Outline - April 8, 2010 Cosmological Principle - what is it, when is it valid? Space and Time as physical quantities Meaningless Cosmological Questions More implications of Hubble’s Law Using Hubble’s Law to estimate the age of the universe Big Bang predictions - motivation for the remaining chapters in the book

The Cosmological Principle On a large enough scale, the universe is both isotropic and homogeneous ISOTROPY: There is no preferred direction in space. (All directions are alike.) HOMOGENEITY: One randomly-chosen large volume of the universe will have the same physical properties (and identical physical laws) as another randomly-chosen large volume of the universe. (All places are alike.)

2-Dimensional Examples of Isotropy and Homogeneity Surface of plain white“cue” ball used for playing pool (billiards) Infinite forest of identical trees Warning: at some level all analogies fail…

Isotropic Forest

Anisotropic Forest (trails = “preferred direction”)

Four 2-Dimensional Universes Which are isotropic on a large scale (same in all directions)? Which are anisotropic on a large scale (have preferred directions)? Which are homogeneous on a large scale (all places are alike)? Which are inhomogeneous on a large scale? Which satisfy the Cosmological Principle?

Obviously isotropic and homogeneous Satisfies the Cosmological Principle

Homogeneous on a large scale (the pattern repeats over and over), but is obviously anisotropic Does not satisfy Cosmological Principle

Homogeneous on a large scale, but is actually anisotropic (look at the diagonals!) Does not satisfy the Cosmological Principle

On a large scale, this is both homogeneous and isotropic Satisfies the Cosmological Principle This is actually a computer simulation of how galaxies may have formed in the universe after the Big Bang. It matches the real, observed universe remarkably well. Yellow = lots of galaxies present, black = very few galaxies present.

How Large is “Large Enough”? Answer: 1.5 billion light years (or more!!) You need to cut out a cube of the universe that’s 1.5 billion ly X 1.5 billion ly x 1.5 billion ly to guarantee the same number of galaxies and same number of galaxy clusters that you would find in a cube of the same size, but very far away from the original cube.

Cosmological Principle, II 1.Maps of the locations of galaxies in the universe (same average number of galaxies, etc. as long as length scale is at least 1.5 billion light years) 2.Cosmic Microwave Background Radiation (CMBR) will turn out to be a strong proof of isotropy of universe Is the cosmological principle nothing more than a convenient assumption without which we would be unable to make progress in studying the entire universe? NO!! (it’s a testable hypothesis)

Unsettling Consequence of Cosmological Principle There can be NO CENTER to the universe! Where is the “center” of the SURFACE of an unmarked sphere?

Working Definition for “The Universe” The universe is that which contains and subsumes 1 all of the laws of nature and everything subject to those laws. (John Hawley & Katharine Holcomb, “Foundations of Modern Cosmology”) 1 subsume: to classify within a larger category or under a general principle The universe contains all that is physical, including space and time.

Space & Time as Physical Quantities Recall from General Relativity: Time is affected by gravity (clock on a high-gravity planet runs slower than clock high up in the atmosphere) Space is affected by gravity (“warped” by massive objects, causes light rays to bend around on curved paths, causes planets to orbit about sun on “natural” curves) Anything that is affected by the physical laws must, itself, be physical!

Meaningless Question What is the color of the sound of a trumpet?

Meaningless Cosmology Questions What happened before the Big Bang? What exists outside the universe? These questions assume attributes (“before”, “outside”) which posit space and time as being distinct from the universe. This is not acceptable in the context of “physical cosmology”. Time did not exist before the universe and space does not exist outside it!

Hubble’s Law Revisited v = H 0 d H 0 is called “Hubble’s constant” This is a text box H 0 = 71 km/s/Mpc or H 0 = 22 km/s/Mly (these are really weird units)

Hubble’s Law Revisited Hubble’s law does not say that “the universe is expanding into space”. What Hubble’s Law does say is that space itself is expanding. Implications of Hubble’s Law / expansion of space: 1.The universe was smaller and denser in the past 2.The universe was hotter in the past 3.The universe had a specific beginning

How old should the universe be? Run the “Hubble Movie” backwards: some time in the past all galaxies would have been on top of each other! if universe has been expanding at constant rate for all time, then all galaxies would have been on top of each other at time equal to 1/H 0 Distance between any two galaxy clusters at the present day: distance = speed x time (the standard formula) speed = H 0 x distance (Hubble’s Law, specifically) Substitute Hubble’s Law into distance formula: distance = H 0 x distance x time Cancel factors of distance and rearrange: time = 1/H 0

“Hubble Age” = 1/H 0 The time on the previous slide is the time it would have taken the galaxy clusters to reach their current separation, having started from being on top of each other, and is an estimate of the age of the universe known as the Hubble Age. Current best measurements of H 0 give a value of 71 km/s/Mpc, which translates into a Hubble Age of about 14 billion years, which agrees reasonably well with the estimated ages of the oldest stars in the universe.

The Cosmological Redshift A subtle point: the cosmological redshift is not a simple Doppler shift Doppler shifts result from sources of waves moving through space; cosmological redshift happens because distances between sources of waves (i.e., galaxies emitting light) increase due to space itself getting bigger as photons travel through the universe they are STRETCHED as the fabric of spacetime expands

Stretched Photon (Balloon Analogy) This has a profound consequence for the relative importance of mass and light for the total “energy budget” of the universe.

Density When we talk about “density” what we usually mean is the “mass density”: mass density = (amount of mass) / volume We can equally define an “energy density” which is just energy density = (amount of energy) / volume Note: the “energy” could be the energy of light (E = hc /  ) or the energy of matter (E = mc 2 )

Affect of Expansion on Energy Density of Mass Consider a 1-meter cube of space that contains 1 kg of hydrogen gas. The mass-energy density of the hydrogen gas is D = mc 2 / volume = (1 kg)x(3x10 8 m/s) 2 / (1 m) 3 = 9x10 16 J/m 3 Now let the cube expand so that its sides are 2 meters long, and don’t let any of the gas escape from the cube. The mass-energy density of the hydrogen gas in the bigger cube is D = mc 2 / volume = (1 kg)x(3x10 8 m/s) 2 / (2 m) 3 = 1.125x10 16 J/m 3 So, the mass-energy density has decreased by a factor of 8 = 2 3 (i.e., three factors of length). What about the energy density of light?

What happens to the energy of a photon when you increase the wavelength? The photon loses energy! Consequence: as the universe expands, the density of “radiant energy” (i.e., light) decreases faster than the density of “mass energy”. For light, the energy density decreases by FOUR factors of length, not three!! There are fewer photons per unit volume, but the energy of each individual photon has also decreased from its original value.

Matter Dominated vs. Radiation Dominated Universe At the present day, there is roughly 20,000 times more energy density in the form of mass/matter. We live at a time when the universe is “Matter Dominated”. In the far distant past there would have been a time when there was more energy density in the form of light/radiation. This would have been a time when the universe was “Radiation Dominated”.

The Big Bang Concept The universe began in an extremely hot, extremely dense state A fantastic notion, but also a truly testable hypothesis.

Big Bang Predictions The universe had a specific beginning The sky should be dark at night Initially, the universe was extremely hot, dense, and opaque The universe has evolved/changed over time The universe is expanding Cosmic objects (such as stars) should have a chemical composition that is roughly 75% hydrogen and 25% helium What evidence exists?

Darkness at Night (a.k.a Olbers’ Paradox) Consider a universe that is: infinite in extent infinite in age filled uniformly with identical stars (or galaxies) that have been shining for all time Result: the sky should ALWAYS be bright

Suppose the stars extend to infinite distances Consider stars at distance, d. What will the combined brightness of all stars at distance, d, be?

All lines of sight eventually intersect a star At a given distance, the brightness of each individual star decreases as (1/distance) 2 BUT the number of stars at that distance increases as (distance) 2 Total amount of light coming from a given distance is the number of stars at that distance times the brightness of each star at that distance, so amount of light you see coming from any particular distance is a CONSTANT value because the factors of distance cancel each other!! The sky should NEVER be dark (but we all know it’s dark at night, so at least one of our starting assumptions was WRONG)!