New method to constrain the relativistic free-streaming gas in the Universe

We discuss a method to constrain the fraction density $f$ of the relativistic gas in the radiation dominant stage, by their impacts on a relic gravitational waves and the cosmic microwave background (CMB) $B$ polarization power spectrum. We find that the uncertainty of $f$ strongly depends on the noise power spectra of the CMB experiments and the amplitude of the gravitational waves. Taking into account of the CMBPol instrumental noises, an uncertainty $\Delta f=0.046$ is obtained for the model with tensor-to scalar ratio $r=0.1$. For an ideal experiment with only the reduced cosmic lensing as the contamination of $B$-polarization, $\Delta f=0.008$ is obtained for the model with $r=0.1$. So the precise observation of the CMB $B$-polarization provides a great opportunity to study the relativistic components in the early Universe.


I. INTRODUCTION
Understanding the cosmic components in the Universe is one of the main tasks for cosmology. The current observations from cosmic microwave background, large scale structure, Type Ia supernova, and etc., have already indicated ∼ 72% dark energy, ∼ 23% dark matter, ∼ 5% baryons and ∼ 0.005% photons as the main components in the present Universe. [1,2,3].
With the upcoming of the more precise observations, it becomes possible and necessary to determine other components. In this Letter, we shall focus on the determination of the relativistic components in the Universe. In addition to the photons and the gravitational wave background [4], these components also include the massless (or tiny massive) neutrinos, the possible scalar field, the Yang-Mills field dark energy in the scaling stage [5,6,7], and some unknown massless (or tiny massive) particles, such as the sterile neutrinos [8]. As known, a large relativistic component in the Universe during the big bang nucleosynthesis (BBN) stage can enhance the expansion rate of the Universe, leading to a change the primordial abundances of the light elements. Thereby, one can constrain the total energy density of the relativistic components during the BBN stage [9], but unable to distinguish each component, as the expansion rate is determined by the total of all the relativistic components.
If a relativistic component behaves as a free-streaming gas of massless particles at the photon decoupling, they will also affect the growth of density perturbations, in addition to the change of the expansion rate. So by the observation of CMB spectra, especially the temperature * Electronic address: Wen.Zhao@astro.cf.ac.uk anisotropy spectrum and the matter perturbation, one can constrain the fraction density f of the relativistic free-streaming gas among all the relativistic components [10,11,12]. However, there are various degeneracies between f and other cosmological parameters, which need to be broken for the method to work.
The stochastic gravitational waves backgrounds, generated in the very early Universe due to the superadiabatic amplication of zero point quantum fluctuations of the gravitational field [4], provide a much cleaner way to study the evolution of the Universe. The effect of the neutrino free-streaming gas on the spectrum of the relic gravitational waves (RGWs) has been examined in the previous works [16,17,18,19]. In particular, it has been found that the neutrino free-streaming gas causes a reduction of the spectral amplitude by 20% in the range (10 −16 ∼ 10 −10 )Hz, and leaves the other portion of the spectrum almost unchanged [19].
This reduced RGWs leave observable imprints on the CMB temperature and polarization anisotropies power spectra [13,14]. Especially, the B-polarization power spectrum, only generated by RGWs, is reduced by (20% ∼ 35%) when ℓ > 200. In Ref. [15] it is pointed out that the similar effect can also be generated by other relativistic free-streaming gas. In this Letter, we introduce a new method to constrain the fraction energy density f of the relativistic free-streaming gas by the future CMB Bpolarization observations. It will be shown that the value of ∆f , the uncertainty of f in the radiation-dominant stage, strongly depends on the value of tensor-to-scalar ratio r, and is limited by the noise power spectra of the CMB experiments. For the model with r = 0.1, CMBPol experiment can give ∆f = 0.046. If considering the ideal case, where only the reduced cosmic lensing effect on the B-polarization is included, then one has ∆f = 0.008.

II. EFFECTS OF FREE-STREAMING GAS ON RGWS AND CMB POLARIZATIONS
Incorporating the perturbations to the spatially flat Friedmann-Lemaitre-Robertson-Walker space-time, the metric is where the perturbations of space-time h ij is a 3 × 3 symmetric matrix. The gravitational wave field is the tensorial portion of h ij , which is transverse-traceless h ij,j = 0, h i i = 0. Since the relic gravitational waves are very weak, |h ij ≪ 1|, one needs to just study the linearized field equation: The relativistic free-streaming gas gives rise to an anisotropic portion π ij , which is also transverse and traceless. By the Fourier decomposition of h ij and π ij , for each mode k and each polarization, Eq.(2) can be put into the form (see for instance [16]) where the overdot denotes a conformal time derivative d/dη. This equation can be modified to the following integro-differential equation [16] h k + 2ȧ where the kernel function in Eq.(4) is f ≡ ρ f /ρ 0 is the fractional density of the relativistic freestreaming gas in the radiation-dominant stage, and η rd is the decoupling time of the relativistic free-streaming gas. One has f = 0.41 for the decoupled neutrino background with the number of species N ν = 3 as the relativistic free-streaming gas. However, if the other relativistic freestreaming gases also exist in the early Universe, the value of f should be larger than 0.41. On the other hand, if the neutrinos do not free-stream, due to some possible couplings [20], then the value of f should be smaller than 0.41. So the determination of f provides a chance to study the relativistic components in the early Universe.
In the analytic approach, Eq.(4) is approximately reduced to the following form [14]: When f = 0, this equation returns to the evolution equation of gravitational waves in the vacuum,ḧ k + 2ȧ aḣ k + k 2 h k = 0 [21,22,23], which only depends on the evolution of the scale factor a(η). Eq.(5) has been solved by perturbations, yielding the full analytic solution h k (η), from the inflation up to the present accelerating stage [14,19], and it has been found that the relativistic freestreaming gas causes a damping of h k by ∼ 20% in the frequency range ν ≃ (10 −16 , 10 −10 )Hz. The RGWs can generate the CMB temperature and polarization anisotropies power spectra C XX ′ ℓ (XX ′ = T T, T E, EE, BB), by the Sachs-Wolfe effect [13,14,24,25,26,27,28,29,30,31,32]. As shown in Ref [14], the mode functions h k (η d ) andḣ k (η d ) at the photon decoupling time η d , i.e. z ∼ 1100, appear in the integral expressions of the spectra of CMB temperature and polarization anisotropies. In Fig.1, we plot the quantities |h k (η d )| and |ḣ k (η d )| as a function of kη 0 , where η 0 is the present conformal time. The conformal wavenumber k is related to the frequency by ν = k/2π, by setting the present scale factor a(η 0 ) = 1. We find that the neutrino free-streaming shifts the peaks of h k (η d ) anḋ h k (η d ) to the right side. In addition, when kη 0 > 200, the amplitudes of h k (η d ) andḣ k (η d ) are obviously reduced by ∼ 20%, due to the existence of the neutrino free-streaming.
The modifications on h k (η d ) andḣ k (η d ) by this relativistic free-streaming gas leave observable imprints in the spectra of CMB. To demonstrate this, the spectra C BB ℓ with and without neutrino free-streaming gas are plotted in Fig.2. The ℓ < 200 portion of the spectra is not affected much by neutrino free-streaming gas. Only on the scales of ℓ > 200, the spectra are modified effectively, i.e. the reduction of amplitude of C BB ℓ by neutrino freestreaming gas is noticeable only starting from the second peak. Given the current precision level of observations on CMB, these small modifications caused by neutrino free-streaming gas are difficult to detect. However, as will be shown in the next section, this modification is expected to be detected by the future CMB experiments, such as the CMBPol project [33], which are sensitive for the CMB polarization observations.

III. CONSTRAINT ON THE RELATIVISTIC FREE-STREAMING GAS
As mentioned, in addition to the decoupled neutrino, there may be other relativistic free-streaming gases in the early Universe, which may also modify the RGWs and CMB power spectra. So by the observations of the CMB power spectra, especially the B-polarization power spectrum (which is only generated by RGWs), we can constrain the fraction energy density of all the relativistic free-streaming gases, which is helpful to understand the various components in the Universe.
If all the CMB fields are Gaussian random, the power and cross spectra of the CMB temperature and polarization anisotropies quantify all the information contained in the observation [34]. We can use the Fisher information matrix techniques to compare and contrast the precision, to which various surveys can determine the parameters underlying the power spectra.
The Fisher matrix is a measure of the curvature of the likelihood function around its maximum in a space spanned by the parameters, such that the statistical error on a given parameter p i is: ∆p i ≃ (F −1 ) 1/2 ii [35,36]. Here we consider the simplest case, only the fraction density of the relativistic free-streaming gas, f , is taken as the free parameter, and only the CMB B-polarization power spectrum is employed to constrain f . The other cosmological parameters are assumed to be well determined by the CMB power spectra C T T ℓ , C T E ℓ and C EE ℓ by the future CMB observations, so they will be fixed as their fiducial choices in the data analysis. Thus the Fisher matrix ∆p i = 1/ √ F ii for p i ≡ f can be written as [35] Here ∆D BB ℓ is the standard deviation of the estimator D BB ℓ [34], which is calculable by where f sky is the cut sky factor. For a special experiment, the noise power spectrum is calculated by where ∆ P is the constant noise per multipole and θ F is the full width at a half maximum beam in radians. We shall discuss three kinds of future CMB experiments: the Planck satellite, the planned CMBPol experiment, and an ideal CMB experiment. Reference sensitivity for representative CMB polarization experiments are given in Table I [33,37]. In the ideal case, we have only considered the reduced lensed B-polarization spectrum as the contamination of C BB ℓ , which approximately corresponds to a noise with ∆ P ≃ 0.8µK-arcmin [38].
In Fig.2, we have plotted the noise power spectra N BB ℓ and the uncertainty ∆D BB ℓ compared with the signal C BB ℓ in the model with the ratio r = 0.1, where we have taken our fiducial choice of the cosmological parameters as below: Ω b = 0.0456, Ω c = 0.228, Ω Λ = 0.726, Ω k = 0, h = 0.705, f = 0.41. The perturbation parameters are adopted as follows: A s = 2.445×10 −9 , n s = 0.96, α s = 0, n t = 0. Fig.2 shows that the modification of C BB ℓ by the relativistic free-streaming gas is noticeable only at ℓ > 200. Since the amplitude of C BB ℓ is very small in this range, only the very sensitive CMB experiments are expectable to be able to detect this modification. Fig.2 also shows that, Planck mission is only sensitive for the reionization peak of C BB ℓ . i.e. ℓ < 10. So it will be not expected to be able to constrain on the relativistic free-streaming gas in the Universe. However, for the CMBPol experiment, the signal C BB ℓ is larger than ∆D BB ℓ when ℓ < 300, and a detection of this modification due to the relativistic freestreaming gas becomes possible. By solving Eq. (6), we obtain ∆f = 0.046 for the model with r = 0.1, and this uncertainty reduced to ∆f = 0.008 for the ideal experiment.
As expected, the value of ∆f sensitively depends on the value of tensor-to-scalar ratio r. In Fig.3, we plot the value of ∆f as a function of r for the CMBPol and ideal experiments. It is seen that, with the increasing of r, the value of ∆f becomes smaller. For r = 0.3, one has ∆f = 0.020 for CMBPol experiment, and ∆f = 0.005 for ideal experiment. However, when r = 0.01,one has ∆f = 0.233 for CMBPol experiment, and ∆f = 0.030 for the ideal experiment.

IV. CONCLUSIONS
The relativistic free-streaming gas can modify the spectrum of RGWs and consequently reduce the CMB Bpolarization power spectra at the scale ℓ > 200. In this Letter, by taking into account the noise power spectra of the future CMB experiments, we have presented a constraint on the fraction density f of the relativistic freestreaming gas among all the relativistic components during the radiation-dominant stage. We find the value of ∆f strongly depends on the noise of the experiments and the amplitude of the RGWs. CMBPol experiment is expected to obtain ∆f = 0.046 for the model with r = 0.1, and ∆f = 0.020 for the model with r = 0.3. For an ideal experiment, where only the B-polarization contamination by the reduced cosmic lensing effect is included, we expect to have ∆f = 0.008 for the model with r = 0.1, and ∆f = 0.005 for the model with r = 0.3. Our result shows that the experiments, like CMBPol, can provide a great chance to study the relativistic components in the early Universe.