The pre-inflationary and inflationary fast-roll eras and their signatures in the low CMB multipoles
We study the entire coupled evolution of the inflaton and the scale factor for general initial conditions and at a given initial time . The generic early universe evolution has three stages: decelerated fast-roll followed by inflationary fast-roll and then inflationary slow-roll (an attractor always reached for generic initial conditions). This evolution is valid for all regular inflaton potentials . In addition, we find a special (extreme) slow-roll solution starting at in which the fast-roll stages are absent. At some time , the evolution backwards in time from reaches generically a mathematical singularity where vanishes and the Hubble parameter becomes singular. We determine the general behaviour near the singularity. The classical homogeneous inflaton description turns to be valid for well before the beginning of inflation, quantum loop effects are negligible there. The singularity is never reached in the validity region of the classical treatment and therefore it is not a real physical phenomenon here. Fast-roll and slow-roll regimes are analyzed in detail including the equation of state evolution, both analytically and numerically. The characteristic time scale of the fast-roll era turns to be where is the double-well inflaton potential, is the inflaton mass and the energy scale of inflation. The whole evolution of the fluctuations along the decelerated and inflationary fast-roll and slow-roll eras is computed. The Bunch-Davies initial conditions (BDic) are generalized for the present case in which the potential felt by the fluctuations can never be neglected. The fluctuations feel a singular attractive potential near the singularity (as in the case of a particle in a central singular potential) with exactly the critical strength () allowing the fall to the centre. Precisely, the fluctuations exhibit logarithmic behaviour describing the fall to . The power spectrum gets dynamically modified by the effect of the fast-roll eras and the choice of BDic at a finite time through the transfer function of initial conditions. The power spectrum vanishes at presents a first peak for ( being the conformal initial time), then oscillates with decreasing amplitude and vanishes asymptotically for . The transfer function affects the low CMB multipoles : the change for is computed as a function of the starting instant of the fluctuations . CMB quadrupole observations indicate large suppressions which are well reproduced for the range .
- I Introduction and summary of results
II The pre-inflationary and inflationary fast-roll eras
- II.1 The complete inflaton evolution through the different eras
- II.2 Inflaton and scale factor behaviour near the initial mathematical singularity
- II.3 Quantum loop effects and the validity of the classical inflaton picture
- II.4 The fast-roll regime: analytic approach
- II.5 The fast-roll regime: numerical solution
- III The slow-roll inflationary era
IV Complete Fluctuations evolution and fast-roll
effects on the power spectrum.
- IV.1 Scalar and tensor fluctuations near the initial singularity.
- IV.2 The primordial power spectrum, Scalar curvature fluctuations and the CMB+LSS data.
- IV.3 Accurate numerical computation of the power spectrum and the transfer function of initial conditions.
- IV.4 The effect of the fast-roll stage on the low multipoles of the CMB
- V Analytic formulas for the transfer function .
- VI Fixing the Total Number of Inflation e-folds and the bound from Entropy
I Introduction and summary of results
Since the Universe expands exponentially fast during inflation, gradients are exponentially erased and can be neglected. At the same time, the exponential stretching of spatial lengths classicalizes the physics and allows a classical treatment. One can therefore consider a homogeneous and classical inflaton field which thus determines self-consistently a homogenous and isotropic Friedman-Robertson Walker metric sourced by this inflaton.
This treatment is valid for early times well after the Planck time sec., at which the quantum fluctuations are expected to be large and thus a full quantum gravity treatment is required.
In this paper we study the entire coupled evolution of the inflaton field and the scale factor of the metric for generic initial conditions, fixed by the values of and at a given initial time .
We show that the generic early universe evolution has three stages: a decelerated fast-roll stage followed by an inflationary fast-roll stage and then by a slow-roll inflationary regime which is an attractor always reached for generic initial conditions. This evolution is valid for all regular inflaton potentials. In addition, we find a particular (extreme) slow-roll solution starting from in which the fast-roll stages are absent.
The evolution backwards in time from reachs generically a mathematical singularity at some time where the scale factor vanishes, and the Hubble parameter becomes singular.
We find the general behaviour of the inflaton and the scale factor near the singularity as given by eqs. (25)-(28) and determine the validity of the classical approximation, namely . It must be stressed that such mathematical singularity is attained extrapolating the classical treatment where it is no more valid. The singularity is never reached in the validity region of the classical treatment and therefore such mathematical singularity is not a real physical phenomenon here.
Quantum loops effects turns to be less than 1% for sec and therefore the classical treatment of the inflaton and the space-time can be trusted well before the begining of inflation.
The fast-roll (both decelerated and inflationary) and slow-roll regimes are analyzed in detail, with both the exact numerical evolution and an analytic approximation, and the whole equation of state evolution in the three regimes. We consider here the double well (broken symmetric) fourth order inflaton potential since it gives the best description of the CMB+LSS data mcmc (); biblia () within the Ginsburg-Landau effective theory approach we follow.
The characteristic time scale of the fast-roll era turns to be where is the double well inflaton potential at zero inflaton field, is the inflaton mass and the energy scale of inflation. The time scale of the inflaton in the extreme slow roll solution goes as the inverse of , namely .
We study the whole evolution of the curvature and tensor fluctuations along the three succesive regimes: decelerated fast-roll followed by inflationary fast-roll and then inflationary slow-roll, and compute the power spectrum by the end of inflation. The fluctuations feel a singular attractive potential near the singularity (as in the case of a particle in a central singular potential) with exactly the critical strength () for which the fall to the centre becomes possible. Precisely, the logarithmic behaviour of the fluctuations for eq.(95) describes the fall to for the critical strength of the potential felt by the fluctuations.
We generalize the Bunch-Davies initial conditions (BDic) to the present case in which the potential felt by the fluctuations can never be neglected.
In general, the mode functions for large behave as free modes since the potential becomes negligible in this limit except at the singularity . One can then impose Bunch-Davies conditions for large which corresponds to assume an initial quantum vacuum Fock state, empty of curvature excitations
Here stands for the conformal time: . Eq.(1) fulfils the Wronskian normalization (that ensures the canonical commutation relations)
In asymptotically flat (or conformally flat) regions of the space-time the potential felt by the fluctuations vanishes and the fluctuations exhibit a plane wave behaviour for all (not necesarily large). This is not the case in strong gravity fields or near curvature singularities as in the present cosmological space-time where can never be neglected at fixed . However, we can choose Bunch-Davies initial conditions (BDic) at (or equivalently, ) by imposing
That is, we consider the initial value problem for the mode functions giving the values of and at . This condition combined with the Wronskian condition eq.(2) implies that
which is equivalent to eq.(1) for large .
The power spectrum at the end of slow-roll inflation gets dynamically modified by the effect of the preceding fast-roll eras through the transfer function of initial conditions :
accounts for the effect of both the initial conditions and the fluctuations evolution during fast-roll (before slow-roll). depends on the time at which BDic are imposed.
where is the number of inflation efolds since the pivot CMB scale exits the horizon. We take here .
Actually, BDic can be imposed at if and only if the inflaton evolution also starts at . This only happens for a particular inflaton solution: the extreme slow-roll solution that we explicitly present and analyze in sec. III.1. In the extreme slow–roll case the fast-roll eras are absent, BDic are imposed at (that is ), then and . Only in this case the fluctuation power spectrum at the end of inflation is the usual power spectrum eq.(116).
When BDic are imposed at finite times , the spectrum is not the usual but it gets modified by a non-zero transfer function eq.(113). The power spectrum vanishes at and exhibits oscillations which vanish at large [see figs. 6 and 7]
Generically, the power spectrum vanishes at and we thus have
as shown in sec. V.1. presents a first peak for and then oscillates asymptotically with decreasing amplitude such that
We solved numerically the fluctuations equation with the BDic eq.(99) covering both the fast-roll and slow-roll regimes, namely for different initial times ranging from the singularity till the transition time from fast-roll to slow-roll. That is to say, we solved the fluctuations evolution for BDic imposed at different times in the three eras and we compare the resulting power spectra among them. We computed the corresponding transfer function, for the BDic imposed at the different eras. We depict vs. for the different values of the time where BDic are imposed in figs. 6.
When the BDic are imposed during the fast–roll stage well before it ends, changes much more significantly than along the extreme slow roll solution. This is due to two main effects: the potential felt by the fluctuations is attractive during fast–roll, and , (far from being almost proportional to ), tends to the constant value as and . The numerical transfer functions obtained from eqs.(104) and (113) are plotted in figs. 6.
We have also computed analytically with BDic at finite times , and a simple form is obtained in the scale invariant case, which is the leading term in the slow-roll expansion:
Different initial times lead essentially to a rescaling of in by a factor since the conformal time is almost proportional to during slow-roll [see figs. 6 and below eq.(136)]. By virtue of the dynamical attractor character of slow–roll, the power spectrum when the BDic are imposed at a finite time cannot really distinguish between the extreme slow–roll solution or any other solution which is attracted to slow–roll well before the time .
Using the transfer function we obtained, we computed the change on the CMB multipoles for and as functions of the starting instant of the fluctuations . We plot for vs. in fig. 9. We see that is positive for small and decreases with becoming then negative. The CMB quadrupole observations indicate a large suppression thus indicating that .
The fact that choosing BDic leads to a primordial power and its respective CMB multipoles which correctly reproduce the observed spectrum justifies the use of BDic.
Besides finding a CMB quadrupole suppression in agreement with observations biblia ()-quamc (), we provide here predictions for the dipole and -multipole suppressions. Forthcoming CMB observations can provide better data to confront our CMB multipole suppression predictions. It will be extremely interesting to measure the primordial dipole and compare with our predicted value.
Ii The pre-inflationary and inflationary fast-roll eras
The current WMAP data are validating the single field slow-roll scenario WMAP5 (). Single field slow-roll models provide an appealing, simple and fairly generic description of inflation. This inflationary scenario can be implemented using a scalar field, the inflaton with a Lagrangian density (see for example ref. biblia ())
where is the inflaton potential. Since the universe expands exponentially fast during inflation, gradient terms are exponentially suppressed and can be neglected. At the same time, the exponential stretching of spatial lengths classicalize the physics and permits a classical treatment. One can therefore consider an homogeneous and classical inflaton field which obeys the evolution equation
in the isotropic and homogeneous FRW metric which is sourced by the inflaton
stands for the Hubble parameter. The energy density and the pressure for a spatially homogeneous inflaton are given by
Threfore, the scale factor obeys the Friedmann equation,
In order to have a finite number of inflation efolds, the inflaton potential must vanish at its absolute minimum
Otherwise, inflation continues forever.
We formulate inflation as an effective field theory within the Ginsburg-Landau spirit 1sN (); gl (); biblia (). The theory of the second order phase transitions, the Ginsburg-Landau theory of superconductivity, the current-current Fermi theory of weak interactions, the sigma model of pions, nucleons (as skyrmions) and photons are all successful effective field theories. Our work shows how powerful is the effective theory of inflation to predict observable quantities that can be or will be soon contrasted with experiments.
The effective theory of inflation should be the low energy limit of a microscopic fundamental theory not yet precisely known. The energy scale of inflation should be at the Grand Unified Theory (GUT) energy scale in order to reproduce the amplitude of the CMB anisotropies biblia (). Therefore, the microscopic theory of inflation is expected to be a GUT in a cosmological space-time. Such a theory of inflation would contain many fields of various spins. However, in order to have a homogeneous and isotropic universe the expectation value of the energy-momentum tensor of the fields must be homogeneous and isotropic. The inflaton field in the effective theory may be a coarse-grained average of fundamental scalar fields, or a composite (bound state) of fundamental fields of higher spin, just as in superconductivity. The inflaton does not need to be a fundamental field, for example it may emerge as a condensate of fermion-antifermion pairs in a GUT in the cosmological background. In order to describe the cosmological evolution is enough to consider the effective dynamics of such condensates. The relation between the effective field theory of inflation and the microscopic fundamental GUT is akin to the relation between the effective Ginzburg-Landau theory of superconductivity and the microscopic BCS theory, or like the relation of the sigma model, an effective low energy theory of pions, photons and chiral condensates with quantum chromodynamics (QCD) quir ().
Vector fields have been considered to describe inflation in ref.gmv (). The results for the inflaton should not be very different from the effective inflaton description since the energy-momentum tensor of the vector field is to be taken homogeneous and isotropic. Namely, we are always in the presence of a scalar condensate.
Since the mass of the inflaton is given by GeV biblia (), massless fields alone cannot describe inflation which leads to the observed amplitude of the CMB anisotropies.
The classical inflaton potential gets modified by quantum loop corrections. We computed relevant quantum loop corrections to inflationary dynamics in ref. biblia (); effpot (). A thorough study of the effect of quantum fluctuations reveals that these loop corrections are suppressed by powers of where is the Hubble parameter during inflation biblia (); effpot (). Therefore, quantum loop corrections are very small, a conclusion that validates the reliability of the classical approximation and of the effective field theory approach to inflationary dynamics. In particular, the (small) one-loop corrections to the potential in an inflationary universe are very different from the Coleman-Weinberg form biblia (); effpot ().
We choose the inflaton field initially homogeneous which ensures it is always homogeneous. The fluctuations around are small and give small corrections to the homogeneity of the Universe. The rapid expansion of the Universe, in the inflationary regimes, takes care of the classical fluctuations, quickly flattening an eventually non-homogeneous condensate.
ii.1 The complete inflaton evolution through the different eras
The inflaton potential has then the universal form
where is the energy scale of inflation and is a dimensionless function. Without loss of generality we can set biblia (). Moreover, provided we can set without loss of generality . Namely, we have for small fields,
where the minus sign in the quadratic term corresponds to new inflation and the plus sign to chaotic inflation.
In these dimensionless variables, the energy density and the pressure for a spatially homogeneous inflaton are given from eq.(13) by
These coupled nonlinear differential equations completely define the time evolution of the inflaton field and the scale factor once the initial conditions are given at the initial time . Namely, the initial conditions are fixed by giving two real numbers, the values of and .
It follows from eqs.(20) that
When the expansion of the universe accelerates and it is then called inflationary.
The derivative of the Hubble parameter is always negative:
Therefore decreases monotonically with increasing . Conversely, if we evolve the solution backwards in time from will generically increase without bounds. Namely, at some time can exhibit a singularity where simultaneously vanishes.
In fact, the equations (20) admit the singular solution for ,
where is an integration constant. The energy density and equation of state take the limiting form,
Namely, the limiting equation of state is .
We have in this regime
where is some constant. That is, the geometry becomes singular for . The behaviour near is non-inflationary, namely decelerated, since
The evolution starts thus by this decelerated fast-roll regime followed by an inflationary fast-roll regime and then by a slow-roll inflationary regime biblia (). Recall that the slow-roll regime is an attractor bgzk (), and therefore the inflaton always reaches a slow-roll inflationary regime for generic initial conditions. We display in fig. 1 the inflaton flow in phase space, namely vs. for different initial conditions.
The number of efolds of slow-roll inflation is determined by the time when the inflaton trajectory reaches the red quasi-horizontal line of slow-roll regime [see fig. 1]. We see that decreases steeply with . This implies that is mainly determined by the initial value of with a mild (logarithmic) dependence on the initial value of
The inflaton flow described by eq.(25) results
which well reproduce the almost vertical blue and green lines in fig. 1.
The inflationary regimes are characterized by the slow-roll parameters and biblia ()
The slow-roll behaviour is defined by the condition . Typically, during slow-roll. More generally accelerated expansion (inflation) happens for while we have decelerated expansion for as follows from eqs.(19)-(23) and (30).
The parameter is also of the order during slow-roll and it is generically of order during fast-roll except when the potential vanishes.
The non-inflationary fast-roll stage starting at the singularity and ending when becomes positive [see eq.(23)].
The inflationary fast-roll stage starts when becomes positive and ends at when becomes smaller than [see eq.(30)].
The inflationary slow-roll stage follows, and it continues as long as and . It ends when becomes negative at .
A matter-dominated stage follows the inflationary era.
The four stages described above correspond to the evolution for generic initial conditions or, equivalently, starting from the singular behaviour eqs.(25). In addition, there exists a special (extreme) slow-roll solution starting at where the fast-roll stages are absent. We derive this extreme slow-roll solution in sec. III.1.
provides a very good fit for the CMB+LSS data, while at the same time being particularly simple, natural and stable in the Ginsburg-Landau sense. This is a new inflation model with the inflaton rolling from the vicinity of the local maxima of at towards the absolute minimum .
The inflaton mass and coupling are naturally expressed in terms of the two relevant energy scales in this problem: the energy scale of inflation and the Planck mass GeV,
Here is the number of efolds since the cosmologically relevant modes exit the horizon till the end of inflation and is the quartic coupling.
The MCMC analysis of the CMB+LSS data combined with the theoretical input above yields the value for the coupling mcmc (); biblia (). turns to be order one consistent with the Ginsburg-Landau formulation of the theory of inflation biblia ().
This model of new inflation yields as most probable values: mcmc (); biblia (). This value for is within reach of forthcoming CMB observations. For and in particular for the best fit value , the inflaton field exits the horizon in the negative concavity region intrinsic to new inflation biblia (). We find for the best fit mcmc (); biblia (),
We have two arbitrary real coefficients characterizing the initial conditions. We can choose them as and [see eq.(25)]. A total number of slow-roll inflation efolds permits to explain the CMB quadrupole suppression quadru (); quamc (); biblia (). Such requirement fixes the value of for a given coupling .
We integrated numerically eqs.(20) with eq.(25) as initial conditions. We find that yields 63 efolds of inflation during the slow-roll era for , the best fit to the CMB and LSS data. We find that is a monotonically increasing function of the coupling for fixed number of slow-roll efolds. At fixed coupling, increases with the number of slow-roll efolds.
We display in fig. 2 as a function of and the number of slow–roll inflation efolds .
For this value of and 63 efolds of inflation during the slow-roll, fast-roll ends by . In figures 3, we depict and vs. till a short time after the end of inflation. We define the time when inflation ends by the condition which gives .
Furthermore, we study in this paper the curvature and tensor fluctuations during the whole inflaton evolution in its three succesive regimes: non-inflationary fast-roll, inflationary fast-roll and inflationary slow-roll.
The equation for the scalar curvature fluctuations take in conformal time and dimensionless variables the form biblia ()
In cosmic time , eq.(35) takes the form
and and are given by eq.(30).
We display vs. in fig. 4 for the best fit value of the coupling and 63 efolds of slow-roll inflation.
The equation for the tensor fluctuations take in conformal time and dimensionless variables the form biblia ()
ii.2 Inflaton and scale factor behaviour near the initial mathematical singularity
In order to find the behaviour of and near the initial singularity we write
where stands for .
The asymptotic solution of eqs.(43) for turns to have the dominant form
where and are fourth degree polynomials in their arguments. The polynomials turn to be of fourth degree because the inflaton potential is of fourth degree. Their explicit expressions follow after calculation
As a consequence, the scale factor near the singularity takes the form
ii.3 Quantum loop effects and the validity of the classical inflaton picture
When quantum loop corrections are expected to become very large spoiling the classical description. More precisely, quantum loop corrections are of the order biblia (). From eqs.(16) and (25) the quantum loop corrections are of the order
where we used GeV biblia ().
The characteristic time is here the Planck time
Namely, the quantum loop corrections are less than 1% for times
The use of a classical and homogeneous inflaton field is justified in the out of equilibrium field theory context as the quantum formation of a condensate during inflation. This condensate turns to obey the classical evolution equations of an homogeneous inflaton eri ().
We see from eq.(25) that the inflaton field becomes negative for . But since a condensate field should be always positive, the classical and homogeneous inflaton picture requires
For the best fit coupling and 63 efolds of inflation we have which is consistent with eq.(54). By comparing this value of with eq.(54) we see that the quantum loop corrections are negligible in the stage where the condensate is already formed.
We can obtain a lower bound on since increases with the number of inflation efolds at fixed inflaton potential and since cannot be smaller than the lower bound provided by flatness and entropy biblia ().
Although all inflationary solutions obtained evolving backwards in time from the slow-roll stage do reach a zero of the scale factor, such mathematical singularity is attained extrapolating the classical treatment where it is no more valid. In fact, one never reaches the singularity in the validity region of the classical treatment. In summary, the classical singularity at is not a real physical phenomenon here.
The classical description with the homogeneous inflaton is very good for well before the beginning of inflation.
ii.4 The fast-roll regime: analytic approach
Or, in a compact form,
which has the exact solution
where turns out to be the characteristic time scale
We find for the best fit to CMB and LSS data, and ,
well after the Planck scale .
where the integration constant was chosen to fulfil eq.(27). The scale factor eq.(62) interpolates between the non-inflationary power law behaviour eq.(27) for and the eternal inflationary de Sitter behaviour for . Since we have set equal to constant, slow-roll De Sitter inflation never stops in this approximation. Namely, neither matter-dominated nor radiation-dominated eras are reached in this approximation.