CMB B-mode auto-bispectrum produced by primordial gravitational waves

TL;DR

This paper calculates the polarization bispectra in the CMB caused by primordial gravitational waves, identifying potential detectability of non-Gaussianities from specific scalar field couplings in inflation models.

Contribution

It introduces a method to compute polarization bispectra from non-Gaussian gravitational waves in scalar-tensor theories using Laplace transformation.

Findings

Future experiments can detect the new non-Gaussianity if the coupling parameter is very large.

The new non-Gaussianity does not conflict with current observational data.

Abstract

Gravitational waves from inflation induce polarization patterns in the cosmic microwave background (CMB). It is known that there are only two types of non-Gaussianities of the gravitaional waves in the most general scalar field theories having second-order field equations. One originates from the inherent non-Gaussianity in general relativity, and the other from a derivative coupling between the Einstein tensor and a kinetic term of the scalar field. We calculate polarization bispectra induced by these non-Gaussianities by transforming them into separable forms by virtue of the Laplace transformation. It is shown that future experiments can detect only the new one if the latter coupling parameter takes an extremely large value, which, however, does not cotradict the current observational data.

Figures

Equations

$$A(\hat{n})=\sum _{\ell m}{a}_{\ell m}{Y}_{\ell m}(\hat{n}).$$

$$B_{{\ell }_1{\ell }_2{\ell }_3}^{m_1{m}_2{m}_3}=⟨a_{{\ell }_1{m}_1}{a}_{{\ell }_2{m}_2}{a}_{{\ell }_3{m}_3}⟩.$$

$$B_{\ell_{1}\ell_{2}\ell_{3}}=\sum_{m_{1}m_{2}m_{3}}\left(\begin{array}[]{@{}c@{\; }c@{\; }c@{}}\ell_{1}&\ell_{2}&\ell_{3}\\[-5.0pt] m_{1}&m_{2}&m_{3}\end{array}\right)B^{m_{1}m_{2}m_{3}}_{\ell_{1}\ell_{2}\ell_{3}},$$

$$b_{{\ell }_1{\ell }_2{\ell }_3}=G_{{\ell }_1{\ell }_2{\ell }_3}^{-1}{B}_{{\ell }_1{\ell }_2{\ell }_3},$$

$$G_{{\ell }_1{\ell }_2{\ell }_3}\equiv {\mathcal{I}}_{{\ell }_1{\ell }_2{\ell }_3}^{000},$$

$$\mathcal{I}_{\ell_{1}\ell_{2}\ell_{3}}^{s_{1}s_{2}s_{3}}\equiv\sqrt{\frac{(2\ell_{1}+1)(2\ell_{2}+1)(2\ell_{3}+1)}{4\pi}}\left(\begin{array}[]{@{}c@{\; }c@{\; }c@{}}\ell_{1}&\ell_{2}&\ell_{3}\\[-5.0pt] s_{1}&s_{2}&s_{3}\end{array}\right).$$

$$G_{{\ell }_1{\ell }_2{\ell }_3}\equiv \frac{2\sqrt{{\ell }_2({\ell }_2+1){\ell }_3({\ell }_3+1)}}{[{\ell }_1({\ell }_1+1)-{\ell }_2({\ell }_2+1)-{\ell }_3({\ell }_3+1)]}{\mathcal{I}}_{{\ell }_1{\ell }_2{\ell }_3}^{0-11}.$$

$$\begin{array}{rl}{\mathcal{L}}_2& =K(\varphi ,X),\\ {\mathcal{L}}_3& =-G_3(\varphi ,X)\square \varphi ,\\ {\mathcal{L}}_4& =G_4(\varphi ,X)R+G_{4X}[(\square \varphi {)}^2-({\nabla }_{\mu }{\nabla }_{\nu }\varphi {)}^2],\\ {\mathcal{L}}_5& =G_5(\varphi ,X)G_{\mu \nu }{\nabla }^{\mu }{\nabla }^{\nu }\varphi -\frac{1}{6}{G}_{5X}[(\square \varphi {)}^3-3(\square \varphi )({\nabla }_{\mu }{\nabla }_{\nu }\varphi {)}^2+2({\nabla }_{\mu }{\nabla }_{\nu }\varphi {)}^3].\end{array}$$

$$g_{00}=-1,g_{0i}=0,g_{ij}=a^2(t)(e^h{)}_{ij},$$

$$(e^h{)}_{ij}={\delta }_{ij}+h_{ij}+\frac{1}{2}{h}_{ik}{h}_{kj}+\frac{1}{6}{h}_{ik}{h}_{kl}{h}_{lj}+\cdots ,$$

$$\begin{array}{rl}{S}^{(2)}& =\frac{1}{8}\int dt{d}^{3}x{a}^3\left[{\mathcal{G}}_T{\dot{h}}_{ij}^2-\frac{{\mathcal{F}}_T}{a^2}({\partial }_k{h}_{ij}{)}^2\right],\\ S^{(3)}& =\int dt{d}^{3}x{a}^3\left[\frac{{\mathcal{F}}_T}{4a^2}\left(h_{ik}{h}_{jl}-\frac{1}{2}{h}_{ij}{h}_{kl}\right){\partial }_k{\partial }_l{h}_{ij}+\frac{X\dot{\varphi }{G}_{5X}}{12}{\dot{h}}_{ij}{\dot{h}}_{jk}{\dot{h}}_{ki}\right],\end{array}$$

$$\begin{array}{rl}{\mathcal{F}}_T& \equiv 2[G_4-X(\ddot{\varphi }{G}_{5X}+G_{5\varphi })],\\ {\mathcal{G}}_T& \equiv 2[G_4-2X{G}_{4X}-X(H\dot{\varphi }{G}_{5X}-G_{5\varphi })].\end{array}$$

$$\begin{array}{rl}⟨{\xi }^{(s)}(\mathbf{k}){\xi }^{\ast (s^{\prime })}({\mathbf{k}}^{\prime })⟩& =(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}-{\mathbf{k}}^{\prime }){\delta }_{s{s}^{\prime }}\frac{{\pi }^2}{k^3}{\mathcal{P}}_h(k),\\ {\mathcal{P}}_h(k)& =\frac{2H^2}{{\pi }^2}\frac{{\mathcal{G}}_T^{1/2}}{{\mathcal{F}}_T^{3/2}}.\end{array}$$

$$⟨{\xi }^{(s_1)}({\mathbf{k}}_1){\xi }^{(s_2)}({\mathbf{k}}_2){\xi }^{(s_3)}({\mathbf{k}}_3)⟩=(2\pi {)}^7{\delta }^{(3)}({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3){\mathcal{P}}_h^2\frac{{\mathcal{S}}_{(\mathrm{GR})}^{s_1{s}_2{s}_3}+{\mathcal{S}}_{(5X)}^{s_1{s}_2{s}_3}}{k_1^2{k}_2^2{k}_3^2},$$

$${\mathcal{S}}_{(\mathrm{GR})}^{s_1{s}_2{s}_3}=-\frac{1}{8}\frac{k_t}{k_1{k}_2{k}_3}\left(1-\frac{{\sum }_{i\ne j}{k}_i^2{k}_j+4k_1{k}_2{k}_3}{k_t^3}\right)\left[k_{3k}{e}_{ki}^{\ast (s_1)}{e}_{ij}^{\ast (s_3)}{e}_{jl}^{\ast (s_2)}{k}_{3l}+\frac{1}{2}{k}_{2i}{e}_{ik}^{\ast (s_1)}{k}_{3k}{e}_{jl}^{\ast (s_2)}{e}_{jl}^{\ast (s_3)}\right]$$

$$+\text{2 circulations},$$

$${\mathcal{S}}_{(5X)}^{s_1{s}_2{s}_3}=-\frac{1}{8}{f}_{5X}\frac{k_1{k}_2{k}_3}{k_t^3}{e}_{ij}^{\ast (s_1)}{e}_{jk}^{\ast (s_2)}{e}_{ki}^{\ast (s_3)},$$

$$\begin{array}{rl}& f_{5X}\equiv \frac{-2HX\dot{\varphi }{G}_{5X}}{{\mathcal{G}}_T}.\end{array}$$

$$a_{\ell m}^{(s)}=4\pi (-i{)}^{\ell }\int \frac{k^{2}dk}{(2\pi {)}^3}{\mathcal{T}}_{\ell }^{(s)}(k)\int d{\Omega }_k{}_{-s}{Y}_{\ell m}^{\ast }{\xi }^{(s)}(\mathbf{k}),$$

$$B_{{\ell }_1{\ell }_2{\ell }_3}=\sum _{s_1{s}_2{s}_3}{B}_{{\ell }_1{\ell }_2{\ell }_3}^{(s_1{s}_2{s}_3)},$$

$$\begin{split}B^{(s_{1}s_{2}s_{3})}_{\ell_{1}\ell_{2}\ell_{3}}&=\sum_{m_{1}m_{2}m_{3}}\left(\begin{array}[]{@{}c@{\; }c@{\; }c@{}}\ell_{1}&\ell_{2}&\ell_{3}\\[-5.0pt] m_{1}&m_{2}&m_{3}\end{array}\right)\\ &~{}~{}~{}~{}\times\prod_{j=1}^{3}\left[4\pi(-i)^{\ell_{j}}\int\frac{k_{j}^{2}dk_{j}}{(2\pi)^{3}}\mathcal{T}^{(s_{j})}_{\ell_{j}}(k_{j})\int d\Omega_{k_{j}}{}_{-s_{j}}Y^{*}_{\ell_{j}m_{j}}\right]\langle\xi^{(s_{1})}({\mathbf{k}}_{1})\xi^{(s_{2})}({\mathbf{k}}_{2})\xi^{(s_{3})}({\mathbf{k}}_{3})\rangle.\end{split}$$

$$B_{{\ell }_1{\ell }_2{\ell }_3}=0\mathrm{for}{\ell }_1+{\ell }_2+{\ell }_3=\mathrm{even},$$

$$B_{{\ell }_1{\ell }_2{\ell }_3}=0\mathrm{for}{\ell }_1+{\ell }_2+{\ell }_3=\mathrm{odd}.$$

$$B_{{\ell }_1{\ell }_2{\ell }_3}^{(s_1{s}_2{s}_3)}$$

$$\times \int x^{2}dx\prod _{j=1}^3\left[\frac{2}{\pi }{i}^{L_j-{\ell }_j}\int d{k}_j{\mathcal{T}}_{{\ell }_j}^{(s_j)}(k_j)j_{L_j}(k_{j}x)\right]$$

$$\times (2\pi {)}^4{\mathcal{P}}_h^2\mathcal{K}(k_1,k_2,k_3),$$

$${\mathcal{K}}_{(\text{GR})}=\frac{(4\pi {)}^{3/2}}{80}\left(k_t-\frac{{\sum }_{i\ne j}{k}_i^2{k}_j+4k_1{k}_2{k}_3}{k_t^2}\right)\left[\frac{k_3}{k_1{k}_2}{\delta }_{{\ell }_1^{\prime }2}{\delta }_{{\ell }_2^{\prime }2}\left(\frac{5{\delta }_{{\ell }_3^{\prime }2}}{\sqrt{21}}-\frac{{\delta }_{{\ell }_3^{\prime }4}}{\sqrt{7}}\right)+\text{2 circulations}\right],$$

$${\mathcal{K}}_{(5X)}=\frac{(4\pi {)}^{3/2}}{80}{f}_{5X}\sqrt{\frac{7}{3}}\frac{k_1{k}_2{k}_3}{k_t^3}{\delta }_{{\ell }_1^{\prime }2}{\delta }_{{\ell }_2^{\prime }2}{\delta }_{{\ell }_3^{\prime }2}.$$

$$p^{-\nu -1}=[\Gamma (\nu +1){]}^{-1}{\int }_0^{\infty }{t}^{\nu }{e}^{-pt}dt(\mathrm{Re}[\nu ]>-1),$$

$$F=\sum _{{\ell }_1,{\ell }_2,{\ell }_3\le {\ell }_{\text{max}}}\frac{B_{{\ell }_1{\ell }_2{\ell }_3}^2}{6C_{{\ell }_1}{C}_{{\ell }_2}{C}_{{\ell }_3}}.$$

Full text

††preprint: RESCEU-5/17

I Introduction

Inflation in the early Universe Starobinsky (1980); Guth (1981); Sato (1981); Linde (1982, 1983) produces both primordial density perturbations Mukhanov and Chibisov (1981); Guth and Pi (1982); Starobinsky (1982); Hawking (1982) and gravitational waves Starobinsky (1979); Rubakov et al. (1982) out of quantum fluctuations. (For a review of inflation, see e.g. Sato and Yokoyama (2015).) The simplest model of slow-roll inflation predicts that they are almost Gaussian and that the primordial density perturbations have only small non-Gaussianities of the order of the slow-roll parameters Acquaviva et al. (2003); Maldacena (2003). Deviations from the simplest model may result in non-Gaussianities large enough to be detected through temperature and E-mode polarization bispectra of the cosmic microwave background (CMB), and CMB experiments Bennett et al. (2013); Ade et al. (2016a) targeted such large non-Gaussianities to verify the model of inflation. The non-Gaussianities are classified to several types which have distinct wavenumber dependence mostly for the sake of simplicity of calculations (e.g. Gangui et al. (1994); Verde et al. (2000); Wang and Kamionkowski (2000); Komatsu and Spergel (2001); Maldacena (2003); Babich et al. (2004); Chen et al. (2007); Senatore et al. (2010)). However, up to now even the most pricise CMB observation has not detected any statistically significant deviation from Gaussianity yet Ade et al. (2016a).

On the other hand, primordial non-Gaussianities of the gravitational waves differ from those of the curvature perturbations. They are known to be classified to only two types Gao et al. (2011) in the framework of the generalized G-inflation Kobayashi et al. (2011) which is originally the most general single-field inflation model based on the Horndeski theory Horndeski (1974); Deffayet et al. (2011) with second-order field equations. One is inherent in the general relativity and always appears in the primordial non-Gaussianities of the order of unity. The other is originated from a certain kind of a non-minimal derivative coupling of the scalar field with the Einstein tensor. Detection of the former would provide an evidence that the standard quantum field theory calculation in curved spacetime works well, whereas that of the latter would be a smoking gun of modification of gravity.

Since primordial density perturbations cannot give rise to B-mode polarization Seljak and Zaldarriaga (1997); Zaldarriaga and Seljak (1997); Kamionkowski et al. (1997a, b), its primary auto-bispectrum just reflects the non-Gaussianities of gravitational waves. In this sense, the B-mode auto-bispectrum is suitable for investigating the non-Gaussianities of tensor perturbations and worthy of consideration. So far only the power spectrum of B-mode polarization has been detected which is induced by not primordial gravitational waves but lensing of E-modes and Galactic dust emission Hanson et al. (2013); Ade et al. (2014a); Naess et al. (2014); Ade et al. (2014b, 2015). Thus there has been little study concerning the B-mode auto-bispectrum with a few exception of models with parity violation Shiraishi (2013); Shiraishi et al. (2013).

The rest of the paper is organized as follows. We first state our notation of bispectrum in Section II. In Section III we show the non-Gaussianity of tensor perturbations in the generalized G-inflation with the Lagrangian of the Horndeski theory. In Section IV we calculate the bispectrum under linear evolution. Finally we show the results in Section V and conclude in Section VI.

II CMB bispectrum and reduced bispectrum

We decompose a CMB anisotopy map $A(\hat{n})$ into a set of coefficients of spherical harmonics.

[TABLE]

The CMB bispectrum is the three-point correlation of the $a_{\ell m}$’s.

[TABLE]

If the Universe is statistically isotropic, then the bispectrum can be angle averaged without any loss of information, that is,

[TABLE]

where \left(\begin{array}[]{@{}c@{\; }c@{\; }c@{}}\ell_{1}&\ell_{2}&\ell_{3}\\[-5.0pt] m_{1}&m_{2}&m_{3}\end{array}\right) is the Wigner-$3j$ symbol. We also define the reduced bispectrum Komatsu and Spergel (2001) as

[TABLE]

with

[TABLE]

where we define

[TABLE]

However $G_{\ell_{1}\ell_{2}\ell_{3}}$ vanishes when $\ell_{1}+\ell_{2}+\ell_{3}=\mathrm{odd}$, and in such a case we redifine $G_{\ell_{1}\ell_{2}\ell_{3}}$ by using identities of Wigner-$3j$ symbols as Kamionkowski and Souradeep (2011)

[TABLE]

III ​​​​Primordial non-Gaussianities in the generalized G-inflation

The generalized G-inflation Kobayashi et al. (2011) is the most general single-field inflation based on the Horndeski theory Horndeski (1974), which is the most general single scalar-tensor theory with second-order field equations. The Lagrangian has four arbitrary functions of a scalar field $\phi$ and $X\equiv-\partial_{\mu}\phi\partial^{\mu}\phi/2$:

[TABLE]

where $R$ is the Ricci scalar and $G_{\mu\nu}$ is the Einstein tensor, $G_{iX}=\partial G_{i}/\partial X$, $(\nabla_{\mu}\nabla_{\nu}\phi)^{2}=\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi$, and $(\nabla_{\mu}\nabla_{\nu}\phi)^{3}=\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\nu}\nabla^{\lambda}\phi\nabla_{\lambda}\nabla^{\mu}\phi$. We use the unitary gauge in which $\phi=\phi(t)$ and write the metric as

[TABLE]

where

[TABLE]

to focus on the tensor perturbations. Then the quadratic and cubic actions for $h_{ij}$ are obtained as below.

[TABLE]

where

[TABLE]

From the quadratic action, the primordial power spectrum of gravitational waves, $\xi^{(s)}({\mathbf{k}})\equiv h_{jk}({\mathbf{k}})e^{*(s)}_{jk}({\mathbf{k}})$, of which $(s)$ stands for the helicity, is given by

[TABLE]

where $e^{(s_{i})}_{jk}=e^{(s_{i})}_{jk}({\mathbf{k}}_{i})$ is a transverse and traceless polarization tensor and it obeys the normalization condition $e^{(s)}_{jk}({\mathbf{k}})e^{*(s^{\prime})}_{jk}({\mathbf{k}})=\delta_{ss^{\prime}}$. From the cubic action, we get the bispectrum of primordial gravitational waves with $\mathcal{P}_{h}^{2}$ Gao et al. (2011)

[TABLE]

[TABLE]

where $k_{t}=k_{1}+k_{2}+k_{3}$, and $f_{5X}$ is given by

[TABLE]

IV Calculation of the bispectrum

Under linear evolution of the perturbations, the $a_{\ell m}$’s are given as

[TABLE]

where the superscript $(s)$ stands for the helicity of cosmological perturbations which induce the CMB anisotropies. Then the bispectrum is represented as the sum of all the helicity contributions

[TABLE]

[TABLE]

Since the Horndeski theory does not violate the parity symmetry, the bispectum vanishes when the total multipole moment is even or odd, depending on the parity of the types of the CMB fluctuations. For example, the auto-bispectrum of B-mode polarization is

[TABLE]

and the auto-bispectrum of E-mode polarization is

[TABLE]

Replacing the polarization tensors with the spin-weighted spherical harmonics, we get the bispectrum originated from the non-Gaussianities of primordial gravitaional waves with the Wigner-$9j$ symbol,

[TABLE]

where $\mathcal{K}=\mathcal{K}_{(\textrm{GR})}+\mathcal{K}_{(5X)}$ represents the wavenumber dependence of the non-Gaussianities with

[TABLE]

Due to the complicated form of $\mathcal{K}(k_{1},k_{2},k_{3})$, $k_{j}$ integration in (26) cannot be performed separately. By using the Laplace transformation,

[TABLE]

however, $k_{t}^{-\nu-1}$ becomes a product of functions of $k_{j}$’s, to make each term in (26) a product of functions of $k_{1}$, $k_{2}$ and $k_{3}$. Although a new integration variable must be introduced here, the exponential factor decays so rapidly that the computation cost can be reduced as in separable templates of non-Gaussianities Smith and Zaldarriaga (2011). We emphasize that the laplace transformation enables us to get the exact primary bispectra of B-mode polarization up to the nonlinearity parameter $f_{5X}$, in contrast to the temperature bispectrum from primordial scalar perturbations which have been studied based on a few templates that are not necessarily derived from specific models of inflation Komatsu (2010); Yadav and Wandelt (2010).

We calculated evolution of linear perturbations with the Boltzmann code CLASS Blas et al. (2011) with the cosmological parameters estimated by the latest observation Ade et al. (2016b) to obtain the transfer functions. We assume the spectrum of primordial gravitational waves is scale invariant, and their amplitude is parametrized by the tensor-to-scalar ratio $r$.

V Results

Once the B-mode auto-bispectrum produced by primordial gravitational non-Gaussianity has been calculated, the maximum possible signal-to-noise ratio where observational errors are dominated by the cosmic variance is given by $(S/N)=1/\sqrt{F^{-1}}$, where $F$ is the Fisher information defined by

[TABLE]

First, we supposed that $f_{5X}=0$ and found that the signal is too small to be detected. We show the signal-to-noise ratios in the zero-noise ideal experiment as bold lines in the figure 1.

Next we assume the case in which $f_{5X}$ is large enough for its non-Gaussianity to dominate. If the coefficient $f_{5X}$ is as large as $10^{7}$ or $3\times 10^{7}$, we are able to detect the bispectrum when tensor-to-scalar ratio $r=0.1$ or $0.01$, respectively, by an observation up to $\ell_{\mathrm{max}}=200$.

We also calculated CMB temperature bispectrum induced by this mode and found that even if $f_{5X}$ takes such a large value, the signal-to-noise ratio of the resultant temperature bispectrum is too small to be detected, so that it would not contradict any current observational data.

We show the shapes of the primordial non-Gaussianities and the B-mode reduced bispectrum (see figure 2-3.) Their shapes are similar in both general relativity and the new class of gravity in which $G_{5X}$ does not vanish.

The B-mode bispectrum $B_{\ell_{1}\ell_{2}\ell_{3}}$ is zero along the lines (see figure 4) where at least two of $\ell_{j}$ are equal, since Wigner-$3j$ symbol \left(\begin{array}[]{@{}c@{\; }c@{\; }c@{}}\ell_{1}&\ell_{2}&\ell_{3}\\[-5.0pt] m_{1}&m_{2}&m_{3}\end{array}\right) is antisymmetric to a replacement of columns when $\ell_{1}+\ell_{2}+\ell_{3}=\mathrm{odd}$. This nature comes from the odd parity of B-mode bispectrum. The bispectrum decreases toward the lines and so the signal-to-noise ratio is smaller than that of the E-mode auto-bispectrum which is not suppressed because it has the even parity.

VI conclusion

In the paper, we calculated the B-mode auto-bispectrum induced by tensor perturbations produced during inflation in the early Universe based on the most general single field inflation model Gao et al. (2011). In this theory it is known that the primordial bispectrum of tensor perturbations consist of two distinct terms, one present in general relativity, and the other propotional to $f_{5X}$. We have found that the former contribution, whose nonlinearity parameter is always unity, cannot be detected by the CMB, whereas the new term may be detectable only if $f_{5X}\gtrsim 10^{7}$

As a future prospect, when 21-cm maps of atomic hydrogen distribution in the dark ages become available, the curl mode of gravitational lensing effect Book et al. (2012) may be the key of detection of the non-Gaussianity of primordial gravitational waves, because 21-cm fluctuations be seen up to $\ell\sim 10^{7}$ in principle and there are many statistically independent fluctuations.

Acknowledgements.

We are grateful to Eiichiro Komatsu for a useful comment in the beginning of the work. HWHT was supported by the Advanced Leading Graduate Course for Photon Science (ALPS). This work was supported by JSPS KAKENHI, Grant-in-Aid for Scientific Research 15H02082 and Grant-in-Aid for Scientific Research on Innovative Areas 15H05888.