# Search for Spectral Irregularities due to Photon–Axionlike-Particle Oscillations with the Fermi Large Area Telescope

###### Abstract

We report on the search for spectral irregularities induced by oscillations between photons and axionlike-particles (ALPs) in the -ray spectrum of NGC 1275, the central galaxy of the Perseus cluster. Using six years of Fermi Large Area Telescope data, we find no evidence for ALPs and exclude couplings above for ALP masses neV at 95 % confidence. The limits are competitive with the sensitivity of planned laboratory experiments, and, together with other bounds, strongly constrain the possibility that ALPs can reduce the -ray opacity of the Universe.

###### pacs:

## I Introduction

Axions and axionlike-particles (ALPs) are predicted by a variety of extensions of the Standard Model Jaeckel and Ringwald (2010); Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978); Witten (1984); Ringwald (2014). If produced non-thermally in the early Universe, these particles may account for all or a significant fraction of the cold dark matter (DM) (e.g. Preskill et al., 1983; Abbott and Sikivie, 1983; Marsh, 2011; Arias et al., 2012), and could be detected through their coupling to photons in magnetic fields Sikivie (1983); Raffelt and Stodolsky (1988). While the axion mass is proportional to its coupling to photons, these two parameters are independent in the case of ALPs.

Photon-ALP interactions could leave an imprint on -ray spectra, provided that the ALP mass is sufficiently small, eV. Above a critical energy photon-ALP mixing becomes maximal, leading to a reduction of the photon flux Hooper and Serpico (2007); Hochmuth and Sigl (2007); de Angelis et al. (2008). Around this is accompanied by spectral irregularities that depend on the strength and morphology of the magnetic field Wouters and Brun (2012). Photon-ALP conversions could also reduce the opacity of the Universe caused by pair production of rays with photons of the extragalactic background light (EBL) de Angelis et al. (2007); Simet et al. (2008). Evidence exists that the -ray absorption is indeed lower than expected from state-of-the-art EBL models De Angelis et al. (2009); de Angelis et al. (2011); Horns and Meyer (2012); Rubtsov and Troitsky (2014), and ALPs have been used to explain these observations Simet et al. (2008); Sánchez-Conde et al. (2009); Domínguez et al. (2011a); Meyer et al. (2013); Galanti et al. (2015) (see, however, Biteau and Williams (2015); Domínguez and Ajello (2015)).

Sources embedded in galaxy clusters are promising to search for ALPs due to the strong magnetic fields extending over large spatial scales in these systems. For example, the absence of irregularities above 200 GeV in the spectrum of the blazar PKS 2155-304, associated with a poor galaxy cluster, has been used to constrain the photon-ALP coupling Abramowski et al. (2013). Here, we focus on the search for irregularities in the spectrum of the radio galaxy NGC 1275 with the Fermi Large Area Telescope (LAT). NGC 1275 is the most favorable target since it is a bright -ray emitter detected with a significance exceeding in the third Fermi-LAT source catalog (3FGL) Ackermann et al. (2015a). Its broadband emission can be explained with synchrotron-self Compton models, which predict a smooth -ray spectrum Aleksić et al. (2014); Tavecchio and Ghisellini (2014). It is located at the center of the Perseus cool-core cluster for which rotation measures (RMs) suggest a high central magnetic field Taylor et al. (2006).

Our analysis makes use of the newest Pass 8 event-level analysis for LAT data. Compared to previous Passes, Pass 8 has an improved angular resolution, a broader energy range, larger effective area, as well as reduced uncertainties in the instrumental response functions (IRFs) Atwood et al. (2013).

## Ii LAT data selection

We make use of six years of LAT data taken between Aug. 4, 2008 and Aug. 4, 2014 in the energy range from 100 MeV to 500 GeV. For lower energies, the effective area decreases rapidly and the energy dispersion increases. At energies above 500 GeV we do not expect sufficient photon statistics Ackermann et al. (2016). We only consider events that arrive at a zenith angle in order to minimize the contribution of rays from the Earth limb. Time intervals that correspond to bright solar flares and -ray bursts are excluded. We extract -ray like events within a region of interest (ROI) centered at the position of NGC 1275: , Ackermann et al. (2015a).

Events passing the Pass 8 P8R2_SOURCE selection cuts are analyzed using the P8R2_SOURCE_V6 IRFs.^{1}^{1}1http://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm
An innovation of the Pass 8 IRFs is the possibility to subdivide an event class into
event types according to the quality of the angular
or energy reconstruction (PSF and EDISP event types, respectively).
In this analysis we will use the EDISP types to maximize our sensitivity to spectral irregularities.
Events are classified into one of four types
ranging from EDISP0 to EDISP3, that denote the quality of the energy reconstruction from worst to best.
All EDISP event types have a similar number of events in each logarithmic energy bin and are mutually exclusive.
The energy dispersion matrices are given in the Supplemental Material
supplements ().

## Iii Photon-ALP oscillations

Following (e.g. Grossman et al., 2002; Csáki et al., 2003; Mirizzi et al., 2008; de Angelis et al., 2011; Meyer et al., 2014), we derive the probability for a final state photon in the photon-ALP beam as a function of energy for an initially un-polarized photon beam (see the Supplemental Material). We expect the irregularities to occur around a critical energy Hooper and Serpico (2007),

(1) |

with ALP mass and plasma frequency in units of neV, coupling constant , and magnetic field . We include photon-ALP mixing in the intra-cluster and Galactic magnetic fields Simet et al. (2008); Horns et al. (2012). The field of the Milky Way is modeled with the coherent component of the model described in Jansson and Farrar (2012). We do not include its turbulent component, as the scales on which the turbulence occurs are usually smaller than the photon-ALP oscillation length. The turbulent intra-cluster field is described below. Absorption of rays by the EBL is taken into account through the model of Domínguez et al. (2011b). We neglect any oscillations in the intergalactic magnetic field (IGMF). With current upper limits on the IGMF strength of G and on the photon-ALP coupling, Ayala et al. (2014), we find that GeV only for . For such low masses, is further constrained below from the non-observation of rays from SN1987A Payez et al. (2015). Given this small coupling and the comparatively short distance to NGC 1275 (redshift ), no strong irregularities should be induced by mixing in the IGMF.

### iii.1 Intra-cluster magnetic field

Faraday RM observations and magneto-hydrodynamic simulations suggest that the magnetic field in galaxy clusters is turbulent and that its strength follows the electron density of the intra-cluster medium (ICM), Dolag et al. (2008); Dubois and Teyssier (2008); Feretti et al. (2012). We model the turbulent component as a divergence-free homogeneous isotropic field with Gaussian turbulence with zero mean and a variance Meyer et al. (2014). The energy density follows a power law in wave numbers . It is non-zero only between the minimum and maximum turbulence scales and .

For the Perseus cluster, we use derived from X-ray observations (Eq. (4) in Churazov et al. (2003)) within the inner kpc. Beyond this radius, we conservatively assume a zero magnetic field. RMs currently only probe the innermost region (tens of pc) around NGC 1275. The observations lead to an estimated central magnetic field of G Taylor et al. (2006). An independent lower limit of G for has been derived from MAGIC observations of the Perseus cluster Aleksić et al. (2012). These results motivate our assumptions for G and , which are also in line with observations of other cool-core clusters (e.g. Kuchar and Enßlin, 2011; Vacca et al., 2012).

For the turbulence spectrum, we assume values derived from RMs of the cool-core cluster A 2199 Vacca et al. (2012), which has a comparable number of member galaxies. The fiducial parameter choices are summarized in Tab. 1.

Parameter | Value |
---|---|

G | |

500 kpc | |

0.5 | |

0.7 kpc | |

35 kpc |

## Iv Data Analysis

We perform a binned Poisson likelihood analysis, similar to the DM signal search from dwarf spheroidal galaxies Ackermann et al. (2014, 2015b). Events are binned into sky maps with a resolution of per pixel. The width of the logarithmically spaced energy bins is chosen to be 30 % of the median energy resolution of each EDISP event type (see the Supplemental Material for details). This results in 39, 67, 94, and 145 energy bins for EDISP0-3, respectively. We have tested with simulations that bin sizes below 40 % of the median energy resolution do not affect the results.

For each event type,
we perform a fit over the entire energy range and ROI for all source parameters (nuisance parameters )
using gtlike included in the Fermi-LAT Science Tools version v10r01p01.^{2}^{2}2http://fermi.gsfc.nasa.gov/ssc/data/analysis/
We include all point sources listed in the 3FGL within from the ROI center.
The diffuse backgrounds are modeled with templates for the
Galactic and the isotropic extragalactic -ray emission.^{3}^{3}3http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html
The energy dispersion is taken into account
in the fitting of the point sources
whereas it is already accounted for in the
the data-driven derivation of the diffuse templates.
Normalizations of the diffuse sources and point sources within
from the ROI center are left free to vary.
All spectral indices of the point sources within are also free parameters.
The time-averaged spectrum of NGC 1275 is modeled with a logarithmic (log) parabola,
,
where is fixed to 530 MeV Ackermann et al. (2015a).

Under the assumption that the profiled nuisance parameters do not change when considering each bin separately Ackermann et al. (2014), we extract the likelihood in each reconstructed energy bin , as a function of expected counts of NGC 1275, and observed counts . For NGC 1275 a power law with fixed spectral index is now assumed in each bin. For each tested value of we re-optimize the normalization of the spectrum of the radio galaxy IC 310 which has an angular separation of from NGC 1275.

Under the ALP hypothesis, characterized by for one random turbulent -field realization , the expected number of photons is calculated through

(2) |

where the integration runs over the true energy bin , is the exposure, and is the energy dispersion for event type EDISP. Under the null hypothesis, reduces to the EBL attenuation. The parameters of the intrinsic source spectrum , , and , are further nuisance parameters. For each tested ALP parameter and magnetic field, we determine these parameters by profiling the joint likelihood of all energy bins

(3) |

for each event type separately, using the pre-computed likelihood curves .
In this way, we treat each event type selection as an independent measurement.^{4}^{4}4
This procedure will result in different best-fit estimators for the source parameters for each event type.
In this way, it is possible to speed up the optimization considerably.
We have verified that our results do not change when the parameters of NGC 1275 are tied
over the event types.
The bin-by-bin likelihood curves for the EDISP3 event type are shown in Fig. 1
together with the best-fit spectra.

We simulate random realizations of the turbulent field , . The dependence of the likelihood on the realizations is not easily parametrizable and we cannot assume that the simulations map the space of possible realizations. Therefore, instead of profiling, we sort the -field realizations for each tested pair by increasing values of the product over the likelihoods and use the realization that corresponds to the quantile of the likelihood distribution (profiling would correspond to ). We will denote this realization as and the corresponding expected counts with . Note that might be different for different ALP parameters, so that .

Similar to Wouters and Brun (2013), we evaluate the ALP hypothesis with a likelihood ratio test. The test statistic () for the ALP hypothesis is calculated from the joint likelihood of all event types:

(4) |

where are the expected counts for the null (no ALP) hypothesis with maximized nuisance parameters and are the expected counts under the ALP hypothesis that, together with , maximize the likelihoods of each event type. We test ALP parameters on a logarithmic grid with steps where and . The mass range is chosen such that falls into the analyzed energy range whereas the maximum coupling is motivated by the bound found in Ayala et al. (2014). For the lower bound, the amplitude of the irregularities is too small to be detectable.

In order to convert the value into a significance, we need to know the underlying probability distribution. We derive the null distribution from Monte-Carlo simulations and from it the threshold value, , for which we can reject the null hypothesis (see the Supplemental Material for details). For a rejection of the no-ALP hypothesis at a (global) significance level, we find that .

## V Results

The best-fit ALP parameters are found at and with , and hence the best fit with ALPs is not significantly preferred over the null hypothesis. We set upper limits by stepping over the ALP parameters and calculating the difference between the log-likelihood values for each pair and the best fit. ALP parameters are excluded with confidence if . The threshold value is calculated under the assumption that the probability distribution of the alternative hypothesis follows the null distribution. We have tested this assumption with simulations and found that this choice results in over coverage for ALP parameters causing the strongest irregularities, thus yielding conservative limits.

The excluded parameter space is shown in the left panel of Fig. 2 (black shaded region). Photon-ALP couplings are ruled out between for and for . At high masses, the limits run almost parallel to the lines of constant (shown as dotted lines for ). For lower masses, ALP couplings along the GeV line with are not excluded. Around this “hole”-like feature, exhibits rapid fluctuations for almost the entire Fermi-LAT energy range. Given the Poisson noise in the data, these ALP parameters cannot be excluded. We stress that the fit with ALPs is not preferred over the null hypothesis. For masses below , irregularities still enter the Fermi-LAT energy range allowing to exclude ALP parameters.

The observed limits agree well with the expected exclusion region derived from Monte-Carlo simulations (shaded regions). The “hole” feature is not visible in the expected limits but occurs in certain Monte-Carlo realizations (an example is given in the Supplemental Material). In 5 % of the simulations (yellow shaded region), ALP parameters are excluded for which the GeV. This is expected since we have derived from the null distribution where for 5 % of the simulations one finds . The parameters for which we could detect an ALP signal at a level agree well with the observed limits (gray hatched region; see the Supplemental Material for details).

The results are subject to systematic uncertainties related to the analysis and magnetic field parameters. Concerning the analysis, changing the energy dispersion has the strongest effect on the limits. If we conservatively broaden the energy dispersion by 20 % the area of the tested ALP parameter grid with decreases by 25 %. All other tested effects related to the analysis change the limits at most by %. Concerning the choice of -field parameters, neither the strength, the power spectrum, nor the dependence on the electron density of the magnetic field are well established for Perseus. Therefore, the full analysis is repeated for a magnetic-field strength of G, for a Kolmogorov-type turbulence spectrum, (as found in the cool-core cluster Hydra A, (e.g. Kuchar and Enßlin, 2011)), and by conservatively assuming that the magnetic field is zero beyond kpc. Increasing increases the excluded area by 43 %. In comparison, the other tested parameters have a subdominant effect of maximally 16 %. The dependence of the limits on the particular choice of the EBL model is negligible due to the relative proximity of NGC 1275 (). The absorption is maximally at 500 GeV with significantly smaller relative differences for a number of EBL models Domínguez et al. (2011b); Franceschini et al. (2008); Finke et al. (2010); Kneiske and Dole (2010); Gilmore et al. (2012); Inoue et al. (2013). We provide a comprehensive summary of all tested systematic uncertainties in the Supplemental Material.

The limits derived in this work are compared to other limits and sensitivities of future experiments in Fig. 2 (right). Our results give the strongest constraints to date for and surpass the expected limits for the planned ALPS II experiment Bähre et al. (2013) in that range. They are only a factor of below the exclusion prospects of the planned IAXO experiment Irastorza et al. (2013). We note that the systematic uncertainties of the future experiments are likely to be smaller than the ones that apply to the present analysis. In conjunction with other limits taken at face value Abramowski et al. (2013); Wouters and Brun (2013); Payez et al. (2015), the parameter space where ALPs could explain hints for a lower -ray opacity compared to EBL-model predictions (light blue region, Meyer et al. (2013)) is now strongly constrained. The limits do not constrain ALPs that could make up the entire DM content of the Universe. This corresponds to the region in Fig. 2 (right) below the line, where is a model dependent factor and is the misalignment angle Arias et al. (2012). Our analysis only constrains ALPs that make up less than 4 % of the DM, or equivalently .

Observations with future -ray instruments could improve the reported limits and test ALP DM models. The planned Gamma-400 satellite, with an envisaged energy resolution of 1 % above 10 GeV Cumani et al. (2015), might be able to better resolve the spectra and probe higher ALP masses. Higher masses could also be reached with the future Cherenkov Telescope Array (CTA) Actis et al. (2011).

It will be possible to reduce the uncertainties of the intra-cluster field with the upcoming Square Kilometer Array (SKA) that will conduct a full-sky polarisation survey Gaensler et al. (2004). It is expected that SKA will observe hundreds of RMs of background sources for the most massive clusters, thereby enabling a more precise determination of their magnetic fields Bonafede et al. (2015).

The analysis presented here can be easily extended to other sources that reside in clusters (e.g. M 87 in the Virgo cluster) or in general to any source where ALP-induced spectral irregularities are expected. ALP parameters not constrained in the present analysis (such as those of the “hole”-like feature) could be probed with the different -field configurations in other sources.

###### Acknowledgements.

## Vi Acknowledgments

The Fermi-LAT Collaboration acknowledges support for LAT development, operation and data analysis from NASA and DOE (United States), CEA/Irfu and IN2P3/CNRS (France), ASI and INFN (Italy), MEXT, KEK, and JAXA (Japan), and the K.A. Wallenberg Foundation, the Swedish Research Council and the National Space Board (Sweden). Science analysis support in the operations phase from INAF (Italy) and CNES (France) is also gratefully acknowledged.

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Supplemental Material: Search for Spectral Irregularities due to Photon–Axionlike-Particle Oscillations with the Fermi Large Area Telescope

## Appendix A Energy dispersion and spectral binning

Event reconstructions prior to Pass 8 allowed to perform an independent analysis of -ray events that convert in either the front or back part of the LAT tracker, e.g. Ackermann et al. (2012). With Pass 8 this is now supplemented with additional event types that make it possible to select events based on the quality of the angular reconstruction (PSF event types), or the quality of the energy reconstruction (EDISP event types). Event types within the PSF or EDISP partition are mutually exclusive and each energy energy bin contains about the same number of events of each type. The analysis presented here is performed for each EDISP type independently. The EDISP event types are defined through the cuts on energy dependent boosted decision tree variables used for the event classification that determine the quality of the event reconstruction.

The exposure-averaged energy dispersion matrix for each event type EDISP () transforms the number of counts in true energy () in the -th energy bin to the number of counts in the -th bin of reconstructed energy (), , via

(1) |

We extract for six years of observation of NGC 1275 between 30 MeV and 1 TeV in 160 logarithmic-spaced energy bins and evaluate it at the ROI center, i.e. the position of NGC 1275. The resulting matrices are shown in Fig. 3 (left). Clearly, the energy resolution improves from event type EDISP0 to EDISP3.

From the we also compute the 68 % containment half-width energy resolution in the same way as described in Ackermann et al. (2012) (Fig. 3, right). At 10.2 GeV the energy resolution improves from 16 % for EDISP0 to 4 % for EDISP3 (using the half width of ). From the full-width energy resolution, we choose the number of logarithmic spaced energy bins used in the analysis. The number of bins should be small enough to resolve the photon-ALP spectral features after convolving them with the energy dispersion. As a compromise between computational time and energy resolution, we choose a bin size of 30 % of the median energy resolution . The total number of bins for each event type EDISP is then

(2) |

with MeV and GeV. Since we forward fold the model spectrum with the detector response (Eq. (2) in the main article), bin-by-bin correlations are self-consistently accounted for.

## Appendix B Modelling Photon-ALP oscillations

We describe the photon-ALP beam of energy in terms of the density matrix , where is a photon-ALP state with photon polarizations and the ALP field strength . Assuming the beam to propagate along the direction and neglecting photon absorption, the evolution of the system along is described with the Von-Neumann-like commutator equation, e.g. Mirizzi and Montanino (2009),

(3) |

with the mixing matrix . The photon-ALP mixing is induced by the transversal magnetic field only. For a homogenous field orientated along , , the mixing matrix reads Raffelt and Stodolsky (1988); Mirizzi et al. (2007),

(4) |

where Faraday rotation is neglected. The terms give the momentum differences between the polarization states. The terms and describe the propagation of photons in a plasma with plasma frequency , where . They also include the QED vacuum polarization through , with the critical magnetic field G and the fine-structure constant . For the energies and magnetic fields under consideration, this term is, however, completely negligible. The same holds for an additional photon dispersion term Dobrynina et al. (2015) since we neglect photon-ALP mixing in the intergalactic medium beyond the galaxy cluster. Photon-ALP mixing is induced by the off-diagonal term . The kinetic term for the ALP is . Numerical values for the momentum differences in suitable units are given in e.g. Horns et al. (2012). The oscillation length is given by , where . For a 10 GeV ray and , , and , one finds kpc. For the considered profile of the electron density (Eq. (4) in Churazov et al. (2003)), the plasma frequency is at most 10 % of the minimum ALP mass considered. Therefore, , making the critical energy (Eq. (1) in the main article) essentially independent of .

Equation (3) is solved by means of the transfer matrix , , with the initial condition , e.g. Csáki et al. (2003); de Angelis et al. (2007); Sánchez-Conde et al. (2009); Bassan et al. (2010). Neither the magnetic field in the Perseus cluster nor the Galactic magnetic field are homogeneous. Therefore, the path in the different fields is split up into segments. In each segment, is assumed to be constant and forms an angle , with the axis. The field is modeled as a a divergence-free homogeneous isotropic field with Gaussian turbulence Meyer et al. (2014) The full transfer matrix is then

(5) |

In this setup, the probability to observe a photon of either polarization after the -th domain is given by

(6) |

For the initial polarization, we assume an un-polarized pure -ray beam, . For the fiducial parameters of the Perseus cluster, the Galactic magnetic field of Ref. Jansson and Farrar (2012), and the EBL model of Ref. Domínguez et al. (2011b), the photon survival probability for one ALP mass and coupling is shown in Fig. 4 for 500 random realizations of the cluster magnetic field.

## Appendix C Null distribution and confidence intervals

The usual procedure to convert a value into a significance with which one can reject the null hypothesis using Wilks’ Theorem Wilks (1938) cannot be applied in the present analysis. First of all, the spectral irregularities do not scale linearly with the ALP parameters. Secondly, under the null hypothesis, the likelihood values are independent of the magnetic-field realizations which are additional nuisance parameters under the alternative hypothesis. Thirdly, photon-ALP oscillations are completely degenerate in coupling and magnetic field since the oscillation strength scales with , leading to potential correlations between the different tested ALP parameters. For these reasons, the null distribution is a priori unknown and needs to be derived from Monte-Carlo simulations.

We simulate six year data sets (pseudo experiments, PEs) within a ROI including all 3FGL point sources and the isotropic backgrounds for each event type between 30 MeV and 1 TeV. We perform these simulations with an energy window and ROI larger than the ones used in the analysis in order to account for a possible migration of photons between adjacent energy bins or sky regions due to the PSF and energy dispersion. All point sources, including NGC 1275, are simulated with energy dispersion, whereas we disable it for the diffuse templates. The actual pointing history of the LAT is used in the simulations. No ALP signal is included. We perform the exact same analysis as done on the data on each PE and calculate the values as the difference between the maximum likelihood and the profile likelihood of the fit without ALPs. For one PE, we show the resulting exclusion region in comparison with the data in Fig. 5 (left). The figure illustrates that the complex behavior of the exclusion region is already evident in Monte-Carlo simulations. The full null distribution is shown in Fig. 5 (right). It can be parametrized with a non-central distribution with 10.09 degrees of freedom (d.o.f.) and non-centrality parameter . For these parameters the Kolmogorov-Smirnov test yields a -value of 0.70. All values are larger than zero indicating that all fits including ALPs are preferred. This is expected since we have deliberately chosen a high quantile . Selecting a smaller quantile would shift the distribution towards zero as -field realizations would be selected that lead to a worse fit to the data. In order to avoid false ALP detections we use the null distribution to derive the threshold value, , for which evidence for ALPs could be claimed at a 3 (2) confidence level. From the fit to the null distribution one finds that (). The uncertainties are at 68 % confidence and are derived from bootstrapping the null distribution times.

Since we are deriving from simulations, one could ask why we are choosing instead of profiling, i.e., . The reason is that we can only test a limited number of -field realizations, thereby probably missing the one actually realized in nature. While the profile likelihood () is sensitive to the number of tested -field realizations, we find that with our analysis is insensitive to the ad-hoc choice of number of realizations. Furthermore, we find better coverage for this choice, as discussed below.

We derive the ALP parameters that are detectable at a certain significance using an Asimov data set, i.e. setting the observed counts equal to the expected number of counts Cowan et al. (2011). For each ALP parameter and -field realization we calculate the expected number of counts using Eq. (2) of the main article assuming the spectral parameters of NGC 1275 that are given in the 3FGL. The obtained Asimov data are subsequently fitted with a log parabola without an ALP contribution. As before, for each tested ALP parameter pair, we sort the likelihood values for the different -field realizations and evaluate the test statistic for the realization that corresponds to the quantile,

(7) |

where the subscript A denotes the Asimov data set. An ALP signal could be detectable at a level where , shown as the gray hatched region in Fig. 2 (left) of the main article.

For the same reasons that make it necessary to derive the null distribution from Monte-Carlo simulations,
we cannot assume that it is possible to derive upper limits on the ALP mass and coupling by increasing
the log likelihood by .^{5}^{5}5This value
corresponds to a confidence limit in case of a distribution with 2 d.o.f. (ALP mass and coupling).
The values are given by the log-likelihood ratio test between the likelihood with a certain ALP mass and coupling with
conditionally maximized nuisance parameters and the unconditionally maximized likelihood,

(8) |

In the above equation, , , and denote the maximum likelihood estimators. As before, we have chosen the magnetic field that corresponds to the quantile of the distribution of likelihoods with respect to the -field realizations. Ensuring coverage would in principle require to simulate the distribution of (alternative distribution) for the complete parameter space, which is not feasible. We therefore make the ansatz that the alternative distribution for each tested ALP parameter is equal to the null distribution. From the non-central fit to the null distribution we find that corresponds to a 95 % upper limit (cf. Fig. 5, right). For this choice, the expected limits (median sensitivity) agree well with the ALP parameters for which a signal could be detected (see Fig. 2, left, of the main article). We examine the coverage with simulations where we multiply the intrinsic spectrum of NGC 1275 with the photon survival probability including ALPs. We test seven ALP parameter pairs with masses between 1 and 60 neV and couplings between 0.5 and , and simulate 100 PEs for each parameter pair. By construction, we find correct coverage for ALP parameters that result in weak irregularities or irregularities largely outside the Fermi-LAT energy range. In these cases we expect the alternative distribution to be close to the null distribution. For masses and couplings where the ALP effect is strongest, we consistently find over coverage, thus making our results conservative. The over coverage becomes more severe if we select or if we treat the magnetic field in a Bayesian fashion by taking the average over the -field realizations. For , the trend reverses and we find under coverage.

We have also carried out a Bayesian analysis with flat priors on the magnetic-field realizations and logarithmic flat priors for and . We find the limits corresponding to a confidence level by integrating over the posterior probability where we sort by decreasing likelihood values. Studying the coverage of the limits, we find under coverage for the same injected ALP signals as above. Therefore, we compare the 99.9 % C.L. Bayesian limits (resulting in coverage ) with the frequentist limits which give coverage in Fig. 6. The Bayesian limits exclude 3 % less of the parameter space than the frequentist analysis. The shape of the excluded region is comparable in the two analyses.

## Appendix D Systematic Uncertainties

We investigate how the systematic uncertainties connected to the LAT instrument, to the analysis choices, and to the choices for the magnetic-field parameters affect the derived bounds on the ALP parameters. We quantify the importance of each systematic uncertainty by studying the change of the excluded area , i.e. the area of the tested ALP parameter space for which . All examined systematics and the respective values of are summarized in Tab. 2. In the following, we discuss each source of systematic uncertainty in further detail.

Systematic test | |
---|---|

Bin-by-bin power-law index | % |

Bracketing of the effective area | % |

Broadening of energy dispersion by 20 % | % |

Central magnetic field G | % |

Kolmogorov turbulence spectrum | % |

Maximum spatial extent of magnetic field kpc | % |

#### d.0.1 LAT instrument and analysis

##### Assumed power-law index for bin-by-bin likelihood.

In the extraction of the likelihood curves as a function of the expected counts in each energy bin we have assumed a power law with spectral index for NGC 1275. Since the bin size is chosen as 30 % of the median energy resolution, the bin-by-bin likelihood should essentially be independent of the assumed spectral shape. We test this by repeating the ALP the analysis for power-law indices of and . As expected, the excluded area in the tested parameter space only changes marginally. Indeed, it decreases by for whereas it is unchanged for .

##### Bracketing of the effective area.

The systematic uncertainty for the effective area is assessed
using the IRF bracketing method Ackermann et al. (2012).
The exposure is scaled with , where
is the relative uncertainty of the effective area and we take .
For an analysis making use of the individual event types it is recommended
to assume a relative uncertainty of 10 % between 100 MeV and 100 GeV.
It increases linearly with the logarithm of energy to 20 % at 30 MeV and 1 TeV^{6}^{6}6http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT_caveats.html.
Using this scaling and repeating the analysis, we find a decrease by
for .
For , the results remain unchanged.
The small change of the limits is anticipated since we are interested
in spectral irregularities that are not affected by an overall scaling of the effective area.

##### Small-scale structures in the effective area.

We assess the possibility that unmodeled structures exist in the effective area.
Such structures might mimic spectral irregularities induced by ALPs.
We follow a similar approach as in Ackermann et al. (2015c) and estimate
the strength of such structures from the residuals of a spectral analysis of bright -ray sources.
At low energies (100 MeV–5 GeV) we consider the Vela pulsar and for higher energies (5 GeV–100 GeV) the Earth limb.
The basic data selections are summarized in Tab. 3.^{7}^{7}7For the actual data analysis of the Vela pulsar and the Limb we use slightly broader energy windows.
All other analysis choices are the same as for the main analysis of NGC 1275.

For the Vela pulsar, we calculate the pulsar phases using the standard ephemeris^{8}^{8}8http://www.slac.stanford.edu/~kerrm/fermi_pulsar_timing/ and the tempo2 package^{9}^{9}9http://www.atnf.csiro.au/research/pulsar/tempo2/.
The spectral parameters of the background sources are determined separately for each EDISP event type
in the off-pulse interval using the standard binned analysis pipeline of the Fermi-LAT Science Tools.
The parameters are taken as an input for the phase averaged fit including the Vela pulsar.
As in the 3FGL, the Vela spectrum is modeled with a power law with a super-exponential cutoff.^{10}^{10}10
The spectral model for a power law spectrum with super-exponential cutoff is defined though
.
We show the fit residuals over the entire ROI in Fig. 7.

The Earth limb rays are produced by cosmic-ray interactions in the Earth’s upper atmosphere. Therefore, the limb data set predominantly comprises rays that arrive at a rocking angle and in the zenith angle range (On region). For the astrophysical background, photons within and are considered (Off regions). The zenith angle cuts are chosen such that the solid angle for the On and Off regions is equal to , respectively Ackermann et al. (2012). The expected number of counts in the energy bin of width for each event type is given by

(9) |

where is the exposure averaged over the full sky. The limb spectrum is modeled with a smooth broken power law to account for the effect of the Earth’s geomagnetic cutoff. Following Albert et al. (2014), we fix the parameters controlling the spectrum below and around the cutoff, namely GeV, , and , whereas the normalization and the spectral index above the cutoff are free parameters. The reference energy is fixed to the central energy of the energy bin where the number of excess counts () divided by its uncertainty is maximal. The expected number of counts of the background is calculated in the same way, assuming however a simple power law as the spectral model. We neglect the energy dispersion since both spectra for the limb and the background are smooth functions in energy that do not exhibit a cutoff in the considered energy range. The fit residuals above 5 GeV are shown in Fig. 7.

We use the residuals to estimate the maximum possible effect of small-scale structures in the exposure in the following way. We define a correction function for each event type that is equal to if the amplitude of is smaller than the absolute value of the fractional residual at energy . Otherwise, is equal to the maximum amplitude of the absolute value of the fractional residual at energy . We then divide each exposure by the corresponding . In this way, irregularities with an amplitude smaller than the fractional residuals will be removed and irregularities with larger amplitudes will be reduced. The maximum amplitude of the functions is given by the maximum of the absolute value of the residuals in Fig. 7. However, we find that the limits are practically unchanged. The reason is that ALP parameters that are excluded at % confidence lead to irregularities of a larger amplitude than what can be counter-acted by possible small scale structures in the exposure.

Vela pulsar | Earth limb | |

Time range (MET) | 239846401–403747203 | 239557417–428859819 |

Energy range | 100 MeV–10 GeV | 1 GeV–100 GeV |

LAT rocking angle | – | |

ROI size | Full sky |

##### Broadening of the energy dispersion.

We investigate the effect on the ALP limits in the case that energy resolution is worse than anticipated. The energy dispersion matrix of each event type is broadened along the reconstructed energy axis by the transformation . The ALP analysis is repeated with the degraded energy dispersion for , and . The results are shown in Fig. 8 (left panel). The fraction of excluded ALP parameters on the tested grid decreases by 6 %, 11 %, and 25 %, respectively, for the choices of above. The effect of the degraded energy dispersion is most visible around the “hole”-like feature where fast oscillating irregularities are present over the entire energy range accessible to the LAT. These irregularities are more likely to be flattened out by the degraded energy dispersion.

#### d.0.2 Intra-cluster magnetic field

The magnetic field of the Perseus cluster on scales of tens and hundreds of kpc is unknown due to the lack of Faraday rotation measurements Taylor et al. (2006). We therefore investigate how the limits change if we alter the fiducial assumptions for the -field calculation.

For fixed ALP parameters of and , Fig. 9 shows the difference in the likelihood that is obtained for different magnetic-field parameters. Higher values of correspond to stronger exclusions. In each panel, all other -field parameters are fixed to their fiducial values. We choose again the random -field realization that results in the quantile.

The top right panel shows the dependence on the central magnetic field. Increasing the field from 2 G to 7 G leads to a sharp increase in . Above 7 G, the photon-ALP mixing is maximal and changing the -field changes the shape of the irregularities (but not their amplitude) leading to the fluctuations of . In the top central panel, the dependence of the radial magnetic field on the electron distribution is increased, where higher values of correspond to a stronger decrease of the field with radius. Surprisingly, the likelihood difference increases above . Inspecting the photon survival probability one sees that the very rapid oscillations become suppressed for higher values of whereas larger features are more pronounced leading to stronger exclusions. The situation is similar when one increases the distance from the cluster center up to where we assume a non-zero field (top left panel of Fig. 9). Naively one would assume that the considered ALP parameters should be more constrained for larger cluster radii. However, while the overall amplitude of the oscillation does not change, the frequency decreases with smaller radii leading to potentially broader irregularities that are not washed out by the energy dispersion.

The lower row of Fig. 9 shows the dependence on the parameters of the turbulence spectrum. Changing these parameters will also change the coherence length of the transversal magnetic field Meyer et al. (2014),

(10) |

where is the axis along the line of sight. The fiducial parameter set results in a coherence length of kpc. For softer turbulence spectra, i.e. smaller values of , the coherence length increases as large spatial scales become more dominant. This leads to broader irregularities in energy, and, subsequently, to larger ALP exclusion areas. For , the turbulence spectrum resembles white noise resulting in fast oscillations of with energy.

The situation is less obvious when changing the minimum wave number, (or maximum turbulence scale). Even though slightly decreases with increasing , strong maxima are visible in the likelihood difference (bottom central panel of Fig. 9). These correspond to cases where exhibits strong isolated peaks at the onset of the irregularities that are in strong tension with the data. These peaks are also seen in Fig. 4 between 100 MeV and 1 GeV, however not as isolated. Our fiducial parameter choices do not lead to such features and can be regarded as conservative in this respect.

Increasing the maximum wavenumber above its fiducial value has only a very small effect on the likelihood and the coherence length. With increasing , the frequency of the oscillations around the critical energy is also increased, yet, when folded with the energy dispersion, these features are not detectable anymore.

In conclusion, the derived ALP limits depend sensitively on the assumed -field parameters. We quantify the effect further by repeating the limit calculation for all ALP parameters but changing one -field parameter at a time. The excluded regions are shown in Fig. 8 (right). We limit ourselves to , , and . Increasing to G allows us to exclude smaller values of the photon-ALP coupling thanks to the full degeneracy of and . The higher value of is close to the measurement reported in Taylor et al. (2006) of G. As a result, the excluded region of ALP parameters increases by . Similarly, assuming a Kolmogorov-type turbulence spectrum, , as observed e.g. in Hydra A Kuchar and Enßlin (2011) and Coma Bonafede et al. (2010), increases the excluded parameter region by . In this case, the coherence length increases to kpc. Decreasing the maximal distance up to which the magnetic field is non-zero to 100 kpc leads to a decrease by , even though it might increase the significance of the exclusion for some ALP parameters as noted above.