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Dark energy sound speed
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Antony Lewis

Joined: 23 Sep 2004
Posts: 673
Affiliation: University of Sussex

Posted: December 15 2004

Are there any theoretical bounds on what sound speed (meaning $c_s^2 \equiv \delta p / \delta \rho$) the dark energy could have? In particular $c_s^2 <0$ or $c_s^2 > 1$ are often not consisdered, but I don't see any immediate contradictions from the cosmological perturbation equations?
Anze Slosar

Joined: 24 Sep 2004
Posts: 198
Affiliation: Brookhaven National Laboratory

Posted: December 15 2004

Well, isn't the sound speed just what is says on the can, i.e. the speed at which small perturbations move?     I would expecte that cs2>1 would violate causality, while cs2<0 would just give you evanescent waves, i.e. a dissipative media. So cs2<0 might actually be an interesting option after all.
Adam Amara

Joined: 25 Sep 2004
Posts: 18
Affiliation: ETH

Posted: December 16 2004

I'm about to show my ignorance but I have no idea what an evanescent wave is.  Cs2<0 doesn’t seem to make much sense to me but it would be really interested to hear more if it does.    Adam
Adam Amara

Joined: 25 Sep 2004
Posts: 18
Affiliation: ETH

Posted: December 16 2004

Sorry, I've just been reading around.  Just in case anyone else was as clueless as i was:    "An electromagnetic wave observed in total internal reflection, undersized waveguides, and in periodic dielectric heterostructures. While wave solutions have real wavenumbers k, k for an evanescent mode is purely imaginary. Evanescent modes are characterized by an exponential attenuation and lack of a phase shift."    Again, sounds interesting, but how does this apply to cosmology?    Adam
Antony Lewis

Joined: 23 Sep 2004
Posts: 673
Affiliation: University of Sussex

Posted: December 16 2004

How it applies to cosmology is really the question. If you give me a can of dark energy, then indeed $c_s^2$ is just what you think of as the sound speed: for a constant equation of state    $\delta'' + k^2 c_s^2 \delta = 0$    (if the sound speed is negative, you immediately see that the solution is exponential rather than oscillating).    The question I'm asking is about dark energy in the universe. Here the full equations are relativistic and coupled to other matter by GR:    $\delta' + 3H(\hat{c}_{s}^2-w)(\delta +3H(1+w)v/k)+ (1+w)kv + 3H w' v/k= -3(1+w)h'$  $v' + H(1-3\hat{c}_{s}^2)v + kA = k \hat{c}_{s}^2 \delta/(1+w)$    Much more complicated! (h' is a source from background expansion fluctuations, v is the velocity, A is the acceleration (zero in synchronous gauge), H is the conformal hubble rate, and derivatives are conformal time). The sound speed is defined in the rest frame of the dark energy as $c_s^2 \equiv \delta p/ \delta \rho$, which is not by definition positive despite the notation.    In general the equation of state w is a function of time, but the sound speed can be a function of time and wavenumber k. So whether or not there are physical constraints on it - especially on super-horizon scales - is unclear to me. If the dark energy is some effective stuff, say a GR representation of some brane-world effect, I don't think you can argue constraints on its sound speed as though it were a simple fluid that could exist independent of cosmology?
Anze Slosar

Joined: 24 Sep 2004
Posts: 198
Affiliation: Brookhaven National Laboratory

Posted: December 16 2004

Antony Lewis wrote:
In general the equation of state w is a function of time, but the sound speed can be a function of time and wavenumber k. So whether or not there are physical constraints on it - especially on super-horizon scales - is unclear to me. If the dark energy is some effective stuff, say a GR representation of some brane-world effect, I don't think you can argue constraints on its sound speed as though it were a simple fluid that could exist independent of cosmology?

Yes, I agree with you. My point was that locally the dark energy doesn't know about cosmology and hence $c_s^2$ cannot be greater than one, at least for large k. If you take $c_s^2$ to be a function of k, then it can be anything on superhorizon scales... Do you buy this?

Your complicated equations are perfect for beard stroking at 5pm with a cup of tea!
Anze Slosar

Joined: 24 Sep 2004
Posts: 198
Affiliation: Brookhaven National Laboratory

Posted: December 16 2004

Antony Lewis wrote:
Much more complicated! (h' is a source from background expansion fluctuations, v is the velocity, A is the acceleration (zero in synchronous gauge), H is the conformal hubble rate, and derivatives are conformal time). The sound speed is defined in the rest frame of the dark energy as $c_s^2 \equiv \delta p/ \delta \rho$, which is not by definition positive despite the notation.

Following Amara's suit, I will now ask a stupid question. Isn't it that dp / dρ = w by definition of w (i.e. 1/3 for standard waves, etc...) How can you then define δp / δρ for a given k mode and how does it relate to w?
Antony Lewis

Joined: 23 Sep 2004
Posts: 673
Affiliation: University of Sussex

Posted: December 16 2004

The equation of state is defined as w≡p' / ρ'. This is not the same as the definition of $c_s^2$ so in general they are different (indeed for quintessence $c_s^2=1$ always). They are only equal if you apply adiabaticity: not the case for quintessence.    Some refs are astro-ph/0307100, astro-ph/0307104, astro-ph/0410680.
Pier Stefano Corasaniti

Joined: 11 Nov 2004
Posts: 42
Affiliation: LUTH, Observatoire de Paris-Meudon

Posted: December 16 2004

Antony Lewis wrote:
not the case for quintessence.

This statement is too strong, whether scalar field perturbations are adiabatic or isocurvature depends on the initial conditions, which we think are set by inflation.

If the quintessence field is a decay product of the inflaton the perturbations in the quintessence will be adiabatic, which means that not only the perturbations of the quintessence field relative to the other fluids
(Sq f =0) initially vanish, but also the scalar field intrinsic entropy perturbation (Gamma_q =0) is initially zero.

If these conditions are verified at an initial time the conservation of the energy momentum tensor (which provide the equation of motion for the perturbations) impose that they are verified at any other time (no matter the gauge, actually everything can be formulated in gauge invariant language as S and Gamma are gauge invariant variables).

In other words adiabaticity is preserved by the flow equations and no matter what is the background scalar field evolution or the choice of the gauge, the scalar field perturbations will remain adiabatic (Gamma_q=0 always) as those in the other fluids.

This can be inferred from General Relativity arguments in the context of the Separate Universe Approach (Wands, Malik, Lyth and Liddle, astro-ph/0003278).

More specifically we have shown this by directly looking at the flow equations in the gauge invariant formulation (Bartolo, Corasaniti, Liddle, Malquarti, astro-ph/0311503).

Also we have found that if the initial conditions are isocurvature ones, which could be the case if the quintessence or k-essence or whatever is not a decay product of the inflaton, then the evolution of the isocurvature quintessence mode depends on the dynamics of the background. In particular during kination regimes the isocurvature mode is amplified, while it decays during tracker ones. This implies that only those quintessence scenarios with initial isocurvature perturbations and with the field undergoing a long kination phase followed by a short period of tracking can give rise to a cosmologically relevant isocurvature perturbation that can survive at present time. (The specific time evolution of this isocurvature mode can be different in the case of k-essence, since the two scenarios (Q <-> K) can be mapped one into the other only at the homogeneus level).

I tend to accept the fact that cs2, as a free parameter for an effective fluid description of scalar field perturbations, is a simple way of parametrizing something which depends on initial conditions, however from a pure physical point of view I find this effective description rather misleading, so one should have always in mind the microscopic description of the scalar field whatsoever is its lagrangian.

Pier-Stefano
Antony Lewis

Joined: 23 Sep 2004
Posts: 673
Affiliation: University of Sussex

Posted: December 16 2004

I should probably have said internally adiabatic (following e.g. astro-ph/0410680). The rest frame sound speed of a quintessence field is always identically 1 independent of the initial conditions (a simple proof is given in e.g. astro-ph/0307104). Hence for w≠ - 1 it is never internally adiabatic. I hope this is correct?    I take you point to be that for models in which $c_s^2$ is not fixed, it may depend on the initial conditions. This is an interesting point I'd not considered.
Pier Stefano Corasaniti

Joined: 11 Nov 2004
Posts: 42
Affiliation: LUTH, Observatoire de Paris-Meudon

Posted: December 17 2004

Antony Lewis wrote:
I should probably have said internally adiabatic (following e.g. astro-ph/0410680). The rest frame sound speed of a quintessence field is always identically 1 independent of the initial conditions (a simple proof is given in e.g. astro-ph/0307104). Hence for w≠ - 1 it is never internally adiabatic. I hope this is correct?    I take you point to be that for models in which $c_s^2$ is not fixed, it may depend on the initial conditions. This is an interesting point I'd not considered.

If by internally adiabatic you mean that the intrinsic entropy perturbation Γq = 0, this is true as long as the initial conditions are adiabatic and on the large scales approximation. In fact inside the horizon the modes get mixed up even if you start from initial adiabatic conditions, as can be seen from Eq.(33) and (34) in astro-ph/0311503.

Perhaps it is just a different language, $c_s^2=1$ it is used to define adiabatic scalar field perturbations in the effective fluid description, while $c_s^2\ne 1$ would correspond to isocurvature ones.

But I find this wording misleading anyway since as you say in the scalar field rest frame $c_s^2=1$ independently of the initial conditions, while adiabaticity has to do with the initial conditions independently of the gauge. And personally I prefer the microscopic description rather than the effective one, although the former requires to specify a lagrangian.
Fabio Finelli

Joined: 12 Nov 2004
Posts: 2
Affiliation: INAF-CNR/IASF, Sezione di Bologna

Posted: January 19 2005

Dear all,
I take this chance to communicate my ideas on the topic, which have generated
astro-ph/0211626, astro-ph/0304325, astro-ph/0405041.

My motivation was to understand if you can discriminate within different Dark Energy models with a different speed of sound: this was suggested by K-essence authors, since quintessence has always cx2=1. By cX2 I mean the quantity in the pressure
perturbations

δ px = cx2 δ ρx + f(η,wx) Θx

where the second term denotes the intrinsic non-adiabatic pressure perturbations,
θx is the velocity potential for DE, wx is the equation of state of wx and η is the conformal time.

Within scalar field theories (where f(η) ≠ 0)
it is difficult to discriminate between cx2=1 and generic cx (unless cx is drastically different from 1) as it can be seen in astro-ph/0405041.

For perfect fluid of model of DE (where f(η) = 0) , cx2 is locked in some way to wx
and there is a chance to have stronger effects on CMB and LSS (as can be seen in
astro-ph/0211626, astro-ph/0304325). For instance, the Chaplygin gas is ruled out at more than 3 σ as a dark energy model (see astro-ph/0304325).

Cheers,
Fabio

Antony Lewis wrote:
The equation of state is defined as w≡p' / ρ'. This is not the same as the definition of $c_s^2$ so in general they are different (indeed for quintessence $c_s^2=1$ always). They are only equal if you apply adiabaticity: not the case for quintessence.    Some refs are astro-ph/0307100, astro-ph/0307104, astro-ph/0410680.

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