COSMOLOGY. Cosmological Principle: The Universe is isotropic and homogeneous, appearing the same in all directions and at all locations.

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1 COSMOLOGY Cosmological Principle: The Universe is isotropic and homogeneous, appearing the same in all directions and at all locations. galaxy B To show the expansion is the same everywhere consider the following: galaxy A Doing some vector math: In other words, the Hubble relation is exactly the same for an observer in galaxy A seeing B and observer in galaxy B seeing A. Expansion appears to be the same to all observers.

2 COSMOLOGY Simple Model of the Universe using Newtonian Physics Consider a Spherical shell in a thin dust filled Universe. Dust is everywhere with a uniform density ρ(t). r Mass m Dust As the Universe expands, the dust is carried with it. Let r (t) be the radius at time t of a thin spherical shell containing mass m. This shell expands with the Universe with recessional velocity v(t) = dr(t)/dt

3 COSMOLOGY The mechanical work E of the shell is: K(t) + U(t) = E The total energy E = constant (conservation of energy!). For convenience we will write the total energy in terms of two constants, k and ϖ (pronounced varpi ). k has units of (length) -2 and ϖ has units of length and may be thought of as the present radius of the shell r(t0). Now E = -(1/2)m kc 2 ϖ 2 which gives the equation: Mr is the mass interior to the shell, Mr = (4/3) π r 3 (t) ρ(t). Note that Mr = constant since no mass is created, the volume expands in lock-step with the decrease in density. Thus, we can rearrange the eqn above:

4 COSMOLOGY The physical nature of the constant k decides the fate of the Universe: 1. if k > 0 then the total energy of the shell is negative and the universe is bounded (closed). The expansion must someday halt and reverse. 2. If k = 0 then the total Energy is exactly zero. The expansion will continue for every and asymptote to zero recessional velocity. The Universe is Flat. 3. If k < 0 the total energy is positive and the universe will is unbounded (open). The Expansion will continue forever.

5 COSMOLOGY So far we have dealt only with Newtonian Cosmology, for which spacetime is flat. In reality the mass in the Universe causes spacetime to have curvature. The terms closed, open, flat here describe only the dynamics. Later they describe the curvature of spacetime (when we use General Relativity). Cosmological Principle means that the expansion is the same everywhere. We will now write the expansion of any shell as: Where r(t) is the coordinate distance. ϖ is the comoving distance (does not change with time). R(t) is the scale factor (dimensionless) such that R(t0) = 1. R is related to the redshift by R = 1/(1+z). Recall that r 3 (t) ρ(t) is constant, which means that R 3 ρ is constant. Thus: Because R=(1+z) -1 and R 3 (t0) = 1 this gives:

6 COSMOLOGY Now we can consider the evolution of our Newtonian Universe. Consider the Hubble Law v(t) = H(t)r(t) = H(t) R(t) ϖ. v(t) is the time derivative of r(t) : And thus the Hubble Law becomes: Previously we had: Inserting v=hr this then gives: Inserting our eqn for H above gives: Multiplying through by R 2 gives: RHS is constant, LHS is only function of t.

7 COSMOLOGY When k=0 (the flat ) case, we can solve for ρ0 and ρc(t): The present day value (t=t0) is then: Where H0 = 100 h km/s/mpc (h is the Hubble parameter). Our current measure is h=0.71 (more on this), which gives: Or about 6 Hydrogen atoms per cubic meter. Note that the best estimate of baryonic matter density is about 4% of the critical density, or 2 protons per 2 m x 2 m x 2m box!

8 COSMOLOGY The ratio of any density to the critical density is the density parameter, Ω. The present day cosmic matter density is then

9 COSMOLOGY General characteristics of our Universe can be determined: We can insert our definition for Ω into the above equations to get: or which for t=t0 becomes Thus for Ω0 > 1 the Universe is closed. Equating the above relations and solving for Ω gives: For Ω0 = 1 the Universe if flat. For Ω0 < 1 the Universe is open. This implies that as z infinity Ω 1. The Universe would be very, very flat. This seemed too perfect; too good to be true to physicists...

10 COSMOLOGY For our flat Universe with pressureless dust we can solve our equations assuming a flat Universe by setting k=0: Taking the square root and integrating with R=0 at t=0: Which gives: valid for Ω0 = 1 and where th = 1 / H0 is the Hubble time, and th ~ yrs. If Ω0 1 then the integral above is much more complicated and involves trigonometric and hyberbolic functions. See your book.

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12 COSMOLOGY The Lookback Time is defined as how far back in time we are looking when we view an object with redshift z. It is defined as: tl = t0 - t(z) where t(z) is given by our previous relation (valid for the special case k=0, equiv. to Ω0=1) : Rewriting R = (1+z) -1 we can write the Lookback time as: For the quasar SDSS at z=6.28 assuming Ω0=1 the Lookback time is (tl / th) = (2/3)( ) =0.633 or tl / t0 = Only 5% of the history of the Universe had unfolded when the light from the quasar left.

13 COSMIC MICROWAVE BACKGROUND In 1946 George Gamow made a bold prediction. He was pondering the cosmic abundances. He considered that the early Universe would have been hot enough for nuclear fusion (just like in the center of the Sun). Gamow proposed that a very early Universe could explain all the abundances. George Gamow ( ) The idea was flawed because there are no stable isotopes of N=5 or 8 nucleons, leaving He-4 the heaviest element (with trace amounts of Li-7). At this time there was a bigger problem in that Hubble had published his expansion parameter to be 500 km / s / Mpc, which gave an timescale for the age of the Universe as th = 1/H0 = 10 9 yr, much less than the age of the Earth from Geology. This led some to argue for a Steady State Model for the Universe.

14 COSMIC MICROWAVE BACKGROUND Steady State Model for the Universe In Hermann Bondi ( ), Thomas Gold ( ), and Fred Hoyle considered a steady state Universe where Universe is infinitely old, time has no beginning and no end. The Universe still expands, but new matter is continuously created to fill the void. This changes the interpretation of the Hubble Time, which becomes a characteristic time for the creating of matter. Mass-creation rate required is roughly one H atom per cubic meter per billion years. Fred Hoyle ( ) Hoyle argued that nucleosynthesis takes place in stars to counter Gamow s arguments that nucleosynthesis occurred only during the early Universe. In 1957 Hoyle and collaborators published what is still a seminal work on stellar nucleosynthesis that was the basis for this course!

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16 COSMIC MICROWAVE BACKGROUND In the 1950s you had two Cosmological Theories that were debated: 1. Steady State Model: headed by Hoyle. 2. Big Bang model, headed by Gamow and others. Steady State model could not explain the Helium abundance, but did explain why the Universe was at least 5 billion years old. Big Bang model (term first coined by Hoyle as a derision) implied the Universe was only a few billion years old at best but could explain the Helium abundance. Was there any evidence that such a violent - A Big Bang - ever took place?

17 COSMIC MICROWAVE BACKGROUND Key idea of Big Bang was that the early Universe was very hot and dense, like the nucleus of a Star. Recall that the density evolves like: For a realistic gas with pressure, we need relation between pressure and density. This is the Equation of State. In general we write it as: P = wρc 2 For a blackbody (photon) radiation w=1/3 for blackbody radiation. For pressureless dust we have w=0. (If Dark Energy is a constant, then it has w = -1, a big test for cosmology...)

18 COSMIC MICROWAVE BACKGROUND For our expanding Universe: in the expansion we can use that for a blackbody the radiation pressure is P = u/3 (where w=1/3, from here) and where u is the energy density for a blackbody: u=at 4. The energy density evolves in an expanding Universe by : The energy density today is much, much smaller by a factor of R 4. A factor of R 3 is due to the change in the volume and another factor of R is due to the expansion of the wavelength of light.

19 COSMIC MICROWAVE BACKGROUND Thus, R 4 at 4 = at0 4 and we find that the blackbody temperature must be related to the temperature at an earlier time as RT = T0. Consider that Helium fusion requires T~10 9 K and a density ρb~10-2 kg/m 3. Higher temperatures would photodissociate deuterium and lower temperatures would not allow protons to overcome the Coulomb barrier. Using the present day value of the baryon density we find that for the needed value R was : This gives a present day temperature of: T0 = R T = 3.47 K.

20 COSMIC MICROWAVE BACKGROUND...This gives a present day temperature of: T0 = R T = 3.47 K. A blackbody with this temperature would emit in the microwave region of the electromagnetic spectrum. From Wien s law for Blackbody radiation we have: λmax = m K / T ~ 8.4 x 10-4 m or 360 GHz. Alpher and Herman made this prediction (they came up with a present-day temperature of 5 K). 16 years later two Princeton physicists, Robert Dicke and P. J. E. Peebles came up withe the same value, and they wanted to go detect this signal. They were scooped unwittingly by two engineers...

21 COSMIC MICROWAVE BACKGROUND Working for Bell, in 1964 Wilson and Penzias were building a huge horn antennae to communicate with AT&T s Telstar satellite. Despite all their efforts they detected a continuous hiss from all directions of the sky. The estimated that a blackbody of 3K would produce this hiss, but they did not know the source. Robert Wilson (left) and Arno Penzias (right) with their 6m Microwave antennae (horn). Penzias then learned of Peeble s prediction of a cosmic microwave background. In 1965 he called Dicke and invited them to Holmdel, NJ to show them their result and all the pieces fell into place.

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24 COSMIC MICROWAVE BACKGROUND Microwave detectors still used in modern communications systems. Microwaves are good for transmitting information from one place to another because microwave energy can penetrate haze, light rain and snow, clouds, and smoke. Shorter microwaves are used in remote sensing (including doppler radar used in weather forecasts). Microwave towers can transmit information like telephone calls and computer data from city-to-city.

25 Penzias and Wilson had discovered the leftover glow from the Big Bang. It is everywhere and has a peak wavelength of 1.06 mm. This is the cosmic microwave background (CMB). This result was a death sentence for the Steady State Model. In 1991 the COBE satellite measured the full spectrum of the CMB.

26 Blackbody with T=2.725 K. Mather et al. 1991, ApJ, 354, L37 Penzias and Wilson received the Nobel Prize in John Mather received the Nobel Prize in 2006

27 But, hearing is believing...

28 Our current measurements of the CMB come from WMAP, the Wilkinson Microwave Anisotropy Probe. Launched in 2001, it was designed to study the slight fluctuations in left in the CMB (more on this). It has confirmed small details of the Big Bang. It also gives us the best measurement of H0: H0 = 71 (+4/-3) km/s/mpc This gives us a current Hubble time of : th = 1/H0 = 4.35 x s = 1.38 x yr. The Big Bang happened everywhere. So the CMB comes from everywhere!

29 Origin of CMB Calculate where the average time between photon scatterings by electrons approaches the timescale of the expansion: At times greater than this, the photons were decoupled from the matter. If electrons had remained free decoupling would have occurred when the Universe was 20 Myr old. But when the Universe was 1 Myr, electrons combined (we call this recombination) with protons and He-4 nuclei and the photon opacity dropped to zero. Surface of Last Scattering was the point from where the CMB photons are now arriving. It is the farthest redshift we can see (in reality there is a thickness to this surface ).

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31 Origin of CMB Surface of Last Scattering Use our model of the expanding Universe and what we know about gas ionization to find that when 50% of the Hydrogen was ionized occurred when R~7.3 x 10-4 which occurs for z=1380 and T=3760 K. WMAP finds the following: which corresponds to an age for the Universe of Recombination does not happen at once, and WMAP can estimate the width of the surface of last scattering: kyr

32 BIG BANG NUCLEOSYNTHESIS We can now address the question, why is ~25% of the mass in the Universe Helium (and the rest Hydrogen?) At a time t~10-4 s the cosmic temperature was ~10 12 K. The Universe was filled with photons, electrons-positrons, neutrinos, and a few protons and neutrons. There were ~5 p and n for every photons, therefore the former were constantly being transformed: The mass difference between a p and n is only: (mp - mn)c 2 = MeV. The thermal energy of the Universe is K is kt ~ 100 MeV. At this energy, the # of neutrons and protons are nearly equal.

33 BIG BANG NUCLEOSYNTHESIS As the Universe expanded, the temperature fell. At T~10 10 K the number of neutrons to protons was frozen at There were 223 neutrons for every 1000 protons, and no more neutrons were being created. The half life of a free neutron is 614 s = 10.2 min. For p + n to form deuterium requires T ~ 10 9 K (higher temperatures dissociate deuterium). In this time 176 s passed, so if you started with 223 neutrons, you d now have 183, and number of protons increased to Now deuterium and Helium fusion can occur at T~10 9 K. and No other elements were made except trace amounts of Li-3. The 183 neutrons and 1040 protons could form 92 He-4 nuclei with 867 protons left over. Because He-4 is 4x more massive than Hydrogen, we should have: [ 4(92)/( (92)) ] =

34 BIG BANG NUCLEOSYNTHESIS Big Bang Nucleosynthesis reactions from Nollett & Burles, Phys. Rev. D, 61, , 2000.

35 Calculated mass abundance of He-4, H-2, He-3, and Li-7. Schramm & Turner 1998, Rev. mod. Phys, 70, 303.

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37 Origin of CMB Final thoughts, from Steven Weinberg in The First Three Minutes: This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations... have something to do with the real world... The most important thing [of the] discovery of the 3 K radiation background in 1965 was to force us all to take seriously the idea that there was an early Universe.

38 Euclidean, Elliptic, and Hyperbolic Geometries 19th Century Mathematicians set out to test and disprove Euclid s 5th postulate, and they managed to do it. It all depends on geometry. Benhard Riemann ( ) developed an elliptic geometry (like the surface of an ellipsoid, like a sphere). : Given, in a plane a line L, and a point P not on L, then through P there exists no line parallel to L. Carl Frederich Gauss ( ), Nikolai Lobachevski ( ) and János Bolyai ( ) developed independently a hyperbolic surface, in which: Given, in a plane, a line L and a point P not on L, then through P there exist at least two lines parallel to L.

39 Relativistic Cosmology We can describe it with a metric (remember General Relativity?) The cosmological principle: Universe is isotropic (no special directions) and homogeneous (no spatial places). This leads to curvature in the Universe. Percy Robertson and Arthur Geoffrey Walker came up with a metric to describe an isotropic, homogeneous universe, the Robertson-Walker metric: where ϖ is the comoving distance and k is a constant describing the curvature. Just like for Newtonian cosmology, k > 0 universe is closed, k = 0 universe is flat, k < 0 universe is open. is the differential proper distance for dt=0.

40 Relativistic Cosmology Aleksandr Friedmann ( ) solved Einstein s equations for the dynamical evolution of the Universe, which is now known as the Friedmann equation: Which we derived using only Newton previously. R is the dimensionless scale factor, ρ is the density, everything else is a constant. In 1922 Friedmann obtained this equation for a nonstatic Universe. It was independently derived in 1927 by a Belgian priest nambed Abbé Georges Lemaître ( ). Lemaître is considered the first person to propose that the universe evolved from a highly dense beginning. He is sometimes called the father of the Big bang. Einstein, after hearing him talk about the nonstatic universe, told him your mathematics is excellent, but your physics is abominable. Einstein was wrong. Abbé Georges Lemaître ( )

41 Relativistic Cosmology Einstein had also realized that his equations could not produce a static Universe. At the time, Hubble s data had not been published, and everyone thought the Universe had to be static. To make his solution static, Einstein inserted a Cosmological Constant, which is just a constant of integration, into his solution: This is our Friedmann equation, except that the constant results in an extra potential energy term: The conservation of energy equation then becomes: The force due to the new potential is:

42 Relativistic Cosmology When Hubble showed the Universe was expanding, Einstein removed the Cosmological Constant and called it the biggest blunder of his life. However, by the late 1990s the evidence suggested that there is a Cosmological Constant, but with the effect of accelerating the expansion (rather than slowing it down or keeping it static). Thus, the Cosmological Constant implies there is some Dark Energy associated with the vacuum, and this is pushing the expansion of the Universe. We can then define the mass density of the dark energy:

43 Relativistic Cosmology We can now rewrite the Friedmann equation as: where We can then write the total density parameter as Which makes the Friedmann equation: Or, at t=t0 :

44 Relativistic Cosmology We can then rewrite the evolution of Hubble s constant as: Where the current values from WMAP are: This implies that the total density parameter is ~1. The Universe is nearly flat:

45 Age of the Universe Look at the behavior of the scale factor to get the age of the Universe as a function of R. neglecting the contribution of relativistic particles during the first 55,000 yr (ρrel = 0) then we arrive at an expression : Plugging in R=1 we get And WMAP measured:

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47 Redshift-Magnitude Relation using Supernovae Ia From Perlmutter & Schmidt (2003), data from Perlmutter et al. ApJ, 517, 565 (1999) and Riess et al. AJ, 116, 1009 (1998).