Cosmic Clocks
  In a perturbed Universe, comoving tracers on a two-dimensional surface of
constant observed redshift are at different proper time 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 space-time perturbations.
As an example, we show that the observed temperature perturbations of the
cosmic microwave background (CMB) on large scales are exactly given by these
proper time perturbations. Together with the six ruler perturbations derived in
Schmidt and Jeong (2012), this completes the set of independent observables
which can be measured with standard rulers and candles.
