Searches for Lorentz-Violating Signals with Astrophysical Polarization Measurements

TL;DR

This paper reviews how astrophysical polarization measurements, especially from high-redshift objects and gamma-ray telescopes, are used to constrain Lorentz and CPT invariance violations in the photon sector within the Standard-Model Extension framework.

Contribution

It provides a comprehensive review of methods and recent results in constraining Lorentz and CPT violation using astrophysical polarization data.

Findings

Constraints on vacuum birefringence from polarization measurements.

Limits on Lorentz violation parameters from gamma-ray observations.

Evidence supporting Lorentz invariance at astrophysical scales.

Abstract

Astrophysical observations are a powerful tool to constrain effects of Lorentz-invariance violation in the photon sector. Objects at high redshifts provide the longest possible baselines, and gamma-ray telescopes allow us to observe some of the highest energy photons. Observations include polarization measurements and time-of-flight measurements of transient or variable objects to constrain vacuum birefringence and dispersion. Observing multiple sources covering the entire sky allows the extraction of constraints on anisotropy. In this paper, I review methods and recent results on Lorentz- and CPT-invariance violation constraints derived from astrophysical polarization measurements in the framework of the Standard-Model Extension.

Equations

$$E=\left(1-{\varsigma }^0\pm \sqrt{({\varsigma }^1{)}^2+({\varsigma }^2{)}^2+({\varsigma }^3{)}^2}\right)p,$$

$$\frac{d\mathbf{s}}{dt}=2E\mathbf{\varsigma }\times \mathbf{s},$$

Full text

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1 Introduction

Observer-independence of the speed of light postulated by Einstein’s Theory of Special Relativity implies that the speed of light for a given observer is independent of frequency, direction of propagation, and polarization. Conversely, if Lorentz symmetry is broken, any of these may no longer hold, resulting in a vacuum that effectively acts like an anisotropic, birefringent, and/or dispersive medium. Theories of quantum gravity suggest that Lorentz invariance may be violated at the Planck scale, but effects must be strongly suppressed at attainable energies. Photons propagating over astrophysical distances enable some of the strongest tests as tiny effects will accumulate during propagation of the photons.

The Standard-Model Extension[1, 2] (SME) is an effective-field-theory approach to describe low-energy effects of a more fundamental high-energy theory. It allows a categorization of effects, described in essence by a set of coefficients that are characterized in part by the mass dimension $d$ of the corresponding operator. The SME photon dispersion relation can be written as

[TABLE]

where the $\varsigma^{i}$ can be written as an expansion in $d$ and spherical harmonics. This results in $(d-1)^{2}$ independent coefficients $k_{(V)jm}^{(d)}$ for odd $d$ describing anisotropy, dispersion, and birefringence; for even $d$, $2(d-1)^{2}-8$ independent birefringent coefficients $k_{(E)jm}^{(d)}$ and $k_{(B)jm}^{(d)}$ as well as $(d-1)^{2}$ nonbirefringent coefficients $c_{(I)jm}^{(d)}$ emerge. Effects of mass dimension $d$ are typically assumed to be suppressed by $M_{\text{Planck}}^{d-4}$.

The individual coefficients $k_{(E)jm}^{(4)}$, $k_{(B)jm}^{(4)}$, $k_{(V)jm}^{(5)}$, and $c_{(I)jm}^{(6)}$ have been constrained using astrophysical polarization[3, 4] and time-of-flight measurements,[5] and the $k_{(V)jm}^{(3)}$ have been constrained using CMB measurements.[6] At $d>6$, linear combinations of coefficients have been constrained.[7] Here, we will review some recent results obtained from polarization measurements of astrophysical objects.

2 Birefringence

The polarization of an electromagnetic wave can be described by the Stokes vector $\boldsymbol{s}=(q,u,v)^{T}$, where $q$ and $u$ describe linear polarization and $v$ describes circular polarization. Vacuum birefringence results in a change of the Stokes vector during the propagation of a photon given by[2]

[TABLE]

which represents a rotation of $\boldsymbol{s}$ along a cone around the birefringence axis $\boldsymbol{\varsigma}=(\varsigma^{1},\varsigma^{2},\varsigma^{3})^{T}$.

Most astrophysical emission mechanisms do not result in a strong circular polarization, but linear polarization can be fairly large reaching multiples of ten percent. In general, Eq. (2) results in a change of the linear polarization angle (PA) as the photon propagates, as well as linear polarization partially turning into circular polarization. However, SME operators of odd and even $d$ result in distinctly different signatures.

If only odd-$d$ coefficients are nonzero, $\varsigma^{1}=\varsigma^{2}=0$ and $\boldsymbol{\varsigma}$ is oriented along the circular-polarization axis $\boldsymbol{v}$. In this case, linear polarization remains linear, but the PA makes full [math] rotations as $\boldsymbol{s}$ will rotate in the $q$–$u$ plane. Any circular polarization would remain unaffected. If only even-$d$ coefficients are nonzero, $\varsigma^{3}=0$ and $\boldsymbol{\varsigma}$ is in the $q$–$u$ plane. Then, a Stokes vector with an initial $v=0$ will in general rotate out of this plane acquiring a circular polarization, while at the same time the PA will undergo swings with a magnitude depending on the angle between $\boldsymbol{s}$ and $\boldsymbol{\varsigma}$. However, in this case linear polarization with a polarization angle of $\text{PA}_{0}=\frac{1}{2}\arctan\left(-\varsigma^{2}/\varsigma^{1}\right)$ will remain unaffected.

For all $d\geq 4$, Eq. (2) results in an energy-dependent rate of change of $\boldsymbol{s}$. Hence, birefringent Lorentz-violating effects can be constrained by measuring the (linear) polarization of astrophysical objects as a function of energy. Spectropolarimetric observations are ideally suited for this purpose as the change of PA with energy is measured directly resulting in the strongest constraints.

Constraints can also be derived from broadband measurements where polarization is integrated over the bandwidth of the instrument.[8] In this case, the rotation of the PA results in a net reduction of the observed polarization fraction (PF). Given a set of SME coefficients, the maximum observable PF, $\text{PF}_{\text{max}}$, can be calculated assuming the emission at the source is $100\text{\,}\mathrm{\char 37\relax}$ polarized. Any observed $\text{PF}>\text{PF}_{\text{max}}$ would then rule out this set of SME coefficients. Constraints obtained in this way tend to be significantly weaker than those obtained from spectropolarimetry.

Observations of a single source can be used to constrain linear combinations of SME coefficients. Typically, limits are obtained by restricting the analysis to a particular mass dimension $d$. In the even-$d$ case, additional assumptions must be made due to the existence of the linearly polarized eigenmode of propagation. Observations of multiple objects can be combined to break all of these degeneracies and to obtain constraints on individual coefficients, even in the even-$d$ case.

The strongest individual constraints result from hard x-ray polarization measurements of gamma-ray bursts (GRBs) profiting from the high photon energies and high redshifts. Typical constraints derived from these measurements are on the order of $1\text{\times}{10}^{-34}\text{\,}{\mathrm{GeV}}^{-1}$ or better in the $d=5$ case.[8, 9] However, the number of bursts whose polarization has been measured is limited at this point and in particular the early results suffer from large systematic uncertainties. At least for $d<6$, very strong constraints can also be obtained from optical polarization measurements of distant objects, such as active galactic nuclei and GRB afterglows. Systematic uncertainties of these measurements are smaller, and a much larger number of results is already available.

In two recent papers,[3, 4] we have combined more than 60 optical polarization measurements, both spectropolarimetric and broadband integrated measurements, to constrain individually all birefringent coefficients of mass dimensions $d=4$ and $d=5$. Constraints on the dimensionless coefficients $k_{(E)jm}^{(4)}$ and $k_{(B)jm}^{(4)}$ are $<3\text{\times}{10}^{-34}$, and the $k_{(V)jm}^{(5)}$ are constrained to $<7\text{\times}{10}^{-26}\text{\,}{\mathrm{GeV}}^{-1}$.

3 Summary and outlook

Astrophysical polarization measurements are an extremely powerful tool to constrain Lorentz-invariance violation in the photon sector due to the extremely long baselines.

In the future, new instruments with significantly improved high-energy polarization sensitivity will become available.[10] The IXPE satellite scheduled for launch in early 2021 will measure x-ray polarization in the $28\text{\,}\mathrm{keV}$ range. Compton telescopes, such as the balloon-borne COSI or the proposed AMEGO mission concept, are sensitive to polarization in the $500\text{\,}\mathrm{keV}$ – $5\text{\,}\mathrm{MeV}$ energy range. A slightly lower energy range is covered by the proposed GRB polarimeter LEAP. AdEPT is a concept for a gamma-ray pair-production telescope that would allow measuring the polarization of gamma-rays of tens of $\mathrm{MeV}$. The ability to measure polarization at high energies is crucial in particular for constraining coefficients of $d\geq 6$.