1.8 Forecasts for Euclid

Here we describe forecasts for the constraints on modified gravity parameters which Euclid observations should be able to achieve. We begin with reviewing the relevant works in literature. Then, after we define our “Euclid model”, i.e., the main specifics of the redshift and weak lensing survey, we illustrate a number of Euclid forecasts obtained through a Fisher matrix approach.

1.8.1 A review of forecasts for parametrized modified gravity with Euclid

Heavens et al. [429Jump To The Next Citation Point] have used Bayesian evidence to distinguish between models, using the Fisher matrices for the parameters of interest. This study calculates the ratio of evidences B 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 γ. They find that Euclid can decisively distinguish between, e.g., DGP and dark energy, with |ln B | ≃ 50. In addition, they find that it will be possible to distinguish any departure from GR which has a difference in γ greater than ≃ 0.03. A phenomenological extension of the DGP model [332, 11] has also been tested with Euclid. Specifically, [199Jump To The Next Citation Point] found that it will be possible to discriminate between this modification to gravity from ΛCDM at the 3σ level in a wide range of angular scale, approximately 1000 ≲ ℓ ≲ 4000.

View Image

Figure 11: Marginalized γ − Σ 0 forecast for weak lensing only analysis with Euclid. Here Σ 0 is defined from Σ = 1 + Σ0a and Σ, defined via Eq. 1.3.28View Equation, is related to the WL potential. Black contours correspond to ℓmax = 5000, demonstrating an error of 0.089(1σ) on Σ0, whereas the red contours correspond to ℓmax = 500 giving an error of 0.034. In both cases, the inner and outer contours are 1σ and 2σ respectively. GR resides at [0.55, 0], while DGP resides at [0.68, 0].

Thomas et al. [886] construct Fisher matrix forecasts for the Euclid weak lensing survey, shown in Figure 11View Image. The constraints obtained depend on the maximum wavenumber which we are confident in using; ℓmax = 500 is relatively conservative as it probes the linear regime where we can hope to analytically track the growth of structure; ℓmax = 10000 is more ambitious as it includes nonlinear power, using the [844Jump To The Next Citation Point] fitting function. This will not be strictly correct, as the fitting function was determined in a GR context. Note that γ is not very sensitive to ℓmax, while Σ0, defined in [41Jump To The Next Citation Point] as Σ = 1 + Σ0a (and where Σ is defined in Eq. 1.3.28View Equation) is measured much more accurately in the nonlinear regime.

Amendola et al. [41Jump To The Next Citation Point] find Euclid weak lensing constraints for a more general parameterization that includes evolution. In particular, Σ(z) is investigated by dividing the Euclid weak lensing survey into three redshift bins with equal numbers of galaxies in each bin, and approximating that Σ is constant within that bin. Since Σ1, i.e., the value of Σ in the a = 1 bin (present-day) is degenerate with the amplitude of matter fluctuations, it is set to unity. The study finds that a deviation from unit Σ (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 (a measure of the abruptness of transitioning between these two regimes), α 1 (controlling the k-dependence of the transition) and α 2 (controlling the z-dependence of the transition).

View Image

Figure 12: Constraints on γ, α1, α2 and A from Euclid, using a DGP fiducial model and 0.4 redshift bins between 0.3 and 1.5 for the central cosmological parameter values fitting WMAP+BAO+SNe.

The forecasts for modified gravity parameters are shown in Figure 12View Image for the Euclid lensing data. Even with this larger range of parameters to fit, Euclid provides a measurement of the growth factor γ to within 10%, and also allows some constraint on the α1 parameter, probing the physics of nonlinear collapse in the modified gravity model.

Finally, Song et al. [848] have shown forecasts for measuring Σ and μ using both imaging and spectroscopic surveys. They combine 20,000 square-degree lensing data (corresponding to [550Jump To The Next Citation Point] rather than to the updated [551Jump To The Next Citation Point]) 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 Σ and μ, both quantities can be measured to 20% accuracy.

1.8.2 Euclid surveys

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 z ∼ 1, 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 2 ∼ 15000 deg capable to measure a large number of galaxy redshifts out to z ∼ 2. A promising observational strategy is to target Hα emitters at near-infrared wavelengths (which implies z > 0.5) since they guarantee both relatively dense sampling (the space density of this population is expected to increase out to z ∼ 2) 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 𝜀, i.e., the fraction of successfully measured redshifts. To minimize shot noise one should obviously strive for a low flux. Indeed, [389Jump To The Next Citation Point] found that a limiting flux −16 −2 − 1 fHα ≥ 1 × 10 erg cm s would be able to balance shot noise and cosmic variance out to z = 1.5. However, simulated observations of mock Hα galaxy spectra have shown that 𝜀 ranges between 30% and 60% (depending on the redshift) for a limiting flux fHα ≥ 3 × 10−16 erg cm −2 s− 1 [551Jump To The Next Citation Point]. Moreover, contamination from [O ii] line drops from 12% to 1% when the limiting flux increases from 1 × 10−16 to − 16 −2 −1 5 × 10 erg cm s [389].

Taking all this into account, in order to reach the top-level science requirement on the number density of Hα galaxies, the average effective Hα line flux limit from a 1-arcsec diameter source shall be lower than or equal to −16 − 2 − 1 3 × 10 erg cm s. 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 Hα galaxies at a given redshift, n (z ), estimated using the latest empirical data (see Figure 3.2 of [551Jump To The Next Citation Point]), where the values account for redshift – and flux – success rate, to which we refer as our reference efficiency 𝜀r.

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 𝜀 r and one where it is increased by a factor of 40%. Then we define the following cases:

The total number of observed galaxies ranges from 3 ⋅ 107 (pess.) to 9 ⋅ 107 (opt.). For all cases we assume that the error on the measured redshift is Δz = 0.001(1 + z), independent of the limiting flux of the survey.

Table 3: Expected galaxy number densities in units of (h∕Mpc )3 for Euclid survey. Let us notice that the galaxy number densities n(z) depend on the fiducial cosmology adopted in the computation of the survey volume, needed for the conversion from the galaxy numbers dN ∕dz to n(z).
z n1(z) ×10 −3 n2(z) ×10 −3 n3(z) ×10 −3
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

n(z) = z2 exp(− (z∕z0)3∕2), (1.8.1 )
where z0 = zmean∕1.412 is the peak of n(z) and zmean the median and typically we assume zmean = 0.9 and a surface density of valid images of ng = 30 per arcmin2 [551Jump To The Next Citation Point]). We also assume that the photometric redshifts give an error of Δz = 0.05(1 + z). Other specifications will be presented in the relevant sections.

1.8.3 Forecasts for the growth rate from the redshift survey

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 [308Jump To The Next Citation Point].

Equation of state.
In order to represent the evolution of the equation of state parameter w, we use the popular CPL parameterization [229Jump To The Next Citation Point, 584Jump To The Next Citation Point]

--z--- w (z) = w0 + w1 1 + z. (1.8.2 )
As a special case we will also consider the case of a constant w. We refer to this as the w-parametrization.

Growth rate.
Here we assume that the growth rate, fg, is a function of time but not of scale. As usual, we use the simple prescription [716, 540, 738, 585, 937Jump To The Next Citation Point]

f = Ω γ , (1.8.3 ) g m
where Ωm (z) is the matter density in units of the critical density as a function of redshift. A value γ ≈ 0.545 reproduces well the ΛCDM behavior while departures from this value characterize different models. Here we explore three different parameterizations of f g:

Reference Cosmological Models.
We assume as reference model a “pseudo” ΛCDM, where the growth rate values are obtained from Eq. (1.8.3View Equation) with γ = 0.545 and Ωm (z ) is given by the standard evolution. Then Ωm (z) is completely specified by setting Ωm,0 = 0.271, Ωk = 0, w0 = − 0.95, w1 = 0. When the corresponding parameterizations are employed, we choose as fiducial values γ = 0 1 and η = 0, We also assume a primordial slope ns = 0.966 and a present normalization σ8 = 0.809. One of the goals of this work is to assess whether the analysis of the power spectrum in redshift-space can distinguish the fiducial model from alternative cosmologies, characterized by their own set of parameters (apart from Ω m,0 which is set equal to 0.27 for all of them). The alternative models that we consider in this work are:

For the fiducial values of the bias parameters in every bin, we assume √ ------ b(z ) = 1 + z (already used in [753Jump To The Next Citation Point]) since this function provides a good fit to Hα line galaxies with luminosity 42 −1 −1 −2 LH α = 10 erg s h modeled by [698Jump To The Next Citation Point] using the semi-analytic GALFORM models of [108]. For the sake of comparison, we will also consider the case of constant b = 1 corresponding to the rather unphysical case of a redshift-independent population of unbiased mass tracers.

The fiducial values for β are computed through

ΩF (z)γF fF βF (z) = -mF------= -gF . (1.8.7 ) b (z) b

Now we express the growth function G(z) and the redshift distortion parameter β (z) in terms of the growth rate fg (see Eqs. (1.8.8View Equation), (1.8.7View Equation)). When we compute the derivatives of the spectrum in the Fisher matrix b(z) and fg(z) are considered as independent parameters in each redshift bin. In this way we can compute the errors on b (and fg) 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 2σ 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.

Results for the f-parameterization.
The total number of parameters that enter in the Fisher matrix analysis is 45: 5 parameters that describe the background cosmology (Ω h2,Ω h2, m,0 b,0 h, n, Ω k) plus 5 z-dependent parameters specified in 8 redshift bins evenly spaced in the range z = [0.5,2.1 ]. They are Ps(z), D (z), H (z), fg(z), b(z). However, since we are not interested in constraining D(z) and H (z ), we always project them to the set of parameters they depend on (as explained in [815Jump To The Next Citation Point]) instead of marginalizing over, so extracting more information on the background parameters.

The fiducial growth function G (z) in the (i + 1)-th redshift bin is evaluated from a step-wise, constant growth rate fg(z) as

{ ∫ } ( ) ( ) z dz ∏ 1 + zi fi 1 + z fi+1 G (z) = exp fg(z)1 +-z- = 1-+-z--- 1-+-z- . (1.8.8 ) 0 i i−1 i
To obtain the errors on si and bi 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 13View Image we show the contour plots at 68% and 95% of probability for all the pairs s(zi) − b(zi) in several redshift bins (with √ ------ b = 1 + z), where zi’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 Δz = 0.1 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: b(z) = 1 finding that the precision in measuring the bias as well as the other parameters has a very little dependence on the b(z) form. Given the robustness of the results on the choice of b(z) in the following we only consider the √ ------ b(z) = 1 + z case.

In Figure 14View Image we show the errors on the growth rate fg as a function of redshift, overplotted to our fiducial ΛCDM (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 f(R ) 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.

  1. The ability of measuring the biasing function is not too sensitive to the characteristic of the survey (b(z) 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 b(z) form.
  2. The growth rate fg can be estimated to within 1 – 2.5% in each bin for the Reference case survey with no need of estimating the bias function b(z) from some dedicated, independent analysis using higher order statistics [925Jump To The Next Citation Point] or full-PDF analysis [825].
  3. The estimated errors on fg depend weakly on the fiducial model of b(z).
View Image

Figure 13: Contour plots at 68% and 98% of probability for the pairs s(z ) − b(z ) i i in 7 redshift bins (with √ ------ b = 1 + z). The ellipses are centered on the fiducial values of the growth rate and bias parameters, computed in the central values of the bins, zi.
View Image

Figure 14: Expected constraints on the growth rates in each redshift bin. For each z the central error bars refer to the Reference case while those referring to the Optimistic and Pessimistic case have been shifted by − 0.015 and +0.015 respectively. The growth rates for different models are also plotted: ΛCDM (green tight shortdashed curve), flat DGP (red longdashed curve) and a model with coupling between dark energy and dark matter (purple, dot-dashed curve). The blue curves (shortdashed, dotted and solid) represent the f (R) model by [456Jump To The Next Citation Point] with n = 0.5,1,2 respectively. The plot shows that it will be possible to distinguish these models with next generation data.
View Image

Figure 15: Expected constraints on the growth rates in each redshift bin. For each z the central error bars refer to the Reference case while those referring to the Optimistic and Pessimistic case have been shifted by − 0.015 and +0.015 respectively. The growth rates for different models are also plotted: ΛCDM (green tight shortdashed curve), flat DGP (red longdashed curve) and a model with coupling between dark energy and dark matter (purple, dot-dashed curve). Here we plot again the f(R ) model by [456Jump To The Next Citation Point] with n = 2 (blue shortdashed curve) together with the model by [864] (cyan solid curve) and the one by [904] (black dotted curve). Also in this case it will be possible to distinguish these models with next generation data.

Table 4: 1σ marginalized errors for the bias and the growth rates in each redshift bin.
F b z F fg
  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 γ0 and γ1, in the context of the γ-parameterization and γ, η in the η-parameterization. In both cases the Fisher matrix elements have been estimated by expressing the growth factor as

[ ∫ z dz′ ] G(z) = δ0 exp (1 + η ) Ωm (z ′)γ(z)-----′ , (1.8.9 ) 0 1 + z
where for the γ-parameterization we fix η = 0.
View Image

Figure 16: γ-parameterization. Left panel: 1 and 2σ marginalized probability regions for constant γ and w: the green (shaded) regions are relative to the Reference case, the blue long-dashed ellipses to the Optimistic case, while the black short-dashed ellipses are the probability regions for the Pessimistic case. The red dot marks the fiducial model; two alternative models are also indicated for comparison. Right panel: 1 and 2σ marginalized probability regions for the parameters γ0 and γ1, relative to the Reference case (shaded yellow regions), to the Optimistic case (green long-dashed ellipses), and to the Pessimistic case (black dotted ellipses). Red dots represent the fiducial model, blue squares mark the DGP while triangles stand for the f(R ) model. Then, in the case of γ-parameterization, one could distinguish these three models (at 95% probability).

Table 5: Numerical values for 1σ constraints on parameters in Figure 16View Image and figures of merit. Here we have fixed Ω k to its fiducial value, Ω = 0 k.
  case σγ σw FoM
------ b = √ 1 + z ref. 0.02 0.017 3052
with opt. 0.02 0.016 3509
Ωk fixed pess. 0.026 0.02 2106

View Image

Figure 17: γ-parameterization. 1 and 2σ marginalized probability regions obtained assuming constant γ and w (red solid curves) or assuming the parameterizations (1.8.5View Equation) and (1.8.2View Equation) and marginalizing over γ1 and w1 (black dashed curves); marginalized error values are in columns σ γmarg,1, σwmarg,1 of Table 8. Yellow dots represent the fiducial model, the triangles a f(R ) model and the squares mark the flat DGP.

Table 6: Numerical values for 1σ constraints on parameters γ and w (assumed constant), relative to the red ellipses in Figures 17View Image, 18View Image and figures of merit. Here we have marginalized over Ωk.
bias case σγ FoM
  ref. 0.03 0.04 1342
√ ------ b = 1 + z opt. 0.03 0.03 1589
  pess. 0.04 0.05 864

Table 7: Numerical values for marginalized 1σ constraints on cosmological parameters using constant γ and w.
  case σh σΩmh2 σ Ωbh2 σΩk σns σ σ8
√ ------ b = 1 + z ref. 0.007 0.002 0.0004 0.008 0.03 0.006

Table 8: 1σ marginalized errors for parameters γ and w expressed through γ and η parameterizations. Columns γ ,w 0,marg1 0,marg1 refer to marginalization over γ ,w 1 1 (Figure 17View Image) while columns γ0,marg2,w0,marg2 refer to marginalization over η,w1 (Figure 18View Image).
bias case σγmarg,1 σwmarg,1 FoM σγmarg,2 σwmarg,2 FoM
ref. 0.15 0.07 97 0.07 0.07 216
b = √1-+-z-- opt. 0.14 0.06 112 0.07 0.06 249
pess. 0.18 0.09 66 0.09 0.09 147

Table 9: Numerical values for 1σ constraints on parameters in right panel of Figure 16View Image and figures of merit.
bias case σγ0 σγ1 FoM
  ref. 0.15 0.4 87
√ ------ b = 1 + z opt. 0.14 0.36 102
  pess. 0.18 0.48 58

View Image

Figure 18: η-parameterization. 1 and 2σ marginalized probability regions obtained assuming constant γ and w (red solid curves) or assuming the parameterizations (1.8.6View Equation) and (1.8.2View Equation) and marginalizing over η and w1 (black dashed curves); marginalized error values are in columns σ γmarg,2, σwmarg,2 of Table 9. Yellow dots represent the fiducial model, the triangles stand for a f (R ) model and the squares mark the flat DGP.
View Image

Figure 19: η-parameterization. Left panel: 1 and 2σ marginalized probability regions for the parameters γ and η in Eq. (1.8.6View Equation) relative to the reference case (shaded blue regions), to the optimistic case (yellow long-dashed ellipses) and to the pessimistic case (black short-dashed ellipses). The red dot marks the fiducial model while the square represents the coupling model. Right panel: present constraints on γ and η computed through a full likelihood method (here the red dot marks the likelihood peak) [307Jump To The Next Citation Point].

Table 10: Numerical values for 1σ constraints on parameters in Figure 19View Image and figures of merit.
bias case σγ ση FoM
  ref. 0.07 0.06 554
√ ------ b = 1 + z opt. 0.07 0.06 650
  pess. 0.09 0.08 362

View Image

Figure 20: Errors on the equation of state. 1 and 2σ marginalized probability regions for the parameters w0 and w1, relative to the reference case and bias ∘ - b = (1 + z). The blue dashed ellipses are obtained fixing γ0,γ1 and Ωk = 0 to their fiducial values and marginalizing over all the other parameters; for the red shaded ellipses instead, we also marginalize over Ωk = 0 but we fix γ0,γ1. Finally, the black dotted ellipses are obtained marginalizing over all parameters but w0 and w1. The progressive increase in the number of parameters reflects in a widening of the ellipses with a consequent decrease in the figures of merit (see Table 11).

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 w0, w1 with three different methods: i) fixing γ0,γ1 and Ωk to their fiducial values and marginalizing over all the other parameters; ii) fixing only γ0 and γ1; iii) marginalizing over all parameters but w0, w1. As one can see in Figure 20View Image 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., [945Jump To The Next Citation Point]).

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

  1. If both γ and w are assumed to be constant and setting Ωk = 0, then a redshift survey described by our Reference case will be able to constrain these parameters to within 4% and 2%, respectively.
  2. Marginalizing over Ω k degrades these constraints to 5.3% and 4% respectively.
  3. If w and γ are considered redshift-dependent and parametrized according to Eqs. (1.8.5View Equation) and (1.8.2View Equation) then the errors on γ0 and w0 obtained after marginalizing over γ1 and w1 increase by a factor ∼ 7, 5. However, with this precision we will be able to distinguish the fiducial model from the DGP and f(R ) scenarios with more than 2σ and 1σ significance, respectively.
  4. The ability to discriminate these models with a significance above 2σ is confirmed by the confidence contours drawn in the γ0-γ1 plane, obtained after marginalizing over all other parameters.
  5. If we allow for a coupling between dark matter and dark energy, and we marginalize over η rather than over γ 1, then the errors on w 0 are almost identical to those obtained in the case of the γ-parameterization, while the errors on γ0 decrease significantly.

    However, our ability in separating the fiducial model from the CDE model is significantly hampered: the confidence contours plotted in the γ-η plane show that discrimination can only be performed wit 1– 1.5σ significance. Yet, this is still a remarkable improvement over the present situation, as can be appreciated from Figure 19View Image 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 η to within 0.06. Reminding that we can write 2 η = 2.1β c [307], this means that the coupling parameter βc 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.

Table 11: 1σ marginalized errors for the parameters w0 and w1, obtained with three different methods (reference case, see Figure 20View Image).
  σw0 σw1 FoM
γ0, γ1, Ωk fixed 0.05 0.16 430
γ0,γ1 fixed 0.06 0.26 148
marginalization over all other parameters 0.07 0.3 87

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 w 0, w 1, similar to those in [945Jump To The Next Citation Point, 614Jump To The Next Citation Point] and the errors on constant γ and w [614Jump To The Next Citation Point]. 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., [753Jump To The Next Citation Point]).

1.8.4 Weak lensing non-parametric measurement of expansion and growth rate

In this section we apply power spectrum tomography [448] to the Euclid weak lensing survey without using any parameterization of the Hubble parameter H (z) as well as the growth function G (z). 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 H (z ) and G (z ). Although such a non-parametric approach is quite common for as concerns the equation-of-state ratio w (z) in supernovae surveys [see, e.g., 22Jump To The Next Citation Point] and also in redshift surveys [815Jump To The Next Citation Point], it has not been investigated for weak lensing surveys.

The Fisher matrix is given by [458]

∑ F αβ = fsky (2ℓ +-1)Δ-ℓ-∂Pij(ℓ)C− 1∂Pkm-(ℓ)C −1, (1.8.10 ) 2 ∂p α jk ∂pβ mi ℓ
where fsky is the observed fraction of the sky, C is the covariance matrix, P (ℓ) is the convergence power spectrum and p is the vector of the parameters defining our cosmological model. Repeated indices are being summed over from 1 to N, the number of redshift bins. The covariance matrix is defined as (no summation over j)
2 −1 Cjk = Pjk + δjkγintnj , (1.8.11 )
where γ int is the intrinsic galaxy shear and n j is the fraction of galaxies per steradian belonging to the j-th redshift bin:
( )2 ∫ ∞ 180- nj = 3600 π n𝜃 0 nj(z)dz (1.8.12 )
where n𝜃 is the galaxy density per arc minute and nj (z ) the galaxy density for the j-th bin, convolved with a gaussian around ˆzj, the center of that bin, with a width of σz(1 + zˆj) in order to account for errors in the redshift measurement.

For the matter power spectrum we use the fitting formulae from [337Jump To The Next Citation Point] and for its nonlinear corrections the results from [844Jump To The Next Citation Point]. Note that this is where the growth function enters. The convergence power spectrum for the i-th and j-th bin can then be written as

3 ∫ ∞ 3 2 ( ) 9H-0- Wi-(z)Wj-(z)E-(z)Ω-m(z)- ---ℓ-- Pij(ℓ) = 4 0 (1 + z)4 P δm πr (z) dz. (1.8.13 )
Here we make use of the window function
∫ ∞ d&tidle;z [ r(z)] Wi (z) = ----- 1 − ---- ni[r(&tidle;z)] (1.8.14 ) z H (&tidle;z) r(&tidle;z)
(with r(z) being the comoving distance) and the dimensionless Hubble parameter
[∫ z 3(1 + w(&tidle;z)) ] E2 (z) = Ω(m0)(1 + z )3 + (1 − Ω(m0))exp -----------dz&tidle; . (1.8.15 ) 0 1 + &tidle;z
For the equation-of-state ratio, finally, we use the usual CPL parameterization.

We determine N intervals in redshift space such that each interval contains the same amount of galaxies. For this we use the common parameterization

n(z) = z2 exp(− (z∕z0)3∕2), (1.8.16 )
where z0 = zmean∕1.412 is the peak of n(z) and zmean the median. Now we can define ˆzi as the center of the i-th redshift bin and add hi ≡ log (H (ˆzi)∕H0 ) as well as gi ≡ log G (zˆi) to the list of cosmological parameters. The Hubble parameter and the growth function now become functions of the h i and gi respectively:

This is being done by linearly interpolating the functions through their supporting points, e.g., (ˆzi,exp(hi)) for H (z). Any function that depends on either H (z) or G (z) hence becomes a function of the hi and gi as well.

Table 12: Values used in our computation. The values of the fiducial model (WMAP7, on the left) and the survey parameters (on the right).
ω m 0.1341
ωb 0.02258
τ 0.088
ns 0.963
Ωm 0.266
w0 –1
w1 0
γ 0.547
γppn 0
σ8 0.801
fsky 0.375
zmean 0.9
σz 0.05
n𝜃 30
γint 0.22

ℓmax 3 5 ⋅ 10
Δ log ℓ 10 0.02

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

As for the sum in Eq. (1.8.10View Equation), we generally found that with a realistic upper limit of ℓmax = 5 ⋅ 103 and a step size of Δ lgℓ = 0.2 we get the best result in terms of a figure of merit (FoM), that we defined as

∑ −2 FoM = σi . (1.8.19 )

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.

View Image

Figure 21: Error bars on the Hubble parameter H (z) with five redshift bins. The exact height of the error bars respectively are (0.23,0.072,0.089,0.064, 0.76).
View Image

Figure 22: Error bars on the growth function G (z) with three redshift bins while marginalizing over the his. The exact height of the error bars respectively are (0.029,0.033,0.25).

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

dlog-G(z)- γ d loga = Ωm (z ) , (1.8.20 )
yielding the result in Figure 21View Image. Binning both H (z) and G (z) and marginalizing over the his yields the plot for G (z) seen in Figure 22View Image.

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.

1.8.5 Testing the nonlinear corrections for weak lensing forecasts

In order to fully exploit next generation weak lensing survey potentialities, accurate knowledge of nonlinear power spectra up to ∼ 1% is needed [465, 469]. However, such precision goes beyond the claimed ±3% accuracy of the popular halofit code [844].

[651] showed that, using halofit for non-ΛCDM 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 w (z ). This procedure was dictated by the lack of appropriate extensions of halofit to non-ΛCDM cosmologies.

In this paragraph we quantify the effects of using the halofit code instead of N-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 w0, wa and Ωm and compare the related confidence ellipses. See [215Jump To The Next Citation Point] for further details.

The weak lensing survey is as specified in Section 1.8.2. Tests are performed assuming three different fiducial cosmologies: ΛCDM model (w0 = − 1, wa = 0) and two dynamical DE models, still consistent with the WMAP+BAO+SN combination [526Jump To The Next Citation Point] at 95% C.L. They will be dubbed M1 (w0 = − 0.67, wa = 2.28) and M3 (w0 = − 1.18, wa = 0.89). 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 N-body simulations of models close to the fiducial ones, obtained by considering parameter increments ±5%. For the ΛCDM 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.

N-body simulations are performed by using a modified version of pkdgrav [859] able to handle any DE state equation w(a ), with 3 3 N = 256 particles in a box with side −1 L = 256h Mpc. 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 Ng = 2048. This allows us to obtain nonlinear spectra in a large k-interval. In particular, our resolution allows to work out spectra up to k ≃ 10h Mpc −1. However, for k > 2– 3h Mpc − 1 neglecting baryon physics is no longer accurate [481, 774, 149, 976, 426]. For this reason, we consider WL spectra only up to ℓmax = 2000.

Particular attention has to be paid to matter power spectra normalizations. In fact, we found that, normalizing all models to the same linear σ8(z = 0 ), the shear derivatives with respect to w0, wa or Ωm were largely dominated by the normalization shift at z = 0, σ8 and σ8,nl values being quite different and the shift itself depending on w 0, w a and Ω m. This would confuse the z dependence of the growth factor, through the observational z-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 σ 8,nl, and not its linear counterpart. For these reasons, we choose to normalize matter power spectra to σ8,nl, assuming to know it with high precision.

View Image

Figure 23: Likelihood contours, for 65% and 95% C.L., calculated including signals up to ℓ ≃ 2000 for the ΛCDM fiducial. Here simulations and halofit yield significantly different outputs.

In Figures 23View Image we show the confidence ellipses, when the fiducial model is ΛCDM, in the cases of 3 or 5 bins and with ℓmax = 2000. 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 Ωm estimate is not affected by the passage from simulations to halofit, since we are dealing with ΛCDM models only. On the contrary, using halofit leads to underestimates of the errors on w0 and wa, by a substantial 30 – 40% (see [215] for further details).

This confirms that, when considering models different from ΛCDM, nonlinear correction obtained through halofit may be misleading. This is true even when the fiducial model is ΛCDM itself and we just consider mild deviations of w from − 1.

View Image

Figure 24: On the left (right) panel, 1- and 2-σ contours for the M1 (M3) model. The two fiducial models exhibit quite different behaviors.

Figure 24View Image then show the results in the w0-wa 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 𝒪 (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]

1.8.6 Forecasts for the dark-energy sound speed

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 Q ⁄= 1. The deviation from unity will depend on the degree of DE clustering and therefore on the sound speed cs. In this subsection we try to forecast the constraints that Euclid can put on a constant cs by measuring Q both via weak lensing and via redshift clustering. Here we assume standard Einstein gravity and zero anisotropic stress (and therefore we have Ψ = Φ) and we allow cs 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 ksh = aH ∕cs, which characterizes the growth of the perturbations; then we have at least three regimes with different behavior of the perturbations:

  1. perturbations larger than the causal horizon (where perturbations are not causally connected and their growth is suppressed),
  2. perturbations smaller than the causal horizon but larger than the sound horizon, k ≪ aH ∕cs (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),
  3. perturbations smaller than the sound horizon, k ≫ aH ∕cs (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 Q. The main problem is therefore to find an explicit expression that shows how Q depends on cs. [785Jump To The Next Citation Point] have provided the following explicit approximate expression for Q (k, a) which captures the behavior for both super- and sub-horizon scales:

1 − ΩM,0 (1 + w )a −3w Q (k,a) = 1 + -------------------2----2. (1.8.21 ) ΩM,0 1 − 3w + 3ν(a)
Here ν(a)2 = k2c2a∕(ΩM,0H2 ) s 0 which it is defined through csk ≡ νaH so that ν counts how deep a mode is inside the sound horizon.

Eq. (1.8.21View Equation) 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 (ν ≈ 0), Eq. (1.8.21View Equation) scales as a −3w and for Ωm,0 = 0.25 and w = − 0.8 we have that

Q − 1 ≈ -3-a2.4 ≃ 0.18a2.4. (1.8.22 ) 17
This is not a negligible deviation today, but it decreases rapidly as we move into the past, as the dark energy becomes less important.8 As a scale enters the sound horizon, Q − 1 grows with one power of the scale factor slower (since δDE stops growing), suppressing the final deviation roughly by the ratio of horizon size to the scale of interest (as now ν2 ≫ 1). In the observable range, (k∕H0 )2 ≈ 102– 104. Therefore, if cs ≈ 1, Q → 1 and the dependence on c s is lost. This shows that Q is sensitive to c s only for small values, 2 − 2 cs ≲ 10.

We can characterize the dependence of Q on the main perturbation parameter c2s by looking at its derivative, a key quantity for Fisher matrix forecasts:

∂ log Q x Q − 1 ------2 = − -------------, (1.8.23 ) ∂ log cs (1 + x) Q
where x = 2ν(a )2∕(1 − 3w ) ≃ 0.2ν(a)2 3 (with the last expression being for w = − 0.8). For the values we are interested in here, this derivative has a peak at the present epoch at the sound horizon, i.e., for cs ≈ H0 ∕k, which in the observable range of k is cs ≈ .01 − .001, and declines rapidly for larger cs. This means that the sensitivity of Q 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 Q:

Impact on weak lensing.
Now it is possible to investigate the response of weak lensing (WL) to the dark-energy parameters. Proceeding with a Fisher matrix as in [41], the main difference here being that the parameter Q has an explicit form. Since Q depends on w and c2s, we can forecast the precision with which those parameters can be extracted. We can also try to trace where the constraints come from. For a vanishing anisotropic stress the WL potential becomes:

2 k2(Φ + Ψ ) = − 2Q 3H-0ΩM,0-ΔM (1.8.29 ) 2a
which can be written, in linear perturbation theory as:
2 k2 (Φ + Ψ ) = − 3H (a) a3Q (a,k )ΩM (a)G (a,k) ΔM (k). (1.8.30 )

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

We use the representative Euclid survey presented in Section 1.8.2 and we extend our survey up to three different redshifts: zmax = 2,3,4. We choose different values of c2s and w0 = − 0.8 in order to maximize the impact on Q: values closer to − 1 reduce the effect and therefore increase the errors on cs.

In Figure 25View Image we report the 1 − σ confidence region for w0,c2 s for two different values of the sound speed and z max. For high value of the sound speed (c2= 1 s) we find σ (w ) = 0.0195 0 and the relative error for the sound speed is 2 2 σ (cs)∕cs = 2615. As expected, WL is totally insensitive to the clustering properties of quintessence dark-energy models when the sound speed is equal to 1. The presence of dark-energy perturbations leaves a w and c2s dependent signature in the evolution of the gravitational potentials through ΔDE ∕Δm and, as already mentioned, the increase of the 2 cs enhances the suppression of dark-energy perturbations which brings Q → 1.

Once we decrease the sound speed then dark-energy perturbations are free to grow at smaller scales. In Figure 25View Image the confidence region for w0,c2s for c2s = 10 −6 is shown; we find σ (w0 ) = 0.0286, σ (c2)∕c2 = 0.132 s s; in the last case the error on the measurement of the sound speed is reduced to the 70% of the total signal.

View Image

Figure 25: Confidence region at 68% for three different values of zmax = 2.5,3.5, 4, red solid, green long-dashed and blue dashed contour, respectively. The left panel shows the confidence region when the sound speed is 2 cs = 1; the right panel with the sound speed 2 − 6 cs = 10. The equation of state parameter is for both cases w0 = − 0.8.

Impact on galaxy power spectrum.
We now explore a second probe of clustering, the galaxy power spectrum. The procedure is the same outlined in Section 1.7.3. We use the representative Euclid survey presented in Section 1.8.2. Here too we also consider in addition possible extended surveys to zmax = 2.5 and zmax = 4.

In Figure 26View Image we report the confidence region for w0,c2s for two different values of the sound speed and zmax. For high values of the sound speed (c2 = 1 s) we find, for our benchmark survey: σ (w0 ) = 0.0133, and 2 2 σ(cs)∕cs = 50.05. 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 c2s = 10− 6 then the errors are σ(w0 ) = 0.0125, σ (c2s)∕c2s = 0.118.

View Image

Figure 26: Confidence region at 68% for three different values of zmax = 2.5,3.5, 4, red solid, green long-dashed and blue dashed contour, respectively. The left panel shows the confidence region when the sound speed is c2s = 1; the right panel with the sound speed c2s = 10− 6. The equation of state parameter is for both cases w0 = − 0.8.

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, cs ≤ 0.01. Let us remind that in order to boost the observable effect, we always assumed w = − 0.8; for values closer to − 1 the sensitivity to c2 s is further reduced. As a test, [786] performed the calculation for w = − 0.9 and c2 = 10−5 s and found 2 σc2s∕c s = 2.6 and 2 σc2s∕cs = 1.09 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 cs ≈ 0.01, 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 [258Jump To The Next Citation Point] (see also Section 1.6.2 of this review ). Small cs could be constructed for instance with scalar fields with non-standard kinetic energy terms.

1.8.7 Weak lensing constraints on f(R) gravity

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

∫ √ --- [ f (R) ] Sgrav = − gd4x ------− ℒm , (1.8.31 ) 16πG
where f(R ) is an arbitrary function of the Ricci scalar and ℒm is the Lagrange density of standard matter and radiation.

In principle one has complete freedom to specify the function f(R ), 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 ΛCDM 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 2∕3 a(t) ∝ t [43] and can also be free of ghost and tachyon instabilities [682, 415].

To this subclass belongs the popular f(R ) model proposed by [456] (1.4.52View Equation). [200] demonstrated that Euclid will have the power of distinguishing between it and ΛCDM 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. (1.4.52View Equation) over the ΛCDM model as a function of the extra parameter n. This can be done because the ΛCDM model is formally nested in this f(R ) model, and the latter is equivalent to the former when n = 0. Their results are shown in Figure 27View Image. For another Bayesian evidence analysis of f(R) models and the added value of probing the growth of structure with galaxy surveys see also [850].

View Image

Figure 27: The Bayes factor ln B for the f (R) model of Eq. (1.4.52View Equation) over standard ΛCDM as a function of the extra parameter n. The green, red and blue curves refer to the conservative, bin-dependent and optimistic ℓmax, respectively. The horizontal lines denote the Jeffreys’ scale levels of significance.

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

M 2(a) = -------1------- (1.8.32 ) 3f,RR [Rback(a)]
where R subscripts denote differentiation with respect to R. The function f,RR can be approximated by its standard ΛCDM form,
Rback-≃ 3Ωm0--+ 12 ΩΛ, (1.8.33 ) H20 a3
valid for z ≲ 1000. The mass M (a) is typically a function of redshift which decays from a large value in the early universe to its present day value M0.

Whilst these models are practically indistinguishable from ΛCDM 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 f(R ) gravity is markedly different than in the standard ΛCDM scenario; δm ≡ δρm∕ρm acquires a nontrivial scale dependence at late times. This is due to the presence of an additional scale M (a) in the equations; as any given mode crosses the modified gravity ‘horizon’ k = aM (a), 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

( ) 2 ¯K2 Ψ = 1 + -----¯-2 Φ, (1.8.34 ) 3 +(2 K ) 2 3-+-2K¯2- 2 k Φ = − 4πG 3 + 3K¯2 a ρm δm, (1.8.35 ) ( 2) ¨δ + 2H δ˙ − 4πG 3-+-4K¯- ρ δ = 0, (1.8.36 ) m m 3 + 3K¯2 m m
where ¯ K = k∕(aM (a )). 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 M (a), choosing a form that broadly describes the behavior of viable f(R ) models. A suitable functional form, which takes into account the evolution of M (a ) in both the matter era and the late-time accelerating epoch, is given by [887Jump To The Next Citation Point]
( a− 3 + 4a −3)2ν M 2 = M 20 -------−-∗3- , (1.8.37 ) 1 + 4a∗
where a∗ is the scale factor at matter-Λ equality; a∗ = (Ωm0 ∕Ω Λ)1∕3. There are two modified gravity parameters; M0 is the mass of the scalaron at the present time and ν is the rate of increase of M (a) to the past.
View Image

Figure 28: Left panel: Linear matter power spectra for ΛCDM (solid line; M −1 = 0 0, ν = 1.5) and scalaron (dashed line; −1 28 −1 − 1 M 0 = 375[10 h eV ], ν = 1.5) cosmologies. The modification to gravity causes a sizeable scale dependent effect in the growth of perturbations. The redshift dependence of the scalaron can be seen by comparing the top and bottom pairs of power spectra evaluated at redshifts z = 0.0 and z = 1.5, respectively. Right panel: The environmental dependent chameleon mechanism can be seen in the mildly nonlinear regime. We exhibit the fractional difference (P (k) − PGR (k))∕PGR (k) between the f(R) and GR power spectra for the model (1.8.37View Equation) with parameters − 1 28 −1 −1 M 0 = 375[10 h eV ] and ν = 1.5. The dashed lines represent linear power spectra (P (k) and PGR (k ) calculated with no higher order effects) and the solid lines are the power spectra calculated to second order. We see that the nonlinearities decrease the modified gravity signal. This is a result of the chameleon mechanism. The top set of lines correspond to z = 0 and the bottom to z = 0.9; demonstrating that the modified gravity signal dramatically decreases for larger z. This is due to the scalaron mass being much larger at higher redshifts. Furthermore, nonlinear effects are less significant for increasing z.

In Figure 28View Image the linear matter power spectrum is exhibited for this parameterization (dashed line), along with the standard ΛCDM 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 f (R ) parameter sensitivity of the weak lensing survey. By combining Euclid weak lensing and Planck Fisher matrices, both modified gravity parameters M0 and ν are shown to be strongly constrained by the growth data in Figure 29View Image. The expected 1σ bounds on M 0 and ν are quoted as −30 M0 = 1.34 ± 0.62 × 10 [h eV], ν = 1.5 ± 0.18 when using linear data l < 400 only and −30 M0 = 1.34 ± 0.25 × 10 [h eV], ν = 1.5 ± 0.04 when utilizing the full set of nonlinear modes l < 10000.

View Image

Figure 29: 68% (dark grey) and 95% (light grey) projected bounds on the modified gravity parameters − 1 M 0 and ν for the combined Euclid weak lensing and Planck CMB surveys. The smaller (larger) contours correspond to including modes l = 400 (10000 ) in the weak lensing analysis.

1.8.8 Forecast constraints on coupled quintessence cosmologies

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

The combined Fisher confidence regions are plotted in Figure 30View Image and the results are in Table 13. The main result is that future surveys can constrain the coupling of dark energy to dark matter 2 β to less than 3 ⋅ 10− 4. Interestingly, some combinations of parameters (e.g., Ωb vs α) seem to profit the most from the combination of the three probes.

Table 13: 1-σ errors for the set Θ ≡ {β2, α,Ωc,h, Ωb,nsσ8,log(A )} of cosmological parameters, combining CMB + P (k) (left column) and CMB + P (k) + WL (right column).
Parameter σi CMB + P (k ) σi CMB + P(k) + WL
β2 0.00051 0.00032
α 0.055 0.032
Ωc 0.0037 0.0010
h 0.0080 0.0048
Ωb 0.00047 0.00041
ns 0.0057 0.0049
σ 8 0.0049 0.0036
log(A ) 0.0051 0.0027

View Image

Figure 30: Comparison among predicted confidence contours for the cosmological parameter set Θ ≡ {β2,α, Ω ,h,Ω ,n ,σ ,log(A)} c b s 8 using CMB (Planck, blue contours), P(k) (pink-violet contours) and weak lensing (orange-red contours) with Euclid-like specifications. Image reproduced by permission from [42Jump To The Next Citation Point], copyright by APS.

We can also ask whether a better knowledge of the parameters {α, Ωc,h,Ωb, ns,σ8,log(A )}, obtained by independent future observations, can give us better constraints on the coupling β2. In Table 14 we show the errors on β2 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 β2. CMB alone, instead, improves by a factor 3 when Ωc is known and by a factor 2 when h is known. The power spectrum is mostly influenced by Ω c, which allows to improve constraints on the coupling by more than a factor 2. Weak lensing gains the most by a better knowledge of σ8.

Table 14: 1-σ errors for β2, for CMB, P (k), WL and CMB + P(k ) + WL. For each line, only the parameter in the left column has been fixed to the reference value. The first line corresponds to the case in which we have marginalized over all parameters. Table reproduced by permission from [42], copyright by APS.
Fixed parameter CMB P(k) WL CMB + P (k) + WL
(Marginalized on all params) 0.0094 0.0015 0.012 0.00032
α 0.0093 0.00085 0.0098 0.00030
Ωc 0.0026 0.00066 0.0093 0.00032
h 0.0044 0.0013 0.011 0.00032
Ω b 0.0087 0.0014 0.012 0.00030
ns 0.0074 0.0014 0.012 0.00028
σ8 0.0094 0.00084 0.0053 0.00030
log(A ) 0.0090 0.0015 0.012 0.00032

1.8.9 Extra-Euclidean data and priors

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 z < 2, 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, 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 Hα emission line galaxies at 0.5 < z < 2.0 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., 811Jump To The Next Citation Point].

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.

BOSS9 maps the redshifts of 1.5 million Luminous Red Galaxies (LRGs) out to z ∼ 0.7 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 2.2 < z < 3.5 quasars measures the acoustic oscillations in the correlations of the Lyman-α 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 WiggleZ10 survey is now completed, it measured redshifts for almost 240,000 galaxies over 1000 square degrees at 0.2 < z < 1. 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 0.6 < z < 1. This will extend to higher redshift and extend the sky coverage of the WiggleZ survey. Quasars in the redshift range 1 < z < 2.2 will be used as tracers of the BAO feature instead of galaxies. The BAO LRG measurement will be extended to z ∼ 0.8, and the quasar number density at z > 2.2 of BOSS will be tripled, thus improving the BAO Lyman-α forest measure.

HETDEX is expected to begin full science operation is 2014: it aims at surveying 1 million Lyman-α emitting galaxies at 1.9 < z < 3.5 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 0.6 < z < 1.5 (and LRGs at z < 0.6) over 14,000 square degrees. The SuMIRe wide survey proposes to survey ∼ 2000 square degrees in the redshift range 0.6 < z < 1.6 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 z ∼ 1.3 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 z ∼ 1.5. 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 0.3 < z < 3.6 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 z ∼ 1.5 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 z ∼ 1.5. 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].

View Image

Figure 31: Redshift coverage and volume for the surveys mentioned in the text. Spectroscopic surveys only are shown. Recall that while future and forthcoming photometric surveys focus on weak gravitational lensing, spectroscopic surveys can extract the three dimensional galaxy clustering information and therefore measure radial and tangential BAO signal, the power spectrum shape and the growth of structure via redshift space distortions. The three-dimensional clustering information is crucial for BAO. For example to obtain the same figure of merit for dark-energy properties a photometric survey must cover a volume roughly ten times bigger than a spectroscopic one.

The Figure 31View Image puts Euclid into context. Euclid will survey Hα emission line galaxies at 0.5 < z < 2.0 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 z < 2 where the dark-energy contribution to the density pod the universe is non-negligible (at z > 2 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. The Planck prior

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 [676Jump To The Next Citation Point] to construct a Gaussian approximation to the WMAP data by introducing two extra parameters,

∘ ------- 2 R ≡ ΩmH 0r(zCMB ), la ≡ πr(zCMB )∕rs(zCMB ), (1.8.38 )
where r(z) is the comoving distance from the observer to redshift z, and rs(zCMB ) is the comoving size of the sound-horizon at decoupling.

In this scheme, la 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, la changes in the same way as the peak structure. R 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 R and la 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, R as defined in Eq. (1.8.38View Equation) 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, [676Jump To The Next Citation Point] simulated Planck data as described in [703Jump To The Next Citation Point] and derived constraints on our base parameter set {R, l,Ω h2,n } a b s with a MCMC based likelihood analysis. In addition to R and la they used the baryon density 2 Ωbh, and optionally the spectral index of the scalar perturbations ns, as these are strongly correlated with R and la, which means that we will lose information if we do not include these correlations. As shown in [676Jump To The Next Citation Point], 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 [676Jump To The Next Citation Point], 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 R and la as the results are sensitive to small variations.

Table 15: R, la, Ωbh2 and ns estimated from Planck simulated data. Table reproduced by permission from [676Jump To The Next Citation Point], copyright by APS.
rms variance
Ωk ⁄= 0
R 1.7016 0.0055
la 302.108 0.098
2 Ωbh 0.02199 0.00017
ns 0.9602 0.0038

Table 16: Covariance matrix for (R, la,Ωbh2, ns) from Planck. Table reproduced by permission from [676Jump To The Next Citation Point], copyright by APS.
Ωk ⁄= 0
R 0.303492E–04 0.297688E–03 –0.545532E–06 –0.175976E–04
la 0.297688E–03 0.951881E–02 –0.759752E–05 –0.183814E–03
Ωbh2 –0.545532E–06 –0.759752E-05 0.279464E–07 0.238882E–06
ns –0.175976E–04 –0.183814E-03 0.238882E–06 0.147219E–04

Table 17: Fisher matrix for (w0, wa, ΩDE, Ωk, ωm, ωb, nS) derived from the covariance matrix for (R,l ,Ω h2,n ) a b s from Planck. Table reproduced by permission from [676], copyright by APS.
w0 .172276E+06 .490320E+05 .674392E+06 –.208974E+07 .325219E+07 –.790504E+07 –.549427E+05
wa .490320E+05 .139551E+05 .191940E+06 –.594767E+06 .925615E+06 –.224987E+07 –.156374E+05
ΩDE .674392E+06 .191940E+06 .263997E+07 –.818048E+07 .127310E+08 –.309450E+08 –.215078E+06
Ωk –.208974E+07 –.594767E+06 –.818048E+07 .253489E+08 –.394501E+08 .958892E+08 .666335E+06
ωm .325219E+07 .925615E+06 .127310E+08 –.394501E+08 .633564E+08 –.147973E+09 –.501247E+06
ωb –.790504E+07 –.224987E+07 –.309450E+08 .958892E+08 –.147973E+09 .405079E+09 .219009E+07
nS –.549427E+05 –.156374E+05 –.215078E+06 .666335E+06 –.501247E+06 .219009E+07 .242767E+06

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