Early Universe, Mass–Energy, and the Cosmic Microwave Background

Group assignment notice: Students must inform the instructor of preferred partners before the end of the day; otherwise partners will be assigned.
Homework update:
- Added direct link to the required podcast (≈ 50 min) on iHeartRadio; also free on Spotify and Apple Podcasts.
- Registration is normally not required, but if prompted it is free.
- Recommendation: listen 1–2× (during a walk, gym session, driving, etc.) to ensure comprehension.

Universe began extremely hot and underwent rapid inflation.
Tiny energy fluctuations during inflation produced the clumpy large-scale structure we observe today.
During the electroweak era:
- Energy density high enough for spontaneous appearance of elementary particles.
- Immediate reconversion to energy was common, producing an almost transient particle soup.

Pair production mechanism:
- Two energetic \gamma-ray photons collide → create a particle–antiparticle pair.
- Example: \gamma + \gamma \rightarrow e^- + e^+ (electron + positron).
Antimatter facts:
- Real, routinely created in modern particle accelerators.
- Antiparticles share identical mass & behavior but opposite charge (e.g., electron e^- vs. positron e^+).
Annihilation:
- Opposite charges attract; matter + antimatter collision reconverts mass to energy (mutual annihilation).
Cosmic asymmetry:
- Matter exceeded antimatter by ~1 particle per 10^9, reason still not fully understood.
- Result: present-day universe is matter-dominated, antimatter scarce.

Einstein’s Special Relativity: E = mc^2
- E = energy (J)
- m = mass (kg)
- c = speed of light \approx 3.00\times10^8\,\text{m s}^{-1}
Units:
- Energy: joule (J)
- Power: watt (W) = 1\,\text{J s}^{-1}

Solar power output: P_\odot = 3.88\times10^{26}\,\text{W}
U.S. annual energy use (approx.): E_{US} \approx 1\times10^{20}\,\text{J yr}^{-1} (some slides mention 10^{22}; instructor used 10^{20} in worked solution).
Energy emitted by Sun in 1 s:
E{1\,s} = P\odot(1\,\text{s}) = 3.88\times10^{26}\,\text{J}
How many U.S.-years in one solar second?
N = \frac{E{1\,s}}{E{US}} = \frac{3.88\times10^{26}}{1\times10^{20}} \approx 3.9\times10^{6}

Interpretation: One second of the Sun’s output equals ~3.9 million years of U.S. energy demand.
Ethical/practical takeaway: enormous untapped solar potential vs current fossil-fuel reliance.

Goal: Mass m such that E = mc^2 = 1\times10^{20}\,\text{J}.

m = \frac{E}{c^2} = \frac{1\times10^{20}\,\text{J}}{(3.00\times10^8\,\text{m s}^{-1})^2} \approx 1.1\times10^{3}\,\text{kg}

1.1\times10^{3}\,\text{kg} \approx 2.5\times10^{3}\,\text{lb} (≈ weight of a compact car).
Thought experiment: if we could convert an entire small car’s mass to pure energy with 100 % efficiency, it would power the U.S. for a year.

Time frame: \sim10^{-4}\,\text{s} to a few minutes after Big Bang.
Universe cooled below threshold for fresh proton/neutron creation; existing baryons locked in.
Neutron–proton ratio: n/p \approx 1/7 (neutrons slightly heavier → fewer produced).
Fusion chain:
1. p + n \rightarrow {}^2\text{H} (deuterium)
2. Deuterium + n/p collisions → ^3\text{H} or ^3\text{He}
3. Rapid formation of ^4\text{He} (very stable).
Predicted primordial abundances (by mass):
- Hydrogen ≈ 74 %
- Helium-4 ≈ 24 %
- “Metals” (all heavier elements) ≈ 2 %
Observations match predictions → strong evidence for Big-Bang model.
Heavier elements (>He) arise later via stellar nucleosynthesis (future lecture).

Isolated neutron beta decays:
n \rightarrow p + e^- + \bar\nu_e (half-life ≈ 15 min).
At very high densities/energies, reverse reactions convert protons ↔ neutrons.
Once temperature dropped, neutrons could no longer be recreated rapidly; leftover neutrons became locked in helium.

A black body absorbs all incident radiation and re-emits according to its temperature.
Key properties:
1. Emits at every wavelength.
2. Hotter body → higher intensity at all wavelengths.
3. Peak wavelength \lambda_{\text{max}} shifts shorter (bluer) for hotter temperatures (Wien’s Law).
Graph interpretation:
- 5 000 K curve peaks in visible (~green) and dominates intensity.
- 2 000 K curve peaks in IR; visible output appears deep red.
- Sun ≈ 6 000 K → yellow-white appearance.
Temperatures in Kelvins; 1 K ≈ 1.8 °F (exact conversion T{!F} = 1.8(T{!K} - 273.15) + 32).

Early universe: fully ionized plasma, teeming with free electrons → strong photon scattering → universe opaque.
As expansion cooled gas to ~3 000 K (≈ 380 000 yr post-Big Bang), electrons combined with nuclei ⇒ neutral atoms.
Event known as recombination (or photon decoupling) made universe largely transparent.
Analogy: fog clearing once water droplets settle; photons can now travel freely.

Scale factor (a) quantifies cosmic expansion; by convention a_{today}=1.
Photon wavelength stretches with expansion:
\lambda{obs} = \lambda{emit}\,\frac{a{today}}{a{emit}}
Example: Emitted \lambda{emit}=600\,\text{nm} when a{emit}=0.5 ⇒ \lambda_{obs}=1\,200\,\text{nm} (now in IR).

After ~13.8 Gyr of stretching, original Big-Bang glow now lies in microwave regime (peak ~1 mm, T \approx 2.725\,\text{K}).
Discovered accidentally (1965) by Arno Penzias & Robert Wilson at Bell Labs while troubleshooting antenna “noise”.
Simultaneously predicted by Princeton group (Dicke, Peebles, Roll, Wilkinson) as leftover thermal radiation.
CMB provides one of the strongest empirical pillars supporting Big-Bang cosmology.

Immense solar output vs human energy needs highlights sustainability opportunities and current under-utilization.
Matter–energy interconversion underscores deep unity of physics and challenges simplistic “mass can’t be destroyed” narratives taught in early schooling.
Discovery of CMB via serendipitous engineering work shows value of cross-disciplinary collaboration and open scientific communication.

Upcoming lectures: detailed CMB mapping missions (COBE, WMAP, Planck), formation of heavier elements inside stars, and deeper exploration of cosmological parameters.
Reminder: final-project partner preferences due via email ASAP.