# Searching for Axion-Like Particles with X-ray Polarimeters

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## Abstract

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## 1. Introduction

## 2. Axion-Like Particle Phenomenology

## 3. Methodology

- The null hypothesis ${H}_{0}$ that ALPs exist with coupling ${g}_{a\gamma \gamma}$ or higher to electromagnetism, and mass ${m}_{a}\lesssim {10}^{-12}$ eV.
- The alternative hypothesis ${H}_{1}$ that ALPs with coupling ${g}_{a\gamma \gamma}$ or higher to electromagnetism, and mass ${m}_{a}\lesssim {10}^{-12}$ eV do not exist.

- Randomly generate 1000 different magnetic field realisations ${\mathbf{B}}_{i}$ for the line of sight to NGC1275.
- For each ${\mathbf{B}}_{i}$, generate the ALP induced linear polarisation ${p}_{0}^{i}\left(E\right)$ and polarisation angle ${\psi}_{0}^{i}\left(E\right)$ spectra, by numerically propagating the initial photon vector through the cluster.
- From each $\{{p}_{0}^{i}\left(E\right),{\psi}_{0}^{i}\left(E\right)\}$ pair, generate 10 fake data sets by randomly sampling from Equation (6).
- Fit the no ALP constant model to each of the resulting 10,000 fake data sets, and find the corresponding likelihoods $\left\{{L}_{ga\gamma \gamma}^{i}\right\}$.
- If fewer than $5\%$ of the $\left\{{L}_{ga\gamma \gamma}^{i}\right\}$ are equal to or higher than ${L}_{\mathrm{no}\mathrm{ALP}}^{\mathrm{av}}$ (or ${L}_{\mathrm{no}\mathrm{ALP}}^{90}$ for the more pessimistic case), ${g}_{a\gamma \gamma}$ is excluded at the $95\%$ confidence level.

## 4. Results

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Projected Bounds with a Likelihood Ratio Test

- For each intrinsic source polarisation, simulate 1000 data sets $\left\{{D}_{i}\right\}$ with no ALPs present.
- Simulate transfer matrices for each value of g considered and for 100 different magnetic field configurations $\left\{{B}_{j}\right\}$.
- For each transfer matrix, find the final spectrum including ALPs for a range of different values for the intrinsic source polarisation degree ${p}_{\mathrm{lin}}^{\mathrm{source}}$ and angle ${\psi}^{\mathrm{source}}$. We take ${p}_{\mathrm{lin}}^{\mathrm{source}}=0-10\%$ in steps of $0.1\%$ and ${\psi}^{\mathrm{source}}=0-\pi $ in steps of $\frac{\pi}{100}$, and we use an interpolating function derived from this data for the maximisation procedure later on.
- We now fit the spectra with ALPs generated in the previous step to the fake data generated without ALPs. For each set $(g,{B}_{j},{D}_{i})$ we find the values of ${p}_{\mathrm{lin}}^{\mathrm{source}}$ and ${\psi}^{\mathrm{source}}$ that maximize the likelihood $L(g,{B}_{j},{p}_{\mathrm{lin}}^{\mathrm{source}},{\psi}^{\mathrm{source}}|{D}_{i})={\displaystyle \prod _{\mathrm{bins}}}{L}_{k}(g,{B}_{j},{p}_{\mathrm{lin}}^{\mathrm{source}},{\psi}^{\mathrm{source}}|{D}_{i})$. In each bin k, ${L}_{k}$ is the probability of measuring the ${p}_{\mathrm{lin}}$ and $\psi $ values given by ${D}_{i}$, given that the true values are those predicted by an ALP model with parameters $(g,{B}_{j},{p}_{\mathrm{lin}}^{\mathrm{source}},{\psi}^{\mathrm{source}})$. These are calculated from Equation (6). We thus obtain a set of maximised likelihoods $L(g,{B}_{j}|{D}_{i})$.
- For each value of g and each ${D}_{i}$, sort the $L(g,{B}_{j}|{D}_{i})$ obtained from different magnetic fields, and select the 95th quantile L value, and the corresponding magnetic field. We thus obtain a set of likelihoods $L\left(g\right|{D}_{i})$.
- For each ${D}_{i}$, find the value of g, $\widehat{g}$ that leads to the maximum $L\left(g\right|{D}_{i})$.
- We first consider the discovery potential of the data—i.e., the possibility of excluding a null hypothesis of no ALPs. For each ${D}_{i}$, we construct a test statistic $T{S}_{i}=-2\mathrm{ln}\left({\displaystyle \frac{L(g=0|{D}_{i})}{L(g=\widehat{g}|{D}_{i})}}\right)$.
- We have hence found the distribution of $TS$ under a null hypothesis of no ALPs. We find the threshold $TS$ value $T{S}_{\mathrm{thresh}}$ such that $95\%$ of the $T{S}_{i}$ are lower than $T{S}_{\mathrm{thresh}}$. This value can be used to demonstrate our discovery potential for ALPs, by finding the $TS$ for some of our fake data with ALPs included. We note that this test statistic does not obey Wilk’s theorem as our hypotheses are not nested.
- We now turn to excluding values of g. Our null hypothesis is now that ALPs exist with some coupling g, and the alternative hypothesis ${H}_{1}$ is that $g\le \widehat{g}$. ${H}_{1}$ obviously includes the case where ALPs do not exist, but excluding ALPs with $g\le \widehat{g}$ should not be possible. Our test statistic for each g is now $\lambda (g,{D}_{i})=-2\mathrm{ln}\left({\displaystyle \frac{L\left(g\right|{D}_{i})}{L\left(\widehat{g}\right|{D}_{i})}}\right)$.
- We take the median value of $\lambda (g,{D}_{i})$ over the ${D}_{i}$ to represent that g. So we now have simply $\lambda \left(g\right)$ for our test statistic.
- We now need the null distribution of $\lambda \left(g\right)$ under the hypothesis that ALPs exist with coupling g. Following [40], we assume that $\lambda \left(g\right)$ and the test statistic for a null hypothesis of no ALPs, $TS$ above, have the same distribution, and therefore $\lambda {\left(g\right)}_{\mathrm{thresh}}$ = $T{S}_{\mathrm{thresh}}$. In [40], this assumption is tested with simulations for part of the parameter space. We therefore exclude a value of g if $\lambda \left(g\right)>T{S}_{\mathrm{thresh}}$.

**Table A1.**Projected upper limits on ${g}_{a\gamma \gamma}$ with IXPE using the likelihood ratio method. The columns correspond to different intrinsic polarisations of the AGN.

$0\%$ | $1\%$ | $5\%$ | |
---|---|---|---|

${L}_{\mathrm{no}\mathrm{ALP}}^{\mathrm{av}}$ | $6\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{-1}$ | $9\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{-1}$ | $1.3\times {10}^{-12}\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{-1}$ |

- For zero intrinsic source polarisation, simulate 10 data sets $\left\{{D}_{i,g,B}\right\}$ for each $\{g,B\}$ pair, with g running from 1–13 ×${10}^{-13}$${\mathrm{GeV}}^{-1}$ in steps of $1\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{-1}$ and 5 different magnetic field configurations. We therefore have 50 fake data sets for each g.
- Fit each $\left\{{D}_{i,g,B}\right\}$ with spectra generated with ALPs of different g, as described in steps 2–7 above. (In this case we use 100 magnetic field configurations rather than 1000 in the interests of computational efficiency.) In this way we calculate a test statistic $T{S}_{i}$ for each $\left\{{D}_{i,g,B}\right\}$.
- We compare $T{S}_{i}$ with the $T{S}_{\mathrm{thresh}}$ calculated in step 8 above. If $T{S}_{i}>T{S}_{\mathrm{thresh}}$ we may exclude the no ALP hypothesis at the $95\%$ confidence level in that fake data set.
- For each g, we find the proportional of the 50 corresponding fake data sets for each the no ALP hypothesis is excluded.

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**Figure 1.**The photon survival probability; (

**a**) Stokes parameters (

**b**) and degree of linear polarization (

**c**) for initially unpolarized photons propagating from NGC1275 at the centre of Perseus, assuming the existence of ALPs with ${g}_{a\gamma \gamma}={10}^{-12}\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{-1}$. The spectra are convolved with an energy resolution of 450 eV.

**Figure 2.**Projected bounds on the ALP-photon coupling assuming a $5\%$ initial polarization. Adapted from [3]. We also include bounds set in [41]. These are based on the non-observation of high degrees of linear polarization in magnetic white dwarfs, using simplified models of the strong magnetic fields these objects.

**Table 1.**Projected upper limits on ${g}_{a\gamma \gamma}$ with IXPE. The columns correspond to different intrinsic polarisations of the AGN. The rows correspond to whether the average or 90th percentile likelihood value is used to characterize how well the no ALP model fits the simulated data.

$0\%$ | $1\%$ | $5\%$ | |
---|---|---|---|

${L}_{\mathrm{no}\mathrm{ALP}}^{\mathrm{av}}$ | $1.2\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $1.2\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $6\times {10}^{-13}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ |

${L}_{\mathrm{no}\mathrm{ALP}}^{90}$ | $1.4\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $1.3\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $1.2\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ |

**Table 2.**Projected upper limits on ${g}_{a\gamma \gamma}$ with Polstar. The columns correspond to different intrinsic polarisations of the AGN. The rows correspond to whether the average or 90th percentile likelihood value is used to characterize how well the no ALP model fits the simulated data.

$0\%$ | $1\%$ | $5\%$ | |
---|---|---|---|

${L}_{\mathrm{no}\mathrm{ALP}}^{\mathrm{av}}$ | $1.0\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $9\times {10}^{-13}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ | $7\times {10}^{-13}\phantom{\rule{3.33333pt}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}}^{-1}$ |

${L}_{\mathrm{no}\mathrm{ALP}}^{90}$ |

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**MDPI and ACS Style**

Day, F.; Krippendorf, S. Searching for Axion-Like Particles with X-ray Polarimeters. *Galaxies* **2018**, *6*, 45.
https://doi.org/10.3390/galaxies6020045

**AMA Style**

Day F, Krippendorf S. Searching for Axion-Like Particles with X-ray Polarimeters. *Galaxies*. 2018; 6(2):45.
https://doi.org/10.3390/galaxies6020045

**Chicago/Turabian Style**

Day, Francesca, and Sven Krippendorf. 2018. "Searching for Axion-Like Particles with X-ray Polarimeters" *Galaxies* 6, no. 2: 45.
https://doi.org/10.3390/galaxies6020045