Intensity mapping (IM) of spectral lines has the potential to revolutionize cosmology by increasing the total number of observed modes by several orders of magnitude compared to the cosmic microwave background (CMB) anisotropies. In this paper, we consider IM of neutral hydrogen (HI) in the redshift range $0 \lesssim z \lesssim 3$ employing a halo model approach where HI is assumed to follow the distribution of dark matter (DM) halos. If a portion of the DM is composed of ultralight axions then the abundance of halos is changed compared to cold dark matter below the axion Jeans mass. With fixed total HI density, $Ω_{\rm HI}$, assumed to reside entirely in halos, this effect introduces a scale-independent increase in the HI power spectrum on scales above the axion Jeans scale, which our model predicts consistent with N-body simulations. Lighter axions introduce a scale-dependent feature even on linear scales due to its suppression of the matter power spectrum near the Jeans scale. We use the Fisher matrix formalism to forecast the ability of future HI surveys to constrain the axion fraction of DM and marginalize over astrophysical and model uncertainties. We find that a HIRAX-like survey is a very reliable IM survey configuration, being affected minimally by uncertainties due to non-linear scales, while the SKA1MID configuration is the most constraining as it is sensitive to non-linear scales. Including non-linear scales and combining a SKA1MID-like IM survey with the Simons Observatory CMB, the benchmark "fuzzy DM" model with $m_a = 10^{-22}\text{ eV}$ can be constrained at the 10% level. For lighter ULAs this limit improves below 1%, and allows the possibility to test the connection between axion models and the grand unification scale across a wide range of masses.
We study cross correlations between line intensity and weak lensing maps to search for axion-like particles (ALPs). Radiative decay of eV-mass ALPs can contribute to cosmic background emissions at optical and infrared wavelengths. Line intensity mapping is a unique means of measuring the background emission at a given photon frequency. If ALPs constitute the abundance of cosmic dark matter, line intensity maps can correlate with large-scale structures probed by weak gravitational lensing effects in galaxy imaging surveys. We develop a theoretical framework to predict the cross correlation. We …
Recent studies using New Horizons LORRI images have returned the most precise measurement of the cosmic optical background to date, yielding a flux that exceeds that expected from deep galaxy counts by roughly a factor of two. We investigate whether this excess, detected at $\sim 4σ$ significance, is due to axion-like dark matter that decays to monoenergetic photons. We compute the spectral energy distribution from such decays and the contribution to the flux measured by LORRI. Assuming that axion-like particles make up all of the dark matter, the parameter space unconstrained to date that exp…
We study the impact of virtual axions on the polarization of photons inside a cavity during the interaction of high-power laser pulses. A novel detection scheme for measuring the axion-induced ellipticity signal during the Light-by-Light (LBL) scattering process is investigated. We show that a momentum exchange between photons in a probe laser beam and a high-intensity target beam may lead to a resonance at the physical mass of the axion. Consequently, the resonant enhancement of vacuum birefringence gives rise to a large ellipticity signal. This signal enhancement can be applied in order to d…
Line intensity mapping is rapidly gaining prominence as a tool to understand galaxy evolution and cosmology at high redshift. However, the standard power spectrum analysis for these surveys suffers from a tight degeneracy between the overall mean line intensity and the bias of the emitting galaxies. We outline a new formalism for the Voxel Intensity Distribution (VID), a one-point statistic which is known to contain information beyond the power spectrum. We use this new calculation to show for the first time that the VID can be used to break this key degeneracy in intensity mapping data.
Intensity mapping has attracted significant interest as an approach to measure the properties of the interstellar medium in typical galaxies at high redshift. Intensity mapping measures the statistics of surface brightness as a function of frequency, making it sensitive not only to all line emission of interest but also radiation from all other sources. Significant effort has gone into developing approaches that reject foreground contamination. Additionally, the target galaxies have multiple sources of emission that can complicate the interpretation of the line brightness. We describe the prob…
Line-intensity mapping, being an imperfect observation of the line-intensity field in a cosmological volume, will be subject to various anisotropies introduced in observation. Existing literature in the context of CO and [C II] line-intensity mapping often predicts only the real-space, spherically averaged line-intensity power spectrum, with some works considering anisotropies while examining projection of interloper emission. We explicitly consider a simplified picture of redshift-space distortions and instrumental effects due to limited resolution, and how these distort an isotropic line-int…
21 cm intensity mapping has arisen as a powerful probe of the high-redshift universe, but its potential is limited by extremely bright foregrounds and high source confusion. In this Letter, we propose a new analysis which can help solve both problems. From the combination of an intensity map with an overlapping galaxy survey we construct a new one-point statistic which is unbiased by foregrounds and contains information left out of conventional analyses. We show that our method can measure the HI mass function with unprecedented precision using observations similar to recent 21 cm detections.
Intensity mapping has been attracting increasing interest as a way to study galaxy evolution and the large scale structure of the Universe. Instead of detecting individual galaxies, we measure the integrated emission from a volume of the universe. Contrary to galaxy surveys, it includes light from the faintest objects in the cosmic web. We will start this overview by introducing line intensity mapping and review its current observational status. We will enumerate the most prominent emission lines for intensity mapping and discuss how to model their observed signal. The prospects of using this …
Infrared fine-structure emission lines from trace metals are powerful diagnostics of the interstellar medium in galaxies. We explore the possibility of studying the redshifted far-IR fine-structure line emission using the three-dimensional (3-D) power spectra obtained with an imaging spectrometer. The intensity mapping approach measures the spatio-spectral fluctuations due to line emission from all galaxies, including those below the individual detection threshold. The technique provides 3-D measurements of galaxy clustering and moments of the galaxy luminosity function. Furthermore, the linea…
Using the galaxy catalog built from ELUCID N-body simulation and the semi-analytical galaxy formation model, we have built a mock HI intensity mapping map. We have implemented the Finger-of-God (FoG) effect in the map by considering the galaxy HI gas velocity dispersion. By comparing the HI power spectrum in the redshift space with the measurement from IllustrisTNG simulation, we have found that such FoG effect can explain the discrepancy between current mock map built from N-body simulation and Illustris TNG simulation. Then we built a parameter-free FoG model and a shot-noise model to calcul…
We develop a formalism to describe extensions of existing axion haloscope designs to those that possess directional sensitivity to incoming dark matter axion velocities. The effects are measurable if experiments are designed to have dimensions that approach the typical coherence length for the local axion field. With directional sensitivity, axion detection experiments would have a greatly enhanced potential to probe the local dark matter velocity distribution. We develop our formalism generally, but apply it to specific experimental designs, namely resonant cavities and dielectric disk halosc…
In the last decade, it has become clear that the dust-enshrouded star formation contributes significantly to early galaxy evolution. Detection of dust is therefore essential in determining the properties of galaxies in the high-redshift universe. This requires observations at the (sub-)millimeter wavelengths. Unfortunately, sensitivity and background confusion of single dish observations on the one hand, and mapping efficiency of interferometers on the other hand, pose unique challenges to observers. One promising route to overcome these difficulties is intensity mapping of fluctuations which …
The power spectrum and the voxel intensity distribution (VID) are two of the main proposed summary statistics to study line-intensity maps. We reformulate the derivation of the VID in terms of the local overdensities and derive for the first time an analytic covariance between the VID and the line-intensity mapping power spectrum. We study the features of this covariance for different experimental setups and show that we can recover similar results to simulation-based covariances. With this formalism, we also compute the cosmic variance contribution the VID uncertainty, which we find to be sub…
Line-Intensity Mapping is an emerging technique which promises new insights into the evolution of the Universe, from star formation at low redshifts to the epoch of reionization and cosmic dawn. It measures the integrated emission of atomic and molecular spectral lines from galaxies and the intergalactic medium over a broad range of frequencies, using instruments with aperture requirements that are greatly relaxed relative to surveys for single objects. A coordinated, comprehensive, multi-line intensity-mapping experimental effort can efficiently probe over 80% of the volume of the observable …
The study of molecular gas is crucial for understanding star formation, feedback, and the broader ecosystem of a galaxy as a whole. However, we have limited understanding of its physics and distribution in all but the nearest galaxies. We present a new technique for studying the composition and distribution of molecular gas in high-redshift galaxies inaccessible to existing methods. Our proposed approach is an extension of carbon monoxide intensity mapping methods, which have garnered significant experimental interest in recent years. These intensity mapping surveys target the 115 GHz $^{12}$C…
One of the toughest challenges in modern cosmology is to probe the small scales $k \sim 0.5\,$Mpc$^{-1}$ in the matter power spectrum and clustering. We show that such small scales will be accessible via upcoming line-intensity mapping surveys, with carbon monoxide (CO) emission from star-forming galaxies at high redshifts as an example. While these galaxies cannot be individually detected and the two-point correlations of the intensity fluctuation maps is not accessible at these scales, the voxel intensity distribution (VID) of the highly non-Gaussian intensity maps is sensitive to the integr…
We explore the potential of using intensity mapping surveys (MeerKAT, SKA) and optical galaxy surveys (DES, LSST) to detect HI clustering and weak gravitational lensing of 21cm emission in auto- and cross-correlation. Our forecasts show that high precision measurements of the clustering and lensing signals can be made in the near future using the intensity mapping technique. Such studies can be used to test the intensity mapping method, and constrain parameters such as the HI density $Ω_{\rm HI}$, the HI bias $b_{\rm HI}$ and the galaxy-HI correlation coefficient $r_{\rm HI-g}$.
Primordial non-Gaussianity (PNG) is a key probe of the origins of primordial fluctuations in the early universe. It has been shown that multi-tracer measurements of large-scale structure can produce high-precision measurements of PNG. Future line intensity mapping surveys are well-suited to these measurements owing to their ability to rapidly survey large volumes and access the large scales at which PNG becomes important. In this paper, we explore for the first time how multi-tracer PNG measurements with intensity mapping surveys depend on the sub-galactic scale physics which drives line emiss…
We compile available constraints on the carbon monoxide (CO) 1-0 luminosity functions and abundances at redshifts 0-3. This is used to develop a data driven halo model for the evolution of the CO galaxy abundances and clustering across intermediate redshifts. It is found that the recent constraints from the CO Power Spectrum Survey ($z \sim 3$; Keating et al. 2016), when combined with existing observations of local galaxies ($z \sim 0$; Keres et al. 2003), lead to predictions which are consistent with the results of smaller surveys at intermediate redshifts ($z \sim 1-2$). We provide convenien…
We present the first simulations of the high-redshift Ly$α$ intensity field that account for scattering in the intergalactic medium (IGM). Using a 3D Monte Carlo radiative transfer code, we find that Ly$α$ scattering smooths spatial fluctuations in the Ly$α$ intensity on small scales and that the spatial dependence of this smoothing depends strongly on the mean neutral fraction of the IGM. Our simulations find a strong effect of reionization on $k=0.1-1~{\rm Mpc^{-1}}$, with $P_{\rm Lyα}\propto k^{-1.75}$ for $\bar{x}_{\rm HI} = 0.63$ and $P_{\rm Lyα} \propto k^{-2.2}$ for $\bar{x}_{\rm HI} = …
We explore the possibility of performing an HI intensity mapping survey with the South African MeerKAT radio telescope, which is a precursor to the Square Kilometre Array (SKA). We propose to use cross-correlations between the MeerKAT intensity mapping survey and optical galaxy surveys, in order to mitigate systematic effects and produce robust cosmological measurements. Our forecasts show that precise measurements of the HI signal can be made in the near future. These can be used to constrain HI and cosmological parameters across a wide range of redshift.
The purpose of line-intensity mapping (IM), an emerging tool for extragalactic astronomy and cosmology, is to measure the integrated emission along the line of sight from spectral lines emitted from galaxies and the intergalactic medium. The observed frequency of the line then provides a distance determination allowing the three-dimensional distribution of the emitters to be mapped. Here we discuss the possibility to use these measurements to seek radiative decays or annihilations from dark-matter particles. The photons from monoenergetic decays will be correlated with the mass distribution, w…
Originally proposed as a cosmological probe of the large-scale structure, line intensity mapping (LIM) also offers a unique window into the astrophysics of galaxy evolution. Adding to the astrophysical explorations of LIM technique that have traditionally focused on small, non-linear scales, we present a novel method to study the global star formation law using forthcoming data from large-scale baryonic acoustic oscillation (BAO) intensity mapping. Using the amplitude of the percent-level but scale-dependent bias induced by baryon fraction fluctuations on BAO scales, we show that combining aut…
Current and future generations of intensity mapping surveys promise dramatic improvements in our understanding of galaxy evolution and large-scale structure. An intensity map provides a census of the cumulative emission from all galaxies in a given region and redshift, including faint objects that are undetectable individually. Furthermore, cross-correlations between line intensity maps and galaxy redshift surveys are sensitive to the line intensity and clustering bias without the limitation of cosmic variance. Using the Fisher information matrix, we derive simple expressions describing sensit…
We study a model of interacting dark matter and dark energy, in which the two components are coupled. We calculate the predictions for the 21-cm intensity mapping power spectra, and forecast the detectability with future single-dish intensity mapping surveys (BINGO, FAST and SKA-I). Since dark energy is turned on at $z\sim 1$, which falls into the sensitivity range of these radio surveys, the HI intensity mapping technique is an efficient tool to constrain the interaction. By comparing with current constraints on dark sector interactions, we find that future radio surveys will produce tight an…
We combine recent simulation work on the SFR--[C II] correlation at high redshift with empirical modeling of the galaxy--halo connection (via UniverseMachine) to forecast [C II] auto power spectra from $z\sim4$ to $z\sim8$. We compare these to sensitivities realistically expected from various instruments expected to come on-line in the next decade. If the predictions of our model are correct, [C II] should be detectable up to $z\sim6$ in this generation of surveys, but detecting [C II] past the end of reionization will require a generational leap in line-intensity survey capabilities.
Line-intensity mapping observations will find fluctuations of integrated line emission are attenuated by varying degrees at small scales due to the width of the line emission profiles. This attenuation may significantly impact estimates of astrophysical or cosmological quantities derived from measurements. We consider a theoretical treatment of the effect of line broadening on both the clustering and shot-noise components of the power spectrum of a generic line-intensity power spectrum using a halo model. We then consider possible simplifications to allow easier application in analysis, partic…
21cm intensity mapping is a novel approach aimed at measuring the power spectrum of density fluctuations and deducing cosmological information, notably from the Baryonic Acoustic Oscillations (BAO). We give an update on the progress of BAO from Integrated Neutral Gas Observations (BINGO) which is a single dish intensity mapping project. First we explain the basic ideas behind intensity mapping concept before updating the instrument design for BINGO. We also outline the survey we plan to make and its projected science output including estimates of cosmological parameters.
Line Intensity Mapping Prediction from the Cosmic Dawn (CoDa) III Simulation for H$α$ from Galaxies and the Intergalactic Medium during the Epoch of Reionization
Line intensity mapping (LIM) is a promising probe to study star formation, the large-scale structure of the Universe, and the epoch of reionization (EoR). Since carbon monoxide (CO) is the second most abundant molecule in the Universe except for molecular hydrogen ${\rm H}_2$, it is suitable as a tracer for LIM surveys. However, just like other LIM surveys, CO intensity mapping also suffers strong foreground contamination that needs to be eliminated for extracting valuable astrophysical and cosmological information. In this work, we take $^{12}$CO($\it J$=1-0) emission line as an example to in…
The abundance of cold molecular gas plays a crucial role in models of galaxy evolution. While deep spectroscopic surveys of CO emission lines have been a primary tool for measuring this abundance, the difficulty of these observations has motivated alternative approaches to studying molecular gas content. One technique, line intensity mapping, seeks to constrain the average molecular gas properties of large samples of individually undetectable galaxies through the CO brightness power spectrum. Here we present constraints on the cross-power spectrum between CO intensity maps and optical galaxy c…
Line intensity mapping (LIM) is a promising observational method to probe large-scale fluctuations of line emission from distant galaxies. Data from wide-field LIM observations allow us to study the large-scale structure of the universe as well as galaxy populations and their evolution. A serious problem with LIM is contamination by foreground/background sources and various noise contributions. We develop conditional generative adversarial networks (cGANs) that extract designated signals and information from noisy maps. We train the cGANs using 30,000 mock observation maps with assuming a Gaus…
We combine available constraints on the local CII 158 $μ$m line luminosity function from galaxy observations (Hemmati et al. 2017), with the evolution of the star-formation rate density and the recent CII intensity mapping measurement in Pullen et al. (2018, assuming detection), to derive the evolution of the CII luminosity - halo mass relation over $z \sim 0-6$. We develop convenient fitting forms for the evolution of the CII luminosity - halo mass relation, and forecast constraints on the CII intensity mapping power spectrum and its associated uncertainty across redshifts. We predict the sen…
The cosmological X-ray emission associated to the possible radiative decay of sterile neutrinos is composed by a collection of lines at different energies. For a given mass, each line corresponds to a given redshift. In this work, we cross correlate such line emission with catalogs of galaxies tracing the dark matter distribution at different redshifts. We derive observational prospects by correlating the X-ray sky that will be probed by the eROSITA and Athena missions with current and near future photometric and spectroscopic galaxy surveys. A relevant and unexplored fraction of the parameter…
Intensity mapping, which images a single spectral line from unresolved galaxies across cosmological volumes, is a promising technique for probing the early universe. Here we present predictions for the intensity map and power spectrum of the CO(1-0) line from galaxies at z ~ 2.4 -2.8, based on a parameterized model for the galaxy-halo connection, and demonstrate the extent to which properties of high-redshift galaxies can be directly inferred from such observations. We find that our fiducial prediction should be detectable by a realistic experiment. Motivated by significant modeling uncertaint…
The 21-cm and Lyman Alpha lines are the dominant line-emission spectral features at opposite ends of the spectrum of hydrogen. Each line can be used to create three dimensional intensity maps of large scale structure. The sky brightness at low redshift due to Lyman Alpha emission is estimated to be 0.4 Jy/Steradian, which is brighter than the zodiacal light foreground.
We propose an innovative method for measuring the neutral hydrogen (HI) content of an optically-selected spectroscopic sample of galaxies through cross-correlation with HI intensity mapping measurements. We show that the HI-galaxy cross-power spectrum contains an additive shot noise term which scales with the average HI brightness temperature of the optically-selected galaxies, allowing constraints to be placed on the average HI mass per galaxy. This approach can estimate the HI content of populations too faint to directly observe through their 21cm emission over a wide range of redshifts. Thi…
Imaging surveys of galaxies will have a high number density and angular resolution yet a poor redshift precision. Intensity maps of neutral hydrogen (HI) will have accurate redshift resolution yet will not resolve individual sources. Using this complementarity, we show how the clustering redshifts approach, proposed for spectroscopic surveys can also be used in combination with intensity mapping observations to calibrate the redshift distribution of galaxies in an imaging survey and, as a result, reduce uncertainties in photometric redshift measurements. We show how the intensity mapping surve…
We present a code to generate mock observations of 21 cm intensity mapping experiments. The emphasis of the code is on reducing the computational cost of running a full-blown simulation, trading computational time for accuracy. The code can be used to generate independent realizations of the cosmological signal and foregrounds, which are necessary, for instance, in order to obtain realistic forecasts for future intensity mapping experiments. The code is able to reproduce the correct angular and radial clustering pattern for the cosmological signal, including redshift-space distortions, lightco…
We present a systematic study of the intensity mapping technique using updated models for the different emission lines from galaxies and identify which ones are more promising for cosmological studies of the post reionization epoch. We consider the emission of ${\rm Lyα}$, ${\rm Hα}$, H$β$, optical and infrared oxygen lines, nitrogen lines, CII and the CO rotational lines. We then identify that ${\rm Lyα}$, ${\rm Hα}$, OII, CII and the lowest rotational CO lines are the best candidates to be used as IM probes. These lines form a complementary set of probes of the galaxies emission spectra. We …
We assume that dark matter is comprised of axion-like particles (ALPs) generated by the realignment mechanism in the post-inflationary scenario. This leads to isocurvature fluctuations with an amplitude of order one for scales comparable to the horizon at the time when the ALP field starts oscillating. The power spectrum of these fluctuations is flat for small wave numbers, extending to scales relevant for cosmological observables. Denoting the relative isocurvature amplitude at $k_*$ = 0.05 Mpc$^{-1}$ by $f_{\rm iso}$, Planck observations of the cosmic microwave background (CMB) yield $f_{\rm…
A direct detection of neutral hydrogen intensity mapping on Mpc scales at $z\approx 0.32$ and $z\approx 0.44$
The University of Manchester（曼彻斯特大学）；University of the Western Cape（西开普敦大学）；McGill University（麦吉尔大学）；The University of Edinburgh（爱丁堡大学）；South African Radio Observatory (SARAO)（南非射电天文台）
Intensity mapping has emerged as a promising tool to probe the three-dimensional structure of the universe. The traditional approach of galaxy redshift surveys is based on individual galaxy detection, typically performed by thresholding and digitizing large-scale intensity maps. By contrast, intensity mapping uses the integrated emission from all sources in a 3D pixel (or voxel) as an analog tracer of large-scale structure. In this work, we develop a formalism to quantify the performance of both approaches when measuring large-scale structures. We compute the Fisher information of an arbitrary…
We forecast the ability of future-generation experiments to detect the fine-structure lines of the carbon and oxygen ions, [CII] and [OIII] in intensity mapping (IM) from the Epoch of Reionization ($z \sim 6-8$). Combining the latest empirically derived constraints relating the luminosity of the [OIII] line to the ambient star-formation rate, and using them in conjunction with previously derived estimates for the abundance of [CII] in haloes, we predict the expected auto-correlation IM signal to be observed using new experiments based on the Fred Young Submillimetre Telescope (FYST) and the ba…
Line-intensity mapping surveys probe large-scale structure through spatial variations in molecular line emission from a population of unresolved cosmological sources. Future such surveys of carbon monoxide line emission, specifically the CO(1-0) line, face potential contamination from a disjoint population of sources emitting in a hydrogen cyanide emission line, HCN(1-0). This paper explores the potential range of the strength of HCN emission and its effect on the CO auto power spectrum, using simulations with an empirical model of the CO/HCN--halo connection. We find that effects on the obser…
The Millimeter-wave Intensity Mapping Experiment (mmIME) recently reported a detection of excess spatial fluctuations at a wavelength of 3 mm, which can be attributed to unresolved emission of several CO rotational transitions between $z\sim1-5$. We study the implications of this data for the high-redshift interstellar medium using a suite of state-of-the-art semianalytic simulations which have successfully reproduced many other sub-millimeter line observations across the relevant redshift range. We find that the semianalytic predictions are mildly in tension with the mmIME result, with a pred…
We forecast ability of dedicated 21 cm intensity mapping experiments to constraint primordial non-Gaussianity using power spectrum and bispectrum. We model the signal including the non-linear biasing expansion using a generalized halo model approach. We consider the importance of foreground filtering scale and of the foreground wedge. We find that the current generation intensity mapping experiments like CHIME do not posses sufficient sensitivity to be competitive with the existing limits. On the other hand, upcoming experiments like HIRAX can improve the current constraints and the proposed P…
The nature of the most abundant components of the Universe, dark energy and dark matter, is still to be uncovered. I tackle this subject considering a novel cosmological probe: the neutral hydrogen emitted 21cm radiation, observed with the intensity mapping technique. I analyse competitive and realistic dark energy and dark matter models and show how they produce distinctive and detectable effects on the 21cm signal. Moreover, I provide radio telescope forecasts showing how these models will be distinguishable in an unprecedented way.
Line intensity mapping is a superb tool to study the collective radiation from early galaxies. However, the method is hampered by the presence of strong foregrounds, mostly produced by low-redshift interloping lines. We present here a general method to overcome this problem which is robust against foreground residual noise and based on the cross-correlation function $ψ_{αL}(r)$ between diffuse line emission and Ly$α$ emitters (LAE). We compute the diffuse line (Ly$α$ is used as an example) emission from galaxies in a $(800{\rm Mpc})^3$ box at $z = 5.7$ and $6.6$. We divide the box in slices an…
We study the possibility to use line-intensity mapping (LIM) to seek photons from the radiative decay of neutrinos in the cosmic neutrino background. The Standard Model prediction for the rate for these decays is extremely small, but it can be enhanced if new physics increases the neutrino electromagnetic moments. The decay photons will appear as an interloper of astrophysical spectral lines. We propose that the neutrino-decay line can be identified with anisotropies in LIM clustering and also with the voxel intensity distribution. Ongoing and future LIM experiments will have -- depending on t…
In this paper, we study lensing of 21cm intensity mapping (IM). Like in the cosmic microwave background (CMB), there is no first order lensing in intensity mapping. The first effects in the power spectrum are therefore of second and third order. Despite this, lensing of the CMB power spectrum is an important effect that needs to be taken into account, which motivates the study of the impact of lensing on the IM power spectrum. We derive a general formula up to third order in perturbation theory including all the terms with two derivatives of the gravitational potential, i.e. the dominant terms…
