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It depends on the scales which you are interested in. During the deSitter evolution of the universe (let's assume exact deSitter and completely flat potential for simlicity) the fluctuations of the inflaton field exit the horizon and freeze in with a power spectrum of $H_*^2/(2\pi)^2$ where $*$ denotes that the value of the Hubble parameter is taken at the ...

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There is a limited sense in which this correct. When inflation ended there was a temperature rise known as reheating, and we believe it was at this stage that the standard model particles were first created, or at least the majority of them. If you are measuring temperature by the energies of the standard model particles then this would have been the hottest ...

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It depends entirely on the magnitude of the cosmological constant. Due to dark energy spacetime is already undergoing very, very slow inflation. The doubling time, i.e. the time taken for the scale factor to double due to dark energy, is in the range 10-20 billion years and this is far too slow to have any effect on gravitationally bound systems like the ...

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This is probably too late to be useful, however yes you should assume slow roll and that the inflaton's potential energy is dominating the energy density of the Universe. So solve: $3H\dot{\phi} + V'(\phi)=0$ with: $H^2 = 8\pi G V(\phi) /3$ This slow roll approximation is valid if the slow roll parameters are $\ll 1$ which requires $\alpha \ll 1/m_p$.

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The scalar spectral index, like the above answer states, describes how the density fluctuations vary with scale. As the size of these fluctuations depends upon the inflaton's motion when these quantum fluctuations are becoming super-horizon sized, different inflationary potentials predict different spectral indices. These depend upon the slow roll ...

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So the 'old' model of inflation was based on the idea that a scalar field (the inflaton) was trapped initially in some meta-stable vacuum. If it's energy density dominates the Universe at this time then the Universe inflates. This continues until the field tunnels out of this vacuum and inflation ends. This leads to bubble nucleation as different patches of ...

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Equivalence of the Einstein and Jordan frames No experiment can measure an absolute scale: every dimensionfull quantity has to be compared to some fixed unit scale in order to be measured, and thus only dimensionless quantities are really physical. The Einstein and Jordan frame are related by a conformal transformation of the metric, which ...

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