Einstein’s field equation

equation: R sub mu nu - 1/2 R g sub mu nu - Lambda g mu nu = 8 pi T sub mu nu

curvature

cosmological constant

matter/energy

density

Einstein's equation of general relativity equates the curvature of spacetime with the density of matter/energy, using a factor called the cosmological constant. Einstein originally derived this equation without the cosmological constant, because at the time, he believed that the universe was static. Many physicists thought that basically, the universe was eternal and unchanging. Einstein added the cosmological constant to allow for this, to balance the attractive force due to gravity and ensure a static universe.

 

It was later realized that the universe is not static. Einstein called the inclusion of the cosmological constant his greatest blunder. In essence, he changed the math to fit the beliefs of science at the time, instead of letting it stand on its own merit. Today, we employ the cosmological constant in a different way. We use it as a parameter to adjust the expansion rate of the universe.

plot showing the effect of the cosmological constant on the evolutionary curves of the universe

The above graph depicts various models of the evolutionary trajectory of the universe, showing the effect of the cosmological constant with various values assigned to it. Notice that the value of the cosmological constant has implications about how old the universe is.  A negative cosmological constant implies a younger universe. For a positive cosmological constant, the larger it is, the farther back in time the universe began.

type 1a supernova data for galaxies, plotting redshift vs. distance

The above graph of redshift vs. distance shows the data points gathered from measuring the redshifts of type 1a supernova standard candles. The black line shows the curve for the prediction made using Hubble's law. Remember, Hubble's law relates redshift to distance of galaxies. The supernova standard candles give another, independent way to measure distance. We can also measure the redshifts of the light from the supernovae.

 

Hubble's law is based on a constant expansion rate. The cosmological recessional speed increases by the same amount per Mpc. For the relatively close supernovae, the slope is constant. We use that constant slope to calculate Hubble's constant. What this plot shows is that the redshifts of the standard candles are lower than Hubble's law predicts, for the most distant supernovae.

 

If the expansion rate was decreasing, faraway objects would lie above the Hubble's law prediction on this graph. When the light was emitted from these distant objects, the expansion rate would have been higher, so the redshifts would be higher. Light from faraway objects would have redshifted more in the past than it does now.

 

Since the data points lie below the Hubble's law prediction, it means that the expansion rate of the universe must be speeding up. For a given distance, the redshift is not as high as it would be if the expansion rate was constant. When the light was emitted from these very distant supernovae, space was not expanding as fast, and so the light waves were not stretched as much. The fact that the more recent light is being stretched at a greater rate is indicated by the steeper curve for light that has not traveled as far.

Measurements from galaxy cluster surveys

galaxy cluster map

We have another independent way to measure the expansion rate of the universe, using measurements from galaxy cluster surveys. The main idea here is that the denser the matter of the early universe, the faster the formation of large scale structure would take place. We can determine the rate of galaxy cluster formation, and then look farther away to see farther back in time, to determine how this rate has changed.

Cosmic microwave background radiation

We have a third method of measuring the expansion rate of the universe, using the cosmic microwave background radiation.

Arno Penzias and Robert WIlson

Why are these two men smiling? They are Arno Penzias (right) and Robert Wilson (left) and they are standing in front of a giant radio antenna that Bell Labs had given them to research intergalactic material. While they were trying to take data, they noticed some background noise that they thought must be a problem with the instrument. They thoroughly the electronics and found no problem. They tried taking data at different times and in different directions, but the noise was always still there.

 

There were some pigeons roosting in the antenna, so they scared away the pigeons and scrubbed the antenna clean. The pigeons came back. They trapped the pigeons and transported them far away, but being homing pigeons, they came back. In frustration, Arno and Robert killed the pigeons and scrubbed the antenna. The noise persisted.

 

In a conversation with a friend, Robert Dicke from Princeton University, it was realized that the annoying noise had the same frequency as had been predicted for the afterglow of the big bang itself. Penzias and Wilson had accidentally discovered observational evidence of the big bang. They won the Nobel Prize for this discovery in 1978.

 

This animation begins by showing the cosmic microwave background radiation (CMBR) data taken by the Cosmic Background Explorer (COBE) satellite. Basically, it shows the light coming from the background sky with features like the sun, Milky Way, etc. removed. This early data is shown transitioning into more precise data taken later by the Wilkinson Microwave Anisotropy Probe (WMAP). These images show a temperature range of 200 microKelvin about the mean temperature. This means that the background radiation is very close to the same energy everywhere.

CMBR curve

This graph shows the blackbody radiation graph for the background radiation of the universe. The peak wavelength gives a measure for the temperature of the universe, about 2.7 Kelvin The theory and data agree so well that the observational data points lie so close to the theoretical curve that they cannot be distinguished from each other.

 

The concept behind the theory is that in the very early stages of the universe, the matter in the universe was opaque to light. The matter particles were mostly hydrogen and helium nuclei and electrons. The matter was so hot that the electrons were moving too fast to be bound to the nuclei. As the universe expanded, it cooled, and became cool enough for the electrons to become bound to the nuclei, forming atoms. When this happened, only photons with energies that corresponded to differences in energy levels of excited electrons interacted with the atoms. All other photons were free to move, and the universe changed from being opaque to being transparent.

 

As the universe further expanded, these free, high energy photons redshifted more and more. If the big bang theory was true, we should be able to see this afterglow everywhere in the sky. The theory predicted  what the shape of the blackbody spectrum would be at the present time, agreeing incredibly well with the observed data. This agreement marked the first observational verification of the big bang theory.

 

This animation begins by showing the WMAP spacecraft imaging the background radiation in the universe. Then it zooms out to view our place in the cosmos, traveling further and further back in time, past the quasar epoch to an early time when the universe was made of superheated gas. The gas contains condensed and rarefied ripples, like sound waves. The configuration signature of this early state of the universe is still observable in the rippling of the present-day cosmic microwave background radiation.

 

This simulation illustrates how the signature of the fluid and the energy present can be captured in the ripples, similar to the signature of the state of the early universe captured in the ripples of the cosmic microwave background radiation.

 

This animation illustrates how the structure of the universe evolved from the early ripples, affecting the clumping and ignition of matter, evolving to the features we see in the universe today.

 

The relative size of the ripples in the cosmic microwave background radiation can be analyzed to give information about the curvature of the universe.

(Left) If the universe is closed, with positive curvature, hot spots of the CMBR would appear larger than actual size as predicted by theory. (Center) If the universe is flat, the hotspots seen would appear to be the actual size, as predicted by theory. (Right) An open universe with negative curvature, would exhibit smaller hot spots than the prediction. What we see from this analysis is that our universe is very nearly flat.

Cobe, WMAP and Planck surveys of the CMBR

The Planck survey in 2013 confirmed and increased the resolution of the Cosmic Microwave Background Radiation. The temperature fluctuations of the CMBR provide detailed information about the early state of the universe.

 

This animation illustrates where in the electromagnetic spectrum the CMBR is found, compared to other sources like stars and the Milky Way in various wavelengths of light.

Observational indications of expansion

  • Three independent methods agree the expansion rate of the universe is accelerating
  • Galaxy surveys
    • Analyze growth of large scale structure over time
      • More mass in the universe means it is easier for clusters, voids, etc. to grow
      • Higher density implies faster formation of structure – indicating less structure in the past
  • WMAP
    • Size of fluctuations at present indicate curvature of the universe
  • Type 1a supernovae (standard candles)
    • Redshift mapping deviates from the Hubble's constant curve

Dark energy

Image source

 

This graph charts the results of the supernova standard candles, the galaxy cluster evolution and the cosmic microwave background radiation on a plot of the density of dark energy vs. matter. All three independent sources converge in a region that indicates that the expansion of the universe is accelerating.

  • It appears we have an accelerating universe
  • We assume there must be something working  against gravity; we call it dark energy
  • Adds density but not gravitational attraction
  • Calculate how much energy it would take to drive the  galaxies apart
  • Ωo ≈ 1 – density of the universe appears to be close  to critical
  • We calculate how much energy would be needed to drive the galaxies apart.
  • It appears that the universe is made up of 5% normal matter, 20% dark matter and 75% dark energy.

© Kathryn Hadley PhD 2020

equation: R sub mu nu - 1/2 R g sub mu nu - Lambda g mu nu = 8 pi T sub mu nu
plot showing the effect of the cosmological constant on the evolutionary curves of the universe
type 1a supernova data for galaxies, plotting redshift vs. distance
galaxy cluster map
Arno Penzias and Robert WIlson
CMBR curve
Cobe, WMAP and Planck surveys of the CMBR
equation: R sub mu nu - 1/2 R g sub mu nu - Lambda g mu nu = 8 pi T sub mu nu
plot showing the effect of the cosmological constant on the evolutionary curves of the universe
type 1a supernova data for galaxies, plotting redshift vs. distance
galaxy cluster map
Arno Penzias and Robert WIlson
CMBR curve
Cobe, WMAP and Planck surveys of the CMBR