  Gamma-ray astronomy

Gamma rays constitute a window to the Universe at the highest energies. Deviations from exact Lorentz invariance in the photon sector can appear in many different and independent forms. In the same way as high-energy cosmic rays can be used to search for conventionally forbiden reactions, a generic extension of Maxwell electrodynamics can produce a rich phenomenology that becomes experimentally accessible when studying high-energy photons. Of particular interest is the study of operators of arbitrary dimension in the theory, which can be written in the form

$\mathcal{L} = -\dfrac{1}{4} F_{\mu\nu} F^{\mu\nu} -\dfrac{1}{4}\,F^{\mu\nu} (\color{red}{\hat{k}_F})_{\mu\nu\lambda\rho} F^{\lambda\rho} + \dfrac{1}{2}\epsilon^{\mu\nu\lambda\rho}\,A_\nu (\color{red}{\hat{k}_{AF}})_\mu F_{\lambda\rho},$

where the coefficients $(\color{red}{\hat{k}_F})_{\mu\nu\lambda\rho}$ and $(\color{red}{\hat{k}_{AF}})_\mu$ control CPT-even and CPT-odd deviations from exact Lorentz invariance, respectively. The hats indicate that in general these coefficients correspond to differential operators, where a nonzero number of derivatives will introduce novel effects for high-energy photons.
In general, the effects of these two types of coefficients can be classified according to their spherical properties. In fact, an spherical decomposition can be used to parametrize all possible Lorentz-violating effects using four types of coefficients denoted $\color{blue}{k^{(d)}_{(X)jm}}$ and $\color{red}{c^{(d)}_{(I)jm}}$, where $d$ denotes the dimension of the operator in the theory, the index $X=V,E,B$ encodes the parity properties, and $jm$ are angular momentum indices that characterize the spherical transformation properties. This spherical decomposition is very useful for studies of events distributed in the sky.

Birenfringence
The coefficients $\color{blue}{k^{(d)}_{(X)jm}}$ make the two polarization modes of the photon move at different speed. This effect produces the rotation of the polarization vector, which is enhanced by the photon energy and the propagation distance. For the study of this effect polarimetry studies appear as a powerful method to test Lorentz invariance. These birenfringence effects can be studied by space telescopes via polarimetry measurements.

Dispersion
For a theory with operator of dimension $d>4$, all the coefficients $\color{blue}{k^{(d)}_{(X)jm}}$ and $\color{red}{c^{(d)}_{(I)jm}}$ produce an energy-dependent photon speed, which leads to dispersive effects. Of particular interest are the effects of the coefficients $\color{red}{c^{(d)}_{(I)jm}}$ because the others can be studied with high precision using polarimetry. The experimental signature of $\color{red}{c^{(d)}_{(I)jm}}$ is the wave dispersion: a burst of photons of different energies (indicated by different colors) that are emitted at the same time by an astrophysical source spreads as the photons propagate because they move at different velocities. This wave-packet spread results in an arrival time delay $\Delta t$ between high-energy $\color{blue}{E_1}$ and low-energy $\color{red}{E_2}$ photons from the source to the telescope proportional to $\color{blue}{E_1^{d-4}}-\color{red}{E_2^{d-4}}$. Since the observable signature is a time delay between photons of different frequencies, the use of transient photon emissions offers the advantage of minimizing systematic effects.

The long propagation path of photons from astrophysical sources makes them sensitive probes of Lorentz invariance because a small Lorentz-violating effect is enhanced by the cosmic distances these photons have travelled. Additionally, the sensitivity to the possible effects arising from Lorentz violation increases with the photon energy. For these reasons, gamma-ray photons are very sensitive to possible deviations from exact Lorentz symmetry and gamma-ray telescopes become powerful tools for the study of fundamental physics.

Once the energy of the high- and low-energy photons is measured and the distance from the source, characterized by its redshift $z$, is identified, the different coefficients for Lorentz violation can be determined for a given mass dimension $d$
by measuring the time spread $\Delta t$ of the burst of photons in the form
$$\Delta t =\big(\color{blue}{E_1^{d-4}}-\color{red}{E_2^{d-4}}\big) \int_0^z \frac{(1+z')^{d-4}}{H(z')}\,dz' \sum_{jm} Y_{jm}(\hat{n}) \color{red}{c^{(d)}_{(I)jm}}$$
where $H(z)$ is the Hubble parameter that accounts for the cosmological distance travelled by these photons and $Y_{jm}(\hat{n})$ are the well-known spherical harmonics that parametrize the orientation dependence (for $j\neq 0$) in terms of the location of the source $\hat{n}$ in the sky.

My research interests regarding gamma rays find direct applications for studies of fundamental physics in ground-based gamma-ray observatories. Of particular interest is the future Cherenkov Telescope Array (CTA), which involves as one of the scientific goals the study of Lorentz invariance violation.

Links:
• Cherenkov Telescope Array - CTA
• High Energy Stereoscopic System - H.E.S.S.
• MAGIC Telescope
• Very Energetic Radiation Imaging Telescope Array System - VERITAS

See also:
• Tests of Lorentz invariance with cosmic rays
• Tests of Lorentz invariance with astrophysical neutrinos
• Introduction to violations of Lorentz invariance

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