Reading Assignment: Arny: Chapter 16 & 17, Cosmos: Chapter 10.

** The Cosmological Constant**:

When Einstein first worked out his solutions to the equations of general relativity for the Universe as a whole, he realized that it was unstable to gravity and would collapse under its own weight.

The completely general mathematical solution to the equations includes an arbitrary constant (). Einstein recognized that the constant would act as a kind of pressure in the Universe (an anti-gravity if you will) that acts over large distances (it grows stronger with greater distance!).

So he included it and gave it just the right value to keep the Universe static. Then it was discovered that the Universe was expanding and he no longer needed to consider the Cosmological Constant. He called it his "greatest blunder".

For most of the 20th Century people generally regarded the Cosmological Constant to be unphysical and therefore set its value to zero.

Recently, the study of type I supernovas in distant galaxies has shown that they are too far away for their redshift (exactly the opposite of what was expected). Thus the Universe has not been decelerating but rather *accelerating*! This is the effect that we would expect from a non zero Cosmological Constant.

The results are still highly controversial. STAY TUNED!

If there is a non-zero Cosmological Constant it would contribute an energy density to the Universe and thus help shape the global geometry.

*However*, the fate of the Universe would now be disconnected from the geometry. As the matter density drops the Cosmological Constant would grow stronger causing the Universe to expand faster and faster. Thus it is possible for any geometry (Open, Flat, or Closed) to expand forever and an ever increasing rate.

NOTE: if the Universe is accelerating then the age of the Universe could be *older* than the Hubble Time since the Universe would have been expanding *slower* in the past.

- Try measuring the density (matter + energy) of the Universe and comparing it to the critical density.
= / _{crit}> 1 Closed

= 1 Flat

< 1 Open

- Measure expansion rate long ago (high redshift) and see how fast the Universe is decelerating (doesn't work if there's a non-zero Cosmological Constant).
- Look at geometrical properties:
- Sum of angles in a triangle.

If 180° Flat

If > 180° Positive Curvature (Closed)

If < 180° Negative Curvature (Open)

- Test for the number of parallel lines possible through two points.
NOTE: (a) and (b) are not practical

- Examine the diameter of galaxies. If galaxies are intrinsically the same size, then very distant galaxies will
*appear*too big. - Test the volume contained within a sphere by counting the number of galaxies as a function of distance. V = (4/3)R
^{3}if flat. Example: doubling the distance should yield 8 times the number of galaxies.

If < 8 x Positive Curvature (Closed)

If > 80 x Negative Curvature (Open)

- Sum of angles in a triangle.
- Look at the abundance of He, deuterium, Lithium: (will discuss later).

Plus many other (related) techniques...

All are difficult and uncertain.

**Problems**:

- Locally, space is flat (neglecting planets, stars, galaxies, etc.). Must look very far to see curvature.
- Evolution of galaxies: luminosity, size
- Inhomogeneities on large scales.
- dark matter: is the
*true*matter density =*measured*matter density? - measurement uncertainties: distance, brightness, etc.

Currently, direct measurements indicate that _{Matter} ~ 0.3

The radiation density is several orders of magnitude lower. So if there is no Cosmological Constant then the Universe is **Open**, infinite, and will expand forever.

However, if there is a Cosmological Constant it seems to be just enough to make _{Total} = 1. So the Universe is **Flat**, infinite, and will expand forever but at an ever increasing rate.

The Universe is expanding however and in oder to make it the same at all times matter must be continously created to keep the overall density constant.

Turns out that the needed amount would be about 1 hydrogen atom/cm^{3}/10^{15} years!

The idea clearly conflicts with the notion of the conservation of mass/energy, but it is too slow to ever be observed and refuted.

The discovery of quasars was bad for the Steady State hypothesis: there are more of them in the past, so the Universe looks different in the past. Not consistent with the idea of an unchanging Universe.

The Fatal Blow to the Steady State Hypothesis was the discovery of a 3° Kelvin background radiation permeating the Universe. Predicted by Big Bang theory, unreconscilable in the Steady State hypothesis.

At the same time a group from Princeton University were actually searching for this background static which is the predicted relic of the Hot Big Bang.

Since the Universe was once very hot and has been expanding it should now have a much lower temperature. Therefore it should emit radiation like a perfect blackbody corresponding to that temperature (Wien's Law: _{peak}T = 2.898 x 10^{-3} m·K).

The whole Universe radiates like a blackbody with a temperature of ~ 3K (_{peak} ~ 1 mm). It is the same in all directions!

An alternate view of this radiation is that it is the redshifted relic of the Universe at a very early time when it was so hot that matter and radiation were in thermal equilibrium with each other.

The CMB (Cosmic Microwave Background) shows that the early Universe was very homogeneous and isotropic.

After the Big Bang, the Universe was extremely HOT and ionized. It was **opaque** to radiation dues mostly to free (unbound) electrons (they scatter photons quite strongly).

The Universe cooled as it expanded. Eventually (t ~ 1 million years), the temperature dropped to about 4000K. Now we had p^{+} + e^{-} H **Atoms!**. And the Universe becomes TRANSPARENT. Photons can freely stream.

Those photons have now redshifted and are visible as a T = 3K blackbody.

They came from an opaque wall at z ~ 1300 (very far away).

This opaque wall marks the boundary (horizon) of our **Observable Universe**
! Even if the Universe is infinite we cannot see beyond this opaque wall.

NOTE: The Cosmic Microwave Background Radiation looks slightly higher in one direction and slightly cooler in the opposite direction. (roughly 1 part in 1000). This is due to the Solar System's motion about the Galaxy + the Galaxy's motion within the Local Group + the Local Group's motion with respect to the expanding Universe. **Peculiar Motion** of about 600 km/s in the direction of the Virgo Cluster.

**Major Problem:**

If the galaxies and clusters of galaxies formed by gravitational contraction of matter, where are the inhomogeneities ("clumps") from which they came? They *should* be visible as *temperature* inhomogeneities in the cosmic background radiation -- yet it is **very smooth**.

**Breakthrough (1992)!**: Clumps found with NASA's Cosmic Background Explorer (COBE)!. Tiny inhomogeneities in temperature (roughly 1 part in 10^{5}) --> **clumps**.

The smallest clumps (about 10° across, the resolution of COBE) are the "seeds" from which giant clusters of galaxies formed! The largest ones are the biggest structures ever seen in the Universe!

These small variations are the imprints of tiny ripples in the fabric of spacetime put there during the Big Bang. The Amplitude of the inhomogeneities suggests that there is much **dark matter**, and that = 1 (i.e., Universe is flat).

But since all other studies of matter (including dark matter using gravitational lenses) only show _{M} = 0.3, We are left to wonder if the rest is made up from the energy density of the vacuum which would be the source of the Cosmological Constant (_{} = 0.7).

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