We study in a Brill-Hartle type of approximation the back-reaction of a
superposition of linear gravitational waves in an Einstein-de Sitter background
up to the second order in the small wave amplitudes $h_{ik}$. The wave
amplitudes are assumed to form a homogeneous and isotropic stochastic process.
No restriction for the wavelengths is assumed. The effective stress-energy
tensor $T^{e}_{\mu\nu}$ is calculated in terms of the correlation functions of
the process. We discuss in particular a situation where $T^{e}_{\mu\nu}$ is the
dominant excitation of the background metric. Apart from the Tolman radiation
universe, a solution with the scale factor of the de Sitter universe exists
with $p = -\rho$ as effective equation of state.Comment: 3 pages, Latex (requires mprocl.sty). To appear in the Proceedings of
  the Eighth Marcel Grossmann meeting (Jerusalem, June 22-27, 1977

Dautcourt, G.

English

arXiv

We study in a Brill-Hartle type of approximation the back-reaction of a superposition of linear gravitational waves in an Einstein-de Sitter background up to the second order in the small wave amplitudes $h_{ik}$. The wave amplitudes are assumed to form a homogeneous and isotropic stochastic process. No restriction for the wavelengths is assumed. The effective stress-energy tensor $T^{e}_{\mu\nu}$ is calculated in terms of the correlation functions of the process. We discuss in particular a situation where $T^{e}_{\mu\nu}$ is the dominant excitation of the background metric. Apart from the Tolman radiation universe, a solution with the scale factor of the de Sitter universe exists with $p = -\rho$ as effective equation of state

MPG.PuRe

Self-consistent solutions for low-frequency gravitational background radiation

arXiv:gr-qc/9712042v1  9 Dec 1997SELF-CONSISTENT SOLUTIONS FOR LOW-FREQUENCYGRAVITATIONAL BACKGROUND RADIATIONG. DAUTCOURTMax-Planck-Institut fu¨r Gravitationsphysik (Albert-Einstein-Institut)Schlaatzweg 1, 14473 Potsdam, GermanyWe study in a Brill-Hartle type of approximation the back-reaction of a superposition oflinear gravitational waves in an Einstein-de Sitter background up to the second order inthe small wave amplitudes hik. The wave amplitudes are assumed to form a homogeneousand isotropic stochastic process. No restriction for the wavelengths is assumed. Theeffective stress-energy tensor T eµν is calculated in terms of the correlation functions ofthe process. We discuss in particular a situation where T eµν is the dominant excitation ofthe background metric. Apart from the Tolman radiation universe, a solution with thescale factor of the de Sitter universe exists with p = −ρ as effective equation of state.While the description of cosmic gravitational radiation in linearized relativityis fairly well known, a study of its nonlinear aspects is much harder. Numericalrelativity3 is one way, but there are also some analytical and semianalytical ap-proaches. Already in 1964 Brill and Hartle4 proposed a scheme which takes theback-reaction of linear gravitational waves into account. As it stands, the Brill-Hartle method is considered as a high-frequency approximation for gravitationalradiation. Thus its application to early inflationary stages of the Universe is ques-tionable, since low-frequency modes may be present, which turn into high-frequencymodes (with wavelengths smaller than the temporary horizon) only at later time.As shortly discussed in this note, a slight modification of the Brill-Hartle approachcan remove this shortcoming. The detailed calculations will be published elsewhere.The wave perturbations are considered as random variables, forming a stochasticprocess, which shares the symmetries of the background metric. The method is sim-ilar to the Monin-Yaglom7 approach to statistical fluid dynamics, and is also usedin optical coherence theory6. For simplicity, an Einstein-de Sitter model is cho-sen as background metric. Tensor perturbations are added in synchronous gauge,assuming g00 = −a2, g0i = 0, gik = a2δik + hik, as well as the gauge conditionshii = 0, hik,k = 0. a(η) is the scale factor. The Einstein tensor may be expandedin powers of hik. Retaining terms up to second order and performing a stochastic(ensemble) average on the field equations, they split into linear wave equations forthe hik and the back-reaction equations. The back-reaction equations relate theEinstein tensor for the background metric to the effective stress-energy tensor ofthe waves, which is represented as stochastic average over bilinear expressions inhik and derivatives of hik. As well known, the effective stress-energy tensor is notgauge invariant in general . However, as shown by Abramo, Brandenberger andMukhanov1 (see also 2, 5), the gauges change the background geometry to secondorder, and these changes just compensate the change of the stress-energy tensor.Representing the hik as stochastic Fourier integrals∫γik(k, η)eikxdk+conj.compl.,the amplitudes γik satisfy an ordinary differential equation. The symmetry proper-ties of the problem allow a simple representation of expressions which are bilinear1in hik, e.g.〈hikhlm〉 = 16πa215(3δilδkm + 3δimδkl − 2δikδlm)∫dkk2f, (1)in terms of a single spectral density f(k, η). f satisfies a nonlinear differentialequation (ǫ0 depends only on k = |k|, and a prime denotes the differentiation withrespect to η)2ff ′′ − f ′2 + 4f2(k2 − a′′a)− 4ǫ0 = 0. (2)For high-frequency waves kη ≫ 1, f = √ǫ0/k follows as solution (”high-frequencyapproximation”). It is convenient to write ρ and p in the effective stress-energy ten-sor in terms of four frequency-independent, but in general time-dependent integrals(”moments”) over the spectral density f(k, η):f0 =∫dk k2ǫ0(k)f, f1 =∫dk k2f ′2f, f2 =∫dk k2f, f4 =∫dk k4f. (3)For a general scale factor one obtains (g = f0 + f1/4)ρg =12Ga4(f4 + g + 3a′af ′2− 7a′2a2f2), (4)3pg =12Ga4(7f4 − 5g + 5a′af ′2− 5a′2a2f2). (5)In the high-frequency approximation 3Ga4pg = Ga4ρg =∫dkk3√ǫ0, energy densityand pressure are positive. If low-frequency modes are present, their contribution canbe negative, and also the equation of state can deviate considerably from the high-frequency relation p = ρ/3. If pressure and density of gravitational waves cannot beneglected compared to other forms of matter, the back-reaction on the scale factormust be taken into account. Taking a pure gravitational radiation universe, one hasto solve the field equations6a′2a4= 16πGρg, (6)a′′a+a′2a2= 4πGa2(ρg − pg), (7)with ρg, pg taken from (4) and (5). Note the further equationsg′ = −f ′4+a′′af ′2, (8)f ′′2= 2g + 2a′′af2 − 2f4, (9)which follow from differentiating f1, f2 and using the differential equation for f .The four equations (6)-(9) give the relation(2a′f2 − af ′2) (aa′′ − 2a′2) = 0. (10)2Vanishing of the first factor leads to the Tolman radiation universe, vanishing ofthe second factor gives the scale factor of the de Sitter cosmos. It is easy to find themoments f2, f4, g from the corresponding differential equations. Self-consistencyhowever requires, that the moments found in this way must be compatible withthe expressions following directly from (3), if the solution of the wave equationis inserted. Compatibility can be achieved indeed, it however requires singularinfrared components for some spectral quantities. The general solution of (2) forthe radiation (s = 0, a ∼ η) and de Sitter (s = 1, a ∼ 1/η) cosmos is known to bef = 2npp∗ + (l + im)p2 + (l − im)p∗2, (11)where l,m, n are three functions of k, connected with ǫ0 by ǫ0 = 4k2(n2 − l2−m2),p(x) = (1+is/x)eix is a complex function of x = kη. Whereas n(k) is not restricted,the spectral functions l(k) andm(k) should be understood as generalized functions8and have the form (b is a constant)al(k) = − b2δ′′(k)k2− 2n2 δ(k)k2, (12)m(k) = 0, (13)where δ(k) is the Dirac delta function and n2 =∫dkk2n(k). The spectral densitiesfor the energy density and the pressure are then (again in the Tolman case)a4Gρ(k, η) = n(k)(k2 − 72η2) + δ(k)(7n2η2− b), (14)3a4Gp(k, η) = n(k)(k2 − 52η2) + δ(k)(5n2η2− b). (15)These quantities show infrared singularities, part of the effective energy densityand pressure resides in infrared (k = 0) modes. It is so far not clear whether thesingularities result from using only a second-order approximation. The integratedquantities are finite, however.1. L.R.W. Abramo, R.H. Brandenberger and V.F. Mukhanov, Phys. Rev. D 56,3248 (1997), Phys. Rev. Lett. 78, 1624 (1997), and in: Proc. 18th TexasSymposium on Relativistic Astrophysics, Chicago, 1996, to appear.2. P. R. Anderson and D. R. Brill, Phys. Rev. D 56, 4824 (1997).3. P. Anninos, J. Masso, E. Seidel, W.-M. Suen, and M. Tobias, Phys. Rev. D56, 842 (1997).4. R. D. Brill and J. B. Hartle, Phys. Rev. 135, B271 (1964).5. M. Bruni, S. Matarrese, S. Mollerach and S.Sonego, Class. Quant. Grav. 14,2585 (1997).6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (CambridgeUniversity Press, Cambridge UK, 1995).7. A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Tur-bulence, Vol. 2 (The MIT Press, Cambridge, Massachusetts, 1975).8. I. N. Sneddon, The Use of Integral Transforms (McGraw-Hill Book Company,New York, 1972).afor the Tolman cosmos, the condition is different for the de Sitter case.3

Self-consistent solutions for low-frequency gravitational background radiation

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