Cosmic time

Cosmic time, or cosmological time, is the time coordinate commonly used in the Big Bang models of physical cosmology.^[1]^[2] This concept of time avoids some issues related to relativity by being defined within a solution to the equations of general relativity widely used in cosmology.

Problems with absolute time

Albert Einstein's theory of special relativity showed that simultaneity is not absolute. An observer located halfway between two lighting strikes may believe they occurred at the same time, while another observer close to one of the strikes will claim it occurred first and the other strike came after. This coupling of space and time, Minkowski spacetime, complicates scientific time comparisons.^[3]^: 202

However, Einstein's theory of general relativity provides a partial solution. In general relativity, spacetime is defined in relation to the distribution of mass. A "clock" conceptually linked to a mass will provide a well defined time measurement for all co-moving masses. Cosmic time is based on this concept of a clock.^[3]^: 205

Definition

Cosmic time $t$ ^[4]^: 42^[5] 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.^[3]^: 205 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.

Reference point

There are two main ways for establishing a reference point for the cosmic time.

Lookback time

The present time can be used as the cosmic reference point creating lookback time. This can be described in terms of the time light has taken to arrive here from a distance object.^[6]

Age of the universe

Alternatively, the Big Bang may be taken as reference to define $t$ as the age of the universe, also known as time since the big bang. The current physical cosmology estimates the present age as 13.8 billion years.^[7]

The $t=0$ 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.

In mathematical terms, a cosmic time on spacetime $M$ is a fibration $t\colon M\to R$ . This fibration, having the parameter $t$ , is made of three-dimensional manifolds $S_{t}$ .

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:^[8] $t(z)\approx {\frac {2}{3H_{0}{\Omega _{0}}^{1/2}}}z^{-3/2}\ ,\ z\gg 1/\Omega _{0}.$ Here $H_{0}$ is the Hubble constant and $\Omega _{0}=\rho /\rho _{\text{crit}}$ is the density parameter ratio of density of the universe, $\rho (t)$ to the critical density $\rho _{c}(t)$ for the Friedmann equation for a flat universe:^[9]^: 47 $\rho _{c}(t)={\frac {3H^{2}(t)}{8\pi G}}$ Uncertainties in the value of these parameters make the time values derived from redshift measurements model dependent.

References

^ On the physical basis of cosmic time by S.E. Rugh and H. Zinkernagel
^ D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford University Press. p. 312. ISBN 0-19-859686-3.
^ ^a ^b ^c Smeenk, Chris (2013-02-15). "Time in Cosmology". In Dyke, Heather; Bardon, Adrian (eds.). A Companion to the Philosophy of Time (1 ed.). Wiley. pp. 201–219. doi:10.1002/9781118522097.ch13. ISBN 978-0-470-65881-9.
^ Dodelson, Scott (2003). Modern Cosmology. Academic Press. ISBN 9780122191411.
^ Bonometto, Silvio (2002). Modern Cosmology. Bristol and Philadelphia: Institute of Physics Publishing. pp. 2. ISBN 9780750308106.
^ "lookback time". Oxford Reference. Retrieved 2024-06-07.
^ How Old is the Universe?
^ Longair, M. S. (1998). Galaxy Formation. Springer. p. 161. ISBN 978-3-540-63785-1.
^ Liddle, Andrew R. (2003). An introduction to modern cosmology (2nd ed.). Chichester ; Hoboken, NJ: Wiley. ISBN 978-0-470-84834-0.

[1] On the physical basis of cosmic time by S.E. Rugh and H. Zinkernagel

[2] D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford University Press. p. 312. ISBN 0-19-859686-3.

[Smeenk2013-3] Smeenk, Chris (2013-02-15). "Time in Cosmology". In Dyke, Heather; Bardon, Adrian (eds.). A Companion to the Philosophy of Time (1 ed.). Wiley. pp. 201–219. doi:10.1002/9781118522097.ch13. ISBN 978-0-470-65881-9.

[4] Dodelson, Scott (2003). Modern Cosmology. Academic Press. ISBN 9780122191411.

[5] Bonometto, Silvio (2002). Modern Cosmology. Bristol and Philadelphia: Institute of Physics Publishing. pp. 2. ISBN 9780750308106.

[6] "lookback time". Oxford Reference. Retrieved 2024-06-07.

[7] How Old is the Universe?

[Longair-8] Longair, M. S. (1998). Galaxy Formation. Springer. p. 161. ISBN 978-3-540-63785-1.

[9] Liddle, Andrew R. (2003). An introduction to modern cosmology (2nd ed.). Chichester ; Hoboken, NJ: Wiley. ISBN 978-0-470-84834-0.

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