# Inflationary potentials yielding constant scalar perturbation spectral indices

###### Abstract

We explore the types of slow-roll inflationary potentials that result in scalar perturbations with a constant spectral index, i.e., perturbations that may be described by a single power-law spectrum over all observable scales. We devote particular attention to the type of potentials that result in the Harrison–Zel’dovich spectrum.

###### pacs:

98.80.Cq^{†}

^{†}preprint: FERMILAB-Pub-03/323-A

^{†}

^{†}preprint: LPT-ORSAY/04-13

## I Introduction

Inflation, a cornerstone of the modern framework for understanding the early universe guth81 ; lrreview , predicts the initial conditions for the formation of structure and the cosmic microwave background (CMB) anisotropies. During inflation, the primordial scalar (density) and tensor (gravitational wave) perturbations generated by quantum fluctuations are redshifted beyond the Hubble radius, becoming frozen as perturbations in the background metric muk81 ; hawking82 ; starobinsky82 ; guth82 ; bardeen83 . However, even when there is only one scalar field — the inflaton — the number of inflation models proposed in the literature is large lrreview . Determination of the properties of the scalar perturbations and tensor perturbations from CMB and large-scale structure observations allows one to constrain the space of possible inflation models dodelson97 ; kinney98a ; probes ; wmapinf ; barger ; kkmr ; liddleleach .

It is often adequate to characterize inflationary perturbations in terms of four quantities: the scalar and tensor power spectra, and , and the scalar and tensor spectral indices and . In this paper we focus on the scalar spectral index which, unless explicitly indicated otherwise, we refer to simply as the ‘spectral index’. Successful inflation models predict close to (the so-called Harrison–Zel’dovich spectrum), and typically has a small scale dependence. The best data available to date, combining the Wilkinson Microwave Anisotropy Probe wmap and Sloan Digital Sky Survey SDSS data sets, indicate that the evidence for anything other than a scale-invariant spectra is marginal at best, with no evidence for significant running of the scalar spectral index tegmarksloan . Moreover, one of us has recently argued that when information criteria are used to carry out cosmological model selection based on the current data sets available, then the best present description of cosmological data uses a scale-invariant () spectrum liddleparameters . It therefore makes sense to be considering the inflationary potentials associated with that spectrum.

It is known that inflaton potentials for constant lead to perturbation spectra that are exact power laws, i.e. is a constant lucchin . However, there has not yet been a systematic analysis of the types of inflaton potentials that yield constant . Here we take a first step in that direction, classifying those potentials within the framework of the slow-roll approximation SRref .

In the next section the basic results employed to calculate the properties of the perturbation spectrum using the slow-roll parameterization of the inflaton potential are reviewed. In Sec. III two exact differential equations connecting the potential and the field to the slow-roll parameters are derived and the general method used to calculate all the relevant cosmological quantities is outlined. In Sec. IV this method is applied to the determination of the inflationary potential yielding a -independent density spectral index: both the Harrison–Zel’dovich and the general case are considered to lowest order and to next order in the slow-roll parameter approximation. In Sec. V the flow of is examined to understand the number of solutions that arise. The conclusions are contained in Sec. VI.

## Ii Review of Basic Concepts

### ii.1 Inflationary Dynamics and Slow Roll Parameters

The dynamics of the standard Friedmann–Robertson–Walker (FRW) universe driven by the potential energy of a single scalar field – the inflaton – are usually expressed by the Friedmann equation for flat spatial sections and by the energy conservation equation:

(2) | |||||

where is the inflaton potential, the Planck mass and the Hubble expansion parameter. Once is specified, the field dynamics are determined by solving the coupled equations (2) and (2). Often it is simplest to do this using the Hamilton–Jacobi approach HJref in which is considered the fundamental quantity to be specified. Equations (2) and (2) then become two first-order equations:

(3) | |||||

(4) |

where . Whichever the method, once the dynamics of the inflaton field is known, is obtained by integrating Eq. (2). Without any loss of generality we assume that during inflation. Here we use the Hubble slow-roll parameters , and as defined in Ref. LPB

(5) | |||||

(6) | |||||

(7) |

The parameters and are the first terms in an infinite hierarchy of slow-roll parameters, whose -th member is defined by

(8) |

During slow-roll , and inflation ends when . The potential and its derivatives can be expressed as exact functions of these slow-roll parameters: up to second order in derivatives of one has

(9) | |||||

(10) | |||||

(11) |

### ii.2 A Hierarchy of Approximation Orders

As mentioned in the introduction, the observable quantities of interest are the power spectrum of the curvature perturbation on comoving hypersurfaces and the spectrum of gravity waves . These define and through

(12) | |||||

(13) |

As discussed in Ref. Lidsey:1995np ; Stewart:1993bc , the expressions for these quantities differ depending on the approximation order assumed in the slow-roll expansion. The approximation order is defined in general by considering how many terms in a slow-roll parameter expansion of a generic expression are retained, lowest-order approximation corresponding to retaining only the lowest-order term and next-order approximation corresponding to retaining terms up to the next-to-lowest order term.

For the perturbation power spectra and spectral indices, the lowest-order term is linear in the slow-roll parameters. To order , these expressions will contain the set of slow-roll parameters with where is a term of order . At next-order (), the expressions will contain the parameters as well as all second-order combinations thereof (namely and ). Hence, for order consistency, whenever an exact and an approximate expression are combined (as shall often be the case below) the result is accurate only to the order of the approximate expression, and the result must be expanded in a power series of slow-roll parameters up to and including terms of an overall degree consistent with the level of approximation assumed.

Recalling Lidsey et al. Lidsey:1995np , it is then possible to think of an infinite hierarchy of expressions for the perturbation spectra and for the spectral indices. It is unfortunate that, due to the complexity of the problem, only the first two approximation orders are currently available in general: indeed, at next-to-lowest order,

(14) | |||||

(15) | |||||

(16) | |||||

(17) |

where Lidsey:1995np ; Stewart:1993bc . As in Ref. Lidsey:1995np , the symbol “” is used to indicate that the results are accurate up to the order of approximation assumed. The lowest-order results are obtained by setting all the terms in curly brackets to zero.

## Iii The Parametrization Method

We now focus on the case of constant . To any order in the slow-roll approximation, imposing -independence of endows the problem with the additional set of relations

(18) |

Therefore, since there are slow-roll parameters at this order, the conditions (18) together with the constancy of mean that only one of those is independent: throughout the rest of this paper we take it to be . As we show in this section, it is then possible to determine and to this order.

The method is the following. First we derive two exact differential equations for and which, as we shall see below, only contain the slow-roll parameters and . Then, at a given order , we impose the conditions given in Eq. (18) which yield . As a result the two differential equations can be integrated to obtain and correct to order . Finally, provided can be inverted, we can obtain . This will be done in the next section where we also solve for all the dynamics of the problem, namely , and .

From Eq.(5) it is straightforward to obtain

(19) |

which, together with the definitions of and , yields the exact differential equation

(20) |

Once is specified, integration of this equation yields .

(21) |

which, divided by Eq. (9), produces the following exact differential equation, useful because it is independent of the Hubble parameter:

(22) |

Given , Eq. (22) can be integrated to give

(23) |

where is the integration constant which can be obtained from the observed perturbation amplitude. Finally from Eq. (9) the following expression for can be obtained

(24) |

As noted in the previous section, once the integrations in Eqs. (23) and (24) have been carried out, order consistency requires that the resulting expressions are expanded in powers of and only terms up to and including order are kept.

Once the expressions for and have
been computed, it is then possible to determine all the other
relevant cosmological quantities. Eq. (24) together with
the expression for gives to the given
order . This, together with the equation obtained for
then enables to be calculated using
Eq. (2).^{1}^{1}1Once again, note that the
conservation equation must be truncated to the correct order
in the approximation scheme. Once this step is carried out, the
time evolution of the Hubble parameter can be derived – either
using Eq. (2) or the solution of Eq. (24)
– and its integration then yields the dynamics of the scale
factor .

Before turning to the specific cases of constant spectral index, it is worth commenting on the apparently singular case of . This is nothing other than the usual exact power-law inflation model and is perfectly regular. From Eq. (20), we see that in this case the solution is , a constant independent of . Substituting this value into Eqs. (5) and (9) we obtain

(25) | |||||

(26) |

Substituting this into the Friedmann equation, Eq. (2), we obtain through

(27) |

Hence in Eq. (25) we find where , the usual power-law inflation result.

Finally, we note that it is also possible to address the present
problem using the definitions of the slow-roll parameters in the
expression for the spectral index to obtain a differential
equation for beato . While at lowest-order this
approach yields results which are equivalent to the ones derived
in the next section,^{2}^{2}2It is straightforward to show that
the condition for is solved by
. the differential equation arising at
next-order does not seem to allow an analytical solution and in
that case the parametrization method outlined above proves to be
preferable.

## Iv Applications

In this section the method outlined above is applied to the determination of the inflationary potentials which yield a -independent spectral index. Two cases will be considered: the Harrison–Zel’dovich power spectrum, and the case of a -independent spectral index not equal to unity. For each case, both lowest-order and next-order approximation results will be derived.

### iv.1 The Harrison–Zel’dovich Case

#### iv.1.1 Lowest-order approximation

Imposing in the lowest-order expression for the spectral index, Eq. (16), yields

(28) |

Thus Eqs. (20) and (23) become

(29) | |||||

(30) |

which can be integrated immediately, giving

(31) | |||||

(32) |

and hence

(33) |

Eq. (24) then yields

(34) |

and the constant can be read off from the lowest-order version of Eq. (14) as

(35) |

This, together with the expression for , can then be used in the Friedmann equation which becomes

(36) |

so that

(37) |

where . Eq. (31) can then be used to compute the dynamics of the slow-roll parameter

(38) |

Finally, the time evolution of the Hubble parameter and of the scale factor are given by:

(39) |

Let us now recall the work of Barrow and Liddle on intermediate inflation Barrow:1993zq . Though the present work differs in spirit from that paper (which starts by postulating a specific dynamics and then goes on to derive the corresponding potential), the two approaches share a common point, as we now outline. In Ref. Barrow:1993zq the scale factor is assumed to take the form

(40) |

with , constants. The authors then prove that this is an exact solution of the ‘intermediate’ inflation potential

(41) |

where , and that it is also a solution in the slow-roll approximation for the potential

(42) |

To see how the present results relate to the ones reported in Ref. Barrow:1993zq , we first quote the expressions for the slow-roll parameters obtained in the intermediate inflation case:

(43) |

Exploiting Eq. (43), the equation for the exact intermediate inflation potential can be recast in the form

(44) |

Now, we can think of this expression as a function of the slow-roll parameter instead of the field . In this perspective, neglecting the in the factor is the same as saying that lowest-order slow-roll approximation is assumed and that by order consistency one should retain only the lowest-order term arising from . In other words, the appearing in the factor will generate terms of higher order, all of which can be consistently neglected in a lowest-order calculation.

Note furthermore that imposing the condition in the form consistent with the lowest-order approximation (that is, ) and using Eq. (43) yields and . This is consistent with the previous calculation, since inserting this value of into Eq. (42) produces an expression for the inflaton potential analogous to Eq. (33)

(45) |

thus showing that the present analysis and the one carried out by Barrow and Liddle in Ref. Barrow:1993zq agree on the lowest-order potential able to produce a Harrison–Zel’dovich density power spectrum.

#### iv.1.2 Next-order approximation

As discussed at the beginning of Sec. III, the two conditions given in Eq. (18) must now be imposed in order to determine . The first condition is simply obtained from Eq. (16): imposing at next-order gives

(46) |

The second condition, , yields Lidsey:1995np

(47) |

These expressions then allow us to solve for and as functions of , giving

(48) |

(49) | |||||

(50) |

These can be integrated exactly to yield

(51) | |||||

(52) |

### iv.2 General power-laws

Having determined the inflationary potential generating a Harrison–Zel’dovich spectrum, in this Section we consider the more general case for which

(53) |

We focus primarily on the case: the results for are obtained by analytic continuation, with some care being taken over the number of solutions available in that case.

#### iv.2.1 Lowest-order approximation

Inserting the lowest-order expression for , Eq. (16), into Eq. (53), gives

(54) |

so that Eqs. (20) and (23) become

(55) | |||||

(56) |

Let’s first consider the case. Depending on whether or , integration of Eq. (55) above yields

(57) |

Similarly, integration of Eq. (56) gives

(58) |

where upper (lower) sign refers to the case. Combining these results produces

(59) |

Examples of such potentials for are illustrated in Fig. 2.

When , the corresponding lowest-order results for and are given by

(60) |

and

(61) |

Inverting Eq. (61) we obtain

(62) |

where now only one solution exists because .

#### iv.2.2 Next-order approximation

First it is necessary to express the slow-roll parameters and as functions of and . At next-order the condition (53) gives

(63) |

On imposing the condition we find

so that Eqs. (20) and (23) in this case take the form

(66) | |||||

(67) |

To solve these equations, let and be the two roots of so that

(68) |

Furthermore we assume , so that and . Using

(69) |

one can integrate Eq. (66) to find, in the cases and respectively,

(70) |

Finally, integration of Eq. (67) yields

(71) |

As in Sec. IV.1.2 the potential and the field have been successfully parametrized with respect to : they can be inverted numerically to find .

## V The flow of

As was pointed out in Sec. IV.2, it is interesting that more than one solution arises in the general power-law case. To further explore the reason for this, it is necessary to consider again the evolution of given by Eq. (55), keeping in mind that without loss of generality is assumed.

### v.1 The case

From Fig. 3, which shows as function of , it is possible to note that is positive for and is negative for . One can see that if , the initial value of , is smaller than , then the slow-roll parameter will increase toward , while if the initial value is greater than , then will decrease toward . In the case, then, independent of its initial value , will tend toward the point .

We have already seen that if , then is a constant given by , and that this fixed point corresponds to power-law inflation generating a -independent density spectral index given by . This result also allows one to reconcile the apparent contradictory requirements for the generation of a Harrison–Zel’dovich power spectrum stemming from the lowest-order slow-roll approximation condition, , and by power-law inflation definition . One can see once again that a Harrison–Zel’dovich power spectrum can be generated by power-law inflation in the limit (i.e. ), which corresponds to pure de Sitter expansion Lidsey:1995np .

Turning our attention to the case , it is easier to consider the derivative of with respect to ,

(72) |

which is also shown in Fig. 3. The interesting feature here is that the point represents an asymptote of : integrating it on either side with yields a logarithmically-diverging field. This necessarily implies that the value of the field, parametrized by , will tend to infinity while tends toward . Remembering that Eq. (55) is integrated to yield , it is then possible to note that the three distinct regions , and will give rise to three different dynamical behaviors for , which, once inserted in the expression for , are able to produce the same density perturbation spectral index. The apparent puzzle that arose at the end of Sec. IV.2.1 has therefore been solved: there are in fact two potentials, and both their domains are . It is now possible to understand that each one of them – together with power law inflation – is able to generate the desired power spectrum, depending on the initial condition chosen for the slow-roll parameter.

### v.2 The case

The cases and are similar. From Eq. (55) we see that, independent of , the value of will tend toward zero as inflation proceeds. In the case the solution derived Sec. IV.2 is the only one available, while in the special case (Harrison–Zel’dovich) it is possible to claim that two different inflationary potentials will be able to generate such a power spectrum: the flat one giving rise to the classical de Sitter expansion, and the one derived in Sec. IV.1.1, whose first term is proportional to .

## Vi Discussion

The analysis that has been carried out shows that inflaton potentials yielding the Harrison–Zel’dovich flat spectrum can be determined to lowest-order and next-order approximation in the slow-roll parameters. Similarly, potentials producing a -independent spectral index slightly different from unity have been derived to lowest-order and to next-order.

It is also possible to speculate that the same procedure can be carried out to any order of expansion in the slow-roll parameters. This is because the implications of the spectral index -independence are not as trivial as they may seem at first glance. Notice in fact that every time a higher approximation order is assumed, new slow-roll parameters will appear in the expression for the spectral index: going from lowest-order to next-order, for example, was introduced. This is hardly surprising, though, because these new parameters just correspond to higher derivatives of or (whatever is the degree of freedom chosen to express the slow-roll parameters) and a higher order treatment necessarily needs to take into account more derivative terms of the potential. However, the requirement of the spectral index to be -independent implies not only a particular value for but also that all its derivatives are equal to zero:

(73) |

Furthermore, the expression for the derivative of the spectral index contains slow-roll parameters up to the one. So once the approximation order is chosen, the problem is characterized by parameters and equations of constraint relating them. This allows the expression of all the slow-roll parameters as functions of . The choice of is not arbitrary, because once the expression for appropriate for the approximation level assumed is derived, the exact expressions for and for , Eqs. (20) and (22), can be exploited to compute and as functions of thus yielding the map .

###### Acknowledgements.

This work was supported in part by NASA grant NAG5-10842. A.V. would like to thank the David and Lucile Packard Foundation and Hotel Victoria, Torino, for financial support. E.J.C. thanks the Kavli Institute for Theoretical Physics, Santa Barbara for their support during the completion of part of this work. A.R.L. was supported in part by the Leverhulme Trust and by PPARC. This work was initiated during a visit by E.W.K. to Sussex supported by PPARC. We thank Cesar Terrero-Escalante for extensive comments on the original version of this paper, and also Filippo Vernizzi for useful comments.## References

- (1) A. Guth, Phys. Rev. D23, 347 (1981)
- (2) D. H. Lyth and A. Riotto, Phys. Rep. 314 1 (1999); A. Riotto, hep-ph/0210162; W. H. Kinney, astro-ph/0301448.
- (3) V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981).
- (4) S. W. Hawking, Phys. Lett. 115B, 295 (1982).
- (5) A. Starobinsky, Phys. Lett. 117B, 175 (1982).
- (6) A. Guth and S. Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).
- (7) J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D28, 679 (1983).
- (8) S. Dodelson, W. H. Kinney, and E. W. Kolb, Phys. Rev. D56, 3207 (1997).
- (9) W. H. Kinney, Phys. Rev. D58, 123506 (1998).
- (10) W. H. Kinney, A. Melchiorri, A. Riotto, Phys. Rev. D63, 023505 (2001); S. Hannestad, S. H. Hansen, and F. L. Villante, Astropart. Phys. 17 375 (2002); S. Hannestad, S. H. Hansen and F. L. Villante, Astropart. Phys. 16, 137 (2001); D. J. Schwarz, C. A. Terrero-Escalante, and A. A. Garcia, Phys. Lett. B517, 243 (2001); X. Wang, M. Tegmark, B. Jain and M. Zaldarriaga, Phys. Rev. D68, 123001 (2003).
- (11) H. V. Peiris et al. [WMAP collaboration], Astrophys. J. Supp. 148, 213 (2003).
- (12) V. Barger, H. S. Lee and D. Marfatia, Phys. Lett. B565, 33 (2003).
- (13) W. H. Kinney, E. W. Kolb, A. Melchiorri, and A. Riotto, hep-ph/0305130.
- (14) S. M. Leach and A. R. Liddle, Phys. Rev. D68, 123508 (2003).
- (15) C. L. Bennett et al. [WMAP collaboration], Astrophys. J. Supp. 148, 1 (2003); D. N. Spergel et al. [WMAP collaboration], Astrophys. J. Supp. 148, 175 (2003).
- (16) M. Tegmark et al. [SDSS Collaboration], astro-ph/0310725.
- (17) M. Tegmark et al. [SDSS Collaboration], astro-ph/0310723.
- (18) A. R. Liddle, astro-ph/0401198.
- (19) F. Lucchin and S. Matarrese, Phys. Rev. D32, 1316 (1985); Phys. Lett. B164, 282 (1895).
- (20) P. J. Steinhardt and M. S. Turner, Phys. Rev. D29, 2162 (1984); E. W. Kolb and M. S. Turner, The Early Universe, Addison–Wesley, Redwood City (1990).
- (21) D. S. Salopek and J. R. Bond, Phys. Rev D42, 3936 (1990).
- (22) A. R. Liddle, P. Parsons, and J. D. Barrow, Phys. Rev. D50, 7222 (1994).
- (23) E. D. Stewart and D. H. Lyth, Phys. Lett. B302, 171 (1993).
- (24) J. E. Lidsey, A. R. Liddle, E. W. Kolb, E. J. Copeland, T. Barreiro, and M. Abney, Rev. Mod. Phys. 69, 373 (1997).
- (25) E. Ayón-Beato, A. García, R. Mansilla, and C. A. Terrero-Escalante, Phys. Rev. D62 103513 (2000); C. A. Terrero-Escalante, E. Ayón-Beato, and A. García, Phys. Rev. D64 023503 (2001).
- (26) J. D. Barrow and A. R. Liddle, Phys. Rev. D47, 5219 (1993).