#----------------------------------------------------------------
#  rescaled likelihood ("{\cal R}") in the case of background
#  -> added sensitivity bound
# 
#  GdA, May 2016
#-----------------------------------------------------------------

x <- 1
T <- 1
rb <- 1

Rr <- function(r, x, T, rb) exp(-r*T)*(1+r/rb)^x

r <- 10^seq(-4, 1.5, len=201)
plot(r, Rr(r, x, rb, T), ty='l', col='blue', log='xy',
     ylim=c(1e-3, max( Rr(r, x, rb, T))) )

Rb <- 0.05
abline(h=Rb, lty=2, col='blue')
sb <- optimise(f=function(r) abs(Rr(r, x, T, rb) - Rb), range(r) )$minimum
abline(v=sb, lty=2, col='blue')
title(sprintf("Sensitivity Bound: %.1e", sb))