# Age of the universe

The age of the Universe is defined as the largest possible value of proper time integrated along a timelike curve from the Earth at the present epoch back to the Big Bang. The time that has elapsed on a hypothetical clock which has existed since the Big Bang and is now here on Earth will depend on the motion of the clock. According to the preceding definition, the age of the universe is just the largest possible value of time having elapsed on such a clock.

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## Age based on WMAP

NASA's Wilkinson Microwave Anisotropy Probe (WMAP) project estimates that the universe is about 13.7 billion years old, with an uncertainty of 200 million years (1.37×1010±2.0×108). However, this age is based on the assumption that the project's underlying model is correct; other methods of estimating the age of the universe could give different ages.

This measurement is made by using the location of the first acoustic peak in the microwave background power spectrum to determine the size of the decoupling surface (size of universe at the time of recombination). The light travel time to this surface (depending on the geometry used) yields a pretty good age for the universe. Assuming the validity of the models used to determine this age, the residual accuracy yields a margin of error near one percent.

## Age based on CNO cycle

Some recent studies found the carbon-nitrogen-oxygen cycle to be two times slower than previously believed, leading to the conclusion that the Universe could be a billion years older than previous estimates (via the CNO cycle).

## Age based on temperature

In its most basic form, the Big Bang theory is based on the idea that the universe was smaller, denser and hotter, in the past. The temperature of the universe can thus indicate the age of the expansion, assuming the initial temperature is known and the evolution of the temperate parameter is correctly modelled.

An expanding (or contracting) universe is given a scale factor to indicate its current size in relation to a fixed point. This could be a simple distance, or a dimensionless ratio of one distance (the size now) divided by a size in the past. In the case of a ratio like this, the present scale factor of the universe is one, and for an expanding universe, the scale factor in the past is less than one. The redshift of an object in the universe is simply the inverse of the scale factor minus one, or more commonly [itex]a(t)={1/{(1+z)}}[itex].

The temperature of the universe is inversely proportional to its scale; somewhat analogous to a gas that would cool down if expanded, or heat up if compressed, the temperature of the universe is thus related to redshift as [itex]T=T_0(1+z)[itex]. We can do a quick test by using the current temperature of 2.7K and the redshift of CMB as 1089 to calculate the temperature of the decoupling surface [itex]T= 2.7*1090 = 2943\mathrm{K}[itex] (this is the temperature of the universe when the CMB was emitted - around the dull red glow of a hot poker.)

In a universe like our own, most of the contents is in the form that does not exert a pressure on its surroundings (clouds of hydrogen gas, stars, planets etc). This is a pressureless, or "dust" model. For this kind of cosmological model, the evolution of the universe (scale factor at a specific time) is [itex]t=t_0{(1+z)}^{-3/2}[itex]. Putting in the redshift of 1089 for the cosmic microwave background and a current age of the universe [itex]t_0 \approx 13.7[itex] Gyr gives us around 380,000 years for the age of the universe when the CMB was emitted.

The cosmological model used above was somewhat erroneous, since in the context of quanta, the universe is vastly more dominated by the number of particles in the CMB radiation (over a billion to one). In the case of radiation, the evolution of the universe is [itex]t=t_0{(1+z)}^{-2}[itex], which for performing a measurement related to the temperature of the CMB is the correct formula to use.

The earliest measurable point in the evolution of the universe is the Planck time, when the universe would be the Planck temperature. The Planck temperature represents the maximum attainable temperature in the physical universe and can only be attained via an extreme event, such as the evaporation of a black hole (the Planck temperature is the Hawking temperature of a black hole with a radius of the Planck length [itex]4.5 \times 10^{30}[itex]K). Therefore [itex]Tp=To(1+z_{\mbox{max}})[itex], where [itex]T_0=2.725[itex]K and now [itex]z_{\mathrm{max}}[itex] is the maximum redshift as if seen from the Planck time [itex]t_p[itex]. Using the radiation dominated formula above, we arrive at an age of the universe of 11.667 Gyr.

This is not the end of the story however: First we must take into consideration the matter domination, which in this calculation is equivalent to a change in the value of the Planck time itself. This is a fairly simple integration and results in a age one third greater at 15.556 Gyr. Finally, this model was also simplified by considering that the entropy of the universe is constant. Obviously, the entropy increases (due to graviational in fall) and this has the effect that the universe has become cooler than this simple model predicts. The difference here is around 13.5% (the temperature would be 6.5% higher if entropy had not increased, so the age difference is this squared). This adjustment finally brings the temperature calculated age down to the WMAP age.

### Planck units

There is a simplification where if expressed in Planck units, the age (to/tp) is equal to the inverse square of the temperature (To/Tp) of the universe. Dividing To/Tp gives the current temperature expressed in the amount of the Planck temperature [itex]6 \times 10^{-31}[itex]. Taking the inverse square gives [itex]2.72 \times 10^{60}[itex] which is the age in Planck units. Multiplying by the Planck time gives the 11.667 Gyr again. There are many other simple relations like this one, including the critical density as the Planck temperature raised to the forth power. In Planck units, the critical density is [itex]1.3 \times 10^{-121}[itex], which when multiplied by the Planck density gives [itex]3.3 \times 10^{-30}[itex] g/cm^3.

## Assumption of strong priors

Calculating the age of the universe is only accurate if the assumptions built into the models being used are also accurate. This is referred to as strong priors and essentially involves stripping the potential errors in other parts of the model to render the accuracy of actual observational data directly into the concluded result. Although this is not a totally invalid procedure in certain contexts, it should be noted that the caveat, "based on the fact we have assumed the underlying model we used is correct", then the age given is thus accurate to the specified error (since this error represents the error in the instrument used to gather the raw data input into the model).

The age of the universe based on the "best fit" to WMAP data "only" is 13.4+/-0.3 Gyr (the slightly higher number of 13.7 includes some other data mixed in). This number represents the first accurate "direct" measurement of the age of the universe (other methods typically involve Hubble's law and age of the oldest stars in globular clusters, etc). It is possible to use different methods for determining the same parameter (in this case – the age of the universe) and arrive at different answers with no overlap in the "errors". To best avoid the problem, it is common to show two sets of uncertainties; one related to the actual measurement and other the related to the systematic errors of the model being used.

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