Primordial Non-gaussianity and different inflationary models

See Planck Collaboration: Planck 2013 Results. XXIV. Constraints on primordial NG.
Current observations support a range of inflation models, according to which the probability distribution of primordial fluctuations are very close to Gaussian. Different classes of models have their own characteristic non-Gaussian deviation. They provide us with a method, which will help in isolating the inflation model that accurately describes our Universe.
The simplest inflationary models are based on a set of minimal conditions: (i) a single weaklycoupled neutral single scalar field (the inflaton, which drives inflation and generates the curvature perturbations); (ii) with a canonical kinetic term; (iii) slowly rolling down its (featureless) potential; (iv) initially lying in a Bunch-Davies (ground) vacuum state. In the last few years, an important theoretical realization has taken place: a detectable amplitude of NG with specific triangular configurations (corresponding broadly to well-motivated classes of physical models) can be generated if any one of the above conditions is violated.
The general perturbed metric is given by \begin{equation} d s^{2}=-(1+2 \Psi) d t^{2}+a^{2}(t)\left[(1-2 \Phi) \gamma_{i j}+2 h_{i j}\right] d x^{i} d x^{j} \end{equation} We construct non-Gaussian primordial gravitational potential $\Phi$, which is defined above, as a function of conformal distance by adding a non-linear term as \begin{equation} \Phi(r)=\Phi_{\mathrm{L}}(r)+f_{\mathrm{NL}} \Phi_{\mathrm{NL}}(r) \end{equation} where $\Phi_{\mathrm{NL}}$ is defined as \begin{equation} \Phi_{\mathrm{NL}}(r)=\left(\Phi_{\mathrm{L}}(r)\right)^{2}-\left\langle\left(\Phi_{\mathrm{L}}(r)\right)^{2}\right\rangle \end{equation} and $f_{NL}$ is a measure of the extent of non-Gaussianity in the field. The quantity $\Phi$, sets the initial conditions for the theoretical calculation of CMB temperature and polarization fluctuations. Hence the information about primordial nonGaussianity will be encoded in the CMB fluctuations.

The search for primordial non-Gaussianity requires statistical observables, which are capable of detecting the presence of any non-Gaussian features in the CMB fields. A few examples of this kind of statistical observables in Fourier space are power spectrum, bispectrum, and trispectrum. While the statistical observables such as Betti numbers, Minkowski functionals, Tensor Minkowski Functionals are defined in real space.

Bispectrum

\begin{equation} \left\langle\Phi\left(\boldsymbol{k}_{1}\right) \Phi\left(\boldsymbol{k}_{2}\right) \Phi\left(\boldsymbol{k}_{3}\right)\right\rangle=(2 \pi)^{3} \delta^{(3)}\left(\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3}\right) B_{\Phi}\left(k_{1}, k_{2}, k_{3}\right) \end{equation} The bispectrum $B_{\Phi}\left(k_{1}, k_{2}, k_{3}\right)$ measures the correlation among three perturbation modes. Assuming translational and rotational invariance, it depends only on the magnitudes of the three wavevectors. In general the bispectrum can be written as \begin{equation} B_{\Phi}\left(k_{1}, k_{2}, k_{3}\right)=f_{\mathrm{NL}} F\left(k_{1}, k_{2}, k_{3}\right) \end{equation} Here, $f_{\mathrm{NL}}$ is the so-called "nonlinearity parameter", a dimensionless parameter measuring the $\bf{amplitude}$ of NG. The bispectrum is measured by sampling triangles in Fourier space. The dependence of the function $F\left(k_{1}, k_{2}, k_{3}\right)$ on the type of triangle (i.e., the configuration) formed by the three wavevectors describes the $\bf{shape}$ of the bispectrum (Babich et al. 2004), which encodes much physical information.

Shapes of Bispectrum

Different NG shapes are linked to distinctive physical mechanisms that can generate such non-Gaussian fingerprints in the early Universe.
  1. “local” NG
    2. characterized by a signal that is maximal for “squeezed” triangles with $k_{1} \ll k_{2} \simeq k_{3}$ (or permutations) which occurs, in general, when the primordial NG is generated on super-horizon scales.
  2. “equilateral” NG
    3. peaks for equilateral configurations $k1 \simeq k2 \simeq k3$, due to correlations between fluctuation modes that are of comparable wavelengths, which can occur if the three perturbation modes mostly interact when they cross the horizon approximately at the same time.
  3. “folded” (or flattened) NG
    4. due to correlations between perturbation modes that are enhanced for $k_1 + k_2 \simeq k3$,
  4. “orthogonal” NG
    5. NG that generates a signal with a positive peak at the equilateral configuration and a negative peak at the folded configuration.

Non-primordial sources of NG

  1. instrumental systematic effects (see e.g., Donzelli et al. 2009);
  2. residual foregrounds and point sources;
  3. secondary CMB anisotropies, such as the Sunyaev-Zeldovich (SZ) effect (Zeldovich & Sunyaev 1969), gravitational lensing (see Lewis & Challinor 2006 for a review), the Integrated Sachs-Wolfe (ISW) effect (Sachs & Wolfe 1967) or the Rees-Sciama effect (Rees & Sciama 1968);
  4. and effects arising from nonlinear (second-order) perturbations in the Boltzmann equations (due to the nonlinear nature of General Relativity and the nonlinear dynamics of the photon-baryon fluid at recombination).

Minkowski functionals

Topological concepts

  1. excursion set
  2. simply connected and multiply connected
  3. genus
Table 1. Minkowski functionals $M_{\nu}$ can be related to simple geometric quantities in two and three dimensions. \begin{array}{llll} \hline & $\nu$ & Geometric quantity & Minkowski functional $M_{\nu}$ \\ \hline$d=2:$ & 0 & covered area $F$ & $M_{0}(K)=F(K)$ \\ & 1 & boundary length $U$ & $M_{1}(K)=(1 / 2 \pi) U(K)$ \\ & 2 & Euler characteristic $\chi$ & $M_{2}(K)=(1 / \pi) \chi(K)$ \\ \hline$d=3:$ & 0 & volume $V$ & $M_{0}(K)=V(K)$ \\ & 1 & surface $S$ & $M_{1}(K)=\frac{1}{8} S(K)$ \\ & 2 & mean curvature $H$ & $M_{2}(K)=\left(1 / 2 \pi^{2}\right) H(K)$ \\ & 3 & Euler characteristic $\chi$ & $M_{3}(K)=(3 / 4 \pi) \chi(K)$ \\ \hline \end{array} In two dimensions, the Minkowski functionals are the covered area of the excursion set, the boundary length between the homogeneous domains and the Euler characteristic, i.e. the number difference of connected domains and holes.

The Minkowski functionals of Gaussian fields are known analytically as functions of the threshold.