Everything You Always Wanted to Know About How Causal Set Theory Can Help with Open Questions in Cosmology, but Were Afraid to Ask

How did our universe as we know it begin? The big bang is the conventional answer to this question provided by general relativity. Of course the big bang singularity represents a regime where the concept of spacetime ceases to be meaningful. Causal sets, by virtue of their fundamental discreteness, evade such singularities. The minimum separation between the elements (presumed to be close to the Planck length) tames infinite curvatures and energies.

The early causal sets at the very early universe will be non-geometric³³3In the limit of the number of elements $N\gg 1$ the notion of a continuum spacetime, for example one that can be approximated via a sprinkling, becomes meaningful., but they will nevertheless have features we have come to expect from geometric scenarios. For example, a broad class of CSG models lead to an early structure that mimics the big bang. This big bang analog is a causal set element that causally precedes every other element in the causal set. This original element is an example of what is known as a post (an element causally related to all the other elements, be it a past or future relation). As we will discuss in the next subsection, a large class of models admitting such posts are followed by periods of rapid growth, which further resembles the behavior close to the big bang.

The first moments after the big bang are believed to consist of a period of exponential expansion. This is the best explanation we have for observations such as the large size of the universe, high degree of isotropy, homogeneity, spatial flatness, and the lack of monopoles. However, the origin of this explanation requires explanation itself, and this is a current open question. Inflation[5, 6, 7] is a popular and practical model (based on the existence of a scalar field(s) with various possible potentials) that is designed to admit such a period of rapid expansion. However, inflation is emphatically not an example of the kind of fundamental understanding, mentioned in the introduction, that we wish to obtain of our universe. For example, not only does it not stem from first principles, the framework of cosmic inflation is too flexible, making it difficult to pinpoint what could be direct evidence for it.[8, 9] Inflation may in the future find its missing motivations if a theory of quantum gravity predicts a particular kind of it.

A similar rapid early growth as our universe occurs in one of the simplest families of sequential growth dynamics called transitive percolation[10, 11, 4]. These models have one free parameter $p$; at each stage $n$ of the growth process, the new element picks with probability $p$ each existing element to be in its past. An $n$-element causal set grown by such a dynamics can be modelled using the following algorithm: If we have labelled the elements $1,2,...,n$, the probability to introduce a relation $i\prec j$ between any pair of elements $i\in\{1,2,...,n-1\}$ and $j\in\{i+1,i+2,...,n\}$ is $p$. We must also ensure transitivity and add all the relations that are implied by the new relations through transitivity.

Originary percolation is a version of transitive percolation with the condition that each new element needs to be related to at least one other existing element.[12] These models ensure that the universe begins with a post resembling the big bang, as discussed in the previous subsection.

It is known that for small $p$, originary percolation leads to a tree-like growth until $n\sim 1/p$ elements have accumulated. This tree-like structure, which resembles a rapid growth, prompted investigations into how closely there might be a similarity to a spacetime such as de Sitter. [13] Despite the non-geometric nature of these early causal sets, proper time and spacetime volume estimates are still well-defined and can be used for comparisons with continuum geometries. The causal set analog of proper time between two (causally related) elements is the length of the longest chain between them.[14] A chain between arbitrary elements $x_{1}$ and $x_{n}$ is a totally ordered sequence of (any number of) relations between them that ultimately relate them by transitivity: $x_{1}\prec x_{2}\prec...\prec x_{n}$. The spacetime volume of the causal diamond between two elements can also be approximated by counting the number of elements within it. In Ref. \refciteAhmed:2009qm these analogs of proper time and spacetime volume were used to investigate whether there was a statistical resemblance to de Sitter spacetime (of any dimension). These studies did find a resemblance to de Sitter spacetime⁴⁴4Some parts of their analysis favored $2+1$ dimensions, while the majority favored $3+1$ dimensions. after the tree-like era of the percolation dynamics.

Another known feature of transitive percolation is that it describes a cyclic cosmology, i.e. one that undergoes alternating periods of expansion and contraction.[15] More general sequential growth dynamics that lead to cyclic cosmologies have also been studied in some detail.[16, 15, 17] In some of these models, the dynamics of each new cycle is different from the previous one which we can interpret as a kind of cosmological renormalization group.[17] A major open question is whether and how one of these dynamics eventually evolves into a manifoldlike causal set such as one approximated by a Friedman-Lemaître-Robertson-Walker (FLRW) universe.[11, 18] A hope would be that the emergence of manifoldlike behaviour will be better understood once the quantum dynamics is developed. Some indications of how this may happen are given in Ref. \refciteCarlip:2022nsv, where a suppression of a large class of non-manifoldlike causal sets is demonstrated in a path sum.

A last remark on this topic is that such rapid early expanding behavior has also been observed in two other independent contexts. One is Ref. \refciteGlaser:2014dwa where the Hartle-Hawking wavefunction in two dimensions was studied using the Benincasa-Dowker causal set action[21]. The other context is when a positive initial $\Lambda$ is set for Everpresent $\Lambda$ dark energy models, discussed in Section 5.[22]

There are three main assumptions that form the basis of the Everpresent $\Lambda$ heuristic argument:

The Hubble tension is the name given to a clash in recent measurements of the Hubble parameter $H_{0}$, with disagreements between supernova (together with Cepheid stars and tip of the red giant branch) data and observations from the cosmic microwave background. Both sets of measurements are robust, leaving little room for systematic error to account for the discrepancy. This disagreement forms the basis of the Hubble tension [42, 43, 44]. In light of this disagreement, many believe that we must abandon traditional models where $\Lambda$ is a constant and instead consider dynamical models where it can have an evolving value, thereby relieving the tension. Everpresent $\Lambda$ is precisely such a dynamical model. In particular, early dark energy models where there is extra dark energy in the period after matter-radiation equality but before recombination, are a leading candidate to alleviate this tension.[45, 44, 46] Due to its random fluctuating nature, such a period of early dark energy can be realized with Everpresent $\Lambda$.

Recent observations of large-scale structure have set stringent constraints on the clustering of matter. This has revealed a tension in the strength of clustering inferred from different probes.[44] $S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/0.3}$ is the parameter used to quantify this tension, where $\Omega_{m}$ is the fraction of the total energy density belonging to matter. In particular, CMB data favors a higher value of $S_{8}$ while lower redshift data such as from weak lensing and galaxy clustering prefer a lower value of $S_{8}$.

Models in which the cosmological constant changes sign over the history of the universe have been shown to relax the $S_{8}$ tension.[47, 48] Since Everpresent $\Lambda$ is likely to change sign over the history of the cosmos, it can behave like these models and potentially relax the $S_{8}$ tension in the same manner.

Another class of models that have been shown to ameliorate the $S_{8}$ tension are interacting dark energy models where the energy density can be transferred between dark matter and dark energy.[49, 50] While this scenario does not automatically occur in current Everpresent $\Lambda$ models, it may be an interesting avenue to modify the present models to incorporate such an interaction.

At present, there are only qualitative indications that Everpresent $\Lambda$ possesses some of the features that are believed to be key to resolving the Hubble and $S_{8}$ tensions. Carrying out a dedicated and quantitative study of how much the current Everpresent $\Lambda$ models (or slightly modified versions of them) can alleviate these tensions is one of the most important directions of future work on Everpresent $\Lambda$. The $S_{8}$ and Hubble tensions are correlated tensions, namely, many proposed models that alleviate one tension tend to worsen the other. Therefore it would be very interesting to see if Everpresent $\Lambda$ can alleviate both at the same time.

The entanglement entropy of a quantum field confined to a spacetime subregion, often scales as the spatial area of the subregion. This “area law” is what makes entanglement entropy a natural candidate to be the microscopic source of horizon entropy. It is also then no surprise that the concept of entanglement entropy sprung from the search for a deeper understanding of black hole entropy [61, 54].

The entanglement entropy of a quantum field described by (an initially pure) density matrix $\rho$ is

where $\rho_{\text{red}}$ is the reduced density matrix encoding the field degrees of freedom in the confined subregion (e.g. the region of spacetime on one side of a horizon) only, or alternatively in its causal complement only.⁸⁸8By the complementarity property of entanglement entropy, we would get the same answer in either case. Intuitively, (9) quantifies how much information one loses as a result of the limited access to the quantum field. See Fig. 5 for a sketch of a spatial version of an entangling region. In some sense, the entanglement entropy counts the number of field modes between the inside and outside of an entangling region, straddling the boundary. Some form of UV cutoff must be taken into account when describing the degrees of freedom; in the absence of such a cutoff, one would end up concluding that an infinite amount of information is lost as a result of the confinement (corresponding to information encoding arbitrarily high energy degrees of freedom of the quantum field theory near the boundary). There are a variety of ways to compute (9) and a variety of choices of UV cutoffs to obtain a finite and thereby meaningful entanglement entropy. In causal set theory, the entanglement entropy is expressed as a sum over solutions to a generalized eigenvalue problem involving the two-point correlation function of the theory, $W(x,x^{\prime})$. The discreteness scale furnishes the covariant UV cutoff that makes this entropy finite.

The eigenvalue problem that needs to be solved is

where $W$ (typically chosen to be $W_{SJ}$) and $i\Delta$ are finite-dimensional matrices that we saw earlier in Section 3.2.1. The arguments $x$ and $x^{\prime}$ of $W$ and $i\Delta$ must be restricted to lie in the subregion whose entanglement entropy we are interested in, before solving (10). We must also ensure that the solutions to (10) satisfy $\Delta v\neq 0$. The summation that yields the entropy in terms of the eigenvalues in (10) then is⁹⁹9This is the first formulation of entanglement entropy that admits the use of a covariant cutoff. This admission is a result of the use of spacetime correlation functions in (10).

Two noteworthy advantages we gain by studying entanglement entropy in causal set theory are:

Another approach to understanding horizon entropy is motivated by considering the entanglement between the causal set elements themselves, in the absence of any quantum matter fields. In analogy with a gas and the role played by its molecules to describe its entropy, the information content of a horizon is suggested to be encoded in suitably defined primitive constituents or “horizon molecules”. These horizon molecules consist of linked elements with at least one element inside the horizon and at least one outside it. Assuming each molecule to contribute an equal amount to the horizon entropy, as a result of the link information across the horizon being lost, the horizon entropy is captured by counting the number of these molecules.

Much like in the case of entanglement entropy we saw in the previous subsection, in the absence of a UV cutoff it is difficult to arrive at a finite entropy, as there would generically be an infinite number of horizon molecules in the continuum. It turns out that even in the causal set case one does not always obtain a finite number of any kind of horizon molecule, but there will be a finite number of particular kinds of horizon molecules.

The original work on this topic considered $1+1$ dimensional black holes, including examples both in and out of equilibrium.[66] Aiming for simplicity, a kind of link $x\prec y$ was chosen as the horizon molecule. These links consist of element $x$ outside the event horizon $\mathcal{H}$ and element $y$ inside, along with additional conditions which can be summarized as:

where $\Sigma$ is a null or spacelike hypersurface on which we are interested in evaluating the entropy. This hypersurface can be arbitrarily chosen and is a necessary reference in order to avoid overcounting the molecules. An element is maximal (minimal) in a causal set $\mathcal{C}$ if there are no elements to its future (past). In Ref. \refciteDou:2003af counting of the molecules defined by the conditions above indeed led to a term proportional to the event horizon. Moreover, the coefficients were the same in both equilibrium and non-equilibrium cases, hinting at a universality.¹⁰¹⁰10This horizon molecules program also has the universality and background independence advantages highlighted in the previous subsection. While in the absence of the quantum dynamics for causal sets, this treatment remains classical, it serves as an interesting derivation of the known entropies from first principles, and may reveal correction terms.

Difficulties arising from extending the above simple case to higher dimensions led to the consideration of other kinds of horizon molecules.[67] These difficulties arose as a result of the number of horizon molecules becoming unbounded when the black hole spacetime is unbounded. More complicated definitions of horizon molecules, for example in terms of higher cardinality triads and diamonds, have since also been explored.

In Ref. \refciteBarton:2019okw the original horizon molecule definition was refined and also extended to an entire class of n-element molecules or “n-molecules”. They succeeded in obtaining well-defined results for higher dimensions, using their definition of molecules. However, they needed to require that $\Sigma$ is strictly spacelike rather than spacelike or null. Ref. \refciteMachet:2020uml showed that there are subtleties in the null case that spoil some of the encouraging results. They also proposed an alternative approach to the problem in terms of the spacetime mutual information (of the action). We refer the reader to Ref. \refciteMachet:2020uml for further details on this.