The effective Planck mass and the scale of inflation

Observable quantities in cosmology are dimensionless, and therefore independent of the units in which they are measured. This is true of all physical quantities associated with the primordial perturbations that source cosmic microwave background anisotropies such as their amplitude and spectral properties. However, if one were to try and infer an absolute energy scale for inflation—a priori, one of the more immediate corollaries of detecting primordial tensor modes—one necessarily makes reference to a particular choice of units, the natural choice for which is Planck units. In this note, we discuss various aspects of how inferring the energy scale of inflation is complicated by the fact that the effective strength of gravity as seen by inflationary quanta necessarily differs from that seen by gravitational experiments at presently accessible scales. The uncertainty in the former relative to the latter has to do with the unknown spectrum of universally coupled particles between laboratory scales and the putative scale of inflation. These intermediate particles could be in hidden as well as visible sectors or could also be associated with Kaluza–Klein resonances associated with a compactification scale below the scale of inflation. We discuss various implications for cosmological observables.

Tabulation of multiple responses

Although multiple response questions are quite common in survey research, Stata's official release does not provide much possibility for an effective analysis of multiple response variables. For example, in a study on drug addiction an interview question might be, "Which substances did you consume during the last four weeks?" The respondents just list all the drugs they took if any, e.g., an answer could be "cannabis, cocaine, heroin" or "ecstasy, cannabis" or "none", etc. Usually, the responses to such questions are held as a set of variables and, therefore, cannot be easily tabulated. I will address this issue and present a new module to compute one- and two-way tables of multiple responses. The module supports several types of data structure, provides significance tests, and offers various options to control the computation and display of the results.