**2.5. Transfer functions and characteristic scales**

The transfer function for models with the full above list of ingredients was first computed accurately by Bond & Szalay (1983), and is today routinely available via public-domain codes such as CMBFAST (Seljak & Zaldarriaga 1996). These calculations are a technical challenge because we have a mixture of matter (both collisionless dark particles and baryonic plasma) and relativistic particles (collisionless neutrinos and collisional photons), which does not behave as a simple fluid. Particular problems are caused by the change in the photon component from being a fluid tightly coupled to the baryons by Thomson scattering, to being collisionless after recombination. Accurate results require a solution of the Boltzmann equation to follow the evolution in detail.

Some illustrative results are shown in figure 2.
Leaving aside the isocurvature models, all adiabatic cases have
*T* 1
on large scales - i.e. there is growth at the universal
rate (which is such that the amplitude of potential perturbations
is constant until the vacuum starts to be important at
*z* 1).
The different shapes of the functions can be understood intuitively
in terms of a few special length scales, as follows:

**(1) Horizon length at matter-radiation equality.**
The main bend visible in all
transfer functions is due to the Mészáros effect, which
arises because the universe is radiation
dominated at early times. Fluctuations in the matter can only grow if dark
matter and radiation fall together. This does not
happen for perturbations of small wavelength, because
photons and matter can separate. Growth only occurs for perturbations of
wavelength larger than the horizon distance, where there has been no
time for the matter and radiation to separate.
The relative diminution in fluctuations at high *k* is the amount
of growth missed out on between horizon entry and
*z*_{eq}, which would be
*D*_{H}^{2} in the
absence of the Mészáros effect.
Perturbations with larger *k* enter the horizon
when *D*_{H}
1/*k*; they are
then frozen until *z*_{eq}, at which point they can grow again.
The missing growth factor is just the square of the change in
*D*_{H} during this period,
which is
*k*^{2}. The approximate limits of the CDM
transfer function are therefore

(45) |

This process continues, until *z*_{eq} = 23 900
_{m}
*h*^{2}, where the universe becomes matter dominated.
We therefore expect a characteristic `break' in the
fluctuation spectrum around the comoving horizon length at this time.
Using a distance-redshift relation that ignores vacuum energy at high
*z*,

(46) |

we obtain

(47) |

Since distances in cosmology always scale as *h*^{-1}, this
means that
_{m}
*h* should be observable.

**(2) Free-streaming length.**
This relatively gentle filtering away of the initial
fluctuations is all that applies to a universe dominated
by Cold Dark Matter, in which random velocities are negligible.
A CDM universe thus contains fluctuations in the dark matter on all
scales, and structure formation proceeds via hierarchical process in
which nonlinear structures grow via mergers.

Examples of CDM would be thermal relic WIMPs with masses
of order 100 GeV. Relic particles that were never in
equilibrium, such as axions, also come under
this heading, as do more exotic possibilities such as primordial black
holes. A more interesting case arises when thermal relics have lower masses.
For collisionless dark matter, perturbations can be
erased simply by free streaming: random
particle velocities cause blobs to disperse.
At early times (*kT* > *mc*^{2}), the particles will
travel at *c*, and so any perturbation that has
entered the horizon will be damped.This process switches off when the
particles become non-relativistic, so that perturbations
are erased up to proper lengthscales of
*ct*(*kT* =
*mc*^{2}). This translates to a comoving horizon scale
(2*ct* / *a* during the radiation era) at *kT* =
*mc*^{2} of

(48) |

(in detail, the appropriate figure for neutrinos will
be smaller by (4/11)^{1/3} since they have a smaller
temperature than the photons).
A light neutrino-like relic that decouples while
it is relativistic satisfies

(49) |

Thus, the damping
scale for HDM (Hot Dark Matter) is of order the bend scale.
Alternatively, if the particle decouples sufficiently
early, its relative number density is boosted by annihilations,
so that the critical particle mass to make
_{m} = 1 can
be boosted to around 1-10 keV (Warm Dark Matter).
The existence of galaxies at *z*
6 tells us that the
coherence scale must have been below about 100 kpc, so
WDM is close to being ruled out. A similar constraint is obtained from
small-scale structure in the Lyman-alpha forest
(Narayanan et al. 2000):
*m* > 0.75 keV.

A more interesting (and probably practically relevant) case
is when the dark matter is a mixture of hot and cold
components. The free-streaming length for the hot
component can therefore be very large, but within
range of observations. The dispersal of HDM fluctuations
reduces the CDM growth rate on all scales below
*L*_{free-stream} - or, relative to small
scales, there is an enhancement in large-scale power.

**(3) Acoustic horizon length.**
The horizon at matter-radiation equality also enters in
the properties of the baryon component. Since the sound speed
is of order *c*, the largest scales that can undergo a single
acoustic oscillation are of order the horizon.
The transfer function for a pure baryon universe shows
large modulations, reflecting the number of oscillations
that have been completed before the universe becomes
matter dominated and the pressure support drops.
The lack of such large modulations in real data is
one of the most generic reasons for believing in
collisionless dark matter. Acoustic oscillations
persist even when baryons are subdominant, however, and
can be detectable as lower-level modulations in the transfer
function (e.g.
Goldberg & Strauss
1998;
Meiksin et al. 1999).

**(4) Silk damping length.**
Acoustic oscillations are also damped on small scales, where
the process is called Silk damping: the
mean free path of photons due to scattering by the plasma
is non-zero, and so radiation can diffuse out of a
perturbation, convecting the plasma with it.
The typical distance of a random walk in terms
of the diffusion coefficient *D* is
*x*
*Dt*^{1/2}, which gives a damping length of

(51) |

the geometric mean of the horizon size and the mean free path.
Since = 1 / (*n*
_{T}) = 44.3(1 +
*z*)^{-3}(_{b} *h*^{2})^{-1}
proper Gpc, we obtain a comoving damping length of

(52) |

This becomes close to the horizon length by the time of last
scattering,
1 + *z* 1100.
The resulting damping effect
can be seen in figure 2
*k* ~ 10*k*_{H}.

**Fitting formulae**
It is invaluable in practice to have some
accurate analytic formulae that fit the
numerical results for transfer functions.
We give below results for some common models of particular interest
(illustrated in figure 2 along with other
cases where a fitting formula is impractical).
For the models with collisionless dark matter,
_{b}
<< _{m}
is assumed, so that all lengths
scale with the horizon size at matter-radiation equality,
leading to the definition

(52) |

We consider the following cases: (1) Adiabatic CDM; (2) Adiabatic
massive neutrinos (1 massive, 2 massless); (3) Isocurvature
CDM; these expressions come from Bardeen et al.
(1986; BBKS).
Since the characteristic length-scale in the transfer function
depends on the horizon size at matter-radiation equality,
the temperature of the CMB enters. In the above formulae,
it is assumed to be exactly 2.7 K; for other values, the
characteristic wavenumbers scale
*T*^{-2}.
For these purposes massless neutrinos count as radiation,
and three species of these contribute a total density
that is 0.68 that of the photons.

(53) |

The case of mixed dark matter (MDM: a mixture of massive neutrinos and CDM) is more complex. Ma (1996) gives the following expression:

(54) |

where *x* *k* /
_{},
_{}
*a*^{1/2}
_{} *h*^{2}
and *x*_{0} is the value of *x* at *a* = 1. The
scale-factor dependence is such that the
damping from neutrino free-streaming is less severe at high redshift,
but the spectrum is very nearly of constant shape for
*z* 10. See
Pogosyan & Starobinsky
(1995)
for a more complicated fit of higher accuracy.

These expressions are useful for work at a level of 10% precision, but increasingly it is necessary to do better. In particular, these expressions do not include the weak oscillatory features that are expected if the universe has a significant baryon content. Eisenstein & Hu (1998) give an accurate (but long) fitting formula that describes these wiggles for the CDM transfer function. This was extended to cover MDM in Eisenstein & Hu (1999).