Cosmological constraints on the curvaton web parameters
We consider the mixed inflaton-curvaton scenario in which quantum fluctuations of the curvaton field during inflation lead to a relatively large curvature perturbation spectrum at small scales. We use the model of chaotic inflation with quadratic potential including supergravity corrections leading to a large positive tilt in the power spectrum of the curvaton field. The model is characterized by the strongly inhomogeneous curvaton field in the Universe and large non-Gaussianity of curvature perturbations at small scales. We obtained the constraints on the model parameters considering the process of primordial black hole (PBH) production in radiation era.
pacs:98.80.-k, 04.70.-s arXiv:1212.6529 [astro-ph.CO]
Curvaton mechanism which has been suggested 15 years ago Mollerach:1989hu (); Linde:1996gt (); Lyth:2001nq (); Moroi:2001ct (); Lyth:2002my () now is the object of intense study. It is assumed, in the standard implementation of the curvaton model, that not the inflaton field perturbations are responsible for the primordial density fluctuations and for the cosmic microwave background fluctuations, but instead the (isocurvature) perturbations of the curvaton field . It is assumed that this curvaton field is subdominant during inflation but in post-inflationary epoch when Hubble constant becomes small, (where is the curvaton mass), curvaton starts oscillating in its potential and behaves as nonrelativistic matter. The energy density of the curvaton decreases as ( is the scale factor) whereas the energy density of radiation produced by the inflaton decay decreases as . As a result the curvaton energy density grows relative to radiation energy density until the curvaton contribution becomes significant. If it happens before the curvaton decay one can say that curvaton mechanism is “effective”, in a sense that just the curvaton (rather than inflaton) field perturbations during inflation determine the resulting (adiabatic) curvature perturbations at cosmological scales.
In scenarios with the “effective” curvaton there is the strong constraint on a value of the curvaton mass: it must be much smaller than the Hubble constant during inflation, , otherwise the primordial density perturbations have too large spectral tilt. Moreover, if the ratio is not small, the coherent length of the curvaton field (i.e., the characteristic size of the region inside of which the field is approximately homogeneous) is also too small and, in particular, smaller than the current horizon size. In the latter case, the primordial perturbation spectrum is strongly non-Gaussian, in contradiction with observations.
The condition is too restrictive and prohibits an use, for a description of the curvaton, particle physics models predicting large ratios at inflation (e.g., some variants of supersymmetric theories). In this connection it is reasonable to consider also the mixed curvaton-inflaton scenarios Langlois:2004nn (); Ferrer:2004nv () in which the curvaton perturbations are additional to the usual perturbations produced by the inflaton. Combining two contributions, one can obtain the primordial perturbation spectrum which is in agreement with data at cosmological scales. At the same time, the prediction for smaller scales may be quite unusual: the spectrum can be, e.g., very blue (i.e., the spectral tilt is large and positive) and, besides, the perturbations can be strongly non-Gaussian. In particular, large value of the tilt arises due to non-renormalizable and supergravity corrections to the Lagrangian of some supersymmetric theories inducing mass terms of the order Dine:1983ys (); Coughlan:1984yk (); Goncharov:1984qm (); Bertolami:1987xb (); Dine:1995kz ().
In most curvaton scenarios it is assumed that the curvaton field in the Universe is highly homogeneous and, as a result, the non-Gaussianity is relatively small. According to the alternative hypothesis, after the long inflationary expansion, the average value of the curvaton field is close to zero, and the local value of the field has a Gaussian probability distribution, variance of which is given by the formula BD (); St1982 ()
Here, is the effective curvaton mass which differs from the true curvaton mass Lyth:2004nx (). The corresponding coherent length is
In Eqs. (1) and (2), is the spectral tilt of the perturbation spectrum of the curvaton field, . The assumption that will have real sense if the scale of interest, , will be larger than (both scales are calculated at the end of inflation). The value of is given by the expression
Here, is the scale leaving the horizon at the end of inflation, is the scale factor at the end of inflation (and at the beginning of radiation era), is a number of e-folds after the scale leaves the horizon. The condition
leads to the inequality . It means that if is not small (), and the coherent length is small, one anticipates the blue curvature spectrum (the curvaton contribution) and large non-Gaussianity at small scales. In this case, the data at cosmological scales are described by the inflaton fluctuations only. In the opposite case, if is very small, the number of e-folds , which is necessary for the fulfilment of the condition becomes large, . In particular, if , one has, instead of the inequality (4),
Traditionally, predictions for the primordial curvature perturbation spectrum in a region of small scales are constrained with a help of primordial black holes (PBHs). PBHs are produced in the early Universe, e.g., in radiation era, due to collapses of primordial density inhomogeneities ZelNo (); Hawking:1971ei (); Hawking74 (); Carr:1974nx (); Carr:1975qj (); Polnarev:1986bi (); Khlopov:2008qy (). Experimental limits from PBH overproduction had been studied in many articles, beginning from pioneering works Page:1976wx (); Zeldovich1977 (); for the latest reviews, see Josan:2009qn (); Carr:2009jm ().
In the present work we consider the predictions of the mixed curvaton-inflaton scenario just for the case which is most relevant for the PBH constraining: we assume that i) the average value of the curvaton field in the Universe is zero, and the Eq. (1) holds, and ii) the spectral tilt is relatively large () and positive. In this case adiabatic perturbations at small scales are produced mostly by the curvaton, resulting in a blue curvature spectrum. Large non-Gaussianity follows in this scenario from the quadratic dependence of the curvature on the curvaton field value. In this case, the typical size of the “curvaton domain” Linde:2005yw () is relatively small, it is smaller than the horizon size at the moment of the formation of PBH with a given mass.
Recently, the PBH formation in a curvaton scenario was studied in Kawasaki:2012wr (); Kohri:2012yw (). In contrast with the present work, authors of Kawasaki:2012wr (); Kohri:2012yw () do not use the assumption about a long period of inflation happened well before the observable Universe left the horizon. They assume, instead, that the curvaton field is nearly homogeneous in the whole Universe. The possibility of an essential PBH production at small scales in such models depends on the concrete inflationary scenario used. The authors of Kawasaki:2012wr () use for a curvaton field a variant of the axion model suggested in Kasuya:2009up () which predicts extremely blue spectrum of curvature fluctuations, while the authors of Kohri:2012yw () used the model with a convex potential [as the concrete realization of a “hilltop curvaton” scenario (see, e.g., Matsuda:2007av ())], in which strong scale dependence of the curvature power spectrum arises due to tachyonic enhancement effects.
The plan of the paper is as follows. In the next Section we derive the basic formula for the curvature perturbation spectrum used in the concrete calculations. In Sec. III, the process of PBH production in our curvaton model is considered. The last Section contains the results of the calculation and conclusions. The technical details concerning the calculation of a probability density function (PDF) of the smoothed curvature field are discussed in the Appendix.
Ii Curvature perturbation spectrum formula
Calculations of primordial curvature power spectra in mixed curvaton-inflaton scenario are carried out, in most cases, using the separated universe assumption and -formalism Starobinsky:1982ee (); Starobinsky:1986fxa (); Salopek:1990jq (); Sasaki:1995aw (); Sasaki:1998ug (); Wands:2000dp (); Lyth:2005fi (); Lyth:2004gb (). It had been shown, in particular Lyth:2004gb (), that the nonlinear curvature perturbation on an uniform energy density hypersurface, given by the formula
is conserved on superhorizon scales, for a fluid with an equation of state . In Eq. (6), is the “nonlinear curvature perturbation” entering the expression for the locally defined scale factor
In our case there are two (non-interacting) fluids, radiation from an inflaton decay and an oscillating curvaton which we consider as pressureless matter field. Assuming that the curvaton decays on an uniform total density hypersurface, one has on this surface, and, from Eq. (6), one has
From here, one has for the fluid densities
In the sudden decay approximation Lyth:2001nq (); Malik:2002jb (); Gupta:2003jc (), the sum of densities is, on the decay hypersurface, equal to (i.e., it is homogeneous quantity). It leads to the important relation Sasaki:2006kq ()
The second relation which is necessary for the calculation of the curvature power spectrum is the nonlinear generalization of the formula for the relative entropy perturbation. In linear theory, one has
Neglecting the curvaton density compared with radiation density (at the beginning of the radiation era), one has
The nonlinear extension of Eq. (14) is given by
Using Eqs. (11, 15) one can connect the curvature perturbation with the curvaton field value on super-Hubble scales during inflation. At a beginning of the curvaton oscillations, one has, in a case of the quadratic potential
Here, is the curvaton mass at the moment of the beginning of oscillations. For simplicity, everywhere below we neglect the change of curvaton mass after , and put .
It is convenient to study the evolution of the curvaton field (from the field value at horizon exit during inflation, , to the field value at the beginning of the oscillations, ) separately for the averaged value and perturbation,
The equations determining the evolution are
(the prime and the dot denote and , respectively). Eq. (19) is written for perturbations on superhorizon scales, where the gradient term () is negligible. For a quadratic potential , a fractional perturbation, , remains constant during the evolution.
As is pointed out in the Introduction, we assume that the early Universe follows the scenario considered in BD (); Starobinsky:1982ee () (“the Bunch-Davies case”). In this scenario, is close to zero. As for the , one can neglect its evolution during inflation. When the curvaton field is close to a minimum of the potential, then, due to a competition between the random walk and a (slow) roll, the typical value of the field, as can be easily shown, is , which is consistent with Eq. (1).
After an end of inflation, the evolution of the total curvaton field takes place (of the average value as well as of the perturbation). Following Ref. Lyth:2003dt (), we denote this evolution introducing the notation
where is the average value of the curvaton field at the end of inflation,
(which will be put equal to zero in final formulas). In a case of the quadratic potential the evolution is linear, so
and one has, finally,
Further, expanding exponents in Eq. (11) up to second order, one obtains, using the connection of with , :
Here, is given by the formula
Since, according to the definition of the , there is the proportionality (the proportionality coefficient is derived below, in Sec. III-A), and since , it follows from Eq. (26) that only the term proportional to survives in this Equation in the limit . It leads to the simple formula for the curvaton-generated part of the total curvature perturbation:
Everywhere below we will use for the notation , dropping the brackets in the index.
The power spectrum of is expressed through the power spectrum of the curvaton field perturbation Lyth:2006gd (),
and the power spectrum of the curvaton field is
The spectral tilt is simply connected with a value of the effective mass of the curvaton field, :
The difference is the number of e-folds of “relevant inflation” Lyth:2006gd (), i.e., the number of e-folds passed from the moment when the observable Universe leaves horizon up to the moment when the scale leaves horizon. The scale enters horizon at the radiation era, just when the curvature perturbation is created. The value of determines the value of horizon mass and, correspondingly, the order of magnitude value of PBH mass that can be produced at this moment.
Finally, we obtain for the curvature spectrum the expression
For calculations using this formula, one needs the relation . It is derived in the next Section, for the concrete choice of the potential [see Eq. (41)].
Iii PBH production in the curvaton model
iii.1 Curvaton potential
Recently, a variety of models of chaotic inflation in supergravity, in connection with the curvaton scenario and curvaton web problem, had been introduced and studied Demozzi:2010aj (). Their models and conclusions, however, can not be used in our work straightforwardly because in our curvaton scenario i) there is no degeneracy of masses of the inflaton and curvaton fields, and ii) our curvaton field is a real, single component field, rather than the radial component of a complex field, as in Demozzi:2010aj (). Both these features are not inconsistent with the general theory of chaotic inflation in supergravity Kallosh:2010ug (); Kallosh:2010xz (): for example, the curvaton field can be imaginary part of the complex scalar field Kallosh:2010xz ().
We consider the model with the simple phenomenological potential of the form
The corresponding effective mass of the curvaton field is and the spectral tilt is given by
The evolution equation for the curvaton field is given above [see Eq. (19)]. The calculation of starts at moment corresponding to an end of inflation and the beginning of the radiation-dominated era (the reheating is assumed to be instant).
The derivative is calculated numerically, and the initial conditions are:
According to this condition, after an onset of the oscillation the time scale of a change of the curvaton field is smaller that the expansion time .
According to (33), the energy density of the (average) curvaton field at the moment is 
After the moment , and until the curvaton’s decay at , the curvaton is assumed to behave like a pressureless matter, so a value of the curvaton density at decay time is [, ]:
The radiation density at the moment can be related to by using the Friedmann equation,
In this Equation, we used the equality , to obtain , while is calculated numerically from the condition given by Eq. (36).
iii.2 PDF for the curvature perturbation
It is generally assumed that the perturbations of the curvaton field at Hubble exit during inflation can be well described by a Gaussian random field (correspondingly, the equation (17) for contains, in its right-hand side, no higher-order terms). In our curvaton model, the curvature perturbation depends on the curvaton field quadratically. In this case, the field is chi-squared distributed, so the probability density function for perturbations is strongly non-Gaussian.
A formula for the PDF in the case of chi-square distribution of -filed perturbations, i.e., in the case when
For applications in PBH production calculations (with using the Press-Schechter formalism PS ()) one must derive the PDF for the smoothed field . This problem is thoroughly discussed in the Appendix. It is argued there that the PDF for the smoothed field can be approximately written in the form
Here, is the variance of the smoothed field [it is given by Eq. (91)] and the function is given by Eq. (67). Effects of the smoothing operation enter, in Eq. (44), only through the variance, while the function is the same in smoothing and non-smoothing cases.
iii.3 PBH mass spectrum and constraints
The PBH constraints are obtained using the Press and Schechter formalism generalized for a case of non-Gaussian PDFs. We will follow the Refs. Bugaev:2011wy (); Lyth:2012yp (); Byrnes:2012yx (); Linde:2012bt () working with the curvature perturbation rather than with the density contrast. The basic formula in the Press and Schechter approach is
In this Equation, is the probability that in a region of comoving size one has , where is the threshold value for the PBH formation in the radiation era, is the mass spectrum of the collapsed objects, is the initial energy density. We will use the value of corresponding to the PBH formation criterion for the density contrast, .
where is the horizon mass corresponding to the time when the fluctuation of mass crosses horizon in radiation era, is the horizon mass at the start of the radiation era, . For the constant we will use the value Bugaev:2000bz (); Bugaev:2008gw (). In the approximation of the fast reheating, coincides with the time of the end of inflation.
where is the scale factor at the end of inflation,
and are scale factor and time at matter-radiation equality, respectively. The derivative is given by the expression
This expression is obtained with using the formula (44) for the non-Gaussian PDF. The dependence of the PBH number density on the curvature perturbation power spectrum arises just through the derivative .
If PBHs form at , one can calculate the energy density fraction of the Universe contained in PBHs at the time of formation (at this time, the horizon mass is equal to Bugaev:2011wy ()):
In this formula, is the minimum mass of the PBH mass spectrum, . The PBH mass spectrum is very steep, so, with high accuracy one has
where is, by definition (see, e.g., Carr:2009jm ()), the fraction of the Universe’s mass in PBHs at their formation time,
Iv Results and discussion
The examples of curvaton-generated curvature perturbation power spectra are shown in Fig. 2, and some examples of the PBH mass spectra calculations are given in Fig. 3. For each curve shown in Figs. 2, 3, the model parameter is chosen so that the predicted PBH abundance is of the same order of magnitude as the currently available limits Carr:2009jm () on the parameter in the corresponding PBH mass range. On the vertical axis of Fig. 3 the combination is shown; just this combination is approximately equal to , as it follows from Eq. (III.3).
It is seen from Fig. 3 that for smaller values of , the PBH mass spectra become more wide. The low mass cut-off of the curves shown is determined by the fact that no PBHs are formed before the curvaton decays at , so the minimal PBH mass is .
For the constraining of the curvaton model parameters, we used the limits for from the review work Carr:2009jm (). Demanding that PBHs are not overproduced, i.e., the value of does not exceed the available limits Carr:2009jm (), one may obtain the corresponding constraints on the parameters of the considered cosmological model. Such constraints are shown in Fig. 4 for the case of and in Fig. 5 for .
In particular, in Figs. 4a and 5a we show the limits on the combination of parameters while in Figs. 4b and 5b - on the value of itself. The prohibited (by PBH overproduction) parameter ranges lie below the corresponding lines.
In the sudden decay approximation, there is a very simple approximate connection between and the PBH mass produced. It follows from the relations
Thus, constraints on (see Figs. 4b, 5b) are at the same time constraints on the mass of PBHs that can be produced in this model [this is reflected on the vertical axis of the Figures 4b, 5b; the relation between and is given by Eq. (55)].
Deriving the constraints, we use the condition
in order not to contradict with the data on the cosmological scales.
One must note that the characteristic values of which determine the constraints on the model parameters shown in Figs. 4, 5 are of order of . This is consistent with the PBH constraints on (for non-Gaussian -perturbations) obtained in our previous work Bugaev:2013vba () (see also Lyth:2012yp ()).
as a function of the constrained model parameters. It is seen that for all cases corresponding to the obtained PBH constraints, . As pointed out in the Introduction, this is needed for the validity of the considered model.
One can see from the resulting Figs. 4, 5 that, generally, PBH constraints are very weak. The forbidden region contains too small values of (although the nucleosynthesis limit, , allows such values). The PBH constraint works only for very high values of Hubble constant during inflation, GeV, and for very large values of curvaton masses, . For other values of parameters, the spectrum amplitude, , is too small and cannot be constrained. For illustrative purposes we show in Figs. 4, 5 constraints for a large interval of values, up to GeV, although there is a well-known upper bound on the Hubble parameter during inflation (according to the recent results of Planck collaboration, Ade:2013uln ()). One must note also that in the forbidden region the reheating temperatures are rather high () and, in standard supersymmetric models, gravitinos are overproduced.
The study was supported by The Ministry of education and science of Russian Federation, project 8525.
Appendix A Moments of PDF of -field
It follows from Eq. (28) that in our model the curvature perturbation depends on the Gaussian curvaton field quadratically,
In Eq. (58) the constant term is subtracted such that now , and is the “overcurvature”. Introducing the notation , one has
and the PDF of the field is
PDF of the field is obtained from the PDF of the field using the Chapman-Kolmogorov equation,
Here, are roots of the equation
The final expression for the PDF of the field is
The variance of the is
Using this equation, the distribution (64) can be written in the form:
In this equation, the notation is introduced. Note, that the product doesn’t depend on and separately, i.e.,
The first (central) moments of the are
and the first cumulants, , are given by the relations (see, e.g., Smith95 ())
The reduced cumulants are defined by the relation (see, e.g., Scoccimarro:2000qg ())
For the first non-trivial reduced cumulants, skewness and kurtosis, one has, respectively,
The general formula for is remarkably simple,
To find the PDF of the smoothed curvature fluctuations one must use the smoothed field,
Here, is the window function. In the present paper we use the Gaussian window function, defined by the equations
The general expressions for the cumulants of the PDF of the smoothed field have been derived in Matarrese:2000iz () using the path integral formalism. In this formalism, authors of Matarrese:2000iz () expressed cumulants through the integrals in -space,
Here, is the window function in -space, , is the power spectrum of the field,
As one can see from Eq. (A), values of the cumulants depend on the -dependence of the power spectrum of the field and on the window size . To study qualitatively the -dependence of the cumulants it is more convenient to use the expressions for through the integrals in real (configuration) space Peebles:1998ph (). The corresponding expression is
Here, the connected -point function in real space is given by the product of two-point correlation functions of field,
If the power spectrum of the field has a form
it follows from Eq. (81) that , and
Integrals in Eq. (79) converge, if , and scale with the window size . Therefore, there is the proportionality , and, as a result, the reduced cumulants almost don’t depend on the smoothing scale Peebles:1998ph (),