Axion-Like Particles, Magnetars, and X-ray Astronomy
573
ΔQ(χ 0 , x) = Q(χ 0 , x) − Q(χ 0 , 1) = −ΔI (χ 0 , x)
(15)
characterize the effects of ALP–photon oscillations in the magnetosphere. Indeed,
in the absence of ALPs, the surface-subtracted Stokes parameters are ΔI (χ 0 , x) =
ΔQ(χ 0 , x) = 0. However, non-vanishing surface-subtracted Stokes parameters
satisfying the relation ΔQ(χ 0 , x) = −ΔI (χ 0 , x) can be seen as a smoking gun
signature for the presence of ALPs.
4 ALP-to-Photon Conversion Probability
By studying the normalized surface-subtracted Stokes parameter
R(χ 0 , x) =
ΔI (χ 0 , x)
A 2
(16)
= cos(2χ 0 )
2π
0
dΔφ 0
2π
1
2
1 −
cos
2χ(x)| χ(1)=χ 0 ,Δφ(1)=Δφ 0
cos(2χ 0 )
,
(17)
it is possible to show that R(χ 0 , x) factorizes as
R(χ 0 , x) = P a→γ (x) cos(2χ 0 ),
(18)
where
P a→γ (x) = sin
2
χ(x)| χ(1)=0,Δφ(1)=0
,
(19)
is the ALP-to-photon conversion probability at a distance xr 0 for pure ALP initial
state [6]. This remarkable factorization property can be traced back to the averaging
over the initial phase difference Δφ 0 and would not necessarily occur if the
dominant ALP and photon production mechanisms were related.
Hence, the Stokes parameters (13) and the averaged intensities (12) at a distance
xr 0 from the magnetar are given explicitly in terms of the initial photon and ALP
intensities through A ⊥ , A, and χ 0 (which depend on the production mechanisms)
and the ALP-to-photon conversion probability through P a→γ (x) (which depends on
the ALP and magnetar parameters through the evolution equations). They are given
respectively by
I (χ 0 , x) = A
2
⊥ +
A 2
2
1 + [2P a→γ (x) − 1] cos(2χ 0 )
,
(20)
Q(χ 0 , x) = A
2
⊥ −
A 2
2
1 + [2P a→γ (x) − 1] cos(2χ 0 )
,
(21)
for the Stokes parameters and