Fudge factor

Abstract: The major theoretical limitation for extracting cosmological parameters from the CMB sky lies in the precision with which we can calculate the cosmological recombination process. Uncertainty in the details of hydrogen and helium recombination could effectively increase the errors or bias the values of the cosmological parameters derived from the Planck satellite, for example. Here we modify the cosmological recombination code RECFAST by introducing one more parameter to reproduce the recent numerical results for the speed-up of the helium recombination. Together with the existing hydrogen fudge factor, we vary these two parameters to account for the remaining dominant uncertainties in cosmological recombination. By using the CosmoMC code with Planck forecast data, we find that we need to determine the parameters to better than ten per cent for He I and one per cent for H, in order to obtain negligible effects on the cosmological parameters. For helium recombination, if the existing studies have calculated the ionization fraction to the 0.1 per cent level by properly including the relevant physical processes, then we already have numerical calculations which are accurate enough for Planck. For hydrogen, setting the fudge factor to speed up low redshift recombination by 14 per cent appears to be sufficient for Planck. However, more work still needs to be done to carry out comprehensive numerical calculations of all the relevant effects for hydrogen, as well as to check for effects which couple hydrogen and helium recombinaton through the radiation field.

Abstract: We present RICO, a code designed to compute the ionization fraction of the universe during the epoch of hydrogen and helium recombination with an unprecedented combination of speed and accuracy. This is accomplished by training the machine learning code PICO on the calculations of a multilevel cosmological recombination code which self-consistently includes several physical processes that were neglected previously. After training, RICO is used to fit the free electron fraction as a function of the cosmological parameters. While, for example, at low redshifts (z {approx}< 900), much of the net change in the ionization fraction can be captured by lowering the hydrogen fudge factor in RECFAST by about 3%, RICO provides a means of effectively using the accurate ionization history of the full recombination code in the standard cosmological parameter estimation framework without the need to add new or refined fudge factors or functions to a simple recombination model. Within the new approach presented here, it is easy to update RICO whenever a more accurate full recombination code becomes available. Once trained, RICO computes the cosmological ionization history with negligible fitting error in {approx}10 ms, a speedup of at least 10{sup 6} over the full recombination code that was usedmore » here. Also RICO is able to reproduce the ionization history of the full code to a level well below 0.1%, thereby ensuring that the theoretical power spectra of cosmic microwave background (CMB) fluctuations can be computed to sufficient accuracy and speed for analysis from upcoming CMB experiments like Planck. Furthermore, it will enable cross-checking different recombination codes across cosmological parameter space, a comparison that will be very important in order to assure the accurate interpretation of future CMB data.« less

Abstract: We present Rico, a code designed to compute the ionization fraction of the Universe during the epoch of hydrogen and helium recombination with an unprecedented combination of speed and accuracy. This is accomplished by training the machine learning code Pico on the calculations of a multi-level cosmological recombination code which self-consistently includes several physical processes that were neglected previously. After training, Rico is used to fit the free electron fraction as a function of the cosmological parameters. While, for example at low redshifts (z<~900), much of the net change in the ionization fraction can be captured by lowering the hydrogen fudge factor in Recfast by about 3%, Rico provides a means of effectively using the accurate ionization history of the full recombination code in the standard cosmological parameter estimation framework without the need to add new or refined fudge factors or functions to a simple recombination model. Within the new approach presented here it is easy to update Rico whenever a more accurate full recombination code becomes available. Once trained, Rico computes the cosmological ionization history with negligible fitting error in ~10 milliseconds, a speed-up of at least 10^6 over the full recombination code that was used here. Also Rico is able to reproduce the ionization history of the full code to a level well below 0.1%, thereby ensuring that the theoretical power spectra of CMB fluctuations can be computed to sufficient accuracy and speed for analysis from upcoming CMB experiments like Planck. Furthermore it will enable cross-checking different recombination codes across cosmological parameter space, a comparison that will be very important in order to assure the accurate interpretation of future cosmic microwave background data.

Abstract: We obtain quantified versions of Ingham's classical Tauberian theorem and some of its variants by means of a natural modification of Ingham's own simple proof. As corollaries of the main general results, we obtain quantified decay estimates for $C_0$-semigroups. The results reproduce those known in the literature but are both more general and, in one case, sharper. They also lead to a better understanding of the previously obscure "fudge factor' appearing in proofs based on estimating contour integrals.