The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

Observations of anisotropies in the cosmic microwave background by the Wilkinson Microwave Anisotropy Probe suggest the possibility of oscillations in the primordial curvature perturbation. Such deviations from the usually assumed scale-free spectrum were predicted in the multiple inflation model wherein 'flat direction' fields undergo rapid phase transitions due to the breaking of supersymmetry by the large vacuum energy driving inflation. This causes sudden changes in the mass of the (gravitationally coupled) inflaton and interrupts its slow roll. We calculate analytically the resulting modifications to the curvature perturbation and demonstrate how the oscillations arise.

Detailed analyses of the WMAP data indicate possible oscillatory features in the primordial curvature perturbation, which moreover appears to be suppressed beyond the present Hubble radius. Such deviations from the usual inflationary expectation of an approximately Harrison-Zeldovich spectrum are expected in the supergravity-based 'multiple inflation' model wherein phase transitions during inflation induce sudden changes in the mass of the inflaton, thus interrupting its slow-roll. In a previous paper we calculated the resulting curvature perturbation and showed how the oscillations arise. Here we perform a Markov Chain Monte Carlo fitting exercise using the 3-year WMAP data to determine how the fitted cosmological parameters vary when such a primordial spectrum is used as an input, rather than the usually assumed power-law spectrum. The 'concordance' ΛCDM model is still a good fit when there is just a 'step' in the spectrum. However if there is a 'bump' in the spectrum (due e.g. to two phase transitions in rapid succession), the precision CMB data can be well-fitted by a flat Einstein-de Sitter cosmology without dark energy. This however requires the Hubble constant to be h ≃ 0.44 which is lower than the locally measured value. To fit the SDSS data on the power spectrum of galaxy clustering requires a ∼ 10% component of hot dark matter, as would naturally be provided by 3 species of neutrinos of mass ∼ 0.5 eV. This CHDM model cannot however fit the position of the baryon acoustic peak in the LRG redshift two-point correlation function. It may be possible to overcome these difficulties in an inhomogeneous Lemaître-Tolman-Bondi cosmological model with a local void, which can potentially also account for the SN Ia Hubble diagram without invoking cosmic acceleration.1 A 'hybrid' inflation model wherein the inflaton is coupled to a 'curvaton' field also yields oscillations together with suppressed power on large scales [14]. A similar phenomenon had been noted earlier for the case where the inflaton potential has a jump in its slope [15]; however such a discontinuity has no physical interpretation. The WMAP glitches have also been interpreted as due to the effects of 'trans-Planckian' physics [2,16,17] and due to resonant particle production [18].

Measurements of the Type Ia supernovae Hubble diagram which suggest that the Universe is accelerating due to the effect of dark energy may be biased because we are located in a 200-300 Mpc underdense 'void' which is expanding 20-30 per cent faster than the average rate. With the smaller global Hubble parameter, the Wilkinson Microwave Anisotropy Probe 5 data on cosmic microwave background (CMB) anisotropies can be fitted without requiring dark energy if there is some excess power in the spectrum of primordial perturbations on 100 Mpc scales. The Sloan Digital Sky Survey (SDSS) data on galaxy clustering can also be fitted if there is a small component of hot dark matter in the form of 0.5 eV mass neutrinos. We show however that if the primordial fluctuations are Gaussian, the expected variance of the Hubble parameter and the matter density are far too small to allow such a large local void. Nevertheless, many such large voids have been identified in the SDSS Luminous Red Galaxy survey in a search for the late integrated Sachs-Wolfe effect due to dark energy. The observed CMB temperature decrements imply that they are nearly empty, thus these real voids too are in gross conflict with the concordance cold dark matter model. The recently observed high peculiar velocity flow presents another challenge for the model. Therefore, whether a large local void actually exists must be tested through observations and cannot be dismissed a priori.