Holographic bounce

TL;DR

This paper explores how holographic principles can naturally lead to nonsingular bouncing cosmologies, with UV corrections controlling the bounce and connections to modified gravity models like F(R).

Contribution

It introduces a holographic bounce model using IR cutoffs, UV corrections, and extended cutoffs that can replicate F(R) gravity scenarios.

Findings

Horizon decrease increases holographic energy density at bounce scales.

UV corrections enable nonsingular bounce solutions with controlled minimum scale factor.

Extended IR cutoffs can reproduce F(R) gravity bounce scenarios.

Abstract

We investigate the bounce realization arising from the application of the holographic principle in the early universe, inspired by its well-studied late-time application. We first consider as Infrared cutoffs the particle and future event horizons, and we show that the decrease of the horizons at early times naturally increases holographic energy density at bouncing scales, while we additionally obtain the necessary null energy condition violation. Furthermore, adding a simple correction to the horizons due to the Ultraviolet cutoff we analytically obtain improved nonsingular bouncing solutions, in which the value of the minimum scale factor is controlled by the UV correction. Finally, we construct generalized scenarios, arisen from the use of extended Infrared cutoffs, and as specific examples we consider cutoffs that can reproduce $F(R)$ gravity, and the bounce realization within it.

Equations

$$\rho =\frac{3c^2}{{\kappa }^2{L}_{\mathrm{IR}}^2},$$

$$d{s}^2=-d{t}^2+a^2(t)\left(\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\Omega }^2\right),$$

$$L_p\equiv a{\int }_0^t\frac{dt}{a},L_f\equiv a{\int }_t^{\infty }\frac{dt}{a}.$$

$$H^2=\frac{{\kappa }^2}{3}\rho ,$$

$$H^2=\frac{c^2}{L_{\mathrm{IR}}^2}.$$

$$\frac{d}{dt}\left(\frac{1}{a}\sqrt{\frac{c^2}{H^2}}\right)=\frac{m}{a},$$

$$a(t)=a_0{\left(t-t_0\right)}^{\frac{c}{c\pm m}},$$

$$L_{\mathrm{IR}}\to \sqrt{L_{\mathrm{IR}}^2+\frac{1}{{\Lambda }_{\mathrm{UV}}^2}}.$$

$$\frac{m}{a}=\frac{d}{dt}\left(\frac{1}{a}\sqrt{\frac{c^2}{H^2}-\frac{1}{{\Lambda }_{\mathrm{UV}}^2}}\right),$$

$$\dot{H}=-\frac{H^3}{c^2}\left\{m\sqrt{\frac{c^2}{H^2}-\frac{1}{{\Lambda }_{\mathrm{UV}}^2}}+H\left(\frac{c^2}{H^2}-\frac{1}{{\Lambda }_{\mathrm{UV}}^2}\right)\right\},$$

$$\frac{\sqrt{c^2-\frac{H^2}{{\Lambda }_{\mathrm{UV}}^2}}}{(c^2-1{)}^{\frac{3}{2}}{\Lambda }_{\mathrm{UV}}^2\sqrt{H^2-c^2{\Lambda }_{\mathrm{UV}}^2}}{\tan }^{-1}\left[\frac{mH}{\sqrt{c^2-1}\sqrt{H^2-c^2{\Lambda }_{\mathrm{UV}}^2}}\right]$$

$$+\frac{m\sqrt{c^2-\frac{H^2}{{\Lambda }_{\mathrm{UV}}^2}}-c^2}{c^2(c^2-1)H{\Lambda }_{\mathrm{UV}}^2}+\frac{{\tanh }^{-1}\left[\frac{H}{\sqrt{c^2-1}{\Lambda }_{\mathrm{UV}}}\right]}{(c^2-1{)}^{\frac{3}{2}}{\Lambda }_{\mathrm{UV}}^3}=-\frac{t}{c^2{\Lambda }_{\mathrm{UV}}^2}+C_0,$$

$$a(t)=a_0{e}^{\int H(t)dt},$$

$$L_{\mathrm{IR}}=L_{\mathrm{IR}}\left(L_p,{\dot{L}}_p,{\ddot{L}}_p,\cdots ,L_f,{\dot{L}}_f,{\ddot{L}}_f,\cdots ,a,H,\dot{H},\ddot{H},\cdots \right).$$

$$L_{\mathrm{IR}}=\frac{1}{6\alpha {\dot{H}}^2{a}^6}{\int }^{t}dt{a}^6\dot{H},$$

$$0=-3\left(H^2-H_0^2\right)-\alpha \left(-108H^2\dot{H}+18{\dot{H}}^2-36H\ddot{H}\right).$$

$$0=\frac{F(R)}{2}-3(H^2+\dot{H})F^{\prime }(R)+18(4H^2\dot{H}+H\ddot{H})F^{\prime \prime }(R),$$

$$F(R)=R-\alpha R^2-6H_0^2.$$

$$a(t)=a_B{\mathrm{e}}^{\frac{\lambda }{2}{t}^2},$$

$$\left(H_0^2-6\alpha {\lambda }^2\right)-\left(1-36\alpha \lambda \right){\lambda }^2{t}^2=0,$$

$$H_0^2=\frac{1}{216\alpha }$$

$$\lambda =\frac{1}{36\alpha }=6H_0^2.$$

$$L_{\mathrm{IR}}={\int }^{t}dt\left(\frac{1}{\beta }-\frac{2\beta Ba(t{)}^{-\frac{1}{\beta }}}{H(t{)}^2}\right),$$

$$\dot{H}=2\beta B{a}^{-\frac{1}{\beta }}-\frac{1}{\beta }{H}^2.$$

$$H=\frac{2\beta Bt}{A+B{t}^2},$$

$$a(t)=a_B{\left(A+B{t}^2\right)}^{\beta },$$

$$L_{\mathrm{IR}}\to \frac{1}{2\beta }{\int }_0^{t}dt-{\int }_t^{\infty }dt\left(\frac{1}{2\beta }-\frac{2\beta Ba(t{)}^{-\frac{1}{\beta }}}{H(t{)}^2}\right).$$

$$L_{\mathrm{IR}}=\frac{1}{2\beta }{\int }^{t}dt\left(1-\frac{A}{a^{1/\beta }-A}\right),$$

$$\dot{H}=\frac{H^2}{2\beta }\left(\frac{A}{a^{1/\beta }-A}-1\right),$$

$$L_{\mathrm{IR}}\to \frac{1}{2\beta }\left({\int }_0^{t}dt+{\int }_t^{\infty }dt\frac{A}{a^{1/\beta }-A}\right),$$

Full text

I Introduction

One of the interesting explanations for the late-time acceleration is the holographic dark energy scenario Li:2004rb (for a review see Wang:2016och ). In this framework the dark energy sector arises holographically from the vacuum energy, after one applies at a cosmological level the holographic principle, which originating from black hole thermodynamics and string theory tHooft:1993dmi ; Susskind:1994vu ; Witten:1998qj ; Bousso:2002ju ; Kiritsis:2013gia establishes a connection of the largest distance of this theory (related to causality and the quantum field theory applicability at large distances) Cohen:1998zx with the Ultraviolet cutoff of a quantum field theory (related to the vacuum energy). Holographic dark energy, both in its basic Li:2004rb ; Wang:2016och ; Horvat:2004vn ; Huang:2004ai ; Enqvist:2004xv ; Zhang:2005yz ; Elizalde:2005ju ; Pavon:2005yx ; Guberina:2005fb ; Wang:2005jx ; Nojiri:2005pu ; Setare:2008pc as well as in its various extensions Gong:2004fq ; Ito:2004qi ; Gong:2004cb ; Setare:2008bb ; Saridakis:2007ns ; Gong:2009dc ; BouhmadiLopez:2011xi ; Khurshudyan:2014axa ; Pasqua:2016wrm ; Jawad:2016tne ; Pourhassan:2017cba ; Paul:2019hys , proves to be very efficient in describing the late-time universe, and it is in agreement with observations Huang:2004wt ; Wang:2004nqa ; Zhang:2005hs ; Li:2009bn ; Lu:2009iv .

Although there has been a large amount of research in the late-time application of holographic principle, namely in the dark-energy era, there has been almost no effort in applying it at early times. Recently, we proceeded to such direction and we constructed an inflationary realization of holographic origin Nojiri:2019kkp . The scenario has the advantage that since at early times the largest distance of the theory is small, the holographic energy density is naturally large in order to lie in the regime necessary for inflation.

In the present work we are interested in applying the holographic principle at early times, however investigating the bounce realization. Nonsingular bouncing cosmologies are of significant interest since they offer a potential solution to the cosmological singularity problem and thus an alternative to inflation Novello:2008ra ; Cai:2012va ; Brandenberger:2012zb ; Cai:2014bea ; Brandenberger:2016vhg ; Cai:2016hea ; deHaro:2015wda . In order for the bounce to be realized the null energy condition has to be violated, and this can be obtained by various modified gravity constructions Nojiri:2006ri ; Capozziello:2011et ; Cai:2015emx , such as the Pre-Big-Bang Veneziano:1991ek and the Ekpyrotic Khoury:2001wf scenarios, $f(R)$ gravity Bamba:2013fha ; Pavlovic:2017umo , $f(T)$ gravity Cai:2011tc , loop quantum cosmology Bojowald:2001xe ; Cai:2014zga ; Odintsov:2015uca , Lagrange modified gravity Cai:2010zma , Finsler gravity Minas:2019urp etc. Alternatively, a nonsingular bounce may be acquired through the introduction of exotic matter sectors Cai:2007qw ; Cai:2009zp . Indeed, as we will see in this work, the energy density of holographic origin can lead to the necessary violation of the null energy condition and trigger the bounce. Additionally, the decrease of the horizons at early times naturally increases holographic energy density at bouncing scales.

The plan of the work is the following. In Section II we construct the holographic bounce scenario, extracting the corresponding analytical solutions. In Section III we proceed to the construction of generalized scenarios, with extended Infrared cutoffs. Finally, Section IV is devoted to the conclusions.

II Holographic bounce

In this section we will construct the basic model of holographic bounce. We start by reminding that in general the holographic density is proportional to the inverse squared Infrared cutoff $L_{\mathrm{IR}}$, namely

[TABLE]

where $\kappa^{2}$ is the gravitational constant and $c$ a parameter. If one wishes to apply the above relation in a cosmological framework, he considers the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric

[TABLE]

with $a(t)$ the scale factor and $k=0,+1,-1$ corresponding to flat, close and open spatial geometry respectively. In the following we focus on the flat case, however the generalization to non-flat geometry is straightforward.

Since the Infrared cutoff $L_{\mathrm{IR}}$ is related to causality it must be a form of horizon. The simplest choice is the Hubble radius, however concerning the late-time application its use cannot lead to an accelerating universe and thus it is rejected Hsu:2004ri . Hence, one may use the particle and future event horizons Li:2004rb , the age of the universe Cai:2007us ; Wei:2007ty , the inverse square root of the Ricci curvature Gao:2007ep , or a combination of Ricci and Gauss-Bonnet invariants Saridakis:2017rdo . More generally, one can consider a general Infrared cutoff constructed by an arbitrary combination of all the above quantities and their derivatives Nojiri:2017opc . In this Section, as a first application we will consider the particle horizon $L_{p}$ and the future event horizon $L_{f}$, which are written as

[TABLE]

Although in the late universe the Friedmann equation includes both the matter and holographic dark energy sectors, in the case of the early universe the former can be neglected Nojiri:2019kkp (unless one considers the possibility of a “matter bounce” Brandenberger:2012zb ; Cai:2011tc , which is not the case of the present work). Therefore, at early times the first Friedmann equation is written as

[TABLE]

where $\rho$ is the energy density of the (effective) fluid that consists the universe, originating from a scalar field, from modified-gravity, or from other sources. In this manuscript we consider that the fluid that drives the bounce has a holographic origin, i.e. it arises from the holographic energy density. Thus, imposing that $\rho$ in (4) is $\rho$ of (1) we deduce that

[TABLE]

Hence, inserting the particle horizon $L_{p}$ or the future event horizon $L_{f}$ into (5) we find that

[TABLE]

with $m=1$ corresponding to the particle horizon and $m=-1$ to the future event horizon.

The general solution of equation (6) is

[TABLE]

with $a_{0}$,$t_{0}$ the two integration constants and $\pm$ corresponding to the two solution branches. As we observe the above solution has very interesting expressions for particular values of the parameter $c$. Specifically, in the case where $\frac{c}{c\pm m}$ is an even number we obtain a bouncing scale factor, which is the scope of interest of the present work. This can be obtained taking the $+$ branch in the case of $m=-1$, i.e. using the future event horizon, and taking the $-$ branch in the case where $m=1$, i.e. using the particle horizon. For instance, choosing $c=2$ we can see that in both the above cases we obtain $a(t)=a_{0}\left(t-t_{0}\right)^{2}$ which indeed corresponds to the bounce realization. In summary, as we observe, in the scenario at hand the bounce can be straightforwardly obtained, since the decrease of the horizons at early times naturally increases holographic energy density at bouncing scales.

We proceed by investigating an improved version of the above scenario. In particular, since we apply the holographic principle at early times, i.e. at high energy scales, we should incorporate the Ultraviolet cutoff $\Lambda_{\mathrm{UV}}$ too. Namely, in this regime the Infrared cutoff acquires a correction from the Ultraviolet one, which takes the form Nojiri:2004pf

[TABLE]

If we insert this expression into (5), with $L_{\mathrm{IR}}$ either the particle horizon $L_{p}$ or the future event horizon $L_{f}$, we acquire

[TABLE]

which is the improved version of (6). In order to solve this equation we first transform it to an equation for $H(t)$, namely

[TABLE]

whose general solution for $\Lambda_{\mathrm{UV}}$ not being equal to 0 or infinity is written implicitly as

[TABLE]

with $C_{0}$ an integration constant. Hence, the scale factor can be obtained through

[TABLE]

with $a_{0}$ another integration constant.

The above solution (11) can describe a bounce for particular choices of the model parameters that satisfy the bounce requirements, namely $H<0$, $H=0$ and $H>0$ before, at, and after the bounce respectively, as well as $\dot{H}>0$ throughout the procedure. In Fig. 1 we present the bounce realization arising from (11), for three choices of the model parameters. The interesting feature is that now we can obtain a nonsingular bounce, which was not possible in the simple case without the UV correction, namely solution (7). Furthermore, as we can see, the value of the minimum scale factor is determined by the value of $\Lambda_{\mathrm{UV}}$, going to zero at $\Lambda_{\mathrm{UV}}\rightarrow\infty$ as expected. This feature is very significant, since nonsingular bounces are more physically important since they are the ones that can alleviate the initial singularity issue of standard cosmology. This is one of the main result of the present work and reveals the capabilities of holographic bounce.

We proceed by referring to the perturbations of the above background evolution, which is a necessary task in every bouncing scenario, since they are related to observables such as the spectral index and the tensor-to-scalar ratio. In particular, while in inflation the cosmological fluctuations emerge initially inside the Hubble horizon, then they exit it, and later on they re-enter giving rise to the primordial power spectrum, in bounce cosmology the situation is different. Namely, the quantum fluctuations around the initial vacuum state are generated in the contracting phase before the bounce, and since the Hubble horizon decreases faster than the wavelengths of the primordial fluctuations, they exit the Hubble radius as contraction continues. Then the bounce happens, and hence at later times in the expanding phase the fluctuations re-enter inside the horizon. As one can deduce, the specific background evolution could affect the processing of perturbations from the bounce point, however this effect is important only in the UV regime, where the gravitational modification becomes significant, while the IR regime, which is responsible for the primordial perturbations, remains almost unaffected Battarra:2014tga ; Quintin:2015rta .

Although the generation of perturbations in bouncing models driven by a scalar degree of freedom is well understood and studied Novello:2008ra ; Allen:2004vz , in the case where the driving mechanism is different the perturbation generation has not been studied in detail. Therefore, the analysis of perturbation generation in the present holographic bounce scenario has to be performed in a systematic way, through the full and detailed perturbation analysis, using the perturbation investigation of holographic cosmology Battefeld:2004jh ; Kim:2008pi ; Mehrabi:2015kta . Only such an analysis could reveal whether the present bouncing scenario suffers from the “observational no-go theorem” that widely exists in nonsingular bounces, namely that their predicted amplitude of primordial non-gaussianity and tensor-to-scalar ratio cannot simultaneously fit observations in a general single component bounce cosmology Quintin:2015rta ; Li:2016xjb (the fact hat holographic bounce is not driven by a single component is a promising feature that the above no-go theorem might be evaded). Furthermore, the perturbation analysis would reveal the differences of the present scenario form other scenarios, such as the ekpyrotic Khoury:2001wf and the matter bounce Cai:2011tc ones, since even if the background behavior is similar, the perturbation evolution will be different due to the fact that the underlying theory and the propagating degrees of freedom are of different, namely of holographic, origin. Nevertheless, this detailed analysis of perturbation generation is a separate project that lies beyond the scope of the present work, and it is left for a future study.

One issue that is present in principle in almost every bouncing scenario, is the anisotropy one. Specifically, the bounce realization is in general unstable against anisotropic stress (BKL instability) Belinsky:1970ew , since the effective energy density in anisotropies scales as $a(t)^{-6}$ and thus in a contracting universe it increases faster than the radiation and matter energy densities, leading the bounce to take place in a non-isotropic and non-homogeneous spacetime. Nevertheless, although the anisotropies grow faster than the background evolution, their domination is not necessarily quick, since this is determined by the initial conditions on the anisotropy. Hence, if the anisotropies are of quantum origin, arising from the backreaction of cosmological perturbations, the moment where they will dominate will depend on the energy scale during the matter contraction, which is typically at very high energy scales. Therefore, one may consider reliably an FRW background evolution up to the bounce phase. However, in order to acquire a full removal of the anisotropy problem, one should introduce additional mechanisms (a well-studied such scenario is the ekpyrotic one Khoury:2001wf , in which one introduces a negative exponential potential that leads to an anisotropy-free evolution Cai:2012va ; Cai:2014bea ; Qiu:2013eoa ; Cai:2013vm ). Nevertheless, since in the present scenario of holographic bounce there is no extra degree of freedom, such mechanisms are difficult to be introduced, ad thus a safe removal of the anisotropy problem should be obtained by imposing the bounce scale to be suitably low, in order for the anisotropies to be always sub-dominant.

Finally, let us make a comment on the preheating phase, which is is needed in any realistic cosmological scenario. The preheating phase in bouncing cosmology has been investigated in Cai:2011ci . Nevertheless, it is interesting to note that in the holographic framework there can be a mechanism of “holographic preheating” Cai:2016sdu ; Cai:2016lqa , through which the energy transfer from the inflaton to the matter field is obtained through a model of holographic superconductor. Hence, due to the holographic foundations of the present scenario, the phase of holographic preheating can be naturally embedded to it.

III Generalized scenarios

In this section we construct generalizations of the above holographic bounce scenarios through the use of extended Infrared cutoffs, inspired by similar constructions in the case of the late universe. In principle, such general Infrared cut-offs may be a function of both $L_{p}$ and $L_{f}$ and their derivatives, as well as of the Hubble horizon and its derivatives and of the scale factor Nojiri:2005pu ; Nojiri:2017opc , namely

[TABLE]

Thus, applying the above extended Infrared cutoff at the early universe provides enhanced freedom to acquire bouncing cosmology.

Without loss of generality we start by considering the model

[TABLE]

with $\alpha$ a parameter. Inserting this into the Friedmann equation (5), setting $c=1$ for simplicity, and adding also a cosmological constant term $\Lambda=-\frac{3}{\kappa^{2}}H_{0}^{2}$, we result to

[TABLE]

Recalling that the Ricci scalar is $R=6(2H^{2}+\dot{H})$ in FRW geometry, the above equation is written in the form

[TABLE]

with

[TABLE]

As one can see, Eq.(16) is just the first Friedmann equation in the $R^{2}$-gravity theory in the absence of matter Nojiri:2017ncd ; Nojiri:2010wj ; Nojiri:2006ri ; Capozziello:2010zz ; delaCruzDombriz:2012xy ; Olmo:2011uz .

As it is known, for the $F(R)$ gravity model of (17) there appears a solution describing the bouncing universe Bamba:2013fha . In fact, if we impose the bouncing scale factor

[TABLE]

with a positive constant $\lambda$, we find that $H=\lambda t$, $\dot{H}=\lambda$, and $\ddot{H}=0$ (i.e. the universe is contracting for $t<0$, is expanding for $t>0$, and at $t=0$ we have the nonsingular bounce realization at a scale factor $a_{B}$). In this case Eq. (15) becomes

[TABLE]

which for $\alpha>0$ is satisfied if we choose

[TABLE]

Hence, the scenario at hand indeed accepts the bouncing scale factor (18) as solution, if we consider the model parameters (20). We mention here that the scale factor (18) is just a simple example that shows the capabilities of the theory. In a realistic inflation realization however the imposed scale factor should include an exit from the exponential expansion. Alternatively, one could transform the above $F(R)$ model in the Einstein frame, where the scalar field dynamics in the effective potential could drive the successful inflation exit. In this way we can re-obtain the usual successful inflationary scenarios of $F(R)$ gravity, nevertheless the underlying theory is of different origin, namely it arises from holography.

As another example, we may consider the model where

[TABLE]

with $\beta$ and $B$ two parameters. Inserting this into the Friedmann equation (5), and setting for simplicity $c=1$, we find

[TABLE]

The solution of (22) is

[TABLE]

which in turn leads to

[TABLE]

with $A$ and $a_{B}$ integration constants. Indeed, the solution (24) describes a bounce realization at $t=0$. Moreover, away from the bounce point $a(t)$ behaves in a power-law way, namely $a(t)\sim t^{2\beta}$, as in the case of the perfect fluid with a constant equation-of-state parameter. Note that since for the solution (24) the integrand of (21) diverges at $t=0$ and goes to a constant $\frac{1}{2\beta}$ when $t\to\infty$, it is necessary to add a small UV correction, as we discussed in the previous section, and use

[TABLE]

Finally, as a last example we consider the model with

[TABLE]

with $\beta$ and $A$ constants. inserting into (5) with $c=1$ we obtain

[TABLE]

whose solution is again given by (24) but now $B$ appears as a constant of integration. Since substituting the solution into (26) we find that the second term in the integrand diverges at the bouncing time, the integration in (26) should be modified as

[TABLE]

i.e. we add a UV correction.

IV Conclusions

In this work we obtained a bounce realization of holographic origin. In particular, inspired by the application of the holographic principle at late-time universe, i.e. by the scenario of holographic dark energy, we applied it at early times. We first considered as Infrared cutoffs the particle or future event horizons, and we showed that the decrease of the horizons at early times naturally increases holographic energy density at bouncing scales, while we additionally obtain the necessary null energy condition violation. Hence, even this simple scenario can lead to the bounce realization.

We proceeded to the addition of a simple correction due to the Ultraviolet cutoff, whose role may be taken into account at the high energy scales of early universe. We first extracted analytical solutions that may lead to the bounce realization for suitable choices of the model parameters, and we provided specific numerical examples. The interesting implication of this UV correction is that it can control the value of the minimum scale factor, which is significant since nonsingular bounces are more physically important relating to the alleviation of the initial singularity issue of standard cosmology.

Lastly, we constructed generalized scenarios of holographic bouncing cosmology, arisen from the use of extended Infrared cutoffs. As specific examples we considered cutoffs that can reproduce $F(R)$ gravity, and in particular $R^{2}$ gravity, which can then lead to a nonsingular bounce realization. These features act as an additional advantage in favour of holographic bounce and reveal the capabilities of the model.

Acknowledgements.

This work is partly supported by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas gCosmic Acceleration h No. 15H05890 (S.N.) and the JSPS Grant-in-Aid for Scientific Research (C) No. 18K03615 (S.N.), and by MINECO (Spain), FIS2016-76363-P, and by project 2017 SGR247 (AGAUR, Catalonia) (S.D.O).