# De Sitter stability and coarse graining

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## Abstract

We present a 4-dimensional back reaction analysis of de Sitter space for a conformally coupled scalar field in the presence of vacuum energy initialized in the Bunch–Davies vacuum. In contrast to the usual semi-classical prescription, as the source term in the Friedmann equations we use expectation values where the unobservable information hidden by the cosmological event horizon has been neglected i.e. coarse grained over. It is shown that in this approach the energy-momentum is precisely thermal with constant temperature despite the dilution from the expansion of space due to a flux of energy radiated from the horizon. This leads to a self-consistent solution for the Hubble rate, which is gradually evolving and at late times deviates significantly from de Sitter. Our results hence imply de Sitter space to be unstable in this prescription. The solution also suggests dynamical vacuum energy: the continuous flux of energy is balanced by the generation of negative vacuum energy, which accumulatively decreases the overall contribution. Finally, we show that our results admit a thermodynamic interpretation which provides a simple alternate derivation of the mechanism. For very long times the solutions coincide with flat space.

## 1 Introduction

The de Sitter spacetime is one of the most analytically tractable examples of a genuinely curved solution to Einstein’s field equation. De Sitter space is not only of academic interest since in the current cosmological context the exponentially expanding de Sitter patch is believed to describe the evolution of the Universe soon after the Big Bang during cosmological inflation and at very late times when Dark Energy has begun to dominate over all other forms of energy.

The potential instability of de Sitter space in quantized theories has been investigated in a variety of different approaches and models over a span of more than 30 years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], recently in [36] where we refer the reader for more references. To the best of our knowledge, at the moment the issue still lies unresolved. If de Sitter space were unstable to quantum corrections and could indeed decay, this could provide an important mechanism for alleviating the cosmological constant problem and perhaps also the fine-tuning issues encountered in the extremely flat inflationary potentials that are required by observations. Most definitely, a de Sitter instability would have a profound impact on the fate of the Universe since it rules out the possibility of an eternally exponentially expanding de Sitter space as classically implied by the \(\Lambda \)CDM concordance model.

One of the main motivations behind the original calculation for the evaporation of black holes in [37, 38, 39] was the discovery of their thermodynamic characteristics [40, 41], in particular the connection between the black hole horizon and entropy: the fact that black holes evaporate implies that they can also be ascribed temperature and understood as thermodynamic objects. Like a black hole de Sitter space also possesses a horizon beyond which a local observer cannot see, which was famously in [42] shown to lead to a thermodynamic description of de Sitter space analogously to a black hole. Currently, the thermodynamics of spacetime horizons is established as a mature, well-studied subject [13, 43, 44, 45, 46, 47, 48, 49]. Based on thermodynamic arguments the seminal study [42] concluded that unlike black holes de Sitter space is stable. However quite interestingly, also by invoking thermodynamic concepts in the equally impactful work [13] it was argued that the de Sitter horizon in fact does evaporate.^{1}

As in the original black hole evaporation calculation [37, 38] we make use of semi-classical gravity – often referred to as quantum field theory in curved spacetime [50, 51] – in order to provide a first principle calculation of the stability of de Sitter space. Our approach allows one to study how the quantized matter back reacts on the classical metric by using the semi-classical versions of Einstein’s equation. In situations where the quantum nature of gravity is subdominant this is expected to give reliable results. Specifically we will focus on the cosmologically most relevant coordinate system, the Friedmann–Lemaître–Robertson–Walker (FLRW) line element describing an expanding, homogeneous and isotropic spacetime. This line element results in the Friedmann equations allowing a straightforward analysis of back reaction, studied for example in [33, 52, 53, 54, 55, 56, 57, 58, 59, 60]. The FLRW coordinates are rarely included in discussions of the thermodynamics of horizons, although see [48, 61, 62].

The decoherence program asserts that the ubiquitous disappearance of macroscopic quantum effects – commonly known as the quantum-to-classical transition – stems from the observationally inaccessible environmental sector that in any realistic set-up is always present [63, 64, 65, 66, 67, 68]. Using this mechanism as a motivation recently in [36] a modification to the usual prescription for semi-classical gravity was explored where in the Einstein equation one implements coarse grained expectation values calculated by including only those states that are observable. It was shown that if a part of the density matrix may be characterized as unobservable and is neglected from the quantum averaging this generically leads a qualitatively different behaviour for the expectation value for the energy-momentum in de Sitter space compared to the usual approach: it implies non-trivial back reaction with an evolving Hubble rate, even when as the initial condition one uses the manifestly de Sitter invariant Bunch–Davies vacuum. The procedure of tracing over unobservable states, in addition to decoherence studies, is often implemented in calculations involving spacetimes with horizons such as black holes and Rindler space [69, 70, 71] and is a key element of the information paradox [72, 73].

As a continuation of the work [36] here we explore the gravitational implications from a particular coarse grained density matrix: the cosmological event horizon of de Sitter space splits the Universe into observable and unobservable patches essentially identically to a black hole, which motivates us to disregard all information contained beyond the horizon. By using this density matrix to calculate the expectation values via the Friedmann equations we then perform a complete 4-dimensional back reaction analysis of de Sitter space with a conformal scalar field initialized in the Bunch–Davies vacuum.

Since the event horizon of de Sitter space is an observer-dependent concept, particle creation associated with the cosmological horizon was in [42] argued to lead to an observer dependence of the back reaction and hence of the metric of spacetime. There it was further concluded that the energy-momentum tensor sourcing the semi-classical Einstein equations cannot be defined in an observer-independent manner. Although during the time of writing, in [42] the derivation of this energy-momentum tensor was ’in preparation’ the calculation to the best of our knowledge does not exist in literature. To a degree this gap is filled by the current work and since the implemented coarse graining prescription is defined by the cosmological horizon of de Sitter space it possesses the observer dependence put forward in [42].

We will also discuss our results in the framework of horizon thermodynamics and provide a complete physical picture of particle creation in de Sitter space leading to a consistent definition of the differential of internal energy. With this picture we are able to formulate the first law of thermodynamics in de Sitter space, which is known to be problematic [13], with which we show how the first principle result admits an alternate derivation by using only thermodynamic concepts.

We emphasize that as far as horizons and particle creation are concerned there is no compelling reason to assume the arguments given not to apply also for non-conformal scalar fields, fermions and vector fields.

A 2-dimensional calculation of this mechanism was initially presented in [34] but here we will provide much more detail, perform also the 4-dimensional calculation and make the connection to horizon thermodynamics.

Our conventions are (+,+,+) [74] and \(c\equiv k_{B}\equiv \hbar \equiv 1\).

## 2 The set-up

*n*-dimensions the matter action for a conformally coupled scalar field is written as

*g*is the determinant of the metric and

*R*the scalar curvature. The equation of motion for the scalar field is

*H*and the scale factor

*a*

## 3 General features of back reaction in de Sitter space

*n*-dimensions. A conformally coupled classical field with the action (1) has a vanishing trace

In the quantized case the previous argument is made more complicated by the counter term contribution \(\delta T_{\mu \nu }\), which leads to an anomalous trace [75, 76, 77, 78] and in de Sitter space in even dimensions gives \({T_\mu }^\mu = -\delta {T_\mu }^\mu \ne 0\). This however does not introduce a significant modification compared to the classical case discussed above.

^{2}

^{3}

- (1)
A conformally coupled theory

- (2)
A FLRW line element

- (3)
\(\rho ^S_m\ne 0\)

- (4)
\(\dot{H}=0\)

*H*cannot be strictly constant. More generally, if there exists a density of conformal matter that may be thought to contain any entropy it must be in a non-vacuous state with \(\rho ^S_m\ne 0\) since the vacuum configuration is described by pure state which has strictly zero entropy, see [36] for more discussion.

## 4 De Sitter space in FLRW and static coordinates

The topic of de Sitter space in various coordinates has been extensively studied in literature, for example see chapter 5 of [50] for a detailed discussion.

*n*-dimensional de Sitter manifold can be understood as all points contained in the

*n*-dimensional hyperboloid embedded in \((n+1)\)-dimensional Minkowski space. The 4-dimensional de Sitter space can then be expressed in terms of a 5-dimensional Minkowski line element

*H*

*r*, \(\theta \) and \(\varphi \), respectively

*t*that can be reached by a ray of light emanating from the origin

*A*subscript in (21) is that we also need coordinates covering the spacetime outside the horizon. These we parametrize withwhere Open image in new window and Open image in new window and the line element is precisely as in (22), but with \(A\rightarrow B\). Note that in the 2-dimensional case studied in Sect. 5.2 one also has a second patch beyond the horizon, which we denote with

*C*. The region covered by (23) is shown as the red region in Fig. 3. It is worth pointing out that the coordinates (21) and (23) significantly resemble the Kruskal–Szekeres parametrization of the Schwarzschild black hole. Using the combination of the coordinates (21) and (23) one may cover the same patch as the FLRW system (17) as can be seen by comparing Figs. 2 and 3. The regions covered separately by coordinates of type (21) and (23) we will refer to as regions

*A*and

*B*or the static patches although this is not strictly speaking a good characterization for (23): since Open image in new window , Open image in new window is in fact the time coordinate and hence the metric (22) is explicitly time dependent beyond the horizon.

## 5 The coarse grained energy-momentum tensor

First, we will adopt the cosmologically motivated choice where de Sitter space is described in terms of the expanding FLRW coordinates used for example when calculating the cosmological perturbations from inflation. This is a consistent approach, since for an observer at rest with the expanding FLRW coordinates the contracting patch is not accessible.

Next we need to define the specific state to be used as the initial condition. In the black hole context making the physical choice for the vacuum is an essential ingredient for the understanding of the evaporation process, see [69] for important pioneering work and [47] for a clear discussion. Again conforming to the usual choice made in inflationary cosmology, we will use the Bunch–Davies vacuum [80, 81] as the quantum sate. A compelling motivation for this choice comes from the fact that the Bunch–Davies vacuum is an attractor state in de Sitter space [33], provided we make the natural assumption that the leading divergences of the theory coincide with those in flat space [82, 83].

A very important feature of the Bunch–Davies vacuum is that it covers the entire FLRW de Sitter patch and hence extends also to regions that would be hidden behind the horizon. This is a natural requirement for an initial condition in a case where the Universe was not always dominated by vacuum energy, which from the cosmological point of view is well-motivated: for example, the Universe may start out as radiation or matter dominated and only at late times asymptotically approach the exponentially expanding de Sitter space as in the \(\Lambda \)CDM model. This will turn out to be crucial for our calculation.

In order to obtain well-defined quantum expectation values we must define a regularization and a renormalization prescription for the ultraviolet divergences. Perhaps the most elegant way would be to analytically continue the dimensions to *n* and redefine the constants of the original action to obtain physical results. Dimensional regularization does have a drawback however, which is that consistency requires one to calculate everything in *n* dimensions, which is surely more difficult than to perform the calculation in 2 dimensions, for example. For our purposes the most convenient choice is the adiabatic subtraction technique [84, 85, 86], which in the non-interacting case is a consistent and a covariant approach [87], but where no explicit regularization is needed as the counter terms can formally be combined in the same integral as the expectation value. This also allows us to use a strictly 2- or a 4-dimensional theory.

### 5.1 Tracing over the unobservable states

So far our approach has followed standard lines. If in some given state \(\vert \Psi \rangle \) we were to calculate the relevant expectation values \(\langle \hat{T}_{\mu \nu }\rangle \equiv \langle \Psi \vert \hat{T}_{\mu \nu }\vert \Psi \rangle \) and use them as the sources in the Friedmann equations we would obtain a result that exponentially fast approaches a configuration with \(\dot{H}=0\) and conclude that de Sitter space is a stable solution also when back reaction is taken into account. This is a manifestation of the de Sitter invariance and the attractor nature of the Bunch–Davies vacuum and true as well for the non-conformal case [33]. However, as discussed in Sect. 4 the horizon in de Sitter space splits the FLRW manifold into two patches only one of which is visible to a local observer. This is very much analogous to how a black hole horizon blocks the observational access of an observer outside the horizon [42, 72]. Here is where our approach will differ from what is traditionally done in semi-classical gravity: following [36] when calculating \(\langle \hat{T}_{\mu \nu }\rangle \) we will use a prescription where we average over only those states that are inside the horizon and thus observable. We note that quite generally coarse graining a state is expected to bring about a qualitative change in the results since it often leads to a violation of de Sitter invariance [36].

Importantly, the Bunch–Davies vacuum in de Sitter space before coarse graining is a zero entropy state, but as explained covers also regions that are hidden from a local observer. If tracing over the unobservable states leads to a non-zero entropy it also suggests the presence of a non-zero energy density, which in light of the arguments given in Sect. 3 implies \(\dot{H}\ne 0\) and gives an important link between loss of information from coarse graining and a potentially non-trivial back reaction in our prescription.

Our choice of neglecting the unobservable states from the expectation values can be motivated as follows. First of all it is a standard procedure in branches of physics where having only partial observable access to a quantum state is a typical feature. An important example is the decoherence program: without an unobservable environment the quantum-to-classical transition does not take place [89]. Neglecting unobservable information is crucial also for the inflationary paradigm: in order to obtain the correct evolution of large scale structure as seeded by the inherently quantum fluctuations from inflation one must calculate the gravitational dynamics from the classicalized i.e. coarse grained energy-momentum tensor [90]. Perhaps most importantly, the energy-momentum tensor one obtains after neglecting the unobservable states corresponds to what an observer would actually measure and in this sense has clear physical significance.

A profound feature of our prescription is that since the horizon in de Sitter space is an observer dependent quantity, so is then the back reaction itself. Although a rather radical proposition, this does not imply an immediate inconsistency. After all, observer dependence is a ubiquitous feature in general – and even special – relativity. Furthermore, the well-known observer dependence of the concept of a particle in quantum theories on curved backgrounds was argued to lead to such a conclusion already in the seminal work [42].

Although our prescription of using a coarse grained energy-momentum tensor as the source term for semi-classical gravity deviates from the standard approach making use of \(\langle \hat{T}_{\mu \nu }\rangle \equiv \langle \Psi \vert \hat{T}_{\mu \nu }\vert \Psi \rangle \), we would like to emphasize that at the moment there is no method for conclusively determining precisely which object is the correct one [50]. This stems from the fact that semi-classical gravity is not a complete first principle approach, but rather an approximation for describing some of the gravitational implications from the quantum nature of matter. Before a full description of quantum gravity is obtained it is likely that this state of affairs will remain.

Tracing over inaccessible environmental states that are separated by a sharp boundary from the accessible ones generically leads to divergent behaviour close to the boundary. This is encountered for example in the context of black hole entropy [91] and entanglement entropy in general [92, 93]. Although by introducing a cut-off or a smoothing prescription well-defined results can be obtained [73], there is valid suspicion of the applicability of the semi-classical approach when close to the horizon. However, we can expect reliable results at the limit when the horizon is far away. At this limit there exists a natural expansion in terms of physical distance in units of the horizon radius, or more specifically in terms of the dimensionless quantities Open image in new window , in the notation of Sect. 4. The neglected terms we will throughout denote as Open image in new window . This limit can be expressed equivalently as being far away from the horizon or close to the center of the Hubble sphere and can equally well be satisfied when *H* is large such as during primordial inflation or when it is very small as it is during the late time Dark Energy dominated phase we are currently entering. The limit where the observer is far from the horizon is also the limit taken in the standard black hole analysis [37, 38].

### 5.2 Two dimensions

For completeness we first go through the steps of the 2-dimensional argument presented in [34], before proceeding to the full 4-dimensional derivation.

*A*,

*B*and

*C*. As is clear from the definitions (51) the

*V*and

*U*coordinates can also be conveniently used to split the FLRW patch in terms of the regions

*A*,

*B*and

*C*since they vanish at the horizons 1 /

*H*and \(-1/H\), respectively. This is illustrated in Fig. 4. Furthermore, in the static patches we define the tortoise coordinateswith Open image in new window , Open image in new window and Open image in new window . It is now a question of straightforward algebra to express of the light-cone coordinates

*U*and

*V*in terms of the static ones. The results can be summarized as

*U*or

*V*since these newer mix and can be thought as separate sectors, as is evident by taking into account.

^{4}\([\hat{a}_{-k},\hat{a}^{{\dagger }}_{k}]=0\), \(V=V(v)\) and \(U=U(u)\) from (53–56). Since \(\hat{\phi }_V\) is expressed only in terms of the \(\hat{a}_{-k}\) operators it consists solely of particles moving towards the left and similarly for \(\hat{\phi }_U\) and the right-moving particles.

*A*,

*B*and

*C*. If we choose Open image in new window , Open image in new window and Open image in new window for

*A*,

*B*and

*C*respectively, it is a simple matter of using the vector transformations \(\partial _{\mu }=\frac{{\partial x^{\tilde{\alpha }}}}{{\partial x^\mu }}\partial _{\tilde{\alpha }}\) with (53–56) to show thatwhich with the help of Fig. 4 one may see to be time-like in terms of conformal time and future-oriented in their respective regions.

^{5}

*A*and

*B*, since for \(\hat{\phi }_V\) only the horizon at \(V=0\) is relevant, which we elaborate more below. From (66) we then get

*A*region are to be understood to vanish in region

*B*and vice versa for the modes in

*B*.

As mentioned, only the regions *A* and *B* are relevant for \(\hat{\phi }_V\). The reason why one may neglect the contribution from region *C* is apparent from the relations (53–56) and (66): \(\hat{\phi }_V\) has no dependence on *U* so no mixing of modes is needed in order to obtain analytic behaviour across the horizon \(U=0\). Thus, including all regions *A*, *B* and *C* in (70) would still give (72), which we have also explicitly checked.

So far we have only studied \(\hat{\phi }_V\) i.e. the particles moving to the left. By using (53–56) and (66) the calculation involving \(\hat{\phi }_U\) proceeds in an identical manner resulting also in a thermal density matrix, but in terms of the right-moving particles \(|n_k,A\rangle \).

*A*are neglected we can write the expectation value of the energy-momentum tensor from (5) by expressing \(\hat{\phi }\) as the top line from (66)

*A*labels. The final unrenormalized expression in the FLRW coordinates (47) can be obtained by using the tensor transformation law \(T_{\mu \nu }=\frac{{\partial x^{\tilde{\alpha }}}}{{\partial x^\mu }}\frac{{\partial x^{\tilde{\beta }}}}{{\partial x^\nu }}T_{\tilde{\alpha }\tilde{\beta }}\) with (53–56) and (51). This givesandAs the discussion after Eq. (42) addressed, (76) and (77) have divergent behaviour on the horizons Open image in new window . This is distinct to the usual ultraviolet divergences encountered in quantum field theory, which are also present in (76) and (77) as the divergent integrals. For our purposes the relevant limit of being close to the origin is obtained with an expansion in terms of Open image in new window givingThe last step in the calculation is renormalization. When we neglect the Open image in new window contributions i.e. study only the region far from the horizon the result is precisely homogeneous and isotropic for which the counter terms can be found by calculating the energy-momentum tensor as an expansion in terms of derivatives the scale factor. This is the adiabatic subtraction technique [84, 85, 86], with which the 2-dimensional counter terms were first calculated in [95] giving coinciding results to [96]. This technique gives the counter terms for the energy and pressure components as formally divergent integrals

In the language of Sect. 3 the \(\pm H^2/(24\pi )\) terms are state independent contributions resulting from renormalization and the integral over the thermal distribution is the state dependent contribution \(\rho ^S_m\). The results (81–83) can be seen to be in agreement with the arguments of Sect. 3, in particular the 2-dimensional version of (13) when taking into account of the modifications arising from our choice of not to implement dimensional regularization.

Before ending this subsection we comment on a technical detail regarding the divergences generated in the coarse grained state. A general – or certainly a desirable – feature of quantum field theory is the universality of the generated divergences and renormalization. For a quantum field on a de Sitter background, one should be able to absorb all divergences in the redefinition of the cosmological constant, preferably also in the coarse grained state (74). But from (76) and (77) we can see that this does not hold due to the Open image in new window -dependence of the generated divergences, likely related to coarse graining and the additional divergence when approaching the horizon. We emphasize however that even if in a carefully defined coarse graining divergences Open image in new window are not generated, the renormalized result would coincide with (81–83), because up to the accuracy we are interested in all the needed counter terms could be derived via adiabatic subtraction, which satisfies \(\delta T_{00}=-\delta T_{ii}/a^2\) [33].

### 5.3 Four dimensions

The calculation in four dimensions proceeds in principle precisely as the 2-dimensional derivation of the previous subsection. The main differences are that the solutions in four dimensions are analytically more involved and that there is only one horizon. Quantization of a scalar field in the static de Sitter patch has been studied in [97, 98, 99] to which we refer the reader for more details. Quite interestingly, although the line element for a Schwarzschild black hole and de Sitter space in static coordinates are very similar, the latter has an analytic solution for the modes while the former does not.

*A*and region

*B*The tortoise coordinates and light-cone coordinates can be obtained trivially from the 2-dimensional results (51), (52) and (53–56) with the replacements \({x}\rightarrow {r}\) and Open image in new window .

*B*region then becomesSince the inner product (93) is independent of the choice of hypersurface we can evaluate it at the limit Open image in new window . It is then a simple matter of using (91) to show thatin region

*B*is the correctly normalized positive frequency mode provided thatwith \(D^{\tiny {B}}_{\ell k}=(D^{\tiny {A}}_{\ell k})^*\).

*A*and for the mode in region

*B*where \(\sim \) denotes an equality up to constant factors of modulus one. Comparing the above to (66) reveals that the discontinuity is precisely of the same form as in two dimensions, leading to two non-trivial linear combinations that are continuous across the horizon

*A*labels as irrelevant

*ii*’ components of Einstein tensor in the static coordinates at the centre of the Hubble sphere. This implies that there is non-trivial back reaction since Einstein’s equation with (101) as the source is not solved by the static line element describing de Sitter space (22). We can conclude that coarse graining such that only observable states are left leads to a violation of de Sitter invariance.

For clarity we summarize the arguments of this section here once more: in de Sitter space as described by the expanding FLRW coordinates (18) initialized to the Bunch–Davies vacuum the energy-momentum of a quantum field has a thermal character when in the density matrix one includes only the observable states inside the horizon. The energy density inside the horizon is maintained at a constant temperature by a continuous flux of radiation incoming from the horizon that precisely cancels the dilution from expansion. This results in \(\rho _m+p_m\ne 0 \), which is independent of the details of renormalization and the conformal anomaly due to the symmetries of the counter terms in de Sitter space. Thus even at the limit when the distance to the horizon is very large and the result is isotropic and homogeneous the sum of the energy and pressure densities does not cancel and because of this the dynamical Friedmann equation (9) then implies that a strictly constant Hubble rate *H* is not a consistent solution. This is also visible in the result given in the static coordinates (101), which does not solve Einstein’s equation if the background is assumed to be strictly de Sitter.

## 6 Self-consistent back reaction

If, as the arguments of the previous section imply, de Sitter space is affected by back reaction in the prescription we have chosen, this naturally leads one to investigate how precisely is the strict de Sitter solution modified. Ultimately, this is determined by the semi-classical Einstein equation.

*p*. This is expected, since we have not included the effect of the flux (104), which continuously injects the system with more energy.

As argued in [37, 38] for the analogous black hole case a flux can be seen to imply a change in the size of horizon: a positive flux coming from the horizon is equivalent to a negative flux going into the horizon. Note that when the cosmological horizon or \(\rho _\Lambda \) absorbs negative energy the horizon radius will grow, where as precisely the opposite relation holds for the horizon and mass of a black hole.

*H*

^{6}Furthermore, it also follows from the results of section 10.4 of [13].

The time scale (110) is quite large, at least in the obvious cosmological applications: for inflation with the maximum scale allowed by the non-observation of tensor modes \(H_0\sim 10^{14}\)GeV [100] gives \(t_{1/2}\sim 10^{13}H_0^{-1}\), which corresponds to \(10^{13}\) *e*-folds of inflation. The breakdown of the Dark Energy dominated late time de Sitter phase can be estimated by using the current Hubble rate \(H_0\sim 10^{-42}\)GeV [101] giving \(t_{1/2}\sim 10^{125}H_0^{-1}\), where \(H_0^{-1}\) corresponds to the age of the Universe.

The observation that in an expanding, homogeneous and isotropic space a non-diluting particle density necessitates a decaying vacuum energy was already made in [102, 103] and has since been studied in [31, 104, 105, 106, 107, 108, 109].

*H*should change very gradually. From (109) we see this to be true, \(-\dot{H}/H^2\sim H^2/M_\mathrm{pl}^2\ll 1\) implying that the quantum modes as well as the right hand side of the Friedmann equations can be calculated in the approximation where the derivatives of

*H*are neglected. One may furthermore check the robustness of the cosmological event horizon and our coarse graining prescription under back reaction: from (109) one gets the scale factor

*H*, as required.

As we discuss in the next section, the proposal that vacuum energy is dynamical also has a deep connection with the thermodynamic interpretation of de Sitter space, which gives it a more solid footing.

## 7 Derivation from horizon thermodynamics

*U*, the horizon and the pressure

*P*of de Sitter space [42, 45]. However, if the de Sitter solution is assumed to be determined only by the cosmological constant term \(\Lambda \), a parameter of the Einstein–Hilbert Lagrangian, the change in internal energy

*dU*requires the problematic concept of varying \(\Lambda \) [13, 112]. If however, we adopt the proposal that semi-classical back reaction is sourced by the coarse grained energy-momentum tensor as discussed in 5.1, the derivation of the two previous sections imply that the vacuum energy becomes a dynamical quantity due to the inevitable contribution of quantum fields and this issue is evaded. In fact quite remarkably, allowing the vacuum energy to vary and by using the standard concepts of horizon thermodynamics we can derive the results (109) and (108) in a mere few lines.

Assuming that the thermodynamic features persist even when spacetime has evolved away from de Sitter, as long as the horizon has heat it will continue to radiate and lose energy by dilution and thus to grow without bound. In this case the ultimate fate of the Universe would not be an eternal de Sitter space with finite entropy, but an asymptotically flat spacetime with no temperature, an infinitely large horizon and hence infinite entropy.

## 8 Summary and conclusions

In this work we have studied the stability of de Sitter space in the semi-classical approach for a model with a non-interacting conformally coupled scalar field and a cosmological constant i.e. vacuum energy. Back reaction was derived in a prescription where the expectation values sourcing the semi-classical Einstein equation were calculated via a coarse grained density matrix containing only states that are observable to a local observer. For the chosen initial condition of the Bunch–Davies vacuum this prescription translates as neglecting all degrees of freedom located beyond the cosmological event horizon. As we have shown via a detailed argument, in our approach de Sitter space is not stable and in agreement with [13] (section 10.4) but in disagreement with [42].

Coarse graining over unobservable states in the density matrix is made frequent use in various contexts such as the decoherence program and black hole information paradox, but rarely considered in cosmological applications, in particular semi-classical backreaction via the Friedmann equations as done in this work. Our study indicates that loss of information from coarse graining states beyond the horizon leads from the initial Bunch–Davies vacuum, a pure state, to a thermal density matrix and manifestly breaks de Sitter invariance.

Our result also shows that a local observer who is only causally connected to states inside the horizon will in the cosmologically relevant expanding FLRW coordinates view de Sitter space as filled with a thermal energy density with a constant temperature given by the Gibbons–Hawking relation \(T=H/(2\pi )\) that is maintained by a continuous incoming flux of energy radiated by the horizon. Without such a flux the expansion of space would dilute and cool the system quickly leading to an empty space.

From the semi-classical Friedmann equations we made the simple but nonetheless important observation that space filled with thermal gas, which in our prescription follows from de Sitter space possessing the cosmological event horizon, is not a solution consistent with having a constant *H*. This follows trivially from the fact that thermal particles do not have the equation of state of vacuum energy and is the key mechanism behind the obtained non-trivial back reaction. This can also be seen in the static coordinates, which are not a solution of Einstein’s equation when the energy-momentum tensor describes thermal gas.

By modifying the Friedmann equations to contain gradually decaying vacuum energy we were able to provide a self-consistent solution for the evolution of the Hubble rate. The solution had the behaviour where *H* remained roughly a constant for a very long time, but eventually after a time scale \(\sim M_\mathrm{pl}^2/H^3\) the system no longer resembled de Sitter space. As a physical picture of the process we proposed that a quantum field in curved space may exchange energy with the vacuum making vacuum energy a dynamical quantity instead of a constant parameter fixed by the Lagrangian. In this interpretation particle creation occurs at the expense of creating a negative vacuum energy contribution. This provides a mechanism allowing the overall vacuum energy to decrease, a very closely analogous picture to black hole evaporation where a black hole loses mass due to a negative energy flux into the horizon.

Finally, we presented an alternate derivation of the main result by using the techniques of horizon thermodynamics. In the thermodynamic derivation the concept of dynamical vacuum energy proved a crucial ingredient, as it gives a well-defined meaning to the differential of internal energy in the cosmological setting allowing a clear interpretation of the first law of thermodynamics for de Sitter space. The derivation via horizon thermodynamics turned out to be remarkably simple providing insights also to spacetimes that are not to a good approximation de Sitter. The thermal argumentation implied that the fate of the Universe is in fact an asymptotically flat space instead of eternal de Sitter expansion.

The possible decay or evaporation of the de Sitter horizon seems like a prime candidate for explaining the unnaturally small amount of vacuum energy that is consistent with observations. Importantly, in our prescription for semi-classical gravity a gradual decrease of *H* is recovered. Unfortunately, the predicted change is quite slow. For the Early Universe and in particular inflation the gradual decrease of *H* from back reaction is much smaller than the slow-roll behaviour usually encountered in inflationary cosmology. Of course due to the multitude of various models of inflation, an evaporation mechanism could potentially provide a novel block for model building in at least some cases.

Perhaps the most profound implication of this work is that it suggests that potentially the eventual de Sitter evolution of the Universe as predicted by the current standard model of cosmology, the \(\Lambda \)CDM model, is not eternal. This indicates that at least some of the problems associated with the finite temperature and entropy of eternal de Sitter space and in particular the issues with Boltzmann Brains [115] could be ameliorated.

Of course all of the perhaps rather significant predictions from this work rest on the coarse graining prescription we have introduced in the semi-classical approach to gravity. Quite unavoidably, it results in an inherently observer-dependent approach due to the observer dependence of the de Sitter horizon. This is in accord with the statements of [42], but from a fundamental point of view appears to result in rather profound conclusions such as Everett - Wheeler or many-worlds interpretation of quantum mechanics, as discussed in [42]. In a semi-classical approximation however no obvious inconsistencies seem to arise when one simply includes the additional step of coarse graining the quantum state with respect to the perceptions of a particular observer, although more work in this regard is required.

Coarse graining over unobservable information gives rise to several natural features: it allows for the generation of entropy, the quantum-to-classical transition via decoherence and by definition leads to a result containing only the information an observer may interact with. When tracing over information beyond the event horizon of de Sitter space it also leads to an energy-momentum with a divergence on the horizon, which may signal a breakdown of the semi-classical approach but more investigation is needed. We end by emphasizing that in this work we have not presented a complete analysis of all physical implications of the prescription, which needs to be done in order to ultimately determine its viability.

## Footnotes

- 1.
The argument can be found in section 10.4 of [13].

- 2.
When dimensional regularization is used one simply has \(\langle \hat{T}_{00}\rangle =\rho ^S_m\).

- 3.
- 4.
Here we neglect the \(k=0\) zero mode, whose quantization is a non-trivial issue in 2-dimensional field theory [94], but does not pose problems in four dimensions.

- 5.
For example, from Fig. 4 we see that in region

*C*we have \(-x\ge 0\) and \(UV\le 0~\Leftrightarrow ~ x^2\ge \eta ^2\). - 6.
We thank the authors of [35] for clarifying this issue.

## Notes

### Acknowledgements

We thank Paul Anderson and Emil Mottola for illuminating discussions and Thanu Padmanabhan for valuable comments on the draft. This research has received funding from the European Research Council under the European Union’s Horizon 2020 program (ERC Grant Agreement no. 648680) and the STFC grant ST/P000762/1.

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