Probing spatial variation of the fine-structure constant using the CMB

The fine-structure constant, $\alpha$, controls the strength of the electromagnetic interaction. There are extensions of the standard model in which $\alpha$ is dynamical on cosmological length and time scales. The physics of the cosmic microwave background (CMB) depends on the value of $\alpha$. The effects of spatial variation in $\alpha$ on the CMB are similar to those produced by weak lensing: smoothing of the power spectrum, and generation of non-Gaussian features. These would induce a bias to estimates of the weak-lensing potential power spectrum of the CMB. Using this effect, Planck measurements of the temperature and polarization power spectrum, as well as estimates of CMB lensing, are used to place limits (95% C.L.) on the amplitude of a scale-invariant angular power spectrum of $\alpha$ fluctuations relative to the mean value ($C_L^{\alpha }=A_{\mathrm{SI}}^{\alpha }/[L(L+1)]$) of $A_{\mathrm{SI}}^{\alpha }\le 1.6\times {10}^{-5}$. The limits depend on the assumed shape of the $\alpha$-fluctuation power spectrum. For example, for a white-noise angular power spectrum ($C_L^{\alpha }=A_{\mathrm{WN}}^{\alpha }$), the limit is $A_{\mathrm{WN}}^{\alpha }\le 2.3\times {10}^{-8}$. It is found that the response of the CMB to $\alpha$ fluctuations depends on a separate-universe approximation, such that theoretical predictions are only reliable for $\alpha$ multipoles with $L\lesssim 100$. An optimal trispectrum estimator can be constructed and it is found that it is only marginally more sensitive than lensing techniques for Planck but significantly more sensitive when considering the next generation of experiments. For a future CMB experiment with cosmic-variance limited polarization sensitivity (e.g., CMB-S4), the optimal estimator could detect $\alpha$ fluctuations with $A_{\mathrm{SI}}^{\alpha }>1.9\times {10}^{-6}$ and $A_{\mathrm{WN}}^{\alpha }>1.4\times {10}^{-9}$.

Figure 1

The change to the CMB power spectra as the fine-structure constant is varied, $\alpha ={\alpha }_0(1+\phi )$. As discussed in the text an increased $\alpha$ leads to a shift in the peak of the visibility function to higher redshifts leading to an increase in the distance to the SLS, which, in turn, shifts the angular scale of the CMB anisotropies to smaller scales (higher values of $l$). The earlier decoupling/recombination for higher $\alpha$ gives us a “snapshot” of the CMB anisotropies at a time when they are less damped, leading to an enhanced amplitude for scales above the damping scale.

Figure 2

The change to the visibility function (top) and the damping of the CMB anisotropies (bottom) with a change in the fine-structure constant $\alpha ={\alpha }_0(1+\phi )$. In the top panel we can see that an increase in $\alpha$ leads to stronger electromagnetic interactions shifting the peak of the visibility function (which marks the temperature at decoupling) to early times (i.e., higher redshift). An increase in $\alpha$ increases the Thomson scattering cross section leading to a larger differential cross section at early times. As we approach decoupling the additional changes to the ionization history cause a larger $\alpha$ to give a smaller differential cross section.

Figure 3

The relative contribution of variations in the visibility function (Vis) and in the transfer function (Evo) to the overall derivative power spectra. It is clear that $\alpha$ modulation has a significant impact on both terms.

Figure 4

The change to the CMB power spectra due to stochastic scale-invariant fluctuations in the fine-structure constant (we have set the amplitude, $A_{\alpha }^{\mathrm{SI}}$, to be large enough to see the effects). The spatial variation of $\alpha$ causes a smoothing of the anisotropies and affects the polarization more than it does the temperature power spectrum. Note that the B-mode power spectrum does not include the effects of gravitational waves but does show the B-modes generated by lensing of the E-mode polarization and spatial fluctuations in $\alpha$.

Figure 5

The affects of $\alpha$ modulation on the lensing potential power-spectrum estimator. The top panel shows expectation value of the lensing potential power-spectrum estimator in the presence of a scale-invariant $\alpha$-modulation contribution. The lower panel shows the residual lensing potential power-spectrum estimator in the presence of a white-noise $\alpha$ power spectrum. The amplitudes have been chosen to saturate the 95% C.L. bounds discussed in Sec. 4.

Figure 6

Bias of separate-universe approximation minimum-variance estimator for $\phi$ reconstruction. The horizontal axis is the wave number of the $\alpha$-modulating mode in units of the inverse acoustic horizon.

Figure 7

The one-dimensional marginalized posterior for $A_{\phi }$ using the three combinations of datasets discussed in the text.

Figure 8

Sensitivity of the optimal trispectrum-based estimator to a spectrum of scale-invariant spectrum for $\phi$.

Figure 9

Constraints to $A_{\alpha }^{\mathrm{SI}}$ and $A_L$ from Planck temperature (T), polarization (P), and estimates of the lensing-potential power spectrum (lensing). Without the lensing data, the degeneracy between $A_{\alpha }^{\mathrm{SI}}$ and $A_L$ causes $A_L$ to be consistent with its expected value of unity. Once we include the lensing data this degeneracy is broken and $A_L$ takes on an anomalously large value.

Figure 10

The derivative power spectra used in this paper, which agree (up to changes in fiducial cosmological parameters) with those in Refs. [23, 52].

Figure 11

The ${\chi }^2$ showing the goodness of fit between the finite-difference derivative and the polynomial derivative.