28.03.04 · astronomy / galaxies

Galaxy formation and evolution: the cosmic star-formation history, hierarchical merging, and the Hubble sequence

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Anchor (Master): Hubble 1926; Toomre & Toomre 1972 ApJ 178:623 (Antennae); White & Rees 1978 MNRAS 183:341 (LCDM galaxy formation); Blumenthal et al. 1984 Nature 311:517; Kauffmann-White-Guiderdoni 1993 MNRAS 264:201 (semi-analytic models); Madau-Pozzetti-Dickinson 1998 MNRAS 298:L36 (cosmic SF history); Baldry et al. 2004 ApJ 600:681 (color bimodality); Springel 2005 Nature 435:629 (Millennium); Schaye 2015 MNRAS 446:521 (EAGLE); Madau-Dickinson 2014 ARAA 53:415; Springel 2018 MNRAS 475:226 (IllustrisTNG)

Intuition Beginner

Galaxies have not always looked the way they do today. Looking at distant galaxies means looking back in time, because their light took billions of years to reach us. About ten billion years ago the cosmos was building stars roughly ten times faster than it does now, and most galaxies were small, gas-rich, and crashing into one another. Edwin Hubble's 1926 sorting of galaxies into ellipticals, spirals, and irregulars was the first map of this diversity, but it is a snapshot of the present, not the whole story.

The Hubble Space Telescope's Deep Field images, first taken in 1995, stared at a small patch of dark sky and found thousands of galaxies stretching back more than ten billion years. They showed a universe in ferment: little clumpy galaxies, glowing with newborn stars, merging into bigger ones. The "cosmic star-formation history", the famous Madau plot drawn by Piero Madau in 1998, records how the star-formation rate rose, peaked when the universe was about three billion years old, and has been falling ever since.

Why the decline? In the standard cosmology, small dark-matter halos formed first and merged into larger ones; gas cooled inside them and lit up as stars. The most massive galaxies burned through their gas quickly or had it heated and expelled by the supermassive black hole at their centre. Today the massive ellipticals are "red and dead", while smaller spirals like our Milky Way keep forming stars. Galaxy formation weaves together cosmology, dark matter, gas physics, and black holes, and this is why the concept exists.

Visual Beginner

The diagram summarises four pillars of galaxy evolution. The cosmic star-formation history plots the rate at which the universe builds stars against redshift, rising to a peak near redshift 2 and declining by about a factor of ten to today. Beneath it, the hierarchical-merger ladder shows small dark-matter halos coalescing into larger ones, with gas cooling and forming stars at each rung. A colour-magnitude diagram splits galaxies into a red sequence of old, quenched ellipticals and a blue cloud of star-forming spirals, separated by the sparsely populated green valley. Arrows mark the feedback loops, supernovae and active black holes, that expel gas and quench star formation.

Each panel measures a different facet of the same history; together they constrain every model of how galaxies form.

Worked example Beginner

The Antennae galaxies, catalogued as NGC 4038 and NGC 4039, are a pair of spiral galaxies about 60 million light-years away, caught in the middle of a merger. Their long curved tidal tails, the "antennae", stretch about 360,000 light-years across, roughly four times the visible width of the Milky Way, and were flung out by the gravitational tug of the encounter.

Step 1. Alar and Juri Toomre simulated this merger in 1972 using only two gravitational potentials acting on rings of test particles. With a single close passage about 600 million years ago, their model reproduced the two long tails almost exactly, without any gas or star-formation physics: pure gravity did the work.

Step 2. The collision has compressed the galaxies' shared gas and triggered a massive burst of star formation. Infrared and X-ray observations find dozens of massive young star clusters in the overlap region, and the system is converting gas into stars at several times the rate of the Milky Way.

Step 3. Over the next 400 million years the two cores will spiral together, most of the gas will be consumed or blown out by supernovae and the central black holes, and the remnant will settle into a single elliptical galaxy, red, gas-poor, and largely finished with star formation.

What this tells us: a major merger can rebuild a galaxy's structure, trigger a starburst, and transform a spiral pair into an elliptical, exactly as the hierarchical picture predicts.

Check your understanding Beginner

Formal definition Intermediate+

Following Mo, van den Bosch, and White ^{[Mo-van den Bosch-White 2010]}, the central objects of galaxy-formation theory are the cosmic star-formation history, the specific star-formation rate, galaxy colour bimodality, and the halo-to-stellar-mass relation.

Definition (cosmic star-formation-rate density). The comoving star-formation-rate density is

ψ (z) = \frac{1}{V _{comoving}} i \sum SFR_{i},

the total rate at which gas is converted into stars per unit comoving volume at redshift $z$ , in units of $M_{⊙} yr^{- 1} Mpc^{- 3}$ . The function $ψ (z)$ over cosmic time is the cosmic star-formation history, measured by integrating galaxy luminosity functions in the ultraviolet, the H $α$ recombination line, the far-infrared, and the radio continuum, and applying dust-extinction and incompleteness corrections.

Definition (specific star-formation rate and quiescence). The specific star-formation rate is $sSFR = SFR / M_{*}$ , with units of $yr^{- 1}$ . A galaxy is quenched when its sSFR falls below roughly $1 0^{- 11} yr^{- 1}$ , the level at which residual star formation adds negligibly to the stellar mass over a Hubble time.

Definition (galaxy colour bimodality). The galaxy distribution in a colour-magnitude diagram is bimodal ^{[Baldry2004 ApJ 600:681]}: a red sequence of old, metal-rich, low-sSFR early-type galaxies (ellipticals and lenticulars) and a blue cloud of young, star-forming late-type galaxies (spirals and irregulars), separated by the sparsely populated green valley. Membership of the red sequence is the observational signature of quenching.

Definition (hierarchical merging and the halo-to-stellar-mass relation). In $Λ$ CDM, structure grows by hierarchical merging: smaller dark-matter halos form first and merge into larger ones ^{[WhiteRees1978 MNRAS 183:341]}; ^{[Blumenthal1984 Nature 311:517]}. The galaxy hosted by a halo of virial mass $M_{h}$ has stellar mass $M_{*}$ fixed by the halo-to-stellar-mass relation, peaking near $M_{h} \sim 1 0^{12} M_{⊙}$ with efficiency $M_{*} / M_{h} \sim 0.03$ , and falling at both low mass (supernova feedback) and high mass (AGN feedback and shock heating).

Counterexamples to common slips Intermediate+

"Galaxies form stars at a constant rate." No. The comoving rate peaked near $z \sim 2$ and has fallen by roughly a factor of ten since; the present-day universe is comparatively quiescent.
"Hierarchical merging means the most massive galaxies form last." Not as stellar populations. The most massive ellipticals assembled their stars early and fast (downsizing); only their dark-matter halos, and sometimes their outer stellar envelopes, were assembled late by dry mergers.
"Red galaxies are old, blue galaxies are young." The stellar-population part is right, but a blue galaxy's dark-matter halo may be ancient, and a red galaxy may have formed its stars in a short merger-induced burst before quenching.
"Ellipticals are evolutionary descendants of spirals along the Hubble tuning fork." No. Hubble's tuning fork is a classification, not a life cycle; the modern view is that massive ellipticals are typically the remnants of major mergers ^{[Toomre1972 ApJ 178:623]}.
"Feedback is a free parameter with no empirical support." Partly false. The AGN-merger-starburst-quenching sequence has independent support from X-ray cavities in clusters, the $M_{bh}$ - $σ$ relation, and the colour bimodality itself; the sub-grid recipes in simulations are calibrated to these observables, not invented.
"JWST has overthrown $Λ$ CDM." No. JWST has found more massive early galaxies than the simplest models predicted, an active puzzle, but not yet a definitive failure of the underlying cosmology.

Key result: the cosmic star-formation history (Madau plot) Intermediate+

Result (Madau-Dickinson 2014). The comoving star-formation-rate density $ψ (z)$ rises steeply from the epoch of reionisation to a broad peak at $z \approx 2$ , where it reaches about $0.1 M_{⊙} yr^{- 1} Mpc^{- 3}$ , roughly ten times the present-day value, and has declined since ^{[MadauDickinson2014 ARAA 53:415]}; ^{[Madau1998 MNRAS 298:L36]}.

Defence. Four largely independent observables trace the same curve. (i) Rest-frame ultraviolet luminosity functions measure the light of massive short-lived stars, converted to SFR via the Kennicutt calibration; deep Hubble and Subaru fields supply the $z > 2$ half. (ii) H $α$ emission-line surveys at $z \sim 1$ - $2$ (FMOS, MOSDEF) capture recombination photons from HII regions, comparatively insensitive to the stellar-initial-mass-function assumptions that dominate the UV uncertainty. (iii) Far-infrared and submillimetre luminosity functions from Herschel, SCUBA, and ALMA recover the dust-obscured component, which dominates at the peak; about $70$ - $80%$ of star formation at $z \sim 2$ is obscured. (iv) Radio-continuum (1.4 GHz) luminosity correlates with SFR through cosmic-ray synchrotron, a dust-independent tracer. The four tracers agree to within factors of two and all show the same rise-peak-decline, so the Madau plot is a robust empirical fact, not an artefact of a single band.

Theoretical interpretation. The decline since $z \sim 2$ tracks the decline in the cosmic gas-accretion rate onto halos, which follows the fall-off in cold-mode accretion efficiency as the universe expands; gas is consumed faster than it is replenished. In massive halos ( $M_{h} ≳ 1 0^{12} M_{⊙}$ ) AGN feedback in "radio mode" heats the halo gas and suppresses cooling, while in low-mass halos supernova feedback and reionisation photoheating expel or warm the gas. The declining massive-galaxy contribution is partly offset by continued low-mass star formation, which is why $ψ (z)$ declines gently rather than collapsing: the cosmic budget shifts to smaller systems.

Bridge. This result builds toward 28.03.01, where the galaxy luminosity function records the integrated consequence of this star-formation history, and appears again in 28.04.05, where the same hierarchical halo assembly under Press-Schechter sets the gravitational scaffold on which the cosmic gas supply cools and forms stars. The foundational reason the rate peaked near $z \sim 2$ is that the universal baryon-accretion rate onto halos peaked then, and this is exactly the bridge between the linear halo mass function and the observed galaxy population; putting these together identifies the peak of $ψ (z)$ with the epoch of maximum gas availability before feedback and consumption take hold.

Exercises Intermediate+

Exercise 4 (medium, short answer).

Explain galaxy "downsizing" and why it appears to contradict naive hierarchical assembly. State the resolution.

Hint

Distinguish the assembly of dark-matter halos from the assembly of stellar populations. Which finishes first in massive halos?

Answer

Downsizing is the observation that the most massive ellipticals assembled their stars earliest and fastest (at $z \sim 2$ - $3$ ), while lower-mass galaxies continue forming stars today. This appears to contradict naive hierarchical assembly ("small things first"), in which the largest dark-matter halos are the last to finish growing. The resolution is that "downsizing" refers to stellar assembly, not halo assembly. The dark-matter halos of massive galaxies do assemble hierarchically and late, but in the most massive halos the gas cools and forms stars rapidly early on (high gas fractions, short cooling times), and AGN feedback then quenches them; lower-mass halos form stars inefficiently and keep doing so for a Hubble time. So the apparent paradox dissolves once gas cooling and feedback are layered onto the halo merger trees.

Exercise 5 (medium, short answer).

Name the three principal quenching mechanisms and the halo-mass regime in which each dominates.

Hint

Think small, large, and environmental. What heats or removes the cold gas reservoir in each case?

Answer

(1) Supernova feedback dominates in low-mass halos ( $M_{h} ≲ 1 0^{11} M_{⊙}$ ): repeated supernovae heat and expel gas faster than it cools. (2) AGN feedback dominates in massive halos ( $M_{h} ≳ 1 0^{12} M_{⊙}$ ), in two modes: "quasar mode" (luminous, gas-rich, radiative, in field galaxies at high redshift) and "radio / maintenance mode" (relativistic jets inflating X-ray cavities in the hot intracluster medium, preventing cooling flows in clusters). (3) Strangulation / starvation operates environmentally: when a galaxy falls into a cluster, the hot intracluster medium strips or shuts off the inflow of fresh cold gas, so the galaxy consumes its residual gas and reddens.

Exercise 6 (hard, symbolic).

The Madau-Dickinson 2014 parametric fit is

ψ (z) = \frac{A ( 1 + z ) ^{α}}{1 + ( ( 1 + z ) / C ) ^{β}},

with $A = 0.015 M_{⊙} yr^{- 1} Mpc^{- 3}$ , $α = 2.7$ , $C = 2.9$ , $β = 5.6$ . (a) Show that $ψ (z)$ has a single maximum at $1 + z_{peak} = C (α / (β - α))^{1/ β}$ . (b) Evaluate $z_{peak}$ numerically. (c) Compute the ratio $ψ (z_{peak}) / ψ (0)$ and confirm the decline is roughly a factor of ten.

Hint

Set $u = 1 + z$ , take the logarithmic derivative $d ln ψ / d u$ , and set it to zero. For the ratio, evaluate the denominator at $z = 0$ (where $(1 + z) / C = 1/ C$ ) and at the peak (where $((1 + z) / C)^{β} = α / (β - α)$ ).

Answer

(a) Put $u = 1 + z > 0$ , so $ψ \propto f (u) = u^{α} / (1 + (u / C)^{β})$ . The logarithmic derivative is

\frac{d ln f}{d u} = \frac{1}{u} [α - \frac{β ( u / C ) ^{β}}{1 + ( u / C ) ^{β}}] .

Setting this to zero gives $α (1 + (u / C)^{β}) = β (u / C)^{β}$ , i.e. $α = (β - α) (u / C)^{β}$ , hence $(u / C)^{β} = α / (β - α)$ and $1 + z_{peak} = C (α / (β - α))^{1/ β}$ .

(b) With $α / (β - α) = 2.7/2.9 = 0.9310$ and exponent $1/ β = 1/5.6 = 0.1786$ , $(0.9310)^{0.1786} = exp (0.1786 \cdot ln 0.9310) = exp (0.1786 \cdot (- 0.0715)) = exp (- 0.0128) = 0.9873$ . Thus $1 + z_{peak} = 2.9 \times 0.9873 = 2.863$ , giving $z_{peak} \approx 1.86$ , consistent with the observed peak near $z \sim 2$ .

(c) At the peak, $(u / C)^{β} = 0.9310$ so the denominator is $1.931$ , and $u^{α} = 2.86 3^{2.7} = exp (2.7 \cdot ln 2.863) = exp (2.7 \cdot 1.0518) = exp (2.840) = 17.12$ , giving $ψ_{peak} = A \cdot 17.12/1.931 = 8.87 A$ . At $z = 0$ , the denominator is $1 + C^{- β} = 1 + 2. 9^{- 5.6} = 1 + 0.00257 = 1.00257$ , so $ψ (0) = A /1.00257 = 0.9974 A$ . The ratio is $8.87/0.9974 \approx 8.9 \approx 9$ , i.e. roughly a factor of ten, as advertised.

Exercise 7 (hard, short answer).

Contrast the physics modelled by the Millennium, Illustris, EAGLE, and IllustrisTNG simulations. Explain what "sub-grid" physics means and why it is necessary.

Hint

Consider what each generation added beyond pure gravity. What processes occur on scales smaller than the simulation cell?

Answer

Millennium (Springel 2005) is dark-matter only: it followed about $1 0^{10}$ particles under gravity from $z = 127$ to $0$ , reproducing the large-scale clustering and the halo merger trees, but assigned galaxy properties by external semi-analytic recipes because it contained no gas. Illustris (2014) added magneto-hydrodynamics, radiative cooling, a prescription for star formation, and both supernova and black-hole feedback. EAGLE (Schaye 2015) ran a comparable hydrodynamic suite with feedback recalibrated to the observed galaxy mass function. IllustrisTNG (Springel 2018) is the successor with improved feedback models and magnetic fields, reproducing galaxy sizes, colours, and the colour bimodality. "Sub-grid" physics refers to processes below the simulation's resolution: individual star formation, supernova explosions, and black-hole accretion. These are represented by phenomenological recipes calibrated against observations rather than computed from first principles, because the relevant scales (parsecs and below) are far below the simulation cell size (kiloparsecs).

Exercise 8 (hard, numeric).

The Antennae galaxies lie at redshift $z \approx 0.0055$ . Using the Hubble-law approximation $d \approx cz / H_{0}$ with $H_{0} = 70 km s^{- 1} Mp c^{- 1}$ and $c = 3.0 \times 1 0^{5} km s^{- 1}$ , compute the distance in Mpc, then convert to millions of light-years using $1 Mpc \approx 3.26$ million light-years. Compare with the quoted value near 60 million light-years.

Hint

First obtain the recession velocity $v = cz$ , then divide by $H_{0}$ , then multiply by $3.26$ .

Answer

About $24 Mpc$ , i.e. roughly $77$ million light-years. The recession velocity is $v = cz = (3.0 \times 1 0^{5}) (0.0055) = 1650 km s^{- 1}$ , giving $d = v / H_{0} = 1650/70 = 23.6 Mpc$ . Converting, $23.6 \times 3.26 = 76.9$ million light-years. This sits at the upper end of the commonly quoted range of 60-80 million light-years for the Antennae; the spread reflects peculiar-velocity corrections and the exact distance indicator used, but the Hubble-law estimate confirms the order of magnitude.

Advanced results Master

Theorem 1 (Hubble 1926: the morphological sequence). Hubble ^[Hubble1926] introduced the morphological classification of "extragalactic nebulae" into ellipticals (E0-E7 by flattening), lenticulars (S0), spirals (Sa-Sc by bulge-to-disk ratio and arm tightness), and irregulars. The sequence is a classification of present-day structure, not an evolutionary track.

Theorem 2 (Toomre and Toomre 1972: mergers build ellipticals). Toomre and Toomre ^{[Toomre1972 ApJ 178:623]} showed that a purely gravitational encounter of two disk galaxies reproduces the tidal tails of the Antennae, NGC 4038/4039. Extrapolating, they inferred that major mergers of gas-rich disk galaxies relax into remnants with the morphology and kinematics of ellipticals, identifying ellipticals as merger products.

Theorem 3 (White and Rees 1978; Blumenthal et al. 1984: the $Λ$ CDM galaxy-formation framework). White and Rees ^{[WhiteRees1978 MNRAS 183:341]} proposed that galaxies form by gas cooling and condensing at the centres of dark-matter halos that grow hierarchically, with star formation, feedback, and mergers as sub-processes. Blumenthal, Faber, Primack, and Rees ^{[Blumenthal1984 Nature 311:517]} made the cold-dark-matter version concrete, identifying the dark-matter candidate and the bottom-up hierarchy that defines the modern paradigm.

Theorem 4 (Madau 1998; Madau-Dickinson 2014: the cosmic star-formation history). Madau, Pozzetti, and Dickinson ^{[Madau1998 MNRAS 298:L36]} compiled UV-derived star-formation rates as a function of redshift and drew the curve now called the Madau plot: a rise to a peak near $z \sim 2$ and a decline since. Madau and Dickinson ^{[MadauDickinson2014 ARAA 53:415]} consolidated fifteen years of multi-wavelength data (UV, H $α$ , far-IR, radio) into the canonical review and the parametric fit analysed in the Full proof set.

Theorem 5 (Baldry et al. 2004: galaxy colour bimodality). Baldry et al. ^{[Baldry2004 ApJ 600:681]}, building on Strateva et al. 2001, showed that the galaxy distribution in the colour-magnitude diagram is bimodal: a red sequence and a blue cloud separated by a green valley. The bimodality is the cleanest observational signature of quenching and the target that simulations must reproduce.

Theorem 6 (Kauffmann, White, and Guiderdoni 1993: semi-analytic models). Kauffmann, White, and Guiderdoni ^{[Kauffmann1993 MNRAS 264:201]} embedded analytic recipes for gas cooling, star formation, feedback, and metal enrichment inside Press-Schechter merger trees, inaugurating the semi-analytic tradition that connects the halo population of 28.04.05 to observable galaxy properties at low computational cost.

Theorem 7 (Springel 2005; Schaye 2015; Springel 2018: cosmological hydrodynamic simulations). The Millennium simulation ^{[Springel2005Millennium Nature 435:629]} traced $1 0^{10}$ dark-matter particles from $z = 127$ to $0$ , confirming that $Λ$ CDM reproduces large-scale clustering. EAGLE ^{[Schaye2015EAGLE MNRAS 446:521]} and IllustrisTNG ^{[Springel2018IllustrisTNG MNRAS 475:226]} added magneto-hydrodynamics, radiative cooling, star formation, and black-hole feedback, reproducing galaxy sizes, colours, the colour bimodality, and the stellar mass function.

Theorem 8 (JWST 2022+: massive early galaxies). Early JWST CEERS and NGDEEP imaging has revealed galaxies at $z > 10$ with stellar masses above $1 0^{9} M_{⊙}$ , more massive and more numerous than pre-JWST models predicted. The tension is an active research topic: candidate resolutions include higher early star-formation efficiency, less efficient feedback, a top-heavy initial mass function, or observational systematics (photometric redshift degeneracies), rather than a definitive failure of $Λ$ CDM.

Synthesis. Galaxy formation and evolution is the foundational reason that the observed cosmic star-formation history, galaxy colour bimodality, and Hubble sequence can be unified within a single $Λ$ CDM framework, and the central insight is that hierarchical halo merging plus gas cooling plus feedback converts the primordial density field of 28.04.05 into today's galaxy population. This is exactly the structure that identifies the simulated galaxy catalogues of IllustrisTNG and EAGLE with the observed galaxy distribution, and putting these together generalises the Press-Schechter halo formalism into a full baryonic theory. The pattern appears again in 28.07.02, where the Jeans-collapse physics of molecular clouds sets the small-scale end of the star-formation process, and the bridge is from the cosmic-web gas supply to the stellar initial mass function. Downsizing and quenching, the AGN feedback that empties massive halos of cold gas, are the empirical signatures that any successful model must reproduce, and the JWST massive-early-galaxy puzzle marks the frontier where the theory is still under construction.

Full proof set Master

Proposition 1 (peak of the Madau-Dickinson fit). Let $ψ (z) = A (1 + z)^{α} / [1 + ((1 + z) / C)^{β}]$ with $A, α, C, β > 0$ and $β > α$ . Then $ψ$ has a single maximum at

1 + z_{peak} = C (\frac{α}{β - α})^{1/ β},

and for the Madau-Dickinson 2014 parameters $(α, C, β) = (2.7, 2.9, 5.6)$ this gives $z_{peak} \approx 1.86$ and a peak-to-present ratio $ψ (z_{peak}) / ψ (0) \approx 8.9$ , i.e. roughly a factor of ten.

Proof. Set $u = 1 + z > 0$ , so $ψ \propto f (u) = u^{α} / (1 + (u / C)^{β})$ . The logarithmic derivative is

\frac{d ln f}{d u} = \frac{1}{u} [α - \frac{β ( u / C ) ^{β}}{1 + ( u / C ) ^{β}}] .

Setting $d ln f / d u = 0$ and clearing the denominator gives $α (1 + (u / C)^{β}) = β (u / C)^{β}$ , equivalently $α = (β - α) (u / C)^{β}$ , hence $(u / C)^{β} = α / (β - α)$ and $u_{peak} = C (α / (β - α))^{1/ β}$ . Since $d ln f / d u > 0$ for small $u$ and $< 0$ for large $u$ , this critical point is the unique maximum.

Substituting $α = 2.7$ , $β = 5.6$ , $C = 2.9$ : $α / (β - α) = 2.7/2.9 = 0.9310$ , and $(0.9310)^{1/5.6} = exp ((1/5.6) ln 0.9310) = exp (0.1786 \times (- 0.0715)) = exp (- 0.0128) = 0.9873$ , so $u_{peak} = 2.9 \times 0.9873 = 2.863$ , giving $z_{peak} = 1.863 \approx 1.9$ .

For the ratio, $ψ (0) = A / (1 + C^{- β})$ with $C^{- β} = 2. 9^{- 5.6} = exp (- 5.6 ln 2.9) = exp (- 5.6 \times 1.0647) = exp (- 5.962) = 0.00257$ , so $ψ (0) = A /1.00257 = 0.9974 A$ . At the peak the denominator is $1 + α / (β - α) = β / (β - α) = 5.6/2.9 = 1.931$ , and $u_{peak}^{α} = 2.86 3^{2.7} = exp (2.7 \times 1.0518) = exp (2.840) = 17.12$ , giving $ψ_{peak} = A \times 17.12/1.931 = 8.87 A$ . Hence $ψ_{peak} / ψ (0) = 8.87/0.9974 \approx 8.9 \approx 9$ . $□$

Proposition 2 (gas-depletion time from the Kennicutt-Schmidt relation). If the star-formation law is the Kennicutt-Schmidt relation $SFR = k M_{gas}^{1.4}$ for a constant $k > 0$ , then the gas-depletion time $τ_{dep} = M_{gas} / SFR$ obeys $τ_{dep} = (1/ k) M_{gas}^{- 0.4}$ , so gas-rich galaxies at high redshift consume their reservoirs on gigayear timescales, after which the cosmic star-formation rate must fall.

Proof. By definition $τ_{dep} = M_{gas} / SFR$ . Substituting the Kennicutt-Schmidt law,

τ_{dep} = \frac{M _{gas}}{k M _{gas}^{1.4}} = \frac{1}{k} M_{gas}^{1 - 1.4} = \frac{1}{k} M_{gas}^{- 0.4} .

Thus $τ_{dep} \propto M_{gas}^{- 0.4}$ : the more gas a galaxy holds, the faster it consumes a fixed fraction of it. At $z \sim 2$ , when typical gas fractions were several tens of percent and gas masses an order of magnitude above today's, $τ_{dep}$ was of order $1$ Gyr; once the gas supply was depleted or expelled by feedback, $ψ (z)$ fell, giving the declining branch of the Madau plot derived in Proposition 1. $□$

Connections Master

Galaxies survey 28.03.01. This unit deepens the galaxy census of 28.03.01 by tracing how the present-day population (spirals, ellipticals, dwarfs) was assembled over cosmic time. The Hubble-sequence classification introduced there is the modern snapshot whose origin the Madau plot and hierarchical merging explain; the galaxy luminosity function is the integrated record of the star-formation history derived here.
Large-scale structure 28.04.05. Galaxies live in the dark-matter halos whose mass function is derived in 28.04.05. The Press-Schechter merger trees are the gravitational scaffold on which the gas cooling, star formation, and quenching of this unit operate; halo assembly sets the rhythm of galaxy assembly, and the cosmic star-formation history is the baryonic response to the hierarchical growth of structure.
Cosmology: FLRW, inflation, and the CMB 28.04.01. The $Λ$ CDM cosmology of 28.04.01 fixes the expansion rate $H (z)$ , the baryon density, and the matter-to-dark-energy transition that together set the gas-accretion and cooling rates driving the cosmic star-formation history. Without the dark-matter and dark-energy budget established there, the peak shape and decline of $ψ (z)$ cannot be reproduced.
Molecular clouds and protostellar evolution 28.07.02. The Jeans-collapse physics of 28.07.02 governs the small-scale conversion of galaxy-scale gas reservoirs into stars. The cosmic star-formation rate density of this unit is the volume integral of the Kennicutt-Schmidt star-formation law whose microscopic mechanism is molecular-cloud collapse; the two scales are joined by the gas-depletion time of Proposition 2.

Historical & philosophical context Master

Edwin Hubble ^[Hubble1926] introduced the morphological classification of galaxies in 1926, the tuning-fork of ellipticals, lenticulars, spirals, and irregulars that bears his name, though his diagram was misread for decades as an evolutionary sequence. The modern physical theory of galaxy formation began with Simon White and Martin Rees ^{[WhiteRees1978 MNRAS 183:341]}, who proposed in 1978 that galaxies form by gas cooling inside growing dark-matter halos. George Blumenthal, Sandra Faber, Joel Primack, and Martin Rees ^{[Blumenthal1984 Nature 311:517]} made the cold-dark-matter version concrete in 1984. The decisive demonstration that galaxy interactions reshape morphology was Alar and Juri Toomre's 1972 simulation of the Antennae ^{[Toomre1972 ApJ 178:623]}, which reproduced the tidal tails by gravity alone and argued that major mergers of spirals produce ellipticals.

The cosmic star-formation history was charted by Piero Madau, Bianca Poggianti, and Mark Dickinson ^{[Madau1998 MNRAS 298:L36]}, whose 1998 plot of star-formation rate against redshift became the field's central empirical constraint. The definitive synthesis is the Madau and Dickinson ^{[MadauDickinson2014 ARAA 53:415]} 2014 review, which reconciled ultraviolet, H $α$ , far-infrared, and radio tracers into the parametric fit of Proposition 1. Galaxy colour bimodality was quantified by Ivan Baldry and collaborators ^{[Baldry2004 ApJ 600:681]}, following Strateva et al. 2001. The computational frontier runs from the Millennium simulation ^{[Springel2005Millennium Nature 435:629]} through EAGLE ^{[Schaye2015EAGLE MNRAS 446:521]} to IllustrisTNG ^{[Springel2018IllustrisTNG MNRAS 475:226]}, each generation adding baryonic physics. The semi-analytic modelling that bridges the halo population to observable galaxies was inaugurated by Guinevere Kauffmann, Simon White, and Cesare Guiderdoni ^{[Kauffmann1993 MNRAS 264:201]}.

Bibliography Master

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@article{Blumenthal1984,
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  volume  = {311},
  pages   = {517--525},
  year    = {1984},
  doi     = {10.1038/311517a0},
}

@article{Kauffmann1993,
  author  = {Kauffmann, Guinevere and White, Simon D. M. and Guiderdoni, Bruno},
  title   = {The Formation and Evolution of Galaxies Within Mixed Cold and Hot Dark Matter Haloes},
  journal = {Monthly Notices of the Royal Astronomical Society},
  volume  = {264},
  pages   = {201--218},
  year    = {1993},
  doi     = {10.1093/mnras/264.1.201},
}

@article{Madau1998,
  author  = {Madau, Piero and Pozzetti, Laura and Dickinson, Mark},
  title   = {The Star Formation History of Field Galaxies},
  journal = {Monthly Notices of the Royal Astronomical Society},
  volume  = {298},
  pages   = {L36--L40},
  year    = {1998},
  doi     = {10.1046/j.1365-8711.1998.29813447.x},
}

@article{Baldry2004,
  author  = {Baldry, Ivan K. and Glazebrook, Karl and Brinkmann, J. and Ivezi{\'c}, {\v Z}. and Lupton, R. H. and Nichol, R. C. and Szalay, A. S.},
  title   = {Quantifying the Bimodal Color-Magnitude Distribution of Galaxies},
  journal = {Astrophysical Journal},
  volume  = {600},
  pages   = {681--694},
  year    = {2004},
  doi     = {10.1086/380092},
}

@article{Springel2005Millennium,
  author  = {Springel, Volker and White, Simon D. M. and Jenkins, Adrian and Frenk, Carlos S. and Yoshida, Naoki and Gao, Liang and Navarro, Julio and Thacker, Robert and Croton, Darren and Helly, John and Peacock, John A. and Cole, Shaun and Thomas, Peter and Couchman, Hugh and Evrard, August and Colberg, J{\"u}rgen M. and Pearce, Frazer},
  title   = {Simulations of the Formation, Evolution and Clustering of Galaxies and Quasars},
  journal = {Nature},
  volume  = {435},
  pages   = {629--636},
  year    = {2005},
  doi     = {10.1038/nature03597},
}

@article{MadauDickinson2014,
  author  = {Madau, Piero and Dickinson, Mark},
  title   = {Cosmic Star-Formation History},
  journal = {Annual Review of Astronomy and Astrophysics},
  volume  = {53},
  pages   = {415--486},
  year    = {2014},
  doi     = {10.1146/annurev-astro-081811-125615},
}

@article{Schaye2015EAGLE,
  author  = {Schaye, Joop and Crain, Robert A. and Bower, Richard G. and Furlong, Michelle and Schaller, Matthieu and Theuns, Tom and Dalla Vecchia, Claudio and Frenk, Carlos S. and McCarthy, Ian G. and Helly, John C. and Jenkins, Adrian and Rosas-Guevara, Yetli M. and White, Simon D. M. and Baes, Maarten and Booth, Crystal M. and Camps, Peter and Navarro, Julio F. and Qu, Yichao and Rahmati, Ali and Sawala, Till and Thomas, Peter A. and Trayford, James},
  title   = {The {EAGLE} Project: Simulating the Evolution and Assembly of Galaxies and Their Environments},
  journal = {Monthly Notices of the Royal Astronomical Society},
  volume  = {446},
  pages   = {521--554},
  year    = {2015},
  doi     = {10.1093/mnras/stu2058},
}

@article{Springel2018IllustrisTNG,
  author  = {Springel, Volker and Pakmor, R{\"u}diger and Pillepich, Annalisa and Weinberger, Rainer and Nelson, Dylan and Hernquist, Lars and Vogelsberger, Mark and Genel, Shy and Torrey, Paul and Marinacci, Federico},
  title   = {First Results from the {IllustrisTNG} Simulations: Matter and Galaxy Clustering},
  journal = {Monthly Notices of the Royal Astronomical Society},
  volume  = {475},
  pages   = {226--249},
  year    = {2018},
  doi     = {10.1093/mnras/stx3304},
}

@book{MoVanDenBoschWhite2010,
  author    = {Mo, Houjun and van den Bosch, Frank and White, Simon},
  title     = {Galaxy Formation and Evolution},
  publisher = {Cambridge University Press},
  year      = {2010},
}

Prerequisites

28.03.01

Tier anchors

beginner: Bennett et al., The Cosmic Perspective (8e), Ch. 20-21; the 'cosmic star-formation history' and 'galaxy mergers' framing
intermediate: Mo, van den Bosch, and White, Galaxy Formation and Evolution (Cambridge, 2010), Ch. 9-13; Sparke and Gallagher, Galaxies in the Universe (2e)
master: Hubble 1926; Toomre & Toomre 1972 ApJ 178:623 (Antennae); White & Rees 1978 MNRAS 183:341 (LCDM galaxy formation); Blumenthal et al. 1984 Nature 311:517; Kauffmann-White-Guiderdoni 1993 MNRAS 264:201 (semi-analytic models); Madau-Pozzetti-Dickinson 1998 MNRAS 298:L36 (cosmic SF history); Baldry et al. 2004 ApJ 600:681 (color bimodality); Springel 2005 Nature 435:629 (Millennium); Schaye 2015 MNRAS 446:521 (EAGLE); Madau-Dickinson 2014 ARAA 53:415; Springel 2018 MNRAS 475:226 (IllustrisTNG)

References

Mo, van den Bosch, and White — Galaxy Formation and Evolution (Cambridge, 2010), Ch. 9-13 · the canonical graduate anchor for galaxy formation and evolution
Hubble1926 · Hubble, E. P. — Extragalactic Nebulae, Astrophysical Journal 64:321 (1926); the morphological sequence · source being verified
Toomre1972 · Toomre, A. and Toomre, J. — Galactic Bridges and Tails, Astrophysical Journal 178:623 (1972); the Antennae simulation · source being verified
WhiteRees1978 · White, S. D. M. and Rees, M. J. — Core Condenstation in Large Hierarchical Haloes, MNRAS 183:341 (1978); the LCDM framework · source being verified
Blumenthal1984 · Blumenthal, Faber, Primack, and Rees — Formation of galaxies and large-scale structure with cold dark matter, Nature 311:517 (1984) · source being verified
Kauffmann1993 · Kauffmann, White, and Guiderdoni — The Formation and Evolution of Galaxies in a Mixed Cold+Hot Dark Matter Universe, MNRAS 264:201 (1993); semi-analytic models · source being verified
Madau1998 · Madau, Pozzetti, and Dickinson — The Star Formation History of Field Galaxies, MNRAS 298:L36 (1998); the cosmic star-formation history · source being verified
Baldry2004 · Baldry et al. — Color bimodality of the galaxy distribution, Astrophysical Journal 600:681 (2004) · source being verified
Springel2005Millennium · Springel et al. — Simulations of the Formation, Evolution and Clustering of Galaxies and Quasars, Nature 435:629 (2005); the Millennium simulation · source being verified
Schaye2015EAGLE · Schaye et al. — The EAGLE project: simulating the evolution and assembly of galaxies, MNRAS 446:521 (2015) · source being verified
MadauDickinson2014 · Madau, P. and Dickinson, M. — Cosmic Star-Formation History, Annual Review of Astronomy and Astrophysics 53:415 (2014) · source being verified
Springel2018IllustrisTNG · Springel et al. — First Results from IllustrisTNG, MNRAS 475:226 (2018) · source being verified

Estimated time

beginner: 25m
intermediate: 50m
master: 85m