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Dark Energy, Inflation and Extra Dimensions

Paul J. Steinhardt, Daniel Wesley

TL;DR

The paper investigates how accelerated expansion, from inflation or dark energy, imposes stringent constraints on higher-dimensional theories obtained by compactification under the null energy condition (NEC).Using a macro-to-micro approach with Einstein gravity in all dimensions and RF/CRF metric forms, it derives no-go theorems via $A$-averaging techniques that relate higher-dimensional stress to 4d cosmology, and introduces key quantities like the breathing mode $\\xi$ and warp factor $\\Omega$.For NEC-satisfying models, sustained inflation and ΛCDM-like dark energy are incompatible; the equation-of-state parameter $w$ must lie below a transient bound $w_{\\rm transient}$ (dependent on the number of extra dimensions $k$), making long inflation or near-$-1$ dark energy untenable unless moduli vary with time or NEC is violated.If NEC is violated, the NEC-violating components must be inhomogeneously distributed in the compact dimensions and must vary in time in precise synchrony with 4d energy density, implying significant back-reaction and fine-tuning challenges for viable cosmologies.Overall, the results provide broad, macro-level constraints on higher-dimensional cosmologies and have substantial implications for string/M-theory compactifications, illustrating the tension between extra dimensions and observed cosmic acceleration.

Abstract

We consider how accelerated expansion, whether due to inflation or dark energy, imposes strong constraints on fundamental theories obtained by compactification from higher dimensions. For theories that obey the null energy condition (NEC), we find that inflationary cosmology is impossible for a wide range of compactifications; and a dark energy phase consistent with observations is only possible if both Newton's gravitational constant and the dark energy equation-of-state vary with time. If the theory violates the NEC, inflation and dark energy are only possible if the NEC-violating elements are inhomogeneously distributed in thecompact dimensions and vary with time in precise synchrony with the matter and energy density in the non-compact dimensions. Although our proofs are derived assuming general relativity applies in both four and higher dimensions and certain forms of metrics, we argue that similar constraints must apply for more general compactifications.

Dark Energy, Inflation and Extra Dimensions

TL;DR

The paper investigates how accelerated expansion, from inflation or dark energy, imposes stringent constraints on higher-dimensional theories obtained by compactification under the null energy condition (NEC).Using a macro-to-micro approach with Einstein gravity in all dimensions and RF/CRF metric forms, it derives no-go theorems via $A$-averaging techniques that relate higher-dimensional stress to 4d cosmology, and introduces key quantities like the breathing mode $\\xi$ and warp factor $\\Omega$.For NEC-satisfying models, sustained inflation and ΛCDM-like dark energy are incompatible; the equation-of-state parameter $w$ must lie below a transient bound $w_{\\rm transient}$ (dependent on the number of extra dimensions $k$), making long inflation or near-$-1$ dark energy untenable unless moduli vary with time or NEC is violated.If NEC is violated, the NEC-violating components must be inhomogeneously distributed in the compact dimensions and must vary in time in precise synchrony with 4d energy density, implying significant back-reaction and fine-tuning challenges for viable cosmologies.Overall, the results provide broad, macro-level constraints on higher-dimensional cosmologies and have substantial implications for string/M-theory compactifications, illustrating the tension between extra dimensions and observed cosmic acceleration.

Abstract

We consider how accelerated expansion, whether due to inflation or dark energy, imposes strong constraints on fundamental theories obtained by compactification from higher dimensions. For theories that obey the null energy condition (NEC), we find that inflationary cosmology is impossible for a wide range of compactifications; and a dark energy phase consistent with observations is only possible if both Newton's gravitational constant and the dark energy equation-of-state vary with time. If the theory violates the NEC, inflation and dark energy are only possible if the NEC-violating elements are inhomogeneously distributed in thecompact dimensions and vary with time in precise synchrony with the matter and energy density in the non-compact dimensions. Although our proofs are derived assuming general relativity applies in both four and higher dimensions and certain forms of metrics, we argue that similar constraints must apply for more general compactifications.

Paper Structure

This paper contains 9 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Compactified Models and NEC violation
  3. Assumptions
  4. Detecting NEC violation
  5. No-go Theorems for Models that Satisfy NEC
  6. No-go Theorems for Models that Violate NEC
  7. Constraints on Models violating the GR or metric conditions
  8. Conclusions
  9. Appendix: Some Useful Relations

Figures (1)

  • Figure 1: Plot of $w_{transient}$ vs. the number of extra dimensions $k$ for extra-dimensional models based on RF (circle) and CRF (square) metrics. The dashed horizontal segment represents the current value of $w$ according to WMAP.