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(TT) components of the stress-energy momentum tensor. By perturbing the Einstein’s equation, we obtain the evolution of the tensor perturbations, see [779]: 

<img src='content_image/367240.jpg'>

where ∂$_{i}$ Π$_{ij}$ = Π$_{ii}$ = 0 and ∂$_{i}$h$_{ij}$ = h$_{ii}$ = 0. The TT part of the spatial components of a symmetric anisotropic stress-tensor T$_{µν}$ can be found by using the spatial projection operators, P$_{ij}$ = δ$_{ij}$ − ˆ k$_{i}$ ˆ k$_{j}$ , with ˆ k$_{i}$ = k$_{i}$/k : 

<img src='content_image/367241.jpg'>

where Λ$_{ij,mn}$( ˆ k ) ≡ ( P$_{im}$ ( ˆ k ) P$_{jn}$ ( ˆ k ) − (1 / 2) P$_{ij}$ ( ˆ k ) P$_{mn}$ ( ˆ k ) ) . The TT perturbation is written as h$_{ij}$ ( t, ˆ k ) = Λ$_{ij,lm}$( ˆ k ) u$_{ij}$ ( t, k ), where 

<img src='content_image/367242.jpg'>

The source terms for the energy momentum tensor in our case are just the gradient terms of the scalar field χ involved during preheating. 

<img src='content_image/367243.jpg'>

where χ$_{1}$ and χ$_{2}$ represent the real and imaginary parts of φ , respectively. The gravitational wave (GW) energy density is given by: 

<img src='content_image/367244.jpg'>

where V is the volume of the lattice. As an application, let us consider exciting gravity waves in hybrid inflation model of inflaton, with inflaton, φ , and the Higgs field, χ , see [779]. The coupled evolution equations that have to be solved numerically on a lattice for the hybrid model of inflation are given by [779] $^{85}$: 

where χ is the inflaton and the φ$_{a}$ are the complex water-fields. 

