3.2 Constraining inflation

The spectrum of cosmological perturbations represents an important source of information on the early universe. During inflation scalar (compressional) and tensor (purely gravitational) fluctuations are produced. The shape and the amplitude of the power spectrum of scalar fluctuations can be related to the dynamics of the inflationary phase, providing a window on the inflaton potential. Inflation generically predicts a deviation from a purely scale-invariant spectrum. Together with future CMB experiments such as Planck, Euclid will improve our constraints on the scalar spectral index and its running, helping to pin down the model of inflation.

3.2.1 Primordial perturbations from inflation

It is convenient to describe primordial perturbations using the curvature perturbation on uniform density hypersurfaces ζ 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 ζ from the spatial part of the metric (assuming a flat FRW universe), as [781Jump To The Next Citation Point, 616Jump To The Next Citation Point]

gij = a2(t) exp(2ζ )δij. (3.2.1 )
This definition, where ζ 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

3 ′ ⟨ζkζk′⟩ = (2 π) δ(k + k )Pζ(k), (3.2.2 )
where ⟨...⟩ denotes the average over an ensemble of realizations. It is useful to define a dimensionless spectrum as -k3 𝒫s(k) ≡ 2π2P ζ(k), where the index s 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 ns, defined by (see, for instance, [570])
d-ln-𝒫s- ns ≡ 1 + dlnk , (3.2.3 )
where ns = 1 denotes a purely scale-invariant spectrum. We also define the running of the spectral index αs as
α ≡ -dns-. (3.2.4 ) s dln k
These quantities are taken at a particular pivot scale. For our analysis we chose it to be k ∗ ≡ 0.05 Mpc −1. Thus, with these definitions the power spectrum can be written as
2π2- ns(k∗)− 1+ 1αs(k∗)ln(kβˆ•k∗) P ζ(k ) = k3 As (k∗)(k βˆ•k∗) 2 , (3.2.5 )
where As 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 hij, defined from the spatial part of the metric as

2 j i gij = a (t)(δij + hij), h i,j = 0 = hi. (3.2.6 )
Each mode has 2 polarizations, h+ and h×, each with power spectrum given by
3 ′ ⟨hkhk ′⟩ = (2π) δ(k + k )Ph(k ). (3.2.7 )
Defining the dimensionless power spectrum of tensor fluctuations as 3 𝒫t(k ) ≡ 2 k2π2Ph(k), where the factor of 2 comes from the two polarizations, it is convenient to define the ratio of tensor to scalar fluctuations as
r ≡ 𝒫t(k∗)βˆ•π’«s (k∗). (3.2.8 )

The form of the power spectrum given in Eq. (3.2.5View Equation) 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 H and the first slow-roll parameter 2 πœ– ≡ −HΛ™ βˆ•H, both evaluated at the time when the comoving scale k crosses the Hubble radius during inflation,

| --1--H2--|| 𝒫s(k ) = 8π2πœ– M 2 | . (3.2.9 ) Plk=aH
During slow-roll, πœ– is related to the first derivative of the inflaton potential V (Ο•), M2 ( ′)2 πœ– ≈ -P2l VV-, where the prime denotes differentiation with respect to Ο•. As H and πœ– vary slowly during inflation, this spectrum is almost scale-invariant. Indeed, the scalar spectral index n s in Eq. (3.2.3View Equation) reads
ns = 1 − 6πœ– + 2ηV, (3.2.10 )
where the second slow-roll parameter η ≡ M 2 V′′ V PlV must be small for inflation to yield a sufficient number of e-foldings. The running of the spectral index defined in Eq. (3.2.4View Equation) is even smaller, being second-order in the slow-roll parameters. It is given by αs = 16πœ–ηV − 24πœ–2 − 2ξV where we have introduced the third slow-roll parameter ′′′′ ξV ≡ M P4lVVV2-.

The spectrum of tensor fluctuations is given by

2|| 𝒫t(k) = 2--H--| , (3.2.11 ) π2M 2Pl|k=aH
which shows that the ratio of tensor to scalar fluctuations in Eq. (3.2.8View Equation) is simply related to the first slow-roll parameter by r = 16πœ–.

As a fiducial model, in the next section we will consider chaotic inflation [577], based on the quadratic inflaton potential V = 1m2 Ο•2 2. In this case, the first two slow-roll parameters are both given in terms of the value of the inflaton field at Hubble crossing Ο• or, equivalently, in terms of number of e-folds from Hubble crossing to the end of inflation N, as 2 2 πœ– = ηV = 2M Plβˆ•Ο• = 1 βˆ•2N, while ξV = 0. This implies

2 ns = 1 − 2βˆ•N ∗, r = 8βˆ•N ∗, αs = − 2βˆ•N ∗, (3.2.12 )
where the star denotes Hubble crossing of the pivot scale k∗. Choosing N∗ = 62.5, this yields ns = 0.968, r = 0.128 and αs = 0 as our fiducial model.

3.2.2 Forecast constraints on the power spectrum

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.

Table 19: 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 likelihood β„’ can be approximated by [923Jump To The Next Citation Point]

β„“max [ ( ) ∑ CˆBBβ„“-- CBBβ„“-- − 2 ln β„’ = (2β„“ + 1)fsky − 3 + CBB + ln CˆBB β„“=β„“min β„“ β„“ ( ) ] (3.2.13 ) ˆCT TCEE + ˆCEE CT T − 2 ˆCTE CT E CT T CEE − (CT E)2 + -β„“---β„“--T-T--β„“EE--β„“---TE-2β„“---β„“---+ ln --β„“---β„“-------β„“---- , Cβ„“ Cβ„“ − (C β„“ ) CˆTβ„“T ˆCEβ„“E − ( ˆCTβ„“ E)2
where we assume lmin = 3 and lmax = 2500. Here, C β„“ is the sum of the model-dependent theoretical power spectrum theory Cβ„“ and of the noise spectrum N β„“, which we assume perfectly known. The mock data Cˆβ„“ is C β„“ for the fiducial model, with Cthβ„“eory calculated using the publicly available code camb [559] and N β„“ 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 ([485Jump To The Next Citation Point, 711, 713Jump To The Next Citation Point]; see also discussion in Section 1.7.3)

P (k, z,μ) = (b + f μ2)2G2 (z)P (k;z = 0 )e− k2μ2σ2r, (3.2.14 ) g g matter
where μ is the cosine of the angle between the wavenumber k and the line of sight, G (z) is the linear growth factor defined in Eq. (1.3.21View Equation), fg ≡ dln Gβˆ•d lna is the linear growth rate (see Eq. (1.3.22View Equation)) and Pmatter(k; z = 0) is the matter power spectrum at redshift 0. The term 2 fgμ comes for the redshift distortions due to the large-scale peculiar velocity field [485], which is correlated with the matter density field. The factor 2 22 e−k μσr 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 −1 ∂rβˆ•∂z σz = H σz, where ∫z ′ ′ r(z) = 0 cdz βˆ•H (z ) is the comoving distance of a flat FRW universe and H 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 v p of the order of few hundred kilometers per second. This term can be parameterized as v√p- −1 2H (1 + z) [713]. Taking the geometric mean of the two contributions, we obtain
2 (1-+-z)2( −6 2 ) σr = H2 10 + vpβˆ•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. (3.2.14View Equation) should not be used for wavenumbers k > -H(z)- vp(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 ([883]; see also Eq. (1.7.26View Equation) in Section 1.7.3)

[ ]1βˆ•2 ( ) --------2-(2-π)3--------- 1- ΔPg = (2πk2dkd μ)(4πr2fskydr) Pg + ¯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 ∑ Pgmodel− Pgfiducial − 2 ln β„’ = -------fiducial--- . (3.2.17 ) k,μ,z bins ΔP g

To produce the mock data we use a fiducial ΛCDM model with 2 Ωch = 0.1128, 2 Ωbh = 0.022, h = 0.72, σ8 = 0.8 and τ = 0.09, where τ 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 k = 0.05Mpc −1 ∗, as given by chaotic inflation with quadratic potential, i.e., n = 0.968 s, α = 0 s and r = 0.128. We have checked that for Planck data r is almost orthogonal to ns and αs. Therefore our result is not sensitive to the fiducial value of r.

The fiducial Euclid spectroscopically selected galaxies are split into 14 redshift bins. The redshift ranges and expected numbers of observed galaxies per unit volume ¯nobs are taken from [551Jump To The Next Citation Point] and shown in the third column of Table 3 in Section 1.8.2 (n (z) 2). The number density of galaxies that can be used is ¯n = πœ€¯nobs, where πœ€ is the fraction of galaxies with measured redshift. The boundaries of the wavenumber range used in the analysis, labeled kmin and kmax, vary in the ranges (0.00435 –0.00334 )h Mpc −1 and (0.16004 – 0.23644 )h Mpc −1 respectively, for 0.7 ≤ z ≤ 2. The IR cutoff kmin is chosen such that kminr = 2π, where r is the comoving distance of the redshift slice. The UV cutoff is the smallest between --H--- vp(1+z) and -π- 2R. Here R is chosen such that the r.m.s. linear density fluctuation of the matter field in a sphere with radius R is 0.5. In each redshift bin we use 30 k-bins uniformly in ln k and 20 uniform μ-bins.

For the fiducial value of the bias, in each of the 14 redshift bins of width Δz = 0.1 in the range (0.7 – 2), we use those derived from [698Jump To The Next Citation Point], 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 βˆ•s as the fiducial value in each redshift bin. Then we marginalize over b and vp in the 14 redshift bins, for a total of 28 nuisance parameters.

Table 20: Cosmological parameters
parameter Planck constraint Planck + Euclid constraint
Ωbh2 0.02227+0−.00.000001111 0.02227+0−.00.000000088
Ωch2 0.1116+0−.00.0000088 0.1116+0−.00.0000022
πœƒ 1.0392+0.0002 −0.0002 1.0392+0.0002 −0.0002
τ re 0.085+0.004 −0.004 0.085+0.003 −0.003
ns +0.003 0.966 −0.003 +0.002 0.966 −0.002
αs +0.005 − 0.000 −0.005 +0.003 − 0.000− 0.003
ln (1010As ) 3.078+0−.00.00099 3.077+0−.00.00066
r 0.128+0−.00.01188 0.127+0−.00.01198
Ωm 0.271+0.005 −0.004 0.271+0.001 −0.001
σ 8 0.808+0.005 −0.005 0.808+0.003 −0.003
h +0.004 0.703 −0.004 +0.001 0.703 −0.001

In these two cases, we consider the forecast constraints on eight cosmological parameters, i.e., 2 Ωbh, Ωch2, πœƒ, τ, lnAs, ns, αs, and r. Here πœƒ 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.

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.

Euclid can improve the ‘figure of merit’ on the ns-αs plane by a factor of 2.2, as shown in the left panel of Figure 46View Image. Because the bias is unknown, the LSS data do not directly measure As or σ8. However, Euclid can measure Ωm to 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 46View Image.

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 β„“’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).

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