The modern physical description of the Universe as a whole can be traced back to Einstein, who argued theoretically for the so-called "cosmological principle": that the distribution of matter and energy must be homogeneous and isotropic on the largest scales. Today isotropy is well established (see the review by Wu, Lahav, & Rees 1999 [389]) for the distribution of faint radio sources, optically-selected galaxies, the X-ray background, and most importantly the cosmic microwave background (hereafter, CMB; see, e.g., Bennett et al. 1996 [36]). The constraints on homogeneity are less strict, but a cosmological model in which the Universe is isotropic but significantly inhomogeneous in spherical shells around our special location, is also excluded [155].
In General Relativity, the metric for a space which is spatially homogeneous and isotropic is the Friedman-Robertson-Walker metric, which can be written in the form
(1) |
where a(t) is the cosmic scale factor which describes expansion in time, and (R, , ) are spherical comoving coordinates. The constant k determines the geometry of the metric; it is positive in a closed Universe, zero in a flat Universe, and negative in an open Universe. Observers at rest remain at rest, at fixed (R, , ), with their physical separation increasing with time in proportion to a(t). A given observer sees a nearby observer at physical distance D receding at the Hubble velocity H(t)D, where the Hubble constant at time t is H(t) = d a(t) / dt. Light emitted by a source at time t is observed at t = 0 with a redshift z = 1/a(t) - 1, where we set a(t = 0) 1 for convenience (but note that old textbooks may use a different convention).
The Einstein field equations of General Relativity yield the Friedmann equation (e.g., Weinberg 1972 [376]; Kolb & Turner 1990 [205])
(2) |
which relates the expansion of the Universe to its matter-energy content. For each component of the energy density , with an equation of state p = p(), the density varies with a(t) according to the equation of energy conservation
(3) |
With the critical density
(4) |
defined as the density needed for k = 0, we define the ratio of the total density to the critical density as
(5) |
With _{m}, _{}, and _{r} denoting the present contributions to from matter (including cold dark matter as well as a contribution _{b} from baryons), vacuum density (cosmological constant), and radiation, respectively, the Friedmann equation becomes
(6) |
where we define H_{0} and _{0} = _{m} + _{} + _{r} to be the present values of H and , respectively, and we let
(7) |
In the particularly simple Einstein-de Sitter model (_{m} = 1, _{} = _{r} = _{k} = 0), the scale factor varies as a(t) t^{2/3}. Even models with non-zero _{} or _{k} approach the Einstein-de Sitter behavior at high redshift, i.e. when (1 + z) >> |_{m}^{-1} - 1| (as long as _{r} can be neglected). In this high-z regime the age of the Universe is,
(8) |
The Friedmann equation implies that models with _{k} = 0 converge to the Einstein-de Sitter limit faster than do open models.
In the standard hot Big Bang model, the Universe is initially hot and the energy density is dominated by radiation. The transition to matter domination occurs at z ~ 3500, but the Universe remains hot enough that the gas is ionized, and electron-photon scattering effectively couples the matter and radiation. At z ~ 1100 the temperature drops below ~ 3000K and protons and electrons recombine to form neutral hydrogen. The photons then decouple and travel freely until the present, when they are observed as the CMB [348].
3.2. Composition of the Universe
According to the standard cosmological model, the Universe started at the big bang about 14 billion years ago. During an early epoch of accelerated superluminal expansion, called inflation, a region of microscopic size was stretched to a scale much bigger than the visible Universe and our local geometry became flat. At the same time, primordial density fluctuations were generated out of quantum mechanical fluctuations of the vacuum. These inhomogeneities seeded the formation of present-day structure through the process of gravitational instability. The mass density of ordinary (baryonic) matter makes up only a fifth of the matter that led to the emergence of structure and the rest is the form of an unknown dark matter component. Recently, the Universe entered a new phase of accelerated expansion due to the dominance of some dark vacuum energy density over the ever rarefying matter density.
The basic question that cosmology attempts to answer is:
What are the ingredients (composition and initial conditions) of the Universe and what processes generated the observed structures in it?
In detail, we would like to know:
(a) Did inflation occur and when? If so, what drove it and how did it end?
(b) What is the nature of of the dark energy and how does it change over time and space?
(c) What is the nature of the dark matter and how did it regulate the evolution of structure in the Universe?
Before hydrogen recombined, the Universe was opaque to electromagnetic radiation, precluding any possibility for direct imaging of its evolution. The only way to probe inflation is through the fossil record that it left behind in the form of density perturbations and gravitational waves. Following inflation, the Universe went through several other milestones which left a detectable record. These include: baryogenesis (which resulted in the observed asymmetry between matter and anti-matter), the electroweak phase transition (during which the symmetry between electromagnetic and weak interactions was broken), the QCD phase transition (during which protons and neutrons were assembled out of quarks and gluons), the dark matter freeze-out epoch (during which the dark matter decoupled from the cosmic plasma), neutrino decoupling, electron-positron annihilation, and light-element nucleosynthesis (during which helium, deuterium and lithium were synthesized). The signatures that these processes left in the Universe can be used to constrain its parameters and answer the above questions.
Half a million years after the big bang, hydrogen recombined and the Universe became transparent. The ultimate goal of observational cosmology is to image the entire history of the Universe since then. Currently, we have a snapshot of the Universe at recombination from the CMB, and detailed images of its evolution starting from an age of a billion years until the present time. The evolution between a million and a billion years has not been imaged as of yet.
Within the next decade, NASA plans to launch an infrared space telescope (JWST) that will image the very first sources of light (stars and black holes) in the Universe, which are predicted theoretically to have formed in the first hundreds of millions of years. In parallel, there are several initiatives to construct large-aperture infrared telescopes on the ground with the same goal in mind ^{1, 2, 3}. The neutral hydrogen, relic from cosmological recombination, can be mapped in three-dimensions through its 21cm line even before the first galaxies formed [226]. Several groups are currently constructing low-frequency radio arrays in an attempt to map the initial inhomogeneities as well as the process by which the hydrogen was re-ionized by the first galaxies.
Figure 4. A sketch of the current design for the James Webb Space Telescope, the successor to the Hubble Space Telescope to be launched in 2011 (http://www.jwst.nasa.gov/). The current design includes a primary mirror made of beryllium which is 6.5 meter in diameter as well as an instrument sensitivity that spans the full range of infrared wavelengths of 0.6 - 28 µm and will allow detection of the first galaxies in the infant Universe. The telescope will orbit 1.5 million km from Earth at the Lagrange L2 point. |
The next generation of ground-based telescopes will have a diameter of twenty to thirty meter. Together with JWST (that will not be affected by the atmospheric backgound) they will be able to image the first galaxies. Given that these galaxies also created the ionized bubbles around them, the same galaxy locations should correlate with bubbles in the neutral hydrogen (created by their UV emission). Within a decade it would be possible to explore the environmental influence of individual galaxies by using the two sets of instruments in concert [390].
Figure 5. Artist conception of the design for one of the future giant telescopes that could probe the first generation of galaxies from the ground. The Giant Magellan Telescope (GMT) will contain seven mirrors (each 8.4 meter in diameter) and will have the resolving power equivalent to a 24.5 meter (80 foot) primary mirror. For more details see http://www.gmto.org/ |
The dark ingredients of the Universe can only be probed indirectly through a variety of luminous tracers. The distribution and nature of the dark matter are constrained by detailed X-ray and optical observations of galaxies and galaxy clusters. The evolution of the dark energy with cosmic time will be constrained over the coming decade by surveys of Type Ia supernovae, as well as surveys of X-ray clusters, up to a redshift of two.
On large scales (> 10Mpc) the power-spectrum of primordial density perturbations is already known from the measured microwave background anisotropies, galaxy surveys, weak lensing, and the Ly forest. Future programs will refine current knowledge, and will search for additional trademarks of inflation, such as gravitational waves (through CMB polarization), small-scale structure (through high-redshift galaxy surveys and 21cm studies), or the Gaussian statistics of the initial perturbations.
The big bang is the only known event where particles with energies approaching the Planck scale [( c^{5} / G)^{1/2} ~ 10^{19} GeV] interacted. It therefore offers prospects for probing the unification physics between quantum mechanics and general relativity (to which string theory is the most-popular candidate). Unfortunately, the exponential expansion of the Universe during inflation erases memory of earlier cosmic epochs, such as the Planck time.
3.3. Linear Gravitational Growth
Observations of the CMB (e.g., Bennett et al. 1996 [36]) show that the Universe at recombination was extremely uniform, but with spatial fluctuations in the energy density and gravitational potential of roughly one part in 10^{5}. Such small fluctuations, generated in the early Universe, grow over time due to gravitational instability, and eventually lead to the formation of galaxies and the large-scale structure observed in the present Universe.
As before, we distinguish between fixed and comoving coordinates. Using vector notation, the fixed coordinate r corresponds to a comoving position x = r / a. In a homogeneous Universe with density , we describe the cosmological expansion in terms of an ideal pressureless fluid of particles each of which is at fixed x, expanding with the Hubble flow v = H(t) r where v = dr / dt. Onto this uniform expansion we impose small perturbations, given by a relative density perturbation
(9) |
where the mean fluid density is , with a corresponding peculiar velocity u v - H r. Then the fluid is described by the continuity and Euler equations in comoving coordinates [283, 284]:
(10) (11) |
The potential is given by the Poisson equation, in terms of the density perturbation:
(12) |
This fluid description is valid for describing the evolution of collisionless cold dark matter particles until different particle streams cross. This "shell-crossing" typically occurs only after perturbations have grown to become non-linear, and at that point the individual particle trajectories must in general be followed. Similarly, baryons can be described as a pressureless fluid as long as their temperature is negligibly small, but non-linear collapse leads to the formation of shocks in the gas.
For small perturbations << 1, the fluid equations can be linearized and combined to yield
(13) |
This linear equation has in general two independent solutions, only one of which grows with time. Starting with random initial conditions, this "growing mode" comes to dominate the density evolution. Thus, until it becomes non-linear, the density perturbation maintains its shape in comoving coordinates and grows in proportion to a growth factor D(t). The growth factor in the matter-dominated era is given by [283]
(14) |
where we neglect _{r} when considering halos forming in the matter-dominated regime at z << 10^{4}. In the Einstein-de Sitter model (or, at high redshift, in other models as well) the growth factor is simply proportional to a(t).
The spatial form of the initial density fluctuations can be described in Fourier space, in terms of Fourier components
(15) |
Here we use the comoving wavevector k, whose magnitude k is the comoving wavenumber which is equal to 2 divided by the wavelength. The Fourier description is particularly simple for fluctuations generated by inflation (e.g., Kolb & Turner 1990 [205]). Inflation generates perturbations given by a Gaussian random field, in which different k-modes are statistically independent, each with a random phase. The statistical properties of the fluctuations are determined by the variance of the different k-modes, and the variance is described in terms of the power spectrum P(k) as follows:
(16) |
where ^{(3)} is the three-dimensional Dirac delta function. The gravitational potential fluctuations are sourced by the density fluctuations through Poisson's equation.
In standard models, inflation produces a primordial power-law spectrum P(k) k^{n} with n ~ 1. Perturbation growth in the radiation-dominated and then matter-dominated Universe results in a modified final power spectrum, characterized by a turnover at a scale of order the horizon cH^{-1} at matter-radiation equality, and a small-scale asymptotic shape of P(k) k^{n-4}. The overall amplitude of the power spectrum is not specified by current models of inflation, and it is usually set by comparing to the observed CMB temperature fluctuations or to local measures of large-scale structure.
Since density fluctuations may exist on all scales, in order to determine the formation of objects of a given size or mass it is useful to consider the statistical distribution of the smoothed density field. Using a window function W(r) normalized so that d^{3} r W(r) = 1, the smoothed density perturbation field, d^{3} r (x) W(r), itself follows a Gaussian distribution with zero mean. For the particular choice of a spherical top-hat, in which W = 1 in a sphere of radius R and is zero outside, the smoothed perturbation field measures the fluctuations in the mass in spheres of radius R. The normalization of the present power spectrum is often specified by the value of _{8} (R = 8 h^{-1} Mpc). For the top-hat, the smoothed perturbation field is denoted _{r} or _{m}, where the mass M is related to the comoving radius R by M = 4 _{m} R^{3}/3, in terms of the current mean density of matter _{m}. The variance < _{m} >^{2} is
(17) |
where j_{1}(x) = (sinx - x cosx) / x^{2}. The function (M) plays a crucial role in estimates of the abundance of collapsed objects, as we describe later.
Species that decouple from the cosmic plasma (like the dark matter or the baryons) would show fossil evidence for acoustic oscillations in their power spectrum of inhomogeneities due to sound waves in the radiation fluid to which they were coupled at early times. This phenomenon can be understood as follows. Imagine a localized point-like perturbation from inflation at t = 0. The small perturbation in density or pressure will send out a sound wave that will reach the sound horizon c_{s} t at any later time t. The perturbation will therefore correlate with its surroundings up to the sound horizon and all k-modes with wavelengths equal to this scale or its harmonics will be correlated. The scales of the perturbations that grow to become the first collapsed objects at z < 100 cross the horizon in the radiation dominated era after the dark matter decouples from the cosmic plasma. Next we consider the imprint of this decoupling on the smallest-scale structure of the dark matter.
3.4. The Smallest-Scale Power Spectrum of Cold Dark Matter
A broad range of observational data involving the dynamics of galaxies, the growth of large-scale structure, and the dynamics and nucleosynthesis of the Universe as a whole, indicate the existence of dark matter with a mean cosmic mass density that is ~ 5 times larger than the density of the baryonic matter [189, 348]. The data is consistent with a dark matter composed of weakly-interacting, massive particles, that decoupled early and adiabatically cooled to an extremely low temperature by the present time [189]. The Cold Dark Matter (CDM) has not been observed directly as of yet, although laboratory searches for particles from the dark halo of our own Milky-Way galaxy have been able to restrict the allowed parameter space for these particles. Since an alternative more-radical interpretation of the dark matter phenomenology involves a modification of gravity [253], it is of prime importance to find direct fingerprints of the CDM particles. One such fingerprint involves the small-scale structure in the Universe [158], on which we focus in this section.
The most popular candidate for the CDM particle is a Weakly Interacting Massive Particle (WIMP). The lightest supersymmetric particle (LSP) could be a WIMP (for a review see [189]). The CDM particle mass depends on free parameters in the particle physics model but typical values cover a range around M ~ 100 GeV (up to values close to a TeV). In many cases the LSP hypothesis will be tested at the Large Hadron Collider (e.g. [33]) or in direct detection experiments (e.g. [16]).
The properties of the CDM particles affect their response to the small-scale primordial inhomogeneities produced during cosmic inflation. The particle cross-section for scattering off standard model fermions sets the epoch of their thermal and kinematic decoupling from the cosmic plasma (which is significantly later than the time when their abundance freezes-out at a temperature T ~ M). Thermal decoupling is defined as the time when the temperature of the CDM stops following that of the cosmic plasma while kinematic decoupling is defined as the time when the bulk motion of the two species start to differ. For CDM the epochs of thermal and kinetic decoupling coincide. They occur when the time it takes for collisions to change the momentum of the CDM particles equals the Hubble time. The particle mass determines the thermal spread in the speeds of CDM particles, which tends to smooth-out fluctuations on very small scales due to the free-streaming of particles after kinematic decoupling [158, 159]. Viscosity has a similar effect before the CDM fluid decouples from the cosmic radiation fluid [182]. An important effect involves the memory the CDM fluid has of the acoustic oscillations of the cosmic radiation fluid out of which it decoupled. Here we consider the imprint of these acoustic oscillations on the small-scale power spectrum of density fluctuations in the Universe. Analogous imprints of acoustic oscillations of the baryons were identified recently in maps of the CMB [348], and the distribution of nearby galaxies [119]; these signatures appear on much larger scales, since the baryons decouple much later when the scale of the horizon is larger. The discussion in this section follows Loeb & Zaldarriaga (2005) [228].
Formalism
Kinematic decoupling of CDM occurs during the radiation-dominated era. For example, if the CDM is made of neutralinos with a particle mass of ~ 100 GeV, then kinematic decoupling occurs at a cosmic temperature of T_{d} ~ 10 MeV [182, 87]. As long as T_{d} << 100 MeV, we may ignore the imprint of the QCD phase transition (which transformed the cosmic quark-gluon soup into protons and neutrons) on the CDM power spectrum [321]. Over a short period of time during this transition, the pressure does not depend on density and the sound speed of the plasma vanishes, resulting in a significant growth for perturbations with periods shorter than the length of time over which the sound speed vanishes. The transition occurs when the temperature of the cosmic plasma is ~ 100-200 MeV and lasts for a small fraction of the Hubble time. As a result, the induced modifications are on scales smaller than those we are considering here and the imprint of the QCD phase transition is washed-out by the effects we calculate.
At early times the contribution of the dark matter to the energy density is negligible. Only at relatively late times when the cosmic temperature drops to values as low as ~ 1 eV, matter and radiation have comparable energy densities. As a result, the dynamics of the plasma at earlier times is virtually unaffected by the presence of the dark matter particles. In this limit, the dynamics of the radiation determines the gravitational potential and the dark matter just responds to that potential. We will use this simplification to obtain analytic estimates for the behavior of the dark matter transfer function.
The primordial inflationary fluctuations lead to acoustic modes in the radiation fluid during this era. The interaction rate of the particles in the plasma is so high that we can consider the plasma as a perfect fluid down to a comoving scale,
(18) |
where _{d} = _{0}^{td} dt / a(t) is the conformal time (i.e. the comoving size of the horizon) at the time of CDM decoupling, t_{d}; is the scattering cross section and n is the relevant particle density. (Throughout this section we set the speed of light and Planck's constant to unity for brevity.) The damping scale depends on the species being considered and its contribution to the energy density, and is the largest for neutrinos which are only coupled through weak interactions. In that case N ~ (T / T_{d}^{})^{3} where T_{d}^{} ~ 1 MeV is the temperature of neutrino decoupling. At the time of CDM decoupling N ~ M / T_{d} ~ 10^{4} for the rest of the plasma, where M is the mass of the CDM particle. Here we will consider modes of wavelength larger than _{f}, and so we neglect the effect of radiation diffusion damping and treat the plasma (without the CDM) as a perfect fluid.
The equations of motion for a perfect fluid during the radiation era can be solved analytically. We will use that solution here, following the notation of Dodelson [109]. As usual we Fourier decompose fluctuations and study the behavior of each Fourier component separately. For a mode of comoving wavenumber k in Newtonian gauge, the gravitational potential fluctuations are given by:
(19) |
where = k / 3^{1/2} is the frequency of a mode and _{p} is its primordial amplitude in the limit 0. In this section we use conformal time = dt / a(t) with a(t) t^{1/2} during the radiation-dominated era. Expanding the temperature anisotropy in multipole moments and using the Boltzmann equation to describe their evolution, the monopole _{0} and dipole _{1} of the photon distribution can be written in terms of the gravitational potential as [109]:
(20) |
where x k and a prime denotes a derivative with respect to x.
The solutions in equations (19) and (20) assume that both the sound speed and the number of relativistic degrees of freedom are constant over time. As a result of the QCD phase transition and of various particles becoming non-relativistic, both of these assumptions are not strictly correct. The vanishing sound speed during the QCD phase transition provides the most dramatic effect, but its imprint is on scales smaller than the ones we consider here because the transition occurs at a significantly higher temperature and only lasts for a fraction of the Hubble time [321].
Before the dark matter decouples kinematically, we will treat it as a fluid which can exchange momentum with the plasma through particle collisions. At early times, the CDM fluid follows the motion of the plasma and is involved in its acoustic oscillations. The continuity and momentum equations for the CDM can be written as:
(21) |
where a dot denotes an -derivative, _{c} is the dark matter density perturbation, _{c} is the divergence of the dark matter velocity field and _{c} denotes the anisotropic stress. In writing these equations we have followed Ref. [230]. The term _{c}^{-1} (_{1} - _{c}) encodes the transfer of momentun between the radiation and CDM fluids and _{c}^{-1} provides the collisional rate of momentum transfer,
(22) |
with n being the number density of particles with which the dark matter is interacting, (T) the average cross section for interaction and M the mass of the dark matter particle. The relevant scattering partners are the standard model leptons which have thermal abundances. For detailed expressions of the cross section in the case of supersymmetric (SUSY) dark matter, see Refs. [87, 159]. For our purpose, it is sufficient to specify the typical size of the cross section and its scaling with cosmic time,
(23) |
where the coupling mass M_{} is of the order of the weak-interaction scale (~ 100 GeV) for SUSY dark matter. This equation should be taken as the definition of M_{}, as it encodes all the uncertainties in the details of the particle physics model into a single parameter. The temperature dependance of the averaged cross section is a result of the available phase space. Our results are quite insensitive to the details other than through the decoupling time. Equating _{c}^{-1} / a to the Hubble expansion rate gives the temperature of kinematic decoupling:
(24) |
The term k^{2} c_{s}^{2} _{c} in Eq. (21) results from the pressure gradient force and c_{s} is the dark matter sound speed. In the tight coupling limit, _{c} << H^{-1} we find that c_{s}^{2} f_{c} T / M and that the shear term is k^{2} _{c} f_{v} c_{s}^{2} _{c} _{c}. Here f_{v} and f_{c} are constant factors of order unity. We will find that both these terms make a small difference on the scales of interest, so their precise value is unimportant.
By combining both equations in (21) into a single equation for _{c} we get
(25) |
where x_{d} = k _{d} and _{d} denotes the time of kinematic decoupling which can be expressed in terms of the decoupling temperature as,
(26) |
with T_{0} = 2.7K being the present-day CMB temperature and z_{d} being the redshift at kinematic decoupling. We have also introduced the source function,
(27) |
For x << x_{d}, the dark matter sound speed is given by
(28) |
where c_{s}^{2}(x_{d}) is the dark matter sound speed at kinematic decoupling (in units of the speed of light),
(29) |
In writing (28) we have assumed that prior to decoupling the temperature of the dark matter follows that of the plasma. For the viscosity term we have,
(30) |
Free streaming after kinematic decoupling
In the limit of the collision rate being much slower than the Hubble expansion, the CDM is decoupled and the evolution of its perturbations is obtained by solving a Boltzman equation:
(31) |
where f(r, q, ) is the distribution function which depends on position, comoving momentum q, and time. The comoving momentum 3-components are dx_{i} / d = q_{i} / a. We use the Boltzman equation to find the evolution of modes that are well inside the horizon with x >> 1. In the radiation era, the gravitational potential decays after horizon crossing (see Eq. 19). In this limit the comoving momentum remains constant, dq_{i} / d =0 and the Boltzman equation becomes,
(32) |
We consider a single Fourier mode and write f as,
(33) |
where f_{0}(q) is the unperturbed distribution,
(34) |
where n_{CDM} and T_{CDM} are the present-day density and temperature of the dark matter.
Our approach is to solve the Boltzman equation with initial conditions given by the fluid solution at a time _{*} (which will depend on k). The simplified Boltzman equation can be easily solved to give _{f}(q, ) as a function of the initial conditions _{f}(q, _{*}),
(35) |
The CDM overdensity _{c} can then be expressed in terms of the perturbation in the distribution function as,
(36) |
We can use (35) to obtain the evolution of _{c} in terms of its value at _{*},
(37) |
where k_{f}^{-2} = [(T_{d} / M)]^{1/2} _{d}. The exponential term is responsible for the damping of perturbations as a result of free streaming and the dispersion of the CDM particles after they decouple from the plasma. The above expression is only valid during the radiation era. The free streaming scale is simply given by dt(v / a) dta^{-2} which grows logarithmically during the radiation era as in equation (37) but stops growing in the matter era when a t^{2/3}.
Equation (37) can be used to show that even during the free streaming epoch, _{c} satisfies equation (25) but with a modified sound speed and viscous term. For x >> x_{d} one should use,
(38) |
The differences between the above scalings and those during the tight coupling regime are a result of the fact that the dark matter temperature stops following the plasma temperature but rather scales as a^{-2} after thermal decoupling, which coincides with the kinematic decoupling. We ignore the effects of heat transfer during the fluid stage of the CDM because its temperature is controlled by the much larger heat reservoir of the radiation-dominated plasma at that stage.
To obtain the transfer function we solve the dark matter fluid equation until decoupling and then evolve the overdensity using equation (37) up to the time of matter - radiation equality. In practice, we use the fluid equations up to x_{*} = 10 max(x_{d}, 10) so as to switch into the free streaming solution well after the gravitational potential has decayed. In the fluid equations, we smoothly match the sound speed and viscosity terms at x = x_{d}. As mentioned earlier, because c_{s}(x_{d}) is so small and we are interested in modes that are comparable to the size of the horizon at decoupling, i.e. x_{d} ~ few, both the dark matter sound speed and the associated viscosity play only a minor role, and our simplified treatment is adequate.
In Figure 6 we illustrate the time evolution of modes during decoupling for a variety of k values. The situation is clear. Modes that enter the horizon before kinematic decoupling oscillate with the radiation fluid. This behavior has two important effects. In the absence of the coupling, modes receive a "kick" by the source term S(x) as they cross the horizon. After that they grow logarithmically. In our case, modes that entered the horizon before kinematic decoupling follow the plasma oscillations and thus miss out on both the horizon "kick" and the beginning of the logarithmic growth. Second, the decoupling from the radiation fluid is not instantaneous and this acts to further damp the amplitude of modes with x_{d} >> 1. This effect can be understood as follows. Once the oscillation frequency of the mode becomes high compared to the scattering rate, the coupling to the plasma effectively damps the mode. In that limit one can replace the forcing term _{0}^{'} by its average value, which is close to zero. Thus in this regime, the scattering is forcing the amplitude of the dark matter oscillations to zero. After kinematic decoupling the modes again grow logarithmically but from a very reduced amplitude. The coupling with the plasma induces both oscillations and damping of modes that entered the horizon before kinematic decoupling. This damping is different from the free streaming damping that occurs after kinematic decoupling.
In Figure 7 we show the resulting transfer function of the CDM overdensity. The transfer function is defined as the ratio between the CDM density perturbation amplitude _{c} when the effect of the coupling to the plasma is included and the same quantity in a model where the CDM is a perfect fluid down to arbitrarily small scales (thus, the power spectrum is obtained by multiplying the standard result by the square of the transfer function). This function shows both the oscillations and the damping signature mentioned above. The peaks occur at multipoles of the horizon scale at decoupling,
(39) |
This same scale determines the "oscillation" damping. The free streaming damping scale is,
(40) |
where T_{eq} is the temperature at matter radiation equality, T_{eq} 1 eV. The free streaming scale is parametrically different from the "oscillation" damping scale. However for our fiducial choice of parameters for the CDM particle they roughly coincide.
The CDM damping scale is significantly smaller than the scales observed directly in the Cosmic Microwave Background or through large scale structure surveys. For example, the ratio of the damping scale to the scale that entered the horizon at Matter-radiation equality is _{d} / _{eq} ~ T_{eq} / T_{d} ~ 10^{-7} and to our present horizon _{d} / _{0} ~ (T_{eq} T_{0})^{1/2} / T_{d} ~ 10^{-9}. In the context of inflation, these scales were created 16 and 20 e - folds apart. Given the large extrapolation, one could certainly imagine that a change in the spectrum could alter the shape of the power spectrum around the damping scale. However, for smooth inflaton potentials with small departures from scale invariance this is not likely to be the case. On scales much smaller than the horizon at matter radiation equality, the spectrum of perturbations density before the effects of the damping are included is approximately,
(41) |
where the first term encodes the shape of the primordial spectrum and the second the transfer function. Primordial departures from scale invariance are encoded in the slope n and its running . The effective slope at scale k is then,
(42) |
For typical values of (n - 1) ~ 1/60 and ~ 1/60^{2} the slope is still positive at k ~ _{d}^{-1}, so the cut-off in the power will come from the effects we calculate rather than from the shape of the primordial spectrum. However given the large extrapolation in scale, one should keep in mind the possibility of significant effects resulting from the mechanisms that generates the density perturbations.
Implications We have found that acoustic oscillations, a relic from the epoch when the dark matter coupled to the cosmic radiation fluid, truncate the CDM power spectrum on a comoving scale larger than effects considered before, such as free-streaming and viscosity [158, 159, 182]. For SUSY dark matter, the minimum mass of dark matter clumps that form in the Universe is therefore increased by more than an order of magnitude to a value of ^{4}
(43) |
where _{crit} = (H_{0}^{2} / 8 G) = 9 × 10^{-30} g cm^{-3} is the critical density today, and _{m} is the matter density for the concordance cosmological model [348]. We define the cut-off wavenumber k_{cut} as the point where the transfer function first drops to a fraction 1 / e of its value at k 0. This corresponds to k_{cut} 3.3 _{d}^{-1}.
Figure 8. A slice through a numerical simulation of the first dark matter condensations to form in the Universe. Colors represent the dark matter density at z = 26. The simulated volume is 60 comoving pc on a side, simulated with 64 million particles each weighing 1.2 × 10^{-10} M_{} (!). (from Diemand, Moore, & Stadel 2005 [105]). |
Recent numerical simulations [105, 146] of the earliest and smallest objects to have formed in the Universe (see Fig. 3.4), need to be redone for the modified power spectrum that we calculated in this section. Although it is difficult to forecast the effects of the acoustic oscillations through the standard Press-Schechter formalism [291], it is likely that the results of such simulations will be qualitatively the same as before except that the smallest clumps would have a mass larger than before (as given by Eq. 43).
Potentially, there are several observational signatures of the smallest CDM clumps. As pointed out in the literature [105, 353], the smallest CDM clumps could produce -rays through dark-matter annihilation in their inner density cusps, with a flux in excess of that from nearby dwarf galaxies. If a substantial fraction of the Milky Way halo is composed of CDM clumps with a mass ~ 10^{-4} M_{}, the nearest clump is expected to be at a distance of ~ 4 × 10^{17} cm. Given that the characteristic speed of such clumps is a few hundred km s^{-1}, the -ray flux would therefore show temporal variations on the relatively long timescale of a thousand years. Passage of clumps through the solar system should also induce fluctuations in the detection rate of CDM particles in direct search experiments. Other observational effects have rather limited prospects for detectability. Because of their relatively low-mass and large size (~ 10^{17} cm), the CDM clumps are too diffuse to produce any gravitational lensing signatures (including femto-lensing [161]), even at cosmological distances.
The smallest CDM clumps should not affect the intergalactic baryons which have a much larger Jeans mass. However, once objects above ~ 10^{6} M_{} start to collapse at redshifts z < 30, the baryons would be able to cool inside of them via molecular hydrogen transitions and the interior baryonic Jeans mass would drop. The existence of dark matter clumps could then seed the formation of the first stars inside these objects [66].
Early Evolution of Baryonic Perturbations on Large Scales
The baryons are coupled through Thomson scattering to the radiation fluid until they become neutral and decouple. After cosmic recombination, they start to fall into the potential wells of the dark matter and their early evolution was derived by Barkana & Loeb (2005) [29].
On large scales, the dark matter (dm) and the baryons (b) are affected only by their combined gravity and gas pressure can be ignored. The evolution of sub-horizon linear perturbations is described in the matter-dominated regime by two coupled second-order differential equations [284]:
(44) |
where _{dm}(t) and _{b}(t) are the perturbations in the dark matter and baryons, respectively, the derivatives are with respect to cosmic time t, H(t) = / a is the Hubble constant with a = (1 + z)^{-1}, and we assume that the mean mass density _{m}(t) is made up of respective mass fractions f_{dm} and f_{b} = 1 - f_{dm}. Since these linear equations contain no spatial gradients, they can be solved spatially for _{dm}(x, t) and _{b}(x, t) or in Fourier space for _{dm}(k, t) and _{b}(k, t).
Defining _{tot} f_{b} _{b} + f_{dm} _{dm} and _{b-} _{b} - _{tot} , we find
(45) |
Each of these equations has two independent solutions. The equation for _{tot} has the usual growing and decaying solutions, which we denote D_{1}(t) and D_{4}(t), respectively, while the _{b-} equation has solutions D_{2}(t) and D_{3}(t); we number the solutions in order of declining growth rate (or increasing decay rate). We assume an Einstein-de Sitter, matter-dominated Universe in the redshift range z = 20 - 150, since the radiation contributes less than a few percent at z < 150, while the cosmological constant and the curvature contribute to the energy density less than a few percent at z > 3. In this regime a t^{2/3} and the solutions are D_{1}(t) = a / a_{i} and D_{4}(t) = (a / a_{i})^{-3/2} for _{tot}, and D_{2}(t) = 1 and D_{3}(t) = (a / a_{i})^{-1/2} for _{b-}, where we have normalized each solution to unity at the starting scale factor a_{i}, which we set at a redshift z_{i} = 150. The observable baryon perturbation can then be written as
(46) |
and similarly for the dark matter perturbation,
(47) |
where C_{i} = D_{i} for i = 1,4 and C_{i} = -(f_{b} / f_{dm})D_{i} for i = 2,3. We may establish the values of _{m}(k) by inverting the 4 × 4 matrix A that relates the 4-vector (_{1}, _{2}, _{3}, _{4}) to the 4-vector that represents the initial conditions (_{b}, _{dm}, _{b}, _{dm}) at the initial time.
Next we describe the fluctuations in the sound speed of the cosmic gas caused by Compton heating of the gas, which is due to scattering of the residual electrons with the CMB photons. The evolution of the temperature T of a gas element of density _{b} is given by the first law of thermodynamics:
(48) |
where dQ is the heating rate per particle. Before the first galaxies formed,
(49) |
where _{T} is the Thomson cross-section, x_{e}(t) is the electron fraction out of the total number density of gas particles, and _{} is the CMB energy density at a temperature T_{}. In the redshift range of interest, we assume that the photon temperature (T_{}= T_{}^{0} / a) is spatially uniform, since the high sound speed of the photons (i.e., c / 3^{1/2}) suppresses fluctuations on the sub-horizon scales that we consider, and the horizon-scale ~ 10^{-5} fluctuations imprinted at cosmic recombination are also negligible compared to the smallWe establish the values of _{m}(k) by inverting the 4 × 4 matrix A that relates the 4-vector (_{1}, _{2}, _{3}, _{4}) to the 4-vector that represents the initial conditions (_{b}, _{dm}, _{b}, _{dm}) at the initial time. er-scale fluctuations in the gas density and temperature. Fluctuations in the residual electron fraction x_{e}(t) are even smaller. Thus,
(50) |
where t_{}^{-1} _{}^{0} (8_{T} c / 3 m_{e}) = 8.55 × 10^{-13} yr^{-1}. After cosmic recombination, x_{e}(t) changes due to the slow recombination rate of the residual ions:
(51) |
where _{B}(T) is the case-B recombination coefficient of hydrogen, _{H} is the mean number density of hydrogen at time t, and y = 0.079 is the helium to hydrogen number density ratio. This yields the evolution of the mean temperature, d / dt = - 2 H + x_{e}(t) t_{}^{-1} (T_{} - ) a^{-4}. In prior analyses [284, 230] a spatially uniform speed of sound was assumed for the gas at each redshift. Note that we refer to p / as the square of the sound speed of the fluid, where p is the pressure perturbation, although we are analyzing perturbations driven by gravity rather than sound waves driven by pressure gradients.
Instead of assuming a uniform sound speed, we find the first-order perturbation equation,
(52) |
where we defined the fractional temperature perturbation _{T}. Like the density perturbation equations, this equation can be solved separately at each x or at each k. Furthermore, the solution _{T} (t) is a linear functional of _{b}(t) [for a fixed function x_{e}(t)]. Thus, if we choose an initial time t_{i} then using Eq. (46) we can write the solution in Fourier space as
(53) |
where D_{m}^{T}(t) is the solution of Eq. (52) with _{T} = 0 at t_{i} and with the perturbation mode D_{m}(t) substituted for _{b}(t), while D_{0}^{T}(t) is the solution with no perturbation _{b}(t) and with _{T} = 1 at t_{i}. By modifying the CMBFAST code (http://www.cmbfast.org/), we can numerically solve Eq. (52) along with the density perturbation equations for each k down to z_{i} = 150, and then match the solution to the form of Eq. (53).
Figure 9 shows the time evolution of the various independent modes that make up the perturbations of density and temperature, starting at the time t_{i} corresponding to z_{i} = 150. D_{2}^{T}(t) is identically zero since D_{2}(t) = 1 is constant, while D_{3}^{T}(t) and D_{4}^{T}(t) are negative. Figure 10 shows the amplitudes of the various components of the initial perturbations. We consider comoving wavevectors k in the range 0.01 - 40 Mpc^{-1}, where the lower limit is set by considering sub-horizon scales at z = 150 for which photon perturbations are negligible compared to _{dm} and _{b}, and the upper limit is set by requiring baryonic pressure to be negligible compared to gravity. _{2} and _{3} clearly show a strong signature of the large-scale baryonic oscillations, left over from the era of the photon-baryon fluid before recombination, while _{1}, _{4}, and _{T} carry only a weak sign of the oscillations. For each quantity, the plot shows [k^{3} P(k) / (2 ^{2})]^{1/2}, where P(k) is the corresponding power spectrum of fluctuations. _{4} is already a very small correction at z = 150 and declines quickly at lower redshift, but the other three modes all contribute significantly to _{b}, and the _{T}(t_{i}) term remains significant in _{T}(t) even at z < 100. Note that at z = 150 the temperature perturbation _{T} has a different shape with respect to k than the baryon perturbation _{b}, showing that their ratio cannot be described by a scale-independent speed of sound.
Figure 9. Redshift evolution of the amplitudes of the independent modes of the density perturbations (upper panel) and the temperature perturbations (lower panel), starting at redshift 150 (from Barkana & Loeb 2005 [29]). We show m = 1 (solid curves), m = 2 (short-dashed curves), m = 3 (long-dashed curves), m = 4 (dotted curves), and m = 0 (dot-dashed curve). Note that the lower panel shows one plus the mode amplitude. |
Figure 10. Power spectra and initial perturbation amplitudes versus wavenumber (from [29]). The upper panel shows _{b} (solid curves) and _{dm} (dashed curves) at z = 150 and 20 (from bottom to top). The lower panel shows the initial (z = 150) amplitudes of _{1} (solid curve), _{2} (short-dashed curve), _{3} (long-dashed curve), _{4} (dotted curve), and _{T}(t_{i}) (dot-dashed curve). Note that if _{1} is positive then so are _{3} and _{T}(t_{i}), while _{2} is negative at all k, and _{4} is negative at the lowest k but is positive at k > 0.017 Mpc^{-1}. |
The power spectra of the various perturbation modes and of _{T}(t_{i}) depend on the initial power spectrum of density fluctuations from inflation and on the values of the fundamental cosmological parameters (_{dm}, _{b}, _{}, and h). If these independent power spectra can be measured through 21cm fluctuations, this will probe the basic cosmological parameters through multiple combinations, allowing consistency checks that can be used to verify the adiabatic nature and the expected history of the perturbations. Figure 11 illustrates the relative sensitivity of [P(k)]^{1/2} to variations in _{dm} h^{2}, _{b} h^{2}, and h, for the quantities _{1}, _{2}, _{3}, and _{T}(t_{i}). Not shown is _{4}, which although it is more sensitive (changing by order unity due to 10% variations in the parameters), its magnitude always remains much smaller than the other modes, making it much harder to detect. Note that although the angular scale of the baryon oscillations constrains also the history of dark energy through the angular diameter distance, we have focused here on other cosmological parameters, since the contribution of dark energy relative to matter becomes negligible at high redshift.
Figure 11. Relative sensitivity of perturbation amplitudes at z = 150 to cosmological parameters (from [29]). For variations in a parameter x, we show d log [P(k)]^{1/2} / d log(x). We consider variations in _{dm} h^{2} (upper panel), in _{b} h^{2} (middle panel), and in the Hubble constant h (lower panel). When we vary each parameter we fix the other two, and the variations are all carried out in a flat _{total} = 1 universe. We show the sensitivity of _{1} (solid curves), _{2} (short-dashed curves), _{3} (long-dashed curves), and _{T}(t_{i}) (dot-dashed curves). |
Cosmological Jeans Mass
The Jeans length _{J} was originally defined (Jeans 1928 [187]) in Newtonian gravity as the critical wavelength that separates oscillatory and exponentially-growing density perturbations in an infinite, uniform, and stationary distribution of gas. On scales smaller than _{J}, the sound crossing time, / c_{s} is shorter than the gravitational free-fall time, (G )^{-1/2}, allowing the build-up of a pressure force that counteracts gravity. On larger scales, the pressure gradient force is too slow to react to a build-up of the attractive gravitational force. The Jeans mass is defined as the mass within a sphere of radius _{J} / 2, M_{J} = (4 / 3) (_{J} / 2)^{3}. In a perturbation with a mass greater than M_{J}, the self-gravity cannot be supported by the pressure gradient, and so the gas is unstable to gravitational collapse. The Newtonian derivation of the Jeans instability suffers from a conceptual inconsistency, as the unperturbed gravitational force of the uniform background must induce bulk motions (compare to Binney & Tremaine 1987 [43]). However, this inconsistency is remedied when the analysis is done in an expanding Universe.
The perturbative derivation of the Jeans instability criterion can be carried out in a cosmological setting by considering a sinusoidal perturbation superposed on a uniformly expanding background. Here, as in the Newtonian limit, there is a critical wavelength _{J} that separates oscillatory and growing modes. Although the expansion of the background slows down the exponential growth of the amplitude to a power-law growth, the fundamental concept of a minimum mass that can collapse at any given time remains the same (see, e.g. Kolb & Turner 1990 [205]; Peebles 1993 [284]).
We consider a mixture of dark matter and baryons with density parameters _{dm}^{z} = _{dm} / _{c} and _{b}^{z} = _{b} / _{c}, where _{dm} is the average dark matter density, _{b} is the average baryonic density, _{c} is the critical density, and _{dm}^{z} + _{b}^{z} = _{m}^{z} is given by equation(83). We also assume spatial fluctuations in the gas and dark matter densities with the form of a single spherical Fourier mode on a scale much smaller than the horizon,
(54) (55) |
where _{dm}(t) and _{b}(t) are the background densities of the dark matter and baryons, _{dm}(t) and _{b}(t) are the dark matter and baryon overdensity amplitudes, r is the comoving radial coordinate, and k is the comoving perturbation wavenumber. We adopt an ideal gas equation-of-state for the baryons with a specific heat ratio = 5/3. Initially, at time t = t_{i}, the gas temperature is uniform T_{b}(r, t_{i}) = T_{i}, and the perturbation amplitudes are small _{dm,i}, _{b,i} << 1. We define the region inside the first zero of sin(kr) / (kr), namely 0 < kr < , as the collapsing "object".
The evolution of the temperature of the baryons T_{b}(r, t) in the linear regime is determined by the coupling of their free electrons to the CMB through Compton scattering, and by the adiabatic expansion of the gas. Hence, T_{b}(r, t) is generally somewhere between the CMB temperature, T_{} (1 + z)^{-1} and the adiabatically-scaled temperature T_{ad} (1 + z)^{-2}. In the limit of tight coupling to T_{}, the gas temperature remains uniform. On the other hand, in the adiabatic limit, the temperature develops a gradient according to the relation
(56) |
The evolution of a cold dark matter overdensity, _{dm}(t), in the linear regime is described by the equation (44),
(57) |
whereas the evolution of the overdensity of the baryons, _{b}(t), with the inclusion of their pressure force is described by (see Section 9.3.2 of [205]),
(58) |
Here, H(t) = / a is the Hubble parameter at a cosmological time t, and µ = 1.22 is the mean molecular weight of the neutral primordial gas in atomic units. The parameter distinguishes between the two limits for the evolution of the gas temperature. In the adiabatic limit = 1, and when the baryon temperature is uniform and locked to the background radiation, = 0. The last term on the right hand side (in square brackets) takes into account the extra pressure gradient force in (_{b} T) = (T _{b} + _{b} T), arising from the temperature gradient which develops in the adiabatic limit. The Jeans wavelength _{J} = 2 / k_{J} is obtained by setting the right-hand side of equation (58) to zero, and solving for the critical wavenumber k_{J}. As can be seen from equation (58), the critical wavelength _{J} (and therefore the mass M_{J}) is in general time-dependent. We infer from equation (58) that as time proceeds, perturbations with increasingly smaller initial wavelengths stop oscillating and start to grow.
To estimate the Jeans wavelength, we equate the right-hand-side of equation (58) to zero. We further approximate _{b} ~ _{dm}, and consider sufficiently high redshifts at which the Universe is matter dominated and flat, (1 + z) >> max[(1 - _{m} - _{}) / _{m}, (_{} / _{m})^{1/3}]. In this regime, _{b} << _{m} 1, H 2 / (3t), and a = (1 + z)^{-1} (3H_{0} (_{m})^{1/2} / 2)^{2/3} t^{2/3}, where _{m} = _{dm} + _{b} is the total matter density parameter. Following cosmological recombination at z 10^{3}, the residual ionization of the cosmic gas keeps its temperature locked to the CMB temperature (via Compton scattering) down to a redshift of [284]
(59) |
In the redshift range between recombination and z_{t}, = 0 and
(60) |
so that the Jeans mass is therefore redshift independent and obtains the value (for the total mass of baryons and dark matter)
(61) |
Based on the similarity of M_{J} to the mass of a globular cluster, Peebles & Dicke (1968) [281] suggested that globular clusters form as the first generation of baryonic objects shortly after cosmological recombination. Peebles & Dicke assumed a baryonic Universe, with a nonlinear fluctuation amplitude on small scales at z ~ 10^{3}, a model which has by now been ruled out. The lack of a dominant mass of dark matter inside globular clusters makes it unlikely that they formed through direct cosmological collapse, and more likely that they resulted from fragmentation during the process of galaxy formation.
At z < z_{t}, the gas temperature declines adiabatically as [(1 + z) / ( 1 + z_{t})]^{2} (i.e., = 1) and the total Jeans mass obtains the value,
(62) |
It is not clear how the value of the Jeans mass derived above relates to the mass of collapsed, bound objects. The above analysis is perturbative (Eqs. 57 and 58 are valid only as long as _{b} and _{dm} are much smaller than unity), and thus can only describe the initial phase of the collapse. As _{b} and _{dm} grow and become larger than unity, the density profiles start to evolve and dark matter shells may cross baryonic shells [167] due to their different dynamics. Hence the amount of mass enclosed within a given baryonic shell may increase with time, until eventually the dark matter pulls the baryons with it and causes their collapse even for objects below the Jeans mass.
Even within linear theory, the Jeans mass is related only to the evolution of perturbations at a given time. When the Jeans mass itself varies with time, the overall suppression of the growth of perturbations depends on a time-weighted Jeans mass. Gnedin & Hui (1998) [150] showed that the correct time-weighted mass is the filtering mass M_{f} = (4 /3) (2 a / k_{f})^{3}, in terms of the comoving wavenumber k_{f} associated with the "filtering scale" (note the change in convention from / k_{J} to 2 / k_{f}). The wavenumber k_{f} is related to the Jeans wavenumber k_{J} by
(63) |
where D(t) is the linear growth factor. At high redshift (where _{m}^{z} 1), this relation simplifies to [153]
(64) |
Then the relationship between the linear overdensity of the dark matter _{dm} and the linear overdensity of the baryons _{b}, in the limit of small k, can be written as [150]
(65) |
Linear theory specifies whether an initial perturbation, characterized by the parameters k, _{dm,i}, _{b,i} and t_{i}, begins to grow. To determine the minimum mass of nonlinear baryonic objects resulting from the shell-crossing and virialization of the dark matter, we must use a different model which examines the response of the gas to the gravitational potential of a virialized dark matter halo.
3.6. Formation of Nonlinear Objects
Let us consider a spherically symmetric density or velocity perturbation of the smooth cosmological background, and examine the dynamics of a test particle at a radius r relative to the center of symmetry. Birkhoff's (1923) [44] theorem implies that we may ignore the mass outside this radius in computing the motion of our particle. We further find that the relativistic equations of motion describing the system reduce to the usual Friedmann equation for the evolution of the scale factor of a homogeneous Universe, but with a density parameter that now takes account of the additional mass or peculiar velocity. In particular, despite the arbitrary density and velocity profiles given to the perturbation, only the total mass interior to the particle's radius and the peculiar velocity at the particle's radius contribute to the effective value of . We thus find a solution to the particle's motion which describes its departure from the background Hubble flow and its subsequent collapse or expansion. This solution holds until our particle crosses paths with one from a different radius, which happens rather late for most initial profiles.
As with the Friedmann equation for a smooth Universe, it is possible to reinterpret the problem into a Newtonian form. Here we work in an inertial (i.e. non-comoving) coordinate system and consider the force on the particle as that resulting from a point mass at the origin (ignoring the possible presence of a vacuum energy density):
(66) |
where G is Newton's constant, r is the distance of the particle from the center of the spherical perturbation, and M is the total mass within that radius. As long as the radial shells do not cross each other, the mass M is constant in time. The initial density profile determines M, while the initial velocity profile determines dr / dt at the initial time. As is well-known, there are three branches of solutions: one in which the particle turns around and collapses, another in which it reaches an infinite radius with some asymptotically positive velocity, and a third intermediate case in which it reachs an infinite radius but with a velocity that approaches zero. These cases may be written as [164]:
(67) |
(68) |
(69) |
where A^{3} = GMB^{2} applies in all cases. All three solutions have r^{3} = 9GMt^{2} / 2 as t goes to zero, which matches the linear theory expectation that the perturbation amplitude get smaller as one goes back in time. In the closed case, the shell turns around at time B and radius 2A and collapses to zero radius at time 2 B.
We are now faced with the problem of relating the spherical collapse parameters A, B, and M to the linear theory density perturbation [283]. We do this by returning to the equation of motion. Consider that at an early epoch (i.e. scale factor a_{i} << 1), we are given a spherical patch of uniform overdensity _{i} (the so-called `top-hat' perturbation). If is essentially unity at this time and if the perturbation is pure growing mode, then the initial velocity is radially inward with magnitude _{i} H(t_{i})r / 3, where H(t_{i}) is the Hubble constant at the initial time and r is the radius from the center of the sphere. This can be easily seen from the continuity equation in spherical coordinates. The equation of motion (in noncomoving coordinates) for a particle beginning at radius r_{i} is simply
(70) |
where M = (4 / 3) r_{i}^{3} _{i} (1 + _{i}) and _{i} is the background density of the Universe at time t_{i}. We next define the dimensionless radius x = ra_{i} / r_{i} and rewrite equation (70) as
(71) |
Our initial conditions for the integration of this orbit are
(72) |
(73) |
where H(t_{1}) = H_{0}[_{m} / a^{3}(t_{1}) + (1 - _{m})]^{1/2} is the Hubble parameter for a flat Universe at a a cosmic time t_{1}. Integrating equation (71) yields
(74) |
where K is a constant of integration. Evaluating this at the initial time and dropping terms of O(a_{i}) (but _{i} ~ a_{i}, so we keep ratios of order unity), we find
(75) |
If K is sufficiently negative, the particle will turn-around and the sphere will collapse at a time
(76) |
where a_{max} is the value of a which sets the denominator of the integral to zero.
For the case of = 0, we can determine the spherical collapse parameters A and B. K > 0 (K < 0) produces an open (closed) model. Comparing coefficients in the energy equations [eq. (74) and the integration of (66)], one finds
(77) |
(78) |
where _{k} = 1 - _{m}. In particular, in an = 1 Universe, where 1 + z = (3H_{0} t / 2)^{-2/3}, we find that a shell collapses at redshift 1 + z_{c} = 0.5929_{i} / a_{i}, or in other words a shell collapsing at redshift z_{c} had a linear overdensity extrapolated to the present day of _{0} = 1.686(1 + z_{c}).
While this derivation has been for spheres of constant density, we may treat a general spherical density profile _{i}(r) up until shell crossing [164]. A particular radial shell evolves according to the mass interior to it; therefore, we define the average overdensity
(79) |
so that we may use in place of _{i} in the above formulae. If is not monotonically decreasing with R, then the spherical top-hat evolution of two different radii will predict that they cross each other at some late time; this is known as shell crossing and signals the breakdown of the solution. Even well-behaved profiles will produce shell crossing if shells are allowed to collapse to r = 0 and then reexpand, since these expanding shells will cross infalling shells. In such a case, first-time infalling shells will never be affected prior to their turn-around; the more complicated behavior after turn-around is a manifestation of virialization. While the end state for general initial conditions cannot be predicted, various results are known for a self-similar collapse, in which (r) is a power-law [132, 40], as well as for the case of secondary infall models [156, 165, 181].
The small density fluctuations evidenced in the CMB grow over time as described in the previous subsection, until the perturbation becomes of order unity, and the full non-linear gravitational problem must be considered. The dynamical collapse of a dark matter halo can be solved analytically only in cases of particular symmetry. If we consider a region which is much smaller than the horizon cH^{-1}, then the formation of a halo can be formulated as a problem in Newtonian gravity, in some cases with minor corrections coming from General Relativity. The simplest case is that of spherical symmetry, with an initial (t = t_{i} << t_{0}) top-hat of uniform overdensity _{i} inside a sphere of radius R. Although this model is restricted in its direct applicability, the results of spherical collapse have turned out to be surprisingly useful in understanding the properties and distribution of halos in models based on cold dark matter.
The collapse of a spherical top-hat perturbation is described by the Newtonian equation (with a correction for the cosmological constant)
(80) |
where r is the radius in a fixed (not comoving) coordinate frame, H_{0} is the present-day Hubble constant, M is the total mass enclosed within radius r, and the initial velocity field is given by the Hubble flow dr / dt = H(t) r. The enclosed grows initially as _{L} = _{i} D(t) / D(t_{i}), in accordance with linear theory, but eventually grows above _{L}. If the mass shell at radius r is bound (i.e., if its total Newtonian energy is negative) then it reaches a radius of maximum expansion and subsequently collapses. As demonstrated in the previous section, at the moment when the top-hat collapses to a point, the overdensity predicted by linear theory is _{L} = 1.686 in the Einstein-de Sitter model, with only a weak dependence on _{m} and _{}. Thus a tophat collapses at redshift z if its linear overdensity extrapolated to the present day (also termed the critical density of collapse) is
(81) |
where we set D(z = 0) = 1.
Even a slight violation of the exact symmetry of the initial perturbation can prevent the tophat from collapsing to a point. Instead, the halo reaches a state of virial equilibrium by violent relaxation (phase mixing). Using the virial theorem U = -2K to relate the potential energy U to the kinetic energy K in the final state (implying that the virial radius is half the turnaround radius - where the kinetic energy vanishes), the final overdensity relative to the critical density at the collapse redshift is _{c} = 18^{2} 178 in the Einstein-de Sitter model, modified in a Universe with _{m} + _{} = 1 to the fitting formula (Bryan & Norman 1998 [71])
(82) |
where d _{m}^{z}-1 is evaluated at the collapse redshift, so that
(83) |
A halo of mass M collapsing at redshift z thus has a virial radius
(84) |
and a corresponding circular velocity,
(85) |
In these expressions we have assumed a present Hubble constant written in the form H_{0} = 100 h km s^{-1}Mpc^{-1}. We may also define a virial temperature
(86) |
where µ is the mean molecular weight and m_{p} is the proton mass. Note that the value of µ depends on the ionization fraction of the gas; for a fully ionized primordial gas µ = 0.59, while a gas with ionized hydrogen but only singly-ionized helium has µ = 0.61. The binding energy of the halo is approximately ^{5}
(87) |
Note that the binding energy of the baryons is smaller by a factor equal to the baryon fraction _{b} / _{m}.
Although spherical collapse captures some of the physics governing the formation of halos, structure formation in cold dark matter models procedes hierarchically. At early times, most of the dark matter is in low-mass halos, and these halos continuously accrete and merge to form high-mass halos. Numerical simulations of hierarchical halo formation indicate a roughly universal spherically-averaged density profile for the resulting halos (Navarro, Frenk, & White 1997, hereafter NFW [266]), though with considerable scatter among different halos (e.g., [72]). The NFW profile has the form
(88) |
where x = r / r_{vir}, and the characteristic density _{c} is related to the concentration parameter c_{N} by
(89) |
The concentration parameter itself depends on the halo mass M, at a given redshift z [377].
More recent N-body simulations indicate deviations from the original NFW profile; for details and refined fitting formula see [268].
^{1} http://www.eso.org/projects/owl/ Back.
^{2} http://celt.ucolick.org/ Back.
^{3} http://www.gmto.org/ Back.
^{4} Our definition of the cut-off mass follows the convention of the Jeans mass, which is defined as the mass enclosed within a sphere of radius _{J}/2 where _{J} 2 / k_{J} is the Jeans wavelength [168]. Back.
^{5} The coefficient of 1/2 in equation (87) would be exact for a singular isothermal sphere, (r) 1/r^{2}. Back.