Cosmological gravity on all scales V: MCMC forecasts combining large scale structure and CMB lensing for binned phenomenological modified gravity

Imagine the universe as a giant, invisible fabric called spacetime. For decades, physicists have believed this fabric follows a specific set of rules written by Albert Einstein, known as General Relativity. These rules explain how gravity works: massive objects like stars and galaxies warp the fabric, causing other things to fall toward them.

However, there's a mystery. When we look at how the universe is expanding and how galaxies are clustering together, the math doesn't quite add up unless we invent a mysterious, invisible substance called "Dark Energy" or "Dark Matter." Some scientists wonder: What if the rules of gravity themselves are slightly different than Einstein thought?

This paper is about testing those "what if" scenarios using the biggest, most powerful telescopes we have (or will have soon). Here is the story of what they did, explained simply.

1. The Problem: The "Nonlinear" Jungle

Imagine trying to predict how a crowd of people moves in a quiet room. That's easy; they just walk in straight lines. This is like looking at the universe when it was young and smooth.

But now, imagine that same crowd in a mosh pit at a rock concert. People are bumping, pushing, and swirling in chaotic ways. This is the nonlinear universe today. Galaxies are crashing into each other, and gravity is getting messy.

The problem is that our current computers are too slow to simulate this "mosh pit" for every possible version of gravity. If we want to test a new theory of gravity, we usually have to throw away the messy, complex parts of the data and only look at the "quiet room" parts. This means we throw away a huge amount of valuable information.

2. The Solution: The "Magic Translator" (The Emulator)

The authors of this paper built a translator.

Instead of running a slow, super-computer simulation every time they want to test a new gravity theory, they ran a set of simulations first and then trained a smart computer program (called an emulator) to learn the patterns.

The Analogy: Think of it like a weather forecaster. Instead of building a new giant wind tunnel for every single day to predict the weather, they use a computer model trained on thousands of past storms.
The Result: This "translator" can instantly predict how the cosmic mosh pit would look if gravity were slightly different, with 99% accuracy. This allows them to use all the data, even the messy, complex parts.

3. The Experiment: Two Different "Lenses"

To test these new gravity rules, the team looked at the universe through two different "lenses":

The Galaxy Lens (Large Scale Structure): They looked at how galaxies are clustered together. This is like looking at a map of cities to see how traffic flows. This tells us how matter is growing and clumping over time.
The CMB Lens (Cosmic Microwave Background): They looked at the "baby picture" of the universe (the afterglow of the Big Bang). As this light travels to us, gravity bends it, just like a lens bends light in a camera. This tells us about the shape of the universe's gravitational "bumps" from very far away and long ago.

They combined these two views (along with other measurements) to create a massive dataset they call the "6x2pt" data vector. It's like combining a high-definition map of today's traffic with a satellite view of the entire highway system from space.

4. The Findings: The "Gravity Slip" and the "Slippery Slope"

The team tested a theory where gravity changes strength over time and distance. They used two main variables:

$\mu$ (Mu): How strong gravity is for matter (like galaxies).
$\eta$ (Eta): How much gravity "slips" or bends light differently than it bends matter.

The Big Discovery:
They found that while they couldn't easily tell $\mu$ and $\eta$ apart (they are "degenerate," like trying to tell if a car is slow because of a flat tire or a bad engine), they could measure a specific combination of the two very precisely.

The Analogy: Imagine you are trying to guess the weight of a bag of apples. You can't tell if the bag is heavy because the apples are big or because there are many of them. But you can measure the total weight perfectly.
In their study, the "total weight" is a value called $\Sigma$ (Sigma). This value controls how much gravity bends light (lensing).

The Redshift Surprise:

Nearby Universe (Low Redshift): The galaxy data was great at measuring things close to us.
Faraway Universe (High Redshift): The galaxy data got weak and fuzzy. But when they added the CMB lensing (the "baby picture" lens), it was like turning on a flashlight in a dark room. The CMB data is sensitive to gravity at very high distances (around redshift $z \approx 2$ ), filling in the gaps where the galaxy data failed.

5. Why This Matters

This paper is a "dress rehearsal" for the future. In the next decade, telescopes like LSST (in Chile) and Simons Observatory will map millions of galaxies.

The authors have shown that:

We can now model the messy, complex universe fast enough to analyze this massive amount of data.
We can test if Einstein was wrong without getting stuck in the weeds of specific theories.
By combining galaxy maps with the ancient light of the Big Bang, we can see the universe's gravity in high definition, from our backyard to the edge of time.

In a nutshell: They built a fast, smart tool to simulate how the universe would look if gravity were weird. They used it to show that by combining two different types of cosmic maps, we can finally get a clear picture of whether the laws of gravity are changing as the universe ages. If they find a deviation, it could rewrite our understanding of physics!

Here is a detailed technical summary of the paper "Cosmological gravity on all scales V: MCMC forecasts combining large scale structure and CMB lensing for binned phenomenological modified gravity."

1. Problem Statement

As cosmology enters the "Stage IV" era (e.g., LSST, Simons Observatory), the field faces a critical bottleneck: the accurate modeling of nonlinear scales in modified gravity (MG) scenarios.

The Gap: While standard $\Lambda$ CDM models have robust nonlinear tools (e.g., emulators, N-body simulations), there is a severe lack of validated tools for the vast majority of model-agnostic modified gravity theories.
The Challenge: Current pipelines often rely on linear theory or severe scale cuts to ensure reliability, discarding the majority of the data volume (nonlinear scales). Existing MG-specific tools (like $f(R)$ or nDGP emulators) are not generalizable to arbitrary phenomenological deviations.
The Need: A fast, flexible framework is required to perform full Bayesian Markov Chain Monte Carlo (MCMC) analyses on model-agnostic MG parameters using realistic survey data vectors that include nonlinear structure formation and astrophysical systematics.

2. Methodology

The authors developed a comprehensive forecasting pipeline using the CosmoSIS framework, integrating several novel components:

A. Phenomenological Parameterization

Instead of testing specific theories, the paper uses a model-agnostic approach parameterizing deviations from General Relativity (GR) via two functions, $\mu(z)$ and $\eta(z)$ , defined in the post-Friedmann formalism:

$\mu(a)$ : Modifies the Poisson equation, governing the growth of matter clustering (probed by dark matter N-body simulations).
$\eta(a)$ : Describes the gravitational slip (difference between the two metric potentials), affecting photon geodesics and lensing.
Binning: These parameters are binned in redshift ($0 \le z \le 3$) into five discrete intervals (Table 1) to allow for time-varying deviations without assuming a specific functional form.

B. Nonlinear Emulation (The Core Innovation)

To bypass the computational cost of running full N-body simulations for every MCMC step, the authors built a Gaussian Process (GP) emulator for the modified gravity boost factor ( $B$ ):
$B(z, k) = \frac{P_{MG}(z, k)}{P_{\Lambda CDM}(z, k)}$

Training Data: The emulator is trained on 500 Latin Hypercube samples of COLA (COmoving Lagrangian Acceleration) simulations.
Accuracy: The emulator reproduces the nonlinear boost factor with < 1% accuracy relative to full N-body simulations up to $k = 1 \, h \, \text{Mpc}^{-1}$ .
Dimensionality Reduction: Principal Component Analysis (PCA) was used to compress the power spectra, mapping cosmological parameters to the first five principal components.

C. Data Vectors and Systematics

The pipeline forecasts constraints using two data configurations:

3 $\times$ 2pt: Cosmic shear, galaxy clustering, and galaxy-galaxy lensing (representing LSST Year 10).
6 $\times$ 2pt: The above plus CMB lensing auto- and cross-correlations (representing Simons Observatory).

Systematics Modeled:

Baryonic Feedback: Modeled using the BCemu (Baryonic Correction Emulator) based on the Baryonification model, marginalizing over cut-off mass, gas profile slope, and ejection factor.
Intrinsic Alignments (IA): Modeled via the Non-Linear Alignment (NLA) model.
Scale Cuts & Nulling: To mitigate modeling uncertainties in the highly nonlinear regime, the authors employ the Bernardeau-Nishimichi-Taruya (BNT) transform (k-cut method) to re-weight cosmic shear data, effectively removing sensitivity to poorly modeled small scales. The analysis is restricted to $k_{cut} = 0.5 \, h \, \text{Mpc}^{-1}$ .

D. Inference

Sampler: Nested sampling algorithm Nautilus is used to explore the posterior distribution and estimate evidence.
Screening: A "superscreened" approach is tested in Appendix C, where MG effects smoothly transition to $\Lambda$ CDM in the quasi-linear regime.

3. Key Contributions

First Full MCMC MG Forecast with Nonlinear Emulation: This is the first study to perform a full Bayesian MCMC analysis on a model-agnostic MG parameterization using a fast emulator trained on COLA simulations, rather than relying on Fisher matrices alone.
Integration of CMB Lensing: Demonstrates the specific utility of adding CMB lensing (6 $\times$ 2pt) to break degeneracies in high-redshift bins where large-scale structure (LSS) alone loses constraining power.
Validation of ReACT vs. COLA: The authors tested the ReACT (halo-model reaction) approach but found it unstable for this specific phenomenological parameterization outside the central prior volume, concluding that COLA-based emulation is currently more robust for this use case.
BNT Transform Implementation: Successfully integrated the BNT transform into a full MG pipeline to handle scale cuts consistently.

4. Key Results

Degeneracy Structure: The analysis confirms the well-known degeneracy between $\mu$ $μ$ and $\eta$ $η$ . However, Principal Component Analysis (PCA) reveals that the data is most sensitive to the combination $\Sigma = \mu(1 + \eta)/2$ , which governs the lensing potential (Weyl potential).
- The "tight" eigenmode aligns with $\Sigma$ .
- The "wide" eigenmode (poorly constrained) corresponds to the orthogonal direction where $\Sigma$ is constant.
Redshift Dependence:
- Low Redshift ( $z < 1$ ): LSS (3 $\times$ 2pt) provides strong constraints due to high signal-to-noise and significant nonlinear growth.
- High Redshift ( $z > 2$ ): LSS constraints degrade significantly. The addition of CMB lensing is crucial here, as the CMB lensing kernel peaks at $z \sim 2$ , providing direct sensitivity to the integrated potential at these redshifts.
Impact of CMB Lensing: Including CMB lensing (6 $\times$ 2pt) significantly tightens constraints on the $\Sigma$ direction, particularly for the highest redshift bin (Bin 5: $2.15 < z < 3 $), where 3$ \times$ 2pt alone fails to constrain the parameters effectively.
Screening Effects: Even with screening mechanisms implemented (which suppress MG effects on small scales), the pipeline retains considerable constraining power ( $<5\%$ on $\mu$ , $<1\%$ on $\Sigma$ ) for bins 1–3.

5. Significance

This work represents a major step forward in preparing for Stage IV cosmological surveys. By demonstrating that fast emulation of nonlinear MG effects is feasible within a full Bayesian pipeline, the authors enable:

Model-Agnostic Searches: The ability to search for generic deviations from GR without being biased toward specific theoretical models.
Robust Systematics Handling: A framework that simultaneously handles baryonic feedback, intrinsic alignments, and scale cuts via the BNT transform.
Future Readiness: The pipeline is ready to be applied to real data from LSST and Simons Observatory, allowing for rigorous tests of gravity across the entire redshift range of the universe.

The paper concludes that while nonlinear modeling remains challenging, the combination of COLA-based emulation, CMB lensing, and advanced statistical techniques (BNT, PCA) provides a viable path to extracting precision gravity constraints from the nonlinear regime.