⚛️ phenomenology

Lukewarm inflation, primordial black holes and gravitational waves

This paper proposes that a secondary period of lukewarm inflation can simultaneously dilute unwanted thermal relics and generate a nearly scale-invariant enhancement of small-scale curvature perturbations, thereby producing a stochastic gravitational wave background consistent with the NANOGrav signal and a population of sub-solar mass primordial black holes that could constitute a significant fraction of dark matter.

Original authors: Paulo B. Ferraz, António Torres Manso, João G. Rosa

Published 2026-09-18

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CC BY 4.0

Original authors: Paulo B. Ferraz, António Torres Manso, João G. Rosa

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

The universe began in a moment of unimaginable expansion, a phase known as inflation, where space itself stretched faster than the speed of light. This rapid growth smoothed out the cosmos, leaving behind a nearly uniform sea of energy and matter, yet it also planted tiny seeds of irregularity. Over billions of years, gravity pulled on these seeds, causing them to clump together into the stars, galaxies, and vast cosmic webs we see today. While this story explains the large-scale structure of the universe, it leaves a few puzzles unsolved. For instance, the intense heat generated when inflation ended should have created exotic, unwanted particles that we do not see today. Furthermore, the standard story suggests that the universe's expansion should have been perfectly smooth, yet recent observations hint at ripples in the fabric of space-time that are far too large to be explained by the usual model. Scientists have long wondered if there was a second, quieter chapter in the universe's infancy that could fix these problems and explain the strange signals we are just beginning to detect.

A team of researchers has proposed a new scenario called "lukewarm inflation" to address these mysteries. Unlike the standard model, which assumes the universe was a cold, empty void during its expansion, or "warm inflation," which requires scorching temperatures near the grand unification scale, this new idea suggests a secondary period of expansion occurred at a much lower, more moderate temperature. In this phase, the field driving the expansion interacted with a bath of radiation, creating a kind of friction that slowed its descent. This friction did more than just sustain the expansion; it acted as a magnifying glass for the tiny fluctuations in the early universe. The researchers found that this process could significantly boost the power of these ripples on small scales, creating a nearly flat, consistent spectrum of disturbances that stretches across a vast range of sizes.

The consequences of this amplified activity are profound and observable. When these enhanced ripples re-entered the universe as it cooled, they did not just create stars; they could have collapsed directly into primordial black holes. These are not the massive black holes found at the centers of galaxies, but rather tiny, sub-solar mass objects, some as light as asteroids and others as heavy as planets. The study suggests that these primordial black holes could make up a significant fraction, or perhaps even all, of the dark matter that holds galaxies together. This offers a compelling solution to the dark matter problem, proposing that the invisible mass shaping our universe is actually a sea of ancient, microscopic black holes formed in the first moments of existence.

Simultaneously, the violent formation of these black holes and the intense fluctuations in the early universe would have generated a background hum of gravitational waves. These are ripples in space-time that travel through the cosmos, carrying information about the events that created them. The researchers calculated that this background signal would have a very specific shape: a nearly constant energy density across a broad range of frequencies, followed by a sharp drop-off at a maximum frequency. This distinct signature matches the recent signal detected by the NANOGrav collaboration, which has been observing the timing of pulsars to hunt for gravitational waves. If the signal persists at higher frequencies with the same strength, it would serve as a "smoking gun" for this specific type of lukewarm inflation, distinguishing it from other potential sources of cosmic noise.

The beauty of this proposal lies in how it connects two seemingly unrelated phenomena: the abundance of dark matter and the detection of gravitational waves. The same physical process that creates the black holes also generates the gravitational waves, meaning that observing one provides direct insight into the other. The researchers showed that by measuring the frequency of the gravitational wave signal and the mass of the resulting black holes, scientists could determine the exact conditions of this early inflationary period, such as the temperature at which it occurred and how long it lasted. This creates a testable framework where future experiments, such as space-based gravitational wave detectors, could confirm or rule out this scenario.

Crucially, this model also solves the problem of unwanted thermal relics. In many theories of physics beyond the standard model, the extreme heat of the early universe should have produced stable, heavy particles that would still be around today, yet they are nowhere to be found. The secondary period of lukewarm inflation acts as a cosmic diluter, expanding the universe enough to wash away these unwanted particles while preserving the seeds of the galaxies we see. This dual function—cleaning up the early universe while simultaneously seeding it with dark matter and gravitational waves—makes the scenario highly attractive. The researchers emphasize that while their calculations are robust, the specific details depend on the nature of the particles involved, and they plan to refine their models to include more complex particle physics setups.

Ultimately, this work offers a coherent picture of a universe that underwent a second, milder burst of expansion. It suggests that the dark matter surrounding us might be composed of tiny black holes born from a burst of friction in the early cosmos, and that the gravitational waves we are now detecting are the echoes of that same event. If future observations confirm the predicted sharp cut-off in the gravitational wave spectrum and find evidence for these specific black hole masses, it would validate a new chapter in the history of the universe, one where the warmth of the early cosmos played a decisive role in shaping the world we inhabit today.

Technical Summary: Lukewarm Inflation, Primordial Black Holes, and Gravitational Waves

Problem Statement
The standard inflationary paradigm, while successful in explaining the nearly scale-invariant spectrum of super-horizon curvature perturbations observed in the Cosmic Microwave Background (CMB) and Large Scale Structure (LSS), faces theoretical challenges. These include the fine-tuning required for the scalar potential (the "eta-problem") and the high temperatures attained during the subsequent reheating process. Such high temperatures can lead to the thermal production of unwanted relics predicted by Beyond the Standard Model (BSM) scenarios, such as gravitinos, scalar moduli, or topological defects. Furthermore, standard single-field inflation typically generates small curvature perturbations on small scales, limiting the formation of Primordial Black Holes (PBHs) and the associated stochastic gravitational wave background (SGWB) to negligible levels.

Methodology
The authors propose a scenario involving a secondary period of "lukewarm inflation" occurring at temperatures parametrically below the Grand Unified Theory (GUT) scale. This model utilizes the dynamics of warm inflation, where a scalar inflaton field follows a slow-roll trajectory sustained by thermal friction arising from interactions with a nearly-thermal radiation bath.

The dynamics are described by an effective Langevin equation for the scalar field $\phi$ :
$\ddot{\phi} + 3H\dot{\phi} + \Upsilon\dot{\phi} + V'(\phi) = \xi$
where $H$ is the Hubble parameter, $\Upsilon$ is the dissipation coefficient, $V(\phi)$ is the potential, and $\xi$ is a Gaussian white noise term. The radiation energy density $\rho_R$ evolves according to $\dot{\rho}_R + 4H\rho_R = \Upsilon\dot{\phi}^2$ .

The authors specifically adopt the "Warm Little Inflaton" (WLI) setup, where the inflaton arises from the collective breaking of a $U(1)$ gauge symmetry. In this framework, the dissipation coefficient scales linearly with temperature ( $\Upsilon = C_T T$ ). The analysis assumes a strong dissipation regime ( $Q = \Upsilon/3H \gg 1$ ), which is naturally achieved at lower temperatures compared to GUT-scale warm inflation. The study employs the Press-Schechter formalism to estimate the PBH mass function and utilizes analytical approximations for the induced SGWB spectrum generated by second-order tensor perturbations sourced by large curvature fluctuations.

Key Contributions and Results

Enhanced Curvature Perturbations: The authors demonstrate that in the lukewarm regime ( $T \ll T_{GUT}$ ), the dissipation coefficient $Q$ becomes very large. This leads to a significant enhancement of the primordial curvature power spectrum $\Delta^2_R$ on small scales compared to the primary inflation stage. The spectrum is nearly scale-invariant over an exponentially large range of comoving scales ( $k_{min} < k < k_{max}$ ), with an amplitude that decreases as the temperature of the lukewarm inflation increases.
$\Delta^2_R \propto Q^{3/2} \frac{T}{H} G(Q)$
where $G(Q)$ grows as a power law ( $G(Q) \propto Q^3$ ) in the strong dissipation limit.
Stochastic Gravitational Wave Background (SGWB): The large curvature perturbations source a SGWB at second order. The resulting energy density spectrum $\Omega_{GW}(f)$ is nearly scale-invariant (with a tilt dependent on the spectral index $n_s$ ) over a broad frequency range corresponding to the scales exiting the horizon during lukewarm inflation. The spectrum exhibits a sharp high-frequency cut-off at $f_{max}$ and follows an $f^3$ infrared behavior below $f_{min}$ . The authors show that this scenario can reproduce the amplitude and shape of the signal reported by the NANOGrav collaboration, provided the signal persists at higher frequencies with a comparable magnitude up to the cut-off.
Primordial Black Hole (PBH) Formation: The enhanced curvature perturbations lead to the gravitational collapse of overdense regions upon horizon reentry, forming a population of sub-solar mass PBHs. The mass distribution of these PBHs depends critically on the spectral tilt $n_s$ :
- Blue-tilted ( $n_s > 1$ ): The PBH mass function exhibits a sharp peak at a characteristic mass $M_* \propto f_{max}^{-2}$ .
- Red-tilted ( $n_s < 1$ ): The mass function displays a broad plateau above the mass threshold $M_*$ .
  The transition mass is given by $M_* \simeq 5 \times 10^{-7} M_\odot (100/g_*)^{1/2} (T_R/\text{TeV})^{-2}$ .
Dark Matter and Relic Dilution: The scenario naturally dilutes unwanted thermal relics produced after the first inflationary stage. Furthermore, the authors find that lukewarm inflation can generate a significant fraction of dark matter in the form of PBHs within the asteroid ( $10^{-16} - 10^{-12} M_\odot$ ) and planetary mass windows. This is particularly relevant for explaining potential microlensing candidates observed by HSC-Subaru in the $10^{-7} - 10^{-6} M_\odot$ range.

Significance and Claims
The paper claims that lukewarm inflation offers a unified explanation for several cosmological phenomena:

It provides a mechanism to dilute unwanted BSM relics without requiring a separate reheating phase, as radiation naturally becomes dominant at the end of the lukewarm stage.
It generates distinctive observational signatures: a nearly constant SGWB energy density over a broad frequency range and a specific PBH mass distribution.
It offers a potential explanation for the NANOGrav signal if that signal extends to higher frequencies with a sharp cut-off.
It establishes consistency relations between observables (e.g., the relation between the peak PBH mass $M_*$ and the cut-off frequency $f_{max}$ , and the slopes of the GW and PBH spectra) that allow for the determination of the underlying model parameters ( $A_s, n_s, k_{min}, k_{max}, T_R, N_e, C_T$ ).
The authors note that a lukewarm inflation period at temperatures around a few TeV would signal new physics just above the energy scale currently probed by the LHC.

The authors conclude that while non-Gaussian effects and specific particle physics realizations (such as those involving right-handed neutrinos) require further investigation, the lukewarm inflation scenario presents a viable and testable framework for early universe cosmology with unique, potentially observable consequences.

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