Distinguishing cosmological models through quantum signatures of primordial perturbations

Abstract

We study the evolution of various measures of quantumness of the curvature perturbation by integrating out the inaccessible entropic fluctuations in the multi-field models of inflation. In particular, we discuss the following measures of quantumness, namely purity, entanglement entropy and quantum discord. The models being considered in this work are ones that produce large scale curvature power spectra similar to those produced by single-field models of inflation. More specifically, we consider different multi-field models which generate nearly scale invariant and oscillatory curvature power spectrum and compare their quantum signatures in the perturbations with the corresponding single-field models. We find that, even though different models of inflation may produce the same observable power spectrum on large scales, they have distinct quantum signatures arising from the perturbation modes. This may allow for a way to distinguish between different models of inflation based on their quantum signatures. Intriguingly, this result generalizes to bouncing scenarios as well.

Appendices

Appendices

A Evolution in the Heisenberg picture

In the main text, we have discussed the evolution of the primordial perturbations in the Schrödinger picture. Here we show how the evolution can be obtained in the Heisenberg picture as well, by promoting the fields to operators. In terms of creation and annihilation operators, the operators associated with the Mukhanov–Sasaki variables and the conjugate momenta can be written as,

$$\begin{aligned} {\hat{v}}_m&=\sum _{n}\left( f_{m n } {\hat{a}}_n + f_{m n}^* {\hat{a}}_n^\dagger \right) \equiv \varvec{F}^{\textrm{T}} \, \hat{\varvec{A}} + \varvec{F}^{\mathrm{T*}} \hat{\varvec{A}}^\dagger ~, \end{aligned}$$

(45a)

$$\begin{aligned} {\hat{p}}_m&=\sum _{n}\left( \pi _{nm} {\hat{a}}_n + \pi _{nm}^* {\hat{a}}_n^\dagger \right) \equiv \varvec{\Pi }^{\textrm{T}} \, \hat{\varvec{A}} + \varvec{\Pi }^{\mathrm{T*}} \hat{\varvec{A}}^\dagger ~, \end{aligned}$$

(45b)

where, the elements of the matrices $\varvec{F}$ and $\varvec{\Pi }$ are the mode functions associated with the respective operators. From the Heisenberg equations of motion, one can see that the mode functions, $\varvec{F}$ and $\varvec{\Pi }$ obey the classical equations as

$$\begin{aligned} \varvec{F}'&= \varvec{\Pi } + \varvec{A}\,\varvec{F}~, \end{aligned}$$

(46a)

$$\begin{aligned} \varvec{\Pi }'&= -\varvec{\Pi }\,\varvec{A} -2\,\varvec{F}\,\varvec{B}~. \end{aligned}$$

(46b)

From the commutation relations between the creation and annihilation operators,

$$\begin{aligned} \left[ {\hat{a}}_m,{\hat{a}}_n\right] =\delta _{mn}~, \end{aligned}$$

(47)

we can obtain the Wronskian associated with the mode functions as

$$\begin{aligned} \varvec{F}\varvec{\Pi }^\dag - \varvec{\Pi } \varvec{F}^\dag =i\, \varvec{I}~. \end{aligned}$$

(48)

Therefore, the two-point correlation functions associated with the above fields can be expressed in terms of the mode functions as,

$$\begin{aligned} \left\langle {\hat{v}}_{m}{\hat{v}}_{n}\right\rangle =&\varvec{F}^\dag \varvec{F}~, \end{aligned}$$

(49a)

$$\begin{aligned} \left\langle {\hat{p}}_{m}{\hat{p}}_{n}\right\rangle =&\varvec{\Pi }^\dag \varvec{\Pi }~, \end{aligned}$$

(49b)

$$\begin{aligned} \left\langle {\hat{v}}_{m}{\hat{p}}_{n}+{\hat{p}}_{n}{\hat{v}}_{m}\right\rangle =&\varvec{F}^\dag \varvec{\Pi }+\left( \varvec{F}^\dag \varvec{\Pi }\right) ^*~. \end{aligned}$$

(49c)

In the Heisenberg formalism, one can solve the equations of motion provided in Eq. (46) and obtain the correlation functions using Eq. (49). Recall that, in the Schrödinger formalism, one also solves for the matrix $\varvec{\Omega }$, appearing in the Schrödinger wave function, using Eq. (6) and obtain the correlation function from Eq. (9). The situation in the Heisenberg picture can be obtained by identifying the relation between $\varvec{\Omega }$ and the mode functions defined in Eq. (45). In other words, this is done by using Eq. (7) to identify $\Omega$ as

$$\begin{aligned} i\,\varvec{\Omega }=\left( \varvec{F}^{-1}\, \varvec{\Pi }\right) ^*~, \qquad \textrm{or}\qquad i\, \varvec{\Omega }=\varvec{F}^{-1} \, \varvec{\Pi }~. \end{aligned}$$

(50)

This provides the connection between the evolution in the Schrödinger and the Heisenberg picture.