Together with the six ruler perturbations derived in [1], this completes the set of independent observables which can be measured with standard rulers and candles.

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 [1], this completes the set of independent observables which can be measured with standard rulers and candles.

Together with the six ruler perturbations derived in [1], this completes the set of independent observables which can be measured with standard rulers and candles.

Cosmic Clocks