EL TRABAJO CIENTÍFICO Objetivos de la unidad

when looking at the enhancement of the skewness signal in the case of PNG parametrised by fN L = 100.

Currently, the best constraints on primordial non-Gaussianity are given by CMB experiments and in particular Planck (Planck Collaboration et al.,2016b) which constrained the local type of PNG up to floc

NL = 0.8 ± 5.0 combining both temperature and polarization data. The ratio in Equation 2.79 for fN L = 1

differs from unity by less than 0.5% up to the maximum scales shown in Figure

2.1.

2.5

S

from data

In this section are presented the cosmological parameter constraints derived from the measurement of the 2pt and 3pt auto-correlation functions on the SDSS DR11 BOSS CMASS north sample data. In order to obtain 1 and 2D confidence intervals, the galaxy mocks produced by Manera et al. (2013); Manera et al.

(2015) have been used to estimate the covariance matrix.

The main reason for these measurements was originally to find experimental evidence of the result presented by Juszkiewicz (2013) relative to possibility of observing a BAO signal in the skewness. Indeed even if the signal would have been much smaller and less peaked than in the 2pt correlation function, from the theoretical point of view it would have been almost completely independent from the galaxy bias parametrisation.

In order to check the presence of BAO, Juskiewicz takes the ratio of an S3 with a physical power spectrum to an Ssmooth₃ obtained by using an analytic expression for a no-wiggle power spectrum (Eisenstein and Hu, 1999). In this way for a radius of ∼ 55−60 Mpc/h of the top-hat window function, the ratio plot has the center of an oscillation around unity with an amplitude of approximately the 3% as shown in Figure 2.2 from Juszkiewicz(2013).

In our project, we planned to use a slightly different function in order to check for the presence of BAO in three-point cumulative statistics. In particular it has been considered an approximation of the derivative of S3, ∆S3 which given a set of data points for different radii of the top hat window function S3(Ri) it

is defined using the Richardson’s extrapolation formula as

∆Si₃ = −S

i+2

3 + 8 Si+13 − 8 Si−13 + Si−23

∆R , (2.81)

where ∆R = Ri+1− Ri and it is constant. This quantity has been introduced

2.5. S3 from data

Figure 2.2: Plot from Juszkiewicz (2013) showing the skewness BAO feature. The red solid line represents the volume-averaged skewness as function of top-hat scale. It shows the characteristic skewness BAO feature, a mild shoulder around the characteristic crossing scale. For comparison, the dotted line represents the skewness for a "no-wiggle" power spectrum. The inset zooms in on the shoulder at the crossing scale.

a peak as the one expected in ∆S3 than the crossing point expected in the S3 ratio.

Secondly, since it is not possible to directly probe the matter field distribution. We measure the skewness of the galaxy field Sg₃, from the mocks and the BOSS data, assuming that the galaxies act as tracers of the matter field. The bias relation linking Sg₃ with Sm

3 (Fry and Gaztanaga, 1993) is

Sg₃ = 1

b (S

m

3 + 3 c2) , (2.82)

where b and c2 are the two relevant bias parameters.

From this definition and from Equation 2.82 it is immediate to see the in- dependence of ∆S3 from the bias parameter c2. A total independence from the galaxy-matter bias is then gained by studying the ratio between ∆S3 obtained from the data and the one obtained by smoothing the data. This is analogous to the S3 ratio used byJuszkiewicz (2013) in his paper, with the advantage that

2.5. S3 from data

in this case there is theoretically no dependence from the bias

R∆S3 =

∆Sm₃ ∆Sm,sm.₃ =

∆Sg₃

∆Sg,sm.₃ , (2.83)

where "m", "g" and "sm." stand for "matter", "galaxies" and "smoothed", respec- tively.

Unfortunately, as soon as we measured δ2 _{and δ}3 _{from both data and galaxy} mocks, it was evident that the cosmic variance error component was too large for a detection of the BAO feature to be possible. We used the measurements of the 2pt and 3pt correlation functions joint data vector to constrain model parameters like the linear galaxy bias b1, the normalisation of the dark matter oscillation amplitude σ8 and the primordial non Gaussianities parameter fN L.

2.5.1

BOSS DR11 data and mocks

In this work we use the DR11 CMASS north sample of the SDSS Baryon Os- cillation Spectroscopic Survey, containing 579, 461 observed galaxies covering a total area of 6, 769 deg2. The redshift range is 0.43 < z < 0.7.

In order to compute the covariance matrix for our analysis we have used 600 galaxy mocks and 600 random mocks for DR11 created byManera et al.(2013). The mocks for the galaxies contained also the "true" redshift of the galaxies together with the observed one so that it has been possible to study the effect of the redshift-space distortions of the used statistics. The galaxy mocks have been constructed by placing dark matter halos in a 2LPT field and populating them with galaxies. The cosmology of the mocks is compatible with WMAP5-7: Ωm = 0.274, ΩΛ = 0.726, σ8 = 0.8, ns = 0.95, h = 0.7, Ωb∆h2 = 0.0224. The

galaxy mocks also have a weighting system that corrects for the incompleteness of a real data-set: wcboss for galaxies/randoms reduced by completeness Cboss,

wcp close pairs weight, wred redshift failure weight. The system consists in using

only the galaxies with all three weights > 0 and applying to each one the weight:

wtot = wcp+wred−1. Therefore the total weight for each galaxy is increased if the

closest galaxy had a redshift failure or if the redshift was not measured since both galaxies formed a close pair. The mocks have a survey geometry corresponding to the BOSS DR11.

2.5.2

Statistical estimators

We measured separately hδ3

gi and hδ2gi by randomly arranging an arbitrary num- ber nsph.of spheres inside the survey volume on a regular three-dimensional grid, and then counting the number of galaxies inside each sphere of radius R. The same is done for a contrast field catalog with approximately one hundred times

2.5. S3 from data

the density, distributing the points using a Poisson process. This allows us to define a δg(R) for the galaxy field as a function of the sphere’s radius:

δg(R) = ng(R) nr(R) F − 1 and F = n tot r ntot g , (2.84)

where F is the renormalisation factor. From now on unless specified, all quantities refer to the galaxy field. Since we are dealing with a discrete distribution it is necessary to use estimators for the continuous galaxy distribution field derived from the discrete counting. What it is actually measured from counting ng(R) and nr(R) inside the a sphere of radius R are the quantities:

k2 = 1 nsph. P ng(R) nr(R) F − 1 !2 k3 = 1 nsph. P ng(R) nr(R) F − 1 !3 . (2.85)

In order to relate these quantities with the continuous cumulative two- and three- point functions for the continuous galaxy field hδ2_{i and hδ}3_{i we need to average} the quantities in Equation 2.85 both over the galaxy Poisson sampling and the cosmic variance. Averaging over cosmic variance means to average over different realizations, in our case this is done by averaging over different mocks. For the two-point expression with same redshift but different sampling densities (the density of the points, galaxies or randoms, varies with the redshift z) we have that the second moment of the discrete density distribution hk2i is given by

hk2i = hδ2i + 1 nsph. X 1 nr(x) , (2.86)

where nr(x) is the number of random points (not clustered) inside a sphere of radius R at a position x inside the survey. Analogously the third moment of discrete density distribution hk3i is related to the continuous one by

hk3i = hδ3i + hδ2_i nsph. X 3 nr(x) + 1 nsph. X 1 nr(x)2 . (2.87)

Both expressions agree withGaztanaga(1994). Since by construction the galaxy number density depends only on the redshift, in all the previous expressions can be rewritten with nr(x) = nr(z).

2.8. RSD-corrected expressions for hδ2_{i and hδ}3_i