The Effect of a Chameleon Scalar Field on the Cosmic Microwave Background

Measurements of the current cosmic acceleration of the universe have led to theories of an unknown energy component with negative pressure in the Universe. This dark energy is generally modelled by a scalar field rolling down a flat potential. On Solar-system scales this scalar field should appear effectively massless. The origin of the dark energy scalar field is still highly speculative; some suggestions have involved the moduli fields of string theory. However, if a near-massless scalar field existed on Earth it should have been detected as a fifth force unless the coupling to normal matter was unnaturally suppressed. Khoury and Weltman Khoury04 introduced the chameleon scalar field which has gravitational strength or greater coupling to normal matter but a mass which depends on the density of surrounding matter. On cosmological scales the chameleon successfully acts as a low-mass dark energy candidate, while in the laboratory its large mass allows it to evade detection.

An extension to the original chameleon model is to include an interaction term between the chameleon and electromagnetic (EM) field of the form $\frac{\phi}{M_{\rm eff}}F_{\mu\nu}F^{\mu\nu}$, first suggested by Brax et al. Brax07 . This leads to interconversion between chameleons and photons in the presence of a magnetic field, which alters the intensity and polarization of any radiation entering the magnetic region. A coupling between the chameleon and EM fields gives rise to observational effects that could potentially be detected either astrophysically or in the laboratory Brax07 ; BurrageSN ; Burrage08 . These predictions have lead to constraints on the mass parameter $M_{F}$ describing the coupling strength to the EM field: $M_{F}\gtrsim 2\times 10^{6}\,\mathrm{GeV}$ Brax07 and $M_{F}\gtrsim 1.1\times 10^{9}\,\mathrm{GeV}$ Burrage08 .

It was noted in Ref. Burrage08 that in low density backgrounds, such as the interstellar and intra-cluster mediums, the mixing between the chameleon field and photons is indistinguishable from the mixing of any very light scalar field with a linear coupling to $F_{\mu\nu}F^{\mu\nu}$ or a very light pseudo-scalar with a linear coupling to $\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$. Such fields are collectively referred to as axion-like particles (ALPs). Standard ALPs have the same mass and photon-coupling everywhere. The best constraints on their parameters come from limits on their production in relatively high density regions such as stellar cores or in supernova 1987A ALPrev . Chameleon scalar fields represent one example of a subclass of ALPs that we term chameleonic ALPs. The defining feature of chameleonic ALPs is that the astrophysical constraints from high density regions are evaded or exponentially weakened. This may be because as the ambient density increases the field mass rises as in the chameleon model, or because the effective photon coupling weakens Jaeckel07 ; Burrage08 . This latter possibility is realized in the Olive–Pospelov model for a density-dependent fine-structure constant Olive08 . The results we derive in this paper technically apply to all ALPs, however the parameter range for which they apply is strongly ruled out for all but chameleonic ALPs. The model for chameleon-like theories is described in detail in subsection I.1 below.

In this paper we consider the propagation of the Cosmic Microwave Background (CMB) radiation through the magnetic field of a galaxy cluster. We predict the modification to the CMB intensity and polarization in the direction of galaxy clusters arising from mixing with a chameleonic ALP.

One of the significant foreground effects acting on the CMB as it passes through a galaxy cluster, which has been extensively verified by experiment, is the Sunyaev–Zel’dovich (SZ) effect. This arises from scattering of CMB photons off electrons in the galaxy cluster atmosphere, causing a change to the CMB intensity in the direction of the cluster. Measurements of the SZ effect can be compared to predictions of the CMB intensity decrement arising from the chameleon, and thus constrain the chameleon model parameters.

This paper is organized as follows: in §II, the conversion between photons and a general scalar ALP in a magnetic field is analysed, and note how it depends on the structure of the magnetic field. In §III, we briefly review the observational evidence for magnetic fields in the intra-cluster medium (ICM) and detail their typical properties. In §IV, we describe the standard SZ effect and show how this is modified when photons are converted to (chameleonic) ALPs in the intra-cluster medium. We note that the frequency and electron density dependence of the chameleonic SZ effect differs from that of the thermal and other standard contributions to the SZ effect. In §V, we exploit this difference to constrain the probability of photon-scalar conversion in the ICM. Under certain reasonable assumptions about the magnetic field structure, we convert these constraints to bounds on the effective photon to scalar coupling. In §VI we discuss the modification to the CMB polarization from chameleon-mixing. We summarize our results in §VII.

In the presence of an axion-like particle (ALP), the properties of electromagnetic radiation propagating through a magnetic field are altered as a result of mixing between the ALP and photons. Over the past 25 years or so, there has been a great deal of work on this subject; for example Refs. Silkvie84 ; Raffelt88 ; Harai92 ; Burrage08 or see Ref. ALPrev for a recent review.

In this section, we present the equations which describe the mixing of a chameleon-like particle with the photon field in the presence of a magnetic field. We also present the solution of these equations, in the limit where the mixing is weak, for two different models of the magnetic field structure in the ICM.

Varying the action in Eq. (I.1) with respect to both $\phi$ and $A_{\mu}$, we find Eq. (2) and

$\displaystyle\nabla_{\mu}\left[B_{F}(\phi/M)F^{\mu\nu}\right]$

$\displaystyle=$

$\displaystyle J^{\nu},$

where $J^{\mu}$ is the background electromagnetic 4-current; $\nabla_{\mu}J^{\mu}=0$.

We are concerned with the propagation of light in sparse astrophysical backgrounds containing a magnetic field $\mathbf{B}(x^{\mu})$. The background values of the chameleon and photon fields are assumed to be slowly varying over length and time scales of ${\cal O}(1/\omega)$ where $\omega$ is the proper frequency of the electromagnetic radiation being considered. We define the perturbations in the photon and chameleon fields about their background values to be, $a_{\mu}=A_{\mu}-\bar{A}_{\mu}$ and $\varphi=\phi-\bar{\phi}$, respectively. When electromagnetic radiation propagates through a plasma with electron number density $n_{\rm e}$, an effective photon mass-squared of $\omega_{\rm pl}^{2}=4\pi\alpha_{\rm em}n_{e}/m_{e}$ is induced; $\omega_{\rm pl}$ is the plasma frequency. Following Ref. Burrage08 in ignoring terms that are second order in the perturbations $\varphi$ and $a_{\mu}$, and assuming $\omega\gg H$ where $H$ is the Hubble parameter, we find

	$\displaystyle-\ddot{\mathbf{a}}+\nabla^{2}\mathbf{a}$	$\displaystyle\simeq$	$\displaystyle\frac{\boldsymbol{\nabla}\varphi\times\mathbf{B}}{M_{\rm eff}}+\omega_{\rm pl}^{2}\mathbf{a},$
	$\displaystyle-\ddot{\varphi}+\nabla^{2}\varphi$	$\displaystyle\simeq$	$\displaystyle\frac{\mathbf{B}\cdot(\boldsymbol{\nabla}\times\mathbf{a})}{M_{\rm eff}}+m_{\phi}^{2}\varphi,$

where $1/M_{\rm eff}\equiv B_{F,\phi}(\bar{\phi})$ and $m_{\phi}^{2}\equiv V_{{\rm eff},\phi\phi}(\bar{\phi},T_{\rm m},\bar{A}_{\mu})$.

We consider a photon field propagating in the $\hat{\mathbf{z}}$ direction, so that in an orthonormal Cartesian basis $(\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}})$ we have $\mathbf{a}=(\gamma_{x},\gamma_{y},0)^{\rm T}$. The above equations of motion for the chameleon and photon polarization states can then be written in matrix form:

$\displaystyle\left[-\partial_{t}^{2}-\partial_{z}^{2}-\left(\begin{array}[]{ccc}\omega_{\rm pl}^{2}&0&-\frac{B_{y}}{M_{\rm eff}}\partial_{z}\\ 0&\omega_{\rm pl}^{2}&\frac{B_{x}}{M_{\rm eff}}\partial_{z}\\ \frac{B_{y}}{M_{\rm eff}}\partial_{z}&-\frac{B_{x}}{M_{\rm eff}}\partial_{z}&m_{\phi}^{2}\end{array}\right)\right]\left(\begin{array}[]{c}\gamma_{x}\\ \gamma_{y}\\ \varphi\end{array}\right)=0.$

(11)

Following a similar procedure to that in Ref. Raffelt88 , we assume the fields vary slowly over time and that the refractive index is close to unity which requires $m_{\rm\phi}^{2}/2\omega^{2}$, $\omega_{\rm pl}^{2}/2\omega^{2}$ and $|B|/2\omega M_{\rm eff}$ all $\ll 1$. Defining $\gamma_{i}=\tilde{\gamma}_{i}(z)e^{i\omega(z-t)+i\beta(z)}$ and $\varphi=\tilde{\varphi}e^{i\omega(z-t)+i\beta(z)}$ where $\beta_{,z}=-\omega_{\rm pl}^{2}(z)/2\omega$, we approximate $-\partial_{t}^{2}\approx\omega^{2}$ and $\omega^{2}+\partial_{z}^{2}\approx 2\omega(\omega+i\partial_{z})$. In addition to this, we define the state vector

$\displaystyle\mathbf{u}=\left(\begin{array}[]{c}\tilde{\gamma}_{x}(z)\\ \tilde{\gamma}_{y}(z)\\ e^{2i\Delta(z)}\tilde{\varphi}(z)\end{array}\right).$

(15)

where

$$\Delta(z)=\int_{0}^{z}\frac{m_{\rm eff}^{2}(x)}{4\omega}{\rm d}x,$$

and $m_{\rm eff}^{2}=m_{\phi}^{2}(z)-\omega_{\rm pl}^{2}(z)$, which simplifies the above mixing field equations for $\mathbf{a}$ and $\varphi$ to

$\displaystyle\mathbf{u}_{,z}=\frac{\mathcal{B}(z)}{2M_{\rm eff}}\mathbf{u},$

(16)

where

$\displaystyle\mathcal{B}(z)=\left(\begin{array}[]{ccc}0&0&-B_{y}e^{-2i\Delta}\\ 0&0&B_{x}e^{-2i\Delta}\\ B_{y}e^{2i\Delta}&-B_{x}e^{2i\Delta}&0\end{array}\right).$

(20)

To solve this system of equations we expand, $\mathbf{u}=\mathbf{u}_{0}+\mathbf{u}_{1}+\mathbf{u}_{2}+\ldots$, such that

$\displaystyle{\mathbf{u}}_{i+1}^{\prime}=\frac{\mathcal{B}(z)}{2M_{\rm eff}}\mathbf{u}_{i}\,;\;{\mathbf{u}}_{0}^{\prime}=0\,,$

and neglect the higher order terms from mixing. We define

$$\mathcal{M}_{1}=\int_{0}^{z}\frac{\mathcal{B}(x)}{2M_{\rm eff}}{\rm d}x.$$

We say that mixing is weak when

$\displaystyle{\rm tr}\left(\mathcal{M}_{1}^{\dagger}(z)\mathcal{M}_{1}(z)\right)\ll 1.$

Thus Eq. (16) can be solved in the limit of weak-mixing:

$\displaystyle\mathbf{u}(z)\simeq\left[\mathbb{I}+\mathcal{M}_{1}(z)+\mathcal{M}_{2}(z)\right]\mathbf{u}(0),$

where $\mathcal{M}_{2}=\int_{0}^{z}\mathcal{M}_{1}^{\prime}(x)\mathcal{M}_{1}(x){\rm d}x$, which can be written

$\displaystyle\mathbf{u}(z)\simeq\left[\mathbb{I}+\mathcal{M}_{1}(z)+\frac{1}{2}\left(\mathcal{M}_{C}(z)+\mathcal{M}_{1}^{2}(z)\right)\right]\mathbf{u}(0),$

where explicitly

	$\displaystyle\mathcal{M}_{1}$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}0&0&-A_{y}^{\ast}\\ 0&0&A_{x}^{\ast}\\ A_{y}&-A_{x}&0\end{array}\right),$		(24)
	$\displaystyle\mathcal{M}_{C}$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}-C_{yy}&-C^{\ast}_{xy}&0\\ C_{xy}&-C_{xx}&0\\ 0&0&C_{xx}+C_{yy}\end{array}\right),$		(28)

and

	$\displaystyle A_{i}$	$\displaystyle=$	$\displaystyle\int_{0}^{z}{\rm d}x\frac{B_{i}(x)e^{2i\Delta(x)}}{2M_{\rm eff}},$
	$\displaystyle C_{ij}$	$\displaystyle=$	$\displaystyle\int_{0}^{z}{\rm d}x\,\left(A_{i}^{\ast\prime}(x)A_{j}(x)-A_{i}^{\ast}(x)A_{j}^{\prime}(x)\right).$

We note that $\mathcal{M}_{n}^{\dagger}=-\mathcal{M}_{n}$.

The polarization state of radiation is described by its Stokes parameters: intensity $I_{\gamma}(z)=|\gamma_{x}(z)|^{2}+|\gamma_{y}(z)|^{2}$, linear polarization $Q(z)=|\gamma_{x}(z)|^{2}-|\gamma_{y}(z)|^{2}$ and $U(z)=2{\rm Re}\left(\gamma_{x}^{\ast}(z)\gamma_{y}(z)\right)$, and circular polarization $V(z)=2{\rm Im}\left(\gamma_{x}^{\ast}(z)\gamma_{y}(z)\right)$. Assuming there is no initial chameleon flux, $I_{\phi}=|\varphi|^{2}=0$, then to leading order the final photon intensity is given by

	$\displaystyle I_{\gamma}(z)$	$\displaystyle=$	$\displaystyle I_{\gamma}(0)\left(1-\mathcal{P}_{\gamma\leftrightarrow\phi}(z)\right)$
			$\displaystyle+Q(0)\mathcal{Q}_{\rm q}(z)+U(0)\mathcal{Q}_{\rm u}(z)+V(0)\mathcal{Q}_{\rm v}(z),$

where we have defined,

	$\displaystyle\mathcal{P}_{\gamma\leftrightarrow\phi}(z)$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(|A_{x}(z)|^{2}+|A_{y}(z)|^{2}\right),$
	$\displaystyle\mathcal{Q}_{\rm q}(z)$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(|A_{x}(z)|^{2}-|A_{y}(z)|^{2}\right),$
	$\displaystyle\mathcal{Q}_{\rm u}(z)$	$\displaystyle=$	$\displaystyle{\rm Re}(A_{x}^{\ast}A_{y}),$
	$\displaystyle\mathcal{Q}_{\rm v}(z)$	$\displaystyle=$	$\displaystyle{\rm Im}(A_{x}^{\ast}A_{y}).$

In this article we consider mixing with the CMB, the intrinsic polarization of which is small, i.e. $\sqrt{Q^{2}(0)+U^{2}(0)+V^{2}(0)}/I_{\gamma}(0)\ll 1$. It follows that to leading order the modification to the Stokes parameters of the CMB radiation, from the effect of photon-ALP mixing, is

$\displaystyle\Delta\mathbf{S}(z)=\Delta\left(\begin{array}[]{c}I_{\gamma}(z)\\ Q(z)\\ U(z)\\ V(z)\end{array}\right)$

$\displaystyle\approx$

$\displaystyle\left(\begin{array}[]{c}-\mathcal{P}_{\gamma\leftrightarrow\phi}(z)\\ \mathcal{Q}_{\rm q}(z)\\ \mathcal{Q}_{\rm u}(z)\\ \mathcal{Q}_{\rm v}(z)\end{array}\right)I_{\gamma}(0).$

(37)

To evaluate $\mathcal{P}_{\gamma\leftrightarrow\phi}$ and the $\mathcal{Q}_{i}$ we need a model for the spatial variation of the magnetic field and effective mass. Typically $n_{\rm e}\sim 10^{-3}\text{--}10^{-2}\,{\rm cm}^{-3}$ in galaxies and galaxy clusters and so $\omega_{\rm pl}\sim{\cal O}(10^{-12})\,{\rm eV}$. For the chameleon-like-theories detailed in §I.1, it follows that $\omega_{\rm pl}^{2}\gg m_{\phi}^{2}$. In what follows we assume that the scalar field is very light so that $m_{\rm eff}^{2}\approx-\omega_{\rm pl}^{2}\propto-n_{\rm e}$. Spatial variations in $m_{\rm eff}^{2}$ are then entirely due to spatial variations in $n_{\rm e}$.

The magnetic fields of galaxies and galaxy clusters have been measured to have a regular component $\mathbf{B}_{\rm reg}(z)$, with a typical length scale of variation similar to the size of the object, and a turbulent component $\delta\mathbf{B}(z)$ which undergoes ${\cal O}(1)$ variations and reversals on much smaller scales. It is common, both in the literature concerning photon-ALP conversion in galaxy and cluster magnetic fields and in that concerning the measurement of those fields, to use a simple cell model to describe $\delta\mathbf{B}$. In this cell model one defines $L_{\rm coh}$ to be the length scale over which the random magnetic field is coherent, i.e. the typical scale over which field reversals of $\delta\mathbf{B}$ occur. A given path of length $L$ is then divided into $N$ magnetic domains, $N=L/L_{\rm coh}$. The magnetic field strength and electron density are assumed to be constant along the path, but the random magnetic field component, $\delta\mathbf{B}_{x}$, is taken to have a different random orientation in each of the $N$ magnetic domains. Although the cell model for $\delta\mathbf{B}$ is common in the literature and reveals most of the key features associated with astrophysical photon-ALP mixing, it is not accurate especially when $|\Delta(z=L)|\gg 1$. Additionally, as was first noted in Ref. Ensslin03 , in this model we do not have $\nabla\cdot\delta\mathbf{B}=0$ as we should. A more realistic model for $\mathbf{B}$ is to assume a spectrum of fluctuations running from some very large scale (e.g. the scale of the galaxy or cluster) down to some very small scale ($\ll L_{\rm coh}$). We describe these fluctuations by a correlation function $R_{\rm B}(\mathbf{x})=\left\langle\delta\mathbf{B}(\mathbf{y})\delta\mathbf{B}(\mathbf{x}+\mathbf{y})\right\rangle$ and the associated power spectrum $P_{\rm B}(k)$; here the angled brackets indicate the expectation of the quantity inside them. We can, and should, also allow for a similar spectrum of fluctuations in the electron number density with some power spectrum $P_{\rm N}(k)$. We now evaluate $\mathcal{P}_{\gamma\leftrightarrow\phi}$ and the $\mathcal{Q}_{i}$ in both this power spectrum model and the cell model.

We saw in the previous section that the mixing of photons with light scalars depends crucially on the structure of the magnetic field along the line of sight. As the CMB photons propagate towards Earth they pass through clusters of galaxies. The space between individual galaxies in the cluster is filled with a hot plasma which supports a magnetic field. In this article, we are focused on the mixing of CMB photons and light (chameleonic) scalars in the intra-cluster (IC) medium. In this section we briefly review the observed properties of IC magnetic fields.

Intra-cluster magnetic fields have been observed to have field strengths as high as $30\,\mu\hbox{G}$ Taylor93 . The galaxy clusters can be divided into two types: non-cooling core clusters for which the magnetic field in the ICM is generally found to be a few to $10\,\mu\mathrm{G}$ and ordered on scales of $10-20\,\mathrm{kpc}$; while in cooling core systems the magnetic field strength is generally higher (around $10-40\,\mu\mathrm{G}$) and ordered on smaller scales of a few to $10\,\mathrm{kpc}$ Clarke04 . In Table 1 we list the typical ranges for some of the galaxy cluster properties.