1. What supplies the acceleration being measured with supernovae? Two guesses: 1.Cosmological Constant 2.Vacuum Energy Density of the Universe

2. The cosmological constant L ! Einstein’s “biggest blunder” but it can be added with a + or – Einstein adds a term:

3. “Virtual” particles and anti-particles are constantly being produced out of the vacuum of space! 2. Vacuum Energy Density of the Universe Current high energy theories predict a sea of particles:

4. This is allowed because of the Heisenberg uncertainty principle It is a statement about certain conjugate variables and the ability to know them with infinite certainty. In this particular case it is a statement about Energy and Time. This allows us to create a certain amount of energy for a certain amount of time – provided we don’t violate the principle stated above.

5. In classical mechanics, Newton's second law is used to make a mathematical prediction as to what path a given system will take following a set of known initial conditions. A wave must follow a derivable equation which is different. It is a partial differential equation which is cyclic in the derivative and gives classic trig functions as solutions (e.g. sin, cos…) Wave Particle Duality

6. In quantum mechanics, the classical wave equation is replaced by a new wave equation. An example is the Schrödinger equation which is a partial differential equation that describes how the quantum state of a system changes with time. It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function"). is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation). Wave Particle Duality

7. The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field): Time-dependent Schrödinger equation (single non- relativistic particle) Wave Particle Duality

8. Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals of one another,[

9. The Repulsion may be gotten from the Vacuum Energy Density of the Universe! In order to check this and make predictions, we need a theory of Quantum Gravity! String Theory? Replace points with strings

10. Geometry = Physics 0 – Dimensions 0 – Dimensions 1 – Dimension 1 – Dimension 2 – Dimensions 2 – Dimensions 3 – Dimensions 3 – Dimensions

14. Cosmological Parameters Cosmological models are typically defined through a few parameters: 1. The Hubble parameter is the normalized rate of expansion: Note that the Hubble parameter is not a constant! The Hubble constant is the Hubble Parameter measured today – we denote its value by H 0. Current estimates are in the range of. We will discuss these efforts in more detail shortly.

15. HubbleConstant Defines theScale the Universe of H O = slope at tOtO RORO tOtO 0 c / H O = Hubble length 1 / H O = Hubble time {

16. 2. The Deceleration Parameter This parameter can be related to the critical density Cosmological Parameters q=1/2 q>1/2 q<1/2

17. 3. The Matter Density Parameter If we rewrite the Friedmann Equation using the Hubble parameter and set the cosmological constant to zero for now: The universe is flat if k=0 so: Cosmological Parameters

18. we see that it has a critical density of We define the Matter Density Parameter

19. 4. The “dark energy” Density Parameter A density parameter for  can be expressed by using the Friedmann equation and setting  =0 we get: the total density parameter is Cosmological Parameters

20. Lets look at the behavior of the scale factor in these Friedmann cosmologies. First, do a Taylor series expansion of a(t) about t=t 0 To exactly reproduce an arbitrary function a(t) for all values of t, an infinite number of terms is required in the expansion. However, the usefulness of a Taylor series expansion resides in the fact that if a doesn’t fluctuate wildly with t, so using only the first few terms of the expansion may give a good approximation in the immediate vicinity of t 0. The scale factor a(t) is a good candidate for a Taylor expansion since there’s no evidence that the real universe has a wildly oscillating scale factor.

21. Keeping the first three terms of the Taylor expansion and dividing by the current scale factor a(t 0 ), gives the scale factor in the recent past and the near future: Using the normalization a(t 0 ) = 1, this expansion for the scale factor is customarily written in the form q 0 is a dimensionless number – the deceleration parameter

22. Parameters Cosmological H O defines the spatial and temporal scale The other parameters ofthe universe (Ωx)(Ωx) determine the shape of the R(t) curves RORO tOtO O c / H O = Hubble length 1 / H O = Hubble time {

23. Cosmological Parameters A few notes: The Hubble parameter is usually constant (even though it changes written as: called the Hubble in time!) and it is often s -1 Mpc -1 ), or s -1 Mpc -1 ) h = H 0 / (100 km h 70 = H 0 / (70 km Thecurrent physical value of the criticaldensity is g cm = 0.921 X 10 -292-3 h 70 p 0,crit The density parameter(s) can bewritten as: O m + O k + O A = 1 Where  k is a fictitious curvature density

24. Recall the definitions of the cosmological parameters: Cosmological Parameters If  k = 0: The Friedmann Eqn.is now:

25. The Hubble ’ s Constant Has a History … LongandDisreputable (1931,ApJ74,43) Since then, the value of the H O has shrunk by an order of magnitude, but the errors were always quoted to be about 10% … Generally, Hubble was estimating H O - 600 km/s/Mpc. This implies for the age of the universe - 1/ H O < 2 Gyr - which was a problem! H O = 560 km/s/Mpc

26. The History of H 0 Majorrevisionsdownwards happened as a resultofrecognizing somemajorsystematicerrors 1200 1000 800 600 400 200 0 1920193019401950 1960 Date 1970198019902000 H 0 (km/s/Mpc) Jan Oort Compilation by John Huchra Baade identifies Pop. I and II Cepheids “Brightest stars” identified as H II regions

27. History of H 0, Continued … The Buteveninthemodern era, measured valuesdiffered covering afactor-of-2spread! Values favored by de Vaucouleurs, van den Bergh Values favored by Sandage, Tamman

28. … from The History of H 0, Continued Notethatthespreadgreatly exceededthequotederrors everygroup! (from R. Kennicutt)