Astronomy 218O b s e r v a t i o n a l

C o s m o l o g y

Cosmological PrincipleThe central tenet of cosmology is the cosmological principle, Viewed on a sufficiently large scale, the properties of the Universe are the same for all observers.

Cosmology is generally not concerned with the study of the myriad of astronomical objects that make up the Universe, except to the extent that they tell us about the behavior of the Universe as a whole.

The cosmological principle is the ultimate flowering of the revolution started by Copernicus, we do not occupy a special place in the Universe.

In the 1980s, Kirshner, Oemler, Schechter & Schechtman took a complementary approach to the CfA redshift survey of Gellar & Huchra, the pencil-beam survey.

This provided a measure of large-scale structure that was narrow but deep. It revealed the first void (the void in Boötes) but no structure larger than 200–300 Mpc in their survey which reached >2 Gpc.

Pencil-Beam Surveys

Obviously, the cosmological principle fails at small scales. Even at the scale of 50 Mpc, the perspective of an observer in the center of a supercluster differs markedly from that of an observer in a void.

The largest structure known in the Universe, the Sloan Great Wall (~300 Mpc) tells us that the cosmological principle only applies on scales of Gpc.

Scales below ~1 Gpc are the realm of astronomy.

Largest Scales

The statement that the Universe appears the “same to all observers” is catchy, but not quantifiable.

A more precise statement is that universe is homogeneous — the same at every location — and isotropic — the same in every direction.

Redshift and pencil surveys support the description of the Universe as homogenous (every Gpc-cube block appears much like the local Gpc-cube block) and isotropic (the Gpc-cube block to our left appears much like the Gpc-cube block to our right).

The notion of the Universe that developed after the Copernican revolution was of a homogenous and isotropic Universe, infinite in space and static in time.

Isotropy & Homogeneity

The first observation that leads to modern cosmology is Olbers’ Paradox.

As pointed out by numerous astronomers over time (Thomas Digges in 1576, Johannes Kepler in 1610, Edmund Halley in 1721, …) observations contradict this infinite model of the Universe because the night sky is mostly dark.

As Heinrich Olbers elucidated most clearly in 1826, if the universe is homogeneous, isotropic, infinite, and unchanging, the entire sky would be as bright as the surface of the Sun.

Olbers’ Paradox

How does an infinite, unchanging homogeneous and isotropic universe prevent a dark sky at night?

For a single star of luminosity L at a distance r, its flux is

the result of the inverse square law for geometric dilution.

If the star has a radius, R✭, it will subtend a solid angle in steradians of

the fraction of a sphere of radius r covered by a circle of radius R✭.

The surface brightness of the star is independent of r.

Surface Brightness

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Bright SkyIf the surface brightness Σ✭ is independent of distance, than distant stars contribute as much as nearby stars.

If we assume a uniform density of stars in the Universe, n✭, then at a radius r, a thin shell of thickness dr contains dN✭ = n✭ 4π r2 dr stars.

The fraction of the shell’s area covered by stars is

Summing over all shells with radius less than r tells the fraction of the sky covered by stars to this radius,

f → 1, making the night sky bright, as r → rOlb ≈ 1/n✭ π R2✭

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Finite UniverseSince the sky is largely dark at night, one of the assumed characteristics of the Universe must be wrong. It was possible that the errant assumption was homogeneity. In Olbers’ day, distances to stars were unknown (Bessel first measured parallax in 1838), so n✭ and/or R✭ could grow much smaller with distance. Infinite size could also be the errant assumption. If the radius of the Universe (or at least the distribution of stars), r0, is smaller than rOlb, then f ≈ n✭ π R2✭ r0 < 1. The average surface brightness is Σ✭ f ≈ n✭ L r0 / 4π. Lord Kelvin in 1901 removed the assumption of infinite age. Since the speed of light had been measured as early as 1677 (by Römer in 1676), stars of age t0 or in a Universe of age t0 can only be seen to a distance r0 = ct0.

Kelvin’s assertion was ultimately proven by Hubble’s 1929 observation of Universal Recession.

From Hubble’s law, ʋ = H0 d, we can estimate the age of the Universe as the time at which all galaxies are at zero distance, t0 = d/ʋ = d / (H0d) = 1/H0

For H0 = 70 km s−1 Mpc −1, t0 = 14.0 Gyr. Conveniently, the age of the Universe is older than the oldest star.

The light from a star further than the horizon distance, r0 = c/H0 ~ 4300 Mpc, is unable to reach us.

The Age of the Universe

The Dark SkyBy integrating the observed luminosity function for galaxies, we determined the total luminosity density.

≈ 2 × 108 L☉ Mpc−3 ≈ 10 W AU−3

Since the light of most galaxies comes from stars, this luminosity density is a sum over all stars,

≈ n✭ L

The average surface brightness of the sky is

≈ 8 × 108 L☉ Mpc−2 ster−1 ≈ 3 × 10−8 W m−2 ster−1

In contrast, Σ✭ ≈ 2 × 107 W m–2 ster–1 so Σsky ~ 10–15 Σ✭ telling what we already know, the night sky is black.

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Preserving homogeneity and isotropy in the face of Universal Recession requires the entire Universe to be expanding, making Hubble’s law the same everywhere.

Any distance in the Universe can be written in terms of a scale factor, a(t), and the current distance, r0 = r(t0).

r(t) = a(t) r0 The distance between any two point increases at a rate

H(t) is the Hubble parameter, H0 = H(t0) is Hubble’s constant

The Expanding Universe

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The idea of expansion that does not originate at a central point is contrary to our common experience.

The analogy of a balloon with coins stuck to it (or ants living on it) can be helpful. As the balloon expands, the coins all move farther and farther apart, but no coin or other point on the surface of the balloon is the “center” of expansion.

Center-Less Universe

Just as distances between locations grow, so to do lengths increase. Thus the same ballon analogy can be used to explain the cosmological redshift, as the wavelengths also grow according to the scale factor a(t).

Cosmological redshift is not due to the motion of a galaxy through space, but due to the motion (stretching or contraction) of space.

Stretching Light

One Quarter HeliumWe described the composition of the Sun in terms of the hydrogen mass fraction, X = 0.71, the helium mass fraction, Y = 0.27 and the mass fraction of metals, Z = 0.02.

Further, we discussed that stars in the Galaxy can have much smaller metallicities Z < 10−4.

If you look however at the helium mass fraction, Y, it approaches 0.25.

In 1946, this led Alpher, Bethe & Gamow, to propose that the Universe started out in a “Big Bang”, hot enough to fuse protons and neutrons into helium.

In 1964, Arno Penzias and Robert Wilson at Bell Labs were trying to get rid of the last bit of “noise” in the radio antenna used to communicate with the new Telstar satellite. They found that this “noise” was identical in all directions in the sky, and was not time dependent. The signal peaked at λ = 1.06 mm, in the microwave band. In consultation with Robert Dicke and James Peebles, this was found to be the anticipated remnant of the “Big Bang”.

Cosmic Microwave Background

Penzias & Wilson found the CMB spectrum was well fit by a 3K blackbody.

Later measurements have refined the temperature (TCOBE = 2.725 ± 0.001 K) but do not find significant departures from a pure blackbody.

The current energy density of the CMB is

= 4.17 × 10–14 J m–3

Using the average photon energy, ε = 2.7 kBT0 = .634 meV, the number density is

Blackbody Background

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A map of the microwave sky shows a distinct dipole pattern, due not to any property of the radiation itself, but to our peculiar motion.

A moving observer detects a different temperature as function of the angle between the line of sight and the direction of motion.

Overall, this dipole anisotropy indicates a 371 km s−1 motion in the direction of Leo (α,δ = 11.2h, −7°).

Compensating for the Sun’s orbit in the Galaxy and the Milky Way’s motion in the local group, this implies ʋ = 627 km s−1 in the direction of Hydra (α, δ = 11.1h, −27°).

Dipole Anisotropy

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Because the blackbody radiation is cosmological, we know that the expansion of space has stretched its wavelength.

A region of volume expands with the Universe, so V(t) ∝ a(t)3. A photon gas in this volume obeys the first law of thermodynamics dQ = dE +PdV.

In a homogeneous universe there is no heat flow, this dQ = 0, thus

For the photon gas

E(t) = u(t)V(t) = (4σSB/c)T(t)4 V(t)

P(t) = u(t) = (4σSB/sc)T(t)4

therefore

Expanding Blackbody

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With V(t) ∝ a(t)3 this can be re-written as

or

Thus T(t) ∝ a(t)−1, which implies that the mean photon energy ε(t) ∝ a(t)−1

and λ(t) ∝ a(t).

Temperature over Time

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Cosmological ObservationsAny model of the Universe must explain:

1) Large scale structures shows that the Universe is homogeneous & isotropic, as long as you ignore scales less than ~ a few hundred Mpc. 2) The dark sky, indicating a Universe that is finite. 3) Hubble’s law, indicating that the Universe has been expanding for ~ 14 Gyr. 4) The cosmic microwave background, which tells us the Universe was hot in the past and is cooling as it expands. 5) The primordial helium abundance, indicating that the Universe was once millions of degrees. Until the past decade, these 5 points summed up the available observations.

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