Sphere amplitudes and observing the universe's size

This paper proposes that sine dilaton gravity, holographically dual to DSSYK, provides a UV-complete framework for 2d quantum cosmology where the sphere amplitude yields a finite, normalizable no-boundary state that predicts a flat distribution for the universe's size, thereby resolving the non-normalizability and small-universe contradictions found in standard slow-roll inflation and dS JT gravity.

The Gist

The Big Picture: Fixing the Universe's "Birth Certificate"

Imagine you are trying to understand how a baby is born, but you only have a blurry, broken photograph of the moment of birth. In physics, this "birth" is the Big Bang, and the "photograph" is a mathematical tool called the No-Boundary Wavefunction. This tool is supposed to tell us the probability of the universe being a certain size when it was born.

For a long time, physicists have had a problem with this tool. When they used it to calculate the size of our universe, it gave two terrible answers:

1. It said the universe should be tiny. (Like a speck of dust).
2. It said the math breaks down. (The answer is "infinity," which means the math is useless).

This paper, written by Andreas Blommaert and Adam Levine, proposes a new way to fix the math. They introduce a new theory called "Sine Dilaton Gravity." Think of this as upgrading the blurry, broken camera to a high-definition, 4K lens that doesn't break when you zoom in too close.

The Main Characters

To understand the paper, let's meet the cast of characters:

• The Universe (The Baby): We want to know how big it is.
• The Observer (You): A person standing inside the universe, trying to measure it.
• The Old Theory (dS JT Gravity): The old camera. It's good, but it gets "glitchy" (divergent) when the universe is very small or very large.
• The New Theory (Sine Dilaton Gravity): The new camera. It's a "UV completion," which is a fancy physics way of saying it works perfectly even at the tiniest, most extreme scales (the "ultraviolet" end of the spectrum).
• DSSYK: A complex quantum system (like a super-complex puzzle) that acts as a hologram. The paper suggests this puzzle is the "code" running the universe.

The Problem: The "Tiny Universe" Glitch

In the old theory, if you ask, "What is the chance the universe is size X?", the math screams, "It's most likely size ZERO!"

The Analogy: Imagine you are trying to guess the weight of a newborn baby. The old math says, "There is a 99.9% chance the baby weighs 0 grams." This contradicts reality (we know the universe is huge). The math also says, "The chance of it being any specific size is infinite," which is like saying the probability of rolling a 6 on a die is 1,000,000%. It's a broken calculator.

The Solution: The "Sine" Upgrade

The authors switch to Sine Dilaton Gravity. They treat the universe not as a smooth, endless sheet, but as something with a built-in "pixel limit" or a "speed bump" at the very smallest scale.

The Analogy: Imagine the old theory was like a video game with infinite resolution. If you zoom in too far, the pixels disappear, and the game crashes. The new theory adds a "pixel limit." You can zoom in all the way to the Big Bang, but instead of crashing, the game just shows you the smallest possible pixel.

The Result:

1. No More Zero: In the new theory, the probability of the universe being size zero is actually zero. The universe cannot be a point; it has a minimum size. This fixes the "infinity" glitch.
2. No More Tiny Bias: The new math stops screaming that the universe must be tiny. It becomes much more neutral.

The Twist: The Observer Changes Everything

Here is the most creative part of the paper. The authors ask: Does it matter who is looking at the universe?

In the old view, we looked at the universe from "outside" (God's eye view). But in reality, we are inside the universe. We are the observers.

The Analogy:

• The God's Eye View: Imagine looking at a giant, empty room from a drone. You might think, "This room is huge, but it's also empty, so maybe it's small?"
• The Observer's View: Now imagine you are a person standing in that room. You can't see the whole room at once. You can only see what's in front of you.

The paper argues that when you include the Observer (a person with a mass, a clock, and a brain) in the calculation, the rules change completely.

• Without an observer: The math still prefers small universes (or breaks).
• With an observer: The math becomes flat.

What does "flat" mean?
It means the universe is equally likely to be small, medium, or huge. The "Observer's No-Boundary State" is like a fair coin flip. It doesn't care if the universe is the size of a marble or the size of a galaxy. It treats all sizes as equally probable.

The "Sphere Amplitude": Counting the Shapes

The paper spends a lot of time calculating something called the Sphere Amplitude.

• The Analogy: Imagine you are a baker trying to figure out how many different shapes of bread (spheres, donuts, etc.) you can bake.
• In the old theory, the number of "spheres" (simple, round universes) was infinite and broken.
• In the new theory, the number is finite. It's like the baker has a limited number of molds. This finiteness is crucial because it suggests the universe has a finite amount of "information" or "memory" (a finite Hilbert space), which is a very comforting idea for physicists.

The "Hologram" Connection

The paper mentions DSSYK.

• The Analogy: Think of the universe as a 3D movie. DSSYK is the 2D code running on the projector.
• The authors show that the "Sine Dilaton" theory (the 3D movie) is perfectly matched to the DSSYK code (the 2D screen). This is a "holographic" relationship. It means the complex physics of the Big Bang can be understood by studying this specific quantum code.

The Conclusion: What Does This Mean for Us?

1. The Big Bang isn't a singularity: The universe didn't start as a mathematical point of infinite density. The new math suggests it had a minimum size, avoiding the "crash."
2. We don't live in a tiny universe: The old math said the universe should be tiny. The new math says, "Actually, it could be any size, and we are just as likely to be here as anywhere else."
3. The Observer matters: You can't talk about the universe without talking about the people inside it. When you include the observer, the weird biases disappear, and the distribution of universe sizes becomes fair and flat.

In a nutshell: The authors fixed a broken calculator that was telling us the universe was too small and too weird. By adding a "pixel limit" to the math and remembering that we are inside the universe looking out, they found that the universe is actually a very normal, fair, and finite place.

Technical Summary

1. Problem Statement

The paper addresses two interconnected problems in theoretical cosmology and quantum gravity:

1. Microscopic Holography for Cosmology: While the holographic duality between SYK models and 2D JT gravity is well-established for asymptotically AdS spacetimes, a precise microscopic description for Big-Bang cosmologies (asymptotically de Sitter, dS) remains elusive. Specifically, the relationship between the Double-Scaled SYK (DSSYK) model and 2D dS gravity needs clarification.
2. The No-Boundary State Pathology: The Hartle-Hawking no-boundary wavefunction, when applied to slow-roll inflation or dS JT gravity, predicts a probability distribution for the size of the universe that is non-normalizable and heavily favors infinitely small universes (diverging as size $\ell \to 0$). This contradicts observational evidence of a large, flat universe and suggests a breakdown in the theoretical framework regarding the Big-Bang singularity.

2. Methodology

The authors employ a combination of canonical quantization, path integral techniques, and holographic duality:

• Sine Dilaton Gravity: They utilize "sine dilaton" gravity as a UV completion of dS JT gravity. The Euclidean action is given by:
  $$I = \frac{1}{2\hbar} \int d^2x \sqrt{g} (\Phi R + 2\sin(\Phi)) + \frac{1}{\hbar} \int du \sqrt{h} \Phi K$$
  This theory is holographically dual to DSSYK.
• Canonical Quantization: They quantize the theory in minisuperspace, solving the Wheeler-DeWitt (WDW) constraint to find exact wavefunctions $\psi(\ell, \Phi)$, where $\ell$ is the spatial size and $\Phi$ acts as a time-like dilaton coordinate.
• No-Boundary Wavefunction Construction: They construct the no-boundary state $\psi_{NB}$ by integrating over the spectral density $\rho(E)$ of DSSYK, matching the boundary conditions to the thermal partition function of DSSYK.
• Sphere Amplitude Calculation: They compute the norm squared of the no-boundary state ($\langle \psi_{NB} | \psi_{NB} \rangle$) in two ways:
  1. Gravitational Side: Using the Klein-Gordon inner product on the space of solutions.
  2. Holographic Side: Using the on-shell action of a dual matrix integral (finite-cut matrix model).
• Observer Inclusion: They introduce a point-like observer with dynamical mass $q$ into the closed universe to analyze the "observer's no-boundary state," distinguishing between global states and local observations.

3. Key Contributions and Results

A. Sine Dilaton as 2D Quantum Cosmology

• The authors interpret sine dilaton gravity as a theory of 2D quantum cosmology describing Big-Bang/Big-Crunch spacetimes.
• Classical Solutions: The classical solutions exhibit a Big-Bang at $\Phi = -\theta$, exponential expansion (resembling dS JT for small $\hbar$), a maximum size, and a Big-Crunch. Unlike pure dS JT, the expansion is finite, providing a natural UV cutoff.
• UV Completion: Sine dilaton gravity acts as a UV completion of dS JT. The periodic potential ($\sin \Phi$) ensures the spectral density $\rho(E)$ has compact support (finite energy range), unlike the exponential growth in standard JT gravity.

B. The Sphere Amplitude and Matrix Duality

• Finite Sphere Amplitude: The authors calculate the sphere amplitude (the norm of the no-boundary state).
  • In dS JT gravity, this amplitude diverges due to the UV behavior of the spectrum.
  • In sine dilaton gravity, the amplitude is finite.
• Holographic Match: The gravitational calculation yields:
  $$Z_{sphere} = -D^2 \int_{-2}^{2} dE_1 \rho(E_1) \int_{-2}^{2} dE_2 \rho(E_2) \log|E_1 - E_2|$$
  This matches exactly the on-shell action of the dual finite-cut matrix integral. This confirms the duality between sine dilaton gravity and DSSYK (or a $q$-deformed ETH matrix model) for cosmological observables.

C. Resolving the "Small Universe" Problem

The paper analyzes the probability distribution $P(\ell|\Phi)$ for the universe's size $\ell$ at a fixed dilaton time $\Phi$.

1. UV Divergence Resolution: In dS JT, $P(\ell) \sim \ell^{-4}$ as $\ell \to 0$, leading to non-normalizability and a preference for small universes. In sine dilaton gravity, the compact spectral support regulates this. The authors show that $P_{sphere}(\ell) \to 0$ as $\ell \to 0$. This resolves the divergence and suggests a quantum resolution of the Big-Bang singularity (the universe never shrinks to a point).
2. Torus/Bra-Ket Contribution: They consider the contribution from "bra-ket wormholes" (cylinder topologies connecting the bra and ket boundaries). This contribution yields a distribution that is approximately flat for large $\ell$. While this competes with the sphere contribution only for exponentially large universes ($\ell \sim D$), it indicates that topology change does not inherently favor small universes.

D. The Observer's Perspective

The most significant conceptual result concerns the inclusion of an observer:

• No-Boundary State for an Observer: Following recent arguments (e.g., [49]), the authors argue that the Hartle-Hawking sphere geometry does not contribute to an observer's state because the observer's worldline cannot smoothly end on the sphere.
• Identity Operator: The observer's no-boundary state is dominated by the cylinder (bra-ket wormhole) geometry. In 2D dilaton gravity, this path integral computes the identity operator on the physical Hilbert space ($L^2(\ell \otimes \Phi)$).
• Flat Distribution: Consequently, the probability distribution for the universe's size, from the perspective of an observer, is flat ($P(\ell) \sim \text{const}$).
• Implication: The observer's state prefers neither small nor large universes. This contrasts sharply with the global no-boundary state in slow-roll inflation, which strongly favors small universes.

4. Significance

• Microscopic Hologram for Cosmology: The work provides a concrete, calculable framework where a specific quantum mechanical system (DSSYK) describes a Big-Bang cosmology, moving beyond the AdS/CFT paradigm.
• Resolution of Singularities: It demonstrates that UV-complete dilaton gravity models (like sine dilaton) naturally regulate the $\ell \to 0$ divergence, offering a mechanism for the avoidance of the Big-Bang singularity in quantum cosmology.
• Observer-Dependent Cosmology: The paper suggests that the "problem of the small universe" in the no-boundary proposal may be an artifact of ignoring the observer. When an observer is included, the preference for small universes disappears, potentially reconciling theoretical predictions with the observed large size of our universe.
• Finite Dimensionality: The finiteness of the sphere amplitude and the compact support of the spectrum suggest an underlying finite-dimensional Hilbert space for the universe, a feature consistent with the thermodynamics of DSSYK.

In summary, the paper establishes sine dilaton gravity as a viable UV completion of dS JT gravity, proves its duality to DSSYK via sphere amplitudes, and argues that an observer's perspective resolves the pathological predictions of the standard no-boundary wavefunction regarding the size of the universe.