Tomo Takahashi- Non-Gaussianity in the Curvaton Scenario

8/3/2019 Tomo Takahashi- Non-Gaussianity in the Curvaton Scenario

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Non-Gaussianity in the Curvaton Scenario

Tomo Takahashi1

Department of Physics, Saga University, Saga 840-8502, Japan

AbstractWe discuss the signatures of non-Gaussianity in the curvaton model where the poten-

tial includes also a non-quadratic term. In such a case the non-linearity parameter

fNL can become very small, and we show that non-Gaussianity is then encoded in

the non-reducible non-linearity parameter gNL of the trispectrum, which can be very

large. Thus the place to look for the non-Gaussianity in the curvaton model may be

the trispectrum rather than the bispectrum. We also show that gNL measures directly

the deviation of the curvaton potential from the purely quadratic form.

1 Introduction

The high precision of cosmological observations such as WMAP [1, 2] and the forthcoming Planck SurveyorMission [3] will soon make it possible to probe the actual physics of the primordial perturbation ratherthan merely describing it. As is well known, the primordial scalar and tensor power spectra, characterizedby spectral indices the and tensor-to-scalar ratio, can be used to test both models of inflation and othermechanisms for the generation of the primordial perturbation such as the curvaton [4, 5, 6]. However,many models can imprint similar features on the primordial power spectrum. Thus a possible deviationfrom Gaussian fluctuations, which may provide invaluable implications on the physics of the early universe,has been the focus of much attention recently.

The simplest inflation models generate an almost Gaussian fluctuation. In contrast, in the curvaton

scenario there can arise a large non-Gaussianity. However, in most studies on the curvaton so far, onesimply assumes a quadratic curvaton potential. Since the curvaton cannot be completely non-interacting(it has to decay), it is of interest to consider the implications of the deviations from the exactly quadraticpotential, which represent curvaton self-interactions. Such self-interactions would arise e.g. in curvatonmodels based on the flat directions of the minimally supersymmetric standard model (MSSM). Evensmall deviations could be important for phenomenology, as was pointed out in [7] where it was shownthat the non-Gaussianity predicted by the curvaton model can be sensitive to the shape of the potential.In particular, the nonlinearity parameter fNL which quantifies the bispectrum of primordial fluctuationcan, in contrast to the case of the quadratic curvaton potential, be very small in some cases. However, thesignatures of non-Gaussianity can be probed not only with the bispectrum but also with the trispectrum.In the following, we will show that even when fNL is negligibly small, gNL can be very large, indicatingthat the first signature of the curvaton-induced primordial non-Gaussianity may not come from the

bispectrum but rather from the trispectrum.

2 Non-linearity parameters

To discuss non-Gaussianity in the scenario, we make use of the non-linearity parameters fNL and gNLdefined by the expansion

= 1 +3

5fNL

21 +

9

25gNL

31 + O(

41 ), (1)

where is the primordial curvature perturbation. Writing the power spectrum as

k1k2 = (2)3P(k1)(k1 + k2), (2)

1E-mail:[email protected]

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the bispectrum and trispectrum are given by

k1k2k3 = (2)3B(k1, k2, k3)(k1 + k2 + k3). (3)

k1k2k3k4 = (2)3T(k1, k2, k3, k4)(k1 + k2 + k3 + k4), (4)

where B and T are products of the power spectra and can be written as

B(k1, k2, k3) =6

5fNL (P(k1)P(k2) + P(k2)P(k3) + P(k3)P(k1)) , (5)

T(k1, k2, k3, k4) = NL (P(k13)P(k3)P(k4) + 11 perms.)

+54

25gNL (P(k2)P(k3)P(k4) + 3 perms.) . (6)

Note the appearance of the independent non-linearity parameter NL. However, in the scenario we are

considering in the following, NL is related to fNL by

NL =36

25f2NL . (7)

To evaluate the primordial curvature fluctuation in the curvaton model, we need to specify a potentialfor the curvaton. Here we go beyond the usual quadratic approximation and consider the followingpotential:

V() =1

2m2

2 + m4

m

n, (8)

which contains a higher polynomial term in addition to the quadratic term. For later discussion, wedefine a parameter s which represents the size of the non-quadratic term relative to the quadratic one:

s 2 m

n2. (9)

Thus the larger s is, the larger is the contribution from the non-quadratic term.When the potential is quadratic, the fluctuation evolves exactly as the homogeneous mode. However,

when the curvaton field evolves under a non-quadratic potential, the fluctuation of the curvaton evolvesnon-linearly on large scales. In that case the curvature fluctuation can be written, up to the third order,as [8]

= N =2

3roscosc

+1

9

3r

1 +

oscosc2osc

4r2 2r3

oscosc

2()

2

+4

81

9r

4

2osc

osc

3osc+ 3

oscosc2osc

9r2

1 +

oscosc2osc

+r3

2

1 9

oscosc2osc

+ 10r4 + 3r5

oscosc

3()

3 , (10)

where osc is the value of the curvaton at the onset of its oscillation and the prime represents the derivativewith respect to . r roughly represents the ratio of the energy density of the curvaton to the total densityat the time of the curvaton decay. The exact definition is given by

r 3

4rad + 3

decay

. (11)

Notice that osc/osc = 1/ for the case of the quadratic potential. With this expression, we can writedown the non-linearity parameter fNL as

fNL =5

4r

1 +

oscosc2osc

5

3

5r

6. (12)

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-500

-400

-300

-200

-100

0

100

200

2 3 4 5 6 7 8

fNL

n

s = 0.2

s = 0.1

s = 0.05

-100000

-90000

-80000

-70000

-60000

-50000

-40000

-30000

-20000

-10000

0

-40 -20 0 20 40 60 80 100

gNL

fNL

s = 0.2s = 0.1s = 0.05

Figure 1: (Left) Plot of fNL as a function ofn for several values of s. (Right) Plot ofgNL as a functionof fNL for several values of s. Notice that fNL and n have one-to-one correspondence. In both panels,

r = 0.01.

Also notice that, although the curvaton scenario generally generates large non-Gaussianity with fNL >O(1), the non-linearity parameter fNL can be very small in the presence of the non-linear evolution ofthe curvaton field which can render the term 1 + (osc

osc)/2osc 0 [7, 8].

In the curvaton model, the non-linearity parameter gNL can be written as

gNL =25

54

9

4r2

2osc

osc

3osc+ 3

osc

osc

2osc

9

r

1 +

osc

osc

2osc

+

1

2

1 9

osc

osc

2osc

+ 10r + 3r2

. (13)

As one can easily see, even if the non-linear evolution of cancels to give a very small fNL, such a

cancellation does not necessarily occur for gNL. This indicates an interesting possibility where the non-Gaussian signature of the curvaton may come from the trispectrum rather than from the bispectrum.

3 Signatures of non-Gaussianity

Let us now consider non-Gaussianity in curvaton models, paying particular attention to the non-linearityparameters fNL and gNL. We show the values of fNL and gNL as a function of the power n in Fig. 1. Inthe left panel of the figure, the value of fNL is plotted as a function of the power n for several values ofs. There we have fixed the value ofr to r = 0.01. As can be read off from Eq. (12), when r is small, fNLcan be well approximated as

fNL 5

4r

1 +

osc

osc

2osc

. (14)

Notice that the combination osc

osc/2osc is zero when n = 2. As the value ofn becomes larger, the

above combination yields a negative contribution. Thus, fNL decreases to zero as the potential deviatesaway from a quadratic form and then becomes zero for some values of n and s. For small values of s,which correspond to the cases where the size of the non-quadratic term is relatively small compared tothe quadratic one, the power n should be large to make fNL very small. It should also be mentionedthat, for a fixed s, if we take larger values ofn beyond the cancellation point fNL = 0, fNL becomesnegative.

However, as already discussed, even if we obtain very small values for fNL, it does not necessarilyindicate that non-Gaussianity is small in the model but may show up in the higher order statistics.Indeed, this appears to be a generic feature of the curvaton model: the trispectrum cannot be suppressedand gNL can be quite large and is always negative for small values of r. Interestingly, even iffNL is zero,the value of |gNL| can be very large, as can be seen Fig. 1. This is because a cancellation which can occur

for fNL does not take place for gNL, which is a smooth function of n. When r is small, gNL is mainly

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determined by the first term in Eq. (13):

gNL

25

54 9

4r2

2osc

osc

3osc + 3

osc

osc

2osc

. (15)

In fact, one can show that the terms (osc

osc)/2osc and (

2osc

osc)/3osc give negative contributions for

the cases being considered here, which indicates that the cancellation between these terms never occurs[9]. Therefore gNL is always negative for small values of r even when fNL is very small. Thus, it mayturn out that the best place to look for non-Gaussianity in curvaton models is the trispectrum and gNLin particular.

4 Summary

We have discussed the signatures of non-Gaussianity in the curvaton model with the potential includinga non-quadratic term in addition to the usual quadratic term. When the curvaton potential is not purely

quadratic, fluctuations of the curvaton field evolve nonlinearly on superhorizon scales. This gives riseto predictions for the bispectrum, characterized by the non-linearity parameter fNL, which can deviateconsiderably from the quadratic case. However, by studying the trispectrum which is characterized bygNL along with fNL, we find that even when fNL is negligibly small, the absolute value of gNL can bevery large. Thus the signature of non-Gaussianity in the curvaton model may come from the trispectrumrather than from the bispectrum if its potential deviates from a purely quadratic form, as one wouldexpect in realistic particle physics models.

Here we discussed the case where a non-quadratic term is also taken into account in the curvatonpotential, however the cases where fluctuations from the inflaton can also be responsible for cosmicdensity perturbations today would also be interesting since such a case can arise in general. Such mixedfluctuation scenarios have been investigated in Refs. [10, 11, 12].

References

[1] E. Komatsu et al. [WMAP Collaboration], arXiv:0803.0547 [astro-ph].

[2] J. Dunkley et al. [WMAP Collaboration], arXiv:0803.0586 [astro-ph].

[3] [Planck Collaboration], arXiv:astro-ph/0604069;Planck webpage, http://www.rssd.esa.int/index.php?project=planck

[4] K. Enqvist and M. S. Sloth, Nucl. Phys. B 626, 395 (2002) [arXiv:hep-ph/0109214];

[5] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002) [arXiv:hep-ph/0110002];

[6] T. Moroi and T. Takahashi, Phys. Lett. B 522, 215 (2001) [Erratum-ibid. B 539, 303 (2002)]

[arXiv:hep-ph/0110096].

[7] K. Enqvist and S. Nurmi, JCAP 0510, 013 (2005) [arXiv:astro-ph/0508573].

[8] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D 74, 103003 (2006) [arXiv:astro-ph/0607627].

[9] K. Enqvist and T. Takahashi, JCAP 0809, 012 (2008) [arXiv:0807.3069 [astro-ph]].

[10] K. Ichikawa, T. Suyama, T. Takahashi and M. Yamaguchi, Phys. Rev. D 78, 023513 (2008)[arXiv:0802.4138 [astro-ph]].

[11] K. Ichikawa, T. Suyama, T. Takahashi and M. Yamaguchi, Phys. Rev. D 78, 063545 (2008)[arXiv:0807.3988 [astro-ph]].

[12] T. Moroi and T. Takahashi, arXiv:0810.0189 [hep-ph].

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Tomo Takahashi- Non-Gaussianity in the Curvaton Scenario