We consider in this section the Higgs-like potential,
V = λ(φ2− σ2)2, (5.2.1)
where σ is the energy scale of symmetry breaking and λ is the coupling con-stant. The value that the parameter σ can take is constrained by how much freedom we would like the initial conditions to have, but given that constrain we should also regulate its value so as to generate enough e-folding for a viable inflation, we took σ = 20MP [20]. The coupling constant λ is constrained using the scalar power spectrum from the Planck results, its value is found to be λ ≈ 10−14, which is in the range of the accepted region [21].
The potential slow-roll parameters and η are,
= MP2
N ≈ (1 + Q)
where we neglected the φ2 term in the second line since φ σ during the slow-roll stage. From (4.2.16),
From (5.2.4) we can produce an expression for φ,
φ ≈ φende
From (4.3.7) we can calculate the tensor to scalar ratio for N = 60 before the end of inflation,
Figure 5.13: The tensor to scalar ratio vs. Q for Q < 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.
The value of the tensor to scalar ratio also depends on the value of σ where we chose σ = 20 since it produces enough inflation in our numerical calculation but it can be seen in figure 5.13 that for low Q the tensor to scalar ratio exceeded the bound given by the Planck data, although we could have chosen a different σ in order to suppress the tensor to scalar ratio, this is just to compare to the result of the numerical calculation, while in figure 5.14 the tensor to scalar ratio is well inside the bound. It should be taken note that there is an accepted region where we can choose the value of σ with range 15MP≤ σ ≤ 40MP [20].
Figure 5.14: The tensor to scalar ratio vs. Q for Q > 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.
The scalar spectral index (4.3.10) for N = 60 before the end of inflation can be calculated as,
ns≈ 1 − 10MP2 1 + Q
3φ2+ σ2
(φ2− σ2)2 (5.2.9)
From figure 5.15 it can be seen that for low dissipation the scalar spectral index is well within the bound given by the Planck data but as the dissipation gets stronger the scalar spectral index deviates tremendously from 1, while in figure 5.16 the scalar spectral index continues to deviate farther from 1 which can be seen to be extremely beyond the 95% confidence region.
Figure 5.15: The tensor to scalar ratio vs. ns for Q < 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confi-dence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
Figure 5.16: The tensor to scalar ratio vs. ns for Q > 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confi-dence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
Numerical Calculation
The initial conditions used in this model were ˜φ0 = 3 and ˜φ00 = 0.0029 (See Appendix C).
Figure 5.17: The temperature evolution for Q = 10−5 and N ≈ 75.
In figure 5.17, for Q = 10−5 the temperature went below the Hubble pa-rameter but dominated it later to produce a total duration of warm inflation around N ≈ 75, while in figure 5.18 for Q = 30 the dissipation is strong enough to induce a rebound temperature above the Hubble parameter with a total duration of warm inflation around N ≈ 885.
Figure 5.18: The temperature evolution for Q = 30, N ≈ 885.
Figure 5.19: The evolution of the inflaton and radiation energy density for Q = 10−5.
Figure 5.20: The evolution of the inflaton and radiation energy density for Q = 30.
As with the chaotic inflation, the total duration of inflation increases as the dissipation increases. In figure 5.19 it can be seen that for Q = 10−5 the radiation energy density is barely comparable to the energy density of the inflaton at the end of inflation while in figure 5.20 for Q = 30 the radiation energy density is very comparable to the energy density of the inflaton.
Figure 5.21: The tensor to scalar ratio as a function of Q (Q < 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.
Figure 5.22: The tensor to scalar ratio as a function of Q (Q > 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.
Figure 5.23: The tensor to scalar ratio as a function of the scalar spectral index ns (Q < 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
Figure 5.24: The tensor to scalar ratio as a function of the scalar spectral index ns (Q > 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
In figure 5.21 it can be seen that the tensor to scalar ratio starts of well below the bound given by the Planck data as is the case in the standard cold inflation. This is in contrast to the slow-roll result in figure 5.13, while in figure 5.22 the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger. In figure 5.23, the tensor to scalar ratio vs. scalar spectral index data points fall within the 95% confidence level region given by the Planck data but starts to enter the 68% confidence level region as Q increases. This is in contrast to the result in figure 5.15 where as Q increases the scalar spectral index shifts towards values 1. In figure 5.24 the scalar spectral index data points are well inside the 68% confidence region.