Modified Gravity Theories on a Nutshell: Inflation, Bounce and Latetime Evolution
Abstract
We systematically review some standard issues and also the latest developments of modified gravity in cosmology, emphasizing on inflation, bouncing cosmology and latetime acceleration era. Particularly, we present the formalism of standard modified gravity theory representatives, like , and gravity theories, but also several alternative theoretical proposals which appeared in the literature during the last decade. We emphasize on the formalism developed for these theories and we explain how these theories can be considered as viable descriptions for our Universe. Using these theories, we present how a viable inflationary era can be produced in the context of these theories, with the viability being justified if compatibility with the latest observational data is achieved. Also we demonstrate how bouncing cosmologies can actually be described by these theories. Moreover, we systematically discuss several qualitative features of the dark energy era by using the modified gravity formalism, and also we critically discuss how a unified description of inflation with dark energy era can be described by solely using the modified gravity framework. Finally, we also discuss some astrophysical solutions in the context of modified gravity, and several qualitative features of these solutions. The aim of this review is to gather the different modified gravity techniques and form a virtual modified gravity “toolbox”, which will contain all the necessary information on inflation, dark energy and bouncing cosmologies in the context of the various forms of modified gravity.
pacs:
04.50.Kd, 95.36.+x, 98.80.k, 98.80.Cq,11.25.wContents
 I Introduction
 II Modified Gravities and Cosmology
 III Inflationary Dynamics in Modified Gravity

IV Latetime Dynamics and Dark Energy
 IV.1 CDM Epoch from Gravity
 IV.2 CDM Epoch from Modified GaussBonnet Gravity
 IV.3 Unification of Inflation with Dark Energy Era in Gravity
 IV.4 Unification of Inflation with Dark Energy Era in Modified GaussBonnet Gravity
 IV.5 Phantom Dark Energy Era
 IV.6 Dark Energy Oscillations in Gravity Theories and Growth Index
 V Astrophysical Applications
 VI Bouncing Cosmologies from Modified Gravity
 VII Conclusion
I Introduction
With this work we shall try to provide a concise pedagogical review on the latest developments in modified gravity aspects of inflation, dark energy and bouncing cosmology. The attempt is challenging and it is conceivable that it is impossible to cover all the different approaches in the field, since the number of these is vast. Hence we focus on modified gravity aspects of early, latetime acceleration and bounce dynamics and we shall try to provide a pedagogical text accessible by nonexperts but also useful to experts.
One of the most fundamental questions in modern theoretical cosmology is, whether the genesis of the Universe was singular or nonsingular. This question is equivalent in asking if the Big Bang theory or the Big Bounce theory actually describes the evolution of our Universe. Naturally thinking, the initial singularity described by the Big Bang theory, is a mentally more convenient description, since we can easily imagine a zero sized Universe, with infinite temperature and energy density, and also in which all fundamental interactions are unified under the yet unknown same theoretical framework. However, no one can actually exclude a cyclic cosmological evolution, in which the Universe never shrinks to zero. Actually the latter perspective seems to be supported by quantum cosmologies, as we discuss later on. One of the main purposes of this review article is to present the tools to study these two types of cosmological scenarios, in the context of modified gravity.
During the last 25 years, the cosmologists community has experienced great surprises, since the observations indicated in the late 90’s Riess:1998cb that the Universe is expanding, but in an accelerating way. This observation utterly changed the cosmologists way of thinking and also it revived theories containing a cosmological constant. It was Einstein that had firstly proposed a theory of cosmological evolution with a cosmological constant, and later on he admitted that this was the greatest scientific mistake of his life. However, it seems that nature can actually incorporate such a cosmological constant, so it seems eventually that Einstein was not wrong to some extent. We need to note that finding the correct model for the latetime acceleration is not an easy task at all. The latetime acceleration era is known as the dark energy era Carroll:2003wy ; Peebles:2002gy ; Durrer:2008in ; Caldwell:2009ix ; Caldwell:2009zzb ; Li:2011sd ; Bamba:2012cp ; Lobo:2008sg ; Gubitosi:2012hu ; Bloomfield:2012ff ; Gleyzes:2013ooa ; Li:2012dt ; Sami:2009jx ; Balakin:2016cbe (see also Weinberg:2002rd ; Perivolaropoulos:2006ce ; Dobado:2008xn ; Dunsby:2015ers ), and up to date there are many proposals that try to model this era, with other using a scalar field, known as quintessence models Zlatev:1998tr ; Carroll:1998zi ; Wang:1998gt ; Chiba:1999wt ; Barreiro:1999zs ; Chiba:1999ka ; Haiman:2000bw ; Capozziello:2002rd ; Sahni:2002kh ; Capozziello:2003gx ; Hao:2004ky ; Bassett:2004wz ; Perivolaropoulos:2004yr ; Zhang:2005rg ; Olivares:2005tb ; Guo:2005ata ; Caldwell:2005tm ; Scherrer:2007pu ; Creminelli:2008wc , while other models use modified gravity in its various forms. With regards to quintessence, there exist various alternative proposals on this topic, see for example Capozziello:2003tk ; Kamenshchik:2001cp ; Banerjee:2000mj ; Sahni:1999qe ; Perrotta:1999am , and also approaches with nonminimal coupling, see Faraoni:2000wk ; Torres:2002pe ; Pettorino:2008ez ; Bertolami:1999dp . This field of research is still developing rapidly up to date.
Apart from the latetime acceleration era, the Universe experienced another acceleration era during its first stages of evolution, and the latest observational data have actually constrained this era Ade:2015lrj ; Array:2015xqh , which is known as inflation Linde:2007fr ; Gorbunov:2011zzc ; Lyth:1998xn ; Linde:1983gd ; Linde:1985ub ; Albrecht:1982wi ; Linde:1993cn ; Sasaki:1995aw ; Turok:2002yq ; Linde:2005dd ; Kachru:2003sx ; Brandenberger:2016uzh ; Bamba:2015uma ; Martin:2013tda ; Martin:2013nzq ; Baumann:2014nda ; Baumann:2009ds ; Linde:2014nna ; Pajer:2013fsa ; Yamaguchi:2011kg ; Byrnes:2010em . During the inflationary era, the Universe increased its size at an exponential rate, so it expanded quite quickly and gained a large size in a relatively small amount of time. From a theoretical point of view, the inflationary description of the early time Universe, solved quite many theoretical problems that the original BigBang theory had, for example the horizon problem etc. The inflationary evolution had originally two different forms, which now are known as the old inflationary scenario Guth:1980zm , and the new inflationary scenario Linde:1983gd ; Albrecht:1982wi . From the time that the new inflationary scenario was introduced, many models where proposed that could actually describe in a successful way the inflationary era, with many of the models using a scalar field or multiple scalar fields, see Byrnes:2010em for details. Also for supergravity models of inflation, see the review Yamaguchi:2011kg , and for string cosmology consult Refs. Linde:2005dd ; Kachru:2003sx . There also exist alternative scenarios for inflation, involving the axion field, see for example Pajer:2013fsa for a review. In addition for inflationary theories in the context of modified gravity in its various forms, see Bamba:2015uma .
However, a successful model at present time (2017), has to be confronted with the observational data coming from the Planck satellite, which severely constraints the inflationary era Ade:2015lrj . The Planck data have excluded many inflationary models from the viable inflationary models list, and nowadays these data are a benchmark with regard to the viability of an inflationary model. Thus, it is compelling that a model has to be compatible with the Planck data, before it can be considered as a candidate for inflation. Apart from the scalartensor models of inflation, there are alternative proposals in the literature that use modified gravity in its various forms Bamba:2015uma ; Nojiri:2013zza ; Nojiri:2006ri ; Capozziello:2011et ; Capozziello:2010zz ; Capozziello:2009nq ; Nojiri:2010wj ; Clifton:2011jh , in order to describe this earlytime era. We need to note that in this review we shall adopt the metric modified gravity formalism, but there also exists the Palatini formalism, for which we refer the reader to Refs. Ferraris:1992dx ; Meng:2003en ; Amarzguioui:2005zq ; Flanagan:2003rb ; Koivisto:2005yc ; Capozziello:2012ny ; Capozziello:2013uya ; Olmo:2011uz ; Olmo:2009xy ; Harko:2011nh ; Makarenko:2014lxa .
Most of the metric modified gravity descriptions provide a consistent framework for the description of the earlytime acceleration, and also the compatibility of the model with the data can be achieved in many of these models. Moreover, each modification of Einstein’s theory of general relativity eventually is confronted with the successes of general relativity. Hence all the constraints on modified gravity imposed by local astrophysical data but also from global constraints, have to be satisfied to an adequate level. Therefore it is conceivable that a modified gravity has many challenges and obstacles to overcome in order to be considered as a viable cosmological theory. Apart from the constraints, the ultimate goal in modified gravity is to offer a selfconsistent theoretical framework in the context of which the earlytime and latetime acceleration will be described by the same theory. In this direction there are many works using various theoretical contexts, that provide such a framework Capozziello:2005tf ; Nojiri:2005pu ; Carter:2005fu ; Liddle:2006qz ; Chen:2006qy ; Nojiri:2007as ; Appleby:2007vb ; Nojiri:2007cq ; Cognola:2007zu ; Cognola:2008zp ; Koivisto:2009fb ; Koivisto:2008xf ; Xia:2007me ; Liddle:2008bm ; Elizalde:2009gx ; Elizalde:2010ep ; Makarenko:2014nca ; deHaro:2016hsh ; Beltran:2015hja . Also dark energy Elizalde:2007kb ; Nojiri:2007te ; Huterer:2006mva ; Nojiri:2006jy ; Capozziello:2005ku ; Borowiec:2006qr ; Nojiri:2005am ; Szydlowski:2006ay ; Borowiec:2006hk ; Allemandi:2004wn and other aspects of cosmological evolution and cosmological implications Easson:2004fq ; Carloni:2004kp ; Clifton:2005aj ; Sanyal:2006wi ; Appleby:2008tv ; Capozziello:2008qc ; Evans:2007ch ; Capozziello:2007ec ; Li:2007jm ; Bertolami:2007gv ; Li:2007xn ; Song:2006ej ; Arbuzova:2011fu are also explained in an appealing way by modified gravity. In Refs. Nojiri:2007as ; Appleby:2007vb ; Nojiri:2007cq ; Cognola:2007zu ; Artymowski:2014gea ; Fay:2007uy the unification of Cold Dark Matter model (CDM hereafter) with inflation was developed, in the context of gravity, while in Refs. Capozziello:2005tf ; Nojiri:2005pu , the same issue was addressed in the context of phantom cosmology. In principle the possibility of phantom inflation always exists in various cosmological contexts Yurov:2003zt ; Piao:2004tq ; GonzalezDiaz:2004df ; Izumi:2010wm ; Feng:2010ya ; Liu:2012iba .
The first unified description of the inflation with dark energy in modified gravity was proposed in Ref. Nojiri:2003ft , while in Ref. Liddle:2006qz the unified description of dark energy and inflation was developed in the context of string theory, and in Nojiri:2005pu ; Chen:2006qy a holographic approach to the same issue was performed. Also in Refs. Xia:2007me the problem under discussion was addressed in the context of a dynamical involving dark energy component, and in Elizalde:2010ep the problem was addressed in the context of modified HoravaLifshitz gravity. The most successful theory will be consistent with observations, with the local and global constraints and will describe simultaneously all or at least most of the evolution eras of our Universe. Finally for some different scenarios see Koivisto:2009fb ; Koivisto:2008xf ; Xia:2007me ; Liddle:2008bm ; Rinaldi:2014yta ; Nojiri:2016vhu ; Borowiec:2008js ; Elizalde:2013paa ; Cai:2008gk ; Geng:2011aj ; Vacaru:2015iga . Also modified gravity usually brings along various exotic features, for example the equivalence principle can be different in comparison to the ordinary Einstein gravity Olmo:2006zu , or the energy conditions may differ Santos:2007bs and in addition the thermodynamic interpretation is different Bamba:2016aoo , see also Zubair:2016bpi . But inevitably modified gravity has eventually to confront the successes of the EinsteinHilbert gravity, so stringent rules are imposed from solar system tests. Some relevant studies with regards to solar system implications and constraints of modified gravity can be found in Refs. Nojiri:2007as ; Olmo:2005zr ; Faraoni:2006hx ; Zhang:2007ne ; Olmo:2006eh ; Allemandi:2006bm ; Erickcek:2006vf ; Lin:2010hk ; Iorio:2010tp ; Olmo:2005hc , while the Newtonian limit of gravity was studied in Ref. Capozziello:2007ms . Also some studies on the stability of can be found for example in Faraoni:2005vk ; Faraoni:2006sy ; Cognola:2007vq ; Sawicki:2007tf and a recent study on the observational constraints on gravity from cosmic chronometers can be found in Nunes:2016drj . Finally, we need to note that the phase space of modified gravity theories is quite richer in geometric structures, in comparison to the Einstein gravity phase space, see for example Faraoni:2005vc ; Carloni:2017ucm .
An alternative description to the inflationary paradigm and to the Big Bang cosmology, is offered by bouncing cosmologies, see Ref. Tolman:1931zz for the original idea of a bounce cosmology, and for an important stream of reviews see Refs. Brandenberger:2012zb ; Brandenberger:2016vhg ; Battefeld:2014uga ; Novello:2008ra ; Cai:2014bea ; deHaro:2015wda ; Lehners:2011kr ; Lehners:2008vx ; Cheung:2016wik ; Cai:2016hea . Particularly, in Ref. Brandenberger:2012zb , the study was focused on the matter bounce scenario and various ways that may realize this scenario were discussed. Also several observational signatures that may distinguish the matter bounce from the standard inflationary paradigm were also discussed. A more focused review on bouncing cosmologies is Ref. Brandenberger:2016vhg , were the origin of primordial perturbations was discussed in the context of bouncing cosmologies, and also several examples that realize bounce cosmologies, including prebigbang scenarios, ekpyrotic scenarios, string gas cosmology, bouncing cosmologies from modified gravity and string theory, were also discussed. Moreover, the observational signatures that may distinguish a bounce from the inflationary paradigm were examined too, like for example the existence of nonGaussianities. In Ref. Battefeld:2014uga , the bouncing cosmologies were discussed and in addition certain potentially fatal effects that undermine nonsingular bouncing models were pointed out. Also, the unstable growth of curvature fluctuations and the growth of the quantum induced anisotropy, in conjunction with the study of various gravitational instabilities, were discussed too. In Ref. Novello:2008ra , the study performed covered the topics of higherorder gravitational theories, theories with a scalar field, bounces in the braneworld scenarios and several quantum cosmology scenarios. Also the cyclic Universes were discussed and the issue of perturbations in the context of bouncing Universes were addressed too. In Ref. Cai:2014bea , the matterekpyrotic bounce scenario was extensively studied, and also various realizations of this scenario were presented, like for example the two scalar field realization. Also the observational constraints were thoroughly discussed and also several mechanisms for generating a red tilt for primordial perturbations were presented too. In Ref. deHaro:2015wda , the matter bounce scenario was also presented, and its realization was achieved by using a single scalar field with a nearly exponential potential. The main result was that the rolling of the scalar field leads to a running of the spectral index, and specifically a negative running is obtained. Also possible theories that realize such a scenario are discussed, such as holonomy corrected loop quantum cosmology theories and also teleparallel gravity. In addition, and insightful study on the reheating process is discussed too. In Ref. Lehners:2008vx , several issues concerning cyclic cosmologies were discussed, including, the ekpyrotic phase of a bounce, how to avoid chaos in such models, the Milne Universe and finally several ekpyrotic models were presented. Also, the scalar and tensor perturbations were addressed, and in addition the link of these theories to a more fundamental theory, like heterotic Mtheory was discussed too. Also in Ref. Cai:2016hea , matter bounce scenarios in which the matter content consists of dark energy and dark matter were reviewed. Specifically, the CDM bounce scenario was discussed, and also theories with interacting dark matter and dark energy were addressed too. Moreover, the observational signatures that may distinguish bounces from the inflationary paradigm were discussed, and also several theories that may realize a bounce were also presented, including, loop quantum cosmology, string Cosmology, gravity, kinetic gravity braiding theories and finally the Fermi bounce mechanism. Furthermore, some interesting information for the occurrence of bounces can be found in Cattoen:2005dx and for a pioneer version of the nonsingular bounce in the context of modified gravity see Mukhanov:1991zn . The Big Bounce cosmology Li:2014era ; Brizuela:2009nk ; Cai:2013kja ; Quintin:2014oea is an appealing alternative to inflation, since the initial singularity which haunts the Big Bang cosmology is absent, hence these cosmologies are essentially nonsingular Cai:2013vm ; Poplawski:2011jz ; Koehn:2015vvy . However, other types of singular bounces appeared in the literature, in which case the singularity which occurs at the bouncing point is a soft type singularity Odintsov:2015zza ; Nojiri:2016ygo ; Oikonomou:2015qha ; Odintsov:2015ynk . In the context of bouncing cosmologies there are various scenarios in the literature, and bouncing cosmologies are often studied in ekpyrotic scenarios of some sort Koehn:2013upa ; Battarra:2014kga ; Martin:2001ue (see Refs. Khoury:2001wf ; Buchbinder:2007ad for the ekpyrotic scenario per se). In the case of a bounce, the Universe is described by a repeating cycle of evolution, in which initially the Universe contracts, until a minimal radius is reached, which is called the bouncing point. After that point, the Universe starts to expand again and this cycle is continuously repeated. Hence bouncing cosmologies are essentially cyclic cosmologies or equivalently oscillating cosmologies Brown:2004cs ; Hackworth:2004xb ; Nojiri:2006ww ; Johnson:2011aa . Cosmological perturbations in bouncing cosmologies are generated usually during the contracting phase Brandenberger:2016vhg , however this is not always true, see for example the singular bounce of Ref. Odintsov:2015ynk , where the primordial perturbations are actually generated during the expanding phase after the Type IV singular bouncing point. For some very relevant studies of perturbations in bouncing cosmologies, see Refs. Peter:2002cn ; Gasperini:2003pb ; Creminelli:2004jg ; Lehners:2015mra . In principle, a scale invariant or nearly scale invariant power spectrum can be generated by a bounce cosmology Brandenberger:2016vhg ; Odintsov:2015ynk , and also the recent observational data can be consistent with cyclic cosmologies Mielczarek:2010ga ; Lehners:2013cka ; Cai:2014xxa . There are many bouncing cosmologies in the literature and some of these scenarios naturally occur in Loop Quantum Cosmology Laguna:2006wr ; Corichi:2007am ; Bojowald:2008pu ; Singh:2006im ; Date:2004fj ; deHaro:2012xj ; Cianfrani:2010ji ; Cai:2014zga ; Mielczarek:2008zz ; Mielczarek:2008zv ; Diener:2014mia ; Haro:2015oqa ; Zhang:2011qq ; Zhang:2011vi ; Cai:2014jla ; WilsonEwing:2012pu . One quite well known bounce cosmology is the matter bounce scenario deHaro:2015wda ; Finelli:2001sr ; Quintin:2014oea ; Cai:2011ci ; Haro:2015zta ; Cai:2011zx ; Cai:2013kja ; Haro:2014wha ; Brandenberger:2009yt ; deHaro:2014kxa ; Odintsov:2014gea ; Qiu:2010ch ; Oikonomou:2014jua ; Bamba:2012ka ; deHaro:2012xj ; WilsonEwing:2012pu , which is known to provide a scale invariant spectrum during the contracting phase, see for example Brandenberger:2016vhg . Also for some alternative scenarios in the context of cosmological bounces, see the informative Refs. Cai:2007qw ; Cai:2010zma ; Avelino:2012ue ; Barrow:2004ad ; Haro:2015zda ; Elizalde:2014uba . An interesting bouncing cosmology scenario appeared in Cai:2007qw , called quintom scenario, see Ref. Cai:2009zp for a comprehensive review on quintom cosmology. The quintom scenario is highly motivated by the current observations which indicate that the dark energy equation of state crosses the phantom divide line. In order that the quintom scenario is realized two scalar fields are required, since a nogo theorem forbids the single scalar field realization of the quintom scenario Cai:2009zp . In the two scalar field realization of the quintom scenario, one scalar field is quintessential and the other scalar field is a phantom one, and the drawback of these theories is the existence of ghost degrees of freedom. Also the quintom scenario may be obtained from string theory motivated higher derivative scalar field theories and from braneworld scenarios Cai:2009zp . The quintom bounce can be realized if the null energy condition is violated, and as we already mentioned, the nogo theorem in quintom cosmology makes compelling to use two scalar fields. One important feature of the two scalar field realization of the quintom bounce is the fact that for each scalar component, the effective equation of state needs not to cross the phantom divide line, and thus the classical perturbations remain stable.
Some bouncing cosmologies scenarios have been proposed to describe the preinflationary era Cai:2015nya ; Piao:2003zm ; Piao:2005ag , and thus in these scenarios the inflationary paradigm is combined with cosmological bounces. So bouncing cosmologies can produce an exact scale invariant power spectrum of primordial curvature perturbations, for example the matter bounce scenario Brandenberger:2016vhg during the contraction era. However, in such cases, during the expansion era, entropy is produced and the perturbation modes grow with the cosmic time Finelli:2001sr . Such a continuous cycle of cosmological bounces can be stopped if a crushing singularity occurs at the end of the expanding era, as for example in the deformed matter bounce scenario studied in Ref. Odintsov:2016tar , where it was shown that the infinite repeating evolution of the Universe stops at the final attractor of the theory, which is a Big Rip singularity Caldwell:2003vq ; McInnes:2001zw ; Nojiri:2003vn ; Nojiri:2005sr ; Gorini:2002kf ; Elizalde:2004mq ; Faraoni:2001tq ; Singh:2003vx ; Csaki:2004ha ; Wu:2004ex ; Nesseris:2004uj ; Sami:2003xv ; Stefancic:2003rc ; Chimento:2003qy ; Chimento:2004ps ; Hao:2004ky ; Babichev:2004qp ; Zhang:2005eg ; Dabrowski:2004hx ; Lobo:2005us ; Cai:2005ie ; Aref'eva:2005fu ; Lu:2005qy ; Godlowski:2005tw ; Guberina:2005mp ; Dabrowski:2006dd ; Chimento:2015gga ; Barrow:2009df ; Yurov:2007tw ; BouhmadiLopez:2007qb ; BouhmadiLopez:2006fu ; Briscese:2006xu . Modified gravity in general offers a consistent theoretical framework in the context of which bouncing cosmologies can be realized, without the need to satisfy specific constraints which are compelling in the case of the standard general relativity approach. Hence the study of bouncing cosmologies in the context of modified gravity is important, and these cosmologies need to be critically examined with regards to their observational consequences. For some recent studies on bouncing cosmologies in the context of modified gravity, see for example Bamba:2013fha ; Barragan:2009sq ; Farajollahi:2010pn ; Escofet:2015gpa , see also Refs. Barragan:2010qb ; Koivisto:2010jj ; Komada:2014asa for bouncing cosmologies in the context of Palatini gravity.
The modified gravity description of our Universe cosmological evolution is one physically appealing theoretical framework, which can potentially explain the various evolution eras of the Universe, for the simple reason that it can provide a unified and theoretically consistent description. There exist a plethora of modified gravity models, that can potentially describe our Universe evolution and the most important criterion for the viability of a modified gravity theory is the compatibility of the theory with present time observations. The observations related to our Universe, mainly consist of large scale observations and astrophysical observations, related to compact gravitational objects or gravitationally bound objects. In both cases, a successful modified gravity theory, should pass all the tests related to observations. But there is also another feature that may render a modified gravity theoretical description as viable, namely the fact that the theory will be able to predict new, yet undiscovered phenomena. We believe that by studying alternative modified gravity models, even if we do not succeed in finding the ultimate modified gravity theory, at least we will pave the way towards finding the most successful theory. In the context of modified gravity, the new Lagrangian terms, introduce new degrees of freedom beyond the standard General Relativity, and the Standard Model of particle physics. When these new terms are applied at a cosmological level, the extra degrees of freedom alter the evolution of the Universe, and may have as an effect the desired behavior of the Universe. These theories may however introduce certain pathologies or extra instabilities, as it happens for example in the case of gravity, where superluminal extra modes appear, which are absent in the gravity and the gravity, or modified GaussBonnet gravity cases, and of course these are absent in General Relativity.
Also with regards to astrophysical solutions, like black holes, neutron stars, and wormholes, modified gravity utterly changes the conditions that are needed to be satisfied in order for the solutions to be consistent. For example, in the case of wormholes in the ordinary Einstein gravity, an exotic matter fluid needs to be present, in order for the wormhole solution to be selfconsistent. On the contrary, in the modified gravity case, the modified gravity part can offer a theoretically appealing and simple remedy to this problem, see for example Lobo:2009ip .
Our aim with this review is to present the latest developments in the description of the inflationary era, dark energy and also in bouncing cosmology, in the context of modified gravity. Our motivation is the fact that up to date the CDM model is successful but does not provide a complete description of the Universe. Also we shall provide all the necessary information for the models we shall study, that make these models consistent both at astrophysical and at cosmological level. Our presentation will have a pedagogical and introductive character, in order to make this review a pedagogical tool available to experts and nonexperts. It is conceivable that our work does not cover all the modified gravity applications on inflation, dark energy and bouncing cosmologies, this task would be very hard to be materialized, since the subject is vast. However we shall provide the most important tools that will enable one to study in some detail inflation and bounces with modified gravity. Also for completeness and in order to render this review an autonomous study on modified gravity, we shall discuss some astrophysical applications of modified gravity and specifically of gravity, which is the most sound representative theory of modified gravity. In most cosmological applications, the background spacetime geometry will be that of the flat FriedmannRobertsonWalker (FRW) geometry. We will start our presentation with chapter II, where we shall present the most important modified gravity descriptions. Specifically we shall provide the theoretical framework of each modified gravity version, and also we also present some illustrative examples in each case. Specifically, in section IIA, we discuss the most important representative theory of modified gravity, namely gravity. We present in detail the equations of motion of the FRW Universe and also we discuss the criteria that render an gravity theory a astrophysical and cosmologically viable theory. We also discuss the stability, in terms of the scalaron mass, and we present the corresponding scalartensor description in the Einstein frame. Several viable gravities are presented, and the conditions of their viability are also quoted. Also we discuss in brief the case that a nonminimal coupling between the gravity sector and the matter Lagrangian exists. In section IIB, we present the GaussBonnet modified gravities. We present the basic equations of motion, which can be used as a reconstruction method in order to realize various cosmologies. Also we calculate the corrections to the Newton law due to the GaussBonnet gravity corrections. The string inspired gravity follows in section IIC, along with several characteristic examples. In section IID, we discuss an important modified gravity theory, namely that of an gravity, with being the torsion scalar. Both the and gravity will be thoroughly discussed in this review article, since these theories have appealing characteristics and are important alternative theories to gravity. In section IIE we discuss massive gravity and bigravity theories, and in sections IIF and IIG, we discuss mimetic and mimetic gravity. The mimetic framework has offered a quite interesting appealing theoretical framework, so we present various versions of the mimetic framework. In section IIH we present another gravity extension, namely that of unimodular gravity, and in sections III, IIJ, we discuss some combinations of unimodular and mimetic gravity, which are accompanied by illustrative examples.
Chapter III is devoted on the study of inflationary dynamics in various theoretical contexts. Traditionally, inflation was firstly studied in the context of scalartensor theory in its simplest form, namely that of a canonical scalar field, the inflaton, but in this chapter we also present some modified gravity descriptions of the inflationary era. For completeness in IIIA we first study the canonical scalar field inflationary paradigm, and in order to illustrate the methods, we calculate the spectral index of primordial curvature perturbations and also the scalartotensor ratio. In the same section we study the noncanonical scalar field case and we also discuss in brief the form of the inflationary dynamics in multiscalar theories of gravity. In section IIIB we present the inflationary dynamics formalism for general theories of gravity. We provide detailed calculations for the slowroll indices and also for the observational indices. Also we focus on interesting subcases of gravity, and particularly , mimetic gravity and gravity, and we use various illustrative examples in order to better support the theoretical formalism. In the same section we present a very useful reconstruction technique which offers the possibility of obtaining the gravity from the Einstein frame theory with a given potential. In section IIIC we present the study of inflationary dynamics in the context of the modified GaussBonnet gravity of the form , where is the GaussBonnet scalar. We use a different approach in comparison to the previous sections of this chapter, so we calculate directly the power spectrum of primordial curvature perturbations, and from this we calculate the spectral index of the primordial curvature perturbations. As a peripheral study we discuss how the perturbations evolve after the horizon crossing, by exploiting a specifically chosen example. We also briefly present the formalism of inflationary dynamics in the context of and one loop quantum gravity. Particularly, in the case we realize the intermediate inflation scenario in section IIID, while section IIIE is devoted to the singular inflation scenario and various phenomenological implications of this cosmological evolution. Finally,in section IIIF we present a useful theoretical approach for the graceful exit issue and in IIIG the reheating era in the context of modified gravity is studied. Usually in modified gravity the graceful exit instance is identified with the moment that the first slowroll index becomes of the order one. However, the growing perturbations that are caused by the unstable de Sitter points can also provide a mechanism for graceful exit from inflation. So we present in brief all the essential information for this alternative approach to the graceful exit issue.
In chapter IV we address the dark energy issue in the context of modified gravity. Particularly, we discuss how the dark energy era can be realized by and gravity, and also we discuss various phenomena related to the latetime era realization with modified gravity. But firstly, we will show in sections IVA and IVB, how the successful CDM model can be realized by and gravity respectively. In sections IVC and IVD we will demonstrate how it is possible to provide a unified description of early and latetime acceleration with and gravity. In section IVE we discuss how a phantom cosmological evolution can be realized in the context of gravity. Finally, in section IVF we will discuss an important feature of gravity when the latetime era is studied. Particularly, we present the dark energy oscillations issue in gravity, and we discuss how this may affect the latetime era. We shall use various matter fluids present, from perfect matter fluids to collisional, and we critically discuss how the fluid viscosity may affect the dark energy oscillations issue.
In chapter V we discuss some astrophysical applications of modified gravity, emphasizing in gravity applications. We shall present some neutron stars and quark stars solutions in gravity, and we briefly examine the astrophysical consequences of these solutions, for general equations of state for the neutron star. Also we discuss the possibility of antievaporation from ReissnerNordström black holes in gravity and finally we present some wormhole solutions from gravity.
In chapter VI we present the general reconstruction techniques which can be used in order to realize bouncing cosmologies with , and gravity. We shall present how to realize bounces by vacuum modified gravity, but also in some cases we shall present how the results are modified in the presence of perfect matter fluids. In the beginning of the chapter we provide an informative overview of bouncing cosmologies which is necessary in order to understand the fundamental characteristics of a cosmological bounce. In section VIA we discuss how a specific bounce ca be realized with vacuum gravity, but we also present the case that perfect matter fluids are present. In section VIB we discuss the realization of the same bounce cosmology as before, and finally in section VIC we discuss the gravity realization. In all three cases, we examine the stability of the modified gravity solutions we found, at the level of the equations of motion. Specifically, we study the stability of the equations of motion, if these are viewed as a dynamical system, and we demonstrate in which cases stability can be ensured.
Finally the conclusions follow at the end of this review, were we summarize the successes of modified gravity and we outline its shortcomings as a theory. Also we discuss in brief various challenging problems in cosmology, which still need to be incorporated successfully to a future theory.
Ii Modified Gravities and Cosmology
ii.1 Gravity
The theory of gravity could be considered as the most popular among modified gravity theories. In this section, a general review of the gravity theory is given. In the literature there are various reviews also discussing this topic, see Bamba:2015uma ; Nojiri:2013zza ; Nojiri:2006ri ; Capozziello:2011et ; Capozziello:2010zz ; Nojiri:2010wj ; Clifton:2011jh .
ii.1.1 General properties
The action of the gravity Nojiri:2006ri is given by replacing the scalar curvature in the EinsteinHilbert action which is,^{1}^{1}1We use the following convention for the curvatures and connections:
(1) 
by using some appropriate function of the scalar curvature, as follows,
(2) 
ndequation In Eqs. (1) and (2), is the matter Lagrangian density. We now review in brief the general properties of gravity.
For later convenience, we define the effective equation of state (EoS) parameter for the gravity theory. The expression can be used in any other modified gravity context. We start with the FRW equations, which in the Einstein gravity coupled with perfect fluid are:
(3) 
Then, the EoS parameter can be given by using the Hubble rate , in the following way,
(4) 
In principle we can use the expression given in Eq. (4) even for modified gravity theories, since it is useful for a generalized fluid description of modified gravity.
By varying the action (2) with respect to the metric, we obtain the equation of motion for the gravity theory as follows,
(5) 
For the spatially flat FRW Universe, in which case the metric is given by,
(6) 
Eq. (5) gives the FRW equations,
(7)  
(8) 
where the Hubble rate is equal to . In terms of the Hubble rate , the scalar curvature is equal to .
We can find several (in many cases exact) solutions of Eq. (7). Without the presence of matter, a simple solution is given by assuming that the Ricci tensor is covariantly constant, that is, . Then Eq. (5) is simplified to the following algebraic equation Nojiri:2003ft (see also Cognola:2005de ):
(9) 
If Eq. (9) has a solution the (anti)de Sitter and/or Schwarzschild (anti)de Sitter space
(10) 
or the Kerr  (anti)de Sitter space is an exact solution in vacuum. In Eq. (10), the minus and plus signs in correspond to the de Sitter and antide Sitter space, respectively. In Eq. (10), is the mass of the black hole, , and is the length parameter of (anti)de Sitter space, which is related to the curvature as follows (the plus sign corresponds to the de Sitter space and the minus sign to the antide Sitter space).
We now consider the perfect fluid representation and scalartensor representation of the gravity. For convenience, we write as the sum of the scalar curvature and the part which expresses the difference from the Einstein gravity case,
(11) 
Eqs. (7) and (II.1.1) indicate that we can express the effective energy density and also the effective pressure including the contribution from gravity as follows, (see, for instance, Nojiri:2009xw )
(12)  
(13) 
which enables us to rewrite the equations (7) and (II.1.1) as in the Einstein gravity case (3),
(14) 
The fluid representation for the FRW equations, however, often leads to an unjustified treatment. For example, it is often ignored that the generalized gravitational fluid contains higherderivative curvature invariants. The viable dark energy models of gravity are discussed in Refs. Nojiri:2003ni ; Nojiri:2006be ; Starobinsky:2007hu ; Nojiri:2007jr ; Nojiri:2003wx , while the unified description of inflation with dark energy are discussed in Refs. Nojiri:2006gh ; Nojiri:2003ft , for a review see Nojiri:2010wj .
ii.1.2 Scalartensor description
We should note that we can also rewrite gravity in a scalartensor form. We introduce an auxiliary field and rewrite the action (2) of the gravity in the following form:
(15) 
We obtain by the variation of the action with respect to and by substituting the obtained equation into the action (15), we find that the action in (2) is reproduced. If we rescale the metric by a kind of a scale transformation (canonical transformation),
(16) 
we obtain the action in the Einstein frame ^{2}^{2}2Note that the difference between the (Jordan) frame and the scalartensor (Einstein) frame description, may lead to a number of issues, for example the Universe in one frame may be accelerating, while decelerating in the other frame Capozziello:2006dj ; Bahamonde:2017kbs , or the singularity types changes from frame to frame Briscese:2006xu ; Bahamonde:2016wmz .,
(17) 
Here is given by solving the equation as . Due to the scale transformation (16), a coupling of the scalar field with usual matter is introduced. The mass of the scalar field is given by
(18) 
and if the mass is not very large, there appears a large correction to the Newton law. Since we would like to explain the accelerating expansion of the current Universe by using the gravity, we may naively expect that the order of the mass should be that of the Hubble rate, that is, . Since the mass is very small, very large corrections to the Newton law could appear. In order to avoid the above problem in the Newton law, a socalled “realistic” model was proposed in Hu:2007nk . In the model, the mass becomes large enough in the regions where the curvature is large or under the presence of matter fluids, as in the Solar System, or in the Earth. Therefore the force mediated by the scalar field becomes shortranged, which also introduces a screening effect in which only the surface of the massive objects like the planets can contribute to the correction to the Newton law even in the vacuum. This is called the Chameleon mechanism Khoury:2003rn , which prevents the large correction to the Newton law.
For example, we may consider the following model, Cognola:2007zu (see also Linder:2009jz ; Bamba:2010ws ),
(19) 
In the Solar System, where , if we choose , we obtain , which is ultimately heavy. In the atmosphere of the Earth, where , and if we choose , again, we obtain . Then, the correction to the Newton law looks to be extremely small in this kind of model. The corresponding Compton wavelength, however, is very small and much shorter than the distance between the atoms. Since the region between the atoms can be regarded as a vacuum except several quantum corrections and we cannot use the approximation to regard the matter fluids as continuous fluid, the Chameleon mechanism does not apply. Therefore, the Compton length cannot be shorter than the distance between the atoms and the scalar field cannot become so large, although this is not in conflict with any observation nor experiment.
We now also need to mention the problem of antigravity. Eq. (15) tells that the effective gravitational coupling is given by . Therefore when is negative, it is possible to have antigravity regions Nojiri:2003ft . Then, we need to require
(20) 
We should note that from the viewpoint of the field theory, the graviton becomes ghost in the antigravity region.
It should be noted that the de Sitter or antide Sitter space solution in (9) corresponds to the extremum of the potential . In fact, we find,
(21) 
Therefore, if Eq. (9) is satisfied, the scalar field should be on the local maximum or local minimum of the potential and can be a constant. When the condition (9) is satisfied, the mass given by (18) has the following form:
(22) 
Therefore, in the case that the condition (20) for avoiding the antigravity holds true, the mass squared is positive, showing that the scalar field is on the local minimum if,
(23) 
On the other hand, the scalar field is on the local maximum of the potential if,
(24) 
In this case the mass squared is negative. The condition (23) is nothing but the stability condition of the de Sitter space.
Note that in the Einstein frame, the Universe is always in the nonphantom phase, where the effective EoS in (4) is larger than although in the Jordan frame, the Universe can be, in general, in a phantom phase. This is because the scale transformation in (16) changes the time coordinate. We should note that we observe the expansion of the Universe via matter. In the Einstein frame, the metric for the matter is not given by but by , which is nothing but the metric in the Jordan frame. Therefore the observations in the Einstein frame is not changed from the observation in the Jordan frame. This indicates that in the Einstein frame, the metric is not physical but is the physical metric Capozziello:2006dj .
We now shortly mention on the matter instability issue pointed out in Dolgov:2003px . The instability may appear when the energy density or the scalar curvature is large compared with that in the Universe, as for example on the Earth. Let the background scalar curvature and separate the scalar curvature as a sum of and the perturbed part as . Then we obtain the following perturbed equation:
(25)  
(26) 
If we assume and are uniform, we can replace the d’Alembertian the second derivative with respect to the time coordinate and therefore Eq. (II.1.2) has the following structure:
(27) 
Then, if , becomes exponentially large with time , we have and the system is rendered unstable. We should note, however, that the scalar field in (II.1.2) is nothing but the scalar curvature and therefore the above matter instability occurs when the mass of the scalar field is negative, . Conversely, if we choose for large in such a way, so that the mass becomes positive, , then the instability does not occur.
ii.1.3 Viable gravities
Using the previous arguments, we summarize the conditions that need to hold true in order for an gravity to be a viable cosmological model, which unifies the accelerating expansion of the present Universe and the earlytime acceleration of the Universe. The first unified inflation dark energy gravity model appeared in Ref. Nojiri:2003ft , and in Refs. Nojiri:2007as ; Nojiri:2007cq ; Cognola:2007zu ; Cognola:2008zp ; Sokolowski:2007pk ; Brookfield:2006mq ; Abdelwahab:2007jp ; Oikonomou:2013rba , various viable models of gravity unifying both of the latetime and the earlytime acceleration were proposed by requiring several conditions, which we list here:

A condition to generate the inflationary era is given by,
(28) Here, is an effective cosmological constant characterizing the early Universe.

In order to generate the accelerating expansion of the Universe at present time, the current value of should be a small constant,
(29) Here, expresses the present time curvature . Note that because we need to take into account the contribution from matter, since the trace part of Eq. (5) indicates that . Here, is the trace part of the energymomentum tensor of all matter fluids. We should note that the quantity needs not to vanish completely but instead it should satisfy . This is due to the fact that we consider the time scale to be years.

The last condition is given by
(30) that is, a flat spacetime solution (Minkowski spacetime) should exist.
Let us consider the powerlaw model where behaves as
(31) 
when is large. Here and are arbitrary constants. The constant may vanish but should not, . Then, the trace equation, which is the trace part of Eq. (5),
(32) 
indicates that,
(33) 
We now assume that the Hubble rate has a structural singularity as follows,
(34) 
where and are arbitrary constants suitably chosen so that the Hubble rate is real. Then the scalar curvature behaves as follows,
(35) 
In Eqs. (34) and (35), the case corresponds to a Type I (Big Rip) singularity, see Refs. Caldwell:1999ew ; McInnes:2001zw ; Nojiri:2003vn ; Nojiri:2003ag ; Faraoni:2001tq ; GonzalezDiaz:2003rf ; GonzalezDiaz:2004as ; Singh:2003vx ; Csaki:2004ha ; Wu:2004ex ; Nesseris:2004uj ; Sami:2003xv ; Stefancic:2003rc ; Chimento:2003qy ; Hao:2004ky ; Babichev:2004qp ; Zhang:2005eg ; Elizalde:2005ju ; Dabrowski:2004hx ; Lobo:2005us ; Cai:2005ie ; Aref'eva:2004vw ; Aref'eva:2005fu ; Lu:2005qy ; Godlowski:2005tw ; Guberina:2005mp ; Dabrowski:2006dd ; Barbaoza:2006hf . The case corresponds to a Type III singularity, the case corresponds to a Type II, and finally the case (but ) corresponds to a Type IV singularity.
The above classification of the finitetime future singularities was proposed in Ref. Nojiri:2005sx . Particularly, the finitetime singularity classification is the following:

Type I (“Big Rip”) : This type of singularity occurs for , , and . Its manifestations in various models and theoretical contexts, have been studied in Ref. Nojiri:2005sx .

Type II (“sudden”) Barrow:2004he ; Barrow:2004hk ; Barrow:2004xh ; FernandezJambrina:2004yy ; Dabrowski:2004bz ; Lake:2004fu ; Nojiri:2004ip ; deHaro:2012wv ; BouhmadiLopez:2006fu ; BouhmadiLopez:2007qb ; BeltranJimenez:2016dfc : This type of singularity occurs for , , and .

Type III : This type of singularity occurs for , , and .

Type IV : This type of singularity occurs for , , , and higher derivatives of diverge. This also includes the case in which () or both and tend to some finite values, whereas higher derivatives of the Hubble rate diverge. This type of singularity was proposed in Nojiri:2005sx .
Here, and are defined in Eq. (3).
Substituting Eq. (35) into Eq. (33), one finds that there are two classes of consistent solutions. One solution is given by and (but ), which corresponds to the Big Rip ( and ) or Big Bang ( and ) singularity at . Another solution is , and (), which corresponds to the Type II future singularity. We should note that when , that is, , there is no singular solution.
Therefore, if we add the above term , where , to the action of and if the added term dominates for large , the modified gravity part of the gravity, namely , behaves as and the future singularity could disappear. On the other hand, if we add the term it dominates as it behaves as an term with for large . Then the singularity appears because this case corresponds to , that is, the case of the Big Rip singularity. In case of , the future singularity does not appear Nojiri:2008fk .
In order to avoid the occurrence of a finitetime future singularity, an additional term is needed to be added in the gravity Lagrangian. The addition of this term was first proposed first in Ref. Abdalla:2004sw in order for the Big Rip singularity to disappear. Furthermore, the term, which effectively eliminates the future singularity, generates the earlytime acceleration simultaneously. In other words, adding the term to a gravitational dark energy model, may lead to the emergence of an inflationary phase in the model as first observed in Ref. Nojiri:2003ft and also the singularity is removed. In Refs. Nojiri:2008fk ; Bamba:2008ut ; Capozziello:2009hc ; Bamba:2009uf , it has been investigated that the term could cancel all types of future singularities. The term often solves other phenomenological problems Kobayashi:2008wc ; Sami:2009jx ; Thongkool:2009js ; Babichev:2009fi ; Appleby:2010dx of dark energy. Even for the models of dark energy besides the gravity, e.g., the models using the perfect fluid approach or in a scalar field context, there could often appear a singularity in the finite future Universe, but by the addition of the term, the singularity can be eliminated Nojiri:2009pf .
Summarizing the above analysis, in order to obtain a realistic and viable gravity model:

In the limit of , the Einstein gravity should be recovered,
(36) If this condition is satisfied, a flat space (Minkowski) is also an solution as in (30).

As we will discuss later, there should appear the stable de Sitter solution, which corresponds to the latetime acceleration, where the curvature is small . This requires that, when ,
(37) Here, and are positive constants and is a positive integer. In some cases this condition may not always be necessary.

As we will also discuss later on, the quasistable de Sitter solution that corresponds to the inflationary era, in the early Universe should appear. In this case, the curvature is large . The de Sitter space should not be exactly stable so that the curvature decreases very slowly. This requires the following condition,
(38) Here, and are positive constants and is a positive integer.

As shown in (20), in order to avoid the antigravity regime, we require the following condition,
(41) which can be rewritten as follows,
(42)
The conditions (1) and (2) indicate that an extra, unstable solution describing the de Sitter spacetime always appears at . Due to the fact that the de Sitter solution is stable, the evolution of the Universe will stop at and therefore the curvature will not become smaller than , which indicates that the extra de Sitter solution is not realized. An example of such gravity is given in Nojiri:2010ny (the models in Nojiri:2007as ; Nojiri:2007cq ; Cognola:2007zu partially satisfy the above conditions). Also for a study on the constraints of gravity coming from large scale structures, see Lombriser:2010mp ; Schmidt:2009am .
We may consider the nonstandard nonminimal coupling of modified gravity with the matter Lagrangian Nojiri:2004bi ; Allemandi:2005qs ; Bertolami:2008zh (see also Capozziello:1999xt ),
(44) 
Here, is the Lagrangian density similar to that of a standard matter fluid. In a more generalized theoretical context, we may extend the model of Eq. (44) in the form of an gravity,
(45) 
Then by varying of the action (45) with respect to the metric we obtain,
(46) 
where is the effective energy momentum tensor defined as follows,
(47) 
We should note that although is conserved, that is, , in contrast the tensor is not conserved in general. We can define this model by specifying the Lagrangian density to be that of a free massless scalar,
(48) 
In the FRW Universe (6), the component and component in Eq. (46) give the following equations,
(49)  
(50) 
Here and have the following expressions,
(51)  
(52) 
where and are the energy density and the pressure given by . This model may generate the accelerating expansion of the Universe, due to the nontrivial coupling of the curvature, although there are problems with geodesics Bertolami:2008ab .
As a variation of the gravity, we may also consider gravity Harko:2011kv , whose action is given by,
(53) 
Here stands for the trace of some “energymomentum” tensor in some sense. Then, by varying the action with respect to the metric, we obtain,
(54) 
Due to the presence of the term including , the theory cannot be correctly defined without specifying the metric dependence of . Usually, the energy momentum tensor is given by the variation of some action with respect to the metric. If the action is given in terms of the fields, the action (53) can be expressed in terms of the fields directly. We should also note that due to the existence of the term in the action of Eq. (53), the conservation of is violated in general. For the FRW Universe (6), the component and component in Eq. (54) give the following equations,
(55)  
(56) 
If we require the conservation laws for , the additional constraints must be imposed on the model. Furthermore by imposing additional assumptions just for the solvability of the theory, several solutions have been studied Momeni:2011am ; Sharif:2012zzd ; Houndjo:2011tu ; Alvarenga:2013syu ; Haghani:2013oma ; Odintsov:2013iba ; Houndjo:2011fb .
ii.2 Modified GaussBonnet Gravity
We consider another class of models in modified gravity in which an arbitrary function of the topological GaussBonnet invariant is added to the action of General Relativity. We call this class of modified gravity theory “modified GaussBonnet gravity”. This class of modified gravity could be closely related with (super)string theory.
ii.2.1 General properties
The starting action is given by Nojiri:2005jg ; Nojiri:2005am ; Cognola:2006eg ; Elizalde:2010jx ; Izumi:2014loa :
(57) 
By varying the action with respect to the metric , we obtain the following equations of motion,