Optimal estimator for resonance bispectra in the CMB

We propose an (optimal) estimator for a CMB bispectrum containing logarithmically spaced oscillations. There is tremendous theoretical interest in such bispectra, and they are predicted by a plethora of models, including axion monodromy models of inflation and initial state modifications. The number of resolved logarithmical oscillations in the bispectrum is limited due to the discrete resolution of the multipole bispectrum. We derive a simple relation between the maximum number of resolved oscillations and the frequency. We investigate several ways to factorize the primordial bispectrum, and conclude that a one-dimensional expansion in the sum of the momenta $\sum k_i=k_t$ is the most efficient and flexible approach. We compare the expansion to the exact result in multipole space and show for ${\omega }_{\text{eff}}=100$ that $\mathcal{O}(10^3)$ modes are sufficient for an accurate reconstruction. We compute the expected ${\sigma }_{f_{\mathrm{NL}}}$ and find that within an effective field theory (EFT) the overall signal to noise scales as $S/N\propto {\omega }^{3/2}$. Using only the temperature data we find $S/N\sim \mathcal{O}(1–10^2)$ for the frequency domain set by the EFT.

Figure 1

Examples of oscillating primordial bispectra $\sin \left(\omega \log \frac{{\ell }_t}{x_{\text{rec}}}\right)$ for $\omega =5$ (red, dashed) and $\omega =12$ (blue, solid).

Figure 2

Lowest multipole ${\ell }_{\min }$ above which the oscillations can be resolved and number of oscillations $N_{\text{osc}}$ resolved between ${\ell }_{\min }$ and ${\ell }_{\max }=3000$.

Figure 3

The integrand of the expansion for $\sin (\omega \log a_t)$. Although this function appears integrable, numerically it is very unstable and not usable for an efficient KSW estimator, prompting us to explore other approaches.

Figure 4

The ratio between the analytical SW approximation and the numerical equivalent. As pointed out by Fergusson and Shellard, one gets a deviation at high $\ell$ due to early truncation. For our purposes, these errors are small enough, as we only need these computations to compare with our expansion. This spectrum was generated with 1000 steps in the $\alpha -\beta$ plane.

Figure 5

The ratio between the oscillating shape for $\omega =100$ and the constant shape. For large frequencies the ratio is always $<1$, making this shape hard to observe unless the amplitude $f_{\text{NL}}$ is large.

Figure 6

Example of the shape function expansion for $\omega =10$. Top: Shape function as a function of the effective multipole scale $\ell =k{x}_{\text{rec}}$. The red dashed line is the original shape function; the black line is the expansion in terms of 600 Fourier modes. Bottom: Corresponding Fourier coefficients $a_n$ and $b_n$.

Figure 7

Comparison between the modal expansion result and the exact result. Top: $\ell ={\ell }_1={\ell }_2={\ell }_3$. Bottom: ${\ell }_1=100,\ell ={\ell }_2={\ell }_3$ with $\omega =50$. Other frequencies work with comparable precision.

Figure 8

Fisher forecast for the precision on the amplitude $f_{\text{NL}}$ on the shape in Eq. (5), assuming an optimal CMB experiment with ${\ell }_{\text{max}}=2000$. We see that approximately ${\sigma }_{\omega }\propto \omega$.

Figure 9

Correlation matrix of logarithmic oscillations. Left: Sine to sine correlation. The frequency resolution is about $\mathrm{\Delta }\omega =1$. Right: Sine to cosine correlation. A strong degeneracy between neighboring frequencies and their phase is observed.

Figure 10

Signal-to-noise ratio $S/N=f_{\text{NL}}\sqrt{F_{BB}}$ for different oscillation parameters ${\epsilon }_{\text{osc}}$ as a function of the oscillation frequency $\omega$, based on the study in [36]. The dashed black line indicates the asymptotic scaling behaviour. Frequencies of order $\mathcal{O}(10^1)$ are observable, depending on ${\epsilon }_{\text{osc}}$.