# 17.23 Mass and Energy in the Early Universe

The cosmological red shift is related to the scale of the universe by R_{0}/R = 1 + z. Since the wavelength of the cosmic radiation stretches with the general expansion, λ ∝ R. Combining these two relationships, λ ∝ (1+z)-1. Wein’s law gives us λ_{max} ∝ T-1, where λ_{max} is the peak wavelength of thermal radiation and T is the temperature of the radiation. Combining these expressions gives the simple result:

T ∝ 1 + z

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

The temperature of the radiation in the universe is proportional to the red shift. Scaling this relationship to the observed temperature of the cosmic background radiation gives T = 2.7 (1+z). If you insert any red shift into this last equation, you get the temperature of the universe at that red shift. There is a simple relationship between red shift and temperature. At the present day, z = 0 and T = 2.7 K. We observe the peak wavelength of the cosmic background radiation at around 1 mm. At the redshift of the earliest galaxies and quasars, z = 5, then T = 2.7 x 6 = 16 K, which is still bitterly cold. The wavelength of the peak of the radiation was a factor of (1+z) smaller, or 1/(1+z) mm, or 0.17 mm.

Our view of the cosmic background radiation dates back to about 300,000 years after the big bang, or a red shift of z = 1100. At this red shift, the temperature of the radiation was T = 2.7 x 1100 ≈ 3000 K (once z is very large, the difference between z and 1+z is negligible). Just as a star has a photosphere, you can think of this radiation as the photosphere of the entire universe! The wavelength of the peak of the radiation was 1/1100 = 0.001 mm, or 1 micron, which is an infrared wavelength. Astronomers can project the universe back to the time 10,000 years after the big bang when radiation was equal in importance to matter in governing the behavior of the universe. The red shift was z ≈ 20,000. At this redshift, the temperature of the radiation was T = 2.7 x 20,000 ? 55,000 K, and the wavelength of the peak of the radiation was 1/20,000 = 5 x 10^{-5} mm, or 50 nm. At this early time, the universe was awash with energetic ultraviolet photons.

Following this progression of increasing temperature back even earlier into the universe, there is an important consequence of Einstein’s equivalence between mass and energy, E = mc^{2}. Particles and antiparticles can be created out of pure radiation. The mechanism is called pair creation, and it dictates how physicists (and nature) create antimatter on Earth. Antimatter is created momentarily in the upper atmosphere by cosmic rays, or expensively and in minute quantities by enormous particle accelerators. The opposite process — annihilation — is defined as the disappearance of a particle/antiparticle pair, releasing high-energy radiation or gamma rays. The pairing up of particles and anti-particles, called pair annihilation, is a perfectly efficient process; 100% of the mass-energy is liberated.

Measured blackbody curve for the cosmic background radiation. Click here for original source URL.

When did pair creation occur in the universe? About one minute after the big bang, when the temperature was 5 × 10^{9} K, radiation had enough energy to create the lightest particle-antiparticle pairs. Electrons and their antiparticles — positrons — could be created in profusion out of the high-energy gamma rays. Even earlier, at an amazing 10^{-4} seconds after the big bang, the temperature was 10^{13} K, or ten trillion degrees! At this time, protons and antiprotons, and neutrons and antineutrons, could be created in pairs. Thus, the early universe was an amazing factory for producing matter (and antimatter) out of radiation.

The very early universe provides an extraordinary aspect of the quantum theory of nature. There is a "fuzziness" in the physical world expressed by Heisenberg’s uncertainty principle. This principle manifests itself in the properties of a tiny subatomic particle. We cannot simultaneously measure the position and motion of a subatomic particle with perfect accuracy. If we know one quantity, we must be uncertain about the other quantity. There is a second version of Heisenberg’s uncertainty principle that is entirely equivalent to the version that involves position and momentum. It takes the form:

ΔE x Δt ≥ h / 2π

In this expression, ΔE is the uncertainty in energy, Δt is the time interval over which the energy is measured, and h is the tiny Planck’s constant. Over a small enough time interval, the energy of a particle or a system is uncertain by an amount related to the Planck constant. But Einstein’s principle says that energy is related to mass, so the uncertainty in energy corresponds to uncertainty in mass. Following the logic of the previous discussion, particle-antiparticle pairs can be created out of quantum fluctuations. The pairs are created (and disappear again) incredibly quickly. Electrons and positrons can be spontaneously created for no more than 10^{-22} seconds, and protons and antiprotons can be spontaneously created for no more than 10^{-25} seconds! Their ephemeral existence leads them to be called virtual pairs.

How can pairs of particles — one made of matter and the other made of antimatter — be temporarily created out of pure energy? It is as if you could borrow an enormous amount of money from your bank as long as you paid it back extremely quickly. The quicker you paid it back, the more you could borrow. Even though your balance is low (or perhaps zero!), you can borrow the money. The quantum fluctuations we have just described can violate the law of conservation of energy, as long as it is done for a very short time.

Virtual pairs cannot be observed directly, but their effects have been detected on normal particles. Physicists have strong evidence that "empty space" is not really empty; it is filled with a seething froth of virtual pairs. All this goes on at such tiny levels of energy and for such fleeting intervals that is unseen by us. However, in the early phases of the universe, these quantum fluctuations were very significant. The fact that the vacuum is not boring and it has energy is intriguing; it suggests a connection to the dark energy that is causing the universal expansion to accelerate. However, simple calculations of vacuum energy are not able to explain why our universe is accelerating. New physics is required.