1. Lecture 27: The Shape of Space Astronomy 1143 - Spring 2014

2. Key Ideas: Curvature of space Critical density Described by the parameter  Determines the fate of the Universe – will it expand forever? Adding up contents of Universe   =1 Measuring the curvature of the Universe Measuring geometry on large scales Observations of angular sizes

3. Inventory of the Universe

4. Photons Negligible amount “Normal” matter Observe amount of stars/gas/dust/planets in the Universe Dark Matter Observe gravitational force on visible matter Dark Energy Observe expansion history of the Universe

5. Importance of Curvature Astronomers are interested in measuring the curvature of space for several reasons Fundamental fact about the Universe Confirms our measurements of the matter and energy density of the Universe The balance of matter & energy determines the fate of our Universe Intriguing thoughts for the multiverse idea How special is our value for  If our Universe’s curvature is flat/open – is it a smaller part of an infinite Universe?

6. This critical density depends only on the gravitational constant G and on the Hubble constant H 0. The Friedman equation states that space is flat (Euclidean) if its density equals a critical density ρ crit. The Critical Density Spacetime is definitely curved; is space?

7. With H 0 = 72 km/sec/Mpc, the critical density is:  crit ~ 10 -26 kg/m 3 is Yes, this is a very low density! Water: 1000 kg/m 3 Air: 1 kg/m 3 The Critical Density

8. 1 m 3 of the universe 10 -26 kg The Critical Density

9. 20 pc 3 of the universe Earth’s mass The Critical Density

10.  for this,  for that In general,  total  means that the Universe is flat

11. Totaling it up  m = Density of Matter and Energy    = Density of the Vacuum Energy If there is a Cosmological Constant, , the Density Parameter becomes:  total  determines the geometry of the Universe Balance of m and  determines expansion history of Universe

12.  =1 (or very close) In general, we know that the Universe isn’t too far off from  =1, otherwise we wouldn’t live in a relatively old, relatively full Universe Given all the numbers  could be, it would be odd if it picked a number close to, but not equal to 1 (see inflation) Adding up the matter and energy density of the Universe, we also get a value of  =1 (within the errors) We can measure the curvature of the Universe directly as well

13. Histories of the Universe

14. The Accelerating Universe The SNIa results combined with constraints from the cosmic background radiation and galaxy clusters give:  m  0.3 ± 0.1    0.7 ± 0.1 Taken together:  total  1 We live in a spatially flat, accelerating Universe.

15. We only experience 3 dimensions of space While other dimensions can be mathematically described, they cannot be observed. We cannot point to the center of the expansion what the Universe is expanding into Observationally, the Big Bang happened everywhere and space is being created We can still measure the curvature of space

16. 2-D Analogs for 3-D Space

17. Measuring Curvature Flat: angles of triangle add to 180° >180° Positive: angles add to >180° <180° Negative: angles add to <180°

18. Curvature is hard to detect on scales smaller than the radius of curvature. good Flat = good approximation bad Flat = bad approximation

19. Measuring parallax (flat space) June January p p a = 1 AU p in arcseconds, d in parsecs p = 1/d d

20. Parallax (positive curvature) p < 1/d June January p p

21. Parallax (negative curvature) p > 1/d As d →infinity, p→1/R June January p p radius of curvature

22. The smallest parallax you measure puts a lower limit on the radius of curvature of negatively curved space. at least Hipparcos measured p as small as 0.001 arcsec; radius of curvature is at least 1000 parsecs.

23. Bigger We need Bigger triangles to measure the curvature accurately!  L d  =L/d (flat)  >L/d (positive)  <L/d (negative)

24. Positivelymagnifying large Positively magnifying lens; distant galaxies appear anomalously large. Negatively demagnifying small Negatively curved space is a demagnifying lens; distant galaxies appear anomalously small. magnifying demagnifying Measuring Curvature

25. Ripples in the Gas The state of matter in the early Universe is a plasma -- hot, ionized gas Not completely smooth – some regions overdense, some regions underdense Gravity pull is stronger towards overdense regions Compressed gas heats up But photons are still interacting with matter – radiation pressure pushing back

26. Analogy – Ripples in a Pond

27. Density Changes  Temperature Changes Bottom line – sound waves start passing through the gas This leads to temperature changes in the gas. Wien’s law – hotter temperatures=shorter waves So when the photons are released to free-stream into the Universe ~400,000 years after the Big Bang, areas that were denser emitted slightly shorter photons on average Can calculate how big these areas should be in physical units

28. Size of Ripples = Sound Horizon How far can these pressure/density waves travel before the photons start free-streaming? Distance=speed x time Time= time until recombination and the release of the cosmic microwave background radiation Speed = speed of sound in the plasma ~ 1/2 the speed of light Can use this number to predict how big in angular size the biggest hot or cold patches should appear for different geometries of the Universe

29. Waves in the Early Universe Gas in the early Universe was being sloshed around by sound waves, leading to hot areas and cool areas Physics tells us how large those fluctuations are in physical units Now just need to measure their angular size

30. The Path of Light Rays

31. Temperature of CMB

32. And the answer is… Distant galaxies are neither absurdly small in angle nor absurdly large. Temperature fluctuations are neither absurdly small in angle nor absurdly large If If the universe is curved, radius of curvature is bigger than the Hubble distance (c/H 0 = 4300 Mpc). We have other lengths that we can use to test the curvature of the Universe All agree that the Universe is flat (or close to)