⚛️ high-energy theory

One-point holographic correlator in the expanding universe

This paper calculates time-dependent thermal one-point functions of massive operators in expanding matter-dominated universes by applying the Grinberg-Maldacena geodesic formula within a Randall-Sundrum II braneworld model containing a $p$ -brane gas, deriving results from the brane's evolving radial position determined by the Israel junction conditions.

Original authors: Souvik Paul, Gopinath Guin, Sunandan Gangopadhyay

Published 2026-07-24

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CC BY 4.0

Original authors: Souvik Paul, Gopinath Guin, Sunandan Gangopadhyay

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe not just as a vast, empty stage where stars play out their drama, but as a hologram. This is the wild idea at the heart of a field called the AdS/CFT correspondence. Think of it like a 3D movie projector. The "film" is a flat, two-dimensional surface (a brane) where all the action happens, but the "movie" we see is a three-dimensional reality with depth. In this cosmic setup, the extra dimension of depth isn't just empty space; it's like a control knob for time and energy. Scientists use this trick to study things that are incredibly hard to understand, like how particles behave when they are squished together tightly. Usually, they look at how two particles talk to each other (a two-point function), but what if we just wanted to know the "vibe" or the average state of a single particle in this expanding, stretching universe? That's the question this paper tackles. It asks: if our universe is expanding like a balloon, how does the "temperature" or energy state of a single massive object change as the balloon gets bigger, and can we use the holographic movie projector to figure it out?

The authors, Souvik Paul, Gopinath Guin, and Sunandan Gangopadhyay, dive into this by treating our universe as a giant, four-dimensional sheet (a "brane") floating inside a five-dimensional ocean. To make this sheet expand and behave like the real universe—with its mix of radiation, matter, and weird "exotic" stuff—they fill the ocean with a gas of invisible, higher-dimensional strings (called p-branes). These strings push and pull on the sheet, making it move. The movement of the sheet is what we experience as the expansion of the universe.

Using a clever mathematical shortcut developed by other physicists (Grinberg and Maldacena), the team calculates how a single, heavy object on this sheet "feels" the heat of the universe. They find that this feeling isn't static; it changes as the universe grows. In the very beginning, when the universe is young and the sheet is zooming away fast, the "vibe" of the object actually grows stronger. But as time goes on and the sheet slows down near the edge of a cosmic black hole (the horizon), the vibe starts to fade away, decaying like a signal getting lost in the distance.

The paper breaks this down into different scenarios, like a cosmic recipe book. First, they look at a universe made only of radiation (like light), only of matter (like dust), or only of "exotic matter" (a strange stuff that acts like negative gravity). They find that in the early days, the signal grows at different speeds depending on the recipe: it grows with the square root of time for radiation, with time to the power of two-thirds for matter, and linearly with time for exotic matter. However, in the late days, no matter what the recipe is, the signal eventually fades away. The paper suggests that this fading happens because the black hole horizon "swallows" the information of the object, acting like a cosmic vacuum cleaner that slowly erases its presence.

They also mix the ingredients, creating universes where radiation and matter coexist, or radiation and exotic matter share the stage. Here, the story follows the thermal history of our real universe: in the early, hot days, radiation rules the show and dictates how the signal grows. But as the universe cools and ages, matter (or exotic matter) takes the wheel, and the signal begins its slow, steady decline. The authors even draw graphs showing this dance, proving that the heavier the object (the higher its "conformal dimension"), the slower it fades away, though it still fades.

Ultimately, this paper doesn't claim to have solved the mystery of the universe's expansion, but it offers a new way to calculate how single objects behave in an expanding, holographic cosmos. It suggests that the "thermal one-point function"—a fancy way of saying the average energy state of a single thing—is a dynamic, time-dependent story that shifts from growth to decay, driven by the invisible strings in the cosmic ocean and the relentless pull of the black hole horizon.

Technical Summary: One-point holographic correlator in the expanding universe

Problem Statement
The AdS/CFT correspondence provides a framework to study strongly interacting quantum field theories (QFT) via weakly interacting gravity in one higher dimension. While two-point correlation functions for boundary operators in bulk geometries (such as black holes) are well-understood through geodesic approximations, the thermal one-point function of massive operators in an expanding universe remains less explored. Specifically, there is a need to understand how the thermal one-point function of massive operators evolves over time in various matter-dominated cosmological scenarios (radiation, matter, and exotic matter) within a holographic framework. Previous studies have addressed two-point functions in expanding universes using braneworld models but have not extended this analysis to the thermal one-point function in the presence of various matter sources.

Methodology
The authors employ the Randall-Sundrum II (RS-II) braneworld model, where a four-dimensional Friedmann-Lemaître-Robertson-Walker (FLRW) universe is localized on a 3-brane embedded in a five-dimensional Anti-de Sitter (AdS) bulk. The expansion of the universe is modeled as the time-dependent radial motion of the brane.

Bulk Geometry and Matter Sources: The bulk spacetime contains a black brane geometry coupled with a $p$ -brane gas. The backreaction of this $p$ -brane gas on the brane generates different cosmological scenarios:
- $p=0$ : Radiation-dominated universe.
- $p=1$ : Matter-dominated universe.
- $p=2$ : Exotic matter-dominated universe.
- Combinations of these parameters yield multi-component universes (e.g., radiation + matter).
Brane Dynamics: The time evolution of the brane's radial position, $r(\tau)$ , is derived using the second Israel junction condition. This condition relates the extrinsic curvature of the brane to its energy-momentum tensor, allowing the authors to solve for the brane's trajectory in both early and late-time regimes for single and multi-component universes.
Holographic Calculation: To compute the thermal one-point function $\langle O \rangle$ , the authors utilize the geodesic formula proposed by Grinberg and Maldacena. This formula relies on the saddle-point approximation of a path integral for a massive scalar field with higher-derivative couplings (specifically to the Weyl tensor squared, $W^2$ ). The one-point function is expressed as:
$\langle O \rangle \sim e^{-\Delta(l_{hor} + i T_s)}$
where $\Delta$ is the conformal dimension of the boundary operator, $l_{hor}$ is the regularized spacelike geodesic length from the boundary to the horizon, and $T_s$ is the proper time for a particle to travel from the horizon to the singularity. The time dependence enters the calculation through the time-dependent radial position of the brane, $\bar{z}(\tau) = 1/r(\tau)$ .

Key Contributions and Results

Derivation of Time-Dependent Brane Positions: The authors explicitly derive the time-dependent radial positions of the brane for single-component (radiation, matter, exotic) and multi-component (radiation+matter, radiation+exotic matter) universes. They provide asymptotic behaviors for both early ( $\tau \to 0$ ) and late ( $\tau \to \infty$ ) times.
- Early Time: The brane position scales as $\tau^{1/2}$ (radiation), $\tau^{2/3}$ (matter), and linearly (exotic matter).
- Late Time: The inverse radial coordinate $\bar{z}(\tau)$ scales as $\tau^{-1/2}$ , $\tau^{-2/3}$ , and $\tau^{-1}$ respectively.
Thermal One-Point Functions: Using the derived brane trajectories, the authors calculate the time-dependent thermal one-point functions for massive operators.
- Single-Component Universes:
  - Early Time: The one-point function exhibits polynomial growth proportional to $\tau^{1/2}$ (radiation), $\tau^{2/3}$ (matter), and $\tau$ (exotic matter).
  - Late Time: The function decays as a power law: $\tau^{-\Delta/2}$ (radiation), $\tau^{-2\Delta/3}$ (matter), and $\tau^{-\Delta}$ (exotic matter).
- Multi-Component Universes:
  - In universes with coexisting radiation and matter, the early-time behavior is dominated by radiation density ( $\propto \tau$ ), while the late-time decay is governed by matter density ( $\propto \tau^{-2\Delta/3}$ ).
  - In universes with coexisting radiation and exotic matter, early-time dynamics are radiation-dominated, while late-time decay is driven by exotic matter ( $\propto \tau^{-\Delta}$ ).
Physical Interpretation: The authors interpret the decay of the one-point function at late times as a result of information absorption by the black brane horizon. Although the brane's motion slows down, the absorption effect dominates, leading to a power-law decay. Conversely, the initial growth is attributed to the rapid shrinking of the spacelike geodesic length as the brane begins to move away from the initial position.

Significance and Claims
The paper claims to provide the first calculation of the time-dependent thermal one-point function for massive operators in an expanding universe within the RS-II braneworld framework. By combining the geodesic formula with the dynamics of $p$ -brane gases, the work bridges the gap between holographic correlation functions and realistic cosmological evolution.

The authors emphasize that their results are consistent with the thermal history of the universe, where radiation dominates early epochs and matter (or exotic matter) dominates later epochs. The graphical analysis confirms that the decay rates depend on the conformal dimension $\Delta$ , with larger $\Delta$ values corresponding to slower decay. The study serves as a foundational step for understanding holographic correlations in time-dependent backgrounds and suggests future directions involving higher curvature corrections, anisotropies, or the Karch-Randall braneworld model, but does not propose specific experimental tests or applications beyond these theoretical extensions.

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