Vorticity generation in cosmological perturbation theory
Save This Post as PDFAdam Christopherson is a PhD student at Queen Mary, U. of London working with Karim Malik on cosmological perturbation theory. Dmitry.
In this blog post, I will summarize recent work on vorticity generation in cosmological perturbation theory, undertaken by Karim Malik, David Matravers and myself. The main result of the paper this is based on, arxiv:0904.0940, is that at second order in perturbation theory, vorticity generation is sourced by entropy gradients.
Vorticity is a common phenomenon in nature and is defined, in classical fluid dynamics, as the curl of the fluid velocity. It will arise in most situations involving real fluids. However, despite its prevalence, vorticity has rarely been studied in the early universe and in cosmology.
The ’standard cosmological model’ is an expanding homogeneous and isotropic model, described by the Friedmann-Robertson-Walker (FRW) solution of General Relativity. But this is an approximation, and in reality things are not so simple.
Observational evidence indicates that the universe is not exactly homogeneous and isotropic. Since General Relativity is highly nonlinear, finding an exact inhomogeneous solution is extremely difficult, so cosmologists resort to using a powerful method called Cosmological Perturbation Theory. Starting with a FRW universe as the background spacetime, small inhomogeneous perturbations are added, order by order. In this work we consider only scalar and vector perturbations, since tensor perturbations are not crucial to the result. After gauge choice, the perturbed metric is
![ds^2=a^2(\eta)\left[-(1+2A)d\eta^2+B_id\eta dx^i+\delta_{ij}dx^idx^j\right] \,,](http://www.nonequilibrium.net/latexrender/pictures/8b8b9fd098e03603322f2e0d59396778.gif "ds^2=a^2(\eta)\left[-(1+2A)d\eta^2+B_id\eta dx^i+\delta_{ij}dx^idx^j\right] \,,")
where 
is the lapse function, and 
is the shear. Since Einstein’s equations connect the geometry of the universe to its matter content, perturbations in the metric imply perturbations in the energy-momentum tensor.
Fluids in the early universe can be well modeled as perfect fluids. Like any thermodynamical system, this can be characterized by three variables, two of which are independent. A natural choice for the independent variables are the energy density, 
, and the entropy, 
, with the pressure 
being 
. The pressure perturbation can then be expanded in a Taylor series as
where we have defined the non-adiabatic pressure (or entropy) perturbation, as 
. This can be readily extended to second order by simply not truncating the Taylor series after linear order terms.
Vorticity Evolution
The vorticity tensor is defined as the projected, anti-symmetrised covariant derivative of the fluid four velocity:
![\omega_{\mu\nu}=\mathcal{P}_\mu^{~\alpha}\mathcal{P}_\nu^{~\beta}u_{[\alpha;\beta]},\,](http://www.nonequilibrium.net/latexrender/pictures/0e8fe332bb4eff324ff431023ceda369.gif "\omega_{\mu\nu}=\mathcal{P}_\mu^{~\alpha}\mathcal{P}_\nu^{~\beta}u_{[\alpha;\beta]},\,")
where 
is the projection tensor into the instantaneous fluid rest space. This definition is equivalent to 
in standard fluid mechanics. We can then calculate the evolution of the vorticity, making use of energy momentum conservation equations and constraint equations. At first order, we reproduce the well known result that vorticity decays in the absence of anisotropic stress, and that if it is initially zero, it will remain zero. Our novel result comes at second order. We find the evolution equation for the second order vorticity to be
![\omega_{2ij}^\prime -3\mathcal{H}c_{\rm{s}}^2\omega_{2ij}=\frac{2a}{\rho_0+P_0}\left\{3\mathcal{H} V_{1[i}\delta P_{\rm{nad}{1,j]}} +\frac{\delta\rho_{1,[j} \delta P_{\rm{nad}1,i]}}{\rho_0+P_0}\right\}\,,](http://www.nonequilibrium.net/latexrender/pictures/c173883ec20bad575c565e3595b9f52b.gif "\omega_{2ij}^\prime -3\mathcal{H}c_{\rm{s}}^2\omega_{2ij}=\frac{2a}{\rho_0+P_0}\left\{3\mathcal{H} V_{1[i}\delta P_{\rm{nad}{1,j]}} +\frac{\delta\rho_{1,[j} \delta P_{\rm{nad}1,i]}}{\rho_0+P_0}\right\}\,,")
even in the case of vanishing vorticity. Thus, at second order in cosmological perturbation theory, vorticity is sourced by entropy gradients. We note that for barotropic fluids, where the entropy perturbation is zero, we recover the result of Lu et al in arxiv:0812.1349.
It is important to consider the possible observational signatures that vorticity in the early universe will have. As we know, the Cosmic Microwave Background (CMB) is polarized, and the polarization can be classified into E-modes (curl-free) and B-modes (divergence-free). The latter is produced (at linear order) only by vector and tensor perturbations. Tensor perturbations correspond to gravitational waves, and are predicted by inflation, whereas inflation does not produce vector perturbations. However, vector perturbations at second order generated by density perturbations, as discussed in this work, exhibit B-mode polarization as an observational signature. This will prove important for current and future experiments, such as Planck or CMBPol.
Furthermore our result, that vorticity is non-zero at second order in the presence of non-adiabatic or entropy perturbations, has immediate implications on the study of magnetic fields in the early universe, since Biermann showed that the generation of magnetic fields is intimately related to vorticity.
Further Reading:
References to relevant literature can be found in our paper, arxiv:0904.0940. For a recent review on Cosmological Perturbation Theory, see Malik and Wands, arxiv:0809.4944.
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