Reliable gravitational production rate of perturbative string states at the end of inflation and during reheating

Feng, Gao, Li, Song and Song (Nucl. Phys. B 800 (2008) 190) estimated, using the de Sitter string spectrum of Li, Song and Song, that gravitational pair production of perturbative string states at the end of slow-roll inflation and during reheating is exponentially suppressed, so that slow-roll inflation alone offers no cosmic-string production mechanism. That verdict rests on uncontrolled approximations — leading-order steepest descent with $|\beta|\ll1$, a near-saddle expansion of the string-state degeneracy, an undetermined evaluation point $H_r$, unknown inflaton–string couplings, neglected backreaction of the produced strings on the Friedmann dynamics, and validity only for $H\sqrt{\alpha'}\ll1$ — and on a string spectrum whose correctness has since been formally disputed. The open problem is the full quantitative program these authors left open: reliable (numerical or higher-order analytic) production rates and energy densities for concrete inflationary models, the role of direct inflaton–string couplings, and self-consistent backreaction on the cosmological background.

Background

Cosmic strings are one-dimensional topological defects that could survive from the early universe; if they were fundamental strings stretched to cosmological scales (cosmic superstrings), their detection would be direct evidence for string theory. Since 2002 the dominant production mechanism studied is the formation of strings as topological defects when a brane–antibrane pair annihilates at the end of brane inflation, a brane being an extended, membrane-like object of string theory (Sarangi and Tye, Phys. Lett. B 536 (2002) 185). A more traditional alternative is gravitational pair production: a time-dependent spacetime metric at the end of inflation amplifies the vacuum of every quantum degree of freedom, an effect central to reheating, the post-inflationary transfer of the inflaton's energy into ordinary particles (Kofman, Linde and Starobinsky, Phys. Rev. D 56 (1997) 3258; Chung, Phys. Rev. D 67 (2003) 083514). For strings this requires knowing the spectrum of a free string in a curved, near-de Sitter background — de Sitter space is a spacetime of constant positive curvature, the local approximation to an inflationary universe — which was computed by Li, Song and Song (JHEP 04 (2007) 042) by quantizing the bosonic string worldsheet, the two-dimensional surface swept out by a propagating string, in de Sitter space: each physical string state of oscillator level $N$ obeys an oscillator equation with a Hubble-dependent effective frequency

$$W(\eta)^2 = k^2 + (4N + 2E_0)a(\eta)^2 - C^2\,a(\eta)^2 H(\eta)^2,$$

where $a(\eta)$ is the scale factor, $H$ the Hubble parameter, $E_0$ the zero-point energy and $C^2\equiv\sum_{m,i}(3-\epsilon+N^i_m+\tilde N^i_m+2N^i_m\tilde N^i_m)$ a combination of the string oscillators' occupation numbers $N^i_m$, $\tilde N^i_m$ and of the slow-roll parameter $\epsilon\equiv-\dot H/H^2$. Earlier estimates by Gubser (Phys. Rev. D 69 (2004) 123507) used an effective field theory with a toy string spectrum, and Tolley and Wesley (Phys. Rev. D 72 (2005) 124009) and Frey, Mazumdar and Myers (Phys. Rev. D 73 (2006) 026003) studied related stringy effects during inflation, but none used the true de Sitter string spectrum.

Feng, Gao, Li, Song and Song then computed the string energy density produced at the end of slow-roll inflation — a phase of quasi-exponential expansion driven by an inflaton field rolling gently down its potential — and during reheating (arXiv:0707.0908, Nucl. Phys. B 800 (2008) 190). Production occurs only while $H$ varies (in exact de Sitter the Bogoliubov coefficient vanishes); for each string mode they compute the Bogoliubov coefficient $\beta$ — the amplitude mixing positive- and negative-frequency solutions of the oscillator equation, whose modulus squared gives the produced occupation number — by a steepest-descent (saddle-point) evaluation of a contour integral about a complex-time saddle $\eta^\star=r-iu$ with $u>0$, assuming $|\beta|\ll1$, with the evaluation data $H_r$, $a_r$, $R_r$ at the saddle fixed by the coupled saddle-point equations (eq. (3.21)). Summing over the string-state degeneracy $D_{n,l}\sim n^{-5/4}e^{(2\pi/3)\sqrt{6n}}$ — Hagedorn-like growth, the characteristic exponential proliferation of string states with oscillator level $n$ — truncated at the maximum producible level $n_{\max}\sim H^{-2}/4$ set by the on-shell constraint, gives an energy density

$$\rho(\eta)\sim{\alpha'^{-2}\over a(\eta)^3A^{3/2}}\left(e^{-A}-{e^{-A\sqrt{n_{\max}}}\over n_{\max}^{1/4}}\right),\qquad A=\left({8-12a_r^2H_r^2\alpha'/\pi^2\over\sqrt{H_r^2\alpha'+R_r\alpha'/6}}-{2\sqrt{6}\pi\over3}\right),$$

with $H_r$, $a_r$, $R_r$ the Hubble parameter, scale factor and Ricci scalar at the real part $r$ of the complex saddle, and $\alpha'$ the string slope parameter, whose square root sets the fundamental string length. Since the semiclassical spacetime description requires the Hubble scale to be far below the string scale, $H\sqrt{\alpha'}\ll1$, one has $A\gg1$ and $\rho$ is exponentially suppressed: highly excited strings are hardly produced, and slow-roll inflation alone does not provide a cosmic-string production mechanism. (An earlier 2007 version of the paper had reached the opposite conclusion; the published version reversed it.) This verdict, however, is an estimate with uncontrolled errors, and the authors explicitly flag what would be needed to make it reliable — the content of the present problem.

Problem Statement

Determine, quantitatively and with controlled errors, whether gravitational pair production of perturbative string states at the end of slow-roll inflation and during reheating is cosmologically negligible: whether the produced string energy density $\rho(\eta)$ is too small — relative both to the inflaton and radiation densities with which it is jointly evolved and to the density required to seed an observable population of cosmic (super)strings — for this channel to constitute a cosmic-string production mechanism, the sense in which the source concluded that slow-roll inflation alone does not offer one. This requires completing the program explicitly left open by Feng, Gao, Li, Song and Song: (1) compute the mode production probabilities $|\beta_{k,n,l}|^2$ and the resulting time-dependent string energy density $\rho(\eta)$ for concrete inflationary models — numerically or analytically beyond the leading-order steepest-descent evaluation, which assumes $|\beta|\ll1$, expands the string-state degeneracy generating function near $w=1$, and truncates the expansion about the complex-time root — this includes determining the evaluation data $H_r$, $a_r$, $R_r$; (2) determine or bound how direct inflaton–string couplings, which are currently unknown and which, per the source's Appendix A, the usual perturbative reheating picture cannot easily accommodate, modify or add to the purely gravitational production channel; and (3) compute the production rate per unit time and include the resulting time-dependent string energy density self-consistently in the Friedmann equations solved jointly with the inflaton and radiation, replacing the present treatment in which the produced strings' backreaction is neglected. All three parts are required and jointly enable one overarching objective: a reliable verdict, with error control, on whether slow-roll inflation and reheating produce a cosmologically significant density of highly excited strings, a verdict that requires all three components together. That verdict concerns slow-roll inflation and reheating generally, as does the source's own estimate, whose Appendix A treats "most of popular inflation scenarios" and derives $1.5<\epsilon<2$; the computations for particular models are the evidence on which it rests.

References

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