Closed universes, de Sitter space, and inflation

We present a new approach to constructing inflationary models in closed universes. Conformal embedding of closed-universe models in a de Sitter background suggests a quantization condition on the available conformal time. This condition implies that the universe is closed at no greater than the 10% level. When a massive scalar field is introduced to drive an inflationary phase this figure is reduced to closure at nearer the 1% level. In order to enforce the constraint on the available conformal time we need to consider conditions in the universe before the onset of inflation. A formal series around the initial singularity is constructed, which rests on a pair of dimensionless, scale-invariant parameters. For physically acceptable models we find that both parameters are of order unity, so no fine-tuning is required, except in the mass of the scalar field. For typical values of the input parameters we predict the observed values of the cosmological parameters, including the magnitude of the cosmological constant. The model produces a very good fit to the most recent cosmic microwave background (CMB) data. The primordial curvature spectrum provides a possible explanation for the low-$\ell$ falloff in the CMB power spectrum observed by Wilkinson microwave anisotropy probe. The spectrum also predicts a falloff in the matter spectrum at high $k$, relative to a power law. A further prediction of our model is a large tensor mode component, with $r\approx 0.2$.

Figure 1

The Poincaré disk. Points inside the disk represent points in a two-dimensional non-Euclidean (hyperbolic) space. A set of $d$-lines are also shown. These are (Euclidean) circles that intersect the unit circle at right angles. Given a $d$-line and point $A$ not on the line, one can find an infinite number of lines through $A$ that do not intersect the line.

Figure 2

Two-dimensional de Sitter Space. The timelike direction is vertical, and spatial sections are closed. The right-hand diagram shows a null geodesic, which is a straight line in the embedding space.

Figure 3

The Lorentzian view of two-dimensional de Sitter space. The boundary is defined by two hyperbolae (shown with thick lines). All geodesics through the origin are straight lines, and null geodesics are always straight lines at $45°$. Two further geodesics, one spacelike and one timelike, are also shown. These are hyperbolae which do not pass through the origin. The timelike geodesic intersects the boundary in such a way that the two tangent vectors have a vanishing Lorentzian inner product.

Figure 4

Flat and open cosmological models. The de Sitter space contains sections representing both flat and open spacetimes. A flat spacetime (left) consists of the space contained inside a light cone located on past infinity. An open spacetime fits inside the light cone from the origin. Both pictures can be used to illustrate cosmological models, with the initial singularity represented by a hyperbolic spacelike surface. An example of these surfaces is shown on each diagram.

Figure 5

Evolution curves in the ${\Omega }_M\mathrm{\text{−}}{\Omega }_{\Lambda }$ plane. The left-hand plot is for dust, and the right-hand plot is for radiation. In both cases the curves converge to ${\Omega }_{\Lambda }=1$, representing a late-time de Sitter phase.

Figure 6

Critical paths as predicted by the de Sitter embedding. The solid line represents a matter-dominated universe, and the broken line shows radiation, for comparison. The straight line is the critical case of spatial flatness. By the time we reach ${\Omega }_M\approx 0.3$ the universe has been driven close to critical density.

Figure 7

Hubble function entry into the inflationary regime. The Hubble function emerges from the big bang going as $1/3t$. As $H$ falls it quickly enters the inflationary region, during which it falls linearly. Time $t$ is measured in units of $t_p$, and $H$ in units of $t_p^{-1}$. The input parameters were set to $b_0=2.48$, $b_4=-0.51$ and $m=m_p$ ($\mu =1$). These values of $b_0$ and $b_4/{\mu }^{4/3}$ are illustrative of physically realistic models. The cosmological constant term has no effect in this region and can be set to zero.

Figure 8

Hubble function exit from the inflationary regime. As the inflationary era ends, the density and pressure start oscillating around values for a matter-dominated cosmology. The effective singularity for this dust cosmology is displaced from $t=0$. The broken line plots the natural log of $2/3(t-14.56)$. The parameters in this model are $b_0=2.48$, $b_4=-0.51$ and $m=m_p$.

Figure 9

Evolution of the conformal time $\eta (t)$ as a function of $\ln (t)$. The parameters for the model are as given in Fig. 8 and again $t$ is measured in units of the Planck time. This plot sets $\mu =1$, so the time variable must be scaled to correspond to more physical values of $\mu \approx {10}^{-6}$.

Figure 10

The horizon problem for a closed universe in the conformal picture. The lower solid line represents the surface of last scattering, and the broken line just above it is the current epoch. The thick line is the future boundary ($t=\infty$). Null geodesics lie at $45°$, so there is no problem with points in the observable part of the surface of last scattering having once been in causal contact.

Figure 11

Matter-dominated trajectory in the ${\Omega }_M\mathrm{\text{−}}{\Omega }_{\Lambda }$ plane. The trajectory (solid line) is picked out by demanding that the total conformal time elapsed throughout the evolution is 0.6. This evolution joins onto an inflationary model which takes up around 0.97 units of conformal time, so that the boundary condition of a total of $\pi /2$ is obeyed.

Figure 12

Evolution of $1/(RH)$. The natural log of the dimensionless ratio of the Hubble radius to the curvature radius is plotted against log time (in units of $t_p$). The parameters for the model are as described in the text. During the inflationary period, $1/(RH)$ is a decreasing function of time, corresponding to an accelerating phase.

Figure 13

Evolution of the scale factor. The evolution of logarithm of the scale factor $R/l_p$ is shown as a function of log time ($t$ in units of $t_p$).

Figure 14

Evolution of the total $\Omega$ during inflation. The total density is quickly driven close to flat, after which $\Omega$ only changes slowly.

Figure 15

Evolution of $H^2/\dot{\varphi }$ during inflation. The function plotted is $-{10}^5{H}^2/\dot{\varphi }$ as a function of $\ln (t/t_p)$. This function controls the magnitude of the curvature perturbation.

Figure 16

The power spectrum of curvature perturbations. The function plotted is ${10}^{10}(2\pi {)}^2{\mathcal{P}}_{\mathcal{R}}(k)$ as a function of $y=\ln (1/x)=\ln (k)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. The solid line represents the numerical predictions from our model, and the broken line is the best-fit power law with an exponential cutoff. The parameters are described in the text.

Figure 17

Comparison of the scalar power spectrum with power law models. The function plotted is ${10}^{10}(2\pi {)}^2{\mathcal{P}}_{\mathcal{R}}(k)$ as a function of ${log}_{10}(k)$, assuming $h=0.65$. The solid line represents the numerical predictions from our model. The long dashes represent the best-fit power law ($n_s=0.96$) and the short dashes are the WMAP running spectral index best fit. Notice that the vertical scale runs from 800 to 1300, so the differences are slightly exaggerated.

Figure 18

Power spectrum of tensor perturbations. The quantity plotted is ${10}^{10}(2\pi {)}^2{\mathcal{P}}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(k)$ versus $y=\ln (k)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. The solid line is the numerical predictions. The dashed line is the best-fit power law with exponential cutoff, offset by 20 units vertically for clarity. All parameters are described in the text.

Figure 19

Predicted CMB power spectrum. These predictions are for a model with ${\Omega }_0=1.02$, as discussed in the text. The experimental points shown with $1\sigma$ error bars are the WMAP results.

Figure 20

The linear matter power spectrum. The solid line is the prediction from CAMB using as input our scalar primordial power spectrum. Long dashes are the 2dF measurements [30] convolved with the survey window function. The dot-dashed line is a comparison fiducial model with $n_s=1$, ${\sigma }_8=1$, $h=0.7$, ${\Omega }_{m}h=0.2$, and ${\Omega }_b=0$.

Figure 21

CMB power spectrum for a model with ${\Omega }_0=1.04$ (I). The parameters are discussed in the text. The experimental points shown with $1\sigma$ error bars are the WMAP results [1] and the dashed curve corresponds to the best-fit $\Lambda$ cold dark matter power law CMB power spectrum as distributed in the WMAP data products.

Figure 22

CMB power spectrum for a model with ${\Omega }_0=1.04$ (II). A similar plot to Fig. 21, but now plotted on a linear scale, with the latest Very Small Array data points also included to extend out to higher $l$ values (open circles).