Validity of semiclassical limit to quantum gravity in two-mode oscillating quantized massive scalar field quantum cosmology

Eur. Phys. J. C (2022) 82:333 https://doi.org/10.1140/epjc/s10052-022-10248-6 Regular Article - Theoretical Physics Validity of semiclassical limit to quantum gravity in two-mode oscillating quantized massive scalar field quantum cosmology Meghna Rathorea , Renu Dhayalb , K. K. Venkataratnamc Department of Physics, Malaviya National Institute of Technology Jaipur, Jaipur 302017, India Received: 3 February 2021 / Accepted: 23 March 2022 © The Author(s) 2022 Abstract Semiclassical Einstein equations are used to describe the interaction of the back-reaction of the classical gravitational field with quantum matter fields in semiclassical gravity. We in our previous studies have made use of the semiclassical approximation to demonstrate the phenomenon of particle production, often called preheating/reheating of the universe, which occurs after the inflationary epoch during the oscillatory phase of two-mode quantized scalar field of chaotic inflationary model. During this oscillatory phase, back-reaction effects from the created particles, on account of the quantum nature of the states considered, could be significant and one might be concerned about the validity of the semiclassical approximation in these two-mode quantum optical states. The validity of the semiclassical approximation in these states is examined and it is presented how the magnitude of states parameter draws limit on the applicability and reliability of semiclassical theory of gravity. It is argued that semiclassical theory to gravity is a good approximation for states which are closer to coherent states i.e., with coherent parameters greater than unity and with squeezed parameter much smaller than unity. 1 Introduction In standard cosmology, the description of the early universe is based on the Friedmann equations with scalar field(s). The Friedmann equations involve the classical description of the gravity and scalar field equation are defined on the Friedmann–Robertson–Walker (FRW) metric, which suggests that the background scaling is examined as classical and the corresponding source of gravity as unquantized scalar field, assuming its validity holds even at the earliest stage of a e-mail: b e-mail: c e-mail: (corresponding author) 0123456789().: V,-vol the universe. Although, effects of quantum gravity are negligible during this time but still quantum fluctuations and quantum implications of the matter fields are believed to contribute significantly. Therefore, a complete description of a cosmological model would require both the gravity and the matter field(s) being treated quantum mechanically. There are several difficulties that emerge when attempting to combine General Theory of Relativity (GTR) and Quantum Field Theory (QFT) to form a complete theory of gravity. Definitely, a consistent quantum theory would involve quantization of metric together with the matter fields, which would certainly alter the concept of spacetime and demands a completely different theory from classical general relativity. However, in the regime where the curvature is small, spacetime is assumed to be a classical entity, although matter field in spacetime is quantum. Therefore, the semiclassical estimation to the quantum gravity [1] is usually considered to be sufficient as it would provide some insight into the structures of the full, elusive theory and in an appropriate limit would also reproduce the notion of classical spacetime. Thus, description of the early universe with an appropriate cosmological model can be studied in terms of the semiclassical Friedmann equations, where gravity can be treated as classical with quantized matter field(s). In semiclassical approximation and inflationary scenario quantum properties of a single homogeneous massive scalar field, inflaton, responsible for the accelerated expansion of the universe has been a work of great interest among researchers for the last two decades [2–9]. The previously mentioned studies have showed that there are significant dissimilarities between the result obtained in classical approximation to gravity from that obtained in the semiclassical approach to gravity (SG), thereby showing quantum implications and phenomenon play a prominent role in the inflationary scenarios and related problems. These studies also reveal that quantum optics nonclassical state formalisms, coherent and squeezed state, are exceptionally helpful to investigate 123 333 Page 2 of 38 chaotic inflationary scenario in connection to the SG and reveals that a large set of initial quantum states are probable for an inflationary scenario to occur [10–22]. We in one of our previous papers [23,24] have made a similar attempt to study the development of a coherently oscillating massive scalar field minimally coupled to the flat FRW Universe using the semiclassical quantum gravity derived from the canonical quantum gravity by the application of two-mode (TM) quantum optical states formalism. Our findings in the paper showed that in the oscillatory phase of the quantum scalar field, the quantum states (TM coherent and squeezed states formalism) obeying the time-dependent Schrödinger equation leads the same power-law expansion of the universe as 2 that of the matter-dominated era i.e., τ 3 [23]. However, one striking dissimilarity is that the SG does not show any oscillatory behaviour of the Hubble constant, in strong contrast with the oscillatory behaviour of classical gravity, thus, with an implication that entangled TM coherent (ETMC) states and TM squeezed entangled coherent (TMSEC) state can also be the possible states of the fields residing the universe at the time of an oscillatory phase of the scalar field [23,24]. Recently, we have studied cosmological particle production due to the quantum fluctuations in an oscillatory phase of the massive scalar field [24]. The semiclassical limit for the gravity was considered, whereas the scalar field is treated quantum mechanically and their dynamics were studied for TM nonclassical states (entangled and non-entangled TM coherent and squeezed state formalism) of the latter. The back-reaction of the quantum field process was included and the dynamics of spacetime are driven by the expectation value of the energy–momentum tensor (EMT) operator of the quantum matter field. However, this semiclassical approach to quantum gravity, wherein the gravitational field is described by the semiclassical Einstein equation which has as a source the expectation value in some quantum state of the matter stress tensor operator, has limits for its validity and applicability. It is limited in the sense that it does not describe quantum fluctuations of gravity. These fluctuations can directly arise from the dynamical degrees of freedom of the gravitational field itself and are termed as active (or spontaneous) fluctuations [25–27]. This is one aspect of the interaction of gravity with quantum matter field(s). Moreover, when scales involved are far away fr (...truncated)