1

15th International Seminar on High Energy Physics QUARKS-2008

Sergiev Posad, Russia, 23-29 May, 2008

Large curvature perturbations near horizon crossing in single-field inflation models

Edgar Bugaev, Peter Klimai(pklimai@gmail.com)

Institute for Nuclear Research, RAS

2

Summary

We fill focus on single scalar field inflation models, which produce power spectrum of curvature perturbations PR(k) with untrivial behavior (far from PR(k) =const), when the wide range of cosmological scales is considered.

We will speak about

Scalar field inflation Slow roll approximation Numerical calculation of PR(k) Models with double well and Coleman-Weinberg potentials Running mass inflation model Possibilities of Primordial Black Hole (PBH) production

3

Inflation, i.e. accelerated expansion in the early Universe, is known to solve several problems of standard cosmology (flatness, homogeneity, etc).

It also provides the mechanism of primordial perturbations generation.

To date, hundreds models of inflation have been proposed. Simplest of such models have only one scalar field φ, with the potential V(φ), which form is model-dependant (should be extracted from the theory, but it does not exist yet). Energy density and pressure for the spatially homogenous field are

and, if the kinetic term is small compared to potential, the equation of state is just p=-ρ , which is required for inflation. The evolution φ(t) is given by

(1) (2)

In fact, two parameters are commonly introduced:

and if they are much smaller than 1, “slow roll” approximation, i.e., neglecting 1st term in (1) and kinetic term in (2), is valid.

Scalar field driven inflation

4

Standard paradigm states that quantum fluctuations in the inflaton field are responsible for producing classical energy density/curvature perturbations, which are observed today.For the perturbed scalar field, we write simply

but it is convenient to work with Mukhanov gauge-independent variables u and z:

For them, equation for perturbations is very simple:

(prime is the d/dτ derivative, dτ=dt/a is the conformal time). It is easy to show that z’’/z=2a2H2 + terms, that are small during the slow-roll inflation. So, when k>>aH (mode is inside the horizon), the solution is

Normalization is dictated here by the quantum origin of fluctuations.For the case k<<aH (mode is far outside the horizon), the growing solution is just u~z, so that the comoving curvature perturbation,

is constant.

5

To obtain the spectrum PR(k),

we must, strictly speaking, integrate the equation for perturbations, starting from time when the mode is deep inside the horizon, because the initial condition for this regime is available. Doing numerical calculations, we start at k=Nund aH, Nund~100, and proceed through the epoch of horizon exit k~aH to the time when the mode is far outside, k<<aH, and R(t) is effectively constant. This value should be taken, and the same procedure should be done for every k.We can rewrite the equation in terms of R and τ:

The approximate formula for the power spectrum exists, which works fine in many cases. Its derivation is based on the assumption that R does not change significantly after horizon exit, i.e. after k=aH. It is very simple:

This formula will break down for some k-modes, if “friction“ term z’/z will become negative at some epoch. In this case, amplification of the perturbation amplitude can generally be expected.

6

The value of power spectrum of curvature perturbations on large (‘cosmological’) scales is known:

The first derivative of PR(k) over ln(k) , i.e. the spectral index n, is also known, and the running of n is constrained (WMAP 5-year data, arXiv:0803.0586):

However, all present observations cover only rather narrow region of comoving wave number k ~ 10-3 – 10 Mpc-1, the behavior at smaller scales (large k) being unknown. It is possible, due to peculiarities of inflationary dynamics, that PR(k) has much larger value on some smaller scale. Specifically, if PR(k)~10-2, the significant amount of primordial black holes (PBHs) will be produced in the early Universe at the epoch of horizon entry for this scale, i.e. when k=aH. The PBH mass will be of the order of horizon mass at that moment. Depending on k, it can be in the very wide range, e.g. of the order of Solar mass, or much smaller.Detection of PBHs would provide valuable information about the Universe. Though PBHs haven’t yet been discovered, their non-observation still gives some info, e.g. constraining PR(k) on the wide range of otherwise unobservable scales.

Motivation

7

The argumentation was simple: slow-roll formula for the power spectrum written in terms of V and V’ gives

and in the region of small V’, high value of PR will be achieved.

The idea of PBHs was first proposed by Zeldovich and Novikov (1966). In the context of inflationary models, the possibility of PBH formation was first studied by Ivanov, Naselsky and Novikov (1994). To achieve a high value of PR, they proposed an inflationary potential with a ‘plateau’ feauture.

Several other ‘toy’ potentials have been studied by Bullock and Primack (1997), e.g. ‘wiggle’ and ‘cliff’ potentials.

Yokoyama (1998) has proposed a scenario with multiple inflation stages supported by one scalar field, and showed that PBH formation in such case is possible.

The power spectrum of perturbations was only estimated using approximate formulas at that time.

8

As our first example, we will take the double-well potential,

It has been first proposed by Yokoyama (1998) to support chaotic new inflation, i.e. two-component inflation, which starts as a chaotic-type inflation at large value of φ, and then proceeds as new inflation, rolling down from the origin φ=0 to one of the minimums φ = ±v . For this to happen, the value of vmust be finely-tuned.

Model with double-well potential

In this example, we took

In the figure, exact solution of the background equation is presented. The initial condition is taken φ(0) = 5 MPl .

9

“Slow roll” parameters and the comoving curvature perturbation for several values of k are shown.When friction term in equation for R

changes sign, R on super-horizon scales is significantly modified.

10

We can estimate the mass of produced PBHs as follows:The reheat temperature for such model is

The horizon mass at the beginning of radiation-dominated epoch is

The PBH mass is estimated by

Power spectrum PR(k) obtained by numerical calculation is shown in the figure. It has a strong peak, so PBH production is possible.

And in our case, MBH~107 g (in the peak). Such PBHs have already evaporated, but still could, in principle, lead to observable effects.

11

If we calculate the power spectrum for this model in the slow roll approximation, we will not obtain a peak.It arises only when exact numerical calculation of the perturbation spectrum is done, mode-by-mode.

In the figure, PR(k) is shown, calculated at the end of inflation (END) and at horizon exit (HE). Slow roll results (SR) are also shown for smaller values k. Near the end of inflation, this approximation is badly violated.

12

This potential has the form

It looks much like the double well potential, but the important difference is its behavior near the origin. For both cases, V(φ) ~ A - Bφp ,but p=4 for CW case and p=2 for double well.

The Coleman-Weinberg potential

In the CW case, new inflation can enter the slow roll phase and last for rather long time, actually being capable to last for so long that the perturbations produced in the chaotic epoch will be inflated away to currently unobservable scales. Because of this, Yokoyama (1998) concluded that this potential is incapable of producing PBHs.

13

Still, power spectrum for CW potential model has a peak, though for parameters we have taken PR~10-5 in the peak, which is not enough for significant PBH production (under standard assumptions of production threshold).Here, for the left peak, v=1.113MG, λ=5.5x10-13, corresponding horizon mass ~100MO.

For the right peak, v=1.112 MG, λ=2.4x10-13, corresponding horizon mass ~1027g.

But recently, it has been shown by Saito, Yokoyama and Nagato [see arXiv:0804.3470] that CW potential is capable of producing significant number of PBHs in the case when v is chosen in such a way that field makes several oscillations between the two minima of the potential before it climbs to the origin and new inflation starts.

14

We’ve seen that large values of PR are possible in models with temporary slow-roll violation, but they generally require fine-tuning of the potential form.

Another interesting question is whether large PR values are accessible under assumption of no slow roll violation? Kohri et al (arXiv:0711.5006) have shown that PBH formation can always be achieved with the suitable form of the single-field model potential. They showed how potential form can be reverse-engineered from the requirement of giving large values of spectrum at some small scale.The suitable potential was found to be of a ‘hilltop’-type. However, simplest hilltop potentials, considered by them (Kohri et al, arXiv:0707.3826),

with p=4 and p=6, were shown not to satisfy observational constraint n’<0.01. They require n’~0.1 for significant production of PBHs. The spectral index in such models is typically growing monotonously from values n<1 at large scales to n>1 at smaller ones.

A well-motivated example of analytically given potential of a ‘hilltop’ type which, with appropriately chosen parameters, leads to formation of PBHs is the potential of the running mass model.

15

The model was proposed by E.D. Stewart (1996,1997). It invokes softly broken global supersymmetry during inflation, with the potential

The dependence of m(φ) is determined by RGEs. Leach et al (2000) have shown that PBH formation is possible in this model. Covi et al (2004), using the Taylor expansion for m(φ) around scale k0, have obtained the potential in the form

The running mass inflation model

Here, φ* is the maximum of V(φ). We will denote the value of φ which inflaton has when scale with wave number k0 leaves the horizon with φ0.The parameter c is related to the gauge coupling constant during inflation, α :

16

The parameters defining the potential shape (s and c) are connected with n and n’ observed at scale k0 :

Thus, if we know n and n’, we can calculate s and c, and then the spectrum PR(k) for all values of k.

It is convenient to introduce another dimensionless parameter

The normalization to large scales is:

17

For our calculation, we used such set of parameters: n=0.95; n’=5x10-3.

For such choice of n and n’, parameters c and s are:c = 0.062 ; s = 0.040.

This is in the allowed region of experimental data, see figure on the right taken from Covi et al, 2004. In this paper, running mass model was compared to observation (WMAP year one data was used).

In the lower figure, time evolution φ(ln a) is shown, values of HI is 1 GeV (lower curve) and 10TeV (upper curve).

18

On the figure, time evolution of parameter δ is shown for the case HI= 10TeV. Slow roll inflation ends near δ=1, but our calculation proceeds further.

We can expect that in the region of δ<<1, slow roll approximation results will coincide with numerical calculation.

We also note that apart from the standard slow roll approximation for the power spectrum, an extended version was derived by Stewart and Lyth:

19

For the power spectrum, the simple approximate analytic formula is available:

In the figure, we show numerically obtained spectrum (solid blue line), spectrum given by this approximate formula (dashed red line) and spectrum calculated using the Stewart-Lyth extended slow roll approximation (dotted orange line).

Arrow shows the scale k at which parameter δ=1.

We see that for large k, power spectrum approaches large values ~ 1. This will lead to PBH production, in this case their mass beingM~1015 g (1016 / k)2 ~ 1015 g.

This is in the interesting region of PBHs which are just evaporating today, and they can be sought for experimentally.

20

Conclusions

1. Single scalar field inflationary models are capable of producing power spectrum of curvature perturbations which is far from trivial n=const when wide range of scales is considered.

2. In cases when slow roll conditions are not satisfied, the approximate slow roll formulae break down or become unreliable. Numerical calculation of the power spectrum is required.

3. Several models exist, in which large power on small scales is produced, even with the ‘red’ (n<1) spectrum on large scales. For the running mass model, this can be acheived even with n0.95 and n’~10-3 on large scales.

4. In this models, the significant number of PBHs is produced in the early Universe, so their existence remains an open possibility.