !3 ). As a consequence, Lorentz symmetry breaking formalisms are of particular interest for this type of phenomenology.
Amongst the processes allowed with both approaches, a notable effect is the modification of the pair production ( ! e+e−) energy threshold. The LIV-modified (first order) energy threshold for pair production from a head-on collision with both leptons at rest is given by: th = (mec2)2 E  » 1± 1 4 E ELIV n+2# .

Astrophysical probes

As mentioned above, the quantum gravity and related phenomenological effects are expected to be small. It is of interest to choose sources of photons which help maximise them. The measurable quantity associated to time-of-flight studies is given in Equation 1.8, which can be maximised for sources emitting photons:
• over large distances, ideally cosmological;
• distributed over a high-valued and large energy band, to maximise the term En. Moreover, the measurement of time delays can only be performed over photon emissions that show some variability. The higher the variability, the better the precision on that measurement. Transient or periodic astrophysical sources emitting VHE photons (E > 1 GeV) are therefore of particular interest for LIV time-of-flight studies. Three candidate sources fit this description: gamma-ray bursts, pulsars and flaring active galactic nuclei. This section gives a brief overview of these objects. A review on the state of art for LIV studies performed with such sources can be found in Chapter 6.

Flaring active galactic nuclei

Active galactic nuclei (AGN) are the core of distant galaxies hosting a central supermassive black hole fed by a surrounding accretion disc mainly made up of gas and dust. About 10% of the observed galaxies possess such a system which is referred as the central engine. It is further surrounded by a larger torus of gas and dust, along with the so-called broad and narrow line region (BLR and NLR) made up of ionised gas. The accretion disc which emits light in the optical/ UV spectrum can be obscured by the surrounding torus, which emits in the IR spectrum, depending on the viewing angle. The BLR and NLR reprocess the disc emission into spectral emission lines. The central engine emission is intense enough (typically between 1051 and 1054 erg.s−−2) to overshadow the light emitted by the galaxy itself. In roughly 10% of AGNs, a fraction of the accreted gas infalling into the black hole is ejected in the form of two luminous extended jets of relativistic and magnetised plasma. Such AGNs are labeled as radio loud, as opposed to radio quiet AGNs where the jet, if any, would emit very low fluxes which cannot be observed. The core where the jets originate can be detected in the radio spectrum. In addition, the jets end into so-called radio-lobes (emitting in the radio spectrum) created from the interaction between the jet plasma and the interstellar medium. The jet itself on the other hand generates a beamed emission covering a large fraction of the electromagnetic spectrum. The jets structure and underlying processes at the origin of this beamed emission will be covered in extended details in Chapter 2.

LIV vs source intrinsic effects

Probing LIV by studying the time-of-flight of photons emitted by astrophysical sources relies on the crucial assumption that there is no correlation between photons energy and the times at which they are emitted. However, there is no absolute certainty on the sources emission mechanisms and nothing guarantees the simultaneity of emission for photons with different energies. Quite on the contrary, there is now rising evidence that source physical processes can induce energy-dependent time delays between photons at the moment of emission. These intrinsic time delays will be added to LIV-induced ones leading to a total measured time delay decomposing as follows:
tn(total) = tn(LIV)+tn(source)(z+1) (1.14) or equivalently.
n(total) = n(LIV)+n(source)(z+1). (1.15).
The LIV term incorporates a correction for the cosmological effects induced by the source distance or redshift. While LIV-induced time delays are a consequence of a propagation effect which cumulates with the distance travelled by the test particles, intrinsic time delays are generated at the source and stay constant whichever distance the test particles travel (see Figure 1.6).

Superluminal motion and relativistic beaming

From consecutive observation campaigns targeting specific blazars, it is possible to track and monitor the evolution of knots within a jet. Over the course of several months or a few years, one can witness the birth and disappearance of one or several knots, travelling several kpc from the jet base (nuclear jet) to the lobes2. The knots usually appear to move in the nuclear jet with apparent superluminal velocity, which arise from relativstic and geometrical effects induced by the jet orientation as illustrated in Figure 2.1.
Consider an observer positionned such that its line of sight makes a small angle with respect to the jet orientation. Let a knot move along the direction at speed v and emit two photons at coordinates (t1,d1) and (t2,d2). Within the time window t=t2−t1, the knot travels a distance d = d2 −d1 = vt along the direction while the two photons travel along the line of sight.


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.

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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