Figure 1:
Left: the cosmic microwave background angular power spectrum for TeVeS (solid) and CDM (dotted) with WMAP 5year data [689]. Right: the matter power spectrum for TeVeS (solid) and CDM (dotted) plotted with SDSS data. 

Figure 2:
The evolution of as a function of the comoving scale , using only the 5year 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 flowequation formalism. From the outside inward, the colored regions are red, yellow, blue, and purple. Image reproduced by permission from [475]; copyright by APS. 

Figure 3:
The complete evolution of , from the flowequation results accepted by the CMB likelihood. Inflation is made to end at where corresponding to . For our choice of priors on the slowroll parameters at , we find that decreases rapidly towards (see inset) and stays close to it during the period when the observable scales leave the horizon (). Image reproduced by permission from [475]; copyright by APS. 

Figure 4:
Required accuracy on to obtain strong evidence against a model where as compared to a cosmological constant model, . For a given , models to the right and above the contour are disfavored with odds of more than 20:1. 

Figure 5:
Ratio of the total mass functions, which include the quintessence contribution, for and at (above) and (below). Image reproduced by permission from [258]; copyright by IOP and SISSA. 

Figure 6:
Extrapolated linear density contrast at collapse for coupled quintessence models with different coupling strength . For all plots we use a constant . We also depict for reference CDM (dotted, pink) and EdS (doubledashed, black) models. Image reproduced by permission from [962]; copyright by APS. 

Figure 7:
Extrapolated linear density contrast at collapse vs. collapse redshift for growing neutrinos with (solid, red), (longdashed, green) and (shortdashed, blue). A reference EdS model (doubledashed. black) is also shown. Image reproduced by permission from [962]; copyright by APS. 

Figure 8:
Extrapolated linear density contrast at collapse vs. collapse redshift for EDE models I (solid, red) and II (longdashed, green), as well as CDM (doubledashed, black). Image reproduced by permission from [962]; copyright by APS. 

Figure 9:
Left: The velocity contribution as a function of for various redshifts. Right: The standard contribution as a function of for various redshifts. 

Figure 10:
Matter power spectrum form measured from SDSS [720] 

Figure 11:
Marginalized forecast for weak lensing only analysis with Euclid. Here is defined from and , defined via Eq. 1.3.28, is related to the WL potential. Black contours correspond to , demonstrating an error of 0.089 on , whereas the red contours correspond to giving an error of 0.034. In both cases, the inner and outer contours are and respectively. GR resides at [0.55, 0], while DGP resides at [0.68, 0]. 

Figure 12:
Constraints on , , and 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. 

Figure 13:
Contour plots at 68% and 98% of probability for the pairs in 7 redshift bins (with ). The ellipses are centered on the fiducial values of the growth rate and bias parameters, computed in the central values of the bins, . 

Figure 14:
Expected constraints on the growth rates in each redshift bin. For each the central error bars refer to the Reference case while those referring to the Optimistic and Pessimistic case have been shifted by and respectively. The growth rates for different models are also plotted: CDM (green tight shortdashed curve), flat DGP (red longdashed curve) and a model with coupling between dark energy and dark matter (purple, dotdashed curve). The blue curves (shortdashed, dotted and solid) represent the model by [456] with respectively. The plot shows that it will be possible to distinguish these models with next generation data. 

Figure 15:
Expected constraints on the growth rates in each redshift bin. For each the central error bars refer to the Reference case while those referring to the Optimistic and Pessimistic case have been shifted by and respectively. The growth rates for different models are also plotted: CDM (green tight shortdashed curve), flat DGP (red longdashed curve) and a model with coupling between dark energy and dark matter (purple, dotdashed curve). Here we plot again the model by [456] with (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. 

Figure 16:
parameterization. Left panel: 1 and 2 marginalized probability regions for constant and : the green (shaded) regions are relative to the Reference case, the blue longdashed ellipses to the Optimistic case, while the black shortdashed 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 2 marginalized probability regions for the parameters and , relative to the Reference case (shaded yellow regions), to the Optimistic case (green longdashed 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 model. Then, in the case of parameterization, one could distinguish these three models (at 95% probability). 

Figure 17:
parameterization. and marginalized probability regions obtained assuming constant and (red solid curves) or assuming the parameterizations (1.8.5) and (1.8.2) and marginalizing over and (black dashed curves); marginalized error values are in columns , of Table 8. Yellow dots represent the fiducial model, the triangles a model and the squares mark the flat DGP. 

Figure 18:
parameterization. 1 and 2 marginalized probability regions obtained assuming constant and (red solid curves) or assuming the parameterizations (1.8.6) and (1.8.2) and marginalizing over and (black dashed curves); marginalized error values are in columns , of Table 9. Yellow dots represent the fiducial model, the triangles stand for a model and the squares mark the flat DGP. 

Figure 19:
parameterization. Left panel: 1 and 2 marginalized probability regions for the parameters and in Eq. (1.8.6) relative to the reference case (shaded blue regions), to the optimistic case (yellow longdashed ellipses) and to the pessimistic case (black shortdashed ellipses). The red dot marks the fiducial model while the square represents the coupling model. Right panel: present constraints on and computed through a full likelihood method (here the red dot marks the likelihood peak) [307]. 

Figure 20:
Errors on the equation of state. and marginalized probability regions for the parameters and , relative to the reference case and bias . The blue dashed ellipses are obtained fixing and to their fiducial values and marginalizing over all the other parameters; for the red shaded ellipses instead, we also marginalize over but we fix . Finally, the black dotted ellipses are obtained marginalizing over all parameters but and . 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). 

Figure 21:
Error bars on the Hubble parameter with five redshift bins. The exact height of the error bars respectively are . 

Figure 22:
Error bars on the growth function with three redshift bins while marginalizing over the s. The exact height of the error bars respectively are . 

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

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

Figure 25:
Confidence region at 68% for three different values of , red solid, green longdashed and blue dashed contour, respectively. The left panel shows the confidence region when the sound speed is ; the right panel with the sound speed . The equation of state parameter is for both cases . 

Figure 26:
Confidence region at 68% for three different values of , red solid, green longdashed and blue dashed contour, respectively. The left panel shows the confidence region when the sound speed is ; the right panel with the sound speed . The equation of state parameter is for both cases . 

Figure 27:
The Bayes factor for the model of Eq. (1.4.52) over standard CDM as a function of the extra parameter . The green, red and blue curves refer to the conservative, bindependent and optimistic , respectively. The horizontal lines denote the Jeffreys’ scale levels of significance. 

Figure 28:
Left panel: Linear matter power spectra for CDM (solid line; , ) and scalaron (dashed line; , ) 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 and , respectively. Right panel: The environmental dependent chameleon mechanism can be seen in the mildly nonlinear regime. We exhibit the fractional difference between the and GR power spectra for the model (1.8.37) with parameters and . The dashed lines represent linear power spectra ( and 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 and the bottom to ; demonstrating that the modified gravity signal dramatically decreases for larger . This is due to the scalaron mass being much larger at higher redshifts. Furthermore, nonlinear effects are less significant for increasing . 

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

Figure 30:
Comparison among predicted confidence contours for the cosmological parameter set using CMB (Planck, blue contours), (pinkviolet contours) and weak lensing (orangered contours) with Euclidlike specifications. Image reproduced by permission from [42], copyright by APS. 

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 threedimensional clustering information is crucial for BAO. For example to obtain the same figure of merit for darkenergy properties a photometric survey must cover a volume roughly ten times bigger than a spectroscopic one. 

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 . 

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 . Marked are different masses of thermalrelic WDM particles in keV (black solid lines). Notice that the differences between different WDM models increases towards higher redshift. 

Figure 34:
The central logslope of the density distribution 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 nonrelativistic CDM, and that do not include any baryonic matter. 

Figure 35:
Full hydrodynamical simulations of massive clusters at redshift . Total projected mass is shown in blue, while Xray 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. 

Figure 36:
Constraints from neutrino oscillations and from cosmology in the  plane. Image reproduced by permission from [480]; copyright by IOP and SISSA. 

Figure 37:
Left: region in the  parameter space allowed by oscillations data. Right: Weak lensing forecasts. The dashed and dotted vertical lines correspond to the central value for given by oscillations data. In this case Euclid could discriminate NI from IH with a . Image reproduced by permission from [480]; copyright by IOP and SISSA. 

Figure 38:
DM halo mass function (MF) as a function of and redshift. MF of the SUBFIND haloes in the CDM body simulation (blue circles) and in the two simulations with (magenta triangles) and (red squares). The blue, magenta and red lines show the halo MF predicted by [824], where the variance in the density fluctuation field, , at the three cosmologies, , has been computed with the software camb [559]. 

Figure 39:
DM halo mass function (MF) as a function of and redshift. Real space twopoint autocorrelation function of the DM haloes in the CDM body simulation (blue circles) and in the simulation with (red squares). The blue and red lines show the DM correlation function computed using the camb matter power spectrum with and , respectively. The bottom panels show the ratio between the halo correlation function extracted from the simulations with and without massive neutrinos. 

Figure 40:
Real space twopoint autocorrelation function of the DM haloes in the CDM body simulation (blue circles) and in the simulation with (red squares). The blue and red lines show the DM correlation function computed using the camb matter power spectrum with and , respectively. The bottom panels show the ratio between the halo correlation function extracted from the simulations with and without massive neutrinos. 

Figure 41:
Mean bias (averaged in ) as a function of redshift compared with the theoretical predictions of [824]. Here the dashed lines represent the theoretical expectations for a CDM cosmology renormalized with the value of the simulations with a massive neutrino component. 

Figure 42:
Twopoint autocorrelation function in real and redshift space of the DMhaloes in the CDM body simulation (blue circles) and in the simulation with (red squares). The bottom panels show the ratio between them, compared with the theoretical expectation. 

Figure 43:
Bestfit values of , as a function of and redshift (points), compared with the theoretical prediction (grey shaded area). The blue dotted lines show the theoretical prediction for and with . 

Figure 44:
The matter power spectrum arising in UDM models with a Lagrangian given by Eq. (2.10.4). CDM is solid, and UDM models with are shown from bottom to top. Image reproduced by permission from [201]. 

Figure 45:
Marginalized uncertainty in 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 CDM++. Image reproduced by permission from [633], copyright by APS. 

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. 

Figure 47:
For illustration purposes this is the effect of a local of on the power spectrum of halos with mass above . 

Figure 48:
Constraints on possible violation of the Etherington relation in the form of deviations from a perfectly transparent universe (). 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. 

Figure 49:
Constraints on the simplest Axionlike 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 is the conversion probability per unit length and is the relevant parameter for (see [65]). 

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. 

Figure 51:
Supernovae and CMB constraints in the ,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 (black ellipses). The disk and diamond represent the absolute bestfit models respectively for the standard FLRW model and the averaged effective model. 

Figure 52:
Upper panel: Evolution of as a function of redshift for the absolute bestfit averaged model represented by the diamond in Figure 51. One can see that all positively curved FLRW models () and only highly negatively curved FLRW models () can be excluded by the estimation of . Central panel: Evolution of the coordinate distance for the bestfit averaged model (solid line), for a CDM model with , and (dashed line), and for the FLRW model with the same parameters as the bestfit averaged model (dasheddotted line). Lower panel: Evolution of the Hubble parameter for the bestfit averaged model (solid line), the FLRW model with the same parameters as the averaged bestfit model (dasheddotted line), and for the same CDM model as in the central panel (dashed line). The error bars in all panels correspond to the expectations for future large surveys like Euclid. 

Figure 53:
Relative errors on for our benchmark survey for different redshifts. 

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 highredshift range. 

Figure 55:
Projected cosmological 8parameter space for a 20,000 square degrees, median redshift of , 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 levels using analytic marginalization over the extra parameters, solid blue ellipses are the 1 contours using the Fishermatrix approximation to the projected likelihood surface, solid red ellipses are the 1 fully marginalized. Image reproduced by permission from [878]. 

Figure 56:
Gaussian approximation (Laplace approximation) to a 6dimensional 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 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 nonGaussianity in the posterior PDF. Image reproduced from 2005 preprint of [894]. 
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