Euclid: Constraints on f(R) cosmologies from the spectroscopic and photometric primary probes

This study forecasts that the Euclid mission will significantly constrain the Hu-Sawicki $f(R)$ modified gravity parameter $f_{R0}$, achieving approximately 1% precision when combining spectroscopic and photometric probes and enabling the distinction of these models from the standard $\Lambda$CDM cosmology at greater than 3$\sigma$ confidence.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: A Cosmic Detective Story

Imagine the Universe is a giant, expanding balloon. For decades, astronomers have been trying to figure out why it's expanding faster and faster. The standard story (called $\Lambda$CDM) says there is a mysterious, invisible force called "Dark Energy" pushing the balloon apart. It's like an invisible hand inflating the balloon.

But what if the hand isn't there? What if the rules of the game have changed? What if Gravity itself is acting differently on the largest scales, like a rubber band that gets stretchier the further you pull it? This is the idea behind Modified Gravity (specifically the Hu-Sawicki $f(R)$ model).

The paper you asked about is a "forecast" from the Euclid mission, a massive space telescope launching soon. The authors are asking: "If we build this super-powerful telescope, how good will we be at catching the culprit? Will we be able to tell if the Universe is being inflated by Dark Energy or if Gravity is just playing tricks on us?"

The Detective Tools: Euclid's Two Eyes

Euclid isn't just one camera; it's a detective with two different eyes, looking at the sky in two different ways:

1. The "Spectroscopic" Eye (The Precision Ruler):

   • What it does: It measures the exact distance to millions of galaxies by looking at their light spectra.
   • The Analogy: Imagine you are at a concert. This eye can tell you exactly how far away every single person in the crowd is standing, down to the inch. It gives you a 3D map of the crowd.
   • The Catch: It's very precise, but it can only see a few thousand people (galaxies) because the process is slow and detailed.

2. The "Photometric" Eye (The Wide-Angle Lens):

   • What it does: It takes pictures of over a billion galaxies, measuring their shapes and positions roughly.
   • The Analogy: This is like taking a high-resolution photo of the whole concert crowd from the back of the hall. You can't measure the exact distance to every person, but you can see the shape of the crowd and how they are squished or stretched (a phenomenon called Weak Lensing).
   • The Catch: It sees everyone, but the distance measurements are fuzzy.

The Mystery: The "Fifth Force"

In the standard story, gravity is a constant rule. In the $f(R)$ model, gravity has a secret "Fifth Force" that kicks in only on very large scales or in very specific conditions. It's like a rubber band that behaves normally when you pull it gently, but suddenly snaps and stretches wildly when you pull it hard.

The paper focuses on a specific "knob" on this theory called $f_{R0}$.

• If the knob is turned all the way to zero, the theory is just standard gravity (no mystery).
• If the knob is turned slightly, the rubber band behaves weirdly.

The goal of the paper is to predict how accurately Euclid can measure the position of that knob.

The Simulation: Running the Numbers

Since Euclid hasn't finished collecting all its data yet, the authors ran a massive computer simulation. They asked: "If the Universe actually works like this $f(R)$ model, what will Euclid see?"

They tested three scenarios for the "knob":

1. HS5 (The Loud Signal): The knob is turned up high. The weird gravity effects are strong and obvious.
2. HS6 (The Middle Ground): The knob is turned up a medium amount. This is the "baseline" they expect to be realistic.
3. HS7 (The Whisper): The knob is barely turned. The weird gravity effects are tiny, almost indistinguishable from the standard story.

The Results: How Good is Euclid?

The authors found that Euclid is going to be an incredible detective, but it needs to use both eyes to solve the case.

• Using just the "Ruler" (Spectroscopic): It can narrow down the knob's position to about 3% accuracy. It's good, but not perfect.
• Using just the "Wide-Angle Lens" (Photometric): It can get the knob to about 1.4% accuracy.
• Using Both Together (The "3x2pt" Combo): This is the magic combination. By cross-referencing the precise distances with the shapes of the galaxies, Euclid can pin down the knob to 1% accuracy.

The "Aha!" Moment:
If the Universe follows the "Middle Ground" scenario (HS6), Euclid will be able to say with high confidence: "Yes, the knob is at this specific spot, and it is definitely NOT zero." This would prove that gravity is modified.

Even more impressively, if the knob is barely turned on (HS7), Euclid might still be able to distinguish it from the standard "no-knob" theory, provided the data is clean and the models are perfect.

The Catch: The "Non-Linear" Jungle

There is a hurdle. The paper admits that the "deep" parts of the Universe (where galaxies are crowded together) are messy. This is called the non-linear regime.

• The Analogy: Imagine trying to predict the weather. Predicting the wind in an empty field is easy (linear). Predicting the wind inside a hurricane where everything is swirling chaotically is hard (non-linear).
• The authors had to invent special mathematical "recipes" to guess how gravity behaves in these chaotic, crowded regions. They tested these recipes against supercomputer simulations to make sure they weren't lying.

The Conclusion: A New Era of Physics

In simple terms, this paper says: "Get ready, because Euclid is going to be the ultimate test for whether Einstein's theory of gravity is the whole story or just part of it."

If the data matches the predictions in this paper, we will know that gravity changes its behavior depending on the scale, just like a rubber band that gets stretchier the further you pull. If the data matches the standard model perfectly, then the "Fifth Force" doesn't exist, and Dark Energy remains the only explanation.

Either way, Euclid is going to tell us the truth about the invisible rules governing our Universe.

Technical Summary

Here is a detailed technical summary of the paper "Euclid: Constraints on f(R) cosmologies from the spectroscopic and photometric primary probes."

1. Problem Statement

The origin of the Universe's accelerated expansion remains a major challenge in cosmology. While the standard $\Lambda$CDM model (with a cosmological constant) fits current data, the theoretical value of vacuum energy is vastly different from observations. An alternative is to modify General Relativity (GR). The Hu-Sawicki $f(R)$ model is a popular modification where the Ricci scalar $R$ in the Hilbert-Einstein action is replaced by $R + f(R)$.

This model introduces a "fifth force" that is scale-dependent and screened in high-density regions (like the Solar System) via the chameleon mechanism. The key free parameter is $f_{R0}$ (the value of the derivative $df/dR$ at redshift $z=0$), which determines the strength of the modification. Current constraints allow $|f_{R0}|$ to be as low as $10^{-7}$, making it difficult to distinguish from standard GR.

The Core Question: How well will the upcoming Euclid space mission be able to constrain the parameter $f_{R0}$ and distinguish Hu-Sawicki $f(R)$ models from the standard $\Lambda$CDM model using its primary probes?

2. Methodology

The authors extend the forecasting pipeline established in the Euclid Collaboration (EC19) to accommodate the specific physics of $f(R)$ gravity.

A. Theoretical Framework

• Modified Gravity Formalism: They utilize the Hu-Sawicki functional form. Unlike $\Lambda$CDM, $f(R)$ gravity induces a scale-dependent growth of structure.
• Boltzmann Solvers: They validated predictions using two codes: MGCAMB (quasi-static approximation) and EFTCAMB (full evolution). They found sub-percent agreement between the two for the relevant scales ($k < 10 \, h \text{ Mpc}^{-1}$), justifying the use of the quasi-static approximation for forecasts.
• Scale-Dependent Growth: The growth rate $f(k, z)$ and the matter power spectrum $P(k, z)$ become dependent on the wavenumber $k$. This requires generalizing the recipes for galaxy clustering and weak lensing.

B. Observables and Probes

The study forecasts constraints using four primary Euclid probes:

1. Spectroscopic Galaxy Clustering (GCsp): Measures 3D clustering. The authors modified the theoretical prediction to account for scale-dependent growth rates in Redshift Space Distortions (RSD) and Baryon Acoustic Oscillations (BAO). They used a Savitzky-Golay filter to generate "no-wiggle" power spectra, replacing the standard Eisenstein-Hu formula which is less accurate for $f(R)$.
2. Photometric Galaxy Clustering (GCph): Measures 2D projected clustering.
3. Weak Lensing (WL): Measures cosmic shear. The window functions were updated to include the modified gravitational potential sum ($\Phi + \Psi$) via the function $\Sigma(k, z)$.
4. Cross-Correlation (XCph): The cross-correlation between GCph and WL.

C. Non-Linear Modelling

Since $f(R)$ effects are most prominent in the non-linear regime, the authors employed a fitting formula (Winther et al. 2019) calibrated against N-body simulations (DUSTGRAIN and ELEPHANT). This formula captures the enhancement of the non-linear matter power spectrum relative to $\Lambda$CDM as a function of $f_{R0}$.

D. Analysis Setup

• Fisher Matrix Formalism: Used to forecast parameter constraints.
• Fiducial Models: Three scenarios were tested:
  • HS5: $|f_{R0}| = 5 \times 10^{-5}$ (Large deviation).
  • HS6: $|f_{R0}| = 5 \times 10^{-6}$ (Baseline, currently allowed).
  • HS7: $|f_{R0}| = 5 \times 10^{-7}$ (Close to GR limit).
• Scenarios:
  • Optimistic: High $k_{max}$ (small scales) and $\ell_{max}$, minimal systematics.
  • Pessimistic: Lower $k_{max}$/$\ell_{max}$, stricter cuts.
  • Quasi-linear: A conservative cut for GCsp ($k_{max} = 0.15 \, h \text{ Mpc}^{-1}$) to avoid non-linear modelling uncertainties.

3. Key Contributions

1. Validated Forecasting Pipeline: This is the first study to produce validated forecasts for the Hu-Sawicki model using the full suite of Euclid primary probes (GCsp, GCph, WL, XCph) combined.
2. Generalized Recipes: The authors successfully generalized the EC19 recipes to handle scale-dependent growth factors and modified lensing kernels, ensuring accurate theoretical predictions for $f(R)$ cosmologies.
3. Non-Linear Integration: They integrated a specific fitting formula for the deeply non-linear power spectrum, allowing the analysis to probe scales where $f(R)$ deviations are most significant.
4. Parameter Transformation: They demonstrated how constraints on the logarithmic parameter $\log_{10}|f_{R0}|$ (used for Fisher matrix stability) translate into asymmetric constraints on the linear parameter $|f_{R0}|$.

4. Key Results

Constraints on $f_{R0}$ (Optimistic Scenario, Baseline HS6)

For a fiducial value of $|f_{R0}| = 5 \times 10^{-6}$:

• GCsp alone: Constrains $\log_{10}|f_{R0}|$ at 3.0% level.
• Photometric probes (WL + GCph + XCph) alone: Constrains at 1.4% level.
• Full Combination (GCsp + WL + GCph + XCph): Constrains at 1.0% level.
  • This corresponds to an absolute error on $|f_{R0}|$ of $\approx 6 \times 10^{-7}$ ($1\sigma$).

Model Discrimination

• HS5 vs. HS6: The full Euclid combination can distinguish these models at $>3\sigma$.
• HS7 vs. $\Lambda$CDM: Even for the model closest to GR ($|f_{R0}| = 5 \times 10^{-7}$), the full combination of probes can distinguish it from $\Lambda$CDM at $>3\sigma$ in the optimistic scenario.
• Degeneracy Breaking: The spectroscopic probe (GCsp) is crucial for breaking degeneracies in parameters like the Hubble constant ($h$) and baryon density ($\Omega_b$), which photometric probes alone struggle to constrain tightly.

Impact of Scenarios

• Pessimistic Scenario: Constraints degrade, particularly for photometric probes (errors increase by a factor of ~1.7 to 2). However, the full combination still provides strong constraints (e.g., ~1.7% relative error for HS6).
• Quasi-linear Cut: Restricting GCsp to linear scales ($k_{max}=0.15$) has a minimal impact on the final constraints when combined with other probes, suggesting the non-linear information from photometric probes dominates the $f_{R0}$ constraint.

5. Significance and Conclusion

• Euclid's Power: The study confirms that Euclid will be a powerful probe for modified gravity. By combining spectroscopic and photometric data, it can constrain the scale-dependence of perturbations and the non-linear regime of structure formation with unprecedented precision.
• Systematics are Key: The results rely heavily on the accurate modelling of the matter power spectrum in the mildly and deeply non-linear regimes. The paper emphasizes that controlling systematic effects (like baryonic feedback and intrinsic alignments) is essential to realize these forecasted constraints.
• Future Prospects: Even for models very close to General Relativity ($|f_{R0}| \sim 10^{-7}$), Euclid has the potential to detect deviations if the data follows the $f(R)$ model, provided the theoretical modelling is robust. This highlights the importance of the non-linear regime in testing gravity theories beyond $\Lambda$CDM.

In summary, the paper demonstrates that the synergy between Euclid's spectroscopic and photometric surveys will allow for stringent tests of the Hu-Sawicki $f(R)$ model, potentially ruling out or confirming deviations from General Relativity at the $1%$level on the parameter$f_{R0}$.