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Primordial Black Hole Evaporation and Spontaneous Dimensional Reduction Physics Faculty Works Seaver College of Science and Engineering 9-17-2012 Primordial Black Hole Evaporation And Spontaneous Dimensional Reduction Jonas R. Mureika Loyola Marymount University, [email protected] Follow this and additional works at: https://digitalcommons.lmu.edu/phys_fac Part of the Physics Commons Recommended Citation Mureika J. Primordial black hole evaporation and spontaneous dimensional reduction. Physics Letters B. 2012;716:171-175. This Article is brought to you for free and open access by the Seaver College of Science and Engineering at Digital Commons @ Loyola Marymount University and Loyola Law School. It has been accepted for inclusion in Physics Faculty Works by an authorized administrator of Digital Commons@Loyola Marymount University and Loyola Law School. For more information, please contact [email protected]. Physics Letters B 716 (2012) 171–175 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Primordial black hole evaporation and spontaneous dimensional reduction J.R. Mureika Department of Physics, Loyola Marymount University, Los Angeles, CA 90045, United States article info abstract Article history: Several different approaches to quantum gravity suggest the effective dimension of spacetime reduces Received 15 May 2012 from four to two near the Planck scale. In light of such evidence, this Letter re-examines the Received in revised form 6 August 2012 thermodynamics of primordial black holes (PBHs) in specific lower-dimensional gravitational models. Accepted 15 August 2012 Unlike in four dimensions, (1 + 1)-D black holes radiate with power P ∼ M2 , while it is known no Available online 17 August 2012 BH (2 + 1)-D (BTZ) black holes can exist in a non-anti-de Sitter universe. This has important relevance to the Editor: T. Yanagida PBH population size and distribution, and consequently on cosmological evolution scenarios. The number of PBHs that have evaporated to present day is estimated, assuming they account for all dark matter. Entropy conservation during dimensional transition imposes additional constraints. If the cosmological constant is non-negative, no black holes can exist in the (2 + 1)-dimensional epoch, and consequently a (1 + 1)-dimensional black hole will evolve to become a new type of remnant. Although these results are conjectural and likely model-dependent, they open new questions about the viability of PBHs as dark matter candidates. © 2012 Elsevier B.V. Open access under CC BY license. 1. Introduction “hides” structure of the manifold at scales approaching the quan- tum regime, turning gravity into an effective lower-dimensional It has long been known theories of gravitation have a much theory. simpler formulation in (2 + 1)-D [1–10] and (1 + 1)-D [11–28], From a string theory perspective, it has been shown that an where associated quantum theories are exactly solvable [12]. energy-dependent dimension emerges from a smooth transforma- A resurgence of interest in lower-dimensional physics has been tion of a three-brane to a one-brane [51]. Similarly, this idea has spurred by a confluence of evidence that the effective dimen- been extended to model a three-brane as a collection of one- sionality of spacetime may depend on the energy scale at which branes at every point [52]. Alternate dimensional reduction sce- interactions take place [29–44]. Instead of revealing extra dimen- narios include fractal spacetimes [39,50,56–59], with additional sions at very short distances [45,46], it is conceivable that the approaches concerning new techniques in gauge coupling unifica- number of spatial dimensions decreases as the Planck length is tion [34] and a strong coupling expansion of the Wheeler–DeWitt approached. equation [35]. Dynamical or spontaneous dimension reduction has been stud- A geometric dimensional reduction framework was recently ied in various contexts, mostly focusing on the energy-dependence proposed wherein a (d + 1)-dimensional spacetime is a recur- of the spacetime’s spectral dimension ds. The latter is the ef- sive lattice-network of lower-dimensional substructures [30,31,29, fective dimension seen by a diffusion process on the manifold 32,33]. Each has a fundamental length scale Lk that becomes rel- ∼ −1 over some time σ , characterized by a probability return func- evant at the energy Ek Lk . This concept naturally addresses tion P(σ ) [47,48]. The spectral dimension is formally defined as the hierarchy problem, and provides a range of phenomenolog- =− d log P(σ ) = ical signatures — including dimensionally-dependent scattering ds 2 d log σ , which for flat space is ds 4 [47], indicating this quantity is a probe of the underlying geometry. The causal dy- cross-sections and gravitational wave frequency thresholds — that namical triangulation approach was the first to demonstrate the could be observable in present or future experiments. The idea is motivationally-similar to, but formally distinct from, the cascading spectral dimension decreases to ds = 2 as the energy scale in- creases [47]. This effect is replicated in a noncommutative-inspired DGP scenarios previously discussed in the literature [53,54]. geometry [49], as well as through the anisotropic scaling factors A lower-dimensional Planckian arena for gravity is thus nat- in Lifshitz gravity [48]. In each model, the described mechanism ural and attractive. It is therefore important to fully understand the roles of the spectral and geometric dimensions as they re- late to gravitational phenomenology. Since the characteristics of E-mail address: [email protected]. spacetime are unknown at quantum scales, one is tempted to take 0370-2693 ©2012ElsevierB.V. Open access under CC BY license. http://dx.doi.org/10.1016/j.physletb.2012.08.029 172 J.R. Mureika / Physics Letters B 716 (2012) 171–175 advantage of this ambiguity and interpret the spectral dimension The solution to (4) is as the geometric dimension of the manifold. This presents sev- 1 eral intriguing questions: is the universe itself effectively lower- 2 =− − 2 + | |− 2 ds1 Λx 2G1 M x C dt dimensional at high energies? If so, how does the transition from 2 one dimension to another affect the dominant physics, and ideally dx2 is it possible to observe evidence of such transitions? + (5) (− 1 Λx2 + 2G M|x|−C) Whatever the underlying framework, it will be assumed that 2 1 the quantum geometry is described semiclassically by an effective where R and T are the Ricci and energy–momentum scalars, G1 is + + (1 1)-D or (2 1)-D metric. In this Letter, the former case will be the one-dimensional gravitational constant, r1 ≡|xH | and C is an represented by a dimensionally-reduced limit of Einstein gravity, arbitrary constant of integration [16]. The black hole’s entropy and and the latter will employ the three-dimensional BTZ metric. Pri- Hawking temperature are respectively mordial black holes play a critical role in a range of early-universe processes, from baryogenesis [60–62] to large-scale structure for- 2 2 − + 2π G1 MBH Λ G1 MBH mation [63–66], and even potentially determining the entropy con- S1 = ln , (6) h¯ M tent of the universe [67]. Since PBHs are possible dark matter can- 0 didates [66,68–73], understanding their evolution and abundances ¯ h 2 CΛ T1 = M − . (7) in dimensionally-reduced spacetimes can shed new light on this 2π BH 2 dilemma of modern cosmology. Here, M0 is an arbitrary constant of integration with dimensions of mass. When the product CΛ is reasonably small, the temperature 2. Lower-dimensional black hole thermodynamics runs linearly with the mass: T1 ∼ MBH. In (d + 1) dimensions, the relation between the radiative power A number of models have addressed gravity in two-dimensional of a black hole of mass MBH and temperature Td is described by spacetimes, all of which require the additional presence of a cou- the generalized Stefan–Boltzmann law [75] pled scalar field (see e.g. [26] for a comprehensive review). It has been demonstrated that such general models will exhibit slightly dM d+1 P = =−σ A − T , (8) different temperatures depending in part on the nature of the dila- d d d 1 d dt d ton coupling. A generic dilaton gravitational theory in two dimen- ∼ d−1 ∼ d+1 sions can be derived from the action where Ad−1 rH is the horizon area and σd kBoltzmann.The √ black hole decay time is 1 2 −2φ 2 2(1−a−b)φ S = d x −ge R + 4a(∇φ) + Be (1) 0 2 dM τd = . (9) where the coefficients a, b, and B depend on the model in question Pd (see [26] for details). For minimally-coupled fields, one finds the MBH Hawking temperature to be It is well known that d = 3 black holes have a Hawking tempera- − − ture T ∼ M 1 and emit radiation as P ∼ M 2. a − 1 3 BH 3 BH ∼ α = There is a pathological issue in (1 + 1)-D that hinders the T (α) MBH, α (2) a calculation of (8). The radiative power is a function of the hori- so a variety of possible temperature profiles are possible depend- zon area, which in this case is ill-defined. It has recently been ing on the value of α. shown this problem may be circumvented by re-interpreting the = For the purposes of the present discussion, however, the theory d-dimensional area in terms of holographic information bits Ad of choice is one whose action is [19] NbitsGd+1, where Nbits is an intrinsic bit-count on the horizon [55]. In the case of a two-dimensional black hole, the horizon √ 2 1 2 consists of antipodes and the bit-count is constant. The energy S2 = d x −g ψ R − (∇ψ) + Lm − 2Λ , (3) + 2 radiated from a generic (1 1)-D black hole with temperature 2α (2) is thus P1(α) ∼ M , which in the case considered herein which can be derived as a dimensionally-reduced form of D- P ∼ 2 is 1 σ1G1 M1.
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