Observations of the neutral Hydrogen (HI ) 21-cm signal hold the potential of allowing us to map out the cosmological large scale structures (LSS) across the entire postreionization era (z 6). Several experiments are planned to map the LSS over a large range of redshifts and angular scales, many of these targeting the Baryon Acoustic Oscillations. It is important to model the HI distribution in order to correctly predict the expected signal, and more so to correctly interpret the results after the signal is detected. In this paper we have carried out semi-numerical simulations to model the HI distribution and study the HI power spectrum P HI (k, z) across the redshift range 1 z 6. We have modelled the HI bias as a complex quantityb(k, z) whose modulus squared b 2 (k, z) relates P HI (k, z) to the matter power spectrum P (k, z), and whose real part b r (k, z) quantifies the cross-correlation between the HI and the matter distribution. We study the z and k dependence of the bias, and present polynomial fits which can be used to predict the bias across 0 z 6 and 0.01 k 10 Mpc −1 . We also present results for the stochasticity r = b r /b which is important for crosscorrelation studies.

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The anisotropy of the redshift space bispectrum contains a wealth of cosmological information. This anisotropy depends on the orientation of three vectors k 1 , k 2 , k 3 with respect to the line of sight. Here we have decomposed the redshift space bispectrum in spherical harmonics which completely quantify this anisotropy. To illustrate this we consider linear redshift space distortion of the bispectrum arising from primordial non-Gaussianity. In the plane parallel approximation only the first four even ℓ multipoles have non-zero values, and we present explicit analytical expressions for all the non-zero multipoles i.e. upto ℓ = 6, m = 4. The ratio of the different multipole moments to the real space bispectrum depends only on β 1 the linear redshift distortion parameter and the shape of the triangle. Considering triangles of all possible shapes, we have studied how this ratio depends on the shape of the triangle for β 1 = 1. We have also studied the β 1 dependence for some of the extreme triangle shapes. If measured in future, these multipole moments hold the potential of constraining β 1 . The results presented here are also important if one wishes to constrain f NL using redshift surveys.

The post-reionization HI 21-cm signal is an excellent candidate for precision cosmology, this however requires accurate modelling of the expected signal. Sarkar et al. (2016) have simulated the real space HI 21-cm signal, and have modelled the HI power spectrum as P HI (k) = b 2 P (k) where P (k) is the dark matter power spectrum and b(k) is a (possibly complex) scale dependent bias for which fitting formulas have been provided. This paper extends these simulations to incorporate redshift space distortion and predict the expected redshift space HI 21-cm power spectrum P s HI (k ⊥ , k ) using two different prescriptions for the HI distributions and peculiar velocities. We model P s HI (k ⊥ , k ) assuming that it is the product of P HI (k) = b 2 P (k) with a Kaiser enhancement term and a Finger of God (FoG) damping which has σ p the pair velocity dispersion as a free parameter. Considering several possibilities for the bias and the damping profile, we find that the models with a scale dependent bias and a Lorentzian damping profile best fit the simulated P

Measurement of fluctuations in diffuse Hi 21 cm background radiation from the post-reionization epoch (z ≤ 6) is a promising avenue to probe the large-scale structure of the universe and understand the evolution of galaxies. We observe the European Large Area ISO Survey-North 1 (ELAIS-N1) field at 300–500 MHz using the upgraded Giant Meterwave Radio Telescope (uGMRT) and employ the “foreground avoidance” technique to estimate the Hi 21 cm power spectrum in the redshift range z = 1.96–3.58. Given the possible systematics that may remain in the data, we find the most stringent upper limits on the spherically averaged 21 cm power spectra at k ∼ 1.0 Mpc−1 are (58.87 mK)2, (61.49 mK)2, (60.89 mK)2, and (105.85 mK)2 at z = 1.96, 2.19, 2.62, and 3.58, respectively. We use this to constrain the product of neutral Hi mass density (ΩHI) and Hi bias (b HI) to the underlying dark matter density field, [ΩHI b HI], as 0.09, 0.11, 0.12, and 0.24 at z = 1.96, 2.19, 2.62, and 3.58, respectively. To the best of our knowledge these are the first limits on the Hi 21 cm power spectra at the redshift range z = 1.96–3.58 and would play a significant role to constrain the models of galaxy formation and evolution.