Large-scale Fluctuations of Dark Energy and the Inhomogeneous Anisotropic Expansion of the Universe

Abstract

The observable effects of large-scale (non-quantum) fluctuations of dark energy (DE) are discussed within the framework of classical general relativity. The paper is focused on different local rates of the cosmological expansion (the phenomenon known as the Hubble constant tension) and the contribution of DE fluctuations to the anisotropy of the cosmic microwave background radiation; other potentially observable effects of DE fluctuations are also indicated, e.g., a criterion for the existence of a structure larger than the observable universe.

APPENDIX: COSMOLOGICAL EXPANSION IN THE DE-DOMINATED UNIVERSE

1.1 (a) Cosmological Expansion without DE Fluctuations

In the homogeneous, isotropic, spatially flat universe, devoid of baryonic matter, dark matter and relativistic particles (the de Sitter model), the evolution of the universe is determined exclusively by the uniformly distributed DE represented by the cosmological constant $\Lambda_{0}$. In this situation, Einstein’s equations (or Friedmann’s equations with the FLRW metric) reduce to

$$\frac{\dot{a}_{0}(t)}{a_{0}(t)}=\sqrt{\frac{\Lambda_{0}c^{2}}{3}},$$

(A.1)

which gives the classical de Sitter solution for the cosmological scale factor $a_{0}(t)$,

$$a_{0}(t)=\exp\Bigg{[}\sqrt{\frac{\Lambda_{0}c^{2}}{3}}(t-t_{0})\Bigg{]},$$

(A.2)

where $t$ is time and $t_{0}$ is the reference time; the natural choice in the late universe (during the past 4 billion years) is $t_{0}\equiv\text{now}$ (i.e., $t_{0}=13.8$ billion years), and $a_{0}(t_{0})\equiv 1$ by definition (by convention). Thus, in the classical de Sitter universe (without fluctuations of DE) the Hubble constant is

$$H_{0}\equiv\frac{\dot{a}_{0}(t)}{a_{0}(t)}=\sqrt{\frac{\Lambda_{0}c^{2}}{3}}.$$

(A.3)

1.2 (b) Cosmological Expansion in the Late Universe with DE Fluctuations

If the density of DE is not perfectly homogeneous, but its distribution does not (appreciably) change with time, i.e., $\Lambda({\mathbf{r}})=\Lambda_{0}+\delta\Lambda({\mathbf{r}})$, one can divide the entire universe into volumes $\Delta V_{i}$, where the value of $\delta\Lambda({\mathbf{r}})$ and consequently $\Lambda({\mathbf{r}})$ can be considered constant in the sense of the spatial average within $\Delta V_{i}$. Einstein’s equations work irrespective of the size of the considered system; therefore, one can solve the differential equation

$$\frac{\dot{a}({\mathbf{r}},t)}{a({\mathbf{r}},t)}=\sqrt{\frac{\Lambda({\mathbf{r}})c^{2}}{3}}$$

(A.4)

in each of the volumes $\Delta V_{i}$, where $\Lambda({\mathbf{r}})$ is constant, obtaining

$$a({\mathbf{r}},t)=\exp\Bigg{[}\sqrt{\frac{\Lambda({\mathbf{r}})c^{2}}{3}}(t-t_{0})\Bigg{]}$$

(A.5)

for all r’s in the universe, because each r belongs to some volume $\Delta V_{i}$.

Einstein’s equations work irrespective of the considered time period. If the distribution of DE changes with time, i.e., $\Lambda({\mathbf{r}},t)=\Lambda_{0}+\delta\Lambda({\mathbf{r}},t)$, one can divide the entire time span of the DE-dominated epoch into periods $\Delta t_{j}$, where the value of $\delta\Lambda({\mathbf{r}},t)$ and consequently $\Lambda({\mathbf{r}},t)$ can be considered constant (in the sense of the spatial and temporal average within $\Delta V_{i}$ and $\Delta t_{j}$). In such a situation, we can solve the differential equation

$$\frac{\dot{a}({\mathbf{r}},t)}{a({\mathbf{r}},t)}=\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}$$

(A.6)

in each of the space-time volumes $\Delta V_{i}\times\Delta t_{j}$ (i.e., ${\mathbf{r}}\in\Delta V_{i}$ and $t\in\Delta t_{j}$), where $\Lambda({\mathbf{r}},t)$ is constant, which gives

$$a({\mathbf{r}},t)=\text{const}_{ij}\times\exp\Bigg{[}\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}t\Bigg{]}$$

$${}\equiv\exp\Bigg{[}\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}(t-t_{ij})\Bigg{]},$$

(A.7)

where the indices ij indicate that the integration constant $\textrm{const}_{ij}$ (or the integration constant expressed in terms of the time $t_{ij}$) can be different in each space-time volume $\Delta V_{i}\times\Delta t_{j}$ .

However, for $\delta\Lambda({\mathbf{r}},t)\to 0$ and consequently $\Lambda({\mathbf{r}},t)\to\Lambda_{0}$, the scale factor in Eq. (A.7) must converge to the scale factor in Eq. (A.2), i.e., $a({\mathbf{r}},t)\to a_{0}(t),$ and $t_{ij}\to t_{0}$. Therefore, for “small” values of $\delta\Lambda({\mathbf{r}},t)$ one can assume $t_{ij}=t_{0}$, which gives an approximate expression for the cosmological scale factor in the de Sitter-like universe (with DE fluctuations)

$$a({\mathbf{r}},t)=\exp\Bigg{[}\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}(t-t_{0})\Bigg{]}$$

(A.8)

for arbitrary ${\mathbf{r}}$ and t. The question is, under which circumstances the approximate formula given in Eq. (A.8) works well; in other words, we need to determine the error associated with the assumed approximation and examine when and where the approximation is acceptable. One can do it by substituting Eq. (A.8) into the differential equation (A.6), which gives

$$\frac{\dot{a}({\mathbf{r}},t)}{a({\mathbf{r}},t)}=\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}$$

$${}\times\Bigg{[}1+\frac{(t-t_{0})}{2\Lambda({\mathbf{r}},t)}\cdot\frac{d\Lambda({\mathbf{r}},t)}{dt}\Bigg{]}.$$

(A.9)

The magnitude of the extra (second) term in the square brackets is the measure of how good (if the second term is much less than 1) or bad (if it is ${\sim}1$ or larger) the approximate expression for the cosmological scale factor in Eq. (A.8) actually is, and whether it is consistent with the definition of the Hubble variable $H({\mathbf{r}},t)$

$$H({\mathbf{r}},t)\equiv\frac{\dot{a}({\mathbf{r}},t)}{a({\mathbf{r}},t)}=\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}.$$

(A.10)

The approximation in Eq. (A.8) is plausible if

$$\left|\frac{\Delta\Lambda}{\Lambda}\right|\ll\left|\frac{2\Delta t}{t-t_{0}}\right|,$$

(A.11)

where the variables ${\mathbf{r}}$ and $t$ were suppressed and the (infinitesimal) differentials $dt$ and $d\Lambda$ were replaced with finite increments $\Delta t$ and $\Delta\Lambda$, because the criterion (A.11) concerns changes in the cosmological scale.

The typical values of $\Delta t$ are in the range $1.6\times 10^{7}\text{ y}\leq\Delta t\leq 4\times 10^{9}\text{ y}$, because the smallest unit on the cosmological scale is a cluster of galaxies of the average diameter $L_{\text{clus}}\approx 1.6\times 10^{7}\text{ ly}$, and $\Delta t$ cannot be larger than the duration of the DE-dominated epoch. The time difference $\lvert t-t_{0}\rvert$ can be of any value from the range $0<|t-t_{0}|\leq 4\times 10^{9}\text{ y}$, where the lower limit corresponds to the processes that started just “recently,” and the upper limit corresponds to the processes that started with the advent of the DE-dominated epoch and continue to the present.

Therefore, the most restrictive condition on the applicability of the approximate expression for the cosmological scale factor in Eq. (A.8) occurs in the case of small-scale processes that lasted for nearly all of the (late) DE-dominated epoch (4 billion years),

$$\Big{|}\frac{\Delta\Lambda}{\Lambda}\Big{|}\ll\frac{2\times 1.6\times 10^{7}\text{ y}}{4\times 10^{9}\text{ y}}=8\times 10^{-3}.$$

(A.12)

The condition (A.11) is much less restrictive for large-scale processes ($\Delta t\approx 4\times 10^{9}\text{ y}$):

$$\left|\frac{\Delta\Lambda}{\Lambda}\right|\ll\frac{2\times 4\times 10^{9}\text{ y}}{4\times 10^{9}\text{ y}}=2.$$

(A.13)

If a given process (that led to the redistribution of DE) lasted for only a small fraction of the DE-dominated epoch (i.e., $|t-t_{0}|\ll 4\times 10^{9}\text{ y}$ in the denominator), the condition in Eq. (A.11) can be satisfied even if the net (relative) inflow of DE $|\Delta\Lambda/\Lambda|$ is of the order of 1 (or even larger).

1.3 (c) Cosmological Expansion in the Very Early (Inflationary) Universe with DE Fluctuations

The classical de Sitter solution of Einstein’s equations for the homogeneous isotropic universe applies also to the DE-dominated universe in the inflationary epoch (i.e., within a small fraction of one second after the Big Bang). If DE, represented by the cosmological constant, could also fluctuate in the inflationary epoch, $\Lambda({\mathbf{r}},t)=\Lambda_{\text{infl}}+\delta\Lambda({\mathbf{r}},t)$, one can follow the reasoning given above to get the approximate de Sitter-like solution of Einstein’s equations in the inflationary epoch

$$a({\mathbf{r}},t)=\exp\Bigg{[}\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}}(t-t_{\text{ref}})\Bigg{]},$$

(A.14)

and the corresponding Hubble variable

$$H({\mathbf{r}},t)\equiv\frac{\dot{a}({\mathbf{r}},t)}{a({\mathbf{r}},t)}=\sqrt{\frac{\Lambda({\mathbf{r}},t)c^{2}}{3}},$$

(A.15)

where the natural choice for the reference time is the time of the Big Bang $t_{\text{ref}}\equiv 0$, the reference cosmological scale factor is $a_{\text{ref}}(t_{\text{ref}})\equiv 1$ by convention, and $\Lambda_{\text{infl}}$ is a parameter that may be very much different from $\Lambda_{0}$ in the late universe, because the processes and laws that govern creation, annihilation and transformation of DE, dark matter and baryonic matter at the end of the inflationary epoch reach far beyond the purely general-relativistic treatment due to their quantum nature.

The condition $|\Delta\Lambda/\Lambda|\ll|2\Delta t/t|$ for validity of the de Sitter-like approximate solution in the inflationary epoch (given in Eq. (A.14)), for fluctuations comparable to the size of the then existing universe (i.e., $c\Delta t\approx ct$), is satisfied provided that $|\Delta\Lambda/\Lambda|\ll 2$.

The spatial size of fluctuations $\delta\Lambda({\mathbf{r}},t)$ that originated in the inflationary period grew rapidly (by up to 26 orders of magnitude) with the cosmological expansion, and so their diameter is comparable to the diameter of the universe also at the later epochs. Simultaneously, the magnitude of the primordial fluctuations $\delta\Lambda({\mathbf{r}},t)$ was damped by up to 78 orders of magnitude during inflation, because (contrary to the constant $\Lambda_{\text{infl}}$ that scales as $a^{0}$) the density of fluctuations $\delta\Lambda({\mathbf{r}},t)$ scales as $a^{-3}$ due to the (local) energy conservation law. Therefore, the only primordial fluctuations $\delta\Lambda({\mathbf{r}},t)$ that had a chance to survive inflation are those that were generated at the end of the inflationary epoch.

Similarly, the diameter of fluctuations in the density of matter (dark matter and baryonic matter created at the end of the inflationary epoch) grew rapidly with the scale factor $a$ during inflation and soon afterwards, and simultaneously their magnitude decreased by a factor of $a^{-3}$. The remnants of the primordial fluctuations of matter contribute to the currently observed largest-scale structures in the distribution of matter that are clearly visible in the CMB maps, gamma-ray burst maps and quasar grouping maps. Those largest structures in the universe must have appeared in the early universe (at the end of the inflationary epoch or soon afterwards, i.e., at most within seconds after the Big Bang), because otherwise no causal interaction could link so enormously distant parts of the universe with the same generating factor (due to the finite speed of light). The processes that violated the weak cosmological principle (which postulates spatial homogeneity and isotropy of the universe at the large scale) could have occurred only in the early universe.

1.4 (d) The Problem of Constant DE Density in the Expanding Universe

The situations considered earlier in this Appendix deal only with the redistribution of DE (the fluctuations do not change the average density of DE). Although the origin of DE, its physical nature, its creation and possible annihilation are not within the scope of this paper and shrouded in deep mystery, it is worth noting that the increase in the total amount of DE in the expanding universe (necessary to keep its density constant) is not contradictory to the fundamental physical laws.

In classical mechanics, Noether’s theorem states that the conservation laws in a given physical system result from the symmetries of that system. In particular, conservation of the total energy, total momentum and total angular momentum results from the symmetry of the considered system under time translation, space translation and rotation, respectively. Noether’s theorem also works in classical field theory, such as general relativity, where it takes a somewhat different form (instead of separate conservation laws for energy, momentum and angular momentum there is one conservation law for the canonical energy-momentum-pressure-stress tensor).

The issue of the total energy in the universe (in cosmology) is very much different from the (intuitively evident) issue of the total energy in local structures. If we consider the entire universe, it is evidently not invariant under space and time translations, because there is no space-time beyond the universe (by definition). Therefore, the total energy of the universe (DE in particular) is not conserved, and the interpretation of the cosmological constant in terms of the constant (average) density of DE (so that the total amount of DE in the universe increases with time as $a^{3}(t)$) does not contradict the fundamental physical laws, although it is highly counterintuitive.