Conclusion of Last Class: Hubble’s Law (that far away galaxies recede faster) implies that the universe is expanding uniformly in all directions, like a photocopy enlargement. Important Points: It is space itself that is expanding, NOT the galaxies rushing outward into previously existing empty space. Things held together by gravity, like the galaxies themselves, do not expand. The redshift observed in Hubble’s Law is technically due to wavelengths expanding with the expanding space, not the Doppler effect, but the formulas are the same as long as the redshift is small.

Why expanding? First part of this class: Can the expansion be explained by our best theory for the structure of space and time, Theory of General Relativity? How mathematically to describe the expansion?

The Einstein equation is not generally solvable! Postulate of GR: Mass and energy curve spacetime. Maybe the expansion is an aspect of this spacetime curvature. Quantitative result of GR: Einstein equation (1915): Very hard math describing spacetime curvature = Very hard math describing distribution of matter & energy The Einstein equation is not generally solvable!

Einstein equation first simplified for our universe by Friedmann, using a simplification called the cosmological principle. Cosmological Principle: The universe is homogeneous and isotropic. Homogeneous: roughly the same in every location. Isotropic: roughly the same in every direction.

Slightly hard math describing scale factor & curvature = Friedmann equation: Slightly hard math describing scale factor & curvature = Slightly hard math describing average density of matter & dark energy Usually solvable: Measure curvature & change in scale factor -> compute average density of matter & dark energy Measure average density of matter & dark energy -> compute change in scale factor & curvature Note: “slightly hard” = integral calculus (MATH 141B) “very hard” = tensor calculus (grad school)

What is this “Dark Energy” term that pops up? It is an extra constant term allowed in the solution to the Friedmann equation (, capital lambda) If it is nonzero, it is a constant “extra” energy possessed by empty space, called the cosmological constant. If it is positive, it would act repulsively (opposite to gravity), and accelerate the expansion of the universe.

BUT, what is allowed often exists… History of Solution: Friedmann equation solved by Einstein (1922). Result: a should be growing or shrinking, NOT constant. BUT: Astronomers at that time (before Hubble) thought that the universe was static, so Einstein set  positive to balance gravity and force a static universe. The expansion of the universe is discovered by Hubble (1929). Einstein realizes his original solution was OK after all. He resets =0 in the solution, and calls  his “greatest blunder”. BUT, what is allowed often exists…

In 1999, several groups of astronomers attempt to measure the slow down in the expansion due to gravity…