Analytic formula for the dynamics around inflation end and implications on primordial gravitational waves
KKOBE-COSMO-20-14
Analytic formula for the dynamics around inflation end and implications onprimordial gravitational waves
Asuka Ito, ∗ Jiro Soda, † and Masahide Yamaguchi ‡ Department of Physics, Kobe University, Kobe 657-8501, Japan Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
We argue that primordial gravitational waves have a spectral break and its information is quiteuseful for exploring the early universe. Indeed, such a spectral break can be a fingerprint of the endof inflation, and the amplitude and the frequency at the break can tell us the energy scale of inflationand the reheating temperature simultaneously. In order to investigate the spectral break, we give ananalytic formula for evolution of the Hubble parameter around the end of inflation where the slowroll approximation breaks down. We also evaluate the spectrum of primordial gravitational wavesaround the break point semi-analytically using the analytic formula for the inflation dynamics.
I. INTRODUCTION
Inflation, the accelerated expansion in the early uni-verse, generates primordial density and tensor pertur-bations as well as solving the flatness and the horizonproblems [1–3]. The experiments of cosmic microwavebackground (CMB) anisotropies have directly observedprimordial density perturbations and their amplitudeswere shown to be ∼ − on large scales [4, 5], whichyield important information on the dynamics of infla-tion, that is, δρ/ρ ∼ H / ˙ φ [6–9]. Unfortunately, thisquantity depends on the velocity of an inflaton as well ason the Hubble parameter during inflation, which deter-mines the energy scale of inflation. Thus, the detectionof primordial tensor perturbations, whose amplitudes aresolely given by the Hubble parameter [10, 11], is press-ing. The primordial tensor perturbations (gravitationalwaves) can be probed in two ways. One way is to observethem through the B-mode polarizations of the CMB indi-rectly. The other way is to directly detect them through,for example, interferometer experiments. In fact, recentobservations of the gravitational waves sourced by bi-nary blackholes through the LIGO/VIRGO collabora-tions open gravitational wave astronomy [12]. There areseveral ongoing and planned experiments to directly de-tect the primordial gravitational waves.The primordial gravitational waves produced duringinflation have fruitful information on the dynamics ofinflation. Since they have almost scale invariant spec-trum on horizon exit in general, their amplitudes give usthe energy scale of inflation. While they are frozen andtheir amplitudes are kept on superhorizon scales, they be-have as damped oscillations on subhorizon scales. Thus,the (would-be) observed spectrum of primordial gravi-tational waves depend on cosmic history much. That is,even if they were exactly scale invariant when they exitedthe horizon, the (would-be) observed one can be scale-dependent. By use of this kind of scale dependence, one ∗ [email protected] † [email protected] ‡ [email protected] can probe the change of the number of relativistic de-grees of freedom [13, 14], the reheating temperature ofthe universe [15–17], for example.An important feature in the spectrum of primordialgravitational waves is a break due to the end of inflation.Indeed, we will show that the frequency and the ampli-tude at the break tells us the energy scale of inflationand the reheating temperature simultaneously. So far,very rough analytic estimates or detailed numerical cal-culations have been done to clarify the spectral shape ofthe (would-be) observed primordial gravitational wavesaround the break point. However, in order to properlyincorporate the history of reheating in the analysis of thespectrum, it is desired to have a more accurate analyticformula. For example, Ema et al. recently pointed outthat, even after inflation end, primordial gravitationalwaves might be produced as a result of inflaton annihila-tion into graviton pairs [18]. Thus, in order to smoothlyconnect the spectral shape at the frequency which ex-its the horizon just at the end of inflation, we need thedetailed information around this frequency. For this pur-pose, we will derive an analytic formula to approximatethe dynamics of inflation around its end, which also yieldsa semi-analytic formula of the spectral of the primordialgravitational waves around the frequency which exits thehorizon at the end of inflation. It should be noted that,typically, the frequency of the spectral break is around10 Hz (see Eq. (7)). In this frequency range, new gravi-tational wave detectors are being proposed and developedintensively [19–23]. Therefore, our study is also impor-tant for the future gravitational wave experiments.The paper is organized as follows. In Sec. II, after giv-ing the basic expression of the amplitude of primordialgravitational waves generated during inflation, we ex-plain how to determine the Hubble parameter and thereheating temperature from the information of primor-dial gravitational waves on the break scale. In Sec. III,an analytic formula for evolution of the Hubble param-eter around the end of inflation is derived. In Sec. IV,we first discuss the spectral index of the spectrum at theend of inflation. An implication to reconstruct the infla-ton potential from the observation of the spectral indexwith our formula is given. We also semi-analytically give a r X i v : . [ a s t r o - ph . C O ] S e p an example of the spectrum around the break point atpresent. Final section is devoted to conclusion. II. THE SPECTRAL BREAK
Inflation predicts a nearly scale invariant spectrum ofprimordial gravitational waves [10, 11]. More explicitly,the dimensionless power spectrum is given by P h ( k ) = 2 π H M (cid:12)(cid:12)(cid:12) k = aH , (1)where H is the Hubble parameter, M pl is the reducedPlanck mass, and a is the scale factor. Here, we evalu-ated the power spectrum at the horizon crossing, k = aH .The amplitude of the power spectrum is determined by H and nearly scale invariant because H is almost con-stant during inflation. Therefore, if we observe the scaleinvariant spectrum of primordial gravitational waves, wesee the energy scale of inflation through the parameter H . Indeed, upper bounds on the energy scale of inflationat the pivot scale is given by the observation of cosmicmicrowave background [5].A feature in the primordial gravitational wave spec-trum produced during inflation is the break around theend of inflation. Above the break frequency, productionof primordial gravitational waves should be exponentiallysuppressed though Ema et al. recently pointed out that,even above such break frequency, the production of pri-mordial gravitational waves might happen as a result ofinflaton annihilation into graviton pairs [18]. Even in thiscase, this frequency still represents the break of the spec-tral shape and can become a finger print. The detectionof the break of the spectrum would be a smoking gunproving the existence of inflation. Also, it would tells usthe energy scale of inflation because the break frequencyis determined by the Hubble parameter at the end ofinflation: f ∗ = H end π , (2)where f ∗ and H end are the break frequency and the Hub-ble parameter at the end of inflation. After inflation, thiscutoff frequency is red-shifted due to the expansion of theuniverse to the frequency f break : f break = a end a f ∗ , (3)where a end and a are the scale factor at the end of in-flation and today, respectively.In general, an inflaton field begins to oscillate aroundthe bottom of the potential after inflation and particleproduction occurs [24, 25]. In this reheating phase, theevolution of the universe mimics that of matter domi-nated phase approximately [26]. As a result of (light)particle production during the reheating, radiation dom-inated phase follows. Therefore, the Hubble parameter evolves during the reheating as H reh H end = (cid:18) a reh a end (cid:19) − / , (4)where a reh is the scale factor at the end of the reheatingphase and thus at the beginning of the radiation domi-nated phase. The Hubble parameter at the end of thereheating phase can be parameterized by the reheatingtemperature, T reh : H = π g ( T reh )90 M T , (5)where g ( T ) stands for effective degrees of freedom for theenergy density at a temperature T [26, 27]. From Eqs. (4)and (5), one can estimate the expansion rate a end /a reh by using parameters, H end , T reh and g ( T reh ).In order to estimate the ratio a reh /a , we use the factthat the entropy S conserves in an adiabatic universe: S ∝ g s ( T ) a ( T ) T = const . . (6) g s ( T ) represents effective degrees of freedom for the en-tropy at a temperature T [26, 27].Using Eqs. (2)-(5), we eventually obtain f break (cid:39) . × Hz (cid:18) H end − M pl (cid:19) / (cid:18) T reh GeV (cid:19) / . (7)Here we used the values for each parameter: g s ( T reh ) = g ( T reh ), g s ( T ) (cid:39) . T (cid:39) . H end and T reh ,it means we can determine not only the energy scale ofinflation but also the reheating temperature simultane-ously.Finally, we mention that the maximum value of thebreak frequency is given by the case of the instantaneousreheating, i.e., H end = H reh , that is f break (cid:39) . × Hz (cid:18) g ( T reh ) (cid:19) / (cid:18) H end − M pl (cid:19) / . (8) III. HUBBLE PARAMETER AROUND THEBREAK
As mentioned in the section II, the spectrum of pri-mordial gravitational waves is determined by the Hub-ble parameter. However, gravitational waves around thebreak frequency (8) was produced just before the end ofinflation where the slow roll approximation gets worse.Therefore, we need a prescription away from the slowroll approximation to discuss the spectrum of primordialgravitational waves around the break frequency. In thissection we give an analytic expression for the Hubble pa-rameter near the end of inflation.Let us consider an inflationary universe which can becharacterized by the scale factor a ( t ) defined by ds = − dt + a ( t ) d x . (9)Exponential expansion of the universe is driven by aninflaton φ ( t ) whose potential is V ( φ ). Then, one canobtain Einstein equations: H = 13 M (cid:20)
12 ˙ φ + V (cid:21) , (10)˙ H = − M ˙ φ , (11)where a dot represents derivative with respect to t and H = ˙ a/a is the Hubble parameter. It is more conve-nient to express the equations by taking φ as a time vari-able [28, 29]. Dividing Eq. (11) by ˙ φ yields H (cid:48) ( φ ) = − M ˙ φ , (12)where a dash stands for derivative with respect to φ .Note that we assume ˙ φ > H (cid:48) <
0) throughout ourdiscussion. Using Eq. (12) to eliminate ˙ φ in Eq. (10), one obtains H (cid:48) ( φ ) = 32 M H ( φ ) − M V ( φ ) . (13)This is the equation for the Hubble parameter with thetime variable φ . It should be stressed that the Hubbleparameter H ( φ ) and the potential V ( φ ) are independent.Indeed, one can not neglect ˙ φ in Eq. (10) near the end ofinflation and thus H and V are not related directly incontrast to the case of the slow roll approximation [30].However at the break point, they are related directly aswe will see soon.Differentiating the above equation with respect to φ ,we obtain H (cid:48)(cid:48) ( φ ) = 12 H (cid:48) (cid:34) HH (cid:48) M − M V (cid:48) (cid:35) . (14)Here, H (cid:48) is given by H and V in Eq. (13), so that H (cid:48)(cid:48) can be regarded as a function of H, V and V (cid:48) . Similarly,differentiating Eq. (14), we get H (cid:48)(cid:48)(cid:48) ( φ ) as a function of H, V, V (cid:48) and V (cid:48)(cid:48) . One can repeat the procedure and n derivatives of the Hubble parameter, H ( n ) , can be writ-ten as a function of H, V, V , · · · , V ( n − . The formalexpression is H ( n ) = 12 H (1) (cid:34) − n − (cid:88) r =1 (cid:18) n − r (cid:19) H ( r +1) H ( n − r ) + 32 M n − (cid:88) r =0 (cid:18) n − r (cid:19) H ( r ) H ( n − r − − M V ( n − (cid:35) , (15)for n ≥ (cid:0) nr (cid:1) = n ! r !( n − r )! .We now define the break point φ by the equation (cid:15) ( φ ) ≡ M H (cid:48) H = 1 . (16)On the break point, the Hubble parameter and the po-tential are linked, H ( φ ) = (cid:115) V ( φ )2 M . (17)Then, we can expand the Hubble parameter: H ( φ ) = ∞ (cid:88) n =0 n ! H ( n ) ( φ ) ( φ − φ ) n = ∞ (cid:88) n =0 n ! H ( n ) (cid:16) V ( φ ) , · · · , V ( n − ( φ ) (cid:17) × ( φ − φ ) n . (18)We used Eqs. (15) and (17) to get the second equality.Eq. (18) shows that the evolution of the Hubble parame-ter, which is needed to calculate the spectrum of primor-dial gravitational waves, can be described by the inflaton potential and its derivatives, which are evaluated at φ .In Fig. (1), we plot Eq. (18) (red line) at second order, n = 2, from the end of inflation with the exact numer-ical value (black doted line) and results from the slowroll approximation [30] (blue, green, orange, pink lines).We see that our formula (18) to the second order expan-sion ( n = 2) shows good agreement with the numericalone between, φ = − M pl ∼ − M pl , where the slow rollapproximation gets worse. It should be mentioned thatthe slow roll approximation seems to break down nearthe end of inflation, namely higher order corrections donot make the result better. Therefore, Eq. (18) is com-plementary to the slow roll approximation . In [30], the Pad´e approximant was used and a better resultwas obtained compared with simple slow roll approximations.However, even in that case, the deviation is unavoidable aroundthe end of inflation. /M pl H/M pl FIG. 1. The Hubble parameters near the end of inflation inthe case of V ( φ ) ∝ φ are depicted. φ/M pl = − n = 2 and ex-panded from the end of inflation, i.e., φ /M pl = −
1. Theblack dotted line is a numerical solution of Eq. (13). The blue,green, orange, pink lines shows approximated solutions of 1st,2nd, 3rd and 4th slow roll approximations, respectively [30].
IV. PRIMORDIAL GRAVITATIONAL WAVESAROUND THE BREAK
In the previous section, we gave an analytic expression(18) to describe the evolution of the Hubble parameternear the end of inflation. It can be used to derive thespectrum of primordial gravitational waves, which is ex-pected to be given by Eq. (1). In this section, we firstexplicitly calculate the spectral index of the spectrum atthe end of inflation. Next, we give a spectrum aroundthe spectral break at present.
A. Spectral index
We define the spectral index n T at the end of inflationby n T = d ln( P h ( k )) d ln k = d ln( P h ( k )) d ln a × d ln ad ln k . (19)Together with Eq. (1), we have d ln( P h ( k )) d ln a = 2 d ln Hd ln a = − M H (cid:48) H . (20)On the other hand, d ln ad ln k = (cid:18) d ln kd ln a (cid:19) − = (cid:18) d ln Hd ln a (cid:19) − = (cid:32) − M H (cid:48) H (cid:33) − . (21) Note that we used the relation k = aH . Therefore, fromEqs. (19)-(21), we obtain n T = − M H (cid:48) ( H, V ) H − M H (cid:48) ( H, V ) , (22)where we explicitly indicated that H (cid:48) is a function of H and V through Eq. (13).Observing that the scale factor can be expressed by anintegral, ln a = (cid:90) − H M H (cid:48) ( H, V ) dφ , (23)one can rewrite the wave number k (or the frequency f )as k = exp (cid:32)(cid:90) − H M H (cid:48) ( H, V ) dφ (cid:33) H (= 2 πf ) . (24)We see that Eqs. (22) and (24) are expressed by H and V . Replacing H in the equations by the analytic expres-sion (18), we can calculate the spectral index against k semi-analytically. In Fig. 2, we depicted the spectral in- n T f /f ⇤ FIG. 2. The spectral index near the end of inflation in the caseof V ( φ ) ∝ φ is depicted. The plot runs from φ/M pl = − φ/M pl = − f ∗ . The numerical result is depicted by the blackdotted line. The red line represents Eq. (18) with n = 2 andexpanded from the end of inflation. The blue and green linesare for n = 4 and n = 6, respectively. dex near the end of inflation for the case of n = 2 (redline), n = 4 (blue line) and n = 6 (green line) in Eq. (18).The plots run from φ/M pl = − φ/M pl = − n T in powers of( φ − φ ), we have n T = (cid:32) √ M pl − V (cid:48) ( φ ) V ( φ ) (cid:33) − ( φ − φ ) − + M − √ M pl V (cid:48) ( φ ) V ( φ ) + 2 (cid:16) V (cid:48) ( φ ) V ( φ ) (cid:17) + V (cid:48)(cid:48) ( φ ) V ( φ ) (cid:16) √ M pl − V (cid:48) ( φ ) V ( φ ) (cid:17) + · · · . (25)We note that the denominators, ( √ /M pl − V (cid:48) ( φ ) /V ( φ ), are not zero and, furthermore, notnecessarily small because the slow roll approximationsis broken at φ . This analytic expression can be usedas a fitting function for data analysis. Since the fittingfunction includes information of the inflaton potentialat the break point, it is useful to discriminate inflationmodels. Indeed, for example, the sign of V (cid:48)(cid:48) ( φ ) isdifferent from a model to a model, so that determiningthe sign of V (cid:48)(cid:48) ( φ ) from observations is important todistinguish inflation models. Therefore, our analyticexpression is useful. B. Spectrum at present
In this subsection, we give an example of the pri-mordial gravitational wave spectrum around the spec-tral break at present. The dimensionless power spectrumaround the end of inflation P h ( k ) is given by Eq. (1). Af-ter inflation, in the super horizon regime, k < aH , thepower spectrum keeps the initial amplitude. When amode k reenter the Hubble horizon, namely in the subhorizon regime k > aH , the power spectrum starts todecay proportional to a − . It means the spectrum atpresent depends on the background evolution of the uni-verse through the scale factor.It would be useful to define the energy density param-eter instead of the power spectrum and evolve it to thepresent time: d ln Ω d ln k = k P h ( k )12 a H × (cid:18) a reenter ( k ) a (cid:19) . (26)where H = 70 km / s Mpc is the Hubble parameter atpresent and a reenter is the scale factor when a mode reen-ters the Hubble horizon, i.e., k = a reenter H .We now assume that, as was done in Sec. II, a reheat-ing phase where the inflaton oscillates around the bot-tom of its potential follows after inflation. In the phase,the Hubble parameter evolves as H ∝ a − / and then a reenter ( k ) ∝ k − . Therefore, the energy density param-eter would proportional to k − . More explicitly, the en- d ln ⌦ d ln f f [Hz] f reh f break FIG. 3. The energy density parameter (27) between f reh and f break is depicted. f reh is the frequency which reenters theHubble horizon at the end of the reheating phase and f break is given by Eq. (7). The black dot line represents a numericalresult where P h is evaluated numerically. The blue and thegreen lines are for approximated results in the case of the1st order slow roll approximation and Eq. (18) with n = 6,respectively. ergy density parameter for the modes which reenter dur-ing the inflaton oscillation is d ln Ω d ln f = π f P h ( f )3 H (cid:18) ff break (cid:19) − × (cid:18) g s ( T ) T g s ( T reh ) T (cid:19) / (cid:18) H reh H end (cid:19) / . (27)Note that H reh is related to T reh by Eq. (5) and f break is determined by T reh and H end in Eq. (7). Therefore, We have not considered the effect of anisotropic stress of neu-trinos [13, 14] here. It would give an overall suppression factorto Eq. (27). for a set of free parameters T reh , H end , one can calculateEq. (27) immediately with the semi-analytic formula weinvestigated in the previous sections.Let us give an example of the spectrum in the case of V ( φ ) ∝ φ where we set T reh = 10 GeV and H end =10 − M pl . The result is depicted in Fig. 3. We see that oursemi-analytic method (green line) well fits the numericalresult (black dot line) around the spectral break wherethe slow roll approximation (blue line) gets worse. Notethat we used the 1st order slow roll approximation inFig. 3 because higher order collections make the resultworse around the spectral break (see Fig. 1). Moreover,Fig. 3 shows that our formula can be connected to theslow roll approximation at a crossing point. Therefore,our formulation with the slow roll approximation enablesus to complete the spectrum in full frequency range inanalytic ways. V. CONCLUSION
We studied a spectral break of primordial gravitationalwaves as a fingerprint of the end of inflation. Indeed, theamplitude and the frequency on the break can tell usthe energy scale of inflation and the reheating tempera-ture simultaneously. This spectral break scale may existaround 10 Hz (see Eq. (7)) and thus the theoretical in-vestigation is crucial for new gravitational wave detectorsdeveloped intensively [19–23].We gave an analytic formula for describing the evolu- tion of the Hubble parameter near the end of inflationin Sec. III. Because the slow roll approximation breaksdown near the end of inflation, our analytic formula-tion is useful and complementary to it. In Sec. IV, wesemi-analytically investigated the spectral index of thespectrum at the end of inflation. We showed how to ex-tract important information of the inflaton potential fromthe observation of the spectral index. Furthermore, wepresented an example of the spectrum around the breakpoint at present in Fig. 3 for specific values of parametersin Eq. (27).It should be mentioned that probing the reheating tem-perature with primordial gravitational waves was alsoproposed in [15–17]. However, the paper focused on abend of the spectrum due to the transition of backgroundspacetime from the reheating to the radiation dominantphase. Then, the possibility to observe the bend withgravitational wave interferometers was discussed. Thissituation is realized when the reheating temperature islow enough. In the present paper, we intend to observethe spectral break due to the end of inflation and thenhigher reheating temperature is favorable compared withthe above case.
ACKNOWLEDGMENTS
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