Stochastic dark energy from inflationary quantum fluctuations
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Abstract
We study the quantum backreaction from inflationary fluctuations of a very light, nonminimally coupled spectator scalar and show that it is a viable candidate for dark energy. The problem is solved by suitably adapting the formalism of stochastic inflation. This allows us to selfconsistently account for the backreaction on the background expansion rate of the Universe where its effects are large. This framework is equivalent to that of semiclassical gravity in which matter vacuum fluctuations are included at the one loop level, but purely quantum gravitational fluctuations are neglected. Our results show that dark energy in our model can be characterized by a distinct effective equation of state parameter (as a function of redshift) which allows for testing of the model at the level of the background.
1 Introduction

matter condensates or physical DE, of which the simplest representatives are scalar condensates (quintessence models);

modified gravity or geometrical DE, which mimics dark energy by changing the relation between geometry and matter or by supplying additional geometric fields to general relativity.
The question of naturalness of initial conditions is not addressed in traditional approaches. For example, in quintessence models typically a quintessence field starts running from a value which is not a (local or global) minimum of the potential. Criticisms are often brushed away by noting that similar malady plagues most of inflationary models. Arguably the main benefit of this work is in that we construct a theory that naturally explains the initial field value – which is accounted for by the calculable amplitude of infrared field fluctuations during inflation – thus addressing this fundamental criticism. The program advocated here can be thought of as a third way for understanding DE, in that in our class of models a link is established between primordial inflation and dark energy. This link, among other things, can be exploited when designing tests for these models.
Observers have devoted a lot of effort (and observational time) to nail down as accurately as possible the amount (and distribution) of dark energy. Since its discovery in 1998 [11, 12] a lot of progress has been made in improving the accuracy of DE measurements [13, 14, 15]. At this moment the Planck satellite [16, 17] and the Dark Energy Survey (DES) [19] collaborations provide the most stringent bounds on dark energy. Presently \(\varLambda \)CDM, which assumes a cosmological constant equation of state \(p=\,\rho \) for dark energy, is consistent with all astronomical data. Assuming a general constant equation of state parameter \(p\!=\!w\rho \) (the socalled wCDM model), combined Planck and Type Ia supernovae data provides constraints \(w \!=\! \,1.006\pm 0.045\) (68% CL).
Considering dynamical dark energy models – where w that vary with redshift z – yields significantly relaxed constraints [17]. Still no statistically significant deviation of DE from an exact cosmological constant has been found.^{1} The upcoming measurements of Large Synoptic Survey Telescope (LSST) [20], ESA’s Euclid satellite mission [21] and the European Extreme Large Telescope (EELT) [22] will further constrain dynamical DE models. For example, the accuracy of Euclid [21] is projected to be a few percent for simple (1 or 2 parameter models) and weaker for more involved models. Furthermore, these missions will be able to test some prominent DE models, including clustering of DE, growth of Universe’s structure, interactions of DE with itself and with other cosmological fluids, and – last but not least – the class of models presented here.
In the regime where the backreaction becomes comparable to the background it can no longer be treated perturbatively, but rather its effects have to be taken into account properly by solving the semiclassical Friedmann equations selfconsistently, with the quantum backreaction as a source. In general, these are rather complicated integrodifferential equations. In principle these equations can be solved numerically by discretizing both the time evolution and the integral over the modes for the backreaction. An example of such computation was done in the context of reheating in [31], but it still presents a challenging numerical problem. Instead of the numerical approach we make use of the observation from the analytic computations [26, 28, 30] that the quantum backreaction for very light fields is dominated by the infrared (IR) modes not only during inflation, but throughout the history, and that the spectrum of these modes is inherited from the inflation era. The evolution of IR modes (which are largely amplified) during inflation (for small enough nonminimal coupling) is very accurately described by the formalism of stochastic inflation fully formulated in [33] (though some of its simplest applications were already used in [32]). Here we adapt this method so that it becomes applicable to the evolution of IR modes in subsequent radiation and matter dominated eras of the Universe, and reproduce all the perturbative results of [30]. Such an approach to the problem was already advocated long ago in [3], just after the original inception of the idea [23, 24], but – to our knowledge – it has not been carried out. We then use this method to approximate the semiclassical Friedmann equations in the regime where quantum backreaction is very large and write them as a closed set of differential equations which are much simpler to solve than the original integrodifferential equations of semiclassical gravity. This approximation captures accurately the backreaction effects. These equations are then solved numerically. We find that the backreaction indeed accelerates the Universe, driving it towards a late de Sitter phase, very much like the observed behaviour of the Universe today, representing thus a novel dark energy candidate.
This paper is organized as follows. Section 2 presents the definition of the scalar field model and the standard cosmological history. In Sect. 3 the stochastic formalism is derived for the model at hand, and in Sect. 4 the stochastic equations are solved in the regime of small backreaction and the results of [30] are reproduced. Section 5 presents the numerical solution of the full semiclassical Friedmann equations at late times, when backreaction is no longer small. In Sect. 6 we summarize and discuss the principal results.
2 Scalar field in FLRW
In this section we define the cosmological background with the standard expansion history, on which our spectator scalar field model is defined and quantized.
2.1 FLRW background
Large portions of the expansion history of the Universe are dominated by a single cosmological fluid with a constant equation of state. These periods are characterized by constant deceleration, or by a constant parameter \(\epsilon \!=\! {\dot{H}}/H^2\), where \(\epsilon \!\approx \!0\) during inflation (where it is known as the slowroll parameter), \(\epsilon \!=\!2\) during radiationdominated period, and \(\epsilon \!=\!3/2\) during matter dominated period (see Fig. 1 for a schematic depiction of the evolution of \(\epsilon \) parameter).
2.2 Nonminimally coupled massive scalar
3 Stochastic formalism
Here we briefly introduce the stochastic formalism of [33] for scalar fields in expanding cosmological space. The dominant contribution to the light scalar field correlators in inflation comes from the superhorizon modes (of wavelengths \(k\!>\!1/aH\)). This is also true for the very light or massless scalar fields in subsequent evolution of the Universe, namely during radiation and matter periods [28, 30]. The slowroll approximation usually employed to derive the stochastic equations need not be correct in situations when the scalar becomes very massive, meaning \(m\!\gtrsim \!H\). Since during decelerating periods of expansion the Hubble rate decays, this can eventually become true. This is why we do not use the slowroll approximation in the stochastic formalism, but rather derive the equations for all three IR correlators. These reduce to the standard equations when the slowroll hierarchy between the correlators is present.
3.1 Equations of motion for field operators
3.2 Equations of motion for IR correlators
4 Comparison with field theoretic calculations
Here we employ the stochastic formalism of the preceding section to calculate the backreaction of the scalar field quantum fluctuations during the three relevant cosmological periods and solve them perturbatively. These solutions are applicable in the regime where the quantum backreaction (43, 44) is still negligible in comparison with the classical sources \(\rho _c\) and \(p_c\) in (4). The sources for the equations of motion (33–35) are determined using the mode functions from [30]. This section serves to reaffirm the results of [30] as well as to test the (perturbative) correctness of the stochastic formalism.
4.1 De Sitter inflationary period
The results presented in this subsection were already derived in [34, 35], both for test fields and for cosmological perturbations, and on the more general slow roll inflationary background.
There are two interesting limits we discuss briefly in the two following subsections.
4.1.1 Minimally coupled limit
4.1.2 Massless limit
Note that in this case the leading order behavior of the correlators at late times in inflation is independent of the stochastic source, namely their growth is dominated by the instability of IR modes during inflation. Nevertheless, the stochastic sources, which in this case contribute significantly only during the beginning of inflationary period, fix the amplitude and the hierarchy of the correlators, accounting for the fact that all the contributing modes are of UV origin.
4.2 Radiation period
4.3 Matter period
There are two competing contributions in (90), one behaving like nonrelativistic matter (with negative energy), and the other one like a cosmological constant (CC). If the parameters of the model are such that the CClike term starts dominating over the matter term before late times in matter era it should also eventually dominate over the background fluid and accelerate the expansion. This regime is beyond the (perturbative) regime discussed in the present section, where the backreaction on the expansion rate was neglected. It requires solving the set of semiclassical Friedman equations (4) (with \(\rho _c\rightarrow \rho _c+\rho _Q\) and \(p_c\rightarrow p_c+p_Q\)) together with the matter era correlator equations selfconsistently, as described at the end of Sect. 3. The expansion dynamics during that regime are determined in the next section.
5 Selfconsistent solution and results
More specifically, initial conditions are set at redshift \(z \!=\! z_{in} \!=\! 10\) (corresponding to about 2 efoldings in the past from today, \(N(z) \!=\!\ln (1\!+\!z)\)) in such a way that the energy density and pressure of the wouldbe cosmological constant is substituted for the quantum backreaction. The equations are then solved numerically. The solutions for different choices of parameters are given in Figs. 2 and 3 which show how geometric quantities such as the Hubble rate H and \(\epsilon \!=\! (3/2)(1\!+\!w)\) (vertical axis) depend on the number of efoldings \(N=\ln ({1\!+\!z})\) (horizontal axis).
In Fig. 2 we see that, when \(\xi \!=\!0\) and \(m \!=\! 0.20H_0\) [\(m\!=\!0.01H(z\!=\!10)\)], the relative difference between the Hubble expansion rate and \(\epsilon \) in our model and \(\varLambda \)CDM is at a subpercent level, which is not observable by current observations and at the verge of being observable by the near future observatories. However the differences become more pronounced when the scalar mass approaches the Hubble rate today. Indeed, when \(m\simeq 0.61H_0\) [\(m^2=10^{3}H^2(z=10)\)] the difference in the expansion rate and \(\epsilon \) (or equivalently w) reaches several percent (see right panels in Fig. 2), which is testable by the next generation of observatories such as Euclid [21] and ELT [22].
In order to study the effect of the nonminimal coupling \(\xi \) on the geometrical quantities H(z) and \(\epsilon (z)\), in Fig. 3 we show the evolution of the Hubble parameter and \(\epsilon \) as a function of the number of efolding \(N \!=\! \ln ({1\!+\!z})\) for \(\xi \!=\! 10^{3}\). As in Fig. 2 we see that for small masses, when \(m\ll H_0\) (\(m\simeq 0.20H_0\)) the difference between \(\varLambda \)CDM and our model is at a subpercent level and hence barely reachable by planned observatories. However, for a mass comparable to the Hubble rate today (\(m\simeq 0.61H_0\)), the difference becomes significant and well within reach of Euclid and ELT. Comparing Figs. 2 and 3 reveals that increasing \(\xi \) seems to increase the difference between our model and \(\varLambda \)CDM. However, for small values of \(\xi \)’s (\(\xi \!=\! 10^{3}\) in Fig. 3) this increase is rather moderate. Note that a nonvanishing mass term (or a potential term) is crucial for the model to exhibit DE behaviour. (The case \(m\!=\!0\) and \(\xi \!<\!0\) is examined in [28], and that model does not exhibit a late time dark energy.)
The nonminimal coupling is a relevant parameter here more from the point of view of relating the DE period to the primordial inflationary period, more precisely to the total number of efoldings of inflation \(N_I\). In case of zero nonminimal coupling this number has to be huge [29], \(N_I\sim 10^{12}\) for both cases of Fig. 2, while for even rather small nonminimal couplings of Fig. 3 the total number of efoldings of inflation is much smaller [30], \(N_I\sim 10^{5}\), which can be determined from (2). Larger values (in magnitude) for nonminimal coupling are problematic in the sense that one has to worry about large backreaction much earlier, during primordial inflation. Although this is an interesting possibility to study in the context of graceful exit from inflation, it is not considered in our DE model (for a recent study of the backreaction of a heavy nonminimally coupled scalar in de Sitter inflation see [37]).
While Figs. 2 and 3 nicely show the recent Universe dynamics in our model and compare them to those of \(\varLambda \)CDM, it is instructive to show how the dark energy component in our model evolves, and that is illustrated in Fig. 4. The upper two plots show that for the vanishing nonminimal coupling (\(\xi \!=\!0\)), the backreaction behaves like a cosmological constant to a very good approximation in the late Universe, causing the transition from the matterdominated to the quaside Sitter phase. This behaviour tends to persist until the Hubble rate drops below the mass scale of the scalar field (\(m\!\sim \!H\)). After this point the Universe transitions to a decelerated phase where \(\epsilon \) oscillates between 0 and 3.
The lower two plots in Fig. 4 demonstrate the effect of nonminimal coupling on the dark energy equation of state. The shaded regions correspond to the uncertainty we have in estimating its equation of state. These are introduced for the following reason – due to nonminimal coupling the backreaction contains a contribution (whose energy density and pressure we denote as \(\rho _Q^m\) and \(p_Q^m\)) that to a good approximation scales like nonrelativistic matter at higher redshifts (while the backreaction is still small), as is evident from Eq. (90). It is more appropriate not to consider this contribution a part of dark energy, but rather to group it together with the background matter fluid. Hence, we have subtracted the contribution from the backreaction that scales away exactly like mater, and determined the equation of state for the remaining contribution in the backreaction fluid. The uncertainty in \(w_Q\) is due to the fact that the amplitude of this matter like contribution is computed approximately to order \({\mathcal {O}}(\xi )\), and that it does not scale exactly like matter, but its behaviour has corrections of the order of \({\mathcal {O}}(\xi N_M)\), where \(N_M\) is the number of efoldings of the matterdominated period. This means we do not know exactly how much of this matterlike fluid to subtract, which introduces sizeable uncertainties in the equation of state only for higher redshifts, but for smaller redshifts (\(z\lesssim 3\)) we get – up to subpercent uncertainties – unique predictions. This implies that our model of dark energy is in principle testable as it gives a definite prediction for the equation of state parameter \(w_{\mathrm{Q}}\) as a function of the redshift.
Even though we know the subtracted matterlike contribution redshifts like ordinary matter on cosmological scales, we do not know how it clusters. If its clustering turns out to be insignificant, its principal effect would be a finite renormalization (reduction) of the Newton constant. In the opposite case the effect would fall under the class of some interacting dark matter/dark energy models, the precise nature of which requires further investigation, which is left for future work.
Let us now compare with current observations. The Planck collaboration papers [16, 17] constrain \(w_{DE} \!=1.019_{0.080}^{+0.075}\) (\(68\%\) CL when the complete Planck data are combined with available data from lensing, baryonic acoustic oscillations (BAO) and \(H_0\) measurements). On the other hand, the \(68\%\) CL error bars on the parameters of the \(w_0w_a\)CDM model [defined by, \(w_{DE}=w_0+(1a)w_a =w_0+[z/(1+z)]w_a=w_0+(1e^{N})w_a\)] are much weaker, \(w_0=0.6\pm 0.25,w_a=\,0.3\pm 0.8\) (with the Planck, BAO, SNIa & \(H_0\) data included) and \(w_0=\,0.6\pm 0.5,w_a=\,1.2\pm 1.2\) (with the Planck, BAO & redshift space distortions data included) (see figure 4 in Ref. [17]), which shows that the constraints \(w_0\) and \(w_a\) depend significantly on the data sets used and hence the quality of the data is still not good enough to place strong bounds on the model. Since these bounds are only slightly modified when the more recent 1 year DES data are accounted for [19], they suffice for our purpose. It is clear that these bounds are not precise enough to place meaningful constraints on the dark energy models shown in Fig. 4. This can be seen even clearer in Fig. 5, in which we observe that the contours of our dark energy model lie comfortably within one standard deviation contours of the \(w_0w_a\)CDM model (which meaningfully constrains dark energy only at relatively small redshifts, \(z\lesssim 2\), or \(N\gtrsim 1\)).
However, with the incipience of upcoming high precision observatories such as Euclid and LSST, the constraints may become strong enough to curb our model, especially when combined with inhomogeneous probes.
6 Discussion and outlook
This paper is one in a series [26, 28, 30], whose goal is to investigate viability of DE models based on quantum fluctuations during inflation. The attractiveness of these models stems from the fact that – when compared to more conventional DE models – they are amenable to additional tests. For example, since the origin of these models can be traced back to inflation, there is an intricate relationship between DE and inflationary observables. Furthermore, since the origin of DE in these models is in quantum fluctuations of matter fields, in general we expect these models to generate inhomogeneous dark energy with calculable amount of inhomogeneities, which can also be used to test this class of models.
In this paper we reanalyze the dark energy model proposed in Ref. [30]. The model utilizes an ultralight, nonminimally coupled scalar field and its dynamics is governed by the action (6). Even though this field is a spectator field during inflation, quantum fluctuations naturally grow large such that, by the end of inflation, they typically reach superPlanckian values. In fact, due to the assumed negative nonminimal coupling, it can take as little as a thousand of efoldings to achieve this feat. This mechanism addresses the question of naturalness of initial conditions – often raised in quintessence DE models – by dynamically generating superPlanckian fluctuations from initially subhorizon quantum fluctuations.
In order to be able to calculate beyond the perturbative regime of Ref. [30], in Sect. 3 we develop a stochastic framework which permits us to solve the problem selfconsistently, also in the regime where the quantum backreaction dominates the background dynamics. To establish the accuracy of our stochastic model, in Sect. 4 we use the stochastic framework to calculate the one loop contribution to the energy density and pressure of quantum fluctuations and demonstrate agreement with the perturbative calculations of Ref. [30] in all of the relevant epochs: inflation, radiation and matter era. This is sufficient to establish perturbative accuracy of the model. The main advantage of the stochastic formulation is in that a selfconsistent solution of semiclassical gravity comes within reach of even modest programmers, as the original set of integrodifferential equations of semiclassical gravity is reduced to a much simpler set of ordinary differential equations with rather facile stochastic sources.
A notable result of our work is that stochastic sources – which are of essential importance for capturing the correct dynamics during inflation – can be neglected during the postinflationary radiation and matter epochs, such that the field dynamics becomes essentially classical. The memory of stochastic sources is still kept in the initial conditions for radiation era and for matter era.
Armed with this formalism we then study how quantum inflationary fluctuations in our model (6) evolve in time and how they affect the background evolution. In particular in Figs. 2 and 3 we compare the evolution of the Hubble rate H(t) and its dimensionless rate of change, \(\epsilon \!=\! \dot{H}/H^2\) (which is equal to the principal slow roll parameter during inflation) in our model with the same in \(\varLambda \)CDM. Our results show that, as the ultralight mass m is taken to be closer to, but still remains smaller than, the Hubble parameter today, the differences become significant and observable by the near future observatories such as Euclid and ELT. From Figs. 4 and 5 we see that the current data are not constraining enough to meaningfully test our the backreaction DE models. However, the near future data (from Euclid, SST, ELT and other planned observatories) will penetrate deeper into the redshift space and will be collecting much more data and thus will be able to constrain the quantum backreaction DE models. Next, we point out that the dependence of \(w_Q=w_Q(z)\) in our quantum backreaction model is quite different from that in typical quintessence models [6]. This owes to the rather unusual feature of the perturbative initial condition in matter era (79, 80), which is in turn dictated by the standard (IRregulated) Chernikov–Tagirov–Bunch–Davies initial condition in inflation (45). Namely, there is a significant negative energy component that initially scales as the dominant background (dark matter) component (cf. Eqs. (79, 80)). This feature is akin to negative energy scenarios studied in [38] and – once the properties of (particlelike) dark matter are mapped out – can be used to test this class of DE models.
While in this paper we only study the background evolution, keeping in mind the quantum inflationary origin of scalar field dark energy in our model, it would be of utmost importance to investigate the growth of structure in the visible and dark energy sector in our DE model, both at Gaussian and nonGaussian levels. Namely, due to the nonminimal coupling the model is nonGaussian and hence we expect it to leave distinct Gaussian and nonGaussian footprints at late times on Universe’s large scale structure. Determining these will require a suitable adaptation of the stochastic formalism for offcoincident correlation functions in inflation [39] or of further developments in the stochastic \(\delta N\) formalism used in [40, 41].
From the theoretical point of view, the model we presented here addresses the issue of naturalness of initial conditions, but there still remains the issue of finetuning in the form of requiring the scalar to have a tiny mass \(m\le H_0\). This issue is not specific to our model, but is a generic feature of all classical quintessence models. It would be of great interest to investigate whether a similar effect can be reproduced within different (interacting) models, via a mechanism that utilises spontaneous symmetry breaking [42] or ’t Hooft’s naturalness hypothesis [43].
Footnotes
 1.
Recent search for different channels of radioactivetype decay of DE with timeindependent decay rates also resulted in upper limits on these rates less than the inverse present age of the Universe only [18].
 2.
In this paper we work in natural units, in which the Planck constant, \(\hbar =1\) and the speed of light, \(c=1\).
 3.
In the calculation there appears a product \(\theta (x)\delta (x)\!\rightarrow \! (1/2)\delta (x)\), which can be justified by using a limiting procedure on a smooth window function.
 4.
One can namely show that UV contributions to \(\rho _Q\) and \(p_Q\) are suppressed as \(\sim H^4\).
 5.
The tachyonic nature of the IR modes during inflation is not problematic, since one can use e.g. preinflationary radiation era to regulate the infrared sector of the theory by a smooth matching of inflationary modes onto the radiation era modes [36].
 6.
The exact form of the exponent in solutions (53) is \(\bigl [3\bigl (1 \!\! \sqrt{1 \!\! 4X/9} \,\bigr )N\bigr ]\), but we have expanded it under the assumption \(X^2N_I \!\ll \! 1\), where \(N_I\) is the total number of efolding. This will be true in our case.
 7.
This estimate assumes inflation at the scale \(H_I\sim 10^{13}\!~\!\mathrm{GeV}\). Since in general \(N_R\gg 1\), our estimates apply also to lower scale inflationary models.
Notes
Acknowledgements
D. G. was supported by the Grant 2014/14/E/ST9/00152 of the Polish National Science Centre (NCN). T. P. acknowledges the DITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW). A. S. was supported by the RSF Grant 161210401.
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