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

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.


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: 2015rpy9026@mnit.ac.in b e-mail: 2015rpy9062@mnit.ac.in c e-mail: kvkamma.phy@mnit.ac.in (corresponding author) 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][3][4][5][6][7][8][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 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][11][12][13][14][15][16][17][18][19][20][21][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 that of the matter-dominated era i.e., τ 2 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][26][27]. This is one aspect of the interaction of gravity with quantum matter field(s). Moreover, when scales involved are far away from Planck dimensions there comes the second aspect of this interaction which is the quantum back-reaction of the quantum fields on the spacetime. Quantum fluctuations of the quantum stress tensor which drive fluctuations of the spacetime geometry are termed as passive (induced) fluctuations [28][29][30][31][32]. This paper primarily focuses on the latter aspect of the interaction of gravity with the matter field(s). Several authors in the recent years have discussed the fluctuations of quantum stress tensors and their physical effects [33][34][35][36][37][38].
Among the issues of interest is the extent to which these fluctuations of the stress tensor in different TM entangled and non-entangled coherent and squeezed state formalism place limits on the validity of the SG during the oscillatory phase of the scalar field. This is important to analyse because the quantum nature of the TM nonclassical states leads to a large amount of particle creation due to non-adiabatic amplification [24] and as a result the corresponding energy density may fluctuate so widely, resulting in large metric fluctuations.
Since we have used the TM entangled and non-entangled coherent and squeezed states to represent the scalar field in the SG in our previous publications, it would be useful to examine whether these states are valid in the semiclassical regime. Thus, in this paper, by using the framework of the Chaotic inflationary scenario, quantum fluctuations of the stress tensor of a real, massive, minimally coupled scalar field are discussed, as are the resulting spacetime metric fluctuations. These fluctuations of the stress tensor were calculated during the scalar field's oscillatory phase in all the TM nonclassical states formalism. Based on the stress tensor fluctuations, we will examine the limit to the validity of the semiclassical approximation in different quantum states considered.
The outline of the paper is as follows: In Sect. 2, we discuss the massive chaotic scalar field inflationary model, we intend to study, in SG. Then in Sect. 3, we introduce and review the ideas of coherent and squeezed states formalism, for single and two-mode cases. In Sect. 4 we then present the nonclassical TM states representation of the scalar field in the SG. In Sect. 5 we describe the criterion suggested earlier by Kuo and Ford [29] to draw the limits on the validity of the semiclassical theory of gravity. Motivated by this fact we then utilize this criterion to study the reliability and validity of the SG theory for our model, when the TM quantized scalar field modes are assumed to be in entangled and nonentangled coherent and squeezed states formalism. Finally, in Sect. 6, we discuss and present the conclusions that can be drawn from our analysis.

Scalar field dominated cosmology in semiclassical gravity
Semiclassical approximation to gravity is considered to be the first estimation to the theory of quantum gravity where the gravitational field is offered a classical definition (not quantized) in harmony with the axioms of GTR as curvature in the geometry of spacetime and the field which is present in spacetime are treated quantum mechanically in agreement with the notions of QFT. This semiclassical theory is considered as an approximation to the yet unknown but true quantum theory of gravitation interacting with other fields. However, when the gravity's quantum nature becomes important, i.e. at the Planck scale and even at distant from the Planck scale, where the stress tensor fluctuations become important, this theory ceases to work at those places [28]. In semiclassical theory, the Einstein field equation is usually formulated as (withh = c = 1 and G = 1 m 2 p ): here, G μν is the Einstein tensor and T μν , the source of the gravitational field, is the expectation value of the EMT for the matter field (ϕ) in a desirable nonclassical state under examination with quantum state satisfying the time-dependent Schrödinger equation: where |ψ is the matter field's quantum state andĤ m is the Hamiltonian operator guiding the quantum state. The homogeneous and isotropic solution for the gravitational field is the FRW-metric, specified by: where a spatially flat universe was considered for simplicity with a dimensionless parameter S(τ ) as the cosmic scale factor to account for the relative evolution of the universe. We are interested in a problem for which there exists a space-like equal time hypersurface t , on which the field can be decomposed into the homogeneous part ϕ 0 (τ ) and a small fluctuation ζ(x, τ ): here, ϕ 0 (τ ) is the 'classical' (infinite wavelength) field, that is the expectation value of the inflaton field on the initial isotropic and homogeneous state, while ζ(τ, x) represents the small quantum fluctuation around ϕ 0 (τ ).
We are considering small inhomogeneity within the massive scalar field, however, we are considering them to be too small such that the wavelength associated with them is large and can be approximated nearly on the homogeneous and isotropic metric Eq. (3). In this background metric minimally coupled massive scalar field of mass, m satisfies the Klein-Gordon equation: where g μν is the metric tensor for the line element Eq. (3), ∇ μ is the covariant derivative and μ, ν = 0, 1, 2, 3. We have assumed nominal coupling between the gravity and the ϕ and the latter dynamics is governed by the Lagrangian density for the massive scalar field and is given by: Using the metric Eq. (3) on Eq. (6), Eq. (5) becomes: where,φ stands for a derivative with respect to time τ and ∂ j = ∂ ∂ x j represents the spatial derivative. In keeping the analogy with point mechanics, we can define a conjugate momentum π and the corresponding Hamiltonian for the massive scalar field is obtained as [1]: We quantize the scalar field according to the functional Schödinger-picture Eq. (2). Knowing the Fock space of exact quantum states for a time-dependent harmonic oscillator, we decompose the Hamiltonian into a sum of harmonic oscillators. For this purpose, we decompose the field into Fourier-modes as However, it is convenient to decompose the field into Fourier modes and then to redefine the Hermitian Fourier modes as [39][40][41] ζ (+) where ζ (+) k ) are the Fourier-cosine and sine modes, respectively. Space integrations can easily be done for the field and momentum: For simplicity reason a compact notation α will be used for {k, (±)}. Then the actual Hamiltonian of symmetric state modes takes the form [39][40][41]: That is, the Hamiltonian Eq. (12) is a countably infinite sum of time-dependent harmonic oscillators where, The functional Schrödinger equation (2) becomes the ordinary Schrödinger equation for the infinite system. As every mode is decoupled from each other, the wave function to Eq. (2) is now given by the product of the wave functions of each Schrödinger equation The ζ α for each mode satisfies the classical field equations: Also the T μν , energy-momentum tensor, is defined as We see immediately, using the definition of the canonical momentum, π(x) that temporal component of energymomentum tensor T 00 is the Hamiltonian density.
3 Quantum mechanical states formalism

Single-mode (SM) states
Coherent (C) states are quasi-classical states with minimum uncertainty such that their quantum-mechanical uncertainty is equally distributed between the canonically conjugate variables. As a consequence, C state uncertainty region is demonstrated through a circle in the optical phase space. This representation is popularly known as Glauber-Sudarshan Prepresentation [42,43] and the states are known as Glauber states. A Single-mode coherent (SMC) state can originate through the operation of displacement operator D( ) on the vacuum (V) state |0 (a reference state) and is defined as [11]: where here, is a complex number with = | | e iθ and | | and θ corresponds to the amplitude and phase of the C state | and a,â † are the ladder operators. However, there exist other states for whom the optical phase space may have the uncertainty circle squeezed for one quadrature with the other been elongated so that the deformed uncertainty circle forms an ellipse. Such states are generally referred to as squeezed (S) states having reduced quantum uncertainty [44][45][46][47]. Classical physics has no counterpart for these highly non-classical states. Production of a S state is accomplished by employing a squeeze operator defined as: here, r and Φ are the corresponding squeezing parameter and squeezing angle, which determines the strength of squeezing and distribution amongst the conjugate variables respectively, with 0 ≤ r ≤ ∞ and −π ≤ Φ ≤ π . Thus a squeezed vacuum (SV) state is described as: Moreover, Yuen [48] in his studies has mentioned that the operation of squeeze operator on a displaced state yields a different state often called as Yuen states or squeezed coherent (SC) states and is defined as: Alternatively, on reversing the action of the squeeze and displacement operators it is feasible to describe yet a different class of states known as displaced squeezed vacuum states or coherent squeezed state (CS) [49].
3.2 Two-mode (TM) states Quantum optical states of TM are obtained by the operation of the TM D( , δ) and Z ab (r, Φ) operators on the two-mode vacuum (TMV) [50], defined respectively by whereâ † (b † ) is the 1st(2nd) mode creation operator and a(b) is the associated annihilation operator, while r, , δ are complex parameters corresponding to S (|ξ 2 ) and C (| , |δ ) states and is defined as follows: where r , Φ are the S state squeezing parameter and squeezing angle and θ 1 and θ 2 are the phases of C states ( and δ). Similar to the SM case, the action of TM operators on TMV state yields two-mode displaced (TMC), squeezed vacuum (TMSV), squeezed displaced (TMSC), displaced squeezed (TMCS) states. The homologous description of these states (and their SM ones) are encapsulated in Table 1. However, with suitable choices to the radian values of coherent phase angle and squeezing angle, we obtained entangled TM coherent (ETMC) states and TM squeezed entangled coherent (TMSEC) states. These states corresponds to the non-oscillatory behavioural pattern of the scale factor and are obtained corresponding to (θ 1 , θ 2 ) = ( nπ 2 + mτ, lπ 2 + mτ ) and Φ = ( pπ + 2mτ ) choices of radian values of phases angles of two coherent states and squeezing angle of squeezed state respectively [23]. Thus, on considering the simplest case with n − l = 1, we get the two coherent states superimposed with π/2 out of phase with respect to each other. By fixing the relative phase Φ 0 = 0 and amplitude equal (δ = i ), we get the following set of four quasi-Bell entangled coherent states [51][52][53] as: where h 2 1,2 = [1 ± e −2| | 2 ] −1 and h 2 3,4 = [1 ± e −2| | 2 cos(2| | 2 )] −1 are the normalization factors corresponding to the four quasi-Bell states defined above. Similarly, nonclassical TMSEC states |η i 12 which again corresponds to set of four quasi-Bell states in both the modes are obtained by the operation of Z ab (r, Φ) on the above states as: The TM displacement and squeezed operator satisfies the well known identities: The action of ladder operators on their respective modes states is described aŝ

Non-classical TM states representation of scalar field in semiclassical theory
In Sect. 2 by redefining the complex field modeζ k , it gets decompose into two real oscillatorsζ k given by its real and imaginary parts. Thus, each k-th mode corresponds to two real oscillators corresponding to the Fourier-cosine and sine modes, respectively.
We now approximate Eq. (12) to two modes for a massive scalar field just by considering the k-th mode. The 1st mode corresponds to theζ (+) k oscillator and the 2nd mode corresponds to theζ (−) k oscillator with the corresponding frequencies obtained for the two oscillators as ω α (τ ) = m 2 + k 2 S 2 (τ ) . Moreover, as the Hubble rate (H) drops below the scalar field mass (H m) after inflation, therefore, the period of the oscillation is much shorter than the Hubble time; as a result field oscillates many times over a Hubble time. On assuming that the expansion of the Universe can be neglected during each period, we obtained ω α (τ ) ≈ m.
A massive, nominally coupled scalar field (to the spatially flat FRW metric) in a harmonic potential can be defined by the time-dependent harmonic oscillators with the two-mode Hamiltonian (refer Sect. 2) as: The Hamiltonian's expectation value in a considered quantum state is given as T 00 2 ψ = π 4 α ψ 4S 6 (τ ) Then the Fock states are the Hamiltonian eigenstates: is the associated annihilation operators following the boson's canonical commutation relations [â, We are dealing with the Schrödinger picture as defined by Kim [54][55][56][57] in one of his research papers, where he found a pair of the invariant operators, interpreted them as the time-dependent creation and annihilation operators which were defined linear in momentum and position and were then applied to the case of the massive scalar field in the FRW Universe. These ladder operators for TM states can respectively be defined as: where, ζ 1 and ζ 2 are the mode functions of field, corresponding to the k-th 1st and 2nd mode respectively. Each scalar field operator and its conjugate momentum operator has a Fock state representation as: Thus, for the cosine mode the field and momentum operators are given as: and similarly, the field and momentum operators for the α sine mode are given as: It should be noted that here, ζ j and ζ * j satisfy Eq. (16) and the Wronskian condition defined as: Wronskian condition is the boundary condition which fixes the normalization constants of two independent solutions. For massive scalar field, the boundary condition requirement is necessary for the exact quantum back-reaction [54][55][56][57].

Quantum stress tensor fluctuations and limits to the validity of semiclassical theory
Even though quantum states in field theory may or may not be eigenstates of the Hamiltonian, and hence have fixed energy, they are never eigenstates of the energy density or other stress tensor components. This means that all quantum field theories exhibit fluctuations of the stress tensor. These fluctuations can manifest themselves either through fluctuating forces on material bodies or through fluctuations of the gravitational field [33]. As mentioned earlier, in this paper, we will be concerned with stress tensor fluctuations which drive the fluctuations of the spacetime geometry. From the semiclassical Einstein equation (1) it appears explicitly that the SG theory should break down when the considered quantum states have large quantum fluctuations of the stress tensor. Ford [28] was amongst the first to have emphasized the significance of these fluctuations which generate passive fluctuations of spacetime. According to the criterion put forward by Ford and his collaborators [29], for the semiclassical theory to be valid, apart from the fact that the associated energy scales should be far below the Planck scale (10 19 GeV), the associated fluctuations in the quantum field's energy-momentum densities should not be too large either, in different nonclassical states under consideration. To achieve a quantitative estimate of the deviation from the SG theory, Ford studied the energy flux of gravitational radiation in linearised gravity produced by the quantum systems (matter field) [28] and compared the predictions from the linear quantum gravity theory and the semiclassical theory. In the semiclassical theory based upon Eq. (1), the flux depends on products of expectation values of stress tensor operators, whereas in a theory in which the metric perturbations are quantized, it depends upon the corresponding products of expectation values. It was observed that the semiclassical theory gives reliable results when the fluctuations in the stress tensor are not too large, that is, when It fails when the quantum states exhibits fluctuations so that T μν (x)T αβ (y) is very different from T μν (x) T αβ (y) . Thus, one can give an essential condition for the validity of the semiclassical theory: the state of the quantum system must be such that expectation values of operator products can be approximated by the products of expectation values. They proposed a numerical measure for the applicability (or non-applicability) of semiclassical gravity theory in various quantum states. The extent to which the semiclassical approximation is violated can be measured by the dimensionless quantity D μναβ (x, y) defined in terms of the EMT as below: This parameter is a dimensionless measure of the stress tensor fluctuations. Here, : T μν (x) : ψ is the expectation value of the normal ordered stress tensor of a scalar field in some state ψ and : T μν (x) T αβ (y) : ψ is the normal ordered 2-point function. The meaning of :: is that the expectation values are to be evaluated relative to the normal ordering procedure for the field. This is done because the formal expectation value of T μν (x) in any quantum state is divergent 1 [1]. Thus, by defining all the operators as being normal ordered with respect to the vacuum expectation value we can expect them to have finite expectation values. Moreover, if one wishes to make precise statements about the stress tensor's expectation value in some state, he needs to study the probability distribution of the eigenvalues of T μν (x) in that state. Since the eigenvalues of the stress tensor are matter distributions in spacetime, if they are far apart in the probability distribution functional, they correspond to very different geometries. Quantitatively, one expects the fluctuations of T μν to be big and to induce equally big metric fluctuations, leading to a breakdown of the semiclassical approximation.
This means that by assuming the semiclassical gravity equation, each eigenvalue of T μν (x) corresponds to a different spacetime [59], making the expectation value a sum 1 Since the Hamiltonian is quadratic in the field and generally, the vacuum expectation value of any quadratic operator diverges, as the product of distributional valued functions at the same spacetime point is a mathematically ill-defined quantity and has ultraviolet divergence, therefore, it requires renormalization. Now, in an ordinary Minkowski spacetime of the free field theory ultraviolet divergences/infinities due to fluctuations on arbitrary short scales are removed by the subtraction of the vacuum expectation value of the energy, also called normal ordering, the physical justification being that these vacuum contributions are unobservable. A similar prescription is used in order to regularize the infinities in cosmological spacetime. However, one additional problem is the absence of an unambiguous choice of vacuum. In Minkowski space, there exists a well-defined, unique, vacuum state, but in curved spacetime, due to time dependence of the geometry, the vacuum becomes ambiguous and thus, the renormalization done by the merely normal ordering of the stress-energy tensor is not just enough. Fortunately, the divergence is a c-number, so the renormalization is state independent. We have used adiabatic regularization scheme [1,58] of normal ordered operator for obtaining divergence-free expressions. over possible spacetimes multiplied by the probability they have to occur. Our purpose here is to use the probability distribution function, to distinguish between the semiclassical and non-semiclassical states. The most important quantity for this purpose is the variance V T . The variance we shall consider is: Extremely quantum states, always have fluctuations bigger than the mean, leading to a big variance. On the other hand, for semiclassical states, the variance in most cases is close to zero since they should have sharply peaked around the mean probability distributions. Another useful feature of the variance is that it throws away states whose existence leads to a probability distribution of T μν (x) which contains heavy tails. For such distributions, the variance is big, since the peak of the probability distribution is dragged away from the mean, due to the huge expectation values of the tail. Now, treating all the components of the energy-momentum tensor will be extremely cumbersome. Therefore, they suggested taking the coincidence limit, x → y, confining our attention to the density fluctuation only. Hence, we only need to consider the purely temporal component of the EMT, i.e., taking all the above into account the following quantity is defined as a measure of the magnitude of fluctuations of T μν ψ : Note that : T 00 (x) : is the mean energy density at x and : T 00 2 (x) : is the mean squared energy density. Thus, D T (x) is a measure of extend of the energy density fluctuations at point x. The local energy density fluctuations are said to be small in the state of interest when D T (x) < 1, which we take to be a measure of the validity of the semiclassical theory with the state defined as semiclassical. However, for large fluctuations D T (x) ≈ or > 1, the state ψ is very quantum and the corresponding semiclassical Einstein equation does not hold good in that particular nonclassical state [29].
Furthermore, while calculating fluctuations and dimensionless parameters we come across numerical expressions that make it inconvenient to be solved analytically. Therefore, we resort to a numerical approach by transforming the scalar field into a new field. Many research papers [23,24,38,[54][55][56][57] make use of the following approach to deal with the massive scalar field during its oscillatory phase. Let the solutions have the form: Therefore obtaining We now focus on the oscillatory phase of the massive scalar field after inflation. In the parameter region of the scalar field possess an oscillatory solutions of the type where, Moreover, with WKB-type approximation for these modes we further solves them perturbatively by using the following approximation ansatzes: Thus, in this section by making use of the criterion suggested earlier by Kuo and Ford we will be calculating variance of the fluctuations of the stress tensor operator V T (density fluctuations) to draw the limits on the validity of the semiclassical theory (through dimensionless parameter D T ) for different TM entangled and non-entangled coherent and squeezed quantum states of scalar field modes during the oscillatory phase of the massive scalar field. In the following seven subsections, we shall evaluate the quantities V T and D T for the two-mode (i) vacuum, (ii) nth number (iii) coherent (iv) squeezed vacuum (v) squeezed states (vi) entangled coherent and (vii) squeezed entangled coherent states of the two-mode quantized scalar field.
As stated earlier, the semiclassical theory is expected to give a reliable results when the quantum state considered have small fluctuations. In other words, the state |ψ of the quantum system must be such that expectation values of stress tensor operator products can be approximated by the products of expectation values, so that : T 00 2 : |ψ is not very different from : T 00 : 2 |ψ implying, V T |ψ ≈ 0 and D T |ψ < 1. The calculations necessary for drawing limits on the validity of the semiclassical approximation, with the aid of the Eqs. (51), will involve evaluating the expectation values of the normal ordered stress tensor operatorT 00 and its squarê T 00 2 using Eqs. (34) and (35) On substituting in Eq. (35) we get normal ordered mean square energy density as: Again :T 00 : in TMV states can be obtained by taking expectation value of :π 2 α : and :ζ 2 α : in TMV states and then substituting in Eq. (34) : On Squaring Eq. (54) and substituting values of ζ 1 , ζ * 1 ,ζ 1 ,ζ * 1 and ζ 2 , ζ * 2 ,ζ 2 ,ζ * 2 from Eqs. (45)- (49) and applying the approximation ansatz given in Eq. (50) we obtain square of the expectation value of stress tensor operator in TMV states as: When these quantities obtained in Eqs. (53) and (55) are substituted in Eqs. (51), we find that V T |0,0 and D T |0,0 are given by the following expressions: During the later stage (mτ > 1) of evolution after inflation, the expansion law is same as that of the matter dominated era τ 2 3 [54][55][56][57], therefore, in the limit mτ 1 ( the regime of particle creation), the expressions in Eq. (56) further reduces to: as the scalar field mass parameter m is determined to be of the order 10 −6 m p [60], therefore, we get a variance of the fluctuations of the order V T |0,0 = 3.125 × 10 −14 ≈ 0. Thus, the TMV states exhibits small fluctuations with :T 00 2 : |0,0 ≈ :T 00 : 2 |0,0 i.e., the expectation value of a product of stress tensors is approximately equal to the corresponding product of expectation values and also since numerical value of D T |0,0 < 1; the semiclassical theory holds good during the scalar field's oscillatory phase in the TMV states. Thus, with variance approximately close to zero, these states are semiclassical states and with D T |0,0 < 1, the semiclassical theory is valid if the scalar field modes is assumed to be in a states like TMV states (Fig. 1).

For a TM nth number (TMN) states |n 1 , n 2
Again, the variance of the fluctuations of the stress tensor operator V T and the dimensionless quantity D T as a measure of the deviation from the semiclassical theory in TMN states for our massive chaotic inflationary scenario during the oscillating stage of the driving scalar field are obtained as: V T |n 1 ,n 2 = : T 00 2 : |n 1 ,n 2 − : T 00 : 2 |n 1 ,n 2 , D T |n 1 ,n 2 = : T 00 2 : |n 1 ,n 2 − : T 00 : 2 The calculations necessary for drawing limits on the validity of the semiclassical approximation, with the aid of the variance of density fluctuations, will involve evaluating the expectation values of the normal ordered stress tensor oper-atorsT 00 andT 00 2 using Eqs. (34) and (35) in TMN states.
Consequently, for evaluatingT 00 2 in TMN states, we used the  50), we obtained them respectively as: Thus, Again :T 00 : in TMN states can be obtained by taking expectation value of :π 2 α : and :ζ 2 α : in TMN states and then substituting in Eq. (34) On squaring the above Eq. (61), and solving it numerically by substituting the values of scalar field's modes coefficient and applying approximation ansatz specified in Eq. (50), we obtain square of the mean energy density as: Substituting Eqs. (60) and (62) in Eq. (58), we obtain V T |n 1 ,n 2 and D T |n 1 ,n 2 in TMN states as: Now with mτ = x, and then in the limit x 1, the expression for the variance of the density fluctuation in TMN states further reduces to: For examining the validity of the semiclassical equation, we study density fluctuations and dimensionless parameter graphically (3-D plot and corresponding array plot) and numerically with the associated number particles states parameters values, n 1 and n 2 , and the corresponding results are tabulated in matrices below ( Table 2): Variance of Density Fluctuations V T |n 1 ,n 2 with (n 1 , 0, 5, 1) and (n 2 , 0, 5, 1) The plots for the density fluctuations and dimensionless parameter refer Fig. 2 for a given range of parameters values of TMN states reveals that the V T |n 1 ,n 2 and D T |n 1 ,n 2 parameter are respectively equivalent to 0 and smaller than 1 along the axes. However, along with and in the neighbourhood of the diagonal line i.e., with larger values of both n 1 and n 2 the TMN states exhibits large values of variance and hence of the dimensionless parameter. From these analyses, it is clear that the semiclassical Einstein equation during the oscillatory phase of the scalar field in TMN states formalism is valid only provided that the corresponding nth particles states have the number of particles such that either n 1 n 2 or n 2 n 1 . This kind of behaviour may be due to the fact that the density fluctuation in the back-reaction term in the semiclassical equation are large in nth particle states when a large number of particles are being produced by the gravitational background i.e. when both the states have very large particles in them. This may be due to the fact that with n 1 and n 2 very large, the inter-mode correlation between the particles from two different states corresponding to two different modes starts playing a vital role and because of the quantum nature of this correlation the TMN states is considered to be quantum for such parameters choices. Thus, the semiclassical theory holds good in the TMN states representation of the scalar field in the oscillatory phase when the number states have particles distributed in two states such that either n 1 n 2 or n 2 n 1 , for such cases the TMN states may be regarded as a semiclassical state.

For a TM coherent (TMC) states | , δ
Similarly, the variance of the stress tensor fluctuations V T and the dimensionless quantity D T as a measure of the deviation from the semiclassical theory in TMC states are defined as:   (66) On substitution of above calculated quantities in Eq. (35), we get normal ordered mean square energy density as: Again :T 00 : for the considered state can be obtained by taking the expectation value of :π 2 α : and :ζ 2 α : in TMC states and then solving it numerically by making use of the Eqs. (45)- (49) and with the approximations ansatz specified in Eq. (50), we obtain the square of the mean energy density in TMC states as: With substitution of these quantities obtained in Eqs. (67) and (68) in Eq. (65), we find that V T | ,δ and D T | ,δ in TMC states are given by the following expressions: In the limit x 1, In this particular state, the expectation value of a product of stress tensor is approximately equal to the corresponding product of expectation values and the dimensionless parameter is always small than unity. This result is consistent and expected since it is common wisdom that coherent states correspond to classical field excitations. Thus it would be highly unusual if they were incompatible with SG. Furthermore, neither the density fluctuations nor the dimensionless parameter depends on the coherent parameters of the two states involved. Thus, the above study reveals that the SG theory is a good approximation for the TMC states' representation of the scalar field in the oscillatory phase (Fig. 3).

For a TM squeezed vacuum (TMSV) states |ξ 2
Analogously, examining the validity of semiclassical approximation in TMSV states for our inflationary model with the Similarly, 1 + 8 (−1) p sinh 3 r cosh r + 4 sinh 4 r + 4 sinh 2 r + 2 sinh 2 r cosh 2 r Thus, :T 00 : in TMSV states can be obtained by taking expectation value of :π 2 α : and :ζ 2 α : in TMSV state and then again solving it numerically by substituting the values of scalar field's modes coefficient and applying approximation ansatz specified in Eq. (50), we obtain the square of the mean energy density in TMSV states as: Again, with substitution of these quantities obtained in Eqs. (74) and (75) in Eq. (71), we find that V T |ξ 2 and D T |ξ 2 are given by the following expressions corresponding to two different choices made with the value of squeezing angle parameter p, as: With mτ = x and in the limit x 1, For examining the validity of the semiclassical approximation of the Einstein equation in TMSV states for the even case of squeezing angle parameter, we study V T |ξ 2 e and D T |ξ 2 e graphically (corresponding plots are given in Figs. 4 and 5 and numerically with the associated squeezing parameter values and the results are tabulated in Table 3. For the even value of squeezing angle parameter p, the TMSV states exhibits small energy density fluctuations (refer Fig. 4). Consequently, the dimensionless parameter (refer Fig. 5) is smaller than unity for squeezing parameter r that are much smaller than unity. For such a case, the TMSV state is said to be the semiclassical state with semiclassical Ein- On further simplification the above Eq. (79) reduces to:   In the limit x 1 ( Table 4, Fig. 6 and 7), Thus, we conclude that the TMSV states for both the even and odd choices of the phase parameter p value, have local energy density fluctuations very small only when r 1. For these particular cases, the expectation value of a product of stress tensors is approximately equal to the corresponding product of expectation values leading to small variances, very close to zero i.e., V T |ξ 2 ≈ 0 and hence D T |ξ 2 1, implying the state is semiclassical and for r 1 the semiclassical gravity is a good approximation for TMSV states. From these analyses, it is clear that the semiclassical Einstein equation in TMSV state formalism during the oscillatory phase of the nonclassical scalar field is valid only provided the corresponding r parameter takes on value much smaller compared to unity. These density fluctuations arise because of the quantum nature of the TMSV states which leads to large particle creation via non-adiabatic amplification during the oscillatory phase of the scalar field, in the semiclassical gravity [24]. However, these energy density fluctuations enhance in the large squeezing limit due to the enormous particle creation and as a result, the expectation value of the energy-momentum density of the quantum field ceases to account for the backreaction adequately leading to big variances and hence D T |ξ 2 ≈ 1 or > 1, making |ξ 2 state very quantum, wherein the semiclassical approximation to the gravity cannot be trusted. 2 and TM squeezed coherent (TMSC) states Z ab | , δ

For a TM coherent squeezed (TMCS) states D δ |ξ
In this section, we shall examine the validity of semiclassical approximation in TMCS and TMSC states for our inflationary model with the aid of variance of density fluctuations V T and the dimensionless quantity D T defined respectively for these states as: Using TMCS and TMSC states definition refer Table 1), coherent and squeezing operator identities and action of ladder operators (refer Sect. 3.2), we obtained the expression for :π 4 α : , :π 2 Thus, :T 00 : in TMCS and TMSC states are obtained by taking expectation value of :π 2 α : and :ζ 2 α : in their respective states, which are then solved numerically by substituting the values of scalar field's modes coefficient, phase angles values and by applying approximation ansatz specified in Eq. (50), to obtain the square of the mean energy density in TMCS and TMSC states, respectively as: Substituting Eqs. (85) and (86) in Eq. (82), we find that V T D δ |ξ 2 ; Z ab | ,δ and D T D δ |ξ 2 ; Z ab | ,δ i.e., energy density fluctuations and validity of SG theory for the TMCS and TMSC states and are respectively given by the following expressions: on further simplification the above Eq. (87) reduces to: In the limit x 1 i.e., during the later stage of evolution after inflation when the expansion law is same as that of the matter dominated era, the energy density fluctuations in these states are obtained as: The expressions obtained for the variance of density fluctuations V T and the dimensionless quantity D T for the D δ |ξ 2 and Z ab | , δ states of the quantised massive TM scalar field are similar to that obtained in the case of TMSV (|ξ 2 ) state. Neither the density fluctuations nor the dimensionless parameter depends on the coherent parameters of the Fig. 8 a, b are the plots for the density fluctuations and dimensionless quantity in TMCS and TMSC states, plotted as a function of squeezing parameter r . Note that the two states are semiclassical states with V T close to zero and D T 1 only when r 1, otherwise the states are very quantum two states involved, with the only dependency on the associated squeezing parameters. Like |ξ 2 state, the semiclassical Einstein equations, during the oscillatory phase of the nonclassical scalar field, in TMCS and TMSC states formalism are reliable and valid only when the associated squeezing parameter is much smaller compared to unity. And, in the limit when r > 1, we find that D T is of the order of unity for both the states, implying that the fluctuations in the backreaction term in the semiclassical equations obtained corresponding to these states are large, as a result the semiclassical gravity theory for these states cannot be trusted (Fig. 8).
Now, in the following subsections, we shall explore the entangled TM Coherent (ETMC) states, Eq. (29) and TM Squeezed Entangled Coherent (TMSEC) states, Eq. (30) of the quantized TM scalar field to examine the validity of semiclassical approximation during the scalar field's oscillations. These states individually correspond to a set of four quasi-Bell states in both the modes of the scalar field. They resulted from the correction made with the choices of phase and squeezing angles of TM coherent and squeezed state respectively, to account for the non-oscillatory behaviour of the Hubble constant in semiclassical gravity in strong contrast with the oscillatory behaviour in the classical gravity.

For a ETMC states |Ψ i 12
By evaluating the dimensionless quantity D T with the aid of V T , we shall discuss and define the criterion for the validity of a semiclassical theory in all the four quasi-Bell states corresponding to ETMC state formalism of a massive quantised TM scalar field during it's oscillatory phase. Variance and dimensionless measure of the stress tensor fluctuations in the considered nonclassical states are defined as : where |Ψ i 12 with i = 1, 2, 3, 4.
Using TMC states formalism, displacement operator identities and action of ladder operators defined for the two separate modes of the quantised scalar field (refer Sect. 3.2), we obtained the expression for :π 2 α : , :ζ 2 α : and :π 4 α : , :π 2 :T 00 : |Ψ i 12 in |Ψ 1 12 and |Ψ 2 12 states are obtained by taking expectation value of :π 2 α : and :ζ 2 α : in the respective states and then solving it numerically by following the above defined procedure, we obtain the square of the mean energy density in |Ψ 1 12 and |Ψ 2 12 states, respectively as: substituting Eqs. (93) and (94) in Eq. (90), we find that V T |Ψ 1 12 ; |Ψ 2 12 and D T |Ψ 1 12 ; |Ψ 2 12 i.e., fluctuations of the energy density and validity of semiclassical approximation to the gravity for the above stated states are respectively given by the following expressions: In the limit x 1, ETMC states are the result of the superposition of two coherent states which are π 2 out of phase. Unlike TMC states, they have the mean energy density and the mean squared energy density and hence the D T |Ψ 1 12 ; |Ψ 2 12 dependent on the coherent parameters of the two states involved. Fig. 9 Plot for the dimensionless quantity in |Ψ 1 12 and |Ψ 2 12 states, plotted as a function of coherent parameter . Note that the two states are quantum with D T approaching unity for very smaller values of , otherwise the states are semiclassical, with semiclassical theory of gravity as a good approximation Presented in Fig. 9 is the plot of dimensionless parameter D T for |Ψ 1 12 and |Ψ 2 12 states, denoted simply as D |Ψ 1 12 ; |Ψ 2 12 in the plot, as a function of coherent parameter | | 2 and their numerical evaluation is tabulated in the Table 5. It is interesting to note that the entangled two-mode coherent states behaves as nonclassial/quantum states with D T approaching unity for smaller values of coherent parameters, which implies that the fluctuations in the backreaction term in the semiclassical equation are larger in these states for smaller values of coherent parameters. However, with a large value of the quantity D T die down as | | −2 , thus destroying its nonclassical behaviour and makes it a classical analogue i.e., coherent with D T approaching zero ( :T 00 2 : ≈ :T 00 : 2 ), which implies that the |Ψ 1 12 and |Ψ 2 12 states are semiclassical states for larger values of coherent parameter for which we can expect the semiclassical gravity theory to be a good approximation. This result is in consistent with the definition of coherent states which corresponds to classical field excitation. These results then imply that if the scalar field modes is assumed to be in a state like the |Ψ 1 12 and |Ψ 2 12 , then we can expect semiclassical theory to be valid during all the stages of evolution for larger values of .
Again, for the 3rd quasi-Bell state considered ( |Ψ 3 12 ); for : T 00 2 : |Ψ i 12 we evaluated the expression for the expectation values of various operator in the oscillatory phase of the scalar field which are respectively obtained as: where ; further with coherent phase angle as θ 1 = ( nπ 2 + mτ ), Eq. (98) reduces to: Similarly, we obtained the expectation values for the other operators as: Thus, : T 00 2 : |Ψ i 12 in the 3rd quasi-Bell state representation of ETMC states formalism is obtained as: Now, :T 00 : |Ψ i 12 in |Ψ 3 12 state is evaluated by taking expectation value of :π 2 α : and :ζ 2 α : in the considered state and then solving it numerically by following the above defined procedure to obtain the square of the mean energy density in |Ψ 3 12 state, given as: substituting Eqs. (101) and (102) in Eq. (90), we find that V T |Ψ 3 12 and D T |Ψ 3 12 , are respectively given by the following expressions: In the limit x 1, The |Ψ 3 states unlike | , δ , |Ψ 1 and |Ψ 2 states shows quantum properties such as squeezing of the quadrature fluctuations and as a result these states are no longer a smooth disk in quadrature space, but rather its shape depends on the coherent parameter of the states involved [24]. These states like |Ψ 1 and |Ψ 2 states have the mean energy density and the mean squared energy density and hence, the D T |Ψ 3 12 dependent on the coherent parameter . Our primary concern is whether or not D T |Ψ 3 12 1, which is best determined by numerical evaluation of D T |Ψ 3 12 as tabulated in Table 6 and Fig. 10 illustrate the results. For states with sufficiently small magnitude of coherent parameter i.e., for the range 0.1 < | | 2 < 0.5, the |Ψ 3 states have a nature similar to the |ξ 2 states with | | behaving as squeezing parameter r , such that with increasing magnitude of coherent parameter, the energy density fluctuations becomes large with D T |Ψ 3 12 approaching unity and the semiclassical theory is not a good approximation for those regions. However, as the magnitude of the coherent parameter increases beyond, | | 2 > 2, we find that D T |Ψ 3 12 becomes smaller and smaller than unity, with D T |Ψ 3 12 approaching 0.5 such that D T |Ψ 3 12 1. Thus, the |Ψ 3 states is semiclassical for coherent parameter magnitude, | | 2 > 1 or | | < 0.1 otherwise, within the range 0.1 < | | 2 < 1, the states are very quantum and corresponds to the regions where the value of D T quantity approaches unity, which implies a complete breakdown of the semiclassical theory.

For a |Ψ 4 12 state
is the normalization factor corresponding to the 4th quasi-Bell state of ETMC state formalism defined above. For the 4th quasi-Bell state considered ( |Ψ 4 12 ); for : T 00 2 : |Ψ i 12 , we obtained the expression for the expectation values of various operator during the oscillatory phase of the scalar field which are respectively obtained as: where A 1 = 1 + e −2| | 2 cos 2 | | 2 , A 2 = 1 − e −2| | 2 cos 2 | | 2 and B 2 = 1+e −2| | 2 sin 2 | | 2 ; further with coherent phase angle as θ 1 = nπ 2 + mτ , Eq. (107) reduces to: where Thus, the expectation value of the squared stress tensor : T 00 2 : |Ψ i 12 in the 4th quasi-Bell state representation of ETMC states formalism is obtained as: The expectation value of the renormalized stress tensor ( :T 00 : |Ψ i 12 ) in |Ψ 4 12 state is obtained, by evaluating expectation value of :π 2 α : and :ζ 2 α : in the considered state and then by solving it numerically on following the above defined procedure, we obtain the square of the mean energy density in |Ψ 4 12 state as: substituting Eqs. (110) and (111) in Eq. (90), we find that V T |Ψ 4 12 and D T |Ψ 4 12 , are respectively given by the following expressions: In the limit x 1, These quantum states like |Ψ 3 state have slightly squeezed fluctuations in the momentum quadrature [24] and have expectation values of the stress tensor and squared stress tensor and hence the D T |Ψ 4 12 parameter dependent on the coherent parameters of the states involved. Presented in Fig. 11 is the plot of D T |Ψ 4 12 as a function of coherent parameter | | 2 and their numerical evaluation is tabulated in the Table 7. Similar to |Ψ 3 states, for small values of parameter these states have small nonclassical nature similar to that of the |ξ 2 state i.e., for the range 0.5 < | | 2 < 1.5. For such a range of magnitude of coherent parameter values, unlike |Ψ 1 , |Ψ 2 states, with increasing magnitude of coherent parameter, the magnitude of D T |Ψ 4 12 starts increasing, although the D T |Ψ 4 12 is still very small than unity, implying small local energy density fluctuations and hence, validity of semiclassical gravity theory holds for the interest state. However, this small nonclassical characters of |Ψ 4 state gets easily destroyed by the increasing magnitude of parameter. Beyond | | 2 > 2, with increasing magnitude of parameter, the magnitude of D T |Ψ 4 12 starts die down, however unlike |Ψ 1 , |Ψ 2 , the D T does not completely die down but approaches a constant magnitude of 0.52. Thus, we can say that the semiclassical gravity theory is a good approximation for |Ψ 4 state and state can be regarded as a semiclassical state during all the stages of evolution for all the values of coherent parameters. It is also interesting to note that the dimensionless parameter; a measure of the magnitude of fluctuations, in |Ψ 3 and |Ψ 4 state is not a monotonic function of the coherent parameter.
Thus, we conclude that for all the four quasi-Bell states of ETMC states we may expect the semiclassical theory to be a good approximation during all the stages of evolution. If the scalar field modes are assumed to be in states like |Ψ 1 and |Ψ 2 quasi-Bell states, we can expect the states to be semiclassical for all the values of . For such states, when the parameter takes values much smaller than 1 we get D T approaching unity where we may expect fluctuations to be larger. However, for larger values of the quantity D T completely dies down, thus destroying this nonclassical behaviour. If the scalar field modes are assumed to be in states like |Ψ 3 and |Ψ 4 states, we see that D T is not a monotonic function of the coherent parameter. For |Ψ 3 state the semiclassical theory is a good approximation for all the values of | | expect for the range 0.5 < | | 2 < 1.
Moreover, for these states when the parameter takes values much smaller than 1, they behave similar to |ξ 2 states, however this nonclassical nature gets destroyed as the magnitude of coherent parameter increases beyond, but unlike |Ψ 1 and |Ψ 2 states the quantity D T does not die down completely. On the other hand for |Ψ 4 state we find the semiclassical theory of gravity are always valid with D T 1. But unlike |Ψ 1 and |Ψ 2 states, these states like |Ψ 3 states have non-zero value of the D T for larger values of | | parameter.

For a TMSEC states |η i 12
Analogously, by evaluating the dimensionless quantity D T with the aid of V T , we shall be able to understand more concretely the validity of a semiclassical approximations, in all the four squeezed quasi-Bell states derived individually by operation of squeezed operator on each one of the quasi-Bell of ETMC state formalism. Variance and dimensionless measure of the stress tensor fluctuations of the considered nonclassical states of a massive quantised TM scalar field during its oscillatory phase are given as : here, |η i 12 = Z ab (r, Φ)|Ψ i 12 ; where |η i 12 with i = 1, 2, 3, 4 and are respectively given as: where are the normalization factors corresponding to the four squeezed quasi-Bell states defined above. Again, by using TM coherent states and squeezed states formalism, displacement and squeezed operators identities and action of ladder operators defined for the two separate modes of the quantised scalar field (refer Sect. 3.2), we obtained the expression for :π 2 α : , :ζ 2 α : and :π 4 α : , :π 2 where, h 2 1,2 = [1 ± e −2| | 2 ] −1 is the normalization constant. Now, for the squeezed quasi-Bell states considered ( |η 1 12 and |η 2 12 ), for : T 00 2 : |η i 12 the various operator's expectation values during the oscillatory phase of the scalar field are respectively obtained as: :π 4 α : |η 1 12 ; |η 2 12 = where, C 3 = 3 1 + 4 | | 2 cosh 2 r + 4 1 + | | 2 sinh 2 r + 6 1 + 8 | | 2 cosh 2 r sinh 2 r + 2 1 + 2 | | 2 sinh 4 r Thus, : T 00 2 : |η i 12 in the 1st and 2nd squeezed quasi-Bell states representation is obtained as: :T 00 2 : |η 1 12 ; |η 2 12 = 1 :T 00 : |η i 12 in |η 1 12 and |η 2 12 states are evaluated by taking the expectation value of :π 2 α : and :ζ 2 α : in the respective states and then solving it numerically on following the above defined procedure, to obtain the square of the mean energy density in |η 1 12 and |η 2 12 states, respectively as: × 1 + 4 | | 2 cosh 2 r + sinh 2 r + sinh 2 r 2 + 4 | | 2 cosh 2 r + sinh 2 r + 4 sinh 2 r + 4 1 + 2 | | 2 2 sinh 2 r cosh 2 r + 4(−1) p × (1 + 2 | | 2 ) 1 + 2 | | 2 (cosh 2 r + sinh 2 r ) On substituting Eqs. (121) and (122) in Eq. (115), we find that V T |η 1 12 ; |η 2 12 and D T |η 1 12 ; |η 2 12 i.e., fluctuations of the energy density and validity of semiclassical approximation to the gravity for the above-stated states are respectively given by the following expressions corresponding to two different choices made with the value of squeezing angle parameter p: • Thus, the dimensionless quantity, as a measure of the deviation from the semiclassical theory, obtained for the |η 1 12 and |η 2 12 states is given as: To examine the validity of the semiclassical equations in |η 1 12 and |η 2 12 states, we studied the dimensionless parameter graphically Figs. 12 and 13 illustrates the results (3-D plot and corresponding array plot) and numerically with the associated coherent parameter | | and squeezing parameter r and the corresponding results are tabulated in matrices below ( Table 8): The plots for the D T |η 1 12 ; |η 2 12 (e) parameter refer Fig. 13 for the given range of parameters values of |η 1 and |η 2 states reveals that when the associated squeezing parameter r takes values much smaller than unity and the associated coherent parameter | | takes value much greater than r i.e., r | |, we certainly find that fluctuations in the stress tensor are very small such that the expectation value of a product of stress tensors is approximately equal to the corresponding product of expectation values ( :T 00 2 : |η 1 ; |η 2 ≈ :T 00 : 2 |η 1 ; |η 2 ) and hence D T |η 1 12 ; |η 2 12 (e) 1. For such cases the states are semiclassical and their semiclassical Einstein equations during the scalar field's oscillatory phase can be trusted. However, as the magnitude of the squeezing parameter r increases relative to that of | |, we find that local energy density fluctuations are large leading to D T |η 1 12 ; |η 2 12 (e) 1, making the |η 1 and |η 2 states very quantum, wherein the semiclassical  Table 8 showing numerical values of D T |η 1 12 ; |η 2 12 (e) tabulated in the form of a matrix for some range of coherent and squeezing parameters approximation to the gravity cannot be trusted. It is interesting to note that the nonclassical characters of the |η 1 and |η 2 states gets completely destroyed when coherent parameter takes on further large values, for such a case :T 00 2 : |η 1 ; |η 2 = :T 00 : 2 |η 1 ; |η 2 with D T |η 1 12 ; |η 2 12 (e) = 0, the region of classical behaviour.
To examine the validity of the semiclassical equations in |η 1 12 and |η 2 12 states, we studied the dimensionless parameter graphically Figs. 14 and 15 illustrates the results (3-D plot and corresponding array plot) and numerically with the associated coherent parameter | | and squeezing parameter r and the corresponding results are tabulated in matrices below ( Table 9): The plots for the D T |η 1 12 ; |η 2 12 (o) parameter refer Fig. 15 for the given range of parameters values of |η 1 and |η 2 states reveals that when the associated squeezing parameter r takes values much smaller than the associated coherent parameter | | i.e., | | r , we find that fluctuations in the stress tensor are very small such that the expectation value of a product of stress tensors is approximately equal to the corresponding product of expectation values ( :T 00 2 : |η 1 ; |η 2  Table 9 showing numerical values of tabulated in the form of a matrix for some range of coherent and squeezing parameters ≈ :T 00 : 2 |η 1 ; |η 2 ) and hence D T |η 1 12 ; |η 2 12 (o) 1. For such cases the states are semiclassical and their semiclassical Einstein equations during the oscillatory phase of the scalar field can be trusted. However, as the magnitude of the squeezed parameter r increases relative to that of | |, we find that local energy density fluctuations are large leading to D T |η 1 12 ; |η 2 12 (o) ≈ 1or 1, making the |η 1 and |η 2 states very quantum, wherein the semiclassical approximation to the gravity cannot be trusted. It is interesting to note that the  Thus, we conclude that the semiclassical Einstein equation during the oscillatory phase of the non-classical scalar field in the |η 1 and |η 2 states is reliable only when the associated squeezing parameter is much smaller compared to the corresponding coherent parameter. These results then imply that the semiclassical theory for our scalar field model as described by these states is valid, during all the stages of evolution only for the states which are sufficiently close to coherent states, i.e., r | | otherwise in large squeezing limit D T |η 1 12 ; |η 2 12 approaches unity and thus suggesting a breakdown of the semiclassical theory. The density fluctuations arise because of the particle creation due to the quantum nature of these states during the oscillatory phase of the scalar field, in the semiclassical gravity.

For a |η 3 12 state
is the normalization factor corresponding to the 3rd quasi-Bell state of TMSEC state formalism defined above.
Again, for the 3rd quasi-Bell state considered |η 3 12 ; for : T 00 2 : |η i 12 , we obtained the expression for the expectation values of various operator during the phase of the scalar field oscillation which are respectively obtained as: × (−1) p cosh 3 r sinh r + (−1) p sinh 3 r cosh r + 12 | | 2 B 1 (−1) p cosh 3 r sinh r + 3(−1) p sinh 3 r cosh r − 6 A 1 (−1) p sinh 3 r cosh r + 2 4 | | 4 A 1 cosh 4 r + 3 A 1 cosh 2 r sinh 2 r + 24 | | 2 B 1 cosh 2 r sinh 2 r where and A 1 , B 1 and A 2 are respectively defined below Eq. (98). Similarly, the expectation values for the other operators in the considered state are given as: Thus, : T 00 2 : |η i 12 in the 3rd squeezed quasi-Bell state of TMSEC states formalism representation is given by: :T 00 2 : |η 1 12 ; |η 2 12 = 1 16m 2 τ 4 + 1 8τ 2 + m 2 16 :T 00 : |η i 12 in |η 3 12 state is obtained, by taking expectation value of :π 2 α : and :ζ 2 α : in the considered state and then solving it numerically by following the above defined procedure to obtain the square of the mean energy density in |η 3 12 state as: :T 00 : 2 Substituting Eqs. (131) and (132) in Eq. (115), we find that V T |η 3 12 and D T |η 3 12 , are respectively given by the following expressions, corresponding to two different choices made with the value of p: • Case 1: p = Even (e) V T|η 3 12 (e) = e −4r 1 + m 2 τ 2 2 64 m 2 τ 4 0.001 − 0.0025 e 2r + 0.0025 With mτ = x and in the limit x 1, the equation further reduces to Correspondingly, the criterion for validity becomes: We now investigate the applicability of the semiclassical approximation, which is best determined by the numerical evaluation of D T |η 3 12 with the associated | | and r parameters and the results are tabulated in the form of matrix given below ( Table 10): The plots for the dimensionless parameter D T |η 3 12 (2-D and 3-D refer Fig. 16) for the given range and choices of parameters values of |η 3 state reveals that when the associated coherent parameter | | takes values much greater than the associated squeezing parameter r i.e., | | r ; i.e., for states which are sufficiently close to coherent states, we find that fluctuations in the stress tensor are very small and hence D T |η 3 12 (e) 1. For such cases the states are semiclassical and their semiclassical Einstein equations during the phase of the scalar field's oscillatory motion can be trusted. However, for smaller values of coherent parameters relative to that of r , we find that local energy density fluctuations are large and D T |η 3 12 (e) approaching unity, making the |η 3 states very quantum, wherein the semiclassical approximation to the gravity cannot be trusted. It is to be noted that the nonclassical characters of the |η 3 states unlike |η 1 and |η 2 does not completely dies down when coherent parameter takes on further large values relative to r , but becomes smaller and smaller than unity with D T |η 3 12 (e) finally approaching 0.5, Fig. 17. Thus, we conclude that during the oscillatory phase of the non-classical scalar field, the semiclassical Einstein equation in the |η 3 states is valid only when the associated squeezing parameter takes on values which is much smaller in magnitude compared to the corresponding coherent parameter otherwise in large squeezing limit due to the creation of large number of particles the expectation value of the energymomentum density of the quantum field ceases to account for the backreaction adequately reflecting a breakdown of the theory.
• Case 2: p = Odd (o)   with mτ = x and in the limit x 1 the equation further reduces to From the above results, we can form the quantity D T |η 3 12 (o) which is obtained as (Table 11, Fig. 18): where These results show that for both the even and odd choices of the p parameter value in |η 3 states, the semiclassical Einstein equations during the oscillatory phase of the scalar field is valid in these states only when the associated coherent parameter | | takes values much greater than the associated squeezing parameter r . Moreover, the associated squeezing parameter is considerably smaller than unity r 1. For smaller values of | |, the |η 3 state is very quantum, and the nonclassical character of these states enhances with the increasing values of r . However with greater values of coherent parameter the nonclassical character of these states starts decreasing and for states with r | | we find D T |η 3 12 1. For such cases the states are semiclassical and their semiclassical Einstein equations in the scalar field's oscillatory phase can be trusted.
Thus, we conclude that in the oscillatory phase of the non-classical scalar field the semiclassical Einstein equations corresponding to the |η 3 states is valid only when the associated squeezing parameter is much smaller compared to the coherent state parameter otherwise in large squeezing limit due to a large number of particles production the expectation value of the energy-momentum density of the quantum field ceases to account for the backreaction adequately reflecting a breakdown of the theory.

For a |η 4 12 state
is the normalization factor corresponding to the 4th quasi-Bell state of TMSEC state formalism defined above. Finally, for the 4th quasi-Bell state considered (|η 4 12 ); for : T 00 2 : |η i 12 we as previously defined, obtained the expression for the expectation values of various operator in the scalar field's oscillatory phase which are respectively evaluated as: where × sinh 3 r cosh r + 24 | | 4 A 1 (−1) p cosh 3 r sinh r + (−1) p sinh 3 r cosh r + 12 | | 2 B 2 (−1) p cosh 3 r sinh r + 3(−1) p sinh 3 r cosh r − 6 A 2 (−1) p sinh 3 r cosh r + 2 4 | | 4 A 2 cosh 4 r + 3 A 2 cosh 2 r sinh 2 r + 24 | | 2 B 2 cosh 2 r sinh 2 r + 24 | | 4 A 1 cosh 2 r sinh 2 r :T 00 : |η i 12 in |η 4 12 state is obtained, by taking expectation value of :π 2 α : and :ζ 2 α : in the considered state and then solving it numerically to obtain the square of the mean energy density in |η 4 12 state as: :T 00 : 2 With mτ = x and in the limit x 1, the equation further reduces to (Table 12) The plots for the dimensionless parameter D T |η 4 12 (e) refer Fig. 19 for the considered state reveals that these states for very small values of parameter have nonclassical behaviour with behaving similar to r parameter, such that with the increasing magnitude of parameter, the D T increases. However, when | | 2 > 2 beyond that with the increasing magnitude of parameter the D T starts decreasing. Moreover, with the increasing magnitude of r parameter the extend of violation of semiclassical approximation increases.    Thus, the study shows that the semiclassical theory holds good in the |η 4 12(e) state representation of the oscillatory scalar field when the associated squeezing parameter for these states takes on value much smaller than the unity and the associated coherent parameter takes on value much greater than squeezing parameter.

Discussions and final remarks
The study of quantum effects in an early Universe scenario requires both of them, the background metric and matter field(s) residing in the Universe in consideration, to be treated quantum mechanically. However, there are many widely known challenges that surface when attempting to combine QFT and GR into a full quantum theory of gravity. Therefore, we resort to semiclassical approximation to gravity wherein the spacetime metric is treated as a classical c-number field and its quantum fluctuations are neglected, although quan-tum fluctuations of the other fields are taken into account since quantum effects of matter field can play a significant role in the early universe. In fact, a classical gravitational field interacts with other fields through their stress tensors, and the only reasonable c-number stress tensor that one may construct with the stress tensor operator of a quantum field is its expectation value in some quantum state. In a series of papers we have discussed the dynamics and quantum effect of the two-mode quantized massive scalar field in the early universe by considering scalar field modes in two-mode entangled and non-entangled coherent and squeezed states formalism in the context of the semiclassical approach and a considerable body of results have been obtained [23,24]. We have particularly addressed the oscillatory phase of the scalar field after inflation and found that SG consists of a rich array of physical effects which are not found at the classical level. One such effect is that entangled and non-entangled states of the quantized TM scalar field leads to the initial stage of cosmological particle creation of the scalar field itself in the expanding FRW Universe due to non-adiabatic amplification, often termed as preheating/reheating of the Universe, which occurs after the inflationary epoch in chaotic inflationary model. However, there is certainly a domain of applicability of associating a classical gravitational field to a quantum field source. Semiclassical gravity, like any other approximation, holds in a specific regime and as it should, breaks down if one tries to treat something that cannot be treated in this regime. It is assumed that the semiclassical theory should break down at the Planck scale, which is when simple order of magnitude estimates suggests that the quantum effects of gravity cannot be ignored because the gravitational energy of a quantum fluctuation in a Planck size region, as determined    by the Heisenberg uncertainty principle, is comparable to the energy of the fluctuation itself. However, the limitations to the semiclassical Einstein equations even lie outside the Planck scale i.e., when the fluctuation induced to the metric by the fluctuations of the stressenergy momentum tensor in the state, where the expectation value is taken, are large. These fluctuations in the density arise on account of the particle creation due to the quantum nature of these TM states during the time of the oscillatory motion of the scalar field, in the semiclassical gravity. During this stage back-reaction effects from the created particles on the scalar field are important, and one may question the validity of the semiclassical approximation and whether this criterion is satisfied in all the entangled and non-entangled states formalism during the time of oscillatory motion of the scalar field. We have addressed the same in this paper.
According to the criterion put forward by Kuo and Ford [29] we can study the limit and validity of the semiclassical theory of gravity by considering the fluctuations in the energy-momentum tensor for matter field in a given background metric. The applicability of semiclassical theory, especially the semiclassical theory of gravity is restricted by the requirement that the quantum fluctuations of the (gravitational) source should be small compared to the source itself. They defined a dimensionless measure of the deviation from semiclassical theory. It is composed of the normalized dispersion in a stress tensor component. When the dimensionless measure is of the order unity, the quantum fluctuations in the expectation value of EMT are large and the semiclassical theory is no longer valid.
Motivated by this fact, we made use of the criterion suggested by them to draw limits on the validity of SG theory to our quantised TM scalar field model. We then utilize this criterion to study the reliability of the theory of semiclassical gravity for our model when the quantized TM scalar field is assumed to be in (i) TM vacuum state, (ii) TM nth-number state (iii) TM coherent state, (iv) TM squeezed vacuum state, (v) TM squeezed coherent and coherent squeezed states (TM squeezed states), (vi) entangled TM coherent state (vii) TM squeezed entangled coherent states. It was observed that there is clearly a limit to the validity of the SG theory in the above states formalism and is dependent on the magnitudes of the states parameter involved. We examined the dimensionless quantity D T numerically and graphically to study the reliability of the semiclassical Einstein equation during the time of oscillatory motion of the nonclassical scalar field in the above state formalisms for the flat FRW metric. The analysis shows that: For the TM vacuum state, the D T |0,0 quantity take values always small compared to unity. Thus, the semiclassical theory we had treated can be believed in if the scalar field modes are assumed to be in the states like TMV state during all the stages of evolution; as the fractional fluctuations in the local energy density are small and in turn, one may expect the metric fluctuations resulting from density fluctuations also to be small.
For the TM nth number state the D T |n 1 ,n 2 quantity is very small compared to unity only along the axes. However, along with and in the neighbourhood of the diagonal line i.e., with larger values of both n 1 and n 2 the TMN state exhibits large values of variance and hence of the dimensionless parameter. From these observations, it is evident that the semiclassical Einstein equation in the oscillatory phase of the non-classical scalar field in TMN state formalism is valid only when the associated nth particles states have number of particles distributed such that either n 1 n 2 or n 2 n 1 . This kind of a behaviour may be due to the fact that the density fluctuation in the back-reaction term in the semiclassical equation are large in nth particle states when a large number of particles are being produced by the gravitational background and as a result due to the creation of a large number of particles the expectation value of the energy-momentum density of the quantum field in the TMN state ceases to account for the backreaction adequately and consequently, the semiclassical theory can not be relied upon. This may be due to the fact that with n 1 and n 2 very large, the inter-mode correlation between the particles from two different states corresponding to two different modes starts playing a vital role and because of the quantum nature of this correlation the TMN state is considered to be quantum for such parameters choices. Thus, the semiclassical theory holds good in the TMN state representation of the scalar field in the oscillatory phase when the number state have particles distributed in two states such that either n 1 n 2 or n 2 n 1 , for such cases the TMN state may be regarded as a semiclassical state.
For the TM coherent state the variance V T |0,0 is approximately ≈ 0, since they are sharply peaked around the mean probability distributions and consequently D T | ,δ quantity always take values small compared to unity. This result is consistent and expected since it is common wisdom that coherent states correspond to classical field excitations. Furthermore, the dimensionless quantity is independent of the coherent parameters and δ of the two states involved. Thus, the above study reveals that the semiclassical gravity theory is a good approximation for the TMC state representation of the scalar field in the oscillatory phase.
Whereas for the TM squeezed vacuum state the theory holds only when the correlated squeezing parameter r takes on value much smaller compared to unity, otherwise the theory does not hold. From the previous investigation of TM squeezed states in semiclassical gravity theory, [24] we observed that large squeezing limit leads to a large number of particles being produced by gravitational background, and as a result, they exhibit large energy density fluctuations and we expect D T |ξ 2 to be of order unity implying that the expectation value of the energy-momentum density of the quantum field in TMSV state does not describe the back-reaction problem adequately, wherein the semiclassical theory based on equation Eq. (1) is bound to prove rather inadequate. For squeezed states, we have a continuous family of states labelled by the squeeze parameter. We see that the larger the squeeze parameter and thus the farther is the extent of violation of the semiclassical approximation.
If the scalar field modes are assumed to be in states like TM coherent squeezed state and TM squeezed coherent states then we may expect the semiclassical theory of gravity will give reliable results as long as the r parameter takes on values much small compared to unity. Moreover, the expression for the D T D δ |ξ 2 and D T Z ab | ,δ quantities in these states reveals that the measure is independent of the coherent parameters of the two states involved and is only dependent on the squeezing parameter. Now, this is to be expected as TM squeezed state are two-parameter families of states which includes the coherent states as one limit, and the squeezed vacuum states as another limit and with large energy density fluctuations in large squeezing limit.
For the case of ETMC states, we have a set of four quasi-Bell states |Ψ i 12 with their D T |Ψ i 12 quantities unlike TMC state are dependent on the coherent parameters of the states involved. Thus, here the magnitude of coherent parameter draws the limit to the extent to which semiclassical theory is valid. In the case of 1st and 2nd quasi-Bell states, we observed these states have small energy density fluctuations for greater values of coherent parameter. However, for smaller values of a coherent parameter the nonclassical behaviour of these states is much resulting in large fluctuations with D T |Ψ 1 , |Ψ 2 approaching unity and therefore, we cannot expect the semiclassical theory to hold for these states with such values of coherent parameters. Although, with large parameter the quantity D T |Ψ 1 , |Ψ 2 die down as | | −2 , thus larger values of parameter reduces the nonclassical behaviour of the states and makes its likely to the coherent states where the expectation value of a product of stress tensor is equivalent to the corresponding product of expectation value and hence D T |Ψ 1 , |Ψ 2 approaching zero and the theory of semiclassical gravity is a good approximation when the quantum states considered are not far from the coherent states. Furthermore, the |Ψ 3 and |Ψ 4 states are much richer in nonclassical behaviour than the |Ψ 1 and |Ψ 2 states. For smaller values of coherent parameter, these states have dimensionless parameter which is not a monotonic function of the parameter; in these states, we have a range for the coherent parameter where the behaves similar to the squeezing parameter r such that with increasing value of coherent parameter D T quantity increases, where we may expect the breakdown of semiclassical approximation. However, beyond that region, we observe for large values of parameter the fluctuations in the local energy density are small and in this limit, the semiclassical equation proves to be quite reliable.
Moreover, unlike 1st and 2nd states, in the 3rd and 4th quasi-Bell states the dimensionless quantity does not vanish completely or approaches zero rather these states have slightly higher values for dimensionless quantity for large values of however, still we can expect D T 1, and the SG theory to hold.
Finally, if the quantum states of the TM quantized scalar field modes are assumed to be TMSEC |η i 12 states then the expression for D T |η i 12 reveals that since |η i 12 states are also two-parameter families of states which includes the ETMC states as one limit and TMSV states as another limit and therefore, the extent to which the semiclassical theory of gravity is a good approximation in these states is dependent on both the coherent and squeezed parameters of the states involved. For the given range of parameters values of |η i 12 states it was observed that when the corresponding squeezing parameter r takes values much smaller compared to the associated coherent parameter | | i.e., | | r furthermore when the squeezing parameter is much smaller than unity, we find that fluctuations in the local stress tensor are small. For such cases the states are semiclassical and their semiclassical Einstein equations during the phase scalar field's oscillatory motion can be trusted. However, as the magnitude of the squeezed parameter r increases relative to that of | |, we find that local energy density fluctuations are large leading to D T |η 1 12 ; |η 2 12 (e) ≈ 1 or > 1, making the states very quantum, wherein the semiclassical approximation to the gravity cannot be trusted. It is interesting to note that the nonclassical characters of the |η 1 and |η 2 states get completely or approximately destroyed when coherent parameter takes on further large values relative to r . However, for the |η 3 and |η 4 states unlike |η 1 and |η 2 the dimensionless quantity does not completely dies down when coherent parameter takes on further large values relative to r , but becomes smaller and smaller than unity with D T |η 3,4 12 finally approaching 0.5. These results are in consistent with the interpretation of squeezed state since for large values of r parameter, r 1 there is a large amount of particle production due to non-adiabatic amplification and as a result, the quantum fluctuations in the expectation value of stress tensor are large resulting into large fluctuations in the metric tensor and hence, the semiclassical theory is no longer valid.
Thus, we conclude that as the applicability of semiclassical theory, especially the semiclassical theory of gravity is restricted by the requirement that the quantum fluctuations of the (gravitational) source should be small compared to the source itself, we should be very careful about the domain of semiclassical gravity in these states. Thus, we see that in our case of TM quantized scalar field model when we consider the states that have a sufficiently small magnitude of squeezing parameter (since larger the squeeze parameter and thus the farther away from classical behaviour) and are much closer to the coherent states (for greater values of parameter), for them we may expect the semiclassical theory we had considered for our model can be trusted upon, during all stages of the evolution.

Data Availability Statement
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