Research Article Testing a Dilaton Gravity Model Using ...downloads.hindawi.com/journals/ahep/2014/282675.pdf · particular type of dilaton gravity models proposed in [ ]. e idea

Research ArticleTesting a Dilaton Gravity Model Using Nucleosynthesis

S Boran and E O Kahya

Department of Physics Istanbul Technical University Maslak 34469 Istanbul Turkey

Correspondence should be addressed to S Boran boransituedutr

Received 7 July 2014 Revised 6 August 2014 Accepted 7 August 2014 Published 27 August 2014

Academic Editor Elias C Vagenas

Copyright copy 2014 S Boran and E O Kahya This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

Big bang nucleosynthesis (BBN) offers one of the most strict evidences for the Λ-CDM cosmology at present as well as the cosmicmicrowave background (CMB) radiation In this work our main aim is to present the outcomes of our calculations related toprimordial abundances of light elements in the context of higher dimensional steady-state universe model in the dilaton gravityOur results show that abundances of light elements (primordial D 3He 4He T and 7Li) are significantly different for some casesand a comparison is given between a particular dilaton gravity model and Λ-CDM in the light of the astrophysical observations

1 Introduction

Thecurrent expansion of the universe is a crucial evidence forthe big bang cosmologymodel It predicts the chemical abun-dances of primordial elements as a result of nuclear reactionswhich began seconds after the big bang and continued forthe next several minutes With the help of inflation one canconsistently solve the well-known problems of the standardmodel such as the observed spatial homogeneity isotropyand flatness of the universe [1]

There are still many unsolved puzzles of this model suchas the origin of dark matter and dark energy cosmologicalconstant problem cosmic coincidence problem and the exactform of the inflation potential [2 3] On the other hand thereare many models which claim solutions to these problemsby modifying Einsteinrsquos general relativity Quintessence 119896-essence phantom quintom and other phenomenologicalmodels are just few examples of alternate gravity models thatoffer a solution to the dark energy problem [4] And alsothere are alternative gravity theories that suggest using extrafields (scalar tensor etc) and higher dimensions (Kaluze-Klein Randall-Sundrum) arising from string theory at thelow-energy limit [5]

In this ocean of models we would like to consideran observable consequence (modified abundances of lightelements) of a new model a higher dimensional dilaton

gravity theory of steady-state cosmological (HDGS) modelin the context of string theory We need to highlight that theoriginal steady-state model [6 7] is unfavorable compared tothe standard big bang scenario But our motivation in thiswork is to suggest a test of a specific higher dimensionaldilaton gravity model which effectively mimics the standardFRW model with the modified Hubble constant Hencean immediate consequence would be the modification ofnucleosynthesis

This modification was investigated in [8] and it wasclaimed that this model gives a better estimate for theprimordial 4He abundance compared to the standard bigbang nucleosynthesis (SBBN) by choosing the number ofdimensions appropriately In this work to further test theirstrong claim we calculated the abundances of the primordialD 3He 4He T and 7Li in the context of this nonstandard(HDGS)model and compared itwith the predictions of SBBNand the astrophysical observations

On the other hand at high energies the quantum grav-itational corrections will start to play an important roleQuantum corrections will modify the dilaton gravity modelsas well and therefore change the whole form of this modelvia action [9ndash12] One would naturally expect to see quantumeffects during the very early universe such as primordial infla-tionary stage During inflation quantum loop effects maylead to very small [13 14] but possibly observable corrections

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 282675 7 pageshttpdxdoiorg1011552014282675

2 Advances in High Energy Physics

to power spectrum [15ndash20] Therefore one might describethe interactions with effective field theories of inflation [21]But in this work we aremainly interested in the consequenceof a geometrical constraint 3+119899-dimensional universe havinga constant volume leading to a modified Hubble parameterduring a later stage nucleosynthesis where quantum gravita-tional corrections are negligible

The paper is organized as follows In Section 2 dynamicsof this particular dilaton gravity model is summarized InSection 3 nucleosynthesis in the context of this model is ana-lyzed In Section 4 the results obtained from our calculationsfor this model and the predictions of SBBN with the help ofPlanck satellite data [22] are compared with the astrophysicalobservations

2 Dynamics of HDGS A DilatonGravity Model

In this section we briefly summarize the dynamics of aparticular type of dilaton gravity models proposed in [8]The idea is introducing a higher dimensional dilaton gravityaction of steady-state cosmology (HDGS) in the string frameTherefore the evolution of the internal 119899-dimensional spaceresults in an evolution of the observed universe to keepthe whole system in a steady state Due to this constraintchoosing particular values for some parameters such as thenumber of extra dimensions leads to possibly observableeffects in our universe Let us start with the action whichstems from the low-energy effective string theory

119878 = int119872

1198891+3+119899

119909radic10038161003816100381610038161198921003816100381610038161003816119890minus2120601

(119877 + 4120596120597120583120601120597120583120601 + 119880 (120601)) (1)

where 119877 is the curvature scalar 119872 stands for manifold 119899corresponds to extra dimensions |119892| is the determinant of119892

120583]metric tensor120601 is the dilaton field taken as space independentreal function of time and 120596 is an arbitrary coupling constant119880(120601) = 119880

0119890120582120601 is a real smooth function of the dilaton field

and corresponds to the dilaton self-interaction potential andboth119880

0and 120582 are real parameters Two interesting cases that

are worth mentioning are 120596 = 1 and 119899 = 6 correspondingto anomaly-free superstring theory and 120596 = 1 with 119899 = 22

corresponds to bosonic string theory The metric is given by

1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112)

+ 1199042(119905) (119889120579

2

1+ sdot sdot sdot + 119889120579

2

119899)

(2)

Here 119905 is the cosmic time and (119909 119910 119911) are the cartesiancoordinates of the 3-dimensional flat space basically theobserved universe The coordinates 120579 are 119899-dimensionalcompact (torodial) internal space coordinates (this representsspace that cannot be observed directly and locally today)While 119886(119905) denotes the scale factor of 3-dimensional externalspace 119904(119905) is the scale factor of 119899-dimensional internal space

This model has the following key properties(i) The (3 + 119899)-dimensional universe has a constant

volume that is 119881 = 1198863119904119899= 1198810 hence steady state But the

internal and external spaces are dynamical (ii) The energy

density is constant in the higher dimensional universe (iii)There is no higher dimensional matter source other than thedilaton field in the action

If the scalar field is redefined as 120573 = 119890120582120601 the relation

between the scalar field and the scale factor of the externalspace turns out to be

(1198861015840

119886)

2

=119899

3 (3 + 119899)

2120596120576

1205732 (3)

where 120576 is a constant of integration Here prime denotesderivative with respect to the ordinary time Imposing theconstant volume condition gives

119886 = 1198860119890plusmn(13)radic3119899(3+119899)radic2120596120576 int 119889119905(1120573)

119904 = 1199040119890∓(1119899)radic3119899(3+119899)radic2120596120576 int 119889119905(1120573)

(4)

where 1198860and 119904

0correspond to the integration constants

Therefore the modified Hubble parameter of the externalspace is obtained as follows

119867119886equiv1198861015840

119886= plusmn

1

3

radic3119899

3 + 119899

radic2120596120576

120573 (5)

Here the physically relevant case is the solution for expand-ing external space with 119867

119886gt 0 The deceleration parameter

for the external space is given by

119902119886equiv minus

11988610158401015840119886

11988610158402= minus1 plusmn 3radic

3 + 119899

3119899

1205731015840

radic2120596120576 (6)

In the case of 120576 = 0 and 1198800

= 0 and with the choice ofappropriate initial conditions it turns out that [8] the earlytime modified deceleration parameter is given by

119902 997888rarr 3radic3 + 119899

3119899

1

radic120596minus 1 (7)

3 Nucleosynthesis in HDGS

We are interested in how abundances of light elements wouldchange in the context of this model Specifically we wouldlike to consider the ratio of themodified expansion rate to thestandard expansion rate during the early radiation dominantepoch This ratio is given by

119878 equiv119867119886

119867SBBN=1 + 119902SBBN1 + 119902119886

(8)

This is true since deceleration parameter stays almost con-stant during primordial nucleosynthesis The value of thedeceleration parameter for standard BBN is 119902SBBN = 1 Since119902119886is given by (7) the so-called standard expansion factor 119878

can be expressed in terms of 120596 and 119899 as

119878 =2

3(radic

3120596119899

3 + 119899) (9)

Advances in High Energy Physics 3

If 119878 = 1 is taken it denotes nonstandard expansion factorThis kind of modification might also arise due to additionallight particles such as neutrinos which would make the ratiobe 119867SBBN = [1 + (743)(119873] minus 3)]

12 In this context ofthe dilaton gravity model that we mentioned it is also goingto occur due to a modification of general relativity We areinterested in the case where 119873] = 3 and therefore the valueof (119878 minus 1) will come only from the modification of generalrelativity

Theprimordial abundances of the light elements (primor-dial D 3He 4He 7Li and T) depend on the baryon densityand the expansion rate of the universe [23 24] The baryondensity parameter [23] is given by

12057810equiv 1010120578119861equiv 1010 119899119861

119899120574

= 2739Ω119861ℎ2 (10)

where 120578119861gives the baryon to photon ratio Ω

119861is dimen-

sionless current critical cosmological density parameter forbaryons and ℎ = ℎ

100equiv 1198670100 kmsminus1Mpcminus1 with119867

0being

the present value of the Hubble parameter Any modificationof the expansion rate would change the time when neutronsfreeze out which will in turn determine the final abundanceof helium-4 as well as all of the other light elements

In the following subsections we will analyze nucleosyn-thesis due to a modification of the expansion rate in thecontext of HDGS models We will express the primordialnuclear abundances of light nuclei in terms of two parametersofHDGSmodels number of extra dimensions119899 and couplingconstant 120596 Particularly we will be interested in the case of120596 = 1 where 119899 = 6 and 119899 = 22 correspond to anomaly-freesuperstring and bosonic string theory respectively

31 4He Abundance in HDGSModels The two body reactionchains of light elements which include deuterium (D)tritium (T) and helium-3 (3He) to produce helium-4 (4He)are more efficient than four body reactions of neutrons andprotons The first step is producing D from 119899 + 119901 rarr D + 120574After that D is converted into 3He and T as follows

D + D 997888rarr3He + 119899 D + D 997888rarr T + 119901 (11)

and finally 4He is produced from D combining with T and3He

T + D 997888rarr4He + 119899 3He + D 997888rarr

4He + 119901 (12)

In order to get precise estimates for abundances of lightelements one should solve nonlinear differential equationsof the nuclear reaction networkThis problem can be studiednumerically and the modern methods are based onWagoneret al [25] code and its updated version by Smith et al [26]The next step is getting a best fit to a numerical work to seehow various abundances depend on 120578

10and other parameters

such as number of extra neutrinos Another venue is applyingsemianalytical methods where one of the earliest works wasdone by Esmailzadeh et al [27] using the method of fixedpoints

In this work we would like to use if there exists the bestfit expressions for certain elements If there is none in the

literature for a certain element then we will use a semianalyt-ical approach that is based on a simple assumption which isthe nuclear reaction network obeying in a quasiequilibriumstate In this state basically one assumes that ldquothe total fluxcoming into each corresponding reservoir must be equal tothe outgoing fluxrdquo [28]

A simple way of estimating of 4He abundance (in generalabundance by weight is related to the ratio of number densityof a particular element to the number density of all nucleons(including the ones in complex nuclei)119883

119860equiv 119860119899119860119899119873 where

119860 is the mass number of a particular element eg 119860 = 4 forhelium) is the following multiply the abundance of neutronsby two at the time when the deuterium bottleneck opens upHere wewill refer to the best fit expression for 4Heabundancethat includes the case of modified expansion rate [29 30]

119884119901= 02485 plusmn 00006 + 00016 ((120578

10minus 6) + 100 (119878 minus 1))

(13)

where 119901 stands for the primordial abundance We will take12057810≃ 6 [31] from here on The SBBN value 119878 = 1 becomes

119884SBBN119901

= 02485 plusmn 00006 Using (9) for the case of HDGSmodels that we are interested in one can get the followingexpression for 4He abundance in terms of 120596 and 119899 as [8]

119884119901= 02485 plusmn 00006 + 016(minus1 +

2

3

radic3120596119899

3 + 119899) (14)

In the case of 120596 = 1 the predicted 119884119901values are obtained as

119884119901= 02393plusmn00006 and 119884

119901= 02618plusmn00006 for 119899 = 6 and

119899 = 22 respectivelyFrom the observational point of view the 4He primordial

abundance119884119901 is determined from the recombination of lines

of the H II from blue compact galaxies (BCGs) [32] Theobservational results of the 4He abundances are given by119884

119901=

02565 plusmn 00060 [33] and 119884119901= 02561 plusmn 00108 [34]

32 Abundances of Other Light Elements in HDGS Models

321 Deuterium Abundance Deuterium is produced by 119901 +119899 rarr D + 120574 and used in four types of reactions (11) (12)Therefore one would expect to solve either numerically oranalytically the equations for this nuclear reaction networkand get the expression for deuterium abundance 119883D equiv

2119899D119899N where 119899D and 119899N are the number densities ofdeuterium and all nucleons respectively

In literature instead of abundances of elements theirabundances relative to hydrogen are given To see why letus look at how deuterium is determined The absorbedprimordial element has more space in the wings of theobserved quasar absorption-line systems (QAS) [35ndash39] thanthe absorbed hydrogen at high redshifts (z) andor at lowmetallicity (Z) Also the observation of the multicomponentvelocities of these absorbed elements is very significant inorder to determine the abundance of deuterium Thereforethe (DH)

119901ratio is more meaningful and is often known as

interstellar mediummeasurement for deuterium abundance


This ratio can be expressed in terms of the abundance byweight of the deuterium as

119910D119901 equiv 105(119899D119899H

)

119901

= 105(13

24119883D119901) (15)

The factor 1324 comes from the fact that mass number ofdeuterium is 2 andhydrogennumber density is equal to 1213of all the nucleons in the universe that is 75 by weight

Let us start with the semianalytical expression forthe abundance of deuterium to calculate (15) Using thequasiequilibrium condition one can get [28]

119883D119901 ≃2119877

exp (11986012057810) minus 1

≃ 487 times 10minus5 (16)

where 119877 ≃ 2 sdot 10minus5 [28] 120578

10≃ 6 and 119860 ≃ 01 Here

the coefficients 119877 and 119860 are related to experimental valuesof nuclear reaction rates of deuterium at temperature oforder 008MeV (We assume that the nuclear interaction ratesare independent of extra dimensions We also assume thatthere are no matter sources in higher dimensions and HDGSis a Kaluza-Klein-type model rather than a brane worldcosmology one) Putting this value in (15) gives 119910SBBND119901 = 263

Let us now use a more precise expression for deuteriumabundance [23] based on a numerical best fit

119910D119901 = 260 (1 plusmn 006) (6

12057810minus 6 (119878 minus 1)

)

16

(17)

From this expression one can get the SBBN value of 119910D119901 (for119878 = 1 and 120578

10≃ 6) as 119910SBBND119901 = 260 plusmn 016 Comparing

this number with the one from the semianalytical method119910SBBND119901 = 263 we can safely assume a quasiequilibrium

condition if necessaryBy using (9) one can express 119910D119901 for HDGS models as

119910D119901

= 260 (1 plusmn 006)(6

12057810minus 6 (minus1 + (23)radic3120596119899 (3 + 119899))

)

16

(18)

Taking 12057810

≃ 6 the predicted values of 119910D119901 are obtained as119910D119901 = 238plusmn 016 and 119910D119901 = 299plusmn 016 for 119899 = 6 and 119899 = 22respectively for 120596 = 1model

Finally the observational results are 119910D119901 = 287 plusmn 022

[40] and 119910D119901 = 254 plusmn 005 [39]

322 Helium-3 Abundance The relevant nuclear reactionsthat involve 3He are

D + D 997888rarr3He + 119899 D + 119901 997888rarr

3He + 120574

3He + 119899 997888rarr T + 119901 3He + D 997888rarr4He + 119901

(19)

The quantity used in the literature to describe 3He is

1199103equiv 105(

1198993He

119899H) = 10

5(13

361198833He) (20)

Making a quasiequilibrium approximation for 3He abun-dance we can express the 3He abundance in terms ofdeuterium abundance after using the experimental values forthe ratios of the related nuclear reaction rates [28]

1198833He ≃02 sdot 119883D + 10

minus5

1 + 4 times 103119883D (21)

From this equation we can see that 3He abundance isnot as sensitive as deuterium since a change in deuteriumabundance would change both parts of the ratio One can alsosee this from the weaker dependence of 119910

3on 12057810 compared

to 119910D119901 for SBBN best fit expression [41] Consider

1199103= 31 (1 plusmn 003) 120578

minus06

10 (22)

Therefore 3He abundance is not a good indicator of amodification of SBBN due to HDGS models

323 Tritium Abundance Using the quasiequilibrium con-dition for tritium119883

119879

119891 [28] is obtained as

119883119879

119891≃ (0015 + 3 sdot 10

21198833He119891)119883D119891 (23)

It is clear from this expression that the value of tritiumabundance will be as sensitive as deuterium abundance toany modification of the expansion rate But the magnitude oftritium abundance is two orders of magnitude smaller thanboth deuterium and helium-3 Therefore observationally itis not very feasible but it should be kept in mind that it canbe used to test for consistency in the future experiments

324 Lithium-7 Abundance Finally we would like to inves-tigate the effects of modified expansion rate on lithiumabundance The 7Li abundance is given by

119910Lip equiv 1010(119899Li119899H

)

119901

(24)

One might think that its smallness would make it irrelevantfor observational purposes But it can actually be measuredin the atmospheres of metal-poor stars in the stellar haloof Milky Way All primordial elements point towards thesame 120578

10parameter except lithiumThe ratio of the expected

SBBN value of lithium-7 abundance to the observed one isbetween 24 and 43 [42]Therefore it should be interesting tocheck if these HDGSmodels offer any solution to the lithiumproblem

The best fit expression to the numerical BBN data of the119910Lip is given in [23] as

119910Lip = 482 (1 plusmn 010) (12057810minus 3 (119878 minus 1)

6)

2

(25)

Taking 119878 = 1 and 12057810

≃ 6 the SBBN value of lithium-7abundance is found as 119910SBBNLip = 482 plusmn 048 In terms of 120596and 119899 the modified form of (25) becomes

119910Lip = 482 (1 plusmn 010) (12057810 minus 3(minus1 +2

3

radic3120596119899

3 + 119899))

2

(26)


Table 1 The abundances of He-4 deuterium and Li-7 for different models

Models and data 119884119901

119910119863119901

119910Lip

SBBN model 02485 plusmn 00006 260 plusmn 016 482 plusmn 048

119899 = 6 dilaton gravity model 02393 plusmn 00006 238 plusmn 016 510 plusmn 051


Observational data 02561 plusmn 00108 [34] 288 plusmn 022 [40] 11 minus 15 [43]02565 plusmn 00060 [33] 254 plusmn 005 [39] 123

+068

minus032[44]

By using (26) the predicted 119910Lip values are found as 119910Lip =

510 plusmn 051 for 119899 = 6 and 119910Lip = 443 plusmn 044 for 119899 = 22 for thecase of 120596 = 1

4 Discussion

We have shown in this work that one gets a considerablemodification to the primordial abundances of light elementsin the case of a higher dimensional steady-state universe indilaton gravity (there are other ways to modify BBN based onscalar-tensor theories for details see [45ndash47] and referencestherein) Although there is a huge class of models that onecan consider with two free parameters 120596 (dilaton couplingconstant) and 119899 (number of internal dimensions) Here wefocused on two interesting cases 120596 = 1 with 119899 = 6 (anomaly-free superstring theory) and 120596 = 1 with 119899 = 22 (bosonicstring theory)

The main idea behind the calculation is modifying theexpansion rate during the nucleosynthesis to get differentabundances for light elementsOne can think of themodifica-tion as being similar to addingmore relativistic particles suchas extra neutrinos into the standard big bang model WhenHubble parameter gets modified all the nucleosynthesis willget modified as well The question is the following is thismodification large enough to observe and if it is then is itcompatible with the data

To answer these questions one should analyze how thenuclear reactions get modified with the modification of theexpansion rate It is well-known that the complete analysis ofthe nuclear reactions governing the primordial abundancesof light elements can be done using numerical methods Weused the results of the previous works where we can whichwere obtained by getting best fit expressions to numerical datarelated to the abundances of these elements And if there areno known best fit expressions in the literature we proceededour analysis based on semianalytical methods

The primordial abundance of helium-4 was already stud-ied in the context of these models It was pointed out that119899 = 22 case is more compatible with helium-4 data comparedto the standard big bang scenario We made a more extensiveanalysis of other light elements and checked the compatibilityof this model with astrophysical observables The results aresummarized in Table 1

One can clearly see from Table 1 that 120596 = 1 and 119899 = 6

dilaton gravity model is incompatible with helium-4 dataand is incompatible with deuterium as well Helium-4 datafavoured the case of 120596 = 1 and 119899 = 22 compared to SBBNas was noted In the case of deuterium earlier measurements

favour (with almost being inside the error bars) dilatongravity model whereas the more recent measurements rulethem out and point towards SBBN Therefore it is fair to saythat one needs more observations and data analysis to seewhich model is favoured

We also showed that helium-3 and tritium abundancesare not very convenient to see a modification of the standardmodel in the context of the dilaton gravity model consideredhere And for the case of lithium-7 one gets almost a tenpercent decrease for the expected abundance comparedto SBBN but it is still far from explaining the observedabundance So these models do not offer a solution to thelithium problem therefore the existence of this problem stillpreserves its place in the literature and leaves an openwindowto new physics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Ozgur Akarsu Ali Kayaand Subir Sarkar for the helpful discussions This work wassupported by TUBITAK-1001 Grant no 112T817

References

[1] A Linde ldquoInflationary cosmologyrdquo in Inflationary Cosmologyvol 738 of Lecture Notes in Physics pp 1ndash54 2007

[2] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[3] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 pp 565ndash586 1999

[4] R Durrer and RMaartens ldquoDark energy andmodified gravityrdquoin Dark Energy Observational and Theoretical Approaches PRuiz-Lapuente Ed pp 48ndash91 Cambridge University Press2010

[5] L Randall and R Sundrum ldquoLargemass hierarchy from a smallextra dimensionrdquo Physical Review Letters vol 83 no 17 pp3370ndash3373 1999

[6] H Bondi andTGold ldquoThe steady-state theory of the expandingUniverserdquo Monthly Notices of the Royal Astronomical Societyvol 108 no 3 pp 252ndash270 1948


[7] F Hoyle ldquoA new model for the expanding universerdquo MonthlyNotices of the Royal Astronomical Society vol 108 pp 372ndash3821948

[8] O Akarsu and T Dereli ldquoLate time acceleration of the 3-spacein a higher dimensional steady state universe in dilaton gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2013 no 2article 050 2013

[9] S Nojiri O Obregon and S D Odintsov ldquo(Non)-singularbrane-world cosmology induced by quantum effects in five-dimensional dilatonic gravityrdquo Physical Review D vol 62Article ID 104003 2000

[10] S Nojiri and S D Odintsov ldquoQuantum dilatonic gravity ind = 24 and 5 dimensionsrdquo International Journal of ModernPhysics A vol 16 no 6 pp 1015ndash1108 2001

[11] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[12] E Elizalde S Jhingan S Nojiri S D Odintsov M Sami andI Thongkool ldquoDark energy generated from a (super-) stringeffective action with higher-order curvature corrections and adynamical dilatonrdquoTheEuropean Physical Journal C vol 53 no3 pp 447ndash457 2008

[13] S Wienberg ldquoQuantum contributions to cosmological correla-tionsrdquo Physical Review D vol 72 no 4 Article ID 043514 19pages 2005

[14] S Weinberg ldquoQuantum contributions to cosmological correla-tions II Can these corrections become largerdquo Physical ReviewD vol 74 Article ID 023508 2006

[15] D Seery ldquoOne-loop corrections to a scalar field during infla-tionrdquo Journal of Cosmology and Astroparticle Physics vol 711 p25 2007

[16] D Seery ldquoOne-loop corrections to the curvature perturbationfrom inflationrdquo Journal of Cosmology and Astroparticle Physicsvol 2008 p 006 2008

[17] D Marolf and I A Morrison ldquoInfrared stability of de Sitterspace loop corrections to scalar propagatorsrdquo Physical ReviewD vol 82 Article ID 105032 2010

[18] S B Giddings and M S Sloth ldquoSemiclassical relations andIR effects in de Sitter and slow-roll space-timesrdquo Journal ofCosmology and Astroparticle Physics vol 1101 article 023 2011

[19] E O Kahya and V K Onemli ldquoQuantum stability of a w lt minus1phase of cosmic accelerationrdquo Physical ReviewD vol 76 ArticleID 043512 2007

[20] E O Kahya and R P Woodard ldquoScalar field equations fromquantum gravity during inflationrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 8 2008

[21] C Cheung P Creminelli A L Fitzpatrick J Kaplan and LSenatore ldquoThe effective field theory of inflationrdquo Journal of HighEnergy Physics vol 2008 no 3 article 014 2008

[22] P A R Ade N Aghanim C Armitage-Caplan et alldquoPlanck 2013 resultsmdashXVI Cosmological parametersrdquohttparxivorgabs13035076

[23] G Steigman ldquoNeutrinos and big bang nucleosynthesisrdquoAdvances in High Energy Physics vol 2012 Article ID 26832124 pages 2012

[24] V Simha and G Steigman ldquoConstraining the early-Universebaryon density and expansion raterdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 6 article 16 2008

[25] R V Wagoner W A Fowler and F Hoyle ldquoOn the synthesis ofelements at very high temperaturesrdquoThe Astrophysical Journalvol 148 p 3 1967

[26] M S Smith L H Kawano and R A Malaney ldquoExperimentalcomputational and observational analysis of primordial nucle-osynthesisrdquo The Astrophysical Journal Supplement Series vol85 no 2 pp 219ndash247 1993

[27] R Esmailzadeh G D Starkman and S Dimopoulos ldquoPrimor-dial nucleosynthesis without a computerrdquo Astrophysical JournalLetters vol 378 no 2 pp 504ndash518 1991

[28] V Mukhanov The Physical Foundation of Cosmology Cam-bridge University Press Cambridge Mass USA 2005

[29] J P Kneller and G Steigman ldquoBBN for pedestriansrdquo NewJournal of Physics vol 6 article 117 2004

[30] G Steigman ldquoPrimordial nucleosynthesis in the precisioncosmology erardquo Annual Review of Nuclear and Particle Sciencevol 57 pp 463ndash491 2007

[31] E Komatsu K M Smith J Dunkley (WMAP Collabo-ration) et al ldquoSeven-year Wilkinson microwave anisotropyprobe (WMAPlowast) observations cosmological interpretationrdquoThe Astrophysical Journal Supplement Series vol 192 no 2article 18 2011

[32] B E J Pagel E A Simonson R J Terlevich and M GEdmunds ldquoThe primordial helium abundance from observa-tions of extragalactic HII regionsrdquoMonthly Notices of the RoyalAstronomical Society vol 255 pp 325ndash345 1992

[33] Y I Izotov and T X Thuan ldquoThe primordial abundance of4He evidence for non-standard big bang nucleosynthesisrdquoAstrophysical Journal Letters vol 710 no 1 pp L67ndashL71 2010

[34] E Aver K A Olive and E D Skillman ldquoA new approach tosystematic uncertainties and self-consistency in helium abun-dance determinationsrdquo Journal of Cosmology and AstroparticlePhysics vol 2010 no 5 article 3 2010

[35] D Kirkman D Tytler N Suzuki J M OrsquoMeara and DLubin ldquoThe cosmological baryon density from the deuterium-to-hydrogen ratio in QSO absorption systems DH towardQ1243+3047rdquoThe Astrophysical Journal Supplement Series vol149 no 1 pp 1ndash28 2003

[36] J M OrsquoMeara S Burles J X Prochaska G E Prochter RA Bernstein and K M Burgess ldquoThe deuterium-to-hydrogenabundance ratio toward the QSO SDSS J15581016-0031200 1rdquoAstrophysical Journal Letters vol 649 no 2 pp L61ndashL65 2006

[37] M Pettini B J Zych M T Murphy A Lewis and C C SteidelldquoDeuterium abundance in themostmetal-poor damped Lymanalpha system converging on Ω

1198870h2rdquo Monthly Notices of the

Royal Astronomical Society vol 391 no 4 pp 1499ndash1510 2008[38] M Fumagalli J M OrsquoMeara and J X Prochaska ldquoDetection

of pristine gas two billion years after the big bangrdquo Science vol334 no 6060 pp 1245ndash1249 2011

[39] M Pettini and R Cooke ldquoA new precise measurementof the primordial abundance of Deuteriumrdquohttparxivorgpdf12053785pdf

[40] F Iocco G Mangano G Miele O Pisanti and P D SerpicoldquoPrimordial nucleosynthesis from precision cosmology to fun-damental physicsrdquo Physics Reports vol 472 no 1ndash6 pp 1ndash762009

[41] G Steigman ldquoPrimordial nucleosynthesis successes and chal-lengesrdquo International Journal of Modern Physics E vol 15 no 1pp 1ndash36 2006

[42] B D Fields ldquoThe primordial lithium problemrdquo Annual Reviewof Nuclear and Particle Science vol 61 pp 47ndash68 2011

[43] M Asplund D L Lambertm P E Nissen et al ldquoLithiumisotopic abundances inmetal-poor halo starsrdquoTheAstrophysicalJournal vol 644 p 229 2006


[44] S G Ryan T C Beers K A Olive B D Fields and J ENorris ldquoPrimordial lithium and big bang nucleosynthesisrdquoTheAstrophysical Journal Letters vol 530 pp L57ndashL60 2000

[45] T Damour and B Pichon ldquoBig bang nucleosynthesis andtensor-scalar gravityrdquo Physical Review D vol 59 Article ID123502 1999

[46] A Coc K A Olive J P Uzan and E Vangioni ldquoBig bangnucleosynthesis constraints on scalar-tensor theories of gravityrdquoPhysical Review D vol 73 Article ID 083525 2006

[47] J A R Cembranos K A Olive M Peloso and J-P UzanldquoQuantum corrections to the cosmological evolution of con-formally coupled fieldsrdquo Journal of Cosmology and AstroparticlePhysics vol 907 article 025 2009

Submit your manuscripts athttpwwwhindawicom

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to power spectrum [15ndash20] Therefore one might describethe interactions with effective field theories of inflation [21]But in this work we aremainly interested in the consequenceof a geometrical constraint 3+119899-dimensional universe havinga constant volume leading to a modified Hubble parameterduring a later stage nucleosynthesis where quantum gravita-tional corrections are negligible

The paper is organized as follows In Section 2 dynamicsof this particular dilaton gravity model is summarized InSection 3 nucleosynthesis in the context of this model is ana-lyzed In Section 4 the results obtained from our calculationsfor this model and the predictions of SBBN with the help ofPlanck satellite data [22] are compared with the astrophysicalobservations

2 Dynamics of HDGS A DilatonGravity Model

In this section we briefly summarize the dynamics of aparticular type of dilaton gravity models proposed in [8]The idea is introducing a higher dimensional dilaton gravityaction of steady-state cosmology (HDGS) in the string frameTherefore the evolution of the internal 119899-dimensional spaceresults in an evolution of the observed universe to keepthe whole system in a steady state Due to this constraintchoosing particular values for some parameters such as thenumber of extra dimensions leads to possibly observableeffects in our universe Let us start with the action whichstems from the low-energy effective string theory

119878 = int119872

1198891+3+119899

119909radic10038161003816100381610038161198921003816100381610038161003816119890minus2120601

(119877 + 4120596120597120583120601120597120583120601 + 119880 (120601)) (1)

where 119877 is the curvature scalar 119872 stands for manifold 119899corresponds to extra dimensions |119892| is the determinant of119892

120583]metric tensor120601 is the dilaton field taken as space independentreal function of time and 120596 is an arbitrary coupling constant119880(120601) = 119880

0119890120582120601 is a real smooth function of the dilaton field

and corresponds to the dilaton self-interaction potential andboth119880

0and 120582 are real parameters Two interesting cases that

are worth mentioning are 120596 = 1 and 119899 = 6 correspondingto anomaly-free superstring theory and 120596 = 1 with 119899 = 22

corresponds to bosonic string theory The metric is given by

1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112)

+ 1199042(119905) (119889120579

2

1+ sdot sdot sdot + 119889120579

2

119899)

(2)

Here 119905 is the cosmic time and (119909 119910 119911) are the cartesiancoordinates of the 3-dimensional flat space basically theobserved universe The coordinates 120579 are 119899-dimensionalcompact (torodial) internal space coordinates (this representsspace that cannot be observed directly and locally today)While 119886(119905) denotes the scale factor of 3-dimensional externalspace 119904(119905) is the scale factor of 119899-dimensional internal space

This model has the following key properties(i) The (3 + 119899)-dimensional universe has a constant

volume that is 119881 = 1198863119904119899= 1198810 hence steady state But the

internal and external spaces are dynamical (ii) The energy

density is constant in the higher dimensional universe (iii)There is no higher dimensional matter source other than thedilaton field in the action

If the scalar field is redefined as 120573 = 119890120582120601 the relation

between the scalar field and the scale factor of the externalspace turns out to be

(1198861015840

119886)

2

=119899

3 (3 + 119899)

2120596120576

1205732 (3)

where 120576 is a constant of integration Here prime denotesderivative with respect to the ordinary time Imposing theconstant volume condition gives

119886 = 1198860119890plusmn(13)radic3119899(3+119899)radic2120596120576 int 119889119905(1120573)

119904 = 1199040119890∓(1119899)radic3119899(3+119899)radic2120596120576 int 119889119905(1120573)

(4)

where 1198860and 119904

0correspond to the integration constants

Therefore the modified Hubble parameter of the externalspace is obtained as follows

119867119886equiv1198861015840

119886= plusmn

1

3

radic3119899

3 + 119899

radic2120596120576

120573 (5)

Here the physically relevant case is the solution for expand-ing external space with 119867

119886gt 0 The deceleration parameter

for the external space is given by

119902119886equiv minus

11988610158401015840119886

11988610158402= minus1 plusmn 3radic

3 + 119899

3119899

1205731015840

radic2120596120576 (6)

In the case of 120576 = 0 and 1198800

= 0 and with the choice ofappropriate initial conditions it turns out that [8] the earlytime modified deceleration parameter is given by

119902 997888rarr 3radic3 + 119899

3119899

1

radic120596minus 1 (7)

3 Nucleosynthesis in HDGS

We are interested in how abundances of light elements wouldchange in the context of this model Specifically we wouldlike to consider the ratio of themodified expansion rate to thestandard expansion rate during the early radiation dominantepoch This ratio is given by

119878 equiv119867119886

119867SBBN=1 + 119902SBBN1 + 119902119886

(8)

This is true since deceleration parameter stays almost con-stant during primordial nucleosynthesis The value of thedeceleration parameter for standard BBN is 119902SBBN = 1 Since119902119886is given by (7) the so-called standard expansion factor 119878

can be expressed in terms of 120596 and 119899 as

119878 =2

3(radic

3120596119899

3 + 119899) (9)





12057810equiv 1010120578119861equiv 1010 119899119861

119899120574

= 2739Ω119861ℎ2 (10)


119861is dimen-



0being







4He + 119901 (12)







119860equiv 119860119899119860119899119873 where


119884119901= 02485 plusmn 00006 + 00016 ((120578

10minus 6) + 100 (119878 minus 1))

(13)


119884SBBN119901


119884119901= 02485 plusmn 00006 + 016(minus1 +

2

3

radic3120596119899

3 + 119899) (14)


119884119901= 02393plusmn00006 and 119884

119901= 02618plusmn00006 for 119899 = 6 and




119901=










119910D119901 equiv 105(119899D119899H

)

119901

= 105(13

24119883D119901) (15)



119883D119901 ≃2119877

exp (11986012057810) minus 1


where 119877 ≃ 2 sdot 10minus5 [28] 120578

10≃ 6 and 119860 ≃ 01 Here



119910D119901 = 260 (1 plusmn 006) (6

12057810minus 6 (119878 minus 1)

)

16

(17)





119910D119901

= 260 (1 plusmn 006)(6


)

16

(18)

Taking 12057810



[40] and 119910D119901 = 254 plusmn 005 [39]


D + D 997888rarr3He + 119899 D + 119901 997888rarr

3He + 120574


(19)


1199103equiv 105(

1198993He

119899H) = 10

5(13

361198833He) (20)


1198833He ≃02 sdot 119883D + 10

minus5

1 + 4 times 103119883D (21)




1199103= 31 (1 plusmn 003) 120578

minus06

10 (22)



119879


119883119879

119891≃ (0015 + 3 sdot 10

21198833He119891)119883D119891 (23)



119910Lip equiv 1010(119899Li119899H

)

119901

(24)






6)

2

(25)

Taking 119878 = 1 and 12057810



3

radic3120596119899

3 + 119899))

2

(26)




119910119863119901

119910Lip





+068

minus032[44]



4 Discussion











Acknowledgments


References

























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







12057810equiv 1010120578119861equiv 1010 119899119861

119899120574

= 2739Ω119861ℎ2 (10)


119861is dimen-



0being







4He + 119901 (12)







119860equiv 119860119899119860119899119873 where


119884119901= 02485 plusmn 00006 + 00016 ((120578

10minus 6) + 100 (119878 minus 1))

(13)


119884SBBN119901


119884119901= 02485 plusmn 00006 + 016(minus1 +

2

3

radic3120596119899

3 + 119899) (14)


119884119901= 02393plusmn00006 and 119884

119901= 02618plusmn00006 for 119899 = 6 and




119901=










119910D119901 equiv 105(119899D119899H

)

119901

= 105(13

24119883D119901) (15)



119883D119901 ≃2119877

exp (11986012057810) minus 1


where 119877 ≃ 2 sdot 10minus5 [28] 120578

10≃ 6 and 119860 ≃ 01 Here



119910D119901 = 260 (1 plusmn 006) (6

12057810minus 6 (119878 minus 1)

)

16

(17)





119910D119901

= 260 (1 plusmn 006)(6


)

16

(18)

Taking 12057810



[40] and 119910D119901 = 254 plusmn 005 [39]


D + D 997888rarr3He + 119899 D + 119901 997888rarr

3He + 120574


(19)


1199103equiv 105(

1198993He

119899H) = 10

5(13

361198833He) (20)


1198833He ≃02 sdot 119883D + 10

minus5

1 + 4 times 103119883D (21)




1199103= 31 (1 plusmn 003) 120578

minus06

10 (22)



119879


119883119879

119891≃ (0015 + 3 sdot 10

21198833He119891)119883D119891 (23)



119910Lip equiv 1010(119899Li119899H

)

119901

(24)






6)

2

(25)

Taking 119878 = 1 and 12057810



3

radic3120596119899

3 + 119899))

2

(26)




119910119863119901

119910Lip





+068

minus032[44]



4 Discussion











Acknowledgments


References

























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119910D119901 equiv 105(119899D119899H

)

119901

= 105(13

24119883D119901) (15)



119883D119901 ≃2119877

exp (11986012057810) minus 1


where 119877 ≃ 2 sdot 10minus5 [28] 120578

10≃ 6 and 119860 ≃ 01 Here



119910D119901 = 260 (1 plusmn 006) (6

12057810minus 6 (119878 minus 1)

)

16

(17)





119910D119901

= 260 (1 plusmn 006)(6


)

16

(18)

Taking 12057810



[40] and 119910D119901 = 254 plusmn 005 [39]


D + D 997888rarr3He + 119899 D + 119901 997888rarr

3He + 120574


(19)


1199103equiv 105(

1198993He

119899H) = 10

5(13

361198833He) (20)


1198833He ≃02 sdot 119883D + 10

minus5

1 + 4 times 103119883D (21)




1199103= 31 (1 plusmn 003) 120578

minus06

10 (22)



119879


119883119879

119891≃ (0015 + 3 sdot 10

21198833He119891)119883D119891 (23)



119910Lip equiv 1010(119899Li119899H

)

119901

(24)






6)

2

(25)

Taking 119878 = 1 and 12057810



3

radic3120596119899

3 + 119899))

2

(26)




119910119863119901

119910Lip





+068

minus032[44]



4 Discussion











Acknowledgments


References

























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119910119863119901

119910Lip





+068

minus032[44]



4 Discussion











Acknowledgments


References

























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Testing a Dilaton Gravity Model Using ...downloads.hindawi.com/journals/ahep/2014/282675.pdf · particular type of dilaton gravity models proposed in [ ]. e idea