List of Figures

View Image Figure 1:
Left: the cosmic microwave background angular power spectrum l(l + 1)Cl ∕(2π) for TeVeS (solid) and ΛCDM (dotted) with WMAP 5-year data [689]. Right: the matter power spectrum P (k) for TeVeS (solid) and ΛCDM (dotted) plotted with SDSS data.
View Image Figure 2:
The evolution of w as a function of the comoving scale k, using only the 5-year WMAP CMB data. Red and yellow are the 95% and 68% confidence regions for the LV formalism. Blue and purple are the same for the flow-equation formalism. From the outside inward, the colored regions are red, yellow, blue, and purple. Image reproduced by permission from [475]; copyright by APS.
View Image Figure 3:
The complete evolution of w (N ), from the flow-equation results accepted by the CMB likelihood. Inflation is made to end at N = 0 where w(N = 0) = − 1∕3 corresponding to 𝜖H (N = 0) = 1. For our choice of priors on the slow-roll parameters at N = 0, we find that w decreases rapidly towards − 1 (see inset) and stays close to it during the period when the observable scales leave the horizon (N ≈ 40 –60). Image reproduced by permission from [475]; copyright by APS.
View Image Figure 4:
Required accuracy on weff = − 1 to obtain strong evidence against a model where − 1 − Δ − ≤ we ff ≤ − 1 + Δ+ as compared to a cosmological constant model, w = − 1. For a given σ, models to the right and above the contour are disfavored with odds of more than 20:1.
View Image Figure 5:
Ratio of the total mass functions, which include the quintessence contribution, for c = 0 s and cs = 1 at z = 0 (above) and z = 1 (below). Image reproduced by permission from [258]; copyright by IOP and SISSA.
View Image Figure 6:
Extrapolated linear density contrast at collapse for coupled quintessence models with different coupling strength β. For all plots we use a constant α = 0.1. We also depict δc for reference ΛCDM (dotted, pink) and EdS (double-dashed, black) models. Image reproduced by permission from [962]; copyright by APS.
View Image Figure 7:
Extrapolated linear density contrast at collapse δc vs. collapse redshift zc for growing neutrinos with β = − 52 (solid, red), β = − 112 (long-dashed, green) and β = − 560 (short-dashed, blue). A reference EdS model (double-dashed. black) is also shown. Image reproduced by permission from [962]; copyright by APS.
View Image Figure 8:
Extrapolated linear density contrast at collapse δc vs. collapse redshift zc for EDE models I (solid, red) and II (long-dashed, green), as well as ΛCDM (double-dashed, black). Image reproduced by permission from [962]; copyright by APS.
View Image Figure 9:
Left: The velocity contribution Cveℓl as a function of ℓ for various redshifts. Right: The standard contribution Cst ℓ as a function of ℓ for various redshifts.
View Image Figure 10:
Matter power spectrum form measured from SDSS [720]
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].
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.
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 [456] 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 [456] 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.
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).
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.
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) [307].
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).
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).
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.
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.
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.
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.
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.
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.
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.
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 [42], copyright by APS.
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.
View Image Figure 32:
The baryonic mass function of galaxies (data points). The dotted line shows a Schechter function fit to the data. The blue line shows the predicted mass function of dark matter haloes, assuming that dark matter is cold. The red line shows the same assuming that dark matter is warm with a (thermal relic) mass of mWDM = 1 keV.
View Image Figure 33:
The fraction of mass in bound structures as a function of redshift, normalized to a halo of Milky Way’s mass at redshift z = 0. Marked are different masses of thermal-relic WDM particles in keV (black solid lines). Notice that the differences between different WDM models increases towards higher redshift.
View Image Figure 34:
The central log-slope α of the density distribution ρ ∝ rα for 9 galaxies/groups and 3 lensing clusters as a function of the enclosed lensing mass. Marked in red is the prediction from structure formation simulations of the standard cosmological model, that assume non-relativistic CDM, and that do not include any baryonic matter.
View Image Figure 35:
Full hydrodynamical simulations of massive clusters at redshift z = 0.6. Total projected mass is shown in blue, while X-ray emission from baryonic gas is in red. The preferential trailing of gas due to pressure from the ICM, and its consequent separation from the non interacting dark matter, is apparent in much of the infalling substructure.
View Image Figure 36:
Constraints from neutrino oscillations and from cosmology in the m-Σ plane. Image reproduced by permission from [480]; copyright by IOP and SISSA.
View Image Figure 37:
Left: region in the Δ-Σ parameter space allowed by oscillations data. Right: Weak lensing forecasts. The dashed and dotted vertical lines correspond to the central value for Δ given by oscillations data. In this case Euclid could discriminate NI from IH with a Δ χ2 = 2. Image reproduced by permission from [480]; copyright by IOP and SISSA.
View Image Figure 38:
DM halo mass function (MF) as a function of Σ and redshift. MF of the SUBFIND haloes in the ΛCDM N-body simulation (blue circles) and in the two simulations with Σ = 0.3 eV (magenta triangles) and Σ = 0.6 eV (red squares). The blue, magenta and red lines show the halo MF predicted by [824], where the variance in the density fluctuation field, σ(M ), at the three cosmologies, Σ = 0,0.3,0.6 eV, has been computed with the software camb [559].
View Image Figure 39:
DM halo mass function (MF) as a function of Σ and redshift. Real space two-point auto-correlation function of the DM haloes in the ΛCDM N-body simulation (blue circles) and in the simulation with Σ = 0.6 eV (red squares). The blue and red lines show the DM correlation function computed using the camb matter power spectrum with Σ = 0 and Σ = 0.6 eV, respectively. The bottom panels show the ratio between the halo correlation function extracted from the simulations with and without massive neutrinos.
View Image Figure 40:
Real space two-point auto-correlation function of the DM haloes in the ΛCDM N-body simulation (blue circles) and in the simulation with Σ = 0.6 eV (red squares). The blue and red lines show the DM correlation function computed using the camb matter power spectrum with Σ = 0 and Σ = 0.6 eV, respectively. The bottom panels show the ratio between the halo correlation function extracted from the simulations with and without massive neutrinos.
View Image Figure 41:
Mean bias (averaged in 10 < r[Mpc ∕h] < 50) as a function of redshift compared with the theoretical predictions of [824]. Here the dashed lines represent the theoretical expectations for a ΛCDM cosmology renormalized with the σ8 value of the simulations with a massive neutrino component.
View Image Figure 42:
Two-point auto-correlation function in real and redshift space of the DM-haloes in the ΛCDM N-body simulation (blue circles) and in the simulation with Σ = 0.6 eV (red squares). The bottom panels show the ratio between them, compared with the theoretical expectation.
View Image Figure 43:
Best-fit values of β-σ12, as a function of Σ and redshift (points), compared with the theoretical prediction (grey shaded area). The blue dotted lines show the theoretical prediction for Σ = 0 and with σ8(z = 0).
View Image Figure 44:
The z = 0 matter power spectrum arising in UDM models with a Lagrangian given by Eq. (2.10.4View Equation). ΛCDM is solid, and UDM models with c = 10 −1,10−2,10− 3 ∞ are shown from bottom to top. Image reproduced by permission from [201].
View Image Figure 45:
Marginalized uncertainty in fax for CMB (green), a galaxy redshift survey (red), weak lensing (blue) and the total (black) evaluated for four different fiducial axion masses, for the cosmology ΛCDM+fax+ν. Image reproduced by permission from [633], copyright by APS.
View Image Figure 46:
The marginalized likelihood contours (68.3% and 95.4% CL) for Planck forecast only (blue dashed lines) and Planck plus Euclid pessimistic (red filled contours). The white points correspond to the fiducial model.
View Image Figure 47:
For illustration purposes this is the effect of a local fNL of ±50 on the z = 0 power spectrum of halos with mass above 1013M ⊙.
View Image Figure 48:
Constraints on possible violation of the Etherington relation in the form of deviations from a perfectly transparent universe (𝜖 = 0). Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample.
View Image Figure 49:
Constraints on the simplest Axion-like particles models. Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample. Here P ∕L is the conversion probability per unit length and is the relevant parameter for τ (z ) (see [65]).
View Image Figure 50:
Constraints on MCP models. Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample.
View Image Figure 51:
Supernovae and CMB constraints in the (Ω ð’Ÿ0 m,n) plane for the averaged effective model with zero Friedmannian curvature (filled ellipses) and for a standard flat FLRW model with a quintessence field with constant equation of state w = − (n + 3)∕3 (black ellipses). The disk and diamond represent the absolute best-fit models respectively for the standard FLRW model and the averaged effective model.
View Image Figure 52:
Upper panel: Evolution of Ωk (z𝒟) as a function of redshift for the absolute best-fit averaged model represented by the diamond in Figure 51. One can see that all positively curved FLRW models (Ω < 0 k,0) and only highly negatively curved FLRW models (Ω > 0.5 k,0) can be excluded by the estimation of Ωk (z𝒟). Central panel: Evolution of the coordinate distance for the best-fit averaged model (solid line), for a ΛCDM model with Ωm,0 = 0.277, ΩΛ = 0.735 and H0 = 73 km ∕s∕Mpc (dashed line), and for the FLRW model with the same parameters as the best-fit averaged model (dashed-dotted line). Lower panel: Evolution of the Hubble parameter H ∕H 0 for the best-fit averaged model (solid line), the FLRW model with the same parameters as the averaged best-fit model (dashed-dotted line), and for the same ΛCDM model as in the central panel (dashed line). The error bars in all panels correspond to the expectations for future large surveys like Euclid.
View Image Figure 53:
Relative errors on ΩK for our benchmark survey for different redshifts.
View Image Figure 54:
Left: same as Figure 53 but now with superimposed the prediction for the Lemaître–Tolman–Bondi model considered by [380]. Right: zoom in the high-redshift range.
View Image Figure 55:
Projected cosmological 8-parameter space for a 20,000 square degrees, median redshift of z = 0.8, 10 bin tomographic cosmic shear survey. Specifications are based on Euclid Yellow book [550] as this figure is representative of a method, rather than on forecast analysis; the discussion is still valid with more updated [551] Euclid specifications. The upper panel shows the 1D parameter constraints using analytic marginalization (black) and the Gaussian approximation (Fisher matrix, blue, dark grey). The other panels show the 2D parameter constraints. Grey contours are 1- 2- and 3-σ levels using analytic marginalization over the extra parameters, solid blue ellipses are the 1-σ contours using the Fisher-matrix approximation to the projected likelihood surface, solid red ellipses are the 1-σ fully marginalized. Image reproduced by permission from [878].
View Image Figure 56:
Gaussian approximation (Laplace approximation) to a 6-dimensional posterior distribution for cosmological parameters, from WMAP1 and SDSS data. For all couples of parameters, panels show contours enclosing 68% and 95% of joint probability from 2 ⋅ 105 MC samples (black contours), along with the Laplace approximation (red ellipses). It is clear that the Laplace approximation captures the bulk of the posterior volume in parameter space in this case where there is little non-Gaussianity in the posterior PDF. Image reproduced from 2005 preprint of [894].