Double power series method for approximating cosmological perturbations

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a non-cosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on sub-horizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well known growing and decaying M\'esz\'aros solutions, these oscillating modes provide a complete set of sub-horizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from Github at https://github.com/AndrewWren/Double-power-series.gi

Free-free and H42alpha emission from the dusty starburst within NGC 4945 as observed by ALMA

We present observations of the 85.69 GHz continuum emission and H42alpha line emission from the central 30 arcsec within NGC 4945. Both sources of emission originate from nearly identical structures that can be modelled as exponential discs with a scale length of ~2.1 arcsec (or ~40 pc). An analysis of the spectral energy distribution based on combining these data with archival data imply that 84% +/- 10% of the 85.69 GHz continuum emission originates from free-free emission. The electron temperature is 5400 +/- 600 K, which is comparable to what has been measured near the centre of the Milky Way Galaxy. The star formation rate (SFR) based on the H42alpha and 85.69 GHz free-free emission (and using a distance of 3.8 Mpc) is 4.35 +/- 0.25 M/yr. This is consistent with the SFR from the total infrared flux and with previous measurements based on recombination line emission, and it is within a factor of ~2 of SFRs derived from radio data. The Spitzer Space Telescope 24 micron data and Wide-field Infrared Survey Explorer 22 micron data yield SFRs ~10x lower than the ALMA measurements, most likely because the mid-infrared data are strongly affected by dust attenuation equivalent to A_V=150. These results indicate that SFRs based on mid-infrared emission may be highly inaccurate for dusty, compact circumnuclear starbursts.Comment: 19 pages, 9 figures, accepted for publication in MNRA

An experimental study of the dual-fuel performance of a small compression ignition diesel engine operating with three gaseous fuels

A dual-fuel engine is a compression ignition (CI) engine where the primary gaseous fuel source is premixed with air as it enters the combustion chamber. This homogenous mixture is ignited by a small quantity of diesel, the ‘pilot’, that is injected towards the end of the compression stroke. In the present study, a direct-injection CI engine, was fuelled with three different gaseous fuels: methane, propane, and butane. The engine performance at various gaseous concentrations was recorded at 1500 r/min and quarter, half, and three-quarters relative to full a load of 18.7 kW. In order to investigate the combustion performance, a novel three-zone heat release rate analysis was applied to the data. The resulting heat release rate data are used to aid understanding of the performance characteristics of the engine in dual-fuel mode. Data are presented for the heat release rates, effects of engine load and speed, brake specific energy consumption of the engine, and combustion phasing of the three different primary gaseous fuels. Methane permitted the maximum energy substitution, relative to diesel, and yielded the most significant reductions in CO2. However, propane also had significant reductions in CO2 but had an increased diffusional combustion stage which may lend itself to the modern high-speed direct-injection engine