Cosmological parameter estimation and the spectral index from inflation

Accurate estimation of cosmological parameters from microwave background anisotropies requires high-accuracy understanding of the cosmological model. Normally, a power-law spectrum of density perturbations is assumed, in which case the spectral index $n$ can be measured to around $\pm 0.004$ using microwave anisotropy satellites such as MAP and Planck. However, inflationary models generically predict that the spectral index $n$ of the density perturbation spectrum will be scale-dependent. We carry out a detailed investigation of the measurability of this scale dependence by Planck, including the influence of polarization on the parameter estimation. We also estimate the increase in the uncertainty in all other parameters if the scale dependence has to be included. This increase applies even if the scale dependence is too small to be measured unless it is assumed absent, but is shown to be a small effect. We study the implications for inflation models, beginning with a brief examination of the generic slow-roll inflation situation, and then move to a detailed examination of a recently-devised hybrid inflation model for which the scale dependence of $n$ may be observable.Comment: 6 pages LaTeX file with one figure incorporated (uses mn.sty and epsf). Important modifications to result

Higher order corrections to primordial spectra from cosmological inflation

We calculate power spectra of cosmological perturbations at high accuracy for two classes of inflation models. We classify the models according to the behaviour of the Hubble distance during inflation. Our approximation works if the Hubble distance can be approximated either to be a constant or to grow linearly with cosmic time. Many popular inflationary models can be described in this way, e.g., chaotic inflation with a monomial potential, power-law inflation and inflation at a maximum. Our scheme of approximation does not rely on a slow-roll expansion. Thus we can make accurate predictions for some of the models with large slow-roll parameters.Comment: 13 pages, 1 figure; section on consistency relations of inflation added; accepted by Physics Letters

Distribution of particles which produces a "smart" material

If $A_q(\beta, \alpha, k)$ is the scattering amplitude, corresponding to a potential $q\in L^2(D)$, where $D\subset\R^3$ is a bounded domain, and $e^{ik\alpha \cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A(\beta):=A_q(\beta, \alpha, k)$, where the unit vector $\alpha$, the incident direction, is fixed, and $k>0$, the wavenumber, is fixed. It is shown that any function $f(\beta)\in L^2(S^2)$, where $S^2$ is the unit sphere in $\R^3$, can be approximated with any desired accuracy by a radiation pattern: $||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon$, where $\epsilon>0$ is an arbitrary small fixed number. The potential $q$, corresponding to $A(\beta)$, depends on $f$ and $\epsilon$, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles $D_m\subset D$, $1\leq m\leq M$, distributed in an a priori given bounded domain $D\subset\R^3$. The geometrical shape of a small particle $D_m$ is arbitrary, the boundary $S_m$ of $D_m$ is Lipschitz uniformly with respect to $m$. The wave number $k$ and the direction $\alpha$ of the incident upon $D$ plane wave are fixed.It is shown that a suitable distribution of the above particles in $D$ can produce the scattering amplitude $A(\alpha',\alpha)$, $\alpha',\alpha\in S^2$, at a fixed $k>0$, arbitrarily close in the norm of $L^2(S^2\times S^2)$ to an arbitrary given scattering amplitude $f(\alpha',\alpha)$, corresponding to a real-valued potential $q\in L^2(D)$.Comment: corrected typo