Inflationary Universe from Anomaly-Free <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 33.3056 15.2545" width="33.3056pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g><g transform="matrix(.017,0,0,-0.017,10.421,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,16.359,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.017,0,0,-0.017,27.118,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Gravity

Beesham, Aroonkumar; Bamba, Kazuharu

doi:https://doi.org/10.1155/2022/9056096

Advances in High Energy Physics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Classical Extensions and Alternative Theories of Gravity

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 9056096 | https://doi.org/10.1155/2022/9056096

Inflationary Universe from Anomaly-Free -Gravity

Aroonkumar Beesham^1,2and Kazuharu Bamba³

Academic Editor: Saibal Ray

Received15 Oct 2021

Revised27 Nov 2021

Accepted27 Jan 2022

Published29 Mar 2022

Abstract

By adding a three-dimensional manifold to an eleven-dimensional manifold in supergravity, we obtain the action of -gravity and find that it is anomaly-free. We calculate the scale factor of the inflationary universe in this model and observe that it is related to the slow-roll parameters. The tensor-scalar ratio is in good agreement with experimental data.

1. Introduction

One of the best models for considering an inflationary universe is suggested by modified theories of gravity like gravity, whose Lagrangian is a function of the scalar curvature in a 4-dimensional universe [1–5]. To date, many models in this subject have been proposed. However, one of the best inflationary models in gravity is the pioneering Starobinsky model, which still remains viable now in contrast to many other models proposed later and produces the best fit to existing observational data. Primordial curvature perturbations and gravitational waves in this model were first calculated in [6]. On the other hand, in these theories, the special property is the existence of the effective cosmological constant in such a way that early-time inflation and late-time cosmic acceleration are naturally unified within a single theory [1].

Some of these gravity theories, especially a power-law mechanism and Yang-Mills gravity, give the best fit values compatible with cosmological parameters like the tensor-to-scalar ratio within the permitted ranges derived by the Planck and BICEP2 observations [2]. In some other versions of these theories, a dynamical scalar field () is coupled to gravity whose dynamics allow for exit from inflation and which gives rise to the correct amount of inflation in agreement with observational data [3]. Also, in some modified gravity theories, is a generic function of the curvature scalar and the Gauss-Bonnet topological parameter. Cosmological dynamical results which are obtained by two effective masses (lengths) correspond to both of these parameters and work, respectively, at early and very early epochs of the evolution of the universe [4]. Furthermore, modified teleparallel gravity leads to second-order equations and solves the particle horizon problem in a spatially flat cosmic space by providing an initial exponential expansion without resorting to an inflaton particle [5]. For more reviews on the connections between inflation, the dark energy problem, and modified gravity theories, see, for instance, [7].

Now, the question that arises is whether -gravity could be anomaly-free or not? Also, it is required to consider the role of anomaly cancellation in the evolution of the universe during the inflationary epoch. Recently, the exact forms of two-dimensional and four-dimensional anomalies from higher derivative gravity and -gravity have been calculated [8]. On the other hand, the conformal-anomaly driven inflation in -gravity without using the scalar-tensor representation has been studied. It has been shown that in -gravity, the curvature perturbations with their high amplitude which are consistent with observations and are created during inflation [6, 9].

Besides these papers, there are some interesting works which connect M-theory to -gravity and inflation. For example, the first attempt to get from strings/M-theory is done long ago in [10]. On the other hand, the study of inflation in with scalars was done in k-essence gravity inflation [11, 12]. Also, the reconstruction of and mimetic gravity from viable slow-roll inflation has been discussed in [13]. And finally, the relation between Geometric Inflation and Dark Energy with Axion -gravity has been argued in [14]. These investigations advise us that there may be a relation between generalized gravity theories, M-theory, and inflation. Each gravity theory may have some anomalies which could be removed in a theory with higher dimensions. For example, all anomalies in string theory have been removed in 11-dimensional M-theory. Thus, we can conclude that anomalies in M-theory may be removed in a theory with -dimensions. In fact, first, we should consider the origin of M-theory and process of its birth on 11-dimensional manifold in a -dimensional manifold and then study its reduction to -gravity in four-dimensional universe. On the other hand, to find the relation between inflation and M-theory and gravity, we have to assume that our universe is located on a manifold which interact and exchange energy and fields with other manifolds in a higher dimensional world. These interactions force to manifolds to become larger and inflate. Motivated by these ideas, we propose a theory which lives on an ()-dimensional manifold with two 11-dimensional manifolds and one 3-dimensional manifold. We will show that our universe is a part of an 11-dimensional manifold which is connected with the other 11-dimensional manifold by an extra 3-dimensional manifold. Two 11-dimensional manifolds interact with each other via exchanging fields which move along the 3-dimensional manifold. These fields are the main cause for the appearance of -gravity in the four-dimensional universe.

This paper consists of two main parts. In Section 2, we show that by adding 3-dimensional manifold to an 11-dimensional one, -gravity emerges. In Section 3, we obtain the cosmological parameters like the tensor to scalar ratio and compare with experimental data.

2. Emergence of -Gravity on an ()-Dimensional Manifold

In this section, we will assume that our universe has been constructed on a D3-brane. Thus, the action of gravity and matter in a 4-dimensional universe should be equal to the action of a D3-brane (see Figure 1). This means that strings which live on a D3-brane produce gravity and different types of matter. Thus, each string () can be expanded in terms of curvatures (), gauge fields (), and scalars () (see Figure 2). According to the Horava-Witten mechanism [15, 16], this D3-brane can be a part of a bigger 11-dimensional manifold on which the action of fields should be anomaly-free (see Figure 3). However, using the relation between strings and fields, we notice that the gauge variation of the action on this manifold is not zero, and some extra anomalies emerge. To remove these anomalies, we have to add an extra 11-dimensional manifold which is connected with the first 11-dimensional manifold by a 3-dimensional manifold. The extra fields which are produced by the interaction of the two manifolds produce -gravity (see Figure 4).

First, let us introduce the action of the D3-brane which is given by [17]: where is the gauge field, is the field strength, is the string, is the metric, is the tension, and is the string length. Substituting and doing some mathematical calculations, the action of the D3-brane in equation (1) is given by [17]:

To construct our universe on a D3-brane, this action should be equal to the action of fields and gravity in a 4-dimensional universe. The action of matter and gravity is given by: where is the scalar field and is the fermionic field. Putting equation (4) equal to equation (5) and doing some mathematical calculations, we obtain the relation between strings and matter fields: where we have assumed and . Using the above relation between matter fields and strings, we can reconsider the Horava-Witten mechanism. We will show that there are also other anomalies that have been ignored in previous considerations. Our goal is to show that by adding a 3-dimensional manifold to 11-dimensional spacetime in the Horava-Witten mechanism, all anomalies can be removed and an action without anomaly can be produced. This action is identical to the action of the modified gravity theory presented in [1–5, 7].

At this stage, we introduce the Horava-Witten mechanism in 11-dimensional spacetime. In this theory, the bosonic part of the action in 11-dimensional supergravity (SUGRA) is given by [15, 16]: where is the rank- Levi-Civita pseudotensor, and CGG is used to denote the product term of the three-form field and four-form field , which are directly related to the gauge field , field strength , and Ricci curvature [16]: where and characterize infinitesimal gauge transformations [16]. Here, is 1 for and for and also is the Dirac delta function. As usual [16], is 1/30th of the trace in the adjoint representation for . The ellipsis () denotes terms that are regular near and hence vanish there [16]. It is clear from the above equation that G-fields can be produced by joining F-fields or R-fields. This means that G-fields are produced by joining two strings, each of which produces one gauge field or curvature () (see Figure 5).

The gauge variation of the CGG term in the action yields the following equation [16]:

where or . The above terms cancel the anomaly of () in 11-dimensional manifolds [16]:

Thus, is necessary for anomaly cancellation. Our goal now is to find a good rationale for its inclusion. We also answer the issue of the relation between CGG terms in 11-dimensional supergravity and -gravity. In fact, we suggest a theory in which -gravity terms appear in the supergravity action without being added by hand. To this end, we choose a unified form for all particles and fields by using Nambu-Poisson brackets and properties of string fields (). Solving equation (7) and assuming that changes of fields with respect to the coordinates be ingnorable, we obtain [18, 19] where we introduced the scalar field . Also, is the gauge field and is the curvature (R). In fact, the origin of all matter is the same and they are different shapes of strings. In the static state, all strings can be described by a unit vector (). When these strings interact with each other or move, the initial symmetry is broken and fields and particles emerge. Using 4-dimensional instead of 2-dimensional brackets, we may derive the GG term in supergravity in terms of strings () [18, 19]:

Equation (21) helps us to extract the CGG terms from the GG terms in supergravity. To this aim, we will add a 3-dimensional manifold to the 11-dimensional manifold which connects it to other 11-dimensional manifold by applying the properties of strings () in the Nambu-Poisson brackets [19]: where ellipses (…) were used to represent higher-order derivatives. The integration is over a 3-dimensional manifold with coordinates (), and consequently, the integration can be shown by ). This result shows that ignoring fluctuations of strings that leads to the production of fields, the result of integration over each 3-dimensional manifold tends to one. When we add one manifold to another, the integration will be the product of integration over each manifold.

Extending the manifold over additional dimensions extends the integration volume element. By extending the 11-dimensional manifold in equation (21) with the 3-dimensional manifold of equation (23), we get [19] where we notice that after making the identification [19]: we recover the CGG action in dimensions [19]:

This equation has three interesting results: (1) the CGG term that appears in the supergravity action is a result of adding a 3-dimensional manifold to an 11-dimensional manifold. This manifold connects the first 11-dimensional manifold to the second one. Combining the 11-dimensional manifold with the 3-dimensional manifold yields 14-dimensional supergravity. The shape of the C-term is now clear in terms of string fields (). It is clear that C-fields are in fact G-fields such that one end of the strings is placed out of the 11-dimensional manifold and on the 3-dimensional manifold and for this reason, C-fields seem to have three ends only (see Figure 6).

To verify that the theory is correct, we should be able to get back the gauge variation of the CGG-action in equation (18) in terms of field strengths and curvature. To this end, using equations (20), (21), (23), and (25), we can calculate the gauge variation of C [19]: where the ellipses (…) represent higher-order derivatives with respect to fields. Using equations (20), (21), (23), (25), and (27), we can calculate the gauge variation of the CGG action in the equation of (26):

The first term in equation (31) cancels the anomaly in equation (18). However, the other terms yield the action of modified gravity and matter. In fact, to remove the anomalies, we have to add two extra actions to equation (9), one is related to -gravity, and the second corresponds to matter. We can write where can be given by:

This -gravity is now free of any anomaly on the ()-dimensional manifold. We can write

The above equation shows that total action is free of any anomaly. This means that any variation in the supergravity action () produces some extra terms as new anomalies. These anomalies could be cancelled by variations in the Chern-Simon action (). Although, variations of these actions also produce some new anomalies, and these could be removed by variation of the f-gravity action (). Thus, the terms help to remove all anomalies.

In fact, this type of gravity is produced by exchanging gravitons, scalars, and fermions between two 11-dimensional manifolds via one 3-dimensional manifold. The order of curvatures depends on the number of dimensions of the manifolds and also on the number of scalars or fermions which are attached to the curvatures. This result is in good agreement with previous predictions in [8, 9].

3. Inflationary Universe in -Gravity

In this section, we construct a 4-dimensional universe on a ()-dimensional manifold. We will show that interaction between two 11-dimensional manifolds in this system has a direct effect on the evolution of the universe during the inflationary epoch. We will obtain the scale factor of the universe in terms of the parameters of the system. Using the action in equation of (36) and the relation between , , , and -terms in equation (17), we can obtain the following field equations:

where denotes the derivative with respect to the curvature and

For a 4-dimensional universe, we have , where is the Hubble parameter and is the scale factor of the universe. Assuming that all fields depend only on time, and using the relation between scalars, gauge fields, curvatures, and strings in equation (17), we can solve equations (39), (41), (43), and (45) simultaneously and obtain where is time of collision between the two 11-dimensional manifolds. This time is much larger than the present age of the universe (). The above result shows the coincidence with the birth of the universe () on the 11-dimensional manifold. The scale factor grows with time, while scalars, fermions, and gauge fields decrease and tend to zero at (). This is because with the passage of time, the two 11-dimensional manifolds come close to each other and all fields which live on the 3-dimensional manifold between them dissolve into the 11-dimensional manifolds. By the disappearance of these fields, gravitons and -gravity emerge that lead to the expansion and inflation of the universe (see Figure 7).

We can test our model by calculating the magnitude of the slow-roll parameters and the tensor-to-scalar ratio () defined in [20] and compare with previous predictions:

Obviously, during the inflationary epoch, the age of the universe () is very smaller with respect to the time of collision between manifolds () and, as a result, () where is an integer. Using equation (53) and expanding the functions ( and ) by applying the Taylor series, we can derive the following results:

This equation indicates that the values of the slow-roll parameters and the tensor-to-scalar ratio are very much smaller than the ones which are consistent with the predictions of the experiments in ref [21]. Thus, this type of -gravity which is produced due to the interaction of two 11-dimensional manifolds creates the correct values for the experimental parameters and can describe cosmological events.

In Figures 8 and 9, we have presented the scalar spectral index and the tensor to scalar ratio parameters in terms of time. It is clear that for the inflationary age of the universe and the colliding time (), the spectral index is less then or around and the tensor to scalar ratio is less than , which is comparable with results of Planck [22].

4. Summary and Conclusion

Motivated by the successes of modified theories of gravity such as -gravity, in this paper, we have investigated how we could make -gravity anomaly-free. To achieve this, we considered the origin of the action of -gravity on an ()-dimensional space-time. However, we found that this was not anomaly-free. So, we proposed a theory which lives on an ()-dimensional manifold with two 11-dimensional manifolds and one 3-dimensional manifold. We showed that our universe is a part of one 11-dimensional manifold which is connected with the other 11-dimensional manifold by an extra 3-dimensional manifold. The two 11-dimensional manifolds interact with each other via exchanging fields which move along the 3-dimensional manifold. These fields are the main cause for the appearance of -gravity in the four-dimensional spacetime. This is because with the passage of time, the two 11-dimensional manifolds come close to each other and all fields which live on the 3-dimensional manifold between them dissolve into the 11-dimensional manifolds. As these fields disappear, gravitons and -gravity emerge that lead to the expansion and inflation of the universe. Hence, the interaction between the two 11-dimensional manifolds has a direct effect on the evolution of the universe during the inflationary era. We have calculated the scale factor of the inflationary universe and examined our model against WMAP and Planck experiments. We have obtained the slow-roll parameters and shown that the tensor-scalar ratio is much smaller than unity (), which is consistent with experimental observations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

An arxiv has previously been published [23]. We have made use of data that is publicly available to support this study.

Acknowledgments

Aroonkumar Beesham acknowledges that this work is based on the research supported wholly/in part by the National Research Foundation of South Africa (Grant number: 118511). The work of KB was supported in part by the JSPS KAKENHI Grant Number JP21K03547. The authors are grateful to Professor Sergey Odintsov for bringing several relevant articles to their attention.

References

S. Nojiri and S. D. Odintsov, “Unifying inflation with ΛCDM epoch in modified gravity consistent with solar system tests,” Physics Letters B, vol. 657, no. 4-5, pp. 238–245, 2007.
View at: Publisher Site | Google Scholar
K. Bamba, S. I. Nojiri, S. D. Odintsov, and D. Saez-Gomez, “Inflationary universe from perfect fluid andF(R)gravity and its comparison with observational data,” Physical Review D, vol. 90, no. 12, article 124061, 2014.
View at: Publisher Site | Google Scholar
R. Myrzakulov, L. Sebastiani, and S. Vagnozzi, “General relativistic, nonstandard model for the dark sector of the Universe,” The European Physical Journal C, vol. 75, no. 1, 2015.
View at: Publisher Site | Google Scholar
M. De Laurentis, M. Paolella, and S. Capozziello, “Cosmological inflation inF(R,G)gravity,” Physical Review D, vol. 91, no. 8, article 083531, 2015.
View at: Publisher Site | Google Scholar
R. Ferraro and F. Fiorini, “Modified teleparallel gravity: inflation without an inflaton,” Physical Review D, vol. 75, no. 8, article 084031, 2007.
View at: Publisher Site | Google Scholar
V. F. Mukhanov and G. V. Chibisov, “Quantum fluctuations and a nonsingular universe,” JETP Letters, vol. 33, p. 532, 1981.
View at: Google Scholar
S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non- invariant models,” Physics Reports, vol. 505, no. 2-4, pp. 59–144, 2011.
View at: Publisher Site | Google Scholar
S.’i. Nojiri and S. D. Odintsov, “On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence,” International Journal of Modern Physics A, vol. 15, no. 3, pp. 413–428, 2000.
View at: Publisher Site | Google Scholar
B. Kazuharu, R. Myrzakulov, S. D. Odintsov, and L. Sebastiani, “Trace-anomaly driven inflation in modified gravity and the BICEP2 result,” Physical Review D, vol. 90, no. 4, article 043505, 2014.
View at: Publisher Site | Google Scholar
S.’i. Nojiri and S. D. Odintsov, “Where new gravitational physics comes from: M-theory?” Physics Letters B, vol. 576, no. 1-2, pp. 5–11, 2003.
View at: Publisher Site | Google Scholar
S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “k -essence f (R) gravity inflation,” Nuclear Physics B, vol. 941, pp. 11–27, 2019.
View at: Publisher Site | Google Scholar
S. V. Chervon, A. V. Nikolaev, T. I. Mayorova, S. D. Odintsov, and V. K. Oikonomou, “Kinetic scalar curvature extended f (R) gravity,” Nuclear Physics B, vol. 936, pp. 597–614, 2018.
View at: Publisher Site | Google Scholar
S. D. Odintsov and V. K. Oikonomou, “The reconstruction of and mimetic gravity from viable slow-roll inflation,” Nuclear Physics B, vol. 929, pp. 79–112, 2018.
View at: Publisher Site | Google Scholar
S. D. Odintsov and V. K. Oikonomou, “Geometric inflation and dark energy with axion F(R) gravity,” Physical Review D, vol. 101, no. 4, article 044009, 2020.
View at: Publisher Site | Google Scholar
P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven dimensions,” Nuclear Physics B, vol. 460, no. 3, pp. 506–524, 1996.
View at: Publisher Site | Google Scholar
P. Horava and E. Witten, “Eleven-dimensional supergravity on a manifold with boundary,” Nuclear Physics B, vol. 475, no. 1-2, pp. 94–114, 1996.
View at: Publisher Site | Google Scholar
G. Grignani, T. Harmark, A. Marini, N. A. Obers, and M. Orselli, “Heating up the BIon,” Journal of High Energy Physics, vol. 2011, no. 6, p. 58, 2011.
View at: Publisher Site | Google Scholar
P.-M. Ho and Y. Matsuo, “M5 from M2,” Journal of High Energy Physics, vol. 2008, no. 6, p. 105, 2008.
View at: Publisher Site | Google Scholar
A. Sepehri and R. Pincak, “The birth of the universe in a new G-theory approach,” Modern Physics Letters A, vol. 32, no. 5, article 1750033, 2017.
View at: Publisher Site | Google Scholar
S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “Unimodular F(R) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2016, no. 5, p. 46, 2016.
View at: Publisher Site | Google Scholar
P. A. R. Ade, N. Aghanim, M. Arnaud et al., “Planck 2015 results-XX. Constraints on inflation,” Astronomy & Astrophysics, vol. 594, 2016.
View at: Google Scholar
BICEP/Keck Collaboration, “BICEP / Keck XIII: improved constraints on primordial gravitational waves using Planck, WMAP, and BICEP/Keck observations through the 2018 observing season,” vol. 127, Article ID 151301, 2021, https://arxiv.org/abs/2110.00483.
View at: Google Scholar
A. Beesham and K. Bamba, “Inflationary universe from anomaly-free F(R)-gravity,” http://arxiv.org/abs/2107.06821.
View at: Google Scholar
S. Nojiri and S. D. Odintsov, “Modified f(R) gravity unifying R^m inflation with the Λ CDM epoch,” Physical Review D, vol. 77, no. 2, article 026007, 2008.
View at: Publisher Site | Google Scholar
S. Capozziello and V. Faraoni, “Beyond Einstein Gravity,” A Survey of Gravitational Theories for Cosmology and Astrophysics, Springer, Dordrecht, 2010.
View at: Google Scholar
S. Capozziello and M. De Laurentis, “Extended theories of gravity,” Physics Reports, vol. 509, no. 4-5, pp. 167–321, 2011.
View at: Publisher Site | Google Scholar
A. de la Cruz-Dombriz and D. Sáez-Gómez, “Black holes, cosmological solutions, future singularities, and their thermodynamical properties in modified gravity theories,” Entropy, vol. 14, no. 9, pp. 1717–1770, 2012.
View at: Publisher Site | Google Scholar
K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, “Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests,” Astrophysics and Space Science, vol. 342, no. 1, pp. 155–228, 2012.
View at: Publisher Site | Google Scholar
A. Joyce, B. Jain, J. Khoury, and M. Trodden, “Beyond the cosmological standard model,” Physics Reports, vol. 568, pp. 1–98, 2015.
View at: Publisher Site | Google Scholar
K. Koyama, “Cosmological tests of modified gravity,” Reports on Progress in Physics, vol. 79, no. 4, article 046902, 2016.
View at: Publisher Site | Google Scholar
K. Bamba and S. D. Odintsov, “Inflationary cosmology in modified gravity theories,” Symmetry, vol. 7, no. 1, pp. 220–240, 2015.
View at: Publisher Site | Google Scholar
A. De Felice and S. Tsujikawa, “f(R) Theories,” Living Reviews in Relativity, vol. 13, no. 1, 2010.
View at: Publisher Site | Google Scholar
Y. F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, “f(T) teleparallel gravity and cosmology,” Reports on Progress in Physics, vol. 79, no. 10, article 106901, 2016.
View at: Publisher Site | Google Scholar
S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “Modified gravity theories on a nutshell: inflation, bounce and late-time evolution,” Physics Reports, vol. 692, pp. 1–104, 2017.
View at: Publisher Site | Google Scholar
E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” International Journal of Modern Physics D, vol. 15, no. 11, pp. 1753–1935, 2006.
View at: Publisher Site | Google Scholar
S. I. Nojiri and S. D. Odintsov, “The minimal curvature of the universe in modified gravity and conformal anomaly resolution of the instabilities,” Modern Physics Letters, vol. 19, no. 8, pp. 627–638, 2004.
View at: Publisher Site | Google Scholar
K. Bamba, S. I. Nojiri, and S. D. Odintsov, “Trace-anomaly driven inflation in f(T) gravity and in minimal massive bigravity,” Physics Letters B, vol. 731, pp. 257–264, 2014.
View at: Publisher Site | Google Scholar
A. Sepehri, arXiv: 1701.04337.
S. D. Odintsov and V. K. Oikonomou, “Bouncing cosmology with future singularity from modified gravity,” Physical Review D, vol. 92, no. 2, article 024016, 2015.
View at: Publisher Site | Google Scholar
P. A. Ade, N. Aghanim, C. Armitage-Caplan et al., “Planck 2013 results. XXII. Constraints on inflation,” Astronomy & Astrophysics, vol. 571, 2014.
View at: Google Scholar
Planck Collaboration-Y, Astronomy & Astrophysics, vol. 641, 2020.
K. Bamba, S. I. Nojiri, and S. D. Odintsov, “Inflationary cosmology and the late-time accelerated expansion of the universe in nonminimal Yang-Mills-F(R)gravity and nonminimal vector-F(R)gravity,” Physical Review D, vol. 77, no. 12, article 123532, 2008.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Aroonkumar Beesham and Kazuharu Bamba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

287

Downloads

475

Citations