Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cosmic Clocks

Donghui Jeong, Fabian Schmidt

TL;DR

This work derives a gauge-invariant, linear expression for the proper-time perturbation $ T( hat)$ on constant-observed-redshift surfaces in a perturbed FRW universe, linking cosmic clocks to observable redshift perturbations. It shows that, on CMB superhorizon scales, the temperature perturbation equals minus the proper-time perturbation, i.e., $ $, providing a unifying view of projection effects within the standard ruler/candle framework. Extending the stdruler formalism to evolving rulers, the authors derive how $ T$ contributes to radial and magnification distortions and compute the associated angular power spectra, showing $ T$ is generally suppressed but can dominate at the largest scales. The paper also establishes the gauge-invariance of $ T$, connects it to line-of-sight integrals of metric perturbations and velocities, and discusses observational implications, including a potential test of cosmic homogeneity (the Copernican principle) via measurements of $ T$ across the sky. Overall, the results provide a principled route to extract proper-time perturbations from large-scale structure and CMB data, clarifying their role relative to other ruler and candle perturbations.

Abstract

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.

Cosmic Clocks

TL;DR

This work derives a gauge-invariant, linear expression for the proper-time perturbation on constant-observed-redshift surfaces in a perturbed FRW universe, linking cosmic clocks to observable redshift perturbations. It shows that, on CMB superhorizon scales, the temperature perturbation equals minus the proper-time perturbation, i.e., , providing a unifying view of projection effects within the standard ruler/candle framework. Extending the stdruler formalism to evolving rulers, the authors derive how contributes to radial and magnification distortions and compute the associated angular power spectra, showing is generally suppressed but can dominate at the largest scales. The paper also establishes the gauge-invariance of , connects it to line-of-sight integrals of metric perturbations and velocities, and discusses observational implications, including a potential test of cosmic homogeneity (the Copernican principle) via measurements of across the sky. Overall, the results provide a principled route to extract proper-time perturbations from large-scale structure and CMB data, clarifying their role relative to other ruler and candle perturbations.

Abstract

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.

Paper Structure

This paper contains 10 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. Notation
  4. Proper time perturbation
  5. Evolving standard ruler
  6. CMB in the superhorizon limit
  7. Discussion
  8. Gauge-invariance of $\mathcal{T}$
  9. Pure-gradient metric perturbation
  10. $\mathcal{T}$ in terms of synchronous-comoving matter density perturbation

Figures (2)

  • Figure 1: Angular power spectrum of $\mathcal{T}$ (blue solid) for scalar perturbations in the standard $\Lambda$CDM cosmology (§ \ref{['sec:not']}). For comparison, we also show the power spectra for the magnification $\mathcal{M}$ (magenta dash-dotted) and longitudinal scalar $\mathcal{C}$ (black dashed), calculated for a non-evolving ruler. All quantities are evaluated for a fixed source redshift of $\tilde{z} = 2$.
  • Figure 2: Angular power spectrum of $\mathcal{T}(\hat{\mathbf{n}})$ (blue solid) and the matter density perturbation $\delta_m^{\rm sc}$ in synchronous-comoving gauge for different sharp source redshifts. From top to bottom, the curves show $\tilde{z} = 1,\,2,\,5,\,10,$ and 100, respectively. $\delta_m^{\rm sc}$ serves as a rough order-of-magnitude estimate of intrinsic tracer density perturbations on a constant-proper-time slice. The dashed green line near the bottom shows the power spectrum of the fractional CMB temperature perturbation $\Theta$ (§ \ref{['sec:CMB']}).