Quantum-bias cosmology: Acceleration from holographic information capacity

I show that a generic quantum phenomenon can drive cosmic acceleration without the need for dark energy or modified gravity. When treating the universe as a quantum system, one typically focuses on the scale factor [of a Friedmann-Robertson-Walker (FRW) spacetime] and ignores many other degrees of freedom. However, the information capacity of the discarded variables will inevitably change as the universe expands, generating quantum bias (QB) in the Friedmann equations. If information could be stored in each Planck volume independently, this effect would give rise to a constant acceleration $10^{120}$ times larger than that observed, reproducing the usual cosmological constant problem. However, once information capacity is quantified according to the holographic principle, cosmic acceleration is far smaller and depends on the past behavior of the scale factor. I calculate this holographic quantum bias, derive the semiclassical Friedmann equations, and obtain their general solution for a spatially flat universe containing matter and radiation. Comparing these QB-CDM solutions (which include standard cold dark matter) to those of $\mathrm{\Lambda }\mathrm{CDM}$, the new theory is shown to be falsifiable, but nonetheless consistent with current observations. In general, realistic QB cosmologies undergo phantom acceleration ($w_{\mathrm{eff}}<-1$) at late times, predicting a big rip in the distant future.

Figure 1

The behavior of the matter density ${\rho }_{\mathrm{m}}$ (black line) and the effective dark energy density ${\rho }_{\mathrm{eff}}$ (colored lines) as the universe expands. For the sake of clarity, we neglect radiation (${\mathrm{\Omega }}_{\mathrm{r}}=0\iff \alpha =0$) and use reference values $\{{\rho }_{\mathrm{m}\star },a_{\star }\}$ such that the early-time asymptote passes through the origin, for each value of $\overline{\gamma }$. [Specifically, ${\rho }_{\mathrm{m}\star }={\rho }_{\mathrm{m}}(u_{\star })$, $a_{\star }=a(u_{\star })$, with $u_{\star }$ solving $F_{\overline{\gamma }}^{\prime }(u_{\star })=4/({\overline{\gamma }}^2-1)$.] In general, the early-time behavior (88) is accurate during the matter-dominated era ${\rho }_{\mathrm{eff}}\ll {\rho }_{\mathrm{m}}$, but breaks down as ${\rho }_{\mathrm{eff}}$ approaches ${\rho }_{\mathrm{m}}$. While ${\rho }_{\mathrm{eff}}\approx {\rho }_{\mathrm{m}}$, the effective equation of state $w_{\mathrm{eff}}$ becomes more negative, and hence the gradient $\mathrm{d}\ln {\rho }_{\mathrm{eff}}/\mathrm{d}\ln a$ increases. Ultimately, quantum bias dominates ${\rho }_{\mathrm{eff}}\gg {\rho }_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ converges on its final value (90). For $\overline{\gamma }\in (1,3)$, the dark energy density ${\rho }_{\mathrm{eff}}$ always has a turning point when ${\rho }_{\mathrm{eff}}\approx {\rho }_{\mathrm{m}}$; the neighborhood of this minimum resembles the current state of our universe: ${\mathrm{\Omega }}_{\mathrm{m}}\approx 1/2$, $w_{\mathrm{eff}}\approx -1$.

Figure 2

The QB-CDM expansion histories (80) are compared to $\mathrm{\Lambda }\mathrm{CDM}$ (92) with ${\mathrm{\Omega }}_{\mathrm{m}0}^{(\mathrm{\Lambda })}=0.31$. As explained in Sec. 6b, the parameters $\{H_0^{(\mathrm{QB})},u_0\}$ have been chosen so that the two models are in exact agreement over the present-day matter density (95a) and conformal age of the universe (95b). The two topmost graphs show the fractional difference in the Hubble expansion rate (97) at each redshift: first for the wide range of values $1.3\le \overline{\gamma }\le 5$ used in Fig. 1; then for a small group $\overline{\gamma }\in \{1.5,1.6,1.7,1.8\}$ that agrees with $\mathrm{\Lambda }\mathrm{CDM}$ most closely. The third graph depicts the fractional difference in the angular diameter distance (98) for the narrow range of $\overline{\gamma }$. Finally, the scale factor is plotted as a function of proper time, for $\overline{\gamma }\in \{1.5,1.6,1.7,1.8\}$ and $\mathrm{\Lambda }\mathrm{CDM}$. Here, vertical dotted lines indicate the proper times ${\overline{\tau }}^{(\mathrm{QB})}/{\tau }_0^{(\mathrm{\Lambda })}\approx \{1.46,1.65,1.89,2.17\}$ at which the respective QB cosmologies undergo a big rip.

Figure 3

Here we depict the past-directed light sheet $\mathcal{L}$ of a simultaneous spherical surface $\mathcal{B}$, within an expanding FRW universe (8). If $\mathcal{B}$ is sufficiently small, we can expand the hologram ($\mathcal{B},\mathcal{L}$) by extending the converging null geodesics of $\mathcal{L}$ backwards through $\mathcal{B}$. (For a past-directed $\mathcal{L}$, this extends the light sheet toward the future.) This produces a new hologram $({\mathcal{B}}^{\prime },{\mathcal{L}}^{\prime })$ that is a strict superset of the former: ${\mathcal{L}}^{\prime }\supset \mathcal{L}$. The new hologram must be considered the more fundamental description, as it contains all the information of the original hologram and more besides. This process of backwards extension can continue until cosmological constraints intervene. The results of this maximization procedure define the natural holograms to cover the FRW spacetime.

Figure 4

Holographic units are spherically symmetric causal diamonds, bounded into the future by a cosmological event horizon, and foliated by the past-directed light sheets of the event horizon at each conformal time $\eta$. On the right, these units are arranged into a self-similar pattern that perfectly tiles an expanding universe with one spatial dimension and final conformal time $\overline{\eta }$. (We generalize this pattern to $D$ spatial dimensions in Sec. pp3-s4.) Each holographic unit begins at $\eta =\overline{\eta }-2^n\mathrm{\Delta }\eta$ for some $n\in \mathbb{Z}$; all reference to the arbitrary scale $\mathrm{\Delta }\eta$ can be removed by a natural averaging procedure described in the main text. On each spatial slice $\eta =\text{const}$, the event horizon is a sphere ${\mathcal{B}}_{\eta }$ of area $A[{\mathcal{B}}_{\eta }]=\mathcal{A}(\overline{\eta }-\eta )[a(\eta ){]}^2$ that encloses a volume $V_{\eta }\equiv \mathcal{V}(\overline{\eta }-\eta )[a(\eta ){]}^3$; each ${\mathcal{B}}_{\eta }$ generates a past-directed light sheet ${\mathcal{L}}_{\eta }$ with information capacity set by the holographic formula (c4). Note that even though the pattern covers the entire ($1+1$)-dimensional spacetime without gaps or overlap, the (cyan shaded) volumes $V_{\eta }$ do not fill each spatial slice: some parts of the slice (magenta dashed line) are occupied by the lower half of a holographic unit (orange triangle), the information capacity of which will be counted on a future slice. Hence the number of spheres ${\mathcal{B}}_{\eta }$ in a large volume $V_{*}$ is ${\mathcal{N}}_{*}=\mu V_{*}/V_{\eta }$, for some “filling factor” $\mu \lesssim 1$.

Figure 5

This cycle generalizes the self-similar pattern of Fig. 4, packing holographic units into an expanding ($2+1$)-dimensional universe without overlap. Each frame represents the state of a comoving square lattice on a sequence of spatial slices $\eta =\text{const}$. The pattern is easiest to follow in reverse chronological order (clockwise) starting from the top-left frame: as $\eta$ decreases, the comoving radii of the event horizons (blue circles) grow, while the initial light sheets (red circles) shrink. Whenever two event horizons touch (frames 1 and 4) every other horizon is transformed into an initial light sheet (frames 2 and 5). These transitions represent the “corner” of a holographic unit, e.g., the $\eta =\overline{\eta }-\mathrm{\Delta }\eta /2$ slice of the unit depicted on the left of Fig. 4. This process prevents any holographic unit from overlapping, but allows small gaps (grey) to appear in the covering. Once we reach the bottom-left frame, the lattice has returned to its starting state, scaled up by a factor of $m=2$. This algorithm is easily generalized to pack holographic units in $D+1$ dimensions or modified to construct (partially overlapping) patterns that cover the entire spacetime.

Figure 6

Over a single scaling cycle (c28) the spatial slices of a self-similar pattern of holographic units undergo two types of evolution. Continuous: As $s$ increases, the comoving radii of the event horizons (blue lines) grow, while the initial light sheets (red lines) shrink. Discrete: At each $s=s_i$, a fraction of the holographic units have corner transitions—their event horizons terminate and become initial light sheets. The diagram above represents a simple example, with two transitions: $s_1<m/2<s_2$. The terms running along diagonal lines indicate the number of such spheres within the integration region $\chi \in [0,{\chi }_{*}]$. Note that the $s_2$ transition produces $n_0{f}_2$ light sheet–bound spheres which still exist at the end of the cycle $s=m$. By the self-similarity of the pattern, there must be $m^D\times (n_0{f}_2)$ similar spheres (smaller by a factor of $1/m$) that survive the previous cycle $s\in [1/m,1)$ and enter the current cycle at $s=1$.