# New Constraints on Lorentz Invariance Violation from Combined Linear and Circular Optical Polarimetry of Extragalactic Sources

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Photon Sector SME

## 3. Optical Polarimetry

#### 3.1. Monochromatic Observations

#### 3.2. Directional Dependence

#### 3.3. Broadband Observations

#### 3.4. Likelihood Model

- linear polarization fraction, ${\prod}_{m}$,
- polarization angle, ${\mathsf{\Psi}}_{m}$, and
- circular polarization fraction, ${V}_{m}$.

## 4. Source Parameters

#### 4.1. Circular Polarization

#### 4.2. Linear Polarization

#### 4.3. Polarization Angle

## 5. Sample Dataset

#### Detection Efficiency Profiles

## 6. SME Constraints

`Python`package

`emcee`[85]. Exploration was carried out by so-called walkers that spawn at some location in the likelihood space (i.e., some combination of SME parameters) and proceed in random Monte Carlo steps. In each step, offsets for the positions of the walkers were drawn from a suitable proposal distribution in all ten dimensions, and the likelihood of compatibility at the new location was estimated.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Polarization Statistics

**Figure A1.**(

**a**) Plots of the expected linear polarization fraction measurement, $E\left[p\right]$ as a function of the “true” polarization fraction $\widehat{p}$, where the expected value is taken as the mean of the corresponding statistical distribution (Equation (A6)). The plots illustrate the linear polarization bias (i.e., the mismatch between expected and true values) due to p being an intrinsically positive quantity. The bias is exacerbated at larger experimental errors $\sigma $. (

**b**) The probability distribution of linear polarization fraction measurements for a variety of “true” values $\widehat{p}$ and $\sigma =0.2$, given in Equation (A5).

## References

- Tanabashi, M.; Hagiwara, K.; Hikasa, K.; Nakamura, K.; Sumino, Y.; Takahashi, F.; Tanaka, J.; Agashe, K.; Aielli, G.; Amsler, C.; et al. Review of Particle Physics. Phys. Rev. D
**2018**, 98, 030001. [Google Scholar] [CrossRef] [Green Version] - Aaij, R.; Beteta, C.A.; Ackernley, T.; Adeva, B.; Adinolfi, M.; Afsharnia, H.; Aidala, C.A.; Aiola, S.; Ajaltouni, Z.; Akar, S.; et al. Test of lepton universality in beauty-quark decays. arXiv
**2021**, arXiv:2103.11769. [Google Scholar] - Myers, R.C.; Pospelov, M. Ultraviolet Modifications of Dispersion Relations in Effective Field Theory. Phys. Rev. Lett.
**2003**, 90, 211601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rizzo, T.G. Lorentz violation in extra dimensions. J. High Energy Phys.
**2005**, 2005, 036. [Google Scholar] [CrossRef] [Green Version] - Amelino-Camelia, G.; Guetta, D.; Piran, T. ICECUBE Neutrinos and Lorentz Invariance Violation. ApJ
**2015**, 806, 269. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Samuel, S. Spontaneous breaking of Lorentz symmetry in string theory. Phys. Rev. D
**1989**, 39, 683–685. [Google Scholar] [CrossRef] [Green Version] - Burgess, C.P.; Cline, J.M.; Filotas, E.; Matias, J.; Moore, G.D. Loop-generated bounds on changes to the graviton dispersion relation. J. High Energy Phys.
**2002**, 2002, 043. [Google Scholar] [CrossRef] - Gambini, R.; Pullin, J. Nonstandard optics from quantum space-time. Phys. Rev. D
**1999**, 59, 124021. [Google Scholar] [CrossRef] [Green Version] - Pospelov, M.; Shang, Y. Lorentz violation in Hořava-Lifshitz-type theories. Phys. Rev. D
**2012**, 85, 105001. [Google Scholar] [CrossRef] [Green Version] - Li, M.; Cai, Y.F.; Wang, X.; Zhang, X. CPT violating electrodynamics and Chern-Simons modified gravity. Phys. Lett. B
**2009**, 680, 118–124. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Potting, R. CPT, strings, and meson factories. Phys. Rev. D
**1995**, 51, 3923–3935. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kostelecký, V.A.; Mewes, M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2009**, 80, 015020. [Google Scholar] [CrossRef] [Green Version] - Adamson, P.; Auty, D.J.; Ayres, D.S.; Backhouse, C.; Barr, G.; Barrett, W.L. A Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector. Phys. Rev. Lett.
**2010**, 105, 151601. [Google Scholar] [CrossRef] - Adamson, P.; Ayres, D.S.; Barr, G.; Bishai, M.; Blake, A.; Bock, G.J. Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector. Phys. Rev. D
**2012**, 85, 031101. [Google Scholar] [CrossRef] [Green Version] - Mattingly, D. Modern Tests of Lorentz Invariance. Living Rev. Relat.
**2005**, 8, 5. [Google Scholar] [CrossRef] [Green Version] - Aaij, R.; Beteta, C.A.; Adeva, B.; Adinolfi, M.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, F.; Ali, S.; et al. Search for violations of Lorentz invariance and CPT symmetry in B
^{0}_{(s)}mixing. Phys. Rev. Lett.**2016**, 116, 241601. [Google Scholar] [CrossRef] [Green Version] - Carle, A.; Chanon, N.; Perries, S. Prospects for Lorentz Invariance Violation searches with top pair production at the LHC and future hadron colliders. Eur. Phys. J. C
**2020**, 80, 128. [Google Scholar] [CrossRef] - Colladay, D.; Kostelecký, V.A. CPT violation and the standard model. Phys. Rev. D
**1997**, 55, 6760–6774. [Google Scholar] [CrossRef] [Green Version] - Colladay, D.; Kostelecký, V.A. Lorentz-violating extension of the standard model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Mewes, M. Signals for Lorentz violation in electrodynamics. Phys. Rev. D
**2002**, 66, 056005. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A. Gravity, Lorentz violation, and the standard model. Phys. Rev. D
**2004**, 69, 105009. [Google Scholar] [CrossRef] [Green Version] - Kislat, F.; Krawczynski, H. Search for anisotropic Lorentz invariance violation with γ-rays. Phys. Rev. D
**2015**, 92, 045016. [Google Scholar] [CrossRef] [Green Version] - Vasileiou, V.; Jacholkowska, A.; Piron, F.; Bolmont, J.; Couturier, C.; Granot, J.; Stecker, F.W.; Cohen-Tanugi, J.; Longo, F. Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations of gamma-ray bursts. Phys. Rev. D
**2013**, 87, 122001. [Google Scholar] [CrossRef] [Green Version] - Boggs, S.E.; Wunderer, C.B.; Hurley, K.; Coburn, W. Testing Lorentz Invariance with GRB 021206. ApJ
**2004**, 611, L77–L80. [Google Scholar] [CrossRef] - Aharonian, F.; Akhperjanian, A.G.; Barres de Almeida, U.; Bazer-Bachi, A.R.; Becherini, Y.; Behera, B.; Beilicke, M.; Benbow, W.; Bernlöhr, K.; Boisson, C.; et al. Limits on an Energy Dependence of the Speed of Light from a Flare of the Active Galaxy PKS 2155-304. Phys. Rev. Lett.
**2008**, 101, 170402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abramowski, A.; Aharonian, F.; Ait Benkhali, F.; Akhperjanian, A.G.; Angüner, E.O.; Backes, M.; Balenderan, S.; Balzer, A.; Barnacka, A.; Becherini, Y.; et al. The 2012 Flare of PG 1553+113 Seen with H.E.S.S. and Fermi-LAT. ApJ
**2015**, 802, 65. [Google Scholar] [CrossRef] [Green Version] - Albert, J.; Aliu, E.; Anderhub, H.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Baixeras, C.; Barrio, J.A.; Bartko, H.; Bastieri, D.; et al. Probing Quantum Gravity using Photons from a flare of the active galactic nucleus Markarian 501 Observed by the MAGIC telescope. Phys. Lett. B
**2008**, 668, 253–257. [Google Scholar] [CrossRef] - Biller, S.D.; Breslin, A.C.; Buckley, J.; Catanese, M.; Carson, M.; Carter-Lewis, D.A.; Cawley, M.F.; Fegan, D.J.; Finley, J.P.; Gaidos, J.A.; et al. Limits to Quantum Gravity Effects on Energy Dependence of the Speed of Light from Observations of TeV Flares in Active Galaxies. Phys. Rev. Lett.
**1999**, 83, 2108–2111. [Google Scholar] [CrossRef] [Green Version] - Ellis, J.; Mavromatos, N.E.; Nanopoulos, D.V.; Sakharov, A.S.; Sarkisyan, E.K.G. Robust limits on Lorentz violation from gamma-ray bursts. Astropart. Phys.
**2006**, 25, 402–411. [Google Scholar] [CrossRef] [Green Version] - Wei, J.J.; Wu, X.F. A Further Test of Lorentz Violation from the Rest-frame Spectral Lags of Gamma-Ray Bursts. ApJ
**2017**, 851, 127. [Google Scholar] [CrossRef] [Green Version] - Wei, J.J.; Wu, X.F. Testing fundamental physics with astrophysical transients. Front. Phys.
**2021**, 16, 44300. [Google Scholar] [CrossRef] - Kislat, F.; Krawczynski, H. Planck-scale constraints on anisotropic Lorentz and C P T invariance violations from optical polarization measurements. Phys. Rev. D
**2017**, 95, 083013. [Google Scholar] [CrossRef] [Green Version] - Kaaret, P. X-ray clues to viability of loop quantum gravity. Nature
**2004**, 427, 287. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Constraints on Relativity Violations from Gamma-Ray Bursts. Phys. Rev. Lett.
**2013**, 110, 201601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Toma, K.; Mukohyama, S.; Yonetoku, D.; Murakami, T.; Gunji, S.; Mihara, T.; Morihara, Y.; Sakashita, T.; Takahashi, T.; Wakashima, Y.; et al. Strict Limit on CPT Violation from Polarization of γ-Ray Bursts. Phys. Rev. Lett.
**2012**, 109, 241104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Laurent, P.; Götz, D.; Binétruy, P.; Covino, S.; Fernandez-Soto, A. Constraints on Lorentz Invariance Violation using integral/IBIS observations of GRB041219A. Phys. Rev. D
**2011**, 83, 121301. [Google Scholar] [CrossRef] [Green Version] - Stecker, F.W. A new limit on Planck scale Lorentz violation from γ-ray burst polarization. Astropart. Phys.
**2011**, 35, 95–97. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Russell, N. Data tables for Lorentz and CPT violation. Rev. Mod. Phys.
**2011**, 83, 11–32. [Google Scholar] [CrossRef] [Green Version] - Friedman, A.S.; Gerasimov, R.; Leon, D.; Stevens, W.; Tytler, D.; Keating, B.G.; Kislat, F. Improved constraints on anisotropic birefringent Lorentz invariance and C P T violation from broadband optical polarimetry of high redshift galaxies. Phys. Rev. D
**2020**, 102, 043008. [Google Scholar] [CrossRef] - Kislat, F. Constraints on Lorentz Invariance Violation from Optical Polarimetry of Astrophysical Objects. Symmetry
**2018**, 10, 596. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Mewes, M. Astrophysical Tests of Lorentz and CPT Violation with Photons. ApJ
**2008**, 689, L1. [Google Scholar] [CrossRef] [Green Version] - Jacob, U.; Piran, T. Lorentz-violation-induced arrival delays of cosmological particles. J. Cosmol. Astropart. Phys.
**2008**, 2008, 031. [Google Scholar] [CrossRef] - McMaster, W.H. Polarization and the Stokes Parameters. Am. J. Phys.
**1954**, 22, 351–362. [Google Scholar] [CrossRef] - Contopoulos, G.; Jappel, A. Transactions of the International Astronomical Union, Volume_XVB: Proceedings of the Fifteenth General Assembly, Sydney 1973 and Extraordinary Assembly, Poland 1973; D. Reidel: Dordrecht, The Netherlands, 1974. [Google Scholar]
- Ma, C.; Arias, E.F.; Eubanks, T.M.; Fey, A.L.; Gontier, A.M.; Jacobs, C.S.; Sovers, O.J.; Archinal, B.A.; Charlot, P. The International Celestial Reference Frame as Realized by Very Long Baseline Interferometry. AJ
**1998**, 116, 516–546. [Google Scholar] [CrossRef] - Fomalont, E. The International Celestial Reference System. The Science of Calibration. In Astronomical Society of the Pacific Conference Series; Deustua, S., Allam, S., Tucker, D., Smith, J.A., Eds.; Astronomical Society of the Pacific: San Francisco, CA, USA, 2016; Volume 503, p. 177. [Google Scholar]
- McKinnon, M.M. Three-Dimensional Statistics of Radio Polarimetry. ApJS
**2003**, 148, 519–526. [Google Scholar] [CrossRef] [Green Version] - Tarantola, A. Elements for Physics: Quantities, Qualities, and Intrinsic Theories; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Planck Collaboration; Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; et al. Planck 2018 results. VI. Cosmological parameters. A&A
**2020**, 641, A6. [Google Scholar] [CrossRef] [Green Version] - Shao, L. Combined search for anisotropic birefringence in the gravitational-wave transient catalog GWTC-1. Phys. Rev. D
**2020**, 101, 104019. [Google Scholar] [CrossRef] - Komatsu, E.; Dunkley, J.; Nolta, M.R.; Bennett, C.L.; Gold, B.; Hinshaw, G.; Jarosik, N.; Larson, D.; Limon, M.; Page, L.; et al. Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation. ApJS
**2009**, 180, 330–376. [Google Scholar] [CrossRef] [Green Version] - Gubitosi, G.; Pagano, L.; Amelino-Camelia, G.; Melchiorri, A.; Cooray, A. A constraint on Planck-scale modifications to electrodynamics with CMB polarization data. J. Cosmol. Astropart. Phys.
**2009**, 2009, 021. [Google Scholar] [CrossRef] [Green Version] - Kahniashvili, T.; Durrer, R.; Maravin, Y. Testing Lorentz invariance violation with Wilkinson Microwave Anisotropy Probe five year data. Phys. Rev. D
**2008**, 78, 123009. [Google Scholar] [CrossRef] [Green Version] - Kaufman, J.P.; Keating, B.G.; Johnson, B.R. Precision tests of parity violation over cosmological distances. MNRAS
**2016**, 455, 1981–1988. [Google Scholar] [CrossRef] [Green Version] - Leon, D.; Kaufman, J.; Keating, B.; Mewes, M. The cosmic microwave background and pseudo-Nambu-Goldstone bosons: Searching for Lorentz violations in the cosmos. Mod. Phys. Lett. A
**2017**, 32, 1730002. [Google Scholar] [CrossRef] [Green Version] - Buzzoni, B.; Delabre, B.; Dekker, H.; Dodorico, S.; Enard, D.; Focardi, P.; Gustafsson, B.; Nees, W.; Paureau, J.; Reiss, R. The ESO Faint Object Spectrograph and Camera / EFOSC. Messenger
**1984**, 38, 9. [Google Scholar] - Hutsemékers, D.; Borguet, B.; Sluse, D.; Cabanac, R.; Lamy, H. Optical circular polarization in quasars. A&A
**2010**, 520, L7. [Google Scholar] [CrossRef] - Matsumiya, M.; Ioka, K. Circular Polarization from Gamma-Ray Burst Afterglows. ApJ
**2003**, 595, L25–L28. [Google Scholar] [CrossRef] [Green Version] - Sagiv, A.; Waxman, E.; Loeb, A. Probing the Magnetic Field Structure in Gamma-Ray Bursts through Dispersive Plasma Effects on the Afterglow Polarization. ApJ
**2004**, 615, 366–377. [Google Scholar] [CrossRef] [Green Version] - Toma, K.; Ioka, K.; Nakamura, T. Probing the Efficiency of Electron-Proton Coupling in Relativistic Collisionless Shocks through the Radio Polarimetry of Gamma-Ray Burst Afterglows. ApJ
**2008**, 673, L123. [Google Scholar] [CrossRef] [Green Version] - Abazajian, K.N.; Adelman-McCarthy, J.K.; Agüeros, M.A.; Allam, S.S.; Allende Prieto, C.; An, D.; Anderson, K.S.J.; Anderson, S.F.; Annis, J.; Bahcall, N.A.; et al. The Seventh Data Release of the Sloan Digital Sky Survey. ApJS
**2009**, 182, 543–558. [Google Scholar] [CrossRef] - Hutsemékers, D.; Lamy, H.; Remy, M. Polarization properties of a sample of broad absorption line and gravitationally lensed quasars. A&A
**1998**, 340, 371–380. [Google Scholar] - Truebenbach, A.E.; Darling, J. The VLBA Extragalactic Proper Motion Catalog and a Measurement of the Secular Aberration Drift. ApJS
**2017**, 233, 3. [Google Scholar] [CrossRef] [Green Version] - Sluse, D.; Hutsemékers, D.; Lamy, H.; Cabanac, R.; Quintana, H. New optical polarization measurements of quasi-stellar objects. The data. A&A
**2005**, 433, 757–764. [Google Scholar] [CrossRef] - Jones, D.H.; Read, M.A.; Saunders, W.; Colless, M.; Jarrett, T.; Parker, Q.A.; Fairall, A.P.; Mauch, T.; Sadler, E.M.; Watson, F.G.; et al. The 6dF Galaxy Survey: Final redshift release (DR3) and southern large-scale structures. MNRAS
**2009**, 399, 683–698. [Google Scholar] [CrossRef] - Impey, C.D.; Tapia, S. The Optical Polarization Properties of Quasars. ApJ
**1990**, 354, 124. [Google Scholar] [CrossRef] - O’Meara, J.M.; Lehner, N.; Howk, J.C.; Prochaska, J.X.; Fox, A.J.; Peeples, M.S.; Tumlinson, J.; O’Shea, B.W. The Second Data Release of the KODIAQ Survey. AJ
**2017**, 154, 114. [Google Scholar] [CrossRef] [Green Version] - Lamy, H.; Hutsemékers, D. Optical polarization of 47 quasi-stellar objects: The data. A&As
**2000**, 142, 451–456. [Google Scholar] [CrossRef] - Ahn, C.P.; Alexandroff, R.; Allende Prieto, C.; Anderson, S.F.; Anderton, T.; Andrews, B.H.; Aubourg, É.; Bailey, S.; Balbinot, E.; Barnes, R.; et al. The Ninth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-III Baryon Oscillation Spectroscopic Survey. ApJS
**2012**, 203, 21. [Google Scholar] [CrossRef] [Green Version] - Véron-Cetty, M.P.; Véron, P. A catalogue of quasars and active nuclei: 13th edition. A&A
**2010**, 518, A10. [Google Scholar] [CrossRef] - Pâris, I.; Petitjean, P.; Aubourg, É.; Myers, A.D.; Streblyanska, A.; Lyke, B.W.; Anderson, S.F.; Armengaud, É.; Bautista, J.; Blanton, M.R.; et al. The Sloan Digital Sky Survey Quasar Catalog: Fourteenth data release. A&A
**2018**, 613, A51. [Google Scholar] [CrossRef] [Green Version] - Schmidt, G.D.; Hines, D.C. The Polarization of Broad Absorption Line QSOS. ApJ
**1999**, 512, 125–135. [Google Scholar] [CrossRef] - Berriman, G.; Schmidt, G.D.; West, S.C.; Stockman, H.S. An Optical Polarization Survey of the Palomar-Green Bright Quasar Sample. ApJS
**1990**, 74, 869. [Google Scholar] [CrossRef] - Sbarufatti, B.; Treves, A.; Falomo, R.; Heidt, J.; Kotilainen, J.; Scarpa, R. ESO Very Large Telescope Optical Spectroscopy of BL Lacertae Objects. I. New Redshifts. AJ
**2005**, 129, 559–566. [Google Scholar] [CrossRef] - Barkhouse, W.A.; Hall, P.B. Quasars in the 2MASS Second Incremental Data Release. AJ
**2001**, 121, 2843–2850. [Google Scholar] [CrossRef] [Green Version] - Beckmann, V.; Gehrels, N.; Shrader, C.R.; Soldi, S. The First INTEGRAL AGN Catalog. ApJ
**2006**, 638, 642–652. [Google Scholar] [CrossRef] - Snellen, I.A.G.; McMahon, R.G.; Hook, I.M.; Browne, I.W.A. Automated optical identification of a large complete northern hemisphere sample of flat-spectrum radio sources with [formmu1]S
_{6cm}>200 mJy. MNRAS**2002**, 329, 700–746. [Google Scholar] [CrossRef] [Green Version] - Visvanathan, N.; Wills, B.J. Optical Polarization of 52 Radio-loud QSOS and BL Lacertae Objects. AJ
**1998**, 116, 2119–2122. [Google Scholar] [CrossRef] [Green Version] - Breger, M. Intracluster dust, circumstellar shells, and the wavelength dependence of polarization in Orion. ApJ
**1977**, 215, 119–128. [Google Scholar] [CrossRef] - Visvanathan, N. An Automatic Fast Digital-Photoelectric Photometer with Polarimeter. PASP
**1972**, 84, 248. [Google Scholar] [CrossRef] - Ebdon, L.; Evans, E.H.; Fisher, A.; Hill, S.J. An Introduction to Analytical Atomic Spectrometry; J. Wiley & Sons: Chichester, UK, 1998. [Google Scholar]
- Noll, S.; Kausch, W.; Barden, M.; Jones, A.M.; Szyszka, C.; Kimeswenger, S.; Vinther, J. An atmospheric radiation model for Cerro Paranal. I. The optical spectral range. A&A
**2012**, 543, A92. [Google Scholar] [CrossRef] - Clough, S.A.; Shephard, M.W.; Mlawer, E.J.; Delamere, J.S.; Iacono, M.J.; Cady-Pereira, K.; Boukabara, S.; Brown, P.D. Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spec. Radiat. Transf.
**2005**, 91, 233–244. [Google Scholar] [CrossRef] - Rothman, L.S.; Gordon, I.E.; Barbe, A.; Benner, D.C.; Bernath, P.F.; Birk, M.; Boudon, V.; Brown, L.R.; Campargue, A.; Champion, J.P.; et al. The HITRAN 2008 molecular spectroscopic database. J. Quant. Spec. Radiat. Transf.
**2009**, 110, 533–572. [Google Scholar] [CrossRef] [Green Version] - Foreman-Mackey, D.; Hogg, D.W.; Lang, D.; Goodman, J. emcee: The MCMC Hammer. PASP
**2013**, 125, 306. [Google Scholar] [CrossRef] [Green Version] - Bagnulo, S.; Cox, N.L.J.; Cikota, A.; Siebenmorgen, R.; Voshchinnikov, N.V.; Patat, F.; Smith, K.T.; Smoker, J.V.; Taubenberger, S.; Kaper, L.; et al. Large Interstellar Polarisation Survey (LIPS). I. FORS2 spectropolarimetry in the Southern Hemisphere. A&A
**2017**, 608, A146. [Google Scholar] [CrossRef] [Green Version] - Siebenmorgen, R.; Voshchinnikov, N.V.; Bagnulo, S.; Cox, N.L.J.; Cami, J.; Peest, C. Large Interstellar Polarisation Survey. II. UV/optical study of cloud-to-cloud variations of dust in the diffuse ISM. A&A
**2018**, 611, A5. [Google Scholar] [CrossRef] [Green Version] - Weisskopf, M.C.; Elsner, R.F.; Hanna, D.; Kaspi, V.M.; O’Dell, S.L.; Pavlov, G.G.; Ramsey, B.D. The prospects for X-ray polarimetry and its potential use for understanding neutron stars. arXiv
**2006**, arXiv:astro–ph/0611483. [Google Scholar] - Vinokur, M. Optimisation dans la recherche d’une sinusoïde de période connue en présence de bruit. Application à la radioastronomie. Ann. D’Astrophysique
**1965**, 28, 412. [Google Scholar]

**Figure 1.**Schematic illustration of the Lorentz invariance violation (LIV) effect on the polarization state of a photon in the Standard-Model Extension (SME) framework. The photon is emitted at the source in an initial polarization state, whose location in the Stokes space (shown here) is indicated with vector ${\mathit{s}}_{z}$. In-flight, the state will precess around the birefringence axis $\mathit{\varsigma}$ through angle $2\mathsf{\Phi}$ until, eventually, the photon arrives at the telescope in the state ${\mathit{s}}_{0}$. The direction of the birefringence axis and the rate of precession are determined by the particular SME configuration. The blue quadrilateral represents the plane of linear polarization where Stokes V is 0. Individual components of $\mathit{\varsigma}=({\varsigma}^{1},{\varsigma}^{2},{\varsigma}^{3})$ and the angles referenced in text ($\xi $, $\zeta $) are labeled as well.

**Figure 2.**The curves show the effect of the broadband operator $\mathcal{T}\left[x\right]$ on the amount of LIV-induced linear polarization in-flight (see Equations (27) and (28)) as a function of $\mathsf{\Phi}$ (see Figure 1). The three cases shown correspond to a monochromatic observation (black, $\mathcal{T}[sin(2\mathsf{\Phi}\left)\right]\to sin\left(2\mathsf{\Phi}\right)$), a broadband observation with an unfiltered Ga-As photomultiplier tube (red), and a broadband observation in the V-band of the EFOSC2 [56] instrument (green). This calculation assumes observations at the zenith. See Section 5 for more details on instruments, bands, and atmospheric effects.

**Figure 3.**(

**a**) Plots of the linear, circular, and total likelihood of compatibility for a single test measurement as a function of the assumed linear polarization fraction at the source. The astrophysical source is assumed to be located at the Vernal equinox ($\alpha =0$, $\delta =0$) and $z=2.0$. For demonstration purposes, the adopted test measurement is ${\prod}_{m}=0.5\pm 0.1$, ${V}_{m}=0.00\pm 0.01$ and ${\mathsf{\Psi}}_{m}=0.0$. The adopted SME configuration has all ten parameters considered in this study set to ${10}^{-34}$ (the order of magnitude of the upper limit derived in [39,40]). The test measurement is assumed to have been taken through the EFOSC2 [56] V filter in zenith (see Section 5). The linear compatibility increases while the circular compatibility decreases with ${p}_{z}$. The most conservative assumption for the value of ${p}_{z}$ is the one maximizing the total compatibility, which, in this case, occurs around ${p}_{z}=0.55$. (

**b**) The effect of SME on the linear and circular polarization fractions, shown here as ${\prod}_{0}$ and ${V}_{0}$ as functions of the redshift of the source. The predictions shown here were derived for a test source at the Vernal equinox ($\alpha =0$, $\delta =0$) observed through the EFOSC2 [56] V filter in zenith. The initial polarization state of the photons was assumed to be ${v}_{z}=0$, ${p}_{z}=1$, and ${\psi}_{z}=-{30}^{\circ}$. The adopted SME configuration is the same as in (

**a**).

**Figure 4.**(

**a**) The atmospheric transmission model employed in this study as a function of wavelength for two airmasses: $Z=1,3$. The most prominent absorption features due to ozone (${\mathrm{O}}_{3}$), oxygen (${\mathrm{O}}_{2}$), and water vapor (${\mathrm{H}}_{2}\mathrm{O}$) are labeled. For clarity, the plots exclude Rayleigh and Mie scattering effects (however, they are included in the analysis). (

**b**) Detection efficiency profiles of the three instruments relevant to this study. For the V band, the transmission of the filter is shown. For both Ga-As and S-20 photomultipliers, the response curves of the detector are used instead. All curves are corrected for the atmospheric transmission. In the figure, the $Z=1$ atmosphere was applied.

**Figure 5.**Probability distributions for the values of the 10 real $d=4$ SME parameters extracted from MCMC chains for the dataset of optical circular and linear polarimetry considered in this study. All parameters are dimensionless. Color-coded are the cases of $Z=1$ (red) and $Z=3$ (blue) atmospheres (Section 5). The distributions are uniformly binned with the bin width of $2.5\times {10}^{-36}$ for the total of 20 bins.

**Table 1.**Coordinates, redshifts, and optical linear/circular polarization measurements of the selected quasars. References (z Ref.) are provided for the redshift values. Transmission bands and references (p Ref.) are provided for the linear polarization measurements. All circular polarization measurements were taken through the Bessel V filter and were reported in [57]. The detection efficiency profiles for the Bessel V filter, Ga-As photomultiplier, and Na-K-Cs-Sb (S20) photomultiplier are shown in Figure 4.

Object Identifier | RAJ2000 ($\mathit{\alpha}$) | DEJ2000 ($\mathit{\delta}$) | z | z Ref. | ${\mathbf{\Pi}}_{\mathbf{m}}\phantom{\rule{3.33333pt}{0ex}}(\%)$ | ${\mathbf{\Psi}}_{\mathbf{m}}\phantom{\rule{3.33333pt}{0ex}}{(}^{\circ})$ | Band | p Ref. | ${\mathbf{V}}_{\mathbf{m}}\phantom{\rule{3.33333pt}{0ex}}(\%)$ |
---|---|---|---|---|---|---|---|---|---|

QSO B1120+0154 | $11\phantom{\rule{3.33333pt}{0ex}}23\phantom{\rule{3.33333pt}{0ex}}20.73$ | $+01\phantom{\rule{3.33333pt}{0ex}}37\phantom{\rule{3.33333pt}{0ex}}47.5$ | $1.47$ | [61] | $1.95\pm 0.27$ | $9\pm 4$ | V | [62] | $-0.02\pm 0.05$ |

QSO B1124-186 | $11\phantom{\rule{3.33333pt}{0ex}}27\phantom{\rule{3.33333pt}{0ex}}04.39$ | $-18\phantom{\rule{3.33333pt}{0ex}}57\phantom{\rule{3.33333pt}{0ex}}17.4$ | $1.05$ | [63] | $11.68\pm 0.36$ | $37\pm 1$ | V | [64] | $-0.04\pm 0.08$ |

QSO J1130-1449 | $11\phantom{\rule{3.33333pt}{0ex}}30\phantom{\rule{3.33333pt}{0ex}}07.05$ | $-14\phantom{\rule{3.33333pt}{0ex}}49\phantom{\rule{3.33333pt}{0ex}}27.4$ | $1.19$ | [65] | $1.30\pm 0.40$ | $23\pm 10$ | Ga-As | [66] | $-0.05\pm 0.05$ |

QSO B1157+014 | $11\phantom{\rule{3.33333pt}{0ex}}59\phantom{\rule{3.33333pt}{0ex}}44.83$ | $+01\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}07.0$ | $2.00$ | [67] | $0.76\pm 0.18$ | $39\pm 7$ | V | [68] | $-0.10\pm 0.08$ |

LBQS 1205+1436 | $12\phantom{\rule{3.33333pt}{0ex}}08\phantom{\rule{3.33333pt}{0ex}}25.38$ | $+14\phantom{\rule{3.33333pt}{0ex}}19\phantom{\rule{3.33333pt}{0ex}}21.1$ | $1.64$ | [61] | $0.83\pm 0.18$ | $161\pm 6$ | V | [68] | $-0.10\pm 0.09$ |

LBQS 1212+1445 | $12\phantom{\rule{3.33333pt}{0ex}}14\phantom{\rule{3.33333pt}{0ex}}40.27$ | $+14\phantom{\rule{3.33333pt}{0ex}}28\phantom{\rule{3.33333pt}{0ex}}59.3$ | $1.63$ | [69] | $1.45\pm 0.30$ | $24\pm 6$ | V | [62] | $0.15\pm 0.09$ |

QSO B1215-002 | $12\phantom{\rule{3.33333pt}{0ex}}17\phantom{\rule{3.33333pt}{0ex}}58.73$ | $-00\phantom{\rule{3.33333pt}{0ex}}29\phantom{\rule{3.33333pt}{0ex}}46.3$ | $0.42$ | [69] | $23.94\pm 0.70$ | $91\pm 1$ | V | [64] | $-0.42\pm 0.40$ |

QSO B1216-010 | $12\phantom{\rule{3.33333pt}{0ex}}18\phantom{\rule{3.33333pt}{0ex}}34.93$ | $-01\phantom{\rule{3.33333pt}{0ex}}19\phantom{\rule{3.33333pt}{0ex}}54.3$ | $0.554$ | [70] | $11.20\pm 0.17$ | $100\pm 1$ | V | [64] | $-0.01\pm 0.07$ |

Ton 1530 | $12\phantom{\rule{3.33333pt}{0ex}}25\phantom{\rule{3.33333pt}{0ex}}27.40$ | $+22\phantom{\rule{3.33333pt}{0ex}}35\phantom{\rule{3.33333pt}{0ex}}13.0$ | $2.05$ | [71] | $0.92\pm 0.14$ | $169\pm 4$ | V | [64] | $0.01\pm 0.10$ |

QSO J1246-2547 | $12\phantom{\rule{3.33333pt}{0ex}}46\phantom{\rule{3.33333pt}{0ex}}46.80$ | $-25\phantom{\rule{3.33333pt}{0ex}}47\phantom{\rule{3.33333pt}{0ex}}49.3$ | $0.63$ | [63] | $8.40\pm 0.20$ | $110\pm 1$ | Ga-As | [66] | $-0.23\pm 0.20$ |

QSO B1246-0542 | $12\phantom{\rule{3.33333pt}{0ex}}49\phantom{\rule{3.33333pt}{0ex}}13.86$ | $-05\phantom{\rule{3.33333pt}{0ex}}59\phantom{\rule{3.33333pt}{0ex}}19.1$ | $2.23$ | [67] | $1.96\pm 0.18$ | $149\pm 3$ | Ga-As | [72] | $0.01\pm 0.03$ |

QSO B1254+0443 | $12\phantom{\rule{3.33333pt}{0ex}}56\phantom{\rule{3.33333pt}{0ex}}59.92$ | $+04\phantom{\rule{3.33333pt}{0ex}}27\phantom{\rule{3.33333pt}{0ex}}34.4$ | $1.02$ | [69] | $1.22\pm 0.15$ | $165\pm 3$ | Ga-As | [73] | $-0.02\pm 0.04$ |

QSO B1256-229 | $12\phantom{\rule{3.33333pt}{0ex}}59\phantom{\rule{3.33333pt}{0ex}}08.46$ | $-23\phantom{\rule{3.33333pt}{0ex}}10\phantom{\rule{3.33333pt}{0ex}}38.7$ | $0.481$ | [74] | $22.32\pm 0.15$ | $157\pm 1$ | V | [64] | $0.18\pm 0.04$ |

QSO J1311-0552 | $13\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}36.56$ | $-05\phantom{\rule{3.33333pt}{0ex}}52\phantom{\rule{3.33333pt}{0ex}}38.6$ | $2.19$ | [75] | $0.78\pm 0.28$ | $179\pm 11$ | V | [62] | $-0.08\pm 0.06$ |

LBQS 1331-0108 | $13\phantom{\rule{3.33333pt}{0ex}}34\phantom{\rule{3.33333pt}{0ex}}28.06$ | $-01\phantom{\rule{3.33333pt}{0ex}}23\phantom{\rule{3.33333pt}{0ex}}49.0$ | $1.78$ | [69] | $1.88\pm 0.31$ | $29\pm 5$ | V | [62] | $-0.04\pm 0.06$ |

[VV96] J134204.4-181801 | $13\phantom{\rule{3.33333pt}{0ex}}42\phantom{\rule{3.33333pt}{0ex}}04.41$ | $-18\phantom{\rule{3.33333pt}{0ex}}18\phantom{\rule{3.33333pt}{0ex}}02.6$ | $2.21$ | [75] | $0.83\pm 0.15$ | $20\pm 5$ | V | [64] | $-0.01\pm 0.07$ |

2E 3238 | $14\phantom{\rule{3.33333pt}{0ex}}19\phantom{\rule{3.33333pt}{0ex}}03.82$ | $-13\phantom{\rule{3.33333pt}{0ex}}10\phantom{\rule{3.33333pt}{0ex}}44.8$ | $0.13$ | [76] | $1.63\pm 0.15$ | $44\pm 3$ | Ga-As | [73] | $0.05\pm 0.06$ |

LBQS 1429-0053 | $14\phantom{\rule{3.33333pt}{0ex}}32\phantom{\rule{3.33333pt}{0ex}}29.25$ | $-01\phantom{\rule{3.33333pt}{0ex}}06\phantom{\rule{3.33333pt}{0ex}}16.1$ | $2.08$ | [69] | $1.00\pm 0.29$ | $9\pm 9$ | V | [62] | $0.02\pm 0.08$ |

QSO J2123+0535 | $21\phantom{\rule{3.33333pt}{0ex}}23\phantom{\rule{3.33333pt}{0ex}}44.52$ | $+05\phantom{\rule{3.33333pt}{0ex}}35\phantom{\rule{3.33333pt}{0ex}}22.1$ | $1.88$ | [77] | $10.70\pm 2.90$ | $68\pm 6$ | Ga-As | [66] | $0.02\pm 0.15$ |

QSO B2128-123 | $21\phantom{\rule{3.33333pt}{0ex}}31\phantom{\rule{3.33333pt}{0ex}}35.26$ | $-12\phantom{\rule{3.33333pt}{0ex}}07\phantom{\rule{3.33333pt}{0ex}}04.8$ | $0.50$ | [67] | $1.90\pm 0.40$ | $64\pm 6$ | S20 | [78] | $-0.04\pm 0.03$ |

QSO B2155-152 | $21\phantom{\rule{3.33333pt}{0ex}}58\phantom{\rule{3.33333pt}{0ex}}06.28$ | $-15\phantom{\rule{3.33333pt}{0ex}}01\phantom{\rule{3.33333pt}{0ex}}09.3$ | $0.67$ | [63] | $22.60\pm 1.10$ | $7\pm 2$ | Ga-As | [66] | $-0.35\pm 0.10$ |

**Table 2.**The derived constraints on the values of the 10 real $d=4$ SME parameters. The upper and lower bounds were taken as the 5th and 95th percentiles of the distributions shown in Figure 5. For each parameter, the most conservative of the $Z=1$ and $Z=3$ cases is shown.

SME Parameter | Upper Bound ($\times {10}^{-35}$) | Lower Bound ($\times {10}^{-35}$) |
---|---|---|

${k}_{\left(E\right)2,0}$ | $\le 2.9$ | $\ge -1.2$ |

$\mathrm{Re}\left[{k}_{\left(E\right)2,1}\right]$ | $\le 1.8$ | $\ge -1.5$ |

$\mathrm{Im}\left[{k}_{\left(E\right)2,1}\right]$ | $\le 0.2$ | $\ge -1.4$ |

$\mathrm{Re}\left[{k}_{\left(E\right)2,2}\right]$ | $\le 3.0$ | $\ge -1.7$ |

$\mathrm{Im}\left[{k}_{\left(E\right)2,2}\right]$ | $\le 1.4$ | $\ge -1.4$ |

${k}_{\left(B\right)2,0}$ | $\le 3.2$ | $\ge -0.7$ |

$\mathrm{Re}\left[{k}_{\left(B\right)2,1}\right]$ | $\le 1.3$ | $\ge -1.8$ |

$\mathrm{Im}\left[{k}_{\left(B\right)2,1}\right]$ | $\le 1.9$ | $\ge -0.8$ |

$\mathrm{Re}\left[{k}_{\left(B\right)2,2}\right]$ | $\le 2.1$ | $\ge -2.1$ |

$\mathrm{Im}\left[{k}_{\left(B\right)2,2}\right]$ | $\le 1.2$ | $\ge -2.3$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gerasimov, R.; Bhoj, P.; Kislat, F.
New Constraints on Lorentz Invariance Violation from Combined Linear and Circular Optical Polarimetry of Extragalactic Sources. *Symmetry* **2021**, *13*, 880.
https://doi.org/10.3390/sym13050880

**AMA Style**

Gerasimov R, Bhoj P, Kislat F.
New Constraints on Lorentz Invariance Violation from Combined Linear and Circular Optical Polarimetry of Extragalactic Sources. *Symmetry*. 2021; 13(5):880.
https://doi.org/10.3390/sym13050880

**Chicago/Turabian Style**

Gerasimov, Roman, Praneet Bhoj, and Fabian Kislat.
2021. "New Constraints on Lorentz Invariance Violation from Combined Linear and Circular Optical Polarimetry of Extragalactic Sources" *Symmetry* 13, no. 5: 880.
https://doi.org/10.3390/sym13050880