Parity violation of primordial magnetic fields in the CMB bispectrum

# Parity violation of primordial magnetic fields in the CMB bispectrum

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###### Abstract

We study the parity violation in the cosmic microwave background (CMB) bispectrum induced by primordial magnetic fields (PMFs). Deriving a general formula for the CMB bispectrum generated from not only non-helical but also helical PMFs, we find that helical PMFs produce characteristic signals, which disappear in parity-conserving cases, such as the intensity-intensity-intensity bispectra arising from . For fast numerical calculation of the CMB bispectrum, we reduce the one-loop formula to the tree-level one by using the so-called pole approximation. Then, we show that the magnetic anisotropic stress, which depends quadratically on non-helical and helical PMFs and acts as a source of the CMB fluctuation, produces the local-type non-Gaussianity. Comparing the CMB bispectra composed of the scalar and tensor modes with the noise spectra, we find that assuming the generation of the nearly scale-invariant non-helical and helical PMFs from the grand unification energy scale () to the electroweak one (), the intensity-intensity-intensity bispectrum for can be observed by the WMAP experiment under the condition that with and being the non-helical and helical PMF strengths smoothed on 1 Mpc, respectively.

a]Maresuke Shiraishi Prepared for submission to JCAP

Parity violation of primordial magnetic fields in the CMB bispectrum

• Department of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan

E-mail: mare@nagoya-u.jp

Keywords: primordial magnetic fields, non-gaussianity, cosmology of theories beyond the SM, cosmological parameters from CMBR

ArXiv ePrint: 1202.2847

## 1 Introduction

The cosmological parity violation is a key feature of ultra-violet completion of general relativity and hence a lot of researchers have extracted their signals from several cosmological phenomena [1, 2, 3, 4, 5, 6, 7]. In particular, the effects on the cosmic microwave background (CMB) have been well-studied and the cosmological parity violation has been verified by analyzing the non-vanishing cross-correlated power spectra between the intensity () and -mode polarization () anisotropies and those between -mode () and -mode polarization anisotropies [8, 9, 10, 11, 12, 13]. Furthermore, beyond the linear-order effects, the impacts of the parity violation on the graviton non-Gaussianities have recently been discussed [14, 15, 16]. According to ref. [16], unlike the parity-conserving non-Gaussianity, the parity-violating one induces the signals arising from in the CMB and bispectra and also those coming from in the CMB and bispectra. In these correlations, the bispectrum from is expected to bring in the detectable information of the parity violation.

On the other hand, if there exists the primordial magnetic field (PMF), which is a favored candidate for the seed field of microgauss-level magnetic fields in galaxies and cluster of galaxies [17, 18, 19], their power spectrum may involve the parity-violating component [20, 21, 22, 23, 24]. Like the above non-magnetic cases, this so-called helical PMF induces the characteristic signals in the CMB and correlations [25, 26, 27, 28, 29]. Although concrete limits on the magnitude of the helical PMF have not obtained yet, these studies imply that the and correlations are detectable if helical PMFs have nanogauss-level magnitudes (at the present time) and these spectra are nearly scale invariant. However, assuming the Gaussianity of the PMF, the beneficial signals are generated also in the CMB bispectra due to the quadratic dependence of the CMB fluctuation on the PMF. In the case where only non-helical PMFs exist, the contributions of PMFs to the primordial non-Gaussianities and the CMB bispectra have been deeply investigated in refs. [30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

In this paper, we newly consider the effects of both non-helical and helical PMFs on the CMB bispectrum. Based on our computation approach [35, 36, 37, 38], we derive a general formula for the CMB bispectrum induced by the non-Gaussianity of the PMF anisotropic stress coming from not only non-helical PMFs but also helical ones. Then, we confirm the existence of the foregoing parity-violating signals such as the bispectrum from . By the pole approximation mentioned in ref. [38], we reduce this formula to a form suitable for the fast calculation in the case where the non-helical and helical PMFs have the nearly scale-invariant spectra. In this process, it is shown that the bispectrum of the PMF anisotropic stresses has the local-type shape even if helical PMFs exist. Computing the CMB bispectra composed of the scalar and tensor modes and estimating the signal-to-noise ratio, we analyze how the helical PMF affects the CMB bispectrum and show how large the PMF strength is required for the detection of the signals from .

This paper is organized as follows. In the next section, we summarize the expressions and statistical properties of both non-helical and helical PMFs. In section 3, the analytic formulae and numerical results of the CMB bispectra, and the signal-to-noise ratio are presented. The final section is devoted to the summary and discussion of this paper. Throughout this paper, we obey the definition of the Fourier transformation as

 f(x)≡∫d3k(2π)3~f(k)eik⋅x , (1.1)

and the rule that the subscripts and superscripts of the Greek characters and alphabets run from 0 to 3 and from 1 to 3, respectively.

## 2 Statistical properties of non-helical and helical magnetic fields

Let us take into account the large-scale primordial magnetic field (PMF), , which is generated in the very early Universe and behaves as a source of the CMB fluctuation, on the homogeneous background and small perturbative Universe as . Here, and denote the scale factor and conformal time, respectively. Neglecting the effects of the back reaction of the fluid on the evolution of magnetic fields and considering the flux conservation, the PMF evolves as . Each component of the energy momentum tensor composed of the PMF is given by

 T0 0(xμ)=−18πa4B2(x) ,T0 c(xμ)=Tb 0(xμ)=0 ,Tb c(xμ)=14πa4[B2(x)2δb c−Bb(x)Bc(x)] . (2.1)

The spatial parts in Fourier space are expressed as

 Tb c(k,τ)≡ργ(τ)[δb cΔB(k)+ΠbBc(k)] ,ΔB(k)=18πργ,0∫d3k′(2π)3Bb(k′)Bb(k−k′) ,ΠbBc(k)=−14πργ,0∫d3k′(2π)3Bb(k′)Bc(k−k′) , (2.2)

where is the photon energy density with being its present value. After this, for simplicity of calculation, we ignore the trivial time-dependence and hence the index can be lowered by .

Conventionally, assuming the Gaussianity of the PMF, the power spectrum is expressed as [25]

 (2.3)

where is a unit vector, is the 3D Levi-Civita tensor normalized by , and is a projection tensor coming from the divergence free nature of PMFs. The first and second terms in the bracket represent non-helical and helical contributions, respectively. For a mathematical relation:

 limk′→−k\BraketB(k)⋅B(k′)≥limk′→−k∣∣\Braket[^k×B(k)]⋅B(k′)∣∣ ,

the power spectra of non-helical and helical PMFs obey such a magnitude relation as

 PB(k)≥|PB(k)| . (2.4)

In order to formulate the CMB bispectrum, it is convenient to use a normalized divergenceless polarization vector in two circular states, , as shown in appendix A. Then, the above expression changes to

 = (2π)32δ(k+k′)∑σ=±1[PB(k)−σPB(k)]ϵ(σ)a(^k)ϵ(−σ)b(^k) (2.5) = (2π)32δ(k+k′)∑σ=±1[PB(k′)−σPB(k′)]ϵ(−σ)a(^k′)ϵ(σ)b(^k′) .

This implies that the second terms of the brackets in the first and second equalities creates the difference of the magnetic power spectra between two circular states as .

To parametrize the magnetic field strengths, we introduce the quantities smoothed on as

 B2r ≡ \BraketB(x)⋅B(x)|r (2.6) = ∫d3k(2π)3∫d3k′(2π)3\BraketBa(k)Ba(k′)e−r2(k2+k′2)/2ei(k+k′)⋅x , B2r ≡ r|\BraketB(x)⋅[∇×B(x)]||r (2.7) = ∫d3k(2π)3∫d3k′(2π)3r|ik′^kbηabc\BraketBa(k)Bc(k′)|e−r2(k2+k′2)/2ei(k+k′)⋅x .

Assuming the simple power-law spectra as

 PB(k)=ABknB ,  PB(k)=ABknB , (2.8)

the spectral amplitudes are written as

 AB=(2π)nB+5B2rΓ(nB+32)knB+3r ,  |AB|=(2π)nB+5B2rΓ(nB+42)knB+3r , (2.9)

where is the Gamma function and . Note that unlike , can take both positive and negative values. Equation (2.4) leads to a constraint on the PMF strengths: if the PMF spectra have nearly scale invariant shapes as .

The PMF anisotropic stress, , induces the CMB anisotropy and therefore we require their bispectrum for the computation of the CMB bispectrum. Considering equation (2.5), this can be straightforwardly calculated as

 \BraketΠBab(k1)ΠBcd(k2)ΠBef(k3)=(−4πργ,0)−3[3∏n=1∫d3k′n∑σn=±1{PB(k′n)−σnPB(k′n)}] ×δ(k1−k′1+k′3)δ(k2−k′2+k′1)δ(k3−k′3+k′2) ×18[ϵ(σ1)a(^k′1)ϵ(−σ1)d(^k′1)ϵ(−σ3)b(^k′3)ϵ(σ3)e(^k′3)ϵ(σ2)c(^k′2)ϵ(−σ2)f(^k′2) +{a↔b or c↔d or e↔f}] , (2.10)

where is the Alfvén-wave damping length scale [40, 41] as and the curly bracket denotes the symmetric seven terms under the permutations of indices: , , or . For the sake of avoiding the IR divergence, we limit the range of the PMF spectral indices as .

Like the discussion in ref. [38], if the tilts of the PMF spectra are enough red as , the shape of the bispectrum of PMF anisotropic stresses depends strongly on the behaviors of the integrands at around three poles, namely, . In this limit, the bispectrum (2.10) reduces to

 \BraketΠBab(k1)ΠBcd(k2)ΠBef(k3)∼(−4πργ,0)−3δ(3∑n=1kn)αABnB+3knB+3∗8π318 ×⎡⎣∑σ2,σ3=±1{PB(k1)−σ3PB(k1)}{PB(k2)−σ2PB(k2)} ×δa,dϵ(σ3)b(^k1)ϵ(−σ3)e(^k1)ϵ(σ2)c(^k2)ϵ(−σ2)f(^k2) +∑σ1,σ2=±1{PB(k1)−σ1PB(k1)}{PB(k3)−σ2PB(k3)} ×ϵ(σ1)a(^k1)ϵ(−σ1)d(^k1)δb,eϵ(−σ2)c(^k3)ϵ(σ2)f(^k3) +∑σ1,σ3=±1{PB(k2)−σ1PB(k2)}{PB(k3)−σ3PB(k3)} ×ϵ(−σ1)a(^k2)ϵ(σ1)d(^k2)ϵ(−σ3)b(^k3)ϵ(σ3)e(^k3)δc,f +{a↔b or c↔d or e↔f}] , (2.11)

where and are a cutoff scale of the integrand and a parameter fixing the uncertainty of the amplitude associated with the approximation, respectively, and we have evaluated an integral at around each pole by following

 ∫d3k′∑σ=±1{PB(k′)−σPB(k′)}ϵ(σ)a(^k′)ϵ(−σ)b(^k′)=αABnB+3knB+3∗8π3δab . (2.12)

In equation (2.12), due to the summation over , the contribution of the second term of the bracket vanishes. This implies that the effects of the helical PMF are tiny at around each pole. From equation (2.11), we can see that the bispectrum of the PMF anisotropic stresses dominates at the squeezed limit such as and has the identical -dependence to the local-type bispectrum of curvature perturbations [42]. Thus, we conclude that with and without helical PMFs, the shape of the non-Gaussianity associated with the PMF anisotropic stress is classified into the local-type configuration.

In the next section, we compute the CMB bispectra generated from the non-Gaussianity of the PMF anisotropic stresses.

## 3 CMB bispectrum from non-helical and helical magnetic fields

In this section, we investigate the effects of both non-helical () and helical () PMFs on the CMB bispectra. At first, on the basis of the formalism presented in refs. [37, 38], we derive their exact and optimal formulae generated from equations (2.10) and (2.11), respectively. Next, through numerical computations, we analyze the magnitudes and shapes of the CMB bispectra and examine whether the parity-violating signals coming from helical PMFs can be detected or not.

### 3.1 Formulation

The CMB intensity and two linear polarization fields () generated from the scalar-, vector- and tensor-mode perturbations () are expanded by the spherical harmonics as

 ΔX(Z)(^n)X=∑ℓma(Z)X,ℓmYℓm(^n) , (3.1)

where is a line-of-sight direction. According to refs. [43, 44], each spherical harmonic coefficient, , is given by

 a(Z)X,ℓm=4π(−i)ℓ∫d3k(2π)3T(Z)X,ℓ(k)∑λ[sgn(λ)]λ+xξ(λ)ℓm(k) ,ξ(λ)ℓm(k)=∫d2^kξ(λ)(k)−λY∗ℓm(^k) , (3.2)

where expresses the helicity of the scalar-, vector- and tensor-mode perturbations, discriminates the parity-even () and -odd () fields, and and are the primordial perturbation and transfer function of each mode, respectively 111Here, we set ..

If there exist large-scale PMFs, their anisotropic stresses generate additional fluctuations in the CMB. The PMF anisotropic stresses survive and become a source of the gravitational potential before neutrinos decouple and they are compensated by the neutrino anisotropic stresses. Thus, gravitational waves and curvature perturbations logarithmically grow even on superhorizon scales prior to neutrino decoupling and produce the CMB tensor- and scalar-mode anisotropies at recombination epoch [45, 46, 47] 222Recently, the solution of the curvature perturbation is being reanalyzed by treating the effects in both the inflationary and the radiation-dominated eras consistently [48, 49]. See however ref. [39].. In contrast, due to the decaying nature of the vector potential, the CMB vector-mode anisotropies are induced by not this mechanism but the vorticities of photons enhanced by the Lorentz force from the PMF anisotropic stresses [50, 51, 45]. Consequently, we can summarize the scalar-, vector- and tensor-mode initial perturbations as

 ξ(0)(k)≈−Rγln(τντB)32O(0)ab(^k)ΠBab(k) ,ξ(±1)(k)≈12O(∓1)ab(^k)ΠBab(k) ,ξ(±2)(k)≈6Rγln(τντB)12O(∓2)ab(^k)ΠBab(k) , (3.3)

where and are the projection tensors decomposing into the scalar-, vector- and tensor-mode variables, respectively, and defined in appendix A. and correspond to the curvature perturbation and gravitational wave on superhorizon scales, respectively [45, 46, 47], and depend on the production time of the PMF, , the epoch of neutrino decoupling, , and the ratio between the energy densities of photons and all relativistic particles, , for . As the upper and lower values of , we take the energy scales of the grand unification () and electroweak symmetry breaking (), corresponding to and , respectively. As and , we use the standard cosmological transfer functions independent of PMFs [52, 53, 54, 43] because the evolution of the cosmological perturbations are little-affected by PMFs posterior to neutrino decoupling [47]. On the other hand, as , we should use the form including the effects of the PMF on the cosmological perturbations shown in refs. [50, 51, 45].

Using equation (3.2), the CMB bispectrum is formulated as

 \Braket3∏n=1a(Zn)Xn,ℓnmn = [3∏n=14π(−i)ℓn∫∞0k2ndkn(2π)3T(Zn)Xn,ℓn(kn)∑λn[sgn(λn)]λn+xn] (3.4) ×\Braket3∏n=1ξ(λn)ℓnmn(kn) .

To obtain the explicit formula for the CMB bispectrum involving the dependence on PMFs, we have to compute the angular bispectrum of the initial perturbations, . Expanding all angular dependence in the delta functions and the contractions of the projection tensors and wave number vectors in by the spin spherical harmonics on the basis of appendix A, and expressing the angular integrals of their spherical harmonics with the Wigner symbols [37], this is obtained as

 \Braket3∏n=1ξ(λn)ℓnmn(kn) = (3.5) ×[3∏n=1∫kD0k′2ndk′n]{PB(k′1)+SPB(k′1)} ×{PB(k′2)+S′PB(k′2)}{PB(k′3)+S′′PB(k′3)} ×fS′′Sλ1L′′Lℓ1(k′3,k′1,k1)fSS′λ2LL′ℓ2(k′1,k′2,k2)fS′S′′λ3L′L′′ℓ3(k′2,k′3,k3),

with

 fS′′SλL′′Lℓ(r3,r2,r1) = ∑L1L2L3∫∞0y2dyjL3(r3y)jL2(r2y)jL1(r1y) (3.6) ×(−1)ℓ+L2+L3(−1)L1+L2+L32I0 0 0L1L2L3I0S′′−S′′L31L′′I0S−SL21LI0λ−λL1ℓ2⎧⎪⎨⎪⎩L′′LℓL3L2L1112⎫⎪⎬⎪⎭ ×⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−2√3(8π)3/2Rγln(τν/τB)(λ=0)23(8π)3/2λ(λ=±1)−4(8π)3/2Rγln(τν/τB)(λ=±2) .

Here, is the Bessel function and the symbol is defined by

 Is1s2s3l1l2l3≡√(2l1+1)(2l2+1)(2l3+1)4π(l1l2l3s1s2s3) . (3.7)

The confinement of and to the Wigner- symbol guarantees the rotational invariance of the CMB bispectrum. If , this is consistent with the corresponding equation for the non-helical case [35, 37, 36, 38]. Note that unlike , associates the spin as . As described in equation (3.8), this differentiates the multipole configurations of non-helical terms from those of helical ones. Via the summations over and , the CMB bispectrum from non-helical and helical PMFs is explicitly written as333Caution about a fact that is determined by , namely, for , respectively.

 \Braket3∏n=1a(Zn)Xn,ℓnmn = (ℓ1ℓ2ℓ3m1m2m3)CZ1CZ2CZ3(−4πργ,0)−3∑LL′L′′{cccℓ1ℓ2ℓ3L′L′′L} (3.8) ×∑L1L2L3L′1L′2L′3L′′1L′′2L′′3(−1)∑3n=1Ln+L′n+L′′n+2ℓn2I0 0 0L1L2L3I0 0 0L′1L′2L′3I0 0 0L′′1L′′2L′′3 ×⎧⎪⎨⎪⎩L′′Lℓ1L3L2L1112⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩LL′ℓ2L′3L′2L′1112⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩L′L′′ℓ3L′′3L′′2L′′1112⎫⎪⎬⎪⎭ ×[3∏n=1(−i)ℓn∫∞0k2ndkn2π2T(Zn)Xn,ℓn(kn)] ×∫∞0A2dAjL1(k1A)∫∞0B2dBjL′1(k2B)∫∞0C2dCjL′′1(k3C) ×∫kD0k′21dk′1[PB(k′1)Q(e)L′3,L2,L−PB(k′1)Q(o)L′3,L2,L]jL2(k′1A)jL′3(k′1B) ×∫kD0k′22dk′2[PB(k′2)Q(e)L′′3,L′2,L′−PB(k′2)Q(o)L′′3,L′2,L′]jL′2(k′2B)jL′′3(k′2C) ×∫kD0k′23dk′3[PB(k′3)Q(e)L3,L′′2,L′′−PB(k′3)Q(o)L3,L′′2,L′′]jL′′2(k′3C)jL3(k′3A) ×8I01−1L′31LI01−1L21LI01−1L′′31L′I01−1L′21L′I01−1L31L′′I01−1L′′21L′′ ×23−NSI0|λ1|−|λ1|L1 ℓ1 2I0|λ2|−|λ2|L′1 ℓ2 2I0|λ3|−|λ3|L′′1 ℓ3 2D(e)L1,ℓ1,x1D(e)L′1,ℓ2,x2D(e)L′′1,ℓ3,x3 ,

where is the number of the scalar modes constituting the CMB bispectrum and

 CZ≡⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−2√3(8π)3/2Rγln(τν/τB)(Z=S)23(8π)3/2(Z=V)−4(8π)3/2Rγln(τν/τB)(Z=T) . (3.9)

Here, we have introduced the filter functions as

 Q(e)L′3,L2,L≡(δL′3,L+1+δL′3,|L−1|)(δL2,L+1+δL2,|L−1|)+δL′3,LδL2,L ,Q(o)L′3,L2,L≡(δL′3,L+1+δL′3,|L−1|)δL2,L+δL′3,L(δL2,L+1+δL2,|L−1|) ,D(e)L1,ℓ1,x1≡(δL1,ℓ1−2+δL1,ℓ1+δL1,ℓ1+2)δx1,0+(δL1,ℓ1−1+δL1,ℓ1+1)δx1,1 , (3.10)

which come from the above summations and selection rules of the Wigner symbols [44] and ensure , and , respectively. Considering these filter functions and a relation derived from the selection rules as

 3∑n=1(ℓn+xn)+L2+L′2+L′′2+L3+L′3+L′′3=even , (3.11)

we can see that the four terms in equation (3.8), which are composed of an even number of the helical PMF power spectra and proportional to or , give the signals under the condition as

 3∑n=1(ℓn+xn)=even . (3.12)

These signals can arise from the parity-even CMB bispectrum as

 \Braket3∏n=1ΔX(Z)(^n)X=\Braket3∏n=1ΔX