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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 inthe 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 ofthe universe will have the same physical properties (andidentical physical laws) as another randomly-chosenlarge 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

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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 preferreddirections)?

• Which are homogeneous on a large scale (all places arealike)?

• Which are inhomogeneous on a large scale?

• Which satisfy the Cosmological Principle?

Obviously isotropic and homogeneous

Satisfies the Cosmological Principle

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Homogeneous on a large scale (the pattern repeatsover and over), but is obviously anisotropic

Does not satisfy Cosmological Principle

Homogeneous on a large scale, but isactually anisotropic (look at the diagonals!)

Does not satisfy the Cosmological Principle

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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 inthe 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 1.5billion ly x 1.5 billion ly to guarantee the same number of galaxies andsame 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 averagenumber of galaxies, etc. as long as length scale is at least 1.5billion light years)

2. Cosmic Microwave Background Radiation (CMBR) will turn out tobe a strong proof of isotropy of universe

Is the cosmological principle nothing more than a convenientassumption without which we would be unable to makeprogress in studying the entire universe?

NO!! (it’s a testable hypothesis)

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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 subsumes1 all of the lawsof nature and everything subject to those laws.

(John Hawley & Katharine Holcomb, “Foundations of Modern Cosmology”)

1subsume: 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 runsslower than clock high up in the atmosphere)

Space is affected by gravity (“warped” by massive objects, causeslight rays to bend around on curved paths, causes planets to orbitabout 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?

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Meaningless Cosmology Questions

• What happened before the Big Bang?

• What exists outside the universe?

These questions assume attributes (“before”, “outside”) which positspace 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 Revisitedv = H0 d

H0 is called “Hubble’s constant”

This is a textbox

H0 = 71 km/s/Mpc or H0 = 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 galaxieswould have been on top of each other at time equal to 1/H0

Distance between any two galaxy clusters at the present day:

distance = speed x time (the standard formula)

speed = H0 x distance (Hubble’s Law, specifically)

Substitute Hubble’s Law into distance formula:

distance = H0 x distance x time

Cancel factors of distance and rearrange:

time = 1/H0

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“Hubble Age” = 1/H0

The time on the previous slide is the time it would have taken thegalaxy clusters to reach their current separation, having startedfrom being on top of each other, and is an estimate of the age ofthe universe known as the Hubble Age.

Current best measurements of H0 give a value of 71 km/s/Mpc, whichtranslates into a Hubble Age of about 14 billion years, which agreesreasonably well with the estimated ages of the oldest stars in theuniverse.

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 ofwaves (i.e., galaxies emitting light) increase due to space itselfgetting bigger

• as photons travel through the universe they are STRETCHED as thefabric of spacetime expands

Stretched Photon (Balloon Analogy)

This has a profound consequence for the relative importance ofmass 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 / λ)or the energy of matter (E = mc2)

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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 = mc2 / volume = (1 kg)x(3x108 m/s)2 / (1 m)3 = 9x1016 J/m3

Now let the cube expand so that its sides are 2 meters long, and don’tlet any of the gas escape from the cube. The mass-energy density of thehydrogen gas in the bigger cube is

D = mc2 / volume = (1 kg)x(3x108 m/s)2 / (2 m)3 = 1.125x1016 J/m3

So, the mass-energy density has decreased by a factor of 8 = 23 (i.e.,three factors of length).

What about the energy density of light?

What happens to the energy of a photon when you increase thewavelength?

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 individualphoton 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 formof 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 energydensity 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.

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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 compositionthat 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 beenshining for all time

Result: the sky should ALWAYS be bright

Suppose the stars extendto infinite distances

Consider stars atdistance, d. What willthe combinedbrightness of all starsat distance, d, be?

All lines of sight eventuallyintersect a star

At a given distance, the brightnessof each individual star decreases

as (1/distance)2 BUT the number ofstars at that distance increases as

(distance)2

Total amount of light coming froma given distance is the number of

stars at that distance times thebrightness of each star at that

distance, so amount of light yousee coming from any particulardistance is a CONSTANT valuebecause the factors of distance

cancel each other!!

The sky should NEVER be dark (but we all know it’s dark at night, so atleast one of our starting assumptions was WRONG)!