(mode is far outside the horizon), the growing solution is just u~z , so that the

comoving curvature perturbation

, is constant.

4

To obtain the spectrum P R (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=N und aH, N und ~100, and proceed through the epoch of horizon exit k~aH to the time when the mode is far outside, 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.

5

Motivation

The value of power spectrum of curvature perturbations on large (‘cosmological’) scales is known: The first derivative of P R (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 P R (k) has much larger value on some smaller scale. Specifically, if P R (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 P R (k) on the wide range of otherwise unobservable scales.

6

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 P R , they proposed an inflationary potential with a ‘plateau’ feauture. 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 P R will be achieved.

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.

7

Model with double-well potential

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 v must be finely-tuned.

In this example, we took In the figure, exact solution of the background equation is presented. The initial condition is taken φ(0) = 5 M Pl .

8

“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.

9

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

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 And in our case, M BH ~10 7 g (in the peak). Such PBHs have already evaporated, but still could, in principle, lead to observable effects.

10

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, P R (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.

11

The Coleman-Weinberg potential

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 - Bφ p , but p=4 for CW case and p=2 for double well. 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) incapable of producing PBHs.

concluded that this potential is 12

Still, power spectrum for CW potential model has a peak, though for parameters we have taken P R ~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=1.113M

G , λ=5.5x10

-13 , corresponding horizon mass ~100M O .

For the right peak, v=1.112 M G , λ=2.4x10

-13 , corresponding horizon mass ~10 27 g.

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.

13

We’ve seen that large values of

P R

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

P R

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 at smaller ones.

n<1

at large scales to

n>1

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.

14

The running mass inflation model

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)

RGEs. Leach et al (2000) is determined by have shown that PBH formation is possible in this model. Covi et al (2004), using the Taylor expansion for

m( φ)

around scale

k 0

, have obtained the potential in the form Here,

φ *

is the maximum of has when scale with wave number

k 0

The parameter

c V( φ).

We will denote the value of leaves the horizon with φ which inflaton

φ 0

.

is related to the gauge coupling constant during inflation,

α

: 15

It is convenient to introduce another dimensionless parameter The normalization to large scales is: The parameters defining the potential shape (

s

and

n’

observed at scale

k 0

: and

c

) are connected with

n

Thus, if we know

n

and

n’, P R (k)

for all values of

k

.

we can calculate

s

and

c

, and then the spectrum 16

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

H I

is

1 GeV

(lower curve) and

10TeV

(upper curve).

17

On the figure, time evolution of parameter

δ

is shown for the case

H I = 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 : 18

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 being

M~10 15 g (10 16 / k) 2 ~ 10 15 g

.

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

19

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 n  0.95 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.

20