Lyα absorption in the nearby Universe: the sightline to Q1821+643

We present the first results of a survey designed to understand the origin of Lyα-forest absorption systems at low redshift. Using the WYFFOS and HYDRA multifibre spectrographs on the William Herschel and Wisconsin Indiana Yale NOAO (WIYN) telescopes, we have identified 51 galaxies brighter than b _j := :18.5 within 30 arcmin of the sightline of the QSO 1821+643. We find three galaxies within 500 h ⁻¹ kpc of the QSO sightline; the nearest galaxy is 104 h ⁻¹ kpc away from the line of sight, and is at the same redshift as a strong ( W _r :=:0.63Å) Lyα absorption line. The remaining two galaxies have no corresponding absorption to extremely low equivalent width limits (< 0.05 Å). Beyond 500 h ⁻¹ kpc, Lyα absorption lines are found at redshifts similar to those of several galaxies, but we show that these coincidences are likely to be accidental.   Half of the Lyα systems for which we could have found at least an L ^* galaxy have no galaxies at the redshifts of the absorbers. For the majority of the remainder, we show that any apparent association with galaxies is probably coincidental. These Lyα systems are characterized by their weak equivalent widths ( W _r :<:0.2Å), and we conclude that this population of absorbers is uncorrelated, or at best weakly correlated, with galaxies.     

The relation between Lyman α absorbers and gas-rich galaxies in the local Universe

We use high-resolution hydrodynamical simulations to investigate the spatial correlation between weak  ( N _{H  i} < 10¹⁵ cm⁻²)  Lyα absorbers and gas-rich galaxies in the local Universe. We confirm that Lyα absorbers are preferentially expected near gas-rich galaxies and that the degree of correlation increases with the column density of the absorber. The real-space galaxy auto-correlation is stronger than the cross-correlation (correlation lengths   r _0,gg= 3.1 ± 0.1 Mpc  h ⁻¹  and   r _0,ag= 1.4 ± 0.1 Mpc  h ⁻¹  , respectively), in contrast with the recent results of Ryan-Weber, and the auto-correlation of absorbers is very weak. These results are robust to the presence of strong galactic winds in the hydrodynamical simulations. In redshift space, a further mismatch arises since at small separations the distortion pattern of the simulated galaxy–absorber cross-correlation function is different from the one measured by Ryan-Weber. However, when sampling the intergalactic medium along a limited number of lines-of-sight, as in the real data, uncertainties in the cross-correlation estimates are large enough to account for these discrepancies. Our analysis suggests that the statistical significance of difference between the cross-correlation and auto-correlation signal in current data sets is ∼1σ only.     

Damped Lyα systems at high redshift and models of protogalactic discs

We employ observationally determined intrinsic velocity widths and column densities of damped Lyman alpha (Lyα) systems at high redshift to investigate the distribution of baryons in protogalaxies within the context of a standard cold dark matter (CDM) model. We proceed under the assumption that damped Lyα systems represent a population of cold, rotationally supported, protogalactic discs, and that the abundance of dark matter haloes is well approximated by a CDM model with critical density and vanishing cosmological constant. Using conditional cross-sections to observe a damped system with a given velocity width and column density, we compare observationally inferred velocity width and column density distributions to the corresponding theoretically determined distributions for a variety of disc parameters and CDM normalizations. In general, we find that the observations cannot be reproduced by the models for most disc parameters and CDM normalizations. Whereas the column density distribution favours small discs with large neutral gas fraction, the velocity width distribution favours large and thick discs with small neutral gas fraction. The possible resolutions of this problem in the context of this CDM model may be (1) an increased contribution of rapidly rotating discs within massive dark matter haloes to damped Lyα absorption, or (2) the abandoning of simple disc models within this CDM model for damped Lyα systems at high redshift. Here the first possibility may be achieved by supposing that damped Lyα system formation occurs only in haloes with fairly large circular velocities, and the second possibility may result from a large contribution of mergers and double discs to damped Lyα absorption at high redshift.     

The properties of Lyα emitting galaxies in hierarchical galaxy formation models

We present detailed predictions for the properties of Lyα-emitting galaxies in the framework of the Λ cold dark matter cosmology, calculated using the semi-analytical galaxy formation model galform . We explore a model that assumes a top-heavy initial mass function in starbursts and that has previously been shown to explain the sub-millimetre number counts and the luminosity function of Lyman-break galaxies at high redshift. We show that this model, with the simple assumption that a fixed fraction of Lyα photons escape from each galaxy, is remarkably successful at explaining the observed luminosity function of Lyα emitters (LAEs) over the redshift range  3 < z < 6.6  . We also examine the distribution of Lyα equivalent widths and the broad-band continuum magnitudes of emitters, which are in good agreement with the available observations. We look more deeply into the nature of LAEs, presenting predictions for fundamental properties such as the stellar mass and radius of the emitting galaxy and the mass of the host dark matter halo. The model predicts that the clustering of LAEs at high redshifts should be strongly biased relative to the dark matter, in agreement with observational estimates. We also present predictions for the luminosity function of LAEs at   z > 7  , a redshift range that is starting to be be probed by near-infrared surveys and using new instruments such as the Dark Ages Z Lyman Explorer (DAzLE).     

The polarization of scattered Lyα radiation around high-redshift galaxies

The high-redshift Universe contains luminous Lyα emitting sources such as galaxies and quasars. The emitted Lyα radiation is often scattered by surrounding neutral hydrogen atoms. We show that the scattered Lyα radiation obtains a high level of polarization for a wide range of likely environments of high-redshift galaxies. For example, the backscattered Lyα flux observed from galaxies surrounded by a superwind-driven outflow may reach a fractional polarization as high as ∼40 per cent. Equal levels of polarization may be observed from neutral collapsing protogalaxies. Resonant scattering in the diffuse intergalactic medium typically results in a lower polarization amplitude (≲7 per cent), which depends on the flux of the ionizing background. Spectral polarimetry can differentiate between Lyα scattering off infalling gas and outflowing gas; for an outflow, the polarization should increase towards longer wavelengths while for infall the opposite is true. Our numerical results suggest that Lyα polarimetry is feasible with existing instruments, and may provide a new diagnostic of the distribution and kinematics of neutral hydrogen around high-redshift galaxies. Moreover, polarimetry may help suppress infrared lines originating in the Earth's atmosphere, and thus improve the sensitivity of ground-based observations to high-redshift Lyα emitting galaxies outside the currently available redshift windows.     

Damped Lyα absorbers from dwarf galaxy ejecta

It is argued that the formation of a dwarf galaxy causes a massive burst of star formation, resulting in the ejection of most of the available gas from the galaxy as a weakly collimated wind. The ejected gas can give rise to a damped Lyα absorber (DLA). Weakly collimated outflows naturally explain the asymmetric profiles seen in low-ionization absorption lines caused by heavy elements associated with DLAs, where absorption is strongest at one edge of the absorption feature. The shape of the distribution of column densities in the model agrees reasonably well with observations. In particular, the break in slope is caused by external photoionization of the wind. A semi-analytical model for galaxy formation is used to show that, for currently acceptable cosmological parameters, dwarf galaxy outflows can account for the majority of DLA systems and their distribution with redshift. This model also predicts a correlation between velocity structure and metallicity of DLA systems, in qualitative agreement with observations. DLAs do not require many large, rapidly rotating disc galaxies to have formed early on, as in other models for their origin.     

ZEN2: a narrow J-band search for z∼ 9 Lyα emitting galaxies directed towards three lensing clusters

We present the results of a continuing survey to detect Lyα emitting galaxies at redshifts   z ∼ 9  : the ' z equals nine' (ZEN) survey. We have obtained deep VLT Infrared Spectrometer and Array Camera observations in the narrow J -band filter NB119 directed towards three massive lensing clusters: Abell clusters 1689, 1835 and 114. The foreground clusters provide a magnified view of the distant Universe and permit a sensitive test for the presence of very high redshift galaxies. We search for   z ∼ 9 Lyα  emitting galaxies displaying a significant narrow-band excess relative to accompanying J -band observations that remain undetected in Hubble Space Telescope ( HST )/Advanced Camera for Surveys (ACS) optical images of each field. No sources consistent with this criterion are detected above the unlensed 90 per cent point-source flux limit of the narrow-band image,   F _NB= 3.7 × 10⁻¹⁸ erg s⁻¹ cm⁻²  . To date, the total coverage of the ZEN survey has sampled a volume at   z ∼ 9  of approximately 1700 comoving Mpc³ to a Lyα emission luminosity of  10⁴³ erg s⁻¹  . We conclude by considering the prospects for detecting   z ∼ 9 Lyα  emitting galaxies in light of both observed galaxy properties at   z < 7  and simulated populations at   z > 7  .     

Spatial fluctuations in the spectral shape of the ultraviolet background at 2 < z < 3 and the reionization of helium

The low-density hydrogen and helium in the intergalactic medium (IGM) probed by quasi-stellar object (QSO) absorption lines is sensitive to the amplitude and spectral shape of the metagalactic ultraviolet (UV) background. We use realistic H  i and He  ii Lyα forest spectra, constructed from state-of-the-art hydrodynamical simulations of a Λ cold dark matter (ΛCDM) universe to confirm the reliability of using line profile fitting techniques to infer the ratio of the metagalactic H  i and He  ii ionization rates. We further show that the large spatial variations and the anticorrelation with H  i absorber density observed in the ratio of the measured He  ii to H  i column densities can be explained in a model where the H  i ionization rate is dominated by the combined UV emission from young star-forming galaxies and QSOs and the He  ii ionization rate is dominated by emission from QSOs only. In such a model the large fluctuations in the column density ratio are due to the small number of QSOs expected to contribute at any given point to the He  ii ionization rate. A significant contribution to UV emission at the He  ii photoelectric edge from hot gas in galaxies and galaxy groups would decrease the expected fluctuations in the column density ratio. Consequently, this model appears difficult to reconcile with the large increase in He  ii opacity fluctuations towards higher redshift. Our results further strengthen previous suggestions that observed He  ii Lyα forest spectra at z ∼ 2–3.5 probe the tail end of the reionization of He  ii by QSOs.     

On the H i content, dust-to-gas ratio and nature of Mg ii absorbers

We estimate the mean dust-to-gas ratio of Mg  ii absorbers as a function of rest equivalent width W ₀ and redshift over the range  0.5 < z < 1.4  . Using the expanded Sloan Digital Sky Survey/ Hubble Space Telescope sample of low-redshift Lyman-α absorbers, we first show the existence of a 8σ correlation between the mean hydrogen column density  〈 N _{H  i} 〉  and W ₀, an indicator of gas velocity dispersion. By combining these results with recent dust-reddening measurements, we show that the mean dust-to-gas ratio of Mg  ii absorbers does not appreciably depend on rest equivalent width. Assuming that, on average, dust-to-gas ratio is proportional to metallicity, we find its redshift evolution to be consistent with that of   L ^★  galaxies from   z = 0.5  to 1.4, and we show that our constraints disfavour dwarf galaxies as the origin of such absorbers. We discuss other scenarios and favour galactic outflows from  ∼ L ^★  galaxies as the origin of the majority of strong Mg  ii absorbers. Finally, we show that, once evolutionary effects are taken into account, the Bohlin et al. relation between A _V and N _H is also satisfied by strong Mg  ii systems down to lower column densities than those probed in our Galaxy.     

Physical effects on the Lyα forest flux power spectrum: damping wings, ionizing radiation fluctuations and galactic winds

We explore several physical effects on the power spectrum of the Lyα forest transmitted flux. The effects we investigate here are not usually part of hydrodynamic simulations and so need to be estimated separately. The most important effect is that of high column density absorbers with damping wings, which add power on large scales. We compute their effect using the observational constraints on their abundance as a function of column density. Ignoring their effect leads to an underestimation of the slope of the linear theory power spectrum. The second effect we investigate is that of fluctuations in the ionizing radiation field. For this purpose we use a very large high-resolution N -body simulation, which allows us to simulate both the fluctuations in the ionizing radiation and the small-scale Lyα forest within the same simulation. We find an enhancement of power on large scales for quasars and a suppression for galaxies. The strength of the effect rapidly increases with increasing redshift, allowing it to be uniquely identified in cases where it is significant. We develop templates that can be used to search for this effect as a function of quasar lifetime, quasar luminosity function and attenuation length. Finally, we explore the effects of galactic winds using hydrodynamic simulations. We find the wind effects on the Lyα forest power spectrum to be degenerate with parameters related to the temperature of the gas that are already marginalized over in cosmological fits. While more work is needed to conclusively exclude all possible systematic errors, our results suggest that, in the context of data analysis procedures, where parameters of the Lyα forest model are properly marginalized over, the flux power spectrum is a reliable tracer of cosmological information.     

The nature of high-redshift galaxies

Using semi-analytic models of galaxy formation set within the cold dark matter (CDM) merging hierarchy, we investigate several scenarios for the nature of the high-redshift     ) Lyman-break galaxies (LBGs). We consider a 'collisional starburst' model in which bursts of star formation are triggered by galaxy–galaxy mergers, and find that a significant fraction of LBGs are predicted to be starbursts. This model reproduces the observed comoving number density of bright LBGs as a function of redshift and the observed luminosity function at     and     with a reasonable amount of dust extinction. Model galaxies at     have star formation rates, half-light radii,     colours and internal velocity dispersions that are in good agreement with the data. Global quantities such as the star formation rate density and cold gas and metal content of the Universe as a function of redshift also agree well. Two 'quiescent' models without starbursts are also investigated. In one, the star formation efficiency in galaxies remains constant with redshift, while in the other, it scales inversely with disc dynamical time, and thus increases rapidly with redshift. The first quiescent model is strongly ruled out, as it does not produce enough high-redshift galaxies once realistic dust extinction is accounted for. The second quiescent model fits marginally, but underproduces cold gas and very bright galaxies at high redshift. A general conclusion is that star formation at high redshift must be more efficient than locally. The collisional starburst model appears to accomplish this naturally without violating other observational constraints.     

On the formation of massive galaxies: a simultaneous study of number density, size and intrinsic colour evolution in GOODS

The evolution of number density, size and intrinsic colour is determined for a volume-limited sample of visually classified early-type galaxies selected from the Hubble Space Telescope /Advanced Camera for Surveys images of the Great Observatories Origins Deep Survey (GOODS) North and South fields (version 2). The sample comprises 457 galaxies over 320 arcmin² with stellar masses above  3 × 10¹⁰ M_⊙  in the redshift range  0.4 < z < 1.2  . Our data allow a simultaneous study of number density, intrinsic colour distribution and size. We find that the most massive systems  (≳3 × 10¹¹ M_⊙)  do not show any appreciable change in comoving number density or size in our data. Furthermore, when including the results from 2dF galaxy redshift survey, we find that the number density of massive early-type galaxies is consistent with no evolution between   z = 1.2  and 0, i.e. over an epoch spanning more than half of the current age of the Universe. We find large discrepancies between the predictions of semi-analytic models. Massive galaxies show very homogeneous intrinsic colour distributions, with nearly flat radial colour gradients, but with a significant negative correlation between stellar mass and colour gradient, such that red cores appear predominantly in massive galaxies. The distribution of half-light radii – when compared to   z ∼ 0  and   z > 1  samples – is compatible with the predictions of semi-analytic models relating size evolution to the amount of dissipation during major mergers.     

The GLARE Survey – II. Faint z≈ 6 Lyα line emitters in the HUDF

The galaxy population at   z ≈ 6  has been the subject of intense study in recent years, culminating in the Hubble Ultra Deep Field (HUDF) – the deepest imaging survey yet. A large number of high-redshift galaxy candidates have been identified within the HUDF, but until now analysis of their properties has been hampered by the difficulty of obtaining spectroscopic redshifts for these faint galaxies. Our 'Gemini Lyman-Alpha at Reionization Era' (GLARE) project has been designed to undertake spectroscopic follow-up of faint  ( z ' < 28.5)  i '-drop galaxies at   z ≈ 6  in the HUDF. In a previous paper we presented preliminary results from the first 7.5 h of data from GLARE. In this paper we detail the complete survey. We have now obtained 36 h of spectroscopy on a single GMOS slitmask from Gemini-South, with a spectral resolution of  λ/Δλ_FWHM≈ 1000  . We identify five strong Lyα emitters at   z > 5.5  , and a further nine possible line emitters with detections at lower significance. We also place tight constraints on the equivalent width of Lyα emission for a further ten i '-drop galaxies and examine the equivalent width distribution of this faint spectroscopic sample of   z ≈ 6  galaxies. We find that the fraction of galaxies with little or no emission is similar to that at   z ≈ 3  , but that the   z ≈ 6  population has a tail of sources with high rest-frame equivalent widths. Possible explanations for this effect include a tendency towards stronger line emission in faint sources, which may arise from extreme youth or low metallicity in the Lyman-break population at high redshift, or possibly a top-heavy initial mass function.     

Light-cone distortion of the clustering and abundance of massive galaxies at high redshifts

Observational surveys of galaxies are not trivially related to single-epoch snapshots from computer simulations. Observationally, an increase in the distance along the line of sight corresponds to an earlier cosmic time at which the properties of the surveyed galaxy population may change. The effect of observing a survey volume along the light cone must be considered in the regime where the mass function of galaxies varies exponentially with redshift. This occurs when the haloes under consideration are rare, that is either when they are very massive or observed at high redshift. While the effect of the light cone is negligible for narrow-band surveys of Lyα emitters, it can be significant for dropout surveys of Lyman-break galaxies (LBGs) where the selection functions of the photometric bands are broad. Since there are exponentially more haloes at the low-redshift end of the survey, the low-redshift tail of the selection function contains a disproportionate fraction of the galaxies observed in the survey. This leads to a redshift probability distribution for the dropout LBGs with a mean less than that of the photometric selection function (PHSF) by an amount of order the standard deviation of the PHSF. The inferred mass function of galaxies is then shallower than the true mass function at a single redshift with the abundance at the high-mass end being twice or more as large as expected. Moreover, the statistical moments of the count of galaxies calculated ignoring the light-cone effect deviate from the actual values.     

Galaxy groups at 0.3 ≤z≤ 0.55 – I. Group properties

The evolution of galaxies in groups may have important implications for the evolution of the star formation history of the Universe, since many processes which operate in groups may suppress star formation and the fraction of galaxies in bound groups grows rapidly between   z = 1  and the present day. In this paper, we present an investigation of the properties of galaxies in galaxy groups at intermediate redshift  ( z ∼ 0.4)  . The groups were selected from the Canadian Network for Observational Cosmology Redshift Survey (CNOC2) redshift survey as described by Carlberg et al., with further spectroscopic follow-up undertaken at the Magellan telescope in order to improve the completeness and depth of the sample. We present the data for the individual groups, and find no clear trend in the fraction of passive galaxies with group velocity dispersion and group concentration. We stack the galaxy groups in order to compare the properties of group galaxies with those of field galaxies at the same redshift. The groups contain a larger fraction of passive galaxies than the field, this trend being particularly clear for galaxies brighter than   M _{B J} < −20  in the higher velocity dispersion groups. In addition, we see evidence for an excess of bright passive galaxies in the groups relative to the field. In contrast, the luminosity functions of the star-forming galaxies in the groups and the field are consistent. These trends are qualitatively consistent with the differences between group and field galaxies seen in the local Universe.     

The chemical evolution of the Universe – I. High column density absorbers

We construct a simple, robust model of the chemical evolution of galaxies from high to low redshift, and apply it to published observations of damped Lyman α quasar absorption line systems (DLAs). The elementary model assumes quiescent star formation and isolated galaxies (no interactions, mergers or gas flows). We consider the influence of dust and chemical gradients in the galaxies, and hence explore the selection effects in quasar surveys. We fit individual DLA systems to predict some observable properties of the absorbing galaxies, and also indicate the expected redshift behaviour of chemical element ratios involving nucleosynthetic time delays.
Despite its simplicity, our 'monolithic collapse' model gives a good account of the distribution and evolution of the metallicity and column density of DLAs, and of the evolution of the global star formation rate and gas density below redshifts z ∼3. However, from the comparison of DLA observations with our model, it is clear that star formation rates at higher redshifts ( z >3) are enhanced. Galaxy interactions and mergers, and gas flows very probably play a major role.     

QSO lensing magnification associated with galaxy groups

We simulated both the matter and light (galaxy) distributions in a wedge of the Universe and calculated the gravitational lensing magnification caused by the mass along the line-of-sight of galaxies and galaxy groups identified in sky surveys. A large volume redshift cone containing cold dark matter particles mimics the expected cosmological matter distribution in a flat universe with low matter density and a cosmological constant. We generate a mock galaxy catalogue from the matter distribution and identify thousands of galaxy groups in the luminous sky projection. We calculate the expected magnification around galaxies and galaxy groups and then the induced quasi-stellar object (QSO)–lens angular correlation due to magnification bias. This correlation is observable and can be used both to estimate the average mass of the lens population and to make cosmological inferences. We also use analytical calculations and various analyses to compare the observational results with theoretical expectations for the cross-correlation between faint QSOs from the 2dF Survey and nearby galaxies and groups from the Automated Plate Measurement and Sloan Digital Sky Survey Early Data Release. The observed QSO–lens anticorrelations are stronger than the predictions for the cosmological model used. This suggests that there could be unknown systematic errors in the observations and data reduction, or that the model used is not adequate. If the observed signal is assumed to be solely due to gravitational lensing, then the lensing is stronger than expected, due to more massive galactic structures or more efficient lensing than simulated.     

Star formation properties of Universidad Complutense de Madrid survey galaxies

We present new near-infrared J and K imaging data for 67 galaxies from the Universidad Complutense de Madrid (UCM) survey used in the determination of the SFR density of the local Universe by Gallego et al. This is a sample of local star-forming galaxies with redshift lower than 0.045, and they constitute a representative subsample of the galaxies in the complete UCM survey. From the new data, complemented with our own Gunn- r images and long-slit optical spectroscopy, we have measured integrated K -band luminosities, r − J and J − K colours, and H α luminosities and equivalent widths. Using a maximum likelihood estimator and a complete set of evolutionary synthesis models, these observations allow us to estimate the strength of the current (or most recent) burst of star formation, its age, the star formation rate and the total stellar mass of the galaxies. An average galaxy in the sample has a stellar mass of 5×10¹⁰ M_⊙ and is undergoing (or has recently completed) a burst of star formation involving about 2 per cent of its total stellar mass. We identify two separate classes of star-forming galaxies in the UCM sample: low-luminosity, high-excitation galaxies (H  ii like ) and relatively luminous spiral galaxies (starburst disc- like ). The former show higher specific star formation rates (SFRs per unit mass) and burst strengths, and lower stellar masses than the latter. With regard to their specific star formation rates, the UCM galaxies are intermediate objects between normal quiescent spirals and the most extreme H  ii galaxies.     

A census of metals at high and low redshift and the connection between submillimetre sources and spheroid formation

Deep surveys in many wavebands have shown that the rate at which stars were forming was at least a factor of 10 higher at redshifts >1 than today. Heavy elements ('metals') are produced by stars, and the star formation history deduced by these surveys implies that a significant fraction of all metals in the Universe today should already exist at   z ∼ 2–3  . However, only 10 per cent of the total metals expected to exist at this redshift have so far been accounted for (in damped Lyman α absorbers and the Lyman forest). In this paper, we use the results of submillimetre surveys of the local and high-redshift Universe to show that there was much more dust in galaxies in the past. We find that a large proportion of the missing metals are traced by this dust, bringing the metals implied from the star formation history and observations into agreement. We also show that the observed distribution of dust masses at high redshift can be reproduced remarkably well by a simple model for the evolution of dust in spheroids, suggesting that the descendants of the dusty galaxies found in deep submillimetre surveys are the relatively dust-free spiral bulges and ellipticals in the Universe today.     

The matter power spectrum from the Lyα forest: an optical depth estimate

We measure the matter power spectrum from 31 Lyα spectra spanning the redshift range of 1.6–3.6. The optical depth, τ, for Lyα absorption of the intergalactic medium is obtained from the flux using the inversion method of Nusser & Haehnelt. The optical depth is converted to density by using a simple power-law relation,  τ∝ (1 +δ)^α  . The non-linear 1D power spectrum of the gas density is then inferred with a method that makes simultaneous use of the one- and two-point statistics of the flux and compared against theoretical models with a likelihood analysis. A cold dark matter model with standard cosmological parameters fits the data well. The power-spectrum amplitude is measured to be (assuming a flat Universe),  σ₈= (0.92 ± 0.09) × (Ω_m/0.3)^−0.3  , with α varying in the range of 1.56–1.8 with redshift. Enforcing the same cosmological parameters in all four redshift bins, the likelihood analysis suggests some evolution in the temperature–density relation and the thermal smoothing length of the gas. The inferred evolution is consistent with that expected if reionization of He  ii occurred at   z ∼ 3.2  . A joint analysis with the Wilkinson Microwave Anisotropy Probe results together with a prior on the Hubble constant as suggested by the Hubble Space Telescope key project data, yields values of Ω_m and σ₈ that are consistent with the cosmological concordance model. We also perform a further inversion to obtain the linear 3D power spectrum of the matter density fluctuations.