1 Introduction

Nowadays, it is generally believed that laws of gravity may be regarded as a manifestation of laws of thermodynamics for large-scale spacetime systems. This idea has been well established over the past three decades, where it has been confirmed that the gravitational field equations can be constructed from the first law of thermodynamics on the boundary of spacetime [1,2,3]. This approach is also reversible, in that one can start from gravitational field equations governing the dynamics of the system in the bulk and show that it can be rewritten in the form of the first law of thermodynamics on the boundary of the system [4]. The deep connection between thermodynamics and gravity has been explored in different setups. In particular, in the context of Friedmann–Robertson–Walker (FRW) cosmology, it has been shown that the Friedmann equations describing the evolution of the Universe can be rewritten as the first law of thermodynamics on the apparent horizon, and vice versa [5,6,7,8,9,10].

Although general relativity (GR) has successfully passed a number of experimental and observational tests, it still suffers from problems such as curvature singularity at the center of black holes or initial singularity in the standard cosmology. Removing these singularities, using first principles and fundamental theories such as string theory and quantum gravity, is one of the main challenges of modern theoretical physics. To relax singularity in the black hole solutions, regular black holes were proposed by combining the nonlinear electrodynamics with the Einstein–Hilbert Lagrangian [11, 12]. An attempt toward constructing singularity-free black holes was performed by averaging noncommutative coordinates on suitable coherent states in string theory [13,14,15]. It was argued that the effects of noncommutativity are equivalent to a nonlocal deformation of the Einstein–Hilbert action [16, 17].

Another approach for dealing with nonsingular black holes comes from the concept of T-duality in string theory. It was argued that the momentum space propagator induced by the path integral duality, for a particle with mass \(m_0\), is given by [18,19,20,21]

$$\begin{aligned} G(p)=-\frac{l_0 }{\sqrt{p^2+m_0^{2}}} K_{1}\left( l_0 \sqrt{p^2+m_0^{2}}\right) , \end{aligned}$$
(1)

where \(p^2=(\hbar k)^2\) is now the squared momentum, \(l_0\) represents the zero-point length of spacetime, and \(K_{1}(x)\) is a modified Bessel function of the second kind. For the massless particles, the above propagator can be written as (\(\hbar =1\)) [22]

$$\begin{aligned} G(k)=-\frac{l_0 }{\sqrt{k^2}} K_{1}\left( l_0 \sqrt{k^2}\right) . \end{aligned}$$
(2)

In the limiting case where \((l_0 k^2)\rightarrow 0\), one finds \(G(k)=-k^{-2}\) for the massless propagator.

The static Newtonian potential corresponding to G(k) at distance r is obtained as [22]

$$\begin{aligned} V(r)=-\frac{M }{\sqrt{r^2+l_0^2}}, \end{aligned}$$
(3)

where M is the mass of the source of the potential. It has been shown that this finite potential changes the metric around an electrically neutral black hole and leads to a regular spacetime at \(r=0\) [22]. In addition, it modifies the Hawking temperature and the entropy associated with the horizon, and hence with the thermodynamics of black holes [22].

In passing, we would like to mention that Eqs. (1) and (2) do not provide the only way to implement a modification of the propagator induced by a minimal length. For instance, another formalism discussed in the literature consists of constructing an effective quantum metric—qmetric—wherein the concept of minimal length is naturally built into the fabric of spacetime itself. Such an approach essentially makes use of tensorial quantities depending on two points (bitensors, such as the squared interval). It was proposed in [23] and later developed in [24,25,26,27], while implications on the propagator were studied in detail in [28]. Although the two approaches are generally expected to yield similar findings, this might not be the case in specific instances. For example, unlike the zero-point length formalism from string T-duality, the qmetric model predicts a discretization of areas due to the modified metric bitensor, which in turn shows singular components in the limit when the two points coincide. An alternative formalism allowing for a minimal length is the generalized uncertainty principle [29], which accounts for an effective deformation of the Heisenberg relation at Planck scale due to quantum gravitational effects (see [30] for a recent review).

In what follows, we shall focus on the zero-point length formalism from string T-duality. In compliance with the entropic force scenario proposed by Verlinde [31], the effects of this model on Newton’s law of gravity were first explored in [32]. Nevertheless, since cosmology can explore physics at higher energy scales, it seems natural that the implications of the zero-point length could be more likely tested in this context. On the cosmological scales, Eq. (3) along with Verlinde’s conjecture lead to correction terms to the Newtonian cosmology, while in the relativistic regime they give rise to modified Friedmann equations in a FRW background [32]. This procedure also leads to a modified expression for the entropy, due to the zero-point length, associated with the horizon, which is useful in studying the thermodynamics of the Universe. We emphasize that the hypothesis of modified horizon entropy is widespread across many extended theories of gravity [33,34,35,36,37,38] (see also [39] for related applications in particle physics).

Using thermodynamics-gravity conjecture and applying the first law of thermodynamics on the apparent horizon, in this work we derive the modified cosmological field equations in the presence of zero-point length from string T-duality. Research in this direction has been active in the last three decades [40,41,42,43], focusing primarily on the development of a minimal-length cosmology in the framework of the generalized uncertainty principle [30, 44,45,46,47,48,49,50,51]. Compared to previous efforts, our analysis advances the understanding of the semiclassical regime of quantum cosmology. Indeed, on the one hand, we offer an alternative perspective by embedding a minimal length from a novel theoretical standpoint. On the other hand, we explore phenomenological consequences of the obtained modified field equations on a wide range of energy scales, including implications for the dynamics of the Universe in the radiation- and matter-dominated eras, the early inflation, and the growth of perturbations and gravitational waves. To the best of our knowledge, such a dedicated analysis has not yet been considered in the literature. On top of that, we extract a constraint on the zero-point length. The result aligns with the original assumption that this cutoff scale is identifiable with the Planck length [18], as well as with the recent estimate of the zero-point length in the context of quantum-corrected black holes [22].

The work is structured as follows. In the next section we develop a modified cosmological scenario based on the zero-point length and study its implications for the evolution of the Universe. Section 3 is devoted to an analysis of inflation, while in Sect. 4 we focus on the growth of perturbations and structure formation. Effects on the spectrum of primordial gravitational waves are explored in Sect. 5. The discussion of closing remarks and perspectives is reserved for Sect. 6. Unless otherwise stated, we shall use Planck units \(\hslash =c=G=k_B=1\).

2 Modified Friedman equations through zero-point length

According to T-duality, the zero-point length affects the gravitational potential [22]. Thus, the gravitational potential of a spherical mass distribution M at distance \(r=R\) is modified as [52, 53]

$$\begin{aligned} \phi (r)=-\frac{M }{\sqrt{r^2+l_0^2}}|_{r=R}, \end{aligned}$$
(4)

where \(l_0\) is the zero-point length, and its value is expected to be of Planck length [22]. Thus, Newton’s law of gravity for a mass point m is modified as

$$\begin{aligned} \vec {F}=-m \vec {\nabla } \phi (r)|_{r=R}=-\frac{Mm}{R^2}\left[ 1+\frac{l_0^2}{R^2}\right] ^{-3/2} \hat{r}. \end{aligned}$$
(5)

Using the entropic force scenario, this leads to correction to the entropy associated with the horizon as [32]

$$\begin{aligned} 1+\left( \frac{1}{2 \pi R}\right) \frac{\mathrm{{d}}{s}}{\mathrm{{d}}R}=\left( 1+\frac{l_0^2}{R^2}\right) ^{-3/2}, \end{aligned}$$
(6)

where s is the correction term to the total entropy of the horizon,

$$\begin{aligned} S_{h}=\frac{A}{4}+s(A), \end{aligned}$$
(7)

and \(A=4\pi R^2\) is the area of the horizon that encloses the volume\(V=4\pi R^3/3\). Combining Eqs. (6) and (7), we arrive at

$$\begin{aligned} \mathrm{{d}}S_h=2\pi R \left( 1+\frac{l_0^2}{R^2}\right) ^{-3/2} \mathrm{{d}}R. \end{aligned}$$
(8)

Before moving on to the cosmological analysis, it is worth noting that deviations from the area-law scaling are very common in quantum gravity, for example, in string theory [54], loop quantum gravity [55], anti-de Sitter/conformal field theory (AdS/CFT) correspondence [56], and generalized uncertainty principle [57]. Additionally, entropic corrections appear from entanglement entropy calculations in four spacetime dimensions at ultraviolet (UV) scale [58]. Therefore, despite the specific string T-duality framework we are considering here, our next arguments are expected to have more general validity in the context of quantum gravity.

Let us assume our Universe is described by the line element

$$\begin{aligned} \mathrm{{d}}s^2={h}_{\mu \nu }\mathrm{{d}}x^{\mu } \mathrm{{d}}x^{\nu }+R^2(\mathrm{{d}}\theta ^2+\sin ^2\theta \mathrm{{d}}\phi ^2), \end{aligned}$$
(9)

where \(R=a(t)r\), \(x^0=t, x^1=r\), and \(h_{\mu \nu }\) = diag \((-1, a^2/(1-kr^2))\) stands for the two-dimensional metric; \(k = -1,0, 1\) represents the curvature of the Universe and a the (time-dependent) scale factor. The apparent horizon is the suitable boundary of the Universe from a thermodynamic viewpoint, which has radius [59]

$$\begin{aligned} R=\frac{1}{\sqrt{H^2+k/a^2}}. \end{aligned}$$
(10)

The temperature associated with the horizon is given by [6, 60]

$$\begin{aligned} T_h=-\frac{1}{2 \pi R}\left( 1-\frac{\dot{R}}{2H R}\right) , \end{aligned}$$
(11)

where the overdot denotes the derivative with respect to the cosmic time t. For an adiabatic expansion, one may assume \(\dot{R}\ll 2HR\) in the infinitesimal interval dt, which physically implies that the apparent horizon radius is kept nearly fixed [5].

The total energy momentum tensor is assumed in the form of the perfect fluid

$$\begin{aligned} T_{\mu \nu }=(\rho +p)u_{\mu }u_{\nu }+pg_{\mu \nu }, \end{aligned}$$
(12)

where \(\rho \) and p are the energy density and pressure, respectively. The conservation of energy for the metric leads to

$$\begin{aligned} \dot{\rho }+3H(\rho +p)=0, \end{aligned}$$
(13)

where \(H=\dot{a}/a\) is the Hubble parameter. The work density due to the volume change of the Universe is [61]

$$\begin{aligned} W=-\frac{1}{2} T^{\mu \nu }h_{\mu \nu }=\frac{1}{2}(\rho -p). \end{aligned}$$
(14)

Taking the differential form of the total energy \(E=\rho (4\pi R^3/3)\) and using the conservation equation (13), we find

$$\begin{aligned} \mathrm{{d}}E=4\pi \tilde{r}_{A}^{2}\rho \mathrm{{d}}\tilde{r}_{A}-4\pi H \tilde{r}_{A}^{3}(\rho +p) \mathrm{{d}}t. \end{aligned}$$
(15)

Inserting Eqs. (8), (11), (14), and (15) into the first law of thermodynamics on the apparent horizon, \(\mathrm{{d}}E = T_h \mathrm{{d}}S_h + W\mathrm{{d}}V\), and using the definition (11) for the temperature, after some calculations, we obtain

$$\begin{aligned} 4\pi H R^3 (\rho +p)\mathrm{{d}}t=\left( 1+\frac{l_0^2}{R^2}\right) ^{-3/2} \mathrm{{d}}R. \end{aligned}$$
(16)

Using the continuity equation (13), we arrive at

$$\begin{aligned} -\frac{2}{R^3}\left( 1+\frac{l_0^2}{R^2}\right) ^{-3/2} \mathrm{{d}}R= \frac{8\pi }{3}\mathrm{{d}}\rho . \end{aligned}$$
(17)

After integrating, we have

$$\begin{aligned} -\frac{2}{l_0^2}\left( 1+\frac{l_0^2}{R^2}\right) ^{-1/2}+C=\frac{8\pi }{3}\rho , \end{aligned}$$
(18)

where C is the constant of integration. Since \({l_0^2}/{R^2}\) is expected to be small, we can expand the left-hand side of the above equations, and we find

$$\begin{aligned} -\frac{2}{l_0^2}\left[ 1- \frac{l_0^2}{2R^2}+\frac{3}{8}\frac{l_0^4}{R^4}+\mathcal {O}\left( \frac{l_0}{R}\right) ^6\right] +C=\frac{8\pi }{3}\rho . \end{aligned}$$
(19)

Clearly, in the limiting case where \(l_0^2\rightarrow 0\), we must restore the standard Friedmann equation. Thus, we set

$$\begin{aligned} C\equiv \frac{2}{l_0^2}-\frac{\Lambda }{3}, \end{aligned}$$
(20)

where \(\Lambda \) is the cosmological constant. The final form of the modified Friedmann equation is obtained as

$$\begin{aligned} H^2+\frac{k}{a^2}-\alpha \left( H^2+\frac{k}{a^2}\right) ^2+\mathcal {O}(\alpha ^2)=\frac{8\pi }{3}\rho +\frac{\Lambda }{3}, \end{aligned}$$
(21)

where \(\alpha =3l_0^2/4\), and we have used relation (10). In a flat Universe (\(k=0\)), the above Friedmann equation reads

$$\begin{aligned} H^2-\alpha H^4=\frac{8\pi }{3}\rho +\frac{\Lambda }{3}. \end{aligned}$$
(22)

As we can see, the use of the modified entropy (7) results in a modified Friedmann equation, with an extra term compared to GR. In passing, we note that such a correction could be re-interpreted as an effective dark energy sector by recasting Eq. (22) as

$$\begin{aligned} H^2=\frac{8\pi }{3}\left( \rho +\rho _{DE}\right) , \end{aligned}$$
(23)

where

$$\begin{aligned} \rho _{DE}=\frac{3}{8\pi }\left( \frac{\Lambda }{3}+\alpha H^4\right) . \end{aligned}$$
(24)

Similarly, the modified second Friedmann equation can be derived by differentiating Eq. (23) with respect to t and using Eq. (13). After some algebra, we obtain

$$\begin{aligned} \dot{H}=-4\pi \left[ \rho +p+\rho _{DE}+p_{DE}\right] , \end{aligned}$$
(25)

where we have defined the pressure of the effective dark energy sector as

$$\begin{aligned} p_{DE}=-\frac{1}{8\pi }\left[ \Lambda +\alpha H^2\left( 4\dot{H}+3H^2\right) \right] . \end{aligned}$$
(26)

Hence, the effective equation of state becomes

$$\begin{aligned} w_{DE}=\frac{p_{DE}}{\rho _{DE}}=-1-\frac{4\alpha H^2 \dot{H}}{3\alpha H^4+\Lambda }=-1-\frac{4\alpha H^2\dot{H}}{\Lambda }+\mathcal {O}(\alpha ^2). \end{aligned}$$
(27)

We would like to note, however, that in the forthcoming analysis we do not conceptualize the zero-point length correction as an effective dark energy contribution. Instead, we adhere to the original interpretation of T-duality and consider the additional term as due to a modified description of gravity.

As expected, in the case \(\alpha =0\), the modified Friedmann equations (23) and (25) reduce to the \(\Lambda \text {CDM}\) paradigm

$$\begin{aligned} H^2= & {} \frac{8\pi }{3}\rho +\frac{\Lambda }{3}, \end{aligned}$$
(28)
$$\begin{aligned} \dot{H}= & {} -4\pi \left( \rho +p\right) . \end{aligned}$$
(29)

2.1 Cosmological solutions

Now we study the cosmological implications of the zero-point length scenario. We shall consider the interesting case where the perfect fluid includes dust matter (\(p_m\approx 0\)) and radiation. From the continuity equation, these components evolve according to

$$\begin{aligned} \rho _m= & {} \rho _{m,0}a^{-3}, \end{aligned}$$
(30)
$$\begin{aligned} \rho _r= & {} \rho _{r,0}a^{-4}, \end{aligned}$$
(31)

where the subscript “0” denotes the values at present time and we set \(a_0=1\).

Dividing both sides of Eq. (22) by \(H_0^2\), we find

$$\begin{aligned} -\beta \left( \frac{H}{H_0}\right) ^4+\left( \frac{H}{H_0}\right) ^2-\Omega _{m,0} a^{-3}-\Omega _{r,0} a^{-4}-\Omega _{\Lambda ,0}=0\,, \end{aligned}$$
(32)

where

$$\begin{aligned} \beta \equiv \alpha H_0 ^2=\frac{3}{4}l_0^2 H_0 ^2\ll 1\,, \end{aligned}$$
(33)

and we have introduced the dimensionless density parameters

$$\begin{aligned} \Omega _i= & {} \frac{8\pi }{3H^2}\rho _i,\quad (i=m,r), \end{aligned}$$
(34)
$$\begin{aligned} \Omega _{\Lambda }= & {} \frac{\Lambda }{3H^2}. \end{aligned}$$
(35)

To convert \(\beta \) in terms of the dimensional zero-point length, it proves convenient to restore for a while the gravitational constant \(G=\ell _p^2=6.7\times 10^{-39}\,\textrm{GeV}^{-2}\), where \(\ell _p\) is the Planck length. For \(H_0\simeq 10^{-42}\,\textrm{GeV}\), we find \(l_0\simeq 1.4\times 10^{61}\sqrt{\beta }\ell _p\). Coming back to our original units, we have

$$\begin{aligned} l_0\simeq 1.4\times 10^{61}\sqrt{\beta }\,. \end{aligned}$$
(36)

Therefore, \(l_0\sim \mathcal {O}(1)\) corresponds to \(\beta \sim \mathcal {O}(10^{-122})\).

Now, solving Eq. (32), we are led to

$$\begin{aligned} \left( \frac{H}{H_0}\right) ^2=\frac{1}{2\beta }\left\{ 1- \sqrt{1-4\beta \left[ \Omega _{m,0} a^{-3}+\Omega _{r,0} a^{-4}+\Omega _{\Lambda ,0}\right] }\right\} . \end{aligned}$$
(37)

Since \(\beta \) is expected to be small, we can expand the right-hand side to the linear order. The result is

$$\begin{aligned} \frac{H^2}{H_0^2}= & {} \left( \Omega _{m,0} a^{-3}+\Omega _{r,0} a^{-4}+\Omega _{\Lambda ,0}\right) \nonumber \\{} & {} \times \left[ 1+\beta \left( \Omega _{m,0} a^{-3}+\Omega _{r,0} a^{-4}+\Omega _{\Lambda ,0}\right) \right] +\mathcal {O}(\beta ^2).\nonumber \\ \end{aligned}$$
(38)

If we define, as usual, the redshift parameter as

$$\begin{aligned} \frac{1}{a}=1+z\,, \end{aligned}$$
(39)

we can rewrite Eq. (38) as

$$\begin{aligned} \frac{H^2(z)}{H_0^2}= & {} \left[ \Omega _{m,0} \left( 1+z\right) ^{3}+\Omega _{r,0} \left( 1+z\right) ^{4}+\Omega _{\Lambda ,0}\right] \nonumber \\ \nonumber{} & {} \times \left\{ 1+\beta \left[ \Omega _{m,0} \left( 1+z\right) ^{3}+\Omega _{r,0} \left( 1+z\right) ^{4}+\Omega _{\Lambda ,0}\right] \right\} \nonumber \\ {}{} & {} +\mathcal {O}(\beta ^2). \end{aligned}$$
(40)

The value of \(\Omega _{\Lambda ,0}\) can be set by imposing the flatness condition \(H(0)/H_0=1\), which gives

$$\begin{aligned} \Omega _{\Lambda ,0}=1-\Omega _{m,0}-\Omega _{r,0}-\beta +\mathcal {O}(\beta ^2)\,. \end{aligned}$$
(41)

A relevant quantity in understanding the dynamics governing the evolution of the Universe is the deceleration parameter

$$\begin{aligned} q=-1-\frac{\dot{H}}{H^2}=-1+\left( 1+z\right) \frac{\partial _z H}{H}\,. \end{aligned}$$
(42)

For \(q>0\), the expansion of the Universe is decelerating, which implies that the gravitational pull of matter dominates over any other influences, causing the expansion to slow down over time. On the other hand, \(q<0\) signifies an accelerating Universe. The discovery of the accelerated expansion, supported by observations of distant type Ia supernovae [62, 63] and the cosmic microwave background (CMB) [64], has led to the acceptance of the existence of some exotic component that exerts negative pressure and counteracts the gravitational attraction of matter.

Implications of the zero-point length on the acceleration of the Universe can be studied by plugging Eq. (37) into (42). We get

$$\begin{aligned} q= & {} \frac{\Omega _{m,0}\left( 1+z\right) ^3+2\Omega _{r,0}\left( 1+z\right) ^4-2\Omega _{\Lambda ,0}}{2\left[ \Omega _{m,0}\left( 1+z\right) ^3+\Omega _{r,0}\left( 1+z\right) ^4+\Omega _{\Lambda ,0}\right] } \nonumber \\{} & {} +\,\frac{\beta }{2}\left[ 3\Omega _{m,0}\left( 1+z\right) ^3+4\Omega _{r,0}\left( 1+z\right) ^4\right] +\mathcal {O}(\beta ^2).\nonumber \\ \end{aligned}$$
(43)

At present time, this reduces to

$$\begin{aligned} q_0=-1+\frac{3}{2}\Omega _{m,0}+2\Omega _{r,0}+\beta \left( 3\Omega _{m,0}+4\Omega _{r,0}\right) +\mathcal {O}(\beta ^2)\,, \end{aligned}$$
(44)

where we have used the condition (41). Note that, in the limiting case where \(\beta =0\), we obtain \(q_0=-1+\frac{3}{2}\Omega _{m,0}+2\Omega _{r,0}\), which correctly reproduces the \(\Lambda \)CDM expression.

It might be interesting to constrain \(\beta \) using the observational value \(q_0=-0.56^{+0.04(0.09)}_{-0.04(0.08)}\) [65]. Substitution into Eq. (44) with \(\Omega _{m,0}\simeq 0.3\) and \(\Omega _{r,0}\simeq 5\times 10^{-5}\) yields \(\beta <\mathcal {O}(10^{-1})\). Additionally, for this range of values of \(\beta \), the transition from deceleration to acceleration phase would occur at redshift \(z_t\in [0.48,0,75]\), which aligns with the estimate \(z_t=0.789^{+0.186}_{-0.165}\) derived from the combined cosmic chronometers, baryonic acoustic oscillations, and supernovae type Ia (CC+BAO+SNeIa) dataset (see also [66]). It should be noted, however, that the above constraint on \(\beta \) provides a very weak condition. This could be somewhat expected, since it is inferred from measurements of cosmic parameters at present time. A more stringent bound will be derived below through the study of inflation.

2.1.1 Matter-dominated era

Let us now investigate the impact of the zero-point length on the dynamics of the scale factor. For the matter-dominated era, neglecting the effects of radiation and cosmological constants, Eq. (38) can be approximated as

$$\begin{aligned} \frac{H^2}{H_0^2}=\Omega _{m,0} a^{-3}\left( 1+\beta \Omega _{m,0} a^{-3}\right) +\mathcal {O}(\beta ^2). \end{aligned}$$
(45)

The explicit form of the scale factor as a function of time is obtained by integrating the above relation. We find

$$\begin{aligned} a(t)=\left[ \frac{9}{4}\Omega _{m,0}\left( H_0^2t^2-\frac{4}{9}\beta \right) \right] ^{\frac{1}{3}}, \end{aligned}$$
(46)

where we have set the integration constant equal to zero. This is indeed the case if we assume \(a\rightarrow 0\) as \(t\rightarrow 0\).

The plot in Fig. 1 shows that the modified scale factor lies below the standard curve for small enough t and vanishes for \(H_0t=2\sqrt{\beta }/3>0\). Below this threshold, Eq. (46) apparently takes complex values. This is, however, a “border” regime, since we are extrapolating our model to scales comparable to or lower than the zero-point length, where the approximation (19) fails. Clearly, this phenomenon indicates a breakdown of the conventional cosmology approaching \(l_0\) and the need for a comprehensive quantum gravity framework. On the other hand, a(t) approaches the standard behavior as the Universe evolves for sufficiently large t, since the zero-point length corrections become increasingly negligible. In the limiting case where \(\beta =0\), one recovers \(a(t)\sim t^{2/3}\), which is the well-known solution in standard cosmology for the matter-dominated Universe.

Fig. 1
figure 1

Plot of a(t) versus \(H_0 t\) for the matter-dominated era and various values of \(\beta \). We set \(\Omega _{m,0}\simeq 0.3\)

Full size image

2.1.2 Radiation-dominated era

For the radiation-dominated era, the Friedmann equation (38) can be rewritten as

$$\begin{aligned} \frac{H^2}{H_0^2}=\Omega _{r,0} a^{-4}\left( 1+\beta \Omega _{r,0} a^{-4}\right) +\mathcal {O}(\beta ^2). \end{aligned}$$
(47)

In this case, the modified scale factor is obtained as

$$\begin{aligned} a(t)=\left[ 4 \Omega _{r,0} \left( H_0^2 t^2 -\frac{\beta }{4}\right) \right] ^{1/4}\,, \end{aligned}$$
(48)

where again we have set the integration constant equal to zero. The curves in Fig. 2 display the same qualitative behavior as discussed in the previous case, for both small and large t. When \(\beta =0\), one recovers \(a(t)\sim t^{1/2}\), which is the expected result in standard cosmology for the radiation-dominated epoch.

For later convenience, we note that the a-dependence in Eq. (47) can be converted to a temperature dependence through the condition \(T a(t)=T_0\), where \(T_0\simeq 3\,\textrm{K}\) is the average temperature of the observable Universe at present time. The latter holds true for any \(\beta \), as it can be checked by using the relation \(\rho \sim T^4\) along with Eqs. (31) and (48). In terms of redshift, we can equivalently write

$$\begin{aligned} \frac{T}{T_0}=z+1\,. \end{aligned}$$
(49)
Fig. 2
figure 2

Plot of a(t) versus \(H_0 t\) for the matter-dominated era and various values of \(\beta \). We set \(\Omega _{r,0}\simeq 5\times 10^{-5}\)

Full size image

2.2 Cosmographic analysis

The depiction of the Universe, as established by the cosmological principle, is referred to as cosmography [67]. Cosmography deals with the study of geometric properties, embedding dimensionless higher-derivative terms of the scale factor [68]. For the FRW Universe, the latter can be expanded as

$$\begin{aligned} a(t)=1+\sum _{n=1}^{\infty }\frac{1}{n!}\frac{\mathrm{{d}}^na}{\mathrm{{d}}t^n}\left( t-t_0\right) ^n\,. \end{aligned}$$
(50)

The coefficients of this Taylor expansion are usually referred to as cosmographic parameters. It is easy to check that the first two parameters are related to the Hubble rate and deceleration parameter, respectively. On the other hand, the third derivative of the scale factor with respect to cosmic time gives the jerk parameter [69, 70]

$$\begin{aligned} j=\frac{1}{a H^3}\frac{\mathrm{{d}}^3a}{\mathrm{{d}}t^3}=q\left( 2q+1\right) +\left( 1+z\right) \frac{\mathrm{{d}}q}{\mathrm{{d}}z}\,, \end{aligned}$$
(51)

which is useful for understanding the acceleration of the Universe and, in particular, the departure of a cosmological model from the \(\Lambda \)CDM.

Fig. 3
figure 3

Plot of j(z) versus z (blue solid curve). The red dashed line sets the \(\Lambda \)CDM value \(j=1\). We set \(\Omega _{m,0}\simeq 0.3\) and \(\Omega _{r,0}\simeq 5\times 10^{-5}\)

Full size image

The evolution of j as a function of redshift is plotted in Fig. 3 (blue solid curve) in comparison with the \(\Lambda \)CDM prediction \(j=1\) (red dashed line). The analytical computation is easily performed by plugging Eq. (43) into (51). As expected, our model mostly deviates from the \(\Lambda \)CDM at higher redshift, while the standard description is recovered approaching the present time. This observation prompts the anticipation of an alternative description of the early Universe in the presence of zero-point length. Furthermore, we find that \(j>0\), which is consistent with an accelerated expansion.

Although the first three cosmographic parameters are sufficient to determine the overall kinematics of the Universe [71], it is sometimes convenient to consider extra terms in the expansion (50). For example, the snap parameter s is a dimensionless quantity that describes the fourth derivative of the scale factor of the Universe with respect to cosmic time. It provides information about the rate of change of acceleration in the expansion of the Universe, thus characterizing the dynamics of cosmic acceleration. Following [72], we have

$$\begin{aligned} s=\frac{1}{a H^4}\frac{\mathrm{{d}}^4a}{\mathrm{{d}}t^4}=-j\left( 2+3q\right) -\left( 1+z\right) \frac{\mathrm{{d}}j}{\mathrm{{d}}z}\,. \end{aligned}$$
(52)

Clearly, for the \(\Lambda \)CDM model \(s=-(2+3q)\) since \(j=1\). Thus, the departure of ds/dq from \(-3\) quantifies the deviation of the evolution of the Universe from the \(\Lambda \)CDM dynamics.

By using Eqs. (43) and (51), we infer the evolution of s in the presence of zero-point length (see Fig. 4). For such small values of \(\beta \), deviations from the \(\Lambda \)CDM model are hardly appreciable in this case even at high redshifts. Observing the behavior of s, it is evident that it assumes negative values in the early Universe, transitioning to positive values as the evolution progresses. At present time \(s_0\simeq -0.2\), which is consistent with the observational estimate of the deceleration parameter \(q_0\simeq -0.6\).

Fig. 4
figure 4

Plot of s(z) versus z (blue solid curve). The red dashed curve gives the \(\Lambda \)CDM prediction. We set \(\Omega _{m,0}\simeq 0.3\) and \(\Omega _{r,0}\simeq 5\times 10^{-5}\)

Full size image
Fig. 5
figure 5

Plot of l(z) versus z (blue solid curve). The red dashed curve gives the \(\Lambda \)CDM prediction. We set \(\Omega _{m,0}\simeq 0.3\) and \(\Omega _{r,0}\simeq 5\times 10^{-5}\)

Full size image

Let us finally consider the lerk parameter, which is given by the fifth derivative of the scale factor as [72]

$$\begin{aligned} l=\frac{1}{a H^5}\frac{\mathrm{{d}}^5a}{\mathrm{{d}}t^5}=-s\left( 3+4q\right) -\left( 1+z\right) \frac{\mathrm{{d}}s}{\mathrm{{d}}z}\,. \end{aligned}$$
(53)

The evolution of l is plotted in Fig. 5, which shows that the lerk parameter maintains exclusively positive values. Similar to the jerk parameter j, both the snap parameter s and the lerk parameter l tend to unity during the later stages of the Universe’s evolution. This trend emphasizes the consistent dynamics exhibited by these parameters in our model.

3 Slow-roll inflation in zero-point length Cosmology

Since the zero-point length is expected to primarily influence the evolution of the early Universe, in this section we investigate its implications for inflation. This phase is usually described in terms of a hypothetical scalar field \(\phi \)—the inflaton—which is thought to have been the driving force behind the inflationary expansion. The dynamics of the inflaton field are governed by the (spatially homogeneous) potential \(V(\phi )\).

Under the canonical scalar field assumption, the model Lagrangian takes the usual form \(\mathcal L=X-V(\phi )\), where \(X=-\frac{1}{2}h^{\mu \nu }\partial _\mu \phi \hspace{0.2mm}\partial _\nu \phi \) is the kinetic term. The energy density and pressure for the inflation are

$$\begin{aligned} \rho _\phi= & {} \frac{\dot{\phi }^2}{2}+V(\phi ), \end{aligned}$$
(54)
$$\begin{aligned} p_\phi= & {} \frac{\dot{\phi }^2}{2}-V(\phi ). \end{aligned}$$
(55)

The continuity equation (13) implies the dynamics

$$\begin{aligned} \ddot{\phi }+ 3H \dot{\phi }+ \partial _\phi V(\phi ) =0\,, \end{aligned}$$
(56)

where \(\partial _\phi V\equiv \frac{\mathrm{{d}}V}{\mathrm{{d}}\phi }\).

The successful models of inflation are typically characterized by a very flat potential along which the inflaton rolls down toward a minimum. If the evolution of the field occurs so slowly that the kinetic energy is negligible compared to the potential term, then the inflaton exhibits a cosmological constant-like behavior, leading to an exponential expansion of the Universe. During this phase, which is referred to as slow-roll inflation, matter and radiation components are diluted away rapidly, allowing us to neglect all sources of energy density other than the inflaton.

Mathematically speaking, the slow-roll approximation is expressed by the conditions [73,74,75]

$$\begin{aligned} \ddot{\phi }\ll & {} H \dot{\phi }, \end{aligned}$$
(57)
$$\begin{aligned} \frac{\dot{\phi }^2}{2}\ll & {} V(\phi ). \end{aligned}$$
(58)

The second relation effectively states that \(V(\phi )\) is nearly constant, since the inflaton changes very slowly in time. This implies that the Hubble rate remains approximately constant. Therefore, it proves useful to introduce the dimensionless parameter

$$\begin{aligned} \epsilon \equiv - \frac{\dot{H}}{H^2}\,, \end{aligned}$$
(59)

where it is understood that \(\epsilon \ll 1\). Additionally, to keep track of the relative change of \(\epsilon \) per Hubble time, we define a second slow-roll parameter as

$$\begin{aligned} \eta \equiv \frac{\dot{\epsilon }}{H \epsilon }\,. \end{aligned}$$
(60)

For a sufficiently long inflation, we likewise need \(\eta \ll 1\).

The total amount of expansion that occurred during inflation is quantified by the number of e-folds [76]

$$\begin{aligned} \mathcal {N}=\int H\mathrm{{d}}t=\int \frac{H}{\dot{\phi }}\mathrm{{d}}\phi \,, \end{aligned}$$
(61)

from the horizon crossing \(\phi _c\) to the end of inflation defined by \(\epsilon (\phi _f)=1\). For typical inflationary models, a sensible working hypothesis is \(\mathcal {N}\) larger than \(\sim 50-60\) [77], which weakly depends on the inflation scale and the thermal history after inflation. However, an extremely long duration of inflation is not excluded, and even appears necessary in certain scenarios. For example, the stochastic axion framework [78, 79], the relaxation scenario [80], and a quintessence model [81] require \(\log _{10}\mathcal {N} \sim \mathcal {O}(10)\), depending on model parameters. A simple single-field model has recently been proposed in the context of stochastic inflation with an average e-fold \(\langle \mathcal {N}\rangle \sim {10^{10}}^{10}\) [82]. In the following, we shall consider more traditional and experimentally favored scenarios involving \(\mathcal {O}(10^2)\lesssim \mathcal {N}\lesssim \mathcal {O}(10^3)\).

According to the inflationary paradigm, the rapid growth of the Universe caused fluctuations of the scalar field to be stretched to large scales and, upon crossing the horizon, to freeze in. In the Fourier space, such fluctuations are characterized by a power spectrum with spectral index [83]

$$\begin{aligned} n_s-1=2\eta -6\epsilon . \end{aligned}$$
(62)

There is another kind of density perturbations, the tensor modes, which are fluctuations in the fabric of spacetime. Similarly to the scalar modes, their spectrum is expected to follow a power-law behavior (see Sect. 5 for more details). The ratio of the tensor to scalar spectra is given by

$$\begin{aligned} r=16\epsilon . \end{aligned}$$
(63)

The parameters (62), (63) play a crucial role, providing for the possibility to constrain or rule out inflationary models by comparison with observations. In particular, the latest Planck Public Release 4, BICEP/Keck Array 2018, Planck CMB lensing and BAO data set the experimental bounds \(r<0.037\) [84] and \(n_s\simeq 0.97\) [85].

To study how the zero-point length affects the above parameters, we now consider the modified Friedmann equation (22), where we neglect for simplicity the cosmological constant. Using Eq. (54) for the inflaton energy density along with the first slow-roll condition (57), we obtain

$$\begin{aligned} H(\phi )\simeq \sqrt{\frac{8\pi V}{3}}\left( 1+\frac{4\pi \beta }{3H_0^2}V\right) +\mathcal {O}(\beta ^2)\,, \end{aligned}$$
(64)

where the \(\phi \)-dependence of V has been omitted in the right-hand side.

On the other hand, by differentiating Eq. (22) with respect to t and using the conditions (54), (55), we find

$$\begin{aligned} \dot{H}\simeq -4\pi \left( 1+\frac{16\pi \beta }{3H_0^2}V\right) \dot{\phi }^2+\mathcal {O}(\beta ^2)\,. \end{aligned}$$
(65)

Note that the time derivative \(\dot{\phi }\) can be expressed in terms of \(\phi \) through Eq. (56), which in the slow-roll approximation simplifies to

$$\begin{aligned} \dot{\phi }\simeq & {} -\frac{1}{3H}\partial _\phi V \nonumber \\= & {} \left( -\frac{1}{2\sqrt{6\pi }}+\sqrt{\frac{2\pi }{27}}\frac{\beta }{H_0^2}V\right) \frac{\partial _\phi V}{\sqrt{V}}\,+\,\mathcal {O}(\beta ^2). \end{aligned}$$
(66)

3.1 Power-law potential

The latter equation indicates that the dynamics of \(\phi \) are ruled by the potential \(V(\phi )\). It should be noted that two different approaches are typically considered in studying inflation. The dynamically motivated approach involves reconstructing V by requiring the Universe to evolve following specific dynamics, which are set by a fixed time dependence of the Hubble rate. From this perspective, inflationary models are classified according to the corresponding Universe evolution. As an example, in [86] the scale factor is set to \(a(t)=t^m\) for a suitable choice of the power term m. Nevertheless, the disadvantage is that the predicted scalar-to-tensor ratio exceeds the constraint set by the BICEP and Planck data. Moreover, the slow-roll parameters turn out to be constant. On the other hand, the potential-motivated method involves setting the inflaton potential first and then solving the ensuing Friedmann equations. Such an approach finds wide application in extended theories of gravity [73, 74, 83, 87,88,89], where deviations of the dynamics of the early Universe from GR are admitted. Inflationary models are in this case categorized based on the form of the potential, which is fixed a priori.

In order to discuss a concrete model of inflation, in the following we consider the potential-motivated approach. Specifically, we assume the power-law form

$$\begin{aligned} V=V_0 \phi ^n\,, \end{aligned}$$
(67)

where the amplitude \(V_0\) and the power term n are both positive constants. Following [73, 83], \(V_0\) is set to unity. Although its actual value may vary in a wide range [90], this assumption is justified by the fact that the magnitude of \(V_0\) is expected not to alter the power-law function during inflation. Furthermore, experimental evidence favors inflationary models with \(n\sim \mathcal {O}\left( 10^{-1}\div 1\right) \), while \(n\ge 2\) tends to be ruled out in the minimal coupling scenario. Here we shall consider \(n=1\).

With the ansatz (67), the dimensionless parameter (59) is found to be

$$\begin{aligned} \epsilon (\phi )\simeq \frac{1}{16\pi \phi ^2}+\mathcal {O}(\beta ^2)\,, \end{aligned}$$
(68)

which also allows us to derive the inflaton at the end of inflation as

$$\begin{aligned} \epsilon (\phi _f)=1 \,\,\Longrightarrow \,\, \phi _f\simeq \frac{1}{4\sqrt{\pi }}+\mathcal {O}(\beta ^2)\,. \end{aligned}$$
(69)

The spectral index \(n_s\) and the tensor-to-scalar ratio r are typically evaluated at the horizon crossing, where the fluctuations of the inflaton freeze and remain nearly constant [73, 74, 83, 91]. From Eq. (61), the inflaton at the horizon crossing is obtained as

$$\begin{aligned} \phi _c\simeq \frac{1}{4}\sqrt{\frac{f(\mathcal {N})}{\pi }}+\frac{\beta }{18H_0^2}\left[ \frac{1-\sqrt{f^3(\mathcal {N})}}{\sqrt{f(\mathcal {N})}}\right] +\mathcal {O}(\beta ^2)\,, \end{aligned}$$
(70)

where we have used the shorthand notation

$$\begin{aligned} f(\mathcal {N})\equiv 1+4\mathcal {N}\,. \end{aligned}$$
(71)

Therefore, the slow-roll parameters (62), (63) become

$$\begin{aligned} r_\beta= & {} 16\epsilon (\phi _c)\nonumber \\\simeq & {} \frac{16}{f(\mathcal {N})}+\frac{64\sqrt{\pi }\beta }{9H_0^2}\,\frac{\sqrt{f^3(\mathcal {N})}-1}{f^2(\mathcal {N})}+\mathcal {O}(\beta ^2), \end{aligned}$$
(72)
$$\begin{aligned} n_s{_\beta }= & {} 1+2\eta (\phi _c)-6\epsilon (\phi _c)\nonumber \\\simeq & {} 1+\frac{2}{f(\mathcal {N})}-\frac{8\sqrt{\pi }\beta }{9H_0^2}\,\frac{5\sqrt{f^3(\mathcal {N})}+1}{f^2(\mathcal {N})}+\mathcal {O}(\beta ^2). \end{aligned}$$
(73)

In order to constrain the parameter \(\beta \) (or, equivalently, the zero-point length \(l_0\)), let us compare the above expressions with experimental bounds on \(n_s\) and r (see Eq. (63)). Assuming the expected-inflation duration \(\mathcal {N}\sim \mathcal {O}(10^3)\), it can be inferred from Fig. 6 that

$$\begin{aligned} r_\beta <0.037 \Longrightarrow {l}_0\lesssim \mathcal {O}(1)\,, \end{aligned}$$
(74)

corresponding to \(\beta \lesssim \mathcal {O}(10^{-122})\). This turns into the more stringent condition \(l_0\lesssim \mathcal {O}(10^{-1})\) \((\beta \lesssim \mathcal {O}(10^{-124}))\) for a total number of e-folds \(\mathcal {N}\sim \mathcal {O}(10^2)\). Direct substitution into Eq. (73) shows that these bounds are also consistent with the observational measurement \(n_s\simeq 0.97\).

Interestingly enough, the above constraints align with the original assumption \(l_0\lesssim 1\) of the zero-point length from quantum fluctuations of the background spacetime [18] and the recent result [22] in quantum-corrected black holes from string T-duality.

Fig. 6
figure 6

3D Plot of \(r_\beta \) versus \(\mathcal {N}\) and \(l_0\). The gray region is excluded by observational constraints [84]

Full size image

4 Structure formation

A key challenge in modern cosmology lies in comprehending the origin and dynamics of perturbation growth during the early Universe, as these fluctuations eventually give rise to the formation of the observed large-scale structures. It is widely accepted that these structures emerge from the amplification of tiny initial density fluctuations through gravitational instability during cosmic evolution. These fluctuations gradually grow until they become sufficiently robust to break away from the background expansion and collapse into gravitationally bound systems (see [92, 93] for a recent review).

An effective framework for investigating the growth of matter perturbations and structure formation in the linear regime is the top-hat spherical collapse (SC) model [94]. This approach considers a uniform and spherically symmetric perturbation in an expanding background, describing the growth of perturbations within a spherical region using the Friedmann equations of gravity [95]. Employing this model, in the following we study the impact of the zero-point length cosmology on the growth of cosmic perturbations and the power spectrum of scalar fluctuations at the early stages of the Universe.

4.1 Growth of perturbations in linear regime

As a first step, by using the relation

$$\begin{aligned} \frac{\ddot{a}}{a}=\dot{H} + H^2\,, \end{aligned}$$
(75)

let us recast the second modified Friedmann equation as

$$\begin{aligned} \frac{\ddot{a}}{a}=-\frac{4\pi }{3}\rho \left( 1+\frac{32\pi \beta }{3H_0^2}\rho \right) \,. \end{aligned}$$
(76)

To apply the top-hat SC model, we focus on a spherical region of radius \(a_p\) and with a homogeneous density \(\rho _c\). Assume that, at time t, \(\rho _c(t)=\rho (t)+\delta \rho \), where \(\delta \rho \) is the density fluctuation. The conservation equation in such simplified spherical region is given by [94]

$$\begin{aligned} \dot{\rho }_c+3h\rho _c=0\,, \end{aligned}$$
(77)

where \(h=\dot{a}_p/a_p\) denotes the local expansion rate inside the spherical region. The second Friedmann equation applied to this region reads

$$\begin{aligned} \frac{\ddot{a}_p}{a_p}=-\frac{4\pi }{3}\rho _c\left( 1+\frac{32\pi \beta }{3H_0^2}\rho _c\right) \,. \end{aligned}$$
(78)

It is now useful to define the density contrast of the fluid by the relation

$$\begin{aligned} \delta \equiv \frac{\rho _c}{\rho }-1=\frac{\delta \rho }{\rho }\,. \end{aligned}$$
(79)

Since the fluctuation \(\delta \rho \) is expected to be smaller than \(\rho \), we can assume \(\delta \ll 1\) (linear regime). This is a reasonable approximation, as the initial stages of the gravitational collapse are well described by the linear regime for all but the very smallest scales [94].

Differentiating Eq. (79) with respect to t, we find

$$\begin{aligned} \dot{\delta }=3\left( 1+\delta \right) \left( H-h\right) , \end{aligned}$$
(80)

where we have used the continuity equation. Further differentiation gives

$$\begin{aligned} \ddot{\delta }=3\left( 1+\delta \right) (\dot{H}-\dot{h})+\frac{\dot{\delta }^2}{1+\delta }\,. \end{aligned}$$
(81)

Equation (81) governs the dynamics of matter perturbations in the top-hat SC model. It can be further manipulated by noting that

$$\begin{aligned} \dot{H}-\dot{h}=h^2-H^2+\frac{4\pi \rho }{3}\left( 1+\frac{64\pi \beta }{3H_0^2}\rho \right) \delta +\mathcal {O}(\delta ^2)\,, \end{aligned}$$
(82)

where we have neglected higher-order terms in \(\delta \). After some algebra, substitution into Eq. (81) gives

$$\begin{aligned} \ddot{\delta }+2H\dot{\delta }-{4\pi \rho }\left( 1+\frac{64\pi \beta }{3H_0^2}\rho \right) \delta +\mathcal {O}(\delta ^2)=0\,. \end{aligned}$$
(83)

Changing the time variable to the scale factor a, we have

$$\begin{aligned} \dot{\delta }= & {} aH\delta ', \end{aligned}$$
(84)
$$\begin{aligned} \ddot{\delta }= & {} a^2H^2\delta ''+aH\left( H+aH'\right) \delta ', \end{aligned}$$
(85)

where the prime denotes derivative with respect to a. Upon substituting into Eq. (83), we obtain

$$\begin{aligned} \delta ''+\delta '\left( \frac{3}{2a}-\frac{4\pi \beta }{H_0^2}\frac{\rho }{a}\right) -\delta \left( \frac{3}{2a^2}+\frac{28\pi \beta }{H_0^2}\frac{\rho }{a^2}\right) +\mathcal {O}(\delta ^2)=0. \end{aligned}$$
(86)

It is easy to check that the standard dynamics are recovered for \(\beta \rightarrow 0\) [94].

Equation (86) has been solved numerically. The solution is plotted in Fig. 7 for various values of \(\beta \) consistent with Eq. (74). As expected, effects of the zero-point length mainly manifest as z increases, with the matter density contrast growing more slowly than the standard behavior (black solid curve) in the very early Universe. Therefore, unlike the classical framework, where small-scale fluctuations evolve unhindered, such fluctuations are smoothed out due to the quantum gravity corrections associated with the zero-point length. This effect limits the growth of fluctuations, leading to a slower increase in the density contrast. A similar suppression has been discussed in [96], assuming a running gravitational constant. Other studies appear in [97,98,99] in the context of full quantum and/or extended gravity.

It should be noted that the above conclusions provide preliminary, but still insightful, results toward understanding the effects of zero-point length on the evolution of the density contrast and structure formation. Indeed, for \(l_0\) around the Planck length, deviations from the classical model could become appreciable at very high redshift, where the linear regime may break down and higher-order corrections should be taken into account. A more refined analysis going beyond the linear regime is reserved for the future.

Before moving on, it is interesting to observe that the evolution of matter overdensity \(\delta \) is closely related to the \(\sigma _8\) tension, which quantifies the gravitational matter clustering from the amplitude of the linearly evolved power spectrum at the scale of \(8h^{-1}\,\textrm{Mpc}\). This tension arises from the fact that the cosmic microwave background (CMB) estimation [100] differs from the Sloan Digital Sky Survey (SDSS)/Baryon Oscillation Spectroscopic Survey (BOSS) direct measurements [101, 102]. The later growth of \(\delta \) in Fig. 7 for nonvanishing \(\beta \) indicates a less efficient clustering of matter over time in the presence of a minimal length, which provides a possible mechanism to alleviate the \(\sigma _8\) tension [103]. A similar outcome in the framework of modified gravity was found in [104] using a Tsallis-like deformation of the Bekenstein–Hawking entropy.

Fig. 7
figure 7

Log-linear plot of \(\delta \) versus z, for various values of \(\beta \). We set the same initial conditions as in [94]

Full size image

4.2 Power spectrum

We now look into the primordial power spectrum for scalar perturbations, which provides an indirect probe of inflation or other structure-formation mechanisms. The simplest inflationary models predict almost purely adiabatic primordial perturbations with a nearly scale-invariant power spectrum given by [105, 106]

$$\begin{aligned} P_{\zeta }(k) = A_S\left( \frac{k}{k_*}\right) ^{n_s-1}\,, \end{aligned}$$
(87)

where \(A_s\simeq 10^{-9}\) is the power amplitude and \(k_*\) a characteristic pivot scale.

The form of Eq. (87) is statistically preferred by Planck, since it yields a good fit to data with a remarkably small number of parameters. However, it does not provide the best observational fit. Improvements can be obtained by allowing for a deviation from the simple power spectrum, considering either a running spectral index or different functional forms [107]. This is mainly due to the possibility of suppressing the primordial power spectrum at large scales.

A paradigmatic modification considered by the Planck Collaboration consists of a broken-power-law (bpl) spectrum of the form [107]

$$\begin{aligned} P^{bpl}_\zeta \propto \left( \frac{k}{k_*}\right) ^{n_s^{bpl}-1+\delta }\,, \end{aligned}$$
(88)

for k below a certain cutoff scale. The best fit is obtained for \(\delta =1.14\) [107].

Interestingly enough, the zero-point length model naturally fits the profile (88). To check this, let us recast Eq. (73) as

$$\begin{aligned} n_s{_\beta }=n_s + g_\beta \,, \end{aligned}$$
(89)

where \(n_s\) is the usual spectral index in standard cosmology (\(\beta =0\)) and

$$\begin{aligned} g_\beta \equiv -\frac{8\sqrt{\pi }\beta }{9H_0^2}\,\frac{5\sqrt{f^3(\mathcal {N})}+1}{f^2(\mathcal {N})}+\mathcal {O}(\beta ^2) \end{aligned}$$
(90)

is the zero-point length correction. The latter term has the same role as the \(\delta \)-shift in the bpl spectrum (88). Said another way, the correction (88) is reproduced in the zero-point length cosmology while still maintaining the simple power spectrum (87). Indeed, upon replacing the modified spectral index (89), the exponent of Eq. (87) becomes

$$\begin{aligned} n_s-1\,\,\longrightarrow \,\, n_s{_\beta }-g_\beta -1\,. \end{aligned}$$
(91)

The right-hand side fits Eq. (88), provided that

$$\begin{aligned} n_s{_\beta }-g_\beta -1=n_s^{bpl}-1+\delta \,. \end{aligned}$$
(92)

Setting \(n_s{_\beta }\) and \(n_s^{bpl}\) equal to the experimental value [85] and using Eq. (33), we find

$$\begin{aligned} l_0=\frac{f(\mathcal {N})}{\pi ^{1/4}}\sqrt{\frac{3\delta }{2+10f^{3/2}(\mathcal {N})}}\,, \end{aligned}$$
(93)

which, for \(\mathcal {O}(10^2)\lesssim N\lesssim \mathcal {O}(10^3)\) and \(\delta =1.14\), gives \(l_0\sim \mathcal {O}(1)\), in compliance with Eq. (74).

Therefore, zero-point length cosmology stands out as an alternative characterization for the bpl spectrum, which is still consistent with latest observations.

5 Primordial gravitational waves

The first-ever detection of gravitational waves (GWs) by the LIGO [108] and VIRGO [109] collaborations has opened up fresh avenues for investigating the Universe. From a theoretical perspective, GWs originate from three main sources categorized on the basis of their generation mechanisms: astrophysical, cosmological, and inflationary sources. Astrophysical GWs are notoriously influenced by the mass of the emitting object, with massive compact objects like black holes expected to produce strong GWs. For the minimum mass \(M_\odot \), for instance, the maximal frequency would be around \(10\, \textrm{kHz}\). On the other hand, various cosmological phenomena in the early Universe, such as first-order phase transitions at \(10^{-2}\,\textrm{GeV}\) or just below the Grand Unified Theory scale, can produce GWs with frequency around \(10^{-5}\,\textrm{Hz}\) or roughly lower than the \(\textrm{GHz}\) range, respectively. Finally, primordial GWs (PGWs) generated during inflation span frequencies from \(10^{-18} \,\textrm{Hz}\) to \(1\,\textrm{GHz}\) (see [110] for more details). While current observatories are primarily focused on astrophysical signals, it is clear that the exploration of high-frequency GWs represents a promising yet challenging test bench for theories beyond the standard cosmology that could not be tested otherwise. Starting from these premises, in what follows we compute the spectrum of PGWs in the zero-point length cosmology and compare it with predictions of the standard scenario.

5.1 Standard cosmology

GWs in the linearized approximation are described as metric perturbations on a curved background. Using the transverse traceless conditions, the equation of motion for tensor perturbation at the first order takes the form [111]

$$\begin{aligned} \ddot{h}_{ij} + 3H\dot{h}_{ij} - \frac{\nabla ^2}{a^2}h_{ij} = 16\pi \Pi _{ij}^{TT}\,, \end{aligned}$$
(94)

where \(\Pi _{ij}^{TT}\) is the transverse-traceless part of the anisotropic stress tensor

$$\begin{aligned} \Pi _{ij} = \frac{T_{ij}-p\hspace{0.3mm} g_{ij}}{a^2}\,, \end{aligned}$$
(95)

and \(T_{ij}, g_{ij}\) and p denote the stress-energy tensor, metric tensor and background pressure, respectively (Latin indices run over the three spatial coordinates of spacetime).

We now focus on GW signals with frequencies higher than \(10^{-11}\,\textrm{Hz}\) [112]. To solve Eq. (94), it proves convenient to work in the Fourier space, where

$$\begin{aligned} h_{ij}(t,\vec {x}) = \sum _{\lambda }\int \frac{\mathrm{{d}}^3k}{\left( 2\pi \right) ^3}\hspace{0.2mm}h^\lambda (t,\vec {k})\hspace{0.2mm}\epsilon ^\lambda _{ij}(\vec {k})\hspace{0.2mm}e^{i\vec {k}\cdot \vec {x}}\,. \end{aligned}$$
(96)

The two independent polarizations have been denoted by \(\lambda =+,\times \), while \(\epsilon ^\lambda \) is the (normalized) spin-2 polarization tensor obeying \(\sum _{ij}\epsilon ^\lambda _{ij}\epsilon ^{\lambda '*}_{ij}=2\delta ^{\lambda \lambda '}\) (it is understood that arrows in Eq. (96) denote three-vectors). In this way, the tensor perturbation \(h^\lambda (t,\vec {k})\) can be factorized as

$$\begin{aligned} h^\lambda (t,\vec {k}) = h_\mathrm{{prim}}^\lambda (\vec {k})X(t,k)\,, \end{aligned}$$
(97)

where X(t, k) and \(h_\mathrm{{prim}}^\lambda \) denote the transfer function and amplitude of the primordial tensor perturbations, respectively.

In this setting, the tensor power spectrum is [113]

$$\begin{aligned} \mathcal {P}_T(k) = \frac{k^3}{\pi ^2}\sum _\lambda \Big |h^\lambda _\mathrm{{prim}}(\vec k)\Big |^2 = \frac{2}{\pi ^2}\hspace{0.3mm}G\hspace{0.3mm} H^2\Big |_{k=aH}\,, \end{aligned}$$
(98)

while the dynamics (94) take the form of a damped harmonic oscillator-like equation

$$\begin{aligned} X'' \ + \ 2\hspace{0.2mm}\frac{a'}{a}X' \ + \ k^2X \ = \ 0\,. \end{aligned}$$
(99)

With abuse of notation, the prime here indicates derivative with respect to the conformal time \(\mathrm{{d}}\tau =\mathrm{{d}}t/a\).

In standard cosmology, the relic density of PGW from first-order tensor perturbation reads [111, 113]

$$\begin{aligned} \nonumber \Omega _{GW} (\tau ,k)= & {} \frac{[X'(\tau ,k)]^2}{12a^2(\tau )H^2(\tau )}\,\mathcal {P}_T(k)\\\simeq & {} \left[ \frac{a_{hc}}{a(\tau )}\right] ^4\left[ \frac{H_{hc}}{H(\tau )}\right] ^2\frac{P_T(k)}{24}, \end{aligned}$$
(100)

where we have averaged over oscillation periods in the second step, i.e.,

$$\begin{aligned} X'(\tau ,k) \simeq k\hspace{0.2mm} X(\tau ,k) \simeq \frac{k\hspace{0.3mm} a_{hc}}{\sqrt{2}a(\tau )} \simeq \frac{a^2_{hc}\hspace{0.3mm}H_{hc}}{\sqrt{2}a(\tau )}\,. \end{aligned}$$
(101)

Moreover, we have used

$$\begin{aligned} k=2\pi f=a_{hc}H(a_{hc}) \end{aligned}$$
(102)

for the horizon crossing moment.Footnote 1 Therefore, the PGW relic density at present time is

$$\begin{aligned} \Omega _{GW}(\tau _0,k)h^2\simeq \left[ \frac{g_*(T_{\textrm{hc}})}{2}\right] \left[ \frac{g_{*s}(T_0)}{g_{*s}(T_{\textrm{hc}})}\right] ^{4/3}\frac{\mathcal {P}_T(k)\Omega _{r}(T_0)h^2}{24}\,, \end{aligned}$$
(103)

where h is the dimensionless (or reduced) Hubble constant, while \(g_*(T)\) and \(g_{*s}(T)\) denote the effective numbers of relativistic degrees of freedom that contribute to the radiation energy density \(\rho \) and entropy density s, respectively, i.e.

$$\begin{aligned} \rho _r=\frac{\pi ^2}{30}g_*(T)T^4\,,\quad \,\, s_r=\frac{2\pi ^2}{45}g_{*s}(T)T^3\,. \end{aligned}$$
(104)

Next, the scale dependence of the tensor power spectrum is given by

$$\begin{aligned} \mathcal {P}_T(k)=A_T\left( \frac{k}{k_*}\right) ^{n_T}\,, \end{aligned}$$
(105)

where the amplitudes of the tensor and scalar perturbations are related via the the tensor-to-scalar ratio r as \(A_T=r A_S\), while \(n_T\) is the tensor spectral index.

The plot in Fig. 8 displays the behavior of the PGW spectrum in Eq. (103) (red dashed line) versus the frequency (we have restored correct units for comparison with literature). The colored regions delineate the projected sensitivities for the Square Kilometre Array (SKA) telescope [117], Laser Interferometer Space Antenna (LISA) interferometer [118], Einstein Telescope (ET) detector [119], and the successor Big Bang Observer (BBO) [120]. Moreover, the Big Bang nucleosynthesis (BBN) bound is set by the constraint on the effective number of neutrinos [121, 122].

Fig. 8
figure 8

Plot of the PGW spectrum versus the frequency f, for \(n_T=0\) and \(A_S\sim 10^{-9}\), consistent with the Planck observational constraints at the CMB scale [115]. The colored regions delineate the projected sensitivities for several GW observatories [116]

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5.2 Zero-point length cosmology

We now study implications of the zero-point length for the PGW spectrum. Toward this end, we make use of the modified Friedmann equation (38) and the relation (102). Following [113], we can write from Eq. (100)

$$\begin{aligned} \Omega _{\textrm{GW}}(\tau ,k)\simeq & {} \left[ \frac{a_{hc}}{a(\tau )}\right] ^4\left[ \frac{H_{hc}}{H_{\textrm{GR}}(\tau )}\right] ^2\left[ \frac{H_{\textrm{GR}}(\tau )}{H(\tau )}\right] ^2\frac{P_T(k)}{24}\nonumber \\= & {} \Omega ^{\textrm{GR}}_{\textrm{GW}}(\tau ,k)\left[ \frac{H_{\textrm{GR}}(\tau )}{H(\tau )}\right] ^2\left[ \frac{a_{hc}}{a_{hc}^{GR}}\right] ^4\left[ \frac{a^{GR}(\tau )}{a(\tau )}\right] ^4\nonumber \\{} & {} \times \left[ \frac{H_{hc}}{H_{hc}^{GR}}\right] ^2 , \end{aligned}$$
(106)

with

$$\begin{aligned} k \ = \ a_{\textrm{hc}}^{\textrm{GR}}H^{\textrm{GR}}(a_{\textrm{hc}}^{\textrm{GR}})\,. \end{aligned}$$
(107)

Here, standard quantities in general relativity have been denoted by GR, e.g., \(\Omega ^{\textrm{GR}}_{\textrm{GW}}(\tau ,k)\) is the PGW relic density

Next, using the flatness condition \(H(z)/H_0=1\), we obtain

$$\begin{aligned} \Omega _{\textrm{GW}}(\tau _0,k) \ \simeq \ \Omega ^{\textrm{GR}}_{\textrm{GW}}(\tau _0,k)\left[ \frac{a_{hc}}{a_{hc}^{GR}}\right] ^4\left[ \frac{H_{hc}}{H_{hc}^{GR}}\right] ^2 \,. \end{aligned}$$
(108)

The modified PGW spectrum for \(\beta \sim \mathcal {O}(10^{-122})\) (blue dot-dashed line) is plotted in Fig. 8. Unfortunately, if the zero-point length is around the Planck scale, no significant deviation from GR (red dashed curve) could be detected in upcoming GW observatories, at least up to frequencies around \(10^{-3}\,\textrm{GHz}\). Nevertheless, it is interesting to note that appreciable effects might still be observed in the frequency range \(f\gtrsim 10^{-4}\,\textrm{GHz}\), provided that \(\beta \gtrsim 10^{-104}\) (\(l_0\gtrsim 10^9\)) (green dot-dashed curve in Fig. 9). If, on the one hand, this range of \(\beta \) values does not fit the above constraint set through the inflation measurements, on the other hand it could be understood by allowing for a running zero-point length parameter. Although not contemplated in the original model, such a scale-dependent behavior would be fully justified, as it is consistent with the typical case occurring in quantum field theory and quantum gravity under renormalization group applications, where all parameters and coupling constants are dynamical in tandem with the energy scale. Moreover, the coupling constant in conventional quantum gravity decreases with increasing distance in such a way that the larger the distance scale, the smaller the coupling, and vice versa (e.g. [123]). Note that a similar framework was investigated in [124] assuming a fluctuating minimal length in the generalized uncertainty principle, and in [125] by setting a varying exponent in Barrow holographic dark energy. Clearly, such a scenario may open up the possibility of new physics to be tested through future measurements of high-frequency GWs above the LIGO/VIRGO range. This analysis goes beyond the scope of the present work and will be explored elsewhere.

Fig. 9
figure 9

Plot of the PGW spectrum versus the frequency f for various values of \(\beta \), \(n_T=0\) and \(A_S\sim 10^{-9}\), consistently with the Planck observational constraints at the CMB scale [115]. The colored regions delineate the projected sensitivities for several GW observatories [116]

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Moreover, a zero-point length exceeding \(l_0\simeq 10^{16}\) (\(\beta \simeq 10^{-90}\), purple dot-dashed curve in Fig. 9) appears disfavored (although not yet excluded) in our model, since it would entail a dramatic departure from GR for even GWs below \(10\,\textrm{kHz}\). This prediction is somewhat consistent with the experimental evidence from particle physics [126] (we remind the reader that subatomic distances have been directly measured up to \(10^{-18}\,\textrm{m}\) and beyond, corresponding to length scales \(l\simeq 10^{17}\) in our units). The same rapid increase of the PGW spectrum at low frequencies has been exhibited in extended scalar–tensor and extra-dimensional gravity [113].

6 Discussion and conclusions

An approach to overcome the singularity problem in black hole physics is presented using a string-inspired T-duality model. In this method, one can remove the singularity at the center of the black hole by taking into account a zero-point length in the gravitational potential, which comes from a momentum space propagator induced by the path integral duality [22]. Based on this, in the present paper we have explored the effects of zero-point length in the cosmological setup. We first derived the modified Friedmann equations due to zero-point length by applying the first law of thermodynamics on the apparent horizon. Then, we explored the cosmological consequences of this model and revealed the influence of the zero-point length on the history of the Universe in both the matter- and radiation-dominated eras. The next effort was devoted to slow-roll solutions for power-law inflation. Comparison of the slow-roll parameters with the observational values yielded the constraint \(l_0\lesssim \mathcal {O}(10^{-1}\div 1)\), depending on the expected-inflation duration. This is consistent with the original assumption [18] and with recent results in quantum-corrected black holes from string T-duality [22].

Using the top-hat spherical collapse model, we explored the impact on the growth of matter fluctuations in the linear regime. In spite of this minimal assumption, it was shown that the zero-point length affects the evolution of the density contrast in a nontrivial way, leading to slower growth of the primordial seed fluctuations into cosmic structures. In this context, we also analyzed implications for the power spectrum of scalar perturbations. Although the simplest (and statistically preferred) inflationary models predict almost purely adiabatic perturbations with a power-law spectrum, a better observational fit was obtained by assuming either a running spectral index or different functional forms [107]. We have found that our model naturally fits the broken-power-law (88), where the role of the \(\delta \)-shift is played by the zero-point length correction (see Eq. (92)).

Finally, we studied the spectrum of primordial gravitational waves in the early Universe. Although Planck scale corrections to GR are hardly detectable in the near future, nontrivial effects might still be observed at frequencies accessible to the next-generation GW experiments if \(l_0\gtrsim 10^9\). Such a scenario is not that speculative, as it could be understood within the context of a dynamical, energy-scale-dependent behavior for the zero-point length. This would align with the typical case observed in quantum field theory and quantum gravity under renormalization group considerations (see e.g. [123]).

Further aspects are yet to be investigated. First, we would like to relax our leading-order assumption (19) and possibly develop exact analytical computations. This is essential for a thorough application of our model to the Planck era, which could provide hints toward the formulation of a modified cosmology in full quantum gravity. Additionally, it would be interesting to adapt the analysis of Sect. 3 to different inflationary models. A paradigmatic case is the Starobinsky inflation (and related generalizations) [127], which is widely studied, for instance, for the theory of the origin of the Universe from a quantum multiverse [128]. In parallel, an intriguing goal is to rephrase the present analysis within the framework of alternative minimal-length approaches to highlight any similarities or differences. Candidate models are the qmetric formalism [23,24,25,26,27,28], which naturally incorporates a minimal length in the fabric of spacetime through a suitable definition of metric distances, and the generalized uncertainty principle, which accounts for a Planck-scale deformation of the Heisenberg inequality due to quantum gravitational effects.

On the other hand, recent advancements in GW experiments offer a fresh perspective on the nature of gravity in the early Universe. By probing different cosmic scales, these experiments provide a window into potential modifications to Einstein’s theory of gravity. For instance, once we detect PGWs, we will gain valuable insights that could help narrow the parameter space for possible modifications to GR during the early Universe. While the present effort provides a preliminary analysis in this area, it is crucial to emphasize that a detailed comparison of various laboratory/astrophysical gravity tests with PGW detection appears essential. Work is in progress in this direction and will be presented elsewhere.

The era between the end of inflation and the onset of radiation dominance in the Universe remains largely unexplored, and there are several UV-complete scenarios that suggest nonstandard cosmology or modifications to gravity. In this sense, the timing of our study is highly pertinent, as the next generation of GW observatories will cover several decades of frequency ranges of various GW amplitudes, allowing for the possibility of uncovering new physics related to the pre-BBN era.