Constraining the Dark Energy Equation of State with Cosmic Voids
CONSTRAINING THE DARK ENERGY EQUATION OF
STATE WITH COSMIC VOIDS
Jounghun Lee and Daeseong Park
Department of Physics and Astronomy, FPRD, Seoul National University, Seoul 151-747,
Korea
jounghun@snu.ac.kr
ABSTRACT
Our universe is observed to be accelerating due to the dominant dark en-
ergy with negative pressure. The dark energy equation of state (w) holds a key
to understanding the ultimate fate of the universe. The cosmic voids behave
like bubbles in the universe so that their shapes must be quite sensitive to the
background cosmology. Assuming a flat universe and using the priors on the
matter density parameter (Ωm) and the dimensionless Hubble parameter (h),
we demonstrate analytically that the ellipticity evolution of cosmic voids may
be a sensitive probe of the dark energy equation of state. We also discuss the
parameter degeneracy between w and Ωm.
Subject headings: cosmology:theory — large-scale structure of universe
Recent observations have revealed that our universe is flat and in a phase of acceler-
ation (Riess et al. 1998; Perlmutter et al. 1999; Spergel et al. 2003). It implies that some
mysterious dark energy fills dominantly the universe at present epoch, exerting anti-gravity.
The nature of this mysterious dark energy which holds a key to understanding the ultimate
fate of the universe is often specified by its equation of state, i.e., the ratio of its pressure to
density: w ≡ Pde/ρde. The anti-gravity of the dark energy corresponds to the negative value
of w. The simplest candidate for the dark energy is the vacuum energy (Λ) with w = −1
that is constant at all times (Einstein 1917). Although all current data are consistent with
the vacuum energy model (e.g., Wang & Tegmark 2004; Jassal et al. 2004; Percival 2005;
Guzzo et al. 2008), the notorious failure of the theoretical estimate of the vacuum energy
density (see Caroll et al. 1992, for a review) has led a dynamic dark energy model to emerge
as an alternative. In this dynamic dark energy models which is often collectively called
quintessence, the dark energy is described as a slowly rolling scalar field with time-varying
equation of state in the range of −1 < w < 0 (Caldwell et al. 1998).
http://arxiv.org/abs/0704.0881v2
– 2 –
The following observables have so far been suggested to discriminate the dark en-
ergy models: the luminosity-distance measure of type Ia supernova (Riess et al. 2004, 2007;
Davis et al. 2007; Kowalski et al. 2008); the abundance of galaxy clusters as a function of
mass (Wang & Steinhardt 1998; Haiman et al. 2001; Weller et al. 2002), the baryonic acous-
tic oscillations in the galaxy power spectrum (Blake & Glazebrook 2003; Hu & Haiman 2003;
Cooray 2004; Seo & Eisentein 2005), and the weak gravitational lensing effect (Hu 1999;
Huterer 2001; Takada & Jain 2004; Song & Knox 2004). True as it is that these observables
can constrain powerfully the value of w, it is still quite necessary and important to find out
as many different observables as possible for consistency tests.
Another possible observable as a dark energy constraint may be the shapes of the cos-
mic voids. As the voids behave like bubbles due to their extremely low densities, their
shapes determined by the spatial distribution of the void galaxies tend to change sensitively
according to the competition between the tidal distortion and the gravitational rarefaction
effect. Therefore, the shape evolution of the voids must depend sensitively on the background
cosmology. In this Letter we study the ellipticity evolution of cosmic voids in the QCDM
(quintessence + cold dark matter) model with the help of the analytic formalism developed
by Park & Lee (2007) and explore the possibility of using it as a complimentary probe of
the dark energy equation of state.
According to Park & Lee (2007), the shape of a void region is related to the eigenvalues
of the local tidal shear tensor as
λ1(µ, ν) =
1 + (δv − 2)ν2 + µ2
(µ2 + ν2 + 1)
, (1)
λ2(µ, ν) =
1 + (δv − 2)µ2 + ν2
(µ2 + ν2 + 1)
, (2)
where {λi}3i=1 (with λ1 > λ2 > λ3) are the three eigenvalues of the local tidal field smoothed
on void scale, δv is the density contrast threshold for the formation of a void: δv =
i=1 λi,
and {µ, ν} (with ν < µ) represents a set of the two parameters that quantify the anisotropic
distribution of the void galaxies. They defined the void ellipticity as ε ≡ 1−ν and evaluated
its probability density distribution as
p(1− ε; z) = p(ν; z, RL) =
p[µ, ν|δ = δv; σ(z, RL)]dµ
10πσ5(z, RL)
2σ2(z, RL)
15δv(λ1 + λ2)
2σ2(z, RL)
× exp
15(λ21 + λ1λ2 + λ
2σ2(z, RL)
(2λ1 + λ2 − δv)
×(λ1 − λ2)(λ1 + 2λ2 − δv)
4(δv − 3)2µν
(µ2 + ν2 + 1)3
. (3)
– 3 –
Here, σ(z, RL)) ≡ D2(z)
∆2(k)W 2(kRL)d ln k is the linear rms fluctuation of the matter
density field smoothed on a Lagrangian void scale of RL at redshift z where D(z) is the
linear growth factor, W (kRL) is a top-hat window function, and ∆
2(k) is the dimensionless
linear power spectrum. Throughout this study, we adopt the linear power spectrum of the
cold dark matter cosmology (CDM) that does not depend explicitly on w (Bardeen et al.
1986).
Equation (3) was originally derived under the assumption of a ΛCDM model (w = −1).
We propose here that it also holds good for the case of a QCDM (quintessence+CDM) model
where the dark energy equation of state changes with time as w(z) = w0 + waz/(1 + z)
(Chevallier & Polarski 2001; Linder 2003) where w0 is the value of w at present epoch and
wa quantifies how the dark energy equation of state changes with time. Then, we employ
the following approximation formula for the linear growth factor, D(z), for a QCDM model
(Basilakos 2003; Percival 2005):
D(z) =
2(z + 1)
Ωαm − ΩQ +
(1 +AΩQ)
. (4)
where
E2(z) = Ωm(1 + z)
3 + ΩQ(1 + z)
−f(z), (5)
f(z) = −3(1 + w0)−
2 ln(1 + z)
, (6)
5− 2/(1− w)
(1− w)(1− 3w/2)
(1− 6w/5)3
[1− Ωm], (7)
A = − 0.28
w + 0.08
− 0.3. (8)
The CDM density parameter Ωm and the dark energy density parameter ΩQ evolve with z
respectively as
Ωm(z) =
Ωm0(1 + z)
E2(z)
, ΩQ(z) =
E2(z)(1 + z)f(z)
, (9)
where Ωm0 and ΩQ0 represent the present values. Equation (3) implies that the mean elliptic-
ity of voids decreases with z. A key question is how the rate of the decrease changes with the
dark energy equation of state. Since most of the recent observations indicate that the dark
energy equation of state at present epoch is consistent with w = −1 (e.g., see Guzzo et al.
2008, and references therein) we focus on how the mean void ellipticity depends on the value
of wa. Even in case that w0 = −1, if wa is found to deviate from zero, it would imply the
dynamic dark energy, disproving the simple ΛCDM model.
To explore how the void ellipticity evolution depends on wa, we evaluate the mean
ellipticity of voids as ε̄(z) =
ε p(ε;RL, z)dǫ for different values of wa through equations
– 4 –
(3)- (9). The other key cosmological parameters are set at Ωm = 0.75,ΩQ = 0.75, h = 0.73,
σ8 = 0.9 and w0 = −1. When the abundance of evolution of galaxy clusters is used to
constrain the dark energy equation of state, the cluster mass is usually set at a certain
threshold, MR, defined as the mass within a certain comoving radius (Wang & Steinhardt
1998). Likewise, we set the Lagrangian scale of a void, RL at 4h
−1Mpc, which is related
to the mean effective radius of a void as R̄E = (1 + δv)
−1/3R̄L/(1 + z). The Lagrangian
scale RL = 4h
−1Mpc corresponds to the mean effective size of a void at present epoch,
RE ∼ 8.5h−1Mpc.
Figure 1 plots ε̄(z) for the four different cases: wa = −1/3, 0, 1/3 and 2/3 (long-dashed,
solid, dashed, and dotted line, respectively). As can be seen, the higher the value of wa is,
the more rapidly ε̄(z) decreases. It also suggests that ε̄(z) is well approximated as a linear
function of z in recent epochs (0 < z < 0.2). Therefore, we fit ε(z) to a straight line as
ε̄(z) ≈ Avz + Bv. Varying the value of wa in the range of [0, 2/3], we compute the best-fit
slope Av. The range, 0 ≤ wa ≤ 2/3, corresponds to the dark energy equation of state range,
−1 ≤ w ≤ −0.9. The result is plotted in Fig. 2. As can be seen, the void ellipticity evolves
more rapidly as the value of wa increases. That is, the void ellipticity undergoes a stronger
evolution when the anti-gravitational effect is less strong in recent epochs. Note that Av
shows a noticeable 30% difference as the dark energy equation of state changes w from −1
to −0.9.
We have so far neglected the parameter degeneracy between w and the other key pa-
rameters. However, as the dependence of the void ellipticity distribution on the dark energy
equation of state comes from its dependence on ∆2(k; Ωm0, σ8, h, w), it is naturally expected
that there should be a strong parameter degeneracy. Here, we focus on the degeneracy be-
tween Ωm0 and w. First, we recompute Av, varying the values of Ωm0 and w0 with setting
wa = 1/3. The left panel of Fig. 3 plots a family of the degeneracy curves in the Ωm0-w0
plane for the three different values of Av. As can be seen, there is a strong degeneracy
between the two parameters. For a given value of Av, the value of w0 increases as the value
of Ωm0 decreases. A similar trend is also found in the Ωm0-wa degeneracy curves that are
plotted in the right panel of Fig. 3 for which the value of w0 is set at −1. It is worth noting
that this degeneracy trend is orthogonal to that found from the cluster abundance evolution
(see Fig.3 in Wang & Steinhardt 1998). Thus, when combined with the cluster analysis, the
void ellipticity analysis may be useful to break the degeneracy between Ωm0 and w.
We have shown that the void ellipticity evolution is in principle a useful constraint of the
dark energy equation of state. We have also shown that it provides a new degeneracy curve
for Ωm0 and w. When combined with the cluster abundance analysis, it should be useful
to break the degeneracy. Furthermore, unlike the mass measurement of high-z clusters
– 5 –
which suffers from considerable scatters, the void ellipticities are readily measured from the
positions of the void galaxies without requiring any additional information.
To use our analytic tool in practice to constrain the dark energy equation of state,
however, it will require to account for the redshift distortion effect since the positions of the
void galaxies are measured in redshift space. In our companion paper (Park & Lee 2009 in
preparation), we have analyzed the Millennium Run Redshift-Space catalog (Springel et al.
2005) and determined the ellipticity distribution of the galaxy voids. From this analysis, it
is somewhat unexpectedly found that the void ellipticity distribution measured in redshift
space is hardly changed from the one in real space. In fact, this result is consistent with
the recent claims of Hoyle & Vogeley (2007) and that of van de Weygaert (2008, private
communication) who have already pointed out that the redshift distortion effect has only
negligible, if any, effect on the shapes of voids. We hope to constrain the dark energy
equation of state by applying our theoretical tool to real observational data and report the
result elsewhere in the near future.
We thank an anonymous referee for a constructive report. J.L. am very grateful to
S.Basilakos for very helpful discussion and comments. This work is financially supported
by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean
Government (MOST, NO. R01-2007-000-10246-0).
– 6 –
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This preprint was prepared with the AAS LATEX macros v5.2.
– 8 –
Fig. 1.— Mean ellipticity of the voids with RL = 4h
−1Mpc as a function of z.
– 9 –
Fig. 2.— Slope of the void ellipticity as a function of wa.
– 10 –
Fig. 3.— Contours of Av in the Ωm0-w0 (left) and in the Ωm-wa (right) plane.
