Big Bang Cosmology Today’s Lecture: Big Bang Cosmology Expansion/Cosmological Redshift Expansion History Geometry of Spacetime Cosmic Microwave Background Homework 9: Due Tuesday, April 22 Help Session: Thursday, May 1, 1:30 pm LGRT 1033 Final Exam: Friday, May 2, 1:30 pm, HASB 138 Reading for today: Chapter 20 Reading for next lecture: Chapter 21

Note: if you are interested in attending, please respond by April 18

The Cosmological Principle Models of our universe are based on several assumptions. The first is the cosmological principle, which states: That on sufficiently large scales, the properties of the Universe are the same for all observers. Therefore not only are the physical structures the same, but so are the physical laws. Sometimes this is stated as saying the Universe is Homogeneous and Isotropic. It follows from these principles that the universe can have no edge or center. The framework of cosmological models is general relativity.

Expansion of the Universe The Hubble law tells us that the Universe is expanding. However, remember that the cosmological redshift is not motion through space, but due to the expansion of space. Expansion animation Expansion animation

A 2-dimensional analogy is the surface of a balloon, where the ballon represents the fabric of space. Imagine being at the top of the ballon, as the ballon expands the galaxies are carried with it and all of the galaxies appear to be moving away. However the same would be seen from anywhere on the ballon. The Universe has no edge nor does the expansion have a preferred “center”. No matter where you are in the universe you will measure the same Hubble Law ( v = H r)

Expansion of the Universe The cosmological principle does not imply that the Universe is constant at all times. Universal expansion points to a beginning of the Universe and implies that the Universe is changing over time. Scale Factor: In discussing the expansion of the universe it makes no sense to talk about size, since the universe could be infinite. Consider the separation between two points in space, the time variation of this separation is given by r(t). Define a fiducial time (now at time t o ), then r o = r(t o ).

We can define a scale factor R(t), such that: R(t) = r(t)/r(t o ) = r(t)/r o. t o We have R(t) = r(t)/r o, so by definition R(t o ) = 1. Note that for an expanding universe: For t < t o, R < 1 For t > t o, R > 1 Hubble's Law: v = dr/dt = H(t) r(t), using the definition of the scale factor, we can rewrite this as: r o dR/dt = H(t) R(t) r o, or dR/dt = H(t) R(t). Hubble's “constant” is unlikely to be a constant, but is likely to change with time. We define: H(t o ) = H o.

Cosmological Redshift In an expanding Universe, the redshift measured for galaxies is NOT due to the Doppler effect. Instead, what is measured is a cosmological redshift. Light emitted at one wavelength is stretched by the expansion of space by the time it reaches the observer. The amount of stretching depends on the light travel time or distance.

Cosmological Redshift Redshift (z) defined as: λ detected – λ emitted )= λ detected /λ emitted - 1 z = (λ detected – λ emitted )/λ emitted = λ detected /λ emitted - 1 Consider radiation emitted by an object at time t (where t < t o ), due to the expansion of the universe, we see: λ detected /λ emitted = R(t o )/R(t) = 1/R(t), therefore z = λ detected /λ emitted - 1 = 1/R(t) - 1 z = λ detected /λ emitted - 1 = 1/R(t) - 1 Thus: z + 1 = 1/R(t), note that R(t) 0 Assume t = t o – Δt, where Δt is very small compared to t o. We can approximate (Taylor expansion): z + 1 = 1/R( t o – Δt) = 1/R(t o ) + Δt dR/dt

Since by definition R(t o ) = 1, we have z + 1 = 1/R( t o – Δt) ≈ 1 + Δt dR/dt z + 1 = 1/R( t o – Δt) ≈ 1 + Δt dR/dt Remember that dR/dt = H(t) R(t). Since Δt is small compared to time t o, we can assume that dR/dt = H o R(t o ) = H o, thus the Hubble constant now. Therefore, light that left an object at time t o – Δt will have a redshift of: z + 1 ≈ 1 + Δt H o. The distance of the object is just d = c Δt, therefore: cz ≈ H o d This is Hubble's law. This simple linear dependence only accurate in local universe (z < 0.3). To derive a general expression, one needs to know the time history of R(t).

Consider the implications of cosmological expansion, in the past, galaxies were closer together. If in fact we run the clock back far enough in time we come to a time when the Universe was infinitely dense (and very hot) This is the beginning of the Universe – the beginning of the expansion (the class of evolving cosmological models are called Big Bang theories). It is interesting that the term big bang was coined by Fred Hoyle to ridicule the notion of a hot and dense early epoch of the Universe – but the term stuck. The Big Bang

Age of the Universe If the universe is expanding at a constant rate, then the age of the Universe is just the inverse of the Hubble constant (see figure). t = 1/H = 1/(71 km s -1 Mpc -1 ) t = 1 Mpc/71 km/sec t = 3.09x10 19 km/71 km/sec t = 4.4x10 17 sec Age is about 13.8 billion years. We need the time evolution of R(t) to do this properly.

Newtonian Cosmology Consider a sphere of radius r centered on O, now consider the motions of points A, B, C and D on surface of sphere relative to O. Thus the force acting on A is only due to the mass internal to radius r, and it acts as a point mass at O. Although not strictly self-consistent, we can gain some valuable insights using just Newtonian dynamics and gravitation. Birkhoff's theorem states that the net gravitational effect of a uniform media external to the sphere on the motion of points A, B, C and D relative to O is zero. r

If the average mass density in the volume defined by radius r is ρ, then the mass within the volume is: M = 4/3 π r 3 ρ. The textbook considers the gravitational force on a particle at A, and it shows that its acceleration relative to O must be non-zero if ρ > 0. We are now going to consider the motion of particle A (with mass m) relative to position O. r

We know that due to the expansion of the universe that mass m is moving away from O at a velocity of H r. Now consider the total energy (both kinetic and gravitational potential energy) of mass m relative to O. kinetic energy = ½ m v 2 gravitational potential energy = -G M m/r If we assume that energy is conserved, then we can set the total energy equal to a constant: E = ½ m v 2 - G M m/r = constant since v = Hr, we have: 22 ½ H 2 r 2 - G M/r = constant

Using our earlier expression for mass, M: M = 4/3 π r 3 ρ we find: 22 ½ H 2 r 2 - 4/3 π r 3 ρ G/r = constant We can rewrite this as: H π ρ G/3 = constant/r 2 The equation is similar to what is commonly known as the Friedmann equation. Assume the density of the universe at t = t o is ρ = ρ o, and the Hubble constant is H = H o. Then: H o π ρ o G/3 = constant/r 2

The total energy can be negative, zero or positive. An analogy to this problem is throwing a ball upward. Depending on the velocity of the ball it will either escape the Earth (positive total energy) or fall back (negative total energy). The dividing line is the escape velocity and this corresponds to a total energy of zero. Cosmology with total energy is zero (constant = 0), then: H o π ρ o G/3 = 0, or o 2 o H o 2 = 8 π ρ o G/3, Solve for the critical density where the total energy is zero: o 2 ρ c = 3 H o 2 /(8 π G)

o 2 With the critical density defined as: ρ c = 3 H o 2 /(8 π G), we have three possibilities (just like the ball analogy): o (1) ρ o > ρ c, the total energy is negative and the expansion of universe is halted by gravity and the universe will eventually collapse (bound universe). o (2) ρ o = ρ c, the total energy is zero and the expansion of universe halted by gravity, but only when R is infinity. o (3) ρ o < ρ c, the total energy is positive, and the universe will expand forever (unbound universe). o We define the density parameter: Ω = ρ o /ρ c

Age of the Universe H o π ρ o G/3 = constant, so if ρ o = 0 then The age depends on the expansion history. Only in a universe with no mass (remember: H o π ρ o G/3 = constant, so if ρ o = 0 then H is constant), is the age 1/H o. 0 < Ω < 1, have an unbound universe with age less than 1/H o. Ω > 1, have a bound universe with an age much less than 1/H o. Ω = 1, have a marginally bound universe with an age in between (2/3H o ).

A Big Bang Universe has another consequence and a solution to: Olber’s Paradox Why is the sky dark at night ? A question that dates to the 17 th century. Just as in a infinite forest, where every line of sight is blocked by a tree, if the Universe is static and infinite then every line of sight would intersect the surface of a star. The night would be as bright as the surface of the Sun !!!

The Big Bang solves the darkness at night problem very simply. The cosmological horizon marks a limit to the observable universe and is limited by time and not space. The universe is finite in time, thus our cosmic horizon is limited (due to the finite speed of light), so we cannot see an infinite number of stars.

Cosmology and General Relativity The universe can have different geometries depending on the density of the universe. Although we have talked about likening the expansion of the Universe to an expanding ballon, that is only one of several possible geometries. Einstein’s general relativity can describe the Universe has a whole. Remember that in general relativity, mass (and energy) curves space/time. The geometry of the Universe depends on the total mass-energy density of the universe.

Closed universe ( Ω > 1): positive curvature - universe has a spherical geometry, and is finite is size. Flat universe ( Ω = 1): no curvature - universe has a flat geometry, and is infinite is size. Open universe ( Ω < 1): negative curvature - universe has a saddle-shaped geometry, and is infinite is size.

The geometry of the universe has a number of consequences (which we will use later):

Big Bang Cosmology The observation that the Universe is expanding has led to the idea of a Big Bang cosmology. Are there other observational consequences of such an Universe ? Answer: The Cosmic Microwave Background (CMB)

Cosmic Microwave Background George Gamow predicted that the young universe had to be very hot and dense and as it expands it cools. The early universe was predicted to be filled with radiation, and that the remnant of that radiation should be detectable, but redshifted to microwave wavelengths. Minutes after the Big Bang the universe had a temperature of ~billion of degrees. The universe was fully ionized. As the universe expanded, it cooled and at about 380,000 years after the Big Bang the Universe cooled to 3000° K and electrons and atomic nuclei could combine into neutral atoms.

When the universe was hot and ionized, it was opaque to light. As it cooled below about 3000 K the universe became transparent to light. Gamow, Alpher, and Hermann predicted observable radiation from this recombination epoch in the universe. Analogy is looking up on a very cloudy day. Can see things clearly until reach cloud layer – see only diffuse light.

The universe expanded by a factor of 1000 since recombination, the emission from this era was redshifted from optical wavelengths to radio wavelengths. Predicted to be a blackbody with a temperature of 3°K. Nobel Prize in Physics) This cosmic microwave background (CMB) was discovered in 1965 by Penzias and Wilson at Bell Labs (received the 1978 Nobel Prize in Physics).

COBE – Cosmic Background Explorer – launched in Confirmed the blackbody nature of the CMB °K CMB Emitted at time of recombination when universe was about 3000 K. Redshifted by about 1000 fold.

COBE imaged the entire sky. Found the CMB to be very uniform and the image of the sky rather boring.

After removing the average emission, see dipole term due to our motion through space (Doppler shift) – ΔT ~ 3.5 mK. After removing the dipole term see the microwave emission from the Milky Way. After removing the foreground Milky Way emission, COBE detected small fluctuations in the CMB – only 1 part in a 100,000 !! Predicted by models. George Smoot and John Mather received the 2006 Nobel Prize in Physics for work on COBE.

The NASA Wilkinson Microwave Anisotropy Probe (WMAP) was launched in The image below provided much more information on the anisotropy of the early universe. Dark blue regions are where the CMB is cooler and red regions warmer than the average. The maximum variation from a 2.73 °K blackbody is only °K.

In 2009 the Planck satellite was launched to provide a more detailed view of the CMB.

Beginning of Structure in the Universe The anisotropies detected in the CMB by WMAP and Planck represent the beginning of the formation of structure in the Universe. Although these only represent deviations from a uniform density Universe at the level of one part in 100,000, these anisotropies eventually will grow into the structure (galaxies, galaxy cluster.... ) that we see today.