We consider in this section the chaotic potential
V = 1
2m2φ2, (5.1.1)
where m is the inflaton mass and φ is the inflaton field. The inflaton mass is constrained using the scalar power spectrum and its value is found to be m ≈ 1013GeV [15].
The potential slow-roll parameters and η are
= MP2
φ2 ≈ φ2end+4N MP2
1 + Q ≈ 2MP2
1 + Q +4N MP2
1 + Q = (2 + 4N )
1 + Q MP2. (5.1.7) Now, we can calculate the tensor to scalar ratio from (4.3.7) for N = 60 before the end of inflation,
r ≈ 16
121(1 + Q)
"
1 + 4
r 36MP2Q 893101π2m2
2√
√ 3πQ 3 + 4πQ
#−1
. (5.1.8)
Figure 5.1: 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.
Figure 5.2: 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.
It can be seen from figure 5.1 that for very low Q, the tensor to scalar ratio exceeds the bound given by the Planck data as is the case in the cold inflation but as the dissipation increases the tensor to scalar ratio is suppressed to values that are within the given bound and it can be seen that this starts to happen around Q ≈ 10−2. In figure 5.2 for high Q the tensor to scalar ratio starts off very low and continues to decrease as the dissipation gets stronger.
The scalar spectral index can be calculated from (4.3.10) for N = 60 before the end of inflation,
ns≈ 1 − 10MP2
(1 + Q)φ2 = 1 − 10
242 = 0.958677. (5.1.9) This shows that for chaotic quadratic potential, the scalar spectral index is approximately independent of the dissipation Γ, in which we plot the tensor to scalar ratio as a function of the scalar spectral index in figure 5.3 and figure 5.4.
Figure 5.3: 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 5.4: 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
One of the problems with the approximations and procedures we have done above is that it does not give any reference if the total duration of inflation for a given Q is enough to solve the problems of the standard Big Bang such that it should also satisfy the condition T > H. The keypoint here is that we just assumed that if we set N = 60 and calculate the observables for a given Q we would be able to get the observables at the horizon exit in warm inflation, but it could also be the case that at N = 60 the temperature is still less than the Hubble constant. So, the best procedure to take is to first numerically solve the evolution of the temperature and Hubble constant during inflation and acquire the values of the corresponding quantities for a given Q at the horizon exit assuring that it is in the region where T > H. The initial conditions used in this model were ˜φ0 = 27 and ˜φ00 = 10−4 (See Appendix C).
Figure 5.5: The temperature evolution for Q = 10−5 and N ≈ 58.
Figure 5.6: The temperature evolution for Q = 30, N ≈ 1020.
It can be seen in figure 5.5 that the rebound temperature can go below the Hubble constant during inflation, but we only consider the region where T > H in which this is the regime of warm inflation; this is where we question if the
duration is enough in order to solve the standard problems of Big Bang. For Q = 10−5, N ≈ 58 so that this can be considered as the lower limit for the model to be a legitimate warm inflation. In figure 5.6 it can be seen that for Q = 30, N ≈ 1020, the rebound temperature is above the Hubble constant.
Figure 5.7: The evolution of the inflaton and radiation energy density for Q = 10−5.
Figure 5.8: The evolution of the inflaton and radiation energy density for Q = 30.
From section 4.1, one of the conditions for warm inflation is that the energy density of the inflaton should dominate over the radiation energy density, but if this is violated or in other words the radiation density is comparable to the inflaton energy density (ρr ≈ ρφ) then inflation is terminated. It can be seen from figure 5.7 that at the end of inflation for Q = 10−5, the radiation energy density is barely comparable to the energy density of the inflaton, this just implies that Q is too small and its behavior is close to the cold inflation case;
while in figure 5.8 for Q = 30, the radiation energy density is comparable to the energy density of the inflaton at the end of inflation.
The effect of Q is that as it increases the number of e-folding also increases so that we should determine the tensor to scalar ratio (4.3.6) and scalar spectral index (4.3.9) only for each Q at the horizon crossing, and we do this for all the Q to be studied.
Figure 5.9: 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 5.10: 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 5.11: 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 5.12: 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.
For the numerical calculation, in figure 5.9 for Q < 1 the tensor to scalar
ratio starts off at values above the bound given by Planck but starts to enter the bound around Q ≈ 10−2, while in figure 5.10 for Q > 1 it is well below the bound. In figure 5.11 and 5.12 it can be seen that the scalar spectral index is approximately independent of the dissipation so that ns ≈ 0.96 which is in the 95% confidence region.