# 17.16 Age of the Universe

The big bang model gives a prediction for the age of the universe. If we imagine the evolving universe as a movie, the birth of the universe is the time in the distant past when all matter and radiation was crushed in a state of infinite temperature and density. The scale factor, R, was zero. Space had not yet begun to unfold. There are three ways to constrain the age of the universe: via the size distribution of hot and cold spots in the cosmic microwave background, via the expansion rate of the universe, and from stellar evolution models.

Temperature map of the universe, as measured by WMAP. Click here for original source URL

In many ways, looking at the expansion rate of the universe is one of the easiest ways to figure out the universe's age. This technique looks at the current expansion rate and then running it backwards to determine when today's universe must have been compressed down to a single point. Mathematically, the age of the universe is:

t_{0} = 1 / H_{0}f(Ω_{r }Ω_{m} Ω_{Λ}...)

Where t_{0} is the current age,H_{0} is the current expansion rate, and f is a correction that takes into consideration the mass, energy, and other characteristics of the universe. For this method to work, you need to not only know the current expansion rate but also how this rate has changed over time due to the effects of gravity (this is related to the mass density, Ω_{m} of the universe) and the effect of dark energy (Ω_{Λ}), which is pushing the universe apart. The best published value for the expansion rate is H_{0}=70.6 ± 3.1 (km/sec) / Mpc. This means that every Mega parsec of space expands roughly 70.6 km each second. This value and the WMAP measured values for the various densities (Ω_{m} Ω_{Λ}) = (0.266, 0.732) yield an age of roughly 13.8 billion years.

Based on stellar evolution models, we estimate the oldest globular clusters are between 11 and 13 billion years old. To get these estimates, astronomers consider stellar evolution models and look for stars at many different, short-lived, transition points. Specifically, they consider which stars have just finished burning hydrogen in their cores and evolved off the main sequence and what is the temperature of the coldest white dwarfs. Stellar evolution models can only put a lower limit on the age of the universe since it is impossible to know how long after the formation of the universe a star cluster formed. Nonetheless, clusters provide an important check on other methods of getting at the universe's age.

The most trusted means of determining the age of the universe comes from very precise measurements of the cosmic microwave background. The hot and cold spots in the CMB reflect the size of "sound waves" (acoustic oscillations) moving through the early universe. The most accurate measurements are from the WMAP and Planck missions. The largest of the hot and cold peaks in the CMB, in combination with theories describing the evolution of the early universe, tell us the size the universe was when the CMB formed roughly 400,000 years after the Big Bang. By measuring the distance to the CMB, the age of the universe can be determined. Based on the WMAP measurements and theoretical models, the universe is estimated to be 13.82 ± 0.12 billion years old. This is the most quoted age used by astronomers and is consistent with both other measurements within their errors.