QSO Absorption Spectra Research Articles

Warm Dark matter constraints from the joint analysis of CMB, Ly α, and global 21 cm data

ABSTRACT With the help of our previously built MCMC (Markov chain Monte Carlo)-based parameter estimation package cosmoreionmc, we investigate in detail the potential of 21 cm global signal, when combined with cosmic microwave background (CMB) and observations related to the Quasar (QSO) absorption spectra, to constraint the mass of warm dark matter (WDM) particle. For the first time, we simultaneously vary all the free parameters (mass of WDM particle, cosmological parameters, and astrophysical parameters) in a joint analysis with CMB, observations related to the QSO absorption spectra and 21 cm global signal, to address the long-overlooked issue of the possible degeneracies between the dark matter particle mass mX and cosmological/astrophysical parameters. From the existing CMB and QSO absorption spectra data, we can rule out mX < 2.8 keV at 95 per cent confidence level. Including a mock 21 cm global signal in the redshift range z = 25−5 expected to be observed with upcoming instruments designed for global signal, the forecasted constraint is found to be much tighter mX > 7.7 keV, assuming that the true dark matter model is the usual cold dark matter. In case the mock 21 cm signal is constructed for dark matter particles having mX = 7 keV, our forecasts indicate that (mX/keV)−1 is in the range [0.1, 0.2] (95 per cent confidence level). This implies that the future 21 cm data should allow detection of the WDM particle mass if mX ∼ 7 keV.

Measuring the photoionization rate, neutral fraction, and mean free path of H i ionizing photons at 4.9 ≤ z ≤ 6.0 from a large sample of XShooter and ESI spectra

ABSTRACT We measure the mean free path ($\lambda _{\rm mfp,H\, \small {I}}$), photoionization rate ($\langle \Gamma _{\rm H\, \small {I}} \rangle$), and neutral fraction ($\langle f_{\rm H\, \small {I}} \rangle$) of hydrogen in 12 redshift bins at 4.85 < z < 6.05 from a large sample of moderate resolution XShooter and ESI QSO absorption spectra. The fluctuations in ionizing radiation field are modelled by post-processing simulations from the Sherwood suite using our new code ‘EXtended reionization based on the Code for Ionization and Temperature Evolution’ (ex-cite). ex-cite uses efficient Octree summation for computing intergalactic medium attenuation and can generate large number of high resolution $\Gamma _{\rm H\, \small {I}}$ fluctuation models. Our simulation with ex-cite shows remarkable agreement with simulations performed with the radiative transfer code Aton and can recover the simulated parameters within 1σ uncertainty. We measure the three parameters by forward-modelling the Lyα forest and comparing the effective optical depth ($\tau _{\rm eff, H\, \small {I}}$) distribution in simulations and observations. The final uncertainties in our measured parameters account for the uncertainties due to thermal parameters, modelling parameters, observational systematics, and cosmic variance. Our best-fitting parameters show significant evolution with redshift such that $\lambda _{\rm mfp,H\, \small {I}}$ and $\langle f_{\rm H\, \small {I}} \rangle$ decreases and increases by a factor ∼6 and ∼104, respectively from z ∼ 5 to z ∼ 6. By comparing our $\lambda _{\rm mfp,H\, \small {I}}$, $\langle \Gamma _{\rm H\, \small {I}} \rangle$ and $\langle f_{\rm H\, \small {I}} \rangle$ evolution with that in state-of-the-art Aton radiative transfer simulations and the Thesan and CoDa-III simulations, we find that our best-fitting parameter evolution is consistent with a model in which reionization completes by z ∼ 5.2. Our best-fitting model that matches the $\tau _{\rm eff, H\, \small {I}}$ distribution also reproduces the dark gap length distribution and transmission spike height distribution suggesting robustness and accuracy of our measured parameters.

The Cosmic Ultraviolet Baryon Survey (CUBS) – III. Physical properties and elemental abundances of Lyman-limit systems at z < 1

ABSTRACT We present a systematic investigation of physical conditions and elemental abundances in four optically thick Lyman-limit systems (LLSs) at z = 0.36–0.6 discovered within the Cosmic Ultraviolet Baryon Survey (CUBS). Because intervening LLSs at z < 1 suppress far-UV (ultraviolet) light from background QSOs, an unbiased search of these absorbers requires a near-UV-selected QSO sample, as achieved by CUBS. CUBS LLSs exhibit multicomponent kinematic structure and a complex mix of multiphase gas, with associated metal transitions from multiple ionization states such as C ii, C iii, N iii, Mg ii, Si ii, Si iii, O ii, O iii, O vi, and Fe ii absorption that span several hundred km s−1 in line-of-sight velocity. Specifically, higher column density components (log N(H i)/cm−2≳ 16) in all four absorbers comprise dynamically cool gas with $\langle T \rangle =(2\pm 1) \times 10^4\,$K and modest non-thermal broadening of $\langle b_\mathrm{nt} \rangle =5\pm 3\,$km s−1. The high quality of the QSO absorption spectra allows us to infer the physical conditions of the gas, using a detailed ionization modelling that takes into account the resolved component structures of H i and metal transitions. The range of inferred gas densities indicates that these absorbers consist of spatially compact clouds with a median line-of-sight thickness of $160^{+140}_{-50}$ pc. While obtaining robust metallicity constraints for the low density, highly ionized phase remains challenging due to the uncertain $N\mathrm{(H\, {\small I})}$, we demonstrate that the cool-phase gas in LLSs has a median metallicity of $\mathrm{[\alpha /H]_{1/2}}=-0.7^{+0.1}_{-0.2}$, with a 16–84 percentile range of [α/H] = (−1.3, −0.1). Furthermore, the wide range of inferred elemental abundance ratios ([C/α], [N/α], and [Fe/α]) indicate a diversity of chemical enrichment histories. Combining the absorption data with deep galaxy survey data characterizing the galaxy environment of these absorbers, we discuss the physical connection between star-forming regions in galaxies and diffuse gas associated with optically thick absorption systems in the z < 1 circumgalactic medium.

Constraints on QSO emissivity using H i and He ii Lyman α forest

The spectrum of cosmic ultraviolet background radiation at He II ionizing energies (E > 4 Ryd) is important to study the He II reionization, thermal history of the intergalactic medium (IGM) and metal lines observed in QSO absorption spectra. It is determined by the emissivity of QSOs at E > 4 Ryd obtained from their observed luminosity functions and the mean spectral energy distribution (SED). The SED is approximated as a power-law at energies E > 1 Ryd, $f_E \propto E^{\alpha}$, where the existing observations constrain the power-law index $\alpha$ only up to ~2.3 Ryd. Here, we constrain $\alpha$ for E > 4 Ryd using recently measured He II Lyman-alpha effective optical depths ($\tau_{HeII}$), H I photoionization rates and updated H I distribution in the IGM. We find that -1.6 > $\alpha$ > -2 is required to reproduce the $\tau_{HeII}$ measurements when we use QSO emissivity obtained from their luminosity function using optical surveys. We also find that the models where QSOs can alone reionize H I can not reproduce the $\tau_{HeII}$ measurements. These models need modifications, such as a break in mean QSO SED at energies greater than 4 Ryd. Even after such modifications the predicted He II reionization history, showing that the He II is highly ionized even at z~5, is significantly different from the standard models. Therefore, the thermal history of the IGM will be crucial to distinguish these models. We also provide the He II photoionization rates obtained from binned $\tau_{HeII}$ measurements.

Calibrating cosmological radiative transfer simulations with Ly α forest data: evidence for large spatial UV background fluctuations atz∼ 5.6–5.8 due to rare bright sources

We calibrate here cosmological radiative transfer simulation with ATON/RAMSES with a range of measurements of the Lyman alpha opacity from QSO absorption spectra. We find the Lyman alpha opacity to be very sensitive to the exact timing of hydrogen reionisation. Models reproducing the measured evolution of the mean photoionisation rate and average mean free path reach overlap at z ~ 7 and predict an accelerated evolution of the Lyman alpha opacity at z > 6 consistent with the rapidly evolving luminosity function of Lyman alpha emitters in this redshift range. Similar to optically thin simulations our full radiative transfer simulations fail, however, to reproduce the high-opacity tail of the Lyman alpha opacity PDF at z > 5. We argue that this is due to spatial UV fluctuations in the post-overlap phase of reionisation on substantially larger scales than predicted by our source model, where the ionising emissivity is dominated by large numbers of sub-L* galaxies. We further argue that this suggests a significant contribution to the ionising UV background by much rarer bright sources at high redshift.

BRINGING SIMULATION AND OBSERVATION TOGETHER TO BETTER UNDERSTAND THE INTERGALACTIC MEDIUM

The methods by which one characterizes the distribution of matter in cosmological simulations is intrinsically different from how one performs the same task observationally. In this paper, we make substantial steps towards comparing simulations and observations of the intergalactic medium (IGM) in a more sensible way. We present a pipeline that generates and fits synthetic QSO absorption spectra using sight lines cast through a cosmological simulation, and simultaneously identifies structure by directly analyzing the variations in HI and OVI number density. We compare synthetic absorption spectra with a less observationally motivated, but more straightforward density threshold-based method for finding absorbers. Our efforts focus on HI and OVI to better characterize the warm/hot intergalactic medium, a subset of the IGM that is challenging to conclusively identify observationally. We find that the two methods trace roughly the same quantities of HI and OVI above observable column density limits, but the synthetic spectra typically identify more substructure in absorbers. We use both methods to characterize HI and OVI absorber properties. We find that both integrated and differential column density distributions from both methods generally agree with observations. The distribution of Doppler parameters between the two methods are similar for Lya and compare reasonably with observational results, but while the two methods agree with each other with OVI systems, they both are systematically different from observations. We find a strong correlation between OVI baryon fraction and OVI column density. We also discuss a possible bimodality in the temperature distribution of the gas traced by OVI.

THE STATE OF STAR FORMATION AND THE INTERGALACTIC MEDIUM ATz∼ 6

In the context of stellar reionization in the standard cold dark matter model, we analyze observations at z ∼ 6 and are able to draw three significant conclusions with respect to star formation and the state of the intergalactic medium (IGM) at z ∼ 6. (1) An initial stellar mass function (IMF) more efficient, by a factor of 10–20, in producing ionizing photons than the standard Salpeter IMF is required at z ∼ 6. This may be achieved by having either (a) a metal-enriched IMF with a lower mass cutoff of ⩾30 M☉ or (b) 2%–4% of stellar mass being Population III massive metal-free stars at z ∼ 6. While there is no compelling physical reason or observational evidence to support (a), (b) could plausibly be fulfilled by continued existence of some pockets of uncontaminated, metal-free gas for star formation. (2) The volume-weighted neutral fraction of the IGM of at z = 5.8 inferred from the SDSS observations of QSO absorption spectra provides enough information to ascertain that reionization is basically complete with at most ∼0.1%–1% of IGM that is unionized at z = 5.8. (3) Barring some extreme evolution of the IMF, the neutral fraction of the IGM is expected to rise quickly toward high redshift from the point of H ii bubble percolation, with the mean neutral fraction of the IGM expected to reach 6%–12% at z = 6.5, 13%–27% at z = 7.7, and 22%–38% at z = 8.8.

Constraining the reionization history with QSO absorption spectra

We use a semi-analytical approach to simulate absorption spectra of QSOs at high redshifts with the aim of constraining the cosmic reionization history. We consider two physically motivated and detailed reionization histories: (i) an early reionization model (ERM) in which the intergalactic medium is reionized by Pop III stars at z≈ 14, and (ii) a more standard late reionization model (LRM) in which overlapping, induced by QSOs and normal galaxies, occurs at z≈ 6. From the analysis of current Lyα forest data at z < 6, we conclude that it is impossible to disentangle the two scenarios, which fit equally well the observed Gunn–Peterson optical depth, flux probability distribution function and dark gap width distribution. At z > 6, however, clear differences start to emerge which are best quantified by the dark gap and peak width distributions. We find that 35 (0) per cent of the lines of sight (LOS) within 5.7 < z < 6.3 show dark gaps of widths >50 Å in the rest frame of the QSO if reionization is not (is) complete at z≳ 6. Similarly, the ERM predicts peaks of width ∼1 Å in 40 per cent of the LOS in the redshift range 6.0–6.6; in the same range, LRM predicts no peaks of width >0.8 Å. We conclude that the dark gap and peak width statistics represent superb probes of cosmic reionization if about ten QSOs can be found at z > 6. We finally discuss strengths and limitations of our method.

The impact of galactic winds from Lyman-break galaxies on the intergalactic medium

An excess of sight-lines close to Lyman-break galaxies (LBGs) with little or no absorption in QSO absorption spectra has been reported, and has been interpreted as the effect of galactic winds on the intergalactic medium. We use here numerical simulations to investigate the flux probability distribution function close to plausible sites of LBGs. We show that a model without galactic winds and a model with a wind bubble size of 0.5 h −1 Mpc (comoving) are consistent with the new determination of the conditional flux distribution by Adelberger et al. Models with the larger bubble sizes ( 1 h −1 Mpc) suggested by the previous observations of Adelberger et al. based on a much smaller sample at higher redshift are not consistent with the new data. The volume filling factor of galactic winds driven by LBGs may thus be much smaller than previously thought, and most of the metals responsible for the metal absorption associated with the low column density Lyα forest appear to be unlikely to have been ejected by LBGs.

Spectral energy distribution of the metagalactic ionizing radiation field from QSO absorption spectra

A computational procedure is presented to estimate the spectral shape of the ionizing background between 1 and 10 Ryd by analyzing optically thin absorption systems in the spectra of high redshift quasars. The procedure is based on the response surface methodology from the theory of experimental design. The shape of the recovered UV background at z ∼ 3 shows a significant intensity decrease between 3 and 4 Ryd compared to the metagalactic spectrum of Haardt & Madau (1996, ApJ, 461, 20). This decrease is interpreted as produced by He  Gunn-Peterson effect. There are no features indicating a contribution from galaxies to the UV background which is therefore dominated by QSOs at z ∼ 3.

Distributions of the Baryon Fraction on Large Scales in the Universe

(Abridged) The nonlinear evolution of a system consisting of baryons and dark matter is generally characterized by strong shocks and discontinuities. The baryons slow down significantly at postshock areas of gravitational strong shocks, which can occur in high overdense as well as low overdense regions. Consequently, the baryon fraction would be nonuniform on large scales. We studied these phenomena with simulation samples produced by the WENO hybrid cosmological hydrodynamic/N-body code. We find that the baryon fraction in high mass density regions is lower on average than the cosmic baryon fraction, and many baryons accumulate in the regions with moderate mass density to form a high baryon fraction phase (HBFP). In dense regions with rho>100, which are the possible hosts for galaxy clusters, the baryon fraction can be lower than the cosmic baryon fraction by about 10%--20% at z ~ 0. Our simulation samples show that about 3% of the cosmic baryon budget was hidden in the HBFP at redshift z=3, while this percentage increases to about 14% at the present day. The gas in the HBFP cannot be detected either by Ly-alpha forests of QSO absorption spectra or by soft X-ray background. That is, the HBFP would be missed in the baryon budget given by current observations.

The Lyman a forest opacity and the metagalactic hydrogen ionization rate at z 2-4

Estimates of the metagalactic hydrogen ionization rate from the Lyα forest opacity in QSO absorption spectra depend on the complex density distribution of neutral hydrogen along the line of sight. We use a large suite of high-resolution hydrodynamical simulations to investigate in detail the dependence of such estimates on physical and numerical parameters in the context of ΛCDM models. Adopting fiducial values for cosmological parameters together with published values of the temperature of the IGM and the effective optical depth, the metagalactic ionization rates that reproduce the Lyα effective optical depth at z=[2, 3, 4] are , respectively. The errors include estimates of uncertainties in the relevant physical parameters and the numerical accuracy of the simulations. We find the errors are dominated by the uncertainty in the temperature of the low-density IGM. The estimated metagalactic hydrogen ionization rate for the neutral hydrogen distribution in the current concordance ΛCDM model is more than four times the value inferred for that in an Einstein–de Sitter model of the same rms-density-fluctuation amplitude σ8. The estimated ionization rate is also more than double that expected from updated estimates of the emissivity of observed QSOs alone. A substantial contribution from galaxies appears to be required at all redshifts.

Erratum: The power spectrum of the flux distribution in the Lyman α forest of a large sample of UVES QSO Absorption Spectra (LUQAS)★

Erratum: The power spectrum of the flux distribution in the Lyman α forest of a large sample of UVES QSO Absorption Spectra (LUQAS)^★

The Reionization History in the Lognormal Model

We study the evolution of baryonic gas before reionization in the lognormal (LN) model of cosmic clustering. We show that the thermal history of the universe around reionization can roughly be divided into three epochs: (1) a cold dark age z > zr, in which the baryon gas is neutral and opaque to Lyα photons; (2) a hot dark age zr > z > zGP, in which a predominant part of the baryon gas is ionized and hot but is still opaque to Lyα photons; and (3) a bright age z < zGP, in which the universe is ionized enough to be transparent to Lyα photons. In the flat cold dark matter cosmological models given by WMAP and COBE, the difference of the two redshifts zr - zGP is found to be as large as ~10, with zr ~ 17 and zGP ~ 7. This reionization history naturally yields a high optical depth of the CMB τe ≃ 0.12-0.19, which is observed in the TE polarization in the WMAP data, and a low redshift zGP for the appearance of the Lyα Gunn-Peterson trough, zGP ≃ 6-8, in QSO absorption spectra. The universe stays so long in an ionized, yet Lyα opaque, stage because the first photoionization heats the intergalactic gas effectively and balances the gravitational clustering for a long period of time. Therefore, a high τe and low zGP is a result of all the considered models. Besides the cosmological parameters, the only free parameter we used in the calculation is Nion, the mean number of ionization photons produced by each baryon in collapsed objects. We take it to be 40-80 in the calculations.

Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra

We have previously presented evidence for a varying fine-str ucture constant, α, in two independent samples of Keck/HIRES QSO absorption spectra. Here we present a detailed manymultiplet analysis of a third Keck/HIRES sample containing 78 absorption systems. We also re-analyse the previous samples, providing a total of 128 absorption systems over the redshift range 0.2 < zabs < 3.7. The results, with raw statistical errors, indicate a small er weighted mean α in the absorption clouds: �α/α = ( 0.574 ± 0.102) × 10 5 . All three samples separately yield consistent and significant values of �α/α. The analyses of low-z (i.e. zabs < 1.8) and high-z systems rely on different ions and transitions with very different dependencies on α, yet they also give consistent results. We identify an addit ional source of random error in 22 high-z systems characterized by transitions with a large dynamic range in apparent optical depth. Increasing the statistical erro rs on �α/α for these systems gives our fiducial result, a weighted mean �α/α = ( 0.543 ± 0.116) × 10 5 , representing 4.7 σ evidence for a varying α. Assuming that �α/α = 0 at zabs = 0, the data marginally prefer a linear increase in α with time rather than a constant offset from the laboratory value: u α/α = (6.40 ± 1.35) × 10 16 yr 1 . The two-point correlation function for α is consistent with zero over 0.2‐13 Gpc comoving scales and the angular distribution of �α/α shows no significant dipolar anisotropy. We therefore have no eviden ce for spatial variations in �α/α. We extend our previous searches for possible systematic errors, giving detailed analyses of potential kinematic effects, line blending, wavelength miscalibration, spectrograph temperature variations, atmospheric dispersion and isotopic /hyperfine structure effects. The latter two are potentially the most significant. However, overall, known systematic errors do not explain the results. Future many-multiplet analyses of indep endent QSO spectra from different telescopes and spectrographs will provide a now crucial check on our Keck/HIRES results.

An Ionizing Ultraviolet Background Dominated by Massive Stars

We discuss the implications of a stellar-dominated UV background at high redshifts for the star formation history of Lyman break galaxies (LBGs) and the thermal and ionization state of the intergalactic medium (IGM). The composite spectrum of 29 LBGs evaluated by Steidel et al. at z = 3.4 can be well fitted by a stellar population with ongoing star formation, a Salpeter initial mass function, modest or negligible dust reddening, and no intrinsic H I photoelectric absorption. Fading starbursts in which star formation has ceased for 107 yr or more cannot reproduce the observed flux shortward of 1 ryd. The small H I optical depth in LBGs suggests that the neutral gas from which stars form is most likely contained in compact clouds of neutral gas with small covering factor. The escape fraction of H-ionizing photons must be close to 100% for the observed sample of LBGs. The spectrum of ionizing photons produced by a stellar population with ongoing star formation is similar to that of QSOs between 1 and 3 ryd but becomes softer between 3 and 4 ryd and drops sharply shortward of 4 ryd. A galaxy-dominated UV background appears inconsistent with the observed He II/H I opacity ratio at z = 2.4 but might be able to explain the Si IV/C IV abundances measured at z > 3 in QSO absorption spectra. A scenario may be emerging where star-forming galaxies reionize intergalactic hydrogen at z > 6 and dominate the 1 ryd metagalactic flux at z > 3, with quasi-stellar sources taking over at lower redshifts. If the large amplitude of the H-ionizing flux estimated by Steidel et al. is correct, hydrodynamical simulations of structure formation in the IGM within the cold dark matter paradigm require a baryon density (to explain the observed opacity of the Ly? forest in QSO absorption spectra) that is similar to or larger than that favored by recent cosmic microwave background experiments and is inconsistent with standard nucleosynthesis values.

Observational signatures of feedback in QSO absorption spectra

Models for the formation of galaxies and clusters of galaxies require strong feedback in order to explain the observed properties of these systems. We investigate whether such feedback has observational consequences for the intergalactic medium, as probed in absorption towards background quasars. A typical quasar sight-line intersects one proto-cluster per unit redshift, and significant feedback from forming galaxies or AGN, heating the proto-cluster gas, will result in a large clearing of reduced absorption in the Lyman alpha-forest. Such a gap could be detected at redshift > 3 when the mean opacity is high. Feedback from Lyman-break galaxies in proto-clusters can be probed by the absorption lines produced in their winds. Strong feedback from galaxies has a major impact on the number and properties of absorption lines with neutral hydrogen column densities of 10^16/ cm^2. This feedback can be probed with multiple sight-lines and by studying the unsaturated higher-order lines of the Lyman series. Galactic winds from dwarf galaxies should break-up into clouds, in order not to over produce the number of absorption lines. These clouds can then coast to large distances.

Non‐Gaussianity and the Recovery of the Mass Power Spectrum from the Lyα Forest

We investigate the eUect of non-Gaussianity on the reconstruction of the initial mass —eld from the Lya forest. We show that the transmitted —ux of QSO absorption spectra are highly non-Gaussian in terms of the statistics, the kurtosis spectrum, and the scale-scale correlation. These non-Gaussianities cannot be completely removed by the conventional algorithm of Gaussianization, and the scale-scale correlations are largely retained in the mass —eld recovered by the Gaussian mapping. Therefore, the mass power spectrum recovered by the conventional algorithm is systematically lower than the initial mass spectrum on scales at which the local scale-scale correlation is substantial. To reduce the non- Gaussian contamination, we present two methods. The —rst is to perform the Gaussianization scale-by- scale using the discrete wavelet transform (DWT) decomposition. We show that the non-Gaussian features of the Lya forest basically will no longer exist in the scale-by-scale Gaussianized mass —eld. The second method is to choose a proper orthonormal basis (representation) to suppress the eUect of the non-Gaussian correlations. In the quasi-linear regime of cosmic structure formation, the DWT power spectrum is efficient for suppressing the non-Gaussian contamination. These two methods signi—cantly improve the recovery of the mass power spectrum from the Lya forest. Subject headings: cosmology: theorydark matterlarge-scale structure of universe ¨ quasars: absorption lines

The Origin of QSO Mg II λ 2796 Lines

We have started a systematic spectroscopic survey of stars in the η Carina nebula using the echelle mode of the Space Telescope Imaging Spectrograph (STIS). The Mg II λ2796/λ2803 lines seen in our R ∼ 100000, spectra of the star CPD–59° 2603 show a remarkable similarity with the intervening Mg II system seen in QSO absorption spectra. Such a comparison can shed light on the nature and origin of lines seen in QSO's.

Primordial abundances of6Li, 9Be,and11Bwith neutrino degeneracy and gravitational constant variation

Recent measurements of deuterium abundances from QSO absorption spectra show two conflicting numbers, which differ by an order of magnitude. Allowing the neutrino degeneracy together with gravitational constant variation at the epoch of primordial nucleosynthesis, the implication of these observations on the ^6Li, ^9Be and ^11B abundances will be discussed. Within the permitted ranges consistent with D, ^4He and ^7Li, we observe the strong \eta dependence of ^11B as in SBBN but no significant dependence of ^6Li and ^9Be on the baryon number density is observed. The predictions of ^6Li and ^9Be for low and high deuterium differ by an order of magnitude and predictions for ^11B differ by several orders of magnitude at most and vary with \eta.