Les Houches on Dark Universe 2025: Elements of cosmology beyond FLRW

This lecture evaluates the performance of the standard FLRW cosmological model in predicting cosmic expansion and light propagation, while presenting results on the challenges posed by backreaction and fitting problems when moving beyond strict homogeneity and isotropy.

The Gist

The Big Picture: The Smooth Map vs. The Bumpy Road

Imagine you are trying to understand the geography of the entire Earth. The standard way cosmologists do this is by using a perfectly smooth, flat map. This map assumes the Earth is perfectly round and uniform everywhere. In physics, this is called the FLRW model. It's a simplified "background" that scientists use to predict how the universe expands and how light travels through it.

However, we know the real Earth isn't smooth. It has mountains, valleys, oceans, and cities. The real universe is full of "lumps" like galaxies, stars, and empty voids.

This lecture asks a simple but profound question: Does the fact that the universe is "lumpy" actually change the rules of the game? Specifically, does the bumpy nature of the universe:

1. Change how fast the universe is expanding? (The Backreaction problem).
2. Change how we measure distances to faraway stars? (The Fitting problem).

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Part 1: The Backreaction Problem (Does the bumpy road change the speed?)

The Analogy: The Traffic Jam
Imagine a highway where cars (galaxies) are driving. The standard model (FLRW) assumes the traffic is perfectly smooth and evenly spaced. It calculates the average speed of the traffic flow based on this smoothness.

But in reality, traffic is chaotic. You have clusters of cars (galaxies) and huge empty stretches of road (voids).

• The Question: Does the fact that cars are clumping together and leaving gaps actually change the overall speed limit of the highway?
• The "Backreaction" Idea: Some scientists wondered if the "clumping" of matter creates a gravitational tug-of-war that speeds up or slows down the expansion of the universe, potentially mimicking the mysterious "Dark Energy" that we think is pushing the universe apart.

The Paper's Findings:
After doing some heavy math (using tools like Buchert's formalism and computer simulations), the paper concludes: No, the bumps don't matter much.

• Think of the universe like a giant ocean. Even though there are waves and ripples (galaxies), the overall level of the water (the expansion rate) isn't significantly changed by the waves themselves.
• Different methods (like patching together small cubes of space or using supercomputers) all agree: The "backreaction" effect is so tiny that it's negligible. The smooth map is still a very good approximation of the real, bumpy road.

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Part 2: The Fitting Problem (Does the bumpy road change the distance?)

The Analogy: The Laser Pointer
Now, imagine you are trying to measure the distance to a lighthouse using a laser pointer.

• The Smooth Model: If the air were perfectly clear and uniform, the laser beam would travel in a straight line, and you could calculate the distance easily.
• The Bumpy Reality: The air is full of heat waves, dust, and turbulence. These act like lenses. Some parts of the air might focus the laser beam (making the lighthouse look brighter and closer), while other parts might spread it out (making it look dimmer and farther).

The Paper's Findings:

1. Individual Measurements are Messy: If you point your laser at one specific lighthouse, the "lumpy" universe might make it look 10% closer or farther than it really is. This is called gravitational lensing.
2. The Average is Perfect: Here is the magic trick. If you point your laser at thousands of lighthouses in every direction and take the average, the errors cancel out perfectly.
   • Some beams get focused (magnified).
   • Some beams get defocused (dimmed).
   • The Result: When you average them all together, the "bumpy" universe gives you the exact same distance as the "smooth" universe.

The "Swiss Cheese" Twist:
The paper discusses a famous idea called the "Swiss Cheese Model." Imagine the universe is a block of cheese (smooth matter) with holes drilled out (empty voids). If light travels through the holes, it shouldn't be focused by the cheese, so it should travel faster/farther.

• The Catch: While the light travels through the holes, it also passes near the edges of the holes where the "cheese" is. The gravity of the cheese edges creates a "shear" (a stretching force) that bends the light back.
• The Conclusion: The stretching caused by the edges perfectly cancels out the lack of focusing in the holes. On average, the distance you measure is the same as if the cheese had no holes at all.

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The Final Verdict

The paper concludes that the standard "smooth" model (FLRW) is actually doing a great job.

• Expansion: The lumps in the universe don't significantly change how fast the universe is growing.
• Distances: While looking at a single object might be tricky because of gravitational lensing, the average distance to objects across the sky is exactly what the smooth model predicts.

Why does this matter?
It means we don't need to throw out our current cosmological theories. The "Cosmological Principle" (that the universe is statistically smooth on large scales) holds up. The universe is bumpy, but the bumps average out to look smooth when we look at the big picture.

One Caveat:
The author mentions a few strange recent observations (like weird differences in the Cosmic Microwave Background in different parts of the sky) that might suggest the universe isn't as uniform as we think. But for now, the "smooth map" remains the best tool we have for navigating the cosmos.

Technical Summary

Technical Summary: Elements of Cosmology beyond FLRW

Problem Statement
Modern cosmology relies on the cosmological principle, which posits that the Universe is statistically homogeneous and isotropic. Strictly applied, this yields the Friedmann–Lemaître–Robertson–Walker (FLRW) model, serving as the background spacetime for describing cosmic expansion and light propagation. However, the real Universe is inhomogeneous. This lecture addresses two fundamental questions regarding the validity of the FLRW approximation in the presence of structure:

1. The Backreaction Problem: Do inhomogeneities (structures) affect the global expansion dynamics of the Universe, potentially mimicking dark energy or altering the Friedmann equations?
2. The Fitting Problem: Is the FLRW model the correct framework for interpreting cosmological observations (specifically the redshift–distance relation), given that light propagates through a "lumpy" Universe rather than a smooth one?

Methodology
The paper employs a combination of analytical general relativity, scalar averaging formalisms, and numerical relativity to investigate these problems.

• Backreaction Analysis:

  • Buchert's Scalar Formalism: The author utilizes a spatial averaging approach for an inhomogeneous dust fluid. By defining a volume-average operator over a comoving domain $D$, the lecture derives an effective Friedmann equation. This introduces a "backreaction term" ($\mathcal{B}$) arising from the variance of the local expansion rate and the mean square of the shear tensor.
  • Relativistic Approaches: The paper examines three specific relativistic frameworks to quantify backreaction:
    1. Clifton & Sanghai's Post-Newtonian (PN) Patchwork: A lattice universe constructed from PN-expanded cells.
    2. Green & Wald's Effective Field Theory: Treating the metric as a smooth background plus small-scale fluctuations, analyzing the effective stress-energy tensor.
    3. GEVOLUTION: A relativistic N-body code that evolves the metric without presupposing Friedmann dynamics, solving Einstein's equations to high order in perturbations.

• Observational Analysis (Fitting Problem):

  • Null Geodesic Formalism: The lecture adopts observational coordinates $(t, \lambda, \phi_a)$ adapted to null geodesics to describe light propagation.
  • Raychaudhuri Equation for Light: The evolution of the angular-diameter distance ($D_A$) is governed by the null Raychaudhuri equation, which includes terms for matter density ($\rho$) and shear ($\Sigma$).
  • Modeling Inhomogeneities: The paper contrasts linear perturbation theory (weak lensing) with the "empty-beam" model (Zel'dovich) and "Swiss-cheese" models (Kantowski, Dyer & Roeder), where light propagates through vacuum regions between mass concentrations.
  • Averaging Theorems: The analysis distinguishes between "source averaging" (averaging over intrinsic source sizes) and "directional averaging" (averaging over lines of sight), utilizing magnification theorems to determine systematic biases.

Key Contributions and Results

• Dynamics (Backreaction):

  • The non-linearity of Einstein's equations implies that smoothing the metric before evolution ($E[\langle g \rangle]$) is not equivalent to evolving then smoothing ($\langle E[g] \rangle$).
  • Buchert's Formalism: The backreaction term $\mathcal{B}$ depends on the competition between the variance of the expansion rate and the shear. In Newtonian limits, this term is a negligible boundary effect.
  • Relativistic Estimates:
    • The PN patchwork model yields a backreaction term scaling as $\epsilon^4$ (where $\epsilon \sim v/c$), which is negligible compared to standard Friedmann terms.
    • Green & Wald's effective field theory demonstrates that small-scale fluctuations produce a correction to the stress-energy tensor that is traceless and negligible for cosmological expansion.
    • GEVOLUTION simulations confirm that the expansion law matches standard Friedmann dynamics to within one part in $10^4$.
  • Conclusion: Current evidence suggests that the backreaction of structures on cosmic expansion is practically negligible within General Relativity.

• Observations (Fitting Problem):

  • Light Propagation: In an inhomogeneous Universe, light beams experience focusing due to matter density and shear. While Zel'dovich's "empty-beam" model suggests light travels mostly through vacuum (leading to larger distances than FLRW predictions), Weinberg's theorem and subsequent analysis show that shear contributions compensate for the lack of matter density.
  • Averaging Bias:
    • Magnification theorems establish that the mean magnification of sources is unity when averaged over source sizes ($\langle \mu \rangle_s = 1$).
    • The bias on the distance–redshift relation is a second-order effect in cosmological perturbations.
    • For Type-Ia Supernovae at $z=1$, the bias is negligible ($\sim 10^{-3}$).
    • For the Cosmic Microwave Background (CMB) at $z \approx 1090$, a naive calculation suggests a 5% bias. However, the paper argues this is an overestimate because finite light beams (corresponding to the sound horizon, $\theta_* \approx 0.6^\circ$) smooth out inhomogeneities smaller than the beam size. When accounting for this cutoff (multipole $\ell \lesssim 300$), the bias drops to $\sim 10^{-3}$.
  • Conclusion: The FLRW distance–redshift relation remains a robust framework for interpreting cosmological data, despite local inhomogeneities.

Significance and Claims
The paper concludes that if the cosmological principle is correct, the FLRW background provides an accurate description of the Universe. The significance of the work lies in rigorously testing the limits of this principle:

1. Validation of Standard Cosmology: It demonstrates that, within General Relativity, neither the expansion dynamics nor the interpretation of standard cosmological observables are significantly compromised by the inhomogeneous nature of the Universe. The "backreaction" and "fitting" problems do not currently invalidate the standard model.
2. Precision Testing: Because the FLRW model holds up so well, the paper argues that the cosmological principle itself must be tested with extreme precision. The author highlights two recent, unexplained anomalies that challenge this principle:
   • Large-scale CMB anisotropy patches where cosmological parameters differ by up to 30%.
   • A quasar dipole that is twice the magnitude expected from the CMB kinematic dipole.
3. Modesty of Claims: The paper explicitly states that while backreaction is negligible in General Relativity, alternative theories of gravity could yield different conclusions. Furthermore, it acknowledges that while average relations hold, significant corrections can occur on individual measurements (e.g., specific lines of sight).

The lecture serves as a comprehensive review of why the FLRW model remains the standard, while identifying the specific observational anomalies that warrant further investigation into the fundamental homogeneity and isotropy of the cosmos.