Trans-Planckian physics from a nonlinear dispersion relation

Trans-Planckian physics from a nonlinear dispersion relation

S. E. Joras

Instituto de Fısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil

G. Marozzi

Dipartimento di Fisica, Universita degli Studi di Bologna and INFN, via Irnerio 46, I-40126 Bologna, Italy(Received 28 August 2008; published 22 January 2009)

We study a particular nonlinear dispersion relation !pðkpÞ—a series expansion in the physical wave

number kp—for modeling first-order corrections in the equation of motion of a test scalar field in a

de Sitter spacetime from trans-Planckian physics in cosmology. Using both a numerical approach and a

semianalytical one, we show that the WKB approximation previously adopted in the literature should be

used with caution, since it holds only when the comoving wave number k � aH. We determine the

amplitude and behavior of the corrections on the power spectrum for this test field. Furthermore, we

consider also a more realistic model of inflation, the power-law model, using only a numerical approach to

determine the corrections on the power spectrum.

DOI: 10.1103/PhysRevD.79.023514 PACS numbers: 98.80.Cq, 98.70.Vc

I. INTRODUCTION

Inflationary models provide answers to many problemsin standard big bang cosmology, in particular, the origin ofdensity fluctuations and the spectrum of cosmic microwavebackground anisotropies. The basis of the whole mecha-nism is the stretching of quantum fluctuations generated atsub-Hubble scales due to the exponential expansion of thespacetime during inflation. This model, however, has aserious ‘‘problem’’: if we consider a plain scalar-field-driven inflationary model—say, chaotic inflation—the pe-riod of inflation lasts so long that the wavelengths of thefluctuations which at present correspond to cosmologicalscales were sub-Planckian at the beginning of the infla-tionary phase. Therefore, the evolution of fluctuations atsuch scales is supposed to follow different rules from thoseprovided by the standard theory of cosmological perturba-tions—which is based on quantum field theory and generalrelativity. The set of rules expected to hold above thisenergy scale is the so-called trans-Planckian physics(TPP from now on). The TPP could lead to deviationsfrom standard predictions on the cosmic microwave back-ground radiation (CMBR), which probes the scales wementioned before. The question to be asked is ‘‘whetherthe predictions of the standard cosmology are insensitive toeffects of TPP.’’ This is the precise statement of the trans-Planckian problem [1,2].

In the present paper we adopt, to mimic TPP, a series-expansion expression for the nonlinear dispersion relation!p [Eq. (19) below], first suggested by [3], to modify the

equation of motion of a test scalar field in a de Sitterspacetime. We argue that, in this framework, the WKBapproximation is valid only for k � aH (or kp � k=a �H with kp the physical wave number and H the Hubble

factor). As a consequence, the perturbative approach basedon theWKB approximation, used in [4] to tackle this issue,should be used with caution.

The outline of the paper is the following. In Sec. II wepresent the standard approach for first-order cosmologicalperturbations and then we restrict ourselves to the case of atest scalar field in a de Sitter spacetime. We also introducethe WKB approximation in this section. In Sec. III weadopt a particular way to approach TPP: a modificationof the equation of motion due to a nonlinear dispersionrelation. The initial conditions for the test scalar field areset in Sec. IV. In the following sections we present anumerical calculation, a 3-piecewise approximation(adopted in Ref. [4]) and a semianalytical approach tosolve the problem. In Sec. VIII we investigate a realisticmodel of inflation, namely, power-law inflation, using thesame nonlinear dispersion relation. We then conclude inSec. IX.

II. STANDARD APPROACH

The spatially flat Friedmann-Robertson-Walker (FRW)metric with first-order cosmological fluctuations is givenby (ds2 ¼ g��dx

�dx�):

g00 ¼ a2ð�1� 2�Þ g0i ¼ �a2

2ð�;i þ BiÞ

gij ¼ a2½�ijð1� 2c Þ þDijEþ ð�j;i þ �i;j þ hijÞ=2�:(1)

with Dij ¼ @i@j � 1=3r2�ij, considering the conformal

time �. To first-order scalar, vector, and tensor perturba-tions evolve independently. Vector perturbations can beomitted because they die away kinematically. The tensorperturbation hij has only two physical degrees of freedom

(polarization states) hþ and h�:

hij ¼ hþeþij þ h�e�ij; (2)

where eþij and e�ij are the polarization tensors having the

PHYSICAL REVIEW D 79, 023514 (2009)

1550-7998=2009=79(2)=023514(12) 023514-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.79.023514

following properties in the Fourier space:

eij ¼ eji; kieij ¼ 0; eii ¼ 0 (3)

eþijeþij ¼ 2; e�ije�ij ¼ 2; eþije�ij ¼ 0: (4)

On expanding in Fourier modes we can define

h�ðt; ~xÞ ¼ 1

að�Þ1

ð2Þ3=2Z

d3k�tkð�Þei ~k� ~x; (5)

with � ¼ þ=� and where �tk will satisfy [5]:

�00tk þ

�k2 � a00

a

��tk ¼ 0: (6)

The scalar sector, in the case of a plain scalar fieldð�; ~xÞ ¼ ð�Þ þ ’ðt; ~xÞ driven inflationary model, canbe reduced to the study of a single gauge-invariant scalarvariable defined by

Q ¼ �s

a¼ ’þ 0

H

�c þ 1

6r2E

�; (7)

where H ¼ a0a . This so-called Mukhanov variable [6]

obeys the following equation of motion in Fourier space:

�00sk þ

�k2 � z00

z

��sk ¼ 0 (8)

with z ¼ a 0H

. Considering an adiabatic vacuum state as

initial condition for those perturbations, all the statisticalproperties are characterized by the two-point correlationfunction, namely, by the power spectrum. The dimension-less power spectrum for scalar and tensor cosmologicalfluctuations are given, respectively, by

PQ ¼ k3

22

��sk

z

��2

; Ph ¼ 2k3

2

��tk

a

��2

: (9)

Their dependence on the mode k is defined by the spectralindex in the following way:

ns � 1 � d lnPQ

d lnk; nt � d lnPh

d lnk(10)

evaluated at a scale k � aH when the mode is outside thehorizon. For ns ¼ 1 one has a scale-invariant spectrum forthe gauge-invariant cosmological scalar fluctuation.

Let us now restrict ourselves to the case of a test scalarfield in a de Sitter spacetime where að�Þ ¼ �1=ðH�Þ. Inthis case, cosmological fluctuations vanish identically.

If we expand the test scalar field ð�; ~xÞ in Fouriermodes

ð�; ~xÞ ¼ 1

að�Þ1

ð2Þ3=2Z

d3k�kð�Þei ~k� ~x; (11)

then the equation of motion for each mode is given by

�00k þ�2ð�Þ�k ¼ 0 (12)

with

�2ð�Þ � k2 � a00

a(13)

which becomes for a de Sitter spacetime

�2ð�Þ ¼ k2 � 2

�2: (14)

We can note that Eq. (12) is the same equation of motionas that of a tensor perturbation (6). In this simple case,Eq. (12) can be exactly solved.The two-point correlation function is given by

h0jð�; ~xÞð�; ~xþ ~rÞj0i ¼Z þ1

0

dk

k

sinkr

krPðkÞ (15)

and the power spectrum is

PðkÞ ¼ k3

22

��k

a

��2

: (16)

For superhorizon modes (k � aH) the spectrum is timeindependent and scale invariant, as one can see from theexact solution in the limit � ! 0�.Equation (12) can also be interpreted as a Schrodinger

equation for a stationary wave function with energy E �!2 ¼ k2 in an effective potential Veffð�Þ � a00=a which isa function of the ‘‘spatial’’ variable �. We might betempted to solve this equation using WKB approximation,just as it is usually done in basic quantum mechanics (QM)[7]. In this approximation the stationary solution ofEq. (12) is given by

�kð�Þ ¼ cþ�1=2

exp

�þi

Z ��ð�0Þd�0

�

þ c��1=2

exp

��i

Z ��ð�0Þd�0

�(17)

as long as the WKB parameter W is much smaller than 1:

W �� 1

�2

�3

4

��0

�

�2 � 1

2

�00

�

�� 1: (18)

Figure 1 shows the behavior of W as a function of theconformal time� in a de Sitter case. According to this plot,the WKB approximation holds for � ! �1 (k � aH,subhorizon scales) but not for values close to zero (k �aH, superhorizon scales), where it equals 0.125 (i.e., notmuch smaller than 1) [8]. It fails exactly where it is

supposed to: near the classical turning point (�tp �� ffiffiffi

2p

=k) and where the effective potential is too steep(� ! 0�). For the subhorizon scales, where E � Veff ,the solution of Eqs. (12) and (14) is a plane wave inconformal time with comoving frequency � ’ k,Eq. (17), as expected.

S. E. JORAS AND G. MAROZZI PHYSICAL REVIEW D 79, 023514 (2009)

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III. TRANS-PLANCKIAN BEHAVIOR:NONLINEAR DISPERSION RELATION

A modification of the linear dispersion relation (d.r.from now on) was proposed by Unruh [9] for describinghigh-energy Physics in the black-hole radiation emission.He was inspired by sound waves, for which a linear d.r.ceases to be valid when the wavelength gets closer to orsmaller than the lattice spacing. Jacobson and Corley [10](see also [2]) also proposed nonlinear terms in the d.r. thatcould be justified by the inclusion of higher-order deriva-tives in the Lagrangian. Other changes can also be intro-duced by arguing that the spacetime symmetries might notsurvive at high energies [11].

Following this approach to mimic TPP, one writes thecomoving frequency ! as ! ¼ að�Þ!pðkpÞ, where !p is

the physical frequency—and assumes that !p is a non-

linear function of kp which differs from the standard

(linear) one only for physical wavelengths closer to orsmaller than the Planck scale. Note that this replacementwould be innocuous if the dispersion relation was linear inkp.

In this paper we focus on the d.r.

!2pðkpÞ ¼ k2p � �k4p þ �k6p; (19)

with �> 0 and �> 0, proposed in Ref. [3]. See Fig. 2 fora sketch of this function. The above expression can be seenas a mere series expansion, but it is also found in solid statephysics and describes the rotons [12]. There has beensuggestions [13] for including a 3-order term in the non-linear d.r. above, coming from an effective-field-theoryapproach. Since such odd-power terms violate CP, we donot consider them here.

Following the discussion in Sec. II, we plot in Fig. 3 theanalogous quantities to W (18), the total energy E � k2,and the effective potential, the latter being given by

Veffð�Þ � ð ��k4Þ�2 � ð ��k6Þ�4 þ 2

�2; (20)

where �� H2 and �� H4 are dimensionless quanti-

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

ωp

k pkp1kp2kp3

FIG. 2 (color online). Nonlinear dispersion relation as a func-tion of the physical wave number kp for a case with three

different solutions (kp1, kp2, and kp3) of the turning-point

equation !2p ¼ 2H2. H ¼ 0:5Mpl and kp is in units of Mpl.

0-9 -6 -3 0

η

FIG. 1 (color online). Total energy (full straight horizontalline), effective potential (dashed blue curve), and WKB parame-ter (full red curve) as a function of the conformal time � for alinear dispersion relation. We have used k ¼ Mpl and � is in

units of 1=Mpl.

0

1

2

-9 -6 -3 0

η

FIG. 3 (color online). Total energy (full straight horizontalline), effective potential (dashed blue curve), and WKB parame-ter (full red curve) as a function of the conformal time � for thenonlinear dispersion relation Eq. (19). We have used �� ¼0:1325, �� ¼ 0:004 375, and H ¼ 0:5Mpl so that there are 3

turning points (indicated by the vertical lines). The WKB curvewas multiplied by 2 � 10�2 so its behavior in the region close tothe turning points can be clearly seen. Note, however, that theWKB approximation is valid only in the region � ! �1 sinceWð� ! 0�Þ ¼ 0:125. We have used k ¼ Mpl and � is in units of

1=Mpl.

TRANS-PLANCKIAN PHYSICS FROM A NONLINEAR . . . PHYSICAL REVIEW D 79, 023514 (2009)

023514-3

ties. As in the linear case, the WKB approximation is validonly when � ! �1.

To compare our results with Ref. [4] we consider only aparticular range of values for � and �, necessary for apositive nonlinear dispersion relation with three real dis-tinct solutions for the classical turning-points equation!2

p ¼ 2H2 (see, also, [14]). The request to have three

real turning-points and a nonlinear !p that differs from

the standard linear one only for physical wavelengthscloser to or smaller than the Planck scale has as conse-quence that H should be comparable with Mpl. In fact,

everywhere in the paper we consider H ¼ 0:5Mpl.

Defining z � 3 ��= ��2 one obtains the following rangesfor z and ��:

3=4< z < 1; gðzÞ< ��< fðzÞ (21)

with

gðzÞ � 1

3z2

�3z

2� 1� ð1� zÞ3=2

�(22)

fðzÞ � 1

3z2

�3z

2� 1þ ð1� zÞ3=2

�; (23)

where z > 3=4 is the physical condition to have a positivenonlinear d.r. The problem of finding the power spectrumfrom this particular nonlinear dispersion relation has beentackled in Ref. [4]. In that paper the authors introduced anapproximate piecewise form of the nonlinear dispersionrelation (see Sec. VI) and apply the WKB approximationfor kp such that !2

p > 2H2. They argue that the amplitude

of the effects of the nonlinearity is proportional to � �ðkp1 � kp2Þ=kp1 where kp1 and kp2 are, respectively, the

first and the second turning points from the right of Fig. 2.Still according to Ref. [4], � measures the ‘‘time’’ spent inthe region where the WKB approximation is not satisfied,but it actually is simply a measure of the distance betweenthe two largest turning points—kp1 and kp2 in Fig. 2. It is

clearly seen in Fig. 3 that � is much smaller that such timeinterval.

We show below the full result (i.e., the exact solutionwithin the three-piecewise approximation) and two otherdifferent approaches—one numerical and the other semi-analytical—for evaluating the nonlinear effects on thepower spectrum.

But before proceeding to proving the claims above, weshall first determine the initial conditions we assume.

IV. THE EQUATION OF MOTION: INITIALCONDITIONS

The equation of motion for the mode function of a testscalar field, following the approach just introduced toconsider the TPP effects, is given by

�00k þ

�a2!2

p � a00

a

��k ¼ 0 (24)

with !p given by Eq. (19).

For k � aH the WKB approximation is valid (seeFig. 3), the term a2!2

p ¼ !2 dominates and the mode

function is given, chosing the adiabatic vacuum (see [4]),by:

�wkbðk; �Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2!ðk; �Þp exp

��i

Z �

�i

!ðk; �Þd��: (25)

To proceed with either a numerical or an analytical ap-proach to find the solution of Eq. (24) we have to find theinitial conditions associated to our nonlinear dispersionrelation in a de Sitter spacetime. We consider a initialtime �i and a fixed value of k (which should be muchlarger than að�iÞH).In this case, for this fixed value k � aH, the comoving

frequency ! becomes nearly equal toffiffiffiffi�

pk3=a2 and the

WKB vacuum is given by

�wkbð�Þ ¼ � 1ffiffiffi2

p�1=4k3=2

1

H�

� exp

��i�1=2k3

H2

3ð�3 � �3

i Þ�; (26)

its derivative by

�0wkbð�Þ ¼

�1ffiffiffi

2p

�1=4k3=21

H�2þ i

1ffiffiffi2

p �1=4k3=2H�

�

� exp

��i�1=2k3

H2

3ð�3 � �3

i Þ�; (27)

and our initial conditions by

�wkbð�iÞ ¼ � 1ffiffiffi2

p�1=4k3=2

1

H�i

(28)

�0wkbð�iÞ ¼ 1ffiffiffi

2p

�1=4k3=21

H�2i

þ i�1=4k3=2ffiffiffi

2p H�i: (29)

To be more accurate the above equations are meaningfulonly when

�2ð�Þ � k2 � �k4

a2þ �

k6

a4� 2

�2’ �

k6

a4(30)

so one obtains the following constraints on �:

� � 2H2 a6

k6; � � a4

k4; � � �

a2

k2: (31)

TheWKB solution is an exact one at the infinite past andthus the choice of the adiabatic vacuum is somewhat‘‘natural.’’ We also recall that all vacua prescriptions areequivalent up to zero-order when the WKB approximationholds [15].


023514-4

The choice of a different set of initial conditions at agiven � ¼ �1, though, can always be seen as the outcomefrom a particular (trans-Planckian) evolution from a differ-ent (and perhaps more consensual) set of initial conditionsdefined at � ¼ �0 <�1. In other words, the choice ofinitial conditions is equivalent to the choice of the physicsthat takes place before the moment when they are set.Nevertheless, that choice does not replace the discussionon the physics that takes place after that moment, whileenergies above the Planck scale are still at play.

V. NUMERICAL APPROACH

In this section we find the correction to the powerspectrum, for our test scalar field in a de Sitter spacetimewith a nonlinear d.r., using a fully numerical approach withinitial conditions given by Eqs. (28) and (29). This hasbeen done using a C code and the GSL library [16], wherewe have set H ¼ 0:5Mpl and considered a fixed value of

k � að�iÞH, stopping the evolution at a time �f for which

k � að�fÞH. We have verified that the result is indepen-

dent from the value of k, namely, that, as expected, thepower spectrum is still scale independent.

We write the power spectrum as

PðkÞ ¼�H

2

�2½1þ C��: (32)

The function C�� represents the correction due to the

nonlinear d.r., which obviously depends on the parameters� and �.

In Fig. 4 we show the correction C�� for different values

of z (going from z ¼ 0:775 to 0.950 with step 0.025) infunction of �. We can clearly see that � is not the onlyparameter that plays a role in the calculation of the non-linear effects on the power spectrum. Indeed, effective

potentials with the same � have different heights, depths,and steepnesses, all of which influence effects on the powerspectrum. Even for small �, the correction can be largedepending on the values of � and �. Therefore, it canhardly be considered a perturbation.From Fig. 4, we can note that for the smaller value of z

one obtains a minimum for C�� at �> 0 (which corre-

sponds to a finite value of � for a fixed z). Besides we haveno correction (C�� ¼ 0) for particular values of z and �.

This means that we could still have no correction in spite offinite � and �. In other words, this nonlinear d.r. yields nocorrection whatsoever to the power spectrum if the pa-rameters happen to be around those values.

VI. LINEAR APPROXIMATION

Since the WKB factor (18) is not much smaller than 1except for k � aH, one cannot use the WKB expression(17) for the solution of (24) (except in the aforementionedregion, of course). Nevertheless, there is indeed an exactsolution of Eq. (24) if we approximate the nonlinear dis-persion relation by a straight lines, as done in Ref. [4].If we write !iðkpÞ ¼ Aikp þ Bi for each of the regions

i ¼ 1, 2, 3 (see Fig. 5), the solutions are

�ið�Þ ¼ ciWM

�iBi

H;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi9

4� �i

s; 2iAik�

�

þ diWW

�iBi

H;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi9

4� �i

s; 2iAik�

�; (33)

where �i � ðBi=HÞ2 and WM;Wð�; �; �Þ are Whittaker func-

tions [17]. We pick Ai and Bi such that the positions of themaximum and minimum of the piecewise d.r. coincide withthose of the full nonlinear case and that the first and thesecond ‘‘turning points’’ kp1 and kp2 are the same as well

-2

0

2

4

6

8

10

12

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cαβ

∆

z=0.775

z=0.950

FIG. 4 (color online). Correction C�� on the power spectrum,with z varying from 0.775 up to 0.950 with step 0.025, as afunction of � from a full numerical calculation using the non-linear d.r. (19).

0

1

0 1 2 3

ωp

k p

FIG. 5 (color online). Three-piecewise linear approximation tothe nonlinear d.r. as suggested in Ref. [4]. The horizontal line isffiffiffi2

pH. Regions 1,2,3 are defined by the vertical lines and labeled

from right to left. kp is in units of Mpl.


023514-5

(see Fig. 5). In this way we have the same � of the exactnonlinear d.r. Of course, we pick A3 ¼ 1 and B3 ¼ 0,corresponding to the linear d.r. in the long-wavelength(small kp) limit.

The coefficients c1, d1 are given by matching the asymp-totic behavior of the previous equation to the initial con-ditions yielded by the WKB approximation (see below).The former is given by [17]

�1ð�Þ e�iA1k�ð2iA1k�ÞþiðB1=HÞ�c1

�ð1þ 2�Þ�ð12 þ �þ Þ þ d1

�

þ eþiA1k�ð2iA1k�Þ�iðB1=HÞ�c1

�ð1þ 2�Þ�ð12 þ �� Þ

�(34)

where � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9=4� �1

pand � �2B1=H. Comparing

Eq. (34) to the expected WKB solution

�wkbð�Þ ¼ 1ffiffiffiffiffiffiffi2!

p exp

��i

Z �

�i

!ð�Þd��

¼ 1ffiffiffiffiffiffiffiffiffiffiffi2A1k

p��

�i

�iðB1=HÞ

e�iA1kð��iÞ (35)

we get

c1 ¼ 0 d1 ¼ 1ffiffiffiffiffiffiffiffiffiffiffi2A1k

p ð2iA1k�iÞ�iB1=H: (36)

Note that the expression for d1 reduces to the expectedform when A1 ¼ 1 and B1 ¼ 0, which correspond to theusual d.r.

The amplitude of the growing mode can be calculated asis usually done in QM. Since we know the exact solution ineach region [Eq. (33)], all we have to do is to match themand their derivatives at the boundaries. In matrix notation,at the boundary between regions 1 and 2, we write:

Wð1ÞM ð�12Þ Wð1Þ

W ð�12ÞW 0ð1Þ

M ð�12Þ W 0ð1ÞW ð�12Þ

!c1d1

� �

¼ Wð2ÞM ð�12Þ Wð2Þ

W ð�12ÞW 0ð2Þ


!c2d2

� �(37)

where the superscripts (1, 2) indicate the value of thesubscript i in Eq. (33), �12 � �kp12=ðHkÞ is the value of

� at that boundary (see Fig. 5) and ð0Þ indicates derivativewith respect to �. We will write the above equation in amore compact form as

M 1 � C1 ¼ M2 � C2 (38)

with an obvious notation. Analogously, we can write thematching at the boundary of regions 2 and 3 as

Wð2ÞM ð�23Þ Wð2Þ

W ð�23ÞW 0ð2Þ


!c2d2

� �

¼ Wð3ÞM ð�23Þ Wð3Þ

W ð�23ÞW 0ð3Þ


!c3d3

� �(39)

and as

M 3 � C2 ¼ M4 � C3: (40)

Such a compact notation allow us to write C3 in terms of C1as

C 3 ¼ M�14 �M3 �M�1

2 �M1 � C1: (41)

The coefficient of the growing mode is given by d3—thesecond component of C3—since it is the coefficient of thedivergent Whittaker function when � ! 0�:

WW

�� iB3

2H;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4�3

p2

; 2iA3k2�

�! �i

k�(42)

while WM ! 0 at the same limit. Therefore,

�3ð�Þ ! �i

k�d3 (43)

as � ! 0�. We can thus write the power spectrum as

PðkÞ ¼ k3

22

��3

a

��2’ H2k

22jd3j2; (44)

in the limit � ! 0� (k � aH). This is, as expected, scaleinvariant.In Fig. 6 we have plotted the correction of the spectrum,

due to this three-piecewise approximation, as a function of�. The reason for such large values is that the nonlinear d.r.can be qualitatively different from the linear one (! k) atlarge j�j even for small �.

-2

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cαβ

∆

z=0.775

z=0.950

FIG. 6 (color online). Correction to the spectrum using the 3-piecewise approximation, with z varying from 0.775 up to 0.950with step 0.025, as a function of �.


023514-6

As it is clear, comparing Fig. 6 to Fig. 4, the resultsobtained with this three-piecewise approximation arepretty different from the numerical one obtained with theexact nonlinear d.r.

VII. SEMIANALYTICAL APPROACH

In this section we use kp as the independent variable.

Since it is proportional to � (kp ¼ k=a ¼ ��Hk), there is

no particular advantage to choose either way, but thisvariable is more ‘‘physical’’ and thus one can rely onher/his physical intuition. For a linear d.r., Eq. (12) isthen written as

d2�k

dk2pþ�1

H2� 2

k2p

��k ¼ 0: (45)

Using a nonlinear d.r. amounts to the substitution

1

H2! 1

H2

!2pðkpÞk2p

(46)

in the previous equation. Although there is no exact solu-tion of Eq. (45) with the substitution (46) when one usesthe d.r. (19), we still can get a fair analytical approximationby writing

�2ðkpÞ � 1

H2

!2pðkpÞk2p

� 2

k2p¼ �

H2k4p � �

H2k2p � 2

k2pþ 1

H2

��2

1ðkpÞ � �H2 k

4p � �

H2 k2p þ 1

H2 � c; kp � k��2

2ðkpÞ � � �H2 k

2p þ 1

H2 � 2k2p� d; kp � k�

;

(47)

where c, d, and the matching point k� are defined byrequiring that

�21ðk�Þ ¼ �2

2ðk�Þ ¼ �2ðk�Þ d�21

dk

��k�¼ d�2

2

dk

��k�;

(48)

which correspond to k� ¼ ðH2=�Þ1=6, d ¼ �ð�=H2Þ1=3,and c ¼ 2ð�=H2Þ1=3. The approximated expressions havethe same limiting behaviors (both at kp ! 0 and kp ! 1)

than the original one. That feature provides another pieceof information on the dependence of the outcome on thedetails of the nonlinear d.r. for intermediate kp, as opposed

to its small- and large-scale limits. See Fig. 7 for a quali-tative comparison.

The calculation itself is carried out by the usual ap-proach in QM, as in the previous section. The requirementsthat the ‘‘wave function’’ and its first derivative are con-tinuous at k� are easily accomplished since the solutions ofthe equation of motion (45) and (46) with the approxima-tions (47) are known analytically. For k > k� we find:

d2f1ðkpÞdk2p

� �21ðkpÞf1ðkpÞ ¼ 0; (49)

f1ðkpÞ ¼ c1 exp

�þ i

H

��

2ffiffiffiffi�

p kp þffiffiffiffi�

p3

k3p

��Ht1ðkpÞ

þ d1 exp

�� i

H

��

2ffiffiffiffi�


p3

k3p

��Ht2ðkpÞ;

(50)

where Ht1ðkpÞ � HeunTðA; 0; B;þi�Þ and Ht2ðkpÞ �HeunTðA; 0; B;�i�Þ are triconfluent Heun functions [18],with

A ��

3

2H2ffiffiffiffi�

p�2=3��1þ cH2 þ �2

4�

�; (51)

B ��

3

2H2�2

�1=3

� (52)

� � ð3 ffiffiffi2

p�1=4HÞ1=3kp: (53)

We can constrain the coefficients ðc1; d1Þ in Eq. (50) bycomparing the asymptotic behavior of the above equationto the expected WKB one, Eq. (26). The asymptotic limitof interest here is taken along a Stokes line ( argði�Þ ¼=2) of the triconfluent Heun function, which means thatthere are two equally dominant terms in the asymptoticexpansion:

HeunTðA; 0; B; i�Þ ��1½aþ b expð�i�3Þ� (54)

as � ! þ1, where a and b are constants. Since neither one

0

2

4

6

8

10

0 0.5 1 1.5 2 2.5 3

kpk*

E

FIG. 7 (color online). Effective potential VeffðkpÞ (thick blackline) and its approximations (thin colored lines) given byVeffðkpÞ ¼ E��2ðkpÞ and Eqs. (48), for �� ¼ 0:1325, �� ¼0:004 375, and H ¼ 0:5Mpl, as a function of kp (in Mpl units).

The straight horizontal line is E � 1=H2. The vertical line marksthe value of k� (see text for definition).


023514-7

is dominant over the other, both must be taken into ac-count. The aforementioned comparison yields

d1b� c1a ¼ 0 � c1bþ d1a ¼ C; (55)

where C � i=½ð3H ffiffiffi2

p�1=4Þ1=3 ffiffiffi

kp �. There are only 2 equa-

tions for 4 unknowns, which should be expected from thelack of a dominant behavior in the asymptotic expansionalong a Stokes line, as mentioned above. One could choosewhichever 2 of the above parameters (say, a and b) to bespecified by comparing the final result to the outcome ofthe numerical calculation using the approximated effectivepotential. Formally, that determination is supposed to bedone at every value of � and �, which would render oursemianalytical approach useless. Nevertheless, we havefound that writing b and d1 in terms of a and c1 yields aqualitatively good behavior for different values of z, i.e.,for different pairs f�;�g when compared to the numericalcalculation (see Fig. 8 below). Such procedure yields

b ¼ c1d1

a (56)

d1 ¼�C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 þ 4a2c21

q2a

: (57)

The sign in the above equation was again numericallydetermined to be the lower one (see below). Followingthis line of reasoning, we used a ¼ 0:1 and c1 ¼ �1:8for all z.

The last step of the semianalytical procedure is theevolution in the second region, when k < k�:

d2f2ðkpÞdk2p

��22ðkpÞf1ðkpÞ ¼ 0; (58)

whose exact solution is

f2ðkpÞ ¼ c21ffiffiffiffiffikp

p WM

�D;

3

4;

ffiffiffiffi�

pk2p

H

�

þ d21ffiffiffiffiffikp

p WW

�D;

3

4;

ffiffiffiffi�

pk2p

H

�; (59)

where D � ð1� dH2Þ=ð4H ffiffiffiffi�

p Þ.The coefficient d2 determines the amplitude of the per-

turbations since the function WWðkpÞ is the growing solu-

tion:

limkp!0

WWðkpÞffiffiffiffiffikp

p ¼ffiffiffiffi

pH1=4

2�1=8�ð�Þ1

kp; (60)

where � � ð54 � 1�dH2

4Hffiffiffi�

p Þ. The above expression allows us to

write the spectrum as

PðkÞ � k3

22

��f2a��2

(61)

’ kffiffiffiffiffiH

p

8�1=4�2ð�Þ jd2j2 (62)

¼�H

2

�2ð1þ C��Þ; (63)

in the superhorizon limit kp � H. The coefficient d2 is

determined by the aforementioned procedure, requiringthat the function and its derivative are continuous at k ¼k�. In matrix notation, it can be written as

g1ðk�Þ h1ðk�Þðdg1ðkpÞdkp

Þkp¼k� ðdh1ðkpÞdkpÞkp¼k�

!c1d1

� �

¼ g2ðk�Þ h2ðk�Þðdg2ðkpÞdkp

Þkp¼k� ðdh2ðkpÞdkpÞkp¼k�

!c2d2

� �(64)

where

g1ðkpÞ � exp

�þi

H

��

2ffiffiffiffi�


p3

k3p

��Ht1ðkpÞ (65)

h1ðkpÞ � exp

��i

H

��

2ffiffiffiffi�


p3

k3p

��Ht2ðkpÞ (66)

g2ðkpÞ � 1ffiffiffiffiffikp

p WM

�D;

3

4;

ffiffiffiffi�

pk2p

H

�(67)

h2ðkpÞ � 1ffiffiffiffiffikp

p WW

�D;

3

4;

ffiffiffiffi�

pk2p

H

�; (68)

which can be written in a more compact form, as in theprevious section, as

M 1 � C1 ¼ M2 � C2: (69)

-5

0

5

10

15

20

25

30

35

40

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cαβ

∆

z=0.775 z=0.950

z=0.950

z=0.775

FIG. 8 (color online). Correction C�� as a function of � whenz varies from 0.775 to 0.950 (as shown by the labels) for thesemianalytical approach (dashed blue lines) and for the numeri-cal evolution with approximated effective potential (47) (full redlines).


023514-8

As before, one can invert such equation and write

C 2 ¼ M�12 �M1 � C1; (70)

which shows once more that d2 is a linear combination ofc1 and d1. Since we have fixed c1 from the beginning, weare not able to find analytically the dependence of d2 on kand, therefore, to say if the spectrum is scale invariant, as itshould be. Nevertheless, we approach the same problemnumerically and we verify that the problem exhibits suchinvariance.

In order to be able to measure the accuracy of the semi-analytical approach, we have used the approximation (47)and numerically evolved the initial conditions (28) and(29). The evolution was split in two pieces: for k > k�and k < k�. Such calculation, besides fixing the sign inEq. (57), also yields the ‘‘best’’ (i.e., more robust withrespect to changes in � and �) values of c1 ¼ �1:8 anda ¼ 0:1.

In Fig. 8 we show the function C�� for different values

of z as a function of �, as given by the numerical evolutionusing the approximated expressions for the effective po-tential and as given by the semianalytical approach withthe same approximation. We can see that the correctiondepends also on z, showing that � is not the only variableto work with.

The semianalytical solution is not expected to work wellfor small ��, which means, for a fixed z, small �. For that,we should have taken into account the next-to-leadingorder terms in the asymptotic expansion of the Heunfunctions, Eq. (54). That would have, however, introducednew parameters that would have had to be fixed by com-paring to the numerical solution once more. Since we seeno advantage in having a large number of such parameters,we did not do so.

We note that we reproduce a minimum C�� at a finite �

as in the previous section. This approximation yields acorrection C�� with a good qualitative behavior as com-

pared to Fig. 4, but a large difference in the magnitude ofthe effect.

VIII. POWER-LAW INFLATION

A numerical approach can also be applied to find thecorrection to the power spectrum in a realistic model ofinflation for the nonlinear dispersion relation under con-sideration. In this section we restrict ourselves to the caseof power-law inflation [19] where, in proper time, aðtÞ tp, with p > 1. This expansion is generated by an expo-nential potential

VðÞ ¼ V0 exp

��

Mpl

ð�iÞ�

(71)

with V0 ¼ M2pl

t2ipð3p� 1Þ and � ¼ ð2pÞ1=2. The scale factor

and the homogeneous solution of the scalar field are givenby

aðtÞ ¼�t

ti

�p; (72)

ðtÞ ¼ i þMplð2pÞ1=2 log tti ; (73)

respectively, and the slow-roll parameters by

�1 �M2

pl

2

�V

V

�2 ¼ 1

p; �2 � M2

pl

V

V¼ 2

p; (74)

where V � dV=d and so on. In conformal time �, the

scale factor and the Hubble parameter become

að�Þ ¼��

�i

�p=ð1�pÞ

; Hð�Þ ¼ � p

p� 1

��

�i

�p=ðp�1Þ 1

�

(75)

with �i ¼ ti=ð1� pÞ. The equation of motion for theMukhanov variable, Eq. (8), becomes (using � as theindependent variable)

�00sk þ

�k2 � a00

a

��sk ¼ 0 (76)

and the canonically normalized solution, associated withthe adiabatic vacuum for k � aH, is given by (see, forexample, [20])

�k ¼��

4

�1=2

Hð1Þ� ð�k�Þ; (77)

with � ¼ 3=2þ 1=ðp� 1Þ. So, in the long-wavelengthlimit, we obtain the following scale dependent powerspectrum

PQðkÞ ¼ 1

ð2Þ2�� 2

�i

�2p=ðp�1Þ 1

�2ð�Þk�ð2=ðp�1ÞÞ (78)

with a spectral index ns ¼ 1� 2p�1 . Using this result,

namely, the dependence on k of the power spectrum, therange p < 60 is disfavored at 2� by the observation (see,for example, [21]).Now we want to see if the introduction of the particular

nonlinear d.r. given in Eq. (19) changes the dependenceof the power spectrum on k. Let us set our background:Eq. (76) can be written, using Eq. (75), as

�00k þ

�k2 � 2p2 � p

ð1� pÞ21

�2

��k ¼ 0: (79)

For p � 1, this can be approximated as

�00k þ ½k2 � 2a2H2��k ¼ 0; (80)

so, in this limit and following the same reasoning as before,to obtain a positive nonlinear d.r. that gives three realdistinct classical turning-points (!2

p ¼ 2H2) at the initial

time [22] we require that


023514-9

3=4< z < 1;gðzÞ

Hð�iÞ2<�<

fðzÞHð�iÞ2

(81)

with gðzÞ and fðzÞ given in Eqs. (22) and (23).The new equation of motion will be given by

�00k þ

�a2�k2

a2� �

k4

a4þ �

k6

a6

�� 2p2 � p

ð1� pÞ21

�2

��k ¼ 0

(82)

and, repeating the calculation in Sec. IV with the newbackground, one obtains the following initial conditions

�wkbð�iÞ ¼ 1ffiffiffi2

p�1=4k3=2

(83)

�0wkbð�iÞ ¼ � 1ffiffiffi

2p

�1=4k3=2p

p� 1

1

�i

� i�1=4ffiffiffi

2p k3=2: (84)

We now proceed with the numerical analysis taking ti ¼ðp� 1Þ=Mpl (and thus �i ¼ �1=Mpl). The restriction to

apply the adiabatic initial condition (� k�i � 1) becomesequivalent to k � Mpl; therefore, we are indeed in the

energy scale of TPP. Besides, with those initial conditionsand in the limit p � 1, we have Hð�iÞ ’ Mpl and the new

d.r. becomes very different from the linear one only for k ¼OðMplÞ. For the numerical analysis we take z ¼ 0:80, � ¼fð0:80Þþgð0:80Þ

2Hð�iÞ2 and the limiting value p ¼ 60, making the

analysis for k between 100Mpl and 300Mpl and going

from �i to a time for which k � aH. We give in Fig. 9the logarithm of the power spectrum calculated for thestandard linear case in function of lnðkÞ, while in Fig. 10the logarithm of the power spectrum calculated for thenonlinear d.r. case in function of lnðkÞ for many differentpoints inside the aforementioned range of k. As one can seefrom this plot, the power spectrum for the nonlinear case

still has a power-law dependence on k, but this d.r. pro-duces a change both in the normalization factor and in thespectral index when compared to the linear case. Fitting thedata in Fig. 10 we obtain a spectral index smaller than thatof the linear case: 0.857 instead of 0.966. The new spectralindex would be almost the same as that of the linear case ifwe have taken p 15. One obtains similar results startingfrom different values of p or with different values for theparameters � and �. Therefore, the introduction of thenonlinear d.r. gives a stronger dependence of the powerspectrum on k and, as a consequence, a possible differentrange of disfavored values for p.A different approach to this problem in a power-law

model is shown in [23]. The authors use the minimum-

-3.845

-3.84

-3.835

-3.83

-3.825

-3.82

-3.815

-3.81

-3.805

4.6 4.8 5 5.2 5.4 5.6 5.8

ln (k

3 P

Q)

ln(k)

FIG. 9 (color online). Logarithm of the power spectrum in apower-law model of inflation in function of lnðkÞ for a linear d.r.k is in units of Mpl.

-2.44

-2.42

-2.4

-2.38

-2.36

-2.34

-2.32

-2.3

-2.28

-2.26

4.6 4.8 5 5.2 5.4 5.6 5.8

ln (k

3 P

Q)

ln (k)

FIG. 10 (color online). Logarithm of the power spectrum in apower-law model of inflation using the nonlinear d.r. (12) infunction of lnðkÞ. k is in units of Mpl.

FIG. 11 (color online). Plot of the function 1=ð1þWÞ in termsof the wave number k (in units of Mpl) and the conformal time �

using the same parameters as in Fig. 10 (see text for the precisevalues). One can clearly see that the WKB approximation holds(W � 1) only for early times.


023514-10

uncertainty principle to fix the initial conditions and alsofind corrections to the power spectrum. We also would liketo underline that our results are not in disagreement withRef. [24]. Indeed, we consider a regime for which H is ofthe order ofMpl while the results of [24] assumeH � Mpl.

Using the same parameters as in Fig. 10, we have plottedin Fig. 11 the behavior of 1=ð1þWÞ, whereW is defined inEq. (18). One can easily see that the WKB approximationdoes not hold at small absolute values of the conformaltime, which stresses our very point [25].

IX. CONCLUSIONS

In this paper we have shown that the nonlinear d.r. (19)presents nonlinear results, i.e., which are not shown in theperturbative approaches used in previous papers in thesubject. In particular, the correction C�� can be almost

as large as 10 depending on the value of the parameters �,�. Therefore, it can hardly be considered a perturbation.

Our results agree, in part, with the literature on theeffects on the power spectrum obtained using differentapproaches to mimic TPP. In fact, also using differentapproaches, the effects are always of the order of H=Mpl

(see [4] and references cited therein) which in our case isOð1Þ.

We have compared previously used approximations andintroduced a new one, which allowed a semianalyticalcalculation.

The fundamental point behind such a large correctionfactor lies in the break of the WKB approximation at early(conformal) times. If the d.r. at this moment is highlynonlinear, as it happens here, the corrections are boundto be large [26].

However, from Fig. 4 we can see that for particular valueof z and � we obtain no correction to the power spectrum.Namely, that we could still have no correction in spite offinite � and �. In other words, this nonlinear d.r. yields nocorrection whatsoever to the power spectrum if the pa-rameters happen to be around those values.In the last section we have used a power-law model of

inflation as the background in order to understand whatfeatures were particular to the de Sitter (exponential) ex-pansion. It is clear that one would always get a scale-freespectrum in a de Sitter background, but what would happento the spectral index ns in a different one? We have shownthat this particular d.r. (19) yields a even smaller ns ascompared to power-law inflation with a linear d.r., andtherefore, it is strongly disfavored by observational results.On the other hand, this result suggests that another non-linear d.r. may yield also a different correction (perhaps inthe opposite direction) to the value of ns in a power-lawbackground.We have not addressed the question on backreactions

which may play an important role, since the corrections tothe power spectrum are not perturbative ones. Such impli-cations are beyond the scope of this work and deserve aseparate study, which will be published elsewhere.

ACKNOWLEDGMENTS

We would like to thank R. Ansari, A. Sarkar, andM. Schulze for useful discussions and Professor R. H.Brandenberger for useful correspondence. S. E. J. thanksProfessor Duval for her help on the Heun functions andacknowledges financial support from ICTP and fromCNPq.

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123501 (2001); R.H. Brandenberger and J. Martin,

Mod. Phys. Lett. A 16, 999 (2001).[3] M. Lemoine, M. Lubo, J. Martin, and J.-P. Uzan, Phys.

Rev. D 65, 023510 (2001).[4] J. Martin and R. Brandenberger, Phys. Rev. D 68, 063513

(2003).[5] L. P. Grishchuk, Zh. Eksp. Teor. Fiz. 67, 825 (1974) [Sov.

Phys. JETP 40, 409 (1975)].[6] V. F. Mukhanov, Zh. Eksp. Teor. Fiz. 94, No. 7, 1 (1988)

[Sov. Phys. JETP 67, 1297 (1988)].[7] Eugen Merzbacher, Quantum Mechanics (John Wiley &

Sons, New York, 1970), 3rd ed..[8] J. Martin and D. J. Schwarz, Phys. Rev. D 67, 083512

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[19] F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316(1985); D. Lyth and E. Stewart, Phys. Lett. B 274, 168(1992).

[20] G. Marozzi, Phys. Rev. D 76, 043504 (2007).[21] F. Finelli, M. Rianna, and N. Mandolesi, J. Cosmol.

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calculated the spectral index with the nonlinear d.r. using,in a strict and forward way, the WKB solutions (17) andthe same parameters as before, for the sake of comparison.The spectrum turns out to be strongly (blue) shifted fromthe correct (numerical) value: nwkbs ¼ 1:22.

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Trans-Planckian physics from a nonlinear dispersion relation

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