Modified Gravity with Cross-Correlation

Note: This project is ongoing and subject to continuous advancements and modifications.

A project of Dunlap Institute in collaboration with CCDS and CASSA to forecast signatures of modified gravity theories through cross-correlation analyses.

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Overview

Why Modified Gravity?

While the ΛCDM framework fits many observations, it relies on undetected dark energy and dark matter. Modified gravity theories offer alternative explanations for cosmic acceleration and large-scale structure formation without invoking unknown components. These theories modify the Poisson equation and gravitational potentials, creating observable signatures testable through galaxy clustering and weak lensing.

Why Cross-Correlation?

Cross-correlation techniques break degeneracies between cosmological parameters and reduce systematic errors by combining uncorrelated datasets.

graph TD
    A[Single Probe] --> B[Parameter Degeneracies]
    A --> C[Systematic Uncertainties]
    A --> D[Cosmic Variance]
    
    E[Cross-Correlation] --> F[Break Degeneracies]
    E --> G[Decorrelate Systematics]
    E --> H[Cancel Sample Variance]
    E --> I[Enhance S/N]
    
    F --> J[Improved Constraints]
    G --> J
    H --> J
    I --> J
    
    J --> K[Tighter Cosmological Parameters]

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Key Advantages

Advantage	Description
Break degeneracies	Between cosmological parameters
Mitigate systematics	Through uncorrelated noise cancellation
Amplify signals	Via joint analysis of multiple tracers

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Modified Gravity Theories

Models

Theory	Description	Key Parameters
f(R) Gravity	Scalar-tensor theory replacing the Ricci scalar (R) in the Einstein-Hilbert action with f(R). Predicts scale-dependent growth rates and modified lensing potentials.	f_R0, scale-dependent growth
DGP Gravity	Braneworld scenario where gravity leaks into an extra dimension at large scales, leading to cosmic acceleration without a cosmological constant.	Ω_rc (or γ growth index), self-accelerating branch

Key Observables

Observable	Type	Notation	Description
Galaxy–Galaxy Clustering	Auto-correlation	$C_\ell^{gg}$ (GG)	Measures galaxy clustering
Galaxy–CMB Lensing Cross-Correlation	Cross-correlation	$C_\ell^{\kappa g}$ (GCMB)	Measures galaxy-lensing correlation

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Project Goals

This project calculates and analyzes galaxy–galaxy power spectra ($C_\ell^{gg}$) and galaxy–CMB lensing cross-power spectra ($C_\ell^{\kappa g}$) using theoretical models including nDGP, e-mantis, and Bacco emulators.

Forecasting Pipeline

The framework forecasts the ability of future cosmological surveys to constrain modified gravity theories using:

Component	Description
Synthetic Observables	Theoretical power spectra from MG models using nDGP, e-mantis, and Bacco
Survey Modeling	Incorporation of survey characteristics (e.g., sky coverage, galaxy density) from LSST/DESC and Simons Observatory (to be added)
Fisher Matrix Analysis	Quantifies the precision of cosmological and MG parameter constraints
Bias and Degeneracy Evaluation	Assesses degeneracies between MG and ΛCDM parameters

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Methodology

Theoretical Framework

The project uses a multi-tracer approach, combining data from different sources to enhance signal-to-noise and reduce systematics through power spectra analysis.

Statistical Analysis Methods

Method	Purpose
Fisher Matrix Forecasting	Quantifies parameter constraints and breaks degeneracies
Emulator-Based Acceleration	Uses nDGP, e-mantis, and Bacco for rapid computation across parameter grids

Emulator Selection

Multiple emulators were explored (MGemu, fRemu, Cosmopower). Ultimately, nDGP, e-mantis, and Bacco were selected based on performance, model coverage, and suitability for the analysis.

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Emulator Parameter Ranges

Current Working Parameters

Parameter	Bacco (ΛCDM)	e-MANTIS (f(R))	nDGP (DGP)
Cold matter density $\Omega_{cb}$	0.23–0.40	0.155–0.465	0.28–0.36
Primordial amplitude $A_s$	~1.7×10⁻⁹ (from $\sigma_8$)	~(1.5–2.0)×10⁻⁹	(1.7–2.5)×10⁻⁹
Hubble parameter $h$	0.60–0.80	0.55–0.85	0.61–0.73
Modified Gravity	None (ΛCDM)	$|f_{R_0}|$: 10⁻⁷–10⁻⁴	$H_0 r_c$: 0.2–20

Full Parameter Comparison

Parameter	Bacco (ΛCDM)	e-MANTIS (f(R))	nDGP (DGP)
Cold matter density $\Omega_{cb}$	0.23–0.40	0.155–0.465	0.28–0.36
Baryon density $\Omega_b$	0.04–0.06	0.037–0.062	0.04–0.06
Primordial amplitude $A_s$	~1.7×10⁻⁹ (tuned from $\sigma_8$)	~(1.5–2.0)×10⁻⁹ (internal)	(1.7–2.5)×10⁻⁹
Spectral index $n_s$	0.92–1.01	0.72–1.20	0.92–1.00
Hubble parameter $h$	0.60–0.80	0.55–0.85	0.61–0.73
Neutrino mass $\Sigma m_\nu$	0.0–0.4 eV	—	—
Dark Energy $w_0$	−1.15 to −0.85	—	—
Dark Energy $w_a$	−0.30 to +0.30	—	—
Modified Gravity param.	None (ΛCDM)	$|f_{R_0}|$: 10⁻⁷–10⁻⁴	$H_0 r_c$: 0.2–20
Scale factor $a$ (z-range)	0.4–1.0 (z: 0–1.5)	0.25–1.0 (z: 0–3)	No explicit $a$ param.
k-range $[h \text{ Mpc}^{-1}]$	$[10^{-2}, 5]$	$[0.03, 10]$	$[0.01, 5]$
Redshift range	0–1.5	0–3	0–2
Accuracy	~1%	~1–3%	~2–3%
Primary Output	Full $P(k)$ [ΛCDM]	Boost factor $B(k)$	Boost factor $B(k)$

Key Differences

Bacco provides the most extensive cosmological parameter space, including neutrino masses and dynamical dark energy, but only for ΛCDM gravity. Outputs the full nonlinear power spectrum.

e-MANTIS specializes in f(R) gravity with broader redshift coverage (z: 0–3) and wavenumber range. Outputs a boost factor relative to ΛCDM.

nDGP provides predictions for nDGP gravity with the tightest parameter constraints but extends to lower wavenumbers (k = 0.01). Returns a boost factor for the modified gravity signature.

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Analysis & Interpretation

Analysis Type	Description
Parameter Sensitivity	Assess the impact of MG parameters on observables
Survey Specifications	Simulate realistic measurements with survey details
Bias Evaluation	Identify biases and systematics in parameter estimation due to MG effects

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References

Emulators • HEALPix/healpy • Angular Power Spectra

Emulator	Links	HEALPix / healpy	Links	Angular Power Spectra	Links
nDGP	Fiorini 2023 • Docs	healpy GitHub	healpy	NaMaster	GitHub • Docs
e-mantis	Sáez-Casares 2023 • Docs	healpy Docs	readthedocs	NaMaster Covariances	Covariances
Bacco	Aricò 2020 • Docs	Tutorial	healpy-sims.ipynb	CCL	Docs

Statistical Tools • LSSTDESC Tutorials

Statistical / Sampling Tool	Docs	LSSTDESC Tutorial	Notebook
emcee	Docs	C_ℓ in pyccl	CellsCorrelations.ipynb
pocoMC	Docs	Emulators (Bacco)	Cosmological_Emulator.ipynb
Corner	Docs	Tomographic bins	Redshift_Distributions.ipynb
GetDist	Docs	emcee + pyccl	MCMC Likelihood Analysis.ipynb

Astropy Cosmology

Category	Resource	Link
General	Overview	docs.astropy.org/cosmology
	Base API	astropy.cosmology.Cosmology
	Units	cosmology/units
Models	LambdaCDM	astropy.cosmology.LambdaCDM
	FlatLambdaCDM	astropy.cosmology.FlatLambdaCDM
	FlatwCDM	astropy.cosmology.FlatwCDM
	FLRW	astropy.cosmology.FLRW
Utilities	Planck18	astropy.cosmology.realizations.Planck18
	Redshift-Distance Units	cosmology.units.redshift_distance

More references can be found in the extended reference list: here

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Contact

Adrita Khan
Email | LinkedIn | Twitter

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This repository offers a comprehensive resource for understanding, testing, and contributing to the Modified Gravity project. It includes theoretical models and tools for computing cross-correlated power spectra, focusing on testing theories like f(R) and DGP using advanced computational methods and survey simulations.