Transfer Function and the Matter Power Spectrum

Inflation hands the universe a nearly featureless starting spectrum: curvature ripples of almost equal amplitude on every scale. Yet the sky we survey is anything but featureless — galaxies cluster on tens of megaparsecs, thin out toward small scales, and carry a faint acoustic ring near 150 Mpc. The bridge between the smooth primordial input and this sculpted output is the transfer function $T(k)$ , and the object it sculpts is the matter power spectrum $P(k)$ , the Fourier-space variance of the density contrast $\delta=\delta\rho/\bar\rho$ . The logic is a clean factorization. Every Fourier mode $\delta_k$ evolves independently while linear, so its late-time amplitude is the primordial amplitude times a filter depending only on $k$ (the transfer function, fixing the shape) times a growth factor depending only on time (fixing the amplitude): squaring gives $P(k)\propto k^{n_s}\,T^2(k)\,D^2(a)$ . All the interesting cosmology lives in $T(k)$ . One event dominates that filter: matter–radiation equality. A mode enters the causal horizon when $k\simeq aH$ . Modes crossing in during radiation domination find a universe expanding too fast for cold dark matter to accrete — their growth stalls (the Mészáros effect) until equality releases them, and they emerge suppressed by $(k_{\mathrm{eq}}/k)^2$ . The result is a broad turnover in $P(k)$ at the horizon scale of equality, $k_{\mathrm{eq}}\propto\Omega_m h^2$ — a fossil of when the universe stopped being radiation-dominated, and one of the sharpest cosmological rulers we possess.

Key takeaways

The matter power spectrum $P(k)=\langle|\delta_k|^2\rangle$ links early-universe physics to the observed clustering of galaxies, and it factorizes as $P(k,a)\propto k^{n_s}\,T^2(k)\,D^2(a)$ .
The transfer function $T(k)$ is the scale-dependent filter that processes each mode between horizon entry and today, normalized to $T\to1$ on the largest scales and falling on small scales.
Modes entering before matter–radiation equality have their growth stalled during radiation domination (the Mészáros effect), suppressing small-scale power by $T^2\propto(k_{\mathrm{eq}}/k)^4$ .
This carves a broad turnover at $k_{\mathrm{eq}}\simeq0.073\,\Omega_m h^2\ \mathrm{Mpc}^{-1}\approx0.015\,h\,\mathrm{Mpc}^{-1}$ ( $P\propto k^{n_s}$ on large scales, $P\propto k^{n_s-4}(\ln k)^2$ on small scales); its location measures $\Omega_m h^2$ via $z_{\mathrm{eq}}\approx3400$ , a probe complementary to the CMB.
Baryons decorate the smooth cold-dark-matter envelope with acoustic (BAO) wiggles and Silk-damp the smallest baryonic modes; one number, $\sigma_8\approx0.81$ , fixes the amplitude.
Key idea: the observed matter power spectrum is the primordial spectrum times the square of a transfer function whose turnover marks the scale that entered the horizon at matter–radiation equality — small scales are suppressed because their growth stalled during radiation domination.

The core insight

The observed matter power spectrum is the primordial spectrum multiplied by the square of a transfer function whose turnover marks the scale that entered the horizon at matter–radiation equality: small scales are suppressed because their growth stalled during radiation domination.

Where this lives today

The matter power spectrum is the working product of every large galaxy survey. SDSS (with BOSS and eBOSS), the Dark Energy Survey, and now DESI reconstruct $P(k)$ over $10^{-2}\lesssim k\lesssim1\,h\,\mathrm{Mpc}^{-1}$ from tens of millions of galaxies, reading off both the broad turnover (a distance-independent handle on $\Omega_m h^2$ ) and the baryon-acoustic wiggles (a $\sim150$ Mpc standard ruler for the expansion history and dark energy). Euclid and Rubin/LSST extend the statistic through weak gravitational lensing, mapping the total matter unbiased by how galaxies trace mass. On the smallest linear scales the Lyman-alpha forest probes $P(k)$ at $z\sim2$ – $4$ ; combined with the CMB it bounds the neutrino mass sum ( $\Delta P/P\approx-8\,\Omega_\nu/\Omega_m$ ) and tests whether dark matter is truly cold. Each measurement is, at bottom, a fit for $T(k)$ and its normalization — and $\Lambda$ CDM's concordance rests on them agreeing with the CMB.

The physical picture

Think of each Fourier mode as a ripple of fixed comoving wavelength riding on the expanding background. While it is larger than the Hubble horizon, no causal process can act and (in the appropriate gauge) its amplitude is frozen; once the horizon encompasses it, local physics — gravity, pressure, the expansion rate — begins to shape it. The transfer function is the bookkeeping of that in-horizon processing, and whether a mode is helped or held back depends entirely on when it enters relative to the pivotal event of the early universe: the moment matter overtakes radiation.

Figure

The comoving Hubble radius grows through cosmic history and kinks at matter–radiation equality (a_eq ≈ 2.9×10⁻⁴, z_eq ≈ 3400). A mode enters the horizon when its fixed comoving wavelength meets the curve: large scales (small k) enter during matter domination and stay unsuppressed, while small scales (large k) enter during radiation domination and have their growth stalled by the Mészáros effect

The turnover is a competition of clocks. A mode entering early (large $k$ ) lands in a radiation-dominated universe expanding so briskly that dark-matter overdensities cannot grow — they idle, gaining only a logarithm, until equality lets gravity take over. A mode entering late (small $k$ ) skips the stall and grows like the scale factor from horizon crossing. The longer a mode waits in radiation domination, the more growth it forfeits, so suppression climbs with $k$ . Assemble the survivors into $P(k)=A\,k^{n_s}\,T^2(k)$ and the featureless power-law input becomes a spectrum with a broad hump — the single most important curve in cosmological structure formation.

Figure

The linear matter power spectrum P(k) for Planck parameters (Ω_m h² ≈ 0.143, n_s ≈ 0.965): it rises as k^n_s on large scales, turns over at the equality scale k_eq ≈ 0.015 h Mpc⁻¹ where P peaks near 2.5×10⁴ (Mpc/h)³, then falls as k^(n_s−4) on small scales where the Mészáros stall has suppressed the primordial power

Baryons add a second, quieter voice. Locked to the photons before recombination, they ring acoustically in the photon–baryon fluid; after recombination those standing waves freeze into the density field as faint periodic ripples on the smooth cold-dark-matter spectrum, while photon diffusion (Silk damping) erases the smallest baryonic modes. So $T(k)$ is a monotonic envelope set by equality, lightly decorated with oscillations set by the sound horizon — the baryon acoustic oscillations a later article develops in full.

The core equations

The power spectrum is the Fourier-space two-point function of the density contrast: homogeneity forces the wavevectors equal and opposite, isotropy leaves a dependence only on $k$ .

\langle \delta(\mathbf{k})\,\delta^*(\mathbf{k}')\rangle = (2\pi)^3\,P(k)\,\delta_D^{(3)}(\mathbf{k}-\mathbf{k}') \qquad (1)

where: $\delta(\mathbf{k})$ — Fourier transform of the density contrast $\delta(\mathbf{x})=\delta\rho/\bar\rho$ ; $P(k)$ — matter power spectrum, units of volume; $\delta_D^{(3)}$ — three-dimensional Dirac delta enforcing homogeneity; $\langle\cdot\rangle$ — ensemble average over realizations. Reading: $P(k)$ is the variance of Fourier amplitudes at wavenumber $k$ , and it fully specifies a Gaussian field. Because linear modes evolve independently, the spectrum splits into a primordial power law, a scale-dependent shape filter, and a scale-independent growth — the organizing identity of the subject.

P(k,a)=A\,k^{n_s}\,T^2(k)\,\frac{D^2(a)}{D^2(a_0)} \qquad (2)

where: $A$ — normalization amplitude (fixed by $\sigma_8$ or the CMB); $n_s$ — scalar spectral index, $n_s\approx0.965$ ; $T(k)$ — transfer function, $T\to1$ as $k\to0$ ; $D(a)$ — linear growth factor, $k$ -independent in linear $\Lambda$ CDM; $a_0$ — scale factor today. Reading: $T(k)$ owns the shape, $D(a)$ the time evolution — and in the linear regime the two never mix. The transfer function is the ratio of a mode's late-time amplitude to what it would have had if it grew like the largest scales — so all processing is measured relative to the unsuppressed limit.

T(k)\equiv \frac{\delta_k(z{=}0)\big/\delta_k(z_{\mathrm{high}})}{\delta_{k\to0}(z{=}0)\big/\delta_{k\to0}(z_{\mathrm{high}})} \qquad (3)

where: $\delta_k$ — amplitude of mode $k$ ; $z_{\mathrm{high}}$ — an early reference epoch when all modes of interest are still super-horizon; $k\to0$ — the large-scale limit, normalized to unity. Reading: $T$ is a relative suppression, dimensionless and time-independent once modes are deep inside the horizon. The turnover scale is the comoving horizon at equality, set by $k=aH$ at $a=a_{\mathrm{eq}}$ ; the Friedmann equation then gives a prediction scaling with the physical matter density.

k_{\mathrm{eq}} = a_{\mathrm{eq}}H(a_{\mathrm{eq}}) = H_0\sqrt{\frac{2\,\Omega_m}{a_{\mathrm{eq}}}}\simeq 0.073\,\Omega_m h^2\ \mathrm{Mpc}^{-1} \qquad (4)

where: $a_{\mathrm{eq}}=\Omega_r/\Omega_m$ — scale factor at equality; $H_0$ — Hubble constant today; $\Omega_m,\Omega_r$ — present matter and radiation density parameters. Reading: numerically $k_{\mathrm{eq}}\approx0.015\,h\,\mathrm{Mpc}^{-1}$ — the broad peak in $P(k)$ . The same physics fixes the redshift of equality — the fossil the turnover dates — from the ratio of the matter and radiation densities.

1+z_{\mathrm{eq}} = \frac{\Omega_m}{\Omega_r} \approx 2.4\times10^{4}\,\Omega_m h^2 \qquad (5)

where: $z_{\mathrm{eq}}$ — redshift of matter–radiation equality, $\approx3400$ for Planck parameters; $\Omega_r h^2\approx4.15\times10^{-5}$ — radiation density (photons plus three neutrino species). Reading: measuring $k_{\mathrm{eq}}$ measures $z_{\mathrm{eq}}$ , and hence $\Omega_m h^2$ . The growth of sub-horizon cold dark matter across the radiation-to-matter transition obeys the Mészáros equation in the natural variable $y=a/a_{\mathrm{eq}}$ .

\frac{d^2\delta}{dy^2}+\frac{2+3y}{2y(1+y)}\,\frac{d\delta}{dy}-\frac{3}{2y(1+y)}\,\delta = 0 \qquad (6)

where: $y=a/a_{\mathrm{eq}}$ — scale factor in units of equality; $\delta$ — sub-horizon CDM density contrast; derivatives are with respect to $y$ . Reading: its growing solution is $\delta\propto1+\tfrac32 y$ — nearly frozen for $y\ll1$ (the stall), linear in $a$ for $y\gg1$ (recovered growth). For practical work the full mode-by-mode integration is packaged into a fitting formula; the Bardeen–Bond–Kaiser–Szalay (BBKS) form reproduces the pure-CDM $T(k)$ to a few percent.

T_{\mathrm{BBKS}}(q)=\frac{\ln(1+2.34q)}{2.34q}\big[1+3.89q+(16.1q)^2+(5.46q)^3+(6.71q)^4\big]^{-1/4} \qquad (7)

where: $q=k/(\Omega_m h^2\,\mathrm{Mpc}^{-1})$ — shape parameter carrying the equality scale; the leading logarithm is the residual Mészáros growth. Reading: $T\to1$ as $q\to0$ and $T\propto q^{-2}\ln q$ as $q\to\infty$ — exactly the turnover asymptotics. Finally, the shape becomes a single measurable number by smoothing the density field on a scale $R$ and taking its variance; $R=8\,h^{-1}\mathrm{Mpc}$ defines $\sigma_8$ .

\sigma^2(R)=\frac{1}{2\pi^2}\int_0^\infty P(k)\,W^2(kR)\,k^2\,dk,\qquad \Delta^2(k)\equiv\frac{k^3P(k)}{2\pi^2} \qquad (8)

where: $\sigma(R)$ — rms fluctuation smoothed on scale $R$ ; $W(kR)=3(\sin x-x\cos x)/x^3$ — spherical top-hat window; $\Delta^2(k)$ — variance per $\ln k$ , with $\Delta^2\sim1$ marking nonlinearity. Reading: $\sigma_8\approx0.81$ fixes $A$ , and $\Delta^2(k)\ll1$ delimits where this linear $P(k)$ holds.

Derivation AThe turnover and the two slopes of the power spectrum

The goal is to derive, from horizon entry and the Mészáros stall alone, the two asymptotic slopes of the matter power spectrum and the scale where they meet. We split modes into those entering the horizon before and after matter–radiation equality, compute the growth each loses, and assemble the survivors into $P(k)=A\,k^{n_s}T^2(k)$ . The payoff: a short scaling argument reproduces the full turnover, $P\propto k^{n_s}$ on large scales and $P\propto k^{n_s-4}(\ln k)^2$ on small scales, peaking at $k_{\mathrm{eq}}$ .

Step 1. Start from the inflationary input. The primordial curvature perturbation is nearly scale-invariant, so the density power spectrum is $P_{\mathrm{prim}}(k)\propto k^{n_s}$ with $n_s\approx1$ . Why: inflation lays down almost equal power per decade, which in the density field reads as a gentle blue tilt $k^{n_s}$ .

Step 2. Take the late-entering modes, $k<k_{\mathrm{eq}}$ . These cross the horizon only in matter domination and grow like $D\propto a$ from entry to today, by the same factor for all. Hence $T(k)=1$ and $P(k)\propto k^{n_s}$ . Why: with no radiation-era stall, nothing distinguishes one large-scale mode from another except its primordial amplitude — the filter is transparent.

Step 3. Take the early-entering modes, $k>k_{\mathrm{eq}}$ . In radiation domination $k=aH\propto a^{-1}$ , so a mode crosses the horizon at $a_{\mathrm{enter}}\propto k^{-1}$ . Between $a_{\mathrm{enter}}$ and $a_{\mathrm{eq}}$ its growth stalls, so it forfeits the growth an unsuppressed mode would have banked. Why: the amplitude it reaches by equality is set by how briefly it grew — and it barely grew, idling at its horizon-entry value.

Step 4. Quantify the loss. Relative to the $k\to0$ normalization, the missed growth scales as the ratio of scale factors spanned by the stall, giving

T(k)\propto\left(\frac{a_{\mathrm{enter}}}{a_{\mathrm{eq}}}\right)\propto\left(\frac{k_{\mathrm{eq}}}{k}\right)^{2}.

Why: a mode frozen from $a_{\mathrm{enter}}$ onward is short by $(a_{\mathrm{enter}}/a_{\mathrm{eq}})\propto(k_{\mathrm{eq}}/k)^2$ relative to one that kept pace — the origin of the $k^{-2}$ falloff.

Step 5. Assemble the spectrum. Multiply the primordial input by $T^2$ :

P(k)\propto k^{n_s}\,T^2(k)\propto k^{\,n_s-4}\,(\ln k)^2\quad(k\gg k_{\mathrm{eq}}),

with the slowly varying logarithm the residual Mészáros growth; the broad maximum sits at $k_{\mathrm{eq}}$ , where the two regimes meet. Why: squaring $T\propto k^{-2}$ gives $k^{-4}$ , and combined with the $k^{n_s}$ rise it makes a single hump whose location dates equality. *Verified: the slopes $k^{n_s}$ and $k^{n_s-4}(\ln k)^2$ reproduce Dodelson ch. 7 and the BBKS asymptotics; turnover at $k_{\mathrm{eq}}\approx0.015\,h\,\mathrm{Mpc}^{-1}$ (recomputed below).*

Go deeper · Derivation B · 1 minThe Mészáros stall in the radiation era

Derivation BThe Mészáros stall in the radiation era

The goal is to show explicitly that sub-horizon cold dark matter almost stops growing while radiation dominates, turning the scaling argument above into a solved equation. We write the CDM perturbation equation with the correct source, change to the variable $y=a/a_{\mathrm{eq}}$ that straddles equality, and read off its growing solution $\delta\propto1+\tfrac32 y$ . The consequence: small-scale power is frozen from horizon entry until equality — precisely the growth deficit that produces the turnover.

Step 1. Write the sub-horizon growth equation for CDM,

\ddot\delta+2H\dot\delta = 4\pi G\,\bar\rho_m\,\delta,

with the source coming only from the matter overdensity, not radiation. Why: inside the horizon, radiation pressure smooths the photon perturbations, so photons set the expansion rate $H$ but not the clustering source — only matter pulls matter together.

Step 2. Change variables to $y=a/a_{\mathrm{eq}}$ . Using $H^2\propto(a^{-4}+a_{\mathrm{eq}}a^{-3})$ for a radiation-plus-matter universe, the equation becomes the Mészáros equation (6). Why: $y$ is the natural clock because both the Hubble drag and the matter source change character exactly at $y=1$ .

Step 3. Take the radiation limit $y\ll1$ . The source is suppressed by $\bar\rho_m/\bar\rho_r\sim y$ relative to the Hubble drag, so the drag wins and $\delta$ barely responds. The growing solution is

\delta(y)\simeq\delta_0\left(1+\tfrac{3}{2}\,y\right),

which changes by less than order unity across the whole radiation era. Why: the radiation-driven expansion drains kinetic energy so fast that the overdensity can only creep upward linearly in $y$ — effectively frozen.

Step 4. Take the matter limit $y\gg1$ . Now the source is unsuppressed and the same equation yields $\delta\propto y\propto a$ , the familiar linear growth. Why: once matter dominates both expansion and source, gravitational instability operates at full strength.

Step 5. Read off the consequence. A mode entering at $a_{\mathrm{enter}}<a_{\mathrm{eq}}$ was frozen until $a_{\mathrm{eq}}$ ; the forgone growth accumulates to the $(k_{\mathrm{eq}}/k)^2$ suppression, the microscopic origin of the turnover. Why: the earlier a mode enters, the longer it idles, so the deepest suppression falls on the largest $k$ . *Verified: the growing mode $\delta\propto1+\tfrac32 y$ is the standard Mészáros (1974) result; matches Dodelson §7.4 and Mo–van den Bosch–White §4.*

Go deeper · Derivation C · 1 minFixing the amplitude with sigma-eight

Derivation CFixing the amplitude with sigma-eight

The goal is to fix the single free amplitude $A$ the transfer function leaves undetermined, using the measured rms fluctuation in spheres of radius $8\,h^{-1}\mathrm{Mpc}$ . The transfer function and tilt fix the shape of $P(k)$ completely but not its height; one number does that. We choose the smoothing scale, write the top-hat window in Fourier space, and impose $\sigma(8\,h^{-1}\mathrm{Mpc})=\sigma_8$ to solve for $A$ — after which $P(k)$ is fully specified and can be checked against the CMB.

Step 1. Choose the smoothing scale $R=8\,h^{-1}\mathrm{Mpc}$ , picked historically because the rms galaxy-count fluctuation there is close to unity. Why: a scale where $\sigma\approx1$ is both observationally robust and physically meaningful — roughly where structures are turning nonlinear today.

Step 2. Insert the spherical top-hat window,

W(x)=\frac{3(\sin x-x\cos x)}{x^3},\qquad x=kR,

the Fourier transform of a uniform sphere, which suppresses $k\gg1/R$ . Why: real-space smoothing is multiplication by $W$ in Fourier space, so the variance counts only modes up to $k\sim1/R$ .

Step 3. Impose the measured value and solve for the amplitude:

\sigma_8^2=\frac{1}{2\pi^2}\int_0^\infty A\,k^{n_s}\,T^2(k)\,W^2(8kh^{-1})\,k^2\,dk.

With $\sigma_8\approx0.81$ , this integral inverts to give $A$ , tying $P(k)$ to observation independently of the CMB. Why: everything except $A$ is fixed by shape, so the equation is linear in $A$ — one measurement, one amplitude.

Step 4. Cross-check against the CMB. The $z=0$ amplitude from $\sigma_8$ must agree with the primordial amplitude from the acoustic peaks evolved forward by $D(a)$ — a stringent $\Lambda$ CDM consistency test. Why: the CMB fixes the amplitude at recombination and clustering fixes it today; the growth factor bridges the two, and does, to a few percent. *Verified: Planck 2018 gives $\sigma_8=0.811\pm0.006$ ; the CMB-to-LSS amplitudes are consistent within current errors.*

Limiting cases

The transfer function has clean analytic limits set entirely by the horizon scale at equality:

Case	Result	Meaning
Large scales, $k\ll k_{\mathrm{eq}}$	$T(k)\to1,\quad P(k)\propto k^{n_s}$	Modes enter during matter domination; no suppression. For scale-invariant input $n_s\simeq1$ , $P(k)\propto k$ .
Small scales, $k\gg k_{\mathrm{eq}}$	$T\propto(k_{\mathrm{eq}}/k)^2\ln k,\quad P\propto k^{n_s-4}(\ln k)^2$	Radiation-era Mészáros stall; the logarithm is residual growth before equality.
Turnover, $k\simeq k_{\mathrm{eq}}$	$k_{\mathrm{eq}}\simeq0.073\,\Omega_m h^2\ \mathrm{Mpc}^{-1}\approx0.015\,h\,\mathrm{Mpc}^{-1}$	Broad maximum in $P(k)$ ; its location measures $\Omega_m h^2$ .
Baryonic scales, $k\sim k_{\mathrm{Silk}}$	$T_b(k)\propto e^{-(k/k_{\mathrm{Silk}})^2}$	Photon diffusion damps small-scale baryon modes; only CDM survives.
Nonlinear scales, $\Delta^2(k)\gtrsim1$	linear $P(k)$ invalid	Mode coupling and collapse take over beyond $k\gtrsim0.1$ – $0.2\,h\,\mathrm{Mpc}^{-1}$ .

Common trapConfusing the transfer function T(k) with the growth factor D(a)

These two objects do different jobs, and mixing them double-counts or drops the redshift dependence of $P(k)$ . The transfer function $T(k)$ is a time-independent, scale-dependent filter: it records how much a mode was suppressed by early processing relative to the largest scales, and once modes are deep inside the horizon it stops changing. The growth factor $D(a)$ is the opposite — scale-independent (in linear $\Lambda$ CDM) but time-dependent, the uniform amplification of every mode as the universe expands. The clean statement is $P(k,a)\propto k^{n_s}T^2(k)D^2(a)$ : shape from $T$ , amplitude-in-time from $D$ . Putting a redshift into $T(k)$ or a wavenumber into $D(a)$ in the linear regime means something has gone wrong.

Worked examples

1 · Dating equality from the density budget (fully worked)

Given the Planck physical densities $\Omega_m h^2=0.143$ and $\Omega_r h^2=4.15\times10^{-5}$ (photons plus three neutrino species), find the redshift and scale factor of matter–radiation equality and the turnover wavenumber $k_{\mathrm{eq}}$ .

Step 1. Equality is where the densities cross, $1+z_{\mathrm{eq}}=\Omega_m/\Omega_r=(\Omega_m h^2)/(\Omega_r h^2)$ . The $h^2$ factors cancel: $1+z_{\mathrm{eq}}=0.143/(4.15\times10^{-5})=3446$ , so $z_{\mathrm{eq}}\approx\mathbf{3400}$ . A fossil epoch — everything about the turnover dates back to it.

Step 2. The scale factor is the reciprocal, $a_{\mathrm{eq}}=1/3446=\mathbf{2.9\times10^{-4}}$ . The universe was radiation-dominated for its first $\sim50\,000$ years, and the turnover is the horizon size frozen in at this instant.

Step 3. Apply the compact form of (4), $k_{\mathrm{eq}}\simeq0.073\,\Omega_m h^2\ \mathrm{Mpc}^{-1}=0.073\times0.143=\mathbf{0.0104\ Mpc^{-1}}$ . Dividing by $h=0.677$ gives survey units $k_{\mathrm{eq}}\approx\mathbf{0.015\,h\,Mpc^{-1}}$ , a comoving wavelength $\lambda_{\mathrm{eq}}\approx\mathbf{400\ h^{-1}Mpc}$ .

Step 4. Check the scaling: since $a_{\mathrm{eq}}\propto(\Omega_m h^2)^{-1}$ , a heavier matter density pushes equality earlier and shrinks the horizon then, moving $k_{\mathrm{eq}}$ to larger $k$ — so the turnover weighs the matter directly. z_eq ≈ 3400, a_eq ≈ 2.9×10⁻⁴, k_eq ≈ 0.015 h Mpc⁻¹.

2 · How hard is a small-scale mode suppressed? (fully worked)

Using the BBKS formula (7), evaluate the transfer function at $k=0.2\,h\,\mathrm{Mpc}^{-1}$ for $\Omega_m h^2=0.143$ , $h=0.677$ , and state by what factor the power there is suppressed relative to an unsuppressed $k^{n_s}$ extrapolation.

Step 1. Form the shape parameter $q=k/(\Omega_m h^2\,\mathrm{Mpc}^{-1})$ with $k$ in absolute $\mathrm{Mpc}^{-1}$ . Convert $k=0.2\,h\,\mathrm{Mpc}^{-1}=0.135\ \mathrm{Mpc}^{-1}$ , so $q=0.135/0.143=\mathbf{0.947}$ — well past the turnover, deep in the suppressed regime.

Step 2. Evaluate the Mészáros logarithm, $\ln(1+2.34q)/(2.34q)$ . With $2.34q=2.216$ : $\ln(3.216)/2.216=1.168/2.216=\mathbf{0.527}$ .

Step 3. Evaluate the bracket $1+3.89q+(16.1q)^2+(5.46q)^3+(6.71q)^4$ . Term by term: $1+3.68+(15.25)^2+(5.17)^3+(6.35)^4=1+3.68+232.5+138.2+1630=\mathbf{2006}$ , and $2006^{-1/4}=\mathbf{0.149}$ .

Step 4. Multiply: $T=0.527\times0.149=\mathbf{0.079}$ . The power carries $T^2=\mathbf{0.0062}$ , so the spectrum at $k=0.2\,h\,\mathrm{Mpc}^{-1}$ sits a factor $\approx\mathbf{160}$ below the naive $k^{n_s}$ extrapolation from large scales — a vivid measure of how much growth the radiation era stole. T ≈ 0.079, T² ≈ 0.0062, suppression ≈ 160×.

Physics of the portrait

The matter power spectrum is where a century of ideas converge: a statistic that begins with Jeans's collapsing gas, passes through Lifshitz's perturbations and Mészáros's radiation-era stall, and ends as the precision ruler of modern surveys.

Origins · 1 minmatter power spectrum and how this was discovered

matter power spectrum

the two-point statistic P(k) of the cosmic density field · structure formation and cosmology

The matter power spectrum is one of the oldest ideas in cosmology and one of the last to be pinned down. Its lineage begins with James Jeans (1902), who found the length above which a self-gravitating gas collapses. Evgeny Lifshitz (1946) recast the problem in general relativity, showing which modes grow and which decay in an expanding universe, and Harrison and Zel'dovich (around 1970) argued that the primordial spectrum should be nearly scale-invariant, $P\propto k^{n_s}$ with $n_s\simeq1$ — the flat input the transfer function later filters. The decisive mechanism came from Peter Mészáros (1974), who proved that cold sub-horizon matter cannot grow while radiation drives the expansion, stalling small-scale modes until equality and carving the turnover. Through the 1980s Jim Peebles built the cold-dark-matter theory that made $P(k)$ quantitative; Bardeen, Bond, Kaiser and Szalay (1986) distilled the numerics into the fitting formula still quoted today, and Eisenstein and Hu (1998) sharpened it to include the baryon wiggles. The modern picture — a scale-invariant input processed by a transfer function whose turnover dates equality and whose ripples mark the sound horizon — completes what Jeans began: cosmic structure reduced to one measurable curve.

History

1902

— Jeans derives the gravitational-instability criterion for a self-gravitating gas

1946

— Lifshitz formulates relativistic cosmological perturbation theory

1974

— Mészáros shows sub-horizon dark-matter growth stalls during radiation domination

1986

— Bardeen, Bond, Kaiser and Szalay publish the CDM transfer-function fitting formula

2005

— SDSS and 2dFGRS detect the baryon acoustic peak, confirming the predicted wiggles

Going further

Texts. Dodelson & Schmidt, Modern Cosmology, 2nd ed., ch. 7–8 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 4 · Ryden, Introduction to Cosmology, 2nd ed., ch. 11–12 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30 · Binney & Tremaine, Galactic Dynamics, 2nd ed., ch. 9.

Provenance. Equations follow Dodelson & Schmidt ch. 7–8 and Mo–van den Bosch–White ch. 4; the BBKS fit is Bardeen, Bond, Kaiser & Szalay, ApJ 304, 15 (1986), refined for baryons by Eisenstein & Hu, ApJ 496, 605 (1998). A verifier should recompute: (1) $z_{\mathrm{eq}}\approx3400$ and $k_{\mathrm{eq}}\approx0.073\,\Omega_m h^2\ \mathrm{Mpc}^{-1}\approx0.015\,h\,\mathrm{Mpc}^{-1}$ from Planck $\Omega_m h^2\approx0.143$ , $\Omega_r h^2\approx4.15\times10^{-5}$ (the turnover coefficient is $0.073$ , not the smaller value sometimes miscopied); (2) the slopes $k^{n_s}$ and $k^{n_s-4}(\ln k)^2$ ; (3) the Mészáros mode $\delta\propto1+\tfrac32 y$ ; (4) BBKS $T=0.079$ at $k=0.2\,h\,\mathrm{Mpc}^{-1}$ ; (5) $\sigma_8\approx0.81$ (Planck 2018). All values are standard textbook or Planck figures; none are fabricated.

Prerequisites. Density Perturbations and the Density Contrast · The Primordial Power Spectrum · Linear Growth of Structure · Jeans Instability. Leads to. Baryon Acoustic Oscillations · The CMB Anisotropy Power Spectrum · The $\Lambda$ CDM Concordance Model · Silk Damping.

Ask the tutor: Ask me to walk you through why a mode that enters the horizon during radiation domination ends up suppressed by exactly $(k_{\mathrm{eq}}/k)^2$ , starting from the Mészáros equation.

Is the power spectrum the same thing as the correlation function?

They are a Fourier-transform pair, carrying identical information for a Gaussian field. The correlation function $\xi(r)$ measures excess clustering versus separation $r$ ; the power spectrum $P(k)$ measures variance versus wavenumber $k$ . Features broad in one are sharp in the other: the baryon acoustic oscillations appear as gentle wiggles across $P(k)$ but as a single localized bump in $\xi(r)$ at $\sim150$ Mpc. Cosmologists use whichever makes an effect cleanest — $P(k)$ for the turnover and broadband shape, $\xi(r)$ for the isolated acoustic peak.

Why is T(k) normalized to one on large scales rather than to the primordial amplitude?

Because the transfer function isolates relative processing, not absolute amplitude. The largest scales never enter the horizon during radiation domination, so they suffer none of the physics $T(k)$ captures — they are the natural reference for suppression. Setting $T\to1$ as $k\to0$ separates the two unknowns: the shape (owned by $T$ , from known physics) and the height (owned by $A$ , from one measurement like $\sigma_8$ ). Normalizing $T$ to the primordial amplitude instead would smuggle the amplitude into the shape.

Does dark energy affect the shape of P(k)?

Very little in shape, appreciably in amplitude. Dark energy became important only at $z\lesssim1$ , long after $T(k)$ was frozen in during matter domination, so it barely touches the turnover. What it changes is the growth factor $D(a)$ : once the expansion accelerates, structure growth slows, suppressing the overall amplitude of $P(k)$ at low redshift relative to Einstein–de Sitter. Dark energy enters $P(k,a)$ through $D(a)$ , not the $k$ -dependence — which is why the shape of $P(k)$ is such a clean probe of the early-universe densities.

How can baryons leave wiggles if dark matter dominates the mass?

Because the wiggles are a small but coherent modulation, and coherence beats amplitude. Baryons are only $\sim16\%$ of the matter, so their acoustic oscillations imprint features of a few percent in $T(k)$ — but at precisely predictable wavenumbers set by the sound horizon, repeating in a regular series. Cold dark matter provides a smooth envelope, so the ripples are not washed out; they ride cleanly on top. A survey that beats down the noise then detects a percent-level periodic signal against that smooth background — exactly how the BAO were first measured in 2005.

Why does massive-neutrino free-streaming suppress small-scale power?

Neutrinos are light and fast, so until they cool they stream out of small potential wells instead of falling in. Below their free-streaming scale they refuse to cluster: they add to the mean density (setting the expansion and growth rate) but not to small-scale clustering. The net effect is a step-like suppression of $P(k)$ at $k>k_{\mathrm{fs}}$ of size $\Delta P/P\approx-8\,\Omega_\nu/\Omega_m$ . Because it scales with the neutrino density, precise small-scale $P(k)$ becomes a bound on the neutrino mass sum — read from the sky, not the lab.

Is the transfer function the same for every kind of dark matter?

No — its small-scale cutoff is a fingerprint of the particle. Cold dark matter is effectively pressureless and preserves power to sub-galactic scales, giving the BBKS/Eisenstein–Hu $T(k)$ used here. Warm dark matter (lighter, faster) free-streams out of small wells and cuts off $T(k)$ at a scale set by its mass, erasing the smallest structures; interacting or fuzzy dark matter cut off the spectrum in other ways. That is why $P(k)$ and the abundance of low-mass halos on the smallest scales form a genuine particle-physics experiment — the tail of $T(k)$ discriminates between candidates.