58. Stability of de Sitter space: dS as a perfect interferometer

Posted by Dmitry

June 13, 2008

Let us now show that QFT of a massive scalar field in de Sitter space features instabilities if the number of dimensions is odd. The expression for the two-point function found in the previous post will be of no help, so we will have to switch to the language of Bogolyuov coefficients and modes.

In the global coordinates the metric of the $dS_{d}$ space is given by

$ds^{2}=-d\tau^{2}+{\rm cosh}^{2}\tau\, d\Omega_{d-1}^{2},$ (1)

where $d\Omega_{d-1}^{2}$ is the metric of (d-1) -dimensional sphere. Let us consider a free massive scalar field on the de Sitter background. General solution of the Klein-Gordon equation

$(\nabla^{2}-m^{2})\phi=0$ (2)

can be represented as a sum over spherical harmonics

$\phi(t,x)=\sum_{L,j}y_{L}(\tau)Y_{Lj}(\Omega),$ (3)

where the functions $y_{L}(\tau)$ satisfy the equation

$\ddot{y}_{L}+(d-1){\rm tanh}\tau\dot{y}_{L}+\left(m^{2}+\frac{L(L+d-2)}{{\rm cosh}^{2}\tau}\right)y_{L}=0.$ (4)

Let us introduce

$\mu=\sqrt{m^{2}-\frac{(d-1)^{2}}{4}}.$ (5)

In this post, we only consider the case when is large enough for the expression above to be real (i.e., the case of heavy scalar field).

By substitution

$\sigma=-e^{-2\tau},\, y_{L}=e^{\left(L+\frac{d-1}{2}-i\mu\right)\tau}x$ (6)

this equation is transformed into the hypergeometric form

$\sigma(1-\sigma)x'{}'+\left(1-i\mu-\left(2L+d-i\mu\right)\sigma\right)x'-$
$-\left(L+\frac{d-1}{2}\right)\left(L+\frac{d-1}{2}-i\mu\right)x=0.$ (7)

In-modes (corresponding to the absence of particles at ${\cal I}_{-}$ , i.e., at $\tau\to-\infty$ ) are given by

$y_{L}^{{\rm in}}(\tau)=\frac{1}{\sqrt{N}}{\rm cosh}^{L}\tau\cdot{}e^{\left(L+\frac{d-1}{2}-i\mu\right)\tau}\cdot$
$\cdot{}F\left(L+\frac{d-1}{2};\, L+\frac{d-1}{2}-i\mu;\,1-i\mu;-e^{-2\tau}\right).$ (8)

At $\tau\to-\infty$ one has

$y_{L}^{{\rm in}}(\tau)\sim e^{\tau\left(\frac{d-1}{2}-i\mu\right)},$ (9)

so that in-modes are positive frequency modes at $\tau\to-\infty$ .

Since the equation (4) is symmetric with respect to the transformation $\tau\to-\tau$ , out-modes (corresponding to the absence of particles at ${\cal I}_{+}$ , i.e., at $\tau\to+\infty$ ) can be immediately identified as

$y_{L}^{{\rm out}}(\tau)=y_{L}^{{\rm in}*}(-\tau),$ (10)

so that

$y_{L}^{{\rm out}}(\tau)=\frac{1}{\sqrt{N}}{\rm cosh}^{L}\tau\cdot{}e^{-\left(L+\frac{d-1}{2}+i\mu\right)\tau}\cdot$
$\cdot{}F\left(L+\frac{d-1}{2};\, L+\frac{d-1}{2}+i\mu;\,1+i\mu;-e^{-2\tau}\right).$ (11)

At $\tau\to+\infty$ one has

$y_{L}^{{\rm out}}(\tau)\sim e^{-\tau\left(\frac{d-1}{2}+i\mu\right)},$ (12)

so that out-modes are positive frequency modes at $\tau\to+\infty$ . Normalization of both in- and out-modes is easily found to be

$N=\frac{\mu}{2^{2L+d-2}}$ . (13)

As one may notice, both in- and out-modes are divergent at $\tau=0$ : according to the Raabe criterion, hypergeometric series defining the functions

$F\left(L+\frac{d-1}{2};\, L+\frac{d-1}{2}\pm i\mu;\,1\pm i\mu;-e^{\pm2\tau}\right)$ (14)

diverge there for any and $\mu$ . Therefore, strictly speaking, we are not allowed to calculate matrix elements between in- and out-modes directly; instead, we have to introduce some modes $\phi^{0\pm}$ regular at $\tau=0$ to calculate the Bogolyubov coefficients $(\phi^{0-},\phi^{{\rm in}})$ and $(\phi^{0+},\phi^{{\rm out}})$ separately. As we see, something interesting happens near the throat $|\tau|\lesssim H^{-1}$ of the de Sitter hyperboloid.

(Note that Strominger, Bousso and Maloney calculate Bogolyubov coefficients between $y_{L}^{in}$ and $y_{L}^{out}$ directly, which is not fare way to do the caclulation from my point of view.)

To show how particles are created between ${\cal I}_{-}$ and ${\cal I}_{+}$ infinities, any choice of modes $\phi^{0\pm}$ is suitable. However, to demonstrate the physical essence of instability in de Sitter space, we choose Euclidean modes as $\phi^{0\pm}$ , since they have a remarkable property of CPT-invariance

$y_{L}^{E}(\tau)=y_{L}^{E*}(-\tau)$

(compare it with the condition (10).

Normalized Euclidean modes have the form

$\phi_{L}^{E}(x)=\frac{2^{L+d/2-1}i^{-L+\frac{d-1}{2}}}{\sqrt{\mu}f_{L}\sqrt{e^{2\pi\mu}-1}}{\rm cosh}^{L}\tau\cdot$
$e^{\left(L+\frac{d-1}{2}+i\mu\right)\tau}\cdot{}F\left(L+\frac{d-1}{2};\,{}L+\frac{d-1}{2}+i\mu;\,2L+d-1;\,1+e^{2\tau}\right),$ (11)

where

$f_{L}=\frac{\Gamma(2L+d-1)}{\Gamma\left(L+\frac{d-1}{2}\right)}\left|{}\frac{\Gamma(i\mu)}{\Gamma\left(L+\frac{d-1}{2}-i\mu\right)}\right|$ . (12)

Using properties of the hypergeometric functions, we immediately find that

$\phi_{L}^{E}=\alpha_{L}\phi_{L}^{{\rm in}}+\beta_{L}\phi_{L}^{{\rm in}*}$ , (13)

where the Bogolyubov coefficients are

$\alpha_{L}=(\phi_{L}^{E},\phi_{L}^{{\rm in}})=\frac{e^{i\theta_{L}}}{\sqrt{1-e^{-2\pi\mu}}},$ (14)

$\beta_{L}=-(\phi_{L}^{E},\phi_{L}^{{\rm in}*})=\frac{i^{d-1}e^{-\pi\mu}e^{-i\theta_{L}}}{\sqrt{1-e^{-2\pi\mu}}},$ (15)

where

$e^{-2i\theta_{L}}=(-1)^{L-\frac{d-1}{2}}\frac{\Gamma(-i\mu)\Gamma\left(L+\frac{d-1}{2}+i\mu\right)}{\Gamma(i\mu)\Gamma\left(L+\frac{d-1}{2}-i\mu\right)}$ (16)

and the scalar product is defined as usual:

$(\phi_{1},\phi_{2})=-i(y_{1}\partial_{\tau}y_{2}^{*}-y_{2}^{*}\partial_{\tau}y_{1}^{*})$ (17)

(we integrated over angles of $d\Omega_{d-1}$ and used orthogonalitity of spherical harmonics).

The Bogolyubov coefficients between Euclidean and out-modes

$\gamma=(\phi_{L}^{{\rm out}},\phi_{L}^{E}),\,\,\,\delta=(\phi_{L}^{{\rm out}},\phi_{L}^{E*})$ (18)

are simply related to (14), (15). Indeed, one finds

$\alpha=\gamma,\,\,\,\beta=-\delta^{*}.$ (19)

After a trivial calculation we conclude that

$\phi_{L}^{{\rm out}}=(\gamma\alpha+\delta\beta^{*})\phi_{L}^{{\rm in}}+(\gamma\beta+\delta\alpha^{*})\phi_{L}^{{\rm in}*}=$
$=(\alpha^{2}-\beta^{*2})\phi_{L}^{{\rm in}}+(\alpha\beta-\alpha^{*}\beta^{*})\phi_{L}^{{\rm in}*}$ . (20)

Therefore, there is no particle production in de Sitter space (in- and out-vacua coincide) if $\alpha\beta=\alpha^{*}\beta^{*}$ . (21)

From the expressions for Bogolyubov coefficints we immediately see that

$\alpha\beta-\alpha^{*}\beta^{*}=\frac{e^{-\pi\mu}}{1-e^{-2\pi\mu}}(i^{d-1}-(-i)^{d-1})=$
$=\frac{e^{-\pi\mu}i^{d-1}}{1-e^{-2\pi\mu}}(1-(-1)^{d-1}).$ (22)

Therefore, if is odd, interference between $\tau\in(-\infty,0)$ and $\tau\in(0,+\infty)$ parts of de Sitter is desctructive, in- and out-vacua coincide (there is no overall particle production), and de Sitter space is stable. On the other hand, if d is even, interference between $\tau\in(-\infty,0)$ and $\tau\in(0,+\infty)$ part of de Sitter is constructive, and de Sitter space should be unstable. The distribution of particles produced in the throat is independent of the angular momentum and is given by

$n_{L}=\frac{4e^{-2\pi\mu}}{(1-e^{-2\pi\mu})^{2}}$ , (23)

so that the total number of produced particles strongly diverges. It would be nice to see the instability of odd-dimensional de Sitter space at the level of Green’s functions though, and we will show it but not this time :-)

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Cosmological perturbations and large scale structure, Cosmology, Journal club, Lectures on de Sitter spacetime, Master Postdoc, Quantum field theory

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Comments

Comment by wandering.the.cosmos (Check me out!) on June 15, 2008 @ 6:45 am

Dear Dmitry,

Your equation (4) has contains a single tau derivative (d/d tau)y_L. I’m not sure why it is symmetric in the transformation tau -> -tau — could you explain?

I’ve found instead, if I first substitute y_L(tau) = exp(-tau(d-1)/2) z_L(tau), then the equation takes the form

z_L”(tau) - ( (1/4)(d(d - 2) + 1) - m^2 - L (d + L - 2) sech^2(t) ) z_L(tau) = 0,

which can also be solved in terms of hypergeometric functions (probably similar/same answer as yours? I’ve not checked…) and is symmetric under tau -> - tau.

I’d also like to ask you: I’ve heard before that de Sitter spacetime does not admit a S-matrix. Is this true? And is it related to this divergence at tau = 0 that you’re discussing here?

Thank you!

P.S. I should say I “discovered” your blog a week or two ago and I hope to read it on a regular basis and also try to learn some QFT from your older posts.

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Comment by wandering.the.cosmos (Check me out!) on June 15, 2008 @ 9:17 am

I’m reading your posts backwards. Having read your post #57, I think there is simply a slight typo in your equation (4): the term with one tau derivative should be (d?1) tanh[tau] y_L’[tau], cf. eq. 3.9 of Bousso, Maloney, and Strominger [arXiv: hep-th/0112218].

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Comment by Dmitry (Check me out!) on June 15, 2008 @ 9:20 am

Dear WTC

> Your equation (4) has contains a single tau derivative (d/d > tau)y_L. I?m not sure why it is symmetric in the
> transformation tau -> -tau ? could you explain?

Indeed, (4) as I wrote it is not invariant w.r.t. \tau->-\tau. This is because I forgot to add \tanh \tau in the second term :-) Fixed now and thanks for finding misprint.

> I?d also like to ask you: I?ve heard before that de Sitter > spacetime does not admit a S-matrix. Is this true? And is
> it related to this divergence at tau = 0 that you?re
> discussing here?

You probably had in mind AdS, didn’t you? Quantum gravity in AdS does not have a description in terms of S-matrix, instead, one has to describe it in terms of boundary degrees of freedom (AdS/CFT).

Actually, dS admits in-out S-matrix for odd dimensions; what I calculate is exactly S-matrix elements between in- and out-states. When number of dimensions is even, de Sitter admits in-in S-matrix (singularity at \tau=0 basically means that there is particle production and in-out S-matrix is of no good).

Your comments and questions are very much welcome.
Cheers.

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Comment by Dmitry (Check me out!) on June 15, 2008 @ 10:06 am

Hi again WTC

You figured out everything yourself while I was posting my answer.

Cheers

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58. Stability of de Sitter space: dS as a perfect interferometer

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58. Stability of de Sitter space: dS as a perfect interferometer

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