Perturbative quantum damping of cosmological expansion

Perturbative quantum gravity in the framework of the Schwinger-Keldysh formalism is applied to compute lowest-order corrections to the actual expansion of the Universe described in terms of the spatially flat Friedman-Lematre-Robertson-Walker solution. The classical metric is approximated by a third order polynomial perturbation around the Minkowski metric. It is shown that the quantum contribution to the classical expansion, although extremely small, has damping properties (quantum friction), i.e. it slows down the expansion.


Introduction
The aim of our work is to explicitly show the appearance of quantum generated damping or quantum friction, i.e. slowing down, in the present (accelerating) expansion of the Universe. In general, quantum corrections to classical gravitational field can be perturbatively calculated in a number of ways. First of all, it is possible to directly calculate quantum (one-loop) corrections to classical gravitational field from the graviton vacuum polarization (self-energy), in the analogy to the case of the Coulomb potential in QED (for example, see Berestetskii et al. [1]), the so-called Uehling potential. Such a type of calculations has been already performed for the Schwarzschild solution by Duff [6], as well as for the spatially flat Friedman-Lemaître-Robertson-Walker (FLRW) metric (Broda [4]). Another approach refers to the energy-momentum tensor calculations, and it has been applied to the Newton potential (for example, see Bjerrum-Bohr et al. [3], and the references therein), to the Reissner-Nordström and the Kerr-Newman solutions (see Donoghue et al. [5]), and to the Schwarzschild and the Kerr solution metrics (see Bjerrum-Bohr et al. [2]). Yet another approach uses the Schwinger-Keldysh (SK) formalism to the case of the Newton potential (e.g., see Park and Woodard [8]). It is argued that only the SK formalism is adequate for time-dependent potentials, hence in particular, in the context of cosmology (e.g., see Weinberg [9], and references therein). As we aim to perturbatively calculate corrections to the spatially flat FLRW metric, we should use the SK formalism, because this is exactly that case (time-dependence of gravitational field) the SK approach has been devised for.
The corrections we calculate are a quantum response to the spatially flat FLRW solution which is described by a small perturbation around the Minkowski metric. Since such a type of calculations is usually plagued by infinities, we confine ourselves to the classical perturbation given by a polynomial of the third degree. Moreover, to avoid infinities on intermediate stages of our calculations, time derivative of the convolution of the time propagator with the perturbation of the metric should be performed in a suitable order. The final result is given in terms of the present time quantum correction q Q 0 to the deceleration parameter q 0 . Interestingly, it appears that q Q 0 is positive, although obviously, it is extremely small.

Quantum damping
Our starting point is a general spatially flat FLRW metric with the cosmological scale factor a (t). To satisfy the condition of weakness of the perturbative gravitational field h µν near our reference time t = t 0 (where t 0 could be the age of the Universe -the present moment) in the expansion the metric should be normalized in such a way that it is exactly Minkowskian for t = t 0 , i.e. a 2 (t) = 1 + h (t) , h (t 0 ) = 0.
Let us note the analogy to the Newton potential (∼ 1/r), where the "reference radius" is in spatial infinity, i.e. r 0 = +∞. Then, in the block diagonal form, To obtain quantum corrections to the classical gravitational field h C µν (x), we shall use the one-loop effective field equation derived by Park and Woodard [8] , and the mass scale µ is coming from the renormalization procedure (see Ford and Woodard [7]). Here κ 2 = 16πG N , where G N is the Newton gravitational constant. The operator D (the Lichnerowicz operator in the flat background) is of the form whereas for the minimally coupled massless scalar field For conformally coupled fields we have D instead of D, where Since the metric depends only on time, we can explicitly perform the spatial integration in (5) with respect to x ′ obtaining the integral kernel (time propagator) For the time-dependent metric of the form the action of the operators D, D and D is given by whereas respectively. There are no mixing of diagonal and non-diagonal terms, and the empty blocks mean expressions which can be non-zero, but they are inessential in our further analysis. Thus, (5) assumes the simple form where the integral kernel K is given in (7), and the convolution "⋆" is standardly defined by One should note that due to "diagonality" of (4) and (8)(9)(10), no non-diagonal terms of the metric enter (11). Now, observing that also the limit of integration in (12) depends on t, one can easily derive the following differentiation formula for the convolution Using symmetry between K and F (see (12)), it is possible to distribute differentiation in (13) in several different ways. For practical purposes, i.e. elimination of possible singular terms on intermediate stages of our calculations, the most convenient form of the eighth derivative is the symmetric one, i.e.
To finally prevent the appearance of possible infinities, i.e. primary UV infinities in the propagator, signaled by µ, as well as divergences in the convolution, which could come from singularities in the kernel (time propagator) K, we assume the following third order polynomial form of the classical metric Henceforth, for simplicity, we use the dimensionless unit of time, τ ≡ t/t 0 , instead of t. Non-singularity of Eq.16 proofs that (14) and (15), have been properly selected. In fact, our choice is unique. First of all, let us observe that for k > 2, is singular, and d k dt k K (t) , for k < 4, is µ-dependent. Then, the only possibility to avoid such troublesome terms admits exactly the products in the second part of the sum in (14). In turn, to nullify the unwanted first part of the sum in (14), h C (τ ) should be of the form (15). The term before the summation sign in (14) vanishes, and thus, it is inessential. Actually, the classical metric (15) does not belong to any favorite family of cosmological solutions, perhaps except for the linear case (h 0 = −1, h 1 = 1, h 2 = h 3 = 0), corresponding to radiation. In fact, a physically realistic metric is not precisely given, for example, by the matter-dominated cosmological scale factor a (τ ) = τ 2/3 , because firstly, the character of cosmological evolution depends on the epoch (time τ ), and secondly, it is "contaminated" by other "matter" components, e.g. radiation, and possibly, dark energy. Therefore, we should consider (15) as a phenomenological description, approximating actual cosmological evolution on the finite time interval τ ∈ [τ 0 , 1], 0 ≤ τ 0 < 1.
Inserting (7) and (15) to (14) we derive, by virtue of (11), the second order differential equation which can be easily integrated out with respect to τ , yieldinġ and where λ ≡ κ 2 /32π 2 t 2 0 ≈ 1 2 · 10 −46 . As a physical observable we are interested in, we take the deceleration parameter The quantum contribution to the deceleration parameter, namely, the lowest order contribution of (16-18) to (19), i.e. q (τ ) = q C (τ ) + q Q (τ ) + O λ 2 , reads To approximate the cosmological evolution by the (four-parameter) phenomenological metric (15), we need four conditions. First of all, we impose the following two obvious boundary conditions h C (0) = −1 and h C (1) = 0, corresponding to a 2 (0) = 0 and a 2 (1) = 1, In this place various different further directions of proceedings could be assumed, depending on the question we pose. Then, let us study the quantum contribution to the actual cosmological evolution. By virtue of (7) and (11), the "effective" time propagator determined by the sixth order derivative of the kernel K, behaves as (∆t) −3 , which follows from, e.g., dimensional analysis. Thus, the largest contributions to the quantum part of the metric h Q (τ ) are coming from integration (12) in the vicinity of τ ≈ 1, because of the large value of (τ − 1) −3 . Therefore, we impose the next two additional conditions at the dominating point τ = 1. Namely, h C is supposed to yield the observed value of the Hubble constant and the observed deceleration parameter q 0 = q C (1). Solving (19), (21) and (22) for h k (k = 1, 2, 3), we obtain To estimate only the order and the qualitative behavior of the present time quantum contribution to the accelerating expansion of the Universe, it is sufficient to insert to (23) the following crude approximation: H 0 = 1 and q 0 = − 1 2 . Now Finally, by virtue of (16-18) and hence (see (20))

Summary
In the framework of the SK (one-loop) perturbative quantum gravity, we have derived the formula (24) expressing the (approximated) value of the present time quantum contribution q Q 0 to the deceleration parameter q 0 . The present time quantum contribution, q Q 0 ∼ +10 −46 , is positive but it is negligibly small in comparison to the observed (negative) value of the deceleration parameter, q 0 ≈ − 1 2 . Therefore, we deal with an extremely small damping (slowing down) of the expansion of the Universe, which is of quantum origin (quantum friction).
One should also stress, that in the course of our analysis, because of some technical difficulties, we have been forced to confine our work to a particular case: a conforming with the FLRW form polynomial (of the third degree) perturbation around the Minkowski metric -to avoid short distance infinities; minimally coupled massless scalar field and conformally coupled fields (trivial contributions) only, without (virtual) gravitons -to avoid calculational limitations.
Finally, it would be desirable to compare our present result to our earlier computation (see Broda [4]), where we have obtained an opposite result, i.e. repulsion instead of damping. First of all, one should note that non-SK approaches are acausal, in general, for finite time intervals, as they take into account contributions coming from the future state of the Universe. This follows from the fact that the Feynman propagator has an "advanced tail", which is not dangerous for (infinite time interval) S-matrix elements. Besides, the present work concerns scalar field contributions, whereas the result of the previous paper is determined by graviton contributions. Nevertheless, in the both approaches, quantum contributions are trivial for conformal fields, which well corresponds to conformal flatness of the FLRW metric.
Supported by the University of Łódź grant.