Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect

The gist

We show that the null geodesic equation for photons is modified in the presence of a charged scalar field, with quantum fluctuations acting as an effective mass term that changes the null paths to timelike curves.

Vacuum Polarization and Photon Propagation

The paper investigates how vacuum polarization, arising from virtual charged particles, modifies photon propagation within a curved spacetime by considering a minimally coupled massive test scalar field (Page 2). The construction involves the Lagrangian for a charged (complex) scalar field and the electromagnetic field in a curved spacetime, which leads to modified field equations (Page 3). In the geometric optics approximation, the second term in equation (8) appears like an effective mass term in equation (7), which should modify photon propagation (Page 4). This modification is analyzed by assuming a highly oscillating phase and deriving the leading order result in equation (10), which shows that photons now follow non-geodesic timelike curves, behaving as if they propagate through a medium (Page 4).

Sachs-Wolfe Effect Modification

The resulting contribution to the Sachs-Wolfe effect varies with photon frequency, leading to frequency-dependent corrections to the cosmic microwave background (CMB) blackbody spectrum in the form of a µ-distortion, as well as modifications to the CMB power spectrum (Page 1). The frequency measured by a comoving observer is given by equation (22), which can be written as ω = ω0 / a [1 + e 2ω 20ϕ∗ϕ] - 1/2γij l i l j - Φ + e 2a 2ω 20ϕ∗ϕ (Page 6). This equation implies that the standard Sachs-Wolfe terms give the standard effect, while the last term is the contribution of the charged scalar field (Page 6).

µ-Distortion Calculation

The distortion is determined by comparing photon frequencies at the last scattering surface and today, leading to a temperature correction T = T0 [1 + e 2ω20ϕ∗ϕ(η0) - e 2a 2Lω20ϕ∗ϕ(ηL)] (Page 7). Defining the µ-distortion in the Boltzmann factor as exp(ωobs/T) = exp(ωobs/T0 + µ), one finds that µ = e 2ω0 T0 / a2L⟨ϕ∗ϕ⟩L -⟨ϕ∗ϕ⟩0 (Page 8). The vacuum expectation values are estimated within the free theory, and for m ∼ TeV, one finds µ ≈ 10-5 e squared, which can be within the reach of upcoming measurements if e ≈ 1/10 (Page 8).

Power Spectrum Analysis

The two-point angular correlation function is related to the correlation functions of the fields Φ, Ψ, γij and ϕ (Page 9). The contribution of δki H in equation (22) is completely negligible since the fields are evaluated today (the time argument of the perturbations in (18) is η0) (Page 9). The largest contribution to the power spectrum comes from Φ and e 2a 2ϕ∗ϕ/ω20 terms in equation (22) (Page 9). The contribution of ϕ∗ϕ in equation (22) to the CMB power spectrum is scale free at superhorizon distances because the correlation function does not depend on x1 − x2 when x1 and x2 are separated at superhorizon distances from equation (36) (Page 10).

Conclusion

The contribution of the scalar field to the superhorizon CMB power spectrum diminishes due to the decay of superhorizon mode functions, which contrasts with the case of µ-distortion where high-energy modes also contribute, leading to a more significant effect (Page 12). The scalar field mass m must be larger than energy scales accessible to current accelerators for this contribution, and an effective Higgs-photon coupling of the form H FµνFµν arises at one loop (Page 12). The vacuum polarization effects are expected to be more significant at earlier times when the mean free path is not very short (Page 12).

--- Page 1 ---

Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect

Ali Kaya∗

Department of Physics and Astronomy,

Texas A&M University, College Station, TX 77843, USA

--- Page 2 ---

I. INTRODUCTION

The quantum vacuum exhibits remarkable properties. It effectively behaves as a medium due to the creation of virtual charged particles, a phenomenon known as vacuum polarization (Page 2). In flat spacetime, vacuum polarization gives rise to several observable effects including charge screening, running of the fine-structure constant, Lamb shift, light-light scattering and spectral line shifts in exotic atoms (Page 2). In an expanding universe, vacuum polarization effects have also been studied to investigate whether the large scale micro-Gauss magnetic fields observed in galaxies and galaxy clusters could be generated by vacuum fluctuations during inflation [1, 2] (Page 2).

--- Page 3 ---

II. PHOTON PROPAGATION IN A SCALAR FIELD MEDIUM

Let us first review how the geodesic equation follows from the field equations in the geometric optics approximation. While this is typically discussed for massless fields, we aim to generalize the result to massive particles (the following discussion for does not offer a genuine geometrical optics derivation of Maxwell’s equations, but it captures the main idea) (Page 3). The construction is most clearly illustrated for a minimally coupled massive test scalar field on a curved background obeying ∇µ∇µφ − m2φ = 0, (1) where ∇µ is the (unique torsion free and metric compatible) covariant derivative of gµν. To obtain the geodesic equation, one takes a solution in the form φ = A exp(iS) (2) and supposes that the phase is highly oscillating. This amounts to assuming that ∇µS is much larger than any other relevant scale, such as the spacetime curvature. One can then setup a perturbation theory based on the number of ∇µS factors. Defining kµ ≡ ∇µS (3) the leading order term in (1) implies kµkµ = −m2, (4) which shows kµ is a time-like vector. Using ∇ν(kµkµ) = 0, one can also see that kµ is a geodesic tangent kµ∇µkν = 0 (Page 3). Therefore, in this approximation, the surfaces of constant phase are generated by time-like geodesics (Page 3). The next order equation dictates how the amplitude A changes along the geodesics kµ∇µ ln A = −1/2∇µkµ. (Page 3).

--- Page 4 ---

FIG. 1: An illustration of photon propagation from point A to point B in two-dimensional spacetime. The actual path is (piece-wise) null (between collisions) while the mean path becomes timelike (Page 4). Imposing the gauge∇µAµ = 0, which implies at the lowest order kµaµ = 0, one can obtain1 kµkµ = −2e 2ϕ∗ϕ, kµ∇µkν = −e 2∇ν(ϕ∗ϕ), where, as in (3), kµ = ∇µS. Eq. (10) shows that photons now follow non-geodesic timelike curves, behaving as if they propagate through a medium (Page 4). Note that the two equations in (10) are consistent; indeed the first one can be seen to hold through a path if it is satisfied at an initial point. In deriving (10) from the field equations, one uses kµ = ∇µS, i.e. kµ is hypersurface orthogonal. In solving (10) below we will ignore this fact. Indeed, one can show that when (∇µkν − ∇νkµ) = 0 at a point P, then kρ∇ρ(∇µkν − ∇νkµ)(P) = 0. As a result, if a congruence of curves defined by (10) is initially hypersurface orthogonal, it will remain so, at least within an initial neighborhood. The same holds true when the right-hand side of (10) is zero (Page 4).

--- Page 5 ---

We are interested in the lowest order impact of the scalar field (vacuum) fluctuations on the motion of photons created, say during reheating after inflation. The first term in (8) is a non-homogeneous source term that produces a particular field AµP = -ie R Gµν(ϕ∂νϕ∗−ϕ∗∂νϕ), where Gµν is a Green function in an appropriate gauge obeying certain boundary conditions. At this linear order, AµP does not interact with the photons generated by other sources (Page 5). Obviously, the second term in (8) appears like an effective mass term in (7), which should modify photon propagation. To determine its effect in the geometric optics approximation, one can proceed as in the case of massive Klein-Gordon field discussed above. After introducing the phase and the amplitude as Aµ = aµ exp(iS), (9) the field equations can be analyzed order by order by assuming the phase is highly oscillating (Page 5).

--- Page 6 ---

III. SACHS-WOLFE EFFECT

The equations (12), (13), and (14) describe the perturbed photon trajectory in the spacetime (11) under the influence of the charged scalar field ϕ and the metric perturbations.

1. Improve cosmic microwave background spectrum modeling by incorporating frequency dependent corrections to CMB blackbody spectra in the form of a mu distortion derived from vacuum polarization effects. This allows AI systems to predict observable spectral deviations from a perfect blackbody based on scalar field properties and regularization choices.

2. Enhance cosmological parameter inference by allowing AI models to constrain the effective scalar field mass m and coupling constant e through comparisons between the predicted two point functions of gravitational perturbations and vacuum polarization contributions. This enables better determination of fundamental physics parameters related to early universe inflation.

3. Develop simulations for CMB power spectrum analysis that account for frequency dependent corrections, leading to more accurate predictions of anisotropies in the CMB. AI systems can then distinguish between standard Sachs-Wolfe effects and these new vacuum polarization signatures, improving data interpretation.

4. Create a framework for estimating vacuum expectation values of the scalar field at different cosmological epochs by utilizing adiabatic regularization techniques to overcome UV divergences. This allows AI to estimate the magnitude of terms like ⟨phi∗phi⟩L and ⟨phi∗phi⟩0, which are crucial for calculating distortion parameters like mu.

5. Enhance particle phenomenology simulations by allowing AI to investigate the impact of different scalar field mass regimes (m H) on superhorizon mode evolution and its contribution to the Sachs-Wolfe effect. This guides searches for new physics by predicting which mass scales are detectable in current or future CMB observations.