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Register a new userA Brachistochrone Approach to Reconstruct the Inflaton Potential
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We propose a new way to implement an inflationary prior to a cosmological dataset that incorporates the inflationary observables at arbitrary order. This approach employs an exponential form for the Hubble parameter $H(phi)$ without taking the slow-roll approximation. At lowest non-trivial order, this $H(phi)$ has the unique property that it is the solution to the brachistochrone problem for inflation.
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We place functional constraints on the shape of the inflaton potential from the cosmic microwave background through a variant of the generalized slow roll approximation that allows large amplitude, rapidly changing deviations from scale-free conditions. Employing a principal component decomposition of the source function G~3(V/V)^2 - 2V/V and keeping only those measured to better than 10% results in 5 nearly independent Gaussian constraints that maybe used to test any single-field inflationary model where such deviations are expected. The first component implies < 3% variations at the 100 Mpc scale. One component shows a 95% CL preference for deviations around the 300 Mpc scale at the ~10% level but the global significance is reduced considering the 5 components examined. This deviation also requires a change in the cold dark matter density which in a flat LCDM model is disfavored by current supernova and Hubble constant data and can be tested with future polarization or high multipole temperature data. Its impact resembles a local running of the tilt from multipoles 30-800 but is only marginally consistent with a constant running beyond this range. For this analysis, we have implemented a ~40x faster WMAP7 likelihood method which we have made publicly available.
Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1sim -2/N$, where $N$ is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs inflation or $alpha$-attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(phi)$. We find that for $n_s-1=-2/N$, $V(phi)$ is either $tanh^2(gammaphi/2)$ (T-model) or $phi^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $gamma$ is the ratio of $1/V$ at $Nrightarrow infty$ to the slope of $1/V$ at a finite $N$ and is related to $alpha$ in the $alpha$-attractors by $gamma^2=2/3alpha$. The tensor-to-scalar ratio $r$ is $r=8/N(gamma^2 N +1) $. The implications for the reheating temperature are also discussed. We also derive formulas for $n_s-1=-p/N$. We find that if the potential is bounded from above, only $p>1$ is allowed. Although $r$ depends on a parameter, the running of the spectral index is independent of it, which can be used as a consistency check of the assumed relation of $n_s(N)$.
By using a usual instanton method we obtain the energy splitting due to quantum tunneling through the triple well barrier. It is shown that the term related to the midpoint of the energy splitting in propagator is quite different from that of double well case, in that it is proportional to the algebraic average of the frequencies of the left and central wells.
We discuss features of the inflaton potential that can lead to a strong enhancement of the power spectrum of curvature perturbations. We show that a steep decrease of the potential induces an enhancement of the spectrum by several orders of magnitude, which may lead to the production of primordial black holes. The same feature can also create a distinctive oscillatory pattern in the spectrum of gravitational waves generated through the scalar perturbations at second order. We study the additive effect of several such features. We analyse a simplified potential, but also discuss the possible application to supergravity models.
The quantum mechanical brachistochrone system with PT-symmetric Hamiltonian is Naimark dilated and reinterpreted as subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultra-fast brachistochrone regime of [C. M. Bender et al, Phys. Rev. Lett. {bf 98}, 040403 (2007)] in an entangled two-spin system.
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