The role of the slope in the the multi-measure cosmological model
aa r X i v : . [ g r- q c ] A p r The role of the slope in the themulti-measure cosmological model
Denitsa Staicova
Abstract
In this work, we report some results on the numerical exploration ofthe model of Guendelman-Nissimov-Pacheva. This model has been previouslyapplied to cosmology, but there were open questions regarding its parameters.Here we demonstrate the existence of families of solutions on the slope of theeffective potential which preserve the duration of the inflation and its power.For this solutions, one can see the previously reported phenomenon of theinflaton scalar field climbing up the slope, with the effect more pronouncedwhen starting lower on the potential slope. Finally we compare the dynamicaland the potential slow-roll parameters for the model and we find that thelatter describe the numerically observed inflationary period better.
Some of the most defining features of the Universe we live in are that it isisotropic, homogeneous and flat. They have been confirmed to great precisionby cosmological probes (WMAP, Planck). Another important observation isthat the universe is currently expanding in an accelerated way (confirmedby the data from SNIa and the Cepheids) which requires the introduction ofdark energy. A model which describe all of those fundamental properties isthe Λ − CDM model, in which different components of the energy densitycontribute to the evolution of the universe as different powers of the scalefactor.
Denitsa StaicovaInstitute for Nuclear Research and Nuclear EnergyBulgarian Academy of Sciences, Sofia 1784, Tsarigradsko shosse 72, Bulgaria, e-mail: [email protected]
Explicitly, in the Friedman-Lemaitre-Robertson-Walker (FLRW) metric˜ g µν = diag {− , a ( t ) , a ( t ) , a ( t ) } , we have for the first Friedman equation: H = ˙ aa = H √ Ω m a − + Ω rad a − + Ω Λ Here H = ˙ a ( t ) /a is the Hubble parameter and a ( t ) is the scale factorparametrizing the expansion of the Universe. H is the current Hubble con-stant, Ω m is the critical matter density (dark matter and baryonic matter), Ω rad is the critical radiation density, and Ω Λ is the critical density of the cos-mological constant (i.e. dark energy). In our units ( G = 1 / π ), ρ crit = 6 H ,therefore Ω x = ρ x /ρ crit = ρ x / (6 H ) for X = { m, rad, Λ } .While the Λ − CDM model offers a rather simple explanation of the evo-lution of the Universe (the minimal Λ − CDM has only 6 parameters), it stillhas its problems. Some of the oldest ones – the horizon problem, the flatnessproblem, the missing monopols problem and the large-structures formationproblem, require the introduction of a new stage of the development of theUniverse – the inflation. The inflation is an exponential expansion of the Uni-verse lasting between 10 − s and 10 − s after the Bing Bang, which howeverincreases the volume of the Universe 10 times.The simplest way to produce inflation [1] is to introduce a scalar field φ which is moving in a potential V infl ( φ ). Inflation is generated by the ex-change of potential energy for kinetic energy. In this case, the evolution ofthe Universe will be described by two differential equations: H = 8 π m P l ( V infl ( φ ) + 12 ˙ φ ) (1)¨ φ + 3 H ˙ φ + V ′ infl ( φ ) = 0 , (2)where the first one is the Friedman equation and the second is the inflatonequation. Inflation occurs when ¨ a ( t ) > φ < V ( φ ), i.e. when the potential energy dominates over the kineticone. The pressure and the energy density are: p φ = ˙ φ / − V infl ( φ ) , ρ φ = ˙ φ / V infl ( φ ) . One can consider different forms for the effective potential, but those sim-plistic inflationary theories have the problem of not being able to reproducethe graceful transition from inflation to the other observed epochs.
There are different ways to obtain a model with richer structure. Here wefollow the model developed by Guendelman, Nissimov and Pacheva [2, 3, 4,5, 6, 7, 8] (also for some more recent applications of the model [9]). Theidea is to couple two scalar fields (the inflaton φ and the darkon u ) to both he slope of the multi-measure model 3 standard Riemannian metric and to another non-Riemannian volume form,so that the model can describe simultaneously early inflation, the smoothexit to modern times, and the existence of dark matter and dark energy.The action of the model: S = S darkon + S inflaton is (for more details [8, 10]): S darkon = Z d x ( √− g + Φ ( C )) L ( u, X u ) S inflaton = Z d xΦ ( A )( R + L (1) ) + Z d xΦ ( B ) (cid:18) L (2) + Φ ( H ) √− g (cid:19) where Φ i ( Z ) = ǫ µνκλ ∂ µ Z νκλ for Z = A, B, C, H , are the non-Riemannianmeasures, constructed with the help of 4 auxiliary completely antisymmetricrank-3 tensors and we have the following Lagrangians for the two scalar fields u and φ : L ( u ) = − X u − W ( u ) L (1) = − X φ − V ( φ ) , V ( φ ) = f e − αφ L (2) = − b e − αφ X φ + U ( φ ) , U ( φ ) = f e − αφ where X c = g µν ∂ µ c∂ ν c are the standard kinetic terms for c = u, φ .Trough the use of variational principle, for this model, it has been foundthat there exists a transformation ˜ g µν = Φ ( A ) √− g g µν (3) ∂ ˜ u∂u = ( W ( u ) − M ) − , (4)for which for the Weyl-rescaled metric ˜ g , the action becomes S ( eff ) = Z d x p − ˜ g ( ˜ R + L ( eff ) ) . (5)For the rescaled metric ˜ g and the derived effective Lagrangian, L eff , theEinstein Field equations are satisfied. The action in the FLRW metric becomes ( v = ˙ u ): S ( eff ) = Z dt a ( t ) (cid:16) − a ( t ) a ( t ) + ˙ φ − v (cid:16) V + M − χ b e − αφ ˙ φ / (cid:17) + v χ ( U + M ) − M ) (cid:17) . from which one can obtain the equations of motion in the standard way.Explicitly, the equations of motion are: Denitsa Staicova v + 3 a v + 2 b = 0 (6)˙ a ( t ) = r ρ a ( t ) , (7) ddt (cid:16) a ( t ) ˙ φ (1 + χ b e − αφ v ) (cid:17) + a ( t ) ( α ˙ φ χ b e − αφ + V φ − χ U φ v v a = − V ( φ )+ M − χ be − αφ ˙ φ χ ( U ( φ )+ M ) − M , b = − p u a ( t ) ( χ ( U ( φ )+ M ) − M ) and ρ = 12 ˙ φ (1 + 34 χ be − αφ v ) + v V + M ) + 3 p u v a ( t ) is the energy density. One can see that the parameters of this system are 12: 4 free parameters { α, b , f , f } , 5 integration constants { M , M , M , χ , p u } and 3 initial con-ditions { a (0) , φ (0) , ˙ φ (0) } .We use the following initial conditions: a (0) = 10 − , φ (0) = φ , ˙ φ (0) = 0 . (9)To narrow down the parameter-space, we add also { a (1) = 1 , ¨ a (0 .
71) = 0 } .The consequences of these choices are as follow:1) The initial condition a (0) = 0 introduces a singularity at the beginningof the evolution.2) The normalization a (1) = 1 fixes the age of the Universe.3) The condition ¨ a (0 .
71) = 0 sets the end of the matter-domination epoch.Defined like this, we have an initial value problem (Eqs.9), which we solveusing the shooting method, starting the integration from t = 0.It is possible to also start the integration backwards, from t = 1, using asinitial conditions: a (1) = 1 , φ (1) = φ end , ˙ φ (1) = 0 and aim for a (0) = 0. Here˙ φ (1) = 0 guarantees that the evolution of the inflaton field has stopped andthe universe is expanding in an accelerated fashion. While both approacheswork, integrating forward has the benefit of dealing with the singularity at a (0) = 0 at the beginning of the integration, rather than at its end. Movingour initial point of integration away from a (0) = 0 decreases the significanceof the term p u /a ( t ) . This effectively means putting p u = 0, which we do notwant, because p u is the conserved Noether charge of the “dust” dark mattercurrent (see [8]). he slope of the multi-measure model 5 The initial velocity of the scalar field ˙ φ (0) is not a free parameter of thesystem, because its value is quickly fixed by the inflaton equation, i.e theresults do not depend on ˙ φ (0) in a very large interval.An important feature of the model, is that the type of evolution one wouldobtain, depends critically on the starting position on the effective potential.We consider as physically “realistic” only the evolution with four epochs— short first deceleration epoch (FD), early inflation (EI), second deceler-ation (SD) which we interpret as radiation and matter dominated epochstogether and finally — slowly accelerating expansion (AE). In terms of theequation of state(EOS) parameter w ( t ) = p/ρ , those are solutions for which:1) w F D → /
3, corresponding to the EOS of ultra-relativistic matter, 2) w EI → − w SD > − / w AE < − / a ( t i ) =0 for t i = t EI , t SD , t AE . In the units we use, t SD ∼ − and t AE ∼ . t SD . Here we will discuss some additional features of the model.In [12] we used the parameter b to set t AE ∼ .
71 and parameter f toensure a (1) = 1. Changing f however changes the effective potential definedby: U eff ( φ ) = 14 ( f e − αφ + M ) χ ( f e − αφ + M ) − M . (10)thus making it harder to study how the solutions depend on the startingposition on the slope.In the current article, we will go a different route and we will use b toset t AE ∼ .
71 and p u to ensure a (1) = 1. This will simplify our problemsignificantly, since now we will have only 3 parameters to consider { b , p u , φ } .It will also enable us to study how the solutions depend on φ , as the effectivepotential does not depend on b and p u .Numerically, we work with the following solution: { χ = 1 , M = − . , M = 0 . , M = 0 . , α = 2 . , f = 5 , f =10 − } .For these values of the parameters, the effective potential is step-like, asseen on Fig.1 a) . The effective potential reaches its asymptotic values for theplateaus for φ − < − . , φ + > . U ′ eff → φ ∈ ( − . , − . φ = − . φ (in this case φ ∈ [ − . , − . Denitsa Staicovaa b
Fig. 1: On the panels one can see: (a) the effective potential (d) the plane p u ( b ), the different branches correspond to different φ , the points are solu-tions, the diamonds are solutions with t AE = 0 . − . a ′′ (0 .
71) = 0,i.e. t AE = 0 .
71. The points on Fig. 1b) do that with precision of 10 − ,the diamonds show the points which satisfy it with precision of 10 − andthe different branches correspond to different φ . From this plot, one cangain insight on the dependence of b ( p u ) for the solutions (here differentpoints on the branches correspond to t AE = 0 . .. . φ depend on t AE which increases with the decreaseof b along each branch.On Fig. 2 a) we show the dependence φ ( b ) for solutions with t AE = 0 . − . We do not show the dependence φ ( p u ) as it appearschaotic. a b Fig. 2: On the panels is (a) the dependence φ ( b ) (b) the dependence N ( φ ) he slope of the multi-measure model 7 A very interesting feature of these solutions, plotted on Fig.2 a) and b), isthat they keep t EI and t SD approximately constant. I.e. while moving up anddown the slope, we do not change the physical properties of our solutions.This can be seen on Fig. 2 b), where we have plotted the e-folds parameterdefined as N = ln ( a SD /a EI ) for the solutions corresponding to different φ .One can see that it remains more or less the same under the precision we areworking with. In our results, only φ ( t ) is sensitive to the changes in φ .On Fig. 3 a) and b) we have plotted how w ( t ) and φ ( t ) vary for φ = {− . , − . , − } and we have zoomed on the interval t = 0 .. . w ( t ) = − φ ( t ). This, however leads to a very small devi-ation in φ ( t = 1) – less than 5% from the lowest to the highest point on theslope. This result is highly unexpected since one could expect that the powerof the inflation and end values of a ( t ) , φ ( t ) will depend more strongly on φ ,which we do not observe here. Note, we could not integrate further up than φ = − .
05 which is far from the upper end of the slope φ = − .
5. This isbecause after this point, ¨ φ ( t ) becomes infinite and the numerical system hitsa singularity.Finally, an important note to make is that on Fig. 3a) one can see thereported before ([12, 13]) climbing up the slope. Somewhat unexpectedly, itis strongest for points with lowest φ . For them, the highest value of φ ( t )is reaching φ = − .
10 which corresponds to the upper plateau. This is afurther confirmation of our observation in [12] that the movement of theinflaton doesn’t correspond to the classical exchange of potential energy forkinetic one, but instead it is closer to the the stability of the L4 and L5Lagrange points. a b
Fig. 3: Zoomed at t ∈ (0 , .
1) are the evolutions of: (a) the inflaton field φ ( t ),(b) the equation of state w ( t ) = p ( t ) /ρ ( t ). Here φ = − . , − . , − Denitsa Staicova
This follows from the fact that the effective potential does not bring thekinetic energy in standard form (Eq. 2). In the slow roll approximation (ne-glecting the terms ∼ ˙ φ , ˙ φ , ˙ φ ) the inflaton equation has the form:( A + 1) ¨ φ + 3 H ( A + 1) ˙ φ + U ′ eff = 0 , (11)where A = b e − αφ V + M U + M .For this reason, we find it interesting to compare the kinetic slow-rollparameters (also called dynamical) defined as: ǫ H = − ˙ HH , η H = − ¨ φH ˙ φ (12)with the potential slow-roll parameters which can be derived to be [14, 4](Note that our A is different from the one used in [4]): ǫ V = 11 + A (cid:18) U ′ eff U eff (cid:19) , η V = 21 + A U ′′ eff U eff (13) b Fig. 4: A zoom in on the slow-roll parameters ((a) ǫ and (b) η ) in the intervalwhere infation occurs ( t = 0 . .. . { b , p u , φ } are { . × − , . × − , − . } On Fig. 4 we present an example of the evolution of the “slow-roll” param-eters for both the kinetic and the potential definitions [14, 4]. One can seefrom the plots that the two definitions in this interval are very similar – bothgive mostly very small values of the slow-roll parameters. One also noticesthat the intervals on t for which the slow-roll parameters of both kinds arevery small (say, | < . | ) are shorter than the numerically obtained one, given he slope of the multi-measure model 9 by t EI ..t SD . In general, it seems that the potential slow-roll parameters giveintervals closer to the numerical ones. However, the conclusion is that if theslow-roll parameters are used to estimate inflation theoretically, those smalldeviations in the intervals may lead to misestimations of N .Finally, a note on the e-folds parameter, which measures the power of theinflation. The theoretical estimation for the number of e-folds needed to solvethe horizon problem is model-dependent but is N >
70. In our example, weget N ≈
15. It is important to note, that there is a numerical maximumof the number of e-folds of about N ≈
22, due to the fact we are startingour integration at a (0) ∼ − . In order to get a higher N , one needs tostart with smaller a (0), but to do so, we need to improve significantly theprecision of the integration. Parameters-wise, the best way to get powerfulearly inflation is trough increasing α or decreasing f [12]. In this work, we have explored numerically the model of Guendelman-Nissimov-Pacheva in a specific part of its parametric space related to thedifferent initial conditions on the slope of the effective potential. Even thoughit is impossible to study the entire parameter space, we have shown some im-portant properties of the model.Most importantly, we have shown that there exist families of solutions onthe slope which preserve the initial and the ending times of the inflation andalso, that they give similar number of e-folds. Furthermore, one can see thatthe main difference between starting at the top of the slope and at its bottomis the behavior of the inflaton scalar field, which climbs up all the way to thetop of the slope before entering in inflation regime. This mechanism requiresfurther study.Finally, we have considered the dynamical and the potential slow-roll pa-rameters for the model and we have shown that the potential slow-roll pa-rameters seem to describe the inflationary period better. They, however, donot match entirely with the numerically obtained duration of the inflation.
Acknowledgments
It is a pleasure to thank E. Nissimov, S. Pacheva and M. Stoilov for thediscussions. To be precise, the intervals are: for ǫ V : 0 . .. .. ǫ H : 0 . .. . η V := 0 . .. .
041 for η H : 0 . .. . The work is supported by Bulgarian NSF grant DN-18/1/10.12.2017 andby Bulgarian NSF grant 8-17. D.S. is also partially supported by COSTActions CA18108.
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