T3 COSMOLOGY VALIDATION

29/07/2025

T3 Cosmology Cheatsheet

1. Stellar Parallax & Distance Measurement

Definition: Parallax is the apparent shift in the position of an object when viewed from different perspectives.

How it works:

Move your head left/right while looking at your finger — the background shifts.
Astronomers do this using Earth’s orbit as the “head movement.”

Formula:

$d = \frac{b}{2\tan(p)}$ Where:

$d$ = distance to the object (in AU or meters)
$b$ = baseline (distance between viewing positions — e.g., Earth’s orbit = 2 AU)
$p$ = parallax angle (half the shift observed)

Definition: Parallax angle is half the angle formed by a nearby star’s shift against a distant background when viewed from opposite sides of Earth’s orbit.

2. Astronomical Units (AU) & Other Distances

Definition: 1 AU (Astronomical Unit) is the average distance between the Earth and the Sun, approximately 150 million kilometers.

Best used for: Solar System objects
1 AU = 1.5 × 10¹¹ m

Definition: A parsec is the distance at which 1 AU subtends an angle of 1 arcsecond.

Parsecs:

$d(\text{pc}) = \frac{1}{p(\text{″})}$

1 parsec ≈ 3.26 light-years
Used for distances to stars

Megaparsec:

Definition: 1 Megaparsec (Mpc) = 1 million parsecs

Used for galaxies

3. Redshift & Expanding Universe

Definition: Redshift is the stretching of light waves, causing them to appear more red when an object is moving away.

Redshift formula:

$z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$

$\lambda_{\text{obs}}$ = observed wavelength
$\lambda_{\text{emit}}$ = emitted/original wavelength

Recessional velocity:

$v = c \cdot z$

$c = 3 \times 10^5,\text{km/s}$

Definition: Recessional velocity is the speed at which an object (e.g. galaxy) is moving away from us due to cosmic expansion.

4. Hubble’s Law

Definition: Hubble’s Law states that the recessional velocity of a galaxy is directly proportional to its distance from Earth.

Equation:

$v = H_0 \cdot d$

$H_0 \approx 70,\text{km/s/Mpc}$

Definition: The Hubble constant (H₀) is the rate at which the Universe is expanding.

5. Estimating the Age of the Universe

Hubble time:

$t_0 \approx \frac{1}{H_0}$

Using: $H_0 = 70,\text{km/s/Mpc}$ ⇒ $t_0 \approx 14,\text{billion years}$

Definition: Hubble time is an estimate of the age of the Universe, calculated as the inverse of the Hubble constant.

6. Investigation Setups

Parallax Method (Experiment):

Measure position of a nearby object (e.g. pencil) from two distant points (baseline).
Calculate angle shift ⇒ $d = \frac{b}{2\tan(p)}$

Tips:

Longer baseline ⇒ more accuracy
Angle should be small (use tangent approx)
Error = from misalignment

Redshift + Hubble’s Law (Experiment):

Measure redshifts of distant galaxies
Use $v = c \cdot z$
Plot v vs d
Slope of line = $H_0$

Definition: A linear relationship between redshift and distance supports the theory that the Universe is expanding.

7. Must-Know Constants

Concept	Value
1 AU	1.5 × 10¹¹ m
1 pc	≈ 3.26 light-years
Speed of Light (c)	$3 \times 10^5,\text{km/s}$
Hubble Constant (H₀)	$70,\text{km/s/Mpc}$

Memorize:

1 AU = Sun–Earth
1 pc = 1 / p(″)
Redshift = fractional wavelength change
Hubble’s Law = v = H₀d
Universe age ≈ 1 / H₀