Quadratic energy-momentum squared gravity: constraints from big bang nucleosynthesis

This paper extends the standard cosmological model within the quadratic energy-momentum squared gravity framework to analyze Big Bang Nucleosynthesis dynamics, deriving stringent constraints on the interaction parameter $\alpha$ from primordial helium abundance measurements that either align with or deviate from standard cosmology depending on the dataset used.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, physicists have had a very specific recipe for how that balloon inflates, how it cools down, and how the first tiny bits of matter (like hydrogen and helium) were baked into existence. This recipe is called the Standard Model of Cosmology.

But, just like a chef might wonder, "What if I added a secret spice?" a group of physicists asked: What if gravity doesn't just pull on matter, but also interacts with a hidden "partner" field?

This paper explores a new theory called Quadratic Energy-Momentum Squared Gravity (qEMSG). Here is the breakdown of what they did, using simple analogies.

1. The New Ingredient: The "Shadow Partner"

In our standard understanding, matter (like stars, gas, and you) has energy and momentum. In this new theory, every piece of matter has a "shadow partner" (called the qEMSF).

• The Analogy: Imagine you are walking down the street (matter). Usually, you just walk. But in this theory, you are also dragging a heavy, invisible backpack (the shadow partner) that is connected to you by a spring.
• The Interaction: Sometimes, you give energy to the backpack (it gets heavier). Sometimes, the backpack gives energy back to you (you get a boost). This exchange is controlled by a dial called $\alpha$ (alpha).
  • If $\alpha$ is positive, the backpack pushes energy into you.
  • If $\alpha$ is negative, you are constantly losing energy to the backpack.

2. The Test Kitchen: Big Bang Nucleosynthesis (BBN)

To test if this "shadow backpack" theory is real, the authors looked at the universe's "kitchen" during the first few minutes after the Big Bang. This era is called Big Bang Nucleosynthesis (BBN).

• The Recipe: During this time, the universe was a hot soup of protons and neutrons. They were swapping places rapidly (like a dance) until the universe cooled enough to freeze them into the first atoms, mostly Helium-4.
• The Critical Factor: The amount of Helium-4 created depends entirely on how fast the universe was expanding and how fast the protons and neutrons were dancing before the "freezing" happened.

3. The Experiment: Changing the Expansion Rate

The authors calculated what would happen to this "Helium recipe" if the shadow backpack existed.

• The Twist: Because the matter is exchanging energy with its shadow partner, the "heat" (energy density) of the universe doesn't cool down exactly the same way it does in the standard model.
• The Result:
  • If $\alpha$ is positive, the universe expands slightly differently, keeping the "dance" between protons and neutrons going a bit longer. This results in more Helium being made.
  • If $\alpha$ is negative, the dance stops sooner, resulting in less Helium.

4. The Verdict: Comparing the Recipe to Reality

The scientists then compared their new "Helium predictions" against real-world measurements of how much Helium exists in the oldest parts of the universe.

They found two interesting outcomes based on two different sets of measurements:

1. The "Aver" Measurement: When they used data from Aver et al., the amount of Helium observed matched the standard model perfectly. This means the "shadow backpack" dial ($\alpha$) is likely set to zero. In other words, the standard model is still the winner here.

   • Constraint: The dial must be between -8.81 and 8.14 (in tiny units). This range includes zero.

2. The "Fields" Measurement: When they used data from Fields et al. (which relies on data from the Planck satellite), the observed Helium was slightly higher than the standard model predicts.

   • The Exciting Part: The authors found that if they turn the dial to a positive value (specifically between 3.48 and 4.43), their new theory perfectly explains the extra Helium!
   • Implication: This suggests the "shadow partner" might actually exist and is helping to create the extra Helium we see.

5. Why This Matters

This paper is a detective story. It doesn't prove the theory is definitely true, but it shows that:

• The theory is mathematically consistent and doesn't break the laws of physics.
• It offers a potential solution to a small mismatch in our current understanding of the universe (the "Helium problem").
• It acts as a filter: If future measurements of Helium confirm the "Fields" data, this theory gains massive credibility. If they confirm the "Aver" data, the theory is likely just a fancy way of saying "nothing special is happening."

The Bottom Line

Think of the universe as a cake. The Standard Model says the recipe makes a perfect cake. This paper asks, "What if we added a secret ingredient?"

• They baked the cake with the secret ingredient.
• They tasted it.
• One taste test said, "It tastes exactly like the original recipe."
• The other taste test said, "It tastes slightly sweeter, and this secret ingredient explains why!"

The authors have now set strict limits on how much of this "secret ingredient" (the parameter $\alpha$) can exist in our universe without ruining the cake. They found that if the ingredient exists, it's a very small amount, but just enough to potentially explain some cosmic mysteries.

Technical Summary

1. Problem Statement

The paper addresses the need to constrain extensions of General Relativity (GR) that introduce non-minimal interactions between matter fields and geometry. Specifically, it investigates Quadratic Energy-Momentum Squared Gravity (qEMSG), a theory where the gravitational action includes a term proportional to the square of the energy-momentum tensor ($T^2 = T_{\mu\nu}T^{\mu\nu}$).

While previous studies have constrained the coupling parameter $\alpha$ of this theory using astrophysical data (e.g., neutron stars) or late-time cosmology, there is a lack of rigorous constraints derived from the Big Bang Nucleosynthesis (BBN) epoch. The authors aim to determine how the qEMSF (quadratic energy-momentum squared field) interaction modifies the early universe's radiation dynamics, specifically the expansion rate and temperature evolution, and how these modifications affect the primordial abundance of Helium-4 ($^4$He).

2. Methodology

Theoretical Framework

The authors adopt a non-minimal matter interaction perspective. Instead of treating the modification as a new geometric term, they interpret the theory as a two-fluid system:

• Standard Material Field ($T_{\mu\nu}$): The usual matter/radiation.
• Partner Field ($T^{qEMSF}_{\mu\nu}$): A field generated by the function $f(T^2) = \alpha T^2$.
• Interaction: The two fields exchange energy-momentum, violating local conservation for individual components ($\nabla_\mu T^{\mu\nu} = -Q^\nu$) while conserving the total system.

Cosmological Dynamics

• Friedmann Equations: The authors derive modified Friedmann equations for a flat FLRW universe containing dust and radiation. The radiation sector acquires a quadratic correction term ($\propto \alpha \rho_r^2$).
• Thermodynamics of Open Systems: Since radiation is not conserved in this model, the authors apply thermodynamics of open systems to analyze entropy transfer. They determine that the direction of energy/entropy transfer depends on the sign of $\alpha$:
  • $\alpha > 0$: Energy flows from the qEMSF partner to radiation (entropy increases).
  • $\alpha < 0$: Energy flows from radiation to the partner (entropy decreases).
• Modified Evolution: The energy density of radiation evolves differently than the standard $\rho_r \propto a^{-4}$, leading to a modified relationship between the scale factor $a$ and temperature $T$.

BBN Analysis

The authors employ a semi-analytical approach to calculate the impact on BBN:

1. Expansion Rate: The Hubble parameter $H$ is modified relative to the standard model ($H_{std}$) by a factor $\sqrt{1 + 4\alpha\rho_r}$.
2. Freeze-out Temperature ($T_f$): The modified expansion rate alters the temperature at which weak interactions (neutron-proton interconversion) freeze out.
3. Time-Temperature Relation: Crucially, the non-conservation of radiation alters the $t-T$ relation, affecting the duration between freeze-out and nucleosynthesis.
4. Helium Abundance ($Y_p$): The authors derive a modified expression for the primordial Helium mass fraction $Y_p$, accounting for changes in:
   • The neutron-to-proton ratio at freeze-out.
   • The decay of neutrons during the interval between freeze-out and nucleosynthesis (due to altered timing).

3. Key Contributions

• Thermodynamic Interpretation: The paper provides a clear thermodynamic interpretation of qEMSG as an open system where the sign of $\alpha$ dictates the direction of entropy transfer between radiation and its partner field.
• Analytical Derivation of $Y_p$: Unlike previous numerical-only approaches, the authors derive an explicit analytical formula (Eq. 61) for the modified Helium abundance, capturing the complex interplay between the modified expansion rate and the modified time-temperature relation.
• Bidirectional Constraints: The study demonstrates that the model can accommodate both positive and negative values of $\alpha$, with distinct physical consequences (e.g., $\alpha > 0$ mimics dark radiation, while $\alpha < 0$ can counteract extra relativistic degrees of freedom).

4. Results

The authors derive stringent constraints on the coupling parameter $\alpha$ (units: $\text{eV}^{-4}$) based on recent observational data for primordial Helium-4:

• Constraint from Aver et al. (2021):
  Using direct measurements of metal-poor H II regions ($Y_p = 0.2453 \pm 0.0034$):
  $$-8.81 \leq \alpha \leq 8.14 \times 10^{-27} \, \text{eV}^{-4} \quad (68\% \text{ CL})$$
  Result: This range includes $\alpha = 0$, meaning the standard model is consistent with these observations.

• Constraint from Fields et al. (2020):
  Using the Planck-CMB baryon density within the standard $\Lambda$CDM framework ($Y_p = 0.2469 \pm 0.0002$):
  $$3.48 \leq \alpha \leq 4.43 \times 10^{-27} \, \text{eV}^{-4} \quad (68\% \text{ CL})$$
  Result: This range excludes $\alpha = 0$, suggesting a statistical preference for a non-zero $\alpha$ (specifically positive) if the Fields et al. estimation is taken as the ground truth. This lends support to the qEMSF interaction model.

• Reconciling Extra Relativistic Degrees of Freedom:
  The authors show that if an extra relativistic species (e.g., a sterile neutrino, $\Delta N_{\nu} \approx 1$) exists, it would typically overproduce Helium. However, a negative $\alpha$ can counterbalance this effect.

  • With one extra species, the constraints shift to negative values: $(-2.17 < \alpha < -0.278) \times 10^{-26} \, \text{eV}^{-4}$ (Aver) and $(-7.96 < \alpha < -6.90) \times 10^{-26} \, \text{eV}^{-4}$ (Fields).
  • This implies the qEMSF model can reconcile deviations from the Standard Model of particle physics with observational data.

5. Significance

• Tightest Cosmological Bounds: The derived constraints ($|\alpha| \sim 10^{-27} \, \text{eV}^{-4}$) are significantly tighter than previous cosmological bounds and comparable to, though slightly weaker than, the tightest astrophysical bounds from neutron stars ($|\alpha| \sim 10^{-35} \, \text{eV}^{-4}$).
• Model Viability: The study validates the qEMSG framework as a viable extension of GR that can be tested against precision cosmology. The fact that one dataset (Fields et al.) favors a non-zero $\alpha$ suggests that qEMSG could potentially resolve tensions or explain anomalies in early universe physics.
• Flexibility: The model's ability to mimic dark radiation (via positive $\alpha$) or suppress its effects (via negative $\alpha$) makes it a powerful tool for exploring physics beyond the Standard Model, particularly regarding the effective number of neutrino species ($N_{\text{eff}}$).
• Future Directions: The authors note that future constraints using primordial Deuterium (which is measured with higher precision than Helium) and CMB-S4 data will further refine these limits.

In summary, this paper establishes Big Bang Nucleosynthesis as a critical laboratory for testing quadratic energy-momentum squared gravity, providing the most stringent cosmological limits on the interaction parameter $\alpha$ to date and demonstrating the model's potential to explain deviations from standard cosmological predictions.