AHL Further cosmology

Summary

In the Cosmology and Age of the universe SL pages we considered cosmological observations and how these act as evidence for the origin and evolution of the universe in the past. Now we will instead look forwards to the future of the universe.

Get the basics

Describing the universe

The cosmological principle

The cosmological principle states that the spatial distribution of matter in the universe is homogeneous (consistent physical properties at any observer's location) and isotropic (consistent physical properties when observed in any direction) when viewed on a large enough scale. This assumes that forces act uniformly throughout the universe and that there are no irregularities in the matter that was initially laid down by the Big Bang.

We will return to cosmological principle and its role when we consider critical density and models of the universe.

Fluctuations in the CMB

While StudyIB Physics geeks will know that we prefer to show the cosmic microwave background as being homogeneous and isotropic, students at AHL must also be prepared to discuss the observed anisotropies in the CMB.

There are three sets of results for you to get familiar with:

  • COBE (NASA Cosmic Background Explorer launched 1989) - took precise measurements of the spectrum, the anisotropy of the radiation and the angular distribution of the diffuse radiation between 1 μm and 1 cm over the whole celestial sphere. The variations in the light are indicative of clusters of galaxies and empty space. In mapping the oldest light in the universe, the Big Bang theory of the universe was confirmed.
  • WMAP (NASA Wilkinson Microwave Anisotropy Probe launched June 2001) - improved the precision of the measurements to produce a map of higher resolution. Found evidence for inflation (the early rapid expansion of the universe), found the temperature of the cosmic background to be 2.7 K, determined the age of the universe and determined the proportions of matter, dark matter and dark energy.
  • Planck space observatory (ESA launched 2009) - the most advanced satellite images the sky with more than 2.5 times greater resolution than WMAP. Reveals patterns in the cosmic background and created the sharpest all-sky map ever made of the universe's cosmic microwave background.

Cosmological origin of redshift

Where did the cosmic microwave background come from? The beginning of the universe!

When the universe was formed, it was densely packed with photons and fundamental particles. The temperature was too high for atom formation and so instead was radiation-dominated. As the universe has expanded, the photons have not disappeared but have instead stretched so that their wavelengths have converted from gamma to microwave radiation.

The anisotropies in the CMB images today show that the early universe was not perfectly uniform, which is why we have a contrast between galaxies and empty space. As the universe expanded and cooled, had the density been perfectly uniform, the gravitational attraction between all particles would have been balanced in all directions, so none would have come together.

Dark matter

Rotation curves and the mass of galaxies

What is the dark matter detected by the probes? First, let's consider the evidence.

Consider a star a distance \(r\) from the centre of the spherical galaxy as shown below. The whole galaxy is rotating so we can consider each star to be in orbit about the centre.

The gravitational force experienced by the star is in two directions: inwards (blue) and outwards (pink):

  • Assuming the density of the galaxy (\(\rho\)) is constant, the outward force is zero
  • The mass of the inner part of the galaxy is \(\rho V={4\over 3}\pi\rho r^3\)
  • Therefore, the gravitational force on the star is \(gm_\text{star}={GM_\text{blue}\over r^2}m_\text{star}={4\over 3}\pi G \rho rm_\text{star}\)
  • Assuming the orbit is circular, the centripetal force is equal to the gravitational force, \({m_\text{star}v^2\over r}={4\over 3}\pi G \rho rm_\text{star}\) 
  • The velocity of the star is proportional to the distance from the centre, \(v^2\propto r^2\Rightarrow v\propto r\)

The equation that appears in your Data Booklet is:

\(v=\sqrt{4\pi G\rho\over3}r\)

  • \(v\) is the velocity of the star (ms-1)
  • \(G\) is the universal gravitational constant (6.67 × 10-11 m3kg-1s-2)
  • \(\rho\) is the density of the galaxy (kgm-3)
  • \(r\) is the distance of the star from the centre of the galaxy (m)

Let's now consider a star on the outside of the galaxy:

The gravitational force on the star is all inwards:

  • Considering all the mass to be at the centre, the gravitational force is \(GM_\text{pink}m_\text{star}\over r^2\)
  • Equating to the centripetal force, \({GM_\text{pink}m_\text{star}\over r^2}={m_\text{star}v^2\over r}\)
  • The velocity of the start is inversely proportional to the root of the distance from the centre, \({1\over r^2}\propto{v^2\over r}\Rightarrow v\propto{1\over \sqrt r}\)

Combining these rules, the predicted variation of velocity and radius is shown here:

The gradient changes from positive to negative when the star is at the edge of the galaxy.

Evidence for dark matter

The actual velocity of different stars can be measured by the redshift of the light they emit. The experimental results for seven galaxies are shown on the next graph:

Figure from Rubin, Ford, and Thonnard (1978), Ap. J. Lett., 225, L107.

We can see that the velocity of stars does not decrease as expected when the distance increases beyond the edge of the visible galaxy. Instead, velocities remain high. This suggests that additional, invisible mass is present. This mass cannot be at the centre of the galaxy, or the graphs for velocities would be scaled but still following the expected pattern. Instead, the results can only be explained if there is a large amount of invisible mass on the edge of the galaxy.

This invisible mass is called dark matter. Calculations predict that there must be about five times more matter than we can see, perhaps made up of:

  • weakly interacting massive particles (WIMPs) that don't interact with regular matter
  • massive compact halo objects (MACHOs) such as black holes that form a ring around the galaxy

Stretch for 7

Critical density

Definition

We know from Hubble's observations of receding galaxies that the universe continues to expand. We also know, however, that the universe is not infinite as an infinite universe would have an infinite number of stars and have existed for infinite time, meaning that the night sky would not be dark but instead bright in every direction. Therefore, there must be a net gravitational force acting between all of the galaxies. Is this strong enough to eventually stop the expansion and commence universal contraction?

The tipping point is referred to as critical density:

  • If the density of the universe exceeds the critical density, the universe will stop expanding and contract. We call this a closed universe. This would lead to a 'big crunch'!
  • If the density of the universe is equal to the critical density, the rate of expansion of the universe would tend towards zero as time tends towards infinity. The universe is considered flat.
  • If the density of the universe is less than the critical density, the universe will expand forever; an open universe.

Derivation

We can derive this critical density from Newtonian gravitation.

If the universe has critical density, a mass at the edge of the expanding universe would have all of its kinetic energy be converted into potential energy at infinity.

  • At present, its kinetic energy is \({1\over 2}mv^2\) and its potential energy is \(-GMm\over r\)
  • At infinity, its kinetic energy and its potential energy are zero (by definition)
  • According to conservation of energy: \({1\over 2}mv^2-{GMm\over r}=0\)
  • The mass of the universe can be determined from the critical denisty: \(M={4\over 3}\pi r^3\rho_c\)
  • And we know from Hubble's law that \(v=H_0r\)

By combination, rearranging and substitution we can calculate a value for the critical density:

\(\rho_c={3{H_0}^2\over 8\pi G}\)

  • \(\rho_c\) is the critical density of the universe (~10-26 kgm-3)
  • \(H_0\) is the Hubble constant (=68 kms-1Mpc-1)
  • \(G\) is the universal gravitational constant (6.67 × 10-11 m3kg-1s-2)

The critical density of approximately 10-26 kgm-3 is equivalent to 6 hydrogen atoms per cubic metre.

In this simulation you can see how the kinetic and potential energies vary as the cloud expands:

Models of the universe

If mass curves space-time then the universe must be curved. The way it curves depends on its density and what exactly the universe contains. The problem is we are not sure about the exact composition of the universe due to the presence of dark matter and dark energy.

In Einstein's time all evidence pointed towards a static universe (in which there is no expansion or contraction). In seeking to quantify properties of the universe, Einstein added a cosmological constant to the equations. Once the redshift of distant galaxies showed that the universe was expanding, Einstein realised the static model was incorrect and abandoned the cosmological constant. but more recent measurements have shown that the rate of expansion is not only constant but increasing due to the presence of dark energy! Einstein's cosmological constant has been reintroduced.

Dark energy

Evidence

According to gravitation, the rate of expansion of the universe should be decreasing. This risked cognitive bias in researchers when analysing experimental data. However, the detailed results from the 1998 (and subsequent) observations on distant supernovae showed that the opposite was in fact true. The expansion of the universe is experimentally verified as accelerating, and yet is still an unexplained phenomenon.

One theory is the presence of dark energy. Dark energy is the name given to a substance that fills all space and causes an outward pressure that counteracts the inward force of gravity. Instead of a constant rate of expansion, or a flat or closed universe, we now predict that the rate of change of the size of the universe will increase over time, faster than an open universe.

Impact on cosmic scale factor

We recall that the cosmic scale factor is defined as the size of the universe at a given time relative to its size now:

\(z={R\over R_0} -1\)

  • \(z\) is the factor by which the universe has expanded in size (dimensionless)
  • \(R\) is the size of the universe at time \(t\)
  • \(R_0\) is the size of the universe now

In a closed universe (orange), \(R\) would eventually fall to less than \(R_0\) and the cosmic scale factor would become negative. In a flat universe, \(R\) would reach a constant value larger than \(R_0\). In an open universe (green), \(R\) would continue to increase but at a decreasing rate so that the cosmic scale factor continues to rise. In the dark energy-dominated universe (red), \(R\) and the cosmic scale factor would continue to increase at an increasing rate. The blue line is a reference for a constant rate of expansion.

Temperature

The temperature of the universe is inversely proportional to the size of the universe: \(T\propto {1\over R}\)

Therefore, the temperature of the universe decreases as the cosmic scale factor increases.

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