# Single Field Inflation models allowed and ruled out by the three years WMAP data

###### Abstract

We study the single field slow-roll inflation models that better agree with the available CMB and LSS data including the three years WMAP data: new inflation and hybrid inflation. We study these models as effective field theories in the Ginsburg-Landau context: a trinomial potential turns out to be a simple and well motivated model. The spectral index of the adiabatic fluctuations, the ratio of tensor to scalar fluctuations and the running index are studied in detail. We derive explicit formulae for and and provide relevant plots. In new inflation, and for the three years WMAP and 2dF central value , we predict and . In hybrid inflation, and for , we predict and . Interestingly enough, we find that in new inflation is bounded from above by and that is a two valued function of in the interval . In the first branch we find . In hybrid inflation we find a critical value for the mass parameter of the field coupled to the inflaton. For , where is the cosmological constant, hybrid inflation is ruled out by the WMAP three years data since it yields a blue tilted behaviour. Hybrid inflation for fullfills all the present CMB+LSS data for a large enough initial inflaton amplitude. Even if chaotic inflation predicts values compatible with the data, chaotic inflation is disfavoured since it predicts a too high value for the ratio of tensor to scalar fluctuations. The model which best agrees with the current data and which best prepares the way to the expected data , is the trinomial potential with negative mass term: new inflation.

###### pacs:

98.80.Cq,05.10.Cc,11.10.-z###### Contents

- I Introduction and Results
- II The Inflaton Potential and the Slow Roll Expansion
- III Spectral index , ratio and running index for New Inflation with the Trinomial Potential
- IV New Inflation with the Trinomial potential confronted to the three years WMAP data
- V Limiting Cases of the Trinomial Potential in New Inflation
- VI Hybrid Inflation
- VII Inflaton Dynamics in Hybrid Inflation
- VIII Spectral index , ratio and running index in Hybrid Inflation

## I Introduction and Results

Inflation was introduced to solve several outstanding problems of the standard Big Bang model guth and became an important part of the standard cosmology. At the same time, it provides a natural mechanism for the generation of scalar density fluctuations that seed large scale structure, thus explaining the origin of the temperature anisotropies in the cosmic microwave background (CMB), as well as that of tensor perturbations (primordial gravitational waves)mukyotr ; libros .

A distinct aspect of
inflationary perturbations is that these are generated by quantum
fluctuations of the scalar field(s) that drive inflation. After
their wavelength becomes larger than the Hubble radius, these
fluctuations are amplified and grow, becoming classical and
decoupling from causal microphysical processes. Upon re-entering
the horizon, during the matter era, these classical perturbations
seed the inhomogeneities which generate structure upon
gravitational collapsemukyotr ; libros . A great
diversity of inflationary models predict fairly generic features:
a gaussian, nearly scale invariant spectrum of (mostly) adiabatic
scalar and tensor primordial fluctuations, making the inflationary
paradigm fairly robust. The gaussian, adiabatic and nearly scale
invariant spectrum of primordial fluctuations provide an excellent
fit to the highly precise wealth of data provided by the Wilkinson
Microwave Anisotropy Probe (WMAP)WMAP ; WMAP3
Perhaps the most striking validation of inflation as a mechanism for generating
*superhorizon* (‘acausal’) fluctuations is the
anticorrelation peak in the temperature-polarization (TE) angular
power spectrum at corresponding to superhorizon
scalesWMAP . The confirmation of many of the robust predictions of
inflation by current high precision observations places
inflationary cosmology on solid grounds.

Amongst the wide variety of inflationary scenarios, single field
slow roll models provide an
appealing, simple and fairly generic description of inflation. Its
simplest implementation is based on a scalar field (the inflaton)
whose homogeneous expectation value drives the dynamics of the
scale factor, plus small quantum fluctuations. The inflaton
potential, is fairly flat during inflation. This flatness not only
leads to a slowly varying Hubble parameter, hence ensuring a
sufficient number of e-folds, but also provides an explanation for
the gaussianity of the fluctuations as well as for the (almost)
scale invariance of their power spectrum. A flat potential
precludes large non-linearities in the dynamics of the
*fluctuations* of the scalar field.

The current WMAP data seem to validate the simpler one-field slow roll scenario WMAP ; WMAP3 . Furthermore, because the potential is flat the scalar field is almost massless, and modes cross the horizon with an amplitude proportional to the Hubble parameter. This fact combined with a slowly varying Hubble parameter yields an almost scale invariant primordial power spectrum. The slow-roll approximation has been recently cast as a expansion 1sN , where is the number of efolds before the end of inflation when modes of cosmological relevance today first crossed the Hubble radius.

The observational progress permit to start to discriminate among different inflationary models, placing stringent constraints on them. The upper bound on the ratio of tensor to scalar fluctuations obtained by WMAP WMAP ; WMAP3 rules out the massless model and necessarily implies the presence of a mass term in the inflaton potential 1sN ; WMAP3 .

Besides its simplicity, the trinomial potential is a physically well motivated potential for inflation in the grounds of the Ginsburg-Landau approach to effective field theories (see for example ref.quir ). This potential is rich enough to describe the physics of inflation and accurately reproduce the WMAP data WMAP ; WMAP3 .

The slow-roll expansion plus the WMAP data constraints the inflaton potential to have the form 1sN

(1) |

where is the inflaton field, is a dimensionless, slowly varying field

(2) |

and is the energy scale of inflation which is determined by the amplitude of the scalar adiabatic fluctuations WMAP to be

Following the spirit of the Ginsburg-Landau theory of phase transitions, the simplest choice is a quartic trinomial for the inflaton potential nos ; 1sN :

(3) |

where the coefficients and are dimensionless and of order one and the signs correspond to large and small field inflation, respectively (chaotic and new inflation, respectively). Inserting eq.(3) in eq.(1) yields,

(4) |

where the mass and the couplings and are given by the following see-saw-like relations,

(5) |

where . Notice that guarantee that and without any fine tuning as stressed in ref. 1sN . That is, the smallness of the couplings directly follow from the form of the inflaton potential eq.(1) and the amplitude of the scalar fluctuations that fixes 1sN .

The small coupling limit of eqs.(3)-(4) corresponds to a quadratic potential while the strong coupling limit yields the massless quartic potential. The extreme asymmetric limit yields a massive model without quadratic term. In such limit the product must be kept fixed since it is determined by the amplitude of the scalar fluctuations.

We study here new inflation with the trinomial potential eqs.(3)-(4) and hybrid inflation [see below], the two models fulfill the observational constraints. We compute in both scenarios and the running as functions of the parameters of the models, derive explicit formulae for and and provide relevant plots. Moreover, we plot the ratio and the running as functions of the scalar index . Since the value of is now known WMAP3 -Teg , these plots allow us to predict the values of and for the different inflationary models considered. These predictions and plots are solely produced from theory and not from any fitting of the data.

The three years WMAP data indicate a red tilted spectrum () with a small ratio of tensor to scalar fluctuations WMAP3 . The present data do not permit to find the precise values neither of the ratio nor of the running index , only upper bounds are obtained WMAP ; WMAP3 . We therefore think that the value of [eq.(50)] obtained through a fit of the data assuming is more precise than the values of obtained through fits allowing both and to vary. Notice that was independently found from the 2dF data under similar assumptions 2dF . More precisely, from the three years WMAP data WMAP3 as well as ref. 2dF we take

(6) |

We find that for and any value of the asymmetry [see figs. 4 and 5], new inflation with the trinomial potential eqs.(3)-(4) predicts

We find for the lower value of the three years WMAP data band:

Moreover, in new inflation with the trinomial potential, we find that is bounded from above by

For we have in this model (see figs. 4 and 6). Interestingly enough, there exists two values (two branches) of for one value of in the interval [see fig. 4]. The value is the maximun in the first branch. The values correspond to a second branch of as a function of in the interval . In the first branch we have

The absolute maximun value belongs to the second branch and corresponds to the quadratic monomial potential obtained from eq.(3) at .

These predicted values of the ratio fullfil the three years WMAP bound including SDSS galaxy survey WMAP3

(7) |

Moreover, one can see from fig. 14 in ref. WMAP3 that from WMAPSDSS.

Chaotic inflation with the trinomial potential eq.(3)-(4) yields larger values of than new inflation for a given value of nos . More precisely, for we find for the binomial potential nos (the trinomial potential introduces very small changes).

Therefore, although the WMAP value for [eq.(6)] is compatible both with chaotic and new inflation, the WMAP bounds on clearly disfavour chaotic inflation. New inflation easily fulfils the three years WMAP bounds on and prepares the way for the expected data on the ratio of tensor/scalar fluctuations .

In the inflationary models of hybrid type, the inflaton is coupled to another scalar field with mass term through a potential of the type lin

(8) | |||

(9) | |||

(10) |

where plays the role of a cosmological constant and couples with .

The initial conditions are chosen such that and are very small (but not identically zero) and therefore inflation is driven by the cosmological constant plus the initial value of the inflaton . The inflaton field decreases with time while the scale factor grows exponentially with time. The field has an effective classical mass square

(11) |

Since the inflaton field decreases with time, becomes negative at some moment during inflation. At such moment, spinodal (tachyonic) unstabilities appear and the field starts to grow exponentially. Inflation stops when both fields and are comparable with and and close to their vaccum values.

We find that the time when the effective mass of the field eq.(11) becomes negative depends on the values of and . For low values of the field starts to grow close to the end of inflation. On the contrary, for higher values of the field starts to grow well before the end of inflation. This is explained by the fact that the scale of time variation of goes as . evolves slowly for small and fastly for large [see figs. 7-11].

Only at hybrid inflation becomes chaotic inflation with the monomial potential . For any value of even very small, the features of hybrid inflation remain.

We compute and for hybrid inflation as functions of the parameters in the potential eq.(8) and the initial value of the inflaton field [see figs. 12-23].

The results of our extended numerical investigation of hybrid inflation can be better expressed in terms of the dimensionless variables

We depict in figs. 12-23 the observables and the running index as functions of and . We present a complete picture for hybrid inflation covering two different, blue tilted and red tilted, regimes. We find that for all the observables, the shape of the curves depends crucially on the mass parameter of the field and the (rescaled) initial amplitude of the inflaton field.

We find a blue tilted spectrum () for while for we can have either or depending on the initial conditions: for we have , and for we have . The value of grows with : for , we find and for , we find .

We see that happens when the cosmological constant is large enough compared with . More precisely, for using . That is, for we have either red or blue tilted spectrum as explained above.

For large and always tend asymptotically to zero whatever be and .

We see from our calculations that all blue tilted values of in the domain can be realized by the hybrid inflation model eq.(8). However, at the light of the three years WMAP data ref. WMAP3 the blue tilted regime in hybrid inflation is ruled out.

The situation is totally different in the red tilted regime in hybrid inflation. The possible values of for such regime of hybrid inflation are in the upper-right quadrant as shown in fig. 24.

Hybrid inflation in the red tilted regime and fulfills the three years WMAP value for [see eq.(50)] as well as the bound on the ratio [eq.(51)]. We can read from fig. 20 and 22 that

Notice that hybrid inflation in the red tilted regime yields a too large ratio for .

At the central three years WMAP value both new and hybrid inflation are allowed. However, for hybrid inflation is in trouble () while for new inflation is excluded.

The potential which best agree with the present red tilted spectrum and which best prepares the way to the expected data (a small ) is the trinomial potential eqs.(3)-(4) with negative mass term, that is small field (new) inflation. Hybrid inflation with a trinomial potential can also reproduce the present data in the red tilted regime and .

All calculations presented in this paper stem from the inflaton potential in the slow roll approximation (dominant order in ). They do not use observational data as input. The analytical formulas and plots provided in the paper allow to read directly the predicted values of and as functions of . In order to make illustrative predictions, we take the value , as a judicious choice. The reader can see directly from the plots presented here our predictions for and for future observational values of .

## Ii The Inflaton Potential and the Slow Roll Expansion

The description of cosmological inflation is based on an isotropic and homogeneous geometry, which assuming flat spatial sections is determined by the invariant distance

(12) |

The scale factor obeys the Friedman equation

(13) |

where .

In single field inflation the energy density is dominated by a
homogeneous scalar *condensate*, the inflaton, whose dynamics
is described by an *effective* Lagrangian

(14) |

The inflaton potential is a slowly varying function of in order to permit a slow-roll solution for the inflaton field .

We showed in ref. 1sN that combining the WMAP data with the slow roll expansion yields an inflaton potential of the form

(15) |

where is a dimensionless, slowly varying field

(16) |

is the number of efolds since the cosmologically relevant modes exited the horizon till the end of inflation and is the energy scale of inflation

The dynamics of the rescaled field exhibits the slow
time evolution in terms of the *stretched*
dimensionless time variable,

(17) |

The rescaled variables and change slowly with time. A large change in the field amplitude results in a small change in the amplitude, a change in results in a change . The form of the potential, eq.(15) and the rescaled dimensionless inflaton field eq.(16) and time variable make manifest the slow-roll expansion as a consistent systematic expansion in powers of 1sN .

The inflaton mass around the minimum is given by a see-saw formula

The Hubble parameter when the cosmologically relevant modes exit the horizon is given by

where we used that . As a result, and . A Ginsburg-Landau realization of the inflationary potential that fits the amplitude of the CMB anisotropy remarkably well, reveals that the Hubble parameter, the inflaton mass and non-linear couplings are see-saw-like, namely powers of the ratio multiplied by further powers of . Therefore, the smallness of the couplings is not a result of fine tuning but a natural consequence of the form of the potential and the validity of the effective field theory description and slow roll. The quantum expansion in loops is therefore a double expansion on and . Notice that graviton corrections are also at least of order because the amplitude of tensor modes is of order . We showed that the form of the potential which fits the WMAP data and is consistent with slow roll eqs.(15)-(16) implies the small values for the inflaton self-couplings 1sN .

The equations of motion in terms of the dimensionless rescaled field and the slow time variable take the form,

(18) | |||

(19) | |||

(20) |

The slow-roll approximation follows by neglecting the terms in eqs.(18). Both and are of order for large . Both equations make manifest the slow roll expansion as an expansion in .

The number of e-folds since the field exits the horizon till the end of inflation (where takes the value ) can be computed in close form from eqs. (18) in the slow-roll approximation (neglecting corrections)

(21) |

The amplitude of adiabatic scalar perturbations is expressed as libros ; WMAP ; 1sN ; barrow ; hu

(22) |

The spectral index , its running and the ratio of tensor to scalar fluctuations are expressed as

(23) | |||

(24) | |||

(25) | |||

(26) | |||

(27) |

In eqs.(21)-(23) the field is computed at horizon exiting. We choose .

Since, and are of order one, we find from eq.(22)

(28) |

where we used and the WMAP value for WMAP . This fixes the scale of inflation to be

This value pinpoints the scale of the potential during inflation to be at the GUT scale suggesting a deep connection between inflation and the physics at the GUT scale in cosmological space-time.

We see that as well as the ratio turn out to be of order . This nearly scale invariance is a natural property of inflation which is described by a quasi-de Sitter space-time geometry. This can be understood intuitively as follows: the geometry of the universe is scale invariant during de Sitter stage since the metric takes in conformal time the form

Therefore, the primordial power generated is scale invariant except for the fact that inflation is not eternal and lasts for . Hence, the primordial spectrum is scale invariant up to corrections. The values and correspond to a critical point as discussed in ref.1sN . This a gaussian fixed point around which the inflation model hovers in the renormalization group sense with an almost scale invariant spectrum of scalar fluctuations during the slow roll stage.

The WMAP results favoured single inflaton models and among them new and hybrid inflation emerge to be preferable than chaotic inflation nos .

We analyze in the subsequent sections new inflation and hybrid inflation in its simple physical realizations within the Ginzburg-Landau approach (the trinomial potential)nos .

## Iii Spectral index , ratio and running index for New Inflation with the Trinomial Potential

We consider here the trinomial potential investigated in ref.nos

(29) |

where and and are dimensionless couplings.

The corresponding dimensionless potential takes the form

(30) |

where the quartic coupling is dimensionless as well as the asymmetry parameter . The couplings in eq.(29) and eq.(30) are related by,

(31) |

and the constant is related to by

The constant ensures that at the absolute minimum of the potential . Thus, inflation does not run eternally. is given by

The parameter reflects how asymmetric is the potential. Notice that is invariant under the changes . Hence, we can restrict ourselves to a given sign for . Without loss of generality, we choose and shall work with positive fields .

Notice that guarantee that and without any fine tuning as stressed in ref. 1sN .

New inflation is obtained by choosing the initial field in the interval . The inflaton slowly rolls down the slope of the potential from its initial value till the absolute minimum of the potential .

Computing the number of efolds from eq.(21), we find the field at horizon crossing related to the parameters and . It is convenient to define the field variable :

We obtain by inserting eq.(30) for into eq.(21) and setting ,

(32) | |||

(33) | |||

(34) |

turns to be a monotonically decreasing function of : decreases from till when increases from till . When vanishes quadratically,

We obtain in analogous way from eqs.(22) and (23) the spectral index, its running, the ratio and the amplitude of adiabatic perturbations,

(35) | |||

(36) | |||

(37) | |||

(38) | |||

(39) | |||

(40) | |||

(41) | |||

(42) | |||

(43) | |||

(44) | |||

(45) | |||

(46) | |||

(47) | |||

(48) |

## Iv New Inflation with the Trinomial potential confronted to the three years WMAP data

We plot , its running and in figs. 1, 2 and 3 as functions of for various values of the asymmetry of the potential being the dimensionless quartic coupling. Figs. 4 and 5 depict and the running as functions of for various values of the asymmetry .

We see that generically and for new inflation for all values of the couplings.

In new inflation we have the absolute upper bound

(49) |

which is attained by the quadratic monomial potential obtained from eq.(30) at . On the contrary, in chaotic inflation is bounded as

This bound holds for all values the asymmetry parameter . The lower and upper bounds for are saturated by the quadratic and quartic monomials, respectively.

We see from fig. 1 that exhibits a single maximun as a function of the quartic coupling for fixed asymmetry . In fig. 6 we plot as a function of . monotonically increases with and rapidly reaches its limiting value . The corresponding value for is . Values cannot be described by new inflation with the trinomial potential eqs.(29)-(30).

We see from fig. 4 and 5 that both and the running are two-valued functions of in the interval . That is, for each in this range there are two possible values for and for the running . Therefore, we can cover the whole range of values choosing the lower branch for . We find for this branch

This maximun value is well below the absolute maximun in new inflation [eq.(49)] which belongs to the second branch.

The plots of the ratio and the running as a function of show that these quantities are not very sensitive to the asymmetry for a given value of .

The three years WMAP WMAP3 data as well as ref. 2dF yield for the value (see also refs. SDSS and Teg )

(50) |

For and any value of the asymmetry [see fig. 4], new inflation with the trinomial potential eqs.(29)-(30) yields

New inflation with the trinomial potential always yield below the maximun value . For we have in this model . These values of the ratio fullfil the three years WMAP bound including SDSS galaxy survey WMAP3

(51) |

Moreover, one can see from fig. 14 in ref. WMAP3 that from WMAPSDSS.

Chaotic inflation with the trinomial potential eq.(29)-(30) yields larger values of than new inflation for a given value of nos . More precisely, we find for for the binomial potential in chaotic inflation nos (the trinomial potential introduces very small changes).

Therefore, although the WMAP value for [eq.(50)] is compatible both with chaotic and new inflation, the WMAP bounds on clearly disfavour chaotic inflation. New inflation easily fulfils the three years WMAP bounds on .

The present data do not permit to find the precise values neither of the ratio nor of the running index ; only upper bounds are obtained WMAP ; WMAP3 . We therefore think that the value of [eq.(50)] obtained through a fit of the data assuming is more precise than the values of obtained through fits allowing both and to vary. Notice that was independently found from the 2dF data under similar assumptions 2dF .

Ref. WMAP3 reports fits yielding negative values for of the order . Notice that the order of magnitude of the running is just fixed by the fact that it is a second order quantity in slow-roll: . Still, the negative sign of the running reported by ref.WMAP3 agrees with the sign prediction of new inflation with the trinomial potential [see fig. 2 and 5].

## V Limiting Cases of the Trinomial Potential in New Inflation

Let us now consider the limiting cases: the shallow limit (), the steep limit and the extremely asymmetric limit of the trinomial potential for new inflation eqs.(29)-(30).

### v.1 The shallow limit of the Trinomial Potential

In the shallow limit tends to , which is the minimum of in eq.(32). We find from eqs.(32)-(48),

(52) | |||

(53) | |||

(54) |

which coincide with for the monomial quadratic potential. That is, the limit is -independent except for . For fixed and the inflaton potential eq.(30) becomes purely quadratic: and

(55) |

where . Notice that the amplitude of scalar adiabatic fluctuations eq.(52) turns out to be proportional to the square mass of the inflaton in this regime which we read from eq.(55): . The shift of the inflaton field by has no observable consequences.

### v.2 The steep limit of the Trinomial Potential

In the steep limit tends to zero for new inflation. We find from eq.(32)