Testable solution of the cosmological constant and coincidence problems

We present a new solution to the cosmological constant (CC) and coincidence problems in which the observed value of the CC, $\Lambda$, is linked to other observable properties of the Universe. This is achieved by promoting the CC from a parameter that must be specified, to a field that can take many possible values. The observed value of $\Lambda \approx (9.3\mathrm{Gyrs}{)}^{-2}$ [$\approx {10}^{-120}$ in Planck units] is determined by a new constraint equation which follows from the application of a causally restricted variation principle. When applied to our visible Universe, the model makes a testable prediction for the dimensionless spatial curvature of ${\Omega }_{\mathrm{k}0}=-0.0056({\zeta }_{\mathrm{b}}/0.5)$, where ${\zeta }_{\mathrm{b}}\sim 1/2$ is a QCD parameter. Requiring that a classical history exist, our model determines the probability of observing a given $\Lambda$. The observed CC value, which we successfully predict, is typical within our model even before the effects of anthropic selection are included. When anthropic selection effects are accounted for, we find that the observed coincidence between $t_{\Lambda }={\Lambda }^{-1/2}$ and the age of the Universe, $t_{\mathrm{U}}$, is a typical occurrence in our model. In contrast to multiverse explanations of the CC problems, our solution is independent of the choice of a prior weighting of different $\Lambda$ values and does not rely on anthropic selection effects. Our model includes no unnatural small parameters and does not require the introduction of new dynamical scalar fields or modifications to general relativity, and it can be tested by astronomical observations in the near future.

Figure 1

An illustration of the manifold $\mathcal{M}$ and its boundary $\partial \mathcal{M}=\partial {\mathcal{M}}_u\cup \partial {\mathcal{M}}_I$ for the general cosmology setup considered in Sec. 3. Here, $\mathcal{M}$ is the causal past of the observer, $\partial {\mathcal{M}}_u$ is the null boundary given by the observer’s past light cone and $\partial {\mathcal{M}}_I$ is a spacelike boundary which represents the initial hypersurface. The model remains well defined in the limit where $\partial {\mathcal{M}}_I$ is the initial singularity. We pick orthogonal null coordinates $u$ and $w$ such that $\partial {\mathcal{M}}_u$ corresponds to $u={\tau }_0$, and on $\partial {\mathcal{M}}_u$, $-{\tau }_0<w<{\tau }_0$ for some fixed ${\tau }_0$, and the time coordinate given by $\tau =(u+w)/2$ vanishes on $\partial {\mathcal{M}}_I$. We also define a radial coordinate $r=(u-w)/2$ and choose $w={\tau }_0$ at the observer’s position, so that there $\tau ={\tau }_0$ and $r=0$. In the figure, constant $u$ surfaces are shown as dotted red line (except $\partial {\mathcal{M}}_u$ which is a solid red line), and correspond to past light cones of points on the line $r=0$ (the dashed black line) which connects the observer with $\partial {\mathcal{M}}_I$. The $w=\mathrm{const}$ surfaces are future-directed light cones of points on $r=0$ and are shown above as dot-dashed blue lines. Surfaces of constant $u$ and $w$ are closed 2-surfaces $S_{(u,w)}$ with intrinsic coordinates ${\theta }^i$; $i=1$, 2.

Figure 2

(a) The relationship between the value of the spatial curvature $k$ and the value of $\Lambda$ that dominants the classical history. We show $k$ in units of $a_{\star }^2{H}_{\star }^2$ where $1/(a_{\star }{H}_{\star })$ is a fixed comoving length scale that is equal to $1/(a_0{H}_0)$ for the observed value of $\Lambda$. $\Lambda$ is shown relative to the value of $\Lambda$ that we observed, ${\Lambda }_{\mathrm{obs}}$. Here the $k-\Lambda$ relationship predicted by our model is for observations at a time $t=t_{\mathrm{U}}\approx 13.77\mathrm{Gyrs}$. We have also fixed the $\Lambda$ independent properties of matter by fixing the matter energy per photon, $\xi$, and the baryon energy per photon, ${\xi }_{\mathrm{b}}$, to their observed values: $\xi =3.43\mathrm{eV}$, ${\xi }_{\mathrm{b}}=0.54\mathrm{eV}$. We note that smaller values of $k$ correspond to larger values of $\Lambda$. (b) The relationship between ${\Omega }_{\mathrm{m}0}\approx 1-{\Omega }_{\Lambda 0}$ and $-2{\Omega }_{\mathrm{k}0}/{\zeta }_{\mathrm{b}}{\Omega }_{\mathrm{b}0}=\mathcal{N}({\tau }_0;\Lambda )$ predicted by our model. When ${\Omega }_{\mathrm{m}0}\approx 1-{\Omega }_{\Lambda 0}$, we find that when $\mathcal{N}({\tau }_0;\Lambda )$ is expressed as a function of ${\Omega }_{\Lambda 0}$, it is almost independent of the observation time determined by ${\tau }_0$. For fixed ${\Omega }_{\mathrm{b}0}/{\Omega }_{\mathrm{m}0}$ and ${\zeta }_{\mathrm{b}}$, ${\Omega }_{\mathrm{k}0}$ corresponds to a specific value of ${\Omega }_{\Lambda 0}$ independently of ${\tau }_0$. As in (a), we have taken $\xi =3.43\mathrm{eV}$, ${\xi }_{\mathrm{b}}=0.54\mathrm{eV}$.

Figure 3

Our model’s prediction for the (prior) probability distribution, $f_{\mathrm{model}}(\Lambda )$, for the value of cosmological constant $\Lambda$ that one would measure at an observation time $t=t_{\mathrm{U}}=13.77\mathrm{Gyrs}$. In our theory, the observed value of $\Lambda$ is given as a function of the spatial curvature, $k$, inside the observer’s past light cone. The curvature parameter $k$ depends on $N$, the number of e-folds of inflation. Hence, ultimately, the probability of astronomers observing a given value of $\Lambda$ depends on the prior probability of living in a bubble universe where the universe underwent $N$ e-folds of inflation. The probability distribution function of $N$ is $f_{\mathrm{N}}(N)$. Given an $f_{\mathrm{N}}(N)$, our model completely determines the prior probability distribution function, $f_{\mathrm{model}}(\Lambda )$. We have plotted $f_{\mathrm{model}}(\Lambda )$ for two different choices of $f_{\mathrm{N}}(N)$. Case I is where $f_{\mathrm{N}}(N)=c(N)$ where $c(N)\approx \mathrm{const}$ for changes, $\Delta N$, in $N$ over less than $\approx 2.5\%$ (e.g. a power-law $N^{-p}$ for $|p|<10$). Case II is where $f_{\mathrm{N}}=c(N)e^{-3N}$ where again $c(N)\approx \mathrm{const}$ for $\Delta N/N\lesssim 2.5\%$. This latter choice is the one calculated in Ref. 40 for slow-roll single-field inflation. In our model this prediction for the prior probability is independent of the fundamental prior weighting of different values of $\Lambda$ in the partition function. Our $f_{\mathrm{model}}(\Lambda )$ does not include any observer-dependent selection effects, apart from the requirement that a classical solution exists. The observed value of $\Lambda$ is shown by a dotted black line, and the whole shaded region is the symmetric 95% confidence interval. The darker shaded area is the symmetric 68% confidence interval. In case I, we have calculated the confidence intervals by sharply cutting off $f_{\mathrm{model}}$ for $\Lambda >{10}^3{\Lambda }_{\mathrm{obs}}$, since such large values of the CC are incompatible with the existence of galaxies. For $\Lambda <{10}^3{\Lambda }_{\mathrm{obs}}$ we have not included any weighting by probability of living in a universe with a given $\Lambda$. For these natural choices of $f_{\mathrm{N}}(N)$, the observed value of $\Lambda$ is within the 95% confidence interval in both cases, and well within it for $f_{\mathrm{N}}=c(N)e^{-3N}$. In both cases, $\Lambda ={\Lambda }_{\mathrm{obs}}$ is not atypical whatever precise form of the observer-conditioned selection effect parametrized by $f_{\mathrm{selec}}(\Lambda )$. When selection effects are included the probability of larger values $\Lambda \gtrsim 100{\Lambda }_{\mathrm{obs}}$ is further suppressed, and the observed value moves within the $1\mathrm{\text{−}}\sigma$ confidence interval for case II, and just outside this interval in case I. This is shown in FIG 4 below. We have taken the matter energy per photon to be $\xi =3.43\mathrm{eV}$, and the baryon energy per photon is ${\xi }_{\mathrm{b}}=0.54\mathrm{eV}$, as is observed.

Figure 4

The predicted posterior probability distribution, $f_{\Lambda }(\Lambda )=f_{\mathrm{selec}}(\Lambda )f_{\mathrm{model}}(\Lambda )$, for the value of cosmological $\Lambda$ that one would measure at an observation time $t=t_{\mathrm{U}}=13.77\mathrm{Gyrs}$. Here, ${\Lambda }_{\mathrm{obs}}$ is the particular value of $\Lambda$ that we have observed. In our theory, the observed value of $\Lambda$ is given as a function of the spatial curvature, $k$, inside the observer’s past light cone. The curvature parameter $k$ depends on $N$, the number of e-folds of inflation. Hence, ultimately, the probability of astronomers observing a given value of $\Lambda$ depends on the prior probability of living in a bubble universe where the universe underwent $N$ e-folds of inflation. The probability distribution function of $N$ is $f_{\mathrm{N}}(N)$. Given an $f_{\mathrm{N}}(N)$, our model completely determines the probability distribution function, $f_{\mathrm{model}}(\Lambda )$, of $\Lambda$ prior to the inclusion of selection effects. In the above plots we have included the limits on $\Lambda$ due to observational selection effects using the prescription for $f_{\mathrm{selec}}$ given by Tegmark et al. in 39 which uses the number of galaxies as a proxy for the number of observers. When these are included, the full posterior probability of living in a bubble universe where one observes a given value of $\Lambda$ is $f_{\Lambda }(\Lambda )$. Additionally, we find that the inclusion of selection effects makes $f_{\Lambda }(\Lambda )$ only relatively weakly dependent on the form of $f_{\mathrm{N}}(N)$ because for allowed values of $\Lambda$, the required $k$ (and hence $N$) vary only over a small range. We have plotted $f_{\Lambda }(\Lambda )$ for two different choices of $f_{\mathrm{N}}(N)$. Case I is where $f_{\mathrm{N}}(N)=c(N)$ where $c(N)\approx \mathrm{const}$ for changes, $\Delta N$, in $N$ over less than $\approx 2.5\%$ (e.g. a power-law $N^{-p}$ for $|p|<10$). Case II is where $f_{\mathrm{N}}=c(N)e^{-3N}$ where again $c(N)\approx \mathrm{const}$ for $\Delta N/N\lesssim 2.5\%$. This latter choice is the one calculated in Ref. 40 for slow-roll single-field inflation. In both cases we see that the observed value of $\Lambda$ (dotted black line) is well inside the 95% confidence interval (the shaded areas), and near the boundary of the 68% confidence interval (the more darkly shaded area). In case I, $\Lambda ={\Lambda }_{\mathrm{obs}}$ is just outside this interval, whereas in case II it is just inside it. In both cases, it is clear that the observed value of $\Lambda$ is typical and can be explained without the need for fine tuning. We have taken the matter energy per photon to be $\xi =3.43\mathrm{eV}$, and the baryon energy per photon is ${\xi }_{\mathrm{b}}=0.54\mathrm{eV}$ for all values of $\Lambda$.