Tener un término medio de autoestima

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 20. We take the detector sensitivities and the values of the full width half maximum from the Planck “Blue Book” [975]. 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.

Table 20: Instrument specifics for the Planck satellite with 30 months of integration. Channel Frequency (GHz) 70 100 143

Resolution (arcmin) 14 10 7.1 Sensitivity - intensity (µK) 8.8 4.7 4.1 Sensitivity - polarization (µK) 12.5 7.5 7.8

For a nearly full-sky CMB experiment (we use fsky = 0.75), the likelihoodL can be approxi-

mated by [1202] −2 lnL= `max X `=`min (2`+ 1)fsky " −3 + ˆ CBB ` CBB ` + ln C BB ` ˆ CBB ` ! + ˆ CT T ` C EE ` + ˆC EE ` C T T ` −2 ˆC T E ` C T E ` CT T ` C EE ` −(C T E ` )2 + ln C T T ` C EE ` −(C T E ` ) 2 ˆ CT T ` Cˆ`EE−( ˆC`T E)2 !# , (3.2.13)

where we assumelmin= 3 andlmax= 2500. Here,C`is the sum of the model-dependent theoretical

power spectrumC<sub>`</sub>theoryand of the noise spectrumN`, which we assume perfectly known. The mock

data ˆC`isC`for the fiducial model, withC theory

` calculated using the publicly available codecamb

[743] and N` calculated assuming a Gaussian beam. We use the model described in [1202, 147]

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 ([649, 933, 935]; see also discussion in Section 1.7.3)

Pg(k, z, µ) = b+fgµ2

2

G2(z)Pmatter(k;z= 0)e−k

2_µ2_σ2

r_{, (3.2.14)}

whereµis the cosine of the angle between the wavenumberkand the line of sight,G(z) is the linear growth factor defined in Eq. (1.3.21),fg≡dlnG/dlnais the linear growth rate (see Eq. (1.3.22))

and Pmatter(k;z = 0) is the matter power spectrum at redshift 0. The term fgµ2 comes for the

redshift distortions due to the large-scale peculiar velocity field [649], which is correlated with the matter density field. The factor e−k2_µ2_σ2

r 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 σz = 0.001(1 + z), this turns into a contribution to σr given by

∂r/∂z σz=H−1σz, wherer(z) =

Rz

0 cdz

0_/H(z0_{) is the comoving distance of a flat FRW universe}

andH 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 dispersionvp of the order of few hundred kilometers per second. This term can

be parameterized as _√vp

2H

−1_{(1 +z) [935]. Taking the geometric mean of the two contributions, we}

obtain σ2_r= (1 +z) 2 H2 10 −6_+v2 p/2 , (3.2.15)

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 wavenumbersk > _vH(z)

p(1+z), 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 ([1154]; see also Eq. (1.7.26) in Section 1.7.3)

∆Pg= 2(2π)3 (2πk2_{dk dµ)(4πr}2_f skydr) 1/2 Pg+ 1 ¯ n . (3.2.16)

Here fsky is the observed fraction of sky, r the comoving distance defined above, and ¯n is the

expected number density of galaxies that can be used.

Finally, we ignore the band-band correlations and write the likelihood as

−2 lnL= X k,µ,zbins Pmodel g −Pgfiducial ∆Pfiducial g !2 . (3.2.17)

To produce the mock data we use a fiducial ΛCDM model with Ωch2= 0.1128, Ωbh2= 0.022,

h= 0.72, σ8 = 0.8 andτ = 0.09, whereτ 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 scalek∗= 0.05 Mpc−1, as given by chaotic inflation with quadratic potential, i.e.,ns= 0.968,

αs= 0 andr= 0.128. We have checked that for Planck dataris almost orthogonal tons andαs.

Therefore our result is not sensitive to the fiducial value ofr.

The fiducial Euclid spectroscopically selected galaxies are split into 14 redshift bins. The redshift ranges and expected numbers of observed galaxies per unit volume ¯nobsare taken from [732]

and shown in the third column of Table 3 in Section 1.8.2 (n2(z)). The number density of galaxies

that can be used is ¯n = ε¯nobs, where ε is the fraction of galaxies with measured redshift. The

boundaries of the wavenumber range used in the analysis, labeledkminandkmax, vary in the ranges

(0.00435 – 0.00334)hMpc−1and (0.16004 – 0.23644)hMpc−1respectively, for 0.7≤z≤2. The IR cutoffkminis chosen such that kminr= 2π, whereris the comoving distance of the redshift slice.

The UV cutoff is the smallest between_v H

p(1+z)and

π

2R. HereRis chosen such that the r.m.s. linear

density fluctuation of the matter field in a sphere with radiusRis 0.5. In each redshift bin we use 30k-bins uniformly in lnkand 20 uniformµ-bins.

For the fiducial value of the bias, in each of the 14 redshift bins of width ∆z= 0.1 in the range (0.7 – 2), we use those derived from [919], 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 vp is redshift dependent choosing

vp= 400 km/s as the fiducial value in each redshift bin. Then we marginalize overbandvp 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., Ωbh2, Ωch2,θ,τ, lnAs,ns,αs, andr. Hereθis the angle subtended by the sound horizon on the

last scattering surface, rescaled by a factor 100. We use the publicly available code_CosmoMC [740] to perform Markov Chain Monte Carlo calculation. The nuisance parameters are marginalized

Table 21: Cosmological parameters

parameter Planck 2015 constraint EUCLID fiducial cosmology Planck + Euclid constraint Ωbh2 0.02238±0.00027 0.022 0.02227+0.00008−0.00008 Ωch2 0.1180±0.0021 0.12 0.1116+0.0002−0.0002 100θ 1.04111±0.00047 1.041 1.0392+0.0002_−0.0002 τre 0.071±0.018 0.09 0.085+0.003−0.003 ns 0.9690±0.0063 0.96 0.966+0.002−0.002 αs −0.008+0.009−0.008 0 −0.000 +0.003 −0.003 ln(1010As) 3.073±0.033 3.098 3.077+0.006−0.006 r0.05 <0.16(95%CL) 0 0.127+0.019−0.018 Ωm 0.304±0.013 0.32 0.271+0.001−0.001 σ8 0.816±0.010 0.83 0.808+0.003−0.003 h 0.682±0.010 0.67 0.703+0.001_−0.001

over in the final result. The marginalized 68.3% confidence level (CL) constraints on cosmological parameters for Planck only (Planck TT + lowP + lensing, for LCDM with r and running of the spectral index [967]), and Planck and Euclid forecast are listed in the second and third columns of Table 21, respectively.

Euclid can improve the ‘figure of merit’ on the ns-αsplane by a factor of 2.2, as shown in the

left panel of Figure 52. Because the bias is unknown, the LSS data do not directly measureAsor

σ8. However, Euclid can measure Ωmto a much better accuracy, which can break the degeneracy

between Ωm and σ8 that one typically finds using CMB data alone. This is shown in the right

panel of Figure 52.

A more extensive and in depth analysis of what constraints on inflationary models a survey like Euclid can provide is presented in [617]. 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., [448, 911] and references therein).