Dynamics of interacting scalar fields in expanding space-time

The effective equation of motion is derived for a scalar field interacting with other fields in a Friedmann-Robertson-Walker background space-time. The dissipative behavior reflected in this effective evolution equation is studied both in simplified approximations as well as numerically. The relevance of our results to inflation are considered both in terms of the evolution of the inflaton field as well as its fluctuation spectrum. A brief examination also is made of supersymmetric models that yield dissipative effects during inflation.

Figure 1

The cut in the one-loop self-energy diagram and the amplitude of the decay process associated to it. Full lines stand for the scalar $\varphi$ and dotted lines to the spinors $\psi$, $\overline{\psi }$. An analogous process follows for the decay $\varphi \to \chi \chi$.

Figure 2

The lowest order terms of the effective propagator for the scalar $\chi$ after integrating over the fermion fields coupled to it. The black ellipse stands for the spinors quantum corrections and the dashed lines to the $\chi$ propagator.

Figure 3

Some of the lowest order diagrams contributing to the action of $\varphi$ after integrating over the scalar $\chi$ and spinors ${\psi }_d$, ${\overline{\psi }}_d$.

Figure 4

The cut in the lowest order nontrivial diagram appearing in the effective action of $\varphi$ that has an imaginary part and the main amplitude associated to it.

Figure 5

The kernel $K_{\chi }(t,0)$ of Eq. (6.1) for various interaction couplings $g$ and $h=g$, with $m_{\chi }=g\phi (0)$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, $\lambda ={10}^{-13}$, and $\xi =0$.

Figure 6

The integrated kernel $I_{\chi }(t,0)$ of Eq. (6.4) for various interaction couplings $g$ and $h=g$, with $m_{\chi }=g\phi (0)$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, $\lambda ={10}^{-13}$, and $\xi =0$.

Figure 7

The kernel compared between the expanding Eq. (6.1) and nonexpanding cases for $g=h=0.37$, with $m_{\chi }=g\phi (0)$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, $\lambda ={10}^{-13}$, and $\xi =0$.

Figure 8

The integrated kernel of Fig. 7 compared between the (a) expanding and (b) nonexpanding cases for $g=h=0.37$, with $m_{\chi }=g\phi (0)$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, $\lambda ={10}^{-13}$, and $\xi =0$.

Figure 9

Evolution of $\phi (t)$ for $g=h=0.1$, $\lambda ={10}^{-13}$, $\xi =0$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, and $\dot{\phi }(0)=0$.

Figure 10

Evolution of $\phi (t)$ for various interaction couplings $g$ and $h=g$, with $\lambda ={10}^{-13}$, $\xi =0$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, and $\dot{\phi }(0)=0$.

Figure 11

Evolution of radiation density ${\rho }_r$, plotted as the ratio $R\equiv {\rho }_r^{1/4}/H$ from various approximations and various interaction couplings $g$, with $h=g$, $\lambda ={10}^{-13}$, $\xi =0$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, and $\dot{\phi }(0)=0$.

Figure 12

Checking the adiabatic approximation Eqs. (4.25), for various interaction couplings $g=0.1$, 0.37, 0.4, and 0.5, with $h=g$, $\lambda ={10}^{-13}$, $\xi =0$, $\phi (0)=m_{\mathrm{P}\mathrm{l}}$, and $\dot{\phi }(0)=0$.