20 Acta Polytechnica CTU Proceedings 1(1): 20–26, 2014 20 doi: 10.14311/APP.2014.01.0020 Gravitational Waves and Dark Energy Peter L. Biermann1,2,3,4,5, Benjamin C. Harms1 1Department of Physics and Astronomy, The University of Alabama, Box 870324, Tuscaloosa, AL 35487-0324, USA 2MPI for Radioastronomy, Bonn, Germany 3Karlsruhe Institute of Technology (KIT) - Institut für Kernphysik, Germany 4Department of Physics, University of Alabama at Huntsville, AL, USA 5Department of Physics & Astronomy, University of Bonn, Germany Corresponding author: bharms@bama.ua.edu Abstract The idea that dark energy is gravitational waves may explain its strength and its time-evolution. A possible concept is that dark energy is the ensemble of coherent bursts (solitons) of gravitational waves originally produced when the first generation of super-massive black holes was formed. These solitons get their initial energy as well as keep up their energy density throughout the evolution of the universe by stimulating emission from a background, a process which we model by working out this energy transfer in a Boltzmann equation approach. New Planck data suggest that dark energy has increased in strength over cosmic time, supporting the concept here. The transit of these gravitational wave solitons may be detectable. Key tests include pulsar timing, clock jitter and the radio background. Keywords: cosmology - dark energy - black holes - gravitational waves. 1 Introduction Dark energy was originally detected as accelerated ex- pansion seen in the distance scale for supernovae of type Ia (Schmidt et al., 1998, Riess et al. 1999, Perlmutter et al. 1999; for a review see Frieman et al. 2008). Many suggestions have been made about what dark energy is, what its strength is, what its time evolution is, and what possible further observational results are. The idea that dark energy is gravitational waves may explain its strength and its time-evolution. One possible concept is that dark energy is the ensemble of coherent bursts (solitons) of gravitational waves orig- inally produced when the first generation of super- massive back holes was formed (Caramete & Biermann 2010); the energy density of such solitons would suf- fice within the uncertainties. These solitons get their initial energy as well as keep up their energy density throughout the evolution of the universe by stimulat- ing emission from a background (Biermann & Harms 2013). Our model of the background metric resem- bles the Randall-Sundrum ideas (1999a, b) but is time- dependent, and describes the energy flow from the background (strong-gravity) brane to our world (weak- gravity) brane. Planck data suggest that dark energy has increased in strength over cosmic time (Planck 2013 XVI), as predicted by our model. Gravitational waves were far below today’s dark energy at the epoch of early nucleosynthesis and of the formation of the microwave background ripples (as summarized in Ligo+Virgo-Coll. 2009), both much earlier than the likely formation epoch of the first generation of super-massive black holes. Our model is also consistent with early star formation (Biermann et al. 2014), as we argue be- low. The transit of the gravitational wave solitons pos- tulated here may be detectable. We discuss the pre- dictions briefly below and elsewhere. We focus on the Boltzmann equation approach, working out the energy transfer from the strong gravity background in stimu- lated emission. 1.1 Gravitational solitons from black hole mergers Inspired by Bekenstein’s (1973) considerations we posit: When the first generation of super-massive black holes was formed, each produced a coherent burst of soliton- like gravitational waves which combine to give a total energy of order ∼ 1 2 NBH,0 MBH c 2 (1 + z?) 3 . (1) In the following we also call this an ensemble of soli- ton waves, or shell fronts. NBH,0 is the original comov- ing density of super-massive black holes. Today super- massive black holes have a density of 10−1.7±0.4 Mpc−3 20 http://dx.doi.org/10.14311/APP.2014.01.0020 Gravitational Waves and Dark Energy above MBH = 3·106 M� (Caramete & Biermann 2010); assuming that they grow by merging, and allowing for statistical and systematic errors, an original comoving density of NBH,0 = 1 Mpc −3 seems possible. This co- moving density is the density black holes had at the beginning, so transposed to today without change in their numbers per comoving volume. The data sug- gest that there was a generation of first super-massive black holes with a mass between MBH ∼ 106 M� and MBH ∼ 107 M�. The original black hole mass may be ∼ 3 · 106 M� considering (i) the black hole mass function (Greene et al. 2006, Caramete & Biermann 2010), (ii) the instability of massive stars (Appenzeller & Fricke 1972a, b) in an agglomeration picture (Spitzer 1969, Sanders 1970), and (iii) the observed black hole in our Galactic Center (e.g. Eckart et al. 2005). The red- shift of creation z? may be large, as formation of massive stars may begin at redshift 80 (Biermann & Kusenko 2006). Redshifts z? from about 30 to 50 allow a quanti- tative interpretation of the data of dark energy. At the original density of black holes adopted here redshift 50 is consistent with the mass of MBH ∼ 3 ·106 M�, and redshift 30 would imply MBH ∼ 107 M�, in either case to make the estimate consistent with dark energy today. What is the motivation for considering gravitational waves? Bekenstein (1973) wrote about the entropy of the universe: “... we must regard black hole entropy as a genuine contribution to the entropy content of the universe”. However, entropy is also information, and information must have a carrier. A natural suggestion is that this carrier is gravitational waves, with an energy commensurate with the black hole scale. This sugges- tion is consistent with the fact that cosmological black holes are not in thermodynamic equilibrium, and there- fore the entropy associated with such black holes should be described by statistical mechanics as advocated in Harms and Leblanc (1992, 1993). This speculation im- mediately gives S kB = NGW,0 = 4π ( MBH mPl )2 , for zero spin , (2) where NGW,0 is the number of gravitons at the forma- tion of the black hole. For MBH = 3 · 106 M� this is NGW,0 ' 1090. MBH is the original mass of the black hole, mPl is the Planck mass, c is the speed of light, and GN is Newton’s constant of gravity. EGW is the average graviton energy given by EGW = 1 8π h̄c3 GNMBH = c2 8π m2Pl MBH . (3) This gives a graviton energy of EGW ' 10−30 erg for this black hole mass. The entire energy content then is NGW,0 EGW = 1 2 MBH c 2 . (4) We picture this as a coherent burst of gravitational waves, or a soliton wave, ejected at formation of the black hole. It is clear from the considerations above that we are not using the weak-field approximation. Multi- plying with the original density of super-massive black holes reproduces our estimate above. 1.2 Five-dimensional background model In our model for the background, which has some sim- ilarity to the Randall-Sundrum (1999a, b) ideas, we identify a possible local metric to describe a 5D world with a 4D strong gravity brane and our 4D world weak- gravity brane. ds2 = −e(u/l) m t/ψ c2 dt2 + e( u l ) p t/β du2 + e(1−b( u l ) n ) 2 t/αtH e−( u l ) k (1− tφ ) dxi dx i , (5) where i = 1, 2, 3, τH is the Hubble time, l = lPl is the Planck length, τPl = lPl/c is the Planck time, u is the coordinate in the fifth dimension, and the re- maining, non-coordinate quantities are arbitrary pa- rameters. Although the five-dimensional covariant di- vergence of the energy-momentum tensor does not van- ish everywhere, it vanishes on the weak-gravity (weak) brane (u = 0), and it is approximately zero on the strong-gravity (strong) brane (u = l) for very small dimensionless ratios β/τH and ψ/τH with ψ < 0 and β > 0, with the conditions a) τH(ψ + β)/(ψβ) >> 1, b) 2(b−1)/α > τH/φ−1, and c) α2 = 3. g00 in the metric above then defines the confining potential for the strong brane. Gravitons from our brane stimulate transitions between bound states on the strong brane, resulting in the emission of gravitons onto our brane. This metric describes a weak brane for our world which is expand- ing with time, and a strong brane which is contracting with time, albeit very slowly for the latter brane. The 5D-cosmological constant measured on the weak brane is Ωweak = − ( τPl τH )2 . Figure 1: The stimulated emission of gravitons in a shell In our model the gravitons on the strong-gravity brane obey a Planck-like distribution with Planck tem- perature. We adopt the point of view that the Planck scales are limits: nothing can go below Planck time 21 Peter L. Biermann, Benjamin C. Harms and Planck length, and no single particle can go be- yond Planck energy, in any frame. The strong brane is stable against collapse, since given a Planck spectrum for any wavelength λ the free-fall time scale τff is al- ways either equal or longer than the pressure wave time scale τs. Figure 2: The strong-gravity (Planck brane) and weak-gravity (Tev brane) branes τff = τPl ( λ lP l )3/2 ≥ τPl ( λ lP l ) = τs. 2 Stimulate Emission of Energy from the Strong-Gravity Brane In the following we use a particle-wave duality for gravi- tons at high energy, which thus associates a wavenum- ber ~k/h̄ and a corresponding length-scale λ to each spa- tial direction, and we assume localization is possible to about a wavelength. 2.1 Rate of energy transfer The distribution function N(k,t) is the distribution of occupied allowed states on our brane for gravitons with momenta k = |~k| and p = |~p| at time t on the shell and satisfies the equation( ∂ ∂t − Ṙ(t) R(t) k ∂ ∂k ) N(k,t) = 1 k ∫ d3 k′ (2π)3 2 k′ ∫ cd3 p (2π)3 2 E(p) ∫ cd3 p′ (2π)3 2 E(p′) ∫ d3 k′′ (2π)3 2 k′′ γ2 |M|2 (2 π)4δ4(K + Q−K′ −K′′ −Q′) (2 π)6 δ3(~k′ − ~k′′) δ3(~k − ~k′) [gb(p ′, t)(1 + N(k,t)) ((N(k′, t) + 1)(N(k′′, t) + 1) − 1) −N(k,t)gb(p,t)(1 + N(k′, t))(1 + N(k′′, t))] . (6) where K = (k,~k), γ = k3ref,1, and Q = (E,~p). kref,1 is a reference momentum to be determined below. The δ-functions, δ3(~k′ − ~k′′) and δ3(~k − ~k′), have been in- serted to impose coherence of the outgoing gravitons. |M|2 is the matrix element squared for the quadrupole emission of a graviton of 4-momentum k′′, and has the dimensions of (momentum)−3 (time)−1. We will as- sume that this matrix element squared is proportional to k5. gb(p,t) is the occupation number distribution of the background particle sea. R(t) = (1 + z?)/(1 + z) is the scale factor for an expanding universe. The follow- ing analysis is done in the observer frame. The Boltzmann equation for N(k,t) to lowest order in the expansion of the 4-dimensional δ-function is( ∂ ∂t − Ṙ(t) R(t) k ∂ ∂k ) N(k,t) ' +κ k N(k,t) (N(k,t) + 1) , (7) where the factor κ is given, after integration over all the δ-functions, by κ = k2ref,2 H(z) 24πkBH |M|2 ln{ kBH+ k } . (8) In the equation above kBH+ is the maximum momen- tum at which stimulated emission of gravitons occurs, just above the momentum of the peak of N(k,t). Since the log-term in eq.8 varies very slowly over the range of k of interest we approximate this term with a constant and set β = {ln(kBH+/k)}/(24 π). |M|2 is related to |M|2 by extracting the factors (k/kref,1)3/k3ref,1, H(z). A threshold function of 2 gbN/(gb + N) arises from the normalized interaction between the gravitons on the background brane and our brane; this function connects to the strong brane only if N > gb, which is the con- dition for stimulated emission. These choices do not introduce new constraints or additional assumptions. Next we redefine (k/kref,2) 2 κ = κ to extract the k-dependence from |M|2. Making the change of variables k = k̃/R(t), eq.7 can be written as ∂N(k̃, t) ∂t ' H(z) k̃ β kBH R(t) N(k̃, t) (N(k̃, t) + 1) . (9) In terms of the frequency of the wave at emission this equation is ∂N(ν0, t) ∂t ' H(z) hν0 β kBH R(t) c N(ν0, t)(N(ν0, t) + 1) . (10) Introducing the dimensionless variables x = hν0 kB Tg0 , and y = ∫ t 0 kBTg0 H(z)β kBH cR(t) dt′ , (11) where kBHc = kBTg0 = mPlc 2 mPl 8 π MBH . Eq.(10) be- comes ∂N ∂y ' +xN(x,y) (N(x,y) + 1) . (12) The solution of this equation is N(x,y) = 1 ex(a−y)+b − 1 , (13) 22 Gravitational Waves and Dark Energy where a and b are constants, to be determined later. This distribution (eq.13) is Planck-like with a time- dependent normalized temperature 1/a. The rate at which energy is created can be calculated from the ex- pression for N in eq.10. The rate at which energy is created can be calcu- lated from the expression for N in eq.(10). The rate of energy creation per existing graviton (of the total number NGW,0 R(t) 4) is d < E > dt = βH(z) R(t) × (14)∫ x3 hν0 N(ν0, t) (N(ν0, t) + 1) dx. The total rate of energy creation is d < ET > dt = NGW,0 R(t) 3 kBH cH(z) β A , (15) where A = ∫ x4 N(x,t) (N(x,t) + 1) dx (16) Inserting all these constants into the integral for y demonstrates that y approaches a constant for the red- shift z? being large, and integrating down to today or even into the future, when y approaches a constant of β << 1. a is equivalent to an inverse temperature, and should be of order unity. Without loss of generality we can set b = 0. This integral strongly depends on the exact value of a − y, and is of order 30 for a − y ' 1, and b approaching zero. The matrix element |M| does not evolve with time, and scales as momentum |M| = �M ( k mPlc ) . (17) Above we have used �M = 1; we now generalize and allow �M to be different from unity. Writing |M| in this way suggests that the interaction between the gravi- tons on our brane and the gravitons on the background brane comes down to a fundamental coupling constant. This behavior is consistent with the idea [22], that the gravitational coupling strongly increases with energy to approach the other three coupling constants at near Planck energies. This allows the expression for d dt to be consis- tent with the observed energy density under the condi- tion that Aβ �M = 3. For the constant a = 1 above, β of order 0.1, and �M = 1, the quantity {Aβ �M} is in fact close to 3. However, if we were to require that the k-range be very large, then β would be larger, and �M would be required to be smaller than unity accordingly. Inserting this parameter dependence into eq.(15) then leads back, to within the approximation that Aβ �M = 3, to the result we were seeking, 3 2 MBH c 2 H(z) ( 1 + z? 1 + z )3 . (18) After integrating we obtain with this redshift depen- dence a constant dark energy density as in eq.( 1) by multiplying by the redshift evolution of black holes NBH,0 (1 + z) 3. The factor of 4 multiplying ρDE in eq.22 below derives from the sum of dark energy den- sity and pressure, and corresponds to 3 (ρDE +PDE/c 2). Therefore the rate of change of dark energy with time is 3 (ρDE + PDE/c 2) H(t) and today 3ρDEH(z = 0) = 3 2 MBH c 2 H(z = 0) (1 + z?) 3 . (19) This justifies the ρDE H(t) term in eqs.22 and 25. This shows that dark energy remains at the level of eq.(1) throughout the evolution of the universe, in the approx- imation that most early super-massive black holes were formed over a short span of time. 2.2 Equation of state We define ρDE(t,u) as the dark energy density, ρ(t,u) as the total energy density, P(t,u) as the total pres- sure, and we use the equation of state PDE(t,u) = ρDE(t,u) c 2/3 . The dark energy density ρ(t,u) is scaled to the value of the dark energy density observed on our brane today ( redshift z = 0 ). The Friedmann-Robertson-Walker form of the Ein- stein equations on the weak-gravity brane must be (H(t)) 2 = ( ˙R(t) R(t) )2 = 8 π GN 3 ρ(t, 0) (20) and R̈(t) R(t) = − 4 π GN 3 ( ρ(t, 0) + 3 P(t, 0) c2 ) + 16π GN 3 ρDE(t, 0) + 4π GN Sinj 3 H(t) , (21) where the rate of change of the energy density used is ρ(t, 0) = − 3 ( ρ(t, 0) + P(t, 0) c2 ) H(t) + 4 ρDE(t, 0) H(t) + Sinj . (22) We emphasize that in the second equation, eq.21, the term 16 π GN ρDE(t, 0)/3 corresponds to the con- tinuous energy transfer by stimulated emission on the basis of existing dark energy; the additional term (4 π GN Sinj)/(3 H(t)) describes new formation of dark energy. This allows a different equation of state for ex- actly the same cosmological observations, as now for PDE = ρDEc 2/3 the modified equation 21 becomes identical to the canonical version of this equation for 23 Peter L. Biermann, Benjamin C. Harms PDE = −ρDEc2. The corresponding set of equations for the strong-brane are HSB(t) 2 = 8 π GN 3 (ρDE(t, lSB) − ΛSB) (23) where ΛSB is the cosmological constant on the strong brane at the beginning of the epoch of black hole for- mation and R̈SB(t) RSB(t) = − 16 π GN 3 ρDE(t, 0) H(t) HSB(t) (24) + 8 π GN 3 ( 3 ρDE(t, lSB) − ΛSB) − 4 π GN Sinj 3 HSB(t) , The rate of change of the energy density on the strong- brane is correspondingly ρ̇DE(t, lSB) = 3 ( ρDE(t, lSB) + 1 c2 PDE(t, lSB) ) HSB(t) −4 ρDE(t, 0)H(t) −Sinj . (25) Eq.24 is derived from eq.23 using the conservation of energy-momentum equations; eqs. 22 and 25 insure that energy is conserved between the two branes, the 4-dimensional boundaries of the 5-dimensional universe described by our model. For epochs before the energy transfer started ρDE(t, 0) = 0, and Sinj = 0, and we can set ρDE(t, lSB) = ΛSB for t < t(z = z?). 3 Comparison to Experimental Limits Our model can be tested by several different types of experiment. We discuss two of these below. 3.1 Pulsar timing experiments Figure 3: Gravitational wave background limit from pulsar timing (dashed line) , and our inferred gravi- tational wave background from stimulated emission of gravitational waves from the background Planck sea constituting dark energy (straight line). The ordinate is the fraction of closure density Ω per log bin of fre- quency, and the abscissa is the frequency of the gravi- tational waves f. The gravitational waves in our model arise from the production of black holes in the early universe at a redshift of z ' 50. The observed constant dark en- ergy density is maintained by the continuous production of gravitational waves by black-hole interactions with the Planck sea background. For an idealized model in which all black holes were created at the same time, and with the same mass, the gravitational wave background peaks near fGW,max ' 10−4.5 Hz M−1BH,6.5 (1 + 50)/(1 + z?). A soliton comes past a given point in space-time on the order of every 20 seconds. Wave forms are cre- ated by uncorrelated solitons passing by a given point. Ultra-precise timing experiments which test the steadi- ness of timing over time scales of order a few seconds to a few minutes would show these variations in the energy density if the precision is high enough. The precision to detect such a signal has to correspond to seconds in the expansion rate of the universe, which requires the precision to be of order 10−17.5, or a few 10−18. This precision is expected to be reached in the next genera- tion of clocks [16]. t ρ 1 Figure 4: The sequential passing of solitons (sharp peaks) can be approximated by a sinusoidal wave form. 4 Conclusions If the validity of our model is proven by experimental tests such as the pulsar timing experiments (Fig.3), the detection of time jitter (Fig.4), or the detection by an observatory such as LIGO or VIRGO of the passage of a shell front by a dedicated type of data analysis, the implications for cosmology are great. The basis of our model is that the source of dark energy is the creation of gravitational waves by the interaction of surfaces at crit- ical density, e.g. the Planck surfaces surrounding black holes, with a strong-gravity brane located a few Planck lengths from our weak-gravity brane. Our model is con- sistent with the big-bang theory after the first Planck time. However, in our model the universe starts from a Lemaitre-like ‘atom’ or ‘seed’ [5]. Our model also has implications for quantum grav- ity theory. Experimental evidence for the validity of 24 Gravitational Waves and Dark Energy our model would imply the existence of a strong-gravity brane and extra dimensions as well as a smallest dis- tance and impenetrable Planck surfaces rather than horizons. Although our model describes several observed cos- mological phenomena, it is largely heuristic. We are currently working on an exact solution of the five- dimensional space-time metric and a more formal math- ematical description of the stimulated emission ampli- tude for the creation of the solitons. Acknowledgement Discussions with Lou Clavelli greatly contributed to the development of the paper; discussions with Laurenţiu Caramete (Bucharest, Romania), Roberto Casadio (Bologna, Italy), Marco Cavaglia (Oxford, MS), Laszlo Gergely (Szeged, Hungary), Shaoqi Hou (Tuscaloosa, AL), Pankaj Joshi (Mumbai, India), Gopal-Krishna (Pune, India), Octavian Micu (Bucharest, Romania), Piero Nicolini (Frankfurt, Germany), Norma Sanchez (Paris, France), Joe Silk (Oxford, United Kingdom), Allen Stern (Tuscaloosa, AL), Dijan Stoikovic (Buffalo, NY), and Hector de Vega (Paris, France) are gratefully acknowledged. Helpful comments on earlier versions of the manuscript were received from R. Casadio, M. Cavaglia, L. Gergely, A. Graham, P. Joshi, O. Micu, P. Nicolini, and D. Stoikovic. This research was supported in part by the DOE under grant DE-FG02-10ER41714. 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AJ 116, 1009 (1998) [19] Sanders, R.H., ApJ 162, 791 (1970) [20] Schmidt, B.P., et al., ApJ 507, 46 (1998) [21] Spitzer, L., Jr., ApJL 158, L139 (1969) doi:10.1086/180451 [22] Dimopoulos, S, Raby, S.A., Wilczek, F. Phys. To- day Oct., p. 25 (1991) DISCUSSION MOSHE ELITZUR: Are there predictions for the ef- fect of your model on the imprint of fluctuations on CMB and baryon acoustic oscillations? BENJAMIN HARMS : Dark energy in our mode is due to the merging of black holes at a ' 50, so well after the formation of the CMB. There may be some effect on the propagation of the CMB photons, but we have not yet worked out the exact nature of this effect. 25 http://dx.doi.org/10.1103/PhysRevD.7.2333 http://dx.doi.org/10.1103/PhysRev.82.863 http://dx.doi.org/10.1103/PhysRevLett.96.091301 http://dx.doi.org/10.1093/mnras/stu541 http://dx.doi.org/10.1146/annurev.astro.46.060407.145243 http://dx.doi.org/10.1016/j.newar.2006.06.080 http://dx.doi.org/10.1038/nature08278 http://dx.doi.org/10.1086/180451 Peter L. Biermann, Benjamin C. Harms JIM BEALL: Can you comment on the effect these ’seed’ black holes have on galaxy formation? BENJAMIN HARMS: The main difference between our model and the ’Big Bang’ theory is that our model allows for large assemblies of stars which have never merged and do not have an AGN, which have appar- ently been observed. 26 Introduction Gravitational solitons from black hole mergers Five-dimensional background model Stimulate Emission of Energy from the Strong-Gravity Brane Rate of energy transfer Equation of state Comparison to Experimental Limits Pulsar timing experiments Conclusions