From negative to positive cosmological constant through decreasing temperature of the Universe: connection with string theory and spacetime foliation results

This paper proposes a modified thermal Renormalization Group framework, where a constant dimensionless temperature drives the cosmological constant to transition from negative (consistent with string theory at high temperatures) to positive (consistent with observations at low temperatures), thereby resolving the discrepancy between theoretical predictions and the observed accelerating expansion of the Universe.

The Gist

The Big Problem: The Universe's "Engine" is Broken

Imagine the Universe is a giant car. For a long time, physicists have been trying to figure out what kind of engine (the Cosmological Constant) is driving it.

• Observation: When we look at the Universe today, it's speeding up. The engine seems to be pushing forward with a positive force.
• String Theory: This is a famous theory that tries to explain how the universe works at the smallest levels. But when String Theory calculates the engine, it says the force should be negative (like a brake or a vacuum sucking everything in).

This is a huge conflict. One says "Go," the other says "Stop." Previous attempts to fix this were like trying to tape a "Go" sticker over a "Stop" sign—it worked on paper, but it felt fake (artificial) or unstable.

The New Idea: Temperature is the Key

The authors of this paper propose a clever solution: The engine changes based on how hot the car is.

Think of the Universe like a block of ice melting into water.

• The Early Universe (The Ice): When the Universe was born, it was incredibly hot and dense. In this "hot" state, the laws of physics (specifically Asymptotically Safe Quantum Gravity) say the engine should be negative. This matches what String Theory predicts.
• The Present Universe (The Water): As the Universe expanded, it cooled down. As the temperature dropped, the engine flipped. It switched from a "brake" (negative) to a "thrust" (positive).

The paper argues that we don't need to force the sign to change; it happens naturally as the Universe cools down, just like water naturally turns to ice when it gets cold.

The Secret Sauce: A New Way to Measure Heat

To prove this, the scientists used a mathematical tool called Renormalization Group (RG) flow. Think of this as a microscope that lets you zoom in and out on the Universe to see how physics changes at different scales.

Usually, when scientists use this microscope with heat, they keep the "temperature setting" fixed. But the authors realized that in the real Universe, the "zoom level" and the "temperature" are linked.

The Analogy of the Spinning Top:
Imagine a spinning top.

• High Speed (High Temperature): When it spins super fast (early Universe), it behaves one way (negative force).
• Slow Speed (Low Temperature): As it slows down (cooling Universe), it wobbles and changes behavior (positive force).

The authors created a new rule for their microscope: instead of keeping the temperature fixed, they kept a ratio constant. They treated the "heat" not as a fixed number, but as something that shrinks along with the size of the universe. This allowed them to see the transition clearly.

What They Found

They tested this idea on three different versions of the "Quantum Gravity" engine:

1. The Basic Model: Just gravity.
2. The Model with Matter: Gravity plus particles (like the stuff we are made of).
3. The "Ghost" Model: A more complex version that accounts for invisible quantum effects.

The Result: In all three cases, the math showed the exact same pattern:

• Hot Universe (Early times): The Cosmological Constant is Negative.
• Cool Universe (Today): The Cosmological Constant is Positive.

Why This Matters

This is a "Goldilocks" solution. It connects two worlds that didn't get along:

1. String Theory: It's happy because the early Universe was negative, just like the theory predicted.
2. Observations: We are happy because the Universe is positive today, allowing for the expansion we see.

It suggests that the Universe didn't need a "magic switch" to flip from negative to positive. It just needed to cool down. The transition happened naturally as the Universe grew older and colder.

The "Folding" Connection

The paper also mentions a connection to Spacetime Foliation. Imagine the Universe as a loaf of bread.

• Standard view: Time is just a line running through the loaf.
• Foliation view: Time is the act of slicing the loaf. Each slice is a "moment" in time.

The authors found that their new math (linking temperature to the size of the universe) looks very similar to how these "slices" of time behave in advanced physics. This suggests their idea isn't just a random guess; it fits into the deeper geometry of how space and time are built.

The Bottom Line

The Universe isn't broken; it's just evolving.

• Then (Hot): The engine was a brake (Negative).
• Now (Cool): The engine is a thruster (Positive).

By understanding how the Universe cools down, we can finally make peace between the predictions of String Theory and the reality of our expanding cosmos.

Technical Summary

1. Problem Statement

The paper addresses a fundamental conflict in theoretical physics regarding the sign of the cosmological constant ($\Lambda$):

• String Theory: Naturally predicts a negative cosmological constant (Anti-de Sitter, AdS space).
• Observational Cosmology: Requires a positive cosmological constant (de Sitter, dS space) to explain the current accelerated expansion of the Universe.
• Existing Solutions & Limitations:
  • String Theory Fixes: Attempts to wrap antibranes to achieve a small positive $\Lambda$ often result in unstable worlds.
  • Ad Hoc Models: The $\Lambda_s$CDM model introduces a sign-switching cosmological constant ($\Lambda \to \Lambda_s$) to fit data, but the sign flip is artificial and lacks a natural phase transition mechanism.
  • Asymptotically Safe (AS) Quantum Gravity: Previous Renormalization Group (RG) studies connect UV (string theory) and IR (low energy) regimes but typically omit temperature, a crucial parameter in early universe cosmology. Furthermore, standard finite-temperature RG approaches often fail to yield non-trivial fixed points or require an artificial separation of scales.

2. Methodology

The authors propose a Modified Thermal Renormalization Group (RG) framework to resolve the sign discrepancy by linking the evolution of $\Lambda$ to the cooling of the Universe.

• Core Hypothesis: Instead of treating temperature ($T$) as a fixed parameter linked to a fixed momentum scale, the authors treat the dimensionless temperature ratio $\tau = T/k$ as a constant over the RG flow, where $k$ is the running RG scale.
  • Relation: $T = \tau k$.
  • As $k \to 0$ (infrared limit), the physical temperature $T$ also approaches zero, simulating the cooling of the expanding Universe.
• Formalism:
  • The study extends the Functional Renormalization Group (FRG) to finite temperatures using Euclidean spacetime (Wick rotation).
  • Time is compactified with radius $R = 1/T$.
  • The momentum integral is replaced by a Matsubara summation over discrete frequencies $\omega_m = 2\pi m T$.
  • Threshold Functions: The standard zero-temperature threshold functions ($\Phi, \tilde{\Phi}$) are generalized to include the dimensionless temperature $\tau$ (Eqs. 12 and 13), effectively integrating out thermal fluctuations alongside quantum fluctuations.
• Models Analyzed: The authors apply this framework to three variants of Asymptotically Safe (AS) Quantum Gravity:
  1. Conformally Reduced Einstein-Hilbert (CREH) gravity: Quantizing only the conformal factor of the metric.
  2. QEG with Matter: Einstein gravity coupled to an $N$-component scalar field.
  3. Ghost-Improved Einstein-Hilbert (EH) gravity: Including wave-function renormalization of ghost fields ($Z_c$).

3. Key Contributions

• Novel RG Scheme: Introduction of the "fixed $\tau$" approach, where the dimensionful temperature $T$ runs with the scale $k$. This contrasts with standard approaches where $T$ is fixed, allowing for the existence of non-trivial fixed points and phase transitions.
• Natural Sign Switching: The framework provides a mechanism where the cosmological constant naturally transitions from negative (AdS) at high temperatures to positive (dS) at low temperatures without ad hoc assumptions.
• Connection to Spacetime Foliation: The authors demonstrate a formal equivalence between their thermal RG parameter $\tau$ and the dimensionless Matsubara mass $m$ used in Arnowitt-Deser-Misner (ADM) formalism (spacetime foliation). This bridges thermal field theory and the geometric structure of foliated spacetimes.
• Universality: The results are shown to be robust across different truncations of AS gravity (CREH, matter-coupled, and ghost-improved).

4. Key Results

• High-Temperature Limit ($\tau \to \infty$):
  • The $g$-coordinate of the Reuter fixed point (the dimensionless Newton constant, $g_* = k^2 G_k$) vanishes ($g_* \to 0$).
  • The cosmological constant $\lambda$ (dimensionless) remains negative.
  • Conclusion: The early Universe, dominated by high temperatures, naturally resides in an AdS phase ($\Lambda < 0$).
• Low-Temperature Limit ($\tau \to 0$):
  • As the Universe cools (RG flow proceeds to IR), the system undergoes a phase transition.
  • The RG trajectories spiral away from the fixed point, driving the cosmological constant to become positive ($\Lambda > 0$).
  • Conclusion: The present Universe resides in a dS phase ($\Lambda > 0$), consistent with observations.
• Phase Diagrams:
  • The authors construct Quantum Phase Transition (QPT) and Classical Phase Transition (CPT) diagrams (Fig. 3).
  • These diagrams show a critical line separating the $\Lambda < 0$ and $\Lambda > 0$ phases, dependent on the dimensionless temperature $\tau$.
  • The transition is driven by the decreasing temperature of the Universe, moving the system from the AdS basin of attraction to the dS basin.

5. Significance

• Resolution of the String Theory vs. Observation Conflict: The paper offers a theoretical pathway where String Theory (predicting AdS) and current Cosmology (requiring dS) are not contradictory but represent different thermal epochs of the same universe.
• Validation of $\Lambda_s$CDM: The results provide a first-principles justification for the phenomenological $\Lambda_s$CDM models (which assume a sign-switching $\Lambda$). The sign flip is shown to be a natural consequence of thermal RG flow in AS gravity, rather than an arbitrary insertion.
• Cosmological Tensions: The authors suggest that this mechanism could help resolve the Hubble tension ($H_0$) and the $S_8$ discrepancy, as the sign-switching $\Lambda$ models have been shown to alleviate these observational tensions.
• Theoretical Bridge: By linking thermal RG flow to spacetime foliation (ADM formalism), the work unifies concepts from quantum field theory at finite temperature with the geometric decomposition of spacetime, suggesting that temperature fluctuations can be treated analogously to quantum fluctuations in the RG flow.

In summary, the paper proposes that the cosmological constant's sign is not a fundamental constant but a running parameter determined by the thermal history of the Universe. As the Universe cools, Asymptotically Safe Quantum Gravity naturally evolves the cosmological constant from the negative values predicted by String Theory to the positive values observed today.