A Photon-Scalar Mixing in Multiple Magnetic Domains

# Chameleon-Photon Mixing in a Primordial Magnetic Field

## Abstract

The existence of a sizable, , cosmological magnetic field in the early Universe has been postulated as a necessary step in certain formation scenarios for the large scale magnetic fields found in galaxies and galaxy clusters. If this field exists then it may induce significant mixing between photons and axion-like particles (ALPs) in the early Universe. The resonant conversion of photons into ALPs in a primordial magnetic field has been studied elsewhere by Mirizzi, Redondo and Sigl (2009). Here we consider the non-resonant mixing between photons and scalar ALPs with masses much less than the plasma frequency along the path, with specific reference to the chameleon scalar field model. The mixing would alter the intensity and polarization state of the cosmic microwave background (CMB) radiation. We find that the average modification to the CMB polarization modes is negligible. However the average modification to the CMB intensity spectrum is more significant and we compare this to high precision measurements of the CMB monopole made by the far infrared absolute spectrophotometer (FIRAS) on board the COBE satellite. The resulting 95% confidence limit on the scalar-photon conversion probability in the primordial field (at ) is . This corresponds to a degenerate constraint on the photon-scalar coupling strength, , and the magnitude of the primordial magnetic field. Taking the upper bound on the strength of the primordial magnetic field derived from the CMB power spectra, , this would imply an upper bound on the photon-scalar coupling strength in the range to , depending on the power spectrum of the primordial magnetic field.

## I Introduction

The origin of the observed large-scale magnetic fields of order found in nearly all galaxies and galaxy clusters is still largely unknown. It is generally believed that the galactic magnetic field develops by some form of amplification from a pre-galactic cosmological magnetic field. The two popular formation scenarios that are considered are either some exponential dynamo mechanism which amplifies a very small seed field of order as the galaxy evolves, or the adiabatic collapse of a larger existing cosmological field of order . There are a number of pros and cons to both scenarios. See (1) for reviews on the subject. Theories explaining the origin of this primordial magnetic (PMF) field are still highly speculative. It has been suggested that a large-scale magnetic field could be produced during inflation if the conformal invariance of the electromagnetic field is broken; see for example (2) or more recently (3).

To date, there is no astrophysical evidence for the existence of a large-scale cosmological magnetic field, and only upper bounds on its magnitude have been derived. So far the strongest constraints have come from measurements of the cosmic microwave background (CMB) and big-bang nucleosynthesis (1); (4); (5); (6); (7). The CMB bounds are derived by considering the effects of Faraday rotation induced by the PMF on the CMB power spectra. In (5) an upper limit in the range to was derived for the mean-field amplitude of the PMF at a comoving length scale of by comparison to the WMAP 5-year data. More recent work (6); (7) analysing the WMAP data in combination with other CMB experiments such as ACBAR, CBI and QUAD have placed tighter constraints on the amplitude of the PMF with an upper bound on the mean-field amplitude at of .

The existence of a primordial magnetic field would induce mixing between CMB photons and axion-like particles (ALPs). ALPs refer collectively to any very light scalar or pseudo-scalar with a linear coupling to or respectively. In this paper we consider non-resonant mixing of scalar ALPs and CMB photons, with specific reference to the chameleon scalar field model (8); (9); (10). Standard ALPs have constant mass and photon-scalar coupling everywhere, while the chameleon model has a density dependent mass. In sparse environments, such as the primordial plasma, the chameleon acts as a very light scalar field and would be indistinguishable from a standard scalar ALP. However in dense environments the chameleon is very heavy and evades the standard ALP constraints (11). The current best constraints on the coupling strength between the chameleon and electromagnetic fields are: (10) and (12). Other chameleon-like theories exist such as the Olive-Pospelov model (12); (13) which have a density-dependent coupling strength, but we do not discuss these further here.

The resonant mixing of photons with scalar and pseudo-scalar ALPs in a primordial magnetic field has been analysed by Mirizzi, Redondo and Sigl (14). They find a constraint on the combined magnetic field strength, , and ALP-photon coupling strength, : , for ALP masses between and . We believe that these results will not necessarily be applicable to the chameleon because its mass evolves as the density of the Universe decreases.

In the following analysis we assume a stochastic primordial magnetic field with a power-law power spectrum, similar to the treatment in (5). We assume fluctuations in the magnetic field are damped on small scales due to Alfvn wave dissipation (15), and subdivide the magnetic field into multiple domains of length of a comparable size to just above the Alfvn wave damping scale . The magnetic field in each domain is assumed to be approximately constant, and correlated to the other domains according to the magnetic power spectrum. A similar method was applied to correlations in quasar polarization spectra by Agarwal, Kamal and Jain (16).

The degree of conversion between chameleons and photons in a magnetic field is inversely proportional to the electron density in the plasma (12). Hence the dominant contribution to photon-scalar mixing will take place in the region after recombination when the ionization fraction drops significantly and before reionization. This greatly simplifies the mixing equations because we do not need to evolve the photon-scalar mixing equations through the last scattering surface, nor include the density inhomogeneities present after reionization. We model a scenario in which the primary CMB is formed at the last scattering surface and then evolves through a primordial magnetic field extending from recombination () to the epoch of reionization ().

This paper is organized as follows: in section II the chameleon model is introduced and the calculations describing photon-scalar mixing in a magnetic field, living in a Friedman-Robertson-Walker (FRW) spacetime, are presented. The power spectrum of the primordial magnetic field is discussed in more detail in section III. In section IV we analyse the evolution of the photon and chameleon states as they propagate through the multiple magnetic domains, and predict the average modification to the CMB intensity and polarization. In section V, our predictions are compared to precision measurements of the CMB monopole made by the far infrared absolute spectrophotometer (FIRAS) on board the cosmic background explorer (COBE) satellite. We present a summary of the work and our conclusions in section VI. The appendix contains details of the equations governing the evolution of the CMB Stokes parameters.

## Ii Chameleon-Photon Mixing in an Expanding Universe

The chameleon scalar field has been suggested as a candidate for the dark energy quintessence field (8). Standard quintessence scalar fields have coupling strengths to matter which are unnaturally fine-tuned to very low values so as to be compatible with fifth-force experiments. By contrast the chameleon model is constructed so that the chameleon can have a gravitational strength (or greater) coupling to normal matter while at the same time evading fifth-force constraints. This is achieved by introducing a density-dependent term in the effective potential of the chameleon, which causes a change to the minimum of the potential depending on the density of the surrounding matter. In sparse environments the chameleon behaves as an effectively massless scalar field while in a laboratory on Earth it is much heavier and evades detection.

In addition to the matter coupling, the chameleon can have a non-zero coupling, , to the electromagnetic (EM) field. The mass parameter, , describing the strength of the photon coupling is best constrained by observations of radiation passing through astrophysical magnetic fields. The two best current constraints in terms of the coupling strength were presented in section I. In terms of , the lower limit from considering constraints on the production of starlight polarization in the galactic magnetic field is (10). Measurements of the Sunyaev–Zel’dovich effect in galaxy clusters places a lower bound on in the range to , depending on the model assumed for the cluster magnetic field (12).

The action describing the chameleon model is that of a generalized scalar-tensor theory:

 S = ∫d4x√−g(12M2PlR−12gμν∂μϕ∂νϕ −V(ϕ)−14BF(ϕ/M)FμνFμν) +Smatter(ψ(i),g(i)μν),

where is the self-interaction potential of the scalar field , and is the matter action, excluding the kinetic term of electromagnetism, which contains the coupling of the matter fields to the conformal metric . The determine the coupling of the scalar field to different matter species, and determines the photon-scalar coupling. For simplicity a universal matter coupling, , is assumed. We take the Universe to be described by a spatially-flat Friedman-Robertson-Walker (FRW) metric: where is the time-dependent scale factor describing the expansion, normalized to today. The coordinates are comoving, related to the physical coordinates by , and is the proper time.

The equation of motion for the field is found by varying with respect to :

 □ϕ ≡ −∂2tϕ−3H(t)∂tϕ+1a2∇2ϕ = V′(ϕ)+14F2B′F(ϕ)−B3i(ϕ)B′i(ϕ)T(i)m,

where is the Hubble expansion rate. is the trace of the stress-energy tensor in the conformal frame described by the metric , for which particle masses are constant and independent of . This is related to the stress-energy tensor in the physical frame by . In general, corresponding to the physical, measured density and pressure. If we assume the chameleon does not couple to the energy density of dark matter or any other exotic particles, then , the baryonic matter density. Hence

 □ϕ=V′eff(ϕ;F2,ρb),

where,

 Veff(ϕ;F2,ρb) ≡ V(ϕ)+14F2BF(ϕ)+ρbBm(ϕ).

The form of the self-interaction potential determines whether a general scalar-tensor theory is chameleon-like or not. For a chameleon field we require the potential to be of runaway form. A typical choice of potential can be described by

 V(ϕ)≈Λ0+Λn+4nϕn, (1)

where for reasons of naturalness . If the chameleon is to be a suitable candidate for dark energy, we require . The chameleon mass is defined by

 m2ϕ≡Veff,ϕϕ(ϕmin;¯F2,ρb),

where is the background value of the electromagnetic field tensor.

The photon equation of motion comes from varying with respect to :

 ∇ν(BF(ϕ)Fμν)=0.

Any electromagnetic components in would introduce electromagnetic currents on the right-hand side of the equation. We neglect these at this stage in the calculation. The primordial plasma is a good conductor and so any currents will be small (17). To a good approximation the induced currents arising from photon propagation will be described by the plasma frequency which is included later in this calculation.

In standard chameleon theories we assume and approximate and , where this defines and . We expect these two coupling strengths to be of similar magnitude but do not require it. Under this approximation, and splitting the electromagnetic field into the background EM field and the photon field , the above equations of motion become

 □φ ≃ m2ϕφ+14Meff(fμν¯Fμν+fμν¯Fμν), ∇μfμν ≃ −1Meff(∇μφ)¯Fμν,

where is the perturbation in the scalar field about its background value, and we neglect terms that are and .

In an inertial, locally Minkowskian frame, with metric , the electromagnetic field tensor has the form

 ^Fμν=⎛⎜ ⎜ ⎜ ⎜⎝0ExEyEz−Ex0Bz−By−Ey−Bz0Bx−EzBy−Bx0⎞⎟ ⎟ ⎟ ⎟⎠.

The coordinate transformation from the locally Minkowski metric, , into the FRW metric, , is

 Fμν=ΛμαΛνβ^Fαβ,

with

 Λμν=∂xμ∂^xν=diag(1,1/a,1/a,1/a).

Thus in an FRW expanding Universe the EM field tensor is given by

 Fμν=⎛⎜ ⎜ ⎜ ⎜⎝0Ex/aEy/aEz/a−Ex/a0Bz/a2−By/a2−Ey/a−Bz/a20Bx/a2−Ez/aBy/a2−Bx/a20⎞⎟ ⎟ ⎟ ⎟⎠.

We take the background to be a large-scale magnetic field , and assume the contributions from electric fields in the plasma are negligible. The photon field can be described by the quantum polarization states where . Taking the Lorentz gauge condition , we find the equations of motion become

 −∂2tφ−3H(t)∂tφ+1a2∇2φ = m2ϕφ+B⋅(∇×a)Meff,

and

 −∂2ta−5H∂ta+2qH2a−4H2a+1a2∇2a = Extra open brace or missing close brace

where we define the deceleration parameter, .

In this analysis we have assumed that the background values of the chameleon and photon fields are slowly varying over length and time scales of where is the proper frequency of the electromagnetic radiation being considered. We further assume that the frequency of the radiation satisfies and that . The equations describing mixing between the photons and scalar field are then

 −¨a+1a2∇2a ≃ ∇φ×Ba2Meff+ω2pla, −¨φ+1a2∇2φ ≃ B⋅(∇×a)Meff+m2ϕφ,

where we have included the plasma frequency, , as an effective photon mass. Whenever electromagnetic radiation propagates through a plasma with electron number density it displaces the electrons slightly and induces oscillations at their natural frequency . This interaction of the EM wave with the electron density hinders its progress and acts as an effective mass for the photons.

Taking the radiation to be propagating in the direction of an orthonormal Cartesian basis , with , the equations of motion for the chameleon and photon can be written in matrix form:

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−∂2t+∂2za2−⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ω2pl0−By∂za2Meff0ω2plBx∂za2MeffBy∂zMeff−Bx∂zMeffm2ϕ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎛⎜⎝|γx⟩|γy⟩|φ⟩⎞⎟⎠=0.

Note how the chameleon scalar field only mixes with the component of photon polarization aligned perpendicular to the transverse magnetic field. The magnitude of the magnetic field aligned parallel to the photon path () plays no part in the mixing.

Following a similar procedure to that in (12); (18) we assume the fields vary slowly over time compared to the frequency of the radiation and that the refractive index is close to unity, which requires , and all . Defining and , we approximate and such that . Thus,

 ⎡⎢ ⎢ ⎢ ⎢ ⎢⎣iωa∂z−⎛⎜ ⎜ ⎜ ⎜ ⎜⎝ω2pl0−iByω2aMeff0ω2pliBxω2aMeffiByωa2Meff−iBxωa2Meffm2ϕ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥⎦⎛⎜ ⎜⎝|^γx⟩|^γy⟩|^φ⟩⎞⎟ ⎟⎠=0.

We further simplify the mixing matrix by defining and with . The magnetic field and frequency can be expressed in terms of their comoving values: and . In the subsequent analysis we drop the subscript-zero notation for comoving magnetic field values. Thus we can write

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ia∂z−⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝00−By2Meff00Bx2Meff−By2MeffBx2Meffa3m2eff2ω0−i∂za⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎛⎜ ⎜⎝|¯γx⟩|¯γy⟩1a|i¯φ⟩⎞⎟ ⎟⎠=0, (2)

where .

To solve this system of equations we must make various simplifying assumptions. We neglect spatial fluctuations in the electron density along the path length, and only consider a simple redshift scaling: , where is the ionization fraction along the path. The ionization fraction drops rapidly at recombination from to its final freeze-out value (19), and only increases again at the epoch of reionization. The conversion between photons and light scalar particles in a magnetic plasma scales inversely with the electron number density in the plasma (12). Thus the conversion will be suppressed for high values of the ionization fraction. We assume the dominant contribution to photon-scalar mixing in a primordial magnetic field occurs between redshift (by which time the ionization fraction has already dropped to ) and redshift (onset of reionization). We approximate the ionization fraction as being at a constant value of in this region, and neglect contributions from other sections of the path length.

The chameleon mass for the generalized self-interaction potential of Eq. (1) is

 m2ϕ ≃ (n+1)Λ−n+4n+10(ρb0a3M+|B|22a4Meff)n+2n+1 ≈ 8.4×10−57a−9/2(Meff109GeV)−32GeV2,

where the second line assumes , a magnetic field of less than , a chameleon coupling strength of less than , and that the average baryonic density is 4% of the critical density. We compare this to the plasma frequency,

 ω2pl = 4παemneme=(4παemXeρb0memb)a−3≡p0a−3 ≈ 1.73×10−49a−3GeV2

over the redshift range to 750, where we have taken the average mass per baryon to be (20). It is clear then that along the path from recombination to reionization. Our subsequent analysis requires the chameleon to be sufficiently light for the plasma frequency to dominate over the path length, and thus is not dependent on the specific form of the chameleon potential. It applies to any scalar ALP satisfying this condition.

The comoving distance, , travelled by the photons is related to the value of the scale factor at by

 dz=da/a2H(a).

 a(z)≃(a1/2(0)+12H0Ω1/2m0z)2

in the matter dominated era between recombination and reionization, where (21). Over the total path length from a redshift of to , the comoving distance covered is .

Changing the integration variable in Eq. (2) to

 ξ(z)≡−2H0Ω1/2m0a−12(z),

such that , and noting , we find

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣i∂ξ−⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝00−By(ξ)2Meff00Bx(ξ)2Meff−By(ξ)2MeffBx(ξ)2Meff−p02ω0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎛⎜ ⎜⎝|¯γx⟩|¯γy⟩|¯χ⟩⎞⎟ ⎟⎠=0 (3)

where . The –dependence of the mixing matrix is entirely due to fluctuations in the magnetic field.

## Iii The Primordial Magnetic Field

The generic model for the primordial magnetic field is of a stochastic field parameterized by a power-law power spectrum up to a cut-off scale ,

 P(k)=ABknB,k

where is some normalization constant. The spectral index must be greater than to prevent infrared divergences in the integral over the power spectrum at long wavelengths. Assuming a statistically homogeneous and isotropic field, the power spectrum is defined by

 ⟨Bi(k)B⋆j(k′)⟩≡(2π)3PijP(k)δ(k−k′),

where is the projector onto the transverse plane imposed by the divergence-free nature of the magnetic field. We adopt the Fourier transform convention,

 Bi(k)=∫d3xBi(x)e−ik⋅x.

The two-point correlation function is then

 ⟨Bi(x)Bj(y)⟩=1(2π)3∫d3kPijP(k)eik⋅(x−y).

Following the prescription in (5); (22), normalization of the magnetic field is achieved by convolving the field (in real space) with a Gaussian smoothing kernel of comoving radius . Defining the Fourier transform of the Gaussian smoothing kernel to be , the Fourier transform of the convolved field is

 Bi(k)|λB=Bi(k)⋅f(k).

The mean-field amplitude of the smoothed field, , is given by

 B2λ = ⟨Bi(x)|λBBi(x)|λB⟩ = 2AB(2π)21λnB+3Γ(nB+32).

Hence the normalization constant,

 Missing or unrecognized delimiter for \left

where .

Constraints on the magnitude of the primordial magnetic field from a comparison of Faraday rotation effects in the CMB with the WMAP 5-year data were given in (5). They found that the upper limit on the mean-field amplitude of the magnetic field on a comoving length scale of was in the range to () for a spectral index to . This range for the spectral index was based on the likely formation scenarios for the primordial magnetic field (2) and current exclusion bounds on the spectral index (23). More recent results analysing the WMAP 5-year data in combination with other CMB experiments (6) constrained the mean-field amplitude and the spectral index (). An analysis of the latest WMAP 7-year data (7) derived upper bounds of and ().

The magnetic field power spectrum will be damped at small length-scales. We assume a cut-off to the power spectrum at the Alfvn wave damping scale (15). This damping scale is approximated (22) as

 (kDMpc−1)nB+5≈2.9×104h(Bλ10−9G)−2(kλMpc−1)nB+3

where .

Using these definitions, we can calculate the correlations in the magnetic field along the line of sight:

 ⟨Bi(x^z)Bj(y^z)⟩ = 1(2π)32π∫0π∫0∞∫0k2sinθkdkdθkdϕk ⋅(δij−kikjk2)P(k)ei(x−y)kcosθk,

where we have written the wave vector in spherical polar coordinates, . Thus

 ⟨Bx(x^z)By(y^z)⟩=⟨By(x^z)Bx(y^z)⟩=0,

and

 ⟨Bx(x^z)Bx(y^z)⟩=⟨By(x^z)By(y^z)⟩≡RB(x−y),

where

 RB(z)=2(2π)2∞∫0k2P(k)dk[sin(zk)zk+cos(zk)(zk)2−sin(zk)(zk)3].

Substituting the assumed form for the magnetic power spectrum, and changing variables, we find

 Missing or unrecognized delimiter for \right (4)

with

 K[θ]≡1∫0tnB+2dt[sin(θt)θt+cos(θt)(θt)2−sin(θt)(θt)3]. (5)

## Iv Evolution of the Stokes Parameters

Following a similar procedure to that in (16) we subdivide the path length into many small domains of length , each with a constant magnetic field strength and direction, and take correlations in the field across the different domains to be described by the magnetic power spectrum. The inverse power-law form assumed for the primordial magnetic power spectrum imposes a rapidly decreasing amplitude to the field fluctuations and an even faster decreasing slope to the magnetic power spectrum with increasing . At very large the amplitude of the fluctuations are damped to zero by Alfvn wave dissipation. However slightly above the Alfvn wave damping scale, , the amplitude of the field is non-zero yet the slope of the power spectrum will be approximately flat. In what follows we assume on scales slightly greater than the magnetic field can be approximated as constant and take . The exact value of is only necessary in determining the number of magnetic domains that the photons traverse along the path. Some variation in will not affect the order of magnitude of the final results.

### iv.1 Evolution through a Single Domain

For a single domain we define to be the magnitude of the transverse component of the magnetic field, and to be the angle it makes with the -axis. Within the domain these stay approximately constant. Returning to Eq. (3), we rotate the basis so as to reduce the problem to two-component mixing. Defining

 |¯γa⟩ = cosσ|¯γx⟩+sinσ|¯γy⟩, |¯γb⟩ = −sinσ|¯γx⟩+cosσ|¯γy⟩,

we find is constant over the domain and

 [i∂ξ−(0B/2MB/2M−p0/2ω0)](|¯γb⟩|¯χ⟩)=0.

Diagonalisation of this two-component mixing leads to

 |¯γnewb(L)⟩ = |¯γnewb(0)⟩e−i(Δ−Δ/cos2θ), |¯χnew(L)⟩ = |¯χnew(0)⟩e−i(Δ+Δ/cos2θ),

where

 |¯γnewb⟩ = cosθ|¯γb⟩+sinθ|¯χ⟩, |¯χnew⟩ = −sinθ|¯γb⟩+cosθ|¯χ⟩,

and

 tan2θ ≡ 2ω0Bp0Meff, (6) Δ ≡ −p04ω0(ξ(L)−ξ(0))≈−p0L4aω0. (7)

 A ≡ sin2θsin(Δcos2θ), tanψ ≡ cos2θtan(Δcos2θ),

the chameleon and photon polarization states in the rotated basis after passing through a single domain of length , are

 |¯γa(L)⟩ = |¯γa(0)⟩, |¯γb(L)⟩ = eiα√1−A2|¯γb(0)⟩+ie−iΔA|¯χ(0)⟩, |¯χ(L)⟩ = e−iβ√1−A2|¯χ(0)⟩+ie−iΔA|¯γb(0)⟩,

where and . The probability of conversion between chameleons and photons in a single domain is .

The intensity and polarization state of radiation is best described by its Stokes parameters: intensity, , linear polarization, and , and circular polarization, . These are defined in terms of the photon polarization states by

 I = ⟨γx|γx⟩+⟨γy|γy⟩, Q = ⟨γx|γx⟩−⟨γy|γy⟩, U+iV = 2⟨γx|γy⟩.

We additionally define chameleon ‘Stokes parameters’ to close the evolution equations,

 J+iK = 2eiψ⟨γx|χ⟩, L+iM = 2eiψ⟨γy|χ⟩, Iχ = ⟨χ|χ⟩.

The evolution of the Stokes parameters through a single magnetic domain, in the rotated basis, is then

 ~Iγ → (1−A22)~Iγ+A22~Q+A2~Iχ +A√1−A2(sin2ψ~L−cos2ψ~M), ~Q → (1−A22)~Q+A22(~Iγ−2~Iχ) −A√1−A2(sin2ψ~L−cos2ψ~M), ~U+i~V → eiα√1−A2(~U+i~V) +ie−iβA(~J+i~K),

and

 ~J+i~K → e−iβ√1−A2(~J+i~K) +ieiαA(~U+i~V), ~L+i~M → e−2iψ(1−A2)(~L+i~M) +e2iψA2(~L−i~M) +iA√1−A2(~Iγ−~Q−2~Iχ).

is found by requiring that the total flux of photons and chameleons along the path length is conserved, .

### iv.2 Evolution through Multiple Magnetic Domains

The evolution equations above can be extended to many domains following a similar procedure to that in (10).

For CMB photons in the range , passing through a magnetic field of less than , and assuming a photon-scalar coupling strength no greater than , the mixing parameters defined in Eqs. (6) and (7) satisfy and in a domain of length . There is a region towards the end of the path at larger for which , but the approximations that follow involving averaging terms in and along the path are still valid in this instance. Thus on average over the total path length,

 A2≃2θ2≪1.

This places us in the regime of weak-mixing which can be solved analytically if we neglect terms smaller than , where is the number of magnetic domains along the path. Details of the mixing equations in multiple domains are presented in the Appendix. Here we quote the average, over many lines of sight, of the modification to the Stokes parameters. We assume there is no initial flux of chameleons and define to be the location of the domain. Then,

 ⟨δIγ⟩⟨I0⟩ ≃ −(2ω0p0Meff)2(12NRB(0) +N−1∑n=0n−1∑r=0cos2Δ(n−r)RB(zr−zn)),
 ⟨δQ⟩⟨Q0⟩ ≃ ⟨δU⟩⟨U0⟩ ≃ −(2ω0p0Meff)2(