firstname = "Anna"
lastname = "Yaparova"
email = "lisa74@yandex.ru"
affiliation = "Immanuel Kant Baltic federal university"
city = "Kaliningrad"
country = "Russia"
passportname = ""
birthday = ""
citizenship = ""
passportnumber = ""
passportissued = ""
passportexpire = ""
workplace = ""
workaddress = ""
visadates = ""
transfer = "yes"
talktitle = "Superpotential and $U(\phi)$ Method for Brane and Friedman Cosmology"
section = "C - Gravitation and cosmology"
talkabstract = "The first part is dedicated to realization of inflation on brane by application of the method of superpotential. It is considered a universe on brane filled with a single real scalar field $\phi$. A rate of change for the field $\dot{\phi}$ is set as a function of the field $\dot{\phi}=U(\phi)$. Using superpotential $W=1/2U^2+V(\phi)$ we have a pair of equation. After its integration, we can get a full energy density $\rho$ of the field, its potential $V$ and Hubble parameter $H$ as functions of the field $\phi$. There are two models under consideration: $U(\phi)=\phi^{-s}$ (for different $s=\mathrm{const}$) and $U(\phi)=-k^2\exp{(-\phi/\phi_s)}$ ($k=\mathrm{const}$, $\phi_s=\mathrm{const}$).\par
The second part is dedicated to a study of inflation in a flat FRLW universe filled with a single real scalar field $\phi$ by reduction of Friedman equations to Abel equation of $1^{\mathrm{st}}$ kind. With given potential $V(\phi)$ this method allows to get a rate of change for the field as function of $\phi$ and derive an energy density of the field, its pressure, Hubble parameter as functions of $\phi$. In some cases it is possible to get exact solutions for this parameters and scale factor as explicit functions of time or in a parametric form. Such case is, for example, the case of potential $V(\phi)=C^2\exp(\kappa\sqrt{2}\phi)$ ($C^2=\mathrm{const}$, $\kappa=\pm1$). This potential gives several types of solutions, depending on sign of an integration constant in solutions of corresponding Able equation. But in general all this solutions is similar: a scale factor grows with acceleration from finite value to infinity for infinite time (the energy density and pressure tend to zero), or it undergoes decelerated decrease to zero for finite time, or it goes down to zero, with deceleration at first and then with acceleration, for finite time (in last two cases it is possible to change a sign of time and get an increase from a finite value to infinity instead of a decrease). 
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