Newtonian cosmology with a quantum bounce

It has been known for some time that the cosmological Friedmann equation deduced from general relativity can also be obtained within the Newtonian framework under certain assumptions. We use this result together with quantum corrections to the Newtonian potentials to derive a set a of quantum corrected Friedmann equations. We examine the behavior of the solutions of these modified cosmological equations paying special attention to the sign of the quantum corrections. We find different quantum effects crucially depending on this sign. One such a solution displays a qualitative resemblance to other quantum models like Loop quantum gravity or non-commutative geometry.


Introduction
It must have come as a surprise to the physics community when McCrea and Milne [1] derived the cosmological Friedmann equations known from general relativity from Newtonian mechanics assuming the existence of expansion. The interest in this derivation has persisted over years [2][3][4][5][6][7][8][9][10] paying attention to refine the Newtonian set up and conclusions. In this paper we go one step further and put forward the question of what kind of modified Friedmann equations would emerge if we include quantum corrections to the Newtonian potential which, of course, is one of the main ingredients in the Newtonian derivation of the Friedmann equations. Such corrections have been known for some time [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. As we will show below, it is straightforward to repeat the McCrea-Milne derivation including these quantum corrections and to arrive at the modified Friedmann equations with terms proportional toh. Depending on the sign of the quantum corrections (and also on the equation of state) different quana e-mail: p.bargueno@uniandes.edu.co b e-mail: s.bravo58@uniandes.edu.co c e-mail: mnowakos@uniandes.edu.co d e-mail: davide.batic@uwimona.edu.jm tum effects emerge with one of them resembling qualitatively those of other quantum models. This similarity consists in the behavior of the scale factor R describing a universe which has been contracting in the past, reaching a minimal value of R and expanding again after the bounce. For the other sign of the quantum effect, the behavior is qualitatively different as the universe spontaneously appears at R min close to the Planck length and starts expanding from this point. The primary expansion is accelerated, reminding us of inflation. We do not claim that our modified Friedmann equations give necessarily the correct description of a quantum universe, but it is certainly worthwhile to consider them. For one they give the right Friedmann equation when no quantum corrections are included and as such could contain the right clues and hindsights when we include the latter. Second, we think it is timely to venture one step more in the area of Newtonian cosmologies.
The paper is organized as follows. In the next section we give a brief account of the derivation of the standard Friedmann equation within the Newtonian framework. Next we introduce the quantum corrections and derive the modified Friedmann equations. In Sect. 4 we study the behavior of these new cosmological equations varying the sign of the quantum corrections and choosing different equations of state.

Friedmann equations from Newtonian dynamics
There exist different derivations of the cosmological Friedmann equation from the Newtonian dynamics [1][2][3][4][5][6][7][8][9][10] and although conceptually such derivations differ [3,10], in the end they all arrive at the same Friedmann equations. We therefore take here the simplest and original point of view which starts by taking into account the expansion of the universe. This is done by writing where H is the Hubble parameter and R is a measure of distance. The next step considers the total energy for an object (say, a galaxy) of mass m, which reads Writing the mass inside the sphere of radius R as M = 4 3 π R 3 ρ where ρ is the density of the universe, Eq. (2) takes the form 2E Since E and m are constants we define k ≡ 2E m and obtain the first Friedmann equation To get the second Friedmann equation the argument goes as follows: when the volume V of the universe expands by dV , the pressure does work equal to pdV, which decreases the energy in V by that amount. Using energy-mass equivalence one obtains On the other hand this is the well-known conservation law which can be cast in the convenient form If we write the first Friedmann equation as By taking a derivative of this equation and replacing R dρ dt from the conservation equation one arrives at the second Firedmann equation From our point of view the crucial ingredient is how the Newtonian potential enters the derivation. Quantum corrections to the latter are known and it makes some sense to try to rederive the cosmological equations by taking this correction into account.

Quantum corrected Friedmann equations
Several authors have obtainedh corrections to the Newtonian potential by taking gravity as an effective theory and performing one-loop graviton calculations [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Sometimes the results of the quantum corrected Newtonian potential is given in a different form, where λ andγ are parameters which take different values depending on the author(s). Partly, we can attribute the reason for these discrepancies to the precise coordinate definition used in the calculation [27]. The question about the ambiguity of this potential due to the lack of clarity on the coordinates has also been risen in some related articles [16,27,29]. It is argued that a redefinition r → r = r (1 + aG M/r ) would change the parameter λ without affecting the observables. The general consensus is that we can write the corrected potential as given in Eq. (9). The aforementioned reparametrization freedom still cannot account for all the discrepancies of the different γ q 's found in the literature. A number of errors have been identified [11,16], but it is not clear if this accounts for all the different values available. It is therefore fair to list some of the results (see Table 1). In the table we have collected the different values for γ q , which also vary in sign (we will see that the sign plays the most important role in the cosmology derived from these corrections).
Having established the quantum correction we can proceed as before. The total energy receives a new contribution due to the quantum correction in the Newtonian potential, i.e., Introducing again the density ρ and the Planck length l p = Gh c 3 the above equation is equivalent to With the help of (1) the first Friedmann equation with an h-correction can be given as The second corrected Friedmann equation follows from the fact that the conservation law (6) remains unchanged. We can proceed as before to obtain We consider Eqs. (13) and (14) as the quantum corrected Friedmann equations derived withing the framework of Newtonian mechanics. We will show below that they imply a quantum bounce or in other words the initial singularity at R = 0 is avoided. For the sake of comparison with other models and a better understanding of similarities and differences between the standard Friemdann equations and Eqs. (13) and (14) we can re-cast the latter in different forms. By re-introducing the cosmological constant and taking an initially flat universe with k = 0. We have then We note that it is sufficient to puth → 0 to recover the Friedmann equations from general relativity. Making use of the standard definitions ρ crit (t) = 3H 2 8π G and ρ vac = 8π G the first Friedmann equation (with the cosmological constant and theh corrections) is simply We could also define a new ρ crit , namelỹ as well as˜ m = ρ ρ crit . Then we simply have If we assume the equation of state of radiation the conservation law gives us with a = R/R 0 . Then it is easy to see that the first Friedmann equation becomes with β = γ q l 2 p /R 2 0 . In the case of positive γ q (positive β ) it would make sense to introduce a critical densitỹ such that H = 0 when ρ =ρ cr . Although we will make a detailed comparison with other models at the end of the paper we notice already here that in loop quantum gravity the expression is similar, i.e., This does not imply that there is quantum bounce only if γ q is positive. Indeed, in the next section by solving explicitly the Friedmann equations that even in the case of γ q < 0 the universe has no singularity at R = 0. Different scenarios are possible, mostly depending on the sign of γ q and the equation of state. Some mathematical features of classical and quantum universes are common. In the following steps we will briefly discuss two solutions of the standard Friedmann equations without quantum corrections. First let us consider a toy universe with = 0, γ q = 0 and k = 0 filled with radiation. It is an easy exercise to show that the solution to the Friedman equations for a = R/R 0 reads (a 2 − 1)/2 = ±τ ≡ √ 8π Gρ 0 /3(t − t 0 ). The two branches correspond to with a + (0) = a − (0). The branch a − is decreasing whereas a + is growing in time (see Fig. 1). It would be incorrect to try to avoid the singularity by gluing the two branches at τ = 0 discarding the rest. This would lead to an ambiguity in the solution as we would have four possible solutions. This tells us that we can only glue the two branches if we arrive at a unique smooth solution. Second, we take the radiation case with = 0, γ q = 0 and k = 1. Due to the Hubble constant can be zero, but this corresponds to a local maximum as we will see. Indeed, the solutions are which we plotted in Fig. 1. There is a restriction on R 0 in form ρ 0 and on R given us the position of the maximum of a. The latter is R ≤ R max = 8π G 3 ρ 0 R 2 0 making the Hubble parameter vanish. In the case of quantum universes as derived in this paper we will see that H = 0 will either indicate a local minimum or an absolute minimum.

Newtonian quantum universes
To find the effect of the new term proportionalh in the Friedmann equations we start from the energy conservation equation and use first an equation of state (EOS) of the form where γ is not to be confused with γ q . We can solve for ρ in terms of R, namely, the standard solution is such that ρ(R 0 ) = ρ 0 . Inserting this into the first Friedmann equation with k = 0, i.e., In the integral form this reads The behavior of the solution depends strongly on the sign of β (which is the same as the sign of γ q ) and on the equation of state (γ ). It therefore makes sense to discuss the different cases separately.
In this case the solution can be given in terms of standard functions, namely where D takes care of the initial value R(t 0 ) = R 0 . After implementing the initial value we obtain  Figure 2 shows the solutions according to Eq. (31). To the right and left of the straight line τ = 0 we have the different branches due to the ± signs in (31). As long as |β | < 1 we get always a non-singular universe: the expanding universe starts at a nonzero value R min < R 0 (below the line a = 1) determined by the single-valuedness of a. The mirror universe in such a case is a contracting one ending at the same R min This is shown in Fig. 2 for β = −0.6. For |β | > 1 we obtain a singular universe starting at R = 0 and ending at some R max > R 0 following one of the signs (the other sign gave the mirror collapsing universe starting at R max and contracting to zero. The critical point seems to be β = −1. If we choose the solution according to one sign and the single-valuedness of the solution we would end up with two expanding universes, one starting from zero and expanding up to R 0 , the other starting from R 0 and expanding up to infinity (a similar picture emerges for the collapsing branch). However, these two curves merge smoothly at R 0 and therefore we can construct a unique forever expanding universe starting at zero (and similarly the mirror image). We conclude that for negative β with |β | < 1 the quantum effect is that the universe starts at a finite value of R 0 . In Sect. 5 we solve these equations by including the cosmological constant. The next section is devoted to the comparison with other quantum models. In the last section we draw our conclusions.

Dust (γ = 1)
The integral to solve is now which can be rewritten in terms of a = R/R 0 as In order to solve the integral appearing in the above expression we first rewrite it as follows: Using 230.1 in [30] yields At this point, it is convenient to split the integral above as Both integrals on the r.h.s. of the above expression can be computed by means of 239.00 in [30] and we find that a 1 dτ τ (τ 2 + |β |) where F(ϕ, k) denotes the elliptic integral of the first kind with amplitude and modulus represented by ϕ and k, respectively. Finally, we obtain The results are plotted in Fig. 3. Following one branch, i.e. one sign, we conclude that all universe are singular as they start at zero (or end at zero). We conclude that in order to get a non-singular universe in the case of negative β the equation of state plays a crucial role. Needless to say that at the beginning of the universe a relativistic equation of state is preferred. Fig. 3 The same as in Fig. 2, but for the case of dust. See text for a detailed discussion The solution is now given by where C is a constant. In terms of a = R/R 0 and implementing the initial value explicitly it reads We see that in general the case with γ q > 0 imposes a certain limit upon the value of R, namely R 2 ≥ R 2 min ≡ β or, equivalently, a 2 ≥ β . This is clearly reflected in Fig. 4 where we have plotted the solutions.
Since according to (40) we obtain a solution if β ≤ 1 we note that all universes start (or end) at R min as expected as long as β is smaller than one. In the case that β = 1 R min is the position of the local minimum. This minimum joins the two branches with different signs smoothly and gives a unique solution. This universe is then different from the others as it "comes" from infinity, reaches a minimum and expands again. We end up with the computation of the following integral: If we apply 230.1 form [30] to I (a), we find that Since the integrand is real, we must require that 0 < β ≤ 1.
It is convenient to rewrite the integral above as follows: The integrals appearing on the r.h.s. of the above expression can be evaluated by means of 237.00 in [30] and we obtain where F is the elliptic integral of the first kind. Hence, the integral I (a) can be computed to be The results are presented in Fig. 5. In the case of positive β there is not much difference if we change the equation of state. Therefore, the interpretations are similar to the radiation case and β = 1 is again a special case. In passing, we make a remark on the curvature scalar. The general theory of relativity is defined by the Einstein equations. Our derivation of the modified Friedmann equation lies outside these equations, but we can still interpret a(t) derived from the Newtonian scheme as a model for a(t) appearing in the Friedmann-Robertson-Walker-Lemaitre (FRWL) metric The latter is a metric based on symmetries as homogeneity and isotropy. In such a case, we are also allowed to calculate the curvature scalar R. Alone from the FRWL metric one obtains [31] Given our modified Friedmann equations for k = 0 and = 0 for the radiation case (ρ(a) = ρ 0 a −4 ), the Ricci scalar takes the simple form where β = . As long as a = 0 we do not have an initial singularity. We know that the case that reproduces a smooth bounce is the one in which we have β > 0 (β > 0 and thus γ q > 0) and this particular case has a bound for the value of β , namely β ≤ 1, thus we have an upper bound on the curvature given by An upper bound expression for the curvature is also found in loop quantum cosmology [32] where the curvature can never be larger than 31/l 2 p .

Cosmological perturbations
The question how to handle cosmological perturbations with the Newtonian (or related) scheme is an important one. Indeed, it has been addressed in a number of papers [33][34][35][36][37][38] without invoking theh-correction present in our Friedmann equations. The adequate theory of cosmological perturbations consistent with general relativity requires a set up which goes beyond the Newtonian approach presented in this paper even in the caseh → 0. This is to say, the derivation of the Friedmann equation does not proceed as we have outlined it in the beginning of our paper, but is based on a set of three equations [39]. The first one is a version of the standard Poisson equation for the gravitational potential, the second one a version of the continuity equation for the mass density and supplemented by an equation of motion (Euler equation). In the above we used the phrase "version" since several different schemes seem to co-exist (see [38] for a brief review). What we could call the purely Newtonian approach starts with ∂ρ ∂t and upon setting ∇ p = 0 (justified by homogeneity and isotropy) leads to the Friedmann equations for matter as known from general relativity. However, the perturbation theory does not agree with general relativity [34,40]. For this reason a Neo-Newtonian approach has been attempted which is also not unique. The Neo-Newtonian type I approach is based on the following equations: which again, given the fact that ∇ p = 0 give rise to the same standard Friedmann equations with k = 0 as well as the continuity equation. When perturbation theory is applied with this scheme the results agree with corresponding ones in general relativity only if we restrict ourselves to the large scale regime [36]. A better prediction is provided by the Neo-Newtonian theory of type II based on a slightly different set of equations: which, given our argument for the choice of ∇ p = 0, just yields the classical Friedmann and continuity equations for k = 0. When perturbation theory is applied to the type II Neo-Newtonian approach, the results are consistent with general relativity under the condition of adiabatic pressure perturbations and a constant equation of state parameter [37]. Finally, the Hwang-Noh approach starting from [33] ρ leads to the standard continuity equation and a modified Friedmann equation of the forṁ There exists also a modified Hwang-Noh [36] approach whose Friedmann equation readṡ The results of perturbation theory in the Hwang-Noh schemes match the general relativistic perturbations in the small scale limit [36].
Having briefly outlined the state of art of the cosmological perturbations within a Newtonian or Neo-Newtonian scheme the question arises which one is suitable when we switch on theh-corrections. Moreover, it is apriori not obvious how we could implement these quantum corrections within the Neo-Newtonian theory. Since we are talking here about corrections we suspect that the outcome of the analysis of the perturbations will be similar to the results obtained without the quantum corrections. But a detailed analysis which is beyond the scope of the present paper would be in order.

The case with a cosmological constant
It is of some interest to treat the full Friedmann equation with the quantum corrections and spatial flatness. Including the cosmological constant , the Friedmann equations read We will solve this case perturbatively.

Radiation (γ = 4/3)
The integral to be solved in the radiation case and nonzero positive cosmological constant is which we can also rewrite this in terms of a(t) and ρ vac ≡ 8π G as follows: Since 1, the integrand appearing in I (a) can be expanded in powers of the small parameter . Hence, we have Taking into account that and we find that a(t) is given at the first order in by the following expression: 6.2 Dust (γ = 1) In this case the integral to be solved has the form 8π Expanding the integrand in powers of the small parameter yields First of all, observe that with I (a) given by (A.9). Let Then the solution can be written as The way to compute G is long and we give all the details in the appendix.

Loop quantum gravity and Friedmann equations
For this section we will follow closely Refs. [41,42]. Let us consider the mini-superspace approach to classical general relativity for the k = 0 case. After defining appropriate Ashtekar variables, c and p, 1 which inherit the Poisson bracket given by {c, p} = 8 3 π Gβ B I (β B I is the Barbero-Immirzi parameter 2 and G is Newton's constant), the gravitational Hamiltonian constraint acquires the usual form The contribution for a massless and free scalar field with a Hamiltonian constraint given by Therefore, defining the Hubble parameter as H =ṗ/(2 p) and the matter density for the scalar field as ρ = p 2 φ /(2| p| 3 ), we get the total Hamiltonian constraint as 16π G H G + H φ , from which the usual Friedmann equation, which predicts the usual big-bang singularity (the volume of the universe goes to zero at t = 0) can be recovered.
To proceed with quantization we have to promote H G to a quantum operator. The impossibility lies in the fact that there is not quantum operator associated to c. The usual way to circumvent this problem is called polymerization (see the [42] and references therein for technical details on the procedure).
The important point is that the equations of motion derived from certain H eff , given byṗ = {p, H eff }, can be expressed as a modified Friedmann equation, in the form where the critical density is given by The key point is that, in essence, the modified Friedmann equation leads to a non-singular evolution. Moreover,ȧ vanishes at ρ crit and the universe bounces. In the limit μ 0 → 0, which corresponds to Gh → 0, the critical density becomes infinity and the classical singularity appears. At this point some comments are in order. First of all, let us recall that theh−corrections to the Friedmann equation for the k = 0 case can be written as Therefore, some similarities and differences with the LQGcorrected Friedmann equation, given by Eq. (73), are present. For positive γ q , as commented before, a constant critical density can be obtained within our model considering radiation, although in this case the Friedmann equation is not completely similar to the LQG case. But qualitative similarities persist. For negative γ q given the different sign which appears in Eq. (75) compared to that of Eq. (73), an immediate comparison between both predictions for the critical density is not evident. Nevertheless we find a quantum effect for negative γ q as the expanding universe starts at a finite nonzero value for R. The crucial sign of γ q will get also reflected in comparison with models other than loop quantum gravity.

The generalized uncertainty principle, Snyder-deformed algebra and Friedmann equations
As we have briefly commented, quantum corrections to the Friedmann equation can be implemented by considering Planck-scale modifications to the Hamiltonian constraint, which lies at the heart of LQG. However, a different approach can be considered. What is the effect, if there is any, of deforming the usual Poisson brackets structure instead of deforming the Hamiltonian constraint. Without introducing Ashtekar variables, interestingly, the generalized uncertainty principle (GUP) provides a theoretical framework where this deformation appears and a consequence of the existence of a minimum length [43].
The starting point is the formulation of ordinary canonical dynamics in FRW geometries. This dynamics is summarized in the scalar constraint Here, N = N (t) is the lapse function, λ is a Lagrange multiplier and is the momenta conjugate to N . In the GUP framework, up to the first order in the deformation parameter, α, the new Poisson bracket is {a, p a } = 1 − 2αp a . Using the new Hamilton's equations and again Eq.
In particular, for the flat case (k = 0), the modified Hubble equation reads At this point, it is important to recall that GUP gives place to a minimum length which, is this case, and taking α > 0, is associated with the scale factor, a(t). Therefore the critical density given by remains finite (this situation is reminiscent of the appearance of a remnant mass in the GUP case for α > 0). At this point it is interesting to consider some specific models for matter in Eq. (79). A similar result can be obtained by invoking Snyder's noncommutative space, which gives place also to a deformed Heisenberg algebra. In particular, the authors of Ref. [45], after replacing the usual Poissonian structure between a and p a by {a, p a } = 1 − αp 2 a , obtained the following modified Friedmann equation: Again considering the flat case the authors deduce where ρ c = 2π G 3α ρ p . In this last step it is also assumed, as a consequence of the deformed algebra, the existence of a minimum length. Let us note again that α > 0 is necessary to smooth out the singularity.
After considering Eq. (81) for radiation and dust matter, we obtain Apart from comparing the critical density of our model, both in the dust and radiation cases, with those present in the previously mentioned approaches, it would also be interest- The plausibility of the matter content is usually addressed with the help of the energy conditions. Introducing the variable ω = γ − 1, the energy conditions corresponding to the fluids with an equation of state of the form p = ωρ, for which ω = c 2 s (the sound speed associated with this equation of state), are [46]: In particular, our model reproduces Snyder's corrections to the Friedmann equation when ω = 1. This corresponds to a ultrastiff or incompressible fluid which has been proposed as a possible description of the very early universe [46]. Moreover, this fluid is equivalent to a free massless scalar [47].
For this fluid, all the energy conditions are satisfied. In case of dealing with a ω = 5/3 fluid, the GUP case is reproduced. In this situation, the dominant condition is violated. Finally, our model reproduces the LQG-corrected Friedmann equation when ω = −1/3. Interestingly, again in this situation, which corresponds to certain dark-energy model [48], all the energy conditions are satisfied.

Entropy corrections and Friedmann equations
In recent years, quantum corrections to the Bekenstein-Hawking entropy have been shown to be either logarithmic or power-law. While the first kind of corrections usually arises from a minimum length scenario (such as LQG, GUP, etc.) (see [43] and references therein), the second one deals with the entanglement of quantum fields inside and outside the horizon [49]. Moreover, the deep connection between gravity and thermodynamics, reinforced by Jacobson [50] and Padmanabhan [51], made some authors [52] derive modified Friedmann equations by using corrections to the entropy in addition with the ideas explored in Refs. [50] and [51]. In addition, the entropy approach developed by Verlinde [53] has been employed [54], assuming power-law corrections to the entropy, to obtain corrections to Friedmann equations.
Although the authors of [52] derive corrections to the Friedmann equations, they depend on the detailed gravitythermodynamics connection. Even more, their main result (regarding our work), which is for the flat case, can be expressed as their Eq. (10), which reads where g is a complicated function of α, which is either the parameter that goes with the log-correction (for instance, α = −1/2 in LQG) or the power of the entropy correction, and H . However, in spite of the formal similitude between Eqs. (75) and (83), the dependence of Eq. (83) on H makes the comparison between both approaches very difficult to establish, unless some specific matter contents are considered.
In the case of power-law entropic corrections [54], the key point is to notice that the Newtonian force gets corrected as where r c is some crossover scale model-dependent and α is, as in the previous case, the power of the entropy correction. For the flat case, the authors of Ref. [54] obtain where, assuming again an equation of the state of the form p = ωρ, β P L is given by β P L = α spite of the similarities, Eqs. (75) and (85) are not equivalent under any circumstances (α = 0 → β P L = 0).

Conclusions
In this paper we have attempted a quantum cosmology based on the quantum corrections to the Newtonian potential and repeating the derivation of Friedmann equation within the Newtonian formalism. The latter is well known to reproduce the correct Friedmann equations. This is one or the reasons why we believe that the quantum corrected equations might hint toward what one might call the full fledged quantum cosmology. Indeed, with a certain choice of the sign of the quantum correction we qualitatively agree with other models of a quantum universe. In such a case a collapsing universe bounces off a minimum length proportional to the Planck length and begins to expand again. Other quantum effects, for the opposite sign of this correction, manifest themselves in a spontaneously created universe at nonzero scale factor again close to the Planck length. We believe that, at least qualitatively, these results go in the right and expected direction.

Appendix: Evaluation of integrals
We need to compute five integrals. Employing 230.01 in [30] yields Applying 230.01 in [30] to the last integral in the above expression leads to Concerning the last integral appearing on the r.h.s. in the expression above, we rewrite it as follows: and by applying 237.04 in [30] we find that Here, ncu = 1/cnu where cn is one of the Jacobian elliptic functions and the associated amplitudes and moduli are given by Invoking 313.02 in [30] we find that du nc 2 u = 1 where E denotes the elliptic integral of the second kind, k = 1 − k 2 is the complementary modulus, dn u and tn u = snu/cnu are the Jacobi elliptic functions. Taking into account that 111.00 and 122.01 in [30] imply that E(0, k) = 0 = E(0, k), dn0 = 1, tn 0 = 0 and, moreover, k = k = 1/ √ 2, we obtain On the other hand, 121.01 in [30] implies that u 1 = F(ϕ, k) and u 1 = F( ϕ, k) with F denoting the elliptic integral of the first kind. Moreover, 120.01 allows also to find that dnu 1 = 1 − k 2 sin 2 ϕ = a + β 2a , Furthermore, by means of 121.00 in [30] we obtain Hence, we conclude that with u 1 and u 1 defined as in (A.5). By means of 316.02 in [30] we find that This completes the derivation.