Finite Larmor radius effects in the magnetosphere

This article reviews theories and observations related to effects produced by finite (and large) Larmor radii of charged particles in the magnetosphere. The FLR effects depend on =r _H /L, wherer _H is the Larmor radius andL is the spatial scale for field/plasma inhomogeneity. The parameter is a basic expansion parameter for most equations describing plasma dynamics in the magnetosphere. The FLR effects enter naturally the drift approximation for particle motion and represent also non-ideal MHD terms in the fluid formalism. The linear and higher order terms in lead to charge separation, energization of particles, and produce viscosity without collisions. The FLR effects introduce also important corrections to the dispersion relations for MHD waves and drift instabilities. Expansion of plasma into magnetic field leads to filamentation of the plasma boundary and to creation of structures with thickness less than an ion gyroradius. Large Larmor radius effects (1) in curved magnetic field geometry lead to stochastic behaviour of particle trajectories and to deterministic chaos. The tiny scale of the electron and ion gyroradii does not necessarily mean that FLR/LLR phenomena have negligible effect on the macroscopic dynamics and energetics of the whole magnetosphere. On the contrary, the small scale gyro-effects may provide the physical mechanism for gyroviscous coupling between the solar wind and the magnetosphere, the mechanism for triggering disruption of the magnetotail current layer, and the mechanism for parallel electric field that accelerate auroral particles.     

Some results of ionospheric investigations by means of coherent radiowaves emitted by satellites

In this paper we discuss theoretical expressions, determining the difference of Doppler shifts of various coherent radiowave frequencies emitted by a radiator moving in the ionosphere or interplanetary medium. The rotating Doppler effect (Faraday effect) caused by the Doppler shifts ±H of the ordinary and extraordinary waves is also considered. In a three-dimensional inhomogeneous ionosphere, stationary in time (N/t = 0), is determined in the general case, by an equation with three variables. The equation for proper depends only on the local value of the electron concentration N _c around the radiator and on integral values, determining, by means of additional calculations, the angle of refraction or its components, the horizontal gradients of electron concentration N/x and N/y, and in some cases, the integral electron concentration ₀ ^zcN dz. We describe the analysis of the measurements, made with the satellites Cosmos I, II and partially XI, assuming that N/t = N/y = 0, with a two variables equation. The expected errors are considered. The results coincide well for different points (Moscow, The Crimea, Sverdlovsk) and thus agree with the measurements of _H and with height-frequency ionospheric characteristics. The curve giving electron concentration versus height N (z) in the outer ionosphere (above the maximum of F2), shows a new maximum higher than the main maximum of the ionosphere N _MF2 at 120–140 km. At this maximum the value of N (z) is (0.9–0.95) N _MF2. The new data on the large-scale horizontal inhomogeneities of the ionosphere, exceed the previous ones by about a factor 10. By means of the irregular variations of the spectrum W() of the inhomogenous formation is determined. Three unknown constant maxima with values 16 to 18 km, 28 to 32 km and 100 to 120 km are found. The spectrum W () mainly characterizes the local properties of the ionosphere along the orbit of the satellite.     

The synthesis of helium and CNO isotopes in massive stars

We present helium and CNO isotopic yields for massive mass-losing stars in the initial mass range 15M M _i 50M . We investigate their dependence on assumptions about mass loss rates, internal mixing processes, and metallicity, and specify the contributions from stellar winds and from supernova ejecta.     

Energy coupling between the solar wind and the magnetosphere

This paper describes in detail how we are led to the first approximation expression for the solar wind-magnetosphere energy coupling function , which correlates well with the total energy consumption rate U _T of the magnetosphere. It is shown that is the primary factor which controls the time development of magnetospheric substorms and storms. The finding of this particular expression indicates how the solar wind couples its energy to the magnetosphere; the solar wind and the magnetosphere constitute a dynamo. In fact, the power P generated by the dynamo can be identified as by using a dimensional analysis. Furthermore, the finding of indicates that the magnetosphere is closer to a directly driven system than to an unloading system which stores the generated energy before converting it to substorm and storm energies. Therefore, the finding of and its implications have considerably advanced and improved our understanding of magnetospheric processes. The finding of has also led us to a few specific future problems in understanding relationships between solar activity and magnetospheric disturbances, such as a study of distortion of the solar current disk and the accompanying changes of . It is also pointed out that one of the first tasks in the energy coupling study is an improvement of the total energy consumption rate U _T of the magnetosphere. Specific steps to be taken in this study are suggested.     

The importance of anisotropic interstellar turbulence and molecular-cloud magnetic mirrors for galactic cosmic-ray propagation

Recent studies suggest that when magnetohydrodynamic (MHD) turbulence is excited by stirring a plasma at large scales, the cascade of energy from large to small scales is anisotropic, in the sense that small-scale fluctuations satisfy the inequality k k , where k and k are, respectively, the components of a fluctuations wave vector and to the background magnetic field. Such anisotropic fluctuations are very inefficient at scattering cosmic rays. Results based on the quasilinear approximation for scattering of cosmic rays by anisotropic MHD turbulence are presented and explained. The important role played by molecular-cloud magnetic mirrors in confining and isotropizing cosmic rays when scattering is weak is also discussed.     

A New Mechanism of Coronal Heating

The interaction between network magnetic fields and emerging intranetwork fields may lead to magnetic reconnection and microflares, which generate fast shocks with an Alfvén Mach number M _A<2. Protons and less abundant ions in the solar corona are then heated and accelerated by fast shocks. Our study of shock heating shows that (a) the nearly nondeflection of ion motion across the shock ramp leads to a large perpendicular thermal velocity (v _th), which is an increasing function of the mass/charge ratio; (b) the heating by subcritical shocks with 1.1 M_A 1.5 leads to a large temperature anisotropy with T/T 50 for O⁵⁺ ions and a mild anisotropy with T_/T 1.2 for protons; (c) the large perpendicular thermal velocity of He⁺⁺ and O⁵⁺ ions can be converted to the radial outflow velocity (u) in the divergent coronal field lines; and (d) the heating and acceleration by shocks with 1.1 M_A 1.5 can lead to u(O⁵⁺) v _th(O⁵⁺) 460 km s^–1 for O⁵⁺ ions, u(He⁺⁺) v _th(He⁺⁺) 360 km s^–1 for He⁺⁺ ions, and u(H⁺) v _th(H⁺) 240 km s^–1 for protons at r=3–4 R . Our results can explain recent SOHO observations of the heating and acceleration of protons and heavier ions in the solar corona.     

Characteristics of nearby interstellar matter

There is a warm tenuous partially ionized cloud (T10⁴ K,n(HI)0.1 cm^–3,n(Hii 0.22–0.44 cm^–3) surrounding the solar system which regulates the environment of the solar system, determines the structure of the heliopause region, and feeds neutral interstellar gas into the inner solar system. The velocity (V–20 km s^–1 froml335°,b0° in the local standard of rest) and enhanced Caii and Feii abundances of this cloud suggest an origin as evaporated gas from cloud surfaces in the Scorpius-Centaurus Association. Although the soft X-ray emission attributed to the Local Bubble is enigmatic, optical and ultraviolet data are consistent with bubble formation caused by star formation epochs in the Scorpius-Centaurus Association as regulated by the nearby spiral arm configuration. The cloud surrounding the solar system (the local fluff) appears to be the leading region of an expanding interstellar structure (the squall line) which contains a magnetic field causing polarization of the light of nearby stars, and also absorption features in nearby upwind stars. The velocity vectors of the solar system and local fluff are perpendicular in the local standard of rest. Combining this information with the low column densities seen towards Sirius in the anti-apex direction, and the assumption that the cloud velocity vector is parallel to the surface normal, suggests that the Sun entered the local fluff within the historical past (less than 10 000 years ago) and is skimming the surface of the cloud. Comparison of magnesium absorption lines towards Sirius and anomalous cosmic-ray data suggest the local fluff is in ionization equilibrium.Reason has moons, but moons not hers, Lie mirror'd on her sea, Confounding her astronomers, But, O! delighting me.Ralph Hodgson     

The magnetogram inversion technique: Applications to the problem of magnetospheric substorms

The magnetogram inversion technique (MIT) is based upon recordings of geomagnetic variations at the worldwide network of ground-based magnetometers. MIT ensures a calculation of a global spatial distribution of the electric field, currents and Joule heating in the ionosphere. Variant MIT-2 provides, additionally, continuous monitoring of the following parameters: Poynting vector flux from the solar wind into the magnetosphere (); power, both dissipated and accumulated in the magnetosphere; magnetic flux in the open tail; and the magnetotail length (l _T) (distance between the dayside and nightside neutral points in the Dungey model). Using MIT-2 and data of direct measurements in the solar wind, an analysis is made of a number of substorms, and a new scenario of substorms is suggested. The scenario includes the convection model, the model with a neutral line and the model of magnetosphere-ionosphere coupling (outside the current sheet), i.e., the three known models. A brief review is given of these and some other substorms models. A new element in the scenario is the strong positive feedback in the primary generator circuit, which ensures growth of the ratio = / _Aby an order of magnitude or more during the substorms. Here _Ais the Pointing vector flux in the Akasofu-Perrault approximation, i.e., without the feedback taken into account. The growth of during the substorm is caused only by the feedback effect. It is assumed that the feedback arises due to an elongation of the magnetotail, i.e., a growth of l _Tby a factor of (23) during the substorm.In the active phase of substorm, a part (the first active phase) has been identified, where the principal role in the energetics is played by the feedback mechanism and the external energy source (although the internal source plus reconnection inside the plasma sheet make a marked contribution). In the second active phase (expansion) the external generator (solar wind) is switched off, and the main role is now played by the internal energy source (the tail magnetic field and ionospheric wind energy).Models of DP-2 DP-1 transitions are also considered, as well as the magnetospheric substorm-solar flare analogy.     

Latitude and solar-cycle dependence of radial IMF intensity

I describe a simple procedure for extrapolating the observed solar magnetic field into the heliosphere, which averages the asymptotic fields computed using the standard source surface and current sheet models. The resultant field is characterized by strong latitudinal gradients (maintained by volume currents outside the source surface) and by abrupt reversals in direction at the current sheets. The model yields good agreement with the observed long-term variation of the radial IMF component in the ecliptic, and is used to predict the variation of |B _r | along the latitudinal trajectory of Ulysses during 1990–1994. As found in earlier studies, the magnitude ofB _r at any latitude is determined largely by the strength and relative orientation of the Sun's dipole moment.     

The Edison infrared space observatory

For five years, theEdison program has had the goal of developing new designs for infrared space observatories which will break the cost curve by permitting more capable missions at lower cost. Most notably, this has produced a series of models for purely radiative and radiative/mechanical (hybrid) cooling which do not use cryogens and optical designs which are not constrained by the coolant tanks. Purely radiatively-cooled models achieve equilibrium temperatures as low as about 20 K at a distance of 1 AU from the sun. More advancedEdison designs include mechanical cooling systems attached to the telescope assembly which lower the optical system temperature to 5 K or less. Via these designs, near-cryogenic temperatures appear achievable without the limitations of cryogenic cooling. OneEdison model has been proposed to the European Space Agency as the next generation infrared space observatory and is presently under consideration as a candidate ESA Cornerstone mission. The basic design is also the starting point for elements of future infrared space interferometers.     

Probing the solar wind acceleration region using spectroscopic techniques

Measurements of the intensities and profiles of UV and EUV spectral lines can provide a powerful tool for probing the physical conditions in the solar corona out to 8 R and beyond. We discuss here how measurements of spectral line radiation in conjunction with measurements of the white light K-corona can provide information on electron, proton and ion temperatures and velocity distribution functions; densities; chemical abundances and mass flow velocities. Because of the fundamental importance of such information, we provide a comprehensive review of the formation of coronal resonance line radiation, with particular emphasis on the H i L line, and discuss observational considerations such as requirements for rejection of stray light and effects of emission from the geocorona and interplanetary dust. Finally, we summarize some results of coronal H i L and white light observations acquired on sounding rocket flights.Paper presented at the IX-th Lindau Workshop The Source Region of the Solar Wind.     

Pulsars as gamma ray sources: Nebular shocks and magnetospheric gaps

I summarize the results of recent research on the structure and particle acceleration properties of relativistic shock waves in which the magnetic field is transverse to the flow direction in the upstream medium, and whose composition is primarily electrons and positrons with an admixture of heavy ions. Shocks which contain heavy ions that are a minority constituent by number but which carry most of the energy density in the upstream medium put 20% of the flow energy into a nonthermal population of pairs downstream, whose distribution in energy space is N(E) E ^-2, where N(E)dE is the number of particles with energy between E and E+dE. Synchrotron maser activity in the shock front, stimulated by the quasi-coherent gyration of the whole particle population as the plasma flowing into the shock reflects from the magnetic field in the shock front, provides the mechanism of thermalization and non-thermal particle acceleration. The maximum energy achievable by the pairs is _± m _± c ² = m _i c ² ₁/Z _i, where ₁ is the Lorentz factor of the upstream flow and Z _i is the atomic number of the ions. The shock's spatial structure contains a series of overshoots in the magnetic field, regions where the gyrating heavy ions compress the magnetic field to levels in excess of the eventual downstream value. These overshoots provide a new interpretation of the structure of the inner regions of the Crab Nebula, in particular of the wisps, surface brightness enhancements near the pulsar. The wisps appear brighter because the small Larmor radius pairs are compressed and radiate more efficiently in the regions of more intense magnetic field. This interpretation suggests that the structure of the shock terminating the pulsar's wind in the Crab Nebula is spatially resolved, and allows one to measure ₁ 4 × 10⁶, the upstream magnetic field B ₁ to be 3 × 10^-5 Gauss, as well as to show that the total ion flow is 3 × 10³⁴ elementary charges/sec, in good agreement with the total current flow predicted by the early Goldreich and Julian (1969) model. The total pair outflow is shown to be about 5 × 10³⁷ pairs per second, in good agreement with the particle flux required to explain the nebular X—ray source.The energetics of particle acceleration within the magnetospheres of rotation powered pulsars and the consequences for pulsed gamma ray emission are also briefly discussed. The gamma ray luminosity above 100 MeV is shown to scale in proportion to _R ^1/2 , as is in accord with some of the simplest ideas about polar cap models. Models based on acceleration in the outer magnetosphere are also briefly discussed.     

Nonlinear theory of the two-point correlation function

We consider the influence of the nonlinear stage of gravitational instability on the two-point correlation functions of gravitationally bound objects. Based on the theory of nonlinear gravitational contraction of a single density peak of dissipationless matter (Gurevich and Zybin, 1988a,b; 1990) we develop a method for calculating the two-point correlation functions of different objects of any mass. The method works good in the region of strong correlations and can be easily extended to calculate higher correlation functions. We show that the main contribution to the correlation function _i in the region of strong correlations _i 1 is made by pair systems located outside large clusters of objects. In this region the shape of _i is determined only by the nonlinear dynamics of gravitational contraction of dissipationless matter and has the form _i C ^–, where 1.8 is a universal parameter.     

Determination of the characteristics of the gamma-ray telescope Gamma-1

The Gamma-1 telescope has been developed through a collaboration of scientists in the USSR and France in order to conduct -ray astronomical observations within the energy range from 50 to 5000 MeV. The major characteristics of the telescope were established by Monte-Carlo simulations and calibrations made with the aid of electron and tagged -ray beams produced by an accelerator, and these have been found to be as follows: the effective area for photons coming along the instrument's axis varies from about 50 cm² at E = 50 MeV to approximately 230 cm² at E 300 MeV; the angular resolution (half opening of the cone embracing 68% events) is equal to 2.7° at E = 100 MeV, and 1.8° at E = 300 MeV; the energy resolution (FWHM) varies from 70% to 35% as the energy of the detected photons increases from 100 to 550 MeV; the telescope's field-of-view at the half-sensitivity level is 300–450 square degrees depending upon the spectrum of the detected radiation, and the event selection logic. Proceeding from the thus obtained characteristics it is demonstrated that a point source producing a photon flux J (E 100 MeV) = 3 × 10^-7 cm^-2 s^-1, can be detected with a 5 significance by observing it during 10⁶ s at the level of the Cygnus background, and a source having intensity J (E 100 MeV) = 10^-6 cm^-2 s^-1 can be detected to within a mean square positional accuracy of about 15.     

Concerning the interpretation of the ultraviolet radiation from B stars

Measurements of the shape of the ultraviolet spectrum from B stars are compared with the theoretical spectra predicted from a homogeneous series of eight model atmospheres which are known to be close to a state of radiative equilibrium and to give a good representation of the ordinarily observed spectral region. The broad-band photometer measurements of Byram, Chubb, and Friedman in the region 1314 indicate that the stars become brighter in the ultraviolet as their temperature increases. The theoretical spectra reproduce this trend. However, the theoretical spectra are about three times as bright at 1314 relative to their brightness at 5560 as is observed.The spectral observations at 50Å resolution of Stecher and Milligan of six absorption-line stars are compared in detail with theoretical spectra. The observed shape of the spectrum is reproduced well by the models from 2600 to longer wavelengths. At wavelengths shorter than 2600 Å, the observed fluxes from B stars are less than the predicted fluxes. At 2000 the deficiency is between a factor two and a factor four. The spectrum of Canis Majoris is observed to have a different shape from that found for four other early-type stars. In the case of Canis Majoris the deficiency at 2000 is about a factor 13.The proper manner in which to compare theory and observation is discussed and some astrophysical terminology is explained. Theoretical fluxes, , are given in Table 1 for eight early B type model atmospheres at wavelengths between the Lyman limit and 6251. These fluxes have been computed without consideration of the opacity due to line blanketing. It is shown that line blanketing can probably account for the differences noted between predicted and observed ultra-violet spectra of B stars. It is not necessary at present to invoke unusual sources of opacity in the stellar atmosphere or in the space between the star and the earth in order to explain the observations. Spectra of B stars in the 2000 region at sufficient resolution to show the line spectrum would clarify the problem.     

Characteristics and interpretation of the photometric variability of eta carinae and its nebula

A short review is given on the history of the peculiar variable object Car and on a number of relevant references describing and discussing its physical characteristics and behaviour, based on different types of observational techniques. The star is known to be variable since the 17th century. The excessive mass loss to which it was subject during the 19th century is now visible as an ellipsoidal reflection nebula of 15 diameter: the so-called homunculus. The remainder of the paper is spent on different kinds of problems partly based on the results of a decade of photometric monitoring in the VBLUW photometric system of Walraven. Foreground reddening and reddening by dust in the homunculus are determined and amount to E(B - V) _J = 0 50 and < 6, respectively. Scanning of the homunculus revealed an estimate for the photometric characteristics of the central object, which presumably consists of a massive hot star surrounded by a cooler gas envelope. The total luminosity is derived using fluxes of various sources in the wavelength region 0.15 < < 175 n resulting in M _bol = - 12 3 ± 0 2. The total observed flux corrected for foreground extinction corresponds to a star with R 96 R if T _eff 30 000 K. The mass may be near 150 M . The excess luminosity in 1843, when the star was presumably bolometrically at least 2 5 brighter than at present, may have been caused by envelope-energized pulsations when the star's luminosity was close to its Eddington limit. The temperature should then have been 50 000 K. The mass loss rate, during the excess luminosity phase lasting 30 yr, is estimated to amount to M 4 × 10^-3 M yr^-1. At present the mass loss may be M 10^-4M yr^-1. Since the homunculus is mainly built up from material expelled in the 30 yr interval (from 1830 to 1860), its total mass amounts to M _hom 0.15 M . The historical observations of the colours of Car and a comparison with the characteristics of S Dor type stars, suggest that Car was subject to a number of S Dor type phases similar to those of P Cyg (in the 17th century), S Dor and others. A satisfactory explanation is found for the complete historical light curve. The decrease in light after the 1843 maximum by 9 ^m , presumably consists of a fading of the luminosity excess and the S Dor effect by 2 5 and 3^m, respectively, and a 3 5 extinction by circumstellar dust. The small amplitude light variations which Car showed during the last decade, were studied with the aid of the variations of the Balmer jump. They are presumably caused by temperature variations of the central star.Percy and Welch (1983) (Publ. Astron. Soc. Pacific 95, 491) have observed P Cyg on a number of nights in 1982 and found for the photometric variations a time scale of 30 to 50 days and an amplitude of 0 _. ^m 15.Based partly on observations collected at the ESO, La Silla, Chile.     

The emission mechanisms for solar radio bursts

Emission mechanisms for meter- solar radio bursts are reviewed with emphasis on fundamental plasma emission.The standard version of fundamental plasma emission is due to scattering of Langmuir waves into transverse waves by thermal ions. It may be treated semi-quantitatively by analogy with Thomson scattering provided induced scattering is unimportant. A physical interpretation of induced scattering is given and used to derive the transfer equation in a semi-quantitative way. Solutions of the transfer equation are presented and it is emphasized that standard fundamental emission with brightness temperatures 10⁹ K can be explained only under seemingly exceptional circumstances.Two alternative fundamental emission mechanisms are discussed: coalescence of Langmuir waves with low-frequency waves and direct conversion due to a density inhomogeneity. It is pointed out for the first time that the coalescence process (actually a related decay process) can lead to amplified transverse waves. The coalescence process saturates when the effective temperature T ^t of the transverse waves reaches the effective temperature T ^l of the Langmuir waves. This saturation occurs provided the energy density in the low-frequency waves exceeds a specific value which is about 10^-9 of the thermal energy density for emission from the corona at 100 MHz. It is suggested that direct emission has been dismissed as a possible alternative without adequate justification.Second harmonic plasma emission is discussed and compared with fundamental plasma emission. It also saturates at T ^t T ^l , and this saturation should occur in the corona roughly for T ^l 10¹⁵ K. If fundamental plasma emission is attributed to coalescence with low-frequency waves, then for T ^l 10¹⁵ K the brightness temperatures at the two harmonics should be equal and equal to T ^l . This offers a natural explanation for the approximate equality of the two brightness temperature often found in type II and type III bursts.Analytic treatments of gyro-synchrotron emission are reviewed. The application of the mechanism to moving type IV bursts is discussed in view of bursts with 10¹⁰ K at 43 MHz.     

The primordial Helium-4 abundance determination: systematic effects

By extrapolating to O/H = N/H = 0 the empirical correlations Y–O/H and Y–N/H defined by a relatively large sample of 45 Blue Compact Dwarfs (BCDs), we have obtained a primordial ⁴Helium mass fraction Y _p=0.2443±0.0015 with dY/dZ=2.4±1.0. This result is in excellent agreement with the average Y _p=0.2452±0.0015 determined in the two most metal-deficient BCDs known, I Zw 18 (Z /50) and SBS 0335–052 (Z /41), where the correction for He production is smallest. The quoted error (1) of 1% is statistical and does not include systematic effects. We examine various systematic effects including collisional excitation of hydrogen lines, ionization structure and temperature fluctuation effects, and underlying stellar Hei absorption, and conclude that combining all systematic effects, our Y _p may be underestimated by 2–4%. Taken at face value, our Y _p implies a baryon-to-photon number ratio =(4.7^+1.0 _–0.8)×10^–10 and a baryon mass fraction _b h ² ₁₀₀=0.017±0.005 (2), consistent with the values obtained from deuterium and Cosmic Microwave Background measurements. Correcting Y _p upward by 2–4% would make the agreement even better.     

Non-BBN Constraints on the Key Cosmological Parameters

Since the baryon-to-photon ratio 10 is in some doubt at present, we ignore the constraints on 10 from big bang nucleosynthesis (BBN) and fit the three key cosmological parameters (h, M, 10) to four other observational constraints: Hubble parameter (ho), age of the universe (to), cluster gas (baryon) fraction (fo fGh3/2), and effective shape parameter (o). We consider open and flat CDM models and flat CDM models, testing goodness of fit and drawing confidence regions by the 2 method. CDM models with M = 1 (SCDM models) are accepted only because we allow a large error on ho, permitting h < 0.5. Open CDM models are accepted only for _M 0.4. CDM models give similar results. In all of these models, large 10 ( 6) is favored strongly over small 10 ( 2), supporting reports of low deuterium abundances on some QSO lines of sight, and suggesting that observational determinations of primordial 4He may be contaminated by systematic errors. Only if we drop the crucial o constraint are much lower values of M and 10 permitted.     

Large scale structure of the universe

The current situation with the cosmological model and fundamental constants is briefly reviewed. Here, we concentrate on evolutionary effects of large-scale structure formation, in particular, the relationship with the quasar distribution and dynamics is discussed. We argue that groups of bright quasars with few or more than dozen of members within regions l _LS(100–150)h ^–1 Mpc found atz<2 may belong to concentrations of young rich clusters of galaxies, and thus be distant Great Attractors like the local GA or the Shapley concentration. These early large-scale galactic structures (i) provide a natural way to bias the distribution of Abell clusters, and (ii) suggest that the spectrum of primordial density perturbations is nearly flat on scales encompassing both the cluster and GAs,l=k ^–1(10,100)h ^–1 Mpc: _k ² k ³ P(k) k , =1 _–0.4 ^+0.6 , whereP(k) is the power spectrum of density perturbations.