7. Non-perturbative production of gauge fields L = 1 2 ∂$_{µ}$φ∂ $^{µ}$φ + 1 2 D$_{µ}$χD µ $_{χ}$∗ + 1 2 m 2 $_{φ}$φ 2 + 1 2 m 2 T | χ | 2 ( φ 2 − φ 2 0 ) + λ 4 | χ | 4 − 1 4 F 2 (408) Π$_{ij}$ = F$_{iC}$F C j − 1 3 δ$_{ij}$F$_{kC}$F kC − D$_{i}$χD$_{j}$$_{χ}$∗ + 1 3 δ$_{ij}$D$_{k}$χD k $_{χ}$∗ − ∂$_{i}$φ∂$_{j}$$_{φ}$∗ + 1 3 δ$_{ij}$∂$_{k}$φ∂ k $_{φ}$∗ (409) Let us now consider non-perturbative effects on gauge fields. For simplicity, let us consider an example where the tachyonic field, χ , is charged under U (1) × U (1), which arises quite naturally in a brane-anti-brane inflation [727]. Here F + and F − are the gauge fields that live in the world volume of the brane and anti-brane respectively. The tachyon, χ , is a bi- fundamental field that couples only to a linear combination of the two gauge fields: D$_{µ}$χ = ∂$_{µ}$χ − ( A + µ − A − µ ) χ , and φ is the inflaton field $^{87}$. The Lagrangian is given by: 1 1 µ 1 2 2 1 2 2 2 2 λ 4 1 2 The tachyon mass is fixed in terms of the string scale, m$_{T}$ = M$_{s}$ , while the mass of the inflaton is determined by the amplitude of the temperature anisotropy, which turns out to be: m$_{φ}$ = 0 . 01 M$_{s}$ . The stress-energy tensor now gets contributions from both the charged tachyon, χ and uncharged inflaton, φ , as well as the gauge fields, which is given by [727]: The energy density of the gravitational waves is simply given by the ‘ t ′ 00 component of the energy-momentum tensor of the gravitational waves calculated in a synchronous gauge, i.e. h$_{0}$$_{i}$ = 0 , h$_{00}$ = 0. In Fig. (19), the energy pumped into the gauge fields and the gravity waves are shown for λ = 1, m$_{φ}$ = 0 . 01 M$_{S}$ and the tachyon mass is given by M$_{S}$ = 10 − $^{4}$M$_{P}$ .. Of course, larger the value of a tachyon mass, greater is the instability, and the growth in the respective perturbations, but compare the energy densities in gauge fields and gravity waves. In Figs. (20), 4 snap-shots of the iso-surface of the constant energy density of gauge field, gravity waves, tachyon and inflaton are depicted. All the fields except the inflaton show a remarkable departure in the homogeneity. Except the inflaton, all fields undergo long wavelength excitations (they all look relatively smooth on small scales), while the inflaton obtains the largest inhomogeneity on the smaller scales. This is due to the fact that the there is no long wavelength amplification for the inflaton in this case [727]. 87 The physical situation is quite similar to that of a hybrid inflation. The tachyon field in brane-anti-brane inflation plays a dynamical role of a water-fall field or the Higgs field. 189