⚛️ phenomenology

Dark matter in scale-invariant gravity with hidden-sector condensation

This paper proposes a unified framework based on classically scale-invariant quadratic gravity coupled to a strongly interacting hidden sector, where hidden-sector confinement dynamically generates the Planck and electroweak scales while enabling Starobinsky inflation and the gravitational freeze-in production of various dark matter candidates.

Original authors: Juan P. Garcés, Jisuke Kubo, Manfred Lindner, Markus Reinig

Published 2026-08-14

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan P. Garcés, Jisuke Kubo, Manfred Lindner, Markus Reinig

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 puzzle where the pieces don't quite fit together. Scientists have two very successful instruction manuals: one for the tiny world of particles (the Standard Model) and one for the massive world of stars and galaxies (the Lambda Cold Dark Matter model). But when you try to combine them, you hit a wall. Why is the force of gravity so weak compared to the other forces? What is "dark matter" that holds galaxies together but refuses to show up in our detectors? And how did the universe suddenly inflate to its current size in a fraction of a second?

Usually, scientists treat these as three separate mysteries, solving them with three different, unrelated theories. But what if they are all connected? What if the rules that govern the very beginning of the universe are the same rules that create the mass of particles and hide the dark matter? This is the big question this paper tackles. It explores a theory where the universe started with no built-in "rulers" or "weights"—no pre-set sizes for anything. Instead, everything we see, from the weight of an atom to the expansion of the cosmos, had to be generated dynamically, like a machine building its own parts as it runs.

The Cosmic Self-Assembly Kit

In this paper, the authors propose a grand unification of these cosmic mysteries using a framework they call "classically scale-invariant quadratic gravity." That's a mouthful, so let's break it down. Imagine the universe as a blank canvas where the artist (nature) refuses to use any pre-measured paint tubes. There are no "1-inch" or "1-kilogram" labels on the tubes. Everything is just a ratio. In this world, mass and size don't exist until something happens to create them.

The paper suggests that the universe has a hidden "engine room" called a hidden sector. Think of this as a secret, ultra-dense neighborhood of particles that we can't see directly. This neighborhood is governed by a force similar to the strong nuclear force that holds protons together, but it's even more intense. In our visible world, this force creates protons and neutrons. In this hidden neighborhood, it creates a whole new zoo of particles.

Here is the magic trick: The authors suggest that the Planck mass (the scale that determines how heavy gravity is) and the electroweak scale (the scale that determines how heavy particles like electrons are) are not fundamental constants written in the stars. Instead, they are generated by the hidden sector "condensing," much like steam turning into water. When this hidden gas cools and clumps together, it spontaneously creates the "rulers" for the universe. It's as if the universe woke up, looked at its own hidden engine, and said, "Okay, now that we have this clump, let's define what a 'kilogram' is based on it."

The Inflaton: The Cosmic Balloon Pump

Once the hidden sector creates these scales, the paper brings in a star character: the scalaron. In the theory of "quadratic gravity," the math includes a term involving the square of the curvature of space ( $R^2$ ). This term acts like a hidden spring or a balloon pump.

The authors show that this scalaron is the perfect candidate for the inflaton—the thing that blew up the universe in the first fraction of a second (inflation). As the scalaron rolls down its energy hill, it inflates the universe, solving the mystery of why the cosmos is so big and smooth. Crucially, this isn't a new, made-up particle; it's a natural consequence of the gravity theory itself. It's like finding that the engine of a car naturally has a turbocharger, rather than having to bolt one on separately.

The Aftermath: Reheating and the Freeze-In

After the inflationary boom, the universe is cold and empty. The scalaron needs to stop inflating and start heating things up. This is called reheating. The scalaron vibrates and decays, dumping its energy into the universe.

Here is where the paper gets really clever. Because the hidden sector and our visible world only talk to each other through gravity (which is very weak), the dark matter particles in the hidden sector are produced in a very specific way called gravitational freeze-in.

Imagine the visible world is a busy party, and the hidden sector is a quiet room next door. The door is made of thick lead (gravity). The scalaron is a loudspeaker that vibrates the wall. Occasionally, a tiny bit of energy leaks through the wall, creating a new particle in the hidden room. Because the wall is so thick, these particles don't interact much; they just "freeze" in place as the universe expands. They don't need to be stable because of some magical symmetry; they are just too heavy and too isolated to decay or interact with us. They are the perfect dark matter candidates.

Three Candidates for the Dark Mystery

The authors don't just stop at the theory; they test three specific "recipes" for what these hidden particles could be, calculating how much of them would be left over today to match the amount of dark matter we observe.

The Hidden $\eta'$ Meson: In the first model, the hidden sector is like a single-flavor version of our own QCD (the theory of quarks). It produces a heavy particle called an $\eta'$ meson. The authors calculate that if this particle has a mass between $10^8$ and $10^{11}$ GeV (which is incredibly heavy, billions of times heavier than a proton), it would perfectly match the dark matter we see. However, they also check if this particle is stable. They find that if it's too heavy, it might decay into gravitons (particles of gravity) and disappear before the universe is old enough. So, they set an upper limit on how heavy it can be to survive until today.
The Hidden Vector Boson: The second model introduces a hidden force similar to the weak force, but with a twist. It creates massive "vector bosons" (like heavy versions of the W and Z bosons). These particles are protected by a "custodial symmetry," a kind of hidden rule that prevents them from decaying. The paper finds that these could be the dark matter, with masses ranging wildly from about 100 GeV up to $10^{10}$ GeV, depending on how strong the hidden force is.
The Charged Pions: The third model is a bit more complex. It creates charged "pions" (light particles) that are stable because of an unbroken hidden electric charge. But this model also produces massless particles (like hidden photons) that act as dark radiation. This is a key prediction: if this model is true, we should see a tiny extra amount of "relativistic degrees of freedom" (a fancy way of saying extra heat or speed) in the early universe, measured as $\Delta N_{eff}$ . The authors calculate that the hidden gauge coupling (how strong the hidden force is) must be small enough to keep this extra radiation within the limits set by current observations.

The Verdict

The paper doesn't claim to have "solved" dark matter. Instead, it suggests a beautiful, unified picture where inflation, the origin of mass, and dark matter are all different sides of the same coin. It proposes that we don't need to invent new, arbitrary rules for the universe; we just need to let the math of scale-invariant gravity and a hidden, strongly interacting sector do the work.

The authors show that this framework is consistent with what we know about the early universe (Starobinsky inflation) and can produce the right amount of dark matter through gravitational freeze-in. They provide specific mass ranges and coupling strengths for three different types of hidden particles that could fit the bill. While the hidden sector remains invisible to our particle colliders, the paper argues that cosmological observations—like measuring the precise amount of dark radiation or the ripples in the cosmic microwave background—could give us the clues we need to confirm if this self-assembling universe is the real story. It's a proposal that turns the "hierarchy problem" (why gravity is so weak) from a bug into a feature, suggesting that the weakness of gravity is actually the key to why the universe has the structure it does.

Technical Summary: Dark Matter in Scale-Invariant Gravity with Hidden-Sector Condensation

Problem Statement
The Standard Model (SM) of particle physics and the $\Lambda$ CDM model of cosmology leave fundamental questions unanswered, specifically the origin of the electroweak scale, the nature of dark matter (DM), and the mechanism driving cosmic inflation. These issues are often treated independently, yet they share a common root: the origin of physical scales. In a four-dimensional Quantum Field Theory (QFT), explicit mass parameters in the Lagrangian lead to the hierarchy problem and lack a dynamical origin. This work proposes a unified framework where the electroweak scale, the Planck scale, inflation, and dark matter all emerge from a single underlying principle: classical scale invariance. The authors argue that by forbidding dimensionful parameters at the classical level, all physical scales must be generated dynamically via quantum effects, analogous to dimensional transmutation in Quantum Chromodynamics (QCD).

Methodology
The authors construct a model based on classically scale-invariant quadratic gravity (QG) coupled to a strongly interacting hidden gauge sector. The framework consists of three sectors:

The SM in the scale-invariant limit: The Higgs mass parameter $\mu_H$ vanishes.
A hidden sector ( $G$ ): A non-abelian gauge theory (e.g., $SU(N_c)$ ) with non-abelian gauge interactions, analogous to QCD, which is completely orthogonal to the SM (no direct Higgs portal or shared fermions).
Quadratic Gravity: The gravitational action includes $R^2$ and Weyl-tensor-squared ( $C^2$ ) terms, containing only dimensionless couplings ( $\gamma$ and $\kappa$ ).

The generation of scales proceeds via a two-step process:

Step 1 (Planck Scale Generation): The hidden sector undergoes confinement and chiral symmetry breaking. A gluon condensate $\langle F^2 \rangle$ forms at a dynamical scale $\Lambda_F$ via dimensional transmutation. In a curved spacetime, this condensate induces the Einstein-Hilbert term, dynamically generating the Planck mass $M_{Pl}$ .
Step 2 (Electroweak Scale Generation): The breaking of scale invariance in the hidden sector is communicated to the SM solely through gravitational interactions. A non-minimal coupling between the Higgs and gravity ( $\xi_H H^\dagger H R$ ) allows the induced Planck mass to generate an effective Higgs mass term via a one-loop gravitational correction. In the semi-conformal limit ( $\xi_H \to -1/6$ ), this mechanism naturally yields the observed electroweak scale without fine-tuning.

Inflationary Dynamics
The $R^2$ term in the gravity sector introduces a scalar degree of freedom, the scalaron ( $\chi$ in the Jordan frame, $\phi$ in the Einstein frame). The authors analyze the multi-field potential involving the scalaron, the SM Higgs, and a composite scalar $\sigma$ (the chiral condensate of the hidden sector).

They demonstrate that in the semi-conformal limit, the Higgs decouples from inflation.
The composite scalar $\sigma$ is stabilized near the minimum of its Nambu-Jona-Lasinio (NJL) effective potential.
Consequently, the system reduces to an effectively single-field Starobinsky inflation scenario. The inflationary potential is $V(\phi) \propto (1 - e^{-\sqrt{2/3}\phi/M_{Pl}})^2$ .
The inclusion of the Weyl-squared term introduces a spin-two ghost, which modifies the tensor-to-scalar ratio $r$ but preserves the spectral index $n_s$ predictions within observational bounds.

Dark Matter Production Mechanism
Post-inflation, the scalaron oscillates around its minimum and reheats the universe. Crucially, the scalaron couples universally to the trace of the energy-momentum tensor ( $T^\mu_\mu$ ).

Gravitational Freeze-in: Since the hidden sector communicates with the SM only gravitationally, dark matter is produced via the gravitational freeze-in mechanism. The scalaron decays into hidden-sector states, which are stable due to the symmetries of the hidden confining theory.
Dark Radiation: The same reheating process produces massless hidden states (dark radiation), contributing to the effective number of relativistic degrees of freedom ( $N_{eff}$ ).
The authors derive general expressions linking the relic abundances of DM and dark radiation to the scalaron decay branching ratios and the reheating temperature, which is itself constrained by the number of inflationary e-folds ( $N_e$ ).

Key Contributions and Results
The paper investigates three representative realizations of the hidden sector to determine the viability of specific dark matter candidates:

Model I: Hidden $\eta'$ Dark Matter
- Setup: A single-flavor $SU(N_c)$ hidden sector. The lightest state is the hidden $\eta'$ meson (a pseudo-Nambu-Goldstone boson with mass generated by the axial anomaly).
- Stability: The $\eta'$ is stable against strong decays but can decay into gravitons via the mixed gravitational anomaly.
- Results: The required mass for the $\eta'$ to match the observed DM abundance ( $\Omega h^2 \approx 0.12$ ) lies in the range $10^8 - 10^{11}$ GeV, depending on $N_e$ . The authors derive an upper bound on the mass based on the requirement that the $\eta'$ lifetime exceeds the age of the universe.
Model II: Hidden Vector Dark Matter
- Setup: A hidden sector with $SU(N_c) \times SU(2)_L$ symmetry. The $SU(2)_L$ gauge bosons acquire mass via the Higgs mechanism (eating NGBs from the chiral condensate).
- Stability: The massive vector bosons ( $Z'$ ) are stabilized by an unbroken custodial $SU(2)_V$ symmetry, which forbids their decay into singlets (like gravitons).
- Results: The mass of the vector DM candidate depends on the gauge coupling $g_{Z'}$ and the decay constant $f_{\pi'}$ . For $g_{Z'} \sim O(1)$ , masses up to $\sim 10^2$ GeV are viable; for smaller couplings ( $g_{Z'} \sim 10^{-2}$ ), masses can reach $\sim 10^{10}$ GeV.
Model III: Hidden Pseudo-NGB Dark Matter
- Setup: Similar to Model II but with a gauged $U(1)_{A'}$ subgroup instead of $SU(2)_L$ .
- Stability & Radiation: The charged hidden pions ( $\pi'^\pm$ ) are stable due to the unbroken $U(1)_{A'}$ charge. The neutral pion ( $\pi'^0$ ) and the gauge boson ( $A'$ ) remain massless, contributing to dark radiation.
- Results: The charged pion mass required for the correct relic abundance is in the $10^8 - 10^{11}$ GeV range. The authors also calculate the contribution to $\Delta N_{eff}$ from the massless states, constraining the hidden gauge coupling $e_{A'}$ .

Significance and Claims
The paper claims to provide a unified origin for the electroweak scale, the Planck scale, inflation, and dark matter without introducing ad hoc mass scales or stabilizing symmetries.

Hierarchy Problem: The framework solves the hierarchy problem by generating all scales dynamically through dimensional transmutation in a hidden sector, communicated to the SM via gravity.
Inflation: It naturally realizes Starobinsky inflation as a consequence of the gravitational sector's $R^2$ term, removing the need for an ad hoc inflaton field.
Dark Matter: It demonstrates that stable dark matter is a generic prediction of hidden confining sectors, emerging naturally from the same dynamics that generate the electroweak scale.
Predictivity: The model links cosmological observables (tensor-to-scalar ratio $r$ , spectral index $n_s$ , dark radiation $\Delta N_{eff}$ , and DM abundance) to the parameters of the hidden sector and the duration of inflation.

The authors emphasize that while the hidden sector may be inaccessible to collider experiments, cosmological observations of inflationary parameters and dark radiation offer a powerful probe of the dynamics responsible for generating the fundamental scales of nature. The framework illustrates that inflation, scale generation, electroweak symmetry breaking, and dark matter can emerge from a single classically scale-invariant theory.

The Cosmic Self-Assembly Kit

The Inflaton: The Cosmic Balloon Pump

The Aftermath: Reheating and the Freeze-In

Three Candidates for the Dark Mystery

The Verdict

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