Dark matter, axions, and polarization quantum channels

Chapter opening

Dark matter and light-field new physics rarely arrive as a direct image of a new particle. They more often appear as small deviations in polarization angle, arrival time, correlation functions, lensing magnification, or image position. Axions and axion-like particles rotate linear polarization or convert into photons in an external magnetic field through the \(aF\tilde F\) coupling. Ultralight dark matter can make that rotation oscillate in time. Black-hole superradiance can build light-boson clouds around Kerr black holes. Cold-dark- matter subhalos perturb multiply imaged lenses. The QCD axion, modern axion cosmology, the CAST helioscope, CMB cosmic birefringence, EHT axion-cloud searches, and strong-lensing substructure measurements provide the main observational entry points [Carroll et al., 1990, CAST Collaboration et al., 2017, Chen et al., 2022, Chen et al., 2020, Dalal and Kochanek, 2002, Komatsu, 2022, Lue et al., 1999, Mao and Schneider, 1998, Marsh, 2016, Minami and Komatsu, 2020, Peccei and Quinn, 1977, Sikivie, 1983, Weinberg, 1978, Wilczek, 1978].

How an axion field enters photon polarization

In the low-energy effective theory, the interaction between an axion-like field \(a\) and electromagnetism is

(291)\[\mathcal L_{a\gamma} = -\frac{1}{4}g_{a\gamma}aF_{\mu\nu}\tilde F^{\mu\nu} = g_{a\gamma}a\,{\bf E}\cdot{\bf B}.\]

\(g_{a\gamma}\) has units of inverse energy, and astrophysics and laboratory papers usually quote it in \({\rm GeV^{-1}}\). \(F_{\mu\nu}\) is the electromagnetic tensor and \(\tilde F^{\mu\nu}\) is its dual. The second equality writes the coupling as \({\bf E}\cdot{\bf B}\), so the effect depends on the relative orientation of electric and magnetic fields and carries chiral information. The QCD axion also approximately obeys the mass-decay-constant relation

(292)\[m_a \simeq 5.7\,\mu{\rm eV} \left(\frac{10^{12}\,{\rm GeV}}{f_a}\right),\]

where \(f_a\) is the Peccei–Quinn symmetry-breaking scale. An ALP need not obey this relation; \(m_a\) and \(g_{a\gamma}\) can be nearly independent. For that reason, parameter-space plots usually show the QCD axion band separately from the wider ALP space [Dias et al., 2014, Marsh, 2016].

In an external magnetic field, the axion mixes with the photon polarization parallel to \(B_\perp\). In the weak-mixing, uniform-field limit with negligible absorption,

(293)\[P_{a\to\gamma} \simeq \left(\frac{g_{a\gamma}B_\perp L}{2}\right)^2 \mathrm{sinc}^2\!\left(\frac{qL}{2}\right), \qquad q\simeq \frac{|m_a^2-m_\gamma^2|}{2E}.\]

\(B_\perp\) is the magnetic field perpendicular to the direction of propagation, \(L\) is the coherent path length, \(E\) is particle energy, and \(m_\gamma\) is the effective photon mass in plasma. For \(qL\ll1\), the conversion amplitude adds coherently along the path and the probability grows as \(L^2\). Once \(qL\gtrsim\pi\), phase mismatch makes different path segments cancel. CAST used a \(9\,{\rm T}\), \(9.26\,{\rm m}\) retired LHC dipole magnet to search for solar axions converting into X-rays. Its 2013–2015 vacuum data gave a 95% C.L. limit \(g_{a\gamma}<0.66\times10^{-10}\,{\rm GeV^{-1}}\), valid in the coherent low-mass regime \(m_a\lesssim0.02\,{\rm eV}\) [CAST Collaboration et al., 2017].

Figure 85 Axion-photon conversion is controlled by both coupling strength and phase. The left panel shows the coherence factor sinc²(qL/2); once qL approaches π, conversion amplitudes from different parts of the path begin to cancel. The right panel gives the order of magnitude of the coherent-limit probability on interstellar scales, using \(g_{11}=g_{a\gamma}/10^{-11}{\rm GeV^{-1}}=1\). In a \(\mu{\rm G}\) field over a kpc path, the single-domain conversion probability is usually far below unity.

The same coupling also rotates the angle of linear polarization. If a photon travels from emission point \(e\) to observer \(o\) through a slowly varying axion background, the geometric-optics limit gives

(294)\[\Delta\psi = \frac{1}{2}g_{a\gamma}\,[a(t_o,{\bf x}_o)-a(t_e,{\bf x}_e)] .\]

This rotation is independent of wavelength, unlike the Faraday rotation \(\psi=\psi_0+{\rm RM}\lambda^2\) discussed in Chapter Propagation effects: plasma, dust, and gravitational lensing. If the axion is local ultralight dark matter,

(295)\[a(t,{\bf x}) \simeq a_0\cos(m_at+\varphi), \qquad a_0=\frac{\sqrt{2\rho_a}}{m_a}.\]

\(\rho_a\) is the local dark-matter energy density; in the Milky Way, a common scale is \(0.3\,{\rm GeV\,cm^{-3}}\). The oscillation period and coherence time are

(296)\[T_a = \frac{2\pi\hbar}{m_ac^2} \simeq 47.9\,{\rm d} \left(\frac{10^{-21}\,{\rm eV}}{m_a}\right), \qquad t_{\rm coh}\sim \frac{T_a}{v^2},\]

where \(v\sim10^{-3}c\) is the Galactic-halo velocity dispersion. For \(m_a=10^{-21}\,{\rm eV}\), the polarization angle can oscillate with a period of tens of days, while the coherence time can reach \(10^5\) yr. CMB searches for axion dark matter use both the common oscillation of the present local field and the washout effect at the last-scattering surface. Fedderke, Graham, and Rajendran emphasized that ordinary static cosmic-birefringence searches are not equivalent to this rapidly oscillating signal [Fedderke et al., 2019, Hui, 2021].

Figure 86 Ultralight dark matter turns polarization rotation into a time series. The left panel shows Δψ(t) for three axion masses; larger mass gives faster oscillation. The right panel converts m_a into the period T_a and coherence time \(t_{\rm coh}\sim T_a/v^2\), using v = 10⁻³c.

CMB birefringence and single-photon polarization event tables

The CMB has fewer polarization foregrounds than many compact sources, but the systematic-error requirements are much tighter. If the linear-polarization Stokes parameters \(Q\) and \(U\) are rotated by an angle \(\beta\),

(297)\[Q\pm iU \rightarrow e^{\pm2i\beta}(Q\pm iU).\]

In harmonic space, this mixes \(E\) and \(B\) modes. If the intrinsic last-scattering \(EB\) signal is small, the observed parity-odd spectra are approximately

(298)\[C_\ell^{EB,{\rm obs}} = \frac{1}{2}\sin(4\beta) \left(C_\ell^{EE}-C_\ell^{BB}\right), \qquad C_\ell^{TB,{\rm obs}} = \sin(2\beta)C_\ell^{TE}.\]

For \(\beta=0.35^\circ\), \(\sin(4\beta)/2\simeq0.012\), so the \(EB\) leakage is at the percent level of the \(EE-BB\) difference. A reanalysis of Planck 2018 used the different frequency and multipole dependence of CMB and Galactic foreground polarization to fit cosmic birefringence angle \(\beta\) and detector-angle miscalibration \(\alpha\) simultaneously, obtaining \(\beta=0.35\pm0.14^\circ\) at 68% C.L. Later WMAP+Planck and Planck PR4 analyses emphasized that dust \(EB\), bandpass mismatch, \(I\to P\) leakage, cross-polarization response, and mask choices remain major systematics [Diego-Palazuelos et al., 2022, Eskilt and Komatsu, 2022, Komatsu, 2022, Minami and Komatsu, 2020].

Figure 87 Cosmic birefringence involves both E/B mixing and instrumental angle calibration. The left panel uses toy EE and BB spectra to show how β produces EB. The right panel shows the degeneracy between α and β: using only the CMB, detector angle error and cosmic rotation are nearly equivalent. Foreground frequency dependence helps break the degeneracy.

A single-photon polarization event table moves the problem from an averaged map into the time domain. A polarization event can be written as

(299)\[e_q=(t_q,\nu_q,\hat{\bf n}_q,p_q,w_q),\]

where \(p_q\) is polarization channel and \(w_q\) is a quality weight. If the polarization angle has a small common oscillation, the event probability can be written

(300)\[P(p_q|t_q,\nu_q,\hat{\bf n}_q) = P_0\!\left[p_q\,\middle|\,\psi_{\rm src}+\psi_{\rm prop}(\nu_q,\hat{\bf n}_q)+\delta\psi(t_q,\hat{\bf n}_q)\right].\]

\(\psi_{\rm src}\) is the intrinsic source polarization, \(\psi_{\rm prop}\) contains Faraday rotation, dust, and the instrumental Mueller matrix, and \(\delta\psi\) is the new-physics term to be tested. The event table can use one likelihood across time, frequency, and polarization channel, avoiding the early compression of the data into one fixed polarization angle. The cost is equally direct: if off-diagonal Mueller-matrix terms are known only to \(10^{-3}\), while the target rotation is \(10^{-3}\,{\rm rad}\), systematic error is already at the signal scale. Strong-field QED vacuum-birefringence candidates, optical polarization of isolated neutron stars, and magnetar X-ray polarization all require source geometry and propagation effects to be modeled before a polarization anomaly can be interpreted as new physics [Mignani et al., 2017].

Pulsars, FRBs, and polarization foregrounds near black holes

Pulsars are useful time-domain arrays because they have stable phases and repeated pulses. The oscillating gravitational potential of ultralight scalar dark matter can produce a narrow-band timing residual in a pulsar timing array. If the potential perturbation is \(\Psi_c\cos(2m_at+\varphi)\), a simplified residual is

(301)\[R_\alpha(t) \simeq \frac{\Psi_c}{2\pi f} \left[ \sin(2\pi ft+\varphi_E) - \sin(2\pi f(t-D_\alpha/c)+\varphi_\alpha) \right],\]

where \(\alpha\) labels pulsar \(\alpha\), \(D_\alpha\) is distance, and \(f\simeq m_a/\pi\) is the potential-oscillation frequency. An axion mass \(m_a\sim10^{-23}\,{\rm eV}\) corresponds to \(f\sim10^{-8}\,{\rm Hz}\), right inside the PTA window of \(10^{-9}\)–\(10^{-7}\,\mathrm{Hz}\). Porayko and Postnov’s analysis of older NANOGrav data gave a 95% C.L. upper limit \(\Psi_c<1.14\times10^{-15}\), corresponding to a scale \(h_c<4\times10^{-15}\) near \(f=1.75\times10^{-8}\,{\rm Hz}\) [Porayko and Postnov, 2014]. In a narrow-band stochastic-background approximation, the correlation between pulsars can be written

(302)\[\zeta_{\alpha\beta}=\frac{1}{2}(1+\delta_{\alpha\beta}),\]

which differs from the Hellings–Downs angular correlation of a stochastic gravitational-wave background. A future “pulsar polarization array” would have to separate the common Earth term, intrinsic magnetospheric polarization, interstellar Faraday rotation, profile mode changes, and receiver polarization leakage.

Figure 88 A common narrow-band signal in a pulsar array. The left panel shows a common Earth term plus each pulsar’s own pulsar term; the curves are vertically offset for readability. The right panel shows the simplified correlation matrix ζ_αβ = (1 + δ_αβ)/2 in the scalar-potential oscillation model, distinct from the angular correlation of a stochastic tensor gravitational-wave background.

FRBs have strong polarization information, but also strong propagation foregrounds. In the \(z=0.193\) host galaxy of FRB 121102, 4–8 GHz bursts showed nearly \(100\%\) linear polarization and a large, variable source-frame Faraday rotation measure: \({\rm RM}_{\rm src}=1.46\times10^5\) and \(1.33\times10^5\,{\rm rad\,m^{-2}}\). The same burst sample contained narrow structure below \(\sim30\,\mu{\rm s}\). If the host contribution is \({\rm DM}_{\rm host}\sim70\)–270 \({\rm pc\,cm^{-3}}\), the corresponding line-of-sight magnetic-field lower limit is at the mG scale, far above the \(\sim5\,\mu{\rm G}\) of the ordinary Galactic ISM [Michilli et al., 2018]. Such sources are laboratories for extremely magnetized plasma. A polarization change that appears wavelength independent still has to be tested against channel-averaging depolarization, time-variable RM, and plasma-lensing selection effects.

Figure 89 FRB polarization foregrounds can easily swamp a small rotation signal. The left panel shows polarization angle wrapping modulo 180^∘ as a function of frequency for different RM values; at \({\rm RM}=10^5\,{\rm rad\,m^{-2}}\), the angle changes rapidly across GHz bands. The right panel shows the retained linear-polarization fraction near \(1.4\,{\rm GHz}\) for finite channel widths. Narrow channels are essential for measuring high RM.

Black-hole superradiance links light boson fields to polarization images. For a Kerr black hole with horizon angular velocity \(\Omega_H\), a boson mode can extract rotational energy if

(303)\[\omega < m\Omega_H .\]

Define

(304)\[\alpha_G=\frac{GM_\bullet m_a}{\hbar c},\]

so that for \(\alpha_G\sim0.1\)–1 the boson Compton wavelength matches the black-hole scale and the cloud grows efficiently. The corresponding axion mass is

(305)\[m_a \simeq 1.34\times10^{-10}\,{\rm eV}\, \alpha_G \left(\frac{M_\odot}{M_\bullet}\right).\]

For Sgr A*, with \(4.3\times10^6M_\odot\), \(\alpha_G=0.4\) corresponds to \(m_a\simeq1.2\times10^{-17}\,{\rm eV}\). For M87*, with \(6.5\times10^9M_\odot\), it corresponds to about \(8\times10^{-21}\,\mathrm{eV}\). Axion-cloud models for EHT polarization images predict polarization-angle oscillations on the axion period: for Sgr A* the period can be \(100\)–\(1000\,{\rm s}\), while for M87* it can be days. Spatial resolution determines whether oscillations from different azimuths around the ring are averaged away [Baryakhtar et al., 2017, Brito et al., 2015, Brito et al., 2015, Chen et al., 2022, Chen et al., 2020].

Figure 90 Black-hole mass selects the superradiant axion mass window. The left panel uses α_G = GM_•m_a/ℏc to show the m_a range corresponding to different black-hole masses; Sgr A* and M87* lie in different ultralight mass ranges. The right panel shows two characteristic timescales for EHT polarization-angle oscillations: M87* can be sampled on day scales, while Sgr A* requires second-to-minute polarization imaging.

Small-scale dark-matter lensing

The existence of dark matter is supported by cluster collisions, weak lensing, and dynamics. The Bullet Cluster is the classic example: weak-lensing mass peaks are separated from the X-ray gas, showing that the dominant mass component is approximately collisionless [Clowe et al., 2006, Kaiser and Squires, 1993]. Small-scale structure gives another test. CDM simulations predict many subhalos and tidal streams in galaxy halos, with a mass function extending far below the visible satellite-galaxy range. Results by Moore, Diemand, and others made “dark substructure” a major candidate explanation for strong-lensing anomalies [Diemand et al., 2008, Flores and Primack, 1994, Moore et al., 1999].

The basic scale of substructure lensing comes from the Einstein angle,

(306)\[\theta_E = \left( \frac{4GM}{c^2} \frac{D_{ls}}{D_lD_s} \right)^{1/2}.\]

For a galaxy strong lens with \(D_l\simeq1\,{\rm Gpc}\) and \(D_s\simeq2\,{\rm Gpc}\), a \(10^8M_\odot\) subhalo has \(\theta_E\) of order tens of mas, while a \(10^6M_\odot\) subhalo has only a few mas. If an image lies at angular distance \(b\gg\theta_E\) from the subhalo, the point-mass astrometric shift is approximately

(307)\[\delta\theta\simeq\frac{\theta_E^2}{b}.\]

This scale makes the resolution requirement explicit: a \(10^6M_\odot\) subhalo at \(b=10\,{\rm mas}\) produces only a \(\mu{\rm as}\)-scale shift. Strong-lens flux-ratio anomalies are sensitive to magnification perturbations. Mao and Schneider showed that low-mass substructure can explain anomalies in some quadruple lenses. Dalal and Kochanek used seven radio lenses to infer a median substructure mass fraction \(f_{\rm sat}\simeq0.02\), with a 90% range of about \(0.006\)–0.07. Vegetti, Hezaveh, and collaborators pushed individual dark substructure detections into spatially resolved modeling with gravitational imaging and ALMA data [Dalal and Kochanek, 2002, Hezaveh et al., 2016, Mao and Schneider, 1998, Vegetti et al., 2010].

Figure 91 The lensing signal of a dark-matter subhalo is limited by angular scale. The left panel shows the Einstein angle as a function of subhalo mass for a representative Gpc lens geometry. The right panel converts δθ ≃ θ_E²/b into an astrometric shift; low-mass subhalos are easiest to see only when the encounter is at mas scales and systematics are controlled at the \(\mu{\rm as}\) level.

Gaia and future astrometric surveys turn this into a time-domain problem. As a subhalo passes near a background source, the positional shift changes with time. A transverse speed \(v_\perp\sim200\,{\rm km\,s^{-1}}\) at kpc distance corresponds to an angular speed of order mas/yr; the signal shape depends on impact parameter, mass profile, and the source’s own proper motion. Mondino et al. discussed astrometric weak-lensing searches with Gaia DR3 and future catalogs, with main limitations from the scanning law, source density, proper-motion covariance, and Solar-System or instrumental systematics [Mondino et al., 2024]. Primordial black-hole dark matter can also be constrained by lensing and dynamics, but the mass function, clustering, and astrophysical foregrounds change the interpretation [Carr et al., 2016].

Turning an anomaly into a test

New-physics searches usually combine more than one observable. Polarization angle, arrival time, image position, and magnification may come from different instruments, with different statistical and systematic errors. After collecting them into a data vector \({\bf d}\), the model can be split into

(308)\[{\bf m}({\boldsymbol\theta}) = {\bf m}_{\rm src} +{\bf m}_{\rm prop} +{\bf m}_{\rm inst} +{\bf m}_{\rm new}({\boldsymbol\theta}),\]

where \({\boldsymbol\theta}\) contains parameters such as \(g_{a\gamma}\), \(m_a\), \(\beta\), the subhalo mass function, or the PBH fraction. In a Gaussian approximation,

(309)\[-2\ln\mathcal L = \left[{\bf d}-{\bf m}({\boldsymbol\theta})\right]^{\mathsf T} \left(C_{\rm stat}+C_{\rm sys}\right)^{-1} \left[{\bf d}-{\bf m}({\boldsymbol\theta})\right] +\ln\det(C_{\rm stat}+C_{\rm sys}).\]

\(C_{\rm stat}\) comes from photon noise, TOA uncertainties, or image noise. \(C_{\rm sys}\) contains the Mueller matrix, absolute polarization angle, Faraday foreground, dust \(EB\), lens macro-model, source structure, and selection function. If \(C_{\rm sys}\) is not in the likelihood, a single outlying point will be overinterpreted.

These signals can be separated by their physical signatures. Faraday rotation varies as \(\lambda^2\), while axion-background rotation is approximately achromatic. Static cosmic birefringence produces global \(EB/TB\) in the CMB; ultralight dark matter oscillates during the observing campaign. A black-hole axion cloud should correlate with black-hole mass, spin, azimuth around the ring, and observing time. Dark-matter subhalo lensing should perturb image position, magnification, and time delay together, whereas stellar surface structure mainly varies with source rotation, wavelength, and baseline direction. A credible result normally has to survive population statistics, blank fields or control sources, frequency reversal tests, time-coherence tests, and instrumental recalibration.

CMB and early-Universe fluctuations push the same requirement to full-sky maps. There the observables are no longer the polarization angle or lens images of a single source, but \(T/E/B\) maps, power spectra, polarization angle, and the systematic-error matrix.