Halo Formation in Warm Dark Matter Models

Paul Bode; Jeremiah P. Ostriker; Neil Turok

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Halo Formation in Warm Dark Matter Models

Paul Bode, Jeremiah P. Ostriker, Neil Turok

arXiv:astro-ph/0010389v3Oct 19, 2000astro-ph

TL;DR

CDM faces small-scale discrepancies involving satellite counts, halo cores, and dwarf-galaxy distributions, motivating WDM as a modification that damps small-scale structure. Using high-resolution N-body simulations and comparisons with CDM, the paper finds that WDM produces less dense and less numerous low-mass halos that form late within a cosmic web, appearing more consistent with the cited observations. These qualitative signatures still require better quantification and higher-resolution simulations.

Problem
CDM predicts too many low-mass satellites, overly concentrated halo cores, and a dwarf-galaxy distribution unlike the observed avoidance of voids and suppression within large halos.
Method
The paper uses high-resolution N-body simulations comparing ΛCDM and ΛWDM across large boxes while balancing large-scale modes against dwarf-scale resolution.
Results
WDM produces less dense and fewer low-mass halos, with formation through top-down fragmentation in sheets and ribbons, later than bright galaxies and concentrated between massive halos.
Takeaways & Limitations
WDM’s halo abundances, spatial distribution, and formation epoch appear in better agreement with dwarf-galaxy observations than CDM’s.
Takeaways & Limitations
The reported effects remain qualitative and require higher-resolution simulations, substantial computation, and better observational quantification for precise statistical tests.

Abstract

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Discrepancies have emerged between the predictions of standard cold dark matter (CDM) theory and observations of clustering on sub-galactic scales. Warm dark matter (WDM) is a simple modification of CDM in which the dark matter particles have initial velocities due either to their having decoupled as thermal relics, or having been formed via non-equilibrium decay. We investigate the nonlinear gravitational clustering of WDM with a high resolution N-body code, and identify a number of distinctive observational signatures. Relative to CDM, halo concentrations and core densities are lowered, core radii are increased, and large halos emerge with far fewer low mass satellites. The number of small halos is suppressed, and those present are formed by `top down' fragmentation of caustics, as part of a `cosmic web' connecting massive halos. Few small halos form outside this web. If we identify small halos with dwarf galaxies, their number, spatial distribution, and formation epoch appear in better agreement with the observations for WDM than they are for CDM.

1. Introduction

The paper motivates WDM as a modest modification of ΛCDM that damps small-scale structure, addressing CDM discrepancies in satellite abundance, halo cores, and dwarf-galaxy distributions. Its simulations identify reduced halo densities and numbers, web-confined and late-forming low-mass halos, while noting that these effects require better quantification.

Motivation: The motivation is that CDM predicts too many low-mass satellites and overly concentrated cores, while observed dwarf galaxies avoid voids and are suppressed within large halos.The paper presents these as related small-scale discrepancies, while emphasizing that ΛCDM succeeds on larger scales.
Results: WDM simulations reduce massive-halo core densities, lower low-mass-halo densities, and suppress low-mass and satellite-halo abundances relative to CDM.The reported effects include smoother massive-halo cores with larger core radii and fewer low-mass halos and satellites.
Results: Low-mass halos form mainly by top-down fragmentation within caustic pancakes or ribbons connecting massive halos, leaving voids in the cosmic web nearly empty.This spatial pattern contrasts with CDM and is presented as a potential explanation for dwarf-galaxy distributions.
Results: WDM predicts late low-mass-halo formation and suppresses halo formation at Z > 5 while increasing evolution at lower redshifts relative to CDM.The paper connects these effects to observational tests of dwarf-galaxy formation at high redshift.
Limitations: The authors caution that all reported effects need better quantification in observations and simulations before precise statistical tests are possible.The numerical setup is explicitly described as a compromise between large-scale coverage and dwarf-galaxy resolution.

2. Warm Dark Matter

WDM is characterized by particle streaming that erases perturbations below a smoothing scale, with particle production mechanisms determining the effective temperature and mass. The paper emphasizes that the relevant streaming history requires an improved scaling beyond commonly used estimates, while viable models must avoid both CDM-like and hot-dark-matter-like limits.

Physical basis: WDM free streaming smooths perturbations below a comoving scale set by particle motion, with the comoving displacement converging during the matter era.The particle momentum redshifts with the scale factor, and the motion in comoving coordinates determines the smoothing length.
Physical basis: The paper derives an improved smoothing-scale scaling because particles become nonrelativistic before matter-radiation equality and perturbations grow during the intervening radiation-era interval.The authors state that earlier scaling relations did not properly account for streaming before equality.
Parameterization: The canonical WDM description uses a relativistically decoupled light fermion, but the physically relevant quantity is its streaming speed rather than particle mass alone.The effective temperature depends on relativistic degrees of freedom at decoupling, while the density fixes the mass parameter used for discussion.
Constraints: WDM requires an intermediate streaming-speed window: insufficient streaming leaves CDM-like structure, whereas excessive streaming delays formation as in hot dark matter.The Gunn-Peterson constraint requires enough early structure to ionize the universe by the highest observed quasar redshifts.
Production mechanisms: Thermal-rel relic scenarios reduce streaming by lowering the effective temperature and increasing particle mass, whereas nonthermal production includes asymmetries, condensate decay, sterile-neutrino resonance, and decays of massive particles or defects.The paper describes multiple mechanisms capable of producing warm particles through modest extensions beyond the standard model.

3. Early Structure Formation in WDM

WDM must preserve sufficiently early structure formation while suppressing small-scale power. The paper finds that particle masses below roughly 500–750 eV are strongly constrained, while WDM collapse occurs near its smoothing scale and can produce dwarf-like objects through late caustic fragmentation.

Early structure constraints: WDM requires a limited streaming-speed window: too little streaming reproduces CDM, while too much delays structure formation like hot dark matter.The Gunn–Peterson constraint requires sufficient early collapse for reionization.
Early structure constraints: A reionization calculation requires mX of order 400 eV for σ8 = 1.0 or 700 eV for σ8 = 0.9, with detailed treatment giving mX ≳ 0.75 keV.The collapse fraction is highly sensitive to σ8 because it enters squared in the Gaussian exponent.
Collapse near the smoothing scale: Below the WDM smoothing scale, fluctuations approach a constant and most mass collapses in objects near that scale, with collapse redshifts rising from roughly 3 to 7 as mX increases from 0.5 to 3 keV.Above the smoothing scale, WDM behaves like CDM.
Early structure constraints: For σ8 = 0.9, warm particle masses below ∼500 eV are strongly constrained because observed bright-galaxy halos must form by high redshift.For masses above 700 eV, more than 1% of dark matter has collapsed by redshift 4.
Dwarf formation: Dwarf-like objects form late through caustic fragmentation near larger galaxies, with an estimated collapse velocity of ∼6.5 km s−1 comparable to observations.Their inferred densities are compatible with the Gunn–Tremaine bound, and the estimate is identified as requiring further study.
Density scaling: Across cluster-to-dwarf scales, CDM predicts a characteristic-density ratio of ∼1400, whereas the observed ratio is 200.The comparison is presented as a possible additional problem for CDM, with a larger observational database desirable.

4. The Phase Space Constraint

The phase-space constraint limits how dense warm-particle halos can become, but primordial thermal velocities mainly affect the innermost halo regions. For warm masses above 1 keV, they are expected to be unimportant on kiloparsec and larger scales.

Phase-space constraint: Liouville’s theorem constrains WDM clustering because collisionless phase-space density is conserved, unlike the effectively unbounded primordial phase-space density of CDM.The constraint is most relevant for dense, low-dispersion systems such as dwarf spheroidal galaxies.
Thermal velocities: By halo-collapse times, gravitational heating greatly exceeds the initial thermal energy because most WDM mass collapses at redshifts Z < 5.For a 1 keV particle at Z = 5, the rms speed is only 0.25 km s−1.
Thermal velocities: Thermal motion mainly gives infalling particles angular momentum and creates a centrifugal barrier that alters structure only inside a small radius.The paper estimates this effect using infall into a pre-existing halo and a simplified halo profile.
Scale dependence: For mX larger than 1 keV, thermal velocities are unimportant for halo structure on scales of a kiloparsec or above, though they may affect very small halos.The possible small-halo effect arises if the characteristic barrier factor is of order unity.
Numerical test: The simulations were designed to separate the effects of suppressed small-scale power from those of increased thermal velocities during massive-halo collapse.The comparison included CDM and WDM runs with the appropriate power spectrum, with and without thermal velocities.
Dwarf-galaxy bound: Draco and Ursa Minor imply mX greater than about 400 eV for warm particles to dominate their cores, a weaker bound than other constraints.The inference uses core radii of ∼200 pc and velocity dispersion σ = 9 km s−1.

5. The Simulations

The simulations compare CDM with low-mass WDM models using matched cosmological setups and suppressed WDM small-scale power. Their resolution limits the smallest resolved objects and requires separate higher-resolution runs for a realistic 1.5 keV model.

Simulation setup: The main suite contains ΛCDM and ΛWDM simulations with mX = 350 eV and 175 eV in 20 h−1 Mpc boxes containing 256^3 particles.All models use ΩX = 0.3, ΩΛ = 0.7, h = 0.67, and σ8 = 0.9; the WDM thermal speeds are v0 = 0.12 and 0.048 km s−1.
Initial conditions: The initial conditions suppress WDM small-scale power through its transfer function and add random velocities drawn from the appropriate Fermi–Dirac distribution.The three simulations use identical displacement and velocity phases for comparison.
Halo identification: Halos are identified with HOP using density smoothing over 32 particles, an outer contour of 100 times the mean, and subsequent removal of unbound particles.The most bound particle defines the halo center for subsequent measurements.
Resolution and scope: The large-box simulations resolve only objects ≳h−1 10^9 M⊙ and are shown only to redshift unity because lower-redshift evolution becomes computationally costly.Additional 128^3-particle, 3 h−1 Mpc simulations reach redshift zero for mX = 1.5 keV.

6. Evolution and Clustering Pattern

WDM preserves large-scale clustering but suppresses small-scale structure, concentrating low-mass halos within thin cosmic-web caustics rather than voids. These halos arise through fragmentation of ribbons and pancakes.

6. Evolution and Clustering Pattern: WDM small halos trace thin caustic sheets in the cosmic web, whereas CDM halos with M < 9 × 10^9h−1M⊙ fill voids.The large-scale clustering pattern is identical across models, but the small-scale spatial distribution differs strongly.
6. Evolution and Clustering Pattern: The median local overdensity of low-mass halos rises from 1.5 in CDM to 4 and 7 in 350 and 175 eV WDM, respectively.Almost all low-mass WDM halos lie above twice the mean density, while half of the corresponding CDM halos lie below it.
6. Evolution and Clustering Pattern: WDM massive halos have roughly five times fewer satellites than CDM massive halos.The satellite count depends on accretion and tidal destruction; later-forming WDM satellites are less dense and more easily disrupted.
6. Evolution and Clustering Pattern: Low-mass halos form mainly through top-down fragmentation of large ribbons and pancakes within a scenario that remains bottom-up on larger scales.The simulations show a late-time instability whose fastest-growing wavelength is comparable to the ribbon thickness.

7. Halo Abundances and Structure

WDM suppresses low-mass halo abundances and produces later-forming, less dense halos with lower concentrations than CDM. The simulations attribute the low-mass population to pancake fragmentation rather than early hierarchical collapse.

7. Halo Abundances and Structure: WDM strongly suppresses low-mass halos below the scale set by the linear-power-spectrum cutoff.The ΛCDM mass function is fit across the simulated mass range, whereas WDM shows a strong low-mass suppression.
7. Halo Abundances and Structure: WDM low-mass halos are formed later by pancake fragmentation and therefore have lower densities than same-mass CDM halos.The mass function steepens again at the lowest masses, but these objects are not early-forming counterparts of CDM halos.
7. Halo Abundances and Structure: Resolution tests reproduce the abrupt mass-function slope change near 10^11h−1M⊙, supporting the reality of caustic fragmentation.The reduced-resolution run yields a similar mass function, although it contains somewhat fewer low-mass halos.
7. Halo Abundances and Structure: Between redshifts 4 and 1, the low-mass ΛCDM abundance changes by less than a factor of two, unlike the evolving WDM population.This difference is used to examine whether WDM small halos form through caustic fragmentation.
7. Halo Abundances and Structure: WDM halos have lower central densities and concentrations than corresponding CDM halos, with differences largest at low masses.The reported profiles use NFW fits; the lower densities agree better with nearby dwarf-galaxy observations.

8. A Higher Warm Particle Mass mX

A 1.5 keV WDM simulation retains the qualitative differences found at lower particle masses. It suppresses low-mass halos and satellites while producing denser environments, lower concentrations, and larger cores than CDM.

8. A Higher Warm Particle Mass mX: At mX = 1.5 keV, WDM suppresses low-mass halos by over a factor of three and places them along caustic structures rather than in voids.Only 20% of WDM halos lie below unit local overdensity, compared with 40% of CDM halos.
8. A Higher Warm Particle Mass mX: The ten largest WDM halos contain 17 satellites, compared with 97 around the ten largest CDM halos.The higher-mass run therefore retains the strong satellite suppression seen in the larger simulations.
8. A Higher Warm Particle Mass mX: In the 1.5 keV run, WDM mean concentration is lower by a factor of 2.5 and mean core radius is larger by a factor of 3 than in CDM.The profiles remain distinct even at a particle mass closer to the stated observational limits.
8. A Higher Warm Particle Mass mX: The authors conclude that the lower-mass simulation trends persist as the warm particle mass increases, while noting that more detailed study is needed.This conclusion is based on the additional pair of 1.5 keV runs.

9. Conclusions

The paper identifies top-down formation below a characteristic mass as the principal WDM distinction from CDM. The resulting low-mass halos are rarer, less dense, concentrated in cosmic-web sheets, and formed later, motivating targeted observational tests.

9. Conclusions: Below a characteristic mass, WDM halos form mainly by fragmentation of pancakes and ribbons, unlike the corresponding early-forming CDM halos.These objects are rarer and considerably less dense than same-mass CDM halos.
9. Conclusions: WDM low-mass halos concentrate in sheets and ribbons between massive halos, matching the observed local-universe distribution of dwarf galaxies more closely than CDM.The paper connects this prediction to the apparent absence of dwarf systems in voids (Peebles 2000, 2001).
9. Conclusions: WDM predicts that dwarf galaxies form later than bright galaxies, in better agreement with observations than the hierarchical CDM picture (Metcalfe et al. 2000).The authors propose testing this through observations of galaxy-formation histories.
9. Conclusions: The proposed tests target reionization and Lyman α clouds, galaxy-formation histories, dwarf abundances and profiles, satellite counts, and inner rotation curves.These observations are intended to distinguish the predicted WDM signatures from CDM.
9. Conclusions: More detailed WDM simulations with higher resolution and substantial computation are needed to quantify the reported qualitative features.This is the paper’s stated limitation and computational scope boundary.

A. Appendix: Linear Theory

The appendix derives the linear-theory suppression of warm-dark-matter perturbations from thermal relic velocities and free streaming, then calibrates the resulting transfer function for N-body inputs. It shows that the smoothing scale depends on particle mass and cosmological parameters, with additional dependence arising from when particles become nonrelativistic.

The free-streaming scale is estimated from the comoving distance traveled after matter-radiation equality, but this crude scaling is corrected for additional streaming during the radiation era.The analytic treatment uses a simplified Gilbert-equation solution to expose the main transfer-function parameter dependence.
Free streaming suppresses small-scale perturbations through the particles’ velocity dispersion, while perturbation growth is absent during the relativistic phase.The Gilbert equation is applied after the maximum Jeans length is reached, when the particles become nonrelativistic.
The calculation specifies an initial non-gravitating perturbation, evolves the Gilbert equation numerically into the matter era, and compares the growing mode with the unaffected k = 0 mode.A simple isotropic ansatz is used for the initial perturbation when only suppression relative to cold dark matter is required.
The Gilbert-equation calculation yields a smoothing-scale fit with α = 0.05 (Ω_X/.4)^.15(h/.65)^1.3(keV/m_X)^1.15(1.5/g_X)^.29.The fitted dependence is expressed with k in h Mpc^-1.
The smoothing-length scaling weakens as m_X increases because higher-mass particles become nonrelativistic sooner and free-stream longer during the radiation era.The additional parameter dependence gives 4β ≈ −0.183.
The warm-dark-matter power spectrum used in N-body simulations is obtained by multiplying the cold-dark-matter spectrum by |T_X(k)|^2.A full Boltzmann calculation provides the transfer-function fit, accurate to a few per cent over the relevant range of k.