Constraining the Dark Energy Equation of State with Cosmic Voids

TL;DR

This paper proposes that the evolution of cosmic void shapes can serve as a sensitive probe for constraining the dark energy equation of state, enhancing understanding of the universe's acceleration.

Contribution

It analytically demonstrates the sensitivity of cosmic void ellipticity evolution to the dark energy equation of state within a flat universe model.

Findings

Void ellipticity evolution depends on dark energy properties

Parameter degeneracy between w and Omega_m discussed

Analytical framework established for using voids as cosmological probes

Abstract

Our universe is observed to be accelerating due to the dominant dark energy 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 (Omega_m) 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 Omega_m.

Figures

Equations

$${\lambda }_1(\mu ,\nu )$$

$${\lambda }_2(\mu ,\nu )$$

$$p(1-\epsilon ;z)$$

$$=$$

$$D(z)=\frac{5{\Omega }_m}{2(z+1)}{\left[{\Omega }_m^{\alpha }-{\Omega }_Q+\left(1+\frac{{\Omega }_m}{2}\right)\left(1+\mathcal{A}{\Omega }_Q\right)\right]}^{-1}.$$

$$E^2(z)$$

$$f(z)$$

$$\alpha$$

$$\mathcal{A}$$

$${\Omega }_m(z)=\frac{{\Omega }_{m0}(1+z{)}^3}{E^2(z)},{\Omega }_Q(z)=\frac{{\Omega }_{Q0}}{E^2(z)(1+z{)}^{f(z)}},$$

Full text

cosmology:theory — large-scale structure of universe

Recent observations have revealed that our universe is flat and in a phase of acceleration (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: $w\equiv P_{de}/\rho_{de}$. 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 ($\Lambda$) 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).

The following observables have so far been suggested to discriminate the dark energy 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 acoustic 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 cosmic 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

[TABLE]

where $\{\lambda_{i}\}_{i=1}^{3}$ (with $\lambda_{1}>\lambda_{2}>\lambda_{3}$) are the three eigenvalues of the local tidal field smoothed on void scale, $\delta_{\rm v}$ is the density contrast threshold for the formation of a void: $\delta_{\rm v}=\sum_{i=1}^{3}\lambda_{i}$, and $\{\mu,\nu\}$ (with $\nu<\mu$) represents a set of the two parameters that quantify the anisotropic distribution of the void galaxies. They defined the void ellipticity as $\varepsilon\equiv 1-\nu$ and evaluated its probability density distribution as

[TABLE]

Here, $\sigma(z,R_{L}))\equiv D^{2}(z)\int_{-\infty}^{\infty}\Delta^{2}(k)W^{2}(kR_{L})d\ln k$ is the linear rms fluctuation of the matter density field smoothed on a Lagrangian void scale of $R_{L}$ at redshift $z$ where $D(z)$ is the linear growth factor, $W(kR_{L})$ is a top-hat window function, and $\Delta^{2}(k)$ 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 (6) was originally derived under the assumption of a $\Lambda$CDM 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 $w(z)=w_{0}+w_{a}z/(1+z)$ (Chevallier & Polarski, 2001; Linder, 2003) where $w_{0}$ is the value of $w$ at present epoch and $w_{a}$ 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):

[TABLE]

where

[TABLE]

The CDM density parameter $\Omega_{m}$ and the dark energy density parameter $\Omega_{Q}$ evolve with $z$ respectively as

[TABLE]

where $\Omega_{m0}$ and $\Omega_{Q0}$ represent the present values. Equation (6) implies that the mean ellipticity 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 $w_{a}$. Even in case that $w_{0}=-1$, if $w_{a}$ is found to deviate from zero, it would imply the dynamic dark energy, disproving the simple $\Lambda$CDM model.

To explore how the void ellipticity evolution depends on $w_{a}$, we evaluate the mean ellipticity of voids as $\bar{\varepsilon}(z)=\int_{0}^{1}~{}\varepsilon~{}p(\varepsilon;R_{L},z)d\epsilon$ for different values of $w_{a}$ through equations (6)- (12). The other key cosmological parameters are set at $\Omega_{m}=0.75$,$\Omega_{Q}=0.75$, $h=0.73$, $\sigma_{8}=0.9$ and $w_{0}=-1$. 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, $M_{R}$, defined as the mass within a certain comoving radius (Wang & Steinhardt, 1998). Likewise, we set the Lagrangian scale of a void, $R_{L}$ at $4h^{-1}$Mpc, which is related to the mean effective radius of a void as $\bar{R}_{E}=(1+\delta_{v})^{-1/3}\bar{R}_{L}/(1+z)$. The Lagrangian scale $R_{L}=4h^{-1}$Mpc corresponds to the mean effective size of a void at present epoch, $R_{E}\sim 8.5h^{-1}$Mpc.

Figure 1 plots $\bar{\varepsilon}(z)$ for the four different cases: $w_{a}=-1/3,0,1/3$ and $2/3$ (long-dashed, solid, dashed, and dotted line, respectively). As can be seen, the higher the value of $w_{a}$ is, the more rapidly $\bar{\varepsilon}(z)$ decreases. It also suggests that $\bar{\varepsilon}(z)$ is well approximated as a linear function of $z$ in recent epochs ($0<z<0.2$). Therefore, we fit $\varepsilon(z)$ to a straight line as $\bar{\varepsilon}(z)\approx A_{v}z+B_{v}$. Varying the value of $w_{a}$ in the range of $[0,2/3]$, we compute the best-fit slope $A_{v}$. The range, $0\leq w_{a}\leq 2/3$, corresponds to the dark energy equation of state range, $-1\leq w\leq-0.9$. The result is plotted in Fig. 2. As can be seen, the void ellipticity evolves more rapidly as the value of $w_{a}$ increases. That is, the void ellipticity undergoes a stronger evolution when the anti-gravitational effect is less strong in recent epochs. Note that $A_{v}$ 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 parameters. However, as the dependence of the void ellipticity distribution on the dark energy equation of state comes from its dependence on $\Delta^{2}(k;\Omega_{m0},\sigma_{8},h,w)$, it is naturally expected that there should be a strong parameter degeneracy. Here, we focus on the degeneracy between $\Omega_{m0}$ and $w$. First, we recompute $A_{v}$, varying the values of $\Omega_{m0}$ and $w_{0}$ with setting $w_{a}=1/3$. The left panel of Fig. 3 plots a family of the degeneracy curves in the $\Omega_{m0}$-$w_{0}$ plane for the three different values of $A_{v}$. As can be seen, there is a strong degeneracy between the two parameters. For a given value of $A_{v}$, the value of $w_{0}$ increases as the value of $\Omega_{m0}$ decreases. A similar trend is also found in the $\Omega_{m0}$-$w_{a}$ degeneracy curves that are plotted in the right panel of Fig. 3 for which the value of $w_{0}$ 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 $\Omega_{m0}$ 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 $\Omega_{m0}$ 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 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).