Modified Friedmann equations and non-singular cosmologies in $d=4$ non-polynomial quasi-topological gravities

This paper demonstrates that non-polynomial quasi-topological gravities in four dimensions can resolve the Big-Bang singularity through three distinct non-singular cosmological scenarios: a de Sitter emergence with finite curvature, a bouncing universe, and an asymptotically Minkowski origin with constant energy density.

The Gist

The Big Problem: The Universe's "Crash"

Imagine the history of our universe as a movie. According to our current best physics (Einstein's General Relativity), if you press "rewind" all the way to the beginning, the movie doesn't just fade to black—it crashes.

At the very start (the Big Bang), the universe becomes infinitely small, infinitely hot, and infinitely dense. In physics terms, this is called a singularity. It's like a computer program trying to divide by zero; the math breaks down, and the laws of physics stop making sense. Scientists want to fix this "crash" so the movie has a smooth beginning instead of a glitch.

The Solution: A New Set of Rules

The authors of this paper propose a new set of rules for gravity, specifically for the very early universe (the "UV" or high-energy regime). They are looking at a specific type of modified gravity called Quasi-Topological Gravity.

Think of General Relativity as a standard recipe for baking a cake. It works great for everyday life, but if you try to bake it at the temperature of a star, the cake turns into a singularity (a burnt, undefined mess). These authors are suggesting a "secret ingredient" (non-polynomial curvature terms) that changes the recipe just enough to prevent the cake from burning, while still tasting exactly the same as the standard cake once it cools down.

The Three Ways to Avoid the Crash

The paper explores three different ways this new gravity could save the universe from the Big Bang singularity. They use a special mathematical tool (a "characteristic function" called $h$) to map out these scenarios.

1. The "Infinite Stretch" (De Sitter Asymptotic)

• The Analogy: Imagine a rubber band being pulled back. In the standard Big Bang, the rubber band snaps back to a single point. In this scenario, the rubber band stretches back forever, getting thinner and thinner, but it never actually reaches a point where it breaks.
• What happens: The universe emerges from a state of constant, rapid expansion (like a de Sitter space). As you go back in time, the universe gets infinitely large in the past, but the density of matter gets infinitely high.
• The Catch: While the shape of the universe is smooth, the matter inside it still gets infinitely dense. It's like a traffic jam that stretches back forever; the road is fine, but the cars are packed so tight they become a singularity. However, this infinite density happens at an "infinite distance" in time, so you never actually reach it.

2. The "Cosmic Bounce" (Bouncing Universe)

• The Analogy: Think of a ball thrown against a trampoline. In the standard Big Bang, the ball falls into a hole and disappears. In this scenario, the ball hits the trampoline, stops, and bounces back up.
• What happens: The universe shrinks down to a tiny, but finite size (it never becomes zero). At this smallest point, it stops shrinking and starts expanding again. This is a "Big Bounce."
• The Catch: To make this work mathematically, the "recipe" (the Lagrangian) has to be multi-valued. Imagine a road that splits into two paths. To get from the shrinking phase to the expanding phase, the universe has to switch lanes on this road. This is a strange feature that hasn't been emphasized much before, but it's necessary for the bounce to happen without breaking the laws of physics.

3. The "Eternal Loiterer" (Minkowski Asymptotic)

• The Analogy: Imagine a car driving on a highway that, as you look further back in time, slows down more and more until it is just sitting still in a parking lot, forever. It never crashes, and it never speeds up to infinite danger.
• What happens: The universe originates from a state that looks like empty, flat space (Minkowski space). It sits there, almost static, for eternity. Then, very slowly, it starts to expand.
• Why it's the best: This is the "Goldilocks" scenario.
  • Unlike the "Infinite Stretch," the matter density never becomes infinite. It stays at a safe, finite limit.
  • Unlike the "Bounce," there is no need for a violent collision or a switch between lanes. The universe just gently wakes up from a long nap.
  • It avoids the "Super-Planckian" regime (where physics gets weird and quantum gravity takes over). The universe stays calm and regular.

The "Magic Function" ($h$)

The authors use a mathematical graph to visualize these three scenarios.

• Standard Gravity: The graph is a straight diagonal line. If you go back far enough, it hits zero (the singularity).
• New Gravity: The graph bends.
  • In the De Sitter case, the graph shoots up to infinity.
  • In the Bounce case, the graph loops back and hits the bottom axis again.
  • In the Minkowski case, the graph hits the bottom axis but does so in a very specific, smooth way (with a vertical slope) that allows the universe to sit still forever before expanding.

The Bottom Line

This paper shows that by tweaking the rules of gravity just a little bit (using non-polynomial terms), we can create a universe that never had a Big Bang singularity.

Instead of a violent explosion from a point of infinite density, the universe could have:

1. Emerged from an eternal, stretching phase.
2. Bounced off a minimum size.
3. Or (the most promising option) gently awakened from an eternal, calm, flat state where nothing ever got too hot or too dense.

This suggests that the "crash" at the beginning of our universe might just be a glitch in our current understanding of gravity, and a slightly modified version of the rules could give us a smooth, non-singular origin story.

Technical Summary

1. Problem Statement

Classical General Relativity (GR) predicts the existence of spacetime singularities (geodesic incompleteness) in both gravitational collapse and cosmological evolution (the Big Bang). While polynomial curvature quasi-topological gravities in $d \geq 5$ dimensions have successfully resolved black-hole singularities, their application to cosmology in the physically relevant $d=4$ dimensions is limited because polynomial higher-curvature extensions of GR do not exist in four dimensions.

The central problem addressed is whether non-polynomial curvature quasi-topological gravities in $d=4$ can resolve the Big-Bang singularity. Specifically, the authors investigate if the mechanisms that regularize black holes can be extended to FLRW cosmologies to produce non-singular early-universe scenarios while preserving the correct infrared (IR) limit of General Relativity.

2. Methodology

The authors employ a dimensional reduction approach to simplify the analysis of higher-curvature theories:

• Reduction to 2D Horndeski Theory: They utilize the fact that the spherical reduction of pure-curvature quasi-topological gravities in $d \geq 4$ dimensions yields specific subclasses of two-dimensional Horndeski theories. This allows them to study the dynamics of spherically symmetric spacetimes (including FLRW) using a 2D effective framework.
• The Master Equation: By evaluating the 2D Horndeski equations of motion on an FLRW background with spatial curvature $k=0$, they derive a generalized Friedmann equation. The core of their analysis rests on a single characteristic function, $h(\psi)$, where $\psi = H^2$ (the squared Hubble parameter).
  The generalized Friedmann equation is given by:
  $$h(H^2) = \varrho$$
  where $\varrho$ is the rescaled energy density of the matter fluid.
• Constraints:
  • IR Limit: The function must satisfy $h(H^2) \to H^2$ as $H^2 \to 0$ to recover standard GR at late times.
  • Acceleration Condition: They derive conditions on $h(H^2)$ required to produce accelerated expansion ($\ddot{a} > 0$) without violating the strong energy condition, which is necessary for singularity resolution.
  • Integrability: The theory must satisfy the integrability condition $\omega(\phi, \chi) = 0$, ensuring the existence of a Lagrangian principle.

3. Key Contributions

The paper makes several theoretical and phenomenological contributions:

1. Classification of Non-Singular Scenarios: The authors identify three distinct geometric scenarios for resolving the Big-Bang singularity based on the behavior of the characteristic function $h(H^2)$ in the ultraviolet (UV) regime.
2. The Role of Multi-Valued Lagrangians: A novel finding is that realizing a bouncing universe in this framework geometrically requires the underlying Lagrangian (and thus the function $h$) to be multi-valued. This feature had not been previously emphasized in this context.
3. Explicit Solutions: The authors provide explicit analytical solutions for the scale factor $a(\tau)$ and energy density evolution for all three scenarios, moving beyond kinematic arguments to dynamical realizations.
4. Distinction between "Failed" Bounces and Minkowski Origins: They clarify the mathematical distinction between a standard bounce and an asymptotic Minkowski origin, linking the latter to a specific divergence in the derivative of the characteristic function.

4. Results: Three Non-Singular Scenarios

The paper details three specific scenarios, each defined by the behavior of $h(H^2)$:

A. Asymptotic de Sitter Universe

• Mechanism: The function $h(H^2)$ possesses a singularity at a finite value $H^2 = H_0^2$. As the universe evolves backward in time, $H^2$ approaches $H_0^2$, and the energy density $\rho$ diverges.
• Behavior: The universe emerges from an eternal de Sitter phase ($a \propto e^{H_0 \tau}$).
• Singularity Status: Curvature invariants remain finite, but the matter density diverges. However, this divergence occurs at infinite affine distance, meaning the spacetime is geodesically complete.
• Example Function: $h(H^2) = \frac{H^2}{1 - H^2/H_0^2}$.

B. Bouncing Universe

• Mechanism: The function $h(H^2)$ is double-valued (multi-branched). It intersects the $H^2=0$ axis at a non-zero energy density $\rho_{max}$.
• Behavior: The universe contracts to a minimum scale factor (where $H=0$) and then re-expands.
• Singularity Status: Completely non-singular. The scale factor never reaches zero, and curvature remains finite.
• Requirement: The multi-valued nature of $h$ is essential to connect the contracting and expanding branches while satisfying the IR limit.
• Example Function: $h(H^2) = H_0^2 \left[ 1 \pm \sqrt{1 - 2H^2/H_0^2} \right]$.

C. Asymptotic Minkowski Universe

• Mechanism: A special case of the bouncing scenario where the function $h(H^2)$ approaches the $H^2=0$ axis with an infinite derivative ($h'(0) \to \infty$).
• Behavior: The universe originates from an asymptotically static Minkowski state ($a \to \text{const}$, $H \to 0$) in the infinite past. It does not undergo a contraction phase; rather, it "loiters" eternally before entering the expansion phase.
• Singularity Status: Completely non-singular. Unlike the de Sitter case, the matter density remains finite (approaching a constant $\rho_0$) even at infinite affine distance. It avoids the super-Planckian regime entirely.
• Significance: This scenario exhibits a form of "asymptotic freedom" where gravity effectively switches off in the UV, yielding a regular near-Minkowski universe even at high densities.

5. Significance

• Theoretical Consistency: The work demonstrates that non-singular cosmologies can be derived from a Lagrangian principle in $d=4$, a feat not guaranteed by effective field equations in other quantum gravity approaches (like Asymptotic Safety or Loop Quantum Cosmology) where the Lagrangian origin is often unclear.
• Resolution of the Trans-Planckian Problem: The asymptotic Minkowski scenario is highlighted as the most physically desirable because it avoids both curvature singularities and the trans-Planckian problem (where energy densities exceed the Planck scale).
• New Paradigm for Early Universe: The paper suggests that the early universe might not have begun with a Big Bang or a standard bounce, but rather as an eternally loitering Minkowski state, offering a new kinematic and dynamic framework for early universe cosmology.
• Connection to Black Holes: It reinforces the deep connection between the resolution of black-hole singularities and cosmological singularities within the framework of quasi-topological gravities, suggesting a unified mechanism for singularity resolution across different spacetime topologies.

In conclusion, the authors successfully map the landscape of non-singular cosmologies in 4D non-polynomial quasi-topological gravity, proving that with specific choices of the characteristic function $h(H^2)$, the Big-Bang singularity can be replaced by geodesically complete, non-singular histories, with the asymptotic Minkowski origin emerging as a particularly robust solution.