Black Hole Remnants in the Early Universe

SLAC-PUB-15599 Black Hole Remnants in the Early Universe

1, 2, 1, 1, 2, 3, Fabio Scardigli, ∗ Christine Gruber, † and Pisin Chen ‡ 1Leung Center for Cosmology and Particle Astrophysics (LeCosPA), National Taiwan University, Taipei 106, Taiwan. 2Department of Physics and Graduate Institute of Astrophysics, National Taiwan University, Taipei 106, Taiwan. 3Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, U.S.A. We consider the production of primordial micro black holes (MBH) remnants in the early universe. These objects induce the universe to be in a matter-dominated era before the onset of inflation. Effects of such an epoch on the CMB power spectrum are discussed and computed both analytically and numerically. By comparison with the latest observational data from the WMAP collaboration, we find that our model is able to explain the quadrupole anomaly of the CMB power spectrum.

PACS numbers: xxx

I. INTRODUCTION sider the possibility of production of micro black holes in the early pre-inflationary Universe, due to quantum Inflation is without doubt the best model to explain fluctuations of the metric field [4, 5], as the seeds for the observed spatially flat and homogeneous Universe. the suppression of the inflaton fluctuations. There are Nevertheless, despite the great successes of the standard two salient features of this MBH nucleation. One is ΛCDM model in explaining almost all the data on CMB that the production rate per unit volume of space and anisotropy as most recently measured by WMAP obser- time is very high at the Planck temperature. To pre- vations, the suppression of the l = 2 quadrupole mode vent unphysical over-production of MBH, we invoke the still remains a puzzle in the framework of the standard holographic principle (HP) to constrain the initial condi- ΛCDM model (for a review on this subject, see e.g. [1]). tion of MBH production. The other is that the rate of such MBH production is a strong function of the back- Recently, several authors [2, 3] have been able to shed ground temperature. In particular, the rate is exponen- some light on this region of the CMB power spectrum, by tially suppressed when the temperature of the universe investigating the possibility of a pre-inflationary epoch, is sufficiently below the Planck temperature. Inflation dominated by radiation, instead of the usual inflationary is in general assumed to start when the temperature of vacuum. They found that a pre inflation radiation era the universe reaches the scale of the GUT energy, about can produce a suppression of the low k modes of the 1015 1016 GeV . Therefore one expects that the MBH primordial power spectrum, and this in turn affects the production− activity would cease long before the onset of low l modes of CMB anisotropy power spectrum. In fact, the inflation, and the MBH would have been totally evap- although inflation has the effect of washing out the initial orated and the universe would turn into radiation era be- conditions of the Universe, it happens that, if the present fore the inflation begins. However, when the Generalized Universe is just comparable to the size of the inflated Uncertainty Principle (GUP) is taken into consideration, region, a pre-inflation era may leave imprints on the CMB the complete decay of the nucleated MBH into radia- power spectrum. tion is prevented, and we have massive, but inert black However, these early attempts suffered of an arbitrary hole remnants [6] populating the pre-inflationary phase initial condition in the pre-inflationary era. Also, the of the Universe. Furthermore, the nucleation of MBH is space of the numerical parameters encoding the initial so efficient and fast that the Universe is put into a mat- radiation density was merely explored, without stating ter dominated era within a few Planck times, just about precise criteria for the choice of specific numerical values. 3 10 tp after the Big Bang (i.e., well before inflation) and In the present paper we propose a pre-inflationary there it stays until the onset of inflation. Such a pre- scenario that is based on the generic micro black hole inflation matter-dominated universe then suppresses the (MBH) production and a minimal set of first princi- initial inflaton fluctuations at the onset of the inflation. ples, namely the generalized uncertainty principle (GUP) and the holographic principle (HP), that can give rise to Accurate numerical simulations allowed us to single the suppression of the CMB quadrupole self-consistently out almost unique numerical values for the relevant ra- without the need of arbitrary inputs. Specifically, we con- diation and matter parameters. We have computed the effects of a pre-inflationary matter epoch on the primordial power spectrum of the quantum fluctuations of a scalar field, both analytically and numerically. Our ana- ∗Electronic address: [email protected] lytical solution, also a new feature of the present attempt †Electronic address: chrisy˙[email protected] with respect to the previous all-numerical investigations, ‡Electronic address: [email protected] has served as a guide for the more precise numerical com-

Published in Phys.Rev.D83:063507,2011 and arXiv:1009.0882. Work supported in part by US Department of Energy under contract DE-AC02-76SF00515. 2 putations. We have considered three alternative scenar- II. BLACK HOLE PHYSICS ios. The main model presented in this paper attempts to explain the suppression of the quadrupole moment of A. Generalized Uncertainty Principle the CMB with a pre-inflationary matter era. In order to isolate the cause of the CMB quadrupole anomaly, we As it is well known from the classical argument of the further examine two variations of this model, one with- Heisenberg microscope [7], the size δx of the smallest out the GUP, where the black holes decay into radiation detail of an object, theoretically detectable with a beam completely, and one without any black hole nucleated at of photons of energy E, is roughly given by all, both models resulting into a radiation-dominated era before inflation. ~c δx , (1) In all three cases the primordial power spectra have ≃ 2E been fed to the CMBFAST code in order to obtain the CMB power spectra, and then compared with each other since larger and larger energies are required to explore and tested against the last WMAP observational data. smaller and smaller details. Boundary conditions have been set in the fully inflation- The research on viable generalizations of the Heisen- ary epoch, and in so doing we avoided any arbitrary as- berg uncertainty principle traces back to many decades sumption on the state of radiation in the pre-inflationary (see for early approaches [8]; see for a review [9], and for era. The pre-inflation matter model seems to be the only more recent approaches [10–12]). In the last 20 years, one, among those studied, which is able to describe the there have been important studies in string theory [13] l = 2 mode suppression, although the radiation model suggesting that, in gedanken experiments on high energy still presents a better fitting of the data at high l values. scattering with high momentum transfer, the uncertainty This conclusion is widely discussed in the last section of relation should be written as the paper, where we also suggest avenues for future re- ~ p δx & + 2 βℓ2 , (2) search. We find it remarkable that, based on our ab initio 2p p ~ model without arbitrary input parameters, our resulting 2 2 suppression of the CMB spectrum agrees well with ob- where ℓp is the Planck length, and βℓp λs, where λs is servations. the characteristic string length. Since in∼ our high energy scattering E cp, the stringy Generalized Uncertainty The paper is organized as follows. In section II we Principle (GUP)≃ can be also written as will provide a brief overview of the black hole physics under the Generalized Uncertainty Principle. The GUP ~c E δx & + 2 βℓ2 , (3) will lead to a new mass-temperature relation and define a 2E p ~c minimum mass and maximum temperature for the black holes. Section III will set up the basic equations gov- where E is the energy of the colliding beams. erning the scenario, on the one hand the absorption and A similar modification of the uncertainty principle has emission processes which determine the black hole mass, been proposed [11, 14], on the ground of gedanken scat- and on the other hand the evolution equations of the tering experiments involving the formation of micro black holes with a gravitational radius of R E. It reads universe depending on its constitution. We will end up S ∼ with a system of four equations, containing black hole ~c 2E for E < p mass, black hole density, radiation density and the scale δx & E (4)  factor as variables. At the end of that section we de- βR (E) for E , rive a condition for a pre inflation matter era, and we  S ≥Ep present inflationary solutions. In section IV we will nu- where RS is the Schwarzschild radius associated with the merically calculate the black hole production and fix the energy E, namely RS = ℓpE/ p. parameters in the evolution equations. Section V will Combining linearly the aboveE inequalities we get deal with the equations needed to be solved to obtain ~c the primordial power spectrum of the quantum fluctua- δx & + βR (E) . (5) tions of a scalar field. First we state some approximate 2E S analytical solutions obtained via the WKB method, and Thus, the GUP originating from micro black hole then we present the numerical result of the equations. In gedanken experiments (MBH GUP) can be written as section VI finally we present the CMB power spectrum of the temperature anisotropies obtained by our model, and ~c E compare it to the two alternative cases with a radiation- δx & + βℓp . (6) 2E p dominated era before inflation. The conclusions of our E work are contained in section VII. Also the stringy inspired GUP (ST GUP, eq.(3)), using the relation pℓp = ~c/2, can be written as Throughout the paper the Planck length is defined as E 2 ~ 3 ~ ~ ℓp = G /c , the Planck energy as pℓp = c/2, and the c E 2 E δx & + βℓp . (7) Planck mass as Mp = p/c . 2E E Ep 3 where β is the deformation parameter, generally believed be fixed soon. With (9) we can rephrase Eq. (8) as to be of O(1). Thus, in 4 dimensions the two principles E coincide. In 4 + n dimensions, however, they lead to 2m Ep + β . (10) remarkably different predictions (see [15]). ≃ E p E According to the equipartition principle the average energy E of unpolarized photons of the Hawking radiation B. From the uncertainty principle to the is linked with their temperature T as mass-temperature relation

E = kBT. (11) Naturally, a modification of the uncertainty relation, In order to fix , we consider the semiclassical limit β i.e. of the basic commutators, has deep consequences → on the quantum mechanics, and on the quantum field 0, and require that formula (10) predicts the standard theory built upon it. The general implementation semiclassical Hawking temperature: of such commutation rules, as regards Hilbert space ~c3 ~c representation, ultraviolet regularization, or modified TH = = . (12) dispersion relations, has been discussed in a vast 8πGkB M 4πkBRS amount of literature (see [16] for an incomplete list). This fixes = π. In the present section, we want to focus on the use Defining the Planck temperature Tp so that p = kBTp/2 of (generalized) uncertainty relations to compute the and measuring all temperatures in Planck unitsE as Θ = basic feature of the Hawking effect, namely the formula T/Tp, we can finally cast formula (10) in the form linking the temperature of the black hole to its mass M. The seminal results of Hawking and Unruh [17, 18] are 1 2m = + ζ 2πΘ , (13) rigorously computed using QFT, based on Heisenberg 2πΘ uncertainty principle, on curved space-time. However, where we have defined the deformation parameter ζ = it has been shown [6, 19, 20] that the full calculation of β/π2. QFT in curved space-time (with standard commutators As already mentioned, in the semiclassical limit both for the ordinary uncertainty principle, or with deformed β and ζ tend to zero and (8) reduces to the ordinary commutators for the GUP) can be safely replaced by Heisenberg uncertainty principle. In this case Eq. (13) a computation employing only the (generalized) uncer- boils down to tainty relation and some basic physical considerations, in order to obtain the mass-temperature formula. 1 m = . (14) 4πΘ The GUP version of the standard Heisenberg formula which is the dimensionless version of Hawking’s formula (1) is (12). ~c E As we have seen, a computation of the mass- δx + βℓ . (8) temperature relation for black holes based on the GUP ≃ 2E p Ep has resulted in a modification of the Hawking formula for high temperatures. In the next subsection, we sum- which links the (average) wavelength of a photon to its marize as this leads also to the remarkable prediction of energy E. Conversely, with the relation (8) one can com- black hole remnants (see [6]). pute the energy E of a photon with a given (average) wavelength λ δx. ≃ Following loosely the arguments of Refs. [6, 15, 19–22], C. Minimum masses, maximum temperatures we can consider an ensemble of unpolarized photons of Hawking radiation just outside the event horizon. From The standard Hawking formula predicts a complete a geometrical point of view, it’s easy to see that the po- evaporation of a black hole, from an initial mass M down sition uncertainty of such photons is of the order of the to zero mass. As we have seen this is a direct conse- Schwarzschild radius R of the hole. An equivalent ar- S quence of the Heisenberg principle. However, when the gument comes from considering the average wavelength mass-temperature relation is derived from the GUP in- of the Hawking radiation, which is of the order of the ge- stead, the formulation immediately leads to a minimum ometrical size of the hole. By recalling that R = ℓ m, S p mass and a maximum temperature for the evaporating where m = M/M is the black hole mass in Planck units p black hole. Precisely we have, for the GUP, (M = /c2), we can estimate the photon positional p Ep uncertainty as 1 Θmax = (15a) 2π√ζ δx 2R = 2ℓ m . (9) ≃ S p

The proportionality constant is of order unity and will mmin = ζ (15b) 4

Note that, as expected, Θmax and mmin 0 in Since p = ~k, the number of quantum states (i.e. station- the Hawking limit β 0. Therefore→ ∞ the use of the→ GUP ary waves) in the volume V, with wave vector in [k, k+dk] eliminates the problem→ of an infinite temperature at the is end of the evaporation process, which is clearly unphysi- V 4πk2dk cal, and leads directly to the prediction of the existence of dN = dn dn dn = (18) x y z 3 2 3 Ω (2π) 4ℓp 2 black hole remnants ([6, 20, 22–24], [52]). In references 1+ β ~2 (~k) [6, 15], it has been shown that also the emission rate (erg/sec) is kept finite by the GUP mass-temperature Since k = ω/c, the number of photons (or gravitons) with formula, in contrast with an infinite output predicted by frequency within ω and ω + dω in a volume V is given by the Hawking formula. 2 V ω Γγ (ω) dnγ = dω . (19) 2 3 2 3 ~ω/kBT π c 4ℓp ~ω 2 e 1 1+ β 2 ( ) III. GOVERNING EQUATIONS ~ c − In case of a perfect black body (perfect emitter) we have In this section we will write down the basic equations for the greybody factor Γγ (ω) = 1 for any ω. The depen- which govern a system of black holes and radiation in the dence of Γγ (ω) from the frequency ω is in general very early universe. We will describe the evolution of a black complicated. It has been studied in many papers (for 4 hole mass as a balance of accretion and evaporation, as dimensional black holes see [30], for emission of gravi- well as consider the dynamical behavior of a universe tons in 4 + n dimensions see [31]), it is in some cases constituted by black holes and radiation. partially unknown, and in many cases can only be com- Then we shall derive a condition for a pre-inflation era puted numerically. In the present model, we neglect the dominated by matter, and the inflationary solutions for frequency dependence of Γγ, and therefore take the value the equations of motion of the scale factor a(t), in both Γγ := Γγ (ω) averaged over all the frequencies. Thus, cases of pre-inflation matter, or radiation dominated eras. for the number of photons (or gravitons) in the interval (ω, ω + dω) in a volume V we write (in 4 dimensions)

2 A. Emission rate equation V ω Γγ dnγ = 2 3 4ℓ2 ~ω/k T dω . (20) π c p ~ω 2 3 e B 1 [1 + β ~2 ( ) ] In this subsection we will describe the evaporation be- c − havior of a micro black hole (in 4 dimensions) taking into Obviously Γγ < 1 for a real non-ideal black body. account the GUP effects. The total energy of photons contained in a volume V (in In the present model we consider only photons or gravi- 4 dimensions) is then tons, nevertheless other kind of gauge or fermionic fields can be added in a straightforward way. γ ∞ ~ ETOT(V )= ω dnγ Before writing down the emission rate equation, we re- 0 view some delicate issues about greybody factors, emit- V (k T )4 = Γ B Γ(4)ζ(4) A(β,T ), (21) ted energy, and the Stefan-Boltzmann constant, in 4 di- γ π2c3~3 mensions with the GUP. The presence of a GUP, i.e. of a minimal length, forces where Γ(s) is the Euler Gamma function, ζ(s) is the Rie- us to take into account the squeezing of the fundamental mann Zeta function, and the function A(β,T ) accounts cell in momentum space (see [21, 26–29]). The squeezing for the cells’ squeezing in momentum space, due to GUP. results in a deformation of the usual Stefan-Boltzmann The function A(β,T ) can be formally written as law. This deformation has to be considered, at least in 3 1 ∞ 1 x principle, since we deal with micro black holes close to A(β,T )= 3 x dx Γ(4)ζ(4) 0 [1 + β(2ℓ k Tx/ ~c)2] e 1 their final evaporation phase, where the predictions of p B − the GUP are expected to differ noticeably from those of (22) the Heisenberg principle. and by this definition we have Due to the deformation of the Heisenberg fundamental inequality, A(β,T ) 1 for β 0 . → → ~ 4ℓ2 Defining the Stefan-Boltzmann constant (in 4 dimen- ∆x∆p 1+ β p ∆p2 , (16) ~2 ≥ 2 sions) as the number of quantum states per momentum space vol- c Γ(4)ζ(4) 4 σ3 = kB, (23) ume (or the invariant phase space volume) is 3 π2c3~3 the total energy can be written as V dp dp dp dn dn dn = x y z (17) x y z ~ 3 2 3 3 σ (2π ) 4ℓp 2 γ 3 4 1+ β 2 p E (V )=Γ VT A(β,T ). (24) ~ TOT γ c 5

The energy dE radiated in photons (or gravitons) from limit the analysis of absorption to the background radia- the black hole, in a time dt, measured by the far observer, tion. The calculations are particularly inspired by Refs. can be written as [34–38]. Absorption terms will appear with a positive sign in Eq.(28). The general form of the absorption term 3σ dE =Γ 3 T 4 A(β,T ), (25) will be γ c V3 dM σ where is the effective volume occupied by photons in = ρeff , (30) 3 dt c the vicinityV of the event horizon, which contains the effective energy density ρeff and the =4π R2 c dt. (26) V3 S appropriate cross section σ for the gravitational capture of relativistic particles in the background by the black Thus, finally, the differential equation of the emission hole. Since we want to consider relativistic background rate is [21, 32, 33] radiation, the effective energy density can be defined as dE = 12π Γ σ R2 T 4 A(β,T ) . (27) ρeff = ρ +3p(ρ) . (31) − dt γ 3 S In the case of radiation with an equation of state param- where the minus sign indicates the loss of mass/energy. eter w = 1 , this results in an effective energy density With the explicit definitions of σ , R , and using Planck 3 3 S of variables m = M/Mp = E/ p,Θ= T/Tp, τ = t/tp 1 E (where = k T and t = ℓ /c), we can rewrite the eff ρrad p 2 B p p p ρ = ρ +3 =2ρ . (32) emissionE rate equation as rad 3 rad

3 Thus, the absorption/accretion term for background dm 8 π Γγ 2 4 = m Θ (β, Θ), (28) radiation reads − dτ 15 A dM σ where we used Γ(4)ζ(4) = π4/15 and = rad 2ρ . (33) dt c rad 3 15 ∞ 1 x (β, Θ) = dx Since the environment is supposed to be isotropic and 4 2 2 3 x A π 0 [1 + 4 β Θ x ] e 1 homogeneous, the cross section for the absorption of rela- − (29) tivistic particles is proportional to the square of the black hole mass [35], In the applications presented in the following sections, 2 2 the GUP will be implemented by considering only the G M σrad = σpart = 27π . (34) cutoff imposed on minimum masses and maximum tem- c4 peratures. In other words, we mimic the cutoff effects Note that a heuristic deduction of such cross section can of the GUP by simply stating that the micro black hole be obtained directly from the spherical geometry of the evaporation stops when T = T or equivalently when max black hole dM =4πR2 ρ dt/c = 16 π G2 M 2ρ dt/c5. M = M , and otherwise using the ”simpler” Hawking S rad rad min In Planckian units the equation for accretion terms form of the mass-temperature relation. This is tanta- reads mount to choose (β, Θ) 1. We adopt this choice in order not to renderA the calculation≃ too tedious, in par- eff dm 2 ρ ticular for those involving the nucleation rate of black = 27πm (35) dτ ρpl holes and the emission rate in the presence of absorption terms. 3 where ρpl := p/ℓp is the Planck energy density. Then, a moreE complete differential equation for the evolution of the mass of a micro black hole can be given B. Absorption terms in the evolution equation for by micro black holes dm 8 π3 Γ ρ = γ m2 Θ4 (β, Θ) + 54 πm2 rad (36) In this section we consider the absorption of radiation dτ − 15 A ρpl by black holes. Therefore, we extend the emission rate equation (28) in order to describe all the processes chang- where we used the background equation of state through eff ing the mass of a black hole. In principle, as our system the effective energy density ρ =2ρrad. consists of radiation and micro black holes, we should also As stated before, for sake of simplicity we assume that take into account the absorption of micro black holes by the black hole evaporation evolves according to the stan- other micro black holes. However, we will see later that dard Hawking mass-temperature relation (12), and thus the black hole density is low enough to neglect scatter- we consider in Eq.(36) the GUP correction function ing processes among black holes themselves, and thus we (β, Θ) 1. We shall keep in mind the cutoff on A ≃ 6 mass/temperature predicted by the GUP, and put it in the evolution of the energy densities ρrad and ρmbh under by hand whenever needed. the cosmic evolution of the scale factor a(t) usually the Then, the differential equation for the evolution of mi- (0) component of the continuity equation is considered, cro black hole mass/energy ε can be written as 0 0 T = G . (45) 2 2 ∇ dε 2 4 54πG ε = 12π Γ σ R T + ρ , (37) 0 dt − γ 3 S c7 rad Here G is a source-sink term that can appear especially in the description of reciprocally interacting subparts of where ε is the average energy content of a single black 2 the whole system, as we shall see in the next section. hole, ε = Mc . Using expression (23) for σ3, and RS = ν 4 T is the energy-momentum tensor of a perfect fluid. 2Gε/c , we can write Specializing this equation to the RW metric, we obtain, for the global energy density ρrad + ρmbh, dε C 2 = 2 + Dε ρrad (38) dt −ε 0 ρ˙rad +ρ ˙mbh +4Hρrad +3Hρmbh = G (46) with where H =a/a ˙ and G0 is a possible source-sink term. We Γ ~ c10 54πG2 C = γ ; D = . (39) shall now compute accurately the form of the continuity 3840 π G2 c7 equation for both the subsystems ”radiation” and ”black holes”, in particular the form of the source-sink term. C. Evolution equations The system we are investigating is a defined mixture of radiation and black holes, where the Hubble radius RH of the universe contains a given fixed total number N of 1. Cosmological equation micro black holes, a given amount of radiation, and the only processes that can happen are exchanges of energy Given the standard RW metric (with Weinberg con- between the black holes and the surrounding radiation. ventions but c = 1) As already mentioned before, in this phase no black hole merging, nor black hole nucleation, is supposed to happen. 2 2 2 2 1 2 2 2 ds = c dt + a (t) dr + r dΩ (40) Let us first focus on the evolution equation for ρmbh. − 1 kr2 − Micro black holes are a particular type of dust: they can where dΩ2 = dθ2 + sin2 θdφ2, and the energy-momentum emit or absorb radiation. As a first step however, we tensor of a perfect fluid suppose that micro black holes have a negligible interaction with radiation (i.e. we treat them as standard dust). Tν = (ρ + p)uuν + pgν (41) Then the continuity equation, without any source term, can be written where ρ is energy density and p is pressure, the (00) com- a˙ ponent of the Einstein equation reads ρ˙ +3Hρ = 0 ; H = (47) mbh mbh a a˙ 2 kc2 8πG + = ρ, (42) This equation takes already into account the variations a a2 3 c2 in mass/energy density due to the simple variation in while from the (ii) components we have volume. In fact, from (47) we have

a¨ a˙ 2 kc2 8πG ρ˙mbh a˙ 2 + + = p. (43) = 3 (48) a a a2 − c2 ρmbh − a In our model, the energy density has contributions of and therefore radiation and matter, and can thus be written as ρ = A ρrad + ρmbh. For simplicity, and following Ref.[3], we ρmbh = 3 (49) now consider a ﬂat metric, i.e. k = 0. The equation is a then written as where A is an integration constant. Hence we see that 3 2 at any time t we should have ρ (t) a(t) = A, so the a˙ 8πG mbh = (ρ + ρ ). (44) integration constant should be written as a 3 c2 rad mbh 3 A = ρmbh(tc) a(tc) , (50) 2. Evolution equations for ρ and ρ mbh rad where tc is the time point when the constant A is determined. It is a characteristic time for the onset of matter We suppose our system to consist of a “soup” of micro era, and will be investigated in section IV. Since a(t) is black holes and radiation. It is well known [see e.g. text- adimensional, A should have the dimensions of an energy books by Weinberg or Landau] that for the description of density. 7

Conversely, considering black holes as dust grains of con- where B is an integration constant. stant mass M, then the link between mass/energy density From the previous two steps, it is then clear that, consid- and volume can be immediately written as ering both the expanding box and the emitting/absorbing black holes, we can write globally forρ ˙ Mc2 N a(t )3 rad ρ = c (51) mbh R (t )3 a3 a˙ Na(t )3 H c ρ˙ = 4 ρ c ε˙ . (59) rad − a rad − R (t )3 a3 where N is the total number of micro black hole in the H c volume a(t)3 at any instant t>t . As will be derived c We see that in the equations (54), (59) the term in section IV, N is considered to be constant, since no Na(t )3 ε/R˙ (t )3 a3, which accounts for the emit- creation, merging, or complete evaporation of micro black c H c ting/absorbing activity by black holes, appears with op- holes are allowed after the time t . All the quantities c posite signs, respectively. This is physically very plausi- M, t , N, R (t ), a(t ) will be computed explicitly via c H c c ble since, if for exampleε> ˙ 0, then that term contributes numerical simulation in section IV. So we have to the accretion of black holes’ masses, while exactly the 2 3 Mc Na(tc) a˙ same amount of energy is taken from the radiation sur- ρ˙mbh = 3 3 4 a˙ = 3 ρmbh . (52) − RH (tc) a − a rounding the black holes. Coherently, we see that the global continuity equation for black holes and radiation which coincides with (47). combined reads If now we suppose that also the mass/energy of the single black hole can change in time, then Eq.(51) reads a˙ a˙ ρ˙mbh + 3 ρmbh +ρ ˙rad +4 ρrad =0 , (60) 3 a a ε(t) N a(tc) ρmbh = 3 3 (53) RH (tc) a(t) that is, it does not contain any source term. This is reasonable, since our system contains only black holes and 2 with ε(t)= M(t)c , and this expression immediately sug- radiation, and therefore the global energy content must gests by derivation the correct source term in the conti- be conserved (only diluted by the cosmic expansion rate nuity equation: H = 0). Systems of equations where one term appears 3 as a source in one equation, and as a sink in another, are a˙ Na(tc) ρ˙mbh + 3 ρmbh = ε.˙ (54) quite common in physics and in cosmology. For example a R (t )3 a3 H c recent models dealing with the interaction between dark Let us now focus on the equation for the radiation matter and dark energy display such features [39]. energy density ρrad. As first step, consider the variation of ρrad due to presence of emitting/absorbing micro black holes, when the system radiation/black holes is contained 3. Complete set of equations in a box of fixed volume. If dε is the variation in a time dt of the energy content of a single black hole, and the We are now able to write down a set of equations that box contains N black holes (all of the same mass), then should describe, hopefully in a complete way, the pri- the variation of the energy of the radiation in the box is mordial ”soup” of radiation and micro black holes, in a temporal interval ranging from the end of black hole pro- dE = Ndε . (55) − duction era (t = tc) to the starting of inflation (t = tinfl). Since the volume of the box scales as V (t) = Considering the equations (38), (54), (59), (44), we can 3 3 3 RH (tc) a(t) /a(tc) , then the variation of the radiation write the system (t is, as before, the comoving time) energy density is dε C = + Dε2 ρ dE Ndε dt −ε2 rad dρrad = = (56) V (t) −V (t) a˙ Na(t )3 ρ˙ + 3 ρ = c ε˙ mbh a mbh R (t )3 a3 which means H c 3 3 a˙ Na(tc) Na(tc) ρ˙rad + 4 ρrad = 3 3 ε˙ ρ˙rad = 3 3 ε˙ (57) a −RH (tc) a −RH (tc) a a˙ 2 8πG This relation is true in the hypothesis of a fixed box. If in = (ρ + ρ ) (61) a 3 c2 rad mbh particular the black hole were inert (neither absorption nor emission) thenε ˙ = 0 and thereforeρ ˙rad = 0. As we see, this is a system of 4 equations for the 4 un- In an expanding box, containing only radiation or just knowns ε(t), ρ (t), ρ (t), a(t). This is a good sign for a few inert black holes ( dust grains), we know that mbh rad ≡ the closure and solvability of the system. However it is from the continuity equation (45) we can write for ρrad clear that this system is strongly coupled, and moreover B nonlinear. Thus, to find an explicit solution is surely hard ρ˙ +4Hρ =0 ρ = (58) rad rad ⇐⇒ rad a4 and perhaps impossible. Nevertheless, the system can 8 be studied in some physically meaningful situations (as We can now wonder how much matter (micro black for example when the micro black holes are very weakly holes = dust) should be present in order to have a matter interacting with radiation, withε ˙ 0, when they essen- dominated universe before the beginning of the inflation. tially behave like dust). In these limits≃ the equations can A condition for this can be easily derived by inspecting yield useful insights on the behavior of the scale factor the exact solution (65) and considering its expansion as a(t), which can be used (via a procedure similar to that of 2 Ref.[3]) to compute the effects of this ”soup” of radiation 2a3/2 3 B 9 B 1 and micro black holes on the successive inflation era, and 3√κA − 2 Aa − 8 Aa possibly on particular features of the power spectrum. B 3/2 B 3 +2 + = t. (69) Aa O Aa D. Pre-inflation matter era Clearly, the universe will be in a matter dominated era 6 We study here the regime just sketched at the end of at the onset of inflation, namely for t = tI 10 tP (the ≃ the previous section, when micro black holes are very time when the temperature of the universe corresponds weekly interacting with radiation (ε ˙ 0). Our primor- to the GUT energy scale), whenever the condition dial ”soup” is therefore composed by radiation≃ and dust. 3 B In this approximation, the second and third equation of 1 (70) system (61) can be immediately integrated to give 2 Aa ≪ is satisfied. An even simpler derivation can be found by A B ρ = , ρ = . (62) writing Eq.(64) in the form mbh a3 rad a4 a˙ 2 A B where the integration constants A and B have dimensions = κ 1+ (71) a a3 Aa as energy densities, and can be written as A = ρ (t ) a(t )3 , B = ρ (t ) a(t )4 . (63) from which we read off that the evolution is matter domi- mbh c c rad r r nated when B/(Aa) 1. In section IV we shall compute ≪ Here tc and tr are the characteristic times for the onset of explicitly via numerical simulation every step of the black matter and radiation eras, respectively. They will both hole nucleation phase, and the associated evolution of be explicitly specified in the next sections. Then the (00) the pre-inflationary radiation and matter eras. We shall equation of system (61) reads conclude that the above matter dominance condition is always met, even well before the onset of inflation. a˙ 2 8πG A B = + . (64) a 3 c2 a3 a4 E. Inflationary solutions Equation (64) is separable, and can be integrated exactly. The solution obeying the boundary condition a(0) = 0for In our equations we now also take into account a con- t = 0 is stant vacuum energy, namely a cosmological constant. In this way, we shall be able to generate inflationary expo- 2 √ 2 (Aa(t) 2B) Aa(t)+ B +2B B = t (65) nential solutions. Following Ref.[3], the (00) component 3√κA − of the Einstein equation in this case reads where κ = (8πG)/(3 c2). Using the binomial expansion 2 a˙ A B = κ + + C . (72) 1 1 a a3 a4 (1 + ǫ)1/2 =1+ ǫ ǫ2 + ... (66) 2 − 8 The constant C in the Friedmann equation results from we find, in the limit A 0 or equivalently a(t) 0 for assuming a power law potential V (φ) for the inflaton t 0, → → field. C and the potential are connected by → 2 3C a = 2√κB t (67) V (φ)= φ2 + C c φ + c , (73) 2 1 2 which is the well know behavior of pure radiation era. where C is the quasi de Sitter parameter in the Fried- In the other regime, when a or A are large, or equivalently mann equation, and c and c are constants to be fixed B 0, we find 1 2 → from the inflationary model. In other words, C is mim- 3 icking the potential for the inflaton field. The previous a3/2 = √κA t (68) 2 equation can be written as 2 which is the the standard result for a matter dominated a˙ A B Ca3 = κ 1+ + (74) era. a a3 Aa A 9 and under the matter era condition, B/(Aa) 1, it It is also useful to derive a condition for the onset of becomes ≪ inflation. From Eq.(72) we can obtain the sign ofa ¨

a˙ 2 A a¨ A B = κ + C . (75) = κ + C , (82) a a3 a −2a3 − a4 Again, this equation is easily separable, and the solution anda> ¨ 0 if obeying the boundary condition a(0) = 0 for t = 0 is A 2B A C > 1+ & , (83) 2a3 Aa 2a3 A 1/3 3 2/3 a(t)= sinh √κC t , (76) C 2 where we used the pre-inflation matter era condition B/(Aa) 1. Therefore we shall be in a full inflationary ≪ which, for vanishing C, or small t, results in solution (68), era,a> ¨ 0, when 1/3 3 2/3 A a(t) (κA)1/3 t2/3, (77) a & . (84) ≃ 2 2C while for large t, it exhibits an exponential (i.e. inflation- IV. BLACK HOLE NUCLEATION: ary) behavior [40], NUMERICAL SIMULATION

A 1/3 a(t) exp √κC t . (78) In this section we numerically simulate the nucleation ≃ 4C of micro black holes in pre-inflation era, and come up with a number of micro black hole remnants sufficient to We can grasp an idea of the overall solution a(t) by nu- make the universe pass from a radiation dominated to a merically integrating Eq.(72). The evolution of the scale matter dominated pre-inflation era. We will express ev- factor is shown in Fig.1. ery quantity in planckian units, meaning e.g. τ = t/tp, Θ = T/T , m = M/M . Thus every quantity is dimen- H L p p a t sionless. 8 In 1982 Gross, Perry and Yaffe [4] investigated the stabil- ity of flat space and the arising gravitational instabilities, 6 which might lead to singularities. They used the formal- ism of path integrals in a quantum version of Einstein’s 4 theory of gravity to analyze these gravitational fluctuations. As a concrete example, they took a box filled with 2 thermal radiation to derive an expression for the probability for the spontaneous formation of black holes out t of the gravitational instabilities of spacetime. Two years 0.5 1 1.5 2 2.5 3 3.5 later, Kapusta [5] gave an alternative heuristic derivation of the nucleation rate, using the classical theory of nu- FIG. 1: Diagram for a(t) versus t, in a model assuming sub- cleation during a thermodynamical phase transition. He sequent radiation and matter eras before inflation. reproduced the rate nearly completely with the classical approach considering the change in free energy of the system during the nucleation of a black hole, and completed In case of radiation dominated pre-inflation era, i.e. no the analogy by inserting by hand quantum corrections matter present (A = 0), equation (72) reads into his classically derived formula. See Appendix 1 for the explicit steps of such derivation. a˙ 2 B = κ + C , (79) The nucleation rate for black holes reads (Eq.(209), Ap- a a4 pendix 1)

2 and the solution obeying the boundary condition a(0) = 8π 167/45 1/16πΘ Γ (Θ) = Θ− e− , (85) 0 for t = 0 is N 15 64π3 B 1/4 1/2 where Θ is the temperature of the universe (the thermal a(t)= sinh 2√κC t , (80) bath), expressed in Planck units, and at the same time C the temperature of the nucleated black holes, connected which for small t or vanishing C is to their mass m by 1 √ 1/4 1/2 Θ= . (86) a(t) = 2 (κB) t . (81) 4πm 10

At a given temperature Θ, all the black holes created To calculate the time of the transition from radiation- to according to this nucleation rate will have mass m. matter-dominance (which evidently takes place after τc), we consider the ratio As stated before, the pre-inflation era is supposed to 43 ρm(τ) ρm(τ) take place from the Planck time tp 10− s to the onset R(τ)= = τ 2. (90) 37 ≃ of inflation, tinf 10− s, when the temperature of the ρr(τ) ρp universe has reached≃ the GUT energy scale. At very early times, when Θ > Θ 1/(4π), the nucleation probabil- At the very beginning there is only radiation. So for a ∗ ≃ while ρm(τ) = 0, the evolution of ρr(τ) is driven by radi- ity is very high, but does not lead to black hole formation 1/2 2 as it is forbidden by the GUP to create smaller than the ation, a(τ) τ , and therefore ρr ρp/τ . This is correct at least∼ before black holes are created,∼ whereas after Planck mass black holes (mmin √ζ Mp where ζ 1). ≃1 ∼ the onset of nucleation, a(τ) evolves in a more complex For temperatures above Θ 4π production of small black holes is not possible.∗ So≃ at least for this very early manner dictated by equations (64, 65). As nucleation time, the universe is simply a chaotic hot sphere that starts, ρm(τ) grows, and some time later R(τ) crosses 1. we suppose to be filled with primordial radiation, follow- R(τ) can only be used for qualitative statements at the ing the approach of Refs.[2, 3]. There might be regions beginning of nucleation, since, in the way it is defined, with larger density than others, but the conditions are it is rigorously valid only until the start of matter nu- too chaotic to allow formation of stable objects like black cleation, and it does not contain any information about holes. The universe can thus be assumed to be radiation- the time-development of the scale factor according to the dominated at the beginning, and will migrate to being full Friedmann equation, when matter and radiation are matter-dominated at a later time, when black holes are both present. So it should only be applied during short starting to be formed. Considering an adiabatically ex- time spans, just after the nucleation starts, when the scale factor doesn’t change significantly. The condition panding universe, we can write T (τ)a(τ)= Tpa(tp), and 3B/(2Aa) 1 is the only significant criterion to fully since during the radiation era the scale factor evolves like ≪ 1/2 determine the radiation to matter transition of the uni- ar(τ) a(tp)τ , and we choose a(tp) = 1, we have ≃ verse at later times, when matter and radiation are both 1 present. Θ = (87) r τ 1/2 during the time when the universe is dominated by radia- B. Nucleation Process tion. For a matter-dominated universe, the temperature evolution is analogously given by Using Eq. (87), in radiation era we can write the nu- 1 cleation rate as a function of time as Θm = 2/3 . (88) τ 1 167/90 τ/16π Γ (τ)= τ e− . (91) N,r 15 8π2 A. State of the universe The temperature Θ of the Universe is linked not only to time, but also to the mass of the nucleated black hole, The parameters in the Friedmann equation contain- since a black hole, at the instant of its creation, is in ing matter and radiation energy densities are defined as thermal equilibrium with the rest of the Universe. In (Eq.(63)) fact

3 1 1 2 2 A = ρm(τc) a (τc), (89a) Θ= and Θ = τ = 16π m . (92) 4πm r τ 1/2 ⇒ r 4 B = ρr(τp) a (τp), (89b) It should be explicitly noted that the relation (92) does and their dimension has to be energy per volume, as the not express the time evolution of the mass of one black scale factor is a dimensionless quantity. hole, but the dependence of the initial mass of the nu- At the Planck time, the universe is in a radiation- cleated black holes on time. The evolution with time of dominated stage, and it is reasonable to assume that the black hole mass, due to evaporation/accretion pro- ρr(τp)= ρp. Assuming a(τp) = 1 (which is an unconven- cesses, is given by relation (36), while equation (92) ex- tional, but convenient choice, and will be converted to presses the evolution with time of the masses of the black the common notion of a(τtoday) = 1 later), the radiation holes at the instant of their creation. Therefore the cutoff parameter can be fixed as B = 1, expressed in Planck from the Generalized Uncertainty Principle, which gives units. For the matter parameter, we have to choose a a minimum mass m 1, can be translated also in terms 2∼ time τc when black holes are starting to be nucleated, of time as τ = 16π 158. This can be seen in Fig. 2 - ∗ ≃ and calculate ρm(τc). This time and the parameter A the curve is truncated at τ (vertical line), which corre- will be derived in the next subsections via numerical sim- sponds to the cutoff at about∗ one Planck mass. This re- ulations. lation also implies that the black holes nucleated at later 11

where L(B) is a so-called light sheet, which deﬁnes a cer- N,rHΤL tain region of space-time B, and A(B) is the codimension 0.8 1 boundary of that region. For our situation, applying the holographic principle simply means that the total en- 0.6 tropy contained in the Hubble sphere cannot exceed the entropy of a black hole of size equal to the Hubble sphere,

0.4 which represents the maximum entropy that can be held in that spacetime region, as black holes are the most en- tropic objects. According to Refs. [42, 43], the expression 0.2 for the entropy of a black hole is A 0.0 time Τ bh 0 100 200 300 400 500 600 S = , bh 4 and so the entropy of a black hole of the size of the Hubble FIG. 2: Nucleation rate, ΓN,r(τ) over time τ. sphere (HS) is given by A S (τ)= HS = πR2 (τ). (96) HS 4 H times have larger masses, according to Eq. (92), while the This can be used to define a cutoff for the nucleation rate. probability of their formation decreases. We demand that at no time point in the evolution of the From the onset of nucleation the number of black holes universe the entropy of the black holes can exceed the nucleated each Planck time per Planck volume is given by total entropy that Hubble sphere can maximally hold, the rate (91), but we have to monitor closely the over- and then we try to find a time point τc, from which this all state of the universe. When the ratio R(τ) crosses condition is fulfilled. If this condition is violated in the unity, then the universe changes to a matter-dominated course of black hole production, then it is simply not stage, in which case the nucleation rate is no longer given allowed to create black holes. The entropy of the black by Eq. (91). As soon as the phase transition happens, the holes within the Hubble sphere is nucleation rate has to be given in terms of the tempera- 2/3 τ ture in a matter-dominated universe (Θm = τ − ): 2 4π 3 S (τ)= Γ (τ ′) πr (τ ′) τ ′ dτ ′, (97) bh N,r s 3 4 3 τ∗ 1 334/135 τ / / 16π ΓN,m(τ)= 2 τ e− . (93) 15 8π where the Schwarzschild radius of the black holes is rs = m = τ 1/2/(4π), from Eq.(92), and we require that at all In order to get the number of black holes nucleated with times time, we have to do a time integration over the nucleation rate, multiplied with the volume of the universe in each Sbh(τ) 6 SHS (τ). time point. Considering that the Hubble radius is defined as RH = c/H, and that for power law expansion rates Again, since the universe has started off in a radiation- (a(t) tα) we have R ct, then, taking as initial dominated era, and the black hole nucleation has not yet ∼ H ∼ condition RH (τp)= ℓp, we can write (in units of Planck begun, the rate as a function of time is defined using 3 1/2 length) RH (τ)= τ. Therefore (since ℓp = Vp = 1) Θr = τ − . It is yet unknown when the production of black holes will change the state of the universe to be 4π V (τ)= τ 3. (94) matter-dominated, but to find out the time of nucleation H 3 start, we are for now bound to the assumption of radiation dominance and thus will use Γ (τ) in the above But before we go on to calculate the black hole number, N,r expression for the entropy. we take into account another principle, which will give The cutoff time turns out to be τ = 993, as can be seen a more stringent cutoff on the nucleation of black holes c in Fig. 3, which shows the entropy of the Hubble sphere than the GUP does. (dashed line) and that of the black holes (full line) as a function of time. C. A Cutoff from the Holographic Principle With τc being determined, we can calculate the number of black holes according to τ The holographic principle [41] places a limit on the 4π 3 information content, or entropy content, in a certain re- Nbh(τ)= ΓN,r(τ ′) τ ′ dτ ′, (98) τ 3 gion of space-time. Quantitatively it states that (in units c where kB = ℓp = 1) and the matter density as τ A(B) 1 3 ρ (τ)= Γ (τ ′) m(τ ′) τ ′ dτ ′. (99) S[L(B)] 6 , (95) m τ 3 N,r 4 τc 12

H L 6 R Τ SbhHΤL, SHHΤL@10 D 10

2 4

1 2

0 time Τ 0 500 1000 1500 2000 2500 3000 0 time Τ 992 993 994 995 996 997 998

FIG. 3: The cutoﬀ as deﬁned by the holographic principle at FIG. 4: Ratio R(τ) of matter to radiation density in the early τ = 993: the curves touch, but the entropy of the black holes universe, after the nucleation starts at τ = 993. (full line) never exceeds the maximum entropy that can be held by the Hubble sphere (dashed line).

With this assumption the estimated number of primordial black holes is given by However, in order to correctly calculate the black hole 4 number and the mass density, we need to know when the N(τinf ) 10 . universe will migrate from being radiation-dominated to ∼ being matter-dominated. We know that black hole nucle- The masses of the black holes nucleated between ation starts when the universe is in a radiation-dominated τc = 993 and τ = 996, according to the mass-time- state. For the very beginning of black hole nucleation, relation Eq. (92), are for a time span of at least one Planck time, we have m(τ ) 2.5. to use the rate ΓN,r to produce black holes. After one c ∼ Planck time, we have to check whether the universe is still in radiation- or already in matter-dominated stage, We can now also calculate and plot the mass den- and take the according production and expansion rates to sity, Eq. (100) in evolution with time. It is shown in Fig. 5. follow the black hole production. A new parameter dr is introduced, which denotes the duration of this radiation- dominated phase of black hole nucleation, until the uni- 6 verse reaches a matter-dominated state. The matter den- mass density ΡmHΤL@10 D 3.0 sity of the black holes can be expressed like 2.5 τc+dr 1 3 2.0 ρm(τ) = 3 ΓN,r(τ ′) mr(τ ′) τ ′ dτ ′ τ τc τ 1.5 1 3 + ΓN,m(τ ′) mm(τ ′) τ ′ dτ ′, (100) τ 3 1.0 τc+dr 1/2 2/3 0.5 where mr = τ /(4π) and mm = τ /(4π). We have 0.0 time Τ separated integrals for the two separate stages of black 5000 10 000 15 000 20 000 25 000 30 000 35 000 40 000 hole production. We can now vary the parameter dr and investigate the ratio we are looking for, (here ρp = 1)

2 R = ρm(τ) τ , (101) FIG. 5: The mass density of the universe as a function of time. to determine at which point the dominant substance in the universe changes. As can be seen from Fig.4, it turns out that the universe very quickly reaches a matter-dominated stage after the onset of black hole production. After only a few Planck D. Collision Rate and Black Hole Thermodynamics times, the ratio is clearly above unity, and so we can safely assume that the universe is in a matter-dominated What is left to do to justify the assumption of a matter- stage at about 3 4 Planck times after the black hole dominated era due to the existence of black holes is to nucleation has begun.− analyze the results in the context of black hole collisions 13

1/2 2/3 and merging. The collision rate of black holes can be Since a(tp) = 1, we can write a(τ) 10− τ and given by starting from a general definition of a scattering ≃ B ρ (t )a4(t ) 100 ρ rate. We consider black holes of mass m(τc), and their ρ (τ) = = r p p = p (104) velocity to be determined by Brownian motion, therefore rad a4(τ) a4(τ) τ 8/3 v = kT . Being n the number of black holes per unit M So the differential equation (36) becomes volume, and σ the scattering cross section, we arrive at dm 1 5400 πm2 = + (105) Γcoll = nσv dτ −480 πm2 τ 8/3 2 Θ = nbh 4πrs c which can be numerically integrated with the initial con- m dition Nbh(τ) 3/2 1/3 = 4πmc τ − (102) m(τ = 998) 2.5 (106) VH (τ) ≃ 2/3 The numerical integration confirms that, despite the ab- where in the last line we used rs = m and Θm = τ − , with c = ℓ = 1. The rate can be seen in Fig.6, together sorption term, the black hole evaporates completely in p a time T (m) 17086 t . Actually, without the ab- with a dashed line denoting the Hubble expansion rate. ∼ p The collision processes are effective as long as the rate sorption term, the lifetime would be about 50% less, is higher than the Hubble expansion rate - otherwise the just T (m) = 7854 tp. In any case, by a cosmic time of τ 2 104 t , well before the inflation starts, all the reaction decouples, and collisions stop because the ex- ∼ p pansion of the universe is faster than the time between black holes have evaporated down to the Planck size rem- collisions. As can be seen clearly from the plot, the colli- nants predicted by the GUP. In Fig. 7 the two lifetime sion rate is at all times lower than the Hubble rate, and diagrams, one for evaporating/absorbing black holes, the thus no collisions or mergings of black holes take place. other for evaporating only black holes, are compared.

mHΤL 3

4 collHΤL,HHΤL@10 D

8 2.5

6 1.5

4 1

0.5 2 Τ 2500 5000 7500 10000 12500 15000 17500 0 time Τ 0 2000 4000 6000 8000 10 000 FIG. 7: Black holes lifetime diagrams: the upper (red) for an evaporating/absorbing black hole; the lower (green) for an evaporating only black hole. FIG. 6: The collision rate of black holes (full line) in development with time, together with the Hubble rate (dashed line).

We can also do some estimations of the black hole ther- E. Fixing the Friedmann Equation modynamics. As we have seen that the black holes are not colliding with each other and don’t have a chance In this subsection we come back to the original goal of to merge, we know that the black hole masses can only this part: to determine the numerical size of the param- change by accretion or evaporation. This process is de- eters in the Friedmann equation that correctly describe scribed by the differential equation (36). In our spe- the universe developed in our model. We have already cific case we consider, as usual, the correction function settled B = 1 by simply assuming that the density of ra- (β, Θ) 1 and the greybody factor Γγ = 1. diation at the Planck time was equal to the Planck den- A ≃ 4 From τc 1000 to the onset of inflation the Universe is sity. For the matter component, we know that about 10 ≃ matter dominated, therefore we can write for the scale black holes are nucleated during a short period around factor τ 103 in the pre-inflation era. A is given by Eq. (89a). The∼ matter density can be simply estimated by 1/2 3 a(tp) τ 1 < τ τc 10 tp ≤ ≃ 4 4 a(τ) (103) 10 black holes 10 √ζǫp ≃  1/6 2/3 6 ρm(τc) 3 9 , a(tp) τc− τ τc τ < τinfl 10 tp ∼ R (τ ) ∼ 10 V  ≤ ≃ H c p  14 where m = √ζ is the minimum mass predicted by short and only lasts from τ to τ 2 104, as we know min c r ∼ the GUP. The scale factor is chosen as a(tp) = 1, and from the previous calculations that the nucleated black 3 4 then evolves in radiation-dominance until τc 10 ; thus holes have an approximate life span of 2 10 tp. During 1/2 3/2 ∼ a(τc) τc 10 . Putting these together, the final this time span the scale factor expands for a factor of result∼ for the∼ matter component is 102/3. Again follows a period of radiation dominance, which lasts from τr to τinf . The temperature at the 4 10 √ζǫp 9/2 1/2 ǫp onset of inflation here again has to match the GUT A = 9 10 10− ζ . (107) 10 Vp ∼ Vp energy scale, and for this reason inflation is starting a 8 little later, at τinf 10 . From τr to τinf there are As stated before (see Sections IIA, IIC), we consider four orders of magnitude∼ in time, and so the scale factor ζ 1. With this number, we can confirm the actual expands for a factor of 102. ∼ validity of the matter-dominance condition for the whole This is the situation we are facing when the GUP is era τc <τ <τinfl. In fact in this era, according to not included in the theory. However, there is a third Eq. (103), the scale factor evolves over the values 103/2 < possibility that can be considered. a(τ) < 107/2, and therefore the matter-dominance ratio spans the values

3 B 1 1 3 G. The Scenario without any black hole 10− 10− 1 , 2 A a(τ) ≃ 10 1/2 a(τ) ∼ − ≪ − which confirms that the Universe is in matter era from After the previous subsection, it is also justified to ask for the case when there are no black holes at all, and no τc to the onset of inflation. With all this information, we are fully equipped to com- black hole nucleation. What happens if we basically ne- mence the calculations of the primordial power spectrum. glect all the black holes nucleation processes in the early It should be noted that C is not yet determined - this will universe, and simply assume that the pre-inflation era be taken care of in the next sections. was completely radiation-dominated? The results from this scenario will then actually correspond to the investigations done before (e.g. [2, 3]), completely cutting out F. The Scenario without the GUP the possibility of black hole formation. In those previous works, a suppression of the CMB power spectrum for low multipole modes, resulting from a suppression in the pri- What if the nucleated black holes wouldn’t be protected by the GUP from evaporating completely but mordial power spectrum, was found. Under these assumptions, the universe is in a radiation- would just transfer their energy to radiation filling the universe? There would be no era of matter-dominance dominated stage from the Planck time until the temperature reached GUT energy scales, at about τ = 107t . before inflation, but there would only be radiation p until the onset of inflation. Or would it? In the During this time span, the scale factor can expand from one Planck length to previous chapter, we have found out that the black hole nucleation is very powerful and the universe is put a = 107/2. (108) into a matter-dominated stage nearly immediately. If inf the GUP is assumed, the evaporation stops when the The power spectra resulting from the scenario introduced mass of the black hole reaches Planck mass, and the here are computed in section V E. Universe stands in matter era until the onset of the inflation. On the contrary, if the GUP is not valid, the black holes, once created, can evaporate down to V. SCALAR FIELD FLUCTUATIONS ON AN zero mass. The universe remains in a matter-dominated EVOLVING BACKGROUND - THE phase for the time of evaporation, but when all black PRIMORDIAL POWER SPECTRUM holes have vanished, radiation is the dominant species again, until inflation starts. That means, there are three In this section we study the quantum fluctuations of a stages during the pre-inflationary era, and the evolution field living in a universe whose background evolves with and growth behavior of the scale factor is changing at a given scale factor a(t). To facilitate comparison with three transition points in time. The parameters of the the case of a pre-inflation radiation dominated universe Friedmann equation have to be evaluated anew, and already considered in Ref.[3], we choose the same con- this leads to a new differential equation for the field vention on the metric, namely perturbation and a new k(a)-relation (see Sec.V A). 2 ν 2 2 2 We can summarize the time frame of this scenario as ds = gν dy dy = dt a(t) dy . (109) follows. The universe starts out radiation-dominated − and remains so until the start of black hole nucleation Here we set c = 1, and we choose a flat metric (for 3 at time τc 10 , where the scale factor has expanded simplicity, and since we deal with an almost flat uni- ∼ 3/2 to about a(τc)=10 . The subsequent matter phase is verse). We consider a zero mass scalar field Φ(t, y) and we 15 perturb the field around its classical expectation value, Another possible strategy to solve Eq.(113) is to change Φ(t, y)=Φ0(t, y)+ ϕ(t, y). The equation of motion for the independent variable from t to a. Then the scalar field perturbation then reads d da d d = =a ˙ , φ˙ =a ˙ φ′ , (121) 2ϕ(t, y)=0, (110) dt dt da da k k where and the equation for φk(a) reads

1 ν 2 2 2 = ∂(√ gg ∂ν ) . (111) 2 a˙ k −√ g − a˙ φ′′ + a¨ +3 φ′ + φk = 0, (122) k a k a2 − In the applications, the a(t) of Eq.(76), or of Eq.(80), will where of coursea ˙ anda ¨ have to be expressed as functions be used in the metric g . ν of a. In the following, we shall use indiﬀerently both Taking the Fourier transform for ϕ procedures.

i k y ϕ(t, y)= φk(t) e− dk, (112) A. The re-entering k-modes we obtain from (110) the equation of motion for φk(t), The k-modes which left the horizon at or shortly af- a˙ k2 φ¨ (t)+3 φ˙ (t) + φ (t) = 0. (113) ter the onset of inflation are those that are just now k a k a2 k re-entering our Hubble radius, and they represent the largest modes of fluctuations currently observable, having In order to solve Eq.(113), different strategies are cus- a size comparable with that of the visible universe. When tomarily used. we solve equation (113), or equivalently Eqs. (120), One possibility is to introduce the conformal time η de- (122), we fix a particular value of the parameter k, i.e. fined by the relation a particular mode, and we obtain the solution for this dt mode, introducing suitable boundary conditions. We are dη = , (114) concerned with modes that leave the horizon just at or a(t) shortly after the beginning of inflation, because they cor- which implies respond to the largest scales observable today, and they could bring imprints of a possible pre inflation era. A pre- d dη d 1 d inflationary era affecting these modes could thus explain = = . (115) dt dt dη a(η) dη the anomaly of the quadrupole moment of the CMB today. Simple geometric considerations will help us to find Then the equation for φk(η) becomes a relation between k and the scale factor. By comparing the FRW metric (109), which is written in comoving a′ 2 coordinates, with the standard euclidean physical metric φ′′(η) +2 φ′ (η) + k φ (η) = 0, (116) k k k 2 2 a dt dYphys, we infer the relation between comoving and physical− coordinates where a ”prime” indicates a derivative with respect to η. Applying now a well known general procedure, valid for dY = a(t)dy . (123) any second order differential equation, we can make the phys com first derivative disappear. In fact, defining Recalling the Hubble law, v = Hd, we get the physical Hubble radius as c = HRH, p, and therefore the comoving φk(η) := vk(η)p(η) (117) Hubble radius as with R c R = H, p = . (124) 1 2a H,c a aH p(η) = exp ′ dη , (118) −2 a Now, the wavelength of a perturbation crossing the hori- we find zon at a time tc (the largest visible perturbation) should be given by the relation 1 p(η)= , (119) a(η) λc =4RH, p(tc), (125) and henceforth the equation for vk(η) reads which means, in comoving coordinates,

2 a′′ λp c v′′ + k vk = 0 . (120) λ = =4 . (126) k − a c a acHc 16

Therefore a particular comoving k-mode at the time of The general plan of our work is to solve equations (131), its horizon crossing obeys the relation (132), regarding k as a parameter, to obtain φ(a, k). Al- ternatively, we can solve Eq.(120) and get φ(η, k), and 2π π acHc kc = = . (127) then, since a = a(η) and η = η(a), make a substitu- λc 2 c tion to φ(a, k). Once the solution φ(a, k) is available, we Since we chose units where c = 1, and moreover π/2 1, can compute the power spectrum P (k) of the quantum we can finally write ≃ fluctuations of the field Φ. Since the power spectrum is usually given as a function of k only, will be necessary to k aH (128) express φ as a function of k only, and this can be done ≃ through the relations (129), (130). The final scope is to as the horizon crossing condition. Besides, reminding obtain the function that H =a/a ˙ , we have for a pre-inflation radiation era P (k)= k3 φ(a(k), k) 2, (133) 1/2 | | B k = a√κ + C (129) a4 which represents the primordial power spectrum of the fluctuations (perturbations) of the field Φ. With P (k) while, for a pre-inflation matter era, then we feed - as in our case - the CMBFAST code to generate the CMB anisotropy power spectra. A 1/2 Unfortunately, analytical solutions of the full equa- k = a√κ + C . (130) a3 tions (131),(132),(120), when we deal with functions a(t) given in (76), (80), are not easily expressible in closed In Fig.8, we give a plot of the (k,a) relation in the case form. A power series solution is in general at hand, but of pre-inflation matter era, with constants chosen arbi- still not much more comfortable than the mere numerical trarily as A = C = 1. one. However, luckily enough, it turns out that analytical solutions of the second order approximations of equa- k tions (131),(132) can be obtained by means of the WKB method, and these solutions are strongly corroborated by 4 numerical insights. Of course, when solving a differential equation, we need 3 boundary conditions in order to fix the solution explicitly. From the mathematical point of view, a boundary 2 condition can be put anywhere in the realm of definition of the solution. From the physical point of view, it is wise 1 to put boundary conditions in regions where the physics is reasonably well known. Since we are almost sure that a 0.5 1 1.5 2 2.5 3 inflation happened, while we know little about a possible pre-inflation era, we choose to put our boundary condi- FIG. 8: Diagram for k versus a (full line), in pre-inflation tions in the full inflationary era. This means that, for matter era large a, or equivalently, for large k (see equations (129), (130)), well after the beginning of inflation, the field φ must generate an almost scale invariant, i.e. flat, primordial power spectrum P (k). To be more precise, the latest observations have shown that the primordial power spec- B. Scalar Field equations trum is not exactly flat, but slightly tilted. The newest analysis of the WMAP data [45] indicate a form like In this subsection we specialize Eq.(122) to particular n 1 a(t) solutions. For the pre inflation matter era case, using P (k) k s− (134) Eq. (75) or (76) to computea, ˙ a¨, we obtain ∼ with 3 5 2 1 4Ca + 2 A k φ′′ + φ′ + φk = 0. k a Ca3 + A k κa(Ca3 + A) ns = 0.963 0.012 (68% CL) . (135) ± (131) Therefore the field φ must behave as In the case of pre inflation radiation era we have, using 1 k 2 (ns 1) Eq. (80), φ(a(k), k) − (136) | |∼ k3/2 2 Ca4 k2 φ′′ + +1 φ′ + φ = 0. for large k. k a Ca4 + B k κ(Ca4 + B) k This condition will allow us to fix properly the arbitrary (132) constants in our solutions. 17

C. Analytical investigations: First order Let’s have a look at the behavior of the solution for a approximation , which means η 0, x 0. Then → ∞ | | → → 2 1 x x3 To begin with, let’s figure out the solution of our dif- v (η) c (k) x2 + c (k) + (143). k ≃ π k 1 2 x 2 − 8 ferential equation for very large a, or k. We use, as first example, the equation for vk(η), (120). We need the ex- 1 Since x a− , we have, for any given k, pression of the conformal time η at large a, i.e. large ∼ cosmic time t. In fact, for t , we have from Eq.(76) φk(a)= (144) → ∞ 1/3 2/3 vk(a) 2 1 1 1 A 3 c1(k) + c2(k) 1+ . a (t) = sinh √κC t a ∼ π k a3 2a2 − 8a4 m C 2 If we require, as usual, φ (a) 0 for a , then this A 1/3 k → → ∞ exp √κC t , (137) necessarily implies c2(k) 0 . So finally the solution is ≃ 4C ≡ vk(η) = η c1(k) J3/2 (k η ) , (145) and for the radiation solution, Eq.(80), | | | | and 1/4 1/2 B vk(a) ar(t) = sinh 2√κC t φ (a)= = (146) C k a B 1/4 2 √κC k 1 k exp √κC t . (138) c1(k) sin cos . ≃ 4C π k k √κCa − a √κCa Note that for large t the solutions are essentially the This is the solution which describes the field φ(a, k) in same, as it should be, since inflation ”washes out” ev- full inflationary era. As we have seen, this solution holds erything (actually, almost everything, as we shall see). for both cases of pre-inflation matter and pre-inflation From dη = dt/a(t), we can express a(t) with the confor- radiation era. The arbitrary integration constant c1(k) mal time η, still has to be fixed, and this can be done by enforc- 1 1 1 ing the boundary condition in inflationary era, namely a(η) = = (139) Eq.(136). Noticing that, for large a, from both equa- √κC η −√κC η | | tions (129),(130), we have with η< 0, and this result holds, in the limit t , for → ∞ k a √κC, (147) both solutions ar(t) or am(t). Thus, the equation for vk ≃ reads and therefore

2 2 2κC c1(k) v′′ + k v =0 . (140) φ(a(k), k) [sin(1) cos(1)] , (148) k − η2 k ≃ π − k3/2 The general solution in terms of Bessel functions is then this means, from (136), 1 2 (ns 1) c1(k) = k − . (149) vk(η) = η c1(k) J3/2 (k η ) + c2(k) J 3/2 (k η ) , | | | | − | | The solution (146) will be employed as boundary condi- (141) tion in inflationary era for the numerical integration of where equations (131), (132). Finally, we note that we can start from equations (131), 2 1 sin x J (x) = cos x (132) for φk(a), instead of equation (120) for vk(η). Tak- 3/2 π √x x − ing the first order approximation for a in the coef- → ∞ 2 1 cos x ficients, we get, from both equations (131), (132) J 3/2(x) = sin x − π √x − x − 4 φk′′ + φk′ = 0 . (150) with a k This has the solution x = k η = . c (k) | | √κC a φ(a, k)= 1 + c (k), (151) a3 2 Then which, for large a, when k a √κC, can be matched 2 1 sin x with the ”almost flat spectrum”≃ boundary condition v (η) = c (k) cos x k π √ 1 x − (136) if k cos x 3 1 + (ns 1) + c2(k) + sin x . (142) c (k) k 2 2 − , c (k) 0 . (152) x 1 ∼ 2 ≡ 18

D. Analytical investigations: Second order and thus approximation 2 k2 9 A2 1 v′′ + + v = 0 (159) −a2 κCa4 − 16 C2 a8 Obviously, a solution like (146) cannot be trusted when we look for information about the behavior of P (k) at Setting low k-modes, since by construction it is built by taking into account the first order approximation only, and re- 1/2 2 k2 9 A2 1 quiring that it reproduces the almost flat spectrum for F (a)= i + , (160) a2 − κCa4 16 C2 a8 high k’s. To have reliable analytical insights on the low k behavior of P (k), one should go to the second order then the WKB ansatz suggests as solution for v(a) approximation. This can be done in principle by starting from equation (120) for vk, but it would involve a quite 1 complicated construction of the conformal time η, and of v(a) = c+(k)exp i F (a)da F (a) the function a(η), in terms of Jacobian elliptic functions (see Appendix 2). A more straightforward path turns +c (k)exp i F (a)da . (161) out to be to start directly from equations (131), (132) − − and develop the coefficients to the second order in a for a . Writing →For ∞ the pre-inflation matter era case, Eq.(131), we have G(a) = F (a)da (162) 4 3A 1 1 φ′′ + + O φ′ + k a − 2C a4 a7 k we see that, for example, for a k2 1 1 → ∞ + + O φ = 0. (153) κ Ca4 a7 k G(a) i √2 log a . (163) ≃ For the pre-inflation radiation era case, Eq.(132), we So, the WKB solution of Eq.(153) explicitly reads have φk(a)= vk(a)p(a)= (164) 4 2B 1 1 4 2 iG(a) iG(a) iπ/4 2 √κC c+(k)e + c (k)e− e− φk′′ + 5 + O 9 φk′ + − a − C a a = 1/4 . [32κC2a6 16k2Ca4 +9κA2] exp[A/ (4Ca3)] k2 1 1 − + + O φ = 0. (154) κ Ca4 a8 k We can always find agreement with the boundary condi- 3/2+(ns 1)/2 tion (136), namely φ k− − for large a or k, Following the usual well know method to get rid of the by defining the arbitrary| |∼ constants c (k) accordingly. In ± first derivative, we write fact for large a, a k/√κC, and we can choose c (k) in a way that ≃ ± φ(a)= v(a)p(a), (155)

iG(a(k)) iG(a(k)) (ns 1)/2 c+(k)e + c (k)e− k − (165) with − ∼ 1 so that the condition (136) is fulfilled. The expression of p(a) = exp f(a)da , (156) −2 φ(a(k), k) when a(k) k/√κC (first approximation) is ≃ φ(a(k), k)= where f(a) is the coefficient of φ′. Then the equation for 4 v(a) reads 2 √κC2 k(ns 1)/2 e iπ/4 − − .(166) [16k6/κ2C +9κA2]1/4 exp[A√κ3C/(4k3)] p′′ p′ v′′(a) + + f + g v(a) = 0, (157) p p We can also consider relation (130) to the second order of approximation, which reads where g(a) is the coefficient of φ. To an equation of the form (157) we can apply the WKB k Aκ a(k) . (167) method. In what follows, we specify the main steps of ≃ √κC − 2k2 the argument for the matter era case, and report only the final result for the radiation era case. Constants c (k) should, and can, still be chosen as in ± 3/2+(n 1)/2 For Eq.(153), pre-inflation matter era, we find (165), in order to have φ k− s− for large k. Once this new expression| | ∼ of a(k) is substituted in 1 A 1 (164), it is instructive to plot the quantities φ(a(k), k) p(a)= exp , (158) 3 2 | | a2 −4C a3 and P = k φ(k) . In Figures (9), (10) we can see these | | 19

ÈΦHkLÈ a k/√κC. Also here we may consider relation (129) 0.8 to≃ the second (or further) order in k,

0.6 k κB√κC a(k) , (172) ≃ √ − 2k3 0.4 κC

0.2 to get a better approximation. In Figures (11), (12) we see plots of the ﬁeld φ and of the primordial power spec- | | k trum P (k). 2 4 6 8 10

È H LÈ FIG. 9: Field |φ(k)| versus k, in pre-inﬂation matter era (ar- Φ k bitrary units). 0.4

PHkL 0.3

1.4 0.2 1.2 1 0.1 0.8 k 0.6 5 10 15 20 0.4 0.2 FIG. 11: Field |φ(k)| versus k, in pre-inﬂation radiation era (arbitrary units). k 2 4 6 8 10

FIG. 10: Primordial power spectrum P (k) versus k, in pre- inﬂation matter era (arbitrary units). PHkL

0.8 qualitative plots, where, for sake of simplicity, we arbi- trarily set the parameters A = B = C = κ = 1. 0.6 For the pre-inﬂation radiation era, we start from Eq.(154). With analogous steps we ﬁnd 0.4

1 B 1 0.2 p(a)= exp (168) a2 −4C a4 k and 5 10 15 20

2 v′′(a) + F (a) v(a) = 0, (169) FIG. 12: Primordial power spectrum P (k) versus k, in pre- inﬂation radiation era (arbitrary units). with

2 k2 9B B2 1/2 F (a)= i + + . (170) Finally, in Figures (13), (14), we compare the plots for a2 − κCa4 Ca6 C2a10 ﬁelds and power spectra, matter and radiation cases, in the same diagrams. The WKB solution of Eq.(154) explicitly reads We see that for both matter and radiation eras there is an exponential suppression of the low k-modes. The φk(a)= vk(a)p(a)= (171) matter diagram presents an interesting cusp, just be- iπ/4 4 2 1/2 iG(a) iG(a) e− √κC a c+(k)e + c (k)e− − fore dropping down, that is absent in the radiation di- 1/4 . [2κC2a8 k2Ca6 +9κBCa 4 + B2κ] exp[B/(4Ca 4)] agram. In the next section the eﬀects of these features − on the CMB spectrum will be further investigated, via Again the functions c (k) should be chosen such that deeper numerical analysis with the help of CMBFAST, a the square bracket’s content± in the numerator goes as code specialized for cosmological simulations of the CMB (n 1)/2 3/2+(n 1)/2 k s− , so that φ k− s− for large power spectrum. ∼ | | ∼ 20

ÈΦHkLÈ When inflation starts, the two competing factors of mat- 0.8 ter and inflation in the Friedmann equation (75) must be of the same order of magnitude. This condition, or more 0.6 rigorously, condition (83), allows us to fix the coefficient C as

0.4 A 1 11 ǫp C = 3 = 10− . (175) 2 ainf 2 Vp 0.2 For the numerical solution of the equation, we have to

k use the horizon crossing condition (130) to write a as a 2 4 6 8 10 function of k. The equation for the scalar field perturbation, Eq. (173), is evaluated numerically for a fixed k , FIG. 13: Field |φ(k)| versus k, for pre-inflation matter era i (red/upper line), and radiation era (green/lower line). repeatedly for many different choices of ki, and the values of φ(a(ki), ki) are assembled to form an evolution of the field perturbations in dependence of k. PHkL The boundary conditions for the numerical solution have

1.4 been obtained in section V B. Using them to solve the diﬀerential equation (173) leads to the result shown 1.2 in Fig. 15. However, the power spectrum as a function 1 of k is only a collection of data points. In order to be of 0.8 any use for the CMBFAST code, it has to be given as an

0.6 analytical function, which is obtained by ﬁtting the data points with an opportune mathematical function. The 0.4 function used for the ﬁtting is 0.2

k b d f 2 4 6 8 10 P (k)= a 2 + 4 6 , (176) − k k − k 1+ c 1+ e 1+ g FIG. 14: Primordial power spectrum P (k) versus k, for pre- inﬂation matter (red/upper line) and radiation (green/lower where the parameters a, ..., g can be found as: line) eras. parameter value a 2.205 · 10−12 E. Numerical computation of the primordial power b 3.233 · 10−12 spectrum c 0.03 d 2.578 · 10−12 1. Matter era 9 e 1.680 · 10− f 1.593 · 10−12 For the numerical solution it is most convenient to use −14 the equation of motion for the scalar ﬁeld perturbation g 6.584 · 10 φ written as a function of the variable a. So we can use equation (131) cast as

2 4 5 3 k Fig. 15 shows the fitting function (full line) together [Aa + Ca ] φ′′ + A +4Ca φ′ + φ = 0. (173) k 2 k κ k with the numerical curve of the power spectrum (dashed line). The field perturbation φk(a) is a function of k and a. In one of the previous sections we determined the param- With these results, it is then possible to continue with eter of the matter contribution in the Friedmann equa- the evaluation of the CMB power spectrum by putting tion, A, to be the fitting function into CMBFAST.

1/2 ǫp A = 10− . (174) Vp 2. Radiation era with totally evaporating black holes After a radiation-dominance era and a subsequent period of matter-dominance before inflation, we can, with The equation of motion for the scalar field perturbation the help of Eq.(103) and the assumption of a(tp) = 1, is Eq.(132), which can be cast in the form compute the scale factor at the onset of inflation 2 4 2B 3 k 7/2 [B + Ca ] φ′′ + +4Ca φ′ + φ = 0. (177) ainf = 10 . k a k κ k 21

The ﬁtting in this case has been done using the func-

12 Pk @10 D tion

3.0 P (k)= a + b Arctan(c k)+ d Arctan2(c k), (182) 2.5

2.0 with the parameters a, ..., d to be adjusted by Mathe- matica. The resulting ﬁtting parameters are given in the 1.5 following table, and the curve (full line) together with 1.0 the numerical result can be seen in Fig. 16.

0.5

h k @ D parameter value 100 200 300 400 500 Mpc a −3.701 · 10−18 b 1.261 · 10−16 c 3.116 FIG. 15: The numerical solution for the primordial power −17 spectrum (dashed line) and its ﬁtting function (full line) in d −5.935 · 10 the pre-inﬂation matter era scenario.

@ 17D The k(a)-relation for the radiation case is given by Pk 10 Eq.(129). We shall compare the situation of the previous 7 section, when massive remnants are left by the black hole 6 evaporation, with the present situation, in which black 5 holes are nucleated but then disappear again completely 4 into radiation. Following the scenario of radiation in pre-inflation era 3 developed in subsection IV F, we can determine the pa- 2 rameters B and C in this case. When τ spans the interval 1 4 h between the end of black hole evaporation, τr 10 , and 0 k @ D 8 ≃ 0.0 0.5 1.0 1.5 2.0 Mpc the onset of inflation, τinf 10 , then the scale factor evolves in full radiation era≃ as 1/6 τr 1/2 FIG. 16: The numerical solution for the primordial power a(τ)= a(tp) τ (178) τc spectrum (dashed line) and its fitting function (full line) in the pre-inflation radiation era scenario, with completely evap- 3 Reminding a(tp) = 1 and τc 10 we get a scale factor orating black holes (no GUP active). at the onset of inflation of ≃ a(τ )=1025/6 104.16. (179) inf ∼ The radiation content of the Universe is fixed at τr, the end of the black hole evaporation era. Therefore, since 3. Radiation era without black holes 4 by then RH = 10 ℓp,

4 B = ρr(τr) a (τr) (180) The equation of motion for the scalar field perturbations can be directly taken over from the previous section 104 √ζǫ ǫ p 3 1/6 4 2/3 4 2/3 p - it describes the universe evolving from a state of radia- = 4 3 [(10 )− (10 ) ] = 10 ζ . (10 ) Vp Vp tion dominance into the inflationary period. Simply the Here, as before, we choose again ζ 1. Thus, the infla- parameters B and C in the equation have to be evaluated tionary parameter C can be fixed by∼ requiring that it is anew. of the same order as the radiation parameter B at the The scale factor at the onset of inflation is, as previously onset of inflation mentioned,

B 16 7/2 a(τinf )=10 . (183) C 4 10− . (181) ≃ ainf ∼ At the Planck time, the radiation had Planck density, Thus we now solve the equation for a scenario without and so GUP in the same way as before, and investigate the dif- ferences in the primordial power spectrum. Fig. 16 shows 4 ǫp B = ρp(τp) a (τp)=1 . (184) the result (dashed line). Vp 22

Thus, the inflationary parameter can be fixed as by following the standard inflationary scenario without any pre-inflation era. B 14 The standard inflation (SI) model is in principle a very C = 4 10− . (185) ainf ∼ good fit to the data, considering all the different features it has to explain. Only the mode with l = 2 in the Again the equation for a scenario without GUP is solved CMB spectrum is very low in comparison to the follow- in the same way as before, and the resulting primordial ing data points. If this is not simply a statistical feature power spectrum is shown in Fig. 17 (dashed line). but the indicator of a new physical phenomenon, then The fitting in this case is done using the same function the SI model has to be modified in order to satisfy the as in the matter case, drop at the low l modes. For now, only the l = 2 mode can be used to construct models with a suppression at b d f P (k)= a 2 + 4 6 , (186) low l modes, but in the far future it will be possible to − k k − k 1+ c 1+ e 1+ g tell whether the tendency of the power spectrum to drop will continue further, whether it will stay at a lower but with the parameters a, ..., g as in the following table: constant level or whether it might even rise up again.

parameter value l Hl 1L 2 15 Cl @ΜK D − 2 Π a −4.566 · 10 6000 b 3.383 · 10−15 c 0.0089 5000 −15 d −6.561 · 10 4000 e 152.074 3000 f 3.783 · 10−16 −7 g 3.564 · 10 2000

1000

0 l 1 5 10 50 100 500 1000 Fit for the power spectrum Pk 12 Pk @10 D

7 FIG. 18: The binned CMB temperature anisotropy spectrum 6 as measured by the WMAP satellite, in comparison with the

5 standard inﬂationary scenario.

3 To produce the CMB power spectrum corresponding to

2 the scenario of standard inﬂation, we assumed a scalar

1 spectral index of ns = 0.963 and a running of the index

h αns = 0.022, i.e. the most recent result of the WMAP 0 k @ D − 0 5 10 15 20 25 30 35 Mpc observations [45]. In the calculation of the CMB power spectrum, there is one parameter that can be varied, the number of e-folds FIG. 17: The numerical solution for the primordial power of inflation. The total number of e-folds of inflation (from spectrum (dashed line) and its fitting function (full line) in the start, when the mode ki left the horizon, to the end the pre-inflation radiation era scenario without black holes. of inflation) can be given (see [46]) as

k0 Ntot = N(k0)+ ln . (187) ki VI. THE CMB POWER SPECTRUM k0 is the currently largest mode within the horizon, 1 k0 =0.002 hMpc− . (188) In this section the previously calculated primordial This is the pivot scale for the wavenumber that the power spectra, in the cases with GUP, without GUP, WMAP team has been using in constraining inﬂation and without black holes, as well as the analytical re- models from their data [45, 47, 48]. N(k ) is the number sults for the approximated equations, will be fed into 0 of e-folds from the time during inﬂation when this mode the CMBFAST code [44] to obtain the CMB tempera- k crossed outside the horizon, ture anisotropy spectrum that can be measured today. 0 We will compare our results to the WMAP seven year k N(k0)= ln . (189) data, and to the result for the CMB spectrum obtained k0 23

Observations can only constrain the number N(k0), as C. Comparison currently the mode k0 reenters the horizon, but no information can be given about whether there were more To have a better impression of how the three scenarios e-folds ∆N of inflation before k0 exited the horizon dur- compare to each other, in Fig.23 there are several graphs ing inflation. The constraint on N(k0) stated in [46] is showing three curves for different values of ∆N. Full lines represent the cases with GUP, dashed lines the cases N(k0)=54 7. (190) without GUP, and the dashed-dotted lines the case of ± pure radiation, where no black holes existed at all. Usually in the power spectrum k is normalized to k0, and the power spectrum is taken at values P k , where k0 2 k = 0.002 hMpc 1. By dividing k in the expression D. χ Calculations 0 − of the power spectrum by numbers smaller than k0, we can add more e-folds to inflation accounting for the time The goodness of the fit of each of the different curves 2 before k0 exited the horizon: can be quantitatively expressed by a χ -value, calculated from the formula k k0 k Ntot = ln + ln = ln . (191) N 2 k k k 1 (Di Ti) 0 i i χ2 = − , (192) N C2 i=1 i Instead of taking the power spectrum as before normal- k ized over k0, we take it as P , where any number where Di is the i th data point, Ti is the correspond- ki ing value calculated− by the model, and C is the error with ki < k0 is possible. i bar of measurement for the i th data point. The theo- So, varying ki is equivalent to adding e-folds ∆N to the retical model, calculated for different− numbers of e-folds experimentally constrained number N(k0)=54 7. ± are compared with the WMAP 7-year-results for the un- binned CMB temperature spectrum. The error bar of the measurement is provided along with the WMAP spec- A. Results from the numerically computed trum data. primordial power spectrum The χ2-value for the SI model calculated according 2 to Eq.(192) is χSI = 1.154. For our model of matter- 2 There are three cases to present from the numerical dominance in the pre-inflation era, the χ -values for dif- calculations. ferent number of e-foldings are given in the following ta- For a matter-dominated era before inflation the CMB ble. power spectrum as obtained by CMBFAST can be seen in Fig. 19. ∆N 0 0.693 1.386 1.792 2.079 2.996 3.584 4.094 2 For the case when the GUP is turned off, the CMB χ 4.801 1.69 1.206 1.353 1.463 1.378 1.171 1.172 power spectrum can be seen in Fig.20. For the case when there are no black holes at all, only For the scenario of radiation-dominance in the pre- pure radiation, the CMB power spectrum can be seen inflation era, with totally evaporating black holes, the in Fig. 21. χ2-values for different number of e-foldings are given as follows.

∆N 0 0.693 1.386 1.792 2.079 2.996 3.584 4.094 B. Results from the analytical solutions for P (k) 2 χ 1.121 1.132 1.141 1.144 1.147 1.151 1.152 1.153

As regard the first and the second case of the list of For the scenario of radiation-dominance without the the previous paragraph, the power spectrum obtained by presence of black holes, the χ2-values for different number WKB solving the approximated differential equation for of e-foldings are given as a pre-inflationary matter and radiation era was fed into CMBFAST as well. The left picture in Fig. 22 shows ∆N 0 0.693 1.386 1.792 2.079 2.996 3.584 4.094 the outcome for the CMB temperature anisotropy spec- 2 trum in the case when the GUP is valid and thus the χ 1.562 1.33 1.17 1.157 1.156 1.155 1.154 1.154 pre-inflation era is matter-dominated (we use the coefficients A, C from Eqs.(174, 175)), whereas the right pic- Fig. 24 should help to make the numbers more under- ture in Fig. 22 shows the results in the case where GUP standable. It shows the χ2-values for the three models does not hold, which corresponds to a pre-inflation ra- over the number of e-foldings ∆N added to the standard diation era with completely evaporating black holes (the number of 54 e-folds. The horizontal line represents the coefficients B, C from Eqs.(180, 181) have been used). value for the SI model. 24

l Hl 1L 2 l Hl 1L 2 Cl @ΜK D C @ΜK D 2 Π 2 Π l 6000 3000

5000 2500

4000 2000

3000 1500

2000 1000

1000 500

0 l 0 l 1 5 10 50 100 500 1000 1 2 5 10 20 50 100

FIG. 19: The CMB power spectrum for a pre-inﬂation matter era, for various cases of ∆N. Overall view and zoom into the lower multipole region.

l Hl 1L l Hl 1L C @ΜK2D C @ΜK2D 2 Π l 2 Π l 6000

2000 5000

4000 1500

3000 1000

2000

500 1000

0 l 0 l 1 5 10 50 100 500 1000 1 2 5 10 20 50

FIG. 20: The CMB power spectrum for a pre-inﬂation radiation era, for various cases of ∆N. Overall view and zoom into the lower multipole region.

l Hl 1L l Hl 1L C @ΜK2D C @ΜK2D 2 Π l 2 Π l 6000 2000

5000

1500 4000

3000 1000

2000

500 1000

0 l 0 l 1 5 10 50 100 500 1000 1 2 5 10 20 50

FIG. 21: The CMB power spectrum for a pre-inﬂation radiation era without any black holes, for various cases of ∆N. Overall view and zoom into the lower multipole region. 25

H L l l 1 2 l Hl 1L 2 Cl @ΜK D C @ΜK D 2 Π 2 Π l

6000 6000

5000 5000

4000 4000

3000 3000

2000 2000

1000 1000

0 l 0 l 1 5 10 50 100 500 1000 1 5 10 50 100 500 1000

FIG. 22: The CMB power spectrum from the analytic solution of the approximated diﬀerential equation with matter dominance and radiation dominance, for various cases of ∆N.

l Hl 1L l Hl 1L C @ΜK2D C @ΜK2D 2 Π l 2 Π l 6000 6000

5000 5000

4000 4000

3000 3000

2000 2000

1000 1000

0 l 0 l 1 5 10 50 100 500 1000 1 5 10 50 100 500 1000

l Hl 1L l Hl 1L C @ΜK2D C @ΜK2D 2 Π l 2 Π l 6000 6000

5000 5000

4000 4000

3000 3000

2000 2000

1000 1000

0 l 0 l 1 5 10 50 100 500 1000 1 5 10 50 100 500 1000

l Hl 1L C @ΜK2D 2 Π l 6000

5000

4000

3000

2000

1000

0 l 1 5 10 50 100 500 1000

FIG. 23: Comparing the CMB power spectrum for pre-inﬂation matter (full lines), radiation with black holes (dashed lines) and radiation without black holes (dashed-dotted lines) eras for ∆N = 0.693147, ∆N = 2.07944, ∆N = 2.99573, ∆N = 3.58352 and ∆N = 4.09434. 26

holes evaporating until zero mass (no GUP), the result is Χ2 5 ç rather poor - only one of the curves shows a suppression in the lower modes, all of them actually lie above the 4 result given by a standard inflationary scenario without ç with GUP á without GUP a pre-inflation era. The model shows a very good accor- ó without black holes 3 dance for the high l region. In contrast to that, the model with only radiation in 2 ç the pre-inflationary era turns out to be a little better; ó ç ó ç ç two of the curves show a suppression, while the others á á çáó áó áó áó çáó çáó 1 lie higher than the curve obtained by the standard inflationary scenario. This can be traced back to the shape of 0 N 0 1 2 3 4 the primordial power spectrum, which in this case rather resembles the matter case than the radiation case with black holes. FIG. 24: The distribution of χ2-values for a pre-inflation mat- The CMB power spectrum, produced with the ana- ter and radiation era, for various cases of ∆N. Circles repre- lytical solutions obtained by approximating the differen- sent the matter model, whereas squares stand for the radia- tial equation, looks very similar to the standard inflation tion model and rotated squares denote the case with radiation picture with only slight deviations. But on the other only. The horizontal line denotes the value of the SI scenario. hand, we know that the WKB analytical solution has been mainly used as a qualitative guide to choose the correct numerical solution among several possibilities, and so we cannot expect from it a perfect fitting of the data. VII. CONCLUSIONS AND OUTLOOK Especially because we pushed the analytical approximation to the second order only. Of course, it is important In this work, we investigated the effects of an era be- to investigate the differential equation analytically as to fore inflation on the CMB power spectrum measured to- obtain the correct boundary conditions for the numerical day. We utilized the phenomenon of black hole nucle- simulation and to firmly support the correctness of the ation from quantum fluctuations of the metric in very numerical result. early times to argue for the existence of several thousand Although the suppression of the lower modes in the micro black holes in the pre-inflation era, which cause the case of pre-inflation matter dominance is there, the over- 3 universe to be matter-dominated from about τ 10 tp, all fit to the current data turns out to be not so successful. until the onset of inflation (τ 106 t ). By setting≃ up ≃ p A drawback of the matter model definitely is the bad fit the Friedmann equation of the universe evolving from for large l when only a few e-folds are added; the behav- matter dominance to an inflationary phase it is possible ior for large l becomes better with increasing ∆N, but to calculate the power spectrum of primordial fluctua- it is not very good for small ∆N. The reason why the tions of a scalar field Φ living in this scenario, and then model without GUP is so much better in the overall fit process this primordial power spectrum by CMBFAST than the other two cases can be found in the fitting pro- code to yield the CMB temperature anisotropy spectrum cess for the primordial power spectrum. The shape of Pk measured today. As an alternative to the matter domi- in the matter model and in the pure radiation is more nance scenario, we also investigated the implications of a eccentric and harder to be fitted than in the case of radi- radiation-dominated era before inflation by relinquishing ation without GUP, where the power spectrum is rather the claim of the Generalized Uncertainty Principle, stat- smooth and the arctan-fit matches the curve quite well. ing that black holes can only evaporate down to Planck In the matter case, the fitting function is more compli- size. These two cases have been calculated both numer- cated and the quality of the fit is definitely worse. This ically and in analytical approximations. A third case of course influences the χ2-value of the model, which is is presented, in which black holes never existed and the quite large for the smaller ∆N. A stronger suppression pre-inflation era is purely radiation-dominated. on several of the low l modes also leads to a larger devia- From the overall analysis that has been done on the tion from the modes with l > 3. The SI model is without three different scenarios, only the model with matter- doubt the best fit in general; however, it doesn’t capture dominance in pre-inflation era is really successful in the the drop at the l = 2 mode. The model with matter in suppression of the l = 2 mode, and incorporates the de- pre-inflation might not be the best fit on all scales, but sired effect on the power spectrum very well. It asymp- in the future, with further modes on larger scales than totes to the standard inflationary model for higher ∆N, l = 2 being suppressed, the model might become more which is expected as with increasing duration of inflation successful than the standard inflation scenario. For sure, the effects of a pre-inflationary era are shifted to larger if the suppression of the l = 2 mode is to be continued scales and thus would be expected to influence the scales with a suppression on even larger scales, the extension that are still to enter the horizon. of standard inflation to having a pre-inflationary era is For the case of radiation in pre-inflation era, with black required, otherwise the drop for the lower modes will re- 27 main unexplained. The probability for a quantum vacuum fluctuation The quality of the results has been put into numbers to produce a black hole of critical mass M is 2 by a χ analysis, which is the best way in the current exp( ∆F/(kBT )), where ∆F is the change in the free situation to give a solid statement about the success of energy− of the system with T and V held fixed. Now the three models today. ∆F = F F , where F is the free energy of the black − g In terms of this analysis, the pre-inflationary matter hole and Fg is the free energy of the thermal gravitons era is disfavored compared to the SI model. We could displaced by the black hole. F is related to the rest en- think that the roles might be really different in the far ergy of the black hole E = Mc2 by future, and the modes that are still to enter the hori- dF zon might be more successfully described by the mat- E = F T (194) ter era scenario. However, even without looking too far − dT into the future, but keeping our mind on the present, This gives us a differential equation for F the pre-inflation matter era model seems to be the only one, among those here studied, able to capture and de- dF F E(T ) = (195) scribe the low l modes suppression. Further refinement dT T − T of the model are obviously in order. And in principle, if a better matching with the observations will be achieved, where, from (193) the model can serve to directly check the validity of the ~c5 Generalized Uncertainty Principle, as relation (107) ex- E(T )= Mc2 = . (196) plicitly suggests, and more indirectly, of the Holographic 8πGkB T Principle. These are surely intriguing avenues for future Integrating F (T ) we find research. Only with time it will become clear whether ′ the model with matter-dominance in pre-inflation era is ~c5 superior to standard inflation in its success to explain the F (T )= . (197) 16πGk T CMB power spectrum. B

To compute the thermal free energy Fg of displaced gravitons we need, as it is clear from Eq.(195), an expression Acknowledgements for the total energy E of such thermal gravitons. This can be obtained from Eq.(24), explicitly rewritten as The authors would like to thank Kin W. Ng and Ron π2 k4 J. Adler for enlightening conversations. F.S. would like Eg (V ) = B VT 4 (198) to thank Mariam Bouhmadi Lopez for conversations and TOT 15 c3 ~3 for having drawn his attention on the wonderful world where we chose the greybody factor for gravitons Γg = 1, of elliptic functions. This research is supported by Tai- and we dropped the correction function A(β,T ), as the wan National Science Council under Project No. NSC effects of GUP are considered only through the cutoff 97-2112-M-002-026-MY3 and by US Department of En- on masses and temperatures. Since the volume of the ergy under Contract No. DE-AC03-76SF00515. We also displaced gravitons coincides with that of the black hole acknowledge the support of the National Center for The- 3 2 (4πR /3, with RS =2GM/c ), we have oretical Sciences of Taiwan. S k T Eg (T )= B . (199) TOT 720 Appendix 1 Therefore Eq.(195) yields Here we compute the nucleation rate n for micro black ∗ kBT T holes, i.e. the number of micro black holes of critical mass Fg(T )= log (200) − 720 T M created in a thermal bath of gravitons, via gravita- p tional instabilities of hot flat space, per unit volume per So, finally, expressing things in Planck units we have unit time. Essentially, we follow the procedure detailed in Ref. [5]. Our discussion, as in that reference, will ∆F 1 1 = 2 + log Θ (201) be based on the standard Heisenberg principle. As said, kB T 16πΘ 720 the GUP will be implemented in our argument only by considering the cutoff imposed on minimum masses and Note that, in the range of interest, namely for 0 < Θ < 1, we have log Θ/720 1/(16πΘ2), so the second term in maximum temperatures. Critical mass and temperature | | ≪ are linked by the relation (14), which in standard units (201) can be neglected. reads Knowing the probability of one statistical fluctuation, exp( ∆F/(kBT )), we have to estimate the density for ~c3 such− fluctuations. Consider the fluctuations on the small- M = . (193) 8πGkB T est scale possible, namely with a wavelength λmin = αℓp, 28

α of order 1. Then, imagining a cubic lattice, the number Appendix 2 of statistical fluctuations in the unit volume is In this Appendix we construct the cosmic scale factor 2 3 n = (202) a as a function of the conformal time η. Once we have 0 λ min a(η), we shall be able to write equation (120) for vk(η). In order to arrive to the function a(η), two different, and the number of fluctuations per unit volume able to but equivalent, procedures can be specified. produce a black hole of critical mass M will be I) When the scale factor a(t) is a known function of ∆F n = n0 exp the cosmic time t, then we can compute (in principle, ∗ −k T B explicitly) the function η(t) 2 3 1 = exp 2 (203) dt λmin −16πΘ η = dη = η = η(t). (210) a(t) ⇒ The number of micro holes with critical mass M, created Hence, inverting the last relation we get per unit time per unit volume, can be therefore computed as t = t(η) a = a(t(η)) = a(η) . (211) ⇒ dn 1 dΘ ∗ = n (204) There exists however also another alternative procedure. dτ ∗ 8πΘ3 dτ II) Sometimes the integral in (210) is not easily doable, The value of dΘ/dτ can be obtained using Eqs.(14) and and moreover the object we are usually interested in is (28), with the correction function A(β,T )=1,andΓγ = the function a(η), and not η(t). Such function can be 1. We have directly computed by re-writing the equation of motion for a in terms of the conformal time η, instead of the dΘ 2π2 = Θ4 (205) cosmic time t. The equation of motion (72) reads, in dτ 15 conformal time, So finally da 2 = κ Ca4 + Aa + B (212) dn π dη ∗ = n Θ (206) dτ ∗ 60 or In the statistical probability of one fluctuation, we should da also include a term for the quantum correction of the free = √κ dη (213) √Ca4 + Aa + B energy of the black hole. It can be shown [49] that for a Schwarzschild metric we have The function a(η) can be obtained directly from the integration of the previous equation. F quantum 212 c2 = log (207) In the following, we shall integrate equation (213) in the k T − 45 k T B B two cases of our interest: pre-inflation radiation era, and pre-inflation matter era. where is a regulator mass of the order of the Planck Pre-inflation radiation era: In this case there is no mass. The final formula for the number of micro holes matter, therefore A = 0. Equation (213) reads with critical mass M, created per unit time per unit volume, reads, in Planck units, da √κ dη = (214) 4 212 √Ca + B dn π ˜ 45 ∗ = n Θ (208) With the substitution dτ ∗ 60 2Θ 212 1/4 2 3 π Θ ˜ 45 1 B = exp a = x (215) λ 60 2Θ −16πΘ2 C min the equation becomes Since in Planck units ℓp = ˜ = 1 the previous formula can be usefully rewritten as dx (κ2BC)1/4 dη = . (216) √ 4 dn 8π 167 1 1+ x ∗ = Θ− 45 exp (209) dτ 15 64 π3 −16πΘ2 We can now make use of the formula (see [50]) which agrees with the numerical pre-factor of Ref.[5] since dx 1 32/135 = F (α, k) (217) we chose α = π 2− 2.66. √1+ x4 2 ≃ 29 where F (α, k) is the elliptic integral of the first kind where capital K is the cosmological perturbation wave number and has nothing to do with the elliptic functions α d F (α, k)= . (218) parameter k. This equation has only a resemblance with 2 2 0 1 k sin the Lame’ equation, but unfortunately, differs from it in − a fundamental way. In our specific case, Eq.(217), we have Pre-inflation matter era: In this case we don’t have 1 radiation, i.e. B = 0, and the equation for the conformal k = √2 scale factor a(η) reads 1 x2 da α = arccos − (219) √κ dη = (228) 1+ x2 √ 4 Ca + Aa Then Using the substitution α d 1/3 2 (κ2BC)1/4 η = . (220) A 2 2 a = x (229) 0 1 k sin C − Inverting the last integral, we get the Jacobi amplitude we have α = am[2 (κ2BC)1/4 η] (221) dx (κ3A2C)1/6dη = (230) 3 and, because of relation (219), x(1 + x ) 1 x2 Again with the help of [51] we can make use of the for- − = cos am[2 (κ2BC)1/4 η] mula 1+ x2 dx 1 2 1/4 1 = cn 2 (κ BC) η, (222) = 4 F (α, k) (231) √ x(1 + x3) √3 2 where cn is the Jacobi cosine-amplitude (see again [50] where F (α, k) is the usual elliptic integral of the first kind for definitions and properties). Reminding the relation (218), but now (215) between a and x, finally we can write 2+ √3 1 2 1/4 1 k = 2 1 cn 2 (κ BC) η, B √2 2 a2(η)= − . (223) C 2 1/4 1 1+(1 √3)x 1+cn 2 (κ BC) η, √ α = arccos − . (232) 2 1+(1+ √3)x It is interesting to check the small η limit (or, equivalently, the small C limit) of the previous relation. Con- Then α sidering the MacLaurin expansion for cn 4 d √3 (κ3A2C)1/6 η = , (233) 2 1 0 1 k2 sin cn(u, k)=1 u2 + O(u4) (224) − − 2 and the Jacobi amplitude reads we have √4 3 2 1/6 1 1 α = am[ 3 (κ A C) η] (234) 2 2 2 2 2 B 2 (κ BC) η + ... 2 a (η) = = κBη (225) Reminding relations (232) and (229), finally we have C 1+1+ ... which is the well known expression for the scale factor the solution a(η) expressed in terms of Jacobi cosine- a(η) in pure radiation era. amplitude

To build equation (120) for vk we need to compute 1 a′′(η)/a(η). It is a bit laborious, but however the result, A 3 1 cn[βη,k] a(η) = − (235) for the parameter k =1/√2, is C (√3 1)+(√3 + 1) cn[βη,k] − 1 2 1 cn βη, where a (η) β √2 ′′ = − , (226) 1 a(η) 2 1+cn βη, 4 3 2 1/6 2+ √3 √2 β = √3 (κ A C) η ; k = . (236) 2 2 1/4 with β = 2(κ BC) . The limit for small η, or small C, can be computed with Therefore the equation for vk reads the help of Eq.(224), and reads 2 2 β 1 cn(βη) κA v′′ + K − v = 0 (227) 2 k − 2 1 + cn(βη) k a(η) = η (237) 4 30 which is the known expression of the conformal scale fac- The equation for vk can be henceforth explicitly writ- tor in pure matter era. ten down, although it still results to be only ”similar” As for the construction of the equation for vk, we have, to the Lame’ equation, but not exactly of that known after a somehow long calculation, kind. Progress on the analytical exact solutions of such equations will be reported in future work. 1 a (η) ′′ = (238) √3β2 a(η) (1 √3)(√3cn(βη) 1)+(1+ √3)(√3 + cn(βη))cn2(βη) − − [(√3 1)+(√3 + 1)]2(1 cn(βη)) − −

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[44] U. Seljak and M. Zaldarriaga, Astrophysical J., 469, nary Survey, edited by S.W.Hawking and W.Israel (Cam- (1996) 437. bridge University Press, London 1979). [45] E. Komatsu, K. M. Smith, and J. Dunkley, Seven [50] I.S. Gradshteyn, I.M.Ryzhik, Table of Integrals, Series, Year WMAP Observations: Cosmological Interpretation, and Products, Academic Press, 1980. See §§ 8.11, 8.14. Ap.J.Supp.S., in press, 2010. In particular, formula 3.143, n.2, p.239. [46] L. Alabidi and D. Lyth, JCAP, 8 (2006) 006. [51] I.S. Gradshteyn, I.M.Ryzhik, Op.cit., formula 3.166, [47] E. Komatsu, J. Dunkley, and M. R. Nolta, Five n.22, p.263. Year WMAP Observations: Cosmological Interpretation, [52] It is interesting to note that analogous results have been Ap.J.Supp.S., 180 (2009) 330. obtained in a completely diﬀerent framework, via non [48] H. Peiris and R. Easther, JCAP, 0607 (2006) 002. commutative geometry inspired black holes. See [25] and [49] S.W.Hawking, in General Relativity: An Einstein Cente- references therein.