Non-local gravity effects in cosmological dynamics probed by IceCube/KM3NeT signals and dark matter relic abundance

This paper investigates how non-local gravity models, reduced to scalar-tensor theories via Noether symmetries, can simultaneously explain the high-energy astrophysical neutrino fluxes observed by IceCube/KM3NeT and the cosmological abundance of dark matter through a specific four-dimensional operator.

The Gist

The Big Idea: Fixing the Universe's "Recipe"

Imagine the universe is a giant, complex cake being baked. For decades, physicists have been using a standard recipe called General Relativity (Einstein's theory of gravity) to describe how the ingredients mix and how the cake rises. This recipe works perfectly for small things (like planets) and medium things (like galaxies).

However, when scientists look at the entire cake (the whole universe) or try to bake it at the very beginning (the Big Bang), the recipe starts to fail. It predicts things that don't match reality, like the mysterious "Dark Matter" and "Dark Energy" that we can't see but know are there.

This paper suggests that the standard recipe is missing a secret ingredient: Non-local gravity.

What is "Non-local Gravity"?

In our daily lives, things usually happen locally. If you push a car, the car moves. The effect is right next to the cause.

Non-local gravity suggests that gravity can act like a "ghostly echo." A change in gravity here might instantly affect the universe over there, or the gravity we feel today might be influenced by what happened billions of years ago. It's like a phone call where the person on the other end hears your voice before you even speak, or a ripple in a pond that somehow knows where the stone will land before it hits the water.

The authors propose that if we add this "echo" effect to our gravity recipe, we can solve two massive mysteries at once.

The Two Mysteries: The "PeV" Neutrinos and the "Ghost" Particles

The paper focuses on two recent, puzzling observations:

1. The IceCube/KM3NeT Neutrinos: Deep under the ice in Antarctica and in the Mediterranean Sea, giant detectors are catching "ghost particles" called neutrinos. Recently, they found some with incredibly high energy (up to 220 PeV). It's like finding a single grain of sand that hits with the force of a speeding train. We don't know where they come from.
2. Dark Matter: We know about 27% of the universe is made of invisible "Dark Matter" that holds galaxies together. But we don't know what it is.

The Connection: A Broken Clock

Here is the problem the authors found:
If you assume the universe follows the standard recipe (General Relativity), the math says:

• To create enough Dark Matter to fill the universe, the "ghost particles" (neutrinos) should be very rare and weak.
• But to explain the super-powerful neutrinos IceCube found, the "ghost particles" should be very common and strong.

These two requirements are like trying to fit a square peg in a round hole. They contradict each other. In the standard universe, you can't have both the right amount of Dark Matter and those super-energetic neutrinos.

The Solution: Changing the Speed of the Universe

The authors say, "What if the universe didn't expand at the speed we think it did in the very early days?"

Imagine the universe is a car driving down a highway.

• Standard Theory: The car drove at a steady speed the whole time.
• This Paper's Theory: In the very beginning (before the universe cooled down enough for atoms to form), the car was driving on a different gear. It was accelerating or decelerating in a weird way because of that "Non-local gravity" echo.

They used a mathematical tool called Noether Symmetry (think of it as a "magic key" that unlocks the correct shape of the universe's expansion) to find these special "gears."

The Result: Everything Fits!

When they plugged this "Non-local" speed into their calculations:

1. The Dark Matter: The universe expanded just fast enough to create the exact amount of Dark Matter we see today.
2. The Neutrinos: The expansion rate also allowed those Dark Matter particles to decay in a way that produces the super-energetic neutrinos IceCube is seeing.

It's as if they found a single setting on a thermostat that makes the room the perfect temperature and smells like fresh coffee at the same time. In the standard universe, you have to choose one or the other. In this "Non-local" universe, you get both.

The "Magic" of the Math

The authors looked at four different ways to describe this "Non-local" gravity (some based on the bending of space, others on the twisting of space). They found that for all of them, the universe likely followed a specific pattern of growth (like a power law, $a(t) \sim t^q$) where the exponent $q$ is different from the standard value.

Essentially, they proved that if gravity has this "long-distance memory" (non-locality), the early universe expanded differently, and that difference perfectly explains the strange signals we are seeing today.

The Bottom Line

This paper suggests that the universe is a bit more "connected" than we thought. Gravity isn't just a local force; it has a memory of the past and a reach across the cosmos.

If this is true, it means:

• We don't need to invent new, weird particles to explain Dark Matter; it's just the standard stuff behaving differently because of how gravity worked in the beginning.
• The crazy high-energy neutrinos aren't a mystery from a distant galaxy; they are the "fingerprint" of Dark Matter decaying, made possible by this special gravity.

It's a beautiful example of how changing one fundamental rule (how gravity works over long distances) can solve multiple puzzles in the cosmic jigsaw puzzle at once.

Technical Summary

1. Problem Statement

The paper addresses a critical incompatibility between the Standard Cosmological Model (based on General Relativity) and two major observational datasets:

1. Dark Matter (DM) Relic Abundance: The observed density of dark matter in the universe ($\Omega_{DM}h^2 \approx 0.1188$).
2. High-Energy Neutrino Fluxes: Recent observations by the IceCube and KM3NeT experiments detecting astrophysical neutrinos with energies up to 220 PeV.

The Conflict:
The authors consider a minimal particle physics model where a PeV-mass Dark Matter particle ($\chi$) decays into neutrinos via a four-dimensional operator $y_{\alpha\chi} \bar{L}_\alpha H \chi$ (involving the Higgs doublet $H$, lepton doublet $L_\alpha$, and Yukawa couplings $y_{\alpha\chi}$).

• To explain the IceCube/KM3NeT neutrino flux, the DM lifetime must be extremely long ($\tau_\chi \sim 10^{28}$ s), implying extremely small Yukawa couplings ($\sum |y_{\alpha\chi}|^2 \sim 10^{-58}$).
• However, in a standard GR cosmological background, these same tiny couplings result in a DM relic abundance that is 33 orders of magnitude too low to match observations. Conversely, to match the relic abundance in GR, the couplings would need to be $\sim 10^{-25}$, which would cause the DM to decay too quickly to explain the neutrino signals.

The paper posits that this discrepancy suggests the early universe did not evolve according to standard General Relativity, but rather under the influence of Non-Local Gravity (NLG) effects.

2. Methodology

The authors employ a multi-step theoretical framework combining modified gravity, dynamical systems analysis, and particle physics phenomenology.

A. Non-Local Gravity Models

The study focuses on Integral Kernel Gravity (IKG) models, where non-local terms involve the inverse d'Alembert operator ($\Box^{-1}$). The authors analyze four specific models:

1. Curvature-based NLG: Action $S = \int d^4x \sqrt{-g} F(R, \Box^{-1}R)$.
2. Teleparallel NLG: Action based on the torsion scalar $T$, $S = \int d^4x \sqrt{-g} F(T, \Box^{-1}T)$.
3. Teleparallel with Boundary Term: Includes the boundary term $B$ (where $R = -T + B$) in the non-local action.
4. Horndeski Gravity: Treated as the scalar-tensor limit of the above non-local theories.

B. Noether Symmetry Approach

To solve the complex, non-linear integro-differential field equations of these models, the authors utilize the Noether Symmetry Approach.

• Scalarisation: The non-local operators are "scalarised" by introducing auxiliary scalar fields (e.g., $\phi = \Box^{-1}R$), converting the theory into a local scalar-tensor theory.
• Symmetry Selection: They search for Noether symmetries in the point-like Lagrangian of the cosmological minisuperspace. The existence of these symmetries guarantees conserved quantities (first integrals of motion), rendering the system integrable.
• Power-Law Solutions: This method yields exact power-law solutions for the scale factor: $a(t) \propto t^q$.

C. Cosmological Expansion Modification

The core mechanism linking gravity to particle physics is the modification of the Hubble expansion rate. In these models, the expansion rate $H(T)$ is related to the standard GR rate $H_{GR}(T)$ by an amplification factor $A(T)$:
$$H(T) = A(T) H_{GR}(T) = \eta \left(\frac{T}{T_*}\right)^\nu H_{GR}(T)$$
where $\nu = \frac{1}{q} - 2$. This factor $A(T)$ alters the thermal history of the universe during the freeze-in epoch (pre-BBN), significantly affecting the production rate of Dark Matter.

D. Freeze-In Calculation

The authors calculate the DM relic abundance using the Boltzmann equation for freeze-in production (where DM is never in thermal equilibrium). They integrate the production rate over the modified expansion history defined by the parameter $q$ (derived from the NLG models) and compare the result against:

1. The observed DM abundance ($\Omega_{DM}h^2 \approx 0.1188$).
2. The decay rate required to match IceCube/KM3NeT neutrino fluxes.

3. Key Contributions

1. Unified Framework: The paper provides a consistent theoretical framework that simultaneously explains the PeV Dark Matter relic abundance and the IceCube/KM3NeT high-energy neutrino flux using a single set of Yukawa couplings ($\sum |y_{\alpha\chi}|^2 \sim 10^{-58}$).
2. Resolution of the GR Incompatibility: It demonstrates that the standard GR cosmology ($q=0.5$) is incompatible with the minimal DM-neutrino decay model, whereas specific non-local gravity models can resolve this tension.
3. Noether Symmetry in NLG: The work extends the Noether Symmetry Approach to non-local gravity models involving torsion and boundary terms, deriving exact power-law cosmological solutions ($a(t) \sim t^q$) that were previously difficult to obtain analytically.
4. Parameter Space Constraints: The study identifies specific constraints on the power-law exponent $q$ required to satisfy observational data.

4. Results

• Required Expansion Rate: To reconcile the DM abundance with the neutrino data, the universe must have expanded at a rate significantly different from standard radiation domination during the freeze-in epoch.
• Constraint on $q$: The analysis reveals that the exponent $q$ in the scale factor $a(t) \sim t^q$ must satisfy:
  $$0.08 \lesssim q \lesssim 0.18$$
  This implies a much slower expansion rate ($H \propto T^{1/q}$) compared to the standard model ($q=0.5$), which enhances the freeze-in production efficiency, allowing the extremely small couplings ($10^{-58}$) to generate the correct relic density.
• Model Viability:
  • Curvature and Teleparallel NLG: Admit solutions with $q \neq 1/3$.
  • Teleparallel with Boundary Term: Requires $q < 1/3$.
  • Horndeski: Allows generic $q$ depending on integration constants.
  • Exclusion: Models with $q > 1/3$ are ruled out as they would require transition temperatures lower than the PeV scale, violating kinematic constraints for the scattering processes.
• KM3NeT/ARCA Consistency: The model is also consistent with the recent detection of a 120 PeV neutrino event by KM3NeT/ARCA, suggesting a potential cosmogenic origin or decay of ultra-heavy DM.

5. Significance

• Probing Non-Local Gravity: The paper suggests that high-energy astrophysical neutrinos serve as a unique probe for non-local gravitational effects in the early universe. The detection of PeV neutrinos could be indirect evidence of the non-local nature of gravity at cosmological scales.
• Alternative to New Physics: It offers a solution to the "PeV DM problem" without invoking complex new particle physics mechanisms (like multi-step decays or heavy mediators); instead, it relies on modifying the gravitational sector (the background cosmology).
• Thermal History Implications: The results highlight that the thermal history of the universe prior to Big Bang Nucleosynthesis (BBN) is not fixed by standard GR. The "amplification factor" $A(T)$ derived from non-local gravity can drastically alter particle production rates, offering a new window to test theories of gravity beyond Einstein's.
• Future Directions: The findings open avenues for investigating non-local cosmologies in the context of astroparticle physics, particularly with upcoming data from KM3NeT and future neutrino telescopes.

In conclusion, the authors demonstrate that Non-Local Gravity provides a viable, minimal, and unified explanation for the coexistence of the observed Dark Matter density and the high-energy neutrino flux, a feat impossible within the standard $\Lambda$CDM framework.