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DEUTSCHES ELEKTRONEN- ( |J^£ Ein Forschungszentrum der Helmholtz-Gemeinschaft v^l DESY 09-080 June 2009

Searching for Axion-Like-Particles in the Sky

C. Burrage Deutsches Elektronen-Synchrotron DESY,

ISSN 0418-98.33

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Searching for axion-like-particles in the sky C. Burragea∗ aTheory Group, Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany

If dark energy couples to the fields of the standard model we can hope to detect or constrain it through non- gravitational effects. If the dark energy field couples to it behaves as an Axion-Like-Particle (ALP). ALPs mix with photons in the presence of magnetic fields and hence affect astronomical observations. We show that empirically established luminosity relations can be used as a new test for ALPs and that when applied to observations of active galactic nuclei this is highly suggestive of the existence of a very light ALP.

DESY 09-080

1. Introduction the allowed Lagrangian terms for axions include a coupling to photons: The remarkable observation that the expan- sion of the universe is accelerating has lead to φ µν λρ φ L ⊃ µνλρF F = − E · B. (1) a wide variety of possible theoretical explana- 4M M tions. Amongst such models it is common to We therefore define an axion-like-particle to be suppose that the acceleration is driven by a new any scalar or pseudo-scalar field with a coupling light scalar ‘quintessence’ field which permeates to two photons where, instead of (1), a scalar ALP the universe. If such a new scalar field exists we has the following interaction term might naively expect it to couple to the fields of φ µν φ 2 2 the standard model. Such a coupling would give L ⊃ Fµν F = (B − E ). (2) 4M 2M rise to a new fifth force mediated by the scalar field and provide new decay modes for standard We no longer demand any relationship between model particles. It is a challenge to quintessence mass and coupling; mφ and M are treated as free models to explain why such effects should be sup- parameters to be constrained. pressed. However a coupling between dark energy There are a number of dark energy candidates and the standard model would mean that dark which behave as axion-like-particles. In coupled energy could be constrained, or detected, by new quintessence models [3] a light scalar fields cou- non-gravitational effects. ples weakly to the fields of the standard model. In this article I will focus on the possibility of There are also axionic dark energy models [4,5] a coupling between the scalar field and photons. in which the dark energy field is a pseudo-scalar. Fields with such couplings are generically known The chameleon model of dark energy allows a as Axion-Like-Particles (ALPs), in analogy with scalar field to couple strongly to the standard the properties of axions. Axions are pseudo-scalar model fields [6–8], without violating fifth force particles first proposed in 1977 to solve the strong constraints because, through self interactions, the CP problem of QCD [1,2]. The mass of an axion field acquires a density dependent mass. The is proportional to the strength of its coupling, and properties of a universal chameleon field evolve with the universe and at late times it behaves as quintessence. ∗I would like to acknowledge support from the German Science Foundation (DFG) under the Collaborative Re- As no axion-like-particle has been observed, search Centre (SFB) 676. any model including such fields is constrained 2 by the results of experiments. In this arti- mixes with the ALP. (3) can be diagonalized and cle we will be interested in ALPs with masses solved to give the mixing matrix −12 mφ . 10 eV, and the constraints on the  ei∆1(z) 0  couplings of such light particles are: For scalar M(z) = U −1 U, (5) −i∆2(z) ALPs, 1026 GeV < M, from laboratory con- 0 e straints on fifth forces and weak equivalence 2 where ∆1(z) = ∆(z) cos θ/ cos 2θ and ∆2(z) = principle violation [9]. For pseudo-scalar ALPs, ∆(z) sin2 θ/ cos 2θ. ∆(z) = m2 z/4ω, tan 2θ = 1011 GeV M, from observations of the neu- eff . 2Bω/Mm2 , m2 = |m2 − ω2 | and trino burst from SN 1987A [10]. For chameleonic eff eff φ P ALPs, 109 GeV M, from observations of the  cos θ − sin θ  . U = . (6) polarization of starlight [11]. sin θ cos θ In the following section I will review the mixing Therefore in the presence of a background mag- of axion-like-particles and photons in the pres- netic field the probability that a suitably polar- ence of magnetic fields. In Section 3 I discuss ized converts into an ALP is how this could affect astronomical observations, focusing on environments in which the mixing is  ∆(z)  P = sin2 2θ sin2 . (7) strongest. I describe, in Section 4, a new test for cos 2θ axion-like-particles which looks for their effects If all other variables are fixed, as the frequency of in the empirically established relations between the light is increased the probability of conversion the high and low energy luminosities of certain increases and becomes frequency independent. astrophysical sources. In Section 5 I show that The mixing has two effects; photon number is when applied to observations of active galactic no longer conserved, instead the total number of nuclei this gives results highly suggestive of the photons and ALPs is conserved, and the polar- existence of a very light axion-like-particle. ization of a light beam is distorted by the mixing of photons and ALPs. These optical effects of 2. Optics with ALPs ALPs can be searched for directly in the labora- The couplings (1) and (2) mean that in the tory, however the constraints from such experi- presence of a background magnetic field a pho- ments are not yet as stringent as those coming ton can oscillate into an ALP (and vice versa). from the astrophysical consequences of ALPs. The wave equation describing photons and scalars traveling through a magnetic field of strength B 3. Astronomy with ALPs oriented orthogonally to the direction of motion Large scale magnetic fields exist in galaxies of the particles is [12] and galaxy clusters, so we may ask whether it is possible to use astronomy to search for ALPs?   −m2 Bω   φ  ω2 + ∂2 + φ M = 0, (3) To model the effects of ALPs on photons pass- z Bω −ω2 A M P ⊥ ing through astronomical magnetic fields we must take into account the variations in the magnetic h 2 2 2i ω − ωP + ∂z Ak = 0, (4) field which occur on many different scales. In what follows it will be sufficient to describe the where A⊥ and Ak are the components of the pho- magnetic field as being made up of cells; within ton polarized perpendicular and parallel to the each cell the magnetic field is constant and its magnetic field. The wave equation for pseudo- magnitude is the same in all cells but its ori- scalars is found by interchanging A⊥ and Ak in entation varies randomly. Astronomical mag- (3) and (4). ω is the photon frequency and ωP netic fields are often weaker than those that can is the plasma frequency, which acts as an effec- be created in the laboratory, however for ALP- tive mass for photons propagating in a plasma. photon mixing this is compensated for by their Notice that only one polarization of the photon much greater extent. 3

!# If an astronomical source is observed through √  2c 1/2 − tan−1 a 1 − − , such a magnetic field photon-ALP oscillations will 1 − p0 affect our observations of both its luminosity and polarization. When this mixing is very strong these effects can be calculated analytically. where a = (1+p0)/(1−p0) and c± = min (c, (1± p0)/2). The probability distribution is plotted for 3.1. Strong mixing various initial polarization fractions in Fig. 1. Strong mixing occurs when [11] NP  1,N∆(L) . π/2. (8) N is the number of magnetic domains of length L that are traversed. In the strong mixing limit the probability of mixing becomes large, and almost frequency independent. In this limit mixing between the ALP and pho- tons fields and between different components of the photon field is so strong that after pass- ing through a large number of randomly ori- ented magnetic domains the initial flux becomes, on average, equally distributed between A1, A2 and φ [13], where we now take an arbitrary ba- sis for the photon polarization states. If the initial photon flux is fully polarized the initial state of the system can be written as u(0) = T (φ(0),A1(0),A2(0)) , with |u(0)| = 1, after strong mixing in a large number of domains this evolves to Figure 1. The probability distribution for the at- p 2 p 2 T tenuation of light which undergoes strong mixing uN = (x, 1 − x cos πΘ, 1 − x sin πΘ) , (9) with ALPs plotted for different initial polariza- where x, Θ ∼ U(−1, 1) [14]. This must be gen- tion fractions. eralized when looking at astronomical sources be- cause there the light emitted is typically partially polarized or unpolarized. A partially polarized state can be written as a superposition of two po- larized states each of which evolves according to The possibility that a large fraction of high energy photons from astronomical sources could (8). Then if the initial flux of photons is I0 of convert into ALPs has been used to explain which a fraction p0 is polarized the final flux of photons after strong mixing is a number of observations of anomalously high  2 2  fluxes of high energy photons [15,16]. Photons of Iobs = 1 − (1 + p0)x /2 − (1 − p0)y /2 I0 (10) such high energy are expected to be highly atten- where x, y ∼ U(−1, 1). We define C = Iobs/I0 uated in the intergalactic medium because they to be the fraction of photons that survive. Then interact with background CMB photons and pair from (10) the probability distribution function for produce. However if photons convert into ALPs C is in magnetic fields close to their source these ALPs 1 can travel relatively unimpeded through the in- f (c; p ) = × (11) C 0 p 2 tergalactic medium and then convert back into 1 − p0 " ! photons in the magnetic field of the Milky Way √  2c −1/2 tan−1 a 1 − + resulting in a higher that expected flux of high 1 + p0 energy photons. 4

4. A new test for ALPs where Lˆ indicates the model with the most likely values of a, b, σ. Then as both models are de- The predicted distribution of the luminosity of scribed by the same number of parameters, r(p0) astronomical sources whose light mixes strongly is equivalent to the Bayesian information crite- with ALPs (11) can be used to test ALP models. rion. Negative r(p0) is evidence against ALP As we do not generally know the expected high strong mixing, and positive r(p0) is evidence for energy flux of astronomical objects, we need a re- ALP strong mixing. Conventionally |r(p0)| > 6 lation that links the high frequency flux to some is considered strong evidence, and |r(p0)| > 10 is other measurable property of the object which is considered very strong evidence. not affected by ALP-photon strong mixing. For a number of compact objects relationships have been empirically established between the high fre- 5. Results from active galactic nuclei quency luminosity of a class of objects and their We shall consider the possible effects of ALP low frequency luminosity. Light at low frequen- mixing on photons propagating through the mag- cies mixes only weakly with ALPs and so this netic fields of galaxy clusters. These magnetic luminosity is not altered at leading order. The fields are chosen because of their great extent and luminosity relations take the form because their properties are well understood [17]. log10 Yi = a + b log10 Xi + Si, (12) Clusters of galaxies are some of the largest objects in the universe; a typical cluster contains ∼ 103 where Yi is the high energy luminosity, and Xi the galaxies in a region ∼ 2 Mpc in radius. Measure- low energy luminosity, of the i-th object. The Si ments of the Faraday rotation of radiation from represent the scatter of individual measurements, intra-cluster radio sources [17] show the presence It is standard to model the scatter as being nor- of magnetic fields with strength B ≈ 1 − 10 µG mally distributed; Si = σδi where δ ∼ N(0, 1). If which are coherent over distances 1 − 100 kpc. the high frequency light has been strongly mixed Typical densities in the diffuse plasma in with ALPs we expect instead −3 −3 the intracluster medium are ne ∼ 10 cm cor- responding to ω ∼ 10−12 eV. We assume that Si = σδi − log10 Ci + µ, (13) P light from a typical source in the cluster travels a where µ, which is included so that the Si still have distance ≈ 0.1 − 1 Mpc through the intracluster zero mean, can be absorbed in a redefinition of a. magnetic field, and in order to make the anal- The ALP strong mixing model can be com- ysis model independent we consider ALPs with pared against the null hypothesis of Gaussian masses mφ . ωP . scatter for a given data set by the likelihood ratio Strong, frequency independent mixing occurs test. Given a set of measurements {Xi,Yi} the when the constraints of (8) are satisfied. For likelihood of each model is light traveling through the magnetic fields of Y 1 11 L (a, b, σ; p ) = √ σ (14) galaxy clusters this requires M . 10 GeV, f 0 (observationally allowed for pseudo-scalars and i 2π chameleonic scalars) and ω 3 − 30 keV. Nu- Z 1 z2 & − i × e 2σ2 fC (c; p0)dc, merical simulations show that the frequency con- 0 straint can be relaxed by an order of magnitude. where zi = log10 Yi − a − b log10 Xi − log10((1 − Therefore if the coupling is sufficiently strong x- f) + fc). f = 1 when the high frequency light ray and γ-ray light mixes strongly with ALPs in is strongly mixed with ALPs, and f = 0 when galaxy clusters. it is not. For each model we fit for a, b, σ by Luminosity relations of the form (12), for which maximizing the likelihood (15). We define the high frequency luminosity is measured at x- or γ-ray frequencies, exist for Gamma Ray ˆ ! L1(p0) Bursts (GRBs), Blazars and Active Galactic Nu- r(p0) = 2 log , (15) Lˆ0 clei (AGN). AGN are the best objects to use for 5 our analysis because there exists a large num- ber of observations which have relatively low in- trinsic scatter. The luminosity relation used re- lates the 2 keV x-ray luminosity and the 250 nm (ω ≈ 4.95 eV) optical luminosity. We use ob- servations of 77 AGN from the COMBO-17 and ROSAT surveys [18], at redshifts z = 0.061−2.54. The likelihood ratio comparing strong ALP mix- ing with the null hypothesis of Gaussian scatter is r(p0 0.4) ≈ 14. (16) . (a) Best fit Gaussian model [14]

For all p0 we find r > 11, but the expectation from AGN physics is that p0 < 0.1 [19]. This is a very strong preference for the ALP strong mixing model over Gaussian scatter. The same analysis applied to the luminosity relations for GRBs and blazars gives, in total, r ≈ 1.6 [14] which is statistically insignificant. We must ask, however, whether (16) is really evidence for ALP strong mixing, or just that Gaussian scatter was a poor choice for the null hypothesis? This null hypothesis was adopted because for other relations, e.g. those of GRBs and blazars, it provides a good fit to the data. (b) Best fit ALP strong mixing model [14] However it may be that a better understanding of the physics of AGN would suggest a better choice for the null hypothesis which could remove the preference for ALP strong mixing. We perform a Figure 2. Simulated ‘fingerprints’ (see text for qualitative check of whether ALP strong mixing details) comparing the second and third moments is really a good fit to the data by plotting ‘finger- of the distribution. Darker regions indicate higher prints’ of the data. To do this we construct 105 density. new data sets, of the same size as the original, by bootstrap resampling (with replacement) of the original data set. For each data set we calculate the statistical moments of the distribution the plot. The equivalent plot for the observations of AGN is shown in Fig. 3. There is a clear qual- !1/m 1 X m itative similarity between the plot for the AGN km(si) = si , (17) Np data and the simulated ALP strong mixing plot. i Although not shown here this similarity also ex- where si = log10 Yi −(a+b log10 Xi). Fingerprints ists when higher moments of the distribution are of the data are then histogram plots of ki vs. kj. plotted. We compare these plots with similar fingerprints produced using simulated data for each model. 6. Conclusions The simulated plots are shown in Fig. 2. The features that distinguish the ALP strong mixing If a dark energy field couples to photons it be- model are a significant asymmetry about the k2 haves as an axion-like-particle. ALPs can mix axis and a long ‘tail’ into the lower right corner of with photons in the presence of a magnetic field. 6

future runs of the CERN Axion Solar Telescope (CAST).

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