Polarization bispectrum for measuring primordial magnetic fields

# Polarization bispectrum for measuring primordial magnetic fields

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###### Abstract

We examine the potential of polarization bispectra of the cosmic microwave background (CMB) to constrain primordial magnetic fields (PMFs). We compute all possible bispectra between temperature and polarization anisotropies sourced by PMFs and show that they are weakly correlated with well-known local-type and secondary ISW-lensing bispectra. From a Fisher analysis it is found that, owing to E-mode bispectra, in a cosmic-variance-limited experiment the expected uncertainty in the amplitude of magnetized bispectra is 80% improved in comparison with an analysis in terms of temperature auto-bispectrum alone. In the Planck or the proposed PRISM experiment cases, we will be able to measure PMFs with strength 2.6 or 2.2 nG. PMFs also generate bispectra involving B-mode polarization, due to tensor-mode dependence. We also find that the B-mode bispectrum can reduce the uncertainty more drastically and hence PMFs comparable to or less than 1 nG may be measured in a PRISM-like experiment.

a,b]Maresuke Shiraishi \affiliation[a]Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy \affiliation[b]INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy \emailAddmaresuke.shiraishi@pd.infn.it

## 1 Introduction

Several cosmological and astrophysical observations support existence of finite magnetic fields in galaxies, cluster of galaxies or large voids (e.g., [1, 2, 3, 4, 5, 6]). There are a variety of studies where these origin is linked with primordial vector field in the very early Universe (e.g., refs. [7, 8, 9, 10]).111At the same time, several papers have also discussed possibilities of magnetic field production in the late-time Universe (e.g., refs. [11, 12, 13, 14, 15]). Despite a fact that such models are strongly constrained by conditions not to contradict inflation or the high energy physics [16, 17, 18, 19, 20], these provide phenomenologically interesting outputs (e.g., refs. [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]).

In this paper, we focus on magnetized non-Gaussian signals in the cosmic microwave background (CMB). Primordial magnetic fields (PMFs) create not only scalar-mode but also vector-mode and tensor-mode CMB anisotropies via energy density and anisotropic stress fluctuations [33, 34, 35, 36, 37, 38, 39, 40, 41]. Recent analyses using CMB power spectra suggest nearly scale-invariant PMFs with strength less than about 3 nG [42, 43, 44, 45]. On the other hand, under an assumption of Gaussianity of PMFs, CMB polyspectra also be generated due to quadratic dependence of the stress fluctuations on Gaussian PMFs [46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. They have diverse shapes unlike CMB bispectra from standard scalar non-magnetized non-Gaussianities since in PMF case the vector-mode and tensor-mode non-Gaussianities can be enhanced [50, 52, 53, 54]. The magnetized bispectrum has provided a new observational constraint on PMFs consistent with bounds from the power spectrum [56].

These previous studies have analyzed effects of temperature auto-bispectrum alone. On the other hand, it is known that polarization bispectra can also help to determine non-Gaussianity parameters [57, 58, 59] and they will be utilized in data analysis of the Planck or the proposed PRISM experiment [60, 61]. In this sense, studying impacts of PMFs on the polarization bispectra will be useful and timely.

On the basis of these motivations, this paper investigates the potential of the polarization bispectra sourced by PMFs. We compute magnetized auto- and cross-bispectra between temperature, E-mode and B-mode anisotropies, and forecast the uncertainty of the amplitude of these bispectra, which depends on PMF strength, via the Fisher analysis. As observations, we assume the Planck and the proposed PRISM experiments. In computation of the CMB bispectra, we consider the dependence on the scalar and tensor modes and ignore the vector mode because of its smallness on scales where we focus on. Then, we confirm that owing to tensor-mode contribution, the polarization bispectra reduce the uncertainty of the magnetized bispectra more drastically in comparison with a forecast from the temperature auto-bispectrum alone. We also find that the existence of local-type non-Gaussianity and secondary ISW-lensing signal does not bias an error estimation of the amplitude of the magnetized bispectra. We follow the formulae and computational procedure in ref. [54].

This paper is organized as follows. In the next section, we analyze signatures of all possible magnetized bispectra composed of the temperature, E-mode and B-mode anisotropies. In section 3, through the Fisher analysis, we discuss the detectability of the magnetized bispectra. The final section is devoted to summary and discussion of this paper. In appendices A and B, we summarize instrumental noise information utilized in section 3 and the uncertainty of the local-type non-Gaussianity.

## 2 Temperature and polarization bispectra originating from primordial magnetic fields

In this section, we examine the dependence of all possible temperature and polarization bispectra generated from PMFs for . The notations and conventions are consistent with refs. [50, 53, 52, 54].

### 2.1 Magnetized CMB fluctuation

Let us start from a cosmological model with large-scale magnetic fields which are created at very early stages of the Universe and stretched beyond horizon by the inflationary expansion. With assumptions of Gaussianity of PMFs and their evolution like radiations: , the PMF power spectrum normalized at the present epoch is given as

 \BraketBi(k)Bj(k′)=(2π)3PB(k)2Pij(^k)δ(k+k′) , (1)

where is a projection tensor which reflects the divergenceless nature of PMFs. The shape of depends strongly on models of primordial magnetogenesis. In order to find observational clues, it is often parametrized as the power-law type:

 PB(k)=ABknB , (2)

where the amplitude depends quadratically on the PMF strength smoothed on as

 AB=(2π)nB+5B2rΓ(nB+32)knB+3r . (3)

PMFs create energy-momentum tensor as

 Ti j(k,τ)≡ργ(τ)[δi jΔB(k)+ΠiBj(k)] ,ΔB(k)=18πργ,0∫d3k′(2π)3Bi(k′)Bj(k−k′) ,ΠiBj(k)=−14πργ,0∫d3k′(2π)3Bi(k′)Bj(k−k′) , (4)

where is energy density of photons. These stress fluctuations can behave as a source of the CMB anisotropies as follows. The first contribution (called passive mode) comes from gravitational interaction via the Einstein equations. In deep radiation dominated era, anisotropic stress fluctuation can enhance superhorizon metric perturbations. After neutrinos decouple, is compensated by anisotropic stress fluctuation of neutrinos and such growth ends. Resulting superhorizon curvature perturbations and gravitational waves are estimated as [40]

 ζ(k)=Rγln(τντB)32O(0)ij(^k)ΠBij(k) ,h(±2)(k)=6Rγln(τντB)12O(∓2)ij(^k)ΠBij(k) , (5)

where , , , and are ratio of divided by total radiation energy density, conformal times of neutrino decoupling and PMF generation, and scalar-mode and tensor-mode projection tensors, respectively [54]. These re-enter horizon just before recombination and generate CMB scalar and tensor fluctuations. Note that vector-mode metric perturbation decays after neutrino decoupling. These passive-mode anisotropies have similar shapes as the CMB fluctuations in non-magnetized standard cosmology [62] since the changes of metric perturbations mentioned above do not affect radiation transfer functions. The second contribution (called compensated mode) is due to Lorentz force at around recombination. The Lorentz force induces baryon velocity via the Euler equations and enhances the CMB scalar and vector fluctuations [33, 35, 36, 38, 40]. Unlike the passive mode, the compensated-mode fluctuations are amplified on small scales and hence they differ from standard CMB patterns. From the analyses of these effects by the CMB power spectra, the PMF strength smoothed on 1 Mpc and the spectral index of the PMF power spectrum have been estimated as and preferred at 95% CL [45].

The tensor and scalar passive modes dominate over the temperature and E-mode fluctuations for [40, 54]. Even in the B-mode fluctuation, the vector compensated mode is hidden by the presence of the tensor passive mode up to . In the following discussions, we are interested in scales which are not so small; therefore we shall take into account the effects of the scalar and tensor passive modes.

### 2.2 CMB bispectra

PMF-induced metric perturbations (5) obey chi-square statistics because of Gaussianity of PMFs. These induce large squeezed-type curvature and tensor bispectra and will be observed as the temperature and polarization bispectra at the present time. In general, these bispectra have very complicated spin and angle dependence due to contraction of , and the bispectrum of [53, 54, 31]. In addition, owing to the dependence of the CMB bispectra on , we are enforced to deal with loop computation. Using a suitable approximation picking up poles, ref. [54] has derived complete formulae applicable to the bispectra composed of not only temperature () but also E-mode () and B-mode () anisotropies.

Computing on the basis of their formalism, we depict the CMB bispectra in figures 1 and 2. Here we distinguish between six parity-even bispectra (, , , , and ) and four parity-odd ones (, , and ) because they are located at completely different multipole configurations, namely and , respectively. The vertical axes express absolute values of reduced bispectra: , where

 \Braket3∏n=1aℓnmn ≡ (ℓ1ℓ2ℓ3m1m2m3)Bℓ1ℓ2ℓ3 , (6) Gℓ1ℓ2ℓ3 ≡ 16[2√ℓ3(ℓ3+1)ℓ2(ℓ2+1)ℓ1(ℓ1+1)−ℓ2(ℓ2+1)−ℓ3(ℓ3+1) (7) ×√∏3n=1(2ℓn+1)4π(ℓ1ℓ2ℓ30−11)+5 perms.⎤⎥⎦.

Note that a relation: holds when [63, 64, 65]. The magnetized bispectrum consists of auto- and cross-correlations between the scalar and tensor anisotropies (i.e., , , , , , , and ). In these figures, we express the magnetized bispectrum composed of every conceivable combination in these eight modes as mode, which means

 total=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩\parbox142.26378pt\flushleft$TTT+TTS+TST+STT$\inner@par$+SST+STS+TSS+SSS$\parbox170.716535pt\flushleft:$\BraketIII$,$\BraketIIE$,$\BraketIEE$,$\BraketEEE$\parbox142.26378pt\flushleft$TTT+TST+STT+SST$\parbox170.716535pt\flushleft:$\BraketIIB$,$\BraketIEB$,$\BraketEEB$\parbox142.26378pt\flushleft$TTT+STT$\parbox170.716535pt\flushleft:$\BraketIBB$,$\BraketEBB$\parbox142.26378pt\flushleft$TTT$\parbox170.716535pt\flushleft:$\BraketBBB$ . (8)

Here the difference in the number of terms by each line is due to a fact that the scalar mode cannot generate the B-mode polarization. We also plot each mode to clarify its contribution to the spectrum.

From these figures, we can confirm that the tensor mode dominates on large scales and the scalar mode catches up with the tensor mode on small scales. These are consistent behaviors with the magnetized power spectra and temperature auto-bispectrum [40, 54]. Especially, we can observe that the modes are times larger than the modes on sufficient large scales. This amplification directly reflects a magnitude relationship between magnetized gravitational waves and curvature perturbations of eq. (5), namely, . Overall behaviors of the magnetized bispectra are consistent with the CMB power spectra predicted by the standard cosmology because their transfer functions are same. In four panels for the temperature and E-mode bispectra, the standard local-type bispectra are also plotted. Furthermore, in panel, we also describe a CMB bispectrum from a correlation between the late-time ISW effect and weak lensing, i.e., the ISW-lensing bispectrum [66, 67, 68, 69, 70, 71, 72, 73, 74]. It is well known that the ISW-lensing bispectrum highly correlates with the local-type bispectrum. We can see that these two types of bispectra resemble the magnetized bispectra, while they are quite different from the spectra because of the tensor-mode contributions. Therefore, the magnetized bispectrum signals will not be biased in a multi-parameter fitting (for details see the next section). In the next section, we evaluate the detectability of these signals.

## 3 Fisher forecast

In this section, through the Fisher analysis, we evaluate the expected error bar of the magnitude of the magnetized bispectra for , which depends on the PMF strength (smoothed on 1 Mpc) and PMF generation epoch:

 Abis=(B13 nG)6[ln(τν/τB)ln(1017)]3 . (9)

We assume noise information of temperature and polarizations in the Planck and PRISM experiments [60, 61] (for details see appendix A).

### 3.1 Temperature and E-mode bispectra

Here we focus on the parity-even signals arising from , , and . The Fisher matrix element of the normalized bispectra including E-mode polarizations is defined as [57, 58]

 Fij=∑X1X2X3X′1X′2X′3∑ℓ1≤ℓ2≤ℓ3≤ℓmax1Δℓ1ℓ2ℓ3~B(i)X′1X′2X′3,ℓ1ℓ2ℓ3[3∏n=1(C−1)XnX′nℓn]~B(j)X1X2X3,ℓ1ℓ2ℓ3 , (10)

where

 Δℓ1ℓ2ℓ3=(−1)ℓ1+ℓ2+ℓ3(1+2δℓ1,ℓ2δℓ2,ℓ3)+δℓ1,ℓ2+δℓ2,ℓ3+δℓ3,ℓ1 , (11)

and and run over eight modes , , , , , , and . The inverse matrix of the power spectrum is explicitly written as

 (C−1)XX′ℓ≡(CIIℓCIEℓCEIℓCEEℓ)−1 , (12)

where is the CMB power spectrum involving information of cosmic variance and instrumental noise . We want to estimate the signals of the magnetized bispectrum () under the contamination of the local-type bispectrum () or the ISW-lensing bispectrum () and accordingly .

Firstly, let us clarify the dependence of the magnetized temperature and polarization bispectra on the scalar and tensor modes under the cosmic-variance-limited ideal experiment. In figure 3 we plot the signal-to-noise ratio, which is given as

 SN=√FMM . (13)

From this figure, we can see that contribution of the tensor mode is quite larger than that of the scalar mode and therefore the mode dominates over the spectrum. However, due to rapidly decaying nature, the tensor mode is saturated for and the scalar mode also contributes to a bit of amplification of the spectrum. Note that the spectrum falls below the mode due to sign difference of each mode. These features have also been observed in the analysis of [54] and are quite different from the local-type bispectrum signatures that behave as simple increasing functions of [57].

Next, to estimate the uncertainty of under the presence of the contamination of the local-type bispectrum, we introduce the Fisher submatrix as

 (2)F = (FMMFMLFLMFLL) . (14)

Then, the errors are given by

 (δAbis,δfNL)=(√(2)F−111,√(2)F−122) . (15)

Numerical results of are described in figure 4. We will also present in appendix A. From this figure, it is found that if we use all information of the temperature and E-mode bispectra, is improved in comparison with the analysis in terms of alone under the ideal case. This is an interesting result since in estimation for the local-type bispectrum is only reduced (See refs. [57, 58] or figure 7). This indicates that the tensor-mode polarization bispectra are quite informative. As described in this figure, measuring with this accuracy is hard in the Planck experiment due to lack of sensitivity of polarizations (see appendix A), while it can be done in the PRISM experiment. In the Planck, PRISM and ideal experiments for , we obtain , and , respectively (table 1).

To quantify resemblance between and , we may compute a shape correlator given by

 rML≡FML√FMMFLL . (16)

A numerical result for , i.e., , guarantees that the magnetized bispectrum is weakly correlated with the local-type bispectrum and its contamination is very small. As this result, of the spectrum in figure 3 coincides with .

Finally, let us evaluate the bias by the ISW-lensing bispectrum. This contaminates only . In the same manner as the above discussion, we compute by following

 δAbis = √(2)F′−111 , (17) (2)F′ = (FMMFMϕFϕMFϕϕ) , (18)

and find that the values for become (Planck), (PRISM) and (ideal), respectively. These are almost identical to the values of from in figure 4 (or table 1) and hence we can conclude that the ISW-lensing bispectrum is also a tiny bias comparable to the local-type bispectrum in the analysis.

### 3.2 B-mode bispectra

In this subsection, we shall consider a possibility of the bispectra including B-mode polarization. Such bispectra are divided into both the parity-even ( and ) and the parity-odd (, , and ) combinations. Although a complete analysis with both these all contributions and the temperature and E-mode bispectra may reduce more drastically, it will be quite complicated. Accordingly, here let us concentrate on the Fisher analysis with alone.

For , the compensated vector mode will exceeds the passive tensor mode. Furthermore, on such scales, lensed CMB fluctuations also generate secondary B-mode fluctuations and may contaminate the magnetized bispectrum [40, 69, 75]. While the consideration of these sources is important, in this paper we work on large scales up to where these are negligible.

Despite the parity-odd case, we can define the Fisher matrix like the parity-even case:

 F≡∑ℓ1≤ℓ2≤ℓ3≤ℓmax~B2BBB,ℓ1ℓ2ℓ3Δℓ1ℓ2ℓ3∏3n=1CBBℓn , (19)

where . Then the error becomes

 δAbis=√F−1 . (20)

Figure 5 describes the numerical results of . As the cosmic-variance spectrum , we adopt non-magnetized tensor-mode power spectrum in the standard cosmology, whose amplitude is determined by the tensor-to-scalar ratio . Especially for the ideal case (), is then simply proportional to and therefore we can write for . Interestingly, unlike the estimation with the temperature and E-mode bispectra, in the ideal experiment does not saturate even for high . This is due to damping behavior of for , which cannot be seen in and (see figure 6). For , owing to this effect, reaches under the PRISM noise level. On the other hand, the Planck experiment is too noisy to reduce the error so much like the case. If , the noise dominates completely and saturates for all in both the experiments.

Finally, we summarize the value of for each case in table 1

## 4 Summary and discussion

In this paper we examined how the polarization bispectra of the CMB anisotropies affect constraining PMFs. Firstly, we confirmed that the tensor-mode signals dominate over the bispectrum for and the scalar mode contributes on very small scales in the auto- and cross-bispectra with the polarizations. Owing to this dependence, the magnetized bispectra are weakly correlated with the standard local-type bispectra and the ISW-lensing bispectrum, and hence from observations the information of PMFs will be able to be extracted efficiently without any contamination.

From the error analyses via the Fisher forecast, we found that potentially, if we utilize all the temperature and E-mode bispectra, the uncertainty of the magnitude of magnetized bispectra can be improved in comparison with the analysis with respect to alone. This is interesting since in the analysis of the local-type non-Gaussianity, the improvement is only . The proposed PRISM experiment will be able to reach this precision, while the Planck experiment cannot. If we assume the GUT-scale generation of PMFs, namely , the expected errors on the PMF strength from all the temperature and E-mode bispectra are given as and in the Planck and PRISM (or ideal) experiments, respectively.

We also considered the possibility of the analysis involving the B-mode bispectrum. In this case, we focused on the Fisher forecast using and found that the uncertainty keeps on reducing as increases due to the damping behavior of the B-mode cosmic-variance spectrum for . In the ideal experiment, we have a relationship with the tensor-to-scalar ratio: for ; therefore we will be able to estimate with nG accuracy if . In practice, the Planck and PRISM instrumental noises relax the value as and for , respectively.

One may be concerned about comparison with bounds from the power spectrum analysis. According to recent literature [42, 43, 44, 45], upper bounds on from the temperature and E-mode power spectra are around 3 nG. As shown above, the bispectrum analysis will provide comparable or tighter constraints on . Concerning the B-mode power spectrum, the magnetized passive-mode signals are indistinguishable from the non-magnetized ones from primordial gravitational waves and hence may be not determined accurately in a multi-parameter fitting. In this sense, the information of the B-mode bispectrum will be more useful.

For , where this paper has not focused on, the vector compensated mode will dominate over the magnetized B-mode bispectrum. To reduce the uncertainty, we must evaluate such vector-mode contribution. Then, more comprehensive analysis including cross-bispectra between temperature, E-mode and B-mode fluctuations will be required. These informative but complex works remain as future issues.

\acknowledgments

We thank Sabino Matarrese for helpful advice and motivational comments. We appreciate helpful advice on estimation of noise spectra given by Bin Hu and Michele Liguori. This work was supported in part by a Grant-in-Aid for JSPS Research under Grant No. 25-573 and the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2.

## Appendix A Noise spectra

Here, we summarize the temperature and polarization noise spectra expected in the Planck and PRISM experiments.

Assuming Gaussian random detector noise, each noise spectrum is estimated as [76, 73, 77]

 NXXl=[∑c1θ2cσ2X,ce−ℓ(ℓ+1)θ2c/(8ln2)]−1 (21)

where is the Full Width Half Maximum (FWHM) per the frequency channel in radians and is the dimensionless sensitivity per . One can find these values (in arcminutes and K) in table 2.

Figure 6 shows numerical results of , and . Here we assume . We can see that in the mode, the cosmic-variance spectrum is comparable to the Planck noise spectrum, i.e., , for . This is a reason why the Planck experiment does not improve so much in the analysis including the -mode polarization as described in figure 4. Likewise, in the PRISM experiment exceeds for and hence never be reduced beyond when (figure 5).

## Appendix B Errors of the local-type non-Gaussianity

In figure 7, we describe the errors of the local-type nonlinearity parameter estimated from the two-dimensional Fisher analysis involving discussed in subsection 3.1. Thanks to the weak correlation with the magnetized bispectrum, is in good agreement with the results from the one-dimensional Fisher analysis fitting alone, i.e., [58]. We confirm that in the ideal experiment, the analysis containing both the temperature and E-mode bispectra reduces the value of to half in comparison with the analysis by alone. In the PRISM experiment, can reach for .

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