Limits on Isocurvature Perturbations from Non-Gaussianity in WMAP Temperature Anisotropies

TL;DR

This work systematically investigates how primordial isocurvature perturbations imprint non-Gaussian signatures in the CMB, via two generic non-linear forms: a Linear Model with a Gaussian term plus a quadratic correction and a Quadratic Model that is purely Gaussian-squared. It derives analytic CMB bispectra and perturbative Minkowski Functionals for these models, and constrains the associated non-Gaussianity using WMAP 5-year data. The key finding is that quadratic isocurvature perturbations can yield detectable non-Gaussianity (up to $f_{\rm NL}\sim 30$) while respecting power-spectrum bounds, leading to a tight upper limit $\alpha<0.070$ (95% CL) for a scale-invariant spectrum; linear-model non-Gaussianity requires extremely large $f_{\rm NL}^{(ISO)}$ to be competitive. The results have implications for axion isocurvature scenarios and motivate complementary analyses with higher-order statistics to robustly probe primordial isocurvature physics.

Abstract

We study the effect of primordial isocurvature perturbations on non-Gaussian properties of CMB temperature anisotropies. We consider generic forms of the non-linearity of isocurvature perturbations which can be applied to a wide range of theoretical models. We derive analytical expressions for the bispectrum and the Minkowski Functionals for CMB temperature fluctuations to describe the non-Gaussianity from isocurvature perturbations. We find that the isocurvature non-Gaussianity in the quadratic isocurvature model, where the isocurvature perturbation S is written as a quadratic function of the Gaussian variable sigma, S=sigma^2-<sigma^2>, can give the same signal-to-noise as f_NL=30 even if we impose the current observational limit on the fraction of isocurvature perturbations contained in the primordial power spectrum alpha. We give constraints on isocurvature non-Gaussianity from Minkowski Functionals using WMAP 5-year data. We do not find a significant signal of the isocurvature non-Gaussianity. For the quadratic isocurvature model, we obtain a stringent upper limit on the isocurvature fraction alpha<0.070 (95% CL) for a scale invariant spectrum which is comparable to the limit obtained from the power spectrum.

Figures (5)

• Figure 1: Angular power spectra $C_l$ for adiabatic $\zeta\zeta$ and isocurvature perturbations ${\cal SS}$. The plotted adiabatic perturbation has the spectral index $n_\phi=0.96$ (solid). For isocurvature perturbations, the spectral index of a Gaussian variable $n_\eta$ is 1 (long-dashed) in the Linear Model and $n_\sigma$ is 1 (short-dashed), 1.5 (dotted) and 2 (dot-dashed) in the Quadratic Model (see § \ref{['sec:quadratic']}). The isocurvature fraction $\alpha$ is set to be 0.067 ($n_\eta=1$ and $n_\sigma=1$), 0.008 ($n_\sigma=1.5$), and 0.001 ($n_\sigma=2$) defined at $k_0=0.002$Mpc$^{-1}$.
• Figure 2: CMB angular bispectra of equilateral configurations $l^2(l+1)^2b_{lll}/(2\pi)^2$ in the Linear Model; the adiabatic component $b_{lll}^{\zeta\zeta\zeta}$ (solid), the mixed components $b_{lll}^{(\zeta\zeta{\cal S})}$ (long-dashed) and $b_{lll}^{(\zeta{\cal SS})}$ (short-dashed) and the isocurvature component $b_{lll}^{\cal SSS}$ (dotted). Upper (Lower) panel shows the positive (negative) side of bispectra plotted in logarithmic scale. The adiabatic perturbation has the power-law index $n_{\phi}=0.96$ and its quadratic amplitude $f_{\rm NL}=50$. The isocurvature perturbation has $n_{\eta}=1$ and $f_{\rm NL}^{(ISO)}=10^4$. The fraction of the isocurvature power spectrum $\alpha$ is set to be an axion-type upper limit $0.067$ for $b_{lll}^{{\cal SSS}}$ and $b_{lll}^{(\zeta{\cal SS})}$ with a weak correlation $|\cos\theta_{\phi\eta}|=0.1$ and a curvaton-type upper limit $0.0037$ for $b_{lll}^{(\zeta\zeta{\cal S})}$ with a strong correlation $|\cos\theta_{\phi\eta}|=1$.
• Figure 3: CMB angular bispectra of equilateral configurations $l^2(l+1)^2b_{lll}/(2\pi)^2$ in the Quadratic Model; the isocurvature components $b_{lll}^{\cal SSS}$ ( Left) and the mixed components $b_{lll}^{(\zeta\zeta{\cal S})}$ ( Right). The power-law index of isocurvature perturbation $n_\sigma$ is set to be 1 (long-dashed), 1.5 (short-dashed) and 2 (dotted). For comparison, the adiabatic bispectrum $b_{lll}^{\zeta\zeta\zeta}$ with $f_{\rm NL}=50$ is plotted in both panels (thin solid lines). The fraction of isocurvature power spectrum $\alpha$ is 0.067 ($n_\sigma=1$), 0.008 ($n_\sigma=1.5$) and 0.001 ($n_\sigma=2$) defined at $k_0=0.002{\rm Mpc}^{-1}$. Thick solid lines in Left panels show the full calculations of isocurvature terms (eq.[\ref{['eq:cbis_full']}]) for each $n_\sigma$ at $l\le 10$. The box-size $L_{\rm max}$ is set to be 30Gpc.
• Figure 4: Skewness parameters $S^{(k)}$ ( left:$k=0$, center:$k=1$, right:$k=2$) of each component in the Quadratic Model plotted as a function of $\theta_s$. The skewness values for a pure isocurvature component are plotted in upper panels, while those for a mixed component are plotted in lower panels. The parameters of isocurvature perturbation are $n_\sigma$=1 with $\alpha=0.067$ (long-dashed), $n_\sigma$=1.5 with $\alpha=0.008$ (short-dashed), and $n_\sigma$=2 with $\alpha=0.001$ (dotted). For comparison, the adiabatic skewness with $n_\phi=0.96$ and $f_{\rm NL}=50$ are plotted with solid lines in all panels.
• Figure 5: Non-Gaussian term of Minkowski Functionals $\Delta v_k$ (eq.[\ref{['eq:delmf_pb']}]) from isocurvature bispectrum in Quadratic Model ( left:$k=0$, center:$k=1$, right:$k=2$). The isocurvature perturbation has a spectral index $n_\sigma$=1 and the fraction $\alpha$ is $0.067$. The smoothing scales are shown by different lines: $\theta_s=10$ (solid), $\theta_s=20$ (long-dashed), $\theta_s=40$ (short-dashed) $\theta_s=70$ (dotted), and $\theta_s=100$ (dot-dashed) arcmin.