Inflationary study of non-Gaussianity using two-dimensional geometrical measures of CMB temperature maps

In this work we effectively calculated the two-dimensional Minkowski functionals for cosmic microwave background (CMB) temperature maps generated by single field models of inflation with a standard kinetic term. We started with a calculation of the bispectrum of initial perturbations and then calculated the two-dimensional configuration space cubic moments for temperature fluctuations. These cubic moments give rise to first order non-Gaussian correction terms to the Minkowski functionals. Thus, we developed a robust mechanism to predict the amount of non-Gaussianity generated by inflation in the CMB temperature maps using Minkowski functionals.

Figure 1

Power spectrum of scalar perturbations $P_{\zeta }(k)$ against momenta $k$ for quadratic potential and step potential with $c=0.002$, $d=0.02M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$.

Figure 2

The AHC and BHC parts of three-point function, or generalized $f_{\mathrm{NL}}$, are plotted against the number of $e$-folds $N=\ln (a)$ while integrating, for equilateral and squeezed triangles. The horizon crossing value ${\tau }^{*}$ corresponds to the two kinks at 6.5 and 8 $e$-folds for the two triangles. The integral in Eq. (14) is divided into two parts as given in Eq. (23). The BHC part, to the left of the kink, is evaluated as the integral from ${\tau }_{*}$ to $\tau \to -\infty$ [$\ln (a)\to 0$] using Cesaro resummation, It plateaus quickly, showing stability of the integration technique. The AHC part is integrated from ${\tau }_{*}$ in the direction of increasing $\tau$, the integration result is shown to the right of the kink, and it also quickly plateaus. The generalized $f_{\mathrm{NL}}$ is the sum of the two integration asymptote (plateau) values, at the left and the right of the plot.

Figure 3

Generalized $f_{\mathrm{NL}}$ for the step potential model plotted against $k=k_1=k_2=k_3$ for the equilateral triangle case with step height and width given by $c=0.002$ and $d=0.02M_{\mathrm{pl}}$, respectively, while the step is located at ${\varphi }_s=15.86M_{\mathrm{pl}}$.

Figure 4

Moments $S_2$, $T_2$, and $U_2$ plotted as a function of smoothing scale $R$ for the quadratic potential model of inflation with transfer function generated by camb code.

Figure 5

Moments $S_2$, $T_2$, and $U_2$ plotted as a function of smoothing scale $R$ for the step model with $c=0.002$, $d=0.02M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ that corresponds to scale $R_s\approx {\tau }_0/2$.

Figure 6

Moments $S_2$, $T_2$, and $U_2$ plotted as a function of smoothing scale $R$ for the step model with $c=0.01$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ that corresponds to scale $R_s\approx {\tau }_0/2$.

Figure 7

Moments $S_2$, $T_2$, and $U_2$ plotted as a function of smoothing scale $R$ for the step model with $c=0.01$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.76M_{\mathrm{pl}}$ that corresponds to scale $R_s\approx {\tau }_0/10$.

Figure 8

Moments $S_2$ are plotted as a function of smoothing scale $R$ for different models and parameters for comparison.

Figure 9

The non-Gaussian correction to filling factor $f_{V_2}$ as a function of threshold $\nu$ for both quadratic and step potentials with $c=0.01$, $d=0.01M_{\mathrm{pl}}$. Left axis for the quadratic potential and right for the step potential while the smoothing scale is $R={\tau }_0/50$.

Figure 10

The non-Gaussian correction to length of isocontours ${\mathcal{N}}_2(\nu )$ as a function of threshold $\nu$ for both quadratic and step potentials. Left axis for the quadratic potential and right for the step potential with $c=0.002$, $d=0.02M_{\mathrm{pl}}$, and ${\varphi }_s=15.66M_{\mathrm{pl}}$ while the smoothing scale is $R={\tau }_0/50$.

Figure 11

The non-Gaussian correction to Euler characteristic ${\chi }_2(\nu )$ as a function of threshold $\nu$ for quadratic potential while the smoothing scale is $R={\tau }_0/50$. Shaded yellow areas are the numerical uncertainty in the value of the moments.

Figure 12

The non-Gaussian correction to Euler characteristic ${\chi }_2(\nu )$ as a function of threshold $\nu$ for step potential with $c=0.002$, $d=0.02M_{\mathrm{pl}}$ in black; $c=0.01$, $d=0.01M_{\mathrm{pl}}$ in blue; and ${\varphi }_s=15.86M_{\mathrm{pl}}$ for both, while the smoothing scale is $R={\tau }_0/50$. Shaded yellow area is the numerical uncertainty in the value of the moments.

Figure 13

The non-Gaussian correction to the length of isocontours ${\mathcal{N}}_2(\nu )$ as a function threshold $\nu$ for step potential with $c=0.01$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ at different smoothing scales $R={\tau }_0/2$ (in black), $R={\tau }_0/10$ (in red), $R={\tau }_0/50$ (in blue), and $R={\tau }_0/100$ (in green).

Figure 14

The non-Gaussian correction to Euler characteristic ${\chi }_2(\nu )$ as a function threshold $\nu$ for step potential with $c=0.01$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ at different smoothing scales $R={\tau }_0/2$ (in black), $R={\tau }_0/10$ (in red), $R={\tau }_0/50$ (in blue), and $R={\tau }_0/100$ (in green).

Figure 15

The non-Gaussian correction to Euler characteristic ${\chi }_2(\nu )$ as a function threshold $\nu$ for step potential with $c=0.1$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ at different smoothing scales $R={\tau }_0/2$ (in black), $R={\tau }_0/10$ (in red), $R={\tau }_0/50$ (in blue), and $R={\tau }_0/100$ (in green).

Figure 16

The non-Gaussian correction to length of isocontours ${\mathcal{N}}_2({\nu }_f)$ as a function of filling factor ${\nu }_f$ for both quadratic and step potentials. Left axis for the quadratic potential and right for the step potential with $c=0.002$, $d=0.02M_{\mathrm{pl}}$, and ${\varphi }_s=15.76M_{\mathrm{pl}}$ while the smoothing scale is $R={\tau }_0/50$.

Figure 17

The non-Gaussian correction to Euler characteristic ${\chi }_2({\nu }_f)$ as a function of ${\nu }_f$ for step potential with $c=0.002$, $d=0.02M_{\mathrm{pl}}$ in black; $c=0.01$, $d=0.01M_{\mathrm{pl}}$ in red; and ${\varphi }_s=15.86M_{\mathrm{pl}}$ for both, while the smoothing scale is $R={\tau }_0/50$. Shaded yellow areas are the numerical uncertainty in the value of the moments.

Figure 18

The non-Gaussian correction to Euler characteristic ${\chi }_2({\nu }_f)$ as a function of filling factor ${\nu }_f$ for step potentials with $c=0.01$, $d=0.01M_{\mathrm{pl}}$, and ${\varphi }_s=15.86M_{\mathrm{pl}}$ at different smoothing scales $R={\tau }_0/2$ (in black), $R={\tau }_0/10$ (in red), $R={\tau }_0/50$ (in blue), and $R={\tau }_0/100$ (in green).