UTTG0203
Damping of Tensor Modes in Cosmology
Steven Weinberg
Theory Group, Physics Department, University of Texas,
Austin, TX, 78712
An analytic formula is given for the traceless transverse part of the anisotropic stress tensor due to free streaming neutrinos, and used to derive an integrodifferential equation for the propagation of cosmological gravitational waves. The solution shows that anisotropic stress reduces the squared amplitude by 35.6 % for wavelengths that enter the horizon during the radiationdominated phase, independent of any cosmological parameters. This decreases the tensor temperature and polarization correlation functions for these wavelengths by the same amount. The effect is less for wavelengths that enter the horizon at later times. At the longest wavelengths the decrease in the tensor correlation functions due to neutrino free streaming ranges from 10.7% for to 9.0% for . An Appendix gives a general proof that tensor as well as scalar modes satisfy a conservation law for perturbations outside the horizon, even when the anisotropic stress tensor is not negligible.
I. Introduction
It is widely expected that the observation of cosmological tensor fluctuations through measurements of the polarization of the microwave background may provide a uniquely valuable check on the validity of simple inflationary cosmologies. For instance, for a large class of inflationary theories with single scalar fields satisfying the “slow roll” approximation, the wavenumber dependence and of the scalar and tensor power spectral functions and the ratio of these spectral functions after horizon exit during inflation are related by[1]
(1) 
But in order to use observations to check such relations, we need to know what happens to the fluctuations between the time of inflation and the present. There is a very large literature on the scalar modes, but ever since the first calculations[2] of the production of tensor modes in inflation, with only one exception[3] known to me, the interaction of these modes with matter and radiation has simply been assumed to be negligible in studies of the cosmic microwave background[4]. It is not included in the widely used computer program of Seljak and Zaldarriaga[5]. As we shall see, the effect is not negligible even at the relatively low values of where the type polarization multipole coefficients are likely to be first measured, and becomes quite significant for larger values of .
II. Damping Effects in the Wave Equation
The interaction of tensor modes with matter and radiation vanishes in the case of perfect fluids, but not in the presence of traceless transverse terms in the anisotropic stress tensor. In general, the tensor fluctuation satisfies
(2) 
where dots indicate ordinary time derivatives. Here the components of the perturbed metric are
(3) 
where is treated as a small perturbation; and is the anisotropic part of the stress tensor, defined by writing the spatial part of the perturbed energymomentum tensor as or equivalently
(4) 
where is the unperturbed pressure. In these formulas we are considering only tensor perturbations, so that
(5) 
For a perfect fluid , but this is not true in general. For instance, in any imperfect fluid with shear viscosity , we have[6] . Nevertheless, as we shall show in the Appendix, even where hydrodynamic approximations are inapplicable, becomes time independent as the wavelength of a mode leaves the horizon, and remains timeindependent until horizon reentry. All modes of cosmological interest are still far outside the horizon at the temperature K where neutrinos are going out of equilibrium with electrons and photons, so can be effected by anisotropic inertia only later, when neutrinos are freely streaming.
We can calculate the contribution of freely streaming neutrinos to exactly[7]. We define a density as
(6) 
with labeling individual neutrino and antineutrino trajectories. The relativistic equations of motion in phase space for any metric with and are
(7) 
It follows then that satisfies a Boltzmann equation
(8) 
it being understood that and are expressed in terms of the independent variable by and . At a time soon after neutrinos started free streaming, had the ideal gas form (assuming zero chemical potentials)
(9) 
where is the number of types of neutrinos, counting antineutrinos separately, and is Boltzmann’s constant. We therefore write
(10) 
so that vanishes for .
In the absence of metric perturbations, Eq. (8) and the initial condition (9) have the solution , where is the zerothorder part of :
(11) 
and . To first order in metric perturbations, Eq. (8) gives
(12) 
where hats denote unit vectors. (In putting the Boltzmann equation in this form, we use that fact that depends on and only through the combination , so that to first order .)
We now suppose that the dependence of is contained in a factor , where is a comoving wave number.^{1}^{1}1Conventionally the comoving coordinate and wave number are normalized by defining so that at present. Here we will leave this normalization arbitrary. Eq. (12) and the initial condition that at then have the solution
(13) 
where we now drop the position argument, and write and as functions of a variable instead of , with defined as the wave number times the conformal time
(14) 
The space part of the neutrino energymomentum tensor is given by
(15) 
This yields terms of first order in from and , a term of first order in from the term in , and a term of first order in from . Collecting all these terms and using Eq. (5) yields a surprisingly simple formula for :
(16) 
where primes now indicate derivatives with respect to or ; is the kernel
(17) 
and is the unperturbed neutrino energy density.
To continue, we use Eq. (16) in Eq. (2) and express timederivatives in terms of derivatives. This gives an integrodifferential equation for [8]:
(18) 
where .
We took the initial time to be soon after neutrinos started free streaming, so interesting perturbations are outside the horizon then, and for some time after. As we show in the Appendix, rapidly became time independent after horizon exit, and remained so until horizon reentry. In terms of , we then have the initial condition
(19) 
The solution of Eq. (18) can therefore be put in the general form
(20) 
where satisfies the same integrodifferential equation as
(21) 
and the initial conditions
(22) 
III. Short Wavelengths
We will first consider wavelengths short enough to have reentered the horizon during the radiationdominated era (though long after neutrino decoupling), and then turn to the general case in the next section. We can take the initial time to be early enough so that it can be approximated as , with the zero of time defined so that during the radiationdominated era we have . Then in Eq. (21) we can set , while for 3 neutrino flavors takes the constant value . Then Eq. (21) becomes
(23) 
Because of the decrease of the factor , the righthand of Eq. (23) becomes negligible for , so deep inside the horizon the solution of Eqs. (22) and (23) approaches a homogeneous solution
(24) 
as compared with the solution for . A numerical solution of Eqs. (22) and (23) shows that follows the solution pretty accurately until , when the perturbation enters the horizon, and thereafter rapidly approaches the asymptotic form (24), with and very small. This asymptotic form provides the initial condition for the later period when the matter energy density becomes first comparable and then greater than that of radiation, so the effect of neutrino damping at these later times is still only to reduce the tensor amplitude by the same factor . Hence, for wavelengths that enter the horizon after electron–positron annihilation and well before radiationmatter equality, all quadratic effects of the tensor modes in the cosmic microwave background, such as the tensor contribution to the temperature multipole coefficients and the whole of the “BB” polarization multipole coefficients , are 35.6% less than they would be without the damping due to free streaming neutrinos. (Photons also contribute to , but this effect is much smaller because at last scattering photons contribute much less than 40% of the total energy.)
IV. General Wavelengths
To deal with perturbations that may enter the horizon after the matter energy density has become important, let us switch the independent variable from to , where is at the time of radiation matter equality. To see how they are related, note that
(25) 
The redshift of matterradiation equality is given by , so Eq. (25) can be simplified to read
(26) 
where
(27) 
Since for , the solution of Eq. (26) is
(28) 
The Hubble constant at matterradiation equality has the value , so Eq. (27) can be written
(29) 
where is the wave number of perturbations that just enter the horizon at the time of radiationmatter equality. (Hence in particular the results of the previous section apply for .)
The fraction of the total energy density in neutrinos is well known
(30) 
where
(31) 
A little algebra then lets us put Eq. (21) in the form
(32) 
where is the same as the given by Eq. (17), but with now given by
(33) 
The initial conditions (22) now read
(34) 
We now have to face the complication that for general the value of at last scattering is not in an asymptotic region where the effect of anisotropic inertia is simply to damp by some constant factor. We therefore now have to consider what feature of is related to observations of the cosmic microwave background. It is that enters into the Boltzmann equation for perturbations to the temperature and Stokes parameters[9], so in the approximation of a sudden transition from opacity to transparency, we expect the BB and other multipole coefficients to depend on only through a factor , where is the value of at last scattering. Hence we will be primarily interested in calculating the value of for various values of , and comparing these values with what they would be in the absence of anisotropic inertia.
For K, we have , so, taking , the parameter is
It will be useful also to have an idea of the value of for which the multipole coefficients in various correlation functions are dominated by perturbations with a given . The dominant contribution to a multipole coefficient of order comes from wave numbers , where is at the time of last scattering, and is the angular diameter distance of the surface of last scattering, which for flat geometries is:
where is the redshift of last scattering. Thus the multipole order that receives its main contribution from wave lengths that are just coming into the horizon at matterradiation equality is
(35) 
where is the redshift of matterradiation equality. For present radiation temperature K and this redshift is . If also and then the integral in Eq. (35) has the value 3.195, and so Eq. (35) gives . Hence for these cosmological parameters, Eq. (29) gives
When referring below to specific values of , it will always be understood that the conversion from to has been made using these cosmological parameters, but it should be kept in mind that the dependence of the function on and is independent of cosmological parameters, and that the value of at last scattering depends only on , . and , not on or .
Let us first consider the case , which for the above cosmological parameters corresponds to . Here the kernel has the constant value , and Eqs. (32) and (34) have a solution of the form
(36) 
where is independent of , and satisfies the inhomogeneous differential equation
(37) 
and the initial conditions
(38) 
According to the above discussion, the streaming of free neutrinos damps the various tensor correlation functions of the cosmic microwave background by a factor , which for becomes , the subscript denoting quantities calculated ignoring this damping, i.e., for , and again equal to the ratio of at last scattering to that at matterradiation equality. Numerical solutions of Eqs. (37) and (38) for and for show that the damping factor is very close to a linear function of and hence of for observationally favored values of , increasing from 0.893 at to 0.910 for .
This damping is relatively insensitive to for small . For instance, numerical integration of Eqs. (32) and (34) shows that for , the damping has only decreased from 9% to 8% for (), and to 7% for (). Matters are more complicated for larger values of and , because the damping factor is the ratio of two oscillating functions with slightly different phases, so that the plot of vs. shows narrow spikes: this ratio becomes infinite at values of for which vanishes and then almost immediately drops to zero at the slightly larger value of for which vanishes. (Even if we average over the small range of values over which last scattering occurs, the plot of vs. still shows finite though high narrow spikes at the zeroes of .) These spikes are not particularly interesting, because they occur at values of where is particularly small, so that the multipole coefficients in the various tensor temperature and polarization correlation functions will be very difficult to measure for the corresponding values of . The values of in the relatively flat regions between the spikes steadily decreases from the value for to a value close to the result found in the previous section for .
The effects considered in this paper will doubtless eventually be taken into account in the computer programs used to analyze data from PLANCK and other future facilities. In the meanwhile, the planning of future observations should take into account that the damping of cosmological gravitational waves is not negligible.
ACKNOWLEDGMENTS
I am grateful for valuable conversations with Richard Bond, Lev Kofman, Eiichiro Komatsu, Richard Matzner and Matias Zaldarriaga. Thanks are due to Michael Trott for advice regarding the numerical solution of Eq. (18), and to Matthew Anderson for checking the numerical results. This research was supported in part by the Robert A. Welch Foundation, by NSF Grant PHY0071512, and by the US Navy Grant No. N00014031 0639, “Quantum Optics Initiative.”
APPENDIX: SUPERHORIZON CONSERVATION LAWS
This Appendix will prove a result quoted in Section II, that in all cases there is a tensor mode whose amplitude remains constant outside the horizon, even where some particles may have mean free times comparable to the Hubble time. The argument is similar to one used previously to show the existence under very general conditions of two scalar modes for which a quantity is constant outside the horizon.[10] It is based on the observation that for zero wave number the Newtonian gauge field equations and the dynamical equations for matter and radiation as well as the condition are invariant under coordinate transformations that are not symmetries of the unperturbed metric.^{2}^{2}2In this respect, the theorem proved here is similar to the Goldstone theorem[11] of quantum field theory. The modes for which or are constant outside the horizon take the place here of the Goldstone bosons that become free particles for long wavelength. The most general such transformations are
(A1) 
where , is an arbitrary function of time, and is an arbitrary constant matrix. Under these conditions we have something like a Goldstone theorem: since the metric satisfies the field equations both before and after the transformation, the change in the metric under these transformations must also satisfy the field equations. This change is simply
(A2) 
This means that for zero wave number we always have a solution with scalar modes
(A3) 
and a tensor mode
(A4) 
(The notation for and is standard, and the same as in Ref. [10].) These are just gauge modes for zero wave number, but if they can be extended to nonzero wave number they become physical modes, since the transformations (A1) are not symmetries of the field equations except for zero wave number. For the scalar modes there are field equations that disappear in the limit of zero wave number, so that conditions and (where is the scalar part of the anisotropic inertia, called in Ref. [10]) and is the perturbation to the velocity potential) must be imposed on the solutions (A3) for them to have an extension to nonzero wave number. It follows then that the zero wave number scalar modes that become physical for nonzero wave number satisfy
(A5) 
Then for zero wave number the quantity has the timeindependent value
(A6) 
For tensor modes there are no field equations that disappear for zero wave number, so the solution constant automatically has an extension to a physical mode for nonzero wave number.
As examples, we note that both the anisotropic stress tensor for an imperfect fluid with shear viscosity and the tensor (16) for freely streaming neutrinos vanish for , so in the limit of zero wave numbers Eq. (2) has a solution with . The above theorem shows that this result applies even when some particle’s mean free time is comparable with the Hubble time, in which case neither the hydrodynamic nor the free streaming approximations are applicable.
The solution with for zero wave number is not the only solution, but the other solutions decay rapidly after horizon exit. There is no anisotropic inertia in scalar field theories, and in the absence of anisotropic inertia, Eq. (2) for zero wave number has two solutions, one with constant, and the other with , for which rapidly becomes constant. The energymomentum tensor of the universe departs from the perfect fluid form later, during neutrino decoupling, and perhaps also during reheating or periods of baryon or lepton nonconservation, but during all these epochs cosmologically interesting tensor fluctuations are far outside the horizon, and hence remain constant.
References

A. Starobinsky, Sov. Astron. Lett. 11, 133 (1985); E. D. Stewart and D. H. Lyth, Phys. Lett. 302B, 171 (1993).

V. A. Rubakov, M. Sazhin, and A. Veryaskin, Phys. Lett. 115B, 189 (1982); R. Fabbri and M.D. Pollock, Phys. Lett. 125B, 445 (1983); L. F. Abbott and M. B. Wise, Nuclear Physics B244, 541 (1984). A. A. Starobinskii, Sov. Astron. Lett. 11, 133 (1985).

The effects of anisotropic inertia due to both neutrinos and photons were included in a large program of numerical calculations reported by J. R. Bond, in Cosmology and Large Scale Structure, Les Houches Session LX, eds. R. Schaeffer, J. Silk, and J. ZinnJustin (Elsevier Science Press, Amsterdam, 1996). Bond concluded from the numerical results that there is an average ‘ 20%’ reduction of the squared tensor amplitude for multipole order larger than about 100, and that this would not be observable in measurements of because according to Eq. (1) tensor modes already make a much smaller contribution to than scalar modes. It is the prospect of cosmic microwave background polarization measurements that makes the effect of anisotropic inertia on the tensor amplitude important.

See, e.g., V. F. Mukhanov, H.A. Feldman, and R. H. Brandenberger, Physics Reports 215, 203 (1992); M. S. Turner, M. White, and J. E. Lidsey, Phys. Rev. D 48, 4613 (1993); M. S. Turner, Phys Rev. D. 55, 435 (1997); D. J. Schwarz, astroph/0303574.

U. Seljak and M. Zaldarriaga, Astrophys. J. 469, 437 (1996).

S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), Eq. (15.10.39). (It should be noted that as defined in this reference is times the used in the present work.) For , this formula for gives the damping of gravitational waves that had been calculated by S. W. Hawking, Astrophys. J. 145, 544 (1966).

Differential equations for both the scalar and tensor parts of the anisotropic stress tensor were given by J. R. Bond and A. S. Szalay, Astrophys. J. 274, 443 (1983), using an orthonormal basis instead of the coordinate basis used here, but the result was applied only for the scalar modes.

For perturbations outside the horizon, where , we can replace with , and the integral in Eq. (8) becomes just . Aside from the term , this equation in the radiationdominated case is then equivalent to Eq. (4.3) of C. W. Misner, Astrophys. J. 151, 431 (1968), which was derived to study a phenomenon different from that considered here: the approach to isotropy of a homogeneous anisotropic cosmology. (This equation was generalized to the case of finite mean free times by C. Misner and R. Matzner, Astrophys. J. 171, 415 (1972).) Misner took (but ), on the ground that a constant term in could be made to vanish by a coordinate transformation, and found a decaying solution. But a constant term in is only a gauge mode when is strictly zero. As remarked in the Appendix, the existence of this gauge mode means that there is a physical mode with for which becomes constant outside the horizon, where is negligible, but which becomes timedependent when the wavelength reenters the horizon. [After the preprint of this work was circulated, I learned of an article by A. K. Rebhan and D. J. Schwarz, Phys. Rev. D 50, 2541 (1994), which obtained an integrodifferential equation like Eq. (18), but with extra terms representing more general initial conditions. No attempt was made to identify the initial conditions that would actually apply cosmologically, or to calculate the damping effect relevant to the cosmic microwave background.]

See, e. g., M. Zaldarriaga and U. Seljak, Phys. Rev. D55, 1830 (1997).

S. Weinberg, Phys. Rev. D67, 123504 (2003).

J. Goldstone, Nuovo Cimento 9, 154 (1961); J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962).