**2.4. Coupled perturbations**

We will often be concerned with the evolution of perturbations in a universe that contains several distinct components (radiation, baryons, dark matter). It is easy to treat such a mixture if only gravity is important (i.e. for large wavelengths). Look at the perturbation equation in the form

(33) |

The rhs represents the effects of gravity, and particles will respond to gravity whatever its source. The coupled equations for several species are thus given by summing the driving terms for all species.

**Matter plus radiation**
The only subtlety is that we must take into account the
peculiarity that radiation and pressureless matter respond
to gravity in different ways, as seen in the equations
of fluid mechanics. The coupled equations for perturbation growth are thus

(34) |

Solutions to this will be simple if the matrix
has time-independent eigenvectors. Only one of
these is in fact time independent: (1, 4/3).
This is the adiabatic mode in which
_{r} =
4_{m} / 3 at all
times. This corresponds to some initial disturbance in which matter
particles and photons are compressed together. The entropy per
baryon is unchanged,
(*T*^{3})
/ (*T*^{3}) =
_{m}, hence
the name `adiabatic'. In this case, the perturbation amplitude
for both species obeys
*L* =
4
*G*(_{m} +
8_{r}
/ 3).
We also expect the baryons and photons to obey this
adiabatic relation very closely even on small scales:
the tight coupling approximation says that Thomson
scattering is very effective at suppressing motion of
the photon and baryon fluids relative to each other.

**Isocurvature modes**
The other perturbation mode is harder to see until we realize
that, whatever initial conditions we choose for
_{r} and
_{m}, any
subsequent changes to matter and radiation on large scales
must be adiabatic (only gravity is acting).
Suppose that the radiation field is initially chosen to be uniform;
we then have

(35) |

where _{i} is
some initial value of
_{m}. The
equation for _{m}
becomes

(36) |

which is as before if
_{i} = 0. The
other solution is therefore a particular integral with
_{i}.
For = 1, the answer
can be expressed most neatly in terms of
*y*
_{m} /
_{r}
(Peebles 1987):

(37) |

At late times,
_{m}
0, while
_{r}
-4_{i} / 3.
This mode is called the isocurvature mode, since it
corresponds to a total density perturbation
/
0 as
*t*_{i}
0.
Subsequent evolution attempts to preserve constant density
by making the matter perturbations decrease while the amplitude
of _{r} increases.
An alternative name for this mode is an entropy perturbation.
This reflects the fact that one only perturbs the initial
ratio of photon and matter number densities.
The late-time evolution is then easily understood:
causality requires that, on large scales, the initial
entropy perturbation is not altered. Hence, as the universe
becomes strongly matter dominated, the entropy perturbation
becomes carried entirely by the photons. This
leads to an increased amplitude of microwave-background
anisotropies in isocurvature models
(Efstathiou & Bond
1986),
which is one reason why such models are not popular.
Of course, a small admixture of isocurvature perturbations
is always going to be hard to rule out (e.g.
Bucher, Moodley &
Turok 2002),
so neglect of this mode
is primarily justified by the fact that the simplest model for
the generation of cosmological perturbations (single-field
inflation) produces pure adiabatic modes.
Models with multiple fields, such as the decaying curvaton of
Lyth & Wands (2002)
tend to generate order-unity isocurvature contributions, which are
impossible to reconcile with CMB data (e.g.
Gordon & Lewis 2002).

**Baryons and dark matter**
This case is simpler, because both components have the same equation of
state:

(38) |

Both eigenvectors are time independent: (1, 1) and
(_{d}, -
_{b}). The
time dependence of these modes is easy to see for an
= 1
matter-dominated universe: if we try
*t*^{n},
then we obtain respectively
*n* = 2/3 or -1 and *n* = 0 or -1/3 for the two modes.
Hence, if we set up a perturbation with
_{b} = 0, this
mixture of the eigenstates will quickly evolve to be dominated
by the fastest-growing mode with
_{b} =
_{d}:
the baryonic matter falls into the dark potential wells.
This is one process that allows universes containing
dark matter to produce smaller anisotropies in the
microwave background: radiation drag allows the dark matter
to undergo growth between matter-radiation equality and
recombination, while the baryons are held back.

This is the solution on large scales, where pressure effects are negligible. On small scales, the effect of pressure will prevent the baryons from continuing to follow the dark matter. We can analyse this by writing down the coupled equation for the baryons, but now adding in the pressure term (sticking to the matter-dominated era, to keep things simple):

(39) |

In the limit that dark matter dominates the gravity,
the first term on the rhs can be taken as an imposed
driving term, of order
_{d} /
*t*^{2}. In the absence of pressure, we saw that
_{b}
and _{d} grow
together, in which case the second term on the rhs is smaller than the
first if *kc*_{S} *t* / *a* << 1.
Conversely, for large wavenumbers (*kc*_{S}
*t*/*a* >> 1),
baryon pressure causes the growth rates in the baryons and
dark matter to differ; the main behaviour of the
baryons will then be slowly declining sound waves,
and we can write the WKB solution.

(40) |

where is conformal time. An alternative way to see that the baryons are damped is to write the coupled equations as

(41) |

where
*k* /
*k*_{J}. In the special case
_{b}
0 and
= constant, a solution is
clearly

(42) |

and this is found to be the asymptotic solution in more general cases (Nusser 2000).

This oscillatory behaviour holds so long as pressure forces
continue to be important. However, the sound speed drops by a large
factor at recombination, and we would then expect the oscillatory
mode to match on to a mixture of the pressure-free growing and
decaying modes. This behaviour can be illustrated in a simple
model where the sound speed is constant until recombination at conformal
time _{r}
and then instantly drops to zero. The behaviour of the density
field before and after *t*_{r} may be written as

(43) |

where
*kc*_{S}.
Matching and its time
derivative on either side of the
transition allows the decaying component to be eliminated, giving
the following relation between the growing-mode amplitude after
the transition to the amplitude of the initial oscillation:

(44) |

The amplitude of the growing mode after recombination depends on the phase of the oscillation at the time of recombination. The output is maximised when the input density perturbation is zero and the wave consists of a pure velocity perturbation; this effect is known as velocity overshoot. The post-recombination transfer function will thus display oscillatory features, peaking for wavenumbers that had particularly small amplitudes prior to recombination. Such effects can be seen at work in determining the relative positions of small-scale features in the power spectra of matter fluctuations and microwave-background fluctuations.