Hemispherical asymmetry from an isotropy violating stochastic gravitational wave background

Measurement of cosmic microwave background (CMB) temperature by Planck has resulted in extremely tight constraints on the $\mathrm{\Lambda }\mathrm{CDM}$ model. However the data indicate evidence of dipole modulated temperature fluctuations at large angular scale which is beyond the standard statistically isotropic (SI) $\mathrm{\Lambda }\mathrm{CDM}$ model. The signal measured by Planck requires a scale dependent modulation amplitude that is beyond the scope of the phenomenological model considered by Planck. We propose a phenomenological model with mixed modulation field for scalar and tensor perturbations which affect the temperature fluctuations at large angular scales. Hence this model is a possible route to explain the scale dependent nature of the modulation field. The salient prediction of this model is the direction dependent tensor to scalar ratio which results in an anisotropic stochastic gravitational wave background (SGWB). This feature is potentially measurable from the $B$-mode polarization map of Planck and BICEP-2 and leads to determination of the modulation strength. Measurability of SI violated polarization field due to this model is estimated for Planck and PRISM. Absence of the signal in the polarization field can restrict the viability of the model.

Figure 1

The figure shows the scalar and tensor metric perturbation contributions to the CMB angular power spectra generated using CAMB [20] with the value of tensor to scalar ratio $(r)$ as 0.2. Note that while $C_l^{TT}$ and $C_l^{EE}$ have large contributions coming from scalar metric perturbations, the $C_l^{BB}$ spectrum on large angular scales is purely sourced by tensors.

Figure 2

Estimation of tensor modulation power $m_1/\pi$ for different values of $r_0$ from the Planck SMICA map. This shows a decrease in $m_1$ with an increase in $r_0$.

Figure 3

(a) In this plot we show the possible range of $r_{\text{BICEP}},⟨r⟩$ for different values of modulation strength $\alpha A$ (shown in the color bar). This shows that with the measured value of $r_{\text{BICEP}}$ in the BICEP patch and all sky average value $⟨r⟩$, we can estimate the value of $\alpha$ uniquely. (b) In this plot we show the determination of $r_0$ (shown in the color bar) from the measurement of $r_{\text{BICEP}}$ and $⟨r⟩$. From the measurement of a particular value of $r_{\text{BICEP}}$ and $⟨r⟩$, both the values of $\alpha A$ and $r_0$ can be determined.

Figure 4

The dipole modulation $\frac{{\alpha }^2{m}_1}{\pi }$ is plotted for different values of $r_0$. In the solid line we plot $2\sigma$ and $3\sigma$ curve of the reconstruction noise $(\sigma =N_1/\pi )$ upto $l_{\text{max}}=64$ as a function of $r_0$ after incorporating Planck instrumental noise for frequency channel $\nu =143\mathrm{GHz}$. In the dashed red line we depict the $3\sigma$ curve of the reconstruction noise for PRISM. The effect of this model is readily measurable in the $B$-mode BipoSH spectra from PRISM for a wide range of $\frac{{\alpha }^2{m}_1}{\pi }$.