1.9 Summary and outlook

This section introduced the main features of the most popular dark energy/modified gravity models. Here we summarize the performance of Euclid with respect to these models. Unless otherwise indicated, we always assume Euclid with no external priors and all errors fully marginalized over the standard cosmological parameters. Here RS denotes the redshift survey, WLS the weak lensing one.

  1. Euclid (RS) should be able to measure the main standard cosmological parameters to percent or sub-percent level as detailed in Table 7 (all marginalized errors, including constant equation of state and constant growth rate, see Table 11 and Figure 20View Image).
  2. The two CPL parameters w ,w 0 1 should be measured with errors 0.06 and 0.26, respectively (fixing the growth rate to fiducial), see Table 11 and Figure 20View Image.
  3. The equation of state w and the growth rate parameter γ, both assumed constant, should be simultaneously constrained to within 0.04 and 0.03, respectively.
  4. The growth function should be constrained to within 0.01 – 0.02 for each redshift bin from z = 0.7 to z = 2 (see Table 4).
  5. A scale-independent bias function b(z) should be constrained to within 0.02 for each redshift bin (see Table 4).
  6. The growth rate parameters γ ,γ 0 1 defined in Eq. 1.8.5View Equation should be measured to within 0.08, 0.17, respectively.
  7. Euclid will achieve an accuracy on measurements of the dark energy sound speed of σ(c2)∕c2= 2615 s s (WLS) and σ(c2)∕c2= 50.05 s s (RS), if c2 = 1 s, or σ(c2)∕c2 = 0.132 s s (WLS) and 2 2 σ(cs)∕cs = 0.118 (RS), if 2 −6 cs = 10.
  8. The coupling β2 between dark energy and dark matter can be constrained by Euclid (with Planck) to less than −4 3 ⋅ 10 (see Figure 30View Image and Table 13).
  9. Any departure from GR greater than ≃ 0.03 in the growth index γ will be distinguished by the WLS [429Jump To The Next Citation Point].
  10. Euclid WLS can detect deviations between 3% and 10% from the GR value of the modified-gravity parameter Σ (Eq. 1.3.28View Equation), whilst with the RS there will be a 20% accuracy on both Σ and μ (Eq. 1.3.27View Equation).
  11. With the WLS, Euclid should provide an upper limit to the present dimensionless scalaron inverse mass μ ≡ H0 ∕M0 of the f(R ) scalar (where the time dependent scalar field mass is defined in Eq. 1.8.37View Equation) as μ = 0.00 ± 1.10 × 10−3 for l < 400 and μ = 0.0 ± 2.10 × 10− 4 for l < 10000
  12. The WLS will be able to rule out the DGP model growth index with a Bayes factor |ln B | ≃ 50 [429Jump To The Next Citation Point], and viable phenomenological extensions could be detected at the 3σ level for 1000 ≲ ℓ ≲ 4000 [199].

At the same time, there are several areas of research that we feel are important for the future of Euclid, both to improve the current analyses and to maximize its science return. Here we provide a preliminary, partial list.

  1. The results of the redshift survey and weak lensing surveys should be combined in a statistically coherent way
  2. The set of possible priors to be combined with Euclid data should be better defined
  3. The forecasts for the parameters of the modified gravity and clustered dark-energy models should be extended to include more general cases
  4. We should estimate the errors on a general reconstruction of the modified gravity functions Σ,μ or of the metric potentials Ψ, Φ as a function of both scale and time.

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