# Constraints on Lorentz Invariance Violation from Optical Polarimetry of Astrophysical Objects

## Abstract

**:**

^{−34}on the dimensionless coefficients.

## 1. Introduction

## 2. Theory

## 3. Astrophysical Polarization Measurements

#### 3.1. Spectropolarimetry

#### 3.2. Polarimetry Integrated over a Broad Bandwidth

## 4. Results

^{7}steps recording each set of selected SME coefficients. The distribution of these sets of coefficients is proportional to $P({k}_{\left(E\right)2m}^{\left(4\right)},{k}_{\left(B\right)2m}^{\left(4\right)})$. From the distribution of values of each individual coefficient, we then find the 5th and 95th percentile as lower and upper limits. The resulting distributions are shown in Figure A1 in Appendix B and the corresponding upper and lower limits on the SME coefficients are listed in Table 2. The constraints turned out to be symmetrical around 0 within at least two-digit precision.

## 5. Discussion

^{−30}. In this scenario, Lorentz invariance violating effects of order unity at the Planck scale due to operators of mass dimension $d=4$ can be ruled out by the constraints presented here. For a similar model to be viable and to result in significant Lorentz invariance violating effects at the Planck scale, one must assume that either $n>2$ or ${M}_{\mathrm{low}}\gg 10\mathrm{TeV}$.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Astrophysical Sources

Source | RA J2000 [${}^{\circ}$] | Dec. J2000 [${}^{\circ}$] | Redshift z | Polarization [%] | Position Angle [${}^{\circ}$] |
---|---|---|---|---|---|

SDSS J0242+0049 | 40.591 | +0.820 | 2.071 | 1.47(24) | −13(5) |

FIRST03133+0036 | 48.328 | +0.606 | 1.250 | 1.48(29) | −55(6) |

FIRST J0809+2753 | 122.256 | +27.895 | 1.511 | 1.75(20) | 73(3) |

PG 0946+301 | 147.421 | +29.922 | 1.220 | 1.65(19) | −66(3) |

PKS 1124−186 | 171.768 | −17.045 | 1.048 | 11.68(36) | 37(1) |

He 1127−1304 | 172.583 | −12.653 | 0.634 | 1.32(13) | 46(3) |

2QZ J114954+0012 | 177.479 | +0.215 | 1.596 | 1.57(22) | −24(4) |

SDSS J1206+0023 | 181.615 | +0.393 | 2.331 | 0.94(15) | −57(5) |

SDSS J1214−0001 | 183.673 | +0.027 | 1.041 | 2.40(32) | −77(4) |

PKS 1219+04 | 185.594 | +4.221 | 0.965 | 5.56(15) | −61(1) |

PKS 1222+037 | 186.218 | +3.514 | 0.960 | 2.51(22) | −82(2) |

TON 1530 | 186.364 | +22.587 | 2.058 | 0.92(14) | −11(4) |

SDSS J1234+0057 | 188.616 | +0.966 | 1.532 | 1.35(23) | 2(5) |

PG 1254+047 | 194.250 | +4.459 | 1.018 | 0.84(15) | −4(5) |

PKS 1256−229 | 194.785 | −22.823 | 1.365 | 22.32(15) | −23(1) |

SDSS J1302−0037 | 195.534 | +0.626 | 1.672 | 1.37(20) | 35(4) |

PKS 1303−250 | 196.564 | −24.711 | 0.738 | 0.91(17) | −75(5) |

FIRST J1312+2319 | 198.056 | +23.333 | 1.508 | 1.10(16) | −14(4) |

SDSS J1323−0038 | 200.769 | +0.649 | 1.827 | 1.13(21) | 15(5) |

CTS J13.07 | 205.518 | −17.700 | 2.210 | 0.83(15) | 20(5) |

SDSS J1409+0048 | 212.328 | +0.807 | 1.999 | 3.91(28) | 30(2) |

HS 1417+2547 | 215.055 | +25.568 | 2.200 | 1.03(18) | −66(5) |

FIRST J1427+2709 | 216.765 | +27.161 | 1.170 | 1.35(25) | 80(5) |

FIRST J21079−0620 | 316.990 | −5.664 | 0.644 | 1.12(22) | −33(6) |

SDSS J2131−0700 | 322.912 | −6.996 | 2.048 | 1.78(32) | 44(5) |

PKS 2204−54 | 331.932 | −52.224 | 1.206 | 1.81(26) | −50(4) |

PKS 2227−445 | 337.735 | −43.725 | 1.326 | 5.26(48) | 18(3) |

PKS 2240−260 | 340.860 | −24.258 | 0.774 | 14.78(21) | −49(1) |

PKS 2301+06 | 346.118 | +6.336 | 1.268 | 3.69(26) | −17(2) |

SDSS J2319−0024 | 349.995 | +0.414 | 1.889 | 1.85(30) | −16(5) |

PKS 2320−035 | 350.883 | −2.715 | 1.411 | 9.56(20) | 90(1) |

PKS 2332−017 | 353.835 | −0.481 | 1.184 | 4.86(19) | −88(1) |

PKS 2335−027 | 354.489 | −1.484 | 1.072 | 3.55(30) | −70(2) |

SDSS J2352+0105 | 358.159 | +1.098 | 2.156 | 1.59(26) | 27(5) |

SDSS J2356−0036 | 359.120 | +0.601 | 2.936 | 1.81(34) | 16(5) |

QSO J2359−12 | 359.973 | −11.303 | 0.868 | 4.12(20) | −29(1) |

Source | Instrument | RA | Dec. | Redshift | Polarization | Position Angle | Refs. |
---|---|---|---|---|---|---|---|

J2000 [${}^{\circ}$] | J2000 [${}^{\circ}$] | z | [%] | [${}^{\circ}$] | |||

GRB 990510 | FORS1 R−band | 204.532 | −80.497 | 1.619 | 1.6(2) | −84(4) | [41,42,43] |

GRB 990712 | FORS1 R-band | 337.971 | −73.408 | 0.430 | 2.9(4) | −59(4) | [44] |

GRB 020813 | FORS1 V-band | 296.674 | −19.601 | 1.25 | 1.42(25) | −43(4) | [47,48] |

GRB 021004 | NOT/ALFOSC | 6.728 | +18.928 | 2.330 | 2.1(6) | −7(8) | [45] |

GRB 030329 | NOT/ALFOSC | 161.208 | +21.522 | 0.169 | 2.4(4) | 65(7) | [46] |

GRB 091018 | FORS2 | 32.192 | −57.55 | 0.97 | 3.25(35) | 57(6) | [49] |

GRB 091208B | HOWPol | 29.410 | 16.881 | 1.06 | 10.4(25) | −88(6) | [50] |

GRB 121024A | FORS2 | 70.467 | −12.268 | 2.298 | 4.83(20) | 173 | [51] ${}^{\u2020}$ |

**Table A3.**Sources selected from the Steward Observatory spectropolarimetric monitoring project [26]. The second column lists the highest observed polarization fraction during cycles 1–7 of the monitoring program. Coordinates have been obtained from the SIMBAD database [59]. Individual references are given for the red shifts. The last two columns give the position angle at the median energy of the linear fit and the differential change in position angle.

Source | ${\mathit{P}}_{\mathbf{max}}$ | RA | Dec. | Redshift | PA at 2.26 eV | $\mathit{\rho}$ | |
---|---|---|---|---|---|---|---|

[%] | J2000 [${}^{\circ}$] | J2000 [${}^{\circ}$] | z | [${}^{\circ}$] | [${}^{\circ}/\mathit{eV}$] | ||

3C 454.3 | 18.83 | 343.491 | +16.148 | 0.859 | [60] | 61.19(23) | −0.3(10) |

4C 14.23 | 20.32 | 111.320 | +14.420 | 1.038 | [61] | −34.42(18) | −0.7(7) |

4C 28.07 | 30.30 | 39.468 | +28.802 | 1.206 | [61] | −62.36(9) | 0.46(32) |

AO 0235+164 | 39.79 | 39.662 | +16.616 | 0.940 | [62] | −20.25(5) | 0.16(17) |

B2 1633+382 | 27.26 | 248.815 | +38.135 | 1.813 | [63] | −3.71(7) | −0.69(26) |

B2 1846+32A | 28.88 | 282.092 | +32.317 | 0.800 | [61] | 2.39(5) | 1.49(18) |

B3 0650+453 | 16.16 | 103.599 | +45.240 | 0.928 | [61] | 79.3(5) | 3.4(18) |

B3 1343+451 | 10.07 | 206.388 | +44.883 | 2.534 | [61] | 35.6(6) | −0.7(26) |

BZU J0742+5444 | 21.73 | 115.666 | +54.740 | 0.723 | [64] | −87.84(19) | 2.2(6) |

CTA 26 | 26.21 | 54.879 | −1.777 | 0.852 | [65] | 67.49(7) | −0.13(24) |

CTA 102 | 23.97 | 338.152 | +11.731 | 1.037 | [66] | 64.81(9) | 1.29(34) |

MG1 J123931+0443 | 33.61 | 189.886 | +4.718 | 1.760 | [63] | −72.28(8) | −0.02(32) |

OJ 248 | 18.09 | 127.717 | +24.183 | 0.941 | [63] | −79.23(11) | −0.9(4) |

PKS 0420−014 | 28.67 | 65.816 | −1.343 | 0.916 | [67] | 10.23(9) | 0.44(31) |

PKS 0454-234 | 35.27 | 74.263 | −23.414 | 1.003 | [60] | 2.49(6) | 0.22(20) |

PKS 0502+049 | 17.59 | 76.347 | +4.995 | 0.954 | [68] | −87.82(13) | 2.1(5) |

PKS 0805−077 | 28.27 | 122.065 | −7.853 | 1.837 | [60] | 41.7(4) | −1.4(11) |

PKS 1118−056 | 22.54 | 170.355 | −5.899 | 1.297 | [60] | 37.26(9) | 1.4(7) |

PKS 1124-186 | 10.49 | 171.768 | −18.955 | 1.048 | [62] | −83.18(19) | 0.2(7) |

PKS 1244−255 | 13.97 | 191.695 | −25.797 | 0.638 | [60] | −14.07(11) | 1.9(4) |

PKS 1441+252 | 37.70 | 220.987 | +25.029 | 0.939 | [61] | −72.34(8) | 0.51(28) |

PKS 1502+106 | 45.16 | 226.104 | +10.494 | 1.839 | [63] | 65.43(12) | 0.1(4) |

PKS 2032+107 | 12.36 | 308.843 | +10.935 | 0.601 | [61] | 68.4(10) | 9.2(34) |

PMN J2345-1555 | 32.69 | 356.302 | −15.919 | 0.621 | [61] | −0.46(4) | 1.30(15) |

S4 1030+61 | 37.71 | 158.464 | +60.852 | 1.400 | [63] | −59.69(12) | −1.3(4) |

SDSS J084411+5312 | 18.72 | 131.049 | +53.214 | 3.704 | [63] | 8.4(4) | −2.0(17) |

Ton 599 | 33.16 | 179.883 | +29.246 | 0.724 | [63] | −52.69(7) | 0.86(26) |

## Appendix B. Coefficient Distributions and Correlations

**Figure A1.**Distributions of all coefficients derived using the MCMC sampling of the coefficients space, marginalized over the nine other coefficients. The constraints listed in each panel are 5th–95th percentile.

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**Figure 1.**Difference of the observed polarization angle $\Delta \psi $ between photons of observer wavelengths of 1033 nm and 443 nm that were emitted with the same linear polarization angle ${\psi}_{z}=0$ (blue) and ${\psi}_{z}=20$ (orange) as a function of source red shift. The photons arrive from the direction of GRB 990510, and $\mathsf{\Re}\left({k}_{\left(E\right)21}^{\left(4\right)}\right)={10}^{-32}$ with all other SME coefficients set to 0, resulting in an angle of the eigenmode in the $Q-U$ plane of the Stokes space of $\xi =-24.8$.

**Figure 2.**Effective polarization of light observed through the HOWPol R-band filter arriving from the direction of GRB 091208B if the light at the source is 100% linearly polarized with a wavelength-independent polarization angle as a function of source redshift. For this illustration, the SME coefficients were set to 0, except for $\mathsf{\Re}\left({k}_{\left(E\right)21}^{\left(4\right)}\right)=2\times {10}^{-32}$. The depolarization is due to averaging over the bandwidth of the filter and the rotation of the polarization angle, as shown for example in Figure 1, as well as the change of linear into circular polarization described by the Müller matrix (Equation (10)). The combination of these two effects leads to the observed “ringing”. As described in Section 2, the effect depends on the linear polarization angle ${\psi}_{z}$ at the source.

**Figure 3.**Example of a spectropolarimetric measurement. The figure shows the polarization angle of 4C 14.23 as a function of photon energy observed on 23 November 2009. The red line is a linear fit in order to determine ${\rho}_{m}$ for comparison with Equation (28) to calculate the probability in Equation (29). All fit results are listed in Table A3.

**Figure 4.**Rotation of the linear polarization angle with energy according to Equation (28) as a function of the SME coefficient ${k}_{\left(E\right)20}^{\left(4\right)}$ while keeping all other SME coefficients at 0. The source was assumed to be at redshift $z=1$ with a codeclination of $\theta =90$°. Results are shown for three different values of the polarization angle at the source, ${\psi}_{z}$. For ${\psi}_{z}=45$°, the Stokes vector rotates out of the $Q-U$ plane, but the linear polarization angle does not change because ${Q}_{z}^{\prime}=0$ (see Equations (26) and (27)).

**Figure 5.**Probability $P(\rho <|{\rho}_{m}\left|\right|\left|\overline{\rho}\right|,{\sigma}_{\rho})$ according to Equation (29) as a function of the SME coefficient ${k}_{\left(E\right)20}^{\left(4\right)}$ while keeping all other SME coefficients at 0. A measured change in polarization of ${\rho}_{m}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}(0\pm 1){}^{\circ}/\mathrm{eV}$ was assumed. Source distance and direction are the same as in Figure 4, and colors have the same meaning. The roots of $\overline{\rho}$ in Figure 4 lead to the spikes in the probability function seen here.

**Figure 7.**Instrument-dependent modulation integrals $\mathcal{F}\left(\vartheta \right)$ according to Equation (33) for the filter transmission curves shown in Figure 6.

**Figure 8.**Source Stokes parameter ${U}_{z}^{\prime}$ as a function of ${\mathsf{\Psi}}_{m}$ calculated by numerically solving Equation (35) for ${\mathsf{\Psi}}^{\prime}={\mathsf{\Psi}}_{m}^{\prime}$ for different values of the function $\mathcal{F}\left(\vartheta \right)$.

**Figure 9.**Maximum observable polarization as a function of measured polarization angle ${\mathsf{\Psi}}_{m}^{\prime}$ and the oscillation integral $\mathcal{F}\left(\vartheta \right)$.

**Figure 10.**Maximum observable polarization fraction of GRB 091208B as a function of the SME coefficient ${k}_{\left(E\right)20}^{\left(4\right)}$ when keeping all other coefficient at 0.

**Figure 11.**Three typical examples of the integral in Equation (41) as function of the upper limit ${\mathsf{\Pi}}_{\mathrm{max}}$. The three cases illustrate a range of measured polarization fractions ${\mathsf{\Pi}}_{m}$ and uncertainties ${\sigma}_{m}$.

**Figure 12.**Probability that the observed polarization fraction of GRB 091208B is smaller than the maximum theoretically possible polarization fraction according to Equation (41) (see Figure 10) as function of the SME coefficient ${k}_{\left(E\right)20}^{\left(4\right)}$ when keeping all other coefficients at 0. The measured polarization fraction is 0.104(25).

Instrument | Filter | $\mathcal{N}$ [10^{−10} GeV] |
---|---|---|

FORS1 | R-band | 3.840 |

FORS1 | V-band | 3.926 |

FORS2 | ${\mathrm{R}}_{\mathrm{Special}}$ | 4.548 |

ALFOSC | R-band | 3.135 |

EFOSCV | V-band | 3.763 |

HOWPol | R-band | 3.611 |

**Table 2.**Limits at the 95% confidence level on all independent SME parameters ${k}_{\left(E\right)2m}^{\left(4\right)}$ and ${k}_{\left(B\right)2m}^{\left(4\right)}$ obtained in this analysis. The dependent parameters ${k}_{\left(E\right)2(-m)}^{\left(4\right)}$ and ${k}_{\left(B\right)2(-m)}^{\left(4\right)}$ can be calculated according to Equation (6).

$|{k}_{\left(E\right)20}^{\left(4\right)}|$ | < 2.4 × 10^{−34} |

$|Re\left({k}_{\left(E\right)21}^{\left(4\right)}\right)|$ | < 1.0 × 10^{−34} |

$|Im\left({k}_{\left(E\right)21}^{\left(4\right)}\right)|$ | < 1.6 × 10^{−34} |

$|Re\left({k}_{\left(E\right)22}^{\left(4\right)}\right)|$ | < 2.4 × 10^{−34} |

$|Im\left({k}_{\left(E\right)22}^{\left(4\right)}\right)|$ | < 2.6 × 10^{−34} |

$|{k}_{\left(B\right)20}^{\left(4\right)}|$ | < 1.5 × 10^{−34} |

$|Re\left({k}_{\left(B\right)21}^{\left(4\right)}\right)|$ | < 2.2 × 10^{−34} |

$|Im\left({k}_{\left(B\right)21}^{\left(4\right)}\right)|$ | < 1.4 × 10^{−34} |

$|Re\left({k}_{\left(B\right)22}^{\left(4\right)}\right)|$ | < 1.9 × 10^{−34} |

$|Im\left({k}_{\left(B\right)22}^{\left(4\right)}\right)|$ | < 2.5 × 10^{−34} |

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Kislat, F.
Constraints on Lorentz Invariance Violation from Optical Polarimetry of Astrophysical Objects. *Symmetry* **2018**, *10*, 596.
https://doi.org/10.3390/sym10110596

**AMA Style**

Kislat F.
Constraints on Lorentz Invariance Violation from Optical Polarimetry of Astrophysical Objects. *Symmetry*. 2018; 10(11):596.
https://doi.org/10.3390/sym10110596

**Chicago/Turabian Style**

Kislat, Fabian.
2018. "Constraints on Lorentz Invariance Violation from Optical Polarimetry of Astrophysical Objects" *Symmetry* 10, no. 11: 596.
https://doi.org/10.3390/sym10110596