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

Administrative & Homework Updates

• 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.

Early-Universe Overview

• 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 & Antimatter

• 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.

Mass–Energy Equivalence Refresher

• 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) $= 1\,\text{J s}^{-1}$

Everyday power example

• A 100 W light-bulb consumes 100 J every second.

Example 1 – Sun’s Power vs U.S. Consumption

1. Solar power output: $P_\odot = 3.88\times10^{26}\,\text{W}$
2. 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).
3. Energy emitted by Sun in 1 s:
   $E<em>{1\,s} = P</em>\odot(1\,\text{s}) = 3.88\times10^{26}\,\text{J}$
4. How many U.S.-years in one solar second?
   $N = \frac{E<em>{1\,s}}{E</em>{US}} = \frac{3.88\times10^{26}}{1\times10^{20}} \approx 3.9\times10^{6}$

• 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.

Example 2 – Converting Mass to Match U.S. Yearly Energy

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.

Big-Bang Nucleosynthesis (BBN)

• 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).

Beta Decay & Particle Conversions

• 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.

Black-Body Radiation Fundamentals

• 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 $T<em>{!F} = 1.8(T</em>{!K} - 273.15) + 32$).

Recombination & Cosmic Transparency

• 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 & Photon Stretching

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

Cosmic Microwave Background (CMB)

• 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.

Ethical, Philosophical & Practical Implications

• 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.

Looking Ahead

• 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.