4.4 Fitting the cross-correlation
4.4.2 Model
In this chapter we measure the cross-correlation between DLAs ant the Lyα forest for a number of datasets (see section 4.2 on page 70). To interpret and to compare these measurements, we use a single model which is described in this section. To do the actual fitting we use the publicly available fitting code baofit (Kirkby et al. 2013). The code has been adapted to include the distortion matrix formalism by Michael Blomqvist and collaborators.
In the limit of large scales the linear theory predicts the shape of the cross-correlations of any two pair of tracers of the LSS. The limit of large scales is broadly defined as the distances at which non-linearities are not important any more. The transition between non-linear and linear scales is not clear and it depends on the type of tracers and the environmental conditions they are subject to. This issue is further discussed in appendix 4.B on page 95.
In Fourier space, linear theory predicts that redshift space distortions of a biased tracer enhance the amplitudes of each Fourier mode by a factor b 1 + βµ2k
, where b is the bias factor of the tracer, β its redshift space distortion parameter, and µkthe cosine of the angle between the Fourier mode and the line of sight. For the DLA-Lyα cross-correlation, the associated power spectrum is then equal to
Pdα
~k, z= bd 1 + βdµ2
k bα 1 + βαµ2k PL(k, z) , (4.29) where the subscript d stands for DLA, the subscript α for Lyα, and PL(k, z) is the linear matter
4. DLA-Lyα forest cross-correlations
in previous studies with BOSS data (e.g. Font-Ribera et al. 2014; Delubac et al. 2015; Blomqvist et al. 2015) nor the MTC correction used in Font-Ribera et al. (2012).The correction is already implemented in the projection of the δ field (see section 4.3 on page 72).
As seen in equation 4.29, we are not sensitive to variations on bdnor bαbut to the product of both. Also, statistics on DLAs are lower than that on the Lyα forest. This makes it difficult to use the DLA-Lyα cross-correlation to provide competitive measurements for the bias and redshift space distortion parameter for both tracers. However, if the parameters of either DLAs or the Lyα forest are independently constrained, then we can give competitive constraints for the other tracer.
Previous analysis of the BOSS Collaboration (see e.g. Blomqvist et al. 2015; Delubac et al. 2015) have studied in detail the Lyα forest autocorrelation and can give such constraints. In particular, for this chapter we will consider both the values for the bias and redshift space distortion parameter for the Lyα forest, and the BAO parameters as those specified in Blomqvist et al. (2015), i.e., we fix βα = 1.39 and bα(1 + βα) = −0.374 at reference redshift zref = 2.3. Also, we fix
βdbd= f (Ω) = 0.968897. To account for the fact that the mean redshift of different bins are not
exactly at zref, we assume the bias factor of the Lyα forest to evolve with redshift as (1 + z) 2.9
, as suggested in McDonald et al. (2006), and the redshift space distortion parameter for the Lyα forest to be constant with redshift. The bias factor of the DLAs is assumed to be constant in redshift as far as the fitting is concerned. The evolution of this parameter will be analysed via the sub-samples Z1, Z2, and Z3.
Summing up, we fix all parameters in equation 4.29 on page 79 but bd. Additionally we allow the symmetry in the cross-correlation to be shifted constant amount ∆v, which is a second parameter of the fit, to account for a possible error in the redshift determination of either the DLAs or the Lyα forest. All fits will be made regarding these specifications unless otherwise noted.
4.5 Results
We have measured the cross-correlation for the all the samples listed in table 4.1 on page 72 with δπ = δσ = 2 h−1Mpc and πmax= σmax = 80 h−1Mpc. The measured biases are summarized in table 4.2 on the next page. The results from the different datasets are organized throughout this section as follows: in section 4.5.1 on page 82 we present the results for the overall sample (sample A, see section 4.2 on page 70) and compare them with the results from samples C1 and C2, in sections 4.5.2 and 4.5.3 on page 85 and on page 86 we explore the redshift dependence and the column density dependence of the parameters using sub-samples Z1 to Z3 and N1 ro N3 respectively (see table 4.1 on page 72 and section 4.2 on page 70). Finally in sections 4.5.4 and 4.5.5 on page 86 and on page 92 we use dataset A to explore the scale dependence of the bias factor and the dependence on the Lyα bias respectively.
All our fits exclude bins with r = π2+ σ21/2 < 5 h−1Mpc or r > 90 h−1Mpc even though the points are shown in the plots. The lower limit on r is placed to avoid including non-linearities present at small scales into the fit. Even though we may already be entering the
4.5. Results Dataset bd χ2(d.o.f.) A 1.87± 0.05 2965.41 (2896-2) C1 1.86± 0.04 3115.52 (2896-2) C2 1.82± 0.04 2974.40 (2896-2) Z1 1.98± 0.09 2918.09 (2896-2) Z2 1.80± 0.09 3015.57 (2896-2) Z3 1.80± 0.10 3074.20 (2896-2) N1 1.91± 0.10 3006.52 (2896-2) N2 1.95± 0.09 3092.98 (2896-2) N3 1.74± 0.09 3004.70 (2896-2)
Table 4.2: Summary of the different DLA datasets. The values of the bias are given at the reference redshift zref = 2.3and for βα= 1.39. See table 4.1 on page 72 for a summary on
the datasets’ properties.
non-linear regime at 5 h−1Mpc, the deviations from the linear theory are small if present at all. The choice of this limit allows for a more direct comparison with the results previously reported by Font-Ribera et al. (2012). This limit is further discussed in appendix 4.B on page 95. The upper limit on r is placed to make sure we stay clear of the BAO peak.
For display purposes alone, in most of the plots the data-points have been averaged in bigger bins. The average is computed as
ξB= P A∈BξA/ √ CAA P A∈B1/ √ CAA , (4.30) πB= P A∈BπA/ √ CAA P A∈B1/ √ CAA , (4.31) 1 CB1B2 = X A1∈B1 X A2∈B2 1 CA1A2 , (4.32)
where the indexes A, A1, and A2stand for the measured bins, the indexes B, B1, and B2stand for the new bins, and the sumPA∈Bis over all bins A that are included in bin B. In this chapter,
as far as the figures are concerned, we will use 11 bins in σ which will be delimited by (0, 4, 8, 12, 16, 20, 28, 32, 40, 48, 64, 80), and 30 bins in π, delimited by (-80, -72, -64, -56, -48, -40, -36, -32, -28, -24, -20, -16, -12, -8, -4, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 48, 56, 64, 72, 80) with average values
computed as described above unless otherwise noted. Note that all delimitations are expressed in h−1Mpc.
4. DLA-Lyα forest cross-correlations