Direct reconstruction of the dark energy scalar-field potential

While the accelerated expansion of the Universe is by now well established, an underlying scalar-field potential possibly responsible for this acceleration remains unconstrained. We present an attempt to reconstruct this potential using recent SN data, under the assumption that the acceleration is driven by a single scalar field. Current approaches to such reconstructions are based upon simple parametric descriptions of either the luminosity distance or the dark energy equation of state. We find that these various approximations lead to a range of derived evolutionary histories of the dark energy equation of state (although there is considerable overlap between the different potential shapes allowed by the data). Instead of these indirect reconstruction schemes, we discuss a technique to determine the potential directly from the data by expressing it in terms of a binned scalar field. We apply this technique to a recent SN data set, and compare the results with model-dependent approaches. In a similar fashion to direct estimates of the dark energy equation of state, we advocate direct reconstruction of the scalar-field potential as a way to minimize prior assumptions on the shape, and thus minimize the introduction of bias in the derived potential.

Figure 1

Hubble diagram for 115 type Ia SNe used in the present analysis. The error bars are on ${\mu }_B$ only. We also include an additional constant error, ${\sigma }_{\mathrm{int}}=0.13$, to account for the SN intrinsic dispersion. For reference we also plot 300 curves drawn uniformly from the $2\sigma$ consistent likelihood fits to the data using the Taylor expansion with $r(z)=z+a_2{z}^2+a_3{z}^3+a_4{z}^4$.

Figure 2

The normalized quintessence potential $\tilde{V}(\varphi )$ vs $\varphi /m_{\mathrm{Pl}}$. The shaded region is allowed at the $2\sigma$ confidence level when using the Taylor expansion for $r(z)$. The solid lines mark the same when using a Padé approximation to the distance. The dashed and dot-dashed lines are for the cases where $w(z)$ is parametrized by $w(z)=w_0+\alpha \ln (1+z)$ and $w(z)=w_0+w_a(1-a)$, respectively. Here $\varphi =0$ corresponds to $z=0$, while $\varphi >0.1$ generally corresponds to $z>1$ (depending on $d\varphi /dz$). The points with error bars show the $1\sigma$ (solid) and $2\sigma$ (dashed) model-independent estimates of the potential described in Sec. 4 [see Eq. (8)]. While there is considerable overlap in the allowed region, there are also significant differences in terms of the redshift evolution of the EOS.

Figure 3

The dark energy EOS, $w(z)$, as a function of redshift. The curves show the $2\sigma$ allowed values and correspond to the potentials shown in Fig. 2. Note that if we impose $\partial \varphi /dz>0$, then $w>-1$, as single scalar-field models do not lead to $w<-1$. However, as shown, most of the parametrizations allow the region below $w<-1$. When constructing the potentials shown in Fig. 2, we apply the condition that $\partial \varphi /dz>0$. Note that the $w(z)$ parametrizations, with two free parameters, are the most restrictive parametrization in the regime $z<0.3$. Over the redshift range probed, the different parametrizations generally agree with each other. The plotted error bars show the $1\sigma$ and $2\sigma$ errors of $w_i(z)$ when the EOS is subdivided into three bins in redshift, with $w_i(z)$ directly measured from data and no restrictions on its values. A Gaussian prior has been taken on ${\Omega }_m$ with a one sigma uncertainty of 0.05 with $w(z)$ parametrizations.

Figure 4

$w$ versus $dw/d\ln a$. The data points show the EOS and its time derivative for 600 model potentials uniformly drawn at the $2\sigma$ confidence level at redshift of 0.1 (filled symbols) and 0.5 (open symbols). The circles, squares, diamonds, and triangle data points show potentials selected under the Taylor expansion, Padé approximation, $w(z)=w_0+\alpha \ln (1+z)$ and $w(z)=w_0+w_a(1-a)$ model fits to the data, respectively. For comparison, we also plot the thawing (dashed) and freezing (solid) potential regions, following Ref. 19. There are considerable differences in $w$ and $dw/d\ln a$ values, and the evolution captured by two redshifts, between the four approaches. These values do not satisfy the expectations under simple model criteria for the dark energy potentials, though all of these potentials, and the $w(z)$ curves, are consistent with the data at the $2\sigma$ confidence level.

Figure 5

The equation of state $w(z)$ (left) and the normalized quintessence potential $\tilde{V}(\varphi )$ (right) for $\mathrm{\text{Essence}}+\mathrm{\text{high-}}z$ type Ia SNe data. The shaded region is allowed at the 2$\sigma$ confidence level when using the Taylor expansion for $r(z)$. The solid lines mark the same when using a Padé approximation to the distance. The dashed and dot-dashed lines are for the cases where $w(z)$ is parametrized by $w(z)=w_0+\alpha \ln (1+z)$ and $w(z)=w_0+w_a(1-a)$, respectively. The points with error bars show the $1\sigma$ model-independent estimates of the potential described in Sec. 4 [see Eq. (8)].