Przemysław Małkiewicz

We summarize our research program on the use of coherent states and covariant integral quantization in quantum cosmology. In particular, we present a recent development within this framework and include new results that shed light on some of its basic properties. Specifically, we investigate the quantum dynamics of a perturbed, fluid-filled Friedmann universe beyond the standard approximation in which the total state factorizes into background and perturbation wave functions. We assume the background geometry to be a superposition of two distinct coherent states—effectively a quantum cat state with no classical counterpart—each coupled to inhomogeneous perturbations. Starting from vacuum initial conditions, we analyze the evolution of a contracting universe through a bounce into the expanding phase. We find that an initially factorized state evolves into a biverse. This state consists of two distinct semiclassical branches, each described by a single coherent state and carrying enhanced perturbations in a slightly non-Gaussian state. We then explore how this dynamics depends on key model parameters, such as the perturbation wavelength and the choice of background solutions, and study their impact on the interaction between branches. The observed universe is assumed to correspond to one branch of this biverse state. This scenario illustrates how genuinely quantum properties of the background geometry may leave observable imprints in the early universe.

We show that the consistent application of the rules of quantum mechanics to cosmological systems inevitably results in the so-called multiverse states in which neither the background spacetime nor the inhomogeneous perturbation are in definite states. We study the multiverse states as perturbations to the usually employed so-called Born-Oppenheimer states that are products of a wave function of the background and a wave function of the perturbation. The obtained corrections involve integrals over \emph{virtual backgrounds} that represent the effect of quantum background fluctuations on the perturbation state. They resemble loop corrections in quantum field theory. This approach demonstrates the inevitable existence of very specific non-Gaussian features in primordial fluctuations. We express the resulting non-Gaussian perturbation as a nonlinear function of the Gaussian perturbation obtained within the Born-Oppenheimer approximation, and compute its trispectrum, to show that the multiverse scenario leads to testable and distinct signatures in cosmological perturbations. Our approach applies both to inflationary and alternative cosmologies.

We propose a new method to investigate signatures of a quantum gravity phase in the primordial state of cosmological perturbations. We formulate and study a quantum model of a perturbed Friedmann-Lemaître-Robertson-Walker universe beyond a tensor-product Born-Oppenheimer-like factorization, that is, without restricting the wave function of the universe to the product of the background and perturbation wave functions. We show that the quantum dynamics generically does not preserve the product form of the universe’s wave function, which spontaneously evolves into a more general entangled state. Upon expanding this state in a suitable basis of background wave functions and setting Gaussian initial conditions for the perturbations, we numerically find that each of these wave functions becomes associated with a non-Gaussian state of an inhomogeneous perturbation.

We study the nonunitary relation between quantum gravitational models defined using different internal times. We show that, despite the nonunitarity, it is possible to provide a prescription for making unambiguous, though restricted, physical predictions independent of specific clocks. To illustrate this result, we employ a model of quantum gravitational waves in a quantum Friedmann universe.

We develop a relativistic framework to investigate the evolution of cosmological structures from the initial density perturbations to the highly nonlinear regime. Our approach involves proposing a procedure to match ``best-fit", exact Bianchi IX (BIX) spacetimes to finite regions within the perturbed Friedmann-Lemaitre-Robertson-Walker universe characterized by a positive averaged spatial curvature. This method enables us to approximately track the nonlinear evolution of the initial perturbation using an exact solution. Unlike standard perturbation theory and exact solutions with a high degree of symmetry (such as spherical symmetry), our approach is applicable to generic initial data, with the only requirement being positive spatial curvature. By employing the BIX symmetries, we can systematically incorporate the approximate effects of shear and curvature into the process of collapse. Our approach addresses the limitations of both standard perturbation theory and highly symmetric exact solutions, providing valuable insights into the nonlinear evolution of cosmological structures.

We develop a relativistic framework to investigate the evolution of cosmological structures from the initial density perturbations to the highly nonlinear regime. Our approach involves proposing a procedure to match “best-fit,” exact Bianchi IX (BIX) spacetimes to finite regions within the perturbed Friedmann-Lemaître-Robertson-Walker universe, characterized by a positive averaged spatial curvature. This method enables us to approximately track the nonlinear evolution of the initial perturbation using an exact solution. Unlike standard perturbation theory and exact solutions with a high degree of symmetry (such as spherical symmetry), our approach is applicable to generic initial data, with the only requirement being positive spatial curvature. By employing the BIX symmetries, we can systematically incorporate the approximate effects of shear and curvature into the process of collapse. Our approach addresses the limitations of both standard perturbation theory and highly symmetric exact solutions, providing valuable insights into the nonlinear evolution of cosmological structures.

A quantum cosmological bouncing model may exhibit an ambiguity stemming from the nonclassical nature of the background evolution: two classically equivalent theories can produce two qualitatively different potentials sourcing the perturbations. We derive explicitly the quantum canonical transformation involving the quantum background to show how it leads to inequivalent theories. We identify the relevant quantum parameter describing the difference and expand upon the ambiguity by calculating the expected power spectra produced for initial quantum vacuum fluctuations in the contracting phase of both potentials. We find that one spectral index corresponds to all values of this parameter but one, while the other thus represents a set of measure zero.

A quantum cosmological bouncing model may exhibit an ambiguity stemming from the nonclassical nature of the background evolution: Two classically equivalent theories can produce two qualitatively different potentials sourcing the perturbations. It reflects the general ambiguity in quantization of the gravitational field at linear order. We derive explicitly the quantum canonical transformation of linear perturbations involving the quantum background to show how it leads to inequivalent theories. We identify the relevant quantum parameter describing the difference and expand upon the ambiguity by calculating the expected power spectra produced for initial quantum vacuum fluctuations in the contracting phase of both potentials. We find that one spectral index corresponds to all values of this parameter but one, while the other thus represents a set of measure zero.