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Physics · Document 8

Cosmology

The expanding universe and the Z2 prediction for dark energy

1. The Expanding Universe

The universe is expanding. Galaxies are moving apart, and spacetime itself is stretching. This expansion is described by the scale factor a(t).

v = H_0 * d
Hubble's Law
H_0: Hubble constant
~70 km/s/Mpc
Scale factor a(t):
Measures the "size" of the universe
🎮 Interactive: Cosmic Expansion Timeline
Time (billions of years)Scale factor a(t)RadMatterDENow
Dark Energy Era
9.8 billion - now
Accelerating expansion takes over

2. The Friedmann Equations

Einstein's equations applied to a homogeneous, isotropic universe give the Friedmann equations governing cosmic expansion:

H^2 = (8 pi G / 3) * rho - k/a^2 + Lambda/3
First Friedmann equation
a_ddot/a = -(4 pi G / 3)(rho + 3p) + Lambda/3
Second Friedmann equation

Density Parameters

We define dimensionless density parameters relative to the critical density:

Omega_M = rho_M / rho_crit, Omega_Lambda = Lambda / (3H^2), Omega_k = -k/(aH)^2
Constraint: Omega_M + Omega_Lambda + Omega_k = 1 (exactly!)
🎮 Interactive: Cosmic Density Parameters
Matter
31.6%
Dark Energy
68.4%
Curvature
Flat
Omega_k = -0.000
Flatness puzzle: Why is Omega_k so close to 0? The Z2 framework naturally produces a flat universe with exactly Omega_M + Omega_Lambda = 1.

3. The Z2 Prediction

The Z2 framework makes a precise prediction for the cosmic composition:

🎮 Interactive: Cosmic Composition: Z2 vs Observation
Z2 PredictionUniverse composition
Dark Energy
13/19
= 68.42%
Matter
6/19
= 31.58%
Agreement: Z2 predicts 68.42%, observed is 68.5% - a match to 0.1%!
The Z2 framework predicts Omega_Lambda = 13/19 and Omega_M = 6/19.
These exact fractions emerge from counting degrees of freedom in the T^3/Z_2 compactification!

Why 13/19 and 6/19?

The numbers 13 and 19 arise from the topology of the T^3/Z_2 orbifold:

  • 19 = Total degrees of freedom from D-branes
  • 13 = Degrees of freedom contributing to vacuum energy
  • 6 = Degrees of freedom giving matter
13 + 6 = 19 (total DOF)
Hence Omega_Lambda + Omega_M = 13/19 + 6/19 = 1 (flat universe)

4. Cosmic History

EraDominant ComponentScale Factora(t) Behavior
RadiationPhotons, neutrinosrho ~ a^-4a ~ t^(1/2)
MatterDark matter, baryonsrho ~ a^-3a ~ t^(2/3)
Dark EnergyCosmological constantrho = consta ~ e^(Ht)
We entered the dark energy era about 4-5 billion years ago. The universe will expand exponentially forever (de Sitter space).

5. Observational Evidence

Type Ia Supernovae

Standard candles showing distant supernovae are fainter than expected, revealing accelerating expansion (1998 discovery, Nobel Prize 2011).

CMB Measurements

Planck satellite measures Omega_Lambda = 0.685 +/- 0.007, remarkably close to the Z2 prediction of 13/19 = 0.6842.

BAO (Baryon Acoustic Oscillations)

Sound waves frozen in the early universe provide a standard ruler, confirming dark energy domination.

Large Scale Structure

Galaxy clustering patterns consistent with Lambda-CDM model with Omega_Lambda ~ 0.7.

6. Connection to Z2 Framework

Z²
Omega_Lambda = 13/19 = 0.6842...
Dark energy fraction predicted from D-brane vacuum energy on T^3/Z_2
Z²
Omega_M = 6/19 = 0.3158...
Matter fraction from D-brane matter content
Z²
Omega_k = 0
Flat universe emerges naturally from the framework

Why Cosmology Validates Z2

  • - 13/19: Matches observed dark energy to 0.1%
  • - Flatness: No fine-tuning needed
  • - Coincidence problem: Why now? Z2 provides a structural answer
  • - Predictive: Not a fit, but a calculation from first principles

Exercises

  1. Calculate 13/19 and 6/19 as decimals. Compare to Planck 2018 values.
  2. If H_0 = 70 km/s/Mpc, what is the Hubble time t_H = 1/H_0 in years?
  3. Show that Omega_M + Omega_Lambda = 1 implies a flat universe (k = 0).
  4. When did the dark energy era begin? (Hint: when did rho_Lambda = rho_M?)
  5. Why does rho_matter ~ a^-3 but rho_radiation ~ a^-4?