Cosmic time

Definition

Cosmic time ${\displaystyle t}$ is a measure of time by a physical clock with zero peculiar velocity in the absence of matter over-/under-densities (to prevent time dilation due to relativistic effects or confusions caused by expansion of the universe). Unlike other measures of time such as temperature, redshift, particle horizon, or Hubble horizon, the cosmic time (similar and complementary to the co-moving coordinates) is blind to the expansion of the universe.

Cosmic time is the standard time coordinate for specifying the Friedmann–Lemaître–Robertson–Walker solutions of Einstein field equations of general relativity. Such time coordinate may be defined for a homogeneous, expanding universe so that the universe has the same density everywhere at each moment in time (the fact that this is possible means that the universe is, by definition, homogeneous). The clocks measuring cosmic time should move along the Hubble flow.

The ${\displaystyle t=0}$ doesn't necessarily have to correspond to a physical event (such as the cosmological singularity) but rather it refers to the point at which the scale factor would vanish for a standard cosmological model such as ΛCDM. For technical purposes, concepts such as the average temperature of the universe (in units of eV) or the particle horizon are used when the early universe is the objective of a study since understanding the interaction among particles is more relevant than their time coordinate or age.

Cosmic time relies on physical concepts like mass that may not be valid for times before approximately 10⁻¹¹ seconds.

Reference point

A value of cosmic time at a distant location can be given relative to the current time at our location, called lookback time, or relative the start of the big bang, called the "age of the universe" for that location.

Lookback time

The lookback time, ${\displaystyle t_L}$, is an age difference: the age of the universe now, ${\displaystyle t_0}$, minus the age of the universe when a photon was emitted at a distant location, ${\displaystyle t_e.}$ The lookback time depends upon the cosmological model: $${\displaystyle t_L(z)=t_H{\int }_0^z\frac{d{z}^{\prime }}{(1+z^{\prime })E(z^{\prime })}}$$ where $${\displaystyle E(z)=\sqrt{{\mathrm{\Omega }}_M(1+z{)}^3+(1-{\mathrm{\Omega }}_M-{\mathrm{\Omega }}_{\mathrm{\Lambda }})(1+z{)}^2+{\mathrm{\Omega }}_{\mathrm{\Lambda }}}}$$ and ${\displaystyle {\mathrm{\Omega }}_M}$ means the present day density parameters for mass and ${\displaystyle {\mathrm{\Omega }}_{\mathrm{\Lambda }}}$ is the cosmological constant. The lookback time at infinite z is the age of the universe at our location and time. This can be described in terms of the time light has taken to arrive here from a distance object.

Age of the universe

Alternatively, the Big Bang may be taken as reference to define ${\displaystyle t}$ as the age of the universe, also known as time since the big bang, at the location of the clock. For an object observed at redshift z, the age of the universe when the observed photons were emitted is:

$${\displaystyle t(z)=t_H{\int }_z^{\mathrm{\infty }}\frac{d{z}^{\prime }}{(1+z^{\prime })E(z^{\prime })}}$$

For every value of redshift, the sum ${\displaystyle t_L(z)+t(z)}$ equals the age at the universe at our location, ${\displaystyle z=0}$. The current physical cosmology estimates the present age as 13.8 billion years.

Relation to redshift

Astronomical observations and theoretical models may use redshift as a time-like parameter. Cosmic time and redshift z are related. In case of flat universe without dark energy the cosmic time can expressed as: $${\displaystyle t(z)\approx \frac{2}{3H_0{{\mathrm{\Omega }}_0}^{1/2}}{z}^{-3/2},z\gg 1/{\mathrm{\Omega }}_0.}$$ Here ${\displaystyle H_0}$ is the Hubble constant and ${\displaystyle {\mathrm{\Omega }}_0=\rho /{\rho }_{\text{crit}}}$ is the density parameter ratio of density of the universe, ${\displaystyle \rho (t)}$ to the critical density ${\displaystyle {\rho }_c(t)}$ for the Friedmann equation for a flat universe: $${\displaystyle {\rho }_c(t)=\frac{3H^2(t)}{8\pi G}}$$ Uncertainties in the value of these parameters make the time values derived from redshift measurements model dependent.