Hamilton-Jacobi formalism to warm inflationary scenario

The Hamilton-Jacobi formalism as a powerful method is being utilized to reconsider the warm inflationary scenario, where the scalar field as the main component driving inflation interacts with other fields. Separating the context into strong and weak dissipative regimes, the goal is followed for two popular functions of $\mathrm{\Gamma }$. Applying slow-rolling approximation, the required perturbation parameters are extracted and, by comparing to the latest Planck data, the free parameters are restricted. The possibility of producing an acceptable inflation is studied where the result shows that for all cases the model could successfully suggest the amplitude of scalar perturbation, scalar spectral index, its running, and the tensor-to-scalar ratio.

Figure 1

Scalar spectral index versus $n$ for $N=60$ and different values of $m$ as $m=1$ (blue color line) and $m=3$ (yellow color line).

Figure 2

The running of the scalar spectral index is shown versus $n_s$ for $n=1.5$ and $m=3$, and for $N=55–65$ in which the small point belongs to $N=55$ and the large point belongs to $N=65$.

Figure 3

The potential versus the scalar field is plotted during the inflationary era, where the constant parameters are taken as $n=1.5$, $m=3$, $N=60$, and $H_0=3\times 10^{-18}$. The parameter $\overline{V}$ is defined by $\overline{V}=\frac{V(\varphi )}{V_0}$, in which $V_0=4.23\times 10^{61}$ is the initial value of the potential.

Figure 4

Energy densities of the scalar field (blue line) and radiation (red line) are depicted in terms of the scalar field during the inflation times, where the constant parameters are taken as $n=1.5$, $m=3$, $N=60$, and $H_0=2\times 10^{-18}$. The parameter ${\overline{\rho }}_i$ is defined by ${\overline{\rho }}_i=\frac{{\rho }_i(\varphi )}{{\rho }_0}$, in which ${\rho }_0$ is the initial value of the scalar field energy density.

Figure 5

Scalar spectral index versus $n$ for $N=60$ and different values of $m$ as $m=-1$ (blue color line) and $m=4$ (yellow color line).

Figure 6

The running of the scalar spectral index is shown versus $n_s$ for $n=1$ and $m=-1$, and for $N=55–65$ in which the small point belongs to $N=55$ and the large point belongs to $N=65$.

Figure 7

The potential versus the scalar field is plotted during the inflationary era, where the constant parameters are taken as $n=1$, $m=-1$, $N=60$, and $H_0=1\times 10^{-8}$. The parameter $\overline{V}$ is defined by $\overline{V}=\frac{V(\varphi )}{V_0}$, in which $V_0=7.26\times 10^{59}$ is the initial value of the potential.

Figure 8

Energy densities of the scalar field (blue line) and radiation (red line) is depicted in terms of the scalar field during the inflation times, where the constant parameters are taken as $n=1$, $m=-1$, $N=60$, and $H_0=1\times 10^{-8}$. The parameter ${\overline{\rho }}_i$ is defined by ${\overline{\rho }}_i=\frac{{\rho }_i(\varphi )}{{\rho }_0}$, in which ${\rho }_0$ is the initial value of the scalar field energy density.

Figure 9

The scalar spectral index versus $n$ for $N=60$ and different values of $m$ as $m=1$ (blue color line) and $m=4$ (yellow color line).

Figure 10

The running of the scalar spectral index is shown versus $n_s$ for $n=2$ and $m=1$, and for $N=55–65$ in which the small point belongs to $N=55$ and the large point belongs to $N=65$.

Figure 11

The potential versus the scalar field is plotted during the inflationary era, where the constant parameters are taken as $n=2$, $m=1$, $N=60$, and $H_0=1\times 10^{-27}$. The parameter $\overline{V}$ is defined by $\overline{V}=\frac{V(\varphi )}{V_0}$, in which $V_0=1.48\times 10^{62}$ is the initial value of the potential.

Figure 12

Energy densities of the scalar field (blue line) and radiation (red line) are depicted in terms of the scalar field during inflation times, where the constant parameters are taken as $n=2$, $m=1$, $N=60$, and $H_0=1\times 10^{-27}$. The parameter ${\overline{\rho }}_i$ is defined by ${\overline{\rho }}_i=\frac{{\rho }_i(\varphi )}{{\rho }_0}$, in which ${\rho }_0$ is the initial value of the scalar field energy density.

Figure 13

Scalar spectral index versus $n$ for $N=60$ and different values of $m$ as $m=1$ (blue color line) and $m=3$ (yellow color line).

Figure 14

The running of scalar spectral index is shown versus $n_s$ for $n=2$ and $m=1.5$, and for $N=55–65$ in which the small point belongs to $N=55$ and the large point belongs to $N=65$.

Figure 15

The potential versus the scalar field is plotted during the inflationary era, where the constant parameters are taken as $n=2$, $m=1.5$, $N=60$, and $H_0=1\times 10^{-28}$. The parameter $\overline{V}$ is defined by $\overline{V}=\frac{V(\varphi )}{V_0}$, in which $V_0=1.48\times 10^{60}$ is the initial value of the potential.

Figure 16

Energy densities of the scalar field (blue line) and radiation (red line) are depicted in terms of the scalar field during inflation times, where the constant parameters are taken as $n=2$, $m=1.5$, $N=60$, and $H_0=1\times 10^{-28}$. The parameter ${\overline{\rho }}_i$ is defined by ${\overline{\rho }}_i=\frac{{\rho }_i(\varphi )}{{\rho }_0}$, in which ${\rho }_0$ is the initial value of the scalar field energy density.

Figure 17

$r-n_s$ diagram for selected free parameters have been plotted for every case in which the small point relates to $N=50$ and the large point corresponds to $N=60$.

Figure 18

Temperature versus the scalar field during the inflationary time has been plotted for all cases, for the corresponding chosen free parameters. The parameter $\overline{T}$ is defined as $\overline{T}=T/H$.