- 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 20).
- The two CPL parameters should be measured with errors 0.06 and 0.26, respectively (fixing the growth rate to fiducial), see Table 11 and Figure 20.
- The equation of state and the growth rate parameter , both assumed constant, should be simultaneously constrained to within 0.04 and 0.03, respectively.
- The growth function should be constrained to within 0.01 – 0.02 for each redshift bin from to (see Table 4).
- A scale-independent bias function should be constrained to within 0.02 for each redshift bin (see Table 4).
- The growth rate parameters defined in Eq. 1.8.5 should be measured to within 0.08, 0.17, respectively.
- Euclid will achieve an accuracy on measurements of the dark energy sound speed of (WLS) and (RS), if , or (WLS) and (RS), if .
- The coupling between dark energy and dark matter can be constrained by Euclid (with Planck) to less than (see Figure 30 and Table 13).
- Any departure from GR greater than in the growth index will be distinguished by the WLS [429].
- Euclid WLS can detect deviations between 3% and 10% from the GR value of the modified-gravity parameter (Eq. 1.3.28), whilst with the RS there will be a 20% accuracy on both and (Eq. 1.3.27).
- With the WLS, Euclid should provide an upper limit to the present dimensionless scalaron inverse mass of the scalar (where the time dependent scalar field mass is defined in Eq. 1.8.37) as for and for
- The WLS will be able to rule out the DGP model growth index with a Bayes factor [429], and viable phenomenological extensions could be detected at the level for [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.

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

Living Rev. Relativity 16, (2013), 6
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