The Maximal Amount of Gravitational Waves in the Curvaton Scenario
aa r X i v : . [ a s t r o - ph ] S e p The Maximal Amount of Gravitational Waves in the Curvaton Scenario
N. Bartolo , S. Matarrese , A. Riotto , and A. V¨aihk¨onen (1) Department of Physics and INFN Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy and(2) D´epartement de Physique Th´eorique, Universit´e de Gen`eve 4, 24 Quai Ansermet, Gen`eve 1211, Switzerland (Dated: October 31, 2018)The curvaton scenario for the generation of the cosmological curvature perturbation on large scalesrepresents an alternative to the standard slow-roll scenario of inflation in which the observed densityperturbations are due to fluctuations of the inflaton field itself. Its basic assumption is that the initialcurvature perturbation due to the inflaton field is negligible. This is attained by lowering the energyscale of inflation, thereby highly suppressing the amount of gravitational waves produced duringinflation. We compute the power-spectrum of the gravitational waves generated at second order inperturbation theory by the curvaton (isocurvature) perturbations between the end of inflation andthe curvaton decay. An interesting property of this contribution to the tensor perturbations is thatit is directly proportional to the amount of non-Gaussianity predicted within the curvaton scenario.We show that the spectrum of gravitational waves may be in the range of future gravitational wavedetectors. PACS numbers: 98.80.cq DFPD 07/A/08
Inflation [1, 2] has become the dominant paradigm forunderstanding the initial conditions for structure for-mation and for Cosmic Microwave Background (CMB)anisotropy. In the inflationary picture, primordial den-sity and gravity-wave fluctuations are created from quan-tum fluctuations “redshifted” out of the horizon duringan early period of superluminal expansion of the universe,where they are “frozen” [3, 4]. These perturbations at thesurface of last scattering are observable as temperatureanisotropy in the CMB. The last and most impressiveconfirmation of the inflationary paradigm has been re-cently provided by the data of the Wilkinson MicrowaveAnisotropy Probe (WMAP) mission which has markedthe beginning of the precision era of CMB measurementsin space [5].Despite the simplicity of the inflationary paradigm,the mechanism by which cosmological adiabatic pertur-bations are generated is not yet established. In the stan-dard slow-roll scenario associated to single-field modelsof inflation, the observed density perturbations are dueto fluctuations of the inflaton field itself when it slowlyrolls down along its potential. When inflation ends, theinflaton φ oscillates about the minimum of its potential V ( φ ) and decays, thereby reheating the universe. As aresult of the fluctuations each region of the universe goesthrough the same history but at slightly different times.The final temperature anisotropies are caused by the factthat inflation lasts different amounts of time in differentregions of the universe leading to adiabatic perturbations.An alternative to the standard scenario is representedby the curvaton mechanism [6, 7] where the final curva-ture perturbations are produced from an initial isocurva-ture perturbation associated to the quantum fluctuationsof a light scalar field (other than the inflaton), the curva-ton, whose energy density is negligible during inflation.The curvaton isocurvature perturbations are transformedinto adiabatic ones when the curvaton decays into radi-ation much after the end of inflation. Contrary to the standard picture, the curvaton mech-anism exploits the fact that the total curvature pertur-bation (on uniform density hypersurfaces) ζ can changeon arbitrarily large scales due to a non-adiabatic pres-sure perturbation which may be present in a multi-fluidsystem. While the entropy perturbations evolve inde-pendently of the curvature perturbation on large scales,the evolution of the large-scale curvature is sourced byentropy perturbations.During inflation, the curvaton energy density is negli-gible and isocurvature perturbations with a flat spectrumare produced in the curvaton field σ , δσ k = (cid:18) H ∗ π (cid:19) , (1)where H ∗ is the value of the Hubble rate during infla-tion. After the end of inflation, the curvaton field oscil-lates during some radiation-dominated era, causing theratio between its energy density and the radiation energydensity to grow, thereby converting the initial isocurva-ture into curvature perturbation. The energy density ρ σ will then be proportional to the square of the oscillationamplitude, and will scale like the inverse of the locally-defined comoving volume corresponding to matter dom-ination. On the spatially flat slicing, corresponding touniform local expansion, its perturbation has a constantvalue δρ σ /ρ σ ≃ ( δσ k /σ ∗ ), where σ ∗ is the value of theclassical curvaton field during inflation.The curvature perturbation ζ is supposed to be negligi-ble when the curvaton starts to oscillate, growing duringsome radiation-dominated era when ρ σ /ρ ∝ a , where a is the scale factor. After the curvaton decays ζ becomesconstant. In the approximation that the curvaton decaysinstantly it is then given by ζ k ≃ r (cid:18) δσ k σ ∗ (cid:19) , (2)where r ≡ ( ρ σ /ρ ) D and the subscript D denotes theepoch of decay. The corresponding spectrum is [6] P ζ ≃ r (cid:18) H ∗ πσ ∗ (cid:19) . (3)It is nearly scale-invariant under the approximation thatthe curvaton field is effectively massless during inflation.The generation of gravity-wave fluctuations is anothergeneric prediction of an accelerated de Sitter expansionof the universe. Gravitational waves, whose possible ob-servation might come from the detection of the B -modeof polarization in the CMB anisotropy [8], may be viewedas ripples of spacetime around the background metric g µν = a ( τ )( dτ − ( δ ij + h ij ) dx i dx j ) , (4)where τ is the conformal time. The tensor h ij is tracelessand transverse and has two degrees polarizations, λ = ± . Since gravity-wave fluctuations are (nearly) frozen onsuperhorizon scales, a way of characterizing them is tocompute their spectrum on scales larger than the horizon.During a de-Sitter stage characterized by the Hubble rate H ∗ , the power-spectrum of gravity-wave modes generatedduring inflation is P h ( k ) = k π X λ = ± | h k | = 8 M p (cid:18) H ∗ π (cid:19) , (5)where M p = (8 πG N ) − / ≃ . × GeV is the Planckscale. Detection of the B -mode of polarization in theCMB anisotropy requires H ∗ ∼ > GeV [9].What about the expected amplitude of gravity-wavefluctuations in the curvaton scenario?The curvaton scenario liberates the inflaton from theresponsibility of generating the cosmological curvatureperturbation. Its basic assumption is therefore that theinitial curvature perturbation due to the inflaton fieldis fully negligible. In the standard slow-roll inflationarymodels where the fluctuations of the inflaton field φ areresponsible for the curvature perturbations, the power-spectrum of the curvature perturbation is given by P ζ ( k ) = 12 M p ǫ (cid:18) H ∗ π (cid:19) (cid:18) kaH ∗ (cid:19) n ζ − , (6)where n ζ ≃ ǫ = ( ˙ φ / M p H ∗ )is the standard slow-roll parameter. Requiring that thecontribution (6) is much smaller than the value requiredto match the CMB anisotropy imposes H ∗ ≪ − M p .This implies that the curvaton scenario predicts the am-plitude of gravitational waves generated during inflation(5) far too small to be detectable by future satellite ex-periments aimed at observing the B -mode of the CMBpolarization (see however [10]). This is not the full story though. Gravitational wavesare inevitably generated at second order in perturba-tion theory by the curvature perturbations [11, 12, 13].This scalar-induced contribution can be computed di-rectly from the observed density perturbations and gen-eral relativity and is, in this sense, independent of the cos-mological model for generating the perturbations. Thegeneration of course takes place after the curvature per-turbation is generated.In the standard scenario, where the curvature pertur-bation is produced during inflation, the production oftensor modes occurs after inflation when the curvatureperturbations re-enter the horizon.In the curvaton scenario, the production of tensormodes through the curvature perturbations may occuronly after the curvaton decays, i.e. after the isocurva-ture perturbations get converted into curvature fluctu-ations. The energy density of gravitational waves (perlogarithmic interval) is given byΩ GW ( k, τ ) = k H ( τ ) P h ( k, τ ) (7)and the one generated by the curvature perturbationsresults to be of order Ω GW ≃ − , for those modes thatre-entered the horizon when the universe was radiationdominated [12, 13].What we will be concerned about in this paper is thegeneration of tensor modes by the curvaton perturbationsbetween the end of inflation and the time of curvatondecay. In other words, we are interested in the tensormodes generated at second order when the perturbationsare still of the isocurvature nature. This contributionmay be larger than the one created by the second-ordercurvature perturbations after the curvaton decay. In thissense, the spectrum of tensor modes computed in thispaper corresponds to the maximal possible amount ofgravity waves within the curvaton scenario.An interesting aspect is that the contribution to thetensor perturbations turns out to be directly proportionalto the possibly large amount of Non-Gaussianity (NG) inthe CMB anisotropies which is predicted within the cur-vaton scenario [14, 15]. NG is usually parametrized interms of the the nonlinear parameter f NL and the lat-ter is predicted to be of the order of 1 /r in the curva-ton scenario; present-day data limit | f NL | to be smallerthan about 10 , that is r ∼ > − [5, 16]. Therefore, thepresent observational bound on the level of NG in theCMB can already put an upper bound on the amountof tensor modes induced by the curvaton perturbations.This relic gravitational radiation may be particularly rel-evant in view of the realization that space-based laser in-terferometers, such as the Big Bang Observer (BBO) andthe Deci-hertz Interferometer Gravitational wave Obser-vatory (DECIGO), operating in the frequency range be-tween ∼ h ′′ ij + 2 H h ′ ij − ∇ h ij = − κ T lmij ∂ l δσ∂ m δσ, (8)where H = a ′ /a is the Hubble rate, the prime stands fordifferentiation with respect to the conformal time and κ = 8 πG N . If we define the Fourier transform of thetensor perturbations as follows h ij ( x , τ ) = X λ = ± Z d k (2 π ) / e i k · x h λ k ( τ ) e λij ( k ) , (9)where the polarization tensors e + ij ( k ) = 1 √ e i ( k ) e j ( k ) + e i ( k ) e j ( k )) ,e − ij ( k ) = 1 √ e i ( k ) e j ( k ) − e i ( k ) e j ( k )) (10)are expressed in terms of orthonormal basis vectors e and e orthogonal to k , the projector tensor in Eq. (8) reads T lmij = X λ = ± Z d k (2 π ) / e i k · x e λij ( k ) e λ lm ( k ) . (11)In Fourier space, the equation of motion for the gravi-tational wave amplitude (for each polarization) then be-comes h ′′ k + 2 H h ′ k + k h k = S ( k , τ ) , S ( k , τ ) = 4 κ Z d p (2 π ) / e + lm ( k ) p l p m δσ p ( τ ) δσ k − p ( τ ) . (12)The solution to this equation can be easily found to be h k ( τ ) = 1 a ( τ ) Z τ dτ ′ g k ( τ ′ , τ ) a ( τ ′ ) S ( k , τ ′ ) , (13)where g k ( τ ′ , τ ) is the appropriate Green function eitherfor a radiation- or a matter-dominated period.We split the perturbations of the curvaton field into atransfer function piece T σ ( k, τ ) and the primordial fluc-tuation δσ k , δσ k ( τ ) = T σ ( k, τ ) δσ k (14)with the primordial power-spectrum defined by h δσ k δσ q i = 2 π k δ ( k + q ) P δσ ( k ) . (15)The power-spectrum of the second-order gravitationalwaves becomes P h ( k, τ ) =16 κ Z ∞ dp Z − d cos θ P δσ ( p ) P δσ ( | k − p | ) sin θa ( τ ) × k p | k − p | (cid:12)(cid:12)(cid:12)(cid:12)Z τ dτ ′ a ( τ ′ ) g k ( τ ′ , τ ) T σ ( p, τ ′ ) T σ ( | k − p | , τ ′ ) (cid:12)(cid:12)(cid:12)(cid:12) , (16) where cos θ = ˆ k · ˆ p . The second-order tensor modes aregenerated when the various modes k enter the horizon.Meanwhile, the production ends when the curvaton de-cays. We will assume in the following that the wholegeneration of tensor modes takes place in the radiation-dominated epoch. This assumption is motivated by re-quiring that the NG induced by the curvaton is sizeable,which requires the curvaton energy density not to dom-inate by the time of decay. Since the Hubble rate isgiven by H = 1 /τ , a given mode k enters the horizon at τ k = 1 /k . Indicating by k D the mode which enters thehorizon at the time of the curvaton decay, k D = a ( τ D )Γ,where Γ is the decay rate of the curvaton, we may writethe evolution of the scale factor as a ( τ ) = (cid:18) k D Γ (cid:19) τ. (17)Trading the curvaton decay rate with the temperature atdecay T D , we obtain k D ≃ − (cid:18) T D GeV (cid:19) Hz . (18)After the end of inflation, the zero mode σ of the curvatonfield starts oscillating at τ m ≡ (1 /k D )(Γ /m ) / , where m is the curvaton mass. Let us first consider those per-turbations which enter the horizon when the zero mode σ of the curvaton decay is already oscillating, that is k ∼ < ( m/ Γ) / k D . In this range of wavenumbers, one canshow that the curvaton perturbations scale as the zeromode, δσ k ( τ ) ∼ σ ∼ a − / . This allows to write δσ k ( τ ) = (cid:18) δσ k σ ∗ (cid:19) (cid:18) k D τ (cid:19) / σ D ≃ ζ k r (cid:18) k D τ (cid:19) / σ D ,T σ ( k, τ ) = (cid:18) σ D σ ∗ (cid:19) (cid:18) k D τ (cid:19) / . (19)where σ D is the value of the curvaton zero mode atthe time of decay. If we introduce the variables x = | k − p | /k and y = p/k and uses the radiation-dominatedGreen function g k ( τ ′ , τ ) = sin( k ( τ − τ ′ )) /k , it is easy torealize that the main tensor mode production happens athorizon entry, that is at τ ≃ τ k . Therefore, the power-spectrum (16) computed at horizon entry is P h ( k, τ k ) ≃ f (cid:18) kk D (cid:19) (cid:18) Γ m (cid:19) Z ∞ dy Z y | − y | dx × y x (cid:18) − (1 + y − x ) y (cid:19) P ζ ( kx ) P ζ ( ky ) , (20)where we have made use of the relations r =( κ / )( m σ D ) and f NL ∼ /r . In the curvaton sce-nario the resulting curvature perturbation is nearly scale-invariant and we can take P ζ ∼ (5 × − ) . The remain-ing integrals in (20) are dominated by the momenta forwhich x ∼ y ∼ ( m/ Γ) / ( k D /k ). We finally obtain anenergy density today of gravitational waves given byΩ GW ≃ − (cid:18) f NL (cid:19) (cid:18) kk D (cid:19) (cid:18) Γ m (cid:19) / , (21)valid for k D ∼ < k ∼ < ( m/ Γ) / k D .For those perturbations which enter the horizon beforethe zero mode of the curvaton field starts oscillating, thatis k ∼ > ( m/ Γ) / k D , the scaling is is δσ k ∼ a − for τ k ∼ <τ ∼ < kτ m , that is till k ∼ > ma . For kτ m ∼ < τ ∼ < τ D thescaling is is δσ k ∼ a − / . Meanwhile, the zero mode σ remains frozen till the mass of the curvaton becomeslarger than the Hubble rate at τ ∼ τ m . Repeating thesteps leading to Eq. (20), we find at horizon entry P h ( k, τ k ) ≃ f (cid:18) Γ m (cid:19) Z ∞ dy Z y | − y | dx × x (cid:18) − (1 + y − x ) y (cid:19) P ζ ( kx ) P ζ ( ky ) , (22)which corresponds to an energy density today ofΩ GW ≃ − (cid:18) f NL (cid:19) (cid:18) Γ m (cid:19) , (23)valid for k ∼ > ( m/ Γ) / k D . Finally, let us note thatthe redshifted gravitational wave at later times is al-ways larger than S /k . Therefore the power-spectrumof gravity waves produced by the curvaton fluctuationsis always bigger than the one generated by the second-order curvature perturbations which is inevitably gener-ated when the cosmological perturbations acquire theiradiabatic nature.There is a simple physical motivation for the fact thatthe amount of gravity waves generated by the curvatondecay is enhanced by powers of 1 /r with respect to the one produced by ordinary second-order curvature pertur-bations. Indeed, being the final adiabatic perturbationsgenerated by the curvaton isocurvature perturbations,the smaller the amount of curvaton energy density atdecay is, the larger the curvaton fluctuations have to be: δσ k ∝ ζ k /r , see Eq. (2). The isocurvature perturbationsgiving rise to gravity waves are therefore parametricallylarger at horizon entry than the second-order curvatureperturbations.From our findings we deduce that the amount of grav-itational waves in the perturbative regime Γ ∼ < m can beas large as Ω GW ≃ − maximazing the observation-ally allowed NG in the CMB anisotropies. This is quiteintriguing since such a spectrum is at the range of, e.g. ,BBO and DECIGO interferometers. To be in the rightfrequency range, between 10 − and 1 Hz, one has to im-pose that the curvaton decays at T D ∼ < GeV. Further-more, the correlated BBO interferometer proposal claimsa sensitivity down to Ω GW ≃ − . This would requireΓ ∼ > − m in units of (cid:0) /f NL (cid:1) . As a final remark,we point out that the expressions (21) and (23) are alsoapplicable to the so-called modulated reheating scenarioin which the curvature perturbations are due to the fluc-tuations of some light field parametrizing the inflatondecay rate [18]. 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