Choose What You Call a Halo, Then Read Feedback

Draw a boundary around a concentration

How large a circle should enclose a halo? The spherical-overdensity method chooses a centre and finds the radius where mean enclosed density reaches a threshold. A mass name must carry that threshold; it is not a wall at the object’s physical surface.

$M_{200\mathrm c}$ uses critical density and $M_{200\mathrm m}$ uses mean matter density at the same epoch. The expansion rate defines the former; the latter also includes the matter density parameter.

\rho_{\rm crit}(z)=\frac{3H(z)^2}{8\pi G},\qquad \bar\rho_m(z)=\Omega_m(z)\rho_{\rm crit}(z)

The mean enclosed thresholds are

200\rho_{\rm crit}

and

200\bar\rho_m

, respectively.

M_{200\mathrm c}=\frac{4\pi}{3}R_{200\mathrm c}^3\,200\rho_{\rm crit}

Solve for radius and mass together, using the same centre and density profile.

Reading: IllustrisTNG official data specifications · Springel et al., First results from IllustrisTNG: matter and galaxy clustering (2018), arXiv:1707.03397v2

Changing the reference changes what is enclosed

At an epoch with $\Omega_m=0.3$ , the thresholds are 200 and 60 times critical density. If mean enclosed density decreases outward, the lower threshold lies at a larger radius and encloses more mass. Multiplying mass by $1/0.3$ is not a conversion: both the radius and enclosed mass change with the profile.

In this Lab’s EdS background, $\Omega_m=1$ , so the two thresholds coincide for the same centre and profile. A simulation with a cosmological constant generally lacks that coincidence.

Friends-of-friends links nearby particles, SUBFIND identifies substructure, and spherical-overdensity mass uses a centre and density threshold. They answer different questions. A density boundary alone proves neither that every member is bound nor that the system is virialised. A bright knot on a 16-node teaching mesh supplies none of those missing tests.

Reading: IllustrisTNG official data specifications · Original finite-mesh calculations and independent small-wave Python replay

Convert units before handling a snapshot

TNG snapshot Coordinates use $\mathrm{ckpc}/h$ . Divide by $h$ for comoving kpc, then multiply by the epoch’s scale factor for physical kpc. For fixed-mass dark matter particles check the header MassTable; its mass unit is $10^{10}M_\odot/h$ . Variable-mass fields of other particle types cannot be substituted indiscriminately. Here $h$ is today’s Hubble constant divided by $100\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ .

As a hypothetical unit exercise, a coordinate value of $100$ with $h=0.6774$ and $a=0.5$ gives about $147.6$ comoving kpc or $73.8$ physical kpc. This is arithmetic, not a TNG particle we inspected.

Snapshot Velocities store peculiar velocity divided by the square root of the scale factor. Multiply by $\sqrt a$ to recover peculiar velocity. This differs from our canonical momentum, and catalog fields such as GroupVel have their own definitions. Three columns of numbers need not describe interchangeable quantities.

v_{\rm pec}=\sqrt a\,v_{\rm stored},\qquad u_{\rm Lab}=\frac{a\,v_{\rm pec}}{H_\star L}

The first equation refers to the TNG snapshot velocity field specified above; the second defines this course’s dimensionless momentum.

Reading: IllustrisTNG official data specifications · Original finite-mesh calculations and independent small-wave Python replay

Separate resolution from feedback

Open Section 2 and Figure 6 of the TNG clustering paper by Springel and colleagues. Start by comparing resolution levels within a box series. TNG100 and TNG300 change both volume and mass resolution. A larger box admits longer waves; finer mass and force resolution help describe smaller structures. These improvements are not interchangeable.

Figure 7 instead compares full-physics and corresponding dark-matter-only calculations. The DMO counterpart represents the total mean matter collisionlessly. It is not the dark matter subset of the full-physics run. A total-matter power comparison needs matching populations in numerator and denominator.

Read the logarithmic axes carefully. A natural-log power ratio of $-0.2$ means a ratio $e^{-0.2}\simeq0.8187$ , about 18.1% lower. The value is chosen for this axis-reading exercise.

Reading: Springel et al., First results from IllustrisTNG: matter and galaxy clustering (2018), arXiv:1707.03397v2 · Original finite-mesh calculations and independent small-wave Python replay

A 2026 preprint: which error is a percentage?

Next read Figures 1 and 4 and Table 1 of Zhang’s preprint 2609.00807v1, submitted on 1 September 2026. The tests use simulation suites with variations in feedback or cosmology. Read Table 1 to identify which choices change between runs. At our reading date, 12 November 2026, this version is marked for submission to JCAP.

Identify the axes first: simulated response horizontally, fitted minus simulated response vertically. Our invented example $S_{\rm sim}=0.80$ and $S_{\rm fit}=0.82$ gives absolute residual $0.02$ . Dividing by the simulated value yields relative error $2.5\%$ . Those are different statements. The left and right panels of Figures 1 and 4 fix $k=1h/\mathrm{Mpc}$ and $3h/\mathrm{Mpc}$ ; a symbol series also appears at several redshifts.

Then compare complexity. Strong-feedback single-parameter examples reach absolute residuals near $0.13$ at $k=3h/\mathrm{Mpc}$ . The three-parameter formula keeps maximum absolute residuals below about $0.03$ over the main tests at $k\leq3h/\mathrm{Mpc}$ and $0\leq z\leq2$ , with the $\Omega_m=0.1$ stress case remaining an exception. This supports a low-dimensional approximation on these simulations and scales, not a universal three-degree-of-freedom theorem.

S(k,z)=\frac{P_{\rm hydro}(k,z)}{P_{\rm DMO}(k,z)},\qquad \Delta S=S_{\rm fit}-S_{\rm sim}

Distinguish response, absolute response residual and relative error

\Delta S/S_{\rm sim}

. In the inspected figures, colour marks redshift and symbols distinguish simulation series.

Reading: Zhang, Baryonic feedback suppression of the matter power spectrum (2026), arXiv:2609.00807v1 · Original finite-mesh calculations and independent small-wave Python replay

Bring the question back to your own calculation

Write a result statement that names the matter population, halo definition, wavenumber units, counterpart in the denominator and error normalisation. Add the tested range and the cases that failed. Those choices determine what the sentence can support.

Our collisionless PM supplies a reproducible gravitational baseline. It contains no gas, star formation or black-hole feedback. Raising initial amplitude cannot stand in for baryonic ejection, and its knots do not predict a TNG halo catalog. That step requires additional physics, appropriate spatial resolution and independent checks.

An accurate power-response fit also does not by itself establish unbiased weak-lensing cosmological parameters. Observation kernels, noise, parameter degeneracies and calibration error must enter a concrete inference. The preprint’s proposed internal mitigation remains a question for that further test.

Reading: Zhang, Baryonic feedback suppression of the matter power spectrum (2026), arXiv:2609.00807v1 · Springel (2005), The cosmological simulation code GADGET-2, astro-ph/0505010v1 · Original finite-mesh calculations and independent small-wave Python replay

Exercises and explained answers

Does knowing $\Omega_m=0.3$ suffice to convert $M_{200\mathrm c}$ into $M_{200\mathrm m}$ ?
Read the explanation
No. The centre and density profile determine the new radius and enclosed mass. Lowering the density threshold does not scale mass by its inverse.
An axis gives $\ln(P/P_{\rm ref})=0.1$ . Is power exactly 10% higher?
Read the explanation
The exact ratio is $e^{0.1}\simeq1.1052$ , about 10.52% higher. Small log differences approximate fractional differences, with an approximation error.
At $S_{\rm sim}=0.5$ , a fit has $\Delta S=0.02$ . What is its relative error?
Read the explanation
It is $0.02/0.5=4\%$ . The absolute response residual remains $0.02$ . Match denominators before comparing percentage claims.

Try the prediction

Name what a knot has not told you

Select Three-way collapse and pause late. Compare its apparent size with the comoving mesh spacing.
List the definitions and resolution checks still required to call it an $M_{200\mathrm c}$ halo.
Read Figures 6 and 7, then write one gravitational claim this Lab supports and one question requiring baryonic physics.

Sources and calculation scope

Springel (2005), The cosmological simulation code GADGET-2, astro-ph/0505010v1
Focused reading: Section 2.1, equations 1–3; Section 3.2/3.2.1, equations 20–23; Section 4.1, equations 26–31, with original pages 3 and 11 viewed. Supplies cosmological momentum, PM windows and time-integration context. The teaching calculation specifies its own cutoff and finite gradient and adds no short-range tree force.
Springel et al., First results from IllustrisTNG: matter and galaxy clustering (2018), arXiv:1707.03397v2
Fixed v2: Methods 2.1–2.2 and original Figures 5–7 read. Figure 6 compares resolution and Figure 7 full-physics/DMO power; the response axes use natural logarithms. Ordered-particle shot-noise limits are discussed in Section 2.2. We inspect published methods and figures, without reproducing TNG runs or downloading snapshots.
IllustrisTNG official data specifications
Header MassTable, dark matter Coordinates/Velocities and group spherical-overdensity definitions checked on 12 November 2026. Snapshot and group velocity conventions differ; each field needs its own conversion. Numerical unit exercises use hypothetical values, not extracted particles. No TNG data or original assets are redistributed.
Zhang, Baryonic feedback suppression of the matter power spectrum (2026), arXiv:2609.00807v1
Submitted 1 September 2026; version labelled for submission to JCAP at the 12 November reading. Focused sections 2–5, Figures 1–4 and Table 1 inspected in the actual PDF/primary HTML. Absolute response residuals, single-/three-parameter differences and the low-matter-density stress exception are retained. No independent reproduction or demonstrated weak-lensing posterior is claimed.
Original finite-mesh calculations and independent small-wave Python replay
Original accepted kernel/linear, nonlinear and five-preset F64 calculations support the teaching figures and calibration. This small public ZIP contains only an exact current-state small-wave fixture, an independent NumPy/SciPy trajectory replay and expected output. The fixture was generated through the frozen model and canonical serializer in Node, not downloaded in a browser. Large raw preset archives remain separately retained; no Engine or source paper is included.