Hubble Expansion Lecture

HUBBLE EXPANSION LECTURE

Eric V. Linder

1 OVERVIEW

Ch 1-3 outline how describe universe:

• Homogeneity, Isotropy --> Metric
• Equation of State --> Components
• Cosmological Tests ($\Omega$) --> Recipe

Ch 4 discusses implications - History

Now dive into detail; breadth and stability (robustness) of Big Bang model.

Three Cornerstones: Hubble Expansion, Prim Nucleosynthesis, CMB

Great successes in detailed foundation; not wallpaper of galaxy formation, CDM, etc.; media reports of doom.

1.2 Historical Perspective

Hubble Expansion cornerstone because:

• Durability - first indications predate Friedmann, Einstein
• Simplicity - arises naturally from atomic physics and special relativity; direct restatement of Principle of Equivalence
• Evidence - abundance of observations, almost million

Remember! NOT an input to theory - surprise

Required by Homogeneity and Isotropy - kinematic, not even GR

Plan of Study: Observational Evidence; Theoretical Background; Comparison, Interpretation, and Consequences

2 OBSERVATIONS

Atmosphere: birth of quantum theory, atomic model, spectroscopy (Comte and parallax)

• Application to Astrophysics natural
• Principle of Equivalence

Result: photons received from distant sources exhibited shifts in wavelength, not just in a single line transition but across the whole spectrum.

Historical Survey of measurements (Table 5.1)

2.2 Observed Properties

Semicoincidence that first spectral shift detected was negative

Vast majority of showed increase in wavelength (to red in optical)

==> Redshift

Real effect? Atomic physics becoming well understood from lab experiments, solar physics (why shine). Hypothesize Equivalence Principle holds

==> Astronomical deviations from either:

• Source environment (physical condition) or
• Difference in reference frame between emitter and observer

Four Properties held the key to interpretation:

• Positivity
• Frequency Independence - $\Delta\lambda/\lambda\sim \lambda^0$ ($10^{12}$ in wavelength)
• Linearity - $\Delta\lambda/\lambda\sim r$
• Isotropy

3 THEORY

Why any shift at all - physical mechanism for changing energy of propagating photon

Equivalence Principle says reference frame or environment. Four cases (general unifying expression in GR):

• Motion of source relative to observer (reference frame)
• Time dependent gravitational effects (frame/environment)
• Static gravitational effects (frame/environment)
• Interaction - e.g. scattering, absorption (environment)

First: Doppler effect - acoustics, waves, SR

Second: Expansion of universe

Third: Black Hole, Pound-Rebka (leave to comparison section)

Fourth: Particle Physics (leave to comparison section)

3.1 Doppler Shift (as \S5.2.1)

Figure 5.3: Doppler Effect of Moving Source

Example 5.1: Story of Battling Armies (CNN)

$z\equiv\dll=\hat n\cdot\vec v/v_s=v_r/v_s$

Radial velocity $v_r=\hat n\cdot \vec v$. (Signal velocity $v_s=1$)

3.2 Expansion Shift (as \S5.2.2)

A few years after Einstein developed GR but before Friedmann came up with the equations of motion for a homogeneous, isotropic universe, Weyl realized that homogeneity and isotropy alone were sufficient to predict numerous effect, including Expansion. These depend only on the background geometry, i.e. the metric, and don't need the equations of motion or even the specific gravitation theory in many cases. In physics, such effects independent of the equations of motion, i.e. not relying on forces, are called Kinematic (the force dependent realm is Dynamics). Given only the RW metric one can calculate the expansion and redshift, to some extent the distance-redshift relations, and many properties of electromagnetic propagation, e.g. the surface brightness law. It is generally accepted that in fact Hubble was aware of Weyl's analysis predicting the redshift (and in fact giving Hubble's Law!) based on the Cosmological Principle.

RW metric quantities: k, a

Scaling $\lambda\sim a$ ==> Redshift

$1+z=a(t_o)/a(t_e)$

COMMENTS

I started the lecture by explaining how the book was divided into three parts and they had just finished the first part, Chapters 1-4. I gave a quick summary of what they should have accomplished. Then I said that the second part, Chapters 5-7, was different in that they were closer examinations of the foundation of the Big Bang model and would teach them not only our current understanding of cosmology but how to interpret new observations and media reports of ``death of Big Bang theory''.

I put strong emphasis on the naturalness of expansion and the wealth of observations. To treat the theory I intended to start with an intuitive approach and leave the equations aside for later lectures or for reading from the book. So I taught the Doppler shift in an elaborated version of the book's ``battling armies'' story. Time ran out as I illustrated how the diagrams for the wave fronts of a moving source and for the retreating/advancing front lines for the general's messengers were the same. Thus I ended just before the radial velocity expression (5.2).

Similarly, I like introducing heuristically the expansion shift by talking about it kinematically and how the scale factor $a$ in the RW metric tells you that photon wavelengths change and hence the redshift is a direct consequence of expansion which is a direct consequence of the Cosmological Principle - no Friedmann or Einstein equations needed. I also sometimes depart from the order of the book by next comparing theory to the four properties of the observations, i.e. doing a quick overview of \S5.4 in such a way as to convince the students that expansion was the proper interpretation.