Future CMB constraints on cosmic birefringence and implications for fundamental physics

TL;DR

Upcoming CMB polarization experiments will greatly enhance constraints on cosmic birefringence, providing insights into new physics beyond standard models, such as pseudoscalar fields, primordial magnetic fields, and Lorentz invariance violations.

Contribution

This paper forecasts the improved bounds on cosmic birefringence from future CMB experiments and explores their implications for fundamental physics.

Findings

LiteBIRD and SO will reduce the isotropic CPR bound to 1.5 and 0.6 arcmin.

CMB-S4-like and PICO will achieve bounds around 0.1 arcmin.

Forecasted bounds will significantly constrain primordial magnetic fields and axion-photon coupling.

Abstract

The primary scientific target of the CMB polarization experiments that are currently being built and proposed is the detection of primordial tensor perturbations. As a byproduct, these instruments will significantly improve constraints on cosmic birefringence, or the rotation of the CMB polarization plane. If convincingly detected, cosmic birefringence would be a dramatic manifestation of physics beyond the standard models of particle physics and cosmology. We forecast the bounds on the cosmic polarization rotation (CPR) from the upcoming ground-based Simons Observatory (SO) and the space-based LiteBIRD experiments, as well as a "fourth generation" ground-based CMB experiment like CMB-S4 and the mid-cost space mission PICO. We examine the detectability of both a stochastic anisotropic rotation field and an isotropic rotation by a constant angle. CPR induces new correlations of CMB observables, including spectra of parity-odd type in the case of isotropic CPR, and mode-coupling correlations in the anisotropic rotation case. We find that LiteBIRD and SO will reduce the 1$\sigma$ bound on the isotropic CPR from the current value of 30 arcmin to 1.5 and 0.6 arcmin, respectively, while CMB-S4-like and PICO will reduce it to $\sim 0.1$ arcmin. The bounds on the amplitude of a scale-invariant CPR spectrum will be reduced by 1, 2 and 3 orders of magnitude by LiteBIRD, SO and CMB-S4-like/PICO, respectively. We discuss implications of the forecasted CPR bounds for pseudoscalar fields, primordial magnetic fields (PMF), and violations of Lorentz invariance. We find that CMB-S4-like and PICO can reduce the 1$\sigma$ bound on the amplitude of the scale-invariant PMF from 1 nG to 0.1 nG, while also probing the magnetic field of the Milky Way. They will also significantly improve bounds on the axion-photon coupling, placing stringent constraints on the string theory axions.

Tables

Table 1. Table 1: Parameters of the CMB experiments considered in our forecast. The values of frequencies in the first column are rounded. Channels with frequencies below 60 and above 150 GHz are dominated by the galactic foregrounds and are not included.

	LiteBIRD		SO SAT		CMB-S4-like		PICO
	space		ground		ground		space
	$f_{sky}=0.6$		$f_{sky}=0.1$		$f_{sky}=0.4$		$f_{sky}=0.6$
target sensitivity333The expected 68% CL upper bound on $r$ assuming it is undetectably small. to $r$	${\sigma }_r=0.001$		${\sigma }_r=0.003$		${\sigma }_r=0.0005$		${\sigma }_r=5\times {10}^{-5}$
de-lensed fraction444Perfect de-lensing corresponds to $f_L=0$, no de-lensing to $f_L=1$.	$f_L=0.5$		$f_L=0.5$		$f_L=0.15$		$f_L=0.1$
(${10}^4\rho$, ${10}^{3}e$, ${10}^2ϵ$)555The beam systematics parameters used in our forecasts as described in Sec. II.3.	(4, 1.5, 1.5)		(1, 0.2, 1.5)		(2, 50, 0.4)		(1, 1.5, 0.4)
Frequency	${\theta }_{\mathrm{fwhm}}$	${\sigma }_P$	${\theta }_{\mathrm{fwhm}}$	${\sigma }_P$	${\theta }_{\mathrm{fwhm}}$	${\sigma }_P$	${\theta }_{\mathrm{fwhm}}$	${\sigma }_P$
(GHz)	′	$\mu$K-′	′	$\mu$K-′	′	$\mu$K-′	′	$\mu$K-′
60	48	19.5	-	-	-	-	$13$	$3.9$
70	43	15.8	-	-	-	-	-	-
75	-	-	-	-	-	-	$11$	$3.2$
78	39	13.3	-	-	-	-	-	-
90	35	11.5	-	-	-	-	$9.5$	$2$
95	-	-	$30$	$2.7$	$2.2$	$2.1$	-	-
100	29	9.0	-	-	-	-	-	-
110	-	-	-	-	-	-	$7.9$	$1.7$
120	25	7.5	-	-	-	- -	-
130	-	-	-	-	-	-	$7.4$	$1.6$
140	23	5.8	-	-	-	-	-	-
150	-	-	$17$	$3$	$1.5$	$2.1$	$6.2$	$1.4$

Table 2. Table 2: Current and forecasted 68 68 68 % CL bounds on the uniform and the anisotropic CPR parameters.

		Current			LiteBIRD			SO			CMB-S4-like			PICO
DL	BS	$\alpha$	$A_{\alpha }$	$\sqrt{\frac{C_2^{\alpha }}{4\pi }}$	$\alpha$	$A_{\alpha }$	$\sqrt{\frac{C_2^{\alpha }}{4\pi }}$	$\alpha$	$A_{\alpha }$	$\sqrt{\frac{C_2^{\alpha }}{4\pi }}$	$\alpha$	$A_{\alpha }$	$\sqrt{\frac{C_2^{\alpha }}{4\pi }}$	$\alpha$	$A_{\alpha }$	$\sqrt{\frac{C_2^{\alpha }}{4\pi }}$
		′	${10}^{-2}$deg2	′	′	${10}^{-3}$deg2	′	′	${10}^{-4}$deg2	′	′	${10}^{-5}$deg2	′	′	${10}^{-5}$deg2	′
yes	no	-	-	-	1.3	2.7	0.9	0.56	3	0.29	0.1	1.4	0.065	0.05	0.4	0.035
yes	yes	-	-	-	1.5	3.3	1.0	0.66	4	0.35	0.11	2.0	0.08	0.06	0.5	0.04
no	no	-	-	-	1.4	3.5	1.0	0.64	5.0	0.4	0.13	2.5	0.09	0.08	1.2	0.06
no	yes	30	2	3	1.6	4.0	1.1	0.71	5.5	0.4	0.15	3.3	0.1	0.09	1.4	0.065

Table 3. Table 3: Current and forecasted 68 68 68 % CL lower bounds on the axion decay constant f a subscript 𝑓 𝑎 f_{a} , in the units of the energy scale of inflation, H I = 10 14 subscript 𝐻 𝐼 superscript 10 14 H_{I}=10^{14} GeV × H 14 absent subscript 𝐻 14 \times H_{14} . These are inferred from the bounds on A α subscript 𝐴 𝛼 A_{\alpha} that include the effects of beam systematics and de-lensing.

	Current	LiteBIRD	SO	CMB-S4-like	PICO
$f_a$ [$H_I$]	50	200	500	2000	4000

Table 4. Table 4: Current and forecasted 68 68 68 % CL lower bounds on the strength of the scale-invariant primordial magnetic field B SI subscript 𝐵 SI B_{\rm SI} derived from mode-coupling correlations induced by Faraday Rotation (FR) compared to those derived from the BB spectra ( C ℓ B ​ B superscript subscript 𝐶 ℓ 𝐵 𝐵 C_{\ell}^{BB} ). The forecast accounts for de-lensing and beam systematics. The f G = 0 subscript 𝑓 𝐺 0 f_{G}=0 case assumes that there is no galactic contribution to FR, while f G = 1 subscript 𝑓 𝐺 1 f_{G}=1 includes the galactic FR based on the rotation measure map of Oppermann  et al. ( 2012 ) .

$B_{\mathrm{SI}}$ [nG]	Current	LiteBIRD	SO	CMB-S4-like	PICO
$C_{\ell }^{BB}$	1.0777This bound is based on fitting all cosmological parameters to TT, EE, ET from Planck and BB from SPT. The forecasts in the remained of the row assume fitting $B_{\mathrm{SI}}$ to BB only with remaining cosmological parameters fixed to their best fit LCDM values.	2.3	1.0	0.55	0.5
${\mathrm{FR}}_{f_G=0}$	-	1.7	0.7	0.16	0.08
${\mathrm{FR}}_{f_G=1}$	50	1.7	0.7	0.18	0.12

Table 5. Table 5: Current CMB birefringence constraints on the isotropic d = 3 𝑑 3 d=3 coefficient for Lorentz violation k ( V ) ​ 00 ( 3 ) subscript superscript 𝑘 3 𝑉 00 k^{(3)}_{(V)00} and projected sensitivities for the future experiments.

$k_{\left(V\right)00}^{\left(3\right)}$ (${10}^{-43}\mathrm{GeV}$)	Data	Ref.
$-0.51\pm 0.75\pm 0.45$	QUaD	Wu et al. (2009)
$0.34\pm 1.16\pm 1.40$	WMAP9	Hinshaw et al. (2013)
$-0.29\pm 0.05\pm 0.26$	Planck	Aghanim et al. (2016)
$\sim 0.017$	LiteBIRD
$\sim 0.007$	SO
$\sim 0.0025$	CMB-S4-like
$\sim 0.0015$	PICO

Equations

$$B_{lm}=2\sum _{LM}\sum _{l^{\prime }{m}^{\prime }}{\alpha }_{LM}{E}_{l^{\prime }{m}^{\prime }}{\xi }_{lm{l}^{\prime }{m}^{\prime }}^{LM}{H}_{l{l}^{\prime }}^L,$$

$${\xi }_{lm{l}^{\prime }{m}^{\prime }}^{LM}\equiv (-1{)}^m\sqrt{(2l+1)(2L+1)(2l^{\prime }+1)/4\pi }$$

$$\displaystyle\times\left(\begin{array}[]{ccc}l&L&l^{\prime}\\ -m&M&m^{\prime}\end{array}\right);\ H_{ll^{\prime}}^{L}\equiv\left(\begin{array}[]{ccc}l&L&l^{\prime}\\ 2&0&-2\end{array}\right)\ ,$$

$$[{\hat{\alpha }}_{B^i{E}^j,LM}{]}_{l{l}^{\prime }}=\frac{2\pi {\sum }_{m{m}^{\prime }}{B}_{lm}^i{E}_{l^{\prime }{m}^{\prime }}^{j\ast }{\xi }_{lm{l}^{\prime }{m}^{\prime }}^{LM}}{(2l+1)(2l^{\prime }+1)C_l^{EE}{H}_{l{l}^{\prime }}^L}$$

$$A\in \{E^i{B}^j,B^i{E}^j,T^i{B}^j,B^i{T}^j\}.$$

$${\sigma }_{\alpha ,L}^{-2}=\sum _{l^{\prime }\ge l}{G}_{l{l}^{\prime }}^L\sum _{A,A^{\prime }}[({\mathcal{C}}^{l{l}^{\prime }}{)}^{-1}{]}_{A{A}^{\prime }}{Z}_{l{l}^{\prime }}^A{Z}_{l{l}^{\prime }}^{A^{\prime }},$$

$$Z_{l{l}^{\prime }}^{X^i{B}^j}=c^2{W}_{l{l}^{\prime }}^{ij}{C}_l^{XE},$$

$$Z_{l{l}^{\prime }}^{B^i{X}^j}=c^2{W}_{l{l}^{\prime }}^{ij}{C}_{l^{\prime }}^{EX},$$

$$[{\mathcal{C}}^{l{l}^{\prime }}{]}_{X^i{B}^j,Y^k{B}^n}={\tilde{C}}_l^{X^i{Y}^k}{\tilde{C}}_{l^{\prime }}^{B^j{B}^n}+{\delta }_{l{l}^{\prime }}{\tilde{C}}_l^{X^i{B}^n}{\tilde{C}}_{l^{\prime }}^{B^j{Y}^k}$$

$$[{\mathcal{C}}^{l{l}^{\prime }}{]}_{B^i{X}^j,B^k{Y}^n}={\tilde{C}}_l^{B^i{B}^k}{\tilde{C}}_{l^{\prime }}^{X^j{Y}^n}+{\delta }_{l{l}^{\prime }}{\tilde{C}}_l^{B^i{Y}^n}{\tilde{C}}_{l^{\prime }}^{X^j{B}^k}$$

$${\tilde{C}}_l^{X^i{Y}^j}=C_l^{XY,\mathrm{prim}}+A_{\mathrm{L}}{C}_l^{XY,\mathrm{WL}}+C_l^{XY,\mathrm{sys}}+N_l^{X^i{Y}^j},$$

$${\left(\frac{S}{N}\right)}^2=\sum _{L=1}^{L_{max}}\frac{(f_{\mathrm{sky}}/2)(2L+1)[C_L^{\alpha }{]}^2}{[C_L^{\alpha }+{\sigma }_{\alpha ,L}^2{]}^2}.$$

$$A_{\alpha }\equiv \frac{L(L+1)C_L^{\alpha }}{2\pi }.$$

$$(S/N{)}^2\approx \sum _{L=1}^{L_S}{f}_{\mathrm{sky}}(2L+1)/2\approx L_S^2/2,$$

$$\frac{S}{N}\approx \frac{\sqrt{\pi A_{\alpha }}}{{\sigma }_{\alpha ,L_S}},$$

$${\sigma }_{\alpha }^2=\frac{{\sigma }_{\alpha ,0}^2}{4\pi f_{\mathrm{sky}}},$$

$$C_l^{EB}=2\alpha C_l^{EE}\mathrm{and}{C}_l^{TB}=2\alpha C_l^{TE}.$$

$$C_l^{BB(\alpha )}=\frac{1}{\pi }\sum _L(2L+1)C_L^{\alpha }\sum _{l_1}(2l_1+1)C_{l_1}^{EE}(H_{l{l}_1}^L{)}^2.$$

$${\left(\frac{S}{N}\right)}^2=\sum _{l=L_{min}}^{L_{max}}\frac{f_{\mathrm{sky}}(2l+1)[C_l^{BB(\alpha )}{]}^2}{2[{\tilde{C}}_l^{BB}+C_l^{BB(\alpha )}{]}^2},$$

$$C_l^{BB(\alpha )}={\alpha }^2{C}_l^{EE}.$$

$${\left(\frac{S}{N}\right)}^2=\sum _{l=L_{min}}^{L_{max}}\frac{f_{\mathrm{sky}}(2l+1)[{\alpha }^2{C}_l^{EE}{]}^2}{2[{\tilde{C}}_l^{BB}+{\alpha }^2{C}_l^{EE}{]}^2}.$$

$$\sqrt{\frac{C_2^{\alpha }}{4\pi }}=\sqrt{\frac{A_{\alpha }}{12}}.$$

$${\sigma }_{C_L^{\alpha }}=\frac{{\sigma }_{\alpha ,L}^2}{\sqrt{f_{\mathrm{sky}}(2L+1)/2}}$$

$${\mathcal{L}}_{a\gamma }=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{\partial }_{\mu }a{\partial }^{\mu }a+\frac{a}{2f_a}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }-\left(\frac{1}{2}{m}_a^2{a}^2\right),$$

$${\ddot{A}}_{\pm }(k,\tau )+\left[k^2\pm \frac{2k}{f_a}(\dot{a}+\hat{n}\cdot \vec{\nabla }a)\right]A_{\pm }(k,\tau )=0,$$

$$m_a<H_{\mathrm{dec}}\sim 10^{-28}eV,$$

$$A_{\alpha }={\left(\frac{H_I}{2\pi f_a}\right)}^2.$$

$$\mathrm{RM}(\hat{n})\equiv c^{-2}{\nu }^2\alpha (\hat{n}).$$

$$A_{\mathrm{RM}}^2=L(L+1)C_L^{\mathrm{RM}}/2\pi$$

$$A_{\mathrm{RM}}\approx 50rad/{\mathrm{m}}^2{B}_{\mathrm{SI}}/\mathrm{nG}.$$

$${\left(\frac{S}{N}\right)}^2=\sum _{L=1}^{L_{max}}\frac{(f_{\mathrm{sky}}/2)(2L+1)[C_L^{\mathrm{RM},\mathrm{PMF}}{]}^2}{[C_L^{\mathrm{RM},\mathrm{PMF}}+f_{\mathrm{G}}{C}_L^{\mathrm{RM},\mathrm{G}}+{\sigma }_{\mathrm{RM},L}^2{]}^2},$$

$${\mathcal{L}}_{\mathrm{LV}}$$

Full text

I Introduction

The impact of the Cosmic Microwave Background (CMB) on our knowledge of the primordial universe has been astounding. In the past quarter of a century, progress has hardly abated. Recent years have witnessed the discovery of temperature anisotropy by COBE Smoot et al. (1992); Bennett et al. (1996), then the first detection of CMB polarization by DASI Kovac et al. (2002) and high resolution full sky CMB maps from WMAP Bennett et al. (2003, 2013), culminating in comprehensive measurements of temperature and E-mode polarization by Planck Adam et al. (2016). The primary focus of current CMB research is the measurements of the so-called B-modes – the parity odd polarization patterns Kamionkowski et al. (1997a); Seljak and Zaldarriaga (1997) that could be created by inflationary gravitational waves (GW) Crittenden et al. (1993); Seljak and Zaldarriaga (1997); Kamionkowski et al. (1997b) as well as a harbinger of potentially new physics Seljak et al. (1997); Avgoustidis et al. (2011); Moss and Pogosian (2014); Lizarraga et al. (2016); Amendola et al. (2014); Raveri et al. (2015). On $\sim 10^{\prime}$ angular scales, or $\ell\sim 1000$, gravitational lensing by large scale structures generates B-modes Zaldarriaga and Seljak (1998) that were measured by POLARBEAR Ade et al. (2014a) and SPTPol Keisler et al. (2015) five years ago. The first measurements of B-modes on larger scales, $\ell\sim 100$, where the inflationary GW are expected to contribute the most, were made by BICEP2/Keck Ade et al. (2014b, 2015a). However, foregrounds, such as polarized dust in our galaxy, are not yet characterized to an accuracy needed to unveil the primordial signal possibly hiding behind.

While foregrounds pose a serious challenge, many experiments are rising to meet it. For example the Simons Observatory Aguirre et al. (2018), currently under construction in the Chilean Atacama desert, has a predicted sensitivity of $\sigma_{r}=0.003$ to the tensor-to-scalar ratio $r$ characterizing the amplitude of gravitational wave (GW) B-modes, which would improve current bounds Ade et al. (2016) by more than an order of magnitude. As the inflationary paradigm is perfectly consistent with $r$ being below the observable range Knox and Song (2002), it is plausible that no GW contribution will ever be seen with B-modes. Fortunately, other fundamental physics will be constrained by improved B-mode measurements (see, e.g., Abazajian et al. (2016) for a review), such as the number of relativistic particle species in the early universe, the sum of the neutrino masses, annihilation rates of dark matter candidates, and possible modifications of gravity. Another effect, which is the subject of this paper, is cosmic birefringence, or cosmic polarization rotation (CPR), that can be caused by parity violating extensions of the standard model Harari and Sikivie (1992); Carroll et al. (1990); Carroll (1998); Pospelov et al. (2009) or primordial magnetic fields Kosowsky and Loeb (1996).

Unlike the inflationary GW B-modes where the target signal is, at best, a few percent of the foreground contribution, most of the signal required for CPR measurements comes from smaller scales that are less affected by galactic foregrounds. Still, foregrounds and instrumental systematic effects, such as the beam asymmetry and imperfect scanning of the sky, play an important role.

CPR is manifested in different types of correlations of CMB observables, depending on whether the rotation angle is uniform or varies across the sky. In either case, the rotation converts some of the E-modes into B-modes, generating a contribution to the B-mode power spectrum. A similar B-to-E conversion also takes place but is negligible, as the primordial B-modes are constrained to be subdominant. A uniform rotation angle leads to parity-odd spectra of EB and TB type. An anisotropic rotation, on the other hand, introduces mode-coupling that leads, in particular, to non-trivial 4-point correlations Kamionkowski (2009); Yadav et al. (2009); Gluscevic et al. (2009, 2012). A detection of CPR would signal new physics beyond the standard models of cosmology and particle physics, and has become an ancillary target of the CMB polarization experiments Xia et al. (2008a); Feng et al. (2006); Ade et al. (2015b); di Serego Alighieri (2015); Gluscevic et al. (2012); Gruppuso et al. (2016, 2012); Gubitosi and Paci (2013); Mei et al. (2015); Molinari et al. (2016); Aghanim et al. (2016); Contreras et al. (2017); Ade et al. (2017). The current upper bound on the constant rotation is $\sim 0.5$ deg Molinari et al. (2016); Contreras et al. (2017). Constraints on the amplitude of the scale-invariant anisotropic rotation spectrum (defined in Eq. (15)) are currently on the order of $0.1$ deg2 Contreras et al. (2017); Ade et al. (2017). As we will show, future CMB experiments, such as LiteBIRD Matsumura et al. (2016), Simons Observatory Aguirre et al. (2018), a CMB-S4-like experiment Abazajian et al. (2016) and PICO Hanany et al. (2019), will improve these bounds by orders of magnitude.

The prospect of accurate measurements of CPR presents an opportunity for probing physics beyond the standard model, such as parity violating axion-photon interactions Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978). These interactions result in different travel speeds of the two photon spin states, causing CPR Harari and Sikivie (1992); Carroll et al. (1990); Carroll (1998); Pospelov et al. (2009). Axion-like parity-violating terms have been discussed in the context of inflation Freese et al. (1990), quintessence Wetterich (1988); Frieman et al. (1995); Carroll (1998); Kaloper and Sorbo (2006); Dutta and Sorbo (2007); Liu et al. (2006); Abrahamse et al. (2008); Lee et al. (2014), baryogenesis Alexander (2016) and neutrino number asymmetry Geng et al. (2007). More generally, CPR probes violations of Lorentz invariance that could emerge in theories of quantum gravity Kostelecký and Samuel (1989), unconventional fields Kostelecký et al. (2003); Bertolami et al. (2004) and theories involving noncommutative spacetime Hayakawa (2000); Carroll et al. (2001). A general self-consistent description of Lorentz violation is provided by the Standard-Model Extension (SME) Colladay and Kostelecký (1997, 1998); Kostelecký (2004). Faraday Rotation (FR) by cosmic magnetic fields is another mechanism Kosowsky and Loeb (1996), where the rotation has a characteristic frequency dependence. As we will show, upcoming and future CMB experiments will significantly improve bounds on primordial magnetic fields (PMF) Yadav et al. (2012a); De et al. (2013); Pogosian (2014); Pogosian and Zucca (2018), providing an important observational handle on theories of inflation and the high energy universe Subramanian (2016).

The rest of the paper is organized as follows. We review the CPR formalism, the relevant observables and systematic effects in Section II. The forecasts for the future CMB polarization experiments, along with brief review of the current bounds, are presented in Section III. We elaborate on implications of improved bounds on CPR for fundamental physics in Section IV. We conclude with a summary in Section V.

II Rotation of CMB polarization

Depending on the underlying physical mechanism (see Section IV), the CPR angle $\alpha$ could be a constant or a function of the line of sight, $\hat{n}$. In this Section, we briefly review the estimator used for both constant and anisotropic CPR.

CMB polarization maps are commonly separated into the so-called E- and B-modes Kamionkowski et al. (1997a, b); Seljak and Zaldarriaga (1997), which are the parity-even and parity-odd patterns of the polarization vector, which we will simply referr to as E and B. While E-modes are produced by Thomson scattering from intensity gradients at first order in cosmological perturbation theory, generating B requires sources with parity-odd components, such as gravitational waves Crittenden et al. (1993), topological defects Seljak et al. (1997) or magnetic fields Seshadri and Subramanian (2001). The weak lensing (WL) of CMB by large scale structures turns some of the E into B, generating the signal measured by POLARBEAR Ade et al. (2014a) and SPT Keisler et al. (2015).

The CPR converts111The CPR discussed in this paper is restricted to rotation of linear polarization along the line of propagation. We do not consider circular polarization which is not expected to be present in the CMB Montero-Camacho and Hirata (2018). E into B, as well as B into E, although the latter effect is small enough to be ignored for very small rotation angles. Expanding $\alpha(\hat{n})$ into spherical harmonics, $\alpha(\hat{n})=\sum_{LM}\alpha_{LM}Y_{LM}(\hat{n})$, the relation between the spherical expansion coefficients of the underlying E- and the induced B-mode can be written as Kamionkowski (2009); Gluscevic et al. (2009)

[TABLE]

where $\xi_{lml^{\prime}m^{\prime}}^{LM}$ and $H_{ll^{\prime}}^{L}$ are related to Wigner $3$-$j$ symbols:

[TABLE]

and the summation is restricted to even $L+l^{\prime}+l$. In contrast, the WL conversion of E into B Hu and Okamoto (2002) couples the odd sums of the modes, making it orthogonal to the CPR effect. Eq. (1) also applies to the case of a constant CPR angle, in which case all $\alpha_{LM}$ are zero, except for $\alpha_{00}$.

II.1 The mode-coupling estimator of the rotation

For an anisotropic $\alpha(\hat{n})$, Eq. (1) implies correlations between different multipoles of E and B. Since the CMB temperature (T) and E are correlated, CPR also correlates T with B. The rotation angle can be extracted from EB and TB correlations Kamionkowski (2009); Gluscevic et al. (2009); Yadav et al. (2009). Given measurements of B and E in frequency channels $i$ and $j$, respectively, the quantity

[TABLE]

provides an unbiased estimator of $\alpha_{LM}$ Pullen and Kamionkowski (2007); Yadav et al. (2009); Gluscevic et al. (2009, 2012); Yadav et al. (2012b). Note that $[{\hat{\alpha}}_{B^{i}E^{j},LM}]_{ll^{\prime}}$ is not symmetrical under interchange of $l$ and $l^{\prime}$, and one should separately consider contributions from BE and EB correlations. Analogous quantities can also be constructed from products of T and B. Hence, given maps of T, E and B from a number of channels (labeled by indices $i,j$), one considers contributions from all quadratic combinations

[TABLE]

The minimum variance estimator ${\hat{\alpha}}_{LM}$ is obtained by combining estimates from all $A$, accounting for the covariance between them.

The variance in ${\hat{\alpha}}_{LM}$ was derived in Gluscevic et al. (2009). For a statistically isotropic CPR, it is defined as $\langle{\hat{\alpha}}^{*}_{LM}{\hat{\alpha}}_{L^{\prime}M^{\prime}}\rangle=\delta_{LL^{\prime}}\delta_{MM^{\prime}}[C_{L}^{\rm\alpha}+\sigma^{2}_{{\rm\alpha},L}]$, where $C_{L}^{\rm\alpha}$ is the CPR power spectrum that receives contributions from the sources of rotation (such as birefringence), while $\sigma^{2}_{{\rm\alpha},L}$ is the combined variance of individual estimators $[{\hat{\alpha}}_{B^{i}E^{j},LM}]_{ll^{\prime}}$. Using a notation similar to that in Gluscevic et al. (2009), we can write

[TABLE]

where the sum is restricted to even $l+l^{\prime}+L$, $G^{L}_{ll^{\prime}}\equiv(2l+1)(2l^{\prime}+1)(H^{L}_{ll^{\prime}})^{2}/\pi$, $A$ and $A^{\prime}$ label the relevant quadratic combinations of E, B and T listed in (8),

[TABLE]

with $X$ denoting either $T$ or $E$, and $W^{ij}_{ll^{\prime}}\equiv\exp[-(l^{2}+l^{\prime 2})\theta^{2}_{ij}/16\ln 2]$ accounts for the finite width of the beam. We take $\theta_{ij}=\max[{\theta^{i}_{\rm fwhm},\theta^{j}_{\rm fwhm}}]$, where $\theta^{i}_{\rm fwhm}$ is the full-width-at-half maximum (FWHM) of the Gaussian beam of the $i$-th channel.

The covariance matrix elements, $[{\cal C}^{ll^{\prime}}]_{AA^{\prime}}$, are

[TABLE]

with $X$ and $Y$ standing for either E or T, and

[TABLE]

is the measured spectrum that includes the primordial contribution $C^{XY,{\rm prim}}_{l}$, the WL contribution $C^{XY,{\rm WL}}_{l}$, the systematic effects $C^{XY,{\rm sys}}_{l}$, and $N^{X^{i}Y^{j}}_{l}$ that includes detector noise, assumed to be uncorrelated between the channels, and the residual contribution of galactic and atmospheric foregrounds. The de-lensing fraction $A_{\rm L}$ is introduced to account for the partial subtraction of the WL contribution. According to Hirata and Seljak (2003), the quadratic estimator method of Hu and Okamoto (2002) can reduce the WL contribution to ${\tilde{C}}_{l}^{BB}$ by a factor of $7$ (implying $A_{\rm L}=0.14$), with iterative methods promising a further reduction Hirata and Seljak (2003).

The signal to noise ratio (SNR) of the detection of the CPR spectrum $C_{L}^{\rm\alpha}$ is given by

[TABLE]

The variance in the rotation estimator is the lowest at small $L$, since a rotation that is uniform over a large patch of the sky affects many E-modes in the same way. Hence, the main contribution to the SNR tends to come from the smaller $L$ modes, at least for the rotation spectra that are close to being scale-invariant.

As will be discussed in Sec. IV, two special cases are of special interest: that of the scale-invariant rotation spectrum Pospelov et al. (2009), and that of the uniform rotation angle Carroll (1998). The amplitude of the scale-invariant spectrum can be conveniently described by a constant parameter $A_{\alpha}$,

[TABLE]

For a noise-dominated measurement, i. e when $C_{L}^{\alpha}<\sigma^{2}_{\alpha,L}$ for all $L$, the SNR is proportional to $A_{\alpha}$. However, if $C_{L}^{\alpha}$ is larger than the variance for $L<L_{S}$, the contribution of these $L$ to the SNR is

[TABLE]

where $L_{S}$ is found by setting $C_{L_{S}}^{\alpha}=\sigma^{2}_{\alpha,L_{S}}$. For a scale-invariant spectrum, this implies $L_{S}(L_{S}+1)\approx L^{2}_{S}\approx 2\pi A_{\alpha}/\sigma^{2}_{\alpha,L}$, or $L_{S}\approx\sqrt{2\pi A_{\alpha}}/\sigma_{\alpha,L_{S}}$. This leads to

[TABLE]

and implies a linear dependence of the SNR on the rotation angle, which is expected, since the rotation estimator can be used to directly reconstruct the rotation map. This translates into a linear dependence of the SNR on the parameters of the underlying theory for the cause of the rotation, such as the axion decay constant or the strength of the primordial magnetic field, which makes the CPR a sensitive probe of fundamental physics.

The case of the uniform rotation angle is a sub-case of a general rotation, with Eq. (9) for the combined variance in the mode-coupling estimator remaining valid for $L=0$. For a full sky, the $00$th multipole of the rotation angle is related to the angle via $\alpha_{00}=\sqrt{4\pi}\alpha$. For a partial sky coverage, the variance in a uniform $\alpha$ is given by

[TABLE]

where $\sigma^{2}_{{\rm\alpha},0}$ is given by Eq. (9) with $L=0$. The uniform rotation generates parity-odd angular spectra of EB and TB type:

[TABLE]

As explicitly shown in Gluscevic et al. (2009), the signal-to-noise of a detection of a uniform $\alpha$ (using the variance in (18)) is equivalent to the signal-to-noise of detection of the EB and TB spectra. We will use (18) in our forecasts as it includes the covariance of multiple frequency channels.

II.2 The B-mode spectrum induced by CPR

In addition to mode-coupling correlations, CPR induces a contribution to the B-mode spectrum. Ignoring the effects of the finite width of the last scattering surface, the B-mode spectrum induced by CPR with a spectrum $C_{L}^{\alpha}$ is

[TABLE]

The signal-to-noise in detecting CPR from the measurement of the BB spectrum is given by

[TABLE]

where the covariance $\tilde{C}_{l}^{BB}$ includes instrumental noise, the systematic effects and the WL contribution.

For a constant CPR angle $\alpha$, we have

[TABLE]

and the signal-to-noise becomes

[TABLE]

Note that the signal-to-noise in BB is quadratic in $\alpha$, while in the case of EB and TB correlations it is essentially linear in CPR. Thus, given a CMB experiment of sufficiently low noise and high resolution, the latter offer a more sensitive probe of the CPR than the B-mode spectrum. This is true for both constant and anisotropic rotation.

II.3 Beam systematics

Optical imperfections in the telescope itself, known as “beam systematics”, are capable of generating spurious CPR Shimon et al. (2008); Pagano et al. (2009); Gubitosi and Paci (2013); Su et al. (2009, 2011) due to the leakage of power from the “standard” correlations of TT, TE and EE type. Beam systematics generate non-zero parity-odd angular spectra $C^{EB}_{l}$ and $C^{TB}_{l}$ in addition to contributing to all parity-even spectra, including $C^{BB}_{l}$. Their multipole-dependence can be modelled Shimon et al. (2008); Su et al. (2011), allowing one to partially separate this non-cosmological signal from the data. The separation cannot be perfect because contributions from the systematics increase the variance and because of the uncertainties associated with the beam model. Moreover, the pixel-rotation systematic effect caused by the misalignment of the telescope is fully degenerate with a uniform (across the sky) CPR angle222In fact, measured TB and EB correlations are sometimes used to correct for the misalignment of the telescope based on the assumption that the underlying CPR is vanishing Yadav et al. (2012b); Keating et al. (2012); Kaufman et al. (2014)..

In our forecasts, we use the parameterized forms of $C^{XY,{\rm sys}}_{l}$ derived in Shimon et al. (2008) for the spurious contributions to CMB spectra caused by the differential pointing, differential ellipticity, and differential rotation in a dual polarized beam. In the formalism of Shimon et al. (2008), these three systematic effects are controlled by the corresponding parameters $\rho$, $e$, and $\epsilon$, which, for simplicity, were assumed to be independent of $l$. Under this simplifying assumption, beam systematics contribute to Eq. (12) only at $l=l^{\prime}$. For each experiment, we evaluate $C^{XY,{\rm sys}}_{l}$ assuming the uncertainties in $\rho$, $e$ and $\epsilon$ are reduced to sufficiently low levels that allow the experiment to achieve its scientific targets, i.e. to exhaust its nominal capacity to detect B-mode polarization. Specifically, two requirements must be satisfied:

Beam systematics should not reduce the experiment’s ability to measure $r$; and 2. 2.

Beam systematics should allow the experiment to measure the lensing B-mode on relevant scales, i.e. we require that $C^{BB,{\rm sys}}_{l*}<N^{BB}_{l*}$ for $l<l_{*}$, where $l_{*}=1150$ corresponds to the peak of the WL spectrum.

When modelling the differential pointing effect, we assume that one of the beams has no pointing error, while the other beam has a pointing error $\rho$. The angle $\theta$ of the second beam is a free parameter that we fix at $\theta=45^{\circ}$. Our calculation of the quadrupole effect assumes that the two beams have the same ellipticity $e$, and the angles that the polarization axes make with the major axes of the two beams are taken to be $\psi_{1}=45^{\circ}$ and $\psi_{2}=0$. The values of $\rho$, $e$ and $\epsilon$, derived for each experiment under the above assumptions are given in Table 1, and we refer the reader to Shimon et al. (2008) for the complete description of the beam model.

III Bounds on the rotation of CMB polarization

Using the formalism presented in the previous section, we perform a forecast of expected bounds on the CPR for the following upcoming and proposed experiments:

• •

Lite (Light) satellite for the studies of B-mode polarization and Inflation from cosmic background Radiation Detection (LiteBIRD) Matsumura et al. (2016) – a proposed small satellite observatory, with channels covering a wide range of frequencies, targeting B-modes in the $2<\ell<200$ range, and aiming to constrain the tensor-to-scalar ratio $r$ at a level of $\sigma_{r}=0.001$. In our forecasts, we only include channels in the 60-150 GHz range, where the galactic foregrounds are relatively weak. For LiteBIRD, we assume that the residual foreground contribution is equal to the noise in the lowest noise channel;

• •

Simons Observatory (SO) Aguirre et al. (2018), a ground-based experiment currently under development, consisting of one 6 meter Large Aperture Telescope (LAT) and three 0.5 meter Small Aperture telescopes (SAT), aiming to achieve $\sigma_{r}=0.003$. For SO, we assume the SAT parameters and the forecasted noise curves from Aguirre et al. (2018)666The SO noise curves are available at https://simonsobservatory.org/assets/supplements/20180822_SO_Noise_Public.tgz that include modelling of atmospheric and galactic foregrounds;

• •

a Stage IV ground based experiment like CMB-S4 Abazajian et al. (2016) covering 40% of the sky at 95 and 150 GHz with $\sim 1$ arcmin resolution and noise levels of $\sim 1$ $\mu$K-arcmin. For CMB-S4-like, we assume that the residual foreground contribution is equal to the noise in the lowest noise channel;

• •

Probe of Inflation and Cosmic Origins (PICO) Hanany et al. (2019), a proposed mid-cost space mission mapping the full sky using multiple channels covering a wide range of frequencies at a resolution of a several arcmin and noise levels of $\sim 1$ $\mu$K-arcmin. In our forecasts, we consider channels in the 60-150 GHz range and use the forecasted noise curves from Hill that were used in Hanany et al. (2019) and include modelling of atmospheric and galactic foregrounds using the methodology described in Aguirre et al. (2018).

The parameters assumed for each of the experiments are summarized in Table 1.

We consider rotation by both a uniform and an anisotropic rotation angle, quantifying the results in terms of the expected 68% confidence level (CL) bounds on the following parameters:

• •

the constant rotation angle $\alpha$, in arcmin (′);

• •

the amplitude of the scale invariant rotation spectrum $A_{\alpha}$ of Eq. (15), in deg2;

• •

the quadrupole moment of the rotation, $\sqrt{C_{2}^{\alpha}/4\pi}$, in arcmin, assuming a scale-invariant spectrum:

[TABLE]

In addition, we plot the statistical uncertainty in $C_{L}^{\alpha}$, under the assumption of no CPR,

[TABLE]

as a function of $L$, in deg2. Before presenting the forecasts, we briefly review the current bounds on CPR.

III.1 Current bounds on CPR

The current bound on the uniform rotation angle is $\alpha<0.5^{\circ}$ at 68% CL derived by Planck Aghanim et al. (2016) from the upper limit on parity-odd two-point correlations of EB and TB type (see also Xia et al. (2010); Gruppuso et al. (2016); Molinari et al. (2016); Contreras et al. (2017)). It improved on the 68% CL bound of $\alpha<1.5^{\circ}$ from WMAP7 Komatsu et al. (2011).

The existing constraints on the anisotropic rotation are based on the assumption of a scale-invariant rotation spectrum. The present bound is $A_{\alpha}<0.07$ deg2 at 95% CL obtained in Contreras et al. (2017) using a pixel based approach to directly estimate the rotation angle on local patches of the Planck polarization maps. According to the scaling in Eq. (17), the corresponding 68% CL bound would be approximately $0.02$ deg2. Expressed in terms of the quadrupole anisotropy, the bound is $\sqrt{C_{2}/4\pi}<5^{\prime}$ at 95% CL or approximately $3^{\prime}$ at 68% CL. A comparable bound, $A_{\alpha}<0.11$ deg2 at 95% CL, was obtained by BICEP2/Keck Ade et al. (2017) using the mode coupling estimator introduced in the previous section. Prior bounds on $A_{\alpha}$ were also derived from WMAP7 Gluscevic et al. (2012) ($A_{\alpha}<12$ deg2 at 68% CL) and POLARBEAR Ade et al. (2015b) ($A_{\alpha}<1$ deg2 at 95% CL). Bounds on $A_{\alpha}$ from POLARBEAR and SPTPol B-mode spectra were also derived in Liu and Ng (2017). The present day bounds are summarized in Table 2.

III.2 Forecasted CPR bounds

Table 2 summarizes the forecasted bounds on the uniform and the anisotropic CPR expected from the four experiments considered in this work.

The ability of a given experiment to constrain CPR is determined primarily by its resolution and the effective noise that includes the residual foreground contributions. Specifically, an optimal experiment for detecting a scale-invariant rotation spectrum would have the resolution to measure most of the $\ell$-modes around the peak of the E-mode spectrum, or $500\lesssim\ell\lesssim 3000$. Having better resolution does not significantly improve constraints on the rotation simply because there is less power in the polarization on smaller scales. However, if the rotation spectra were not scale-invariant but had a significant blue tilt, with most of the power on small scales, having polarization measurements at a higher resolution could be beneficial. We leave investigation of this latter possibility for future work.

From Table 2 one can see that LiteBIRD would lower the bounds on CPR by an order of magnitude, while the Simons Observatory will lower them by two orders. Both CMB-S4-like and PICO are capable of improving them by yet another order of magnitude, with PICO being somewhat more sensitive to CPR thanks to the lower detector noise.

In Fig. 1 we plot the forecasted statistical uncertainty in the rotation spectrum $C_{L}^{\alpha}$ given by Eq. (25). The curves take into account the contribution of beam systematics, and assume partial de-lensing by a fraction $f_{L}$, given for each experiment in Table 1. The plot also shows the forecasted 68% CL bounds on the amplitude of the scale-invariant rotation spectrum $A_{\alpha}$ (the horizontal lines) for each experiment, along with the current bound on $A_{\alpha}$ from BICEP2/Keck Ade et al. (2017).

IV Implications for fundamental physics

IV.1 A pseudoscalar field coupled to electromagnetism

A number of well-motivated extensions of the standard model involve a (nearly) massless axion-like pseudoscalar field coupled to photons via the Chern-Simons (CS) interaction term. The relevant contribution to the Lagrangian can be written as

[TABLE]

where $F_{\mu\nu}$ is the electromagnetic field strength, $\tilde{F}_{\mu\nu}$ is its dual, $a({\bf x},\eta)$ is the pseudoscalar field, $f_{a}$ is the axion decay rate, and $m_{a}$ is the axion mass which is either zero or constrained to be very small. One should think of $a$ as being the phase of a complex scalar field with a spontaneously broken $U(1)$ symmetry, i.e. a (pseudo-) Goldstone boson, with the value of $f_{a}$ set by the symmetry breaking scale. Axions were first introduced in the context of the QCD Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978) as a solution of the strong CP violation problem ’t Hooft (1976). Axion-like fields are ubiquitous in string theory Svrcek and Witten (2006); Fox et al. (2004) and can be relevant in developing models of inflation Freese et al. (1990), quintessence Wetterich (1988); Frieman et al. (1995); Carroll (1998); Kaloper and Sorbo (2006); Dutta and Sorbo (2007); Abrahamse et al. (2008), baryogenesis Alexander (2016) and neutrino number asymmetry Geng et al. (2007). We refer the reader to Marsh (2016) for a recent review of cosmological implications of axion-like fields.

The parity-violating term in (26) makes the right- and left-handed polarization states propagate at different velocities,

[TABLE]

where the vector-potential is decomposed into $A_{\pm}=A_{x}\pm iA_{y}$, a phenomenon known as birefringence. This causes a rotation of the linear polarization of an electromagnetic wave as it propagates Harari and Sikivie (1992). If the wavelength of the radiation is much smaller than the typical scale over which $a$ varies, the rotation angle is independent of the wave’s frequency and is given by $\Delta\alpha=\Delta a/f_{a}$, where $\Delta a$ is the net change in $a$ along the photon’s trajectory Harari and Sikivie (1992); Carroll et al. (1990); Carroll (1998); Pospelov et al. (2009). In order to produce any rotation of the CMB polarization, the axion mass must be smaller than the Hubble scale at decoupling,

[TABLE]

otherwise, the axion will start oscillating around the minimum of the potential giving $\Delta a=0$. Note that the same criterion prevents $a$ from being the dark matter, since being a matter particle of relevance to structure formation requires it to start oscillating prior to decoupling.

A uniform CPR angle is possible if a) $\dot{a}$ is non-zero between the time of decoupling and today, and b) the average value of $a$ is non-zero at decoupling. The first of these conditions requires the mass to be sufficiently large for $a$ to be dynamical between the decoupling and today Carroll (1998), namely, $m_{a}>H_{0}\sim 10^{-33}$ eV. The second condition requires the value of $a$ to be uniform across the universe, which would be the case if its value was set during or prior to inflation. More specifically, the initial value of $a$ is set randomly at the time of the $U(1)$ symmetry breaking. If inflation happened at a scale $H_{I}>f_{a}$, the sky-averaged value of $a$ would be zero, as it would correspond to averaging over its value in many causally disconnected parts of the universe. On the other hand, if $f_{a}\geq H_{I}$, our observable universe would originate from the same patch that was causally connected at the time of symmetry breaking and the initial value of $a$ would be uniform across the sky. Hence, having an observable uniform CPR angle requires $f_{a}>H_{I}$ in addition to $10^{-33}{\rm eV}\lesssim m_{a}\lesssim 10^{-28}$ eV.

As discussed in Sec. II.1, a uniform CPR angle would manifest itself in non-zero $C_{\ell}^{TB}$ and $C_{\ell}^{EB}$ Lue et al. (1999) and imply a global violation of parity in the universe. As our forecasts have shown, future experiments will improve the sensitivity to a constant CPR by over two orders of magnitude. In the context of axion-like fields, they will bound $\dot{a}/f_{a}$ and translate into constraints on a combination of $m_{a}$ and $f_{a}$ that would be complimentary to those from axion dark matter searches. In particular, a detection of a uniform CPR angle would imply a non-zero axion mass. We further discuss the uniform rotation angle in Sec. IV.3 in the context of a general framework of searching for Lorentz violating extensions of the standard model.

Generally, the pseudoscalar $a$ would vary in space and time, with the spatial distribution largely determined by whether the symmetry breaking scale is above or below that of inflation. If $f_{a}<H_{I}$, then $a({\bf x},\eta)$ is expected to be uncorrelated on scales larger than the horizon size at the time of the symmetry breaking, implying a blue rotation spectrum on scales probed by CMB experiments with a cut off at an extremely high value of $L$. Such a CPR spectrum would have practically no power at low $L$ and would be undetectable. However, the breaking of the $U(1)$ symmetry would also produce global cosmic strings Vilenkin and Everett (1982) which would remain topologically stable up to the epoch corresponding to the very small axion mass scale $m_{a}$ (which could be zero). A scaling network of axion strings would act as a continuous source of perturbations, sourcing axion fluctuations on scales corresponding to the horizon at any given time Yamaguchi et al. (1998); Klaer and Moore (2017); Gorghetto et al. (2018); Kawasaki et al. (2018). Detailed properties of such a spectrum and its effect on the CPR could be a subject of a future investigation.

Of special interest is the case when $f_{a}>H_{I}$, in which case stochastic fluctuations in the pseudoscalar field would be generated during the period of inflation Lyth (1992); Lyth and Stewart (1992); Fox et al. (2004). This would result in a scale-invariant spectrum of the CPR angle with an amplitude

[TABLE]

Thus, an upper bound on $A_{\alpha}$ implies a lower bound on the coupling scale $f_{a}$. In Pospelov et al. (2009), the authors studied the CMB B-mode spectrum generated by such a CPR (see Eq. (20)) and derived a 95% CL upper bound of $A_{\alpha}<4.2\times 10^{-3}\ {\rm rad}^{2}=13.8\ {\rm deg}^{2}$ from the upper bound on the BB spectrum from QUaD Pryke et al. (2009), implying $f_{a}>2.4\times 10^{14}\ {\rm GeV}\times H_{14}$, where $H_{14}=H_{I}/10^{14}$GeV. As discussed in Section II, given a CMB experiment of sufficiently low noise and high resolution, the mode-coupling EB and TB correlations offer a more sensitive probe of the rotation angle compared to the BB spectrum. The present 95% CL bound on $A_{\alpha}$ from BICEP2/Keck Ade et al. (2017) is $0.11$ deg2, corresponding to $f_{a}>2.7(5.3)\times 10^{15}\ {\rm GeV}\times H_{14}$ at 95% (68%) CL.

The current and future 68% CL lower CMB bounds on $f_{a}$ are shown in Table 3. They are significantly tighter than those obtained from astrophysical probes of pseudoscalar interactions Galaverni et al. (2015); Kamionkowski (2010); Loredo et al. (1997), assuming that the inflationary scale is not significantly below $14$ GeV. Generally, low-mass particles such as neutrinos and axions would be produced in the interior of stars, and stellar constraints typically require $f_{a}>10^{11}$ GeV Raffelt (1999). The bound obtained by the CERN Axion Solar Telescope (CAST) experiment, which searched for the direct emission of pseudoscalars from the solar interior, is $f_{a}>2\times 10^{10}$ GeV Andriamonje et al. (2007). The bounds from laboratory experiments, such as the Polarization of Vacuum with LASer (PVLAS) experiment Della Valle et al. (2016), are significantly weaker than those from astrophysics.

Experiments such as CMB-S4-like and PICO are able to probe $f_{a}\sim{\rm a\ few}\times 10^{17}\ {\rm GeV}\times H_{14}$, in the range close to the Planck scale of $10^{19}$ GeV. In particular, this would exclude the range of $f_{a}\sim 10^{16}\ {\rm GeV}\times H_{14}$ that is of most interest for string theory, leading to non-trivial bounds on the string theory axions Svrcek and Witten (2006); Pospelov et al. (2009) and implementations of inflation in the related models.

IV.2 Faraday Rotation by a Primordial Magnetic Field

The origin of micro-Gauss ($\mu$G) strength galactic magnetic fields is one of the long standing puzzles in astrophysics Widrow (2002). Producing them with a dynamo mechanism requires a seed field of a certain minimum strength Widrow et al. (2012). Adding to the puzzle is the presence of $\mu$G strength fields in proto-galaxies too young to have gone through the number of revolutions necessary for the dynamo to work Athreya et al. (1998). There is also preliminary evidence for lower limits on primordial magnetic fields (PMF) from observations of cosmic rays for magnetic fields in the intergalactic space coherent over cosmological distances Neronov and Vovk (2010); Takahashi et al. (2012); Dolag et al. (2011); Tavecchio et al. (2011); Taylor et al. (2011); Vovk et al. (2012). PMFs could have been generated in the aftermath of phase transitions in the early universe Vachaspati (1991), during inflation Turner and Widrow (1988); Ratra (1992), or at the end of inflation Diaz-Gil et al. (2008). Once produced, they would be sustained by the primordial plasma in a frozen-in configuration until the epoch of recombination and beyond leaving potentially observable imprints in the CMB. Thus, improved constraints on the PMF are valuable tools for discriminating among different theories of the early universe Barnaby et al. (2012); Long et al. (2014); Durrer and Neronov (2013).

A stochastic PMF contributes to the CMB anisotropy through metric perturbations and the Lorentz force exerted on ions in the pre-recombination plasma Seshadri and Subramanian (2001); Mack et al. (2002); Lewis (2004); Finelli et al. (2008); Paoletti et al. (2009); Shaw and Lewis (2010). It also generates Faraday rotation (FR) of CMB polarization converting E modes into B modes Harari et al. (1997); Scoccola et al. (2004); Kahniashvili et al. (2009); Pogosian et al. (2011) and inducing mode-coupling correlations between E, B and T Yadav et al. (2012a); Pogosian (2014). CMB signatures depend on the shape of the PMF spectrum, which in turns is determined by the generation mechanism of the PMF. The originally proposed simple inflationary models of magnetogenesis Turner and Widrow (1988); Ratra (1992) predict a scale-invariant spectrum, although other values are possible in more complicated models Bonvin et al. (2013). Magnetic fields produced in phase transitions after inflation have blue spectra with most power on very small scale. We will focus on the well-motivated case of the scale-invariant PMF Kahniashvili et al. (2017) that is most likely to have observable CPR Pogosian et al. (2011); Yadav et al. (2012a).

It is conventional to quote limits on the PMF in terms of $B_{\lambda}$, which is the magnetic field strength smoothed over a region of comoving size $\lambda$. For a scale-invariant PMF, this measure is independent of $\lambda$, and is the same as the effective PMF strength obtained by taking the square root of the magnetic energy density Mack et al. (2002), so we will quote the bounds in terms of $B_{\rm SI}=B_{\rm eff}=B_{\lambda}$. The current bound, derived from a combination of the 2015 Planck TT, EE, TE spectra Adam et al. (2016) and the SPT B-mode spectrum Keisler et al. (2015) is $B_{\rm SI}<1.2$ nG at 95% CL, or $<1$ nG at 68% CL Zucca et al. (2017). In particular, the measured B-modes by SPT at small scales play an important role, reducing the bound on $B_{\rm SI}$ by a factor of two.

Because the magnetic contribution to CMB spectra scales as $B^{4}_{\rm SI}$, an orders-of-magnitude improvement in the accuracy of the B-mode spectrum would only result in a modest reduction of the bound on $B_{\rm SI}$. In contrast, Faraday Rotation (FR) scales linearly with $B_{\rm SI}$, promising much tighter bounds on the PMF Pogosian and Zucca (2018). At present, such FR-based PMF bounds are not competitive compared to those from CMB spectra, e.g. the POLARBEAR collaboration obtained $B_{\rm SI}<93$ nG at 95% CL Ade et al. (2015b) based on the analysis of mode-coupling EB correlations in their 150 GHz map, but they will improve dramatically with the lower noise and higher resolutions of future experiments.

To extract the FR signal, one can use the rotation angle estimator (7) after accounting for the $\nu^{-2}$ frequency dependence of the FR angle $\alpha(\hat{n})$. Namely, one can use a combination of channels to constrain the frequency independent rotation measure (RM), defined as

[TABLE]

The details of constructing the multi-frequency RM estimator can be found in Pogosian (2014).

A scale-invariant PMF implies a scale-invariant RM spectrum Pogosian et al. (2011), i. e. the quantity

[TABLE]

is constant over the scales of interest and is related to $B_{\rm SI}$ via De et al. (2013)

[TABLE]

The SNR of the detection of the primordial RM spectrum $C_{L}^{\rm RM,PMF}$ is given by

[TABLE]

where $\sigma^{2}_{{\rm RM},L}$ is the variance in the RM estimator analogous to $\sigma^{2}_{{\alpha},L}$ that takes into account de-lensing and beam systematics, and $f_{\rm G}$ is the fraction of the Milky Way RM spectrum $C_{L}^{\rm RM,G}$ that may be known from other sources and can be subtracted. We use estimates of $C_{L}^{\rm RM,G}$ from De et al. (2013) based on the galactic RM map of Oppermann et al. (2012).

Table 4 shows the 68% CL bounds on the scale-invariant PMF expected from the FR measurements, and compare them to the bounds one would obtain by constraining the (non-FR) vector and tensor mode contributions of the PMF to the BB spectrum. As one can see, while the BB based constraints are stronger today, they will not significantly improve on the present $1$ nG bound. On the other hand, the FR based estimates will eventually do better, thanks for the linear scaling of the SNR with the PMF strenght.

Importantly, experiments like CMB-S4 and PICO can achieve bounds on the PMF strength $\sim 0.1$ nG, which is a critical threshold for ruling out the purely primordial (no dynamo) origin of the $\sim 1-10$ $\mu$G galactic magnetic fields. Namely, a $0.1$ nG field coherent over a $1$ Mpc size region would be adiabatically compressed into a $\sim 1$ $\mu$G field in the galactic halo Grasso and Rubinstein (2001).

The FR caused by a $\sim 0.1$ nG PMF is approximately the same as that due to the magnetic field in the Milky Way near the galactic poles De et al. (2013). Thus, lowering the FR based bound on the PMF below $0.1$ nG would require an independent measurement of the galactic RM. This should be possible in the future with improved versions of the galactic RM maps Oppermann et al. (2012) based on studies of extragalactic radio sources. Regardless of that, experiments like CMB-S4 and PICO will have the sensitivity to use FR to probe the magnetic field in our galaxy. Since FR probes the line-of-sight component of the magnetic field, it is complementary to studies using synchrotron radiation which probe the transverse component.

IV.3 Model-independent constraints on Lorentz-violating physics

The last two decades have seen a resurgence of interest in tests of Lorentz invariance due, in part, to the suggestion that violations of Lorentz invariance could emerge in theories of quantum gravity Kostelecký and Samuel (1989); Kostelecký and Potting (1991, 1995). Of the hundreds of searches for Lorentz violation in particles and in gravity Bluhm (2006); Tasson (2014); Kostelecký and Russell (2011), tests involving astrophysical sources are among the most sensitive since tiny Lorentz-violating defects can accumulate over long propagation times Ellis and Mavromatos (2013). CMB radiation is the oldest light available to observation and provides extreme sensitivity to certain forms of Lorentz violation Feng et al. (2006); Cabella et al. (2007); Kostelecký and Mewes (2007); Xia et al. (2008a, b); Kostelecký and Mewes (2008); Komatsu et al. (2009); Wu et al. (2009); Kahniashvili et al. (2008); Pagano et al. (2009); Brown et al. (2009); Xia et al. (2010); Komatsu et al. (2011); Xia (2012); Gruppuso et al. (2012); Hinshaw et al. (2013); Kaufman et al. (2014); Mei et al. (2015); Galaverni et al. (2015); Zhao et al. (2015); Aghanim et al. (2016); Gruppuso et al. (2016); Molinari et al. (2016).

Tests of Lorentz symmetry are aided by a theoretical framework known as the Standard-Model Extension (SME), which aims at providing a general all-encompassing self-consistent description Lorentz violation in both the Standard Model of particle physics and General Relativity Colladay and Kostelecký (1997, 1998); Kostelecký (2004). The early work on the SME was largely motivated by suggestions that Lorentz invariance may be spontaneously broken in string theory Kostelecký and Samuel (1989); Kostelecký and Potting (1991, 1995). Other possible origins of Lorentz violation include small spacetime variations of physical constants or unconventional fields Kostelecký et al. (2003); Bertolami et al. (2004), theories involving noncommutative spacetime Hayakawa (2000); Carroll et al. (2001) and unconventional coupling to gravity Shore (2005). In the SME, photons are described by the usual Maxwell Lagrangian augmented by an infinite series of Lorentz-violating terms Kostelecký and Mewes (2009),

[TABLE]

Each term in the series gives a different class of Lorentz violation controlled by tensor coefficients ${(k_{AF}^{(d)})_{\kappa}}^{\alpha_{1}\ldots\alpha_{(d-3)}}$ and $(k_{F}^{(d)})^{\kappa\lambda\mu\nu\alpha_{1}\ldots\alpha_{(d-4)}}$. For example, the axion-photon coupling term in Eq. (26) is physically equivalent to $\mathcal{L}_{\text{CS}}=-(\partial_{\kappa}a/2f_{a})\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}F_{\mu\nu}$, yielding a correspondence between the gradient of the axion field and the $d=3$ coefficients for Lorentz violation,

[TABLE]

The label $d=3,4,5,\ldots$ is the mass dimension of the conventional piece appearing with the coefficient, and it is expected that lower-$d$ terms dominate at attainable energies. Consequently, most tests of Lorentz symmetry in photons have focused on the leading-order $d=3$ and $d=4$ violations.

Each term in the Lagrangian (34) leads to vacuum birefringence, which can be tested with extreme precision using polarimetry of astrophysical sources. For $d\geq 4$, the effects on the polarization of light grow with photon energy, and are best constrained using high-energy sources Kostelecký and Mewes (2006). However, the lowest-order $d=3$ term gives energy-independent birefringence, so the CMB provides the ideal source for this class of violations. Lorentz violation of the CS type was first bounded at the level of $10^{-42}\,{\rm GeV}$ three decades ago in a study of polarization in radio galaxies Carroll et al. (1990). Since Lorentz violation generally comes with violations of rotational symmetry, the effects of birefringence are typically direction dependent, and the full-sky CMB can test anisotropic birefringence more effectively than point sources.

The $d=3$ Lorentz violations cause a simple rotation in the linear polarization. Integrating from recombination to today, the CMB polarization rotates about the line of sight $\hat{n}$ by an angle Kostelecký and Mewes (2008)

[TABLE]

where $T\simeq 3.8^{\circ}/10^{-43}$ GeV is the time since recombination in units convenient for studies involving the SME. For convenience, we have expanded the CPR rotation angle $\alpha(\hat{n})$ in spherical harmonics. There are four non-zero spherical coefficients, $k^{(3)}_{(V)00}$, $k^{(3)}_{(V)11}$, $k^{(3)}_{(V)10}$, and k^{(3)}_{(V)1(-1)}=-\big{(}k^{(3)}_{(V)11}\big{)}^{*}, which are linear combinations of the four $d=3$ tensor coefficients $(k_{AF}^{(3)})_{\kappa}$ (note that in the case of the photo-axion coupling, this corresponds to assuming constant gradients in Eq. (35)). While anisotropic birefringence in the CMB has been considered by a number of researchers Kamionkowski (2010); Li and Yu (2013); Zhao and Li (2014); Li et al. (2015a, b); Ade et al. (2015b, 2017), relatively few constraints exist on the three $l=1$ coefficients describing the potential dipole anisotropy in the CPR angle (but see e.g the analysis of the 2003 BOOMERANG data in Kostelecký and Mewes (2007)). A dipole in the CPR angle was recently measured in the analysis of Planck data Contreras et al. (2017), where it was expressed using the form $\alpha(\hat{n})=A_{1}\,\hat{n}\cdot\hat{N}$, where $A_{1}$ is the maximum $\alpha$ and $\hat{N}$ is the direction at which maximum rotation occurs. Ref. Contreras et al. (2017) reports an amplitude of $A_{1}=0.32^{\circ}\pm 0.10^{\circ}\pm 0.08^{\circ}$ and a direction $\hat{N}$ at galactic coordinates $l=295^{\circ}\pm 22^{\circ}\pm 5^{\circ},b=17^{\circ}\pm 17^{\circ}\pm 16^{\circ}$. Neglecting the covariance between the parameters, the corresponding 1$\sigma$ bounds on anisotropic SME coefficients are

[TABLE]

representing an improvement of more than two orders of magnitude over previous bounds.

As one can see from Table 2, the bounds on the quadrupole of anisotropic rotation will improve by a factor of 3 with LiteBIRD, a factor of 10 with SO, a factor of 30 with CMB-S4-like and a factor of 60 with PICO. Correspondingly, a comparable improvement is expected for the dipole contribution and the corresponding SME coefficients.

The remaining coefficient for Lorentz violation $k^{(3)}_{(V)00}$ yields a uniform rotation by angle

[TABLE]

thus, the bounds on uniform CPR from Table 2 can be readily converted into bounds on $k^{(3)}_{(V)00}$. The current bound from Planck Aghanim et al. (2016) limits $k^{(3)}_{(V)00}$ to a few $\times 10^{-44}\,{\rm GeV}$. The projected constraints from LiteBIRD, SO, CMB-S4-like and PICO are given in Table 5. The sub-arcminute sensitivity expected in Stage IV experiments should yield sensitivities better than $10^{-46}\,{\rm GeV}$, representing at least a hundred-fold improvement in our ability to test Lorentz violation.

Overall, the future bounds on CPR will not just improve the CMB bounds on Lorentz violation, but will provide the best overall constraints on the $d=3$ CPT and Lorentz violation in photons, improving on the original Carroll, Field and Jackiw result Carroll et al. (1990) by four orders of magnitude!

V Conclusions

Using the primordial universe as a probe of fundamental physics is not a new idea. Yet, until now, such measurements were beyond the reach of practical investigation. Now, upcoming and future CMB experiments will dramatically improve our ability to constrain cosmic polarization rotation, opening opportunities for probing both conventional aspects of fundamental physics, and so-called “physics beyond the standard model”. In particular, as we have shown, these results will significantly improve the bounds on axion-photon coupling, coming close to excluding the entire class of string theory axions. More generally, they will put new stringent bounds on Lorentz violation in the universe, with important implications for model-building in high energy physics and quantum gravity. Crucially, these bounds are of a completely complementary nature to laboratory-based probes, which will add confidence if these exotic effects are ever discovered.

Experiments like CMB-S4 and PICO will improve bounds on primordial magnetic fields, achieving constraints close to the critical threshold of $0.1$ nG, which would rule out the purely primordial (i.e., no dynamo) origin of the observed $\mu$G level magnetic fields in galaxies. These observations will also open the possibility to use Faraday Rotation of CMB polarization as a probe the line-of-sight component of the magnetic field in our galaxy, complementing information obtained from the galactic synchrotron radiation that probes the transverse component of the field.

Acknowledgements.

We thank Vera Gluscevic and Eric Hivon for useful discussions and Colin Hill for providing the simulated PICO noise curves. This is not an official paper by the Simons Observatory or the CMB-S4 collaborations. L.P. is supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada. The work of M.S. at Tel Aviv University has been supported by a grant from the Jewish Community Foundation (San Diego, CA). M.M. was supported in part by the United States National Science Foundation under grant number PHY-1819412. This research was enabled in part by support provided by WestGrid and Compute Canada.