Cosmology: The Expanding Universe, Big Bang & Dark Sector

Cosmology has become a precision science. Satellites — COBE, WMAP and Planck — have mapped the $2.7\ \text{K}$ microwave background across the whole sky and measured its tiny temperature ripples (one part in a hundred thousand), the seeds from which galaxies later grew. From those maps the composition $5\%/27\%/68\%$ and the age $13.8\ \text{Gyr}$ are pinned down to a per cent or two. Yet the two biggest ingredients remain mysteries: laboratory experiments deep underground hunt for dark-matter particles, while surveys of millions of galaxies probe whether dark energy is truly the unchanging energy of empty space or something that evolves. There is even a live puzzle called the "Hubble tension" — the value of $H_0$ measured from nearby galaxies ($\approx 73$) disagrees slightly but stubbornly with the value inferred from the microwave background ($\approx 67$), and no one yet knows whether it signals new physics or a hidden error. The expanding universe Hubble sketched with a single straight line in 1929 is still being redrawn. To go further — from the algebraic Hubble law to the full curved-spacetime account of an expanding cosmos — the next step is general relativity, named in the seam below.

Start with what we actually see. Point a telescope with a spectrograph at a distant galaxy and spread its light into a spectrum. The dark absorption lines of hydrogen, calcium and other elements are there — the same fingerprints we measure in a laboratory flame — but every one of them sits at a longer wavelength than it should. The whole pattern is shifted toward the red end of the spectrum. This is a redshift, and Hubble's revelation was that nearly every galaxy shows one, and that the shift grows with distance. The naive reading is that galaxies are simply flying away from us through space, like shrapnel from a bomb. That picture is wrong in an important way, and getting it right is the heart of cosmology. The galaxies are (mostly) not moving through space at all. Rather, space itself is expanding, carrying the galaxies along with it and stretching the wavelength of any light in transit. A photon emitted long ago as blue light rides an expanding fabric; by the time it reaches us its wavelength has been stretched into the red. The redshift is therefore not a Doppler shift of an object rushing past — it is a cosmological redshift, the imprint of how much the universe has grown while the light was in flight. The essential consequence is that there is no centre. Picture a loaf of raisin bread rising in the oven. As the dough swells, every raisin moves away from every other raisin, and any raisin you choose to sit on sees all the others receding — the more distant ones faster, because there is more expanding dough between you and them. No raisin is the centre; the expansion happens everywhere at once. Our galaxy is not special: an astronomer in any galaxy sees exactly the same law of recession. The Big Bang did not happen at a point in space that we could fly back to; it happened everywhere, because everywhere was compressed together.

Hubble's measurement can be stated in one line. For every galaxy far enough away that the general expansion dominates its own local wandering, the recession speed $v$ is directly proportional to its distance $d$:

where: $v$ is the recession speed (km/s), $d$ is the galaxy's distance (megaparsecs, Mpc, where $1\ \text{Mpc} = 3.086\times10^{22}\ \text{m} \approx 3.26$ million light-years), and $H_0$ is the Hubble constant, the slope of the line, with the currently accepted value $H_0 \approx 70\ \text{km/s per Mpc}$. The physical reading is that the constant of proportionality is the same for every galaxy in the sky — a straight line through the origin on a plot of speed against distance. A galaxy at $d = 100\ \text{Mpc}$ recedes at $v = 70 \times 100 = 7000\ \text{km/s}$; one twice as far recedes twice as fast. The subscript zero on $H_0$ is a reminder that it is the present-day value of a quantity that has changed over cosmic history — the expansion rate was different in the past. This linear law is exactly what a uniform expansion must produce, and it is what makes the "no centre" claim precise rather than poetic. If the whole fabric of space swells by the same factor everywhere, then a galaxy twice as far has twice as much stretching fabric between us and it, and so recedes twice as fast. We will show in the second derivation below that an observer in any galaxy measures the very same law with the very same $H_0$ — proof that our position is in no way central.

We measure a galaxy's redshift, not its speed, so we need the link between them. The redshift $z$ is defined as the fractional change in wavelength — how much longer the received wavelength $\lambda_{\text{obs}}$ is than the emitted (rest) wavelength $\lambda_0$:

where: $\lambda_0$ is the wavelength the line would have in a laboratory, $\lambda_{\text{obs}}$ is what we receive, $z$ is dimensionless, $v$ is the recession speed and $c = 3.0\times10^8\ \text{m/s}$. The physical reading: for nearby galaxies (small $v$) the redshift is just the recession speed as a fraction of light speed, so a galaxy fleeing at $7000\ \text{km/s}$ has $z = 7000 / 300{,}000 = 0.023$ — every wavelength arrives $2.3\%$ longer than emitted. Deeper still, $z$ measures how much space has grown since the light was emitted: at $z = 1$ the universe has doubled in size since that photon set out. The approximation $z \approx v/c$ is a first step only; once $v$ approaches an appreciable fraction of $c$ (roughly $z \gtrsim 0.1$) the simple proportionality breaks down and the full relativistic relation is needed — the exact treatment of redshift in a curved, expanding spacetime belongs to general relativity and is named at the end, not derived here.

Hubble's law hides a clock. If every galaxy has been receding at a steady speed $v$ since the beginning and now sits at distance $d$, then the time it took to get there is simply distance over speed, $t = d/v$. But Hubble's law says $v = H_0 d$, so the distance cancels and every galaxy gives the same time:

where: $1/H_0$ is the Hubble time, the reciprocal of the Hubble constant. The physical reading is striking: because the distance cancels, all galaxies point back to a single moment of common origin — a beginning. Putting in $H_0 = 70\ \text{km/s per Mpc}$ gives (worked in full in the first derivation) about $14$ billion years. This is an estimate: it assumes a constant expansion rate, whereas gravity slowed the early expansion and dark energy is speeding up the recent one. The two corrections happen to nearly cancel, so the honest Hubble-time estimate of $14\ \text{Gyr}$ lands remarkably close to the precisely measured age of $13.8\ \text{Gyr}$. The exact expansion history follows from the Friedmann equations of general relativity — differential equations solved with calculus that lie in the university seam named at the end; here we use only the algebraic Hubble law.

Rewind the expansion and everything grows hotter and denser. Squeeze all of today's galaxies back together and the universe becomes a searing, uniform plasma; push far enough and ordinary structures cannot survive at all. That hot, dense early state, expanding and cooling ever since, is the Big Bang model. It is emphatically not an explosion of matter into a pre-existing empty space — it is the expansion of space, happening everywhere, that we have already met. Three independent lines of evidence, using entirely different physics, converge on it.

Proof one — the redshift–distance law itself. The universal recession, $v = H_0 d$, is exactly what a uniformly expanding universe predicts, and running it backwards demands a beginning. This is the evidence Hubble supplied.

Proof two — the cosmic microwave background. If the universe began hot, its early radiation should still be here, stretched by the expansion into long wavelengths and cooled to just a few degrees above absolute zero. In 1965 Penzias and Wilson found exactly this: a microwave hiss coming uniformly from every direction, a near-perfect thermal (black-body) glow at $T = 2.7\ \text{K}$. Wien's law places the peak of a $2.7\ \text{K}$ black body at $\lambda_{\text{max}} = b/T = (2.9\times10^{-3})/2.7 = 1.1\times10^{-3}\ \text{m}$, about $1.1\ \text{mm}$ — squarely in the microwave band, which is why it is called the cosmic microwave background. It is the oldest light in existence, released when the cooling universe first turned transparent.

Proof three — the light elements. In its first few minutes the universe was hot enough for nuclear fusion, and the physics of that furnace predicts a definite recipe: roughly $75\%$ hydrogen and $25\%$ helium by mass, with a trace of deuterium and lithium and essentially nothing heavier. This is precisely the composition we measure in the most ancient, unprocessed gas. The heavier elements — carbon, oxygen, iron — came later, forged inside stars (the stellar-physics article, this article's prerequisite). The cosmic quarter-helium abundance is a fossil of the Big Bang's first three minutes.

Weigh the universe and a disturbing accounting emerges. The luminous matter — stars, gas, planets, everything built of protons and neutrons — makes up only about $5\%$ of the total. The rest is dark, and it comes in two utterly different kinds.

Dark matter, about $27\%$ of the total, betrays itself only through gravity. Spiral galaxies rotate far too fast at their edges: the visible stars supply nowhere near enough gravity to hold a galaxy spinning that quickly together — it should fly apart. The stars orbit as if embedded in a vast invisible halo of extra mass, several times the visible amount. The same excess gravity bends light (gravitational lensing) and shapes how galaxies cluster. Dark matter is not ordinary matter that is merely unlit; its total is far more than all the gas and dead stars combined, and it neither emits nor absorbs light. What it is — some new particle beyond the Standard Model — remains one of physics' great open questions.

Dark energy, about $68\%$ of the total, is stranger still. In 1998 two teams measuring distant supernovae found that the expansion is not slowing under gravity as expected but accelerating. Something with negative pressure — a property of empty space itself, in the simplest model — is pushing the expansion faster. We call it dark energy, and it now dominates the cosmic energy budget. Adding the three ingredients, $5\% + 27\% + 68\% = 100\%$: ordinary matter is a rounding error, and $95\%$ of the universe is a dark sector we can measure but not yet identify.

Board level — recession speed, redshift and stretched wavelength at 100 Mpc (IB-HL and A-Level pattern)

A galaxy lies at a distance $d = 100\ \text{Mpc}$. Take $H_0 = 70\ \text{km/s per Mpc}$ and $c = 3.0\times10^{5}\ \text{km/s}$. Find (a) its recession speed, (b) its redshift $z$, and (c) the observed wavelength of a hydrogen line whose rest wavelength is $500.0\ \text{nm}$.

Step 1. Recession speed from Hubble's law: $v = H_0 d = 70 \times 100 = 7000\ \text{km/s}$. That is $7.0\times10^{6}\ \text{m/s}$, about $2.3\%$ of the speed of light — small enough that the simple formulae apply.

Step 2. Redshift for small $v$: $z = v/c = 7000 / (3.0\times10^{5}) = 0.023$. Every wavelength arrives $2.3\%$ longer than it left.

Step 3. Observed wavelength: $\lambda_{\text{obs}} = \lambda_0(1 + z) = 500.0 \times 1.0233 = 511.7\ \text{nm}$, a shift of $\Delta\lambda = 11.7\ \text{nm}$ toward the red. Answer: $v = 7000\ \text{km/s}$, $z = 0.023$, and the $500.0\ \text{nm}$ line is seen at $511.7\ \text{nm}$.

Competitive level — from a measured redshift back to distance and lookback time (IB-HL and A-Level pattern)

A quasar's spectral lines are redshifted by $z = 0.050$. Using $H_0 = 70\ \text{km/s per Mpc}$, $1\ \text{Mpc} = 3.26\times10^{6}\ \text{ly}$ and $c = 3.0\times10^{5}\ \text{km/s}$, estimate (a) its recession speed, (b) its distance in Mpc and in light-years, and (c) roughly how long ago the light we now see set out.

Step 1. Speed from the redshift (small-$z$ regime): $v = zc = 0.050 \times (3.0\times10^{5}) = 15{,}000\ \text{km/s}$. This is $5\%$ of $c$ — near the upper edge where $z \approx v/c$ is still trustworthy, so a quick check that we are allowed to use it matters.

Step 2. Distance from Hubble's law, inverted: $d = v/H_0 = 15{,}000 / 70 = 214\ \text{Mpc}$. Convert to light-years: $d = 214 \times 3.26\times10^{6} = 7.0\times10^{8}\ \text{ly}$, about $700$ million light-years.

Step 3. Lookback time — how long the light has been travelling — is roughly distance over light speed: $t \approx d/c = (7.0\times10^{8}\ \text{ly})/c = 7.0\times10^{8}\ \text{yr}$ (a light-year is, by definition, the distance light covers in a year, so distance in light-years equals travel time in years). Answer: $v \approx 15{,}000\ \text{km/s}$, $d \approx 214\ \text{Mpc} \approx 7.0\times10^{8}\ \text{ly}$, and the light left about $700$ million years ago. Timing note: under two minutes. The chain is $z \to v = zc \to d = v/H_0 \to$ light-years, and the lookback time is free once the distance is in light-years (they are numerically equal). The one trap is unit hygiene — keep $c$ in km/s to match $H_0$'s km/s, and only convert to light-years at the end. Always glance at $z$ first: if $z$ were near or above $1$ you would have to abandon $z = v/c$ and flag the relativistic regime instead of blindly multiplying.

Claiming cosmology is on your board when it is not. Cosmology — Hubble's law, the Big Bang, the dark sector — is enrichment beyond the Indian core boards. It appears in IB Physics HL (HL-only material on galaxies and the expanding universe; formerly the Astrophysics option) and in some A-Level specifications, and nowhere in CBSE, ICSE, JEE (Main or Advanced), NEET or AP Physics 2. Do not cite it as a CBSE/JEE/NEET topic; conversely, if you sit IB-HL or A-Level, know that $v = H_0 d$, the Hubble-time age and the CMB are fair game.

Saying the Big Bang exploded from a point in space. It expanded space itself, everywhere at once, with no centre and no edge. An examiner rewards "expansion of space," not "explosion in space."

Confusing cosmological redshift with a Doppler shift through space. The wavelength stretches because space stretches; $z \approx v/c$ is only the small-$z$ coincidence, and it fails once $z \gtrsim 0.1$. State the mechanism, not just the formula.

Mishandling the units of $H_0$. $H_0 = 70\ \text{km/s per Mpc}$ must be converted to $\text{s}^{-1}$ ($2.27\times10^{-18}$) before you invert it for an age — mixing km, s and Mpc is the commonest slip in the Hubble-time calculation.

Forgetting the age is an estimate. $1/H_0 = 14\ \text{Gyr}$ assumes constant expansion; the measured $13.8\ \text{Gyr}$ comes from a full model. Quote $1/H_0$ as an approximate age and say why.

Calling dark matter "matter we just cannot see yet." It is not ordinary unlit matter — its total dwarfs all the gas and dead stars, and Big-Bang element abundances forbid it from being protons and neutrons. It is some new kind of matter, detected only by gravity.

Texts. For IB-HL: the IB Physics HL cosmology content on galaxies and the expanding universe (Hubble's law, the Big Bang, cosmic scale; formerly the Astrophysics option). For A-Level: the AQA/OCR Astrophysics and Cosmology options. Broader reading: Weinberg, The First Three Minutes (the Big Bang for a general reader) · Liddle, An Introduction to Modern Cosmology (the algebra-and-a-little-calculus next level) · Ryden, Introduction to Cosmology (the full undergraduate treatment).

Provenance. Anchor values from LANE_SPEC: $H_0 = 70\ \text{km/s per Mpc}$; $1\ \text{Mpc} = 3.086\times10^{22}\ \text{m} = 3.26\times10^{6}\ \text{ly}$; $1\ \text{yr} = 3.156\times10^{7}\ \text{s}$; $c = 3.0\times10^{8}\ \text{m/s}$; CMB $T = 2.7\ \text{K}$; Wien $b = 2.9\times10^{-3}\ \text{m}\,\cdot\,\text{K}$; composition $5\%/27\%/68\%$. Recompute list: $H_0 = 2.27\times10^{-18}\ \text{s}^{-1}$; Hubble time $1/H_0 = 4.41\times10^{17}\ \text{s} = 14.0\ \text{Gyr}$ (vs measured $13.8$); Hubble length $c/H_0 = 1.32\times10^{26}\ \text{m} \approx 14$ Gly; $d = 100\ \text{Mpc} \Rightarrow v = 7000\ \text{km/s}$, $z = 0.023$, $\lambda: 500.0 \to 511.7\ \text{nm}$; $z = 0.050 \Rightarrow v = 15{,}000\ \text{km/s}$, $d = 214\ \text{Mpc} = 7.0\times10^{8}\ \text{ly}$, lookback $\approx 700\ \text{Myr}$; CMB Wien peak $\lambda_{\text{max}} = 1.1\ \text{mm}$; composition sum $= 100\%$; the no-centre check $\vec{v}_C - \vec{v}_B = H_0(\vec{r}_C - \vec{r}_B)$.

Prerequisites. Stellar Physics: The H–R Diagram & the Lives of Stars — where luminosity, stellar fusion and the manufacture of the heavy elements are built; cosmology sets the stage on which those stars live and die. Leads to. The honest next step is general relativity: aq-sr-special-to-general-relativity, the university seam where the algebraic Hubble law is replaced by the curved-spacetime Friedmann model of an expanding universe (that treatment uses calculus and lies beyond this article — it is named as a pointer, never a dependency).