Heavens et al. [429] have used Bayesian evidence to distinguish between models, using the Fisher matrices for the parameters of interest. This study calculates the ratio of evidences for a 3D weak lensing analysis of the full Euclid survey, for a dark-energy model with varying equation of state, and modified gravity with additionally varying growth parameter . They find that Euclid can decisively distinguish between, e.g., DGP and dark energy, with . In addition, they find that it will be possible to distinguish any departure from GR which has a difference in greater than . A phenomenological extension of the DGP model [332, 11] has also been tested with Euclid. Specifically, [199] found that it will be possible to discriminate between this modification to gravity from CDM at the level in a wide range of angular scale, approximately .

Thomas et al. [886] construct Fisher matrix forecasts for the Euclid weak lensing survey, shown in Figure 11. The constraints obtained depend on the maximum wavenumber which we are confident in using; is relatively conservative as it probes the linear regime where we can hope to analytically track the growth of structure; is more ambitious as it includes nonlinear power, using the [844] fitting function. This will not be strictly correct, as the fitting function was determined in a GR context. Note that is not very sensitive to , while , defined in [41] as (and where is defined in Eq. 1.3.28) is measured much more accurately in the nonlinear regime.

Amendola et al. [41] find Euclid weak lensing constraints for a more general parameterization that includes evolution. In particular, is investigated by dividing the Euclid weak lensing survey into three redshift bins with equal numbers of galaxies in each bin, and approximating that is constant within that bin. Since , i.e., the value of in the bin (present-day) is degenerate with the amplitude of matter fluctuations, it is set to unity. The study finds that a deviation from unit (i.e., GR) of 3% can be detected in the second redshift bin, and a deviation of 10% is still detected in the furthest redshift bin.

Beynon et al. [132] make forecasts for modified gravity with Euclid weak lensing including [457] in interpolating between the linear spectrum predicted by modified gravity, and GR on small scales as required by Solar System tests. This requires parameters (a measure of the abruptness of transitioning between these two regimes), (controlling the -dependence of the transition) and (controlling the -dependence of the transition).

The forecasts for modified gravity parameters are shown in Figure 12 for the Euclid lensing data. Even with this larger range of parameters to fit, Euclid provides a measurement of the growth factor to within 10%, and also allows some constraint on the parameter, probing the physics of nonlinear collapse in the modified gravity model.

Finally, Song et al. [848] have shown forecasts for measuring and using both imaging and spectroscopic surveys. They combine 20,000 square-degree lensing data (corresponding to [550] rather than to the updated [551]) with the peculiar velocity dispersion measured from redshift space distortions in the spectroscopic survey, together with stringent background expansion measurements from the CMB and supernovae. They find that for simple models for the redshift evolution of and , both quantities can be measured to 20% accuracy.

The Euclid mission will produce a catalog of up to 100 million galaxy redshifts and an imaging survey that
should allow to estimate the galaxy ellipticity of up to 2 billion galaxy images. Here we discuss these
surveys and fix their main properties into a “Euclid model”, i.e., an approximation to the real Euclid survey
that will be used as reference mission in the following.

Modeling the Redshift Survey.

The main goals of next generation redshift surveys will be to constrain the dark-energy parameters and to
explore models alternative to standard Einstein gravity. For these purposes they will need to consider very
large volumes that encompass , i.e., the epoch at which dark energy started dominating the energy
budget, spanning a range of epochs large enough to provide a sufficient leverage to discriminate among
competing models at different redshifts.

Here we consider a survey covering a large fraction of the extragalactic corresponding to capable to measure a large number of galaxy redshifts out to . A promising observational strategy is to target H emitters at near-infrared wavelengths (which implies ) since they guarantee both relatively dense sampling (the space density of this population is expected to increase out to ) and an efficient method to measure the redshift of the object. The limiting flux of the survey should be the tradeoff between the requirement of minimizing the shot noise, the contamination by other lines (chiefly among them the [O ii] line), and that of maximizing the so-called efficiency , i.e., the fraction of successfully measured redshifts. To minimize shot noise one should obviously strive for a low flux. Indeed, [389] found that a limiting flux would be able to balance shot noise and cosmic variance out to . However, simulated observations of mock H galaxy spectra have shown that ranges between 30% and 60% (depending on the redshift) for a limiting flux [551]. Moreover, contamination from [O ii] line drops from 12% to 1% when the limiting flux increases from to [389].

Taking all this into account, in order to reach the top-level science requirement on the number density of H galaxies, the average effective H line flux limit from a 1-arcsec diameter source shall be lower than or equal to . However, a slitless spectroscopic survey has a success rate in measuring redshifts that is a function of the emission line flux. As such, the Euclid survey cannot be characterized by a single flux limit, as in conventional slit spectroscopy.

We use the number density of H galaxies at a given redshift, , estimated using the latest empirical data (see Figure 3.2 of [551]), where the values account for redshift – and flux – success rate, to which we refer as our reference efficiency .

However, in an attempt to bracket current uncertainties in modeling galaxy surveys, we consider two further scenarios, one where the efficiency is only the half of and one where it is increased by a factor of 40%. Then we define the following cases:

- Reference case (ref.). Galaxy number density which include efficiency (column in Table 3).
- Pessimistic case (pess.). Galaxy number density , i.e., efficiency is (column in Table 3).
- Optimistic case (opt.). Galaxy number density , i.e., efficiency is (column in Table 3).

The total number of observed galaxies ranges from (pess.) to (opt.). For all cases we
assume that the error on the measured redshift is , independent of the limiting flux of
the survey.

0.65 – 0.75 | 1.75 | 1.25 | 0.63 |

0.75 – 0.85 | 2.68 | 1.92 | 0.96 |

0.85 – 0.95 | 2.56 | 1.83 | 0.91 |

0.95 – 1.05 | 2.35 | 1.68 | 0.84 |

1.05 – 1.15 | 2.12 | 1.51 | 0.76 |

1.15 – 1.25 | 1.88 | 1.35 | 0.67 |

1.25 – 1.35 | 1.68 | 1.20 | 0.60 |

1.35 – 1.45 | 1.40 | 1.00 | 0.50 |

1.45 – 1.55 | 1.12 | 0.80 | 0.40 |

1.55 – 1.65 | 0.81 | 0.58 | 0.29 |

1.65 – 1.75 | 0.53 | 0.38 | 0.19 |

1.75 – 1.85 | 0.49 | 0.35 | 0.18 |

1.85 – 1.95 | 0.29 | 0.21 | 0.10 |

1.95 – 2.05 | 0.16 | 0.11 | 0.06 |

Modeling the weak lensing survey. For the weak lensing survey, we assume again a sky coverage of 15,000 square degrees. For the number density we use the common parameterization

where is the peak of and the median and typically we assume and a surface density of valid images of per arcmin [551]). We also assume that the photometric redshifts give an error of . Other specifications will be presented in the relevant sections.In this section we forecast the constraints that future observations can put on the growth rate and on a scale-independent bias, employing the Fisher matrix method presented in Section 1.7.3. We use the representative Euclid survey presented in Section 1.8.2. We assess how well one can constrain the bias function from the analysis of the power spectrum itself and evaluate the impact that treating bias as a free parameter has on the estimates of the growth factor. We estimate how errors depend on the parametrization of the growth factor and on the number and type of degrees of freedom in the analysis. Finally, we explicitly explore the case of coupling between dark energy and dark matter and assess the ability of measuring the coupling constant. Our parametrization is defined as follows. More details can be found in [308].

- -parameterization. This is in fact a non-parametric model in which the growth rate itself is modeled as a step-wise function , specified in different redshift bins. The errors are derived on in each -th redshift bin of the survey.
- -parameterization. As a second case we assume where the function is parametrized as As shown by [969, 372], this parameterization is more accurate than that of Eq. (1.8.3) for both CDM and DGP models. Furthermore, this parameterization is especially effective to distinguish between a CDM model (i.e., a dark-energy model with a constant equation of state) that has a negative () and a DGP model that instead, has a positive (). In addition, modified gravity models show a strongly evolving [378, 673, 372], in contrast with conventional dark-energy models. As a special case we also consider constant (only when also is assumed constant), to compare our results with those of previous works.
- -parameterization. To explore models in which perturbations grow faster than in the CDM case, like in the case of a coupling between dark energy and dark matter [307], we consider a model in which is constant and the growth rate varies as where quantifies the strength of the coupling. The example of the coupled quintessence model worked out by [307] illustrates this point. In that model, the numerical solution for the growth rate can be fitted by the formula (1.8.6), with , where is the dark energy-dark matter coupling constant and best fit values and . In this simple case, observational constraints over can be readily transformed into constraints over .

- DGP model. We consider the flat space case studied in [602]. When we adopt this model then we set , [372] or [587] and when and are assumed constant.
- model. Here we consider different classes of models: i) the one proposed in [456], depending on two parameters, and , which we fix to and . For the model with we assume , , values that apply quite generally in the limit of small scales (provided they are still linear, see [378]) or and . Unless differently specified, we will always refer to this specific model when we mention comparisons to a single model. ii) The model proposed in [864] fixing and , which shows a very similar behavior to the previous one. iii) The one proposed in [904] fixing .
- Coupled dark-energy (CDE) model. This is the coupled model proposed by [33, 955]. In this case we assume , (this value comes from putting as coupling, which is of the order of the maximal value allowed by CMB constraints) [44]. As already explained, this model cannot be reproduced by a constant . Forecasts on coupled quintessence based on [42, 33, 724] are discussed in more detail in Section 1.8.8.

For the fiducial values of the bias parameters in every bin, we assume (already used in [753]) since this function provides a good fit to H line galaxies with luminosity modeled by [698] using the semi-analytic GALFORM models of [108]. For the sake of comparison, we will also consider the case of constant corresponding to the rather unphysical case of a redshift-independent population of unbiased mass tracers.

The fiducial values for are computed through

Now we express the growth function and the redshift distortion parameter in terms of the growth rate (see Eqs. (1.8.8), (1.8.7)). When we compute the derivatives of the spectrum in the Fisher matrix and are considered as independent parameters in each redshift bin. In this way we can compute the errors on (and ) self consistently by marginalizing over all other parameters.

Now we are ready to present the main result of the Fisher matrix analysis . We note that in all tables below we always quote errors at 68% probability level and draw in the plots the probability regions at 68% and/or 95% (denoted for shortness as 1 and 2 values). Moreover, in all figures, all the parameters that are not shown have been marginalized over or fixed to a fiducial value when so indicated.

The fiducial growth function in the -th redshift bin is evaluated from a step-wise, constant growth rate as

To obtain the errors on and we compute the elements of the Fisher matrix and marginalize over all other parameters. In this case one is able to obtain, self-consistently, the error on the bias and on the growth factor at different redshifts, as detailed in Table 4. In Figure 13 we show the contour plots at 68% and 95% of probability for all the pairs in several redshift bins (with ), where ’s are the central values of the bins. We do not show the ellipses for all the 14 bins to avoid overcrowding.Table 4 illustrates one important result: through the analysis of the redshift-space galaxy power spectrum in a next-generation Euclid-like survey, it will be possible to measure galaxy biasing in redshift bins with less than 1.6% error, provided that the bias function is independent of scale. We also tested a different choice for the fiducial form of the bias: finding that the precision in measuring the bias as well as the other parameters has a very little dependence on the form. Given the robustness of the results on the choice of in the following we only consider the case.

In Figure 14 we show the errors on the growth rate as a function of redshift, overplotted to our fiducial CDM (green solid curve). The three sets of error bars are plotted in correspondence of the 14 redshift bins and refer (from left to right) to the Optimistic, Reference and Pessimistic cases, respectively. The other curves show the expected growth rate in three alternative cosmological models: flat DGP (red, longdashed curve), CDE (purple, dot-dashed curve) and different models (see description in the figure caption). This plot clearly illustrates the ability of next generation surveys to distinguish between alternative models, even in the less favorable choice of survey parameters.

The main results can be summarized as follows.

- The ability of measuring the biasing function is not too sensitive to the characteristic of the survey ( can be constrained to within 1% in the Optimistic scenario and up to 1.6% in the Pessimistic one) provided that the bias function is independent of scale. Moreover, we checked that the precision in measuring the bias has a very little dependence on the form.
- The growth rate can be estimated to within 1 – 2.5% in each bin for the Reference case survey with no need of estimating the bias function from some dedicated, independent analysis using higher order statistics [925] or full-PDF analysis [825].
- The estimated errors on depend weakly on the fiducial model of .

z | |||||||||

ref. | opt. | pess. | ref. | opt. | pess. | ||||

0.7 | 0.016 | 0.015 | 0.019 | 1.30 | 0.7 | 0.76 | 0.011 | 0.010 | 0.012 |

0.8 | 0.014 | 0.014 | 0.017 | 1.34 | 0.8 | 0.80 | 0.010 | 0.009 | 0.011 |

0.9 | 0.014 | 0.013 | 0.017 | 1.38 | 0.9 | 0.82 | 0.009 | 0.009 | 0.011 |

1.0 | 0.013 | 0.012 | 0.016 | 1.41 | 1.0 | 0.84 | 0.009 | 0.008 | 0.011 |

1.1 | 0.013 | 0.012 | 0.016 | 1.45 | 1.1 | 0.86 | 0.009 | 0.008 | 0.011 |

1.2 | 0.013 | 0.012 | 0.016 | 1.48 | 1.2 | 0.87 | 0.009 | 0.009 | 0.011 |

1.3 | 0.013 | 0.012 | 0.016 | 1.52 | 1.3 | 0.88 | 0.010 | 0.009 | 0.012 |

1.4 | 0.013 | 0.012 | 0.016 | 1.55 | 1.4 | 0.89 | 0.010 | 0.009 | 0.013 |

1.5 | 0.013 | 0.012 | 0.016 | 1.58 | 1.5 | 0.91 | 0.011 | 0.010 | 0.014 |

1.6 | 0.013 | 0.012 | 0.016 | 1.61 | 1.6 | 0.91 | 0.012 | 0.011 | 0.016 |

1.7 | 0.014 | 0.013 | 0.017 | 1.64 | 1.7 | 0.92 | 0.014 | 0.012 | 0.018 |

1.8 | 0.014 | 0.013 | 0.018 | 1.67 | 1.8 | 0.93 | 0.014 | 0.013 | 0.019 |

1.9 | 0.016 | 0.014 | 0.021 | 1.70 | 1.9 | 0.93 | 0.017 | 0.015 | 0.025 |

2.0 | 0.019 | 0.016 | 0.028 | 1.73 | 2.0 | 0.94 | 0.023 | 0.019 | 0.037 |

Next, we focus on the ability of determining and , in the context of the -parameterization and , in the -parameterization. In both cases the Fisher matrix elements have been estimated by expressing the growth factor as

where for the -parameterization we fix .- -parameterization. We start by considering the case of constant and in which
we set and . As we will discuss in the next Section, this
simple case will allow us to cross-check our results with those in the literature. In Figure 16
we show the marginalized probability regions, at and levels, for and . The
regions with different shades of green illustrate the Reference case for the survey whereas the
blue long-dashed and the black short-dashed ellipses refer to the Optimistic and Pessimistic
cases, respectively. Errors on and are listed in Table 5 together with the corresponding
figures of merit [FoM] defined to be the squared inverse of the Fisher matrix determinant and
therefore equal to the inverse of the product of the errors in the pivot point, see [21]. Contours
are centered on the fiducial model. The blue triangle and the blue square represent the flat DGP
and the models’ predictions, respectively. It is clear that, in the case of constant and
, the measurement of the growth rate in a Euclid-like survey will allow us to discriminate
among these models. These results have been obtained by fixing the curvature to its fiducial
value . If instead, we consider curvature as a free parameter and marginalize over,
the errors on and increase significantly, as shown in Table 6, and yet the precision is
good enough to distinguish the different models. For completeness, we also computed the fully
marginalized errors over the other cosmological parameters for the reference survey, given in
Table 7.
As a second step we considered the case in which and evolve with redshift according to Eqs. (1.8.5) and (1.8.2) and then we marginalized over the parameters , and . The marginalized probability contours are shown in Figure 17 in which we have shown the three survey setups in three different panels to avoid overcrowding. Dashed contours refer to the -dependent parameterizations while red, continuous contours refer to the case of constant and obtained after marginalizing over . Allowing for time dependency increases the size of the confidence ellipses since the Fisher matrix analysis now accounts for the additional uncertainties in the extra-parameters and ; marginalized error values are in columns , of Table 8. The uncertainty ellipses are now larger and show that DGP and fiducial models could be distinguished at level only if the redshift survey parameter will be more favorable than in the Reference case.

We have also projected the marginalized ellipses for the parameters and and calculated their marginalized errors and figures of merit, which are reported in Table 9. The corresponding uncertainties contours are shown in the right panel of Figure 16. Once again we overplot the expected values in the and DGP scenarios to stress the fact that one is expected to be able to distinguish among competing models, irrespective on the survey’s precise characteristics.

- -parameterization.
We have repeated the same analysis as for the -parameterization taking into account the possibility of coupling between DE and DM, i.e., we have modeled the growth factor according to Eq. (1.8.6) and the dark-energy equation of state as in Eq. (1.8.2) and marginalized over all parameters, including . The marginalized errors are shown in columns , of Table 8 and the significance contours are shown in the three panels of Figure 18 which is analogous to Figure 17. Even if the ellipses are now larger we note that errors are still small enough to distinguish the fiducial model from the and DGP scenarios at and level respectively.

Marginalizing over all other parameters we can compute the uncertainties in the and parameters, as listed in Table 10. The relative confidence ellipses are shown in the left panel of Figure 19. This plot shows that next generation Euclid-like surveys will be able to distinguish the reference model with no coupling (central, red dot) to the CDE model proposed by [44] (white square) only at the level.

bias | case | FoM | FoM | ||||

ref. | 0.15 | 0.07 | 97 | 0.07 | 0.07 | 216 | |

opt. | 0.14 | 0.06 | 112 | 0.07 | 0.06 | 249 | |

pess. | 0.18 | 0.09 | 66 | 0.09 | 0.09 | 147 | |

Finally, in order to explore the dependence on the number of parameters and to compare our results to previous works, we also draw the confidence ellipses for , with three different methods: i) fixing and to their fiducial values and marginalizing over all the other parameters; ii) fixing only and ; iii) marginalizing over all parameters but , . As one can see in Figure 20 and Table 11 this progressive increase in the number of marginalized parameters reflects in a widening of the ellipses with a consequent decrease in the figures of merit. These results are in agreement with those of other authors (e.g., [945]).

The results obtained in this section can be summarized as follows.

- If both and are assumed to be constant and setting , then a redshift survey described by our Reference case will be able to constrain these parameters to within 4% and 2%, respectively.
- Marginalizing over degrades these constraints to 5.3% and 4% respectively.
- If and are considered redshift-dependent and parametrized according to Eqs. (1.8.5) and (1.8.2) then the errors on and obtained after marginalizing over and increase by a factor 7, 5. However, with this precision we will be able to distinguish the fiducial model from the DGP and scenarios with more than and significance, respectively.
- The ability to discriminate these models with a significance above is confirmed by the confidence contours drawn in the - plane, obtained after marginalizing over all other parameters.
- If we allow for a coupling between dark matter and dark energy, and we marginalize over
rather than over , then the errors on are almost identical to those obtained in the
case of the -parameterization, while the errors on decrease significantly.
However, our ability in separating the fiducial model from the CDE model is significantly hampered: the confidence contours plotted in the - plane show that discrimination can only be performed wit significance. Yet, this is still a remarkable improvement over the present situation, as can be appreciated from Figure 19 where we compare the constraints expected by next generation data to the present ones. Moreover, the Reference survey will be able to constrain the parameter to within 0.06. Reminding that we can write [307], this means that the coupling parameter between dark energy and dark matter can be constrained to within 0.14, solely employing the growth rate information. This is comparable to existing constraints from the CMB but is complementary since obviously it is obtained at much smaller redshifts. A variable coupling could therefore be detected by comparing the redshift survey results with the CMB ones.

It is worth pointing out that, whenever we have performed statistical tests similar to those already discussed by other authors in the context of a Euclid-like survey, we did find consistent results. Examples of this are the values of FoM and errors for , , similar to those in [945, 614] and the errors on constant and [614]. However, let us notice that all these values strictly depend on the parametrizations adopted and on the numbers of parameters fixed or marginalized over (see, e.g., [753]).

In this section we apply power spectrum tomography [448] to the Euclid weak lensing survey without using any parameterization of the Hubble parameter as well as the growth function . Instead, we add the fiducial values of those functions at the center of some redshift bins of our choice to the list of cosmological parameters. Using the Fisher matrix formalism, we can forecast the constraints that future surveys can put on and . Although such a non-parametric approach is quite common for as concerns the equation-of-state ratio in supernovae surveys [see, e.g., 22] and also in redshift surveys [815], it has not been investigated for weak lensing surveys.

The Fisher matrix is given by [458]

where is the observed fraction of the sky, is the covariance matrix, is the convergence power spectrum and is the vector of the parameters defining our cosmological model. Repeated indices are being summed over from to , the number of redshift bins. The covariance matrix is defined as (no summation over ) where is the intrinsic galaxy shear and is the fraction of galaxies per steradian belonging to the -th redshift bin: where is the galaxy density per arc minute and the galaxy density for the -th bin, convolved with a gaussian around , the center of that bin, with a width of in order to account for errors in the redshift measurement.For the matter power spectrum we use the fitting formulae from [337] and for its nonlinear corrections the results from [844]. Note that this is where the growth function enters. The convergence power spectrum for the -th and -th bin can then be written as

Here we make use of the window function (with being the comoving distance) and the dimensionless Hubble parameter For the equation-of-state ratio, finally, we use the usual CPL parameterization.We determine intervals in redshift space such that each interval contains the same amount of galaxies. For this we use the common parameterization

where is the peak of and the median. Now we can define as the center of the -th redshift bin and add as well as to the list of cosmological parameters. The Hubble parameter and the growth function now become functions of the and respectively:This is being done by linearly interpolating the functions through their supporting points, e.g., for . Any function that depends on either or hence becomes a function of the and as well.

The values for our fiducial model (taken from WMAP 7-year data [526]) and the survey parameters that we chose for our computation can be found in Table 12.

As for the sum in Eq. (1.8.10), we generally found that with a realistic upper limit of and a step size of we get the best result in terms of a figure of merit (FoM), that we defined as

Note that this is a fundamentally different FoM than the one defined by the Dark Energy Task Force. Our definition allows for a single large error without influencing the FoM significantly and should stay almost constant after dividing a bin arbitrarily in two bins, assuming the error scales roughly as the inverse of the root of the number of galaxies in a given bin.

We first did the computation with just binning and using the common fit for the growth function slope [937]

yielding the result in Figure 21. Binning both and and marginalizing over the s yields the plot for seen in Figure 22.Notice that here we assumed no prior information. Of course one could improve the FoM by taking into account some external constraints due to other experiments.

In order to fully exploit next generation weak lensing survey potentialities, accurate knowledge of nonlinear power spectra up to is needed [465, 469]. However, such precision goes beyond the claimed accuracy of the popular halofit code [844].

[651] showed that, using halofit for non-CDM models, requires suitable corrections. In spite of that, halofit has been often used to calculate the spectra of models with non-constant DE state parameter . This procedure was dictated by the lack of appropriate extensions of halofit to non-CDM cosmologies.

In this paragraph we quantify the effects of using the halofit code instead of -body outputs for nonlinear corrections for DE spectra, when the nature of DE is investigated through weak lensing surveys. Using a Fisher-matrix approach, we evaluate the discrepancies in error forecasts for , and and compare the related confidence ellipses. See [215] for further details.

The weak lensing survey is as specified in Section 1.8.2. Tests are performed assuming three different fiducial cosmologies: CDM model (, ) and two dynamical DE models, still consistent with the WMAP+BAO+SN combination [526] at 95% C.L. They will be dubbed M1 (, ) and M3 (, ). In this way we explore the dependence of our results on the assumed fiducial model. For the other parameters we adopt the fiducial cosmology of Secton 1.8.2.

The derivatives to calculate the Fisher matrix are evaluated by extracting the power spectra from the -body simulations of models close to the fiducial ones, obtained by considering parameter increments . For the CDM case, two different initial seeds were also considered, to test the dependence on initial conditions, finding that Fisher matrix results are almost insensitive to it. For the other fiducial models, only one seed is used.

-body simulations are performed by using a modified version of pkdgrav [859] able to handle any DE state equation , with particles in a box with side . Transfer functions generated using the camb package are employed to create initial conditions, with a modified version of the PM software by [510], also able to handle suitable parameterizations of DE.

Matter power spectra are obtained by performing a FFT (Fast Fourier Transform) of the matter density fields, computed from the particles distribution through a Cloud-in-Cell algorithm, by using a regular grid with . This allows us to obtain nonlinear spectra in a large -interval. In particular, our resolution allows to work out spectra up to . However, for neglecting baryon physics is no longer accurate [481, 774, 149, 976, 426]. For this reason, we consider WL spectra only up to .

Particular attention has to be paid to matter power spectra normalizations. In fact, we found that, normalizing all models to the same linear , the shear derivatives with respect to , or were largely dominated by the normalization shift at , and values being quite different and the shift itself depending on , and . This would confuse the dependence of the growth factor, through the observational -range. This normalization problem was not previously met in analogous tests with the Fisher matrix, as halofit does not directly depend on the DE state equation.

As a matter of fact, one should keep in mind that, observing the galaxy distribution with future surveys, one can effectively measure , and not its linear counterpart. For these reasons, we choose to normalize matter power spectra to , assuming to know it with high precision.

In Figures 23 we show the confidence ellipses, when the fiducial model is CDM, in the cases of 3 or 5 bins and with . Since the discrepancy between different seeds are small, discrepancies between halofit and simulations are truly indicating an underestimate of errors in the halofit case.

As expected, the error on estimate is not affected by the passage from simulations to halofit, since we are dealing with CDM models only. On the contrary, using halofit leads to underestimates of the errors on and , by a substantial 30 – 40% (see [215] for further details).

This confirms that, when considering models different from CDM, nonlinear correction obtained through halofit may be misleading. This is true even when the fiducial model is CDM itself and we just consider mild deviations of from .

Figure 24 then show the results in the - plane, when the fiducial models are M1 or M3. It is evident that the two cases are quite different. In the M1 case, we see just quite a mild shift, even if they are (10%) on error predictions. In the M3 case, errors estimated through halofit exceed simulation errors by a substantial factor. Altogether, this is a case when estimates based on halofit are not trustworthy.

The effect of baryon physics is another nonlinear correction to be considered. We note that the details of a study on the impact of baryon physics on the power spectrum and the parameter estimation can be found in [813]

As we have seen in Section 1.3.1, when dark energy clusters, the standard sub-horizon Poisson equation that links matter fluctuations to the gravitational potential is modified and . The deviation from unity will depend on the degree of DE clustering and therefore on the sound speed . In this subsection we try to forecast the constraints that Euclid can put on a constant by measuring both via weak lensing and via redshift clustering. Here we assume standard Einstein gravity and zero anisotropic stress (and therefore we have ) and we allow to assume different values in the range 0 – 1.

Generically, while dealing with a non-zero sound speed, we have to worry about the sound horizon , which characterizes the growth of the perturbations; then we have at least three regimes with different behavior of the perturbations:

- perturbations larger than the causal horizon (where perturbations are not causally connected and their growth is suppressed),
- perturbations smaller than the causal horizon but larger than the sound horizon, (this is the only regime where perturbations are free to grow because the velocity dispersion, or equivalently the pressure perturbation, is smaller than the gravitational attraction),
- perturbations smaller than the sound horizon, (here perturbations stop growing because the pressure perturbation is larger than the gravitational attraction).

As we have set the anisotropic stress to zero, the perturbations are fully described by . The main problem is therefore to find an explicit expression that shows how depends on . [785] have provided the following explicit approximate expression for which captures the behavior for both super- and sub-horizon scales:

Here which it is defined through so that counts how deep a mode is inside the sound horizon.Eq. (1.8.21) depends substantially on the value of the sound speed or, to put it differently, on the scale considered. For scales larger than the sound horizon (), Eq. (1.8.21) scales as and for and we have that

This is not a negligible deviation today, but it decreases rapidly as we move into the past, as the dark energy becomes less important.We can characterize the dependence of on the main perturbation parameter by looking at its derivative, a key quantity for Fisher matrix forecasts:

where (with the last expression being for ). For the values we are interested in here, this derivative has a peak at the present epoch at the sound horizon, i.e., for , which in the observable range of is , and declines rapidly for larger . This means that the sensitivity of to the sound speed can be boosted by several orders of magnitude as the sound speed is decreased.There are several observables that depend on :

- The growth of matter perturbations
There are two ways to influence the growth factor: firstly at background level, with a different Hubble expansion. Secondly at perturbation level: if dark energy clusters then the gravitational potential changes because of the Poisson equation, and this will also affect the growth rate of dark matter. All these effects can be included in the growth index and we therefore expect that is a function of and (or equivalently of and ).

The growth index depends on dark-energy perturbations (through ) as [785]

where Clearly here, the key quantity is the derivative of the growth factor with respect to the sound speed: From the above equation we also notice that the derivative of the growth factor does not depend on like the derivative , but on as it is an integral (being the value of today). The growth factor is thus not directly probing the deviation of from unity, but rather how evolves over time, see [786] for more details. - Redshift space distortions
The distortion induced by redshift can be expressed in linear theory by the factor, related to the bias factor and the growth rate via:

The derivative of the redshift distortion parameter with respect to the sound speed is: We see that the behavior versus is similar to the one for the derivative, so the same discussion applies. Once again, the effect is maximized for small . The derivative is comparable to that of at but becomes more important at low redshifts. - Shape of the dark matter power spectrum
Quantifying the impact of the sound speed on the matter power spectrum is quite hard as we need to run Boltzmann codes (such as camb, [559]) in order to get the full impact of dark-energy perturbations into the matter power spectrum. [786] proceeded in two ways: first using the camb output and then considering the analytic expression from [337] (which does not include dark energy perturbations, i.e., does not include ).

They find that the impact of the derivative of the matter power spectrum with respect the sound speed on the final errors is only relevant if high values of are considered; by decreasing the sound speed, the results are less and less affected. The reason is that for low values of the sound speed other parameters, like the growth factor, start to be the dominant source of information on .

Hence, the lensing potential contains three conceptually different contributions from the dark-energy perturbations:

- The direct contribution of the perturbations to the gravitational potential through the factor .
- The impact of the dark-energy perturbations on the growth rate of the dark matter perturbations, affecting the time dependence of , through .
- A change in the shape of the matter power spectrum , corresponding to the dark energy induced dependence of .

We use the representative Euclid survey presented in Section 1.8.2 and we extend our survey up to three different redshifts: . We choose different values of and in order to maximize the impact on : values closer to reduce the effect and therefore increase the errors on .

In Figure 25 we report the confidence region for for two different values of the sound speed and . For high value of the sound speed () we find and the relative error for the sound speed is . As expected, WL is totally insensitive to the clustering properties of quintessence dark-energy models when the sound speed is equal to . The presence of dark-energy perturbations leaves a and dependent signature in the evolution of the gravitational potentials through and, as already mentioned, the increase of the enhances the suppression of dark-energy perturbations which brings .

Once we decrease the sound speed then dark-energy perturbations are free to grow at smaller scales. In Figure 25 the confidence region for for is shown; we find , ; in the last case the error on the measurement of the sound speed is reduced to the 70% of the total signal.

In Figure 26 we report the confidence region for for two different values of the sound speed and . For high values of the sound speed () we find, for our benchmark survey: , and . Here again we find that galaxy power spectrum is not sensitive to the clustering properties of dark energy when the sound speed is of order unity. If we decrease the sound speed down to then the errors are , .

In conclusion, as perhaps expected, we find that dark-energy perturbations have a very small effect on dark matter clustering unless the sound speed is extremely small, . Let us remind that in order to boost the observable effect, we always assumed ; for values closer to the sensitivity to is further reduced. As a test, [786] performed the calculation for and and found and for WL and galaxy power spectrum experiments, respectively.

Such small sound speeds are not in contrast with the fundamental expectation of dark energy being much smoother that dark matter: even with , dark-energy perturbations are more than one order of magnitude weaker than dark matter ones (at least for the class of models investigated here) and safely below nonlinearity at the present time at all scales. Models of “cold” dark energy are interesting because they can cross the phantom divide [536] and contribute to the cluster masses [258] (see also Section 1.6.2 of this review ). Small could be constructed for instance with scalar fields with non-standard kinetic energy terms.

In this section, we present the Euclid weak lensing forecasts of a specific, but very popular, class of models, the so-called models of gravity. As we have already seen in Section 1.4.6 these models are described by the action

where is an arbitrary function of the Ricci scalar and is the Lagrange density of standard matter and radiation.In principle one has complete freedom to specify the function , and so any expansion history can be reproduced. However, as discussed in Section 1.4.6, those that remain viable are the subset that very closely mimic the standard CDM background expansion, as this restricted subclass of models can evade solar system constraints [230, 906, 410], have a standard matter era in which the scale factor evolves according to [43] and can also be free of ghost and tachyon instabilities [682, 415].

To this subclass belongs the popular model proposed by [456] (1.4.52). [200] demonstrated that Euclid will have the power of distinguishing between it and CDM with a good accuracy. They performed a tomographic analysis using several values of the maximum allowed wavenumber of the Fisher matrices; specifically, a conservative value of 1000, an optimistic value of 5000 and a bin-dependent setting, which increases the maximum angular wavenumber for distant shells and reduces it for nearby shells. Moreover, they computed the Bayesian expected evidence for the model of Eq. (1.4.52) over the CDM model as a function of the extra parameter . This can be done because the CDM model is formally nested in this model, and the latter is equivalent to the former when . Their results are shown in Figure 27. For another Bayesian evidence analysis of models and the added value of probing the growth of structure with galaxy surveys see also [850].

This subclass of models can be parameterized solely in terms of the mass of the scalar field, which as we have seen in Eq. (1.4.71) is related to the functional form via the relation

where subscripts denote differentiation with respect to . The function can be approximated by its standard CDM form, valid for . The mass is typically a function of redshift which decays from a large value in the early universe to its present day value .Whilst these models are practically indistinguishable from CDM at the level of background expansion, there is a significant difference in the evolution of perturbations relative to the standard GR behavior.

The evolution of linear density perturbations in the context of gravity is markedly different than in the standard CDM scenario; acquires a nontrivial scale dependence at late times. This is due to the presence of an additional scale in the equations; as any given mode crosses the modified gravity ‘horizon’ , said mode will feel an enhanced gravitational force due to the scalar field. This will have the effect of increasing the power of small scale modes.

Perturbations on sub-horizon scales in the Newtonian gauge evolve approximately according to

where . These equations represent a particular example of a general parameterization introduced in [636, 131, 983]. To solve them one should first parameterize the scalaron mass , choosing a form that broadly describes the behavior of viable models. A suitable functional form, which takes into account the evolution of in both the matter era and the late-time accelerating epoch, is given by [887] where is the scale factor at matter- equality; . There are two modified gravity parameters; is the mass of the scalaron at the present time and is the rate of increase of to the past.In Figure 28 the linear matter power spectrum is exhibited for this parameterization (dashed line), along with the standard CDM power spectrum (solid line). The observed, redshift dependent tilt is due to the scalaron’s influence on small scale modes, and represents a clear modified gravity signal. Since weak lensing is sensitive to the underlying matter power spectrum, we expect Euclid to provide direct constraints on the mass of the scalar field.

By performing a Fisher analysis, using the standard Euclid specifications, [887] calculates the expected parameter sensitivity of the weak lensing survey. By combining Euclid weak lensing and Planck Fisher matrices, both modified gravity parameters and are shown to be strongly constrained by the growth data in Figure 29. The expected bounds on and are quoted as , when using linear data only and , when utilizing the full set of nonlinear modes .

In this section we present forecasts for coupled quintessence cosmologies [33, 955, 724], obtained when combining Euclid weak lensing, Euclid redshift survey (baryon acoustic oscillations, redshift distortions and full shape) and CMB as obtained in Planck (see also the next section for CMB priors). Results reported here were obtained in [42] and we refer to it for details on the analysis and Planck specifications (for weak lensing and CMB constraints on coupled quintessence with a different coupling see also [637, 284]). In [42] the coupling is the one described in Section 1.4.4.4, as induced by a scalar-tensor model. The slope of the Ratra–Peebles potential is included as an additional parameter and Euclid specifications refer to the Euclid Definition phase [551].

The combined Fisher confidence regions are plotted in Figure 30 and the results are in Table 13. The main result is that future surveys can constrain the coupling of dark energy to dark matter to less than . Interestingly, some combinations of parameters (e.g., vs ) seem to profit the most from the combination of the three probes.

Parameter | CMB + | CMB + + WL |

0.00051 | 0.00032 | |

0.055 | 0.032 | |

0.0037 | 0.0010 | |

0.0080 | 0.0048 | |

0.00047 | 0.00041 | |

0.0057 | 0.0049 | |

0.0049 | 0.0036 | |

0.0051 | 0.0027 | |

We can also ask whether a better knowledge of the parameters , obtained by independent future observations, can give us better constraints on the coupling . In Table 14 we show the errors on when we have a better knowledge of only one other parameter, which is here fixed to the reference value. All remaining parameters are marginalized over.

It is remarkable to notice that the combination of CMB, power spectrum and weak lensing is already a powerful tool and a better knowledge of one parameter does not improve much the constraints on . CMB alone, instead, improves by a factor 3 when is known and by a factor 2 when is known. The power spectrum is mostly influenced by , which allows to improve constraints on the coupling by more than a factor 2. Weak lensing gains the most by a better knowledge of .

Fixed parameter | CMB | WL | CMB + + WL | |

(Marginalized on all params) | 0.0094 | 0.0015 | 0.012 | 0.00032 |

0.0093 | 0.00085 | 0.0098 | 0.00030 | |

0.0026 | 0.00066 | 0.0093 | 0.00032 | |

0.0044 | 0.0013 | 0.011 | 0.00032 | |

0.0087 | 0.0014 | 0.012 | 0.00030 | |

0.0074 | 0.0014 | 0.012 | 0.00028 | |

0.0094 | 0.00084 | 0.0053 | 0.00030 | |

0.0090 | 0.0015 | 0.012 | 0.00032 | |

Other dark-energy projects will enable the cross-check of the dark-energy constraints from Euclid. These include Planck, BOSS, WiggleZ, HETDEX, DES, Panstarrs, LSST, BigBOSS and SKA.

Planck will provide exquisite constraints on cosmological parameters, but not tight constraints on dark energy by itself, as CMB data are not sensitive to the nature of dark energy (which has to be probed at , where dark energy becomes increasingly important in the cosmic expansion history and the growth history of cosmic large scale structure). Planck data in combination with Euclid data provide powerful constraints on dark energy and tests of gravity. In the next Section 1.8.9.1, we will discuss how to create a Gaussian approximation to the Planck parameter constraints that can be combined with Euclid forecasts in order to model the expected sensitivity until the actual Planck data is available towards the end of 2012.

The galaxy redshift surveys BOSS, WiggleZ, HETDEX, and BigBOSS are complementary to Euclid, since the overlap in redshift ranges of different galaxy redshift surveys, both space and ground-based, is critical for understanding systematic effects such as bias through the use of multiple tracers of cosmic large scale structure. Euclid will survey H emission line galaxies at over 20,000 square degrees. The use of multiple tracers of cosmic large scale structure can reduce systematic effects and ultimately increase the precision of dark-energy measurements from galaxy redshift surveys [see, e.g., 811].

Currently on-going or recently completed surveys which cover a sufficiently large volume to measure BAO at several redshifts and thus have science goals common to Euclid, are the Sloan Digital Sky Survey III Baryon Oscillations Spectroscopic Survey (BOSS for short) and the WiggleZ survey.

BOSS^{9}
maps the redshifts of 1.5 million Luminous Red Galaxies (LRGs) out to over 10,000 square
degrees, measuring the BAO signal, the large-scale galaxy correlations and extracting information of the
growth from redshift space distortions. A simultaneous survey of quasars measures the
acoustic oscillations in the correlations of the Lyman- forest. LRGs were chosen for their high bias, their
approximately constant number density and, of course, the fact that they are bright. Their spectra and
redshift can be measured with relatively short exposures in a 2.4 m ground-based telescope. The
data-taking of BOSS will end in 2014.

The WiggleZ^{10}
survey is now completed, it measured redshifts for almost 240,000 galaxies over 1000 square degrees at
. The target are luminous blue star-forming galaxies with spectra dominated by patterns of
strong atomic emission lines. This choice is motivated by the fact that these emission lines can be
used to measure a galaxy redshift in relatively short exposures of a 4 m class ground-based
telescope.

Red quiescent galaxies inhabit dense clusters environments, while blue star-forming galaxies trace better lower density regions such as sheets and filaments. It is believed that on large cosmological scales these details are unimportant and that galaxies are simply tracers of the underlying dark matter: different galaxy type will only have a different ‘bias factor’. The fact that so far results from BOSS and WiggleZ agree well confirms this assumption.

Between now and the availability of Euclid data other wide-field spectroscopic galaxy redshift surveys will take place. Among them, eBOSS will extend BOSS operations focusing on 3100 square degrees using a variety of tracers. Emission line galaxies will be targeted in the redshift window . This will extend to higher redshift and extend the sky coverage of the WiggleZ survey. Quasars in the redshift range will be used as tracers of the BAO feature instead of galaxies. The BAO LRG measurement will be extended to , and the quasar number density at of BOSS will be tripled, thus improving the BAO Lyman- forest measure.

HETDEX is expected to begin full science operation is 2014: it aims at surveying 1 million Lyman- emitting galaxies at over 420 square degrees. The main science goal is to map the BAO feature over this redshift range.

Further in the future, we highlight here the proposed BigBOSS survey and SuMIRe survey with HyperSupremeCam on the Subaru telescope. The BigBOSS survey will target [OII] emission line galaxies at (and LRGs at ) over 14,000 square degrees. The SuMIRe wide survey proposes to survey square degrees in the redshift range targeting LRGs and [OII] emission-line galaxies. Both these surveys will likely reach full science operations roughly at the same time as the Euclid launch.

Wide field photometric surveys are also being carried out and planned. The on-going Dark Energy Survey
(DES)^{11}
will cover 5000 square degrees out to and is expected to complete observations in 2017; the
Panoramic Survey Telescope & Rapid Response System (Pan-STARRS), on-going at the single-mirror stage,
The PanSTARSS survey, which first phase is already on-going, will cover 30,000 square degrees with 5
photometry bands for redshifts up to . The second pause of the survey is expected to be competed
by the time Euclid launches. More in the future the Large Synoptic Survey Telescope (LSST) will cover
redshifts over 20,000 square degrees, but is expected to begin operations in
2021, after Euclid’s planned launch date. The galaxy imaging surveys DES, Panstarrs, and
LSST will complement Euclid imaging survey in both the choice of band passes, and the sky
coverage.

SKA (which is expected to begin operations in 2020 and reach full operational capability in 2024) will survey neutral atomic hydrogen (HI) through the radio 21 cm line, over a very wide area of the sky. It is expected to detect HI emitting galaxies out to making it nicely complementary to Euclid. Such galaxy redshift survey will of course offer the opportunity to measure the galaxy power spectrum (and therefore the BAO feature) out to . The well behaved point spread function of a synthesis array like the SKA should ensure superb image quality enabling cosmic shear to be accurately measured and tomographic weak lensing used to constrain cosmology and in particular dark energy. This weak lensing capability also makes SKA and Euclid very complementary. For more information see, e.g., [755, 140].

The Figure 31 puts Euclid into context. Euclid will survey H emission line galaxies at over 20,000 square degrees. Clearly, Euclid with both spectroscopic and photometric capabilities and wide field coverage surpasses all surveys that will be carried out by the time it launches. The large volume surveyed is crucial as the number of modes to sample for example the power spectrum and the BAO feature scales with the volume. The redshift coverage is also important especially at where the dark-energy contribution to the density pod the universe is non-negligible (at for most cosmologies the universe is effectively Einstein–de Sitter, therefore, high redshifts do not contribute much to constraints on dark energy). Having a single instrument, a uniform target selection and calibration is also crucial to perform precision tests of cosmology without having to build a ‘ladder’ from different surveys selecting different targets. On the other hand it is also easy to see the synergy between these ground-based surveys and Euclid: by mapping different targets (over the same sky area and ofter the same redshift range) one can gain better control over issues such as bias factors. The use of multiple tracers of cosmic large scale structure can reduce systematic effects and ultimately increase the precision of dark-energy measurements from galaxy redshift surveys [see, e.g., 811].

Moreover, having both spectroscopic and imaging capabilities Euclid is uniquely poised to explore the clustering with both the three dimensional distribution of galaxies and weak gravitational lensing.

Planck will provide highly accurate constraints on many cosmological parameters, which makes the construction of a Planck Fisher matrix somewhat non-trivial as it is very sensitive to the detailed assumptions. A relatively robust approach was used by [676] to construct a Gaussian approximation to the WMAP data by introducing two extra parameters,

where is the comoving distance from the observer to redshift , and is the comoving size of the sound-horizon at decoupling.In this scheme, describes the peak location through the angular diameter distance to decoupling and the size of the sound horizon at that time. If the geometry changes, either due to non-zero curvature or due to a different equation of state of dark energy, changes in the same way as the peak structure. encodes similar information, but in addition contains the matter density which is connected with the peak height. In a given class of models (for example, quintessence dark energy), these parameters are “observables” related to the shape of the observed CMB spectrum, and constraints on them remain the same independent of (the prescription for) the equation of state of the dark energy.

As a caveat we note that if some assumptions regarding the evolution of perturbations are changed, then the corresponding and constraints and covariance matrix will need to be recalculated under each such hypothesis, for instance, if massive neutrinos were to be included, or even if tensors were included in the analysis [255]. Further, as defined in Eq. (1.8.38) can be badly constrained and is quite useless if the dark energy clusters as well, e.g., if it has a low sound speed, as in the model discussed in [534].

In order to derive a Planck fisher matrix, [676] simulated Planck data as described in [703] and derived constraints on our base parameter set with a MCMC based likelihood analysis. In addition to and they used the baryon density , and optionally the spectral index of the scalar perturbations , as these are strongly correlated with and , which means that we will lose information if we do not include these correlations. As shown in [676], the resulting Fisher matrix loses some information relative to the full likelihood when only considering Planck data, but it is very close to the full analysis as soon as extra data is used. Since this is the intended application here, it is perfectly sufficient for our purposes.

The following tables, from [676], give the covariance matrix for quintessence-like dark energy (high sound speed, no anisotropic stress) on the base parameters and the Fisher matrix derived from it. Please consult the appendix of that paper for the precise method used to compute and as the results are sensitive to small variations.

0.303492E–04 | 0.297688E–03 | –0.545532E–06 | –0.175976E–04 | |

0.297688E–03 | 0.951881E–02 | –0.759752E–05 | –0.183814E–03 | |

–0.545532E–06 | –0.759752E-05 | 0.279464E–07 | 0.238882E–06 | |

–0.175976E–04 | –0.183814E-03 | 0.238882E–06 | 0.147219E–04 | |

.172276E+06 | .490320E+05 | .674392E+06 | –.208974E+07 | .325219E+07 | –.790504E+07 | –.549427E+05 | |

.490320E+05 | .139551E+05 | .191940E+06 | –.594767E+06 | .925615E+06 | –.224987E+07 | –.156374E+05 | |

.674392E+06 | .191940E+06 | .263997E+07 | –.818048E+07 | .127310E+08 | –.309450E+08 | –.215078E+06 | |

–.208974E+07 | –.594767E+06 | –.818048E+07 | .253489E+08 | –.394501E+08 | .958892E+08 | .666335E+06 | |

.325219E+07 | .925615E+06 | .127310E+08 | –.394501E+08 | .633564E+08 | –.147973E+09 | –.501247E+06 | |

–.790504E+07 | –.224987E+07 | –.309450E+08 | .958892E+08 | –.147973E+09 | .405079E+09 | .219009E+07 | |

–.549427E+05 | –.156374E+05 | –.215078E+06 | .666335E+06 | –.501247E+06 | .219009E+07 | .242767E+06 | |

Living Rev. Relativity 16, (2013), 6
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