Unitary quantum matter-bounce in a universe with a positive cosmological constant

Abstract We analyze the Wheeler-DeWitt quantization of a spatially flat Friedmann–Lemaïtre–Robertson–Walker universe containing pressureless dust and a positive cosmological constant ( $$\Lambda &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ). Following relational time framework, we establish a direct mathematical correspondence between the cosmological Hamiltonian and the radial Schrödinger equation for the scattering states of the non-relativistic hydrogen atom. This exact solvability allows us to rigorously construct the physical Hilbert space and ensure the self-adjointness of the Hamiltonian. As a concrete result, we show that the wave packets unitarily evolve depicting a non-singular quantum bounce, systematically replacing the classical Big Bang singularity. Finally, we discuss the physical relevance of this exact solution within the matter-bounce scenario. We demonstrate that this framework provides a robust quantum origin for a bounce during a dust-dominated contracting phase—a necessary prerequisite for generating a scale-invariant spectrum of primordial perturbations—derived from the unitary dynamics of the quantized background.