Enhanced Linear Matter Power Spectra

• Enhanced linear matter power spectra are modified CDM spectra that include localized power bumps to simulate inflationary features and nonstandard dark matter physics.
• They alter the subhalo mass function by amplifying substructure at mass scales linked to bump parameters like amplitude, scale, and width.
• Nonlinear simulations reveal that these enhancements persist over time, offering a cosmological probe into early-universe dynamics and dark matter properties.

Enhanced linear matter power spectra refer to matter power spectra P(k) that deviate from the standard cold dark matter (CDM) case by exhibiting local or global increases in power at specific wavenumbers, as opposed to the canonical smooth, featureless spectrum predicted by inflation and evolved through gravitational instability. These enhancements can be motivated by a range of physical mechanisms, including features in the early-universe transfer function, nonstandard dark matter microphysics, or primordial power spectrum modulations. The astrophysical implications of such enhancements are manifest primarily through their impact on the abundance, structure, and spatial distribution of dark matter subhalos, with distinct signatures observable in galactic substructure (Nadler et al., 22 Jul 2025).

1. Characterization and Generation of Enhanced Linear Power Spectra

Enhanced linear matter power spectra are typically constructed by imposing a localized excess in the transfer function relative to ΛCDM,

$$\mathcal{T}_\text{bump}(k) = 1 + A \exp\left[ - \frac{(\log k - \log k_0)^2}{\sigma_k^2} \right],$$

where $A$ sets the amplitude, $k_0$ the central wave number of the enhancement, and $\sigma_k$ its logarithmic width. The modified linear power spectrum is then

$$P_\mathrm{mod}(k) = \mathcal{T}_\text{bump}^2(k) P_\mathrm{CDM}(k),$$

where $P_\mathrm{CDM}(k)$ is the standard cold dark matter linear theory spectrum. Frequently, a small-scale cutoff is introduced in addition to the bump to mimic effects such as free streaming or collisional damping, enforcing a suppression of power for $k > k_\mathrm{cut}$.

The motivation for such modifications includes:

• Inflationary features: Step-like modulations or bumps from nontrivial inflaton potential dynamics, resonant effects, or transient non-slow-roll episodes.
• Nonstandard dark matter physics: For example, early interactions, decays, or thermal histories that lead to scale-dependent modifications to $P(k)$, including both enhancements and cutoffs.

2. Impact on the Dark Matter Subhalo Mass Function

The subhalo mass function (SHMF), which characterizes the number of gravitationally bound subhalos as a function of their peak mass $M_\mathrm{peak}$, is acutely sensitive to features in the linear matter power spectrum. In simulations with a Gaussian enhancement at $k_0$ (Nadler et al., 22 Jul 2025):

• The SHMF exhibits an amplification at masses $M_\mathrm{peak}$ corresponding to comoving scales associated with the bump ($M \propto k^{-3}$). The enhancement typically peaks near and just below the mass scale seeded by the excess in $P(k)$.
• The amplitude and precise location of the SHMF bump directly follow the parameters $A$, $k_0$, and $\sigma_k$.
• When a cutoff is introduced at $k > k_\mathrm{cut}$, the SHMF is sharply suppressed below the corresponding mass, providing a hard truncation to the spectrum of small subhalos.
• In SHMF ratio plots, $$f_\mathrm{sub}(M_\mathrm{peak}) = \left[ dN/d\log M_\mathrm{peak} \right]_\mathrm{mod} / \left[ dN/d\log M_\mathrm{peak} \right]_\mathrm{CDM}$$, the feature appears as a broad enhancement (bump) followed by a downturn at lower masses (if a cutoff is present).

This effect establishes a direct mapping between features in the linear power spectrum and the emergent subhalo population, allowing in principle for inverse constraints on $P(k)$ from observed or simulated substructure properties.

3. Dependence of Substructure Properties on Bump Scale and Shape

The subhalo properties respond intricately to the detailed location and shape of the enhancement in $P(k)$:

• The central scale $k_0$ of the bump intrinsically sets the characteristic subhalo mass $M_0$ of the SHMF inflection. Lower $k_0$ (larger scales) leads to SHMF enhancement at higher mass; higher $k_0$ targets smaller subhalo masses.
• The presence or absence of a cutoff at small scales dictates whether the SHMF enhancement is followed by suppression (with cutoff) or by a plateau (without cutoff).
• The width parameter $\sigma_k$ determines how diffuse the enhancement is in $M_\mathrm{peak}$, affecting SHMF breadth.
• The amplitude $A$ governs the strength of the amplification, providing a potential lever for early-universe parameter constraints based on subhalo statistics.

Consequently, observations of the SHMF—e.g., via satellite galaxy counts or gravitational lensing—can be employed to probe the detailed structure of the primordial or transfer function power spectrum if systematic uncertainties can be controlled.

4. Role of Nonlinear Evolution and Mode Coupling

The relationship between the initial (linear) $P(k)$ and late-time subhalo abundance is not one-to-one, due to nonlinear mode coupling during structure formation:

• While features in $P(k)$ are imprinted in the initial field, gravitational collapse and mergers transfer power between scales (“mode coupling”), altering the eventual SHMF.
• Detailed zoom-in simulations indicate that, for localized enhancements, the resulting SHMF feature remains relatively sharp after nonlinear evolution (Nadler et al., 22 Jul 2025), implying that nonlinear processes do not completely erase the signature of the initial bump.
• The SHMF enhancement is primarily set at infall: subhalos are formed with excess mass and concentration, and this imprint persists through subsequent evolutionary processes.
• Nevertheless, accurate $P(k)$ reconstruction from substructure properties necessitates explicit modeling of these nonlinearities to avoid misinterpretation or bias, as the mapping is not strictly linear.

This requirement complicates inversion but preserves the possibility of using subhalo statistics as probes of the small-scale primordial universe.

5. Enhancement Effects on Subhalo Structure and Spatial Distribution

Beyond the number abundance, enhanced linear power spectra modify subhalo internal structure and spatial arrangement:

• A pronounced $P(k)$ bump leads to higher initial subhalo masses and, more significantly, increased concentrations. Concentration is typically quantified as $$c_\mathrm{eff} = R_\mathrm{vir} / (R_\mathrm{max} / 2.126)$$, and simulations show a systematic upward shift in $c_\mathrm{eff}$ for the affected subhalos.
• Subhalos assembled under enhanced $P(k)$ are found to reside closer to the host halo center than their CDM counterparts. This centralization may reflect the infall of more massive (hence more tightly bound) subhalos and their greater resilience to tidal disruption, a consequence of their elevated concentrations.
• Nonlinear gravitational evolution preserves the signature of the $P(k)$ enhancement in these internal and spatial properties when the enhancement scale corresponds to well-resolved structures.

The combined signatures in SHMF, concentration, and spatial profile offer a multi-dimensional diagnostic for the detailed form of the linear matter spectrum.

6. Astrophysical and Cosmological Implications

The possibility of enhanced linear power spectra carries multiple implications:

• Detailed measurements of satellite galaxy abundances, subhalo lensing, or other indirect probes of substructure may allow for empirical constraints on $P(k)$ at scales otherwise inaccessible to large-scale structure or CMB data.
• The paper demonstrates that small-scale cosmological data (especially counts and properties of low-mass dark matter subhalos) have the potential to reveal or limit early-universe or dark matter microphysical phenomena that generate localized spectral enhancements.
• The persistence of SHMF features after nonlinear evolution suggests that subhalo statistics are viable tools for reconstructing primordial fluctuations, provided that mode coupling is handled using suitably resolved simulations and analytic models.
• Bounding or detecting nonstandard features in $P(k)$ has implications for the dynamics of galaxy formation, the nature of dark matter, and the interpretation of current and future small-scale structure data.

These results emphasize the necessity of accurate, high-resolution modeling of both the initial power spectrum and the nonlinear dynamics of dark matter, to leverage astrophysical observations for fundamental physics.

7. Summary Table: Effects of Linear Matter Power Spectrum Enhancement

Enhancement Parameter	SHMF Effect	Subhalo Structural Effect
Bump amplitude $A$	Sets size of SHMF amplification	Higher $A$ → stronger concentration
Bump scale $k_0$	Sets $M_\mathrm{peak}$ of SHMF feature	Affects spatial scale of effect
Cutoff (yes/no; $k_\mathrm{cut}$)	SHMF suppression at low $M_\mathrm{peak}$	May affect minimum subhalo mass
Width $\sigma_k$	Controls mass range of enhancement	Broader $\sigma_k$ → wider feature

The mapping from $P(k)$ enhancement features to subhalo observables enables utilization of substructure as a cosmological probe, contingent on precision modeling of both initial conditions and nonlinear structure formation (Nadler et al., 22 Jul 2025).