# Modifications by QCD transition and annihilation on analytic spectrum of relic gravitational waves in accelerating universe

Abstract

As predicted by quantum chromodynamics(QCD), around MeV in the early universe, the QCD transition occurs during which the quarks are combined into the massive hadrons. This process reduces the effective relativistic degree of freedom, and causes a change in the expansion behavior of the universe. Similarly, the annihilation occurred around Mev has the same kind of effect. Besides, the dark energy also drives the present stage accelerating expansion. We study these combined effects on the relic gravitational waves (RGWs). In our treatment, the QCD transition and the annihilation, each is respectively represented by a short period of expansion inserted into the radiation era. Incorporating these effects, the equation of RGWs is analytically solved for a spatially flat universe, evolving from the inflation up to the current acceleration, and the spectrum of RGWs is obtained, covering the whole range of frequency Hz. It is found that the QCD transition causes a reduction of the amplitude of RGWs by in the range Hz, and the annihilation causes a reduction in the range Hz. In the presence of the dark energy, the combination of the QCD transition and the annihilation, causes a larger reduction of the amplitude by for the range Hz, which covers the bands of operation of LIGO and LISA. By analysis, it is shown that RGWs will be difficult to detect by the present LIGO, but can be tested by LISA for certain inflationary models.

PACS numbers: 04.30.-w, 98.80.-k, 04.62.+v

1. Introduction

The existence of a stochastic background of relic gravitational waves (RGWs) is generally predicted in inflationary models [1, 2, 3]. Since the relic gravitons decoupled much earlier than the cosmic microwave background (CMB) photons, the detections of RGWs would open a new window to the very early universe. Unlike gravitational radiations from finite stellar objects, RGWs exist everywhere and anytime, and have a wide spreading spectrum, serving as one of the major scientific goals of the laser interferometers gravity-wave detections, including the current LIGO [4], VIRGO [5], GEO600 [6], TAMA [7], AIGO [8], and the future LISA [9], ASTROD [10], BBO [11], and DECIGO [12]. Moreover, along with the density perturbations, RGWs also contribute to the CMB anisotropies and polarizations [13, 14, 15, 16, 17]. For instance, the B-polarization of CMB on very large scales can only be generated by RGWs. Through the detections of CMB polarizations one may have a chance to obtain the direct evidence of GWs for the first time [18, 19, 20, 21, 22]. Therefore, the precise information of RGWs is much desired.

The spectrum of RGWs depends on the following factors. First, it depends sensitively on the specific inflationary models [1, 3, 23]. After being generated, it will be altered by the subsequent stages of expansion of the universe, among which notably is the currently accelerating expansion of the universe [3, 24]. Finally, it would be further modified by other physical processes occurred in the early universe. For example, the neutrino free-streaming [25] has been shown to affect the spectrum of RGWs in the very low frequency range Hz [26]. Although this would modify the contribution to CMB polarizations, but would not affect the detections of LIGO [4] and LISA [9] that operate effectively at higher frequencies, Hz and Hz, respectively. Other important physical processes include the QCD transition around a temperature MeV [27, 28], and the annihilation around MeV. These two processes will be studied in this work. Leaving aside the detail of the QCD transition, which is notoriously complicated and still under study, we are concerned with only the thermodynamic property that the transition brings to the system, i.e., a significant drop in the relativistic degrees of freedom. Consequently, the constituent components in the stress tensor also change in the Friedmann’s equation. Thereby the expansion of the Universe in terms of the scale factor will subsequently change, and the spectrum of RGWs would be modified. The annihilation has the similar effect. Schwarz [29] studied the QCD transition and the annihilation and estimated the reduction of the energy density spectrum of RGWs. Watanabe and Komatsu [30] investigated these effects with more details, and gave a numerical solution of the energy density spectrum of RGWs. In these works, the important effect of the accelerating expansion of the present universe [3] has not been included, neither the effects of inflation and reheating. Moreover, to show the prospects of detecting RGWs at the laser interferometers gravity-wave detection, an explicit demonstration of the current spectrum of RGWs itself and its direct comparison with the sensitivity curve of gravity-wave detection are also needed.

In this paper, by extending our previous analytical calculation of the spectrum of RGWs, we explore the consequences caused by the QCD transition and the annihilation, and, at the same time, take into account of the accelerating expansion driven by the dark energy [3]. Beside the energy density spectrum , we demonstrate explicitly the spectrum of RGWs itself, and compare it directly with the sensitivity of the ongoing and forthcoming GW detectors, such as LIGO, LISA, etc [4, 9]. Aiming at giving a comprehensive compilation, by using the set of parameters , , , , and , respectively, such important cosmological elements have been explicitly parameterized, as the inflation, the reheating, the QCD transition, the annihilation, the dark energy and the tensor/scalar ratio. This will considerably facilitate further studies on the RGWs and the relevant physical processes. For instance, it can be easily used in the calculation of CMB anisotropies and polarizations generated by RGWs [16, 21].

The organization of this paper is as follows. In section 2, from the inflation up to the acceleration, the scale factor is specified by the continuity conditions for the subsequential stages of expansion. The periods of the QCD transition and the annihilation are also modelled as having a scale factor of power-law form, which is inserted into the radiation era. In section 3, the analytical solution of RGWs in terms of Bessel’s functions is determined with the coefficients being fixed by continuity condition joining two consecutive expansion stages. The impact of the dark energy on the expansion is emphasized. In section 4, we present the resulting spectra of RGWs and discuss the modifications caused by the QCD transition, the annihilation, and other effects. Appendix A supplies the details of our treatment of the period of QCD transition and annihilation by modelling the corresponding scale factor. Appendix B gives an interpretation of the modifications on RGWs due to the QCD transition. In this paper the unit with is used.

2. Expansion history of the universe

From the inflationary up to the currently accelerating stage, the expansion of the universe can be described by the spatially flat () Robertson-Walker spacetime with a metric

(1) |

where is the conformal time. The scale factor for the successive stages can be approximately described by the following forms [31]:

The inflationary stage:

(2) |

where , and . This generic form of scale factor is a simple modelling of inflationary expansion, and the index is a parameter. The special case of is the de Sitter expansion of inflation. If the inflationary expansion is driven by a scalar field, then the index is related to the so-called slow-roll parameters, and [32], as . In this class of scalar inflationary models one usually has . Besides, the observational results of WMAP also indicate that should be slightly smaller than [18]. But, for demonstration purpose, we allow the parameter to take values to examine the possibility of detection by GWS detectors.

The reheating stage:

(3) |

where is the beginning of radiation era and . We will mostly take the model parameter , though other values are also taken to demonstrate the effect of various reheating models.

The radiation-dominant stage:

The QCD transition occurs around Mev for a period, and the process of annihilation into photons starts around MeV and ends up around MeV [30]. These two periods should be included into the radiation stage. Before the QCD transition, one has

(4) |

where is the beginning of the QCD transition and . During the QCD transition, is modelled by

(5) |

where is the ending of the QCD transition and . The power index in Eq.(5) is a model parameter describing the QCD transition. We have found that the spectrum of RGWs does not vary considerably for the values of in the interval . For concreteness we take in calculations. (For more details, see Appendix A). The expansion rate around Mev is plotted in Fig.1, showing a jump-up caused by the QCD transition. After the QCD transition and before the annihilation, one has

(6) |

where is the beginning of the annihilation and . The two slopes in Eq.(4) and in Eq.(6) are related as by considerations of the details of the QCD transition (See Appendix A). Similarly, the period of annihilation is modelled by

(7) |

where is the ending of the annihilation and . The power index in Eq.(7), as a model parameter, can be taken as (see Appendix A). After the annihilation, one has

(8) |

where , and is the beginning of the matter era, which can be taken at a redshift [18]. The two slopes in Eq.(6) and in Eq.(8) satisfy the relation (See Appendix A).

The matter-dominant stage:

(9) |

where and is the beginning of the acceleration era.

The accelerating stage:

(10) |

where is the present time and . The index in Eq.(10) depends on the dark energy . By fitting with the numerical solution of the Friedmann equation [3, 26],

(11) |

where , one can take for , and for . The redshift of the start of this stage depends on the specific models of the dark energy. For instance, in the cosmological constant model with and , it starts at , and, in the quantum effective Yang-Mills condensate dark energy model, it starts at [33].

In the above specifications of , there are nine instances of time, , , , , , , , , and , which separate the different stages. Eight of them are determined by how much increases over each stage based on the cosmological considerations. We take the following specifications: for the reheating stage, for the first radiation stage, for the second radiation stage, for the third radiation stage, for the fourth radiation stage, for the fifth radiation stage (see Appendix A), for the matter stage, and 3, 26]. The remaining time instance is fixed by an overall normalization for the present accelerating stage [

(12) |

There are also 22 constants in the expressions of , among which , , , and are imposed as the model parameters describing the inflation, the reheating, the QCD transition, the annihilation and the acceleration, respectively. Based on the definition of the expansion rate of the present universe , one has . Making use of the continuity conditions of and of at the eight given joining points , , , , , , , , and , all parameters are fixed as the following:

and

(14) |

In the expanding universe, the physical wavelength is related to the comoving wavenumber by

(15) |

and the wavenumber corresponding to the present Hubble radius is

(16) |

There is another wavenumber involved

(17) |

whose corresponding wavelength at the time is the Hubble radius . In the present universe the physical frequency corresponding to a wavenumber is given by

(18) |

3. Analytical solution of RGWs

In the presence of the gravitational waves, the perturbed metric is

(19) |

where the tensorial perturbation is a matrix and is taken to be transverse and traceless

(20) |

The wave equation of RGWs is

(21) |

We decompose into the Fourier modes of the comoving wave number and into the polarization state as

(22) |

where ensuring that be real, is the polarization tensor, and denotes the polarization states . Here is treated as a classical field. In terms of the mode , Eq.(21) reduces to

(23) |

By assumption, for each polarization, , , the wave equation is the same and has the same statistical properties, so that the super index can be dropped from from now on. Since for all the stages of expansion the scale factor is of a power-law form

(24) |

the solution to Eq.(23) is a linear combination of Bessel function and Neumann function

(25) |

where the constants and are completely determined by the continuity of and of at the joining points , , , , , , , and .

In particular, we write down explicitly the solution for the inflationary stage since it give the initial condition for the spectrum of RGWs,

(26) |

where and

(27) |

are taken [34], so that the so-called adiabatic vacuum is achieved: in the high frequency limit [35]. Moreover, the constant in Eq.(26) is independent of , whose value is determined by the initial amplitude of the spectrum. For the -dependence of is given by

(28) |

As will be seen later, this choice will lead to the required scale-invariant initial spectrum in Eq.(32).

It should be mentioned that around the temperature Mev neutrinos decoupled from electrons and photons and started free-streaming in space. This will give rise to an anisotropic part of the energy-momentum tensor as a source of the equation of RGWs. The previous works have shown that the neutrino free-streaming in combination with the dark energy would cause a reduction of the amplitude of RGWs by in the low frequency range Hz [25, 26, 30]. Although this modified RGWs will contribute to the CMB anisotropies and polarizations on very large scales, the frequency range is outside the frequency bands of LIGO and LISA. On the other hand, as will be seen, the impact on RGWs by the QCD transition is in the high frequency range Hz. Since these two frequency ranges are not overlapped, for simplicity of computing, we will not include the neutrino free-streaming here.

In our previous study [26], we have examined the issue of how the RGWs would be, if our current universe were matter-dominated. The amplitudes of RGWs in the CDM (accelerating) and in the CDM universe were compared, and the ratio was found to be . Here a similar examination is extended to the case including the QCD transition and the annihilation. To be specific, we assume that both universes have the same initial and at the time with , when . By numerically solving the Friedmann equation in both models, we plot the scale factor in Fig.2, showing that the ratio of scale factors at present is . As is known [1, 3], for wavelengths shorter than the horizon the modes decay as , so the CDM model would predict an amplitude of RGWs higher than the CDM model. This is indeed confirmed by our calculation including the QCD transition and the annihilation, and the ratio is

(29) |

Moreover, there are some subtleties with the matter-dominant model, regarding to interpreting the current observations. As it stands, the actual universe is CDM, so the observed Hubble constant is properly interpreted as the current expansion rate in the accelerating model, . The virtual matter-dominant universe would have a smaller rate . If the observed Hubble constant were regarded as the expansion rate of the virtual matter-dominant universe [30], one would come up with an amplitude of lower by an extra factor than it should be.

4. Spectrum of relic gravitational waves

The spectrum of RGWs at a time is defined by the following equation [31]:

(30) |

where the right-hand side is the expectation value of the . Calculation yields the spectrum at present

(31) |

where the factor counts for the two independent polarizations.

One of the most important properties of the inflation is that the initial spectrum of RGWs at the time of the horizon-crossing during the inflation is nearly scale-invariant [31]:

(32) |

where , and is a constant to be fixed by the observed CMB anisotropies in practice. The First Year WMAP gives the scalar spectral index , the Three Year WMAP gives [18], while in combination with constraints from SDSS, SNIa, and the galaxy clustering, it would give (68% CL) [20]. The five-year WMAP data give [36], and the WMAP data combined with Baryon Acoustic Oscillations and Type Ia supernovae give ( CL) [37].. From the relation [1, 3], the inflation index for . As mentioned earlier, we will allow the parameter to take values to demonstrate the RGWs spectrum. Note that the constant is directly proportional to in Eq.(26). Since the observed CMB anisotropies [18] is at , which corresponds to anisotropies on scales of the Hubble radius , so, as in Refs.[26], we take the normalization of the spectrum

(33) |

where is defined in Eq.(17), its corresponding physical frequency being Hz. The tensor/scalar ratio can be related to the slow roll parameter in the scalar inflationary model as [32]. However, the value of is model-dependent, and frequency-dependent [16, 17]. This has long been known to be a notoriously thorny issue [38]. In our treatment, for simplicity, is only taken as a constant parameter for normalization of RGWs. Currently, only observational constraints on have been given. The Three Year WMAP constraint is ( CL) evaluated at Mpc, and the full WMAP constraint is (95% CL) [21]. The combination from such observations, as of SDSS, 3-year WMAP, supernovae SN, and galaxy clustering, gives an upper limit ( CL) [20]. The five-year WMAP gives a limit () for power-law models [36], and the WMAP data combined with Baryon Acoustic Oscillations and Type Ia supernovae give ( CL) [37]. For concreteness, we take .

The spectral energy density of the RGWs is given by

(34) |

which follows from the definition of the total energy density of RGWs [31]

(35) |

where is the energy density of RGWs, and is the critical energy density. The integration in Eq.(35) has the lower and upper limits, and , as the cutoffs of the wavenumber. The corresponding frequencies are and . Detailed analyses of these limits are given in Ref.[26]. In past, in the absence of direct detection of RGWs, the constraint on RGWs through the energy density has been commonly used; especially, a bound from the Big Bang nucleosythesis (BBN) [39]

(36) |

has been frequently employed in practice, where being the Hubble parameter [18].

In the following we give the resulting spectra and of RGWs, demonstrate their explicit dependence upon the model parameters , , , and the modifications by the QCD transition and annihilation. Also we will compare it with the sensitivity of detections, such as LIGO and LISA.

Figure 3 gives the spectrum for three values of the inflationary index , , and , respectively, where the fixed , , , , and are taken. It is seen that is very sensitive to . A smaller will generate lower amplitude of RGWs for all frequencies.

Figure 4 shows the influence of reheating stage on RGWs. The spectrum is given for three different values of , , and . It is seen that the reheating process will affect RGWs only in very high frequency range Hz. This covers the frequency range of some very high frequency gravity wave detection systems, such as the Gaussian laser beam detector aiming at the frequency Hz [40], or the circular waveguide detector aiming at Hz [41]. We remark that the portion of predicted spectrum for the frequency range Hz may be not reliable. This is because the energy scale of the conventional inflationary models are usually less than Gev [26], which will gives a cutoff of frequency around Hz.

Presented in Fig.5 is the modification of the spectrum by the QCD transition. There is a critical frequency Hz, below which the spectrum is the same for both models with or without the QCD transition. But, in the high frequency range Hz, the amplitude of is reduced by in comparison with the model without QCD transition.

(37) |

Note that the frequency range of this reduction on RGWs covers those of the major laser interferometers GW detectors, such as LIGO and VIRGO operating effectively in the frequency bands around Hz, and LISA around Hz, respectively. So the modifications on RGWs due to the QCD transition are relevant to these major detections. In Appendix B we will give an interpretation to the origin of this critical frequency Hz and of the reduction in amplitude by for Hz.

Fig.6 gives the spectrum modified by the annihilation, which has a similar reduction effect. The lower frequency portion Hz of the spectrum is not affected by the annihilation. However, in the high frequency range Hz, the amplitude of is reduced by by the annihilation. Also the frequency range of this reduction covers that of LIGO, VIRGO, LISA, etc.

Fig.7 is an enlarged portion of the spectrum around Hz, illustrating the details of reductions of the spectrum by both QCD transition and annihilation. The combination of the QCD transition and the annihilation reduce .

The influence of the dark energy on the spectrum is demonstrated in Fig.8, where , , and are taken respectively. As explained in the last section, over the whole range of frequency Hz, the amplitude of spectrum is suppressed by the presence of , but the slope remains the same. In particular, with is reduced by in comparison with the model . (See Eq.(29))

Figure 9 is a comparison of the LIGO detection with our calculated RGWs with the fixed tensor/scalar ratio and the dark energy . The upper smooth curve is from the LIGO H1 Upper limits ( CL) from PowerFlux best-case [42], and the lower three fluctuating curves are the RGWs spectra of , , and , corresponding to those in Fig.3, respectively. Here the vertical axis is the root mean square amplitude per root Hz, which equals to

(38) |

The plot gives only the frequency range Hz, on which the LIGO works efficiently. The reductions due to the QCD transition and the annihilation have been incorporated in the curve of calculated . Our result shows that there is a gap of about one order of magnitude even for the inflationary model. As it currently stands, the possibility for LIGO to detect the RGWs predicted by the inflationary model is not high, let alone other models with . Other two model parameters, i.e., the tensor/scalar ratio and the dark energy will substantially influence the height of . The five-year WMAP data improve the upper limit on the tensor/scalar ratio ( CL) for power-law inflationary models and ( CL) for models with a running index, and give the value of dark energy [36]. So if we take the upper limit for power-law models, the height of will increase by by Eq.(33), and, furthermore, if we take , will increase by another [3]. These together will allow an increase of by a total . Therefore, the current LIGO with greatly enhanced sensitivity [4] will definitely be able to put a constraint on the inflationary model. However, it is seen from Fig.9 that the curve for , supported by the scalar inflation models and the WMAP data [18] [36] [37], is about five orders below the LIGO sensitivity. So we may say that the RGWs generated by scalar inflation models is unlikely to be directly detected by LIGO at the moment.

Figure 10 is a comparison of the LISA sensitivity curve with the spectra from Fig.3 in the lower frequency range Hz that is also covered by the modifications of the QCD transition and the annihilation. Assume that LISA has one year observation time corresponding to frequency bin Hz (i.e., one cycle/year) around each frequency. To make a comparison with the sensitity curve, we need to rescale the spectrum in Eq.(31) into the root mean square spectrum in the band [31] [39],

(39) |

This r.m.s spectrum can be directly compared with the 1 year integration sensitivity curve that is downloaded from LISA [43]. The plot shows that LISA by its present design will be able to detect the inflationary model of . If the ratio , it is still possible for LISA to detect the model of . However, LISA is unlikely to be able to directly detect the model of , as there is a gap of two orders. One may say that, in regards to detection of RGWs, LISA will give a stronger constraint on the RGWs spectrum than LIGO will do.

Figure 11 shows the -dependence of the spectral energy density defined in Eq.(34). Clearly, is very sensitive to the inflationary parameter . A larger gives a higher . The Advanced LIGO [4] will be able to detect RGWs with at Hz, and it might impose stronger constraints on other inflationary models. On the other hand, the total energy density defined in Eq.(35) has also been used as a constraint on RGWs. Taking the parameters , , , and , we find for the inflationary model of , which is orders higher than the BBN bound [39] in Eq.(36). So this model is disfavored, unless some other mechanism is introduced to reduce its . We also obtain for the model , and for the model ; both models are safely below the BBN bound in Eq.(36).

Fig.12 shows in the modified by the QCD transition and annihilation, where the reduction on RGWs is more noticeable. In the range Hz the QCD transition alone reduces by , and the combination of QCD transition and the annihilation reduces by .

The impact of dark energy on for the model is plotted in Fig.13. Three values of , , are taken. A larger gives a lower . It is seen that the causes a decrease of the amplitude of by a factor of over the whole range of frequencies.

ACKNOWLEDGMENT: We are grateful the referees for valuable suggestions. Y.Zhang’s research work is supported by the CNSF No.10773009, SRFDP, and CAS.

Appendix A: Modelling of QCD transition and annihilation

The detail of the QCD transition is notoriously complicated and still under study. Here we will only consider the change of the effective degree of freedom, and give a simple working model for the scale factor around the QCD transition lasting a short period. Thus the radiation era can be tentatively divided into the three parts with the scale factor been listed in Eqs.(4), (5), and (6). After using the continuity conditions of and of at the two given joining points and , one still needs to determine parameters, , , . Since the QCD transition temperature during the radiation era, as soon as is given, the initial time of QCD transition is determined, so is the parameter . Still and need to be fixed in the following.

Around the QCD transition, the contributions from the matter and the dark energy can be neglected. Only the radiation is important, of which the energy density, the pressure, and the entropy density are given by

(40) |

(41) |

(42) |

respectively, where and denote the effective number of relativistic species contributing to the energy density and entropy, respectively [44]. In an adiabatic universe, the entropy per unit comoving volume is conserved

(43) |

From Eqs. (40), (42) and (43), one has

(44) |

During the QCD transition era, one has [29, 30]. Therefore, here we will not distinguish the difference between and . Thus, Eq.(44) is reduced to

(45) |

which is plugged into the Friedmann equation , yielding

(46) |

Thus, with the decreasing of across the QCD transition the expansion rate in terms of the conformal time increases. As predicted by the Standard Model of particle physics, before the QCD transition, at the time ; and after the QCD transition, it becomes at the time [45, 46]. Applying Eq.(46) to Eqs.(4) and (6) yields the ratio

(47) |

From the expression