Post-reionization H i 21-cm signal: A probe of negative cosmological constant

In this study, we investigate a cosmological model involving a negative cosmological constant (AdS vacua in the dark energy sector). We consider a quintessence field on top of a negative cosmological constant and study its impact on cosmological evolution and structure formation. We use the power spectrum of the redshifted HI 21 cm brightness temperature maps from the post-reionization epoch as a cosmological probe. The signature of baryon acoustic oscillations (BAO) on the multipoles of the power spectrum is used to extract measurements of the angular diameter distance D A ( z ) and the Hubble parameter H ( z ). The projected errors on these are then subsequently employed to forecast the constraints on the model parameters (Ω Λ , w 0 , w a ) using Markov Chain Monte Carlo techniques. We find that a negative cosmological constant with a phantom dark energy equation of state (EoS) and a higher value of H 0 is viable from BAO distance measurements data derived from galaxy samples. We also find that BAO imprints on the 21cm power spectrum obtained from a futuristic SKA-mid like experiment yield a 1 − σ error on a negative cosmological constant and the quintessence dark energy EoS parameters to be Ω Λ = − 0 . 883 0 . 978 − 2 . 987 and w 0 = − 1 . 030 0 . 023 − 0 . 082 , w a = − 0 . 088 0 . 162 − 0 . 343 respectively, which is competitive with other probes reported in the literature.


INTRODUCTION
One of the most significant discoveries of the twenty-first century was the fact that the expansion of the Universe is accelerated (Amendola & Tsujikawa 2010).Several independent observations confirm the counter-intuitive phenomenon of dark energy (Riess et al. 1998;Perlmutter et al. 2003;McDonald & Eisenstein 2007;Scranton et al. 2003;Eisenstein et al. 2005).Observations indicate that about ∼ 64% of the universe's total energy budget is made up of dark energy, which has a large negative pressure and acts as a repulsive force against gravity (Padmanabhan 2003;Ratra & Peebles 1988).In the last few decades, cosmological observations have attained an unprecedented level of precision.The ΛCDM model Carroll (2001); Ratra & Peebles (1988); Bull (2016b) provides a good description towards explaining most properties of a wide range of astrophysical and cosmological data, including distance measurements at high redshifts (Riess et al. 1998;Perlmutter et al. 2003;Padmanabhan & Choudhury 2003), the cosmic microwave background (CMB) anisotropies power spectrum (Spergel et al. 2007), the statistical properties of large scale structures of the Universe (Bull 2016a) and the observed abundances of different types of light nuclei (Schramm & Turner 1998;Steigman 2007;Cyburt et al. 2016).All these observations point towards an accelerated expansion history of the Universe.
A positive cosmological constant is sometimes interpreted as a scalar field at the positive minimum of its potential by moving the term Λgµν to the right-hand side of the Einstein's equation to include it in the energy momentum tensor Tµν .A Quintessence (Carroll 1998;Brax & Martin 1999;Caldwell & Linder 2005;Nomura et al. 2000) scalar field, on the contrary, slowly rolls towards the minimum in the positive part of the potential giving rise to a dynamical dark energy with a time dependent equation of state w(a) = PDE/ρDE.Several reports of the Hubble tension (Di Valentino et al. 2016, 2020;Vagnozzi 2020;Alestas et al. 2020;Anchordoqui et al. 2020;Banerjee et al. 2021;Di Valentino et al. 2021;et.al. 2022) has led to the proposal of a wide range of dark energy models.There are certain proposed quintessence models with an AdS vacuum (Dutta et al. 2020;Calderón et al. 2021;Akarsu et al. 2020;Visinelli et al. 2019; Ye & Piao 2020; Yin 2022) which do not rule out the possibility of a negative Λ.We have considered Quintessence models, with a non zero vacuum, which can be effectively seen as as a rolling scalar field ϕ on top of a cosmological constant Λ ̸ = 0.The combination ρ DE = ρ ϕ + Λ satisfying the energy condition ρDE > 0 drives an accelerated expansion (Sen et al. 2023).
The possibility of 21-cm intensity mapping experiments as a precision probe of cosmology faces several observational challenges.The signal is buried is foregrounds from galactic and extragalactic sources.While the foregrounds are several orders of magnitude brighter than the 21 cm signal, its spectral properties are strikingly different from the 21-cm signal which allows for the two to be separated.Several techniques attempt to remove the foregrounds from the measured visibilities (e.g.(Paciga et al. 2011;Datta et al. 2010;Chapman et al. 2012;Mertens et al. 2018;Trott et al. 2022), by assuming the smooth nature of the foregrounds.The multi-frequency angular power spectrum (MAPS) (Datta et al. 2007) has been proposed as a tool for foreground removal by several groups (Ghosh et al. 2011;Elahi et al. 2023).Some other groups adopt a 'foreground avoidance' strategy where only the region outside the foreground wedge is used to estimate the 21-cm powerspectrum (e.g.Pober et al. (2013Pober et al. ( , 2014)); Liu et al. (2014); Dillon et al. (2015); Pal et al. (2021)) Further, one requires extremely precise bandpass, calibration for detection of the signal.Calibration introduce spectral structure into the foreground signal, making it further difficult to effectively remove foregrounds.This difficulty has led to many proposals for precise bandpass calibration (Mitchell et al. 2008;Kazemi et al. 2011;Sullivan et al. 2012;Kazemi & Yatawatta 2013;Dillon et al. 2020;Kern et al. 2020;Byrne et al. 2021;Sims et al. 2022;Ewall-Wice et al. 2022;Byrne 2023).
In this paper, we have made projections of uncertainties on the dark energy parameters in Quintessence models, with a non zero vacuum, using a proposed future observation of the power spectrum of the post-reionization 21 cm signal.We have used a Fisher/Monte-carlo analysis to indicate how the error projection on the binned power spectrum allow us to constrain dark energy models with a negative Λ.
The paper is organized as follows: In Section-2 we discuss the dark energy models and constraints of observable quantities like the Hubble parameter and growth rate of density perturbations from diverse observations.In Section-3 we discuss the 21-cm signal from the post reionization epoch and noise projections using the futuristic SKA1 -mid observations.We also constrain dark energy model parameters using Markov Chain Monte Carlo (MCMC) simulation.We discuss our results and other pertinent observational issues in the concluding section.

QUINTESSENCE DARK ENERGY WITH NON-ZERO VACUUM
We consider a Universe where the Quintessence field (ϕ) and cosmological constant Λ both contribute to the overall dark energy density i.e. ρDE = ρ ϕ + Λ with the constraint that ρDE > 0 to ensure the late time cosmic acceleration (Sen et al. 2023).Instead of working with a specific form of the Quintessence potential we chose to use a broad equation of state (EoS) parametrization w ϕ (z).It has been shown that at most a two-parameter model can be optimally constrained from observations (Linder & Huterer 2005).We use the CPL model proposed by Chevallier & Polarski (2001) and Linder (2003) which gave a phenomenological model-free parametrization and incorporate several features of dark energy.This model has been extensively used by the Dark Energy Task force (Albrecht et al. 2006) as the standard two parameter description of dark energy dynamics.It has also been shown that a wide class of quintessence scalar field models can be mapped into the CPL parametrization (Pantazis et al. 2016).The equation of state (EoS) is given by This model gives a smooth variation of w ϕ (z) = w0 + wa as z → ∞ to w ϕ (z) = w0 for z = 0 and the corresponding density of the quintessence field varies with redshift as ρ ϕ (a) ∝ a −3(1+w 0 +wa) exp 3waa .In a spatially flat Universe, evolution of the Hubble parameter H(a) is given by with Ωm0 + Ω ϕ0 + ΩΛ = 1.We shall henceforth call this model with Λ along with a scalar field as the CPL-ΛCDM model.We consider two important cosmological observables.Firstly we consider a dimensionless quantifier of cosmological distances (Eisenstein et al. 2005) where rs denotes the sound horizon at the drag epoch and D V (z) is the BAO effective distance DV (Amendola & Tsujikawa 2010) is defined as This dimension-less distance r BAO is a quantifier of the background cosmological model (density parameters) and is thereby sensitive to the dynamical evolution of dark energy.
two separate regions consistent with data: The third quadrant corresponds to Phantom models with negative Λ and the first quadrant which corresponds to non-phantom models with positive Λ.It is also clear that in spatially flat cosmologies with conditions ρm > 0 and ρ ϕ > 0 implies that ΩΛ < 1 which is not supported by data.The addition of a negative cosmological constant to a phantom dark energy model seems viable from the data.We find that the CPL-Λ CDM with a phantom field and negative Λ and H0 = 72 Km/s/Mpc, the observational data as also ΛCDM with H0 = 67.4Km/s/Mpcare all qualitatively consistent.We note that while computing rBAO, the sound horizon distance rs is fixed to the value computed for Ωm and Ω b from Aghanim et al. (2020) since rs does not change much with Ω ϕ ΩΛ.
Figure (2) shows variation of f σ8(z) in the (ΩΛ, w0) plane.The solid red line corresponds to the observational data from SDSS-III BOSS f σ8(z = 0.51) = 0.470 ± 0.041 (Sánchez et al. 2017), f σ8(z = 0.61) = 0.457 ± 0.052 (Chuang et al. 2017) and eBOSS DR16 LRGxELG data f σ8(z = 0.7) = 0.4336 ± 0.05003 (Zhao et al. 2021) respectively.While the mean observational f σ8 data falls in the non-phantom sector with negative Λ, the error bars are quite large and again, the ΛCDM predictions (with H0 = 67.4Km/s/Mpc),observed data and CPL-ΛCDM with phantom field and negative Λ for H0 = 72 Km/s/Mpc are all consistent within 1 − σ errors.The addition of a negative Λ to a phantom dark energy model seems to also push H0 to a higher value.
In models with negative cosmological constants there are regions in the (w0 − ΩΛ) which corresponds to cosmologies which never had an accelerated phase in the past or had a transient accelerated phase or H 2 (z) < 0. These regions are studied in an earlier work Calderón et al. (2021).In the range of (w0 − ΩΛ) shown in the above figures we have shaded these regions  (Sánchez et al. 2017), f σ 8 (z = 0.61) = 0.457 ± 0.052 (Chuang et al. 2017) and eBOSS DR16 LRGxELG data f σ 8 (z = 0.7) = 0.4336 ± 0.05003 (Zhao et al. 2021).The red dotted contour corresponds to f σ 8 (z) computed for a ΛCDM model.The grey sectors correspond to the models for which the Universe did not ever go through an accelerated phase till that redshift.
where the acceleration parameter became negative, corresponding to the fact that in these models, the universe did not ever accelerate upto that redshift.

THE POST-REIONIOZTION H i 21-CM SIGNAL
The epoch reionization epoch is believed to have ended around z ∼ 6 (Gallerani et al. 2006).Subsequently only a small fraction of H i survives the process of ionization and remains housed in the over-dense regions of the IGM.These neutral clumped, dense gas clouds remain neutral and shielded from background ionizing radiation.These are now believed to be the damped Lyman-α systems (DLAs) (Wolfe et al. 2005) associated with galaxies.The predominant source of the 21-cm radiation in epochs z < 6 are these DLA system which stores ∼ 80% of the H i at z < 4 Prochaska et al. (2005) with H i column density greater than 2×10 20 atoms/cm 2 (Lanzetta et al. 1995;Storrie-Lombardi et al. 1996;Peroux et al. 2003).The study of clustering of DLAs indicate their association with galaxies.These gas clumps are hence have a biased presence in regions where matter over densities are highly non-linear Cooke et al. (2006) an intensity mapping of the diffused background in all radio-observations at the observation frequencies less than 1420MHz is believed to give a wealth of cosmological and astrophysical information.Measuring the statistical properties of the fluctuations of the diffuse 21-cm intensity distribution on the plane of the sky and as a function of redshift gives a way to study cosmological structure formation tomographically.Modeling the post-reionization H i signal is based on several simplifying assumptions which are supported by extensive numerical simulations and astrophysical observations.
• Post-reionization 21-cm Spin temperature : In the post-reionization epoch the spin temperature Ts >> Tγ where Tγ is the CMB temperature.This is due to the Wouthheusen field coupling which leads to an enhanced population of the triplet state of H i .Consequently radiative transfer of CMBR through a gas cloud in this epoch shall cause the 21-cm radiation is seen in emission against the background CMBR (Madau et al. 1997;Bharadwaj & Ali 2004;Loeb & Zaldarriaga 2004).Further, the kinetic gas temperature remains strongly coupled to the Spin temperature through Lyman-α scattering or collisional coupling (Madau et al. 1997).
• Peculiar flow of H i : The theory of cosmological perturbation shows that on large sacles the baryonic matter falls into the regions of dark matter overdensities.Thus the non-Hubble H i peculiar flow of the gas is primarily determined by the dark matter distribution on large scales.The H i peculiar velocity manifests as a redshift space distortion anisotropy in the 21-cm power spectrum in a manner similar to the Kaiser effect seen in galaxy surveys (Hamilton 1998).
• Intensity mapping and noise due to discrete clouds: The source of the 21-cm signal are DLA clouds.Intensity mapping ignores the discrete nature of the sources and aims to map the smoothed diffuse intensity distribution (Furlanetto et al. 2006;Pritchard & Loeb 2012;Bull et al. 2015).The discreteness of the source will introduce a Poisson sampling noise.We neglect this noise in our modeling since the number density n of the DLA sources is very large (Bharadwaj & Srikant 2004) and the Poisson noise typically goes as 1/n.
• Gaussian fluctuations: The overdensity field of dark matter distribution is believed to be generated by Gaussian process in the very early Universe leading to a scale invariant primordial power spectrum.We assume that there are no non-gaussianities, whereby the statistics of the random overdensity field is completely exhausted by studying the two-point correlation/power-spectrum.All p-point correlation functions where p is odd, are assumed to be zero in the first approximation.Primordial non-gaussianity and nonlinear structure formation will make the field non-gaussian, but this is neglected as a first approximation.The gas is believed to follow the dark matter and also expected to not show any non-gaussian effects.
• Post-reionization H i as a biased tracer: The distribution of baryonic matter in the form of neutral hydrogen is an unsolved problem in cosmology.The linear theory predictions indicate that on large scales, baryonic matter follows the underlying dark matter distribution.However, at low redshifts, the growth of density fluctuations is likely to be plagued by non-linearilites and it is not apriori meaningful to extrapolate the predictions of linear theory in this epoch where overdensities δ ∼ 1. Galaxy redshift surveys show that the galaxies trace the underlying dark matter distribution (Dekel & Lahav 1999;Mo et al. 1996;Yoshikawa et al. 2001) with a bias.If we model the post-reionization H i to be primarily stored in dark matter haloes, it is plausible to expect that the gas to trace the underlying dark matter density field with a possible bias as well.
We define a bias function bT (k, z) as where PHI(k, z) and Pm(k, z) denote the H i and dark matter power spectra respectively.With this definition of a general function bT (k, z), we merely relocate the lack of knowledge of H i distribution to a scale and redshift dependent function that quantifies the properties of post-reionization H i clustering.
Theoretical considerations show that the bias is scale dependent on small scales below the Jean's length (Fang et al. 1993).However, on large scales the bias is expected to be scale-independent.The scales above which the linear bias approximation is acceptable is however, dependent on the redshift.While the neutral fraction on the post-reionization epoch is believed to be a constant, studies (Wyithe & Loeb 2009) show that small fluctuations in the ionizing background may also contribute to a scale dependency in the bias bT (k, z).The most compelling studies of the post-reionization H i has been through the use of N-body numerical simulations (Bagla et al. 2010;Guha Sarkar et al. 2012;Sarkar et al. 2016;Carucci et al. 2017).These simulations uses diverse rules for populating neutral hydrogen to dark matter halos in a certain mass range and identifying them as DLAs.
Similar to the behaviour of galaxy bias (Fry 1996;Dekel & Lahav 1999;Mo et al. 1996Mo et al. , 1999)), these N-body simulations of the post-reionization H i agree on the generic qualitative behaviour.On large scale the bias is found to be linear (scale independent) and is a monotonically rising function of redshift for 1 < z < 4 (Marín et al. 2010).However, on small scales the bias becomes scale-dependent as rises steeply on small scales.The rise of the bias on small scales owes it origin to the absence of small mass halos as is expected from the CDM power spectrum and consequent distribution of H i in larger mass halos.In this work we use the fitting formula for bT (k, z) obtained from numerical simulations (Sarkar et al. 2016).

The post-reionization H i 21cm power spectrum
Adopting all the modeling assumptions discussed in the last section, the power spectrum of post-reionization H i 21-cm excess brightness temperature field δT b from redshift z (Furlanetto et al. 2006;Bull et al. 2015;Bharadwaj & Ali 2004;Bharadwaj et al. 2009) is given by where The term f (z)µ 2 has its origin in the H i peculiar velocities (Bharadwaj et al. 2001;Bharadwaj & Ali 2004) which, is also assumed to be sourced by the dark matter fluctuations.Since our cosmological model is significantly different from the fiducial one (i.e., ΛCDM), the difference will introduce additional anisotropies in the correlation function through the Alcock-Paczynski effect (Simpson & Peacock 2010;Samushia et al. 2012;Montanari & Durrer 2012).In the presence of the Alcock-Paczynski effect, the redshift-space HI 21-cm power spectrum is given by: (Furlanetto et al. 2006;Bull et al. 2015) where F = α ∥ /α ⊥ , with α ∥ and α ⊥ being the ratios of angular and radial distances between fiducial and real cosmologies, ⊥ is due to the scaling of the survey's physical volume.As the real geometry of the Universe differs from the one predicted by the fiducial cosmology, we introduce additional distortion in the redshift space.The AP test is sensitive to the isotropy of the Universe and can help differentiate between different cosmological models.We note that the geometric factors shall also imprint in the BAO feature of the power spectrum.Since 0 ⩽ µ1 the redshift space 21cm power spectrum can be decomposed in the basis of Legendre polynomials P ℓ (µ) as (Hamilton 1998) The odd harmonics vanish by pair exchange symmetry and non-zero azimuthal harmonics.( as Y ℓ,m 's with m ̸ = 0 vanish by symmetry about the line of sight).Using the standard normalization the first few Legendre polynomials are given by The coefficients of the expansion of the 21cm power spectrum, can be found by inverting the equation ( 8).Thus we have While full information is contained in an infinite set of functions {P ℓ (z, k)}, we shall be interested in the first few of these function which has the dominant information.

3.2
The BAO feature in the multipoles of 21-cm power spectrum The sound horizon at the drag epoch provides a standard ruler, which can be used to calibrate cosmological distances.Baryons imprint the cosmological power spectrum through a distinct oscillatory signature (White 2005;Eisenstein & Hu 1998).The BAO imprint on the 21-cm signal has been studied extensively (Sarkar & Bharadwaj 2013, 2011).The baryon acoustic oscillation (BAO) is an important probe of cosmology (Eisenstein et al. 2005;Percival et al. 2007;Anderson et al. 2012;Shoji et al. 2009;Sarkar & Bharadwaj 2013) as it allows us to measure the angular diameter distance DA(z) and the Hubble parameter H(z) using the transverse and the longitudinal oscillatory features respectively (Lopez-Corredoira 2014).
The sound horizon at the drag epoch is given by where a drag is the scale factor at the drag epoch redshift z d and cs is the sound speed given by cs(a) = c/ 3(1 + 3ρ b /4ργ) where ρ b and ργ denotes the baryonic and photon densities, respectively.The Planck 2018 constrains the value of z d and s(z d ) to be z d = 1060.01± 0.29 and s(z d ) = 147.21± 0.23Mpc (Aghanim et al. 2020).We shall use these as the fiducial values in our subsequent analysis.The standard ruler 's' defines a transverse angular scale and a redshift interval in the radial direction as Measurement of θs and δzs, allows the independent determination of DA(z) and H(z).The BAO feature comes from the baryonic part of P (k).In order to isolate the BAO feature, we subtract the cold dark matter power spectrum from total P (k) as P b (k) = P (k) − Pc(k).Owing to significant deviations between the assumed cosmology and the fiducial cosmology, our longitudinal and tangential coordinates are by α ∥ and α ⊥ respectively, the true power spectrum scaled as (Matsubara & Suto 1996;Ballinger et al. 1996;Simpson & Peacock 2010).
Incorporating the Alcock-Paczynski corrections explicitly in the BAO power spectrum can be written as (Hu & Sugiyama 1996;Seo & Eisenstein 2007) where A is a normalization, s = 1/k silk and s = 1/k nl denotes the inverse scale of 'Silk-damping' and 'non-linearity' respectively.In our analysis we have used The changes in DA(z) and H(z) are reflected as changes in the values of s ⊥ and s ∥ respectively, and the errors in s ⊥ and s ∥ corresponds to fractional errors in DA and H(z) respectively.We use p1 = ln(s −1 ⊥ ) and p2 = ln(s ∥ ) as parameters in our analysis.The Fisher matrix is given by We choose SKA's a Medium-Deep Band-2 survey that covers a sky area of 5,000 deg 2 in the frequency range 0.95−1.75GHz(z = [0 − 0.5]) and a Wide Band-1 survey that covers a sky area of 20,000 deg 2 in the frequency range 0.35 − 1.05GHz (z = [0.35− 3]) (Bacon et al. 2020).We calculate the expected error projections on DA(z) and H(z) in five evenly spaced, non-overlapping redshift bins, in the redshift range [z=0-3] with ∆z = 0.5.Each of the six bins is taken to be independent and is centered at redshifts of z = [0.25,0.75, 1.25, 1.75, 2.25].

Visibility correlation
We use a visibility correlation approach to estimate the noise power spectrum for the 21-cm signal (Bharadwaj & Sethi 2001;Bharadwaj & Ali 2005;McQuinn et al. 2006;Geil et al. 2011;Villaescusa-Navarro et al. 2014;Sarkar & Datta 2015).A radiointerferometric observation measures the complex visibility.The measured visibility written as a function of baseline U = (u, v) and frequency ν is a sum of signal and noise where, δT b ( ⃗ θ, ν) is the fluctuations of the 21-cm brightness temperature and A( ⃗ θ) is the telescope beam.The factor 2k B λ 2 2 converts brightness temperature to intensity (Raleigh Jeans limit).Defining ∆ν as the difference from the central frequency, a further Fourier transform in frequency ∆ν gives us where B(∆ν) is the frequency response function of the radio telescope.
Performing the ⃗ θ and ∆ν integral we have Defining new integration variables as Approximately, we may write where B is the bandwidth of the telescope and where Ae is the effective area of each dish.Hence The noise in the visibilities measured at different baselines and frequency channels are uncorrelated.We then have where where Ae is the effective area of the dishes, t is the correlator integration time and ∆ν is the channel width.If B is the observing bandwidth, there would be B/∆ν channels.The system temperature Tsys can be written as where Now cosidering a total observation time To and a bin ∆U, there is a reduction of noise by a factor √ N p where Np is the number of visibility pairs in the bin where Nvis is the number of visibilities in the bin.We may write where Nant is the total number of antennas and ρ(U) is the baseline distribution function.
where an additional reduction by √ 2 is incorporated by considering visibilities in half plane.The 21 cm power spectrum is not spherically symmetric, due to redshift space distortion but is symmetric around the polar angle ϕ.Because of this symmetry, we want to sum all the Fourier cells in an annulus of constant (k, µ = cos θ = k ∥ /k) with radial width ∆k and angular width ∆θ for a statistical detection.The number of independent cells in such an annulus is where  Thus the full covariance matrix for visibility correlation is (Villaescusa-Navarro et al. 2014;Sarkar & Datta 2015;Geil et al. 2011;McQuinn et al. 2006) We choose δ 2 U = Ae/λ 2 , ∆k = k/10, ∆µ = µ/10.The baseline distribution function ρ(U) is normalized as For uniform baseline distribution Generally Where c is fixed by normalization of ρ(U) and ρant is the distribution of antennae.The covariance matrix in Eq (37) is used in our analysis to make noise projections on the 21-cm power spectrum and its multipoles.Observations with total time time exceeding a limiting value will make the instrumental noise insignificant and the Signal to Noise Ratio is primarily influenced by cosmic variance for such observations.Therefore, by introducing Npoint as the number of independent pointings, the covariance is further reduced by a factor of 1/ Npoint.
We consider 250 dish antennae each of diameter 15m and efficiency 0.7.We assume Tsys = 60K and an observation bandwidth of 128MHz.The k-range between the smallest and largest baselines is binned as ∆k = αk where α = 1/N bin ln(Umax/Umin).The minimum value of k is taken to be 0.005Mpc −1 the maximum value of k is taken to be 0.5Mpc −1 with logarithmically number of bins N bin = 8.We consider a total observation time of 500 × 150hrs with 150 independent pointings, we obtain the 1 − σ errors on P ℓ (k, z).The fiducial model is chosen to be the ΛCDM. Figure (5) shows the multiples of P21(k, z) for selective parameter values of CPL-ΛCDM model.The central dotted line corresponds to ΛCDM.The fiducial redshift is chosen to be 0.2 (top) and 0.57 (bottom).We found that in the k range 0.01Mpc −1 < k < 0.1Mpc −1 phantom models are distinguishable from ΛCDM at a sensitivity of > 3σ.For higher multipoles, they are even more differentiable from fiducial ΛCDM.On the contrary, non-phantom models remain statistically indistinguishable from the ΛCDM model while considering monopole only.They are only distinguishable in higher multipoles.
We see a strong effect of ΩΛ on the multipole components of the power spectrum.A non-trivial ΩΛ introduces additional enhancement of anisotropy in the 21-cm power spectrum through the redshift space distortion factor f µ 2 .Additionally the power spectrum gets further modified through the departure of the factor F = α ∥ /α ⊥ from unity and through the matter power spectrum P (k, µ) though the scalings of k ⊥ and k ∥ .This explains the significant deviation of the 21-cm power spectrum for the CPL-ΛCDM model from its standard ΛCDM counterpart.This is become more prominent in the quadrupole and hexadecapole components cause of the terms with the anisotropy are enhanced by integrals of higher powers of µ in the Legendre polynomials.Table 2.The parameter values, obtained in the MCMC analysis are tabulated along the 1 − σ uncertainty.
The BAO imprint on the monopole P0(z, k) allows us to constrain DA(z) and H(z).We perform a Markov Chain Monte Carlo (MCMC) analysis to constrain the model parameters using the projected error constraints obtained on the binned H(z) and DA(z) from the P0(z, k).  2).While comparing with the projected error limits for the parameters of the CPL-ΛCDM as obtained in Sen et al. ( 2023), we find that 21-cm alone doesn't impose stringent constraints on the values of ΩΛ and wa.However, it does exhibit a reasonably good ability to constrain the parameter w0.To attain more robust constraints on these model parameters, a more comprehensive approach is required.This involves combining the 21-cm power spectrum data with other cosmological observations such as the CMB, BAO, SNIa, galaxy surveys etc.Through the joint analysis, it becomes possible to significantly improve the precision of parameter estimation.

CONCLUSION
In this work, we study the possibility of constraining negative Λ using the post-reionization H i 21-cm power spectrum.We specifically investigate the quintessence models with the most widely used dark energy EoS parameterization and add a non-zero vacua (in terms of a ±Λ).
By the analysis of BOSS (SDSS) data we find that addition of a negative cosmological constant to a phantom dark energy model seems viable.We see that the CPL-ΛCDM with a phantom field and negative Λ and H0 = 72 Km/s/Mpc qualitatively consistent with the data.
Further we study the non-trivial CPL-ΛCDM model with the f σ8 data from the galaxy surveys.We find that the mean observational f σ8 falls in the non-phantom sector with negative Λ. Since the error bars are quite large, both ΛCDM predictions (with H0 = 67.4Km/s/Mpc),and CPL-ΛCDM with phantom field and negative Λ for H0 = 72 Km/s/Mpc are consistent within 1 − σ errors.The addition of a negative Λ to a phantom dark energy model also seems to push H0 to a higher value.
Subsequently, we look into the influence of the Alcock-Packzynski effect on 3D H i 21-cm power spectrum.Using ΛCDM as a fiducial cosmology, we explore the implications of the first few multipoles of the redshift-space 21-cm power spectrum for the upcoming SKA intensity mapping experiments.We find that the multipoles specially the quadrupole and hexadecapole components show significant departure from their standard ΛCDM counterparts.We focus on the BAO feature on the monopole component, and estimate the projected errors on the H(z) and DA(z) over a redshift range z ∼ 0 − 3.
Further, we perform a MCMC analysis to constrain the CPL-ΛCDM model parameters using the projected error constraints obtained on the binned H(z) and DA(z) from the P0(z, k).We find that 21-cm alone doesn't impose stringent constraints on the model parameters.Combining the 21-cm power spectrum data with other cosmological observations such as the CMB, BAO, SNIa, galaxy surveys etc can significantly improve the precision of parameter estimation.
We have not factored in several observational challenges towards detecting the 21-cm signal.Proper mitigation of large galactic and extra-galactic foregrounds and minimizing calibration errors are imperative for the any cosmological investigation.In a largely observationally idealized scenario, we have obtained error projections on the model parameters from the BAO imprint on the post-reionization 21-cm intensity maps.We employ a Bayesian analysis techniques to put constraints on the model parameters.Precision measurement of these parameters shall enhance our understanding of the underlying cosmological dynamics and potential implications of negative Λ values.

Figure 1 .
Figure1.shows r BAO in the (Ω Λ , w 0 ) plane.The red contour line corresponds to the observational data point and the blue shaded region depicts the 1σ errors.The data points in the left two figures come from the 2df galaxy survey at redshifts of z = 0.2 and z = 0.35 respectively(Percival et al. 2007) and the third figure shows the high redshift data at z = 0.57 from BOSS SDSS-III survey(Anderson et al. 2012).The red dotted contour correspond to r BAO computed for a ΛCDM model.The grey sectors correspond to the models for which the Universe did not ever go through an accelerated phase till that redshift.
; Zwaan et al. (2005); Nagamine et al. (2007).The possibility of the presently functioning and upcoming radio telescopes to detect the cosmological 21-cm signal from low redshifts has led to an extensive literature on the post-reionization H i signal (Subramanian & Padmanabhan 1993; Visbal et al. 2009; Bharadwaj & Sethi 2001; Bharadwaj et al. 2001; Bharadwaj & Pandey 2003; Bharadwaj & Srikant 2004; Wyithe & Loeb 2009).Though flux from individual DLA clouds is extremely weak (< 10µJy) to be detected in radio observations, even with the next generation radio arrays, it is possible to detect the collective diffuse radiation without requiring to resolve the individual sources.Such

Figure 3 .
Figure 3. shows the 3D H i 21-cm power spectrum at z = 1 in the (k ⊥ , k ∥ ) space.The asymmetry in the signal is indicative of redshift space distortion: the left figure corresponds to the ΛCDM.In contrast, the right figure represents the CPL-ΛCDM model, where the Alcock-Paczynski effect enhanced the distortions.The colorbar shows the value of the dimensionless quantity ∆ 2 21 = k 3 P 21 (k)/(2π 2 ) in mK 2 .

Table 1 .
Table showing the telescope parameters used in our analysis.