1. Cosmological Models II Connecting Hubble’s law and the cosmological scale factor What determines the kind of Universe in which we live? The Friedman equation The Critical Density and  Some useful definitions Where has relativity gone?

2. I : CONNECTING STANDARD MODELS AND HUBBLE’S LAW New way to look at redshift observed by Hubble and Slipher… Redshift is not due to velocity of galaxies Galaxies are (approximately) stationary in space… Galaxies get further apart because the space between them is physically expanding! The expansion of space also effects the wavelength of light… as space expands, the wavelength expands and so there is a redshift. So, cosmological redshift is due to cosmological expansion of wavelength of light, not the regular Doppler shift from galaxy motions.

3. Relation between z and R(t) Redshift of a galaxy given by Using our new view of redshift, we write So, we have…

4. Co-moving coordinates. Let’s formalize this… Consider a set of co- ordinates that expand with the space. We call these co-moving coordinates.

5. If a galaxy remains at rest once the overall expansion has been accounted for, then it has fixed co-moving coordinates. Consider two galaxies that have fixed co-moving coordinates. Let’s define a “co-moving” distance D Then, the real (proper) distance between the galaxies is… proper distance = R(t) D

7. So.. Hubble’s constant given by

8. II : THE CRITICAL DENSITY When we go through the GR stuff, we get the Friedmann Equation… this is what determines the dynamics of the Universe Which gives (dividing by R 2 )

9. Let’s examine this equation… H 2 must be positive… so the RHS of this equation must also be positive. Suppose density is zero (  =0) Then, we must have negative k (i.e., k=-1) So, empty universes are open and expand forever Flat and spherical Universes can only occur in presence of matter.

10. Now, suppose the Universe is flat (k=0) Friedmann equation then gives So, this case occurs if the density is exactly equal to the critical density…

11. In general case, we can define the density parameter… Can now rewrite Friedmann’s equation yet again using this… we get

12. Can now see a very important result… within context of the standard model:  <1 means universe is hyperbolic and will expand forever  =1 means universe is flat and will (just manage to) expand forever  >1 means universe is spherical and will recollapse Physical interpretation… if there is more than a certain amount of matter in the universe, the attractive nature of gravity will ensure that the Universe recollapses.

13. The deceleration parameter, q The deceleration parameter measures how quickly the universe is decelerating For those comfortable with calculus, actual definition is: Turns out that its value is given by This gives a consistency check for the standard models… we can attempt to measure  in two ways: Direct measurement of how much mass is in the Universe Measurement of deceleration parameter

14. Deceleration shows up as a deviation from Hubble’s law… A very subtle effect – have to detect deviations from Hubble’s law for objects with a large redshift

15. III : SOME USEFUL DEFINITIONS Have already come across… Standard model (Homogeneous & Isotropic GR- based models with  =0) Critical density  c (average density needed to just make the Universe flat) Density parameter  =  /  c We will also define… Cosmic time (time as measured by a clock which is stationary in co-moving coordinates, i.e., stationary with respect to the expanding Universe)

16. Hubble time, t H =1/H (cosmic time since the big bang if the universe were expanding at a constant rate.)

17. Hubble distance, D=ct H (distance that light travels in a Hubble time). This gives an approximate idea of the size of the observable Universe. Age of the Universe, t age (the amount of cosmic time since the big bang). In standard models, this is always less than the Hubble time. Look-back time, t lb (amount of cosmic time that passes between the emission of light by a certain galaxy and the observation of that light by us) Particle horizon (a sphere centered on the Earth with radius ct age ; i.e., the sphere defined by the distance that light can travel since the big bang). This gives the edge of the actual observable Universe.

18. IV: WHERE HAS RELATIVITY GONE? Question: Cosmology is based upon General Relativity, a theory that treats space and time as relative quantities. So, how can we talk about concepts such as… Galaxies being stationary with respect to the expanding Universe? Relativity tells us that there is no such thing as being stationary! The age of the Universe? Surely doesn’t time depending upon the observer? What do you think??