Cosmic clocks

In a perturbed universe, comoving tracers on a two-dimensional surface of constant observed redshift are at different proper times since the big bang. For tracers whose age is known independently, one can measure these perturbations of the proper time. Examples of such sources include cosmic events which only happen during a short period of cosmic history, as well as evolving standard candles and standard rulers. In this paper, we derive a general gauge-invariant linear expression for this perturbation in terms of spacetime perturbations. We show that this perturbation in general contributes a previously overlooked leading order term to observables such as the magnification (although this contribution is generally small). Further, as an illustrative example, we show that the observed temperature perturbations of the cosmic microwave background on large scales (l << 100) are exactly given by these proper-time perturbations. Together with the six ruler perturbations derived in [F. Schmidt and D. Jeong, Phys. Rev. D 86, 083527 (2012)], this completes the set of independent observables which can be measured with standard rulers and candles.