This work presents a universal and revisited formalism for the entropy of the apparent horizon in modified gravity to investigate the validity of the Generalized Second Law (GSL) of thermodynamics. This revisited horizon entropy is constructed directly from the modified Friedmann equations in a Friedmann–Robertson–Walker (FRW) universe. The resulting entropy relation contains, beside the standard Bekenstein–Hawking term, an additional integral contribution that encodes the effective energy density and pressure generated by deviations from general relativity. Using this universal entropy formula, a compact expression for the GSL is derived. This formalism is then applied to some viable $\textit{f}$($\textit{T}$) and $\textit{f}$($\textit{R}$) gravity models, in order to re-evaluate the validity of the GSL as a function of redshift. The analysis demonstrates that including the integral term in the revisited entropy can relatively improve the late-time validity of the GSL for some of these models while living others unchanged, thereby reinforcing the profound connection between thermodynamics and gravity.

Iran

Soma Heydari

Parastoo Askari

Kayoomars Karami

Revisited apparent horizon entropy and GSL in modified gravity

Here, the <math altimg='si3.svg'><mi>&#945;</mi></math>-attractor inflation is investigated within a framework incorporating a minimal measurable length, as implemented by the Generalized Uncertainty Principle (GUP). The GUP modifications to the Friedmann equations and cosmological perturbation parameters are employed to assess the model observational viability against the Atacama Cosmology Telescope (ACT) data. Our results indicate that in the <math altimg='si13.svg'><mrow><mi>r</mi><mo linebreak='goodbreak'>&#8722;</mo><msub><mi>n</mi><mi>s</mi></msub></mrow></math> plane, the predictions of the standard <math altimg='si3.svg'><mi>&#945;</mi></math>-attractor model ( <math altimg='si11.svg'><mrow><mi>&#946;</mi><mo linebreak='goodbreak'>=</mo><mn>0</mn></mrow></math>) lies near the <math altimg='si32.svg'><mrow><mn>2</mn><mi>&#963;</mi></mrow></math> boundary of joint observations. More interestingly enough is that in the presence of GUP effect, the predictions of the model for the GUP parameter <math altimg='si33.svg'><mrow><mi>&#946;</mi><mo>&#8819;</mo><mi>O</mi><mo>(</mo><msup><mn>10</mn><mn>13</mn></msup><mo>)</mo></mrow></math> shifts into the <math altimg='si34.svg'><mrow><mn>68</mn><mspace width='0.16em' /><mo>%</mo></mrow></math> CL interval. This value for <math altimg='si2.svg'><mi>&#946;</mi></math> is in well agreement with upper bounds on the GUP parameter deduced from cosmological analysis as well as quantum and gravitational experiments.

Hana Heidarian

Milad Solbi

<math altimg='si3.svg'><mi>&#945;</mi></math>-attractor inflation modified by GUP in light of ACT observations

Here, we investigate the formation of primordial black holes (PBHs) in non-minimal coupling Gauss–Bonnet inflationary model in the presence of power-law potentials. We employ a two part coupling function to enhance primordial curvatures at small scales as well as satisfy Planck measurements at the CMB scale. Moreover, our model satisfies the swampland criteria. We find PBHs with different mass scales and demonstrate that PBHs with masses around  $$\mathcal {O}(10^{-14})M_{\odot }$$  <math> <mrow> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>14</mn> </mrow> </msup> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>   can account for almost all of the dark matter in the universe. In addition, we investigate the implications of the reheating stage and show that the PBHs in our model are generated during the radiation-dominated era. Furthermore, we investigate the production of scalar-induced gravitational waves (GWs). More interestingly enough is that, for the specific cases  $$D_\textrm{n}$$  <math> <msub> <mi>D</mi> <mtext>n</mtext> </msub> </math>   in our model, the GWs can be considered as a source of PTA signal. Also, we conclude that the GWs energy density parameter at the nano-Hz regime can be parameterized as  $$\Omega _\mathrm{GW_0} (f) \sim f^{5-\gamma }$$  <math> <mrow> <msub> <mi>Ω</mi> <msub> <mi>GW</mi> <mn>0</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>∼</mo> <msup> <mi>f</mi> <mrow> <mn>5</mn> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </mrow> </math>  , where the obtained  $$\gamma $$  <math> <mi>γ</mi> </math>   is consistent with the PTA Observations.

Primordial black holes in non-minimal Gauss–Bonnet inflation in light of the PTA data

Here, generation of PBHs and secondary GWs from non-canonical inflation with quartic potential have been probed. It is illustrated that, quartic potential in non-canonical setup with a generalized power-law Lagrangian density can source a consistent inflationary era with the latest observational data. Besides, we show that our model satisfies the swampland criteria. At the same time, defining a peaked function of inflaton field as non-canonical mass scale parameter  $$M(\phi )$$  <math> <mrow> <mi>M</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </math>   of the Lagrangian, gives rise to slow down the inflaton in a while. In this span, namely Ultra-Slow-Roll (USR) stage, the amplitude of the curvature perturbations on small scales enlarges versus CMB scales. It has been illustrated that, further to the peaked aspect of the chosen non-canonical mass scale parameter, the amount of  $$\alpha $$  <math> <mi>α</mi> </math>   parameter of the Lagrangian has enlarging impact on the amplitude of the scalar perturbations. As a consequence of adjusting three parameter Cases of this model, three Cases of PBHs in proper mass scopes to explain LIGO-VIRGO events, microlensing events in OGLE data and DM content in its totality, could be produced. In the end, power-law behavior of the current density parameter of gravitational waves  $$\Omega _{\mathrm{GW_0}}$$  <math> <msub> <mi>Ω</mi> <msub> <mi>GW</mi> <mn>0</mn> </msub> </msub> </math>   in terms of frequency has been examined. Also, the logarithmic power index as  $$n=3-2/\ln (f_c/f)$$  <math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>-</mo> <mn>2</mn> <mo>/</mo> <mo>ln</mo> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>/</mo> <mi>f</mi> <mo>)</mo> </mrow> </math>   in the infrared regime is obtained.

Primordial black holes and secondary gravitational waves from generalized power-law non-canonical inflation with quartic potential

Here, we study the generalized second law (GSL) of thermodynamics in the framework of massive gravity. To do this, we consider a FRW universe filled only with matter and enclosed by the apparent horizon. In addition, we consider two models including generalized massive gravity (GMG) as well as dRGT massive gravity on de Sitter. For both models, we first study the dynamics of background cosmology and then explore the validity of GSL. We conclude that for the selected values of model parameters the GSL is respected.

Mohammad Beigmohammadi

Generalized second law of thermodynamics in massive gravity

We study the non-Gaussianity and secondary Gravitational Waves (GWs) in the process of the Primordial Black Holes (PBHs) production from inflation. In our work, we focus on the  $$\alpha $$  <math> <mi>α</mi> </math>  -attractor inflation model in which a tiny bump in the inflaton potential enhances the amplitude of the curvature perturbations at some scales and consequently leads to the PBHs production with different mass scales. We implement the computational code BINGO which calculates the non-Gaussianity parameter in different triangle configurations. Our examination implies that in this setup, the non-Gaussianity gets amplified significantly in the equilateral shape around the scales in which the power spectrum of the scalar perturbations undergoes a sharp declination. The imprints of these non-Gaussianities can be probed in the scales corresponding to the BBN and  $$\mu $$  <math> <mi>μ</mi> </math>  -distortion events, or in smaller scales, and detection of such signatures in the future observations may confirm the idea of our model for the generation of PBHs or rule it out. Moreover, we investigate the secondary GWs in this framework and show that in our model, the peak of the present fractional energy density is obtained as  $$\Omega _{\mathrm{GW0}} \sim 10^{-8}$$  <math> <mrow> <msub> <mi>Ω</mi> <mrow> <mi>GW</mi> <mn>0</mn> </mrow> </msub> <mo>∼</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>8</mn> </mrow> </msup> </mrow> </math>   at different frequencies which depends on the model parameters. These results lie well within the sensitivity region of some GWs detectors at some frequencies, and therefore the observational compatibility of our model can be evaluated by the forthcoming data from these detectors. We further provide some estimations for the tilts of the induced GWs spectrum in the different intervals of frequency, and demonstrate that the spectrum obeys the power-law relation  $$\Omega _{\mathrm{GW0}}\sim f^{n}$$  <math> <mrow> <msub> <mi>Ω</mi> <mrow> <mi>GW</mi> <mn>0</mn> </mrow> </msub> <mo>∼</mo> <msup> <mi>f</mi> <mi>n</mi> </msup> </mrow> </math>   in those frequency bands.

Kazem Rezazadeh

Zeinab Teimoori

Saeid Karimi

Non-Gaussianity and secondary gravitational waves from primordial black holes production in  $$\alpha $$  <math> <mi>α</mi> </math>  -attractor inflation

We investigate the generation of primordial black holes (PBHs) with the aid of gravitationally increased friction mechanism originated from the nonminimal field derivative coupling (NMDC) to gravity framework, with the quartic potential. Applying the coupling parameter as a two-parted function of inflaton field and fine-tuning of five parameter assortments we can acquire ultra slow-roll phase to slow down the inflaton field due to high friction. This enables us to achieve enough enhancement in the amplitude of curvature perturbations power spectra to generate PBHs with different masses. The reheating stage is considered to obtain criteria for PBHs generation during radiation dominated era. We demonstrate that three cases of asteroid mass PBHs ( $$10^{-12}M_{\odot }$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>12</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>  ,  $$10^{-13}M_{\odot }$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>13</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>  , and  $$10^{-15}M_{\odot })$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mrow> <mo>)</mo> </mrow> </mrow> </math>   can be very interesting candidates for comprising  $$100\%$$  <math> <mrow> <mn>100</mn> <mo>%</mo> </mrow> </math>   ,  $$98.3\%$$  <math> <mrow> <mn>98.3</mn> <mo>%</mo> </mrow> </math>   and  $$99.1\%$$  <math> <mrow> <mn>99.1</mn> <mo>%</mo> </mrow> </math>   of the total dark matter (DM) content of the universe. Moreover, we analyse the production of induced Gravitational Waves (GWs), and illustrate that their spectra of current density parameter  $$({\Omega _{{\mathrm{GW}}_\mathrm{0}}})$$  <math> <mrow> <mo>(</mo> <msub> <mi>Ω</mi> <msub> <mi>GW</mi> <mn>0</mn> </msub> </msub> <mo>)</mo> </mrow> </math>   for all parameter Cases foretold by our model have climaxes which cut the sensitivity curves of GWs detectors, ergo the veracity of our outcomes can be tested in light of these detectors. At last, our numerical results exhibit that the spectra of  $${\Omega _{{\mathrm{GW}}_\mathrm{0}}}$$  <math> <msub> <mi>Ω</mi> <msub> <mi>GW</mi> <mn>0</mn> </msub> </msub> </math>   behave as a power-law function with respect to frequency,  $${\Omega _{{\mathrm{GW}}_\mathrm{0}}} (f) \sim (f/f_c)^{n} $$  <math> <mrow> <msub> <mi>Ω</mi> <msub> <mi>GW</mi> <mn>0</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>∼</mo> <msup> <mrow> <mo>(</mo> <mi>f</mi> <mo>/</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math>  , in the vicinity of climaxes. Also, in the infrared regime  $$f\ll f_{c}$$  <math> <mrow> <mi>f</mi> <mo>≪</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> </math>  , the power index satisfies the relation  $$n=3-2/\ln (f_c/f)$$  <math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>-</mo> <mn>2</mn> <mo>/</mo> <mo>ln</mo> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>/</mo> <mi>f</mi> <mo>)</mo> </mrow> </math>  .

Primordial black holes in nonminimal derivative coupling inflation with quartic potential and reheating consideration

Within the framework of inflationary model with field-dependent kinetic term for quartic and natural potentials, we investigate generation of the primordial black holes (PBHs) and induced gravitational waves (GWs). In this setup, we consider a kinetic function as  $$G(\phi )=g_I(\phi )\big (1+g_{II}(\phi )\big )$$  <math> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>g</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> </mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>g</mi> <mrow> <mi>II</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> <mrow> <mo>)</mo> </mrow> </mrow> </math>   and show that in the presence of first term  $$g_I(\phi )$$  <math> <mrow> <msub> <mi>g</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </math>   both quartic and natural potentials, in contrast to the standard model of inflation, can be consistent, with the 68% CL of Planck observations. Besides, the second term  $$g_{II}(\phi )$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mi>II</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </math>   can cause a significant enhancement in the primordial curvature perturbations at the small scales which results the PBHs formation. For the both potentials, we obtain an enhancement in the scalar power spectrum at the scales  $$k\sim 10^{12}~{\mathrm{Mpc}}^{-1}$$  <math> <mrow> <mi>k</mi> <mo>∼</mo> <msup> <mn>10</mn> <mn>12</mn> </msup> <mspace width="3.33333pt"></mspace> <msup> <mrow> <mi>Mpc</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>  ,  $$10^{8}~{\mathrm{Mpc}}^{-1}$$  <math> <mrow> <msup> <mn>10</mn> <mn>8</mn> </msup> <mspace width="3.33333pt"></mspace> <msup> <mrow> <mi>Mpc</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>  , and  $$10^{5}~{\mathrm{Mpc}}^{-1}$$  <math> <mrow> <msup> <mn>10</mn> <mn>5</mn> </msup> <mspace width="3.33333pt"></mspace> <msup> <mrow> <mi>Mpc</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>  , which causes PBHs production in mass scales around  $$10^{-13}M_{\odot }$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>13</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>  ,  $$10^{-5}M_{\odot }$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>  , and  $$10 M_{\odot }$$  <math> <mrow> <mn>10</mn> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>  , respectively. Observational constraints confirm that PBHs with a mass scale of  $$10^{-13}M_{\odot }$$  <math> <mrow> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>13</mn> </mrow> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </math>   can constitute the total of dark matter in the universe. Furthermore, we estimate the energy density parameter of induced GWs which can be examined by the observation. Also we conclude that it can be parametrized as a power-law function  $$\Omega _{\mathrm{GW}}\sim (f/f_c)^n$$  <math> <mrow> <msub> <mi>Ω</mi> <mi>GW</mi> </msub> <mo>∼</mo> <msup> <mrow> <mo>(</mo> <mi>f</mi> <mo>/</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math>  , where the power index equals  $$n=3-2/\ln (f_c/f)$$  <math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>-</mo> <mn>2</mn> <mo>/</mo> <mo>ln</mo> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>/</mo> <mi>f</mi> <mo>)</mo> </mrow> </math>   in the infrared limit  $$f\ll f_{c}$$  <math> <mrow> <mi>f</mi> <mo>≪</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> </math>  .

Primordial black holes formation in the inflationary model with field-dependent kinetic term for quartic and natural potentials

We study inflation within the framework of non-canonical scalar field. In this scenario, we obtain the inflationary observables such as the scalar spectral index, the tensor-to-scalar ratio, the running of the scalar spectral index as well as the equilateral non-Gaussianity parameter. Then, we apply these results for the exponential and inverse power-law potentials. Our investigation shows that although the predictions of these potentials in the standard canonical inflation are completely ruled out by the Planck 2015 observations, their results in non-canonical scenario can lie inside the allowed regions of the Planck 2015 data. We also find that in non-canonical inflation, the predictions of the aforementioned potentials for the running of the scalar spectral index and the equilateral non-Gaussianity parameter are in well agreement with the Planck 2015 results. Furthermore, we show that in the context of non-canonical inflation, the graceful exit problem of the exponential and inverse power-law potentials is resolved.

Resurrecting the exponential and inverse power-law potentials in non-canonical inflation

Within the framework of Brans–Dicke gravity, we investigate inflation with the quartic potential, λφ4/4 , in the presence of generalized Brans–Dicke parameter ωGBD(φ) . We obtain the inflationary observables containing the scalar spectral index, the tensor-to-scalar ratio, the running of the scalar spectral index and the equilateral non-Gaussianity parameter in terms of general form of the potential U(φ) and ωGBD(φ) . For the quartic potential, our results show that the predictions of the model are in well agreement with the Planck 2015 data for the generalized Brans–Dicke parameters ωGBD(φ)=ω0φn and ω0ebφ . This is in contrast with both the Einstein and standard Brans–Dicke gravity, in which the result of quartic potential is disfavored by the Planck data.