A Separate Universe Approach to Quintessence Perturbations

Abstract

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 perturbations 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 cosmological 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 amplify them. Comment: 11 pages. v2: References added

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arXiv:astro-ph/0503680v2 6 Jun 2005
A Separate Universe Approach to
Quintessence Perturbations
Christopher Gordon
Kavli Institute for Cosmological Physics, Enrico Fermi Institute and
Department of Astronomy and Astrophysics, University of Chicago, Chicago
IL 60637
Abstract
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
amplify them.
1Introduction
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

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canonical scalar field which is usually referred to as the ‘quintessence’. The
perturbations were evaluated using the metric approach [6]. This requires
specifying a foliation of the space-time into constant time hypersurfaces or
equivalently the choice of a gauge. In reference [4], the Newtonian gauge is
used and in reference [5] 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 [7]. 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 [8], in the comoving gauge [5]
and in a gauge invariant formulism [9]. 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 [7] 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,
ρ,
?da
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
H2≡
dt
1
a
?2
=
1
3M2
p
ρ(1)
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
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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
d2a
dt2= −
1
6M2
p
(ρ + 3p)a.(4)
The continuity equation is given by
dρ
dt+ 3H(1 + w)ρ = 0.(5)
Assuming the equation of state, w, is constant, the solution to the above
equation is
ρ =
ρ0
a3(1+w)
(6)
where ρ0is the density at a = 1. The distance a signal, traveling at the speed
of light, can travel is
?tf
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
ti
1
adt(7)
N ≡ log(a).(8)
Then from the definition of the Hubble parameter
dN
dt
= H(9)
which when substituted into the distance equation (Eq. (7)) gives
D = a
?N
0
1
a
1
HdN.
(10)
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Using the equation for the density (Eq. (6)) and the definition of efolds
(Eq. (8)), the Friedmann equation (Eq. (1)) can be expressed as
H2=
1
3M2
p
e−3(1+w)N. (11)
Substituting this into Eq. (10) and solving the integral gives
D =
2√3
1 + 3w
Mp
√ρ0eN?eN(1+3w)/2− 1?.(12)
The Hubble distance is defined as
DH≡1
H.
(13)
Using the distance equation (Eq. (12)) and the Friedmann equation (Eq. (11))
gives
D
DH
1 + 3w
=
2
?
1 − e−1
2(1+3w)N?
.(14)
From the Friedmann equation (Eq. (11)), the distance equation (Eq. (12))
and the definition of efolds (Eq. (8))
D
aDH|N=0
=
2
(1 + 3w)
?
e
1
2(1+3w)N− 1
?
.(15)
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
[10]. For large N and w < −1/3, the ratio of the red shifting initial Hubble
parameter to the casual distance (Eq. (15)) tends to
D
aDH|N=0
→
−2
1 + 3w.
(16)
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.
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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
D
DH
→
2
1 + 3w.
(17)
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 [6]. 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 [7].
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
δf =
?
i
∂f
∂φi|N=0δφi|N=0.(18)
This equation summarizes the separate Universe approach and will be used
in evaluating matter perturbations in the rest of the paper.
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