arXiv:astro-ph/0503680v2 6 Jun 2005
A Separate Universe Approach to
Kavli Institute for Cosmological Physics, Enrico Fermi Institute and
Department of Astronomy and Astrophysics, University of Chicago, Chicago
There is some observational evidence that the dark energy may
not be smooth on large scales. This makes it worth while to try and
get as simple and as intuitive a picture of how dark energy pertur-
bations behave so as to be able to better constrain possible models
of dark energy and the generation of large scale perturbations. The
separate Universe method provides an easy way to evaluate cosmolog-
ical perturbations, as all that is required is an understanding of the
background behavior. Here, this method is used to show how the size
of the dark energy perturbations, preferred by observations, is larger
than would be expected, and so some mechanism may be required to
One way to rule out a cosmological constant would be to show that the dark
energy in not spatially homogenous. Recently, it was found that the low
quadrupole in the WMAP data [1, 2, 3] favors perturbations in the dark
energy at the two sigma level [4, 5]. The dark energy was taken to be a
canonical scalar field which is usually referred to as the ‘quintessence’. The
perturbations were evaluated using the metric approach . This requires
specifying a foliation of the space-time into constant time hypersurfaces or
equivalently the choice of a gauge. In reference , the Newtonian gauge is
used and in reference  the comoving gauge.
An alternative way of treating perturbations in cosmology is the ‘separate
Universe’ approach, in which the background equations can be perturbed di-
rectly to get the large scale perturbations in the flat gauge . This has the
advantage that the derivation and solution of the perturbation equations fol-
low simply from the background equations. It also provides a more intuitive
picture of how the perturbations behave.
In this article we study the quintessence perturbations using the separate
Universe approach. There are many other treatments of quintessence pertur-
bations, for example in the Newtonian gauge , in the comoving gauge 
and in a gauge invariant formulism . I hope the treatment in this article
will be complementary to those, and help to provide a clearer picture on the
nature of density perturbations in quintessence.
2Separate Universe Method
This method provides a simple way of modeling large scale inhomogenities.
Here we give a pedagological explanation, which emphasizes its casual un-
derpinnings. See references  for more detailed derivations.
If the Universe is spatially smooth and the overall geometry is flat, then
Friedmann equation relates the scale of the Universe, a, to its overall density,
where H is known as the Hubble parameter, t is the time and Mp≡ (8πG)−1/2=
2.436 × 1018GeV is the reduced Planck mass. The pressure, p, is related to
the density by the equation of state parameter
w ≡ p/ρ.(2)
It has the value of zero for non-relativistic matter and a third for radiation.
For a canonical scalar field it is greater than or equal to minus one. Unless
otherwise specified, w will be assumed to satisfy
|w| < 1.(3)
Einstein’s theory of general relativity also relates the acceleration of the scale
factor to the matter density and pressure
(ρ + 3p)a. (4)
The continuity equation is given by
dt+ 3H(1 + w)ρ = 0.(5)
Assuming the equation of state, w, is constant, the solution to the above
where ρ0is the density at a = 1. The distance a signal, traveling at the speed
of light, can travel is
where the units are chosen so that the speed of light is one, and tiand tfare
the initial time and final times respectively. It is convenient to express time
in terms of the number of efolds of expansion
D ≡ a
N ≡ log(a).(8)
Then from the definition of the Hubble parameter
which when substituted into the distance equation (Eq. (7)) gives
D = a
Using the equation for the density (Eq. (6)) and the definition of efolds
(Eq. (8)), the Friedmann equation (Eq. (1)) can be expressed as
Substituting this into Eq. (10) and solving the integral gives
1 + 3w
The Hubble distance is defined as
Using the distance equation (Eq. (12)) and the Friedmann equation (Eq. (11))
1 + 3w
1 − e−1
From the Friedmann equation (Eq. (11)), the distance equation (Eq. (12))
and the definition of efolds (Eq. (8))
(1 + 3w)
As can be seen from Eqs. (14) and (15), the value of w = −1/3 is special.
For w < −1/3, the acceleration equation (Eq. (4)) and the definition of the
equation of state (Eq. (2)) give an accelerating scale factor. This is thought
to have occurred in the early Universe during a period known as inflation
. For large N and w < −1/3, the ratio of the red shifting initial Hubble
parameter to the casual distance (Eq. (15)) tends to
1 + 3w.
It follows that, points that are initially more than of order a Hubble distance
apart are always out of causal contact as long as inflation lasts.
After inflation, there is the radiation dominated era with w = 1/3 followed
by the matter dominated era with w = 0. When w > −1/3, the ratio of the
casual distance to the Hubble distance (Eq. (14)) tends to
1 + 3w.
Thus, scales larger than the Hubble horizon remain out of casual contact
until the Hubble horizon grows to be comparable to them.
It follows that a patch of space whose density or other matter variables
are different from those of the surrounding space, and whose size is larger
than the Hubble distance during inflation, will evolve like a separate homoge-
nous Universe. It will continue to do so until the Hubble distance becomes
comparable to the patch size. The difference between a matter variable in
the patch and outside the patch can be evaluated by solving the background
equations for the background space-time and those for the patch and then
subtracting the difference between the two. The coordinate freedom of the
time surfaces on which to match the patch and background is the same as
the usual gauge freedom in cosmological perturbations . If the coordinate
system is chosen so that the patch and the background have the same scale
factor or equivalently the same efoldings, then, under reasonable assump-
tions, the difference between the matter variables in the patch and in the
background space-time is the same as the perturbation in the flat gauge .
In a homogenous space-time, the state of the Universe at any time can be
totally specified in terms of the degrees of freedom such as the different flu-
ids’ densities and pressures and the values and time derivatives of the scalar
fields. Thus, the large scale, flat gauge perturbation, δf, of a function, f, of
the matter degrees of freedom, φi, can be evaluated as
This equation summarizes the separate Universe approach and will be used
in evaluating matter perturbations in the rest of the paper.
3 Quintessence Background Equations
The Klein-Gordon equation for the quintessence, Q, is given by
∂Q= 0 (19)
where V is the quintessence potential. We can express the Klein-Gordon
equation (Eq. (19)) in terms of N, by using the Friedmann equation (Eq. (1))
and the acceleration equation (Eq. (4)), as
2(1 − w)Q′+
where the prime indicates differentiation with respect to the number of efolds,
N, and w is the total equation of state parameter (Eq. (2)), including the
effects of any other matter present. If the value of Q does not change much
during the period of interest, then its potential may be approximated by a
constant plus a linear term
V ≈ V∗
1 +Q − Q∗
where V∗ and ǫ∗ are constants and correspond to the potential and first
slow roll parameter, at the point about which the expansion is taken, Q∗,
respectively. A class of potentials that do not satisfy this criteria in general
are the ‘tracking’ potentials which require 
In the tracking regime, the solutions are insensitive to changes in the initial
conditions of the quintessence. It follows, from the discussion in Sec. 2, that
a patch with different initial conditions for the quintessence, will quickly ap-
proach the background solution for the quintessence, and so the perturbations
will be suppressed [12, 8, 5].
Substituting the linear form of the potential (Eq. (21)) into the Klein-
Gordon equation (Eq. (20)) and using the Friedmann equation (Eq. (11))
2(1 − w)Q′+ 3√2√ǫ∗MpV∗
Assuming the total equation of state parameter (w) is a constant, less than
one, and denoting the values of Q and its derivative, at N = 0, by Q0and Q′
respectively, the solution to the Klein-Gordon equation (Eq. (23)) for large
3(1 − w)Q′
As ǫ∗< 1 is needed for the quintessence to cause acceleration, and V ? ρ
today, it follows that the last term in the above equation will be negligible
until a redshift of about one. It follows that, the quintessence field is frozen
until about that point. This does not apply to the case where the field starts
rolling in the early Universe due to a steep potential and then rolls into an
area of parameter space where the potential is shallow.
The energy density of the quintessence is
Q = Q0+
3(3 + 4w + w2)
ρQ≡ V +1
= V +1
Substituting the solution for Q (Eq. (24)) and the linear potential (Eq. (21))
into the above equation gives
3(1 − w)
This shows that the density will also be frozen until about z ∼ 1.
ρQ= V |Q=Q0+ V∗
3(1 + w)(3 + w)2
Using the separate Universe approach (Eq. (18)), the density perturbation
of the quintessence, in the flat gauge and on large scales, can be written as
Substituting the result for the quintessence density (Eq. (26)) into the above
equation and using the background density equation (Eq. (6)) gives
3(1 − w)
3(1 + w)(3 + w)2
As can be seen from the above equation, the quintessence will acquire large
scale density perturbations even if it is initially homogenous. This is in
contrast to two fluid components which, as can be seen from the density
equation (Eq. (6)), only depend on their own initial perturbation, and so if
one is initially homogenous it will remain homogenous on large scales. The
reason why the scalar field acquires a perturbation is from the coupling to
the Hubble parameter in the Klein-Gordon equation (Eq. (20)). In the case
of a fluid, this coupling is lost in the continuity equation (Eq. (5)) when the
time parameter is converted to efolds.
Using the solution for the quintessence density (Eq. (26)) and density
perturbation (Eq. (28)), it can be shown that
This shows that the perturbation in ρQbecomes adiabatic if the scalar field
is initially unperturbed.
The magnitude of the adiabatic perturbation can be evaluated from the
solution for the perturbation (Eq. (28)) and background density (Eq. (26))
to give, assuming ǫ∗V∗/ρ ≪ 1,
The value of the perturbation in the matter variable, ρ0, can be evaluated by
its relation to the measured value of the curvature perturbation on constant
density hyper-surfaces . In the flat gauge, this is given by 
3(1 + w)(3 + w)2
3(1 + w)ρ0. (31)
The WMAP CMB measurements constrain ζ  and so give the variance of
the density perturbation to be
3(1 + w)ρ0
≈ 5 × 10−5. (32)
Substituting the above equation into the expression for the adiabatic pertur-
bation (Eq. (30)) gives
< 4 × 10−5. (33)
Using the WMAP data, the quintessence density perturbation is measured
to be 
at the 68% confidence interval. It follows that the adiabatic perturbation
(Eq. (33)) is small in comparison to this value.
quintessence is a light field it will acquire the usual large scale perturbations
which depend on the Hubble parameter,
= (8 ± 4) × 10−4
During inflation, if the
As gravitational waves have not been detected in the WMAP data, this puts
an upper limit on this quantity 
?1/2< 9 × 10−6Mp. (36)
Using the above equation in the solution for the quintessence density per-
turbation and background value (Eqs. (28) and (26)) gives the non-adiabatic
part, from inflation, of the quintessence density perturbation as
which is also too small compared to the observationally preferred value
Using the separate Universe approach, I have investigated the perturbations
in the quintessence by perturbing the background solutions. It was shown
how the quintessence perturbation and background value can be frozen from
inflation until about dark energy domination. Also the way in which a ho-
mogenous quintessence field acquires an adiabatic perturbation was illumi-
Both the adiabatic perturbation and the non-adiabatic perturbation in
the quintessence from inflation where shown to be smaller than the value
preferred by observations. This problem was originally identified in references
[4, 5] using the Newtonian and comoving curvature gauges.
In this paper, the adiabatic perturbation was shown to be determined by
the perturbation in the non-relativistic matter and so fixed by observations.
However, the non-adiabatic perturbation from δQ0 can be made large by
making the quintessence density more sensitive to the initial value of Q0. This
requires some evolution in δQ. A mechanism for amplifying the quintessence
perturbation is given in reference .
Acknowledgments: I thank Wayne Hu and Dragan Huterer for useful
discussions. Also, I was supported by the KICP under NSF PHY-0114422.
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