⚛️ phenomenology

A lower bound on primordial power spectrum from halo substructure

This study demonstrates that to remain consistent with observations of stellar streams and gravitational lensing, the primordial power spectrum must feature a log-normal bump with an amplitude of at least $10^{-9}$ at wavenumbers between 10 and 30 Mpc $^{-1}$ , as this specific enhancement is required to reproduce the observed halo and subhalo structures.

Original authors: Shin'ichiro Ando, Nagisa Hiroshima, Koji Ishiwata, Tomo Takahashi

Published 2026-07-24

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CC BY 4.0

Original authors: Shin'ichiro Ando, Nagisa Hiroshima, Koji Ishiwata, Tomo Takahashi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic ocean. Long ago, right after the Big Bang, this ocean was mostly smooth, but it had tiny, invisible ripples. These ripples were the seeds of everything we see today: stars, galaxies, and even you. Scientists call these ripples "primordial curvature perturbations." Think of them like the initial bumps on a trampoline; where the trampoline is bumpy, gravity pulls harder, gathering matter together to form structures.

For a long time, we've been able to map these ripples on the very largest scales, like looking at the ocean from a satellite. We know they are there and how big they are on those massive scales. But what about the tiny ripples, the ones that would create the small, rocky islands in our cosmic ocean? Those have been much harder to see. It's like trying to spot a single pebble on a beach while standing on a lighthouse. If there were no tiny ripples, the universe would be smooth and boring, with no small galaxies or satellite clusters. But if there were too many or too big ripples, the universe would look very different, perhaps filled with too many tiny black holes or strange, dense clumps. So, figuring out how "bumpy" the universe is on these small scales is a huge mystery. It helps us understand the very first moments of the universe and how the cosmic structures we love came to be.

Now, enter a team of scientists who decided to play detective using a very specific kind of cosmic magnifying glass. They wanted to know: "Is there a minimum amount of 'bumpiness' required to create the small galaxies and star clusters we actually see today?" To solve this, they built a digital simulation of the universe, but with a twist. They imagined the primordial ripples had a specific shape: a smooth background that suddenly stopped (a "cutoff") at a certain size, followed by a sudden, sharp spike or "bump" at even smaller sizes. It's like taking a smooth hill, cutting off the top, and then gluing a giant, jagged mountain peak right onto the flat spot.

The researchers then watched how their digital universe evolved. They asked: "If we have this specific shape of ripples, do we get enough small galaxies and star clusters to match what astronomers see in our own Milky Way?" They looked at two main clues: the number of tiny satellite galaxies orbiting our galaxy, and the way light from distant objects gets bent and distorted by invisible clumps of matter (gravitational lensing), as well as the gaps left in streams of stars that were ripped apart by gravity.

Here is what they found. If the "bump" in the ripples is too small or too weak, the simulation produces a universe that is too empty. It simply doesn't create enough small sub-galaxies to match the real observations. The stars would be too spread out, and the lensing effects would be too weak. The paper suggests that to make the universe look like the one we live in, there must be a significant amount of bumpiness at very small scales. Specifically, the amplitude of this bump needs to be at least around $10^{-9}$ (a very small number, but huge in the context of the early universe) at wavenumbers between 10 and 30 $Mpc^{-1}$ .

In simpler terms, the universe cannot be too smooth on small scales. If the primordial ripples were any quieter than this specific threshold, we wouldn't see the rich tapestry of small galaxies and star streams that we actually observe today. The study rules out models where the universe is too calm on these tiny scales. While they didn't prove exactly how these ripples were made, they successfully set a "floor" for how loud they must have been. If the early universe was quieter than this, our cosmic neighborhood would look very different, and we wouldn't be here to wonder about it.

Problem
While the amplitude of the primordial curvature perturbation is tightly constrained on large scales ( $\sim 10^{-9}$ ) by Cosmic Microwave Background (CMB) and large-scale structure observations, it remains poorly constrained on smaller scales ( $k \gtrsim \mathcal{O}(1) \text{ Mpc}^{-1}$ ). Existing constraints on small-scale power, derived from CMB spectral distortions, ultra-compact minihalos, and primordial black holes, primarily provide upper bounds. However, the observation of small-scale structures in the Milky Way, such as satellite counts, stellar streams, and gravitational lensing anomalies, implies the existence of a lower bound on primordial fluctuations required to source these structures. This paper aims to derive a lower bound on the primordial power spectrum ( $P_\zeta$ ) at small scales by analyzing the consistency of observed halo substructure with theoretical models.

Methodology
The authors model the primordial power spectrum of the curvature perturbation, $P_\zeta$ , as a nearly scale-invariant form with a sharp cutoff at $k_{cut} = 5 h \text{ Mpc}^{-1}$ and an additional log-normal bump at smaller scales ( $k > k_{cut}$ ). The model is defined by:
$P_\zeta = P_{CMB}^\zeta \Theta(k_{cut} - k) + P_{bump}^\zeta$
where $P_{bump}^\zeta$ is characterized by amplitude $A$ , peak wavenumber $k_b$ , and width $\Delta$ .

To connect this primordial spectrum to observable substructure, the authors employ the following steps:

Host Halo Evolution: They compute the time evolution of a Milky Way-like host halo mass ( $M_0 = 1.3 \times 10^{12} M_\odot$ at $z=0$ ) using the Extended Press-Schechter (EPS) formalism. Unlike previous works that relied on stochastic realizations, they utilize an efficient method calculating the expectation value of the host halo mass at each redshift step, validated against random realizations.
Subhalo Mass Function: Using the SASHIMI Python package, they calculate the subhalo mass distribution function. This incorporates the EPS formalism for accretion history and includes tidal stripping effects based on analytical formulations.
Correction for Cutoff: To address the unphysical production of halos in regions where the power spectrum is suppressed (below the cutoff), they introduce a collapse probability function, $P_{coll}(z, M)$ , derived from the variance contributed solely by the bump component. This modifies the subhalo mass function to ensure halos do not form where the primordial power is effectively zero.
Observational Comparison: The theoretical subhalo mass functions are compared against observational data from:
- Stellar streams and Gravitational Lensing: Providing constraints on the subhalo mass distribution function within 300 kpc of the Galactic center.
- Satellite Counts: Using the Dark Energy Survey and PanSTARRS1 data to count dwarf spheroidal galaxies with $V_{max} > 4 \text{ km/s}$ .

Key Contributions

Efficient Calculation: The authors demonstrate that calculating the host halo mass evolution via the expectation value of the EPS formalism yields results consistent with stochastic simulations but significantly reduces computational cost.
Lower Bound Derivation: They successfully derive a lower bound on the amplitude of the primordial power spectrum on small scales ( $k \sim 10-30 \text{ Mpc}^{-1}$ ) by requiring the model to reproduce the observed abundance of subhalos.
Tidal Stripping Analysis: The study explicitly quantifies the impact of tidal stripping on the subhalo mass function, showing that it broadens the distribution and improves the fit to observational data, particularly for narrow bumps ( $\Delta \sim 0.1$ ).
Parameter Constraints: The paper provides 95% confidence level exclusion limits on the parameter space of the bump amplitude ( $A$ ) and peak wavenumber ( $k_b$ ) for various widths ( $\Delta$ ).

Results

Amplitude Requirement: To be consistent with stellar stream and gravitational lensing data, the amplitude of the log-normal bump ( $P_\zeta$ ) must be at least $\sim 10^{-8}$ (for $\Delta = 0.1$ ) at wavenumbers $k_b \sim 20-40 h \text{ Mpc}^{-1}$ . Models with amplitudes significantly lower than this fail to produce sufficient subhalos in the $10^8 - 10^{10} M_\odot$ mass range.
Satellite Counts: Constraints derived from satellite counts are less severe than those from lensing and streams, yielding a lower bound of $P_\zeta \sim 10^{-9}$ . However, satellite counts provide a constraint over a broader range of scales.
Dependence on Width ( $\Delta$ ): For small widths ( $\Delta = 0.1$ ), the mass function is sharp, and tidal stripping is crucial for fitting the data. As $\Delta$ increases, the constraint on $k_b$ becomes less sensitive, and the allowed parameter space broadens.
Exclusion Regions: The analysis excludes regions where the bump amplitude is too low to generate the observed substructure. Specifically, for $k_b \gtrsim 80 h \text{ Mpc}^{-1}$ , the constraints become stringent because the resulting subhalos are too small to be detected as $V_{max} > 4 \text{ km/s}$ satellites.

Significance
The paper establishes that the existence of observed small-scale structures in the Galaxy imposes a strict lower bound on the primordial power spectrum at scales smaller than those probed by CMB. This finding is significant because it complements existing upper bounds, effectively narrowing the allowed parameter space for models of primordial fluctuation generation. The results suggest that mechanisms suppressing small-scale power (e.g., negative spectral running) must be consistent with these lower bounds. Furthermore, the study validates the use of stellar streams and gravitational lensing as powerful probes for constraining the physics of the early universe, specifically the generation mechanism of primordial fluctuations.

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