It is convenient to describe primordial perturbations using the curvature perturbation on uniform density hypersurfaces introduced in [86]. An important property of this quantity is that for adiabatic perturbations – i.e., in absence of isocurvature perturbations, discussed in Section 3.5 – it remains constant on super-Hubble scales, allowing us to connect the early inflationary phase to the late-time universe observations, regardless of the details of reheating. In a gauge where the energy density of the inflaton vanishes, we can define from the spatial part of the metric (assuming a flat FRW universe), as [781, 616]

This definition, where enters the metric in the exponential form, has the advantage that it is valid also beyond linear order and can be consistently used when discussing non-Gaussian fluctuations, such as in Section 3.3.The power spectrum of primordial perturbations is given by

where denotes the average over an ensemble of realizations. It is useful to define a dimensionless spectrum as , where the index stands for scalar, to distinguish it from the spectrum of tensor perturbations, defined below. The deviation from scale-invariance of the scalar spectrum is characterized by the spectral index , defined by (see, for instance, [570]) where denotes a purely scale-invariant spectrum. We also define the running of the spectral index as These quantities are taken at a particular pivot scale. For our analysis we chose it to be . Thus, with these definitions the power spectrum can be written as where is the normalization parameterising the amplitude of the fluctuations.During inflation tensor modes are also generated. They are described by the gauge invariant metric perturbation , defined from the spatial part of the metric as

Each mode has polarizations, and , each with power spectrum given by Defining the dimensionless power spectrum of tensor fluctuations as , where the factor of 2 comes from the two polarizations, it is convenient to define the ratio of tensor to scalar fluctuations asThe form of the power spectrum given in Eq. (3.2.5) approximates very well power spectra of perturbations generated by slow-roll models. In particular, the spectrum of scalar fluctuations is given in terms of the Hubble rate and the first slow-roll parameter , both evaluated at the time when the comoving scale crosses the Hubble radius during inflation,

During slow-roll, is related to the first derivative of the inflaton potential , , where the prime denotes differentiation with respect to . As and vary slowly during inflation, this spectrum is almost scale-invariant. Indeed, the scalar spectral index in Eq. (3.2.3) reads where the second slow-roll parameter must be small for inflation to yield a sufficient number of -foldings. The running of the spectral index defined in Eq. (3.2.4) is even smaller, being second-order in the slow-roll parameters. It is given by where we have introduced the third slow-roll parameter .The spectrum of tensor fluctuations is given by

which shows that the ratio of tensor to scalar fluctuations in Eq. (3.2.8) is simply related to the first slow-roll parameter by .As a fiducial model, in the next section we will consider chaotic inflation [577], based on the quadratic inflaton potential . In this case, the first two slow-roll parameters are both given in terms of the value of the inflaton field at Hubble crossing or, equivalently, in terms of number of -folds from Hubble crossing to the end of inflation , as , while . This implies

where the star denotes Hubble crossing of the pivot scale . Choosing , this yields , and as our fiducial model.

We will now study how much Euclid will help in improving the already very tight constraints on the power spectrum given by the Planck satellite. Let us start discussing the forecast for Planck. We assume 2.5 years (5 sky surveys) of multiple CMB channel data, with instrument characteristics for the different channels listed in Table 19. We take the detector sensitivities and the values of the full width half maximum from the Planck “Blue Book” [735]. In this analysis we use three channels for Planck mock data and we assume that the other channels are used for foreground removal and thus do not provide cosmological information.

Channel Frequency (GHz) | 70 | 100 | 143 |

Resolution (arcmin) | 14 | 10 | 7.1 |

Sensitivity - intensity () | 8.8 | 4.7 | 4.1 |

Sensitivity - polarization () | 12.5 | 7.5 | 7.8 |

For a nearly full-sky CMB experiment (we use ), the likelihood can be approximated by [923]

where we assume and . Here, is the sum of the model-dependent theoretical power spectrum and of the noise spectrum , which we assume perfectly known. The mock data is for the fiducial model, with calculated using the publicly available code camb [559] and calculated assuming a Gaussian beam. We use the model described in [923, 109] to propagate the effect of polarization foreground residuals into the estimated uncertainties on the cosmological parameters. For simplicity, in our simulation we consider only the dominating components in the frequency bands that we are using, i.e., the synchrotron and dust signals. The fraction of the residual power spectra are all assumed to be 5%.Let us turn now to the Euclid forecast based on the spectroscopic redshift survey. We will model the galaxy power spectrum in redshift space as ([485, 711, 713]; see also discussion in Section 1.7.3)

where is the cosine of the angle between the wavenumber and the line of sight, is the linear growth factor defined in Eq. (1.3.21), is the linear growth rate (see Eq. (1.3.22)) and is the matter power spectrum at redshift . The term comes for the redshift distortions due to the large-scale peculiar velocity field [485], which is correlated with the matter density field. The factor accounts for the radial smearing due to the redshift distortions that are uncorrelated with the large-scale structure. We consider two contributions. The first is due to the redshift uncertainty of the spectroscopic galaxy samples. Assuming a typical redshift uncertainty , this turns into a contribution to given by , where is the comoving distance of a flat FRW universe and is the Hubble parameter as a function of the redshift. The second contribution comes from the Doppler shift due to the virialized motion of galaxies within clusters, which typically have a pairwise velocity dispersion of the order of few hundred kilometers per second. This term can be parameterized as [713]. Taking the geometric mean of the two contributions, we obtain where the two velocities in the parenthesis contribute roughly the same. Practically neither the redshift measurement nor the virialized virialized motion of galaxies can be precisely quantified. In particular, the radial smearing due to peculiar velocity is not necessarily close to Gaussian. Thus, Eq. (3.2.14) should not be used for wavenumbers , where the radial smearing effect is important.On large scales the matter density field has, to a very good approximation, Gaussian statistics and uncorrelated Fourier modes. Under the assumption that the positions of observed galaxies are generated by a random Poissonian point process, the band-power uncertainty is given by ([883]; see also Eq. (1.7.26) in Section 1.7.3)

Here is the observed fraction of sky, the comoving distance defined above, and is the expected number density of galaxies that can be used.Finally, we ignore the band-band correlations and write the likelihood as

To produce the mock data we use a fiducial CDM model with , , , and , where is the reionization optical depth. As mentioned above, we take the fiducial value for the spectral index, running and tensor to scalar ratio, defined at the pivot scale , as given by chaotic inflation with quadratic potential, i.e., , and . We have checked that for Planck data is almost orthogonal to and . Therefore our result is not sensitive to the fiducial value of .

The fiducial Euclid spectroscopically selected galaxies are split into 14 redshift bins. The redshift ranges and expected numbers of observed galaxies per unit volume are taken from [551] and shown in the third column of Table 3 in Section 1.8.2 (). The number density of galaxies that can be used is , where is the fraction of galaxies with measured redshift. The boundaries of the wavenumber range used in the analysis, labeled and , vary in the ranges and respectively, for . The IR cutoff is chosen such that , where is the comoving distance of the redshift slice. The UV cutoff is the smallest between and . Here is chosen such that the r.m.s. linear density fluctuation of the matter field in a sphere with radius is 0.5. In each redshift bin we use 30 -bins uniformly in and 20 uniform -bins.

For the fiducial value of the bias, in each of the 14 redshift bins of width in the range (0.7 – 2), we use those derived from [698], i.e. (1.083, 1.125, 1.104, 1.126, 1.208, 1.243, 1.282, 1.292, 1.363, 1.497, 1.486, 1.491, 1.573, 1.568), and we assume that is redshift dependent choosing as the fiducial value in each redshift bin. Then we marginalize over and in the 14 redshift bins, for a total of 28 nuisance parameters.

In these two cases, we consider the forecast constraints on eight cosmological parameters, i.e., , , , , , , , and . Here is the angle subtended by the sound horizon on the last scattering surface, rescaled by a factor 100. We use the publicly available code CosmoMC [558] to perform Markov Chain Monte Carlo calculation. The nuisance parameters are marginalized over in the final result. The marginalized 68.3% confidence level (CL) constraints on cosmological parameters for Planck forecast only, and Planck and Euclid forecast are listed in the second and third columns of Table 20, respectively.

Euclid can improve the ‘figure of merit’ on the - plane by a factor of 2.2, as shown in the left panel of Figure 46. Because the bias is unknown, the LSS data do not directly measure or . However, Euclid can measure to a much better accuracy, which can break the degeneracy between and that one typically finds using CMB data alone. This is shown in the right panel of Figure 46.

A more extensive and in depth analysis of what constraints on inflationary models a survey like Euclid can provide is presented in [460]. In particular they find that for models where the primordial power spectrum is not featureless (i.e., close to a power law with small running) a survey like Euclid will be crucial to detect and measure features. Indeed, what we measure with the CMB is the angular power spectrum of the anisotropies in the 2-D multipole space, which is a projection of the power spectrum in the 3-D momentum space. Features at large ’s and for small width in momentum space get smoothed during this projection but this does not happen for large-scale structure surveys. The main limitation on the width of features measured using large-scale structure comes from the size of the volume of the survey: the smallest detectable feature being of the order of the inverse cubic root of this volume and the error being determined by number of modes contained in this volume. Euclid, with the large volume surveyed and the sheer number of modes that are sampled and cosmic variance dominated offers a unique opportunity to probe inflationary models where the potential is not featureless. In addition the increased statistical power would enable us to perform a Bayesian model selection on the space of inflationary models (e.g., [334, 691] and references therein).

Living Rev. Relativity 16, (2013), 6
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