Age of the universe

The age of the Universe was estimated to be about 13.7 billion (13.7 × 10⁹) years, with an uncertainty of 200 million years, by NASA's Wilkinson Microwave Anisotropy Probe project (WMAP). However this is based on the assumption that the underlying model that was used is correct. Other methods of esimating the age of the universe give different ages.

Some recent studies, found the 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 maximum age of stars, etc). There is a sense of triumphantism in the scientific community surrounding results like this, and therefore a more careful analysis of the methods and assumptions used, tend to be overlooked.

This, of course, is a classic example of how different methods for determining the same parameter (in this case ? the age of the universe) can give different answers with no overlap in the "errors". It is quite common to see two sets of uncertainties, one related to the measurement and other the related to the systematic errors of the model. In some cases, this can not be done.

Calculation of the age from the temperature of the universe

The redshift of an object in a dynamic universe is related to a scale factor of that universe by the relation <math>R=R_0/(1+z)<math>. Where <math>R<math> represents the ?scale? of the universe as seen at the redshift z, where the current scale is <math>R_0<math>. The ?scale? is just a device to measure the size of the universe, it can be thought of as the radius, but most people use the "scale factor" <math>a=R/R_0<math>, which would be dimensionless regardless of how you represented <math>R<math>.

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 T=To(1+z). 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 <math>T= 2.7*1090 = 2943\mathrm{K}<math> (this is the temperature of the universe when the CMB was emitted - around the dull red glow of a hot poker.)

One of the most important cosmological models is based on the Planck time. At this time, the universe had the Planck temperature at a state of essentially zero entropy. The Planck temperature is the maximum attainable temperature in the universe and can be thought of as the Hawking temperature of black hole with a radius of the Planck length.

The Planck temperature Tp comes out to around 4.5x10^30K, and we can state <math>Tp=To(1+z_{\mbox{max}})<math>, where <math>T_0=2.725<math>K and <math>z_{\mathrm{max}}=1.65\times 10^30<math> is the maximum redshift at the Planck time <math>t_p<math>. We know that <math>t_p=t_0(1+z_\mathrm{max})^{-2}<math>, so putting in the Planck time gives us an age of the universe of 11.667 Gyr. This is not the end of the story however: If time was absolute and never changed, then this would be the correct value, but we need to take into consideration the change in time over the age of the universe. This is a fairly simple integration and results in a age one third as much at 15.556 Gyr. The CMB temperature is known to a 2mK accuracy, and with some error in things like the Planck units (mainly from G), the accuracy of this age determination is around 24 Myr.

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 6x10^-31. Taking the inverse square gives 2.72x10^60 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 1.3x10^-121, which when multiplied by the Planck density gives 3.3x10^-30 g/cm^3.

See also

• Science 299 (2003) 1532-1533, available here http://arxiv.org/abs/astro-ph/0303180