Probing Spatial Variation of the Fine-Structure Constant Using the CMB

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Probing Spatial Variation Of The Fine-Structure Constant Using The CMB

Tristan L. Smith Swarthmore College, [email protected]

D. Grin

David B. Robinson , '19

Davy Qi , '19

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Recommended Citation Tristan L. Smith; D. Grin; David B. Robinson , '19; and Davy Qi , '19. (2019). "Probing Spatial Variation Of The Fine-Structure Constant Using The CMB". Physical Review D. Volume 99, Issue 4. DOI: 10.1103/ PhysRevD.99.043531 https://works.swarthmore.edu/fac-physics/374

This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Physics & Astronomy Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. PHYSICAL REVIEW D 99, 043531 (2019)

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

Tristan L. Smith,1 Daniel Grin,2 David Robinson,1,* and Davy Qi1,* 1Department of Physics and Astronomy, Swarthmore College, 500 College Avenue, Swarthmore, Pennsylvania 19081, USA 2Department of Physics and Astronomy, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA

(Received 19 September 2018; published 20 February 2019)

The fine-structure constant, α, controls the strength of the electromagnetic interaction. There are extensions of the standard model in which α is dynamical on cosmological length and time scales. The physics of the cosmic microwave background (CMB) depends on the value of α. The effects of spatial variation in α 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 α fluctuations relative to the mean value α α 1 α ≤ 1 6 10−5 α (CL ¼ ASI=½LðL þ Þ)ofASI . × . The limits depend on the assumed shape of the -fluctuation α α power spectrum. For example, for a white-noise angular power spectrum (CL ¼ AWN), the limit is α ≤ 2 3 10−8 α AWN . × . It is found that the response of the CMB to fluctuations depends on a separate- universe approximation, such that theoretical predictions are only reliable for α multipoles with L ≲ 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., α α 1 9 10−6 α 1 4 10−9 CMB-S4), the optimal estimator could detect fluctuations with ASI > . × and AWN > . × .

DOI: 10.1103/PhysRevD.99.043531

I. INTRODUCTION of the Universe. The BSBM model makes predictions for how α will vary in time and space [10,11]. Variations and Ever since Paul Dirac hypothesized the Law of Large extensions of the BSBM model exist that consider other Numbers [1], physicists have explored the possibility that effects such as density inhomogeneities [12],aswellasmore constants of nature are not in fact constant. Dirac proposed complicated scalar field couplings and potentials [13,14], time variation of the gravitational constant G to ensure that including a quintessence field [15], among others. certain large numbers in cosmology would be the same There has also been growing interest in models that order of magnitude throughout time [1,2], and Gamow then “disformally” couple electromagnetism to a scalar field suggested that time variation of the electric charge e could [16,17], as well as string-inspired “runaway dilaton” models explain the same coincidences [3,4]. The time dependence with dynamical extra dimensions that are stabilized by matter required to explain these coincidences has been ruled out couplings in a way that yields potentially observable time by stellar evolution and a variety of anthropic arguments evolution and spatial fluctuations in α [18–21],aswellas [5], but others have since explored more subtle variations in models in which a light scalar dark matter component or dark these and other fundamental constants, which emerge as energy itself induces α fluctuations [22–24]. predictions of theories with large extra dimensions [6–8]. On the observational side, claims have been made that There are several theories that naturally incorporate a the absorption spectra of distant quasars support cosmo- dynamical fine-structure constant, α.Bekensteinproposeda logical time variation and a dipole in α [25–29]. More model for a varying α that suppresses violations of the weak recent observations and analyses have failed to reproduce equivalence principle to undetectable levels [9].Thefull such a result consistently [30,31], calling into question the theory, known as the Bekenstein-Sandvik-Barrow-Magueijo method (in particular, the spatial stability of the wavelength (BSBM) model, places Bekenstein’s scalar field in a cosmo- calibration) used to obtain the spatial dipole result [32]. logical context, allowing it and α to evolve with the expansion Future efforts at the Very Large Telescope [21] could improve sensitivity to α variations by an additional 2 orders *These authors contributed equally to this work. of magnitude.

α α ≤ 1 3 Other observational techniques have been used to attempt in at recombination was directly limited to CL¼2 . × to constrain the magnitude of time variation in α.For 10−2 at 68% C.L. For a scale-invariant power spectrum this example, the rare-earth element abundance data from can be translated into a constraint to the amplitude of α 2 α 2 6 10−2 Oklo (a naturally occurring uranium fission reactor from ASI ¼ CL¼1 < . × . Here, we use 2015 Planck approximately 2 billion years ago in Gabon), which is satellite data (which includes small-scale polarization completely independent of cosmological models, places measurements) to test for spatial variation of the fine- constraints on the possible temporal variations of α to −6.7× structure constant on smaller scales (α multipoles L ≥ 8). 10−17 yr−1 <α_=α<5.0×10−17 yr−1 at the 2σ level [33]. We find that the amplitude of a scale-invariant spectrum α α 8 10−6 As affects the recombination history and diffusion must have ASI < × at 68% C.L. The dramatic damping of sound waves in the baryon-photon plasma, improvement in the overall order of magnitude of the models with varying α can be probed using cosmic sensitivity results from the use of many more multipoles microwave background (CMB) anisotropies, now charac- (8 ≤ L ≤ 100) to search for α variations. terized at ∼0.1% precision using data from the Planck All attempts at using the CMB to search for the spatial satellite [34,35] as well as a variety of ground-based variation of the fine-structure constant rely on a separate- experiments like the South Pole Telescope [36,37] and universe (SU) approximation (for the response of CMB Atacama Cosmology Telescope [38]. These measurements fluctuations to α variations) [23,46,52]. Here we show that require that the difference between the fine-structure this approximation breaks down if the length scale of α constant today and at recombination obeys the limit δα=α ≤ fluctuations is smaller than the sound horizon at the surface 7.3 × 10−3 at 68% C.L. [39–48]. of last scattering (SLS), analogous to an effect that occurs Measurements of the CMB anisotropies by Planck for compensated isocurvature perturbations [53]. This, in provide an additional motivation for considering a spatially turn, implies that a more complete treatment of these types varying α. Gravitational lensing smooths the CMB power of effects may lead to additional sensitivity. spectrum while also inducing a non-Gaussian contribution This paper is organized as follows. In Sec. II, we explore to the trispectrum. It has been noted that the level of the effect of varying α on the visibility function, and briefly smoothing in the power spectrum is larger than what is describe how these changes propagate to the CMB power expected given the measured amplitude of the non- spectrum and then lay out the details of the relevant Gaussian part of the trispectrum [34,35]. Analogous to calculation using the codes HYREC [54] and CAMB [55]. the effects of weak gravitational lensing, the spatial We then determine how the additional non-Gaussian variation of α smooths the CMB power spectrum and correlations induced by α fluctuations can be detected contributes to the non-Gaussian part of the trispectrum. using an optimal estimator or existing CMB weak-lensing It is thus possible that the measured smoothing/ data products. trispectrum discrepancy points towards modulation of In Sec. III, we use a toy model to show that the SU the primordial CMB anisotropies beyond weak gravita- approximation should break down for α modulation on tional lensing at the level of about 3 standard deviations. In angular scales smaller than the acoustic horizon at the fact this possibility was extensively explored in Ref. [35]. SLS (L ≳ 100). There, the Planck collaboration considered the effects of In Sec. IV, we compare this theory to data to search for compensated isocurvature perturbations (CIPs) to explain spatial variation in α and generate constraints. At the this anomaly. A spatially varying α produces effects very current level of experimental precision, these (lensing-data similar to CIPs and may provide an alternative explanation. derived) constraints are essentially optimal. In Sec. V,we At a similar level of significance, the Planck measure- forecast the sensitivity of future experiments to α fluctua- ments confirm a previously identified deviation from tions using an optimal estimator. In Sec. VI, we explore the isotropy [49]—a “hemispherical asymmetry” in the CMB possibility that α fluctuations could explain apparent power spectra. The presence of a large-scale spatially anomalies between theory and the observed amplitude of varying α, possibly correlated with the primordial anisot- weak gravitational lensing the CMB. There, we also ropies, may explain such an apparent deviation from estimate the implications of our work for specific vary- isotropy in the CMB [50]. ing-α models, with an eye towards those that could explain Spatial fluctuations in α modulate the recombination the claimed dipole in α seen in observations of quasar history and rate of diffusion damping of baryon-photon spectra. We conclude in Sec. VII, summarizing our con- plasma perturbations, and thus induce higher-order (and straints and discussing how sensitivity to α fluctuations non-Gaussian) correlations in the CMB. The ability of the could be improved with novel cosmological observables. CMB to test models of spatially varying α (in the Sachs- We present some of the detailed expressions used in Wolfe limit) was first pointed out and used to obtain this paper in Appendix A. The second derivatives of constraints in Ref. [51]. The effect on anisotropies at all CMB power spectra, needed to construct α-induced cor- scales was computed in Ref. [23] and then applied to data in rections to the observed power spectra, are described in Ref. [52], followed by Ref. [46], in which the spatial dipole Appendix B.

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II. THE CMB AND THE SPATIAL VARIATION where η is the conformal time, τ is the optical depth, X refer OF THE FINE-STRUCTURE CONSTANT to temperature or E/B-mode polarization (X ∈ T; E; B), respectively, η0 is the conformal time today, and j ðxÞ is a In this section we summarize how the physics that l spherical Bessel function of order l. The measured power produces the CMB depends on α. After that we focus spectrum is then given by on how a spatially varying α affects the CMB. Z 2 2 2 A. An overview of how the CMB depends Cl ¼ π k dkPζðkÞjXlðkÞj ; ð2Þ on the value of the fine-structure constant We briefly review how the spatial variation of the fine- where PζðkÞ is the primordial power spectrum. The terms structure constant affects the CMB. Readers interested in that appear in the source functions take the form more details may find them in prior work, such as Refs. [23,48,52]. The fine-structure constant sets the rates τ ∼ F τ Δ η; τ ST;Pðk; Þ ð Þ xðk; Þ; ð3Þ of processes relevant for hydrogen recombination [54] as well as the Thomson scattering cross section (which in turn sets the where FðτÞ is some function of the optical depth and baryon-photon diffusion damping scale), and thus affects Δxðk; η; τÞ is some linear perturbation either to a fluid observable properties of the CMB [56] asshowninFig.1. component or to the gravitational potentials. Very roughly α The impact of modulation may be separated into speaking we are separating the physics that determines effects induced by changes to the time dependence of the what we “see” in the CMB from the physics that dictate the decoupling process and effects induced by changes to evolution of the photon perturbations of the CMB. perturbation evolution. This division is inspired by the line A modulation of the fine-structure constant affects both of sight integral used to compute the CMB anisotropies of these aspects of the observed CMB. [57]. The line of sight integral allows us to write the A spatial modulation in the fine-structure constant causes anisotropy in temperature or polarization today in terms of τ η τ a modulation to ð Þ through its effects on recombination a transfer function, SXðk; Þ,as [23,48]. As summarized in Ref. [48] several physical Z quantities that play essential roles in the physics of η 0 τ η − η η recombination depend on the fine-structure constant in a XlðkÞ¼ SXðk; Þjl½kð 0 Þd ; ð1Þ 0 variety of ways,

FIG. 1. The change to the CMB power spectra as the fine-structure constant is varied, α ¼ α0ð1 þ φÞ. As discussed in the text an increased α 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 α 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.

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σ ∝ α2 ∝ α8 ∝ α6 T A2γ PSA1γ in the peak of the visibility to earlier times. In this sense the spatial modulation of α causes the surface of last scattering α ∝ α2 β ∝ α5 T ∝ α−2; ð4Þ rec phot eff to become “wrinkled.” The rate of change of the optical depth leads to a σ where T is the Thomson scattering cross section; A2γ is the damping of anisotropies on scales below the diffusion α β two-photon decay rate of the second shell; rec and phot are scale. This damping is controlled by the differential optical the effective recombination and photoionization rates, depth, τ_ ≡ anHXeσT, where ne is the electron density and respectively; Teff is the effective temperature at which X is the ionization fraction. As we discuss further in α β e rec and phot are evaluated; and PSA1γ is the effective Sec. III, the evolution of the temperature perturbations dipole transition rate for the main resonances. These effects during the time when the differential optical depth is large combine to yield a scaling at the time of recombination compared to the Hubble rate, τ_ ≫ H (i.e., while baryons α ∝ α3.44 β rec , with a scaling of the photoionization rate phot and photons are tightly coupled), gives a damped-driven α related to rec by detailed balance [48,58]. The overall harmonic oscillator equation of motion with a damping effect of a modulation of α on recombination is to shift the proportional to 1=τ_. An increase in the fine-structure peak and broaden the width of the visibility function, g ¼ constant leads to a decrease in the damping and hence τ_eτ as shown in Fig. 2. There we show the change to the an increase in the overall amplitude of the power spectra. visibility function when the fine-structure constant is We show the change in the level of diffusion damping for shifted by a multiplicative factor, α ¼ α0ð1 þ φÞ. We can different values of α in the bottom panel of Fig. 2. see that for values of α larger than the standard value the The α modulation of the distance to the SLS and the electromagnetic interactions are stronger leading to a shift diffusion damping contribute nearly equally to the full modulation. We can see this in Fig. 3, which shows their relative contribution to the cross-power spectrum between the temperature or polarization and the derivative of the temperature or polarization with respect to φ: DX;dY ≡ 1 X;dY 2π α l lðl þ ÞCl =ð Þ. The effect of modulation on the evolution of the perturbations we see in the CMB has an important impact on the way in which we construct estimators for the α modulation as we discuss in detail in Secs. II D and III.

B. Characterizing spatial fluctuations in the fine-structure constant We parametrize the fluctuations in the fine-structure constant as

αðx⃗Þ¼α0½1 þ φðx⃗Þ: ð5Þ

In theories that promote the fine-structure constant to a dynamical quantity, the field φ also depends on time. Previous work has used the CMB to constrain this time evolution and has found that the fine-structure constant cannot vary appreciably during the process of recombina- Δη α_ α ≲ 10−3 tion. (In Ref. [48], it is shown that j = j where Δη is the duration of decoupling, i.e., the width of FIG. 2. The change to the visibility function (top) and the the visibility function.) Therefore we take α_ ¼ 0 here damping of the CMB anisotropies (bottom) with a change in the α α 1 φ and leave simultaneous constraints to both temporal and fine-structure constant ¼ 0ð þ Þ. In the top panel we can α see that an increase in α leads to stronger electromagnetic spatial variation of to future work. interactions shifting the peak of the visibility function (which We write the Fourier transform of φ as marks the temperature at decoupling) to early times (i.e., higher Z α 3 redshift). An increase in increases the Thomson scattering cross d k ⃗ φ x⃗ φ k⃗eik·x⃗; section leading to a larger differential cross section at early times. ð Þ¼ 2π 3 ð Þ ð6Þ As we approach decoupling the additional changes to the ð Þ ionization history cause a larger α to give a smaller differential cross section. and define its power spectrum in the usual way,

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FIG. 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 α modulation has a significant impact on both terms.

φ ⃗φ ⃗0 2π 3δð3Þ ⃗− ⃗0 As has been pointed out, when calculating the effects of h ðkÞ ðk Þi¼ð Þ D ðk k ÞPαðkÞ: ð7Þ φ ≠ 0 on the CMB we truncate the sum over the φ 100 In the limit of instantaneous decoupling, the α-modulation multipoles at Lmax ¼ since the modulation of CMB angular distribution at the SLS is given by fluctuations is suppressed for φ modes with wavelengths Z smaller than the acoustic horizon at the SLS (for details see 4πiL Sec. III). φ 3 φ ⃗ χ ˆ LM ¼ 3 2 d k ðkÞjLðk ÞYLMðkÞ; ð8Þ The spatial fluctuations in the fine-structure constant ð2πÞ = could have different statistical properties, such as a which gives rise to an angular power spectrum at the SLS Gaussian power spectrum [23], or a white-noise (constant) (which is located at a comoving distance χ )of power spectrum [52]. In this work, we focus on the scale- invariant case, but also place constraints on the white-noise α φ φ case to allow a comparison with Ref. [52]. CL ¼h LM LMið9Þ

nα AαΓðL þ 2 Þ C. The effects of a spatially varying fine-structure ¼ nα ; ð10Þ ΓðL − 2 þ 2Þ constant on the CMB power spectrum As discussed in Sec. II the spatial modulation of the fine- Aα ≃ ; for L ≳ 1; ð11Þ structure constant affects both the location of the SLS as ½LðL þ 1Þ1−nα=2 well as the evolution of the perturbations. A full accounting for the effects of φðx⃗Þ requires a solution to the modified set nα 3 nα where Aα ≡ AαΓð1 − 2 Þ=Γð2 − 2 Þ and where we assume of second-order evolution equations. As a first approxima- that the α-modulation power spectrum is a power law: tion to this, we use the SU approximation and take the A χ nα 3 α PαðkÞ¼ αðk Þ =k . Note that for a scale-invariant solution to the original evolution equations (where does α SI 1 not vary in space) and then expand that solution in a power spectrum we have CL ¼ Aα =½LðL þ Þ; a white-noise α WN series in φðx⃗Þ. The SU approximation has been used to angular power spectrum with CL ¼ Aα corresponds to the limit nα → 2.Ifφ couples to the inflaton field or calculate the effects of CIPs in which the initial baryon standard-model particles (as it does in most theories, i.e., number density fluctuates in space with an equal and Ref. [13]), then we expect that its power spectrum is scale opposite CDM fluctuation [53,59–62]. invariant. The estimators that we construct using the SU approxi- Using the power spectrum we can compute the relation- mation are limited in that they effectively filter the data and ship between the variance of α on the sky and the amplitude remove information about the modulation on small scales. of its power spectrum, The exact cutoff in this filter is determined by a comparison between the power series approximation and a perturbative Z σ2 1 XLmax 2 1 solution to the full dynamical equations in the presence of α φ ˆ 2 2Ω L þ α α2 ¼ 4π h ðnÞ id ¼ 4π CL: ð12Þ the modulation. For example, CIPs effectively cause a 0 L¼1 modulation in the baryon-photon sound speed, causing a

043531-5 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) spatial modulation in the sound horizon of the CMB. A TT;obs ˜ TT δ TT;ϕ δ TT;α Cl ¼ Cl þ Cl þ Cl ; ð16Þ careful analysis using a simplified model for the evolution equations describing the temperature perturbations in ˜ XX0 0 where Cl denotes the true primordial XX power Ref. [53] showed that the resulting estimator filtered out spectrum (without corrections from noise, φ fluctuations, information on the CIPs on scales smaller than the sound or gravitational lensing) of the quantity X, which can horizon of the CMB. We perform a similar analysis for the denote temperature or E/B-mode polarization moments α spatial modulation of in Sec. III and find that the power (X ∈ fT; E; Bg). The standard lensing correction to the law expansion is accurate for L

043531-6 PROBING SPATIAL VARIATION OF THE … PHYS. REV. D 99, 043531 (2019) of B-mode polarization through lensing). In this case the Eqs. (16) and (22) give us an efficient way to compute the induced B-mode polarization is [60] effects of a spatially modulated fine-structure constant on the CMB power spectra. When computing the modulated LþXl0þl odd power spectrum for the Markov chain Monte Carlo BB ≃ BB; lensed α dE;dE L 2 Cl Cl þ CLCl0 ðKll0;2Þ : ð23Þ (MCMC), we use a finite difference to compute the second L;l0 derivative of the power spectra with respect to φ with a step size of Δφ ¼ 0.012. We show that this step size gives the The expression for the induced B-mode polarization cannot best estimate of the second derivative in Appendix B. be written in the form of Eq. (22): The second term in Computing these derivatives requires the evaluation of Eq. (17) arises because there is a nonzero value for the CMB power spectra CXX0 for various global φ values. To temperature anisotropy even in the absence of CIPs (and l compute these, we modify the CAMB [55] code. We set analogously so for the E-mode polarization anisotropy). On CAMB to use the recombination code HYREC [54,68],in the other hand, there is no B mode in the unlensed CMB in order to include completely the rich ensemble of effects the absence of primordial gravitational waves, and thus no relevant for adequately modeling cosmic recombination in term like the second term of Eq. (17) in the expression for the precision cosmology era. HYREC has the added the CIP-induced B-mode power spectrum. The absence advantage that φ may be readily changed by changing a of this term prevents the algebraic simplifications that single argument, ensuring that all the relevant derivatives yield Eq. (22). are correctly computed without neglecting any relevant We show the unmodulated and modulated power spectra physical effects. in Fig. 4. There we can see that a spatially varying fine- structure constant leads to a smoothing of the CMB power spectra and has a larger effect on the polarization than it D. Additional effects of a spatially varying does on the temperature anisotropies. Also note that we fine-structure constant on the CMB SI −4 have chosen Aα ¼ 6 × 10 for this figure in order to The spatial modulation of the fine-structure constant also highlight the power-spectrum modulation; when saturating produces a contribution to correlations beyond the CMB the upper limit using the T and E spectra we have power spectrum. In particular, a nonzero realization of φðx⃗Þ SI −5 Aα < 5.2 × 10 (see Sec. IV). induces off-diagonal correlations between CMB multipole Since the power-spectrum constraints are driven by a moments with l ≠ l0 and m ≠ m0 [23,52]. Using these, an modulation of the temperature and E-mode polarization optimal estimator for φLM may be constructed, which in

FIG. 4. The change to the CMB power spectra due to stochastic scale-invariant fluctuations in the fine-structure constant (we have set SI the amplitude, Aα , to be large enough to see the effects). The spatial variation of α 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 α.

043531-7 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) turn can be used to construct an optimal estimator of noise bias and non-Gaussian lensing bias contributions to Aα. The formalism is very similar to that used in weak the CMB four-point correlation [70–72]. See Appendix A lensing of the CMB [66]. We review these correlations in for detailed formulas. Sec. II D 1, discuss the related optimal estimators in Mutatis mutandis the off-diagonal correlations induced Sec. II D 2, and then in Sec. II D 3 obtain a method for by φ in Sec. II D 1 may be used to obtain a minimum- applying existing CMB lensing measurements to probe variance estimator of φLM (as derived for CIPs in spatial variations in α. Refs. [53,60] and in different notation for α fluctuations in Ref. [52]). The derivation in Ref. [53] closely follows the 1. Off-diagonal correlations treatment in Ref. [69] and generalizes to α fluctuations. Deflections of CMB photons and higher-order modu- lations of the transfer functions produce off-diagonal CMB 3. Contribution of fine-structure constant fluctuations correlations for fixed lens and φ realizations, given by to CMB lensing estimators [59,60,66] As discussed, estimates of the lensing-potential power ϕϕ spectrum, C , are built out of the (non-Gaussian) connected 0 ˜ XX0 m L hX X 0 0 ij α ¼ C δ 0 δ − 0 ð−1Þ lm l m lens; l ll m m part of the CMB trispectrum [66,71,72]. In the presence of a X ll0 L spatially varying α, the estimator used to reconstruct the þ ð−1ÞM lensing-potential power spectrum gains an additional con- mm0 −M LM tribution proportional to Aα, a bias that can itself be used to ϕ XX0 φ XX0 × ½ LMflLl0 þ LMhlLl0 ; ð24Þ estimate Aα using existing lensing data products. The method presented here closely follows the method used XX0 XX0 α in Ref. [62] with the CIP modulation field, Δ, replaced by the where flLl0 and hlLl0 are the lensing= response functions for different quadratic pairs (see Table IV in Appendix A) α-modulation field φ. We summarize the important points and are defined in terms of the unmodulated power here and direct the reader to Ref. [62] for details. 0 α spectrum, C˜ XX , the appropriately weighted Wigner coef- With a nonzero spatial modulation of it is straightfor- l ward to show that the standard lensing estimator gains a ficients and the derivative power spectra α Z contribution from CL, 0 0 2 2 dX ðkÞ X;dX ≡ l ϕϕ ϕϕ α Cl k dkPζðkÞXlðkÞ : ð25Þ ˆ F π dφ hCL i¼CL þ CL L; ð28Þ

The multipole moments of the lensing-potential are where denoted by ϕLM. This formalism was first developed for X α F ≡ ω β fluctuations in Ref. [52]. Here we use the equivalent L wωwβQLQL; ð29Þ notation of Ref. [62]. ω;β P ω ω ϕ φ 0 h 0 g 0 2. Minimum-variance estimators for LM and LM ω ≡ Pll lLl lLl QL ω ω ; ð30Þ 0 f g Under the null hypothesis (i.e., no α variation), the ll lLl0 lLl0 minimum-variancepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi estimator for the “deflection field” ω α d ≡ LðL þ 1Þϕ from a single pair ω ¼ XX0 of TABLE I. The -modulation contribution to the lensing po- LM LM 4F observables is [66,69] tential power-spectrum estimator, L L, defined in Eq. (28). The α-modulation contribution is different for the two instruments X ll0 L because its affect on the lensing potential power-spectrum ˆ ω ω M 0 ω d A −1 X X 0 0 g 0 ; 26 estimator depends on the noise properties of the instrument. LM ¼ L ð Þ lm l m 0 ll L ð Þ 0 0 mm −M lm;l m L Planck CMB-S4 ω ω 1 2.83 0.384 where AL is the normalization and gll0L is the optimal weights, which are defined in Appendix A. This in turn can 2 2.3 1.06 be used to derive (in the absence of a nonzero φ) an optimal 3 2.49 1.23 estimator for the lensing-potential power spectrum 4 2.67 1.35 5 2.81 1.44 X XL ˆ ω ˆ β 6 2.93 1.51 ϕϕ 1 β d d Cˆ ¼ vωv LM LM − B ; ð27Þ 7 3.02 1.56 L 2 1 L L 1 L 8 3.09 1.61 L þ ω;β M¼−L LðL þ Þ 9 3.15 1.64 ω 10 3.2 1.68 where v are weights chosen to yield an optimal estimator L >10 3.29 1.21L0.143 for the deflection field and BL are the standard Gaussian

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ω ω and as before, hlLl0 , flLl0 are listed in Table IV in III. ASSESSING THE SEPARATE Appendix A. The brackets in Eq. (28) denote an average UNIVERSE APPROXIMATION over realizations of the primordial CMB, realizations of the We have investigated the effects of spatial variation of lensing potential ϕ, and realizations of the fine-structure φ F the fine-structure constant by using the SU approximation, modulation . The values of L for Planck and CMB-S4 where we calculate the effects of the spatial variation of the are shown in Table I. φ fine-structure constant on the CMB by computing a power The optimal weights in Eq. (A10) show that the series of the CMB maps in φ. contribution to the lensing potential power spectrum This power series would be accurate to all scales if the depends on the noise properties of the relevant CMB modulation due to φðx⃗Þ only affected the visibility function, experiment. The full Planck lensing analysis [72], which which codifies the projection of dynamical perturbations onto includes both CMB temperature and polarization maps, a fixed-redshift “screen,” but as we now discuss, the dynami- computes a minimum variance (mv) estimator from all cal equations are also affected. In the case of compensated possible CMB map auto- and cross-correlations, as dis- isocurvature perturbations in Ref. [53], such complications cussed in detail in Sec. II D 2. The Planck lensing estimator imply that the power law expansion is only accurate down to relies on maps constructed from the 143 and 217 GHz some cutoff scale Lmax. Here we determine this cutoff for channels. The Planck analysis also uses a bandpass filter in spatial modulation of the fine-structure constant. harmonic space to restrict the power-spectrum multipoles To a first approximation the observed CMB provides a 100 ≤ ≤ 2048 to l . We also list the noise parameters snapshot of the temperature and polarization perturbations associated with the fourth-generation CMB experiment, around the SLS, CMB-S4 [73]. We compute the sensitivity of CMB-S4 to a spatially varying fine-structure constant in Sec. V. We show Z η0 the α-modulation contribution to the lensing potential TðnˆÞ¼ Tðχ; χn;ˆ ηÞgðηÞdη; ð31Þ Dˆ ϕϕ ≡ 1 2 ϕϕ 2π 0 power-spectrum estimator, l ½lðl þ Þ Cl =ð Þ, in Fig. 5. Z _ η0 Tðχ; χn;ˆ ηÞ EðnˆÞ¼− gðηÞ dη; ð32Þ 0 τ_ðηÞ

where χ ¼ η0 − η is the comoving distance, gðηÞ is the visibility function, T are the temperature perturbations, and E are the E-mode polarization perturbations. We consider the evolution of the temperature and polarization perturbations using a tight-coupling approximation, where τ_ ≫ a=a_ . In this case the temperature perturbations follow an equation of motion (neglecting superhorizon terms, polarization source terms, and other complications),

̈ 2 2 2 _ T − cs∇ T − 2β∇ T ¼ 0; ð33Þ

2 where cs ≡1=½3ð1þRÞ R ≡ 3ρb=4ργ, β ≡ 2=½45ð1 þ RÞτ_, and we have ignored the effects of the gravitational potentials (which are small during radiation domination). We consider a simplified case (which has all the salient features of the full problem), in which the damping, β, is treated as constant in pffiffiffi time and cs ≃ 1= 3. In this case the equation of motion, Eq. (33), is a damped, undriven, harmonic oscillator. We FIG. 5. The affects of α modulation on the lensing potential model the effects of the spatial variation in the fine-structure power-spectrum estimator. The top panel shows expectation value constant by writing β ¼ β0½1 þ φðx⃗Þ∂β=∂φ. of the lensing potential power-spectrum estimator in the presence As in Ref. [53], we solve the dynamical equation of a scale-invariant α-modulation contribution. The lower panel perturbatively, writing T ¼ T0 þ T1, where T1 is first order shows the residual lensing potential power-spectrum estimator in ⃗ ⃗ in φ. Imposing the initial conditions T0ðk; 0Þ¼−ζðkÞ=5 α the presence of a white-noise power spectrum. The amplitudes ζ ⃗ have been chosen to saturate the 95% C.L. bounds discussed [where ðkÞ is the initial curvature perturbation] and _ ⃗ in Sec. IV. T0ðk; 0Þ¼0 the zeroth-order solution is

043531-9 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) β ∂ ⃗ −βk2η ⃗ k SU T T0 k; η −e ζ k =5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s k; η c k; η ; T ðx;⃗ηÞ ≃ Tðx;⃗η; φ ¼ 0Þþφðx⃗Þ ; ð41Þ ð Þ¼ ð Þ 2 2 2 ð Þþ ð Þ ∂φ k β − cs

ð34Þ SU ¼ T0ðx;⃗ηÞþT1 ðx;⃗ηÞ; ð42Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where cðk; ηÞ ≡ cosh kη k β − cs ; ð35Þ Z β τ_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SU ⃗ dln 3 ⃗ ⃗ ⃗ T1 ðk;ηÞ¼ d k1φðk−k1Þζðk1Þ η ≡ η 2β2 − 2 ð2πÞ3 dφ sðk; Þ sinh k k cs ; ð36Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 −βηk1 2 2 2 βk1cse ½ηk1 β k1 −cs cðk1;ηÞ−sðk1;ηÞ × 2 2 2 3=2 : where the hyperbolic trigonometric functions take into 5ðβ k1 −csÞ account the transition between a strongly damped system ð43Þ when k>cs=β to a genuinely oscillatory system when k< c =β via the identities cosh ðixÞ¼cos x, sinh ðixÞ¼i sin x. s It is straightforward to verify that in the squeezed limit The first-order solution, T1, satisfies the dynamical ⃗→ ⃗ ⃗η → SU ⃗η equation (k k1), T1ðk; Þ T ðk; Þ. In a fixed realization of φ, correlations between observed ̈ 2 2 2 _ ⃗ modes are then induced after ensemble averaging over the T1 c k T1 2βk T1 F k; η ; 37 þ s þ ¼ ð Þ ð Þ primordial curvature fluctuation ζ, and so evaluating the η η temperature perturbation at ¼ (the conformal time at where the SLS) Z τ_ 3 ⃗ d ln d k1 2 _ ⃗ ⃗ ⃗ ⃗ ⃗ ≃ ⃗ ⃗ φ ⃗ Fðk; ηÞ ≡ 2β k1T0ðk1; ηÞφðk1 − kÞ; ð38Þ hTðkÞT ðk1Þi hT0ðkÞT0ðk1Þi þ Rðk; k1Þ ðKÞ; ð44Þ dφ ð2πÞ3 ⃗ ⃗ δð3Þ ⃗− ⃗ hT0ðkÞT0ðk1Þi ¼ PTTðkÞ ðk k1Þ; ð45Þ so that T1 behaves as a driven, damped, harmonic oscillator. The Green’s function of the left-hand side of Eq. (38) A kβ 2 is obtained by replacing the right hand with δðη − ζÞ and PðkÞ¼ cðk; ηÞþpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðk; ηÞ ; ð46Þ 25 3 2β2 − 2 solving to obtain k k cs 8 −βηk2 < 0 if η ≤ ζ; 2Aβdlnτ_ e 1 Rðk;k1Þ¼ 2 Gðη − ζÞ¼ sðk;ηÞ −β 2 η−ζ ð39Þ 25 dφ ðk − k1Þðk þ k1Þc : pffiffiffiffiffiffiffiffiffiffiffiffi e k ð Þ if η > ζ: s 2β2− 2 2 2 2 k k cs − 2β η −βηðk2−k2Þ ðcs k Þsðk; Þ × k1e 1 2βcðk;ηÞ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2 2 − 2 ’ k k cs The solution to Eq. (37) is then obtainedR via the usual Green s η 2 − 2β2 2 η ⃗η ζ η − ζ F ⃗ζ ðcs k1Þsðk1; Þ function expression, T1ðk; Þ¼ 0 d Gð Þ ðk; Þ, þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2βk1cðk1;ηÞ yielding β2k2 − c2 1 s kβ Z 3 3 −βη 2 −pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η − η τ_ φ ⃗− ⃗ ζ ⃗ k1 × sðk1; Þ cðk1; Þ ⃗ dln d k1 k1 ðk k1Þ ðk1Þe 2β2 − 2 T1 k;η 2β k cs ð Þ¼ φ 2π 3 5 − 2 d ð Þ ðk k1Þðk þ k1Þcs ⃗ ⃗ þðk ↔ k1Þ; ð47Þ 2 −2β2 2 η −βηðk2−k2Þ ðcs k Þsðk; Þ × k1e 1 2βcðk;ηÞ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k β2k2 −c2 ⃗ ⃗− ⃗ s where K ¼ k k1 and we have assumed a scale-invariant 2 2 2 ðcs −2β k1Þsðk1;ηÞ power spectrum of primordial curvature perturbations þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2βk1cðk1;ηÞ : ð40Þ 3 ⃗ ⃗ 2 2 2 PζðkÞ¼A=k . The interchange k ↔ k1 indicates that β k1 − cs to the first term of Eq. (47) we must add the same term It is straightforward to then check that Eq. (40) satisfies with the swap performed. These two first-order terms arise ⃗ ⃗ ⃗ the necessary boundary conditions, T1ðk; 0Þ¼0 and from the cross terms in the product TðkÞT ðk1Þ¼ _ ⃗ ⃗ ⃗ ⃗ ⃗ T1ðk; 0Þ¼0. ½T0ðkÞþT1ðkÞ½T0ðkÞþT1ðkÞ. To obtain the SU approximation, we start with the In the null hypothesis of no α fluctuations, only the unmodulated solution for the temperature perturbations zeroth-order solution T0 contributes, while in the presence in Eq. (36), transform it to real space, and expand it in a of a fixed α modulation with wave vector K⃗, the isotropy- power series in φ so that breaking response Rðk; k1Þ function codifies the imprint of

043531-10 PROBING SPATIAL VARIATION OF THE … PHYS. REV. D 99, 043531 (2019) a fixed α fluctuation on off-diagonal correlations of the A value b ¼ 1 indicates that φ reconstruction based on the temperature. SU response is robust, while deviations indicate the The response may also be calculated using the separate- limitations of this approximation. universe approximation [Eq. (43)], to obtain In Ref. [53], this integral was evaluated in three different ways: a fully analytic result valid in the k ≫ K limit, an β τ_ SU A dln evaluation in which the oscillatory (acoustic) features in the R ðk;k1Þ¼ 25 dφ power spectrum are averaged out analytically prior to a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 −βη 2 2 2 2 ⃗ ⃗ ⃗ ⃗ k1 numerical integral over the cosine k · K=ðjkjjKjÞ, and a βcse ½ηk1 β k1 − cscðk1;ηÞ − sðk1;ηÞ × 2 2 2 2 3 2 direct numerical evaluation of Eq. (52) including numerical k ðβ k − c Þ = 1 1 s noise. The response function here is much more compli- β −pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik η − η cated, and so we go directly to a fully numerical integral. × 2 2 2 sðk; Þ cðk; Þ k β − cs We assume a Poisson-noise power spectrum with 1% the amplitude of P at k π= η c , commensurate with the ⃗↔ ⃗ TT ¼ ð sÞ þ k k1: ð48Þ ∼1%-level noise of modern CMB experiments. The noise ’ ⃗→ ⃗ term also regulates the effect of unphysical 0 s in the power It is straightforward to verify that in the limit k1 k, that occur in the toy model, but are “filled” in by the ⃗ SU which implies that K → 0, that R ðk; k1Þ¼Rðk; k1Þ. Doppler term and other effects in a more complete The analysis tools used to search for α variations in this calculation. We check that once noise rises above a critical paper [e.g., the CMB off-diagonal correlations represented threshold (well below our chosen NTT), the resulting curves by Eq. (A1)] are derived in the SU limit of a much more for bðKÞ become independent of the noise level to 0.1% complete model, one that includes baryons, scattering accuracy. terms, time-dependent gravitational potentials, neutrinos, The result is shown in Fig. 6 (we use realistic values for and so forth. In this more complete model, a dynamical the primordial baryon-photon plasma at decoupling: β ¼ model would be much more challenging to obtain. For this 2 −1 η 280 −1 Mpc h and ¼ Mpc h ), withffiffiffi K normalized toy model, where we have both SU and dynamical response p relative to the acoustic horizon s ¼ η= 3. We see that the functions, we can quantitatively assess how biased our φ minimum-variance estimator based on the SU approxima- inferences about the field will be when the SU approxi- tion is unbiased for Ks ≲ 2, and is then biased as the SU mation is used. approximation overestimates the response for the Fourier- φ K⃗ To infer the Fourier transform ð Þ of the modulating space triangles dominating the estimate. This is only a toy field from the data assuming the SU response, we may use model, and there are many complicating factors that could the minimum-variance estimator change the result at the order-unity level. This onset of bias Z 3 ⃗ ⃗ SU d k TðkÞTðk1ÞR ðk; k1Þ φˆ K⃗ NSU ; ð Þ¼ K 2π 3 ˜ ˜ ð49Þ ð Þ PTTðkÞPTTðk1Þ Z 3 SU 2 SU −1 d k ½R ðk; k1Þ ðNK Þ ¼ 2π 3 ˜ ˜ ; ð50Þ ð Þ PTTðkÞPTTðk1Þ where the observed power spectrum ˜ PTTðkÞ¼PTTðkÞþNTTðkÞð51Þ includes the additional effect of a Poisson noise term, that is, NTTðkÞ¼const. The SU response does not perfectly reproduce the full dynamical response. If the dynamical response for some ⃗ ⃗ ⃗ Fourier-space wave-vector triangle (a triplet k, k1, K)is lower than the SU response, the absence of correlations for this triplet would lead to an erroneously low estimate φˆ ðK⃗Þ from this triangle, and vice versa. In our toy model, we can compute this bias, FIG. 6. Bias of separate-universe approximation minimum- φ hφˆ ðK⃗Þi variance estimator for reconstruction. The horizontal axis is the ≡ α bðKÞ ⃗ : ð52Þ wave number of the -modulating mode in units of the inverse φðKÞ acoustic horizon.

043531-11 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) near the acoustic scale motivates us to proceed (conserva- TABLE II. Planck sensitivity in the 143 and 217 GHz channels tively), as was done in Ref. [53], and impose a cutoff of to temperature and polarization at the two frequencies used to L ¼ 100 in our trispectrum forecasts and lensing-based estimate the lensing potential [72,76]. The last line gives the reconstructions of the φ power spectrum. sensitivity for CMB-S4, a proposed next-generation CMB telescope [73]. θ μ μ IV. CURRENT CONSTRAINTS Channel (arcmin) wT ð K arcminÞ wPð K arcminÞ To use Planck data as a test for spatial variation of the 143 GHz 7 30 60 217 GHz 5 40 95 fine-structure constant, we modified the publicly available 2 CMB-S4 3 1 1.4 Boltzmann solver CAMB to compute the α-modulated CMB power spectra and lensing-potential estimator given by Eq. (28). In particular, we modified CAMB to compute SI The one-dimensional marginalized posterior on Aα the sum of the lensing-potential power spectrum and the using the different combinations of datasets is shown in α-modulation contribution, given in Table II. We compared ϕϕ Fig. 7. We also used Cˆ to search for a white-noise power these theoretical predictions to the Planck data using the L α α WN publicly available Planck likelihood code [74] and the spectrum of fluctuations, that is, CL ¼ Aα , and find that WN ≤ 2 3 10−8 MCMC code cosmomc3 [75]. Aα . × at 95% C.L. The Planck data have been divided up into a large- angular-scale dataset (low multipole number) and a small- V. THE OPTIMAL ESTIMATOR angular-scale dataset (high multipole number) [74]. For all constraints we use the entire range of measurements for the The constraints obtained in this work from the observed TT power spectrum as well as the low multipole polari- CMB trispectrum rely on the contribution of spatial α zation (TE and EE) data, which we denote as “T+LowP.” fluctuations in to the lensing-potential estimator. There φ We also compute constraints using the entire multipole is, however, an optimal estimator that relies on the range of polarization measurements, denoted by “T þ P.” distinct (from lensing) off-diagonal CMB multipole corre- φ The division between these two datasets is the multipole lations induced by the field, as shown in Refs. [53,59,60] number l ¼ 29, which approximately corresponds to an and summarized in Sec. II D 2. An analogous estimator was angular scale of ≃5°. In addition to the temperature and used to obtain the WMAP constraints to CIPs in Ref. [61]. polarization power spectra we use the Planck estimate of The Fisher information F (which yields the minimum the lensing-potential power spectrum [72]. We use the theoreticallypffiffiffiffiffiffiffiffiffi possible theoretical uncertainty in Aα, σ 1 “aggressive” estimate of the lensing-potential power spec- A ¼ =F) is given by 8 trum, which extends down to Lmin ¼ . We used the plik likelihood [74] and varied all 27 Planck nuisance X 2 1 ∂ ΔΔ 2 ð L þ Þ CL ΔΔ −2 parameters. F ¼ fsky ðNL Þ ; ð53Þ 2 ∂Aα Our results are shown in Table III. The T þ LowP L SI datasets favor a nonzero α modulation with Aα ¼ −5 XX0 ð4.7 1.8Þ × 10 . As demonstrated in Fig. 1, polarization where NL are defined in Ref. [61] and computed under data can break degeneracies present in a temperature-only the null hypothesis. We use Eq. (53) to forecast the analysis. Indeed, when we include the full polarization sensitivity of Planck and CMB-S4 (although Planck data measurements from Planck (i.e., T þ P) the best-fit are public, we wish to compare our constraints to an SI value for the α power spectrum decreases to Aα ¼ optimal trispectrum analysis) to a scale-invariant angular 2 7þ1.2 10−5 power spectrum for φ. ð . −1.5 Þ × . If we additionally include estimates of the lensing-potential power spectrum (i.e., T þ Pþ If we assume the null hypothesis, and that the posterior lensing), the upper limit to the α modulation improves likelihood for Aα is Gaussian, we find that the optimum SI −5 SI −5 estimator has a 95% C.L. sensitivity of Aα ≃ 1.0 × 10 for to Aα ≤ 2 × 10 at 95% C.L. As noted in Secs. II C and Appendix III, the response of the CMB to α fluctuations is Planck noise parameters, offering a slight improvement α not known for φ multipole L>100, due to the breakdown over the constraint from the contribution to the lensing- of the SU approximation. For a conservative test of the potential estimator (see Fig. 8 for an illustration as well varying α hypothesis, we also run MCMC chains with only as Sec. IV). We repeated this analysis [using Eq. (53)] with α WN the lensing power spectrum, and obtain the upper limit a shot-noise power spectrum (CL ¼ Aα ) and found SI −5 that the optimal estimator has a 95% C.L. sensitivity of Aα < 1.6 × 10 at 95% C.L. WN −8 Aα ≃ 1.3 × 10 , again slightly lower than the limit obtained in Sec. IV. In other words, the constraints 2http://camb.info (obtained using the α contribution to the lensing-potential 3http://cosmologist.info/cosmomc/ estimator) in Sec. IV are nearly optimal using Planck data.

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TABLE III. Best-fit values and standard deviations for cosmological parameters with the three different Planck datasets as described in the text. All upper limits to Aα show 95% C.L.

Parameter T þ LowP T þ P Lensing T þ P þ lensing

ωb……… 0.02268 0.00031 0.02237 0.00018 0.02222 0.00015 0.02226 0.00016 ωc……… 0.1156 0.0027 0.1185 0.0016 0.1193 0.0014 0.1190 0.0014 ns……… 0.9761 0.0076 0.9675 0.0051 0.9643 0.0047 0.9652 0.0047 10 3 045 0 041 3 048 0 040 3 050 0 024 3 049 0 025 log ð10 AsÞ . . . . . . . . τ………: 0.060 0.021 0.058 0.020 0.059 0.013 0.059 0.014 H0……… 69.5 1.367.93 0.74 67.46 0.62 67.60 0.63 SI 104… 0 47 0 18þ0.35 0 28þ0.12þ0.24 <0.16 <0.21 Aα × . . −0.36 . −0.15−0.26

It is also interesting to consider the sensitivity of a future, in which lensing-induced correlations have been filtered nearly cosmic-variance limited (CVL) experiment, like the out. We defer an analysis that includes these complications CMB-S4 concept [73]. We use the noise parameters in to future work, and simply note that Eq. (53) quantifies the Table II, and reconstruction noise as given in Eq. (A10),with best α reconstruction we could achieve using the CMB. ω → ω the replacement fl0Ll hl0Ll. We then use Eq. (53) and α find that CMB-S4 will be sensitive to scale-invariant VI. DISCUSSION SI −6 WN −9 fluctuations with Aα ≥ 1.9 × 10 and Aα ≥ 1.4 × 10 α (at 95% C.L. or greater). This difference, illustrated in Fig. 8 We now explore if our results are consistent with for the scale-invariant case, is driven by the constraining fluctuations being responsible for the anomalous smooth- power of a nearly CVL polarization experiment. ing of the CMB power spectra (e.g., Ref. [35]), or with the α Given the fact that the trispectrum is so much more putative detection of an angular dipole in seen in quasar constraining than the φ-induced smoothing of the CMB spectra (e.g., Ref. [26]). power spectrum, we neglect primary power-spectrum constraints in this Fisher analysis. For futuristic experiments A. The variation of the fine-structure constant (like CMB-S4), the reconstruction noise for both lensing and the anomalous smoothing of the and α fluctuations may be low enough that lensing CMB power spectrum could introduce a significant bias [77] to the estimators Weak gravitational lensing by clustered matter between described in Sec. II D 2, requiring either a debiased us and the SLS causes two main effects on the CMB: it minimum-variance estimator (as discussed in Ref. [78]) smooths the CMB power spectra (both temperature and “ ” – or a delensed CMB map (as discussed in Refs. [79 81]), polarization) and it generates correlations between different

FIG. 7. The one-dimensional marginalized posterior for Aφ FIG. 8. Sensitivity of the optimal trispectrum-based estimator using the three combinations of datasets discussed in the text. to a spectrum of scale-invariant spectrum for φ.

043531-13 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) multipoles leading to a (non-Gaussian) connected part of degeneracy between these two amplitudes and they SI −4 the CMB trispectrum (see Ref. [82] and references therein). must take on values Aα < 1.8 × 10 at 95% C.L. and The CMB trispectrum can, in turn, be used to estimate the 0 98þ0.18þ0.28– AL ¼ . −0.11−0.32 so that AL is fully consistent with unity. lensing potential power spectrum and the amplitude of this When we include the estimates of the lensing-potential SI power spectrum predicts the level of smoothing of the CMB power spectrum (T þ P þ Lensing) the value of Aα is SI −5 power spectra [83]. The internal consistency of these two much more constrained (Aα < 1.7 × 10 at 95% C.L.) effects has been used to explore possible deviations from and is unable to account for the anomalous smoothing of the standard cosmological model, and any discrepancy may the CMB power spectra so that AL ¼ 1.14 0.056, about be due to physical processes that modulate the CMB 2.5 standard deviations larger than its expected value. anisotropies, such as spatial variations in α. Recent This shows that the anomalous smoothing of the CMB CMB observations by the Planck satellite have found that power spectra is unlikely to be explained by a spatial the level of smoothing of the CMB power spectra is 3 variation of α. standard deviations larger than what is expected from the amplitude of the lensing-potential power spectrum. B. The consequences for dynamical models As shown in Fig. 4, in the presence of a spatially varying of fine-structure constant variation α the CMB power spectra are smoothed. The observed anomalous smoothing of the CMB power spectra leads to With limits in hand it is interesting to explore the SI α the preference for a nonzero value of Aα when using the implications of our constraints for specific dynamical Planck temperature and polarization data shown in models. This allows us to propagate our constraints at the Table III. CMB to late times in order to compare to the putative α In order to explore whether the additional modulation due measurement of the angular dipole in from quasar to a spatially varying α might explain any additional spectra. SI In one scenario [23], α fluctuations are sourced by a smoothing we ran an MCMC with both Aα and AL, where scalar field with low (but non-negligible) mass mϕ, with AL is a parameter that controls the level of the smoothing of 4 the power spectra due to weak lensing (and does not affect quadratic couplings to standard-model gauge fields. It is the amplitude of the lensing-potential power spectrum) and interesting to ask if this model can simultaneously stay σ α has an expected value of A ¼ 1 in the standard cosmo- within our constraints to ð α= 0Þ, but explain the claimed L α logical model [83–85]. Figure 9 shows that with just dipole of in analyses of quasar spectra [25]. To answer φ the temperature and polarization (T þ P) data there is a this question, we must consider the time evolution of , given the vastly disparate redshifts of the two observables involved. The scalar field equation of motion is η ϕ̈þ 3Hϕ_ þ m2 þ ϕ ¼ 0: ð54Þ ϕ t2

Here η is proportional to the fraction of the total matter density that contributes to the expectation value hE2 − B2i for the electromagnetic-field Lagrangian density and the coupling of ϕ to matter, H is the usual Hubble parameter, and mϕ is the mass of the light field, while E and B are the electric and magnetic field. When η is negligible, the background field evolves according to ϕ0ðtÞ ∝ sin ðmϕtÞ=ðmϕtÞ,asdo perturbations δϕðx;⃗ tÞ, so long as they are still outside the horizon. μν In this model, the usual Maxwell Lagrangian (FμνF ) 2 μν acquires an additive correction that scales as ϕ FμνF . When canonical field normalization is imposed on the photon (Aμ), we see that a spatially varying α arises, with

SI the lowest-order contribution from spatial fluctuations FIG. 9. Constraints to Aα and AL from Planck temperature (T), δα α φ ∝ ϕ δϕ polarization (P), and estimates of the lensing-potential power given by = ¼ 0ðtÞ ðtÞ. For particle masses spectrum (lensing). Without the lensing data, the degeneracy SI 4 between Aα and AL causes AL to be consistent with its expected In a full high-energy theory of varying α, large quantum value of unity. Once we include the lensing data this degeneracy corrections to mϕ would result. We consider this scenario to be an is broken and AL takes on an anomalously large value. effective theory and neglect quantum corrections.

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≳ ∼ 10−28 −5 mϕ Hrec eV, the power spectrum (and variance) armsα dipole of 9 × 10 , closer to the QSO hints of α thus evolves as t−4 ∝ ð1 þ zÞ6, where the last scaling of Ref. [25]. emerges because most of the redshift (z) interval between As α fluctuations are correlated with primordial density decoupling and today occurs primarily during matter fluctuations, perhaps an additional improvement in sensi- domination. In related models, this scalar can even be a tivity could be achieved by correlating α fluctuations with significant component of the dark matter [24]. CMB observables. This would allow us to use the observed Evolving the 95% C.L. limits to a scale-invariant φ bispectrum (rather than the trispectrum) to search for spectrum from Planck and forward in time to z ∼ 2 yields a variations in the fine-structure constant. Furthermore, all rms dipole value of the estimates in this section compare a rms dipole signal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the observed QSO dipole. A more rigorous analysis could (in the context of a specific model and its equations of L L 1 Cα = 2π ∼ 10−12; 55 ð þ Þ LjL¼1 ð Þ ð Þ motion for α perturbations) map this dipole to a predicted pattern of isotropy breaking in CMB maps, perhaps α many orders of magnitude below the dipole inferred in improving sensitivity, even bringing a runaway-dilaton −5 5 Ref. [25] (0.97 0.88 × 10 at 95% C.L.). In other explanation for the QSO results to be empirically tested words, the model of Ref. [23] cannot simultaneously using Planck and other data. In any of these models, accommodate CMB measurements and hints from QSO depending on the details, the scalar field could be the spectra of a spatial dipole in α. cosmological dark energy or just an unrelated scalar; in In contrast, the total electromagnetic Lagrangian of the either case, the time evolution of ϕ0 and δϕ is related to the −2ϕ μν BSBM theory [9,10] is ∝e FμνF , with a homogeneous evolution of the dark-energy density in a predictable scalar field equation of motion of the form way [21].

ϕ̈ 3Hϕ_ ∝ ρ e−2ϕ: 56 þ m ð Þ VII. CONCLUSIONS The resulting superhorizon spatial variations in α behave as A variety of theoretical ideas motivate the consideration δα=α0 ¼ 2δϕ ¼ const during matter domination, while of a spatially varying fine-structure constant. By modulat- subhorizon fluctuations grow [11]. As a result, there is ing the recombination history and Thomson scattering rate negligible decay in the rms fluctuation α fluctuations on of the early universe, such a scenario would alter CMB the scales of interest, and existing limits from the CMB statistics. The mathematical formalism is similar to that σ α ∼ 10−3 [ð α= 0Þθ>10° ] do not rule out a putative QSO used in studies of CIPs and weak gravitational lensing of −5 α dipole of σα=α0 ≃ 10 for this theory. Thus, even a the CMB, but with a specific response to the physics of futuristic experiment like CMB-S4 would be unable to modulation. Using a toy model, we find that this response rule out the BSBM explanation for the QSO dipole. falls off for scales L ≥ 100, just as for CIPs. Another interesting possibility is the runaway dilaton Here, we used measurements of CMB trispectra (as model [19,20]. Light scalars (dilatons) controlling the captured by the optimal estimator of the weak-lensing volume of extra dimensions appear in some variants of power spectrum) and power spectra to test for the presence string theory. To prevent dilatons from causing highly of a scale-invariant power spectrum of α fluctuations. This constrained violations of the weak equivalence principle, is an interesting possibility as any fluctuation in α sourced one can posit a large dilaton mass, or rely on the matter by a massless field present during inflation would naturally couplings of the dilaton (∝e−ϕ) and a noncanonical kinetic have a scale-invariant spectrum. Using just the α contri- term to dynamically drive it towards weak coupling via bution to the lensing potential power spectrum for a scale- α SI 1 cosmological evolution [18]. The latter option is the run- invariant power spectrum (CL ¼ Aα =½LðL þ Þ), we find SI −5 away dilaton scenario. the constraint (at 95% C.L.) Aα < 1.6 × 10 , which One interesting feature of this model is that the ampli- implies a fractional variation in α on tens of degrees or α σ α 2 5 10−3 tude of spatiotemporal fluctuations is related to the larger of ð α= 0Þθ>10° < . × [constraints to the amplitude of primordial density fluctuations (and thus variance, a derived parameter, are obtained using 2 ≤ ≤ 20 As) [18,19]. Spatial fluctuations evolve as φ ∝ ln ð1 þ zÞ Eq. (12) but with the multipole range L ]. For a WN in this model [19,21]. With this scaling, a scale-invariant constant (white-noise) power spectrum CL ¼ Aα , we find α WN 2 3 10−8 σ α 8 9 10−4 spectrum of fluctuations saturating our CMB limits that Aα < . × and ð α= 0Þθ>10° < . × , all would decay to a rms α dipole of ∼2 × 10−4 at z ∼ 2.At at 95% C.L. This is an improvement over the constraints CMB-S4 sensitivity levels, however, this would improve to found using the 2013 Planck data [52]. Furthermore, we find that at Planck noise levels, the 5Note that these limits are much more stringent than those sensitivity of our estimator (based on lensing data products) estimated in Ref. [52], where it is assumed that δα=α0 ∝ δϕðtÞ ∼ is nearly optimal, as shown by the Fisher analysis for scale- 2 1=t rather than δα=α0 ∼ 1=t . invariant α fluctuations in Sec. V; we performed the same

043531-15 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019) analysis for white-noise power spectrum, and again found in future work, which will also update our analysis to that our lensing-based constraint is comparable in sensi- include power spectra and lensing [87] results from the tivity to a full trispectrum analysis for Planck noise levels. Planck 2018 data release [35], which indeed contain In this paper we have considered the effects of a spatially statistically marginal hints for CIPs, which could also be varying fine-structure constant on both the CMB power- caused by α fluctuations [88]. Indeed, as noted in Ref. [89], and trispectra. Other constants that play central roles in there are still systematic (but unsubtracted) biases contrib- determining the physics of recombination may also be uting to estimators of non-Gaussianity in the CMB. These associated with fields, which, in turn, may have scale- could also affect observable signatures of α modulation; invariant spatial fluctuations, such as the electron mass. The a full trispectrum analysis including these biases and effects of the spatial variation of other constants would be a variety of interesting theoretical possibilities (CIPs, α similar to what we have found here: the detailed effects on fluctuations, etc.) is thus in order. the power spectra will be slightly different since those Looking beyond CMB anisotropies, the full network of effects depend on how the variation of the constant affects bound-bound and bound-free transition during the recombi- the shape of the spectra; however, the effects on the nation era will produce spectral distortions of the CMB away trispectra should be nearly indistinguishable; the contribu- from a perfect thermal spectrum (see Refs. [58,90,91], and tion to the trispectra will scale approximately as L−2 references therein). The rates of the relevant transitions regardless of the field that is being varied; the modification depend very sensitively on α, and so a futuristic measurement due to the detailed physics of the response induces a mild of spatially dependent CMB spectral distortions from correction to this scaling (see, e.g., Ref. [62]). recombination lines would offer an interesting (and more Future experiments (e.g., CMB-S4 [73]) will achieve primordial) test of the possibilities explored here. nearly cosmic-variance limited measurements of CMB Furthermore, the rate of diffusion damping /efficiency of polarization, thus pushing the sensitivity to scale-invariant generating CMB spectral distortions all depends sensitively SI −6 α α fluctuations as low as Aα ¼ 1.9 × 10 (or in the white- on [92,93]. Anisotropies of continuum CMB spectral WN −9 distortions could thus also be an interesting test of spatial noise case, Aα ¼ 1.4 × 10 ), or variances as low as σ α 8 6 10−4 variations in α (as well as to time evolution of the background ð α= 0Þθ>10° ¼ . × [or in the white-noise case, −4 value, as noted in Ref. [48]). ðσα=α0Þθ 10 ¼ 2.2 × 10 ]. > ° In coming decades, observations of absorption in the We considered the possibility that α may alleviate the 21-cm (hyperfine) transition of neutral hydrogen may help anomalously large smoothing of the CMB power spectra us to finally understand the “dark ages,” the epoch between relative to the amplitude of the lensing-potential power CMB decoupling at z ∼ 1090 and the formation of the first spectrum. In the end, scale-invariant α fluctuations cannot stars near z ∼ 10–20 (see Ref. [94] and references therein resolve this tension, due to trispectrum estimates of the for a more comprehensive discussion). As noted in lensing-potential power spectrum. This conclusion depends Refs. [95], the 21-cm line rest frame frequency scales as on the shape of the modulating field’s power spectrum and 4 ν21 ∝ α , the Einstein rate coefficient for the relevant decay precise response of observables to the modulating field. In scales as A ∝ α13, and the spin-changing collisional cross future work, it will be interesting to explore what type of α long-wavelength modulation could explain anomalous sections of hydrogen also depend sensitively on .Asa result, 21-cm cosmology should provide a new probe of smoothing of the CMB power spectra while satisfying α trispectrum constraints. spatial fluctuations in , with the added advantage that Above, we used our phenomenological limits to estimate measurements (by experimental efforts like HERA [96] and SKA [97]) at many redshifts should facilitate stringent tests constraints to actual dynamical theories of varying α, of the time evolution of perturbations in different models of appropriating a CMB analysis that treated α as spatially spatially varying α. varying but constant in time. We thus remind the reader that the translations of our constraints to limits on specific models of varying α are just order-of-magnitude estimates. ACKNOWLEDGMENTS Robust tests require a proper evolution of the background α D. R. and D. Q. made equal contributions to this work. value, a proper relativistic treatment of perturbation evo- We thank the Provosts’ Offices of Swarthmore and lution, and a computation of the imprint of these dynamics Haverford Colleges for supporting this research. We thank on observables using a Boltzmann code like CAMB [55] or Jens Chluba, Marc Kamionkowski, Wayne Hu, Yevgeny CLASS [86], with appropriate modifications for the model Stadnik, Victor Flambaum, Asantha Cooray, and John of interest. O’Bryan for stimulating discussions. We thank Yacine Additional improvements could also follow from ana- Ali-Haïmoud for thorough comments on this manuscript lyzing CMB trispectra directly (rather than a lensing power- and stimulating discussions. Part of this work was com- spectrum based estimator) and doing a map-level analysis pleted at the Aspen Center for Physics, which is supported for the time-evolved imprint of the claimed QSO dipole. by National Science Foundation Grant No. PHY-1607611. We will pursue a more complete analysis along these lines This work was supported in part by the National Science

043531-16 PROBING SPATIAL VARIATION OF THE … PHYS. REV. D 99, 043531 (2019) Z 2 0 Foundation under Grant No. NSF PHY-1125915 at the X;dX0 2 dXlðkÞ C ≡ k dkPζðkÞX ðkÞ : ðA4Þ Kavli Institute for Theoretical Physics (KITP) at UC Santa l π l dφ Barbara. D. G. thanks KITP for its hospitality during the completion of this work. The multipole moments of the lensing potential are denoted by ϕLM. This formalism was first developed for α fluctuations in Ref. [52]. Here we use the equivalent notation APPENDIX A: DETAILED EXPRESSIONS of Ref. [62]. FOR THE α-INDUCED CMB Under the null hypothesis (i.e., no α variation), the OFF-DIAGONAL CORRELATIONS ω ≡ pminimum-varianceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi estimator for the deflection field dLM 0 Deflections of CMB photons and higher-order modu- LðL þ 1ÞϕLM from a single pair ω ¼ XX of observables lations of the transfer functions produce off-diagonal CMB is [66,69] φ correlations for fixed lens and realizations, given by [59,60,66] X ll0 L ˆ ω ω M 0 ω d ¼ A ð−1Þ X X 0 0 g 0 ; ðA5Þ LM L lm l m mm0 −M ll L 0 ˜ XX0 m lm;l0m0 hX X 0 0 ij α ¼ C δ 0 δ − 0 ð−1Þ lm l m lens; l ll m m X ll0 L ω ω where A and g 0 are þ ð−1ÞM L ll L 0 − LM mm M X ω ω ω −1 XX0 XX0 A ¼ LðL þ 1Þð2L þ 1Þf g f g ; ðA6Þ ϕ φ L l1l2 l1Ll2 l1Ll2 × ½ LMflLl0 þ LMhlLl0 ; ðA1Þ

XX0 XX0 α XX;t X0X0;t ω − −1 lþLþl0 XX0;t XX0;t ω where flLl0 and hlLl0 are the lensing= response functions ω Cl0 Cl flLl0 ð Þ Cl Cl0 fl0Ll g 0 ≡ : ðA7Þ for different quadratic pairs (see Table IV) and are defined lLl XX;t XX;t X0X0;t X0X0;t XX0;t XX0;t 2 C C 0 C C 0 −ðC C 0 Þ ˜ XX0 l l l l l l in terms of the unmodulated power spectrum, Cl , the appropriately weighted Wigner coefficients, From this we can construct an optimal estimator for the lensing-potential power spectrum ≡ 1 0 0 1 − 1 sGlLl0 ½LðL þ Þþl ðl þ Þ lðl þ Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω β 0 1 X XL ˆ ˆ 2L 1 2l 1 2l0 1 lLl ϕϕ ω β dLMd ð þ Þð þ Þð þ Þ Cˆ ¼ v v LM − B ; ðA8Þ × 16π ; L 2 1 L L 1 L s 0 ∓s L þ ω;β M¼−L LðL þ Þ ðA2Þ where the optimal weights for the minimum-variance rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi estimator are given by [66] ð2L þ 1Þð2l þ 1Þð2l0 þ 1Þ lLl0 H 0 ≡ ; s lLl 4π X s 0 ∓s ω ≡ mv N−1 ωβ vL NL ð L Þ ; ðA9Þ ðA3Þ β

ω β X and the derivative power spectra ωβ A A 0 0 β N ≡ L L fgω ½CXY;tCX Y ;tg L LðL þ 1Þð2L þ 1Þ l1Ll2 l1 l2 l1Ll2 l1l2 TABLE IV. The lensing and α-modulation response functions. 0 0 β −1 Lþl1þl2 CXY ;tCX Y;tg ; A10 “Even” and “odd” indicate that the functions are nonzero only þð Þ l1 l2 l2Ll1 g ð Þ when L þ l þ l0 is even or odd, respectively. To translate from the conventions of Ref. [53] we need to swap l ↔ l0, which leads to X −1 mv ≡ N−1 ωβ a minus sign for the two odd responses, EB and TB. Note that NL ð L Þ : ðA11Þ the B-mode autocorrelation, BB, vanishes at linear order in the ωβ φ field. Using the same construction, we can form an optimal 0 XX0 XX0 0 α ω ω XX f 0 h 0 lþl þL lLl lLl estimator for CL by replacing fl0Ll with hl0Ll in Eqs. (A6) TT ˜ TT ˜ TT ˜ T;dT ˜ T;dT Even and (A7). Cl 0Fl0Ll þCl0 0FlLl0 ðCl þCl0 Þ0HlLl0 TE ˜ TE ˜ TE ˜ T;dE ˜ E;dT Even Cl 2Fl0Ll þCl0 0FlLl0 Cl 2HlLl0 þCl0 0HlLl0 ˜ TE ˜ T;dE TB iC F 0 iC H 0 Odd l 2 l Ll l 2 lLl APPENDIX B: POWER-SPECTRA DERIVATIVES EE ˜ EE ˜ EE ˜ E;dE ˜ E;dE Even Cl 2Fl0Ll þCl0 2FlLl0 ðCl þCl0 Þ2HlLl0 EB ˜ EE − ˜ BB ˜ E;dE ˜ B;dB Odd In order to calculate the second derivative of the power i½Cl 2Fl0Ll Cl0 2FlLl0 iðCl þCl0 Þ2HlLl0 ˜ BB ˜ BB ˜ B;dB ˜ B;dB spectrum we use a finite-difference approximation for the BB C 2F 0 þC 0 2F 0 ðC þC 0 Þ H 0 Even l l Ll l lLl l l 2 lLl second derivative,

043531-17 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019)

FIG. 10. The derivative power spectra used in this paper, which agree (up to changes in fiducial cosmological parameters) with those in Refs. [23,52].

2 ∂ Cl Clð0 þ ΔφÞ − 2Clð0ÞþClð0 − ΔφÞ Then, we compute the finite-difference second derivative ≈ : ∂φ2 Δφ 2 ðB1Þ Δφ φ¼0 ð Þ for a range of step sizes and compare it to the second derivative computed from the polynomial fit. From this χ2 To use this approximation, we must find the step size, Δφ, procedure we identify a minimum in between the finite χ2 that gives the most accurate derivative. To do this, we first difference and polynomial second derivatives. The fit a polynomial to the power spectrum as a function of φ,at between these two methods for the TT spectrum is shown each multipole moment l. Figure 11 shows the χ2 of the in Fig. 11, and is given by finite-difference derivative fit with respect to the actual 2 P power spectrum. Note that at small values of Δφ the χ − 2 χ2 ≡ Plðyl flÞ increases due to residual numerical noise in the derivative. 2 ; ðB2Þ lðylÞ

TT where fl are the finite-difference second derivatives of Cl and yl are the polynomial fit second derivatives. We used a step size of Δφ ¼ 0.01 and confirmed that this same step size allows us to accurately compute the second-order derivative of the EE and TE power spectra. The effects computed in this paper rely on the power spectra between temperature and E-mode polarization and the derivatives of those with respect to φ. We show these derivatives in Fig. 10. X;dX0 In this work, derivative power spectra Cl are computed with a suitably modified version of CAMB (and HYREC [54]), in which finite-difference derivatives of 0 the CMB transfer functions XlðkÞ such that relative FIG. 11. The χ2 showing the goodness of fit between the finite- convergence relative convergence is exhibited at the difference derivative and the polynomial derivative. 1%–10% level.

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−3 We show the derivative power spectra in Fig. 10;they futuristic experiments) are at the σα=α0 ∼ 10 level, the −5 are consistent with the numerical derivatives shown in fractional error in CMB two-point observables (∼10 )iswell Refs. [23,52], up to changes in fiducial values of cosmological below cosmic variance (≲10−3 for the scales of interest); these parameters. Since our constraints (as well as the sensitivity of numerical derivatives are sufficiently accurate for our purpose.

[1] P. A. M. Dirac, Proc. R. Soc. A 165, 199 (1938). [28] M. T. Murphy, J. K. Webb, V. V. Flambaum, and S. J. [2] C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961). Curran, eConf. C020620, FRAT02 (2002). [3] G. Gamow, Phys. Rev. Lett. 19, 913 (1967). [29] M. T. Murphy, J. K. Webb, and V. V. Flambaum, Mon. Not. [4] G. Gamow, Phys. Rev. Lett. 19, 759 (1967). R. Astron. Soc. 345, 609 (2003). [5] E. Teller, Phys. Rev. 73, 801 (1948). [30] A. Songaila and L. L. Cowie, Astrophys. J. 793, 103 (2014). [6] A. Chodos and S. Detweiler, Phys. Rev. D 21, 2167 (1980). [31] C. J. A. P. Martins and A. M. M. Pinho, Phys. Rev. D 95, [7] E. W. Kolb, M. J. Perry, and T. P. Walker, Phys. Rev. D 33, 023008 (2017). 869 (1986). [32] M. T. Murphy and K. L. Cooksey, Mon. Not. R. Astron. [8] P. Nath and M. Yamaguchi, Phys. Rev. D 60, 116004 Soc. 471, 4930 (2017). (1999). [33] T. Damour and F. Dyson, Nucl. Phys. B480, 37 (1996). [9] J. D. Bekenstein, Phys. Rev. D 25, 1527 (1982). [34] P. A. R. Ade et al. (Planck Collaboration), Astron. [10] J. D. Barrow, J. Magueijo, and H. B. Sandvik, Phys. Rev. D Astrophys. 594, A13 (2016). 66, 043515 (2002). [35] N. Aghanim et al. (Planck Collaboration), arXiv:1807.06209. [11] J. D. Barrow and D. F. Mota, Classical Quantum Gravity 20, [36] K. T. Story et al., Astrophys. J. 779, 86 (2013). 2045 (2003). [37] B. A. Benson et al., Proc. SPIE Int. Soc. Opt. Eng. 9153, [12] D. F. Mota and J. D. Barrow, Mon. Not. R. Astron. Soc. 349, 91531P (2014). 291 (2004). [38] T. Louis et al., J. Cosmol. Astropart. Phys. 06 (2017) 031. [13] J. D. Barrow and A. A. H. Graham, Phys. Rev. D 88, 103513 [39] P. P. Avelino, S. Esposito, G. Mangano, C. J. A. P. Martins, (2013). A. Melchiorri, G. Miele, O. Pisanti, G. Rocha, and P. T. P. [14] A. A. H. Graham, Classical Quantum Gravity 32, 015019 Viana, Phys. Rev. D 64, 103505 (2001). (2015). [40] C. J. A. P. Martins, A. Melchiorri, G. Rocha, R. Trotta, P. P. [15] E. J. Copeland, N. J. Nunes, and M. Pospelov, Phys. Rev. D Avelino, and P. T. P. Viana, Phys. Lett. B 585, 29 (2004). 69, 023501 (2004). [41] G. Rocha, R. Trotta, C. J. A. P. Martins, A. Melchiorri, P. P. [16] J. D. Bekenstein, Phys. Rev. D 48, 3641 (1993). Avelino, R. Bean, and P. T. P. Viana, Mon. Not. R. Astron. [17] C. van de Bruck, J. Mifsud, and N. J. Nunes, J. Cosmol. Soc. 352, 20 (2004). Astropart. Phys. 12 (2015) 018. [42] K. Ichikawa, T. Kanzaki, and M. Kawasaki, Phys. Rev. D [18] T. Damour and A. M. Polyakov, Nucl. Phys. B423, 532 74, 023515 (2006). (1994). [43] E. Menegoni, S. Galli, J. G. Bartlett, C. J. A. P. Martins, and [19] T. Damour, F. Piazza, and G. Veneziano, Phys. Rev. D 66, A. Melchiorri, Phys. Rev. D 80, 087302 (2009). 046007 (2002). [44] S. Galli, M. Martinelli, A. Melchiorri, L. Pagano, B. D. [20] T. Damour, F. Piazza, and G. Veneziano, Phys. Rev. Lett. 89, Sherwin, and D. N. Spergel, Phys. Rev. D 82, 123504 081601 (2002). (2010). [21] C. J. A. P. Martins, arXiv:1709.02923. [45] E. Menegoni, M. Archidiacono, E. Calabrese, S. Galli, [22] D. Parkinson, B. A. Bassett, and J. D. Barrow, Phys. Lett. B C. J. A. P. Martins, and A. Melchiorri, Phys. Rev. D 85, 578, 235 (2004). 107301 (2012). [23] K. Sigurdson, A. Kurylov, and M. Kamionkowski, Phys. [46] P. A. R. Ade et al. (Planck Collaboration), Astron. Rev. D 68, 103509 (2003). Astrophys. 580, A22 (2015). [24] Y. V. Stadnik and V. V. Flambaum, Phys. Rev. Lett. 115, [47] I. de Martino, C. Martins, H. Ebeling, and D. Kocevski, 201301 (2015). Phys. Rev. D 94, 083008 (2016). [25] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, [48] L. Hart and J. Chluba, Mon. Not. R. Astron. Soc. 474, 1850 R. F. Carswell, and M. B. Bainbridge, Phys. Rev. Lett. 107, (2018). 191101 (2011). [49] P. A. R. Ade et al. (Planck Collaboration), Astron. [26] J. A. King, J. K. Webb, M. T. Murphy, V. V. Flambaum, Astrophys. 594, A16 (2016). R. F. Carswell, M. B. Bainbridge, M. R. Wilczynska, and [50] L. Dai, D. Jeong, M. Kamionkowski, and J. Chluba, Phys. F. E. Koch, Mon. Not. R. Astron. Soc. 422, 3370 (2012). Rev. D 87, 123005 (2013). [27] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. [51] J. D. Barrow, Phys. Rev. D 71, 083520 (2005). Drinkwater, and J. D. Barrow, Phys. Rev. Lett. 82, 884 [52] J. O’Bryan, J. Smidt, F. De Bernardis, and A. Cooray, (1999). Astrophys. J. 798, 18 (2015).

043531-19 SMITH, GRIN, ROBINSON, and QI PHYS. REV. D 99, 043531 (2019)

[53] C. He, D. Grin, and W. Hu, Phys. Rev. D 92, 063018 (2015). [76] R. Adam et al. (Planck Collaboration), Astron. Astrophys. [54] Y. Ali-Haïmoud and C. M. Hirata, Phys. Rev. D 83, 043513 594, A8 (2016). (2011). [77] C. H. Heinrich, D. Grin, and W. Hu, Phys. Rev. D 94, [55] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 043534 (2016). 473 (2000). [78] M. Su, A. P. S. Yadav, M. McQuinn, J. Yoo, and M. [56] J. Silk, Astrophys. J. 151, 459 (1968). Zaldarriaga, arXiv:1106.4313. [57] U. Seljak and M. Zaldarriaga, Astrophys. J. 469, 437 [79] K. M. Smith et al., AIP Conf. Proc. 1141, 121 (2009). (1996). [80] K. M. Smith, D. Hanson, M. LoVerde, C. M. Hirata, and [58] J. Chluba and Y. Ali-Haimoud, Mon. Not. R. Astron. Soc. O. Zahn, J. Cosmol. Astropart. Phys. 06 (2012) 014. 456, 3494 (2016). [81] P. Larsen, A. Challinor, B. D. Sherwin, and D. Mak, Phys. [59] D. Grin, O. Dore, and M. Kamionkowski, Phys. Rev. Lett. Rev. Lett. 117, 151102 (2016). 107, 261301 (2011). [82] A. Lewis and A. Challinor, Phys. Rep. 429, 1 (2006). [60] D. Grin, O. Dore, and M. Kamionkowski, Phys. Rev. D 84, [83] E. Calabrese, A. Slosar, A. Melchiorri, G. F. Smoot, and O. 123003 (2011). Zahn, Phys. Rev. 77, 123531 (2008). [61] D. Grin, D. Hanson, G. P. Holder, O. Dore, and M. [84] E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Lett. B Kamionkowski, Phys. Rev. D 89, 023006 (2014). 761, 242 (2016). [62] T. L. Smith, J. B. Muñoz, R. Smith, K. Yee, and D. Grin, [85] F. Renzi, E. Di Valentino, and A. Melchiorri, Phys. Rev. D Phys. Rev. D 96, 083508 (2017). 97, 123534 (2018). [63] M. Zaldarriaga and U. Seljak, Phys. Rev. D 59, 123507 [86] J. Lesgourgues, arXiv:1104.2932. (1999). [87] N. Aghanim et al. (Planck Collaboration), arXiv:1807.06210. [64] W. Hu, Phys. Rev. D 62, 043007 (2000). [88] Y. Akrami et al. (Planck Collaboration), arXiv:1807.06211. [65] L. Knox, Astrophys. J. 480, 72 (1997). [89] J. C. Hill, Phys. Rev. D 98, 083542 (2018). [66] T. Okamoto and W. Hu, Phys. Rev. D 67, 083002 (2003). [90] J. A. Rubino-Martin, J. Chluba, and R. A. Sunyaev, Mon. [67] J. B. Muñoz, D. Grin, L. Dai, M. Kamionkowski, and E. D. Not. R. Astron. Soc. 371, 1939 (2006). Kovetz, Phys. Rev. D 93, 043008 (2016). [91] J. Chluba and R. A. Sunyaev, Mon. Not. R. Astron. Soc. [68] Y. Ali-Haimoud and C. M. Hirata, Phys. Rev. D 82, 063521 419, 1294 (2012). (2010). [92] J. Chluba, R. Khatri, and R. A. Sunyaev, Mon. Not. R. [69] T. Namikawa, D. Yamauchi, and A. Taruya, J. Cosmol. Astron. Soc. 425, 1129 (2012). Astropart. Phys. 01 (2012) 007. [93] R. Khatri and R. A. Sunyaev, J. Cosmol. Astropart. Phys. 06 [70] M. Kesden, A. Cooray, and M. Kamionkowski, Phys. Rev. (2013) 026. D 67, 123507 (2003). [94] J. R. Pritchard and A. Loeb, Rep. Prog. Phys. 75, 086901 [71] S. Das et al., Phys. Rev. Lett. 107, 021301 (2011). (2012). [72] P. A. R. Ade et al. (Planck Collaboration), Astron. [95] R. Khatri and B. D. Wandelt, Phys. Rev. Lett. 98, 111301 Astrophys. 594, A15 (2016). (2007). [73] K. N. Abazajian et al., arXiv:1610.02743. [96] D. R. DeBoer et al., Publ. Astron. Soc. Pac. 129, 045001 [74] N. Aghanim et al. (Planck Collaboration), Astron. (2017). Astrophys. 594, A11 (2016). [97] R. Maartens, F. B. Abdalla, M. Jarvis, and M. G. Santos (SKA [75] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002). Cosmology SWG), Proc. Sci. AASKA14 (2015) 016.

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