The extraction of a signal from some observational data sets that contain different contaminant emissions, often at a greater level than the signal itself, is a common problem in Astrophysics and Cosmology. The signal can be recovered, for instance, using a simple Wiener filter. However, in certain cases, additional information may also be available, such as a second observation which correlates to a certain level with the sought signal. In order to improve the quality of the reconstruction, it would be useful to include as well this additional information. Under these circumstances, we have constructed a linear filter, the linear covariance-based filter, that extracts the signal from the data but takes also into account the correlation with the second observation. To illustrate the performance of the method, we present a simple application to reconstruct the so-called Integrated Sachs-Wolfe effect from simulated observations of the Cosmic Microwave Background and of catalogues of galaxies.
We analyse the local variance effect in the standard method for detecting the integrated Sachs-Wolfe effect (ISW) via cross-correlating the cosmic microwave background (CMB) with the large-scale structure (LSS). Local variance is defined as the systematic noise in the ISW detection that originates in the realisation of the matter distribution in the observed Universe. We show that the local variance contributes about 11 per cent to the total variance in the standard method, if a perfect and complete LSS survey up to z ~ 2 is assumed. Due to local variance, the estimated detection significance …
Aims: Implement a matched filter (MF) cross-correlation algorithm in multipole space and compare it to the standard Angular Cross Power Spectrum (ACPS) method. Apply both methods on a Integrated Sachs Wolfe (ISW) - Large Scale Structure (LSS) cross correlation scenario and study how sky masks influence the multipole range where signal arises and its comparison to theoretical predictions. Methods: The MF requires the inversion of a multipole covariance matrix that if $f_{sky} \lt 1$ is generally non-diagonal and singular. We use a SVD approach that focuses on those modes carrying most of the in…
The late-time growth of large scale structures (LSS) is imprinted in the CMBR anisotropy through the Integrated Sachs Wolfe (ISW) effect. This is perceived to be a very important observational probe of dark energy. Future observations of redshifted 21-cm radiation from the cosmological neutral hydrogen (HI) distribution hold the potential of probing the LSS over a large redshift range. We have investigated the possibility of detecting the ISW through cross-correlations between the CMBR anisotropies and redshifted 21-cm observations. Assuming that the HI traces the dark matter, we find that the…
We present a cross-correlation analysis of the WMAP cosmic microwave background (CMB) temperature anisotropies and the SDSS galaxy density fluctuations. We find significant detections of the angular CMB-galaxy correlation for both the flux limited galaxy sample (z~0.3) and the high redshift (z ~ 0.5) color selected sample. The signal is compatible with that expected from the integrated Sachs-Wolfe (ISW) effect at large angles (θ> 3deg) and the Sunyaev-Zeldovich (SZ) effect at small scales (θ< 1 deg). The detected correlation at low-z is in good agreement with a previous analysis using the APM …
We cross-correlate large scale structure (LSS) observations from a number of surveys with CMB anisotropies from WMAP to investigate the Integrated Sachs-Wolfe (ISW) effect as a function of redshift, covering z~0.1-2.5. Our main goal is to go beyond reporting detections towards developing a reliable likelihood analysis that allows one to determine cosmological constraints from ISW observations. With this in mind we spend a considerable amount of effort in determining the redshift-dependent bias and redshift distribution b(z)*dN/dz of these samples by matching with spectroscopic observations whe…
The scattering of temperature anisotropy quadrupole by free electrons in galaxy clusters leads to a now well-known polarization signal in the cosmic microwave background (CMB) fluctuations. Using multi-frequency polarization data, one can extract the temperature quadrupole and separate it from the contaminant polarization associated with the kinematic quadrupole due to transverse motion of clusters. At low redshifts, the temperature quadrupole contains a significant contribution from the integrated Sachs-Wolfe effect (ISW) associated with the growth of density fluctuations. Using polarization …
I discuss several issues that arise when trying to constrain the dark energy equation of state using correlations of the integrated Sachs-Wolfe effect with galaxy counts and lensing of the cosmic microwave background. These techniques are complementary to others such as galaxy shear surveys, and can use data that will already be obtained from currently planned observations. In regimes where cosmic variance and shot noise are the dominant sources of error, constraints could be made on the mean equation of state to within 0.33 and its first derivative to within 1.0. Perhaps more interesting is t…
We discuss the correlation between late-time integrated Sachs-Wolfe (ISW) effect in the cosmic microwave background (CMB) temperature anisotropies and the large scale structure of the local universe. This correlation has been proposed and studied in the literature as a probe of the dark energy and its physical properties. We consider a variety of large scale structure tracers suitable for a detection of the ISW effect via a cross-correlation. In addition to luminous sources, we suggest the use of tracers such as dark matter halos or galaxy clusters. A suitable catalog of mass selected halos fo…
Scattering of the temperature anisotropy quadrupole by free electrons in galaxy clusters leads to a secondary polarization signal in the cosmic microwave background (CMB) fluctuations. At low redshifts, the temperature quadrupole contains a significant contribution from the integrated Sachs-Wolfe (ISW) effect associated with the growth of density fluctuations. Using polarization data from a sample of clusters over a wide range in redshift, one can statistically establish the presence of the ISW effect and determine its redshift evolution. Given the strong dependence of the ISW effect on the ba…
A generic prediction of general relativity is that the cosmological linear density growth factor $D$ is scale independent. But in general, modified gravities do not preserve this signature. A scale dependent $D$ can cause time variation in gravitational potential at high redshifts and provides a new cosmological test of gravity, through early time integrated Sachs-Wolfe (ISW) effect-large scale structure (LSS) cross correlation. We demonstrate the power of this test for a class of $f(R)$ gravity, with the form $f(R)=-λ_1 H_0^2\exp(-R/λ_2H_0^2)$. Such $f(R)$ gravity, even with degenerate expans…
Secondary anisotropies of the cosmic microwave background (CMB) can be detected by using the cross-correlation between the large-scale structure (LSS) and the CMB temperature fluctuations. In such studies, chance correlations of primordial CMB fluctuations with the LSS are the main source of uncertainty. We present a method for reducing this noise by exploiting information contained in the polarisation of CMB photons. The method is described in general terms and then applied to our recently proposed optimal method for measuring the integrated Sachs-Wolfe (ISW) effect. We obtain an expected sig…
We construct a map of the time derivative of the gravitational potential traced by SDSS Luminous Red Galaxies. The potential decays on large scales due to cosmic acceleration, leaving an imprint on cosmic microwave background (CMB) radiation through the integrated Sachs-Wolfe (ISW) effect. With a template fit, we directly measure this signature on the CMB at a 2-sigma confidence level. The measurement is consistent with the cross-correlation statistic, strengthening the claim that dark energy is indeed the cause of the correlation. This new approach potentially simplifies the cosmological inte…
Observations of the Cosmic Microwave Background (CMB), large scale structure (LSS) and standard candles such as Type 1a Supernovae (SN) each place different constraints on the values of cosmological parameters. We assume an inflationary Cold Dark Matter model with a cosmological constant, in which the initial density perturbations in the universe are adiabatic. We discuss the parameter degeneracies inherent in interpreting CMB or SN data, and derive their orthogonal nature. We then present our preliminary results of combining CMB and SN likelihood functions. The results of combining the CMB an…
We evaluate the two point functions of the stress energy from the largest string simulations carried out so far. The two point functions are used to calculate the cosmic microwave background (CMB) and cold dark matter (CDM) power spectra from local cosmic string models for structure formation. We find that our spectra differ significantly from those previously calculated for both global and local defects. We find a higher Doppler peak at $l=400-600$ and a less severe bias problem than for global defects. Spectra were obtained for a variety of network energy-decay mechanisms.
CMB power spectra and the SNIa luminosity-redshift curves have difficulty distinguishing between models with the same average value of the dark energy equation of state. We propose using the cross-correlation of the CMB temperature anisotropy with large scale structure to help break this degeneracy.
In the light of the recent Boomerang and Maxima observations of the CMB which show an anomalously low second acoustic peak, we reexamine the prediction by Adams et al (1997) that this would be the consequence of a 'step' in the primordial spectrum induced by a spontaneous symmetry breaking phase transition during primordial inflation. We demonstrate that a deviation from scale-invariance around $k\sim0.1h$~Mpc$^{-1}$ can simultaneously explain both the feature identified earlier in the APM galaxy power spectrum as well the recent CMB anisotropy data, with a baryon density consistent with the B…
We discuss the possibilities of the simultaneous determination of the neutrino masses and the evolution of dark energy from future cosmological observations such as cosmic microwave background (CMB), large scale structure (LSS) and the cross correlation between them. Recently it has been discussed that there is a degeneracy between the neutrino masses and the equation of state for dark energy. It is also known that there are some degeneracies among the parameters describing the dark energy evolutions. We discuss the implications of these on the cross correlation of CMB with LSS in some details…
The observed apparent acceleration of the universe is usually attributed to negative pressure from a mysterious dark energy. This acceleration causes the gravitational potential to decay, heating or cooling photons travelling through crests or troughs of large-scale matter density fluctuations. This phenomenon, the late-time integrated Sachs-Wolfe (ISW) effect, has been detected, albeit at low significance, by cross-correlating various galaxy surveys with the Cosmic Microwave Background (CMB). Recently, the best evidence has come from the statistical combination of results from multiple correl…
In a universe with a cosmological constant, the large-scale gravitational potential varies in time and this is, in principle, observable. Using an N-body simulation of a $Λ$CDM universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, $\dotΦ$, needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB). The linear part of the $\dotΦ$ power spectrum (the integrated Sachs-Wolfe effect or ISW) drops quickly as the relative importance of $Ω_Λ$ diminishes at high redshift, while the non-linear part (th…
In this paper I investigate the family of mixed three-point correlation functions <tau^q gamma^{3-q}>, q=0,1,2, between the integrated Sachs-Wolfe (iSW) temperature perturbation tau and the galaxy overdensity gamma as a tool for detecting the gravitational interaction of cosmic microwave background (CMB) photons with the potentials of nonlinearly evolving cosmological structures. Both the iSW-effect as well as the galaxy overdensity are derived in hyper-extended perturbation theory to second order and I emphasise the different parameter sensitivities of the linear and non-linear iSW-effect. I …
In the mirror world hypothesis the mirror baryonic component emerges as a possible dark matter candidate. Here we study the behaviour of the mirror dark matter and the differences from the more familiar CDM candidate for structure formation, cosmic microwave background and large scale structure. We show mirror models for CMB and LSS power spectra and compare them with observations, obtaining bounds on the mirror parameter space.
We explore the signature of a cold dark matter compensated void in the quasi non-linear regime. We find that this void is entirely a cold spot on the microwave background, in contrast to a non-linear void. On the last scattering surface(LSS), it appears as either a hot or cold spot depending on where this surface cuts the void. In addition, because the usual cancellations do not occur, the void's LSS signature can be very large as it is proportional to $R/H^{-1}$ rather than $(R/H^{-1})^3$. This implies strict limits for voids on the LSS.
The cross-correlation between cosmic microwave background (CMB) temperature anisotropies and the large scale structure (LSS) traced by the galaxy distribution, or sources at different wavelengths, is now well known. This correlation results from the integrated Sachs-Wolfe (ISW) effect in CMB anisotropies generated at late times due to the dark energy component of the Universe. In a reionized universe, the ISW quadrupole rescatters and contributes to the large-scale polarization signal. Thus, in principle, the large-scale polarization bump in the E-mode should also be correlated with the galaxy…
The effect of quintessence perturbations on the ISW effect is studied for a mixed dynamical scalar field dark energy (DDE) and pressureless perfect fluid dark matter. A new and general methodology is developed to track the growth of the perturbations, which uses only the equation of state (EoS) parameter $w_{\rm DDE} (z) \equiv p_{\rm DDE}/ρ_{\rm DDE}$ of the scalar field DDE, and the initial values of the the relative entropy perturbation (between the matter and DDE) and the intrinsic entropy perturbation of the scalar field DDE as inputs. We also derive a relation between the rest frame soun…
Observations of the Cosmic Microwave Background (CMB) are discussed, with particular emphasis on current ground-based experiments and on future satellite, balloon and interferometer experiments. Observational techniques and the effects of contaminating foregrounds are highlighted. Recent CMB data is used with large scale structure (LSS) data to constrain cosmological parameters and the complementary nature of CMB, LSS and supernova distance data is emphasized.
We present an analysis of the constraining power of future measurements of the Integrated Sachs-Wolfe (ISW) effect on models of the equation of state of dark energy as a function of redshift, w(z). To achieve this, we employ a new parameterization of w, which utilizes the mean value of w(z) (<w>) as an explicit parameter. This helps to separate the information contained in the estimation of the distance to the last scattering surface (from the CMB) from the information contained in the ISW effect. We then use Fisher analysis to forecast the expected uncertainties in the measured parameters fro…
Constraining cosmological parameters from measurements of the Integrated Sachs-Wolfe effect requires developing robust and accurate methods for computing statistical errors in the cross-correlation between maps. This paper presents a detailed comparison of such error estimation applied to the case of cross-correlation of Cosmic Microwave Background (CMB) and large-scale structure data. We compare theoretical models for error estimation with montecarlo simulations where both the galaxy and the CMB maps vary around a fiducial auto-correlation and cross-correlation model which agrees well with th…
We review the relevant 10+ parameters associated with inflation and matter content; the relation between LSS and primary and secondary CMB anisotropy probes; COBE constraints on energy injection; current anisotropy band-powers which strongly support the gravitational instability theory and suggest the universe could not have reionized too early. We use Bayesian analysis methods to determine what current CMB and CMB+LSS data imply for inflation-based Gaussian fluctuations in tilted $Λ$CDM, $Λ$hCDM and oCDM model sequences with age 11-15 Gyr, consisting of mixtures of baryons, cold (and possibly…
We explore the main physical processes which potentially affect the topological signal in the Cosmic Microwave Background (CMB) for a range of toroidal universes. We consider specifically reionisation, the integrated Sachs-Wolfe (ISW) effect, the size of the causal horizon, topological defects and primordial gravitational waves. We use three estimators: the information content, the S/N statistic and the Bayesian evidence. While reionisation has nearly no effect on the estimators, we show that taking into account the ISW strongly decreases our ability to detect the topological signal. We also s…
Dark matter currents in the large-scale structure give rise to gravitomagnetic terms in the metric, which affect the light propagation. Corrections to the weak lensing power spectrum due to these gravitomagnetic potentials are evaluated by perturbation theory. A connection between gravitomagnetic lensing and the integrated Sachs-Wolfe (iSW) effect is drawn, which can be described by a line-of-sight integration over the divergence of the gravitomagnetic vector potential. This allows the power spectrum of the iSW-effect to be derived within the framework of the same formalism as derived for grav…
We show how estimates of parameters characterizing inflation-based theories of structure formation localized over the past year when large scale structure (LSS) information from galaxy and cluster surveys was combined with the rapidly developing cosmic microwave background (CMB) data, especially from the recent Boomerang and Maxima balloon experiments. All current CMB data plus a relatively weak prior probability on the Hubble constant, age and LSS points to little mean curvature (Omega_{tot} = 1.08\pm 0.06) and nearly scale invariant initial fluctuations (n_s =1.03\pm 0.08), both predictions …
A broad class of dark energy models, which have been proposed in attempts at solving the cosmological constant problems, predict a late time variation of the equation of state with redshift. The variation occurs as a scalar field picks up speed on its way to negative values of the potential. The negative potential energy eventually turns the expansion into contraction and the local universe undergoes a big crunch. In this paper we show that cross-correlations of the CMB anisotropy and matter distribution, in combination with other cosmological data, can be used to forecast the imminence of suc…
This is the second paper in the series that confronts predictions of a model of the landscape with cosmological observations. We show here how the modifications of the Friedmann equation due to the decohering effects of long wavelength modes on the wavefunction of the Universe defined on the landscape leave unique signatures on the CMB spectra and large scale structure (LSS). We show that the effect of the string corrections is to suppress $σ_8$ and the CMB $TT$ spectrum at large angles, thereby bringing WMAP and SDSS data for $σ_8$ into agreement. We find interesting features imprinted on the…
Using both cosmic microwave background and large scale structure data, we put stringent bounds on the possible cold dark matter, baryon and neutrino isocurvature contributions to primordial fluctuations in the Universe. Neglecting the possible effects of spatial curvature, tensor perturbations and reionization, we perform a Bayesian likelihood analysis with nine free parameters, and find that the relative amplitude of the isocurvature component cannot be larger than about a few percent for uncorrelated models. On the other hand, for correlated adiabatic and isocurvature components, the ratio c…
We compute CBR anisotropies in mixed models with different hot components, including neutrinos or volatile HDM arising from the decay of heavier particles. The CBR power spectra of these models exhibit a higher doppler peak than CDM, and the discrepancy is even stronger in volatile models when the decay gives rise also to a neutral scalar. CBR experiments, together with Large Scale Structure (LSS) data, are then used to constrain the space parameter of mixed models, when values of the primeval spectral index n > 1 are also considered. Even if n>1 is allowed, however, LSS alone prescribes that …
The past, present and future of cosmic microwave background (CMB) anisotropy research is discussed, with emphasis on the Boomerang and Maxima balloon experiments. These data are combined with large scale structure (LSS) information and high redshift supernova (SN1) observations to explore the inflation-based cosmic structure formation paradigm. Here we primarily focus on a simplified inflation parameter set, {omega_b,omega_{cdm},Omega_{tot}, Omega_Q,w_Q, n_s,tau_C, sigma_8}. After marginalizing over the other cosmic and experimental variables, we find the current CMB+LSS+SN1 data gives Omega_{…
A new inflationary scenario whose exponential potential $V(Φ)$ has a quadratic dependence on the field $Φ$ in addition to the standard linear term is confronted with the tree-year observations of the Wilkinson-Microwave Anisotropy Probe and the Sloan Digital Sky Survey data. The number of e-folds ($N$), the ratio of tensor-to-scalar perturbations ($r$), the spectral scalar index of the primordial power spectrum ($n_s$) and its running ($dn_s/d\ln k$) depend on the dimensionless parameter $α$ multiplying the quadratic term in the potential. In the limit $α\to 0$ all the results of the standard …
We study the integrated Sachs-Wolfe (ISW) effect using a model-independent parameterization of the dark energy equation of state. Cosmic variance severely restricts the class of models distinguishable from Lambda-CDM. In particular if the present value of the equation of state satisfies w_Q^o <-0.8, the only distinguishable models are those with a rapidly varying equation of state, characterized by a late transition occurring at redshifts z<3. Due to the degeneracy with other cosmological parameters, models with a slowly varying behavior cannot be differentiated from each other or from a cosmo…
In the mirror world hypothesis the mirror baryonic component emerges as a possible dark matter candidate. An immediate question arises: how the mirror baryons behave and what are the differences from the more familiar dark matter candidates as e.g. cold dark matter? In this paper we answer quantitatively to this question. First we discuss the dependence of the relevant scales for the structure formation (Jeans and Silk scales) on the two macroscopic parameters necessary to define the model: the temperature of the mirror plasma (limited by the Big Bang Nucleosynthesis) and the amount of mirror …
I review recent constraints on and claimed detections of a time-varying fine-structure constant $α$, and discuss our own work, particularly on the effects of $α$ in the cosmic microwave background and large-scale structure. Our results are consistent with no variation in $α$ from the epoch of recombination to the present day, and restrict any such variation to be less than about 4%. The forthcoming MAP and Planck experiments will be able to break most of the currently existing degeneracies between $α$ and other parameters, and measure $α$ to better than percent accuracy.
Models with late time cosmic acceleration, such as the Lambda-dominated CDM model, predict a freeze out for the growth of linear gravitational potential at moderate redshift z<1, what can be observed as temperature anisotropies in the CMB: the so called integrated Sachs-Wolfe (ISW) effect. We present a direct measurement of the ISW effect based on the angular cross-correlation function, w_{TG}, of CMB temperature anisotropies and dark-matter fluctuations traced by galaxies. We cross-correlate the first-year WMAP data in combination with the APM Galaxy survey. On the largest scales, theta = 4-1…
We present cosmological perturbation theory in neutrinos probe interacting dark-energy models, and calculate cosmic microwave background anisotropies and matter power spectrum. In these models, the evolution of the mass of neutrinos is determined by the quintessence scalar field, which is responsible for the cosmic acceleration today. We consider several types of scalar field potentials and put constraints on the coupling parameter between neutrinos and dark energy. Assuming the flatness of the universe, the constraint we can derive from the current observation is $\sum m_ν < 0.87 eV$ at the 9…
We present and constrain a cosmological model which component is a pressureless fluid with bulk viscosity as an explanation for the present accelerated expansion of the universe. We study the particular model of a constant bulk viscosity coefficient ζ_m. The possible values of ζ_m are constrained using the cosmological tests of SNe Ia Gold 2006 sample, the CMB shift parameter R from the three-year WMAP observations, the Baryon Acoustic Oscillation (BAO) peak A from the Sloan Digital Sky Survey (SDSS) and the Second Law of Thermodynamics (SLT). It was found that this model is in agreement with …
We use cosmic microwave background and large scale structure data to test a broad and physically well-motivated class of inflationary models: those with flat tree-level potentials (typical in supersymmetry). The non-trivial features of the potential arise from radiative corrections which give a simple logarithmic dependence on the inflaton field, making the models very predictive. We also consider a modified scenario with new physics beyond a certain high-energy cut-off showing up as non-renormalizable operators (NRO) in the inflaton field. We find that both kinds of models fit remarkably well…
We use cosmic microwave background (CMB) and large-scale structure data to constrain cosmological models where the primordial perturbations have both an adiabatic and a cold dark matter (CDM) isocurvature component. We allow for a possible correlation between the adiabatic and isocurvature modes, and for different spectral indices for the power in each mode and for their correlation. We do a likelihood analysis with 11 independent parameters. We discuss the effect of choosing the pivot scale for the definition of amplitude parameters. The upper limit for the isocurvature fraction is 18% around…
This is the second paper of a series devoted to the study of the cosmological implications of the existence of mirror dark matter. The parallel hidden mirror world has the same microphysics as the observable one and couples the latter only gravitationally. The primordial nucleosynthesis bounds demand that the mirror sector should have a smaller temperature T' than the ordinary one T, and by this reason its evolution can be substantially deviated from the standard cosmology. In this paper we took scalar adiabatic perturbations as the input in a flat Universe, and computed the power spectra for …
We obtain very stringent bounds on the possible cold dark matter, baryon and neutrino isocurvature contributions to the primordial fluctuations in the Universe, using recent cosmic microwave background and large scale structure data. In particular, we include the measured temperature and polarization power spectra from WMAP and ACBAR, as well as the matter power spectrum from the 2dF galaxy redshift survey. Neglecting the possible effects of spatial curvature, tensor perturbations and reionization, we perform a Bayesian likelihood analysis with nine free parameters, and find that the amplitude…
We explore constraints on extensions to the standard models of cosmology and particle physics which modify the early-Universe expansion rate S = H'/H (parametrized by the effective number of neutrinos N_nu). The constraints on N_nu and the baryon density parameter (eta_B = n_B/n_gamma = 10^(-10)*eta_10) from BBN at 20 minutes are compared with those from the CMB at 400 kyr and LSS at 14 Gyr. BBN provides the strongest constraint on N_nu (1.6 < N_nu < 3.3 at 95% confidence), but a weaker constraint on eta_B. The CMB/LSS best constrain the baryon density (5.9 < eta_10 < 6.4 at 95% confidence), i…
We study systematically which features in the cosmic microwave background (CMB) and large-scale structure (LSS) probe various inhomogeneous properties of the dark sectors (including neutrinos, dark matter, and dark energy). We stress, and quantify by simple formulas, that the primary CMB anisotropies are very susceptible to the gravitational potentials during horizon entry, less at recombination. The CMB thus allows us to scan Φ+Ψand the underlying dark kinetics for all redshifts z~1-10^5. LSS, on the other hand, responds strongest to Φat low redshifts. Dark perturbations are often parameteriz…
Generic models for the origin of structure predict a spectrum of initial fluctuations with a mixture of adiabatic and isocurvature perturbations. Using the observed anisotropies of the cosmic microwave backgound, the matter power spectra from large scale structure surveys and the luminosity distance vs redshift relation from supernovae of type Ia, we obtain strong bounds on the possible cold dark matter/baryon as well as neutrino isocurvature contributions to the primordial fluctations in the Universe. Neglecting the possible effects of spatial curvature and tensor perturbations, we perform a …
We consider the angular power spectrum in a finite universe with different boundary conditions and perform a fit to the CMB, LSS and supernova data. A finite universe could be the consequence of a holographic constraint, giving rise to an effective IR cutoff at the future event horizon. In such a model there is a cosmic duality relating the dark energy equation of state and the power spectrum, which shows a suppression and oscillatory behaviour that is found to describe the low l features extremely well. However, much of the discussion here will also apply if we actually live inside an expandi…
