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Cosmology as Representation: Informational Invariance and the Limits of Scientific Realism

The paper challenges the view that cosmology reveals a unique underlying reality, arguing instead that its objective content lies in cross-representational informational invariants, thereby redefining scientific progress as the refinement of accessible observational structure rather than convergence toward a fixed ontology.

Original authors: Stefano Profumo

Published 2026-06-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Stefano Profumo

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

The Big Idea: The Map vs. The Territory

Imagine you are trying to describe a massive, mysterious forest to someone who has never seen it. You have a few tools: a drone camera, a satellite map, and a set of compass readings.

For a long time, scientists have believed that if they build a perfect mathematical model of the forest (let's call it the "Standard Model"), they have discovered the actual trees, rocks, and rivers inside it. They think, "Our map is so accurate, the forest must be exactly like our map says."

Profumo argues that this is a mistake.

He suggests that while our maps (cosmological models like $\Lambda$ CDM) are incredibly accurate at predicting what we see, they don't necessarily tell us what the forest is made of at a fundamental level. Instead, what we have really discovered are the rules of the map itself—the patterns, the connections, and the limits of what we can see.

The Problem: One Story, Many Tellers

The paper starts by pointing out a problem called underdetermination. This is a fancy way of saying: "The evidence fits many different stories."

The Analogy: Imagine you hear a song played on a piano. You know the notes (the data). You could say, "This is a piano." But you could also say, "This is a synthesizer programmed to sound exactly like a piano," or "This is a recording of a piano played through a specific filter."
The Science: In cosmology, we see the universe expanding and galaxies moving. The "Standard Model" says this is caused by "Dark Matter" and "Dark Energy." But Profumo argues that other mathematical stories could explain the exact same observations without needing those specific "invisible ingredients." Just because the math works doesn't mean the story about what is out there is the only true one.

The Solution: Finding the "Unchangeable" Patterns

If we can't be sure about the "ingredients" (the ontology), what can we be sure about? Profumo proposes we look for Informational Invariants.

Think of these as the skeleton of the data that stays the same no matter how you dress it up.

Correlation Structure (The Rhythm):
- Analogy: Imagine listening to a complex piece of music. You might not know if the band is playing jazz or classical, but you can definitely hear that the bass always hits right after the snare drum. That relationship is real, even if you aren't sure what instruments are making the sound.
- In Cosmology: We know for a fact that the temperature of the Cosmic Microwave Background (the "afterglow" of the Big Bang) has specific bumps and dips that relate to each other in a precise pattern. Whether you call the cause "Dark Matter" or "Modified Gravity," that pattern must exist. That pattern is the invariant.
Distinguishability (What We Can Tell Apart):
- Analogy: Imagine you have two very similar shades of blue paint. If you look at them with your naked eye, they look identical. If you use a microscope, you might see they are slightly different. The "limit" of your vision is a real fact about your situation, not just a flaw in the paint.
- In Cosmology: There are certain combinations of numbers in our models that the universe allows us to measure clearly, and others that are "blurred" together. The fact that we cannot tell two specific things apart is a hard, objective limit of our observation. That limit is an invariant.
Information Bounds (The Fence):
- Analogy: If you are standing in a valley, you can only see the mountains that aren't blocked by the hills in front of you. The "horizon" isn't a trick; it's a physical limit of your position.
- In Cosmology: Because we are stuck inside the universe, we can only see a certain slice of it (our "past light cone"). There are hard limits on how much information we can ever gather. These limits are real features of our situation, not just gaps in our knowledge.

The New Philosophy: "Informational Structural Realism"

Profumo calls his new view Informational Structural Realism.

Old View (Naive Realism): "Our model is true, so Dark Matter particles definitely exist."
Old View (Empiricism): "We can't know what's real, so let's just say the model is useful for making predictions."
Profumo's View: "We can know something real! We know the structure of the information. We know the patterns, the limits, and the relationships that any successful model must follow. We are realists about the data's shape, not necessarily the story's characters."

The "Fisher-Rao" Tool: The Ruler for Reality

To make this idea scientific and not just philosophical, the paper uses a mathematical tool called Information Geometry (specifically the Fisher-Rao metric).

The Analogy: Imagine you have a map of a city. You can draw the streets using miles, kilometers, or "steps." The names of the streets might change, and the units might change, but the distance between two points and the shape of the neighborhood remain the same.
The Math: Profumo uses this geometry to measure the "distance" between different scientific models. It helps scientists see which parts of a model are just arbitrary labels (like calling a street "Main St" vs. "1st Ave") and which parts are the actual, unchangeable structure of the data.

What This Means for Scientific Progress

Finally, the paper redefines what "progress" looks like in cosmology.

The Old Idea: Progress is like walking toward a finish line where we finally find the "True Theory" and the "True List of Particles."
Profumo's Idea: Progress is like increasing the resolution of a camera.
- At first, the picture is blurry. We can't tell if the object is a cat or a dog.
- As we get better tools, the picture gets sharper. We can finally tell it's a dog.
- But even with a 4K camera, we might still debate what kind of dog it is (a Golden Retriever or a Labrador).
- The Point: The "dog-ness" (the informational structure) is what we are getting better at seeing. The specific breed (the specific ontology) might still be up for debate.

Summary

The paper argues that cosmology is incredibly successful, but we shouldn't get too excited about claiming we know exactly what the universe is "made of" down to the smallest particle. Instead, we have successfully mapped the rules of the game: the patterns of connection, the limits of what we can see, and the mathematical shapes that any description of the universe must fit.

We aren't just guessing; we are uncovering the invariant structure of information that the universe reveals to us, even if we don't yet know the full story behind it.

Technical Summary: Cosmology as Representation: Informational Invariance and the Limits of Scientific Realism

Problem Statement
The paper addresses the tension between the empirical success of modern cosmology (specifically the $\Lambda$ CDM model) and the epistemic security of the ontological conclusions drawn from it. While cosmological models provide extraordinarily accurate fits to observational data across diverse probes (CMB, large-scale structure, supernovae), the author argues that empirical success underdetermines not only the ontology of the universe (e.g., the specific nature of dark matter or dark energy) but also the mathematical and conceptual frameworks used to represent that data.

The central problem is the "Galilean program" assumption: that scientific progress consists of recovering a uniquely privileged mathematical structure corresponding to a mind-independent reality. The paper contends that this inference is flawed because:

Underdetermination: Empirical adequacy does not uniquely fix ontology (e.g., the ether, caloric theory, and Newtonian absolute space were empirically successful but ontologically discarded).
Representational Non-Uniqueness: Modern physics exhibits cases where distinct formalisms (e.g., quantum dualities, equivalent formulations of QM) encode the same physical content.
Theory-Laden Inference: Cosmological data are not raw givens but are extracted through complex, theory-mediated pipelines (foreground subtraction, calibration, statistical inference), making the evidential base dependent on representational choices.

Consequently, the paper challenges the standard realist view that successful models reveal the universe's fundamental constituents, proposing instead that the secure content of cosmology lies elsewhere.

Methodology
The paper employs a philosophical analysis grounded in information geometry and structural realism. The methodology proceeds through three main stages:

Conceptual Analysis of Underdetermination: The author distinguishes between empirical adequacy, explanatory coherence, and ontological commitment, arguing that the first two do not guarantee the third. The analysis draws on historical examples (ether, caloric) and contemporary physics (quantum interpretations, AdS/CFT duality) to demonstrate that empirical content can be preserved across formally distinct frameworks.
Formalization via Information Geometry: To operationalize the notion of "invariant structure," the paper utilizes tools from information theory:
- Fisher–Rao Metric: Used to define a Riemannian metric on the space of statistical models. This metric quantifies statistical distinguishability between models independent of arbitrary parameterization. By Čencov's theorem, this metric is the unique Riemannian metric invariant under sufficient statistics.
- Kullback–Leibler (KL) Divergence: Used to define equivalence classes of models. Two models are considered empirically equivalent if their KL divergence falls below the detectability threshold set by the observational configuration.
- Eigenstructure Analysis: The paper analyzes the eigenvalues and eigenvectors of the Fisher information matrix to distinguish between "stiff" directions (well-constrained by data) and "sloppy" directions (degenerate or prior-sensitive).
Bayesian Integration: The framework is reconciled with standard Bayesian inference in cosmology. The author argues that features robust under prior variation (stiff eigenmodes) correspond to the informational invariants, while prior-sensitive features (sloppy modes) are representation-dependent.

Key Contributions
The paper introduces and defines Informational Structural Realism (ISR), a position distinct from both Ontic Structural Realism (OSR) and Constructive Empiricism.

Definition of Informational Invariants: ISR identifies the objective content of cosmology not as specific entities or the internal structure of a single theory, but as cross-representational informational invariants. These are features that persist across all empirically adequate representations. The paper categorizes these into three families:
1. Correlation Structure: Statistical dependencies among observables (e.g., CMB angular power spectrum, matter power spectrum) that any adequate model must reproduce.
2. Distinguishability Structure: The geometry of model space defining which parameter combinations can be resolved by data and which remain degenerate.
3. Information Bounds: Fundamental limits on accessible information imposed by causal horizons, cosmic variance, and entropy bounds.
Operational Criterion for Objectivity: Unlike OSR, which reads structure off a single favored formalism, ISR locates objectivity in the intersection of an equivalence class of models. A feature is objectively significant if it is invariant under admissible transformations of representation (reparameterization, model formulation) and inference pipelines.
Reframing Scientific Progress: The paper redefines scientific progress in cosmology. Rather than convergence toward a uniquely correct ontology, progress is characterized as the progressive refinement of informational structure. As observational precision improves, equivalence classes split, and formerly degenerate directions in model space become distinguishable.

Results and Applications
The paper applies the ISR framework to analyze specific cosmological phenomena:

The Hubble Tension: The discrepancy between $H_0$ inferred from CMB data and local distance-ladder measurements is reframed not merely as a conflict of values, but as a structural mismatch in the informational landscape. The tension indicates that the joint KL divergence between the data distributions of the two probes exceeds the detectability threshold under a single $\Lambda$ CDM model, revealing an informational inconsistency rather than a simple measurement error.
The $S_8$ Tension: The paper illustrates how the status of the $S_8$ tension changes with refinements in analysis pipelines (e.g., intrinsic alignment modeling). The invariant is the joint constraint surface defined by the data; the specific numerical value of $S_8$ is partially representation-dependent.
Dark Energy Evolution (DESI): The paper analyzes recent DESI results suggesting time-varying dark energy. It argues that the invariant is the shape of the distance–redshift relation, while the specific encoding (cosmological constant vs. dynamical scalar field) remains representation-dependent and subject to revision based on catalog choices.
Dark Matter: ISR distinguishes between the invariant pattern of gravitational discrepancies (rotation curves, lensing, CMB peaks) and the specific ontological interpretation (cold dark matter particles vs. modified gravity). The pattern is invariant; the ontology is not.

Significance and Claims
The paper claims to offer a "more defensible locus of objectivity" for cosmology that avoids the pitfalls of naive realism and the limitations of instrumentalism.

Modest Realism: The author argues that cosmology yields objective knowledge, but this knowledge is about the structure of what can be known rather than the fundamental inventory of what exists. The "miracle" of scientific success is explained by the model's ability to efficiently encode invariant informational structure, not by its unique revelation of fundamental ontology.
Epistemic Humility: The framework acknowledges the "embedded observer" problem. Objectivity is not a "view from nowhere" but is indexed to the physical constraints of the observer (causal horizons, cosmic variance). ISR provides a realistic account of objectivity within these epistemic horizons.
Limitations: The paper explicitly states that information geometry does not prove realism; it provides an operational criterion for identifying candidates for objective content. The step from "invariant under reparameterization" to "ontologically fundamental" remains a philosophical inference. The formalism is local (Fisher metric) and assumes a shared data space, limiting its application to comparisons within cosmological practice rather than across radically different theoretical frameworks.

In conclusion, the paper posits that cosmology's greatest achievement is not the identification of the universe's constituents, but the precise mapping of the constraints on the informational structure of the observable universe. Scientific progress is the progressive sharpening of these constraints, not necessarily the convergence to a single, final ontology.